LMFDB - Embedded newform 39.5.g.a.31.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,5,Mod(31,39)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("39.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	39.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.03142856027$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 $ x^20 + 5446x^16 - 1452x^15 + 106320x^13 + 8376897x^12 - 1643220x^11 + 1054152x^10 + 232107192x^9 + 3337033804x^8 + 3105930144x^7 + 2297014848x^6 + 21034904256x^5 + 158658808624x^4 + 235144077504x^3 + 177846480000x^2 + 27353289600*x + 2103506496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.8
Root			$-2.89776 + 2.89776i$ of defining polynomial
Character	$\chi$	$=$	39.31
Dual form			39.5.g.a.34.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.89776 - 2.89776i) q^{2} +5.19615 q^{3} -0.794047i q^{4} +(18.6373 - 18.6373i) q^{5} +(15.0572 - 15.0572i) q^{6} +(-15.2549 - 15.2549i) q^{7} +(44.0632 + 44.0632i) q^{8} +27.0000 q^{9} -108.013i q^{10} +(-59.2741 - 59.2741i) q^{11} -4.12599i q^{12} +(-157.156 + 62.1541i) q^{13} -88.4101 q^{14} +(96.8421 - 96.8421i) q^{15} +268.074 q^{16} +272.906i q^{17} +(78.2396 - 78.2396i) q^{18} +(-54.5892 + 54.5892i) q^{19} +(-14.7989 - 14.7989i) q^{20} +(-79.2668 - 79.2668i) q^{21} -343.524 q^{22} +596.223i q^{23} +(228.959 + 228.959i) q^{24} -69.6962i q^{25} +(-275.292 + 635.507i) q^{26} +140.296 q^{27} +(-12.1131 + 12.1131i) q^{28} -1314.18 q^{29} -561.251i q^{30} +(874.924 - 874.924i) q^{31} +(71.8036 - 71.8036i) q^{32} +(-307.997 - 307.997i) q^{33} +(790.817 + 790.817i) q^{34} -568.619 q^{35} -21.4393i q^{36} +(-403.847 - 403.847i) q^{37} +316.373i q^{38} +(-816.604 + 322.962i) q^{39} +1642.44 q^{40} +(983.966 - 983.966i) q^{41} -459.392 q^{42} -2237.13i q^{43} +(-47.0664 + 47.0664i) q^{44} +(503.206 - 503.206i) q^{45} +(1727.71 + 1727.71i) q^{46} +(-369.867 - 369.867i) q^{47} +1392.95 q^{48} -1935.58i q^{49} +(-201.963 - 201.963i) q^{50} +1418.06i q^{51} +(49.3532 + 124.789i) q^{52} +4399.13 q^{53} +(406.545 - 406.545i) q^{54} -2209.42 q^{55} -1344.36i q^{56} +(-283.654 + 283.654i) q^{57} +(-3808.17 + 3808.17i) q^{58} +(976.401 + 976.401i) q^{59} +(-76.8972 - 76.8972i) q^{60} -6163.60 q^{61} -5070.64i q^{62} +(-411.882 - 411.882i) q^{63} +3873.05i q^{64} +(-1770.57 + 4087.33i) q^{65} -1785.01 q^{66} +(-1708.57 + 1708.57i) q^{67} +216.700 q^{68} +3098.07i q^{69} +(-1647.72 + 1647.72i) q^{70} +(672.586 - 672.586i) q^{71} +(1189.71 + 1189.71i) q^{72} +(-1169.63 - 1169.63i) q^{73} -2340.51 q^{74} -362.152i q^{75} +(43.3464 + 43.3464i) q^{76} +1808.44i q^{77} +(-1430.46 + 3302.19i) q^{78} +8345.50 q^{79} +(4996.17 - 4996.17i) q^{80} +729.000 q^{81} -5702.60i q^{82} +(6513.68 - 6513.68i) q^{83} +(-62.9415 + 62.9415i) q^{84} +(5086.23 + 5086.23i) q^{85} +(-6482.67 - 6482.67i) q^{86} -6828.65 q^{87} -5223.62i q^{88} +(8366.65 + 8366.65i) q^{89} -2916.35i q^{90} +(3345.55 + 1449.24i) q^{91} +473.429 q^{92} +(4546.24 - 4546.24i) q^{93} -2143.57 q^{94} +2034.79i q^{95} +(373.103 - 373.103i) q^{96} +(2271.58 - 2271.58i) q^{97} +(-5608.84 - 5608.84i) q^{98} +(-1600.40 - 1600.40i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 24 q^{5} - 24 q^{7} + 540 q^{9} + 372 q^{11} - 224 q^{13} + 480 q^{14} - 252 q^{15} - 2328 q^{16} - 840 q^{19} + 228 q^{20} + 936 q^{21} + 3536 q^{22} - 1404 q^{24} - 828 q^{26} - 1984 q^{28} - 5064 q^{29}+ \cdots + 10044 q^{99}+O(q^{100}) $ 20 * q + 24 * q^5 - 24 * q^7 + 540 * q^9 + 372 * q^11 - 224 * q^13 + 480 * q^14 - 252 * q^15 - 2328 * q^16 - 840 * q^19 + 228 * q^20 + 936 * q^21 + 3536 * q^22 - 1404 * q^24 - 828 * q^26 - 1984 * q^28 - 5064 * q^29 + 1712 * q^31 - 7260 * q^32 - 540 * q^33 + 8040 * q^34 + 5976 * q^35 - 868 * q^37 - 2952 * q^39 + 6336 * q^40 - 6000 * q^41 + 648 * q^42 + 20868 * q^44 + 648 * q^45 - 1992 * q^46 + 1692 * q^47 + 2592 * q^48 + 8952 * q^50 - 12416 * q^52 - 18816 * q^53 - 3280 * q^55 - 9360 * q^57 + 1256 * q^58 - 6588 * q^59 + 9036 * q^60 - 5512 * q^61 - 648 * q^63 + 33048 * q^65 + 13464 * q^66 - 4704 * q^67 - 28296 * q^68 - 31096 * q^70 + 9756 * q^71 + 13692 * q^73 - 28296 * q^74 - 17072 * q^76 - 23868 * q^78 - 5056 * q^79 + 42228 * q^80 + 14580 * q^81 + 24708 * q^83 + 26496 * q^84 - 4752 * q^85 - 77424 * q^86 + 5256 * q^87 + 36000 * q^89 + 19104 * q^91 + 12912 * q^92 - 18432 * q^93 + 112832 * q^94 - 52344 * q^96 + 50012 * q^97 + 11544 * q^98 + 10044 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

$n$	$14$	$28$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.89776	−	2.89776i	0.724440	−	0.724440i	−0.245066	−	0.969506i	$-0.578810\pi$
$2$	2.89776	−	2.89776i	0.724440	−	0.724440i	0.969506	+	0.245066i	$0.0788096\pi$
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$		−	0.794047i		−	0.0496279i
$5$	18.6373	−	18.6373i	0.745491	−	0.745491i	−0.228138	−	0.973629i	$-0.573264\pi$
$5$	18.6373	−	18.6373i	0.745491	−	0.745491i	0.973629	+	0.228138i	$0.0732637\pi$
$6$	15.0572	−	15.0572i	0.418256	−	0.418256i
$7$	−15.2549	−	15.2549i	−0.311324	−	0.311324i	0.534098	−	0.845422i	$-0.320651\pi$
$7$	−15.2549	−	15.2549i	−0.311324	−	0.311324i	−0.845422	+	0.534098i	$0.820651\pi$
$8$	44.0632	+	44.0632i	0.688488	+	0.688488i
$9$	27.0000			0.333333
$10$		−	108.013i		−	1.08013i
$11$	−59.2741	−	59.2741i	−0.489869	−	0.489869i	0.418396	−	0.908265i	$-0.362592\pi$
$11$	−59.2741	−	59.2741i	−0.489869	−	0.489869i	−0.908265	+	0.418396i	$0.862592\pi$
$12$		−	4.12599i		−	0.0286527i
$13$	−157.156	+	62.1541i	−0.929915	+	0.367776i
$13$	−157.156	+	62.1541i	−0.929915	+	0.367776i
$14$	−88.4101			−0.451072
$15$	96.8421	−	96.8421i	0.430409	−	0.430409i
$16$	268.074			1.04717
$17$			272.906i			0.944312i	0.881515	+	0.472156i	$0.156524\pi$
$17$			272.906i			0.944312i	−0.881515	+	0.472156i	$0.843476\pi$
$18$	78.2396	−	78.2396i	0.241480	−	0.241480i
$19$	−54.5892	+	54.5892i	−0.151217	+	0.151217i	−0.778661	−	0.627445i	$-0.784101\pi$
$19$	−54.5892	+	54.5892i	−0.151217	+	0.151217i	0.627445	+	0.778661i	$0.284101\pi$
$20$	−14.7989	−	14.7989i	−0.0369972	−	0.0369972i
$21$	−79.2668	−	79.2668i	−0.179743	−	0.179743i
$22$	−343.524			−0.709761
$23$			596.223i			1.12708i	0.826090	+	0.563538i	$0.190560\pi$
$23$			596.223i			1.12708i	−0.826090	+	0.563538i	$0.809440\pi$
$24$	228.959	+	228.959i	0.397499	+	0.397499i
$25$		−	69.6962i		−	0.111514i
$26$	−275.292	+	635.507i	−0.407236	+	0.940099i
$27$	140.296			0.192450
$28$	−12.1131	+	12.1131i	−0.0154504	+	0.0154504i
$29$	−1314.18			−1.56263			−0.781317	−	0.624135i	$-0.785452\pi$
$29$	−1314.18			−1.56263			−0.781317	+	0.624135i	$0.785452\pi$
$30$		−	561.251i		−	0.623612i
$31$	874.924	−	874.924i	0.910431	−	0.910431i	−0.0858748	−	0.996306i	$-0.527368\pi$
$31$	874.924	−	874.924i	0.910431	−	0.910431i	0.996306	+	0.0858748i	$0.0273685\pi$
$32$	71.8036	−	71.8036i	0.0701207	−	0.0701207i
$33$	−307.997	−	307.997i	−0.282826	−	0.282826i
$34$	790.817	+	790.817i	0.684098	+	0.684098i
$35$	−568.619			−0.464179
$36$		−	21.4393i		−	0.0165426i
$37$	−403.847	−	403.847i	−0.294994	−	0.294994i	0.544055	−	0.839049i	$-0.316888\pi$
$37$	−403.847	−	403.847i	−0.294994	−	0.294994i	−0.839049	+	0.544055i	$0.816888\pi$
$38$			316.373i			0.219095i
$39$	−816.604	+	322.962i	−0.536886	+	0.212335i
$40$	1642.44			1.02652
$41$	983.966	−	983.966i	0.585346	−	0.585346i	−0.351022	−	0.936367i	$-0.614166\pi$
$41$	983.966	−	983.966i	0.585346	−	0.585346i	0.936367	+	0.351022i	$0.114166\pi$
$42$	−459.392			−0.260426
$43$		−	2237.13i		−	1.20991i	−0.796258	−	0.604957i	$-0.793190\pi$
$43$		−	2237.13i		−	1.20991i	0.796258	−	0.604957i	$-0.206810\pi$
$44$	−47.0664	+	47.0664i	−0.0243112	+	0.0243112i
$45$	503.206	−	503.206i	0.248497	−	0.248497i
$46$	1727.71	+	1727.71i	0.816500	+	0.816500i
$47$	−369.867	−	369.867i	−0.167437	−	0.167437i	0.618415	−	0.785852i	$-0.287775\pi$
$47$	−369.867	−	369.867i	−0.167437	−	0.167437i	−0.785852	+	0.618415i	$0.787775\pi$
$48$	1392.95			0.604581
$49$		−	1935.58i		−	0.806154i
$50$	−201.963	−	201.963i	−0.0807852	−	0.0807852i
$51$			1418.06i			0.545199i
$52$	49.3532	+	124.789i	0.0182519	+	0.0461497i
$53$	4399.13			1.56608			0.783041	−	0.621970i	$-0.213667\pi$
$53$	4399.13			1.56608			0.783041	+	0.621970i	$0.213667\pi$
$54$	406.545	−	406.545i	0.139419	−	0.139419i
$55$	−2209.42			−0.730385
$56$		−	1344.36i		−	0.428686i
$57$	−283.654	+	283.654i	−0.0873050	+	0.0873050i
$58$	−3808.17	+	3808.17i	−1.13204	+	1.13204i
$59$	976.401	+	976.401i	0.280495	+	0.280495i	0.833306	−	0.552812i	$-0.186445\pi$
$59$	976.401	+	976.401i	0.280495	+	0.280495i	−0.552812	+	0.833306i	$0.686445\pi$
$60$	−76.8972	−	76.8972i	−0.0213603	−	0.0213603i
$61$	−6163.60			−1.65644			−0.828219	−	0.560405i	$-0.810646\pi$
$61$	−6163.60			−1.65644			−0.828219	+	0.560405i	$0.810646\pi$
$62$		−	5070.64i		−	1.31911i
$63$	−411.882	−	411.882i	−0.103775	−	0.103775i
$64$			3873.05i			0.945568i
$65$	−1770.57	+	4087.33i	−0.419070	+	0.967416i
$66$	−1785.01			−0.409781
$67$	−1708.57	+	1708.57i	−0.380612	+	0.380612i	−0.871322	−	0.490711i	$-0.836737\pi$
$67$	−1708.57	+	1708.57i	−0.380612	+	0.380612i	0.490711	+	0.871322i	$0.336737\pi$
$68$	216.700			0.0468642
$69$			3098.07i			0.650718i
$70$	−1647.72	+	1647.72i	−0.336270	+	0.336270i
$71$	672.586	−	672.586i	0.133423	−	0.133423i	−0.637241	−	0.770664i	$-0.719925\pi$
$71$	672.586	−	672.586i	0.133423	−	0.133423i	0.770664	+	0.637241i	$0.219925\pi$
$72$	1189.71	+	1189.71i	0.229496	+	0.229496i
$73$	−1169.63	−	1169.63i	−0.219485	−	0.219485i	0.588797	−	0.808281i	$-0.299602\pi$
$73$	−1169.63	−	1169.63i	−0.219485	−	0.219485i	−0.808281	+	0.588797i	$0.799602\pi$
$74$	−2340.51			−0.427412
$75$		−	362.152i		−	0.0643826i
$76$	43.3464	+	43.3464i	0.00750457	+	0.00750457i
$77$			1808.44i			0.305016i
$78$	−1430.46	+	3302.19i	−0.235118	+	0.542767i
$79$	8345.50			1.33720			0.668602	−	0.743620i	$-0.266893\pi$
$79$	8345.50			1.33720			0.668602	+	0.743620i	$0.266893\pi$
$80$	4996.17	−	4996.17i	0.780652	−	0.780652i
$81$	729.000			0.111111
$82$		−	5702.60i		−	0.848096i
$83$	6513.68	−	6513.68i	0.945519	−	0.945519i	−0.0530714	−	0.998591i	$-0.516901\pi$
$83$	6513.68	−	6513.68i	0.945519	−	0.945519i	0.998591	+	0.0530714i	$0.0169011\pi$
$84$	−62.9415	+	62.9415i	−0.00892028	+	0.00892028i
$85$	5086.23	+	5086.23i	0.703976	+	0.703976i
$86$	−6482.67	−	6482.67i	−0.876510	−	0.876510i
$87$	−6828.65			−0.902187
$88$		−	5223.62i		−	0.674537i
$89$	8366.65	+	8366.65i	1.05626	+	1.05626i	0.998320	+	0.0579425i	$0.0184540\pi$
$89$	8366.65	+	8366.65i	1.05626	+	1.05626i	0.0579425	+	0.998320i	$0.481546\pi$
$90$		−	2916.35i		−	0.360043i
$91$	3345.55	+	1449.24i	0.404003	+	0.175008i
$92$	473.429			0.0559345
$93$	4546.24	−	4546.24i	0.525638	−	0.525638i
$94$	−2143.57			−0.242596
$95$			2034.79i			0.225461i
$96$	373.103	−	373.103i	0.0404842	−	0.0404842i
$97$	2271.58	−	2271.58i	0.241426	−	0.241426i	−0.576014	−	0.817440i	$-0.695393\pi$
$97$	2271.58	−	2271.58i	0.241426	−	0.241426i	0.817440	+	0.576014i	$0.195393\pi$
$98$	−5608.84	−	5608.84i	−0.584011	−	0.584011i
$99$	−1600.40	−	1600.40i	−0.163290	−	0.163290i
$100$	−55.3421			−0.00553421
$101$			6490.97i			0.636308i	0.948039	+	0.318154i	$0.103063\pi$
$101$			6490.97i			0.636308i	−0.948039	+	0.318154i	$0.896937\pi$
$102$	4109.20	+	4109.20i	0.394964	+	0.394964i
$103$		−	5614.96i		−	0.529264i	−0.964350	−	0.264632i	$-0.914750\pi$
$103$		−	5614.96i		−	0.529264i	0.964350	−	0.264632i	$-0.0852504\pi$
$104$	−9663.49	−	4186.07i	−0.893444	−	0.387026i
$105$	−2954.63			−0.267994
$106$	12747.6	−	12747.6i	1.13453	−	1.13453i
$107$	9686.14			0.846025			0.423012	−	0.906124i	$-0.360973\pi$
$107$	9686.14			0.846025			0.423012	+	0.906124i	$0.360973\pi$
$108$		−	111.402i		−	0.00955090i
$109$	−16372.1	+	16372.1i	−1.37801	+	1.37801i	−0.530029	+	0.847980i	$0.677819\pi$
$109$	−16372.1	+	16372.1i	−1.37801	+	1.37801i	−0.847980	+	0.530029i	$0.822181\pi$
$110$	−6402.36	+	6402.36i	−0.529121	+	0.529121i
$111$	−2098.45	−	2098.45i	−0.170315	−	0.170315i
$112$	−4089.44	−	4089.44i	−0.326008	−	0.326008i
$113$	−19245.1			−1.50717			−0.753585	−	0.657350i	$-0.771677\pi$
$113$	−19245.1			−1.50717			−0.753585	+	0.657350i	$0.771677\pi$
$114$			1643.92i			0.126495i
$115$	11112.0	+	11112.0i	0.840225	+	0.840225i
$116$			1043.52i			0.0775503i
$117$	−4243.20	+	1678.16i	−0.309972	+	0.122592i
$118$	5658.76			0.406403
$119$	4163.15	−	4163.15i	0.293987	−	0.293987i
$120$	8534.35			0.592664
$121$		−	7614.16i		−	0.520057i
$122$	−17860.7	+	17860.7i	−1.19999	+	1.19999i
$123$	5112.84	−	5112.84i	0.337950	−	0.337950i
$124$	−694.731	−	694.731i	−0.0451828	−	0.0451828i
$125$	10349.4	+	10349.4i	0.662358	+	0.662358i
$126$	−2387.07			−0.150357
$127$		−	26337.0i		−	1.63290i	−0.577417	−	0.816449i	$-0.695939\pi$
$127$		−	26337.0i		−	1.63290i	0.577417	−	0.816449i	$-0.304061\pi$
$128$	12372.0	+	12372.0i	0.755129	+	0.755129i
$129$		−	11624.5i		−	0.698544i
$130$	6713.43	+	16974.8i	0.397245	+	1.00443i
$131$	−3094.41			−0.180316			−0.0901581	−	0.995927i	$-0.528737\pi$
$131$	−3094.41			−0.180316			−0.0901581	+	0.995927i	$0.528737\pi$
$132$	−244.564	+	244.564i	−0.0140361	+	0.0140361i
$133$	1665.51			0.0941549
$134$			9902.03i			0.551461i
$135$	2614.74	−	2614.74i	0.143470	−	0.143470i
$136$	−12025.1	+	12025.1i	−0.650147	+	0.650147i
$137$	−10453.9	−	10453.9i	−0.556978	−	0.556978i	0.371467	−	0.928446i	$-0.378855\pi$
$137$	−10453.9	−	10453.9i	−0.556978	−	0.556978i	−0.928446	+	0.371467i	$0.878855\pi$
$138$	8977.46	+	8977.46i	0.471406	+	0.471406i
$139$	5922.48			0.306531			0.153265	−	0.988185i	$-0.451021\pi$
$139$	5922.48			0.306531			0.153265	+	0.988185i	$0.451021\pi$
$140$			451.511i			0.0230363i
$141$	−1921.89	−	1921.89i	−0.0966695	−	0.0966695i
$142$		−	3897.99i		−	0.193314i
$143$	12999.4	+	5631.13i	0.635698	+	0.275374i
$144$	7238.00			0.349055
$145$	−24492.6	+	24492.6i	−1.16493	+	1.16493i
$146$	−6778.64			−0.318007
$147$		−	10057.6i		−	0.465433i
$148$	−320.674	+	320.674i	−0.0146400	+	0.0146400i
$149$	−22740.7	+	22740.7i	−1.02431	+	1.02431i	−0.0246105	+	0.999697i	$0.507835\pi$
$149$	−22740.7	+	22740.7i	−1.02431	+	1.02431i	−0.999697	+	0.0246105i	$0.992165\pi$
$150$	−1049.43	−	1049.43i	−0.0466414	−	0.0466414i
$151$	−30969.3	−	30969.3i	−1.35824	−	1.35824i	−0.876082	−	0.482161i	$-0.839852\pi$
$151$	−30969.3	−	30969.3i	−1.35824	−	1.35824i	−0.482161	−	0.876082i	$-0.660148\pi$
$152$	−4810.76			−0.208222
$153$			7368.46i			0.314771i
$154$	5240.43	+	5240.43i	0.220966	+	0.220966i
$155$		−	32612.4i		−	1.35744i
$156$	256.447	+	648.422i	0.0105378	+	0.0266446i
$157$	−16868.7			−0.684358			−0.342179	−	0.939635i	$-0.611165\pi$
$157$	−16868.7			−0.684358			−0.342179	+	0.939635i	$0.611165\pi$
$158$	24183.3	−	24183.3i	0.968725	−	0.968725i
$159$	22858.5			0.904178
$160$		−	2676.45i		−	0.104549i
$161$	9095.32	−	9095.32i	0.350886	−	0.350886i
$162$	2112.47	−	2112.47i	0.0804934	−	0.0804934i
$163$	23110.4	+	23110.4i	0.869826	+	0.869826i	0.992453	−	0.122626i	$-0.0391317\pi$
$163$	23110.4	+	23110.4i	0.869826	+	0.869826i	−0.122626	+	0.992453i	$0.539132\pi$
$164$	−781.315	−	781.315i	−0.0290495	−	0.0290495i
$165$	−11480.5			−0.421688
$166$		−	37750.2i		−	1.36994i
$167$	16613.8	+	16613.8i	0.595710	+	0.595710i	0.939168	−	0.343458i	$-0.111598\pi$
$167$	16613.8	+	16613.8i	0.595710	+	0.595710i	−0.343458	+	0.939168i	$0.611598\pi$
$168$		−	6985.50i		−	0.247502i
$169$	20834.7	−	19535.7i	0.729482	−	0.684000i
$170$	29477.3			1.01998
$171$	−1473.91	+	1473.91i	−0.0504056	+	0.0504056i
$172$	−1776.39			−0.0600455
$173$			9230.43i			0.308411i	0.988039	+	0.154206i	$0.0492818\pi$
$173$			9230.43i			0.308411i	−0.988039	+	0.154206i	$0.950718\pi$
$174$	−19787.8	+	19787.8i	−0.653581	+	0.653581i
$175$	−1063.21	+	1063.21i	−0.0347170	+	0.0347170i
$176$	−15889.9	−	15889.9i	−0.512973	−	0.512973i
$177$	5073.53	+	5073.53i	0.161944	+	0.161944i
$178$	48489.1			1.53040
$179$		−	28771.7i		−	0.897965i	−0.893541	−	0.448982i	$-0.851787\pi$
$179$		−	28771.7i		−	0.897965i	0.893541	−	0.448982i	$-0.148213\pi$
$180$	−399.570	−	399.570i	−0.0123324	−	0.0123324i
$181$		−	2334.88i		−	0.0712700i	−0.999365	−	0.0356350i	$-0.988655\pi$
$181$		−	2334.88i		−	0.0712700i	0.999365	−	0.0356350i	$-0.0113454\pi$
$182$	13894.1	−	5495.05i	0.419458	−	0.165893i
$183$	−32027.0			−0.956345
$184$	−26271.5	+	26271.5i	−0.775978	+	0.775978i
$185$	−15053.2			−0.439831
$186$		−	26347.8i		−	0.761586i
$187$	16176.3	−	16176.3i	0.462589	−	0.462589i
$188$	−293.692	+	293.692i	−0.00830953	+	0.00830953i
$189$	−2140.20	−	2140.20i	−0.0599144	−	0.0599144i
$190$	5896.34	+	5896.34i	0.163333	+	0.163333i
$191$	−11500.7			−0.315251			−0.157625	−	0.987499i	$-0.550384\pi$
$191$	−11500.7			−0.315251			−0.157625	+	0.987499i	$0.550384\pi$
$192$			20124.9i			0.545924i
$193$	36690.3	+	36690.3i	0.985002	+	0.985002i	0.999889	−	0.0148869i	$-0.00473884\pi$
$193$	36690.3	+	36690.3i	0.985002	+	0.985002i	−0.0148869	+	0.999889i	$0.504739\pi$
$194$		−	13165.0i		−	0.349797i
$195$	−9200.15	+	21238.4i	−0.241950	+	0.558538i
$196$	−1536.94			−0.0400078
$197$	3775.20	−	3775.20i	0.0972763	−	0.0972763i	−0.656794	−	0.754070i	$-0.728088\pi$
$197$	3775.20	−	3775.20i	0.0972763	−	0.0972763i	0.754070	+	0.656794i	$0.228088\pi$
$198$	−9275.16			−0.236587
$199$			47612.3i			1.20230i	0.799136	+	0.601151i	$0.205291\pi$
$199$			47612.3i			1.20230i	−0.799136	+	0.601151i	$0.794709\pi$
$200$	3071.04	−	3071.04i	0.0767760	−	0.0767760i
$201$	−8877.97	+	8877.97i	−0.219746	+	0.219746i
$202$	18809.3	+	18809.3i	0.460967	+	0.460967i
$203$	20047.6	+	20047.6i	0.486486	+	0.486486i
$204$	1126.01			0.0270571
$205$		−	36676.9i		−	0.872740i
$206$	−16270.8	−	16270.8i	−0.383420	−	0.383420i
$207$			16098.0i			0.375692i
$208$	−42129.4	+	16661.9i	−0.973774	+	0.385122i
$209$	6471.46			0.148153
$210$	−8561.82	+	8561.82i	−0.194146	+	0.194146i
$211$	−75668.4			−1.69961			−0.849806	−	0.527096i	$-0.823281\pi$
$211$	−75668.4			−1.69961			−0.849806	+	0.527096i	$0.823281\pi$
$212$		−	3493.11i		−	0.0777215i
$213$	3494.86	−	3494.86i	0.0770318	−	0.0770318i
$214$	28068.1	−	28068.1i	0.612894	−	0.612894i
$215$	−41694.0	−	41694.0i	−0.901980	−	0.901980i
$216$	6181.90	+	6181.90i	0.132500	+	0.132500i
$217$	−26693.8			−0.566879
$218$			94885.0i			1.99657i
$219$	−6077.59	−	6077.59i	−0.126719	−	0.126719i
$220$			1754.38i			0.0362475i
$221$	−16962.2	−	42888.7i	−0.347295	−	0.878129i
$222$	−12161.6			−0.246766
$223$	−953.201	+	953.201i	−0.0191679	+	0.0191679i	−0.716626	−	0.697458i	$-0.754314\pi$
$223$	−953.201	+	953.201i	−0.0191679	+	0.0191679i	0.697458	+	0.716626i	$0.254314\pi$
$224$	−2190.71			−0.0436606
$225$		−	1881.80i		−	0.0371713i
$226$	−55767.6	+	55767.6i	−1.09186	+	1.09186i
$227$	6110.06	−	6110.06i	0.118575	−	0.118575i	−0.645329	−	0.763905i	$-0.723280\pi$
$227$	6110.06	−	6110.06i	0.118575	−	0.118575i	0.763905	+	0.645329i	$0.223280\pi$
$228$	225.235	+	225.235i	0.00433277	+	0.00433277i
$229$	9576.99	+	9576.99i	0.182624	+	0.182624i	0.792498	−	0.609874i	$-0.208780\pi$
$229$	9576.99	+	9576.99i	0.182624	+	0.182624i	−0.609874	+	0.792498i	$0.708780\pi$
$230$	64399.7			1.21739
$231$			9396.93i			0.176101i
$232$	−57906.8	−	57906.8i	−1.07585	−	1.07585i
$233$			72801.5i			1.34100i	0.741910	+	0.670500i	$0.233920\pi$
$233$			72801.5i			1.34100i	−0.741910	+	0.670500i	$0.766080\pi$
$234$	−7432.88	+	17158.7i	−0.135745	+	0.313366i
$235$	−13786.6			−0.249645
$236$	775.309	−	775.309i	0.0139204	−	0.0139204i
$237$	43364.5			0.772036
$238$		−	24127.7i		−	0.425952i
$239$	72643.5	−	72643.5i	1.27175	−	1.27175i	0.326578	−	0.945170i	$-0.394105\pi$
$239$	72643.5	−	72643.5i	1.27175	−	1.27175i	0.945170	−	0.326578i	$-0.105895\pi$
$240$	25960.9	−	25960.9i	0.450710	−	0.450710i
$241$	64322.5	+	64322.5i	1.10746	+	1.10746i	0.993483	+	0.113978i	$0.0363593\pi$
$241$	64322.5	+	64322.5i	1.10746	+	1.10746i	0.113978	+	0.993483i	$0.463641\pi$
$242$	−22064.0	−	22064.0i	−0.376751	−	0.376751i
$243$	3788.00			0.0641500
$244$			4894.19i			0.0822056i
$245$	−36073.9	−	36073.9i	−0.600981	−	0.600981i
$246$		−	29631.6i		−	0.489649i
$247$	5186.06	−	11971.9i	0.0850048	−	0.196232i
$248$	77104.0			1.25364
$249$	33846.1	−	33846.1i	0.545896	−	0.545896i
$250$	59979.9			0.959678
$251$			49293.7i			0.782427i	0.920300	+	0.391213i	$0.127945\pi$
$251$			49293.7i			0.782427i	−0.920300	+	0.391213i	$0.872055\pi$
$252$	−327.054	+	327.054i	−0.00515013	+	0.00515013i
$253$	35340.6	−	35340.6i	0.552119	−	0.552119i
$254$	−76318.4	−	76318.4i	−1.18294	−	1.18294i
$255$	26428.8	+	26428.8i	0.406441	+	0.406441i
$256$	9733.62			0.148523
$257$			82424.4i			1.24793i	0.781453	+	0.623964i	$0.214479\pi$
$257$			82424.4i			1.24793i	−0.781453	+	0.623964i	$0.785521\pi$
$258$	−33684.9	−	33684.9i	−0.506053	−	0.506053i
$259$			12321.3i			0.183678i
$260$	3245.54	+	1405.92i	0.0480109	+	0.0207976i
$261$	−35482.7			−0.520878
$262$	−8966.85	+	8966.85i	−0.130628	+	0.130628i
$263$	41367.3			0.598062			0.299031	−	0.954243i	$-0.403337\pi$
$263$	41367.3			0.598062			0.299031	+	0.954243i	$0.403337\pi$
$264$		−	27142.7i		−	0.389444i
$265$	81987.7	−	81987.7i	1.16750	−	1.16750i
$266$	4826.24	−	4826.24i	0.0682096	−	0.0682096i
$267$	43474.4	+	43474.4i	0.609833	+	0.609833i
$268$	1356.68	+	1356.68i	0.0188890	+	0.0188890i
$269$	−59626.2			−0.824011			−0.412005	−	0.911181i	$-0.635172\pi$
$269$	−59626.2			−0.824011			−0.412005	+	0.911181i	$0.635172\pi$
$270$		−	15153.8i		−	0.207871i
$271$	−75798.8	−	75798.8i	−1.03210	−	1.03210i	−0.999467	−	0.0326374i	$-0.989609\pi$
$271$	−75798.8	−	75798.8i	−1.03210	−	1.03210i	−0.0326374	−	0.999467i	$-0.510391\pi$
$272$			73159.1i			0.988850i
$273$	17384.0	+	7530.46i	0.233251	+	0.101041i
$274$	−60586.0			−0.806996
$275$	−4131.18	+	4131.18i	−0.0546272	+	0.0546272i
$276$	2460.01			0.0322938
$277$		−	134127.i		−	1.74806i	−0.485876	−	0.874028i	$-0.661499\pi$
$277$		−	134127.i		−	1.74806i	0.485876	−	0.874028i	$-0.338501\pi$
$278$	17161.9	−	17161.9i	0.222063	−	0.222063i
$279$	23623.0	−	23623.0i	0.303477	−	0.303477i
$280$	−25055.2	−	25055.2i	−0.319582	−	0.319582i
$281$	−19968.0	−	19968.0i	−0.252885	−	0.252885i	0.569268	−	0.822152i	$-0.307227\pi$
$281$	−19968.0	−	19968.0i	−0.252885	−	0.252885i	−0.822152	+	0.569268i	$0.807227\pi$
$282$	−11138.3			−0.140063
$283$			151960.i			1.89739i	0.316186	+	0.948697i	$0.397598\pi$
$283$			151960.i			1.89739i	−0.316186	+	0.948697i	$0.602402\pi$
$284$	−534.065	−	534.065i	−0.00662151	−	0.00662151i
$285$			10573.1i			0.130170i
$286$	53986.8	−	21351.4i	0.660017	−	0.261033i
$287$	−30020.6			−0.364465
$288$	1938.70	−	1938.70i	0.0233736	−	0.0233736i
$289$	9043.27			0.108275
$290$			141948.i			1.68784i
$291$	11803.5	−	11803.5i	0.139387	−	0.139387i
$292$	−928.744	+	928.744i	−0.0108926	+	0.0108926i
$293$	22398.8	+	22398.8i	0.260909	+	0.260909i	0.825423	−	0.564514i	$-0.190937\pi$
$293$	22398.8	+	22398.8i	0.260909	+	0.260909i	−0.564514	+	0.825423i	$0.690937\pi$
$294$	−29144.4	−	29144.4i	−0.337179	−	0.337179i
$295$	36394.9			0.418212
$296$		−	35589.6i		−	0.406200i
$297$	−8315.93	−	8315.93i	−0.0942753	−	0.0942753i
$298$			131794.i			1.48410i
$299$	−37057.7	−	93699.8i	−0.414511	−	1.04808i
$300$	−287.566			−0.00319518
$301$	−34127.2	+	34127.2i	−0.376676	+	0.376676i
$302$	−179483.			−1.96793
$303$			33728.1i			0.367372i
$304$	−14634.0	+	14634.0i	−0.158349	+	0.158349i
$305$	−114873.	+	114873.i	−1.23486	+	1.23486i
$306$	21352.1	+	21352.1i	0.228033	+	0.228033i
$307$	36789.1	+	36789.1i	0.390339	+	0.390339i	0.874808	−	0.484469i	$-0.160987\pi$
$307$	36789.1	+	36789.1i	0.390339	+	0.390339i	−0.484469	+	0.874808i	$0.660987\pi$
$308$	1435.99			0.0151373
$309$		−	29176.2i		−	0.305571i
$310$	−94503.0	−	94503.0i	−0.983382	−	0.983382i
$311$			78846.4i			0.815194i	0.913162	+	0.407597i	$0.133633\pi$
$311$			78846.4i			0.815194i	−0.913162	+	0.407597i	$0.866367\pi$
$312$	−50213.0	−	21751.5i	−0.515830	−	0.223450i
$313$	20618.6			0.210460			0.105230	−	0.994448i	$-0.466442\pi$
$313$	20618.6			0.210460			0.105230	+	0.994448i	$0.466442\pi$
$314$	−48881.6	+	48881.6i	−0.495777	+	0.495777i
$315$	−15352.7			−0.154726
$316$		−	6626.72i		−	0.0663627i
$317$	42129.4	−	42129.4i	0.419244	−	0.419244i	−0.465699	−	0.884943i	$-0.654197\pi$
$317$	42129.4	−	42129.4i	0.419244	−	0.419244i	0.884943	+	0.465699i	$0.154197\pi$
$318$	66238.6	−	66238.6i	0.655023	−	0.655023i
$319$	77896.6	+	77896.6i	0.765485	+	0.765485i
$320$	72183.1	+	72183.1i	0.704913	+	0.704913i
$321$	50330.6			0.488453
$322$		−	52712.2i		−	0.508392i
$323$	−14897.7	−	14897.7i	−0.142796	−	0.142796i
$324$		−	578.860i		−	0.00551422i
$325$	4331.90	+	10953.2i	0.0410121	+	0.103698i
$326$	133937.			1.26027
$327$	−85072.0	+	85072.0i	−0.795594	+	0.795594i
$328$	86713.5			0.806007
$329$			11284.6i			0.104254i
$330$	−33267.6	+	33267.6i	−0.305488	+	0.305488i
$331$	68308.6	−	68308.6i	0.623476	−	0.623476i	−0.322943	−	0.946418i	$-0.604672\pi$
$331$	68308.6	−	68308.6i	0.623476	−	0.623476i	0.946418	+	0.322943i	$0.104672\pi$
$332$	−5172.17	−	5172.17i	−0.0469242	−	0.0469242i
$333$	−10903.9	−	10903.9i	−0.0983314	−	0.0983314i
$334$	96285.5			0.863113
$335$			63686.0i			0.567485i
$336$	−21249.4	−	21249.4i	−0.188221	−	0.188221i
$337$		−	87835.7i		−	0.773412i	−0.922203	−	0.386706i	$-0.873613\pi$
$337$		−	87835.7i		−	0.773412i	0.922203	−	0.386706i	$-0.126387\pi$
$338$	3764.29	−	116984.i	0.0329495	−	1.02398i
$339$	−100000.			−0.870165
$340$	4038.70	−	4038.70i	0.0349369	−	0.0349369i
$341$	−103721.			−0.891983
$342$			8542.08i			0.0730317i
$343$	−66154.0	+	66154.0i	−0.562300	+	0.562300i
$344$	98575.2	−	98575.2i	0.833011	−	0.833011i
$345$	57739.5	+	57739.5i	0.485104	+	0.485104i
$346$	26747.6	+	26747.6i	0.223425	+	0.223425i
$347$	98270.4			0.816138			0.408069	−	0.912951i	$-0.366202\pi$
$347$	98270.4			0.816138			0.408069	+	0.912951i	$0.366202\pi$
$348$			5422.27i			0.0447737i
$349$	−17440.9	−	17440.9i	−0.143192	−	0.143192i	0.631877	−	0.775069i	$-0.282285\pi$
$349$	−17440.9	−	17440.9i	−0.143192	−	0.143192i	−0.775069	+	0.631877i	$0.782285\pi$
$350$			6161.85i			0.0503008i
$351$	−22048.3	+	8719.97i	−0.178962	+	0.0707784i
$352$	−8512.19			−0.0686999
$353$	63740.5	−	63740.5i	0.511524	−	0.511524i	−0.403469	−	0.914993i	$-0.632196\pi$
$353$	63740.5	−	63740.5i	0.511524	−	0.511524i	0.914993	+	0.403469i	$0.132196\pi$
$354$	29403.8			0.234637
$355$		−	25070.3i		−	0.198931i
$356$	6643.52	−	6643.52i	0.0524201	−	0.0524201i
$357$	21632.4	−	21632.4i	0.169734	−	0.169734i
$358$	−83373.5	−	83373.5i	−0.650522	−	0.650522i
$359$	−168217.	−	168217.i	−1.30521	−	1.30521i	−0.924831	−	0.380377i	$-0.875794\pi$
$359$	−168217.	−	168217.i	−1.30521	−	1.30521i	−0.380377	−	0.924831i	$-0.624206\pi$
$360$	44345.8			0.342174
$361$			124361.i			0.954267i
$362$	−6765.91	−	6765.91i	−0.0516308	−	0.0516308i
$363$		−	39564.3i		−	0.300255i
$364$	1150.76	−	2656.52i	0.00868527	−	0.0200498i
$365$	−43597.6			−0.327248
$366$	−92806.7	+	92806.7i	−0.692815	+	0.692815i
$367$	4967.29			0.0368797			0.0184399	−	0.999830i	$-0.494130\pi$
$367$	4967.29			0.0368797			0.0184399	+	0.999830i	$0.494130\pi$
$368$			159832.i			1.18023i
$369$	26567.1	−	26567.1i	0.195115	−	0.195115i
$370$	−43620.7	+	43620.7i	−0.318632	+	0.318632i
$371$	−67108.2	−	67108.2i	−0.487560	−	0.487560i
$372$	−3609.93	−	3609.93i	−0.0260863	−	0.0260863i
$373$	−115465.			−0.829914			−0.414957	−	0.909841i	$-0.636203\pi$
$373$	−115465.			−0.829914			−0.414957	+	0.909841i	$0.636203\pi$
$374$		−	93749.9i		−	0.670236i
$375$	53776.8	+	53776.8i	0.382413	+	0.382413i
$376$		−	32595.1i		−	0.230556i
$377$	206530.	−	81681.3i	1.45312	−	0.574698i
$378$	−12403.6			−0.0868088
$379$	−79418.8	+	79418.8i	−0.552898	+	0.552898i	−0.927276	−	0.374378i	$-0.877856\pi$
$379$	−79418.8	+	79418.8i	−0.552898	+	0.552898i	0.374378	+	0.927276i	$0.377856\pi$
$380$	1615.72			0.0111892
$381$		−	136851.i		−	0.942754i
$382$	−33326.2	+	33326.2i	−0.228380	+	0.228380i
$383$	30297.2	−	30297.2i	0.206540	−	0.206540i	−0.596255	−	0.802795i	$-0.703345\pi$
$383$	30297.2	−	30297.2i	0.206540	−	0.206540i	0.802795	+	0.596255i	$0.203345\pi$
$384$	64287.0	+	64287.0i	0.435974	+	0.435974i
$385$	33704.4	+	33704.4i	0.227387	+	0.227387i
$386$	212640.			1.42715
$387$		−	60402.5i		−	0.403304i
$388$	−1803.74	−	1803.74i	−0.0119815	−	0.0119815i
$389$		−	135764.i		−	0.897193i	−0.893734	−	0.448597i	$-0.851924\pi$
$389$		−	135764.i		−	0.897193i	0.893734	−	0.448597i	$-0.148076\pi$
$390$	34884.0	+	88203.7i	0.229349	+	0.579906i
$391$	−162713.			−1.06431
$392$	85287.7	−	85287.7i	0.555028	−	0.555028i
$393$	−16079.0			−0.104106
$394$		−	21879.2i		−	0.140942i
$395$	155537.	−	155537.i	0.996874	−	0.996874i
$396$	−1270.79	+	1270.79i	−0.00810372	+	0.00810372i
$397$	−122596.	−	122596.i	−0.777851	−	0.777851i	0.201614	−	0.979465i	$-0.435381\pi$
$397$	−122596.	−	122596.i	−0.777851	−	0.777851i	−0.979465	+	0.201614i	$0.935381\pi$
$398$	137969.	+	137969.i	0.870996	+	0.870996i
$399$	8654.22			0.0543604
$400$		−	18683.8i		−	0.116774i
$401$	132123.	+	132123.i	0.821653	+	0.821653i	0.986345	−	0.164692i	$-0.0526631\pi$
$401$	132123.	+	132123.i	0.821653	+	0.821653i	−0.164692	+	0.986345i	$0.552663\pi$
$402$			51452.5i			0.318386i
$403$	−83119.1	+	191879.i	−0.511789	+	1.18146i
$404$	5154.14			0.0315786
$405$	13586.6	−	13586.6i	0.0828323	−	0.0828323i
$406$	116186.			0.704860
$407$			47875.4i			0.289017i
$408$	−62484.4	+	62484.4i	−0.375363	+	0.375363i
$409$	2700.68	−	2700.68i	0.0161445	−	0.0161445i	−0.698988	−	0.715133i	$-0.746366\pi$
$409$	2700.68	−	2700.68i	0.0161445	−	0.0161445i	0.715133	+	0.698988i	$0.246366\pi$
$410$	−106281.	−	106281.i	−0.632248	−	0.632248i
$411$	−54320.2	−	54320.2i	−0.321572	−	0.321572i
$412$	−4458.54			−0.0262663
$413$		−	29789.8i		−	0.174650i
$414$	46648.2	+	46648.2i	0.272167	+	0.272167i
$415$		−	242795.i		−	1.40975i
$416$	−6821.45	+	15747.2i	−0.0394176	+	0.0909950i
$417$	30774.1			0.176976
$418$	18752.7	−	18752.7i	0.107328	−	0.107328i
$419$	−94447.5			−0.537975			−0.268988	−	0.963144i	$-0.586689\pi$
$419$	−94447.5			−0.537975			−0.268988	+	0.963144i	$0.586689\pi$
$420$			2346.12i			0.0133000i
$421$	15371.4	−	15371.4i	0.0867262	−	0.0867262i	−0.662413	−	0.749139i	$-0.730468\pi$
$421$	15371.4	−	15371.4i	0.0867262	−	0.0867262i	0.749139	+	0.662413i	$0.230468\pi$
$422$	−219269.	+	219269.i	−1.23127	+	1.23127i
$423$	−9986.42	−	9986.42i	−0.0558122	−	0.0558122i
$424$	193840.	+	193840.i	1.07823	+	1.07823i
$425$	19020.5			0.105304
$426$		−	20254.5i		−	0.111610i
$427$	94025.1	+	94025.1i	0.515689	+	0.515689i
$428$		−	7691.25i		−	0.0419865i
$429$	67546.8	+	29260.2i	0.367020	+	0.158987i
$430$	−241639.			−1.30686
$431$	−42041.0	+	42041.0i	−0.226317	+	0.226317i	−0.811152	−	0.584835i	$-0.801159\pi$
$431$	−42041.0	+	42041.0i	−0.226317	+	0.226317i	0.584835	+	0.811152i	$0.301159\pi$
$432$	37609.8			0.201527
$433$			29381.2i			0.156709i	0.996926	+	0.0783545i	$0.0249666\pi$
$433$			29381.2i			0.156709i	−0.996926	+	0.0783545i	$0.975033\pi$
$434$	−77352.1	+	77352.1i	−0.410670	+	0.410670i
$435$	−127268.	+	127268.i	−0.672572	+	0.672572i
$436$	13000.2	+	13000.2i	0.0683877	+	0.0683877i
$437$	−32547.4	−	32547.4i	−0.170433	−	0.170433i
$438$	−35222.8			−0.183601
$439$			354668.i			1.84032i	0.391542	+	0.920160i	$0.371942\pi$
$439$			354668.i			1.84032i	−0.391542	+	0.920160i	$0.628058\pi$
$440$	−97354.0	−	97354.0i	−0.502862	−	0.502862i
$441$		−	52260.6i		−	0.268718i
$442$	−173434.	−	75128.8i	−0.887747	−	0.384558i
$443$	−22527.8			−0.114792			−0.0573960	−	0.998351i	$-0.518280\pi$
$443$	−22527.8			−0.114792			−0.0573960	+	0.998351i	$0.518280\pi$
$444$	−1666.27	+	1666.27i	−0.00845238	+	0.00845238i
$445$	311863.			1.57487
$446$			5524.30i			0.0277720i
$447$	−118164.	+	118164.i	−0.591384	+	0.591384i
$448$	59082.9	−	59082.9i	0.294378	−	0.294378i
$449$	−53263.7	−	53263.7i	−0.264204	−	0.264204i	0.562556	−	0.826759i	$-0.309818\pi$
$449$	−53263.7	−	53263.7i	−0.264204	−	0.264204i	−0.826759	+	0.562556i	$0.809818\pi$
$450$	−5453.00	−	5453.00i	−0.0269284	−	0.0269284i
$451$	−116647.			−0.573485
$452$			15281.5i			0.0747978i
$453$	−160921.	−	160921.i	−0.784182	−	0.784182i
$454$		−	35411.0i		−	0.171801i
$455$	89361.7	−	35342.0i	0.431647	−	0.170714i
$456$	−24997.4			−0.120217
$457$	94033.6	−	94033.6i	0.450247	−	0.450247i	−0.445189	−	0.895436i	$-0.646864\pi$
$457$	94033.6	−	94033.6i	0.450247	−	0.450247i	0.895436	+	0.445189i	$0.146864\pi$
$458$	55503.7			0.264601
$459$			38287.7i			0.181733i
$460$	8823.43	−	8823.43i	0.0416986	−	0.0416986i
$461$	−220771.	+	220771.i	−1.03882	+	1.03882i	−0.0396051	+	0.999215i	$0.512610\pi$
$461$	−220771.	+	220771.i	−1.03882	+	1.03882i	−0.999215	+	0.0396051i	$0.987390\pi$
$462$	27230.1	+	27230.1i	0.127575	+	0.127575i
$463$	−148393.	−	148393.i	−0.692231	−	0.692231i	0.270491	−	0.962722i	$-0.412814\pi$
$463$	−148393.	−	148393.i	−0.692231	−	0.692231i	−0.962722	+	0.270491i	$0.912814\pi$
$464$	−352296.			−1.63634
$465$		−	169459.i		−	0.783716i
$466$	210962.	+	210962.i	0.971474	+	0.971474i
$467$			89596.5i			0.410826i	0.978675	+	0.205413i	$0.0658537\pi$
$467$			89596.5i			0.410826i	−0.978675	+	0.205413i	$0.934146\pi$
$468$	1332.54	+	3369.30i	0.00608398	+	0.0153832i
$469$	52128.0			0.236987
$470$	−39950.4	+	39950.4i	−0.180853	+	0.180853i
$471$	−87652.6			−0.395114
$472$			86046.8i			0.386234i
$473$	−132604.	+	132604.i	−0.592699	+	0.592699i
$474$	125660.	−	125660.i	0.559294	−	0.559294i
$475$	3804.66	+	3804.66i	0.0168628	+	0.0168628i
$476$	−3305.74	−	3305.74i	−0.0145900	−	0.0145900i
$477$	118776.			0.522028
$478$		−	421007.i		−	1.84261i
$479$	290497.	+	290497.i	1.26611	+	1.26611i	0.948083	+	0.318023i	$0.103019\pi$
$479$	290497.	+	290497.i	1.26611	+	1.26611i	0.318023	+	0.948083i	$0.396981\pi$
$480$		−	13907.2i		−	0.0603613i
$481$	88567.6	+	38366.1i	0.382811	+	0.165828i
$482$	372782.			1.60458
$483$	47260.7	−	47260.7i	0.202584	−	0.202584i
$484$	−6046.00			−0.0258094
$485$		−	84672.0i		−	0.359962i
$486$	10976.7	−	10976.7i	0.0464729	−	0.0464729i
$487$	186744.	−	186744.i	0.787388	−	0.787388i	−0.193677	−	0.981065i	$-0.562041\pi$
$487$	186744.	−	186744.i	0.787388	−	0.787388i	0.981065	+	0.193677i	$0.0620414\pi$
$488$	−271588.	−	271588.i	−1.14044	−	1.14044i
$489$	120085.	+	120085.i	0.502195	+	0.502195i
$490$	−209067.			−0.870750
$491$			26475.0i			0.109818i	0.998491	+	0.0549088i	$0.0174868\pi$
$491$			26475.0i			0.109818i	−0.998491	+	0.0549088i	$0.982513\pi$
$492$	−4059.83	−	4059.83i	−0.0167717	−	0.0167717i
$493$		−	358646.i		−	1.47561i
$494$	−19663.9	−	49719.8i	−0.0805778	−	0.203740i
$495$	−59654.2			−0.243462
$496$	234545.	−	234545.i	0.953372	−	0.953372i
$497$	−20520.4			−0.0830757
$498$		−	196156.i		−	0.790938i
$499$	72703.0	−	72703.0i	0.291979	−	0.291979i	−0.545883	−	0.837862i	$-0.683806\pi$
$499$	72703.0	−	72703.0i	0.291979	−	0.291979i	0.837862	+	0.545883i	$0.183806\pi$
$500$	8217.87	−	8217.87i	0.0328715	−	0.0328715i
$501$	86327.6	+	86327.6i	0.343933	+	0.343933i
$502$	142841.	+	142841.i	0.566822	+	0.566822i
$503$	245797.			0.971496			0.485748	−	0.874099i	$-0.338547\pi$
$503$	245797.			0.971496			0.485748	+	0.874099i	$0.338547\pi$
$504$		−	36297.7i		−	0.142895i
$505$	120974.	+	120974.i	0.474362	+	0.474362i
$506$		−	204817.i		−	0.799955i
$507$	108261.	−	101511.i	0.421167	−	0.394907i
$508$	−20912.8			−0.0810374
$509$	−105992.	+	105992.i	−0.409109	+	0.409109i	−0.881428	−	0.472319i	$-0.843417\pi$
$509$	−105992.	+	105992.i	−0.409109	+	0.409109i	0.472319	+	0.881428i	$0.343417\pi$
$510$	153169.			0.588884
$511$			35685.3i			0.136662i
$512$	−169747.	+	169747.i	−0.647533	+	0.647533i
$513$	−7658.66	+	7658.66i	−0.0291017	+	0.0291017i
$514$	238846.	+	238846.i	0.904049	+	0.904049i
$515$	−104648.	−	104648.i	−0.394561	−	0.394561i
$516$	−9230.37			−0.0346673
$517$			43847.1i			0.164044i
$518$	35704.2	+	35704.2i	0.133064	+	0.133064i
$519$			47962.7i			0.178061i
$520$	−258118.	+	102084.i	−0.954579	+	0.377530i
$521$	408225.			1.50392			0.751959	−	0.659209i	$-0.229109\pi$
$521$	408225.			1.50392			0.751959	+	0.659209i	$0.229109\pi$
$522$	−102820.	+	102820.i	−0.377345	+	0.377345i
$523$	−145126.			−0.530567			−0.265284	−	0.964170i	$-0.585466\pi$
$523$	−145126.			−0.530567			−0.265284	+	0.964170i	$0.585466\pi$
$524$			2457.10i			0.00894872i
$525$	−5524.59	+	5524.59i	−0.0200439	+	0.0200439i
$526$	119873.	−	119873.i	0.433260	−	0.433260i
$527$	238772.	+	238772.i	0.859731	+	0.859731i
$528$	−82566.1	−	82566.1i	−0.296165	−	0.296165i
$529$	−75641.2			−0.270300
$530$		−	475162.i		−	1.69157i
$531$	26362.8	+	26362.8i	0.0934982	+	0.0934982i
$532$		−	1322.49i		−	0.00467271i
$533$	−93478.3	+	215793.i	−0.329046	+	0.759597i
$534$	251957.			0.883576
$535$	180523.	−	180523.i	0.630704	−	0.630704i
$536$	−150570.			−0.524093
$537$		−	149502.i		−	0.518440i
$538$	−172783.	+	172783.i	−0.596947	+	0.596947i
$539$	−114730.	+	114730.i	−0.394910	+	0.394910i
$540$	−2076.22	−	2076.22i	−0.00712011	−	0.00712011i
$541$	−376207.	−	376207.i	−1.28538	−	1.28538i	−0.937560	−	0.347823i	$-0.886921\pi$
$541$	−376207.	−	376207.i	−1.28538	−	1.28538i	−0.347823	−	0.937560i	$-0.613079\pi$
$542$	−439294.			−1.49540
$543$		−	12132.4i		−	0.0411477i
$544$	19595.6	+	19595.6i	0.0662158	+	0.0662158i
$545$			610263.i			2.05459i
$546$	72196.1	−	28553.1i	0.242174	−	0.0957785i
$547$	−110782.			−0.370249			−0.185124	−	0.982715i	$-0.559269\pi$
$547$	−110782.			−0.370249			−0.185124	+	0.982715i	$0.559269\pi$
$548$	−8300.91	+	8300.91i	−0.0276417	+	0.0276417i
$549$	−166417.			−0.552146
$550$			23942.4i			0.0791483i
$551$	71739.8	−	71739.8i	0.236296	−	0.236296i
$552$	−136511.	+	136511.i	−0.448011	+	0.448011i
$553$	−127310.	−	127310.i	−0.416304	−	0.416304i
$554$	−388667.	−	388667.i	−1.26636	−	1.26636i
$555$	−78218.8			−0.253937
$556$		−	4702.73i		−	0.0152125i
$557$	141690.	+	141690.i	0.456696	+	0.456696i	0.897569	−	0.440873i	$-0.145331\pi$
$557$	141690.	+	141690.i	0.456696	+	0.456696i	−0.440873	+	0.897569i	$0.645331\pi$
$558$		−	136907.i		−	0.439702i
$559$	139047.	+	351577.i	0.444977	+	1.12512i
$560$	−152432.			−0.486072
$561$	84054.3	−	84054.3i	0.267076	−	0.267076i
$562$	−115725.			−0.366400
$563$		−	301845.i		−	0.952287i	−0.879368	−	0.476143i	$-0.842034\pi$
$563$		−	301845.i		−	0.952287i	0.879368	−	0.476143i	$-0.157966\pi$
$564$	−1526.07	+	1526.07i	−0.00479751	+	0.00479751i
$565$	−358675.	+	358675.i	−1.12358	+	1.12358i
$566$	440345.	+	440345.i	1.37455	+	1.37455i
$567$	−11120.8	−	11120.8i	−0.0345916	−	0.0345916i
$568$	59272.6			0.183720
$569$			149750.i			0.462532i	0.972891	+	0.231266i	$0.0742867\pi$
$569$			149750.i			0.462532i	−0.972891	+	0.231266i	$0.925713\pi$
$570$	30638.3	+	30638.3i	0.0943006	+	0.0943006i
$571$			477531.i			1.46463i	0.680964	+	0.732317i	$0.261561\pi$
$571$			477531.i			1.46463i	−0.680964	+	0.732317i	$0.738439\pi$
$572$	4471.38	−	10322.1i	0.0136663	−	0.0315484i
$573$	−59759.2			−0.182010
$574$	−86992.5	+	86992.5i	−0.264033	+	0.264033i
$575$	41554.5			0.125685
$576$			104572.i			0.315189i
$577$	108891.	−	108891.i	0.327069	−	0.327069i	−0.524402	−	0.851471i	$-0.675711\pi$
$577$	108891.	−	108891.i	0.327069	−	0.327069i	0.851471	+	0.524402i	$0.175711\pi$
$578$	26205.2	−	26205.2i	0.0784391	−	0.0784391i
$579$	190649.	+	190649.i	0.568691	+	0.568691i
$580$	19448.3	+	19448.3i	0.0578131	+	0.0578131i
$581$	−198731.			−0.588726
$582$		−	68407.2i		−	0.201956i
$583$	−260754.	−	260754.i	−0.767175	−	0.767175i
$584$		−	103076.i		−	0.302225i
$585$	−47805.4	+	110358.i	−0.139690	+	0.322472i
$586$	129813.			0.378026
$587$	379654.	−	379654.i	1.10182	−	1.10182i	0.107632	−	0.994191i	$-0.465673\pi$
$587$	379654.	−	379654.i	1.10182	−	1.10182i	0.994191	−	0.107632i	$-0.0343270\pi$
$588$	−7986.17			−0.0230985
$589$			95522.9i			0.275345i
$590$	105464.	−	105464.i	0.302970	−	0.302970i
$591$	19616.5	−	19616.5i	0.0561625	−	0.0561625i
$592$	−108261.	−	108261.i	−0.308908	−	0.308908i
$593$	20445.7	+	20445.7i	0.0581423	+	0.0581423i	0.735580	−	0.677438i	$-0.236910\pi$
$593$	20445.7	+	20445.7i	0.0581423	+	0.0581423i	−0.677438	+	0.735580i	$0.736910\pi$
$594$	−48195.2			−0.136594
$595$		−	155180.i		−	0.438330i
$596$	18057.1	+	18057.1i	0.0508343	+	0.0508343i
$597$			247401.i			0.694149i
$598$	−378904.	−	164135.i	−1.05956	−	0.458986i
$599$	45314.1			0.126293			0.0631466	−	0.998004i	$-0.479886\pi$
$599$	45314.1			0.126293			0.0631466	+	0.998004i	$0.479886\pi$
$600$	15957.6	−	15957.6i	0.0443267	−	0.0443267i
$601$	−71319.6			−0.197451			−0.0987256	−	0.995115i	$-0.531477\pi$
$601$	−71319.6			−0.197451			−0.0987256	+	0.995115i	$0.531477\pi$
$602$			197785.i			0.545758i
$603$	−46131.3	+	46131.3i	−0.126871	+	0.126871i
$604$	−24591.1	+	24591.1i	−0.0674068	+	0.0674068i
$605$	−141907.	−	141907.i	−0.387698	−	0.387698i
$606$	97736.0	+	97736.0i	0.266139	+	0.266139i
$607$	37835.4			0.102688			0.0513441	−	0.998681i	$-0.483649\pi$
$607$	37835.4			0.102688			0.0513441	+	0.998681i	$0.483649\pi$
$608$			7839.41i			0.0212069i
$609$	104170.	+	104170.i	0.280873	+	0.280873i
$610$			665748.i			1.78916i
$611$	81115.5	+	35138.0i	0.217281	+	0.0941226i
$612$	5850.91			0.0156214
$613$	269915.	−	269915.i	0.718301	−	0.718301i	−0.249956	−	0.968257i	$-0.580416\pi$
$613$	269915.	−	269915.i	0.718301	−	0.718301i	0.968257	+	0.249956i	$0.0804162\pi$
$614$	213212.			0.565555
$615$		−	190579.i		−	0.503877i
$616$	−79685.7	+	79685.7i	−0.210000	+	0.210000i
$617$	128460.	−	128460.i	0.337440	−	0.337440i	−0.517963	−	0.855403i	$-0.673310\pi$
$617$	128460.	−	128460.i	0.337440	−	0.337440i	0.855403	+	0.517963i	$0.173310\pi$
$618$	−84545.6	−	84545.6i	−0.221368	−	0.221368i
$619$	−78942.1	−	78942.1i	−0.206029	−	0.206029i	0.596548	−	0.802577i	$-0.296538\pi$
$619$	−78942.1	−	78942.1i	−0.206029	−	0.206029i	−0.802577	+	0.596548i	$0.796538\pi$
$620$	−25895.8			−0.0673668
$621$			83647.8i			0.216906i
$622$	228478.	+	228478.i	0.590560	+	0.590560i
$623$		−	255265.i		−	0.657680i
$624$	−218911.	+	86577.8i	−0.562209	+	0.222350i
$625$	429328.			1.09908
$626$	59747.7	−	59747.7i	0.152466	−	0.152466i
$627$	33626.7			0.0855360
$628$			13394.6i			0.0339633i
$629$	110212.	−	110212.i	0.278567	−	0.278567i
$630$	−44488.5	+	44488.5i	−0.112090	+	0.112090i
$631$	−138220.	−	138220.i	−0.347146	−	0.347146i	0.511899	−	0.859045i	$-0.328942\pi$
$631$	−138220.	−	138220.i	−0.347146	−	0.347146i	−0.859045	+	0.511899i	$0.828942\pi$
$632$	367729.	+	367729.i	0.920649	+	0.920649i
$633$	−393185.			−0.981271
$634$		−	244162.i		−	0.607435i
$635$	−490850.	−	490850.i	−1.21731	−	1.21731i
$636$		−	18150.7i		−	0.0448725i
$637$	120304.	+	304187.i	0.296484	+	0.749655i
$638$	451451.			1.10910
$639$	18159.8	−	18159.8i	0.0444743	−	0.0444743i
$640$	461162.			1.12588
$641$		−	369011.i		−	0.898098i	−0.893507	−	0.449049i	$-0.851763\pi$
$641$		−	369011.i		−	0.898098i	0.893507	−	0.449049i	$-0.148237\pi$
$642$	145846.	−	145846.i	0.353855	−	0.353855i
$643$	111450.	−	111450.i	0.269561	−	0.269561i	−0.559362	−	0.828923i	$-0.688954\pi$
$643$	111450.	−	111450.i	0.269561	−	0.269561i	0.828923	+	0.559362i	$0.188954\pi$
$644$	−7222.11	−	7222.11i	−0.0174138	−	0.0174138i
$645$	−216648.	−	216648.i	−0.520758	−	0.520758i
$646$	−86340.2			−0.206894
$647$		−	401104.i		−	0.958184i	−0.877765	−	0.479092i	$-0.840966\pi$
$647$		−	401104.i		−	0.958184i	0.877765	−	0.479092i	$-0.159034\pi$
$648$	32122.1	+	32122.1i	0.0764987	+	0.0764987i
$649$		−	115751.i		−	0.274811i
$650$	44292.5	+	19186.8i	0.104834	+	0.0454125i
$651$	−138705.			−0.327288
$652$	18350.8	−	18350.8i	0.0431677	−	0.0431677i
$653$	−341135.			−0.800018			−0.400009	−	0.916511i	$-0.630993\pi$
$653$	−341135.			−0.800018			−0.400009	+	0.916511i	$0.630993\pi$
$654$			493037.i			1.15272i
$655$	−57671.3	+	57671.3i	−0.134424	+	0.134424i
$656$	263776.	−	263776.i	0.612954	−	0.612954i
$657$	−31580.1	−	31580.1i	−0.0731615	−	0.0731615i
$658$	32700.0	+	32700.0i	0.0755259	+	0.0755259i
$659$	32739.6			0.0753881			0.0376941	−	0.999289i	$-0.487999\pi$
$659$	32739.6			0.0753881			0.0376941	+	0.999289i	$0.487999\pi$
$660$			9116.03i			0.0209275i
$661$	20237.5	+	20237.5i	0.0463184	+	0.0463184i	0.729887	−	0.683568i	$-0.239573\pi$
$661$	20237.5	+	20237.5i	0.0463184	+	0.0463184i	−0.683568	+	0.729887i	$0.739573\pi$
$662$		−	395884.i		−	0.903342i
$663$	−88138.3	−	222856.i	−0.200511	−	0.506988i
$664$	574028.			1.30196
$665$	31040.5	−	31040.5i	0.0701916	−	0.0701916i
$666$	−63193.7			−0.142471
$667$		−	783542.i		−	1.76121i
$668$	13192.1	−	13192.1i	0.0295639	−	0.0295639i
$669$	−4952.98	+	4952.98i	−0.0110666	+	0.0110666i
$670$	184547.	+	184547.i	0.411109	+	0.411109i
$671$	365342.	+	365342.i	0.811437	+	0.811437i
$672$	−11383.3			−0.0252074
$673$			151692.i			0.334914i	0.985879	+	0.167457i	$0.0535556\pi$
$673$			151692.i			0.334914i	−0.985879	+	0.167457i	$0.946444\pi$
$674$	−254527.	−	254527.i	−0.560291	−	0.560291i
$675$		−	9778.11i		−	0.0214609i
$676$	−15512.3	−	16543.8i	−0.0339455	−	0.0362027i
$677$	−753102.			−1.64315			−0.821574	−	0.570102i	$-0.806904\pi$
$677$	−753102.			−1.64315			−0.821574	+	0.570102i	$0.806904\pi$
$678$	−289777.	+	289777.i	−0.630383	+	0.630383i
$679$	−69305.3			−0.150323
$680$			448231.i			0.969358i
$681$	31748.8	−	31748.8i	0.0684594	−	0.0684594i
$682$	−300558.	+	300558.i	−0.646189	+	0.646189i
$683$	50909.1	+	50909.1i	0.109132	+	0.109132i	0.759564	−	0.650432i	$-0.225412\pi$
$683$	50909.1	+	50909.1i	0.109132	+	0.109132i	−0.650432	+	0.759564i	$0.725412\pi$
$684$	1170.35	+	1170.35i	0.00250152	+	0.00250152i
$685$	−389666.			−0.830445
$686$			383397.i			0.814705i
$687$	49763.5	+	49763.5i	0.105438	+	0.105438i
$688$		−	599717.i		−	1.26698i
$689$	−691347.	+	273424.i	−1.45632	+	0.575967i
$690$	334631.			0.702858
$691$	−281346.	+	281346.i	−0.589230	+	0.589230i	−0.937423	−	0.348193i	$-0.886795\pi$
$691$	−281346.	+	281346.i	−0.589230	+	0.589230i	0.348193	+	0.937423i	$0.386795\pi$
$692$	7329.40			0.0153058
$693$			48827.9i			0.101672i
$694$	284764.	−	284764.i	0.591243	−	0.591243i
$695$	110379.	−	110379.i	0.228516	−	0.228516i
$696$	−300893.	−	300893.i	−0.621145	−	0.621145i
$697$	268530.	+	268530.i	0.552749	+	0.552749i
$698$	−101079.			−0.207468
$699$			378288.i			0.774227i
$700$	844.238	+	844.238i	0.00172293	+	0.00172293i
$701$			212436.i			0.432307i	0.976359	+	0.216154i	$0.0693512\pi$
$701$			212436.i			0.432307i	−0.976359	+	0.216154i	$0.930649\pi$
$702$	−38622.4	+	89159.2i	−0.0783727	+	0.180922i
$703$	44091.4			0.0892161
$704$	229571.	−	229571.i	0.463204	−	0.463204i
$705$	−71637.5			−0.144133
$706$		−	369410.i		−	0.741138i
$707$	99019.1	−	99019.1i	0.198098	−	0.198098i
$708$	4028.62	−	4028.62i	0.00803693	−	0.00803693i
$709$	27433.3	+	27433.3i	0.0545741	+	0.0545741i	0.733867	−	0.679293i	$-0.237713\pi$
$709$	27433.3	+	27433.3i	0.0545741	+	0.0545741i	−0.679293	+	0.733867i	$0.737713\pi$
$710$	−72647.8	−	72647.8i	−0.144114	−	0.144114i
$711$	225328.			0.445735
$712$			737324.i			1.45445i
$713$	521650.	+	521650.i	1.02613	+	1.02613i
$714$		−	125371.i		−	0.245924i
$715$	347222.	−	137324.i	0.679196	−	0.268618i
$716$	−22846.1			−0.0445641
$717$	377467.	−	377467.i	0.734244	−	0.734244i
$718$	−974903.			−1.89109
$719$		−	537471.i		−	1.03967i	−0.854265	−	0.519837i	$-0.825993\pi$
$719$		−	537471.i		−	1.03967i	0.854265	−	0.519837i	$-0.174007\pi$
$720$	134897.	−	134897.i	0.260217	−	0.260217i
$721$	−85655.6	+	85655.6i	−0.164773	+	0.164773i
$722$	360369.	+	360369.i	0.691310	+	0.691310i
$723$	334229.	+	334229.i	0.639393	+	0.639393i
$724$	−1854.00			−0.00353698
$725$			91593.1i			0.174256i
$726$	−114648.	−	114648.i	−0.217517	−	0.217517i
$727$			330967.i			0.626204i	0.949720	+	0.313102i	$0.101368\pi$
$727$			330967.i			0.626204i	−0.949720	+	0.313102i	$0.898632\pi$
$728$	83557.4	+	211274.i	0.157660	+	0.398642i
$729$	19683.0			0.0370370
$730$	−126335.	+	126335.i	−0.237071	+	0.237071i
$731$	610526.			1.14254
$732$			25431.0i			0.0474614i
$733$	−164154.	+	164154.i	−0.305522	+	0.305522i	−0.843170	−	0.537648i	$-0.819313\pi$
$733$	−164154.	+	164154.i	−0.305522	+	0.305522i	0.537648	+	0.843170i	$0.319313\pi$
$734$	14394.0	−	14394.0i	0.0267171	−	0.0267171i
$735$	−187445.	−	187445.i	−0.346976	−	0.346976i
$736$	42811.0	+	42811.0i	0.0790314	+	0.0790314i
$737$	202547.			0.372899
$738$		−	153970.i		−	0.282699i
$739$	−268903.	−	268903.i	−0.492387	−	0.492387i	0.416670	−	0.909058i	$-0.363197\pi$
$739$	−268903.	−	268903.i	−0.492387	−	0.492387i	−0.909058	+	0.416670i	$0.863197\pi$
$740$			11953.0i			0.0218279i
$741$	26947.6	−	62208.0i	0.0490776	−	0.113295i
$742$	−388927.			−0.706416
$743$	−397022.	+	397022.i	−0.719180	+	0.719180i	−0.968437	−	0.249258i	$-0.919813\pi$
$743$	−397022.	+	397022.i	−0.719180	+	0.719180i	0.249258	+	0.968437i	$0.419813\pi$
$744$	400644.			0.723790
$745$			847648.i			1.52722i
$746$	−334590.	+	334590.i	−0.601223	+	0.601223i
$747$	175869.	−	175869.i	0.315173	−	0.315173i
$748$	−12844.7	−	12844.7i	−0.0229573	−	0.0229573i
$749$	−147761.	−	147761.i	−0.263388	−	0.263388i
$750$	311665.			0.554071
$751$		−	100943.i		−	0.178976i	−0.995988	−	0.0894882i	$-0.971477\pi$
$751$		−	100943.i		−	0.178976i	0.995988	−	0.0894882i	$-0.0285231\pi$
$752$	−99151.9	−	99151.9i	−0.175334	−	0.175334i
$753$			256137.i			0.451734i
$754$	361782.	−	835168.i	0.636361	−	1.46903i
$755$	−1.15437e6			−2.02512
$756$	−1699.42	+	1699.42i	−0.00297343	+	0.00297343i
$757$	−418346.			−0.730035			−0.365018	−	0.931001i	$-0.618937\pi$
$757$	−418346.			−0.730035			−0.365018	+	0.931001i	$0.618937\pi$
$758$			460273.i			0.801083i
$759$	183635.	−	183635.i	0.318766	−	0.318766i
$760$	−89659.4	+	89659.4i	−0.155227	+	0.155227i
$761$	−70616.1	−	70616.1i	−0.121937	−	0.121937i	0.643505	−	0.765442i	$-0.277480\pi$
$761$	−70616.1	−	70616.1i	−0.121937	−	0.121937i	−0.765442	+	0.643505i	$0.777480\pi$
$762$	−396562.	−	396562.i	−0.682969	−	0.682969i
$763$	499510.			0.858015
$764$			9132.06i			0.0156452i
$765$	137328.	+	137328.i	0.234659	+	0.234659i
$766$		−	175588.i		−	0.299253i
$767$	−214134.	−	92759.6i	−0.363995	−	0.157677i
$768$	50577.4			0.0857499
$769$	−642589.	+	642589.i	−1.08663	+	1.08663i	−0.0907545	+	0.995873i	$0.528928\pi$
$769$	−642589.	+	642589.i	−1.08663	+	1.08663i	−0.995873	+	0.0907545i	$0.971072\pi$
$770$	195335.			0.329456
$771$			428290.i			0.720491i
$772$	29133.9	−	29133.9i	0.0488836	−	0.0488836i
$773$	152962.	−	152962.i	0.255990	−	0.255990i	−0.567431	−	0.823421i	$-0.692063\pi$
$773$	152962.	−	152962.i	0.255990	−	0.255990i	0.823421	+	0.567431i	$0.192063\pi$
$774$	−175032.	−	175032.i	−0.292170	−	0.292170i
$775$	−60978.9	−	60978.9i	−0.101526	−	0.101526i
$776$	200186.			0.332438
$777$			64023.3i			0.106046i
$778$	−393412.	−	393412.i	−0.649963	−	0.649963i
$779$			107428.i			0.177028i
$780$	16864.3	+	7305.35i	0.0277191	+	0.0120075i
$781$	−79733.8			−0.130720
$782$	−471503.	+	471503.i	−0.771030	+	0.771030i
$783$	−184374.			−0.300729
$784$		−	518878.i		−	0.844177i
$785$	−314387.	+	314387.i	−0.510183	+	0.510183i
$786$	−46593.1	+	46593.1i	−0.0754183	+	0.0754183i
$787$	210034.	+	210034.i	0.339110	+	0.339110i	0.856032	−	0.516922i	$-0.172922\pi$
$787$	210034.	+	210034.i	0.339110	+	0.339110i	−0.516922	+	0.856032i	$0.672922\pi$
$788$	−2997.68	−	2997.68i	−0.00482762	−	0.00482762i
$789$	214951.			0.345291
$790$		−	901420.i		−	1.44435i
$791$	293581.	+	293581.i	0.469219	+	0.469219i
$792$		−	141038.i		−	0.224846i
$793$	968645.	−	383093.i	1.54035	−	0.609197i
$794$	−710510.			−1.12701
$795$	426021.	−	426021.i	0.674057	−	0.674057i
$796$	37806.4			0.0596677
$797$		−	508211.i		−	0.800069i	−0.916500	−	0.400035i	$-0.868998\pi$
$797$		−	508211.i		−	0.800069i	0.916500	−	0.400035i	$-0.131002\pi$
$798$	25077.9	−	25077.9i	0.0393808	−	0.0393808i
$799$	100939.	−	100939.i	0.158112	−	0.158112i
$800$	−5004.44	−	5004.44i	−0.00781944	−	0.00781944i
$801$	225900.	+	225900.i	0.352087	+	0.352087i
$802$	765719.			1.19048
$803$			138658.i			0.215037i
$804$	7049.52	+	7049.52i	0.0109056	+	0.0109056i
$805$		−	339024.i		−	0.523165i
$806$	315161.	+	796880.i	0.485135	+	1.22666i
$807$	−309827.			−0.475743
$808$	−286013.	+	286013.i	−0.438090	+	0.438090i
$809$	−302513.			−0.462218			−0.231109	−	0.972928i	$-0.574235\pi$
$809$	−302513.			−0.462218			−0.231109	+	0.972928i	$0.574235\pi$
$810$		−	78741.3i		−	0.120014i
$811$	261828.	−	261828.i	0.398083	−	0.398083i	−0.479473	−	0.877557i	$-0.659172\pi$
$811$	261828.	−	261828.i	0.398083	−	0.398083i	0.877557	+	0.479473i	$0.159172\pi$
$812$	15918.7	−	15918.7i	0.0241433	−	0.0241433i
$813$	−393862.	−	393862.i	−0.595886	−	0.595886i
$814$	138731.	+	138731.i	0.209376	+	0.209376i
$815$	861431.			1.29690
$816$			380146.i			0.570913i
$817$	122123.	+	122123.i	0.182959	+	0.182959i
$818$		−	15651.8i		−	0.0233915i
$819$	90329.7	+	39129.4i	0.134668	+	0.0583359i
$820$	−29123.2			−0.0433123
$821$	−141594.	+	141594.i	−0.210068	+	0.210068i	−0.804296	−	0.594228i	$-0.797458\pi$
$821$	−141594.	+	141594.i	−0.210068	+	0.210068i	0.594228	+	0.804296i	$0.297458\pi$
$822$	−314814.			−0.465919
$823$			1.05798e6i			1.56199i	0.624540	+	0.780993i	$0.285287\pi$
$823$			1.05798e6i			1.56199i	−0.624540	+	0.780993i	$0.714713\pi$
$824$	247413.	−	247413.i	0.364392	−	0.364392i
$825$	−21466.3	+	21466.3i	−0.0315390	+	0.0315390i
$826$	−86323.7	−	86323.7i	−0.126523	−	0.126523i
$827$	95483.6	+	95483.6i	0.139610	+	0.139610i	0.773458	−	0.633848i	$-0.218525\pi$
$827$	95483.6	+	95483.6i	0.139610	+	0.139610i	−0.633848	+	0.773458i	$0.718525\pi$
$828$	12782.6			0.0186448
$829$		−	623513.i		−	0.907270i	−0.891188	−	0.453635i	$-0.850127\pi$
$829$		−	623513.i		−	0.907270i	0.891188	−	0.453635i	$-0.149873\pi$
$830$	−703561.	−	703561.i	−1.02128	−	1.02128i
$831$		−	696942.i		−	1.00924i
$832$	−240726.	−	608671.i	−0.347757	−	0.879298i
$833$	528231.			0.761261
$834$	89176.1	−	89176.1i	0.128208	−	0.128208i
$835$	619271.			0.888193
$836$		−	5138.64i		−	0.00735251i
$837$	122748.	−	122748.i	0.175213	−	0.175213i
$838$	−273686.	+	273686.i	−0.389731	+	0.389731i
$839$	−666830.	−	666830.i	−0.947308	−	0.947308i	0.0513717	−	0.998680i	$-0.483641\pi$
$839$	−666830.	−	666830.i	−0.947308	−	0.947308i	−0.998680	+	0.0513717i	$0.983641\pi$
$840$	−130191.	−	130191.i	−0.184511	−	0.184511i
$841$	1.01978e6			1.44182
$842$		−	89085.5i		−	0.125656i
$843$	−103757.	−	103757.i	−0.146003	−	0.146003i
$844$			60084.3i			0.0843482i
$845$	24210.4	−	752395.i	0.0339070	−	1.05374i
$846$	−57876.5			−0.0808652
$847$	−116153.	+	116153.i	−0.161907	+	0.161907i
$848$	1.17929e6			1.63995
$849$			789610.i			1.09546i
$850$	55117.0	−	55117.0i	0.0762864	−	0.0762864i
$851$	240783.	−	240783.i	0.332481	−	0.332481i
$852$	−2775.08	−	2775.08i	−0.00382293	−	0.00382293i
$853$	−515777.	−	515777.i	−0.708866	−	0.708866i	0.257431	−	0.966297i	$-0.417124\pi$
$853$	−515777.	−	515777.i	−0.708866	−	0.708866i	−0.966297	+	0.257431i	$0.917124\pi$
$854$	544925.			0.747172
$855$			54939.3i			0.0751538i
$856$	426802.	+	426802.i	0.582478	+	0.582478i
$857$		−	489846.i		−	0.666957i	−0.942758	−	0.333479i	$-0.891778\pi$
$857$		−	489846.i		−	0.666957i	0.942758	−	0.333479i	$-0.108222\pi$
$858$	280524.	−	110945.i	0.381061	−	0.150707i
$859$	1.37694e6			1.86607			0.933037	−	0.359780i	$-0.117148\pi$
$859$	1.37694e6			1.86607			0.933037	+	0.359780i	$0.117148\pi$
$860$	−33107.0	+	33107.0i	−0.0447634	+	0.0447634i
$861$	−155992.			−0.210424
$862$			243649.i			0.327907i
$863$	−715898.	+	715898.i	−0.961235	+	0.961235i	−0.999276	−	0.0380415i	$-0.987888\pi$
$863$	−715898.	+	715898.i	−0.961235	+	0.961235i	0.0380415	+	0.999276i	$0.487888\pi$
$864$	10073.8	−	10073.8i	0.0134947	−	0.0134947i
$865$	172030.	+	172030.i	0.229918	+	0.229918i
$866$	85139.8	+	85139.8i	0.113526	+	0.113526i
$867$	46990.2			0.0625128
$868$			21196.1i			0.0281330i
$869$	−494672.	−	494672.i	−0.655055	−	0.655055i
$870$			737582.i			0.974477i
$871$	162316.	−	374705.i	0.213957	−	0.493916i
$872$	−1.44282e6			−1.89748
$873$	61332.5	−	61332.5i	0.0804753	−	0.0804753i
$874$	−188629.			−0.246937
$875$		−	315756.i		−	0.412417i
$876$	−4825.90	+	4825.90i	−0.00628883	+	0.00628883i
$877$	−123217.	+	123217.i	−0.160203	+	0.160203i	−0.782657	−	0.622454i	$-0.786136\pi$
$877$	−123217.	+	123217.i	−0.160203	+	0.160203i	0.622454	+	0.782657i	$0.286136\pi$
$878$	1.02774e6	+	1.02774e6i	1.33320	+	1.33320i
$879$	116387.	+	116387.i	0.150636	+	0.150636i
$880$	−592288.			−0.764834
$881$			863892.i			1.11303i	0.830837	+	0.556516i	$0.187862\pi$
$881$			863892.i			1.11303i	−0.830837	+	0.556516i	$0.812138\pi$
$882$	−151439.	−	151439.i	−0.194670	−	0.194670i
$883$			268829.i			0.344790i	0.985028	+	0.172395i	$0.0551505\pi$
$883$			268829.i			0.344790i	−0.985028	+	0.172395i	$0.944850\pi$
$884$	−34055.7	+	13468.8i	−0.0435797	+	0.0172355i
$885$	189114.			0.241455
$886$	−65280.2	+	65280.2i	−0.0831599	+	0.0831599i
$887$	1.04691e6			1.33065			0.665323	−	0.746556i	$-0.268294\pi$
$887$	1.04691e6			1.33065			0.665323	+	0.746556i	$0.268294\pi$
$888$		−	184929.i		−	0.234520i
$889$	−401768.	+	401768.i	−0.508361	+	0.508361i
$890$	903706.	−	903706.i	1.14090	−	1.14090i
$891$	−43210.8	−	43210.8i	−0.0544299	−	0.0544299i
$892$	756.886	+	756.886i	0.000951263	+	0.000951263i
$893$	40381.5			0.0506384
$894$			684822.i			0.856845i
$895$	−536226.	−	536226.i	−0.669425	−	0.669425i
$896$		−	377468.i		−	0.470180i
$897$	−192557.	−	486878.i	−0.239318	−	0.605112i
$898$	−308691.			−0.382800
$899$	−1.14980e6	+	1.14980e6i	−1.42267	+	1.42267i
$900$	−1494.24			−0.00184474
$901$			1.20055e6i			1.47887i
$902$	−338016.	+	338016.i	−0.415456	+	0.415456i
$903$	−177330.	+	177330.i	−0.217474	+	0.217474i
$904$	−847999.	−	847999.i	−1.03767	−	1.03767i
$905$	−43515.7	−	43515.7i	−0.0531311	−	0.0531311i
$906$	−932623.			−1.13619
$907$			293077.i			0.356260i	0.984007	+	0.178130i	$0.0570047\pi$
$907$			293077.i			0.356260i	−0.984007	+	0.178130i	$0.942995\pi$
$908$	−4851.68	−	4851.68i	−0.00588464	−	0.00588464i
$909$			175256.i			0.212103i
$910$	156536.	−	361362.i	0.189031	−	0.436374i
$911$	−1.22775e6			−1.47936			−0.739681	−	0.672958i	$-0.765024\pi$
$911$	−1.22775e6			−1.47936			−0.739681	+	0.672958i	$0.765024\pi$
$912$	−76040.3	+	76040.3i	−0.0914228	+	0.0914228i
$913$	−772185.			−0.926361
$914$		−	544974.i		−	0.652354i
$915$	−596897.	+	596897.i	−0.712946	+	0.712946i
$916$	7604.58	−	7604.58i	0.00906326	−	0.00906326i
$917$	47204.8	+	47204.8i	0.0561368	+	0.0561368i
$918$	110949.	+	110949.i	0.131655	+	0.131655i
$919$	−952406.			−1.12769			−0.563847	−	0.825879i	$-0.690679\pi$
$919$	−952406.			−1.12769			−0.563847	+	0.825879i	$0.690679\pi$
$920$			979259.i			1.15697i
$921$	191162.	+	191162.i	0.225363	+	0.225363i
$922$			1.27948e6i			1.50513i
$923$	−63896.6	+	147504.i	−0.0750023	+	0.173142i
$924$	7461.61			0.00873954
$925$	−28146.6	+	28146.6i	−0.0328960	+	0.0328960i
$926$	−860015.			−1.00296
$927$		−	151604.i		−	0.176421i
$928$	−94362.5	+	94362.5i	−0.109573	+	0.109573i
$929$	865268.	−	865268.i	1.00258	−	1.00258i	0.00258353	−	0.999997i	$-0.499178\pi$
$929$	865268.	−	865268.i	1.00258	−	1.00258i	0.999997	−	0.00258353i	$-0.000822365\pi$
$930$	−491052.	−	491052.i	−0.567756	−	0.567756i
$931$	105662.	+	105662.i	0.121904	+	0.121904i
$932$	57807.8			0.0665511
$933$			409698.i			0.470653i
$934$	259629.	+	259629.i	0.297619	+	0.297619i
$935$		−	602963.i		−	0.689712i
$936$	−260914.	−	113024.i	−0.297815	−	0.129009i
$937$	−829106.			−0.944345			−0.472173	−	0.881506i	$-0.656530\pi$
$937$	−829106.			−0.944345			−0.472173	+	0.881506i	$0.656530\pi$
$938$	151054.	−	151054.i	0.171683	−	0.171683i
$939$	107137.			0.121509
$940$			10947.2i			0.0123894i
$941$	1.18512e6	−	1.18512e6i	1.33839	−	1.33839i	0.440774	−	0.897618i	$-0.354704\pi$
$941$	1.18512e6	−	1.18512e6i	1.33839	−	1.33839i	0.897618	−	0.440774i	$-0.145296\pi$
$942$	−253996.	+	253996.i	−0.286237	+	0.286237i
$943$	586664.	+	586664.i	0.659729	+	0.659729i
$944$	261748.	+	261748.i	0.293724	+	0.293724i
$945$	−79775.1			−0.0893313
$946$			768509.i			0.858750i
$947$	94860.4	+	94860.4i	0.105775	+	0.105775i	0.758014	−	0.652238i	$-0.226170\pi$
$947$	94860.4	+	94860.4i	0.105775	+	0.105775i	−0.652238	+	0.758014i	$0.726170\pi$
$948$		−	34433.4i		−	0.0383145i
$949$	256512.	+	111117.i	0.284823	+	0.123381i
$950$	22050.0			0.0244322
$951$	218911.	−	218911.i	0.242051	−	0.242051i
$952$	366884.			0.404813
$953$		−	313080.i		−	0.344723i	−0.985034	−	0.172361i	$-0.944860\pi$
$953$		−	313080.i		−	0.344723i	0.985034	−	0.172361i	$-0.0551397\pi$
$954$	344186.	−	344186.i	0.378178	−	0.378178i
$955$	−214341.	+	214341.i	−0.235016	+	0.235016i
$956$	−57682.4	−	57682.4i	−0.0631142	−	0.0631142i
$957$	404762.	+	404762.i	0.441953	+	0.441953i
$958$	1.68358e6			1.83444
$959$			318947.i			0.346802i
$960$	375074.	+	375074.i	0.406982	+	0.406982i
$961$		−	607464.i		−	0.657770i
$962$	367824.	−	145472.i	0.397456	−	0.157192i
$963$	261526.			0.282008
$964$	51075.1	−	51075.1i	0.0549610	−	0.0549610i
$965$	1.36762e6			1.46862
$966$		−	273900.i		−	0.293520i
$967$	359201.	−	359201.i	0.384136	−	0.384136i	−0.488454	−	0.872590i	$-0.662439\pi$
$967$	359201.	−	359201.i	0.384136	−	0.384136i	0.872590	+	0.488454i	$0.162439\pi$
$968$	335504.	−	335504.i	0.358053	−	0.358053i
$969$	−77410.9	−	77410.9i	−0.0824431	−	0.0824431i
$970$	−245359.	−	245359.i	−0.260771	−	0.260771i
$971$	−1.68455e6			−1.78668			−0.893338	−	0.449385i	$-0.851643\pi$
$971$	−1.68455e6			−1.78668			−0.893338	+	0.449385i	$0.851643\pi$
$972$		−	3007.85i		−	0.00318363i
$973$	−90346.8	−	90346.8i	−0.0954305	−	0.0954305i
$974$		−	1.08228e6i		−	1.14083i
$975$	22509.2	+	56914.2i	0.0236784	+	0.0598703i
$976$	−1.65230e6			−1.73456
$977$	857213.	−	857213.i	0.898048	−	0.898048i	−0.0972150	−	0.995263i	$-0.530993\pi$
$977$	857213.	−	857213.i	0.898048	−	0.898048i	0.995263	+	0.0972150i	$0.0309934\pi$
$978$	695957.			0.727620
$979$		−	991852.i		−	1.03486i
$980$	−28644.4	+	28644.4i	−0.0298254	+	0.0298254i
$981$	−442047.	+	442047.i	−0.459336	+	0.459336i
$982$	76718.1	+	76718.1i	0.0795564	+	0.0795564i
$983$	821373.	+	821373.i	0.850029	+	0.850029i	0.990136	−	0.140107i	$-0.0447448\pi$
$983$	821373.	+	821373.i	0.850029	+	0.850029i	−0.140107	+	0.990136i	$0.544745\pi$
$984$	450576.			0.465348
$985$		−	140719.i		−	0.145037i
$986$	−1.03927e6	−	1.03927e6i	−1.06899	−	1.06899i
$987$			58636.4i			0.0601912i
$988$	−9506.29	−	4117.98i	−0.00973861	−	0.00421861i
$989$	1.33383e6			1.36366
$990$	−172864.	+	172864.i	−0.176374	+	0.176374i
$991$	−1.01521e6			−1.03374			−0.516868	−	0.856065i	$-0.672902\pi$
$991$	−1.01521e6			−1.03374			−0.516868	+	0.856065i	$0.672902\pi$
$992$		−	125645.i		−	0.127680i
$993$	354942.	−	354942.i	0.359964	−	0.359964i
$994$	−59463.3	+	59463.3i	−0.0601834	+	0.0601834i
$995$	887364.	+	887364.i	0.896305	+	0.896305i
$996$	−26875.4	−	26875.4i	−0.0270917	−	0.0270917i
$997$	590999.			0.594561			0.297280	−	0.954790i	$-0.403920\pi$
$997$	590999.			0.594561			0.297280	+	0.954790i	$0.403920\pi$
$998$		−	421352.i		−	0.423043i
$999$	−56658.2	−	56658.2i	−0.0567717	−	0.0567717i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.5.g.a.31.8	✓	20
3.2	odd	2		117.5.j.b.109.3		20
13.8	odd	4	inner	39.5.g.a.34.8	yes	20
39.8	even	4		117.5.j.b.73.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.8	✓	20	1.1	even	1	trivial
39.5.g.a.34.8	yes	20	13.8	odd	4	inner
117.5.j.b.73.3		20	39.8	even	4
117.5.j.b.109.3		20	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	39.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.89776 - 2.89776i) q^{2} +5.19615 q^{3} -0.794047i q^{4} +(18.6373 - 18.6373i) q^{5} +(15.0572 - 15.0572i) q^{6} +(-15.2549 - 15.2549i) q^{7} +(44.0632 + 44.0632i) q^{8} +27.0000 q^{9} -108.013i q^{10} +(-59.2741 - 59.2741i) q^{11} -4.12599i q^{12} +(-157.156 + 62.1541i) q^{13} -88.4101 q^{14} +(96.8421 - 96.8421i) q^{15} +268.074 q^{16} +272.906i q^{17} +(78.2396 - 78.2396i) q^{18} +(-54.5892 + 54.5892i) q^{19} +(-14.7989 - 14.7989i) q^{20} +(-79.2668 - 79.2668i) q^{21} -343.524 q^{22} +596.223i q^{23} +(228.959 + 228.959i) q^{24} -69.6962i q^{25} +(-275.292 + 635.507i) q^{26} +140.296 q^{27} +(-12.1131 + 12.1131i) q^{28} -1314.18 q^{29} -561.251i q^{30} +(874.924 - 874.924i) q^{31} +(71.8036 - 71.8036i) q^{32} +(-307.997 - 307.997i) q^{33} +(790.817 + 790.817i) q^{34} -568.619 q^{35} -21.4393i q^{36} +(-403.847 - 403.847i) q^{37} +316.373i q^{38} +(-816.604 + 322.962i) q^{39} +1642.44 q^{40} +(983.966 - 983.966i) q^{41} -459.392 q^{42} -2237.13i q^{43} +(-47.0664 + 47.0664i) q^{44} +(503.206 - 503.206i) q^{45} +(1727.71 + 1727.71i) q^{46} +(-369.867 - 369.867i) q^{47} +1392.95 q^{48} -1935.58i q^{49} +(-201.963 - 201.963i) q^{50} +1418.06i q^{51} +(49.3532 + 124.789i) q^{52} +4399.13 q^{53} +(406.545 - 406.545i) q^{54} -2209.42 q^{55} -1344.36i q^{56} +(-283.654 + 283.654i) q^{57} +(-3808.17 + 3808.17i) q^{58} +(976.401 + 976.401i) q^{59} +(-76.8972 - 76.8972i) q^{60} -6163.60 q^{61} -5070.64i q^{62} +(-411.882 - 411.882i) q^{63} +3873.05i q^{64} +(-1770.57 + 4087.33i) q^{65} -1785.01 q^{66} +(-1708.57 + 1708.57i) q^{67} +216.700 q^{68} +3098.07i q^{69} +(-1647.72 + 1647.72i) q^{70} +(672.586 - 672.586i) q^{71} +(1189.71 + 1189.71i) q^{72} +(-1169.63 - 1169.63i) q^{73} -2340.51 q^{74} -362.152i q^{75} +(43.3464 + 43.3464i) q^{76} +1808.44i q^{77} +(-1430.46 + 3302.19i) q^{78} +8345.50 q^{79} +(4996.17 - 4996.17i) q^{80} +729.000 q^{81} -5702.60i q^{82} +(6513.68 - 6513.68i) q^{83} +(-62.9415 + 62.9415i) q^{84} +(5086.23 + 5086.23i) q^{85} +(-6482.67 - 6482.67i) q^{86} -6828.65 q^{87} -5223.62i q^{88} +(8366.65 + 8366.65i) q^{89} -2916.35i q^{90} +(3345.55 + 1449.24i) q^{91} +473.429 q^{92} +(4546.24 - 4546.24i) q^{93} -2143.57 q^{94} +2034.79i q^{95} +(373.103 - 373.103i) q^{96} +(2271.58 - 2271.58i) q^{97} +(-5608.84 - 5608.84i) q^{98} +(-1600.40 - 1600.40i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 24 q^{5} - 24 q^{7} + 540 q^{9} + 372 q^{11} - 224 q^{13} + 480 q^{14} - 252 q^{15} - 2328 q^{16} - 840 q^{19} + 228 q^{20} + 936 q^{21} + 3536 q^{22} - 1404 q^{24} - 828 q^{26} - 1984 q^{28} - 5064 q^{29}+ \cdots + 10044 q^{99}+O(q^{100}) \) 20 * q + 24 * q^5 - 24 * q^7 + 540 * q^9 + 372 * q^11 - 224 * q^13 + 480 * q^14 - 252 * q^15 - 2328 * q^16 - 840 * q^19 + 228 * q^20 + 936 * q^21 + 3536 * q^22 - 1404 * q^24 - 828 * q^26 - 1984 * q^28 - 5064 * q^29 + 1712 * q^31 - 7260 * q^32 - 540 * q^33 + 8040 * q^34 + 5976 * q^35 - 868 * q^37 - 2952 * q^39 + 6336 * q^40 - 6000 * q^41 + 648 * q^42 + 20868 * q^44 + 648 * q^45 - 1992 * q^46 + 1692 * q^47 + 2592 * q^48 + 8952 * q^50 - 12416 * q^52 - 18816 * q^53 - 3280 * q^55 - 9360 * q^57 + 1256 * q^58 - 6588 * q^59 + 9036 * q^60 - 5512 * q^61 - 648 * q^63 + 33048 * q^65 + 13464 * q^66 - 4704 * q^67 - 28296 * q^68 - 31096 * q^70 + 9756 * q^71 + 13692 * q^73 - 28296 * q^74 - 17072 * q^76 - 23868 * q^78 - 5056 * q^79 + 42228 * q^80 + 14580 * q^81 + 24708 * q^83 + 26496 * q^84 - 4752 * q^85 - 77424 * q^86 + 5256 * q^87 + 36000 * q^89 + 19104 * q^91 + 12912 * q^92 - 18432 * q^93 + 112832 * q^94 - 52344 * q^96 + 50012 * q^97 + 11544 * q^98 + 10044 * q^99

Introduction

L-functions

Modular forms

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