LMFDB - Embedded newform 117.5.j.b.109.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,5,Mod(73,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("117.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	117.j (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:	$12.0942856808$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			109.3
Root			$-2.89776 + 2.89776i$ of defining polynomial
Character	$\chi$	$=$	117.109
Dual form			117.5.j.b.73.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.89776 + 2.89776i) q^{2} -0.794047i q^{4} +(-18.6373 + 18.6373i) q^{5} +(-15.2549 - 15.2549i) q^{7} +(-44.0632 - 44.0632i) q^{8} -108.013i q^{10} +(59.2741 + 59.2741i) q^{11} +(-157.156 + 62.1541i) q^{13} +88.4101 q^{14} +268.074 q^{16} -272.906i q^{17} +(-54.5892 + 54.5892i) q^{19} +(14.7989 + 14.7989i) q^{20} -343.524 q^{22} -596.223i q^{23} -69.6962i q^{25} +(275.292 - 635.507i) q^{26} +(-12.1131 + 12.1131i) q^{28} +1314.18 q^{29} +(874.924 - 874.924i) q^{31} +(-71.8036 + 71.8036i) q^{32} +(790.817 + 790.817i) q^{34} +568.619 q^{35} +(-403.847 - 403.847i) q^{37} -316.373i q^{38} +1642.44 q^{40} +(-983.966 + 983.966i) q^{41} -2237.13i q^{43} +(47.0664 - 47.0664i) q^{44} +(1727.71 + 1727.71i) q^{46} +(369.867 + 369.867i) q^{47} -1935.58i q^{49} +(201.963 + 201.963i) q^{50} +(49.3532 + 124.789i) q^{52} -4399.13 q^{53} -2209.42 q^{55} +1344.36i q^{56} +(-3808.17 + 3808.17i) q^{58} +(-976.401 - 976.401i) q^{59} -6163.60 q^{61} +5070.64i q^{62} +3873.05i q^{64} +(1770.57 - 4087.33i) q^{65} +(-1708.57 + 1708.57i) q^{67} -216.700 q^{68} +(-1647.72 + 1647.72i) q^{70} +(-672.586 + 672.586i) q^{71} +(-1169.63 - 1169.63i) q^{73} +2340.51 q^{74} +(43.3464 + 43.3464i) q^{76} -1808.44i q^{77} +8345.50 q^{79} +(-4996.17 + 4996.17i) q^{80} -5702.60i q^{82} +(-6513.68 + 6513.68i) q^{83} +(5086.23 + 5086.23i) q^{85} +(6482.67 + 6482.67i) q^{86} -5223.62i q^{88} +(-8366.65 - 8366.65i) q^{89} +(3345.55 + 1449.24i) q^{91} -473.429 q^{92} -2143.57 q^{94} -2034.79i q^{95} +(2271.58 - 2271.58i) q^{97} +(5608.84 + 5608.84i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) $ 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

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	−0.969506	−	0.245066i	$-0.921190\pi$
$2$	−2.89776	+	2.89776i	−0.724440	+	0.724440i	0.245066	+	0.969506i	$0.421190\pi$
$3$		0			0
$3$		0			0
$4$		−	0.794047i		−	0.0496279i
$5$	−18.6373	+	18.6373i	−0.745491	+	0.745491i	−0.973629	−	0.228138i	$-0.926736\pi$
$5$	−18.6373	+	18.6373i	−0.745491	+	0.745491i	0.228138	+	0.973629i	$0.426736\pi$
$6$		0			0
$7$	−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
$9$		0			0
$10$		−	108.013i		−	1.08013i
$11$	59.2741	+	59.2741i	0.489869	+	0.489869i	0.908265	−	0.418396i	$-0.137408\pi$
$11$	59.2741	+	59.2741i	0.489869	+	0.489869i	−0.418396	+	0.908265i	$0.637408\pi$
$12$		0			0
$13$	−157.156	+	62.1541i	−0.929915	+	0.367776i
$13$	−157.156	+	62.1541i	−0.929915	+	0.367776i
$14$	88.4101			0.451072
$15$		0			0
$16$	268.074			1.04717
$17$		−	272.906i		−	0.944312i	−0.881515	−	0.472156i	$-0.843476\pi$
$17$		−	272.906i		−	0.944312i	0.881515	−	0.472156i	$-0.156524\pi$
$18$		0			0
$19$	−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
$21$		0			0
$22$	−343.524			−0.709761
$23$		−	596.223i		−	1.12708i	−0.826090	−	0.563538i	$-0.809440\pi$
$23$		−	596.223i		−	1.12708i	0.826090	−	0.563538i	$-0.190560\pi$
$24$		0			0
$25$		−	69.6962i		−	0.111514i
$26$	275.292	−	635.507i	0.407236	−	0.940099i
$27$		0			0
$28$	−12.1131	+	12.1131i	−0.0154504	+	0.0154504i
$29$	1314.18			1.56263			0.781317	−	0.624135i	$-0.214548\pi$
$29$	1314.18			1.56263			0.781317	+	0.624135i	$0.214548\pi$
$30$		0			0
$31$	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
$33$		0			0
$34$	790.817	+	790.817i	0.684098	+	0.684098i
$35$	568.619			0.464179
$36$		0			0
$37$	−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
$39$		0			0
$40$	1642.44			1.02652
$41$	−983.966	+	983.966i	−0.585346	+	0.585346i	−0.936367	−	0.351022i	$-0.885834\pi$
$41$	−983.966	+	983.966i	−0.585346	+	0.585346i	0.351022	+	0.936367i	$0.385834\pi$
$42$		0			0
$43$		−	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
$45$		0			0
$46$	1727.71	+	1727.71i	0.816500	+	0.816500i
$47$	369.867	+	369.867i	0.167437	+	0.167437i	0.785852	−	0.618415i	$-0.212225\pi$
$47$	369.867	+	369.867i	0.167437	+	0.167437i	−0.618415	+	0.785852i	$0.712225\pi$
$48$		0			0
$49$		−	1935.58i		−	0.806154i
$50$	201.963	+	201.963i	0.0807852	+	0.0807852i
$51$		0			0
$52$	49.3532	+	124.789i	0.0182519	+	0.0461497i
$53$	−4399.13			−1.56608			−0.783041	−	0.621970i	$-0.786333\pi$
$53$	−4399.13			−1.56608			−0.783041	+	0.621970i	$0.786333\pi$
$54$		0			0
$55$	−2209.42			−0.730385
$56$			1344.36i			0.428686i
$57$		0			0
$58$	−3808.17	+	3808.17i	−1.13204	+	1.13204i
$59$	−976.401	−	976.401i	−0.280495	−	0.280495i	0.552812	−	0.833306i	$-0.313555\pi$
$59$	−976.401	−	976.401i	−0.280495	−	0.280495i	−0.833306	+	0.552812i	$0.813555\pi$
$60$		0			0
$61$	−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
$63$		0			0
$64$			3873.05i			0.945568i
$65$	1770.57	−	4087.33i	0.419070	−	0.967416i
$66$		0			0
$67$	−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
$69$		0			0
$70$	−1647.72	+	1647.72i	−0.336270	+	0.336270i
$71$	−672.586	+	672.586i	−0.133423	+	0.133423i	−0.770664	−	0.637241i	$-0.780075\pi$
$71$	−672.586	+	672.586i	−0.133423	+	0.133423i	0.637241	+	0.770664i	$0.280075\pi$
$72$		0			0
$73$	−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
$75$		0			0
$76$	43.3464	+	43.3464i	0.00750457	+	0.00750457i
$77$		−	1808.44i		−	0.305016i
$78$		0			0
$79$	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
$81$		0			0
$82$		−	5702.60i		−	0.848096i
$83$	−6513.68	+	6513.68i	−0.945519	+	0.945519i	−0.998591	−	0.0530714i	$-0.983099\pi$
$83$	−6513.68	+	6513.68i	−0.945519	+	0.945519i	0.0530714	+	0.998591i	$0.483099\pi$
$84$		0			0
$85$	5086.23	+	5086.23i	0.703976	+	0.703976i
$86$	6482.67	+	6482.67i	0.876510	+	0.876510i
$87$		0			0
$88$		−	5223.62i		−	0.674537i
$89$	−8366.65	−	8366.65i	−1.05626	−	1.05626i	−0.998320	−	0.0579425i	$-0.981546\pi$
$89$	−8366.65	−	8366.65i	−1.05626	−	1.05626i	−0.0579425	−	0.998320i	$-0.518454\pi$
$90$		0			0
$91$	3345.55	+	1449.24i	0.404003	+	0.175008i
$92$	−473.429			−0.0559345
$93$		0			0
$94$	−2143.57			−0.242596
$95$		−	2034.79i		−	0.225461i
$96$		0			0
$97$	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
$99$		0			0
$100$	−55.3421			−0.00553421
$101$		−	6490.97i		−	0.636308i	−0.948039	−	0.318154i	$-0.896937\pi$
$101$		−	6490.97i		−	0.636308i	0.948039	−	0.318154i	$-0.103063\pi$
$102$		0			0
$103$		−	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
$105$		0			0
$106$	12747.6	−	12747.6i	1.13453	−	1.13453i
$107$	−9686.14			−0.846025			−0.423012	−	0.906124i	$-0.639027\pi$
$107$	−9686.14			−0.846025			−0.423012	+	0.906124i	$0.639027\pi$
$108$		0			0
$109$	−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
$111$		0			0
$112$	−4089.44	−	4089.44i	−0.326008	−	0.326008i
$113$	19245.1			1.50717			0.753585	−	0.657350i	$-0.228323\pi$
$113$	19245.1			1.50717			0.753585	+	0.657350i	$0.228323\pi$
$114$		0			0
$115$	11112.0	+	11112.0i	0.840225	+	0.840225i
$116$		−	1043.52i		−	0.0775503i
$117$		0			0
$118$	5658.76			0.406403
$119$	−4163.15	+	4163.15i	−0.293987	+	0.293987i
$120$		0			0
$121$		−	7614.16i		−	0.520057i
$122$	17860.7	−	17860.7i	1.19999	−	1.19999i
$123$		0			0
$124$	−694.731	−	694.731i	−0.0451828	−	0.0451828i
$125$	−10349.4	−	10349.4i	−0.662358	−	0.662358i
$126$		0			0
$127$		−	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
$129$		0			0
$130$	6713.43	+	16974.8i	0.397245	+	1.00443i
$131$	3094.41			0.180316			0.0901581	−	0.995927i	$-0.471263\pi$
$131$	3094.41			0.180316			0.0901581	+	0.995927i	$0.471263\pi$
$132$		0			0
$133$	1665.51			0.0941549
$134$		−	9902.03i		−	0.551461i
$135$		0			0
$136$	−12025.1	+	12025.1i	−0.650147	+	0.650147i
$137$	10453.9	+	10453.9i	0.556978	+	0.556978i	0.928446	−	0.371467i	$-0.121145\pi$
$137$	10453.9	+	10453.9i	0.556978	+	0.556978i	−0.371467	+	0.928446i	$0.621145\pi$
$138$		0			0
$139$	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
$141$		0			0
$142$		−	3897.99i		−	0.193314i
$143$	−12999.4	−	5631.13i	−0.635698	−	0.275374i
$144$		0			0
$145$	−24492.6	+	24492.6i	−1.16493	+	1.16493i
$146$	6778.64			0.318007
$147$		0			0
$148$	−320.674	+	320.674i	−0.0146400	+	0.0146400i
$149$	22740.7	−	22740.7i	1.02431	−	1.02431i	0.0246105	−	0.999697i	$-0.492165\pi$
$149$	22740.7	−	22740.7i	1.02431	−	1.02431i	0.999697	−	0.0246105i	$-0.00783457\pi$
$150$		0			0
$151$	−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
$153$		0			0
$154$	5240.43	+	5240.43i	0.220966	+	0.220966i
$155$			32612.4i			1.35744i
$156$		0			0
$157$	−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
$159$		0			0
$160$		−	2676.45i		−	0.104549i
$161$	−9095.32	+	9095.32i	−0.350886	+	0.350886i
$162$		0			0
$163$	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
$165$		0			0
$166$		−	37750.2i		−	1.36994i
$167$	−16613.8	−	16613.8i	−0.595710	−	0.595710i	0.343458	−	0.939168i	$-0.388402\pi$
$167$	−16613.8	−	16613.8i	−0.595710	−	0.595710i	−0.939168	+	0.343458i	$0.888402\pi$
$168$		0			0
$169$	20834.7	−	19535.7i	0.729482	−	0.684000i
$170$	−29477.3			−1.01998
$171$		0			0
$172$	−1776.39			−0.0600455
$173$		−	9230.43i		−	0.308411i	−0.988039	−	0.154206i	$-0.950718\pi$
$173$		−	9230.43i		−	0.308411i	0.988039	−	0.154206i	$-0.0492818\pi$
$174$		0			0
$175$	−1063.21	+	1063.21i	−0.0347170	+	0.0347170i
$176$	15889.9	+	15889.9i	0.512973	+	0.512973i
$177$		0			0
$178$	48489.1			1.53040
$179$			28771.7i			0.897965i	0.893541	+	0.448982i	$0.148213\pi$
$179$			28771.7i			0.897965i	−0.893541	+	0.448982i	$0.851787\pi$
$180$		0			0
$181$		−	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
$183$		0			0
$184$	−26271.5	+	26271.5i	−0.775978	+	0.775978i
$185$	15053.2			0.439831
$186$		0			0
$187$	16176.3	−	16176.3i	0.462589	−	0.462589i
$188$	293.692	−	293.692i	0.00830953	−	0.00830953i
$189$		0			0
$190$	5896.34	+	5896.34i	0.163333	+	0.163333i
$191$	11500.7			0.315251			0.157625	−	0.987499i	$-0.449616\pi$
$191$	11500.7			0.315251			0.157625	+	0.987499i	$0.449616\pi$
$192$		0			0
$193$	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
$195$		0			0
$196$	−1536.94			−0.0400078
$197$	−3775.20	+	3775.20i	−0.0972763	+	0.0972763i	−0.754070	−	0.656794i	$-0.771912\pi$
$197$	−3775.20	+	3775.20i	−0.0972763	+	0.0972763i	0.656794	+	0.754070i	$0.271912\pi$
$198$		0			0
$199$			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
$201$		0			0
$202$	18809.3	+	18809.3i	0.460967	+	0.460967i
$203$	−20047.6	−	20047.6i	−0.486486	−	0.486486i
$204$		0			0
$205$		−	36676.9i		−	0.872740i
$206$	16270.8	+	16270.8i	0.383420	+	0.383420i
$207$		0			0
$208$	−42129.4	+	16661.9i	−0.973774	+	0.385122i
$209$	−6471.46			−0.148153
$210$		0			0
$211$	−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
$213$		0			0
$214$	28068.1	−	28068.1i	0.612894	−	0.612894i
$215$	41694.0	+	41694.0i	0.901980	+	0.901980i
$216$		0			0
$217$	−26693.8			−0.566879
$218$		−	94885.0i		−	1.99657i
$219$		0			0
$220$			1754.38i			0.0362475i
$221$	16962.2	+	42888.7i	0.347295	+	0.878129i
$222$		0			0
$223$	−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
$225$		0			0
$226$	−55767.6	+	55767.6i	−1.09186	+	1.09186i
$227$	−6110.06	+	6110.06i	−0.118575	+	0.118575i	−0.763905	−	0.645329i	$-0.776720\pi$
$227$	−6110.06	+	6110.06i	−0.118575	+	0.118575i	0.645329	+	0.763905i	$0.276720\pi$
$228$		0			0
$229$	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
$231$		0			0
$232$	−57906.8	−	57906.8i	−1.07585	−	1.07585i
$233$		−	72801.5i		−	1.34100i	−0.741910	−	0.670500i	$-0.766080\pi$
$233$		−	72801.5i		−	1.34100i	0.741910	−	0.670500i	$-0.233920\pi$
$234$		0			0
$235$	−13786.6			−0.249645
$236$	−775.309	+	775.309i	−0.0139204	+	0.0139204i
$237$		0			0
$238$		−	24127.7i		−	0.425952i
$239$	−72643.5	+	72643.5i	−1.27175	+	1.27175i	−0.326578	+	0.945170i	$0.605895\pi$
$239$	−72643.5	+	72643.5i	−1.27175	+	1.27175i	−0.945170	+	0.326578i	$0.894105\pi$
$240$		0			0
$241$	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
$243$		0			0
$244$			4894.19i			0.0822056i
$245$	36073.9	+	36073.9i	0.600981	+	0.600981i
$246$		0			0
$247$	5186.06	−	11971.9i	0.0850048	−	0.196232i
$248$	−77104.0			−1.25364
$249$		0			0
$250$	59979.9			0.959678
$251$		−	49293.7i		−	0.782427i	−0.920300	−	0.391213i	$-0.872055\pi$
$251$		−	49293.7i		−	0.782427i	0.920300	−	0.391213i	$-0.127945\pi$
$252$		0			0
$253$	35340.6	−	35340.6i	0.552119	−	0.552119i
$254$	76318.4	+	76318.4i	1.18294	+	1.18294i
$255$		0			0
$256$	9733.62			0.148523
$257$		−	82424.4i		−	1.24793i	−0.781453	−	0.623964i	$-0.785521\pi$
$257$		−	82424.4i		−	1.24793i	0.781453	−	0.623964i	$-0.214479\pi$
$258$		0			0
$259$			12321.3i			0.183678i
$260$	−3245.54	−	1405.92i	−0.0480109	−	0.0207976i
$261$		0			0
$262$	−8966.85	+	8966.85i	−0.130628	+	0.130628i
$263$	−41367.3			−0.598062			−0.299031	−	0.954243i	$-0.596663\pi$
$263$	−41367.3			−0.598062			−0.299031	+	0.954243i	$0.596663\pi$
$264$		0			0
$265$	81987.7	−	81987.7i	1.16750	−	1.16750i
$266$	−4826.24	+	4826.24i	−0.0682096	+	0.0682096i
$267$		0			0
$268$	1356.68	+	1356.68i	0.0188890	+	0.0188890i
$269$	59626.2			0.824011			0.412005	−	0.911181i	$-0.364828\pi$
$269$	59626.2			0.824011			0.412005	+	0.911181i	$0.364828\pi$
$270$		0			0
$271$	−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
$273$		0			0
$274$	−60586.0			−0.806996
$275$	4131.18	−	4131.18i	0.0546272	−	0.0546272i
$276$		0			0
$277$		−	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
$279$		0			0
$280$	−25055.2	−	25055.2i	−0.319582	−	0.319582i
$281$	19968.0	+	19968.0i	0.252885	+	0.252885i	0.822152	−	0.569268i	$-0.192773\pi$
$281$	19968.0	+	19968.0i	0.252885	+	0.252885i	−0.569268	+	0.822152i	$0.692773\pi$
$282$		0			0
$283$			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
$285$		0			0
$286$	53986.8	−	21351.4i	0.660017	−	0.261033i
$287$	30020.6			0.364465
$288$		0			0
$289$	9043.27			0.108275
$290$		−	141948.i		−	1.68784i
$291$		0			0
$292$	−928.744	+	928.744i	−0.0108926	+	0.0108926i
$293$	−22398.8	−	22398.8i	−0.260909	−	0.260909i	0.564514	−	0.825423i	$-0.309063\pi$
$293$	−22398.8	−	22398.8i	−0.260909	−	0.260909i	−0.825423	+	0.564514i	$0.809063\pi$
$294$		0			0
$295$	36394.9			0.418212
$296$			35589.6i			0.406200i
$297$		0			0
$298$			131794.i			1.48410i
$299$	37057.7	+	93699.8i	0.414511	+	1.04808i
$300$		0			0
$301$	−34127.2	+	34127.2i	−0.376676	+	0.376676i
$302$	179483.			1.96793
$303$		0			0
$304$	−14634.0	+	14634.0i	−0.158349	+	0.158349i
$305$	114873.	−	114873.i	1.23486	−	1.23486i
$306$		0			0
$307$	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
$309$		0			0
$310$	−94503.0	−	94503.0i	−0.983382	−	0.983382i
$311$		−	78846.4i		−	0.815194i	−0.913162	−	0.407597i	$-0.866367\pi$
$311$		−	78846.4i		−	0.815194i	0.913162	−	0.407597i	$-0.133633\pi$
$312$		0			0
$313$	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
$315$		0			0
$316$		−	6626.72i		−	0.0663627i
$317$	−42129.4	+	42129.4i	−0.419244	+	0.419244i	−0.884943	−	0.465699i	$-0.845803\pi$
$317$	−42129.4	+	42129.4i	−0.419244	+	0.419244i	0.465699	+	0.884943i	$0.345803\pi$
$318$		0			0
$319$	77896.6	+	77896.6i	0.765485	+	0.765485i
$320$	−72183.1	−	72183.1i	−0.704913	−	0.704913i
$321$		0			0
$322$		−	52712.2i		−	0.508392i
$323$	14897.7	+	14897.7i	0.142796	+	0.142796i
$324$		0			0
$325$	4331.90	+	10953.2i	0.0410121	+	0.103698i
$326$	−133937.			−1.26027
$327$		0			0
$328$	86713.5			0.806007
$329$		−	11284.6i		−	0.104254i
$330$		0			0
$331$	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
$333$		0			0
$334$	96285.5			0.863113
$335$		−	63686.0i		−	0.567485i
$336$		0			0
$337$		−	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
$339$		0			0
$340$	4038.70	−	4038.70i	0.0349369	−	0.0349369i
$341$	103721.			0.891983
$342$		0			0
$343$	−66154.0	+	66154.0i	−0.562300	+	0.562300i
$344$	−98575.2	+	98575.2i	−0.833011	+	0.833011i
$345$		0			0
$346$	26747.6	+	26747.6i	0.223425	+	0.223425i
$347$	−98270.4			−0.816138			−0.408069	−	0.912951i	$-0.633798\pi$
$347$	−98270.4			−0.816138			−0.408069	+	0.912951i	$0.633798\pi$
$348$		0			0
$349$	−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
$351$		0			0
$352$	−8512.19			−0.0686999
$353$	−63740.5	+	63740.5i	−0.511524	+	0.511524i	−0.914993	−	0.403469i	$-0.867804\pi$
$353$	−63740.5	+	63740.5i	−0.511524	+	0.511524i	0.403469	+	0.914993i	$0.367804\pi$
$354$		0			0
$355$		−	25070.3i		−	0.198931i
$356$	−6643.52	+	6643.52i	−0.0524201	+	0.0524201i
$357$		0			0
$358$	−83373.5	−	83373.5i	−0.650522	−	0.650522i
$359$	168217.	+	168217.i	1.30521	+	1.30521i	0.924831	+	0.380377i	$0.124206\pi$
$359$	168217.	+	168217.i	1.30521	+	1.30521i	0.380377	+	0.924831i	$0.375794\pi$
$360$		0			0
$361$			124361.i			0.954267i
$362$	6765.91	+	6765.91i	0.0516308	+	0.0516308i
$363$		0			0
$364$	1150.76	−	2656.52i	0.00868527	−	0.0200498i
$365$	43597.6			0.327248
$366$		0			0
$367$	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
$369$		0			0
$370$	−43620.7	+	43620.7i	−0.318632	+	0.318632i
$371$	67108.2	+	67108.2i	0.487560	+	0.487560i
$372$		0			0
$373$	−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
$375$		0			0
$376$		−	32595.1i		−	0.230556i
$377$	−206530.	+	81681.3i	−1.45312	+	0.574698i
$378$		0			0
$379$	−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
$381$		0			0
$382$	−33326.2	+	33326.2i	−0.228380	+	0.228380i
$383$	−30297.2	+	30297.2i	−0.206540	+	0.206540i	−0.802795	−	0.596255i	$-0.796655\pi$
$383$	−30297.2	+	30297.2i	−0.206540	+	0.206540i	0.596255	+	0.802795i	$0.296655\pi$
$384$		0			0
$385$	33704.4	+	33704.4i	0.227387	+	0.227387i
$386$	−212640.			−1.42715
$387$		0			0
$388$	−1803.74	−	1803.74i	−0.0119815	−	0.0119815i
$389$			135764.i			0.897193i	0.893734	+	0.448597i	$0.148076\pi$
$389$			135764.i			0.897193i	−0.893734	+	0.448597i	$0.851924\pi$
$390$		0			0
$391$	−162713.			−1.06431
$392$	−85287.7	+	85287.7i	−0.555028	+	0.555028i
$393$		0			0
$394$		−	21879.2i		−	0.140942i
$395$	−155537.	+	155537.i	−0.996874	+	0.996874i
$396$		0			0
$397$	−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
$399$		0			0
$400$		−	18683.8i		−	0.116774i
$401$	−132123.	−	132123.i	−0.821653	−	0.821653i	0.164692	−	0.986345i	$-0.447337\pi$
$401$	−132123.	−	132123.i	−0.821653	−	0.821653i	−0.986345	+	0.164692i	$0.947337\pi$
$402$		0			0
$403$	−83119.1	+	191879.i	−0.511789	+	1.18146i
$404$	−5154.14			−0.0315786
$405$		0			0
$406$	116186.			0.704860
$407$		−	47875.4i		−	0.289017i
$408$		0			0
$409$	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
$411$		0			0
$412$	−4458.54			−0.0262663
$413$			29789.8i			0.174650i
$414$		0			0
$415$		−	242795.i		−	1.40975i
$416$	6821.45	−	15747.2i	0.0394176	−	0.0909950i
$417$		0			0
$418$	18752.7	−	18752.7i	0.107328	−	0.107328i
$419$	94447.5			0.537975			0.268988	−	0.963144i	$-0.413311\pi$
$419$	94447.5			0.537975			0.268988	+	0.963144i	$0.413311\pi$
$420$		0			0
$421$	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
$423$		0			0
$424$	193840.	+	193840.i	1.07823	+	1.07823i
$425$	−19020.5			−0.105304
$426$		0			0
$427$	94025.1	+	94025.1i	0.515689	+	0.515689i
$428$			7691.25i			0.0419865i
$429$		0			0
$430$	−241639.			−1.30686
$431$	42041.0	−	42041.0i	0.226317	−	0.226317i	−0.584835	−	0.811152i	$-0.698841\pi$
$431$	42041.0	−	42041.0i	0.226317	−	0.226317i	0.811152	+	0.584835i	$0.198841\pi$
$432$		0			0
$433$			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
$435$		0			0
$436$	13000.2	+	13000.2i	0.0683877	+	0.0683877i
$437$	32547.4	+	32547.4i	0.170433	+	0.170433i
$438$		0			0
$439$			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
$441$		0			0
$442$	−173434.	−	75128.8i	−0.887747	−	0.384558i
$443$	22527.8			0.114792			0.0573960	−	0.998351i	$-0.481720\pi$
$443$	22527.8			0.114792			0.0573960	+	0.998351i	$0.481720\pi$
$444$		0			0
$445$	311863.			1.57487
$446$		−	5524.30i		−	0.0277720i
$447$		0			0
$448$	59082.9	−	59082.9i	0.294378	−	0.294378i
$449$	53263.7	+	53263.7i	0.264204	+	0.264204i	0.826759	−	0.562556i	$-0.190182\pi$
$449$	53263.7	+	53263.7i	0.264204	+	0.264204i	−0.562556	+	0.826759i	$0.690182\pi$
$450$		0			0
$451$	−116647.			−0.573485
$452$		−	15281.5i		−	0.0747978i
$453$		0			0
$454$		−	35411.0i		−	0.171801i
$455$	−89361.7	+	35342.0i	−0.431647	+	0.170714i
$456$		0			0
$457$	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
$459$		0			0
$460$	8823.43	−	8823.43i	0.0416986	−	0.0416986i
$461$	220771.	−	220771.i	1.03882	−	1.03882i	0.0396051	−	0.999215i	$-0.487390\pi$
$461$	220771.	−	220771.i	1.03882	−	1.03882i	0.999215	−	0.0396051i	$-0.0126100\pi$
$462$		0			0
$463$	−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
$465$		0			0
$466$	210962.	+	210962.i	0.971474	+	0.971474i
$467$		−	89596.5i		−	0.410826i	−0.978675	−	0.205413i	$-0.934146\pi$
$467$		−	89596.5i		−	0.410826i	0.978675	−	0.205413i	$-0.0658537\pi$
$468$		0			0
$469$	52128.0			0.236987
$470$	39950.4	−	39950.4i	0.180853	−	0.180853i
$471$		0			0
$472$			86046.8i			0.386234i
$473$	132604.	−	132604.i	0.592699	−	0.592699i
$474$		0			0
$475$	3804.66	+	3804.66i	0.0168628	+	0.0168628i
$476$	3305.74	+	3305.74i	0.0145900	+	0.0145900i
$477$		0			0
$478$		−	421007.i		−	1.84261i
$479$	−290497.	−	290497.i	−1.26611	−	1.26611i	−0.948083	−	0.318023i	$-0.896981\pi$
$479$	−290497.	−	290497.i	−1.26611	−	1.26611i	−0.318023	−	0.948083i	$-0.603019\pi$
$480$		0			0
$481$	88567.6	+	38366.1i	0.382811	+	0.165828i
$482$	−372782.			−1.60458
$483$		0			0
$484$	−6046.00			−0.0258094
$485$			84672.0i			0.359962i
$486$		0			0
$487$	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
$489$		0			0
$490$	−209067.			−0.870750
$491$		−	26475.0i		−	0.109818i	−0.998491	−	0.0549088i	$-0.982513\pi$
$491$		−	26475.0i		−	0.109818i	0.998491	−	0.0549088i	$-0.0174868\pi$
$492$		0			0
$493$		−	358646.i		−	1.47561i
$494$	19663.9	+	49719.8i	0.0805778	+	0.203740i
$495$		0			0
$496$	234545.	−	234545.i	0.953372	−	0.953372i
$497$	20520.4			0.0830757
$498$		0			0
$499$	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
$501$		0			0
$502$	142841.	+	142841.i	0.566822	+	0.566822i
$503$	−245797.			−0.971496			−0.485748	−	0.874099i	$-0.661453\pi$
$503$	−245797.			−0.971496			−0.485748	+	0.874099i	$0.661453\pi$
$504$		0			0
$505$	120974.	+	120974.i	0.474362	+	0.474362i
$506$			204817.i			0.799955i
$507$		0			0
$508$	−20912.8			−0.0810374
$509$	105992.	−	105992.i	0.409109	−	0.409109i	−0.472319	−	0.881428i	$-0.656583\pi$
$509$	105992.	−	105992.i	0.409109	−	0.409109i	0.881428	+	0.472319i	$0.156583\pi$
$510$		0			0
$511$			35685.3i			0.136662i
$512$	169747.	−	169747.i	0.647533	−	0.647533i
$513$		0			0
$514$	238846.	+	238846.i	0.904049	+	0.904049i
$515$	104648.	+	104648.i	0.394561	+	0.394561i
$516$		0			0
$517$			43847.1i			0.164044i
$518$	−35704.2	−	35704.2i	−0.133064	−	0.133064i
$519$		0			0
$520$	−258118.	+	102084.i	−0.954579	+	0.377530i
$521$	−408225.			−1.50392			−0.751959	−	0.659209i	$-0.770891\pi$
$521$	−408225.			−1.50392			−0.751959	+	0.659209i	$0.770891\pi$
$522$		0			0
$523$	−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
$525$		0			0
$526$	119873.	−	119873.i	0.433260	−	0.433260i
$527$	−238772.	−	238772.i	−0.859731	−	0.859731i
$528$		0			0
$529$	−75641.2			−0.270300
$530$			475162.i			1.69157i
$531$		0			0
$532$		−	1322.49i		−	0.00467271i
$533$	93478.3	−	215793.i	0.329046	−	0.759597i
$534$		0			0
$535$	180523.	−	180523.i	0.630704	−	0.630704i
$536$	150570.			0.524093
$537$		0			0
$538$	−172783.	+	172783.i	−0.596947	+	0.596947i
$539$	114730.	−	114730.i	0.394910	−	0.394910i
$540$		0			0
$541$	−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
$543$		0			0
$544$	19595.6	+	19595.6i	0.0662158	+	0.0662158i
$545$		−	610263.i		−	2.05459i
$546$		0			0
$547$	−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
$549$		0			0
$550$			23942.4i			0.0791483i
$551$	−71739.8	+	71739.8i	−0.236296	+	0.236296i
$552$		0			0
$553$	−127310.	−	127310.i	−0.416304	−	0.416304i
$554$	388667.	+	388667.i	1.26636	+	1.26636i
$555$		0			0
$556$		−	4702.73i		−	0.0152125i
$557$	−141690.	−	141690.i	−0.456696	−	0.456696i	0.440873	−	0.897569i	$-0.354669\pi$
$557$	−141690.	−	141690.i	−0.456696	−	0.456696i	−0.897569	+	0.440873i	$0.854669\pi$
$558$		0			0
$559$	139047.	+	351577.i	0.444977	+	1.12512i
$560$	152432.			0.486072
$561$		0			0
$562$	−115725.			−0.366400
$563$			301845.i			0.952287i	0.879368	+	0.476143i	$0.157966\pi$
$563$			301845.i			0.952287i	−0.879368	+	0.476143i	$0.842034\pi$
$564$		0			0
$565$	−358675.	+	358675.i	−1.12358	+	1.12358i
$566$	−440345.	−	440345.i	−1.37455	−	1.37455i
$567$		0			0
$568$	59272.6			0.183720
$569$		−	149750.i		−	0.462532i	−0.972891	−	0.231266i	$-0.925713\pi$
$569$		−	149750.i		−	0.462532i	0.972891	−	0.231266i	$-0.0742867\pi$
$570$		0			0
$571$			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
$573$		0			0
$574$	−86992.5	+	86992.5i	−0.264033	+	0.264033i
$575$	−41554.5			−0.125685
$576$		0			0
$577$	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
$579$		0			0
$580$	19448.3	+	19448.3i	0.0578131	+	0.0578131i
$581$	198731.			0.588726
$582$		0			0
$583$	−260754.	−	260754.i	−0.767175	−	0.767175i
$584$			103076.i			0.302225i
$585$		0			0
$586$	129813.			0.378026
$587$	−379654.	+	379654.i	−1.10182	+	1.10182i	−0.107632	+	0.994191i	$0.534327\pi$
$587$	−379654.	+	379654.i	−1.10182	+	1.10182i	−0.994191	+	0.107632i	$0.965673\pi$
$588$		0			0
$589$			95522.9i			0.275345i
$590$	−105464.	+	105464.i	−0.302970	+	0.302970i
$591$		0			0
$592$	−108261.	−	108261.i	−0.308908	−	0.308908i
$593$	−20445.7	−	20445.7i	−0.0581423	−	0.0581423i	0.677438	−	0.735580i	$-0.263090\pi$
$593$	−20445.7	−	20445.7i	−0.0581423	−	0.0581423i	−0.735580	+	0.677438i	$0.763090\pi$
$594$		0			0
$595$		−	155180.i		−	0.438330i
$596$	−18057.1	−	18057.1i	−0.0508343	−	0.0508343i
$597$		0			0
$598$	−378904.	−	164135.i	−1.05956	−	0.458986i
$599$	−45314.1			−0.126293			−0.0631466	−	0.998004i	$-0.520114\pi$
$599$	−45314.1			−0.126293			−0.0631466	+	0.998004i	$0.520114\pi$
$600$		0			0
$601$	−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
$603$		0			0
$604$	−24591.1	+	24591.1i	−0.0674068	+	0.0674068i
$605$	141907.	+	141907.i	0.387698	+	0.387698i
$606$		0			0
$607$	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
$609$		0			0
$610$			665748.i			1.78916i
$611$	−81115.5	−	35138.0i	−0.217281	−	0.0941226i
$612$		0			0
$613$	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
$615$		0			0
$616$	−79685.7	+	79685.7i	−0.210000	+	0.210000i
$617$	−128460.	+	128460.i	−0.337440	+	0.337440i	−0.855403	−	0.517963i	$-0.826690\pi$
$617$	−128460.	+	128460.i	−0.337440	+	0.337440i	0.517963	+	0.855403i	$0.326690\pi$
$618$		0			0
$619$	−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
$621$		0			0
$622$	228478.	+	228478.i	0.590560	+	0.590560i
$623$			255265.i			0.657680i
$624$		0			0
$625$	429328.			1.09908
$626$	−59747.7	+	59747.7i	−0.152466	+	0.152466i
$627$		0			0
$628$			13394.6i			0.0339633i
$629$	−110212.	+	110212.i	−0.278567	+	0.278567i
$630$		0			0
$631$	−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
$633$		0			0
$634$		−	244162.i		−	0.607435i
$635$	490850.	+	490850.i	1.21731	+	1.21731i
$636$		0			0
$637$	120304.	+	304187.i	0.296484	+	0.749655i
$638$	−451451.			−1.10910
$639$		0			0
$640$	461162.			1.12588
$641$			369011.i			0.898098i	0.893507	+	0.449049i	$0.148237\pi$
$641$			369011.i			0.898098i	−0.893507	+	0.449049i	$0.851763\pi$
$642$		0			0
$643$	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
$645$		0			0
$646$	−86340.2			−0.206894
$647$			401104.i			0.958184i	0.877765	+	0.479092i	$0.159034\pi$
$647$			401104.i			0.958184i	−0.877765	+	0.479092i	$0.840966\pi$
$648$		0			0
$649$		−	115751.i		−	0.274811i
$650$	−44292.5	−	19186.8i	−0.104834	−	0.0454125i
$651$		0			0
$652$	18350.8	−	18350.8i	0.0431677	−	0.0431677i
$653$	341135.			0.800018			0.400009	−	0.916511i	$-0.369007\pi$
$653$	341135.			0.800018			0.400009	+	0.916511i	$0.369007\pi$
$654$		0			0
$655$	−57671.3	+	57671.3i	−0.134424	+	0.134424i
$656$	−263776.	+	263776.i	−0.612954	+	0.612954i
$657$		0			0
$658$	32700.0	+	32700.0i	0.0755259	+	0.0755259i
$659$	−32739.6			−0.0753881			−0.0376941	−	0.999289i	$-0.512001\pi$
$659$	−32739.6			−0.0753881			−0.0376941	+	0.999289i	$0.512001\pi$
$660$		0			0
$661$	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
$663$		0			0
$664$	574028.			1.30196
$665$	−31040.5	+	31040.5i	−0.0701916	+	0.0701916i
$666$		0			0
$667$		−	783542.i		−	1.76121i
$668$	−13192.1	+	13192.1i	−0.0295639	+	0.0295639i
$669$		0			0
$670$	184547.	+	184547.i	0.411109	+	0.411109i
$671$	−365342.	−	365342.i	−0.811437	−	0.811437i
$672$		0			0
$673$			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
$675$		0			0
$676$	−15512.3	−	16543.8i	−0.0339455	−	0.0362027i
$677$	753102.			1.64315			0.821574	−	0.570102i	$-0.193096\pi$
$677$	753102.			1.64315			0.821574	+	0.570102i	$0.193096\pi$
$678$		0			0
$679$	−69305.3			−0.150323
$680$		−	448231.i		−	0.969358i
$681$		0			0
$682$	−300558.	+	300558.i	−0.646189	+	0.646189i
$683$	−50909.1	−	50909.1i	−0.109132	−	0.109132i	0.650432	−	0.759564i	$-0.274588\pi$
$683$	−50909.1	−	50909.1i	−0.109132	−	0.109132i	−0.759564	+	0.650432i	$0.774588\pi$
$684$		0			0
$685$	−389666.			−0.830445
$686$		−	383397.i		−	0.814705i
$687$		0			0
$688$		−	599717.i		−	1.26698i
$689$	691347.	−	273424.i	1.45632	−	0.575967i
$690$		0			0
$691$	−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
$693$		0			0
$694$	284764.	−	284764.i	0.591243	−	0.591243i
$695$	−110379.	+	110379.i	−0.228516	+	0.228516i
$696$		0			0
$697$	268530.	+	268530.i	0.552749	+	0.552749i
$698$	101079.			0.207468
$699$		0			0
$700$	844.238	+	844.238i	0.00172293	+	0.00172293i
$701$		−	212436.i		−	0.432307i	−0.976359	−	0.216154i	$-0.930649\pi$
$701$		−	212436.i		−	0.432307i	0.976359	−	0.216154i	$-0.0693512\pi$
$702$		0			0
$703$	44091.4			0.0892161
$704$	−229571.	+	229571.i	−0.463204	+	0.463204i
$705$		0			0
$706$		−	369410.i		−	0.741138i
$707$	−99019.1	+	99019.1i	−0.198098	+	0.198098i
$708$		0			0
$709$	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
$711$		0			0
$712$			737324.i			1.45445i
$713$	−521650.	−	521650.i	−1.02613	−	1.02613i
$714$		0			0
$715$	347222.	−	137324.i	0.679196	−	0.268618i
$716$	22846.1			0.0445641
$717$		0			0
$718$	−974903.			−1.89109
$719$			537471.i			1.03967i	0.854265	+	0.519837i	$0.174007\pi$
$719$			537471.i			1.03967i	−0.854265	+	0.519837i	$0.825993\pi$
$720$		0			0
$721$	−85655.6	+	85655.6i	−0.164773	+	0.164773i
$722$	−360369.	−	360369.i	−0.691310	−	0.691310i
$723$		0			0
$724$	−1854.00			−0.00353698
$725$		−	91593.1i		−	0.174256i
$726$		0			0
$727$			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
$729$		0			0
$730$	−126335.	+	126335.i	−0.237071	+	0.237071i
$731$	−610526.			−1.14254
$732$		0			0
$733$	−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
$735$		0			0
$736$	42811.0	+	42811.0i	0.0790314	+	0.0790314i
$737$	−202547.			−0.372899
$738$		0			0
$739$	−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
$741$		0			0
$742$	−388927.			−0.706416
$743$	397022.	−	397022.i	0.719180	−	0.719180i	−0.249258	−	0.968437i	$-0.580187\pi$
$743$	397022.	−	397022.i	0.719180	−	0.719180i	0.968437	+	0.249258i	$0.0801866\pi$
$744$		0			0
$745$			847648.i			1.52722i
$746$	334590.	−	334590.i	0.601223	−	0.601223i
$747$		0			0
$748$	−12844.7	−	12844.7i	−0.0229573	−	0.0229573i
$749$	147761.	+	147761.i	0.263388	+	0.263388i
$750$		0			0
$751$		−	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
$753$		0			0
$754$	361782.	−	835168.i	0.636361	−	1.46903i
$755$	1.15437e6			2.02512
$756$		0			0
$757$	−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
$759$		0			0
$760$	−89659.4	+	89659.4i	−0.155227	+	0.155227i
$761$	70616.1	+	70616.1i	0.121937	+	0.121937i	0.765442	−	0.643505i	$-0.222520\pi$
$761$	70616.1	+	70616.1i	0.121937	+	0.121937i	−0.643505	+	0.765442i	$0.722520\pi$
$762$		0			0
$763$	499510.			0.858015
$764$		−	9132.06i		−	0.0156452i
$765$		0			0
$766$		−	175588.i		−	0.299253i
$767$	214134.	+	92759.6i	0.363995	+	0.157677i
$768$		0			0
$769$	−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
$771$		0			0
$772$	29133.9	−	29133.9i	0.0488836	−	0.0488836i
$773$	−152962.	+	152962.i	−0.255990	+	0.255990i	−0.823421	−	0.567431i	$-0.807937\pi$
$773$	−152962.	+	152962.i	−0.255990	+	0.255990i	0.567431	+	0.823421i	$0.307937\pi$
$774$		0			0
$775$	−60978.9	−	60978.9i	−0.101526	−	0.101526i
$776$	−200186.			−0.332438
$777$		0			0
$778$	−393412.	−	393412.i	−0.649963	−	0.649963i
$779$		−	107428.i		−	0.177028i
$780$		0			0
$781$	−79733.8			−0.130720
$782$	471503.	−	471503.i	0.771030	−	0.771030i
$783$		0			0
$784$		−	518878.i		−	0.844177i
$785$	314387.	−	314387.i	0.510183	−	0.510183i
$786$		0			0
$787$	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
$789$		0			0
$790$		−	901420.i		−	1.44435i
$791$	−293581.	−	293581.i	−0.469219	−	0.469219i
$792$		0			0
$793$	968645.	−	383093.i	1.54035	−	0.609197i
$794$	710510.			1.12701
$795$		0			0
$796$	37806.4			0.0596677
$797$			508211.i			0.800069i	0.916500	+	0.400035i	$0.131002\pi$
$797$			508211.i			0.800069i	−0.916500	+	0.400035i	$0.868998\pi$
$798$		0			0
$799$	100939.	−	100939.i	0.158112	−	0.158112i
$800$	5004.44	+	5004.44i	0.00781944	+	0.00781944i
$801$		0			0
$802$	765719.			1.19048
$803$		−	138658.i		−	0.215037i
$804$		0			0
$805$		−	339024.i		−	0.523165i
$806$	−315161.	−	796880.i	−0.485135	−	1.22666i
$807$		0			0
$808$	−286013.	+	286013.i	−0.438090	+	0.438090i
$809$	302513.			0.462218			0.231109	−	0.972928i	$-0.425765\pi$
$809$	302513.			0.462218			0.231109	+	0.972928i	$0.425765\pi$
$810$		0			0
$811$	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
$813$		0			0
$814$	138731.	+	138731.i	0.209376	+	0.209376i
$815$	−861431.			−1.29690
$816$		0			0
$817$	122123.	+	122123.i	0.182959	+	0.182959i
$818$			15651.8i			0.0233915i
$819$		0			0
$820$	−29123.2			−0.0433123
$821$	141594.	−	141594.i	0.210068	−	0.210068i	−0.594228	−	0.804296i	$-0.702542\pi$
$821$	141594.	−	141594.i	0.210068	−	0.210068i	0.804296	+	0.594228i	$0.202542\pi$
$822$		0			0
$823$			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
$825$		0			0
$826$	−86323.7	−	86323.7i	−0.126523	−	0.126523i
$827$	−95483.6	−	95483.6i	−0.139610	−	0.139610i	0.633848	−	0.773458i	$-0.281475\pi$
$827$	−95483.6	−	95483.6i	−0.139610	−	0.139610i	−0.773458	+	0.633848i	$0.781475\pi$
$828$		0			0
$829$		−	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
$831$		0			0
$832$	−240726.	−	608671.i	−0.347757	−	0.879298i
$833$	−528231.			−0.761261
$834$		0			0
$835$	619271.			0.888193
$836$			5138.64i			0.00735251i
$837$		0			0
$838$	−273686.	+	273686.i	−0.389731	+	0.389731i
$839$	666830.	+	666830.i	0.947308	+	0.947308i	0.998680	−	0.0513717i	$-0.0163593\pi$
$839$	666830.	+	666830.i	0.947308	+	0.947308i	−0.0513717	+	0.998680i	$0.516359\pi$
$840$		0			0
$841$	1.01978e6			1.44182
$842$			89085.5i			0.125656i
$843$		0			0
$844$			60084.3i			0.0843482i
$845$	−24210.4	+	752395.i	−0.0339070	+	1.05374i
$846$		0			0
$847$	−116153.	+	116153.i	−0.161907	+	0.161907i
$848$	−1.17929e6			−1.63995
$849$		0			0
$850$	55117.0	−	55117.0i	0.0762864	−	0.0762864i
$851$	−240783.	+	240783.i	−0.332481	+	0.332481i
$852$		0			0
$853$	−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
$855$		0			0
$856$	426802.	+	426802.i	0.582478	+	0.582478i
$857$			489846.i			0.666957i	0.942758	+	0.333479i	$0.108222\pi$
$857$			489846.i			0.666957i	−0.942758	+	0.333479i	$0.891778\pi$
$858$		0			0
$859$	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
$861$		0			0
$862$			243649.i			0.327907i
$863$	715898.	−	715898.i	0.961235	−	0.961235i	−0.0380415	−	0.999276i	$-0.512112\pi$
$863$	715898.	−	715898.i	0.961235	−	0.961235i	0.999276	+	0.0380415i	$0.0121119\pi$
$864$		0			0
$865$	172030.	+	172030.i	0.229918	+	0.229918i
$866$	−85139.8	−	85139.8i	−0.113526	−	0.113526i
$867$		0			0
$868$			21196.1i			0.0281330i
$869$	494672.	+	494672.i	0.655055	+	0.655055i
$870$		0			0
$871$	162316.	−	374705.i	0.213957	−	0.493916i
$872$	1.44282e6			1.89748
$873$		0			0
$874$	−188629.			−0.246937
$875$			315756.i			0.412417i
$876$		0			0
$877$	−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
$879$		0			0
$880$	−592288.			−0.764834
$881$		−	863892.i		−	1.11303i	−0.830837	−	0.556516i	$-0.812138\pi$
$881$		−	863892.i		−	1.11303i	0.830837	−	0.556516i	$-0.187862\pi$
$882$		0			0
$883$			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
$885$		0			0
$886$	−65280.2	+	65280.2i	−0.0831599	+	0.0831599i
$887$	−1.04691e6			−1.33065			−0.665323	−	0.746556i	$-0.731706\pi$
$887$	−1.04691e6			−1.33065			−0.665323	+	0.746556i	$0.731706\pi$
$888$		0			0
$889$	−401768.	+	401768.i	−0.508361	+	0.508361i
$890$	−903706.	+	903706.i	−1.14090	+	1.14090i
$891$		0			0
$892$	756.886	+	756.886i	0.000951263	+	0.000951263i
$893$	−40381.5			−0.0506384
$894$		0			0
$895$	−536226.	−	536226.i	−0.669425	−	0.669425i
$896$			377468.i			0.470180i
$897$		0			0
$898$	−308691.			−0.382800
$899$	1.14980e6	−	1.14980e6i	1.42267	−	1.42267i
$900$		0			0
$901$			1.20055e6i			1.47887i
$902$	338016.	−	338016.i	0.415456	−	0.415456i
$903$		0			0
$904$	−847999.	−	847999.i	−1.03767	−	1.03767i
$905$	43515.7	+	43515.7i	0.0531311	+	0.0531311i
$906$		0			0
$907$			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
$909$		0			0
$910$	156536.	−	361362.i	0.189031	−	0.436374i
$911$	1.22775e6			1.47936			0.739681	−	0.672958i	$-0.234976\pi$
$911$	1.22775e6			1.47936			0.739681	+	0.672958i	$0.234976\pi$
$912$		0			0
$913$	−772185.			−0.926361
$914$			544974.i			0.652354i
$915$		0			0
$916$	7604.58	−	7604.58i	0.00906326	−	0.00906326i
$917$	−47204.8	−	47204.8i	−0.0561368	−	0.0561368i
$918$		0			0
$919$	−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
$921$		0			0
$922$			1.27948e6i			1.50513i
$923$	63896.6	−	147504.i	0.0750023	−	0.173142i
$924$		0			0
$925$	−28146.6	+	28146.6i	−0.0328960	+	0.0328960i
$926$	860015.			1.00296
$927$		0			0
$928$	−94362.5	+	94362.5i	−0.109573	+	0.109573i
$929$	−865268.	+	865268.i	−1.00258	+	1.00258i	−0.00258353	+	0.999997i	$0.500822\pi$
$929$	−865268.	+	865268.i	−1.00258	+	1.00258i	−0.999997	+	0.00258353i	$0.999178\pi$
$930$		0			0
$931$	105662.	+	105662.i	0.121904	+	0.121904i
$932$	−57807.8			−0.0665511
$933$		0			0
$934$	259629.	+	259629.i	0.297619	+	0.297619i
$935$			602963.i			0.689712i
$936$		0			0
$937$	−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
$939$		0			0
$940$			10947.2i			0.0123894i
$941$	−1.18512e6	+	1.18512e6i	−1.33839	+	1.33839i	−0.440774	+	0.897618i	$0.645296\pi$
$941$	−1.18512e6	+	1.18512e6i	−1.33839	+	1.33839i	−0.897618	+	0.440774i	$0.854704\pi$
$942$		0			0
$943$	586664.	+	586664.i	0.659729	+	0.659729i
$944$	−261748.	−	261748.i	−0.293724	−	0.293724i
$945$		0			0
$946$			768509.i			0.858750i
$947$	−94860.4	−	94860.4i	−0.105775	−	0.105775i	0.652238	−	0.758014i	$-0.273830\pi$
$947$	−94860.4	−	94860.4i	−0.105775	−	0.105775i	−0.758014	+	0.652238i	$0.773830\pi$
$948$		0			0
$949$	256512.	+	111117.i	0.284823	+	0.123381i
$950$	−22050.0			−0.0244322
$951$		0			0
$952$	366884.			0.404813
$953$			313080.i			0.344723i	0.985034	+	0.172361i	$0.0551397\pi$
$953$			313080.i			0.344723i	−0.985034	+	0.172361i	$0.944860\pi$
$954$		0			0
$955$	−214341.	+	214341.i	−0.235016	+	0.235016i
$956$	57682.4	+	57682.4i	0.0631142	+	0.0631142i
$957$		0			0
$958$	1.68358e6			1.83444
$959$		−	318947.i		−	0.346802i
$960$		0			0
$961$		−	607464.i		−	0.657770i
$962$	−367824.	+	145472.i	−0.397456	+	0.157192i
$963$		0			0
$964$	51075.1	−	51075.1i	0.0549610	−	0.0549610i
$965$	−1.36762e6			−1.46862
$966$		0			0
$967$	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
$969$		0			0
$970$	−245359.	−	245359.i	−0.260771	−	0.260771i
$971$	1.68455e6			1.78668			0.893338	−	0.449385i	$-0.148357\pi$
$971$	1.68455e6			1.78668			0.893338	+	0.449385i	$0.148357\pi$
$972$		0			0
$973$	−90346.8	−	90346.8i	−0.0954305	−	0.0954305i
$974$			1.08228e6i			1.14083i
$975$		0			0
$976$	−1.65230e6			−1.73456
$977$	−857213.	+	857213.i	−0.898048	+	0.898048i	−0.995263	−	0.0972150i	$-0.969007\pi$
$977$	−857213.	+	857213.i	−0.898048	+	0.898048i	0.0972150	+	0.995263i	$0.469007\pi$
$978$		0			0
$979$		−	991852.i		−	1.03486i
$980$	28644.4	−	28644.4i	0.0298254	−	0.0298254i
$981$		0			0
$982$	76718.1	+	76718.1i	0.0795564	+	0.0795564i
$983$	−821373.	−	821373.i	−0.850029	−	0.850029i	0.140107	−	0.990136i	$-0.455255\pi$
$983$	−821373.	−	821373.i	−0.850029	−	0.850029i	−0.990136	+	0.140107i	$0.955255\pi$
$984$		0			0
$985$		−	140719.i		−	0.145037i
$986$	1.03927e6	+	1.03927e6i	1.06899	+	1.06899i
$987$		0			0
$988$	−9506.29	−	4117.98i	−0.00973861	−	0.00421861i
$989$	−1.33383e6			−1.36366
$990$		0			0
$991$	−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
$993$		0			0
$994$	−59463.3	+	59463.3i	−0.0601834	+	0.0601834i
$995$	−887364.	−	887364.i	−0.896305	−	0.896305i
$996$		0			0
$997$	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
$999$		0			0

Twists

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

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

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.89776 + 2.89776i) q^{2} -0.794047i q^{4} +(-18.6373 + 18.6373i) q^{5} +(-15.2549 - 15.2549i) q^{7} +(-44.0632 - 44.0632i) q^{8} -108.013i q^{10} +(59.2741 + 59.2741i) q^{11} +(-157.156 + 62.1541i) q^{13} +88.4101 q^{14} +268.074 q^{16} -272.906i q^{17} +(-54.5892 + 54.5892i) q^{19} +(14.7989 + 14.7989i) q^{20} -343.524 q^{22} -596.223i q^{23} -69.6962i q^{25} +(275.292 - 635.507i) q^{26} +(-12.1131 + 12.1131i) q^{28} +1314.18 q^{29} +(874.924 - 874.924i) q^{31} +(-71.8036 + 71.8036i) q^{32} +(790.817 + 790.817i) q^{34} +568.619 q^{35} +(-403.847 - 403.847i) q^{37} -316.373i q^{38} +1642.44 q^{40} +(-983.966 + 983.966i) q^{41} -2237.13i q^{43} +(47.0664 - 47.0664i) q^{44} +(1727.71 + 1727.71i) q^{46} +(369.867 + 369.867i) q^{47} -1935.58i q^{49} +(201.963 + 201.963i) q^{50} +(49.3532 + 124.789i) q^{52} -4399.13 q^{53} -2209.42 q^{55} +1344.36i q^{56} +(-3808.17 + 3808.17i) q^{58} +(-976.401 - 976.401i) q^{59} -6163.60 q^{61} +5070.64i q^{62} +3873.05i q^{64} +(1770.57 - 4087.33i) q^{65} +(-1708.57 + 1708.57i) q^{67} -216.700 q^{68} +(-1647.72 + 1647.72i) q^{70} +(-672.586 + 672.586i) q^{71} +(-1169.63 - 1169.63i) q^{73} +2340.51 q^{74} +(43.3464 + 43.3464i) q^{76} -1808.44i q^{77} +8345.50 q^{79} +(-4996.17 + 4996.17i) q^{80} -5702.60i q^{82} +(-6513.68 + 6513.68i) q^{83} +(5086.23 + 5086.23i) q^{85} +(6482.67 + 6482.67i) q^{86} -5223.62i q^{88} +(-8366.65 - 8366.65i) q^{89} +(3345.55 + 1449.24i) q^{91} -473.429 q^{92} -2143.57 q^{94} -2034.79i q^{95} +(2271.58 - 2271.58i) q^{97} +(5608.84 + 5608.84i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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