LMFDB - Embedded newform 39.5.g.a.34.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [39,5,Mod(31,39)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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			34.7
Root			$-1.66937 - 1.66937i$ of defining polynomial
Character	$\chi$	$=$	39.34
Dual form			39.5.g.a.31.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.66937 + 1.66937i) q^{2} -5.19615 q^{3} -10.4264i q^{4} +(-28.4160 - 28.4160i) q^{5} +(-8.67431 - 8.67431i) q^{6} +(-17.1341 + 17.1341i) q^{7} +(44.1155 - 44.1155i) q^{8} +27.0000 q^{9} -94.8736i q^{10} +(54.2346 - 54.2346i) q^{11} +54.1772i q^{12} +(-168.969 + 3.22849i) q^{13} -57.2062 q^{14} +(147.654 + 147.654i) q^{15} -19.5322 q^{16} -70.4411i q^{17} +(45.0730 + 45.0730i) q^{18} +(194.123 + 194.123i) q^{19} +(-296.276 + 296.276i) q^{20} +(89.0312 - 89.0312i) q^{21} +181.075 q^{22} -716.329i q^{23} +(-229.231 + 229.231i) q^{24} +989.935i q^{25} +(-287.462 - 276.683i) q^{26} -140.296 q^{27} +(178.647 + 178.647i) q^{28} +725.866 q^{29} +492.978i q^{30} +(586.446 + 586.446i) q^{31} +(-738.454 - 738.454i) q^{32} +(-281.811 + 281.811i) q^{33} +(117.592 - 117.592i) q^{34} +973.762 q^{35} -281.513i q^{36} +(575.923 - 575.923i) q^{37} +648.126i q^{38} +(877.990 - 16.7757i) q^{39} -2507.17 q^{40} +(-1810.07 - 1810.07i) q^{41} +297.252 q^{42} -2879.48i q^{43} +(-565.472 - 565.472i) q^{44} +(-767.231 - 767.231i) q^{45} +(1195.82 - 1195.82i) q^{46} +(1064.86 - 1064.86i) q^{47} +101.492 q^{48} +1813.85i q^{49} +(-1652.57 + 1652.57i) q^{50} +366.023i q^{51} +(33.6615 + 1761.74i) q^{52} +2549.76 q^{53} +(-234.206 - 234.206i) q^{54} -3082.26 q^{55} +1511.75i q^{56} +(-1008.69 - 1008.69i) q^{57} +(1211.74 + 1211.74i) q^{58} +(-2264.35 + 2264.35i) q^{59} +(1539.50 - 1539.50i) q^{60} +6434.68 q^{61} +1957.99i q^{62} +(-462.620 + 462.620i) q^{63} -2152.99i q^{64} +(4893.16 + 4709.68i) q^{65} -940.895 q^{66} +(-5472.27 - 5472.27i) q^{67} -734.447 q^{68} +3722.15i q^{69} +(1625.57 + 1625.57i) q^{70} +(4643.32 + 4643.32i) q^{71} +(1191.12 - 1191.12i) q^{72} +(-5605.57 + 5605.57i) q^{73} +1922.86 q^{74} -5143.86i q^{75} +(2024.00 - 2024.00i) q^{76} +1858.52i q^{77} +(1493.70 + 1437.69i) q^{78} -6871.05 q^{79} +(555.027 + 555.027i) q^{80} +729.000 q^{81} -6043.36i q^{82} +(-2647.22 - 2647.22i) q^{83} +(-928.275 - 928.275i) q^{84} +(-2001.65 + 2001.65i) q^{85} +(4806.92 - 4806.92i) q^{86} -3771.71 q^{87} -4785.17i q^{88} +(-6804.23 + 6804.23i) q^{89} -2561.59i q^{90} +(2839.81 - 2950.45i) q^{91} -7468.73 q^{92} +(-3047.26 - 3047.26i) q^{93} +3555.29 q^{94} -11032.4i q^{95} +(3837.12 + 3837.12i) q^{96} +(9498.95 + 9498.95i) q^{97} +(-3027.99 + 3027.99i) q^{98} +(1464.33 - 1464.33i) 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 $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.66937	+	1.66937i	0.417343	+	0.417343i	0.884287	−	0.466944i	$-0.154645\pi$
$2$	1.66937	+	1.66937i	0.417343	+	0.417343i	−0.466944	+	0.884287i	$0.654645\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$		−	10.4264i		−	0.651650i
$5$	−28.4160	−	28.4160i	−1.13664	−	1.13664i	−0.989049	−	0.147591i	$-0.952848\pi$
$5$	−28.4160	−	28.4160i	−1.13664	−	1.13664i	−0.147591	−	0.989049i	$-0.547152\pi$
$6$	−8.67431	−	8.67431i	−0.240953	−	0.240953i
$7$	−17.1341	+	17.1341i	−0.349675	+	0.349675i	−0.859988	−	0.510314i	$-0.829529\pi$
$7$	−17.1341	+	17.1341i	−0.349675	+	0.349675i	0.510314	+	0.859988i	$0.329529\pi$
$8$	44.1155	−	44.1155i	0.689304	−	0.689304i
$9$	27.0000			0.333333
$10$		−	94.8736i		−	0.948736i
$11$	54.2346	−	54.2346i	0.448220	−	0.448220i	−0.446542	−	0.894762i	$-0.647345\pi$
$11$	54.2346	−	54.2346i	0.448220	−	0.448220i	0.894762	+	0.446542i	$0.147345\pi$
$12$			54.1772i			0.376230i
$13$	−168.969	+	3.22849i	−0.999818	+	0.0191035i
$13$	−168.969	+	3.22849i	−0.999818	+	0.0191035i
$14$	−57.2062			−0.291868
$15$	147.654	+	147.654i	0.656239	+	0.656239i
$16$	−19.5322			−0.0762978
$17$		−	70.4411i		−	0.243741i	−0.992546	−	0.121870i	$-0.961111\pi$
$17$		−	70.4411i		−	0.243741i	0.992546	−	0.121870i	$-0.0388892\pi$
$18$	45.0730	+	45.0730i	0.139114	+	0.139114i
$19$	194.123	+	194.123i	0.537736	+	0.537736i	0.922863	−	0.385127i	$-0.125843\pi$
$19$	194.123	+	194.123i	0.537736	+	0.537736i	−0.385127	+	0.922863i	$0.625843\pi$
$20$	−296.276	+	296.276i	−0.740691	+	0.740691i
$21$	89.0312	−	89.0312i	0.201885	−	0.201885i
$22$	181.075			0.374123
$23$		−	716.329i		−	1.35412i	−0.735928	−	0.677059i	$-0.763254\pi$
$23$		−	716.329i		−	1.35412i	0.735928	−	0.677059i	$-0.236746\pi$
$24$	−229.231	+	229.231i	−0.397970	+	0.397970i
$25$			989.935i			1.58390i
$26$	−287.462	−	276.683i	−0.425239	−	0.409294i
$27$	−140.296			−0.192450
$28$	178.647	+	178.647i	0.227866	+	0.227866i
$29$	725.866			0.863099			0.431550	−	0.902089i	$-0.357967\pi$
$29$	725.866			0.863099			0.431550	+	0.902089i	$0.357967\pi$
$30$			492.978i			0.547753i
$31$	586.446	+	586.446i	0.610245	+	0.610245i	0.943010	−	0.332765i	$-0.107981\pi$
$31$	586.446	+	586.446i	0.610245	+	0.610245i	−0.332765	+	0.943010i	$0.607981\pi$
$32$	−738.454	−	738.454i	−0.721147	−	0.721147i
$33$	−281.811	+	281.811i	−0.258780	+	0.258780i
$34$	117.592	−	117.592i	0.101723	−	0.101723i
$35$	973.762			0.794908
$36$		−	281.513i		−	0.217217i
$37$	575.923	−	575.923i	0.420689	−	0.420689i	−0.464752	−	0.885441i	$-0.653856\pi$
$37$	575.923	−	575.923i	0.420689	−	0.420689i	0.885441	+	0.464752i	$0.153856\pi$
$38$			648.126i			0.448840i
$39$	877.990	−	16.7757i	0.577245	−	0.0110294i
$40$	−2507.17			−1.56698
$41$	−1810.07	−	1810.07i	−1.07678	−	1.07678i	−0.996796	−	0.0799854i	$-0.974513\pi$
$41$	−1810.07	−	1810.07i	−1.07678	−	1.07678i	−0.0799854	−	0.996796i	$-0.525487\pi$
$42$	297.252			0.168510
$43$		−	2879.48i		−	1.55732i	−0.627448	−	0.778659i	$-0.715900\pi$
$43$		−	2879.48i		−	1.55732i	0.627448	−	0.778659i	$-0.284100\pi$
$44$	−565.472	−	565.472i	−0.292083	−	0.292083i
$45$	−767.231	−	767.231i	−0.378880	−	0.378880i
$46$	1195.82	−	1195.82i	0.565132	−	0.565132i
$47$	1064.86	−	1064.86i	0.482055	−	0.482055i	−0.423733	−	0.905787i	$-0.639280\pi$
$47$	1064.86	−	1064.86i	0.482055	−	0.482055i	0.905787	+	0.423733i	$0.139280\pi$
$48$	101.492			0.0440505
$49$			1813.85i			0.755455i
$50$	−1652.57	+	1652.57i	−0.661028	+	0.661028i
$51$			366.023i			0.140724i
$52$	33.6615	+	1761.74i	0.0124488	+	0.651531i
$53$	2549.76			0.907710			0.453855	−	0.891076i	$-0.350048\pi$
$53$	2549.76			0.907710			0.453855	+	0.891076i	$0.350048\pi$
$54$	−234.206	−	234.206i	−0.0803177	−	0.0803177i
$55$	−3082.26			−1.01893
$56$			1511.75i			0.482065i
$57$	−1008.69	−	1008.69i	−0.310462	−	0.310462i
$58$	1211.74	+	1211.74i	0.360208	+	0.360208i
$59$	−2264.35	+	2264.35i	−0.650487	+	0.650487i	−0.953110	−	0.302623i	$-0.902138\pi$
$59$	−2264.35	+	2264.35i	−0.650487	+	0.650487i	0.302623	+	0.953110i	$0.402138\pi$
$60$	1539.50	−	1539.50i	0.427638	−	0.427638i
$61$	6434.68			1.72929			0.864644	−	0.502386i	$-0.167544\pi$
$61$	6434.68			1.72929			0.864644	+	0.502386i	$0.167544\pi$
$62$			1957.99i			0.509363i
$63$	−462.620	+	462.620i	−0.116558	+	0.116558i
$64$		−	2152.99i		−	0.525633i
$65$	4893.16	+	4709.68i	1.15815	+	1.11472i
$66$	−940.895			−0.216000
$67$	−5472.27	−	5472.27i	−1.21904	−	1.21904i	−0.967969	−	0.251070i	$-0.919217\pi$
$67$	−5472.27	−	5472.27i	−1.21904	−	1.21904i	−0.251070	−	0.967969i	$-0.580783\pi$
$68$	−734.447			−0.158834
$69$			3722.15i			0.781801i
$70$	1625.57	+	1625.57i	0.331749	+	0.331749i
$71$	4643.32	+	4643.32i	0.921111	+	0.921111i	0.997108	−	0.0759967i	$-0.0242139\pi$
$71$	4643.32	+	4643.32i	0.921111	+	0.921111i	−0.0759967	+	0.997108i	$0.524214\pi$
$72$	1191.12	−	1191.12i	0.229768	−	0.229768i
$73$	−5605.57	+	5605.57i	−1.05190	+	1.05190i	−0.0533222	+	0.998577i	$0.516981\pi$
$73$	−5605.57	+	5605.57i	−1.05190	+	1.05190i	−0.998577	+	0.0533222i	$0.983019\pi$
$74$	1922.86			0.351143
$75$		−	5143.86i		−	0.914463i
$76$	2024.00	−	2024.00i	0.350416	−	0.350416i
$77$			1858.52i			0.313462i
$78$	1493.70	+	1437.69i	0.245512	+	0.236306i
$79$	−6871.05			−1.10095			−0.550477	−	0.834850i	$-0.685554\pi$
$79$	−6871.05			−1.10095			−0.550477	+	0.834850i	$0.685554\pi$
$80$	555.027	+	555.027i	0.0867230	+	0.0867230i
$81$	729.000			0.111111
$82$		−	6043.36i		−	0.898774i
$83$	−2647.22	−	2647.22i	−0.384267	−	0.384267i	0.488370	−	0.872637i	$-0.337592\pi$
$83$	−2647.22	−	2647.22i	−0.384267	−	0.384267i	−0.872637	+	0.488370i	$0.837592\pi$
$84$	−928.275	−	928.275i	−0.131558	−	0.131558i
$85$	−2001.65	+	2001.65i	−0.277045	+	0.277045i
$86$	4806.92	−	4806.92i	0.649935	−	0.649935i
$87$	−3771.71			−0.498311
$88$		−	4785.17i		−	0.617920i
$89$	−6804.23	+	6804.23i	−0.859012	+	0.859012i	−0.991222	−	0.132210i	$-0.957793\pi$
$89$	−6804.23	+	6804.23i	−0.859012	+	0.859012i	0.132210	+	0.991222i	$0.457793\pi$
$90$		−	2561.59i		−	0.316245i
$91$	2839.81	−	2950.45i	0.342931	−	0.356291i
$92$	−7468.73			−0.882412
$93$	−3047.26	−	3047.26i	−0.352325	−	0.352325i
$94$	3555.29			0.402364
$95$		−	11032.4i		−	1.22242i
$96$	3837.12	+	3837.12i	0.416354	+	0.416354i
$97$	9498.95	+	9498.95i	1.00956	+	1.00956i	0.999954	+	0.00960578i	$0.00305766\pi$
$97$	9498.95	+	9498.95i	1.00956	+	1.00956i	0.00960578	+	0.999954i	$0.496942\pi$
$98$	−3027.99	+	3027.99i	−0.315284	+	0.315284i
$99$	1464.33	−	1464.33i	0.149407	−	0.149407i
$100$	10321.5			1.03215
$101$		−	16541.4i		−	1.62155i	−0.585361	−	0.810773i	$-0.699047\pi$
$101$		−	16541.4i		−	1.62155i	0.585361	−	0.810773i	$-0.300953\pi$
$102$	−611.028	+	611.028i	−0.0587301	+	0.0587301i
$103$		−	11148.4i		−	1.05084i	−0.850842	−	0.525421i	$-0.823908\pi$
$103$		−	11148.4i		−	1.05084i	0.850842	−	0.525421i	$-0.176092\pi$
$104$	−7311.73	+	7596.58i	−0.676010	+	0.702347i
$105$	−5059.82			−0.458940
$106$	4256.49	+	4256.49i	0.378826	+	0.378826i
$107$	9291.29			0.811537			0.405769	−	0.913976i	$-0.367004\pi$
$107$	9291.29			0.811537			0.405769	+	0.913976i	$0.367004\pi$
$108$			1462.78i			0.125410i
$109$	1985.75	+	1985.75i	0.167136	+	0.167136i	0.785719	−	0.618583i	$-0.212293\pi$
$109$	1985.75	+	1985.75i	0.167136	+	0.167136i	−0.618583	+	0.785719i	$0.712293\pi$
$110$	−5145.43	−	5145.43i	−0.425243	−	0.425243i
$111$	−2992.59	+	2992.59i	−0.242885	+	0.242885i
$112$	334.666	−	334.666i	0.0266794	−	0.0266794i
$113$	−1799.10			−0.140896			−0.0704478	−	0.997515i	$-0.522443\pi$
$113$	−1799.10			−0.140896			−0.0704478	+	0.997515i	$0.522443\pi$
$114$		−	3367.76i		−	0.259138i
$115$	−20355.2	+	20355.2i	−1.53914	+	1.53914i
$116$		−	7568.17i		−	0.562439i
$117$	−4562.17	+	87.1693i	−0.333273	+	0.00636783i
$118$	−7560.07			−0.542952
$119$	1206.94	+	1206.94i	0.0852300	+	0.0852300i
$120$	13027.6			0.904696
$121$			8758.21i			0.598198i
$122$	10741.9	+	10741.9i	0.721706	+	0.721706i
$123$	9405.40	+	9405.40i	0.621680	+	0.621680i
$124$	6114.52	−	6114.52i	0.397666	−	0.397666i
$125$	10370.0	−	10370.0i	0.663680	−	0.663680i
$126$	−1544.57			−0.0972895
$127$		−	7557.96i		−	0.468594i	−0.972165	−	0.234297i	$-0.924721\pi$
$127$		−	7557.96i		−	0.468594i	0.972165	−	0.234297i	$-0.0752789\pi$
$128$	−8221.12	+	8221.12i	−0.501777	+	0.501777i
$129$			14962.2i			0.899118i
$130$	306.299	+	16030.7i	0.0181242	+	0.948563i
$131$	9705.21			0.565539			0.282769	−	0.959188i	$-0.408747\pi$
$131$	9705.21			0.565539			0.282769	+	0.959188i	$0.408747\pi$
$132$	2938.28	+	2938.28i	0.168634	+	0.168634i
$133$	−6652.22			−0.376065
$134$		−	18270.5i		−	1.01751i
$135$	3986.65	+	3986.65i	0.218746	+	0.218746i
$136$	−3107.54	−	3107.54i	−0.168012	−	0.168012i
$137$	−7306.03	+	7306.03i	−0.389261	+	0.389261i	−0.874424	−	0.485163i	$-0.838760\pi$
$137$	−7306.03	+	7306.03i	−0.389261	+	0.389261i	0.485163	+	0.874424i	$0.338760\pi$
$138$	−6213.66	+	6213.66i	−0.326279	+	0.326279i
$139$	14508.5			0.750921			0.375460	−	0.926838i	$-0.377485\pi$
$139$	14508.5			0.750921			0.375460	+	0.926838i	$0.377485\pi$
$140$		−	10152.8i		−	0.518002i
$141$	−5533.17	+	5533.17i	−0.278314	+	0.278314i
$142$			15502.9i			0.768838i
$143$	−8988.88	+	9339.07i	−0.439576	+	0.456701i
$144$	−527.370			−0.0254326
$145$	−20626.2	−	20626.2i	−0.981032	−	0.981032i
$146$	−18715.6			−0.878005
$147$		−	9425.03i		−	0.436162i
$148$	−6004.81	−	6004.81i	−0.274142	−	0.274142i
$149$	6209.08	+	6209.08i	0.279675	+	0.279675i	0.832979	−	0.553304i	$-0.186633\pi$
$149$	6209.08	+	6209.08i	0.279675	+	0.279675i	−0.553304	+	0.832979i	$0.686633\pi$
$150$	8587.00	−	8587.00i	0.381645	−	0.381645i
$151$	6322.22	−	6322.22i	0.277278	−	0.277278i	−0.554743	−	0.832021i	$-0.687184\pi$
$151$	6322.22	−	6322.22i	0.277278	−	0.277278i	0.832021	+	0.554743i	$0.187184\pi$
$152$	17127.6			0.741327
$153$		−	1901.91i		−	0.0812469i
$154$	−3102.56	+	3102.56i	−0.130821	+	0.130821i
$155$		−	33328.9i		−	1.38726i
$156$	−174.910	−	9154.27i	−0.00718731	−	0.376162i
$157$	26942.7			1.09305			0.546526	−	0.837442i	$-0.315950\pi$
$157$	26942.7			1.09305			0.546526	+	0.837442i	$0.315950\pi$
$158$	−11470.3	−	11470.3i	−0.459475	−	0.459475i
$159$	−13248.9			−0.524067
$160$			41967.8i			1.63937i
$161$	12273.6	+	12273.6i	0.473501	+	0.473501i
$162$	1216.97	+	1216.97i	0.0463714	+	0.0463714i
$163$	22832.2	−	22832.2i	0.859355	−	0.859355i	−0.131907	−	0.991262i	$-0.542110\pi$
$163$	22832.2	−	22832.2i	0.859355	−	0.859355i	0.991262	+	0.131907i	$0.0421101\pi$
$164$	−18872.5	+	18872.5i	−0.701685	+	0.701685i
$165$	16015.9			0.588279
$166$		−	8838.37i		−	0.320742i
$167$	7664.75	−	7664.75i	0.274831	−	0.274831i	−0.556211	−	0.831041i	$-0.687745\pi$
$167$	7664.75	−	7664.75i	0.274831	−	0.274831i	0.831041	+	0.556211i	$0.187745\pi$
$168$		−	7855.31i		−	0.278320i
$169$	28540.2	−	1091.03i	0.999270	−	0.0382000i
$170$	−6683.00			−0.231246
$171$	5241.31	+	5241.31i	0.179245	+	0.179245i
$172$	−30022.6			−1.01483
$173$			9382.63i			0.313496i	0.987639	+	0.156748i	$0.0501011\pi$
$173$			9382.63i			0.313496i	−0.987639	+	0.156748i	$0.949899\pi$
$174$	−6296.39	−	6296.39i	−0.207966	−	0.207966i
$175$	−16961.6	−	16961.6i	−0.553849	−	0.553849i
$176$	−1059.32	+	1059.32i	−0.0341982	+	0.0341982i
$177$	11765.9	−	11765.9i	0.375559	−	0.375559i
$178$	−22717.6			−0.717005
$179$		−	322.023i		−	0.0100503i	−0.999987	−	0.00502517i	$-0.998400\pi$
$179$		−	322.023i		−	0.0100503i	0.999987	−	0.00502517i	$-0.00159957\pi$
$180$	−7999.46	+	7999.46i	−0.246897	+	0.246897i
$181$		−	28479.3i		−	0.869306i	−0.900598	−	0.434653i	$-0.856871\pi$
$181$		−	28479.3i		−	0.869306i	0.900598	−	0.434653i	$-0.143129\pi$
$182$	9666.09	−	184.690i	0.291815	−	0.00557571i
$183$	−33435.6			−0.998405
$184$	−31601.2	−	31601.2i	−0.933400	−	0.933400i
$185$	−32730.8			−0.956343
$186$		−	10174.0i		−	0.294081i
$187$	−3820.35	−	3820.35i	−0.109249	−	0.109249i
$188$	−11102.6	−	11102.6i	−0.314131	−	0.314131i
$189$	2403.84	−	2403.84i	0.0672949	−	0.0672949i
$190$	18417.1	−	18417.1i	0.510170	−	0.510170i
$191$	39049.7			1.07041			0.535206	−	0.844722i	$-0.320234\pi$
$191$	39049.7			1.07041			0.535206	+	0.844722i	$0.320234\pi$
$192$			11187.3i			0.303474i
$193$	−34849.0	+	34849.0i	−0.935570	+	0.935570i	−0.998046	−	0.0624768i	$-0.980100\pi$
$193$	−34849.0	+	34849.0i	−0.935570	+	0.935570i	0.0624768	+	0.998046i	$0.480100\pi$
$194$			31714.5i			0.842665i
$195$	−25425.6	−	24472.2i	−0.668656	−	0.643583i
$196$	18911.9			0.492292
$197$	33755.4	+	33755.4i	0.869784	+	0.869784i	0.992448	−	0.122664i	$-0.0391438\pi$
$197$	33755.4	+	33755.4i	0.869784	+	0.869784i	−0.122664	+	0.992448i	$0.539144\pi$
$198$	4889.04			0.124708
$199$		−	12091.0i		−	0.305322i	−0.988279	−	0.152661i	$-0.951216\pi$
$199$		−	12091.0i		−	0.305322i	0.988279	−	0.152661i	$-0.0487842\pi$
$200$	43671.5	+	43671.5i	1.09179	+	1.09179i
$201$	28434.7	+	28434.7i	0.703813	+	0.703813i
$202$	27613.7	−	27613.7i	0.676740	−	0.676740i
$203$	−12437.0	+	12437.0i	−0.301804	+	0.301804i
$204$	3816.30			0.0917027
$205$			102870.i			2.44782i
$206$	18610.8	−	18610.8i	0.438561	−	0.438561i
$207$		−	19340.9i		−	0.451373i
$208$	3300.34	−	63.0596i	0.0762838	−	0.00145755i
$209$	21056.3			0.482048
$210$	−8446.71	−	8446.71i	−0.191535	−	0.191535i
$211$	51063.6			1.14695			0.573477	−	0.819221i	$-0.305594\pi$
$211$	51063.6			1.14695			0.573477	+	0.819221i	$0.305594\pi$
$212$		−	26584.8i		−	0.591509i
$213$	−24127.4	−	24127.4i	−0.531804	−	0.531804i
$214$	15510.6	+	15510.6i	0.338689	+	0.338689i
$215$	−81823.2	+	81823.2i	−1.77011	+	1.77011i
$216$	−6189.23	+	6189.23i	−0.132657	+	0.132657i
$217$	−20096.4			−0.426775
$218$			6629.90i			0.139506i
$219$	29127.4	−	29127.4i	0.607314	−	0.607314i
$220$			32136.9i			0.663985i
$221$	227.418	+	11902.4i	0.00465630	+	0.243696i
$222$	−9991.47			−0.202733
$223$	7861.80	+	7861.80i	0.158093	+	0.158093i	0.781721	−	0.623628i	$-0.214342\pi$
$223$	7861.80	+	7861.80i	0.158093	+	0.158093i	−0.623628	+	0.781721i	$0.714342\pi$
$224$	25305.4			0.504333
$225$			26728.3i			0.527966i
$226$	−3003.36	−	3003.36i	−0.0588018	−	0.0588018i
$227$	−9642.38	−	9642.38i	−0.187125	−	0.187125i	0.607327	−	0.794452i	$-0.292242\pi$
$227$	−9642.38	−	9642.38i	−0.187125	−	0.187125i	−0.794452	+	0.607327i	$0.792242\pi$
$228$	−10517.0	+	10517.0i	−0.202313	+	0.202313i
$229$	36285.4	−	36285.4i	0.691928	−	0.691928i	−0.270728	−	0.962656i	$-0.587264\pi$
$229$	36285.4	−	36285.4i	0.691928	−	0.691928i	0.962656	+	0.270728i	$0.0872643\pi$
$230$	−67960.7			−1.28470
$231$		−	9657.15i		−	0.180978i
$232$	32021.9	−	32021.9i	0.594938	−	0.594938i
$233$			2365.03i			0.0435637i	0.999763	+	0.0217819i	$0.00693393\pi$
$233$			2365.03i			0.0435637i	−0.999763	+	0.0217819i	$0.993066\pi$
$234$	−7761.47	−	7470.43i	−0.141746	−	0.136431i
$235$	−60518.0			−1.09584
$236$	23609.0	+	23609.0i	0.423890	+	0.423890i
$237$	35703.0			0.635636
$238$			4029.67i			0.0711402i
$239$	−18841.4	−	18841.4i	−0.329850	−	0.329850i	0.522679	−	0.852529i	$-0.324933\pi$
$239$	−18841.4	−	18841.4i	−0.329850	−	0.329850i	−0.852529	+	0.522679i	$0.824933\pi$
$240$	−2884.01	−	2884.01i	−0.0500696	−	0.0500696i
$241$	−27569.6	+	27569.6i	−0.474676	+	0.474676i	−0.903424	−	0.428748i	$-0.858955\pi$
$241$	−27569.6	+	27569.6i	−0.474676	+	0.474676i	0.428748	+	0.903424i	$0.358955\pi$
$242$	−14620.7	+	14620.7i	−0.249653	+	0.249653i
$243$	−3788.00			−0.0641500
$244$		−	67090.5i		−	1.12689i
$245$	51542.3	−	51542.3i	0.858680	−	0.858680i
$246$			31402.2i			0.518907i
$247$	−33427.5	−	32174.0i	−0.547910	−	0.527365i
$248$	51742.7			0.841289
$249$	13755.3	+	13755.3i	0.221857	+	0.221857i
$250$	34622.8			0.553964
$251$		−	52279.9i		−	0.829826i	−0.909861	−	0.414913i	$-0.863812\pi$
$251$		−	52279.9i		−	0.829826i	0.909861	−	0.414913i	$-0.136188\pi$
$252$	4823.46	+	4823.46i	0.0759552	+	0.0759552i
$253$	−38849.8	−	38849.8i	−0.606943	−	0.606943i
$254$	12617.0	−	12617.0i	0.195564	−	0.195564i
$255$	10400.9	−	10400.9i	0.159952	−	0.159952i
$256$	−61896.1			−0.944459
$257$		−	87118.8i		−	1.31900i	−0.751703	−	0.659501i	$-0.770768\pi$
$257$		−	87118.8i		−	1.31900i	0.751703	−	0.659501i	$-0.229232\pi$
$258$	−24977.5	+	24977.5i	−0.375240	+	0.375240i
$259$			19735.8i			0.294209i
$260$	49105.0	−	51018.1i	0.726406	−	0.754705i
$261$	19598.4			0.287700
$262$	16201.6	+	16201.6i	0.236023	+	0.236023i
$263$	−92681.2			−1.33992			−0.669962	−	0.742395i	$-0.733690\pi$
$263$	−92681.2			−1.33992			−0.669962	+	0.742395i	$0.733690\pi$
$264$			24864.5i			0.356756i
$265$	−72453.8	−	72453.8i	−1.03174	−	1.03174i
$266$	−11105.0	−	11105.0i	−0.156948	−	0.156948i
$267$	35355.8	−	35355.8i	0.495951	−	0.495951i
$268$	−57056.0	+	57056.0i	−0.794387	+	0.794387i
$269$	−123465.			−1.70624			−0.853118	−	0.521717i	$-0.825292\pi$
$269$	−123465.			−1.70624			−0.853118	+	0.521717i	$0.825292\pi$
$270$			13310.4i			0.182584i
$271$	60295.3	−	60295.3i	0.821003	−	0.821003i	−0.165249	−	0.986252i	$-0.552843\pi$
$271$	60295.3	−	60295.3i	0.821003	−	0.821003i	0.986252	+	0.165249i	$0.0528428\pi$
$272$			1375.87i			0.0185969i
$273$	−14756.1	+	15331.0i	−0.197991	+	0.205705i
$274$	−24393.0			−0.324910
$275$	53688.8	+	53688.8i	0.709934	+	0.709934i
$276$	38808.7			0.509461
$277$			62673.2i			0.816812i	0.912800	+	0.408406i	$0.133915\pi$
$277$			62673.2i			0.816812i	−0.912800	+	0.408406i	$0.866085\pi$
$278$	24220.1	+	24220.1i	0.313391	+	0.313391i
$279$	15834.0	+	15834.0i	0.203415	+	0.203415i
$280$	42958.0	−	42958.0i	0.547933	−	0.547933i
$281$	17050.4	−	17050.4i	0.215934	−	0.215934i	−0.590848	−	0.806783i	$-0.701207\pi$
$281$	17050.4	−	17050.4i	0.215934	−	0.215934i	0.806783	+	0.590848i	$0.201207\pi$
$282$	−18473.8			−0.232305
$283$			135558.i			1.69259i	0.532716	+	0.846294i	$0.321171\pi$
$283$			135558.i			1.69259i	−0.532716	+	0.846294i	$0.678829\pi$
$284$	48413.1	−	48413.1i	0.600242	−	0.600242i
$285$			57325.9i			0.705767i
$286$	−30596.2	+	584.600i	−0.374054	+	0.00714705i
$287$	62027.7			0.753047
$288$	−19938.3	−	19938.3i	−0.240382	−	0.240382i
$289$	78559.1			0.940590
$290$		−	68865.6i		−	0.818854i
$291$	−49358.0	−	49358.0i	−0.582870	−	0.582870i
$292$	58445.9	+	58445.9i	0.685470	+	0.685470i
$293$	−33642.1	+	33642.1i	−0.391876	+	0.391876i	−0.875356	−	0.483480i	$-0.839373\pi$
$293$	−33642.1	+	33642.1i	−0.391876	+	0.391876i	0.483480	+	0.875356i	$0.339373\pi$
$294$	15733.9	−	15733.9i	0.182029	−	0.182029i
$295$	128687.			1.47874
$296$		−	50814.3i		−	0.579966i
$297$	−7608.91	+	7608.91i	−0.0862600	+	0.0862600i
$298$			20730.5i			0.233441i
$299$	2312.66	+	121037.i	0.0258684	+	1.35387i
$300$	−53631.9			−0.595910
$301$	49337.2	+	49337.2i	0.544555	+	0.544555i
$302$	21108.2			0.231440
$303$			85951.6i			0.936200i
$304$	−3791.65	−	3791.65i	−0.0410281	−	0.0410281i
$305$	−182848.	−	182848.i	−1.96558	−	1.96558i
$306$	3174.99	−	3174.99i	0.0339078	−	0.0339078i
$307$	−121981.	+	121981.i	−1.29425	+	1.29425i	−0.362112	+	0.932135i	$0.617944\pi$
$307$	−121981.	+	121981.i	−1.29425	+	1.29425i	−0.932135	+	0.362112i	$0.882056\pi$
$308$	19377.7			0.204268
$309$			57928.7i			0.606704i
$310$	55638.2	−	55638.2i	0.578962	−	0.578962i
$311$			1370.53i			0.0141699i	0.999975	+	0.00708495i	$0.00225523\pi$
$311$			1370.53i			0.0141699i	−0.999975	+	0.00708495i	$0.997745\pi$
$312$	37992.8	−	39473.0i	0.390295	−	0.405500i
$313$	52462.0			0.535495			0.267748	−	0.963489i	$-0.413721\pi$
$313$	52462.0			0.535495			0.267748	+	0.963489i	$0.413721\pi$
$314$	44977.3	+	44977.3i	0.456178	+	0.456178i
$315$	26291.6			0.264969
$316$			71640.4i			0.717437i
$317$	−101779.	−	101779.i	−1.01284	−	1.01284i	−0.999916	−	0.0129239i	$-0.995886\pi$
$317$	−101779.	−	101779.i	−1.01284	−	1.01284i	−0.0129239	−	0.999916i	$-0.504114\pi$
$318$	−22117.4	−	22117.4i	−0.218715	−	0.218715i
$319$	39367.1	−	39367.1i	0.386858	−	0.386858i
$320$	−61179.4	+	61179.4i	−0.597455	+	0.597455i
$321$	−48279.0			−0.468541
$322$			40978.5i			0.395225i
$323$	13674.2	−	13674.2i	0.131068	−	0.131068i
$324$		−	7600.85i		−	0.0724056i
$325$	−3196.00	−	167269.i	−0.0302580	−	1.58361i
$326$	76230.8			0.717291
$327$	−10318.2	−	10318.2i	−0.0964962	−	0.0964962i
$328$	−159704.			−1.48446
$329$			36490.7i			0.337125i
$330$	26736.5	+	26736.5i	0.245514	+	0.245514i
$331$	90476.9	+	90476.9i	0.825813	+	0.825813i	0.986935	−	0.161122i	$-0.0515112\pi$
$331$	90476.9	+	90476.9i	0.825813	+	0.825813i	−0.161122	+	0.986935i	$0.551511\pi$
$332$	−27600.9	+	27600.9i	−0.250408	+	0.250408i
$333$	15549.9	−	15549.9i	0.140230	−	0.140230i
$334$	25590.6			0.229397
$335$			311000.i			2.77122i
$336$	−1738.98	+	1738.98i	−0.0154034	+	0.0154034i
$337$			48432.3i			0.426457i	0.977002	+	0.213228i	$0.0683979\pi$
$337$			48432.3i			0.426457i	−0.977002	+	0.213228i	$0.931602\pi$
$338$	49465.4	+	45822.8i	0.432981	+	0.401096i
$339$	9348.38			0.0813461
$340$	20870.0	+	20870.0i	0.180537	+	0.180537i
$341$	63611.3			0.547048
$342$			17499.4i			0.149613i
$343$	−72217.5	−	72217.5i	−0.613838	−	0.613838i
$344$	−127030.	−	127030.i	−1.07347	−	1.07347i
$345$	105769.	−	105769.i	0.888625	−	0.888625i
$346$	−15663.1	+	15663.1i	−0.130835	+	0.130835i
$347$	50065.9			0.415799			0.207899	−	0.978150i	$-0.433337\pi$
$347$	50065.9			0.415799			0.207899	+	0.978150i	$0.433337\pi$
$348$			39325.4i			0.324724i
$349$	−36678.7	+	36678.7i	−0.301137	+	0.301137i	−0.841458	−	0.540322i	$-0.818302\pi$
$349$	−36678.7	+	36678.7i	−0.301137	+	0.301137i	0.540322	+	0.841458i	$0.318302\pi$
$350$		−	56630.5i		−	0.462290i
$351$	23705.7	−	452.945i	0.192415	−	0.00367647i
$352$	−80099.5			−0.646465
$353$	−91471.6	−	91471.6i	−0.734069	−	0.734069i	0.237354	−	0.971423i	$-0.423720\pi$
$353$	−91471.6	−	91471.6i	−0.734069	−	0.734069i	−0.971423	+	0.237354i	$0.923720\pi$
$354$	39283.3			0.313474
$355$		−	263889.i		−	2.09394i
$356$	70943.6	+	70943.6i	0.559775	+	0.559775i
$357$	−6271.45	−	6271.45i	−0.0492076	−	0.0492076i
$358$	537.576	−	537.576i	0.00419444	−	0.00419444i
$359$	19904.6	−	19904.6i	0.154442	−	0.154442i	−0.625657	−	0.780098i	$-0.715169\pi$
$359$	19904.6	−	19904.6i	0.154442	−	0.154442i	0.780098	+	0.625657i	$0.215169\pi$
$360$	−67693.5			−0.522327
$361$		−	54953.8i		−	0.421680i
$362$	47542.6	−	47542.6i	0.362799	−	0.362799i
$363$		−	45509.0i		−	0.345370i
$364$	−30762.5	−	29609.0i	−0.232177	−	0.223471i
$365$	318576.			2.39126
$366$	−55816.4	−	55816.4i	−0.416677	−	0.416677i
$367$	13539.5			0.100524			0.0502620	−	0.998736i	$-0.483994\pi$
$367$	13539.5			0.100524			0.0502620	+	0.998736i	$0.483994\pi$
$368$			13991.5i			0.103316i
$369$	−48871.9	−	48871.9i	−0.358927	−	0.358927i
$370$	−54639.9	−	54639.9i	−0.399123	−	0.399123i
$371$	−43687.7	+	43687.7i	−0.317403	+	0.317403i
$372$	−31772.0	+	31772.0i	−0.229593	+	0.229593i
$373$	−76501.0			−0.549857			−0.274928	−	0.961465i	$-0.588654\pi$
$373$	−76501.0			−0.549857			−0.274928	+	0.961465i	$0.588654\pi$
$374$		−	12755.1i		−	0.0911890i
$375$	−53884.1	+	53884.1i	−0.383176	+	0.383176i
$376$		−	93953.5i		−	0.664565i
$377$	−122649.	+	2343.45i	−0.862942	+	0.0164882i
$378$	8025.81			0.0561701
$379$	9873.34	+	9873.34i	0.0687362	+	0.0687362i	0.740639	−	0.671903i	$-0.234523\pi$
$379$	9873.34	+	9873.34i	0.0687362	+	0.0687362i	−0.671903	+	0.740639i	$0.734523\pi$
$380$	−115028.			−0.796592
$381$			39272.3i			0.270543i
$382$	65188.4	+	65188.4i	0.446729	+	0.446729i
$383$	92064.1	+	92064.1i	0.627614	+	0.627614i	0.947467	−	0.319853i	$-0.103634\pi$
$383$	92064.1	+	92064.1i	0.627614	+	0.627614i	−0.319853	+	0.947467i	$0.603634\pi$
$384$	42718.2	−	42718.2i	0.289701	−	0.289701i
$385$	52811.6	−	52811.6i	0.356294	−	0.356294i
$386$	−116352.			−0.780906
$387$		−	77746.0i		−	0.519106i
$388$	99039.8	−	99039.8i	0.657880	−	0.657880i
$389$		−	35489.2i		−	0.234529i	−0.993101	−	0.117265i	$-0.962587\pi$
$389$		−	35489.2i		−	0.234529i	0.993101	−	0.117265i	$-0.0374125\pi$
$390$	−1591.57	−	83298.0i	−0.0104640	−	0.547653i
$391$	−50459.0			−0.330054
$392$	80018.7	+	80018.7i	0.520738	+	0.520738i
$393$	−50429.7			−0.326514
$394$			112701.i			0.725996i
$395$	195248.	+	195248.i	1.25139	+	1.25139i
$396$	−15267.7	−	15267.7i	−0.0973609	−	0.0973609i
$397$	45764.3	−	45764.3i	0.290366	−	0.290366i	−0.546859	−	0.837225i	$-0.684177\pi$
$397$	45764.3	−	45764.3i	0.290366	−	0.290366i	0.837225	+	0.546859i	$0.184177\pi$
$398$	20184.4	−	20184.4i	0.127424	−	0.127424i
$399$	34566.0			0.217121
$400$		−	19335.6i		−	0.120848i
$401$	53143.4	−	53143.4i	0.330492	−	0.330492i	−0.522281	−	0.852773i	$-0.674919\pi$
$401$	53143.4	−	53143.4i	0.330492	−	0.330492i	0.852773	+	0.522281i	$0.174919\pi$
$402$			94936.2i			0.587462i
$403$	−100985.	−	97197.9i	−0.621792	−	0.598476i
$404$	−172467.			−1.05668
$405$	−20715.2	−	20715.2i	−0.126293	−	0.126293i
$406$	−41524.1			−0.251911
$407$		−	62470.0i		−	0.377123i
$408$	16147.3	+	16147.3i	0.0970015	+	0.0970015i
$409$	75294.0	+	75294.0i	0.450105	+	0.450105i	0.895389	−	0.445284i	$-0.146897\pi$
$409$	75294.0	+	75294.0i	0.450105	+	0.450105i	−0.445284	+	0.895389i	$0.646897\pi$
$410$	−171728.	+	171728.i	−1.02158	+	1.02158i
$411$	37963.3	−	37963.3i	0.224740	−	0.224740i
$412$	−116237.			−0.684781
$413$		−	77594.9i		−	0.454918i
$414$	32287.1	−	32287.1i	0.188377	−	0.188377i
$415$			150447.i			0.873546i
$416$	127160.	+	122392.i	0.734791	+	0.707239i
$417$	−75388.6			−0.433544
$418$	35150.8	+	35150.8i	0.201179	+	0.201179i
$419$	−264545.			−1.50686			−0.753429	−	0.657530i	$-0.771601\pi$
$419$	−264545.			−1.50686			−0.753429	+	0.657530i	$0.771601\pi$
$420$			52755.7i			0.299068i
$421$	30687.4	+	30687.4i	0.173140	+	0.173140i	0.788357	−	0.615218i	$-0.210932\pi$
$421$	30687.4	+	30687.4i	0.173140	+	0.173140i	−0.615218	+	0.788357i	$0.710932\pi$
$422$	85244.1	+	85244.1i	0.478673	+	0.478673i
$423$	28751.2	−	28751.2i	0.160685	−	0.160685i
$424$	112484.	−	112484.i	0.625688	−	0.625688i
$425$	69732.1			0.386060
$426$		−	80555.2i		−	0.443889i
$427$	−110252.	+	110252.i	−0.604688	+	0.604688i
$428$		−	96874.7i		−	0.528838i
$429$	46707.6	−	48527.3i	0.253789	−	0.263676i
$430$	−273187.			−1.47748
$431$	138755.	+	138755.i	0.746956	+	0.746956i	0.973906	−	0.226950i	$-0.0728755\pi$
$431$	138755.	+	138755.i	0.746956	+	0.746956i	−0.226950	+	0.973906i	$0.572876\pi$
$432$	2740.30			0.0146835
$433$			192478.i			1.02661i	0.858206	+	0.513305i	$0.171579\pi$
$433$			192478.i			1.02661i	−0.858206	+	0.513305i	$0.828421\pi$
$434$	−33548.3	−	33548.3i	−0.178111	−	0.178111i
$435$	107177.	+	107177.i	0.566399	+	0.566399i
$436$	20704.2	−	20704.2i	0.108914	−	0.108914i
$437$	139056.	−	139056.i	0.728158	−	0.728158i
$438$	97248.9			0.506917
$439$		−	221370.i		−	1.14865i	−0.818626	−	0.574327i	$-0.805264\pi$
$439$		−	221370.i		−	1.14865i	0.818626	−	0.574327i	$-0.194736\pi$
$440$	−135975.	+	135975.i	−0.702352	+	0.702352i
$441$			48973.9i			0.251818i
$442$	−19489.8	+	20249.1i	−0.0997616	+	0.103648i
$443$	354252.			1.80511			0.902557	−	0.430570i	$-0.141687\pi$
$443$	354252.			1.80511			0.902557	+	0.430570i	$0.141687\pi$
$444$	31201.9	+	31201.9i	0.158276	+	0.158276i
$445$	386698.			1.95277
$446$			26248.5i			0.131958i
$447$	−32263.3	−	32263.3i	−0.161471	−	0.161471i
$448$	36889.5	+	36889.5i	0.183801	+	0.183801i
$449$	−187721.	+	187721.i	−0.931152	+	0.931152i	−0.997778	−	0.0666259i	$-0.978777\pi$
$449$	−187721.	+	187721.i	−0.931152	+	0.931152i	0.0666259	+	0.997778i	$0.478777\pi$
$450$	−44619.4	+	44619.4i	−0.220343	+	0.220343i
$451$	−196337.			−0.965270
$452$			18758.1i			0.0918147i
$453$	−32851.2	+	32851.2i	−0.160087	+	0.160087i
$454$		−	32193.4i		−	0.156191i
$455$	−164536.	+	3143.78i	−0.794763	+	0.0151855i
$456$	−88997.8			−0.428006
$457$	−88792.4	−	88792.4i	−0.425151	−	0.425151i	0.461822	−	0.886973i	$-0.347196\pi$
$457$	−88792.4	−	88792.4i	−0.425151	−	0.425151i	−0.886973	+	0.461822i	$0.847196\pi$
$458$	121148.			0.577543
$459$			9882.61i			0.0469079i
$460$	212231.	+	212231.i	1.00298	+	1.00298i
$461$	−56596.4	−	56596.4i	−0.266310	−	0.266310i	0.561302	−	0.827611i	$-0.310301\pi$
$461$	−56596.4	−	56596.4i	−0.266310	−	0.266310i	−0.827611	+	0.561302i	$0.810301\pi$
$462$	16121.4	−	16121.4i	0.0755297	−	0.0755297i
$463$	171067.	−	171067.i	0.798003	−	0.798003i	−0.184777	−	0.982780i	$-0.559156\pi$
$463$	171067.	−	171067.i	0.798003	−	0.798003i	0.982780	+	0.184777i	$0.0591563\pi$
$464$	−14177.8			−0.0658526
$465$			173182.i			0.800934i
$466$	−3948.11	+	3948.11i	−0.0181810	+	0.0181810i
$467$		−	68140.5i		−	0.312443i	−0.987722	−	0.156222i	$-0.950069\pi$
$467$		−	68140.5i		−	0.312443i	0.987722	−	0.156222i	$-0.0499314\pi$
$468$	908.862	+	47567.0i	0.00414960	+	0.217177i
$469$	187524.			0.852534
$470$	−101027.	−	101027.i	−0.457343	−	0.457343i
$471$	−139998.			−0.631074
$472$			199785.i			0.896767i
$473$	−156168.	−	156168.i	−0.698021	−	0.698021i
$474$	59601.6	+	59601.6i	0.265278	+	0.265278i
$475$	−192169.	+	192169.i	−0.851718	+	0.851718i
$476$	12584.1	−	12584.1i	0.0555401	−	0.0555401i
$477$	68843.4			0.302570
$478$		−	62906.5i		−	0.275321i
$479$	−113713.	+	113713.i	−0.495610	+	0.495610i	−0.910068	−	0.414458i	$-0.863971\pi$
$479$	−113713.	+	113713.i	−0.495610	+	0.495610i	0.414458	+	0.910068i	$0.363971\pi$
$480$		−	218071.i		−	0.946489i
$481$	−95453.9	+	99172.6i	−0.412576	+	0.428649i
$482$	−92047.9			−0.396205
$483$	−63775.6	−	63775.6i	−0.273376	−	0.273376i
$484$	91316.6			0.389816
$485$		−	539844.i		−	2.29501i
$486$	−6323.57	−	6323.57i	−0.0267726	−	0.0267726i
$487$	30988.8	+	30988.8i	0.130661	+	0.130661i	0.769413	−	0.638752i	$-0.220549\pi$
$487$	30988.8	+	30988.8i	0.130661	+	0.130661i	−0.638752	+	0.769413i	$0.720549\pi$
$488$	283869.	−	283869.i	1.19201	−	1.19201i
$489$	−118640.	+	118640.i	−0.496149	+	0.496149i
$490$	172086.			0.716728
$491$		−	229476.i		−	0.951861i	−0.879483	−	0.475931i	$-0.842111\pi$
$491$		−	229476.i		−	0.951861i	0.879483	−	0.475931i	$-0.157889\pi$
$492$	98064.4	−	98064.4i	0.405118	−	0.405118i
$493$		−	51130.8i		−	0.210372i
$494$	−2092.47	−	109513.i	−0.00857442	−	0.448759i
$495$	−83221.0			−0.339643
$496$	−11454.6	−	11454.6i	−0.0465604	−	0.0465604i
$497$	−159118.			−0.644179
$498$			45925.5i			0.185181i
$499$	29259.2	+	29259.2i	0.117506	+	0.117506i	0.763415	−	0.645909i	$-0.223521\pi$
$499$	29259.2	+	29259.2i	0.117506	+	0.117506i	−0.645909	+	0.763415i	$0.723521\pi$
$500$	−108122.	−	108122.i	−0.432487	−	0.432487i
$501$	−39827.2	+	39827.2i	−0.158674	+	0.158674i
$502$	87274.5	−	87274.5i	0.346322	−	0.346322i
$503$	432230.			1.70836			0.854180	−	0.519978i	$-0.174060\pi$
$503$	432230.			1.70836			0.854180	+	0.519978i	$0.174060\pi$
$504$			40817.4i			0.160688i
$505$	−470040.	+	470040.i	−1.84311	+	1.84311i
$506$		−	129710.i		−	0.506607i
$507$	−148299.	+	5669.16i	−0.576929	+	0.0220548i
$508$	−78802.3			−0.305360
$509$	−150630.	−	150630.i	−0.581402	−	0.581402i	0.353886	−	0.935289i	$-0.384860\pi$
$509$	−150630.	−	150630.i	−0.581402	−	0.581402i	−0.935289	+	0.353886i	$0.884860\pi$
$510$	34725.9			0.133510
$511$		−	192092.i		−	0.735645i
$512$	28210.4	+	28210.4i	0.107614	+	0.107614i
$513$	−27234.7	−	27234.7i	−0.103487	−	0.103487i
$514$	145434.	−	145434.i	0.550476	−	0.550476i
$515$	−316792.	+	316792.i	−1.19443	+	1.19443i
$516$	156002.			0.585910
$517$		−	115504.i		−	0.432133i
$518$	−32946.4	+	32946.4i	−0.122786	+	0.122786i
$519$		−	48753.6i		−	0.180997i
$520$	423634.	−	8094.37i	1.56669	−	0.0299348i
$521$	−152305.			−0.561096			−0.280548	−	0.959840i	$-0.590516\pi$
$521$	−152305.			−0.561096			−0.280548	+	0.959840i	$0.590516\pi$
$522$	32717.0	+	32717.0i	0.120069	+	0.120069i
$523$	78780.8			0.288016			0.144008	−	0.989577i	$-0.454001\pi$
$523$	78780.8			0.288016			0.144008	+	0.989577i	$0.454001\pi$
$524$		−	101190.i		−	0.368533i
$525$	88135.1	+	88135.1i	0.319765	+	0.319765i
$526$	−154719.	−	154719.i	−0.559208	−	0.559208i
$527$	41309.9	−	41309.9i	0.148742	−	0.148742i
$528$	5504.40	−	5504.40i	0.0197443	−	0.0197443i
$529$	−233286.			−0.833638
$530$		−	241905.i		−	0.861177i
$531$	−61137.3	+	61137.3i	−0.216829	+	0.216829i
$532$			69358.7i			0.245063i
$533$	311690.	+	300002.i	1.09716	+	1.05601i
$534$	118044.			0.413963
$535$	−264021.	−	264021.i	−0.922425	−	0.922425i
$536$	−482823.			−1.68058
$537$			1673.28i			0.00580257i
$538$	−206109.	−	206109.i	−0.712086	−	0.712086i
$539$	98373.3	+	98373.3i	0.338610	+	0.338610i
$540$	41566.4	−	41566.4i	0.142546	−	0.142546i
$541$	−35408.4	+	35408.4i	−0.120979	+	0.120979i	−0.765004	−	0.644025i	$-0.777263\pi$
$541$	−35408.4	+	35408.4i	−0.120979	+	0.120979i	0.644025	+	0.765004i	$0.277263\pi$
$542$	201310.			0.685279
$543$			147983.i			0.501894i
$544$	−52017.5	+	52017.5i	−0.175773	+	0.175773i
$545$		−	112854.i		−	0.379947i
$546$	−50226.5	+	959.676i	−0.168480	+	0.00321914i
$547$	10939.9			0.0365628			0.0182814	−	0.999833i	$-0.494181\pi$
$547$	10939.9			0.0365628			0.0182814	+	0.999833i	$0.494181\pi$
$548$	76175.6	+	76175.6i	0.253662	+	0.253662i
$549$	173736.			0.576429
$550$			179253.i			0.592572i
$551$	140907.	+	140907.i	0.464120	+	0.464120i
$552$	164205.	+	164205.i	0.538899	+	0.538899i
$553$	117729.	−	117729.i	0.384976	−	0.384976i
$554$	−104625.	+	104625.i	−0.340891	+	0.340891i
$555$	170074.			0.552145
$556$		−	151272.i		−	0.489338i
$557$	32717.3	−	32717.3i	0.105455	−	0.105455i	−0.652411	−	0.757866i	$-0.726242\pi$
$557$	32717.3	−	32717.3i	0.105455	−	0.105455i	0.757866	+	0.652411i	$0.226242\pi$
$558$			52865.8i			0.169788i
$559$	9296.38	+	486543.i	0.0297502	+	1.55703i
$560$	−19019.7			−0.0606497
$561$	19851.1	+	19851.1i	0.0630752	+	0.0630752i
$562$	56926.8			0.180237
$563$		−	180795.i		−	0.570388i	−0.958470	−	0.285194i	$-0.907942\pi$
$563$		−	180795.i		−	0.570388i	0.958470	−	0.285194i	$-0.0920580\pi$
$564$	57691.0	+	57691.0i	0.181364	+	0.181364i
$565$	51123.1	+	51123.1i	0.160148	+	0.160148i
$566$	−226296.	+	226296.i	−0.706390	+	0.706390i
$567$	−12490.7	+	12490.7i	−0.0388527	+	0.0388527i
$568$	409685.			1.26985
$569$			500901.i			1.54713i	0.633715	+	0.773566i	$0.281529\pi$
$569$			500901.i			1.54713i	−0.633715	+	0.773566i	$0.718471\pi$
$570$	−95698.2	+	95698.2i	−0.294547	+	0.294547i
$571$			246396.i			0.755722i	0.925862	+	0.377861i	$0.123340\pi$
$571$			246396.i			0.755722i	−0.925862	+	0.377861i	$0.876660\pi$
$572$	97372.9	+	93721.7i	0.297609	+	0.286449i
$573$	−202908.			−0.618002
$574$	103547.	+	103547.i	0.314279	+	0.314279i
$575$	709119.			2.14478
$576$		−	58130.8i		−	0.175211i
$577$	−184815.	−	184815.i	−0.555119	−	0.555119i	0.372794	−	0.927914i	$-0.378400\pi$
$577$	−184815.	−	184815.i	−0.555119	−	0.555119i	−0.927914	+	0.372794i	$0.878400\pi$
$578$	131144.	+	131144.i	0.392549	+	0.392549i
$579$	181081.	−	181081.i	0.540151	−	0.540151i
$580$	−215057.	+	215057.i	−0.639290	+	0.639290i
$581$	90715.2			0.268737
$582$		−	164794.i		−	0.486513i
$583$	138285.	−	138285.i	0.406854	−	0.406854i
$584$			494585.i			1.45016i
$585$	132115.	+	127161.i	0.386048	+	0.371573i
$586$	−112322.			−0.327093
$587$	−402989.	−	402989.i	−1.16954	−	1.16954i	−0.982317	−	0.187227i	$-0.940050\pi$
$587$	−402989.	−	402989.i	−1.16954	−	1.16954i	−0.187227	−	0.982317i	$-0.559950\pi$
$588$	−98269.1			−0.284225
$589$			227685.i			0.656302i
$590$	214827.	+	214827.i	0.617141	+	0.617141i
$591$	−175398.	−	175398.i	−0.502170	−	0.502170i
$592$	−11249.1	+	11249.1i	−0.0320976	+	0.0320976i
$593$	180607.	−	180607.i	0.513601	−	0.513601i	−0.402027	−	0.915628i	$-0.631694\pi$
$593$	180607.	−	180607.i	0.513601	−	0.513601i	0.915628	+	0.402027i	$0.131694\pi$
$594$	−25404.2			−0.0720000
$595$		−	68592.9i		−	0.193751i
$596$	64738.3	−	64738.3i	0.182251	−	0.182251i
$597$			62826.9i			0.176278i
$598$	−198196.	+	205917.i	−0.554233	+	0.575825i
$599$	−461474.			−1.28615			−0.643077	−	0.765801i	$-0.722343\pi$
$599$	−461474.			−1.28615			−0.643077	+	0.765801i	$0.722343\pi$
$600$	−226924.	−	226924.i	−0.630343	−	0.630343i
$601$	−482801.			−1.33666			−0.668328	−	0.743867i	$-0.732990\pi$
$601$	−482801.			−1.33666			−0.668328	+	0.743867i	$0.732990\pi$
$602$			164724.i			0.454532i
$603$	−147751.	−	147751.i	−0.406346	−	0.406346i
$604$	−65918.0	−	65918.0i	−0.180688	−	0.180688i
$605$	248873.	−	248873.i	0.679935	−	0.679935i
$606$	−143485.	+	143485.i	−0.390716	+	0.390716i
$607$	219694.			0.596266			0.298133	−	0.954524i	$-0.403636\pi$
$607$	219694.			0.596266			0.298133	+	0.954524i	$0.403636\pi$
$608$		−	286701.i		−	0.775573i
$609$	64624.8	−	64624.8i	0.174247	−	0.174247i
$610$		−	610481.i		−	1.64064i
$611$	−176490.	+	183366.i	−0.472758	+	0.491176i
$612$	−19830.1			−0.0529446
$613$	193600.	+	193600.i	0.515209	+	0.515209i	0.916118	−	0.400909i	$-0.131306\pi$
$613$	193600.	+	193600.i	0.515209	+	0.515209i	−0.400909	+	0.916118i	$0.631306\pi$
$614$	−407265.			−1.08029
$615$		−	534527.i		−	1.41325i
$616$	81989.4	+	81989.4i	0.216071	+	0.216071i
$617$	234942.	+	234942.i	0.617149	+	0.617149i	0.944799	−	0.327650i	$-0.106257\pi$
$617$	234942.	+	234942.i	0.617149	+	0.617149i	−0.327650	+	0.944799i	$0.606257\pi$
$618$	−96704.5	+	96704.5i	−0.253203	+	0.253203i
$619$	518107.	−	518107.i	1.35219	−	1.35219i	0.468984	−	0.883206i	$-0.344620\pi$
$619$	518107.	−	518107.i	1.35219	−	1.35219i	0.883206	−	0.468984i	$-0.155380\pi$
$620$	−347500.			−0.904006
$621$			100498.i			0.260600i
$622$	−2287.92	+	2287.92i	−0.00591371	+	0.00591371i
$623$		−	233168.i		−	0.600749i
$624$	−17149.1	+	327.667i	−0.0440425	+	0.000841519i
$625$	29362.4			0.0751678
$626$	87578.5	+	87578.5i	0.223485	+	0.223485i
$627$	−109412.			−0.278311
$628$		−	280915.i		−	0.712288i
$629$	−40568.7	−	40568.7i	−0.102539	−	0.102539i
$630$	43890.4	+	43890.4i	0.110583	+	0.110583i
$631$	−335596.	+	335596.i	−0.842866	+	0.842866i	−0.989231	−	0.146365i	$-0.953243\pi$
$631$	−335596.	+	335596.i	−0.842866	+	0.842866i	0.146365	+	0.989231i	$0.453243\pi$
$632$	−303120.	+	303120.i	−0.758892	+	0.758892i
$633$	−265334.			−0.662195
$634$		−	339815.i		−	0.845403i
$635$	−214767.	+	214767.i	−0.532623	+	0.532623i
$636$			138139.i			0.341508i
$637$	−5855.99	−	306484.i	−0.0144318	−	0.755317i
$638$	131437.			0.322905
$639$	125370.	+	125370.i	0.307037	+	0.307037i
$640$	467222.			1.14068
$641$			555098.i			1.35099i	0.737362	+	0.675497i	$0.236071\pi$
$641$			555098.i			1.35099i	−0.737362	+	0.675497i	$0.763929\pi$
$642$	−80595.5	−	80595.5i	−0.195542	−	0.195542i
$643$	263449.	+	263449.i	0.637197	+	0.637197i	0.949863	−	0.312666i	$-0.101222\pi$
$643$	263449.	+	263449.i	0.637197	+	0.637197i	−0.312666	+	0.949863i	$0.601222\pi$
$644$	127970.	−	127970.i	0.308557	−	0.308557i
$645$	425166.	−	425166.i	1.02197	−	1.02197i
$646$	45654.7			0.109401
$647$		−	396185.i		−	0.946433i	−0.880946	−	0.473216i	$-0.843093\pi$
$647$		−	396185.i		−	0.946433i	0.880946	−	0.473216i	$-0.156907\pi$
$648$	32160.2	−	32160.2i	0.0765894	−	0.0765894i
$649$			245612.i			0.583123i
$650$	273898.	−	284569.i	0.648279	−	0.673535i
$651$	104424.			0.246399
$652$	−238058.	−	238058.i	−0.559999	−	0.559999i
$653$	825533.			1.93601			0.968006	−	0.250929i	$-0.0807360\pi$
$653$	825533.			1.93601			0.968006	+	0.250929i	$0.0807360\pi$
$654$		−	34450.0i		−	0.0805440i
$655$	−275783.	−	275783.i	−0.642813	−	0.642813i
$656$	35354.7	+	35354.7i	0.0821560	+	0.0821560i
$657$	−151350.	+	151350.i	−0.350633	+	0.350633i
$658$	−60916.5	+	60916.5i	−0.140697	+	0.140697i
$659$	123474.			0.284317			0.142159	−	0.989844i	$-0.454596\pi$
$659$	123474.			0.284317			0.142159	+	0.989844i	$0.454596\pi$
$660$		−	166988.i		−	0.383352i
$661$	121255.	−	121255.i	0.277522	−	0.277522i	−0.554597	−	0.832119i	$-0.687128\pi$
$661$	121255.	−	121255.i	0.277522	−	0.277522i	0.832119	+	0.554597i	$0.187128\pi$
$662$			302079.i			0.689294i
$663$	−1181.70	−	61846.5i	−0.00268832	−	0.140698i
$664$	−233566.			−0.529754
$665$	189029.	+	189029.i	0.427451	+	0.427451i
$666$	51917.2			0.117048
$667$		−	519959.i		−	1.16874i
$668$	−79915.8	−	79915.8i	−0.179093	−	0.179093i
$669$	−40851.1	−	40851.1i	−0.0912750	−	0.0912750i
$670$	−519174.	+	519174.i	−1.15655	+	1.15655i
$671$	348982.	−	348982.i	0.775101	−	0.775101i
$672$	−131491.			−0.291177
$673$			140601.i			0.310425i	0.987881	+	0.155213i	$0.0496062\pi$
$673$			140601.i			0.310425i	−0.987881	+	0.155213i	$0.950394\pi$
$674$	−80851.5	+	80851.5i	−0.177979	+	0.177979i
$675$		−	138884.i		−	0.304821i
$676$	−11375.5	−	297571.i	−0.0248930	−	0.651174i
$677$	−543307.			−1.18541			−0.592704	−	0.805420i	$-0.701940\pi$
$677$	−543307.			−1.18541			−0.592704	+	0.805420i	$0.701940\pi$
$678$	15605.9	+	15605.9i	0.0339492	+	0.0339492i
$679$	−325511.			−0.706035
$680$			176608.i			0.381937i
$681$	50103.3	+	50103.3i	0.108037	+	0.108037i
$682$	106191.	+	106191.i	0.228307	+	0.228307i
$683$	−475365.	+	475365.i	−1.01903	+	1.01903i	−0.0192116	+	0.999815i	$0.506116\pi$
$683$	−475365.	+	475365.i	−1.01903	+	1.01903i	−0.999815	+	0.0192116i	$0.993884\pi$
$684$	54648.0	−	54648.0i	0.116805	−	0.116805i
$685$	415216.			0.884898
$686$		−	241115.i		−	0.512362i
$687$	−188545.	+	188545.i	−0.399485	+	0.399485i
$688$			56242.7i			0.118820i
$689$	−430830.	+	8231.87i	−0.907544	+	0.0173404i
$690$	353134.			0.741723
$691$	−212706.	−	212706.i	−0.445476	−	0.445476i	0.448371	−	0.893847i	$-0.352004\pi$
$691$	−212706.	−	212706.i	−0.445476	−	0.445476i	−0.893847	+	0.448371i	$0.852004\pi$
$692$	97827.1			0.204290
$693$			50180.0i			0.104487i
$694$	83578.6	+	83578.6i	0.173531	+	0.173531i
$695$	−412274.	−	412274.i	−0.853526	−	0.853526i
$696$	−166391.	+	166391.i	−0.343488	+	0.343488i
$697$	−127503.	+	127503.i	−0.262456	+	0.262456i
$698$	−122461.			−0.251354
$699$		−	12289.1i		−	0.0251515i
$700$	−176849.	+	176849.i	−0.360916	+	0.360916i
$701$		−	56550.4i		−	0.115080i	−0.998343	−	0.0575400i	$-0.981674\pi$
$701$		−	56550.4i		−	0.115080i	0.998343	−	0.0575400i	$-0.0183257\pi$
$702$	40329.8	+	38817.5i	0.0818373	+	0.0787686i
$703$	223600.			0.452439
$704$	−116767.	−	116767.i	−0.235599	−	0.235599i
$705$	314461.			0.632686
$706$		−	305400.i		−	0.612716i
$707$	283421.	+	283421.i	0.567013	+	0.567013i
$708$	−122676.	−	122676.i	−0.244733	−	0.244733i
$709$	284754.	−	284754.i	0.566470	−	0.566470i	−0.364668	−	0.931138i	$-0.618818\pi$
$709$	284754.	−	284754.i	0.566470	−	0.566470i	0.931138	+	0.364668i	$0.118818\pi$
$710$	440529.	−	440529.i	0.873892	−	0.873892i
$711$	−185518.			−0.366985
$712$			600344.i			1.18424i
$713$	420088.	−	420088.i	0.826345	−	0.826345i
$714$		−	20938.8i		−	0.0410728i
$715$	520807.	−	9951.05i	1.01874	−	0.0194651i
$716$	−3357.54			−0.00654931
$717$	97902.7	+	97902.7i	0.190439	+	0.190439i
$718$	66456.3			0.128910
$719$			513980.i			0.994234i	0.867684	+	0.497117i	$0.165608\pi$
$719$			513980.i			0.994234i	−0.867684	+	0.497117i	$0.834392\pi$
$720$	14985.7	+	14985.7i	0.0289077	+	0.0289077i
$721$	191017.	+	191017.i	0.367453	+	0.367453i
$722$	91738.2	−	91738.2i	0.175985	−	0.175985i
$723$	143256.	−	143256.i	0.274054	−	0.274054i
$724$	−296937.			−0.566483
$725$			718561.i			1.36706i
$726$	75971.4	−	75971.4i	0.144138	−	0.144138i
$727$		−	5325.23i		−	0.0100756i	−0.999987	−	0.00503779i	$-0.998396\pi$
$727$		−	5325.23i		−	0.0100756i	0.999987	−	0.00503779i	$-0.00160358\pi$
$728$	−4880.69	−	255440.i	−0.00920912	−	0.481977i
$729$	19683.0			0.0370370
$730$	531821.	+	531821.i	0.997975	+	0.997975i
$731$	−202834.			−0.379582
$732$			348613.i			0.650610i
$733$	471793.	+	471793.i	0.878100	+	0.878100i	0.993338	−	0.115238i	$-0.0367631\pi$
$733$	471793.	+	471793.i	0.878100	+	0.878100i	−0.115238	+	0.993338i	$0.536763\pi$
$734$	22602.4	+	22602.4i	0.0419530	+	0.0419530i
$735$	−267821.	+	267821.i	−0.495759	+	0.495759i
$736$	−528976.	+	528976.i	−0.976518	+	0.976518i
$737$	−593573.			−1.09280
$738$		−	163171.i		−	0.299591i
$739$	383237.	−	383237.i	0.701745	−	0.701745i	−0.263040	−	0.964785i	$-0.584725\pi$
$739$	383237.	−	383237.i	0.701745	−	0.701745i	0.964785	+	0.263040i	$0.0847252\pi$
$740$			341265.i			0.623201i
$741$	173694.	+	167181.i	0.316336	+	0.304474i
$742$	−145862.			−0.264932
$743$	77467.3	+	77467.3i	0.140327	+	0.140327i	0.773781	−	0.633454i	$-0.218363\pi$
$743$	77467.3	+	77467.3i	0.140327	+	0.140327i	−0.633454	+	0.773781i	$0.718363\pi$
$744$	−268863.			−0.485719
$745$		−	352874.i		−	0.635780i
$746$	−127709.	−	127709.i	−0.229479	−	0.229479i
$747$	−71474.9	−	71474.9i	−0.128089	−	0.128089i
$748$	−39832.4	+	39832.4i	−0.0711924	+	0.0711924i
$749$	−159198.	+	159198.i	−0.283774	+	0.283774i
$750$	−179905.			−0.319831
$751$			287709.i			0.510121i	0.966925	+	0.255061i	$0.0820954\pi$
$751$			287709.i			0.510121i	−0.966925	+	0.255061i	$0.917905\pi$
$752$	−20799.1	+	20799.1i	−0.0367797	+	0.0367797i
$753$			271654.i			0.479101i
$754$	−208659.	−	200835.i	−0.367024	−	0.353261i
$755$	−359304.			−0.630330
$756$	−25063.4	−	25063.4i	−0.0438527	−	0.0438527i
$757$	−962044.			−1.67882			−0.839408	−	0.543502i	$-0.817098\pi$
$757$	−962044.			−1.67882			−0.839408	+	0.543502i	$0.817098\pi$
$758$			32964.5i			0.0573731i
$759$	201870.	+	201870.i	0.350419	+	0.350419i
$760$	−486698.	−	486698.i	−0.842622	−	0.842622i
$761$	496273.	−	496273.i	0.856943	−	0.856943i	−0.134034	−	0.990977i	$-0.542793\pi$
$761$	496273.	−	496273.i	0.856943	−	0.856943i	0.990977	+	0.134034i	$0.0427932\pi$
$762$	−65560.0	+	65560.0i	−0.112909	+	0.112909i
$763$	−68047.8			−0.116887
$764$		−	407148.i		−	0.697534i
$765$	−54044.6	+	54044.6i	−0.0923484	+	0.0923484i
$766$			307378.i			0.523860i
$767$	375294.	−	389915.i	0.637942	−	0.662795i
$768$	321621.			0.545284
$769$	326544.	+	326544.i	0.552191	+	0.552191i	0.927073	−	0.374882i	$-0.122317\pi$
$769$	326544.	+	326544.i	0.552191	+	0.552191i	−0.374882	+	0.927073i	$0.622317\pi$
$770$	176324.			0.297393
$771$			452683.i			0.761526i
$772$	363350.	+	363350.i	0.609664	+	0.609664i
$773$	213799.	+	213799.i	0.357804	+	0.357804i	0.863003	−	0.505199i	$-0.168581\pi$
$773$	213799.	+	213799.i	0.357804	+	0.357804i	−0.505199	+	0.863003i	$0.668581\pi$
$774$	129787.	−	129787.i	0.216645	−	0.216645i
$775$	−580544.	+	580544.i	−0.966566	+	0.966566i
$776$	838101.			1.39179
$777$		−	102550.i		−	0.169861i
$778$	59244.6	−	59244.6i	0.0978790	−	0.0978790i
$779$		−	702751.i		−	1.15805i
$780$	−255157.	+	265098.i	−0.419391	+	0.435729i
$781$	503658.			0.825721
$782$	−84234.8	−	84234.8i	−0.137746	−	0.137746i
$783$	−101836.			−0.166104
$784$		−	35428.5i		−	0.0576395i
$785$	−765602.	−	765602.i	−1.24241	−	1.24241i
$786$	−84185.9	−	84185.9i	−0.136268	−	0.136268i
$787$	642953.	−	642953.i	1.03808	−	1.03808i	0.0388320	−	0.999246i	$-0.487636\pi$
$787$	642953.	−	642953.i	1.03808	−	1.03808i	0.999246	−	0.0388320i	$-0.0123637\pi$
$788$	351948.	−	351948.i	0.566795	−	0.566795i
$789$	481586.			0.773606
$790$			651882.i			1.04452i
$791$	30825.8	−	30825.8i	0.0492677	−	0.0492677i
$792$		−	129200.i		−	0.205973i
$793$	−1.08726e6	+	20774.3i	−1.72897	+	0.0330354i
$794$	152795.			0.242364
$795$	376481.	+	376481.i	0.595674	+	0.595674i
$796$	−126066.			−0.198963
$797$			735073.i			1.15721i	0.815606	+	0.578607i	$0.196404\pi$
$797$			735073.i			1.15721i	−0.815606	+	0.578607i	$0.803596\pi$
$798$	57703.4	+	57703.4i	0.0906141	+	0.0906141i
$799$	−75009.8	−	75009.8i	−0.117496	−	0.117496i
$800$	731022.	−	731022.i	1.14222	−	1.14222i
$801$	−183714.	+	183714.i	−0.286337	+	0.286337i
$802$	177432.			0.275857
$803$			608032.i			0.942965i
$804$	296472.	−	296472.i	0.458639	−	0.458639i
$805$		−	697534.i		−	1.07640i
$806$	−6321.36	−	330840.i	−0.00973062	−	0.509270i
$807$	641543.			0.985096
$808$	−729731.	−	729731.i	−1.11774	−	1.11774i
$809$	−388192.			−0.593130			−0.296565	−	0.955013i	$-0.595841\pi$
$809$	−388192.			−0.593130			−0.296565	+	0.955013i	$0.595841\pi$
$810$		−	69162.9i		−	0.105415i
$811$	−163285.	−	163285.i	−0.248259	−	0.248259i	0.571997	−	0.820256i	$-0.306169\pi$
$811$	−163285.	−	163285.i	−0.248259	−	0.248259i	−0.820256	+	0.571997i	$0.806169\pi$
$812$	129674.	+	129674.i	0.196671	+	0.196671i
$813$	−313303.	+	313303.i	−0.474006	+	0.474006i
$814$	104286.	−	104286.i	0.157389	−	0.157389i
$815$	−1.29760e6			−1.95355
$816$		−	7149.24i		−	0.0107369i
$817$	558972.	−	558972.i	0.837426	−	0.837426i
$818$			251387.i			0.375696i
$819$	76674.9	−	79662.0i	0.114310	−	0.118764i
$820$	1.07256e6			1.59512
$821$	−401933.	−	401933.i	−0.596303	−	0.596303i	0.343024	−	0.939327i	$-0.388549\pi$
$821$	−401933.	−	401933.i	−0.596303	−	0.596303i	−0.939327	+	0.343024i	$0.888549\pi$
$822$	126750.			0.187587
$823$		−	129326.i		−	0.190935i	−0.995433	−	0.0954674i	$-0.969565\pi$
$823$		−	129326.i		−	0.190935i	0.995433	−	0.0954674i	$-0.0304346\pi$
$824$	−491816.	−	491816.i	−0.724350	−	0.724350i
$825$	−278975.	−	278975.i	−0.409881	−	0.409881i
$826$	129535.	−	129535.i	0.189857	−	0.189857i
$827$	51675.4	−	51675.4i	0.0755567	−	0.0755567i	−0.668319	−	0.743875i	$-0.732986\pi$
$827$	51675.4	−	51675.4i	0.0755567	−	0.0755567i	0.743875	+	0.668319i	$0.232986\pi$
$828$	−201656.			−0.294137
$829$			984088.i			1.43194i	0.698131	+	0.715970i	$0.254015\pi$
$829$			984088.i			1.43194i	−0.698131	+	0.715970i	$0.745985\pi$
$830$	−251151.	+	251151.i	−0.364568	+	0.364568i
$831$		−	325659.i		−	0.471587i
$832$	6950.92	+	363789.i	0.0100414	+	0.525537i
$833$	127769.			0.184135
$834$	−125852.	−	125852.i	−0.180937	−	0.180937i
$835$	−435603.			−0.624767
$836$		−	219542.i		−	0.314127i
$837$	−82276.1	−	82276.1i	−0.117442	−	0.117442i
$838$	−441624.	−	441624.i	−0.628876	−	0.628876i
$839$	378169.	−	378169.i	0.537232	−	0.537232i	−0.385483	−	0.922715i	$-0.625965\pi$
$839$	378169.	−	378169.i	0.537232	−	0.537232i	0.922715	+	0.385483i	$0.125965\pi$
$840$	−223216.	+	223216.i	−0.316349	+	0.316349i
$841$	−180399.			−0.255060
$842$			102457.i			0.144517i
$843$	−88596.3	+	88596.3i	−0.124670	+	0.124670i
$844$		−	532409.i		−	0.747413i
$845$	−841999.	−	779994.i	−1.17923	−	1.09239i
$846$	95992.8			0.134121
$847$	−150064.	−	150064.i	−0.209175	−	0.209175i
$848$	−49802.4			−0.0692562
$849$		−	704379.i		−	0.977216i
$850$	116409.	+	116409.i	0.161119	+	0.161119i
$851$	−412551.	−	412551.i	−0.569663	−	0.569663i
$852$	−251562.	+	251562.i	−0.346550	+	0.346550i
$853$	−296254.	+	296254.i	−0.407161	+	0.407161i	−0.880747	−	0.473586i	$-0.842959\pi$
$853$	−296254.	+	296254.i	−0.407161	+	0.407161i	0.473586	+	0.880747i	$0.342959\pi$
$854$	−368104.			−0.504724
$855$		−	297874.i		−	0.407474i
$856$	409890.	−	409890.i	0.559396	−	0.559396i
$857$			1.04442e6i			1.42205i	0.703168	+	0.711023i	$0.251768\pi$
$857$			1.04442e6i			1.42205i	−0.703168	+	0.711023i	$0.748232\pi$
$858$	158982.	−	3037.67i	0.215960	−	0.00412635i
$859$	503818.			0.682790			0.341395	−	0.939920i	$-0.389101\pi$
$859$	503818.			0.682790			0.341395	+	0.939920i	$0.389101\pi$
$860$	853122.	+	853122.i	1.15349	+	1.15349i
$861$	−322305.			−0.434772
$862$			463268.i			0.623473i
$863$	−283034.	−	283034.i	−0.380029	−	0.380029i	0.491083	−	0.871113i	$-0.336601\pi$
$863$	−283034.	−	283034.i	−0.380029	−	0.380029i	−0.871113	+	0.491083i	$0.836601\pi$
$864$	103602.	+	103602.i	0.138785	+	0.138785i
$865$	266617.	−	266617.i	0.356332	−	0.356332i
$866$	−321318.	+	321318.i	−0.428449	+	0.428449i
$867$	−408205.			−0.543050
$868$			209533.i			0.278108i
$869$	−372649.	+	372649.i	−0.493470	+	0.493470i
$870$			357836.i			0.472765i
$871$	942311.	+	906977.i	1.24210	+	1.19553i
$872$	175204.			0.230416
$873$	256472.	+	256472.i	0.336520	+	0.336520i
$874$	464271.			0.607783
$875$			355360.i			0.464144i
$876$	−303694.	−	303694.i	−0.395756	−	0.395756i
$877$	86669.2	+	86669.2i	0.112685	+	0.112685i	0.761201	−	0.648516i	$-0.224610\pi$
$877$	86669.2	+	86669.2i	0.112685	+	0.112685i	−0.648516	+	0.761201i	$0.724610\pi$
$878$	369548.	−	369548.i	0.479383	−	0.479383i
$879$	174810.	−	174810.i	0.226250	−	0.226250i
$880$	60203.4			0.0777420
$881$		−	346696.i		−	0.446681i	−0.974741	−	0.223340i	$-0.928304\pi$
$881$		−	346696.i		−	0.446681i	0.974741	−	0.223340i	$-0.0716961\pi$
$882$	−81755.6	+	81755.6i	−0.105095	+	0.105095i
$883$			570534.i			0.731746i	0.930665	+	0.365873i	$0.119230\pi$
$883$			570534.i			0.731746i	−0.930665	+	0.365873i	$0.880770\pi$
$884$	124099.	−	2371.16i	0.158805	−	0.00303428i
$885$	−668678.			−0.853750
$886$	591378.	+	591378.i	0.753352	+	0.753352i
$887$	988202.			1.25603			0.628013	−	0.778203i	$-0.283869\pi$
$887$	988202.			1.25603			0.628013	+	0.778203i	$0.283869\pi$
$888$			264039.i			0.334843i
$889$	129499.	+	129499.i	0.163856	+	0.163856i
$890$	645542.	+	645542.i	0.814976	+	0.814976i
$891$	39537.0	−	39537.0i	0.0498022	−	0.0498022i
$892$	81970.3	−	81970.3i	0.103021	−	0.103021i
$893$	413427.			0.518436
$894$		−	107719.i		−	0.134777i
$895$	−9150.60	+	9150.60i	−0.0114236	+	0.0114236i
$896$		−	281722.i		−	0.350918i
$897$	−12016.9	−	628929.i	−0.0149351	−	0.781658i
$898$	−626753.			−0.777219
$899$	425681.	+	425681.i	0.526702	+	0.526702i
$900$	278680.			0.344049
$901$		−	179608.i		−	0.221246i
$902$	−327759.	−	327759.i	−0.402848	−	0.402848i
$903$	−256364.	−	256364.i	−0.314399	−	0.314399i
$904$	−79368.0	+	79368.0i	−0.0971200	+	0.0971200i
$905$	−809268.	+	809268.i	−0.988088	+	0.988088i
$906$	−109682.			−0.133622
$907$		−	720192.i		−	0.875455i	−0.899108	−	0.437728i	$-0.855783\pi$
$907$		−	720192.i		−	0.875455i	0.899108	−	0.437728i	$-0.144217\pi$
$908$	−100535.	+	100535.i	−0.121940	+	0.121940i
$909$		−	446617.i		−	0.540515i
$910$	−279919.	−	269423.i	−0.338026	−	0.325351i
$911$	963155.			1.16054			0.580269	−	0.814425i	$-0.302947\pi$
$911$	963155.			1.16054			0.580269	+	0.814425i	$0.302947\pi$
$912$	19702.0	+	19702.0i	0.0236876	+	0.0236876i
$913$	−287142.			−0.344473
$914$		−	296455.i		−	0.354867i
$915$	950104.	+	950104.i	1.13483	+	1.13483i
$916$	−378326.	−	378326.i	−0.450895	−	0.450895i
$917$	−166290.	+	166290.i	−0.197755	+	0.197755i
$918$	−16497.7	+	16497.7i	−0.0195767	+	0.0195767i
$919$	−1.29069e6			−1.52823			−0.764117	−	0.645077i	$-0.776825\pi$
$919$	−1.29069e6			−1.52823			−0.764117	+	0.645077i	$0.776825\pi$
$920$			1.79596e6i			2.12188i
$921$	633834.	−	633834.i	0.747233	−	0.747233i
$922$		−	188961.i		−	0.222285i
$923$	−799569.	−	769587.i	−0.938540	−	0.903347i
$924$	−100689.			−0.117934
$925$	570127.	+	570127.i	0.666328	+	0.666328i
$926$	571149.			0.666082
$927$		−	301006.i		−	0.350281i
$928$	−536019.	−	536019.i	−0.622421	−	0.622421i
$929$	620998.	+	620998.i	0.719546	+	0.719546i	0.968512	−	0.248966i	$-0.0800907\pi$
$929$	620998.	+	620998.i	0.719546	+	0.719546i	−0.248966	+	0.968512i	$0.580091\pi$
$930$	−289105.	+	289105.i	−0.334264	+	0.334264i
$931$	−352109.	+	352109.i	−0.406235	+	0.406235i
$932$	24658.8			0.0283883
$933$		−	7121.47i		−	0.00818100i
$934$	113752.	−	113752.i	0.130396	−	0.130396i
$935$			217118.i			0.248354i
$936$	−197417.	+	205108.i	−0.225337	+	0.234116i
$937$	1.00082e6			1.13992			0.569960	−	0.821672i	$-0.306959\pi$
$937$	1.00082e6			1.13992			0.569960	+	0.821672i	$0.306959\pi$
$938$	313048.	+	313048.i	0.355799	+	0.355799i
$939$	−272600.			−0.309168
$940$			630985.i			0.714107i
$941$	902091.	+	902091.i	1.01876	+	1.01876i	0.999821	+	0.0189381i	$0.00602854\pi$
$941$	902091.	+	902091.i	1.01876	+	1.01876i	0.0189381	+	0.999821i	$0.493971\pi$
$942$	−233709.	−	233709.i	−0.263374	−	0.263374i
$943$	−1.29661e6	+	1.29661e6i	−1.45809	+	1.45809i
$944$	44227.7	−	44227.7i	0.0496307	−	0.0496307i
$945$	−136615.			−0.152980
$946$		−	521403.i		−	0.582628i
$947$	−64662.8	+	64662.8i	−0.0721032	+	0.0721032i	−0.742239	−	0.670136i	$-0.766236\pi$
$947$	−64662.8	+	64662.8i	−0.0721032	+	0.0721032i	0.670136	+	0.742239i	$0.266236\pi$
$948$		−	372254.i		−	0.414212i
$949$	929071.	−	965266.i	1.03161	−	1.07180i
$950$	−641603.			−0.710917
$951$	528861.	+	528861.i	0.584764	+	0.584764i
$952$	106490.			0.117499
$953$			1.57625e6i			1.73556i	0.496948	+	0.867781i	$0.334454\pi$
$953$			1.57625e6i			1.73556i	−0.496948	+	0.867781i	$0.665546\pi$
$954$	114925.	+	114925.i	0.126275	+	0.126275i
$955$	−1.10963e6	−	1.10963e6i	−1.21667	−	1.21667i
$956$	−196448.	+	196448.i	−0.214947	+	0.214947i
$957$	−204557.	+	204557.i	−0.223353	+	0.223353i
$958$	−379659.			−0.413679
$959$		−	250364.i		−	0.272229i
$960$	317897.	−	317897.i	0.344941	−	0.344941i
$961$		−	235683.i		−	0.255201i
$962$	−324904.	+	6207.93i	−0.351079	+	0.00670806i
$963$	250865.			0.270512
$964$	287452.	+	287452.i	0.309322	+	0.309322i
$965$	1.98054e6			2.12681
$966$		−	212930.i		−	0.228183i
$967$	−561644.	−	561644.i	−0.600632	−	0.600632i	0.339849	−	0.940480i	$-0.389624\pi$
$967$	−561644.	−	561644.i	−0.600632	−	0.600632i	−0.940480	+	0.339849i	$0.889624\pi$
$968$	386373.	+	386373.i	0.412340	+	0.412340i
$969$	−71053.3	+	71053.3i	−0.0756722	+	0.0756722i
$970$	901199.	−	901199.i	0.957806	−	0.957806i
$971$	−70845.6			−0.0751405			−0.0375703	−	0.999294i	$-0.511962\pi$
$971$	−70845.6			−0.0751405			−0.0375703	+	0.999294i	$0.511962\pi$
$972$			39495.2i			0.0418034i
$973$	−248590.	+	248590.i	−0.262578	+	0.262578i
$974$			103464.i			0.109061i
$975$	16606.9	+	869153.i	0.0174694	+	0.914296i
$976$	−125684.			−0.131941
$977$	−617175.	−	617175.i	−0.646576	−	0.646576i	0.305588	−	0.952164i	$-0.401147\pi$
$977$	−617175.	−	617175.i	−0.646576	−	0.646576i	−0.952164	+	0.305588i	$0.901147\pi$
$978$	−396107.			−0.414128
$979$			738050.i			0.770052i
$980$	−537400.	−	537400.i	−0.559559	−	0.559559i
$981$	53615.2	+	53615.2i	0.0557121	+	0.0557121i
$982$	383080.	−	383080.i	0.397252	−	0.397252i
$983$	665135.	−	665135.i	0.688339	−	0.688339i	−0.273526	−	0.961865i	$-0.588190\pi$
$983$	665135.	−	665135.i	0.688339	−	0.688339i	0.961865	+	0.273526i	$0.0881898\pi$
$984$	829847.			0.857053
$985$		−	1.91839e6i		−	1.97726i
$986$	85356.3	−	85356.3i	0.0877974	−	0.0877974i
$987$		−	189611.i		−	0.194639i
$988$	−335459.	+	348528.i	−0.343658	+	0.357046i
$989$	−2.06265e6			−2.10879
$990$	−138927.	−	138927.i	−0.141748	−	0.141748i
$991$	250003.			0.254565			0.127282	−	0.991867i	$-0.459375\pi$
$991$	250003.			0.254565			0.127282	+	0.991867i	$0.459375\pi$
$992$		−	866127.i		−	0.880153i
$993$	−470132.	−	470132.i	−0.476783	−	0.476783i
$994$	−265627.	−	265627.i	−0.268843	−	0.268843i
$995$	−343579.	+	343579.i	−0.347041	+	0.347041i
$996$	143419.	−	143419.i	0.144573	−	0.144573i
$997$	−1.47712e6			−1.48602			−0.743011	−	0.669279i	$-0.766603\pi$
$997$	−1.47712e6			−1.48602			−0.743011	+	0.669279i	$0.766603\pi$
$998$			97688.8i			0.0980808i
$999$	−80799.8	+	80799.8i	−0.0809617	+	0.0809617i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.5.g.a.34.7	yes	20
3.2	odd	2		117.5.j.b.73.4		20
13.5	odd	4	inner	39.5.g.a.31.7	✓	20
39.5	even	4		117.5.j.b.109.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.7	✓	20	13.5	odd	4	inner
39.5.g.a.34.7	yes	20	1.1	even	1	trivial
117.5.j.b.73.4		20	3.2	odd	2
117.5.j.b.109.4		20	39.5	even	4

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+(1.66937 + 1.66937i) q^{2} -5.19615 q^{3} -10.4264i q^{4} +(-28.4160 - 28.4160i) q^{5} +(-8.67431 - 8.67431i) q^{6} +(-17.1341 + 17.1341i) q^{7} +(44.1155 - 44.1155i) q^{8} +27.0000 q^{9} -94.8736i q^{10} +(54.2346 - 54.2346i) q^{11} +54.1772i q^{12} +(-168.969 + 3.22849i) q^{13} -57.2062 q^{14} +(147.654 + 147.654i) q^{15} -19.5322 q^{16} -70.4411i q^{17} +(45.0730 + 45.0730i) q^{18} +(194.123 + 194.123i) q^{19} +(-296.276 + 296.276i) q^{20} +(89.0312 - 89.0312i) q^{21} +181.075 q^{22} -716.329i q^{23} +(-229.231 + 229.231i) q^{24} +989.935i q^{25} +(-287.462 - 276.683i) q^{26} -140.296 q^{27} +(178.647 + 178.647i) q^{28} +725.866 q^{29} +492.978i q^{30} +(586.446 + 586.446i) q^{31} +(-738.454 - 738.454i) q^{32} +(-281.811 + 281.811i) q^{33} +(117.592 - 117.592i) q^{34} +973.762 q^{35} -281.513i q^{36} +(575.923 - 575.923i) q^{37} +648.126i q^{38} +(877.990 - 16.7757i) q^{39} -2507.17 q^{40} +(-1810.07 - 1810.07i) q^{41} +297.252 q^{42} -2879.48i q^{43} +(-565.472 - 565.472i) q^{44} +(-767.231 - 767.231i) q^{45} +(1195.82 - 1195.82i) q^{46} +(1064.86 - 1064.86i) q^{47} +101.492 q^{48} +1813.85i q^{49} +(-1652.57 + 1652.57i) q^{50} +366.023i q^{51} +(33.6615 + 1761.74i) q^{52} +2549.76 q^{53} +(-234.206 - 234.206i) q^{54} -3082.26 q^{55} +1511.75i q^{56} +(-1008.69 - 1008.69i) q^{57} +(1211.74 + 1211.74i) q^{58} +(-2264.35 + 2264.35i) q^{59} +(1539.50 - 1539.50i) q^{60} +6434.68 q^{61} +1957.99i q^{62} +(-462.620 + 462.620i) q^{63} -2152.99i q^{64} +(4893.16 + 4709.68i) q^{65} -940.895 q^{66} +(-5472.27 - 5472.27i) q^{67} -734.447 q^{68} +3722.15i q^{69} +(1625.57 + 1625.57i) q^{70} +(4643.32 + 4643.32i) q^{71} +(1191.12 - 1191.12i) q^{72} +(-5605.57 + 5605.57i) q^{73} +1922.86 q^{74} -5143.86i q^{75} +(2024.00 - 2024.00i) q^{76} +1858.52i q^{77} +(1493.70 + 1437.69i) q^{78} -6871.05 q^{79} +(555.027 + 555.027i) q^{80} +729.000 q^{81} -6043.36i q^{82} +(-2647.22 - 2647.22i) q^{83} +(-928.275 - 928.275i) q^{84} +(-2001.65 + 2001.65i) q^{85} +(4806.92 - 4806.92i) q^{86} -3771.71 q^{87} -4785.17i q^{88} +(-6804.23 + 6804.23i) q^{89} -2561.59i q^{90} +(2839.81 - 2950.45i) q^{91} -7468.73 q^{92} +(-3047.26 - 3047.26i) q^{93} +3555.29 q^{94} -11032.4i q^{95} +(3837.12 + 3837.12i) q^{96} +(9498.95 + 9498.95i) q^{97} +(-3027.99 + 3027.99i) q^{98} +(1464.33 - 1464.33i) 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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