LMFDB - Embedded newform 625.2.b.c.624.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(624,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("625.624");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.99065012633$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.58140625.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 $ x^8 - 3x^7 + 4x^6 - 7x^5 + 11x^4 + 5x^3 - 10x^2 - 25*x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			624.2
Root			$-0.983224 + 0.644389i$ of defining polynomial
Character	$\chi$	$=$	625.624
Dual form			625.2.b.c.624.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08529i q^{2} +2.19849i q^{3} -2.34841 q^{4} +4.58448 q^{6} +0.992398i q^{7} +0.726543i q^{8} -1.83337 q^{9} +O(q^{10})$	\(q-2.08529i q^{2} +2.19849i q^{3} -2.34841 q^{4} +4.58448 q^{6} +0.992398i q^{7} +0.726543i q^{8} -1.83337 q^{9} +2.00000 q^{11} -5.16297i q^{12} +3.37406i q^{13} +2.06943 q^{14} -3.18178 q^{16} +2.89451i q^{17} +3.82309i q^{18} +2.58448 q^{19} -2.18178 q^{21} -4.17057i q^{22} +4.54963i q^{23} -1.59730 q^{24} +7.03588 q^{26} +2.56484i q^{27} -2.33056i q^{28} +5.38430 q^{29} +0.136538 q^{31} +8.08800i q^{32} +4.39698i q^{33} +6.03588 q^{34} +4.30550 q^{36} -2.14910i q^{37} -5.38938i q^{38} -7.41785 q^{39} +8.63318 q^{41} +4.54963i q^{42} -4.64398i q^{43} -4.69683 q^{44} +9.48728 q^{46} +9.92630i q^{47} -6.99512i q^{48} +6.01515 q^{49} -6.36356 q^{51} -7.92369i q^{52} -7.56521i q^{53} +5.34841 q^{54} -0.721020 q^{56} +5.68196i q^{57} -11.2278i q^{58} -4.91775 q^{59} -2.76972 q^{61} -0.284720i q^{62} -1.81943i q^{63} +10.5022 q^{64} +9.16896 q^{66} +2.18577i q^{67} -6.79751i q^{68} -10.0023 q^{69} +9.64254 q^{71} -1.33202i q^{72} -0.775929i q^{73} -4.48150 q^{74} -6.06943 q^{76} +1.98480i q^{77} +15.4683i q^{78} -15.8508 q^{79} -11.1389 q^{81} -18.0026i q^{82} +1.77110i q^{83} +5.12372 q^{84} -9.68401 q^{86} +11.8373i q^{87} +1.45309i q^{88} -14.5080 q^{89} -3.34841 q^{91} -10.6844i q^{92} +0.300177i q^{93} +20.6992 q^{94} -17.7814 q^{96} -17.0291i q^{97} -12.5433i q^{98} -3.66673 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10}) $ 8 * q - 6 * q^4 + 6 * q^6 - 4 * q^9	$ 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9} + 16 q^{11} - 12 q^{14} - 2 q^{16} - 10 q^{19} + 6 q^{21} - 20 q^{24} + 6 q^{26} - 20 q^{29} + 16 q^{31} - 2 q^{34} - 12 q^{36} - 18 q^{39} + 26 q^{41} - 12 q^{44} + 6 q^{46} + 14 q^{49} - 4 q^{51} + 30 q^{54} + 10 q^{56} - 30 q^{59} + 6 q^{61} + 44 q^{64} + 12 q^{66} - 8 q^{69} + 46 q^{71} - 12 q^{74} - 20 q^{76} - 10 q^{79} - 32 q^{81} + 18 q^{84} - 14 q^{86} - 30 q^{89} - 14 q^{91} + 68 q^{94} - 54 q^{96} - 8 q^{99}+O(q^{100}) $ 8 * q - 6 * q^4 + 6 * q^6 - 4 * q^9 + 16 * q^11 - 12 * q^14 - 2 * q^16 - 10 * q^19 + 6 * q^21 - 20 * q^24 + 6 * q^26 - 20 * q^29 + 16 * q^31 - 2 * q^34 - 12 * q^36 - 18 * q^39 + 26 * q^41 - 12 * q^44 + 6 * q^46 + 14 * q^49 - 4 * q^51 + 30 * q^54 + 10 * q^56 - 30 * q^59 + 6 * q^61 + 44 * q^64 + 12 * q^66 - 8 * q^69 + 46 * q^71 - 12 * q^74 - 20 * q^76 - 10 * q^79 - 32 * q^81 + 18 * q^84 - 14 * q^86 - 30 * q^89 - 14 * q^91 + 68 * q^94 - 54 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 2.08529i	− 1.47452i	−0.675610	−	0.737260i	$-0.736119\pi$
$2$	− 2.08529i	− 1.47452i	0.675610	−	0.737260i	$-0.263881\pi$
$3$	2.19849i	1.26930i	0.772800	+	0.634650i	$0.218856\pi$
$3$	2.19849i	1.26930i	−0.772800	+	0.634650i	$0.781144\pi$
$4$	−2.34841	−1.17421
$5$	0	0
$5$	0	0
$6$	4.58448	1.87161
$7$	0.992398i	0.375091i	0.982256	+	0.187546i	$0.0600533\pi$
$7$	0.992398i	0.375091i	−0.982256	+	0.187546i	$0.939947\pi$
$8$	0.726543i	0.256872i
$9$	−1.83337	−0.611122
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	− 5.16297i	− 1.49042i
$13$	3.37406i	0.935796i	0.883782	+	0.467898i	$0.154989\pi$
$13$	3.37406i	0.935796i	−0.883782	+	0.467898i	$0.845011\pi$
$14$	2.06943	0.553079
$15$	0	0
$16$	−3.18178	−0.795445
$17$	2.89451i	0.702022i	0.936371	+	0.351011i	$0.114162\pi$
$17$	2.89451i	0.702022i	−0.936371	+	0.351011i	$0.885838\pi$
$18$	3.82309i	0.901111i
$19$	2.58448	0.592921	0.296460	−	0.955045i	$-0.404194\pi$
$19$	2.58448	0.592921	0.296460	+	0.955045i	$0.404194\pi$
$20$	0	0
$21$	−2.18178	−0.476103
$22$	− 4.17057i	− 0.889169i
$23$	4.54963i	0.948664i	0.880346	+	0.474332i	$0.157310\pi$
$23$	4.54963i	0.948664i	−0.880346	+	0.474332i	$0.842690\pi$
$24$	−1.59730	−0.326047
$25$	0	0
$26$	7.03588	1.37985
$27$	2.56484i	0.493603i
$28$	− 2.33056i	− 0.440435i
$29$	5.38430	0.999839	0.499919	−	0.866072i	$-0.333363\pi$
$29$	5.38430	0.999839	0.499919	+	0.866072i	$0.333363\pi$
$30$	0	0
$31$	0.136538	0.0245229	0.0122614	−	0.999925i	$-0.496097\pi$
$31$	0.136538	0.0245229	0.0122614	+	0.999925i	$0.496097\pi$
$32$	8.08800i	1.42977i
$33$	4.39698i	0.765417i
$34$	6.03588	1.03515
$35$	0	0
$36$	4.30550	0.717584
$37$	− 2.14910i	− 0.353311i	−0.984273	−	0.176655i	$-0.943472\pi$
$37$	− 2.14910i	− 0.353311i	0.984273	−	0.176655i	$-0.0565278\pi$
$38$	− 5.38938i	− 0.874273i
$39$	−7.41785	−1.18781
$40$	0	0
$41$	8.63318	1.34828	0.674138	−	0.738605i	$-0.264515\pi$
$41$	8.63318	1.34828	0.674138	+	0.738605i	$0.264515\pi$
$42$	4.54963i	0.702024i
$43$	− 4.64398i	− 0.708200i	−0.935208	−	0.354100i	$-0.884787\pi$
$43$	− 4.64398i	− 0.708200i	0.935208	−	0.354100i	$-0.115213\pi$
$44$	−4.69683	−0.708073
$45$	0	0
$46$	9.48728	1.39882
$47$	9.92630i	1.44790i	0.689853	+	0.723950i	$0.257675\pi$
$47$	9.92630i	1.44790i	−0.689853	+	0.723950i	$0.742325\pi$
$48$	− 6.99512i	− 1.00966i
$49$	6.01515	0.859306
$50$	0	0
$51$	−6.36356	−0.891077
$52$	− 7.92369i	− 1.09882i
$53$	− 7.56521i	− 1.03916i	−0.854421	−	0.519581i	$-0.826088\pi$
$53$	− 7.56521i	− 1.03916i	0.854421	−	0.519581i	$-0.173912\pi$
$54$	5.34841	0.727827
$55$	0	0
$56$	−0.721020	−0.0963503
$57$	5.68196i	0.752594i
$58$	− 11.2278i	− 1.47428i
$59$	−4.91775	−0.640237	−0.320118	−	0.947378i	$-0.603723\pi$
$59$	−4.91775	−0.640237	−0.320118	+	0.947378i	$0.603723\pi$
$60$	0	0
$61$	−2.76972	−0.354626	−0.177313	−	0.984155i	$-0.556740\pi$
$61$	−2.76972	−0.354626	−0.177313	+	0.984155i	$0.556740\pi$
$62$	− 0.284720i	− 0.0361595i
$63$	− 1.81943i	− 0.229227i
$64$	10.5022	1.31278
$65$	0	0
$66$	9.16896	1.12862
$67$	2.18577i	0.267035i	0.991046	+	0.133517i	$0.0426272\pi$
$67$	2.18577i	0.267035i	−0.991046	+	0.133517i	$0.957373\pi$
$68$	− 6.79751i	− 0.824319i
$69$	−10.0023	−1.20414
$70$	0	0
$71$	9.64254	1.14436	0.572179	−	0.820128i	$-0.306098\pi$
$71$	9.64254	1.14436	0.572179	+	0.820128i	$0.306098\pi$
$72$	− 1.33202i	− 0.156980i
$73$	− 0.775929i	− 0.0908157i	−0.998969	−	0.0454078i	$-0.985541\pi$
$73$	− 0.775929i	− 0.0908157i	0.998969	−	0.0454078i	$-0.0144587\pi$
$74$	−4.48150	−0.520963
$75$	0	0
$76$	−6.06943	−0.696212
$77$	1.98480i	0.226189i
$78$	15.4683i	1.75144i
$79$	−15.8508	−1.78336	−0.891679	−	0.452667i	$-0.850473\pi$
$79$	−15.8508	−1.78336	−0.891679	+	0.452667i	$0.850473\pi$
$80$	0	0
$81$	−11.1389	−1.23765
$82$	− 18.0026i	− 1.98806i
$83$	1.77110i	0.194404i	0.995265	+	0.0972019i	$0.0309892\pi$
$83$	1.77110i	0.194404i	−0.995265	+	0.0972019i	$0.969011\pi$
$84$	5.12372	0.559044
$85$	0	0
$86$	−9.68401	−1.04425
$87$	11.8373i	1.26909i
$88$	1.45309i	0.154899i
$89$	−14.5080	−1.53785	−0.768923	−	0.639341i	$-0.779207\pi$
$89$	−14.5080	−1.53785	−0.768923	+	0.639341i	$0.779207\pi$
$90$	0	0
$91$	−3.34841	−0.351009
$92$	− 10.6844i	− 1.11393i
$93$	0.300177i	0.0311269i
$94$	20.6992	2.13496
$95$	0	0
$96$	−17.7814	−1.81481
$97$	− 17.0291i	− 1.72904i	−0.502595	−	0.864522i	$-0.667621\pi$
$97$	− 17.0291i	− 1.72904i	0.502595	−	0.864522i	$-0.332379\pi$
$98$	− 12.5433i	− 1.26706i
$99$	−3.66673	−0.368520
$100$	0	0
$101$	−2.54716	−0.253452	−0.126726	−	0.991938i	$-0.540447\pi$
$101$	−2.54716	−0.253452	−0.126726	+	0.991938i	$0.540447\pi$
$102$	13.2698i	1.31391i
$103$	− 10.1654i	− 1.00163i	−0.865555	−	0.500815i	$-0.833034\pi$
$103$	− 10.1654i	− 1.00163i	0.865555	−	0.500815i	$-0.166966\pi$
$104$	−2.45140	−0.240380
$105$	0	0
$106$	−15.7756	−1.53226
$107$	− 4.81720i	− 0.465697i	−0.972513	−	0.232848i	$-0.925195\pi$
$107$	− 4.81720i	− 0.465697i	0.972513	−	0.232848i	$-0.0748046\pi$
$108$	− 6.02330i	− 0.579592i
$109$	−16.2743	−1.55879	−0.779397	−	0.626531i	$-0.784474\pi$
$109$	−16.2743	−1.55879	−0.779397	+	0.626531i	$0.784474\pi$
$110$	0	0
$111$	4.72479	0.448457
$112$	− 3.15759i	− 0.298365i
$113$	− 6.75704i	− 0.635649i	−0.948150	−	0.317825i	$-0.897048\pi$
$113$	− 6.75704i	− 0.635649i	0.948150	−	0.317825i	$-0.102952\pi$
$114$	11.8485	1.10971
$115$	0	0
$116$	−12.6446	−1.17402
$117$	− 6.18589i	− 0.571886i
$118$	10.2549i	0.944041i
$119$	−2.87251	−0.263322
$120$	0	0
$121$	−7.00000	−0.636364
$122$	5.77565i	0.522903i
$123$	18.9800i	1.71137i
$124$	−0.320647	−0.0287950
$125$	0	0
$126$	−3.79403	−0.337999
$127$	− 1.49081i	− 0.132288i	−0.997810	−	0.0661441i	$-0.978930\pi$
$127$	− 1.49081i	− 0.132288i	0.997810	−	0.0661441i	$-0.0210697\pi$
$128$	− 5.72414i	− 0.505948i
$129$	10.2097	0.898918
$130$	0	0
$131$	14.1147	1.23320	0.616602	−	0.787275i	$-0.288509\pi$
$131$	14.1147	1.23320	0.616602	+	0.787275i	$0.288509\pi$
$132$	− 10.3259i	− 0.898757i
$133$	2.56484i	0.222399i
$134$	4.55796	0.393748
$135$	0	0
$136$	−2.10299	−0.180330
$137$	− 0.689447i	− 0.0589035i	−0.999566	−	0.0294517i	$-0.990624\pi$
$137$	− 0.689447i	− 0.0589035i	0.999566	−	0.0294517i	$-0.00937613\pi$
$138$	20.8577i	1.77553i
$139$	16.5719	1.40561	0.702803	−	0.711384i	$-0.251931\pi$
$139$	16.5719	1.40561	0.702803	+	0.711384i	$0.251931\pi$
$140$	0	0
$141$	−21.8229	−1.83782
$142$	− 20.1074i	− 1.68738i
$143$	6.74812i	0.564307i
$144$	5.83337	0.486114
$145$	0	0
$146$	−1.61803	−0.133909
$147$	13.2242i	1.09072i
$148$	5.04699i	0.414860i
$149$	−3.21156	−0.263101	−0.131551	−	0.991309i	$-0.541996\pi$
$149$	−3.21156	−0.263101	−0.131551	+	0.991309i	$0.541996\pi$
$150$	0	0
$151$	17.6863	1.43929	0.719647	−	0.694340i	$-0.244304\pi$
$151$	17.6863	1.43929	0.719647	+	0.694340i	$0.244304\pi$
$152$	1.87774i	0.152305i
$153$	− 5.30670i	− 0.429021i
$154$	4.13887	0.333519
$155$	0	0
$156$	17.4202	1.39473
$157$	1.65512i	0.132093i	0.997817	+	0.0660465i	$0.0210386\pi$
$157$	1.65512i	0.132093i	−0.997817	+	0.0660465i	$0.978961\pi$
$158$	33.0535i	2.62960i
$159$	16.6321	1.31901
$160$	0	0
$161$	−4.51505	−0.355836
$162$	23.2277i	1.82494i
$163$	− 0.892934i	− 0.0699400i	−0.999388	−	0.0349700i	$-0.988866\pi$
$163$	− 0.892934i	− 0.0699400i	0.999388	−	0.0349700i	$-0.0111336\pi$
$164$	−20.2743	−1.58316
$165$	0	0
$166$	3.69325	0.286652
$167$	5.19558i	0.402046i	0.979587	+	0.201023i	$0.0644265\pi$
$167$	5.19558i	0.402046i	−0.979587	+	0.201023i	$0.935573\pi$
$168$	− 1.58516i	− 0.122297i
$169$	1.61570	0.124285
$170$	0	0
$171$	−4.73830	−0.362347
$172$	10.9060i	0.831573i
$173$	5.76465i	0.438278i	0.975694	+	0.219139i	$0.0703248\pi$
$173$	5.76465i	0.438278i	−0.975694	+	0.219139i	$0.929675\pi$
$174$	24.6842	1.87130
$175$	0	0
$176$	−6.36356	−0.479671
$177$	− 10.8116i	− 0.812652i
$178$	30.2534i	2.26758i
$179$	−8.66887	−0.647942	−0.323971	−	0.946067i	$-0.605018\pi$
$179$	−8.66887	−0.647942	−0.323971	+	0.946067i	$0.605018\pi$
$180$	0	0
$181$	14.2909	1.06223	0.531116	−	0.847299i	$-0.321773\pi$
$181$	14.2909	1.06223	0.531116	+	0.847299i	$0.321773\pi$
$182$	6.98240i	0.517570i
$183$	− 6.08920i	− 0.450127i
$184$	−3.30550	−0.243685
$185$	0	0
$186$	0.625955	0.0458972
$187$	5.78902i	0.423335i
$188$	− 23.3111i	− 1.70013i
$189$	−2.54534	−0.185146
$190$	0	0
$191$	−1.26636	−0.0916305	−0.0458153	−	0.998950i	$-0.514589\pi$
$191$	−1.26636	−0.0916305	−0.0458153	+	0.998950i	$0.514589\pi$
$192$	23.0891i	1.66631i
$193$	− 21.1730i	− 1.52406i	−0.647540	−	0.762031i	$-0.724202\pi$
$193$	− 21.1730i	− 1.52406i	0.647540	−	0.762031i	$-0.275798\pi$
$194$	−35.5105	−2.54951
$195$	0	0
$196$	−14.1261	−1.00900
$197$	− 12.2013i	− 0.869308i	−0.900597	−	0.434654i	$-0.856871\pi$
$197$	− 12.2013i	− 0.869308i	0.900597	−	0.434654i	$-0.143129\pi$
$198$	7.64618i	0.543390i
$199$	−10.4065	−0.737695	−0.368848	−	0.929490i	$-0.620248\pi$
$199$	−10.4065	−0.737695	−0.368848	+	0.929490i	$0.620248\pi$
$200$	0	0
$201$	−4.80540	−0.338947
$202$	5.31156i	0.373720i
$203$	5.34337i	0.375031i
$204$	14.9443	1.04631
$205$	0	0
$206$	−21.1978	−1.47692
$207$	− 8.34114i	− 0.579749i
$208$	− 10.7355i	− 0.744375i
$209$	5.16896	0.357545
$210$	0	0
$211$	−8.65769	−0.596020	−0.298010	−	0.954563i	$-0.596323\pi$
$211$	−8.65769	−0.596020	−0.298010	+	0.954563i	$0.596323\pi$
$212$	17.7662i	1.22019i
$213$	21.1990i	1.45253i
$214$	−10.0452	−0.686679
$215$	0	0
$216$	−1.86346	−0.126793
$217$	0.135500i	0.00919832i
$218$	33.9365i	2.29847i
$219$	1.70587	0.115272
$220$	0	0
$221$	−9.76626	−0.656950
$222$	− 9.85253i	− 0.661259i
$223$	− 28.3434i	− 1.89801i	−0.315256	−	0.949007i	$-0.602091\pi$
$223$	− 28.3434i	− 1.89801i	0.315256	−	0.949007i	$-0.397909\pi$
$224$	−8.02652	−0.536295
$225$	0	0
$226$	−14.0904	−0.937277
$227$	22.2415i	1.47622i	0.674682	+	0.738109i	$0.264281\pi$
$227$	22.2415i	1.47622i	−0.674682	+	0.738109i	$0.735719\pi$
$228$	− 13.3436i	− 0.883701i
$229$	−2.47559	−0.163592	−0.0817958	−	0.996649i	$-0.526066\pi$
$229$	−2.47559	−0.163592	−0.0817958	+	0.996649i	$0.526066\pi$
$230$	0	0
$231$	−4.36356	−0.287101
$232$	3.91192i	0.256830i
$233$	− 5.95605i	− 0.390194i	−0.980784	−	0.195097i	$-0.937498\pi$
$233$	− 5.95605i	− 0.390194i	0.980784	−	0.195097i	$-0.0625021\pi$
$234$	−12.8993	−0.843256
$235$	0	0
$236$	11.5489	0.751770
$237$	− 34.8479i	− 2.26362i
$238$	5.99000i	0.388274i
$239$	7.03243	0.454890	0.227445	−	0.973791i	$-0.426963\pi$
$239$	7.03243	0.454890	0.227445	+	0.973791i	$0.426963\pi$
$240$	0	0
$241$	1.17976	0.0759953	0.0379976	−	0.999278i	$-0.487902\pi$
$241$	1.17976	0.0759953	0.0379976	+	0.999278i	$0.487902\pi$
$242$	14.5970i	0.938330i
$243$	− 16.7942i	− 1.07735i
$244$	6.50444	0.416404
$245$	0	0
$246$	39.5787	2.52344
$247$	8.72020i	0.554853i
$248$	0.0992004i	0.00629923i
$249$	−3.89375	−0.246757
$250$	0	0
$251$	4.60867	0.290897	0.145448	−	0.989366i	$-0.453538\pi$
$251$	4.60867	0.290897	0.145448	+	0.989366i	$0.453538\pi$
$252$	4.27277i	0.269159i
$253$	9.09927i	0.572066i
$254$	−3.10877	−0.195062
$255$	0	0
$256$	9.06799	0.566750
$257$	− 9.75542i	− 0.608526i	−0.952588	−	0.304263i	$-0.901590\pi$
$257$	− 9.75542i	− 0.608526i	0.952588	−	0.304263i	$-0.0984102\pi$
$258$	− 21.2902i	− 1.32547i
$259$	2.13277	0.132524
$260$	0	0
$261$	−9.87138	−0.611023
$262$	− 29.4331i	− 1.81838i
$263$	− 0.995828i	− 0.0614054i	−0.999529	−	0.0307027i	$-0.990225\pi$
$263$	− 0.995828i	− 0.0614054i	0.999529	−	0.0307027i	$-0.00977450\pi$
$264$	−3.19460	−0.196614
$265$	0	0
$266$	5.34841	0.327932
$267$	− 31.8958i	− 1.95199i
$268$	− 5.13310i	− 0.313554i
$269$	3.28853	0.200506	0.100253	−	0.994962i	$-0.468035\pi$
$269$	3.28853	0.200506	0.100253	+	0.994962i	$0.468035\pi$
$270$	0	0
$271$	12.1500	0.738063	0.369031	−	0.929417i	$-0.379689\pi$
$271$	12.1500	0.738063	0.369031	+	0.929417i	$0.379689\pi$
$272$	− 9.20970i	− 0.558420i
$273$	− 7.36146i	− 0.445536i
$274$	−1.43769	−0.0868543
$275$	0	0
$276$	23.4896	1.41391
$277$	− 11.8666i	− 0.712993i	−0.934297	−	0.356496i	$-0.883971\pi$
$277$	− 11.8666i	− 0.712993i	0.934297	−	0.356496i	$-0.116029\pi$
$278$	− 34.5571i	− 2.07259i
$279$	−0.250324	−0.0149865
$280$	0	0
$281$	−24.6416	−1.47000	−0.734998	−	0.678070i	$-0.762817\pi$
$281$	−24.6416	−1.47000	−0.734998	+	0.678070i	$0.762817\pi$
$282$	45.5069i	2.70990i
$283$	3.36343i	0.199935i	0.994991	+	0.0999675i	$0.0318739\pi$
$283$	3.36343i	0.199935i	−0.994991	+	0.0999675i	$0.968126\pi$
$284$	−22.6447	−1.34371
$285$	0	0
$286$	14.0718	0.832081
$287$	8.56755i	0.505727i
$288$	− 14.8283i	− 0.873764i
$289$	8.62180	0.507165
$290$	0	0
$291$	37.4383	2.19467
$292$	1.82220i	0.106636i
$293$	8.96340i	0.523647i	0.965116	+	0.261824i	$0.0843239\pi$
$293$	8.96340i	0.523647i	−0.965116	+	0.261824i	$0.915676\pi$
$294$	27.5763	1.60828
$295$	0	0
$296$	1.56142	0.0907555
$297$	5.12967i	0.297654i
$298$	6.69702i	0.387948i
$299$	−15.3507	−0.887756
$300$	0	0
$301$	4.60867	0.265640
$302$	− 36.8811i	− 2.12227i
$303$	− 5.59991i	− 0.321706i
$304$	−8.22325	−0.471636
$305$	0	0
$306$	−11.0660	−0.632600
$307$	9.48133i	0.541128i	0.962702	+	0.270564i	$0.0872102\pi$
$307$	9.48133i	0.541128i	−0.962702	+	0.270564i	$0.912790\pi$
$308$	− 4.66112i	− 0.265592i
$309$	22.3486	1.27137
$310$	0	0
$311$	29.3320	1.66327	0.831633	−	0.555325i	$-0.187406\pi$
$311$	29.3320	1.66327	0.831633	+	0.555325i	$0.187406\pi$
$312$	− 5.38938i	− 0.305114i
$313$	− 18.8901i	− 1.06773i	−0.845570	−	0.533865i	$-0.820739\pi$
$313$	− 18.8901i	− 1.06773i	0.845570	−	0.533865i	$-0.179261\pi$
$314$	3.45140	0.194774
$315$	0	0
$316$	37.2243	2.09403
$317$	22.7893i	1.27998i	0.768385	+	0.639988i	$0.221061\pi$
$317$	22.7893i	1.27998i	−0.768385	+	0.639988i	$0.778939\pi$
$318$	− 34.6826i	− 1.94490i
$319$	10.7686	0.602925
$320$	0	0
$321$	10.5906	0.591109
$322$	9.41516i	0.524687i
$323$	7.48081i	0.416244i
$324$	26.1587	1.45326
$325$	0	0
$326$	−1.86202	−0.103128
$327$	− 35.7789i	− 1.97858i
$328$	6.27237i	0.346334i
$329$	−9.85084	−0.543094
$330$	0	0
$331$	2.96299	0.162861	0.0814304	−	0.996679i	$-0.474051\pi$
$331$	2.96299	0.162861	0.0814304	+	0.996679i	$0.474051\pi$
$332$	− 4.15928i	− 0.228270i
$333$	3.94010i	0.215916i
$334$	10.8343	0.592824
$335$	0	0
$336$	6.94194	0.378714
$337$	− 18.8123i	− 1.02477i	−0.858756	−	0.512385i	$-0.828762\pi$
$337$	− 18.8123i	− 1.02477i	0.858756	−	0.512385i	$-0.171238\pi$
$338$	− 3.36920i	− 0.183261i
$339$	14.8553	0.806829
$340$	0	0
$341$	0.273075	0.0147879
$342$	9.88071i	0.534287i
$343$	12.9162i	0.697410i
$344$	3.37405	0.181916
$345$	0	0
$346$	12.0209	0.646249
$347$	− 22.7382i	− 1.22065i	−0.792151	−	0.610325i	$-0.791039\pi$
$347$	− 22.7382i	− 1.22065i	0.792151	−	0.610325i	$-0.208961\pi$
$348$	− 27.7989i	− 1.49018i
$349$	−1.93849	−0.103765	−0.0518824	−	0.998653i	$-0.516522\pi$
$349$	−1.93849	−0.103765	−0.0518824	+	0.998653i	$0.516522\pi$
$350$	0	0
$351$	−8.65392	−0.461912
$352$	16.1760i	0.862184i
$353$	− 5.24945i	− 0.279400i	−0.990194	−	0.139700i	$-0.955386\pi$
$353$	− 5.24945i	− 0.279400i	0.990194	−	0.139700i	$-0.0446138\pi$
$354$	−22.5453	−1.19827
$355$	0	0
$356$	34.0708	1.80575
$357$	− 6.31519i	− 0.334235i
$358$	18.0771i	0.955402i
$359$	−22.5937	−1.19245	−0.596226	−	0.802817i	$-0.703334\pi$
$359$	−22.5937	−1.19245	−0.596226	+	0.802817i	$0.703334\pi$
$360$	0	0
$361$	−12.3205	−0.648445
$362$	− 29.8005i	− 1.56628i
$363$	− 15.3894i	− 0.807736i
$364$	7.86346	0.412157
$365$	0	0
$366$	−12.6977	−0.663720
$367$	− 7.29872i	− 0.380990i	−0.981688	−	0.190495i	$-0.938991\pi$
$367$	− 7.29872i	− 0.380990i	0.981688	−	0.190495i	$-0.0610093\pi$
$368$	− 14.4759i	− 0.754610i
$369$	−15.8278	−0.823961
$370$	0	0
$371$	7.50770	0.389781
$372$	− 0.704940i	− 0.0365494i
$373$	22.3074i	1.15503i	0.816380	+	0.577516i	$0.195978\pi$
$373$	22.3074i	1.15503i	−0.816380	+	0.577516i	$0.804022\pi$
$374$	12.0718	0.624216
$375$	0	0
$376$	−7.21188	−0.371924
$377$	18.1669i	0.935645i
$378$	5.30776i	0.273002i
$379$	32.9466	1.69235	0.846177	−	0.532903i	$-0.178899\pi$
$379$	32.9466	1.69235	0.846177	+	0.532903i	$0.178899\pi$
$380$	0	0
$381$	3.27754	0.167913
$382$	2.64072i	0.135111i
$383$	20.7002i	1.05773i	0.848706	+	0.528865i	$0.177382\pi$
$383$	20.7002i	1.05773i	−0.848706	+	0.528865i	$0.822618\pi$
$384$	12.5845	0.642199
$385$	0	0
$386$	−44.1516	−2.24726
$387$	8.51410i	0.432796i
$388$	39.9914i	2.03026i
$389$	1.14446	0.0580263	0.0290132	−	0.999579i	$-0.490764\pi$
$389$	1.14446	0.0580263	0.0290132	+	0.999579i	$0.490764\pi$
$390$	0	0
$391$	−13.1690	−0.665983
$392$	4.37026i	0.220731i
$393$	31.0310i	1.56531i
$394$	−25.4432	−1.28181
$395$	0	0
$396$	8.61100	0.432719
$397$	− 4.68513i	− 0.235140i	−0.993065	−	0.117570i	$-0.962490\pi$
$397$	− 4.68513i	− 0.235140i	0.993065	−	0.117570i	$-0.0375105\pi$
$398$	21.7005i	1.08775i
$399$	−5.63877	−0.282292
$400$	0	0
$401$	−24.0851	−1.20275	−0.601376	−	0.798966i	$-0.705381\pi$
$401$	−24.0851	−1.20275	−0.601376	+	0.798966i	$0.705381\pi$
$402$	10.0206i	0.499784i
$403$	0.460687i	0.0229484i
$404$	5.98179	0.297605
$405$	0	0
$406$	11.1424	0.552990
$407$	− 4.29821i	− 0.213054i
$408$	− 4.62340i	− 0.228892i
$409$	1.89934	0.0939165	0.0469583	−	0.998897i	$-0.485047\pi$
$409$	1.89934	0.0939165	0.0469583	+	0.998897i	$0.485047\pi$
$410$	0	0
$411$	1.51574	0.0747661
$412$	23.8726i	1.17612i
$413$	− 4.88037i	− 0.240147i
$414$	−17.3937	−0.854852
$415$	0	0
$416$	−27.2894	−1.33797
$417$	36.4331i	1.78414i
$418$	− 10.7788i	− 0.527207i
$419$	2.32806	0.113733	0.0568666	−	0.998382i	$-0.481889\pi$
$419$	2.32806	0.113733	0.0568666	+	0.998382i	$0.481889\pi$
$420$	0	0
$421$	−23.9501	−1.16725	−0.583627	−	0.812022i	$-0.698367\pi$
$421$	−23.9501	−1.16725	−0.583627	+	0.812022i	$0.698367\pi$
$422$	18.0537i	0.878842i
$423$	− 18.1985i	− 0.884843i
$424$	5.49645	0.266931
$425$	0	0
$426$	44.2060	2.14179
$427$	− 2.74866i	− 0.133017i
$428$	11.3128i	0.546824i
$429$	−14.8357	−0.716274
$430$	0	0
$431$	1.19227	0.0574294	0.0287147	−	0.999588i	$-0.490859\pi$
$431$	1.19227	0.0574294	0.0287147	+	0.999588i	$0.490859\pi$
$432$	− 8.16074i	− 0.392634i
$433$	− 25.6138i	− 1.23092i	−0.788167	−	0.615461i	$-0.788970\pi$
$433$	− 25.6138i	− 1.23092i	0.788167	−	0.615461i	$-0.211030\pi$
$434$	0.282556	0.0135631
$435$	0	0
$436$	38.2187	1.83035
$437$	11.7584i	0.562483i
$438$	− 3.55723i	− 0.169971i
$439$	−19.3741	−0.924676	−0.462338	−	0.886704i	$-0.652989\pi$
$439$	−19.3741	−0.924676	−0.462338	+	0.886704i	$0.652989\pi$
$440$	0	0
$441$	−11.0280	−0.525141
$442$	20.3654i	0.968685i
$443$	2.46263i	0.117003i	0.998287	+	0.0585016i	$0.0186323\pi$
$443$	2.46263i	0.117003i	−0.998287	+	0.0585016i	$0.981368\pi$
$444$	−11.0958	−0.526582
$445$	0	0
$446$	−59.1040	−2.79866
$447$	− 7.06059i	− 0.333955i
$448$	10.4224i	0.492412i
$449$	14.3585	0.677618	0.338809	−	0.940855i	$-0.389976\pi$
$449$	14.3585	0.677618	0.338809	+	0.940855i	$0.389976\pi$
$450$	0	0
$451$	17.2664	0.813041
$452$	15.8683i	0.746384i
$453$	38.8833i	1.82690i
$454$	46.3798	2.17671
$455$	0	0
$456$	−4.12819	−0.193320
$457$	25.1964i	1.17864i	0.807901	+	0.589319i	$0.200604\pi$
$457$	25.1964i	1.17864i	−0.807901	+	0.589319i	$0.799396\pi$
$458$	5.16231i	0.241219i
$459$	−7.42395	−0.346520
$460$	0	0
$461$	28.8255	1.34254	0.671269	−	0.741214i	$-0.265749\pi$
$461$	28.8255	1.34254	0.671269	+	0.741214i	$0.265749\pi$
$462$	9.09927i	0.423336i
$463$	31.8796i	1.48157i	0.671742	+	0.740786i	$0.265547\pi$
$463$	31.8796i	1.48157i	−0.671742	+	0.740786i	$0.734453\pi$
$464$	−17.1316	−0.795317
$465$	0	0
$466$	−12.4201	−0.575348
$467$	− 43.0996i	− 1.99441i	−0.0747039	−	0.997206i	$-0.523801\pi$
$467$	− 43.0996i	− 1.99441i	0.0747039	−	0.997206i	$-0.476199\pi$
$468$	14.5270i	0.671512i
$469$	−2.16916	−0.100162
$470$	0	0
$471$	−3.63877	−0.167666
$472$	− 3.57295i	− 0.164459i
$473$	− 9.28795i	− 0.427060i
$474$	−72.6679	−3.33775
$475$	0	0
$476$	6.74584	0.309195
$477$	13.8698i	0.635054i
$478$	− 14.6646i	− 0.670744i
$479$	−21.0263	−0.960717	−0.480359	−	0.877072i	$-0.659494\pi$
$479$	−21.0263	−0.960717	−0.480359	+	0.877072i	$0.659494\pi$
$480$	0	0
$481$	7.25121	0.330627
$482$	− 2.46014i	− 0.112057i
$483$	− 9.92630i	− 0.451662i
$484$	16.4389	0.747223
$485$	0	0
$486$	−35.0207	−1.58857
$487$	27.9190i	1.26513i	0.774507	+	0.632565i	$0.217998\pi$
$487$	27.9190i	1.26513i	−0.774507	+	0.632565i	$0.782002\pi$
$488$	− 2.01232i	− 0.0910933i
$489$	1.96311	0.0887748
$490$	0	0
$491$	14.9611	0.675183	0.337591	−	0.941293i	$-0.390388\pi$
$491$	14.9611	0.675183	0.337591	+	0.941293i	$0.390388\pi$
$492$	− 44.5728i	− 2.00950i
$493$	15.5849i	0.701909i
$494$	18.1841	0.818142
$495$	0	0
$496$	−0.434433	−0.0195066
$497$	9.56924i	0.429239i
$498$	8.11959i	0.363847i
$499$	44.3253	1.98427	0.992137	−	0.125160i	$-0.0399443\pi$
$499$	44.3253	1.98427	0.992137	+	0.125160i	$0.0399443\pi$
$500$	0	0
$501$	−11.4224	−0.510316
$502$	− 9.61040i	− 0.428933i
$503$	23.6212i	1.05322i	0.850108	+	0.526609i	$0.176537\pi$
$503$	23.6212i	1.05322i	−0.850108	+	0.526609i	$0.823463\pi$
$504$	1.32189	0.0588818
$505$	0	0
$506$	18.9746	0.843522
$507$	3.55211i	0.157755i
$508$	3.50105i	0.155334i
$509$	26.7154	1.18414	0.592070	−	0.805886i	$-0.298311\pi$
$509$	26.7154	1.18414	0.592070	+	0.805886i	$0.298311\pi$
$510$	0	0
$511$	0.770031	0.0340642
$512$	− 30.3576i	− 1.34163i
$513$	6.62877i	0.292667i
$514$	−20.3428	−0.897283
$515$	0	0
$516$	−23.9767	−1.05552
$517$	19.8526i	0.873116i
$518$	− 4.44743i	− 0.195409i
$519$	−12.6735	−0.556306
$520$	0	0
$521$	32.7073	1.43293	0.716466	−	0.697622i	$-0.245759\pi$
$521$	32.7073	1.43293	0.716466	+	0.697622i	$0.245759\pi$
$522$	20.5846i	0.900966i
$523$	− 0.235966i	− 0.0103181i	−0.999987	−	0.00515904i	$-0.998358\pi$
$523$	− 0.235966i	− 0.0103181i	0.999987	−	0.00515904i	$-0.00164218\pi$
$524$	−33.1471	−1.44804
$525$	0	0
$526$	−2.07658	−0.0905434
$527$	0.395210i	0.0172156i
$528$	− 13.9902i	− 0.608847i
$529$	2.30084	0.100037
$530$	0	0
$531$	9.01604	0.391263
$532$	− 6.02330i	− 0.261143i
$533$	29.1289i	1.26171i
$534$	−66.5117	−2.87824
$535$	0	0
$536$	−1.58806	−0.0685936
$537$	− 19.0584i	− 0.822432i
$538$	− 6.85753i	− 0.295649i
$539$	12.0303	0.518181
$540$	0	0
$541$	−33.5572	−1.44274	−0.721369	−	0.692551i	$-0.756487\pi$
$541$	−33.5572	−1.44274	−0.721369	+	0.692551i	$0.756487\pi$
$542$	− 25.3363i	− 1.08829i
$543$	31.4183i	1.34829i
$544$	−23.4108	−1.00373
$545$	0	0
$546$	−15.3507	−0.656951
$547$	38.5125i	1.64668i	0.567552	+	0.823338i	$0.307891\pi$
$547$	38.5125i	1.64668i	−0.567552	+	0.823338i	$0.692109\pi$
$548$	1.61911i	0.0691648i
$549$	5.07790	0.216720
$550$	0	0
$551$	13.9156	0.592825
$552$	− 7.26712i	− 0.309309i
$553$	− 15.7303i	− 0.668922i
$554$	−24.7452	−1.05132
$555$	0	0
$556$	−38.9176	−1.65047
$557$	4.33445i	0.183657i	0.995775	+	0.0918283i	$0.0292711\pi$
$557$	4.33445i	0.183657i	−0.995775	+	0.0918283i	$0.970729\pi$
$558$	0.521996i	0.0220978i
$559$	15.6691	0.662731
$560$	0	0
$561$	−12.7271	−0.537339
$562$	51.3848i	2.16754i
$563$	34.9018i	1.47094i	0.677559	+	0.735468i	$0.263038\pi$
$563$	34.9018i	1.47094i	−0.677559	+	0.735468i	$0.736962\pi$
$564$	51.2492	2.15798
$565$	0	0
$566$	7.01371	0.294808
$567$	− 11.0542i	− 0.464233i
$568$	7.00572i	0.293953i
$569$	41.9646	1.75925	0.879623	−	0.475671i	$-0.157795\pi$
$569$	41.9646	1.75925	0.879623	+	0.475671i	$0.157795\pi$
$570$	0	0
$571$	−13.8332	−0.578900	−0.289450	−	0.957193i	$-0.593472\pi$
$571$	−13.8332	−0.578900	−0.289450	+	0.957193i	$0.593472\pi$
$572$	− 15.8474i	− 0.662613i
$573$	− 2.78408i	− 0.116307i
$574$	17.8658	0.745704
$575$	0	0
$576$	−19.2544	−0.802268
$577$	12.7793i	0.532008i	0.963972	+	0.266004i	$0.0857035\pi$
$577$	12.7793i	0.532008i	−0.963972	+	0.266004i	$0.914296\pi$
$578$	− 17.9789i	− 0.747824i
$579$	46.5486	1.93449
$580$	0	0
$581$	−1.75764	−0.0729191
$582$	− 78.0696i	− 3.23609i
$583$	− 15.1304i	− 0.626638i
$584$	0.563746	0.0233280
$585$	0	0
$586$	18.6912	0.772128
$587$	12.1870i	0.503009i	0.967856	+	0.251505i	$0.0809254\pi$
$587$	12.1870i	0.503009i	−0.967856	+	0.251505i	$0.919075\pi$
$588$	− 31.0560i	− 1.28073i
$589$	0.352879	0.0145401
$590$	0	0
$591$	26.8245	1.10341
$592$	6.83798i	0.281039i
$593$	− 31.2580i	− 1.28361i	−0.766866	−	0.641807i	$-0.778185\pi$
$593$	− 31.2580i	− 1.28361i	0.766866	−	0.641807i	$-0.221815\pi$
$594$	10.6968	0.438896
$595$	0	0
$596$	7.54208	0.308936
$597$	− 22.8785i	− 0.936356i
$598$	32.0107i	1.30901i
$599$	33.3707	1.36349	0.681746	−	0.731589i	$-0.261221\pi$
$599$	33.3707	1.36349	0.681746	+	0.731589i	$0.261221\pi$
$600$	0	0
$601$	−46.8052	−1.90922	−0.954611	−	0.297854i	$-0.903729\pi$
$601$	−46.8052	−1.90922	−0.954611	+	0.297854i	$0.903729\pi$
$602$	− 9.61040i	− 0.391691i
$603$	− 4.00732i	− 0.163191i
$604$	−41.5349	−1.69003
$605$	0	0
$606$	−11.6774	−0.474362
$607$	− 30.7401i	− 1.24770i	−0.781543	−	0.623851i	$-0.785567\pi$
$607$	− 30.7401i	− 1.24770i	0.781543	−	0.623851i	$-0.214433\pi$
$608$	20.9033i	0.847741i
$609$	−11.7473	−0.476027
$610$	0	0
$611$	−33.4919	−1.35494
$612$	12.4623i	0.503760i
$613$	− 38.2895i	− 1.54650i	−0.634103	−	0.773248i	$-0.718631\pi$
$613$	− 38.2895i	− 1.54650i	0.634103	−	0.773248i	$-0.281369\pi$
$614$	19.7713	0.797904
$615$	0	0
$616$	−1.44204	−0.0581014
$617$	− 0.425306i	− 0.0171222i	−0.999963	−	0.00856109i	$-0.997275\pi$
$617$	− 0.425306i	− 0.0171222i	0.999963	−	0.00856109i	$-0.00272511\pi$
$618$	− 46.6032i	− 1.87466i
$619$	−7.51147	−0.301912	−0.150956	−	0.988541i	$-0.548235\pi$
$619$	−7.51147	−0.301912	−0.150956	+	0.988541i	$0.548235\pi$
$620$	0	0
$621$	−11.6691	−0.468263
$622$	− 61.1656i	− 2.45252i
$623$	− 14.3977i	− 0.576833i
$624$	23.6020	0.944834
$625$	0	0
$626$	−39.3912	−1.57439
$627$	11.3639i	0.453831i
$628$	− 3.88691i	− 0.155105i
$629$	6.22061	0.248032
$630$	0	0
$631$	−11.2716	−0.448714	−0.224357	−	0.974507i	$-0.572028\pi$
$631$	−11.2716	−0.448714	−0.224357	+	0.974507i	$0.572028\pi$
$632$	− 11.5163i	− 0.458094i
$633$	− 19.0338i	− 0.756528i
$634$	47.5222	1.88735
$635$	0	0
$636$	−39.0589	−1.54879
$637$	20.2955i	0.804136i
$638$	− 22.4556i	− 0.889025i
$639$	−17.6783	−0.699343
$640$	0	0
$641$	26.0825	1.03020	0.515099	−	0.857131i	$-0.327755\pi$
$641$	26.0825	1.03020	0.515099	+	0.857131i	$0.327755\pi$
$642$	− 22.0844i	− 0.871601i
$643$	− 31.9492i	− 1.25995i	−0.776614	−	0.629977i	$-0.783064\pi$
$643$	− 31.9492i	− 1.25995i	0.776614	−	0.629977i	$-0.216936\pi$
$644$	10.6032	0.417825
$645$	0	0
$646$	15.5996	0.613759
$647$	7.39433i	0.290701i	0.989380	+	0.145351i	$0.0464310\pi$
$647$	7.39433i	0.290701i	−0.989380	+	0.145351i	$0.953569\pi$
$648$	− 8.09286i	− 0.317918i
$649$	−9.83550	−0.386077
$650$	0	0
$651$	−0.297895	−0.0116754
$652$	2.09698i	0.0821240i
$653$	18.6853i	0.731212i	0.930770	+	0.365606i	$0.119138\pi$
$653$	18.6853i	0.731212i	−0.930770	+	0.365606i	$0.880862\pi$
$654$	−74.6091	−2.91745
$655$	0	0
$656$	−27.4689	−1.07248
$657$	1.42256i	0.0554994i
$658$	20.5418i	0.800803i
$659$	−9.80157	−0.381815	−0.190907	−	0.981608i	$-0.561143\pi$
$659$	−9.80157	−0.381815	−0.190907	+	0.981608i	$0.561143\pi$
$660$	0	0
$661$	−28.1585	−1.09524	−0.547619	−	0.836728i	$-0.684466\pi$
$661$	−28.1585	−1.09524	−0.547619	+	0.836728i	$0.684466\pi$
$662$	− 6.17868i	− 0.240141i
$663$	− 21.4710i	− 0.833866i
$664$	−1.28678	−0.0499368
$665$	0	0
$666$	8.21622	0.318372
$667$	24.4966i	0.948511i
$668$	− 12.2014i	− 0.472085i
$669$	62.3127	2.40915
$670$	0	0
$671$	−5.53943	−0.213847
$672$	− 17.6462i	− 0.680718i
$673$	− 39.0253i	− 1.50432i	−0.658983	−	0.752158i	$-0.729013\pi$
$673$	− 39.0253i	− 1.50432i	0.658983	−	0.752158i	$-0.270987\pi$
$674$	−39.2290	−1.51104
$675$	0	0
$676$	−3.79434	−0.145936
$677$	− 5.03533i	− 0.193523i	−0.995308	−	0.0967617i	$-0.969152\pi$
$677$	− 5.03533i	− 0.193523i	0.995308	−	0.0967617i	$-0.0308485\pi$
$678$	− 30.9775i	− 1.18969i
$679$	16.8997	0.648549
$680$	0	0
$681$	−48.8977	−1.87376
$682$	− 0.569440i	− 0.0218050i
$683$	− 30.3312i	− 1.16059i	−0.814406	−	0.580295i	$-0.802937\pi$
$683$	− 30.3312i	− 1.16059i	0.814406	−	0.580295i	$-0.197063\pi$
$684$	11.1275	0.425470
$685$	0	0
$686$	26.9340	1.02834
$687$	− 5.44257i	− 0.207647i
$688$	14.7761i	0.563334i
$689$	25.5255	0.972444
$690$	0	0
$691$	−21.2329	−0.807739	−0.403869	−	0.914817i	$-0.632335\pi$
$691$	−21.2329	−0.807739	−0.403869	+	0.914817i	$0.632335\pi$
$692$	− 13.5378i	− 0.514629i
$693$	− 3.63886i	− 0.138229i
$694$	−47.4156	−1.79987
$695$	0	0
$696$	−8.60032	−0.325994
$697$	24.9888i	0.946520i
$698$	4.04230i	0.153003i
$699$	13.0943	0.495273
$700$	0	0
$701$	32.7698	1.23770	0.618849	−	0.785510i	$-0.287599\pi$
$701$	32.7698	1.23770	0.618849	+	0.785510i	$0.287599\pi$
$702$	18.0459i	0.681098i
$703$	− 5.55432i	− 0.209485i
$704$	21.0045	0.791636
$705$	0	0
$706$	−10.9466	−0.411981
$707$	− 2.52780i	− 0.0950676i
$708$	25.3902i	0.954222i
$709$	−19.8459	−0.745330	−0.372665	−	0.927966i	$-0.621556\pi$
$709$	−19.8459	−0.745330	−0.372665	+	0.927966i	$0.621556\pi$
$710$	0	0
$711$	29.0604	1.08985
$712$	− 10.5407i	− 0.395029i
$713$	0.621196i	0.0232640i
$714$	−13.1690	−0.492836
$715$	0	0
$716$	20.3581	0.760817
$717$	15.4607i	0.577392i
$718$	47.1144i	1.75829i
$719$	−24.2201	−0.903258	−0.451629	−	0.892206i	$-0.649157\pi$
$719$	−24.2201	−0.903258	−0.451629	+	0.892206i	$0.649157\pi$
$720$	0	0
$721$	10.0882	0.375703
$722$	25.6917i	0.956144i
$723$	2.59370i	0.0964608i
$724$	−33.5609	−1.24728
$725$	0	0
$726$	−32.0914	−1.19102
$727$	5.86510i	0.217524i	0.994068	+	0.108762i	$0.0346887\pi$
$727$	5.86510i	0.217524i	−0.994068	+	0.108762i	$0.965311\pi$
$728$	− 2.43277i	− 0.0901643i
$729$	3.50531	0.129826
$730$	0	0
$731$	13.4420	0.497172
$732$	14.3000i	0.528542i
$733$	− 34.5015i	− 1.27434i	−0.770723	−	0.637171i	$-0.780105\pi$
$733$	− 34.5015i	− 1.27434i	0.770723	−	0.637171i	$-0.219895\pi$
$734$	−15.2199	−0.561777
$735$	0	0
$736$	−36.7974	−1.35637
$737$	4.37155i	0.161028i
$738$	33.0054i	1.21495i
$739$	−39.5712	−1.45565	−0.727826	−	0.685762i	$-0.759469\pi$
$739$	−39.5712	−1.45565	−0.727826	+	0.685762i	$0.759469\pi$
$740$	0	0
$741$	−19.1713	−0.704275
$742$	− 15.6557i	− 0.574739i
$743$	29.7058i	1.08980i	0.838501	+	0.544900i	$0.183433\pi$
$743$	29.7058i	1.08980i	−0.838501	+	0.544900i	$0.816567\pi$
$744$	−0.218091	−0.00799562
$745$	0	0
$746$	46.5172	1.70312
$747$	− 3.24708i	− 0.118804i
$748$	− 13.5950i	− 0.497083i
$749$	4.78058	0.174679
$750$	0	0
$751$	26.8870	0.981122	0.490561	−	0.871407i	$-0.336792\pi$
$751$	26.8870	0.981122	0.490561	+	0.871407i	$0.336792\pi$
$752$	− 31.5833i	− 1.15172i
$753$	10.1321i	0.369235i
$754$	37.8833	1.37963
$755$	0	0
$756$	5.97751	0.217400
$757$	44.6792i	1.62389i	0.583731	+	0.811947i	$0.301592\pi$
$757$	44.6792i	1.62389i	−0.583731	+	0.811947i	$0.698408\pi$
$758$	− 68.7031i	− 2.49541i
$759$	−20.0047	−0.726123
$760$	0	0
$761$	20.3080	0.736163	0.368081	−	0.929794i	$-0.380015\pi$
$761$	20.3080	0.736163	0.368081	+	0.929794i	$0.380015\pi$
$762$	− 6.83461i	− 0.247592i
$763$	− 16.1506i	− 0.584690i
$764$	2.97393	0.107593
$765$	0	0
$766$	43.1658	1.55964
$767$	− 16.5928i	− 0.599131i
$768$	19.9359i	0.719375i
$769$	−26.0577	−0.939665	−0.469832	−	0.882756i	$-0.655686\pi$
$769$	−26.0577	−0.939665	−0.469832	+	0.882756i	$0.655686\pi$
$770$	0	0
$771$	21.4472	0.772402
$772$	49.7229i	1.78956i
$773$	− 13.7305i	− 0.493851i	−0.969034	−	0.246926i	$-0.920580\pi$
$773$	− 13.7305i	− 0.493851i	0.969034	−	0.246926i	$-0.0794203\pi$
$774$	17.7543	0.638166
$775$	0	0
$776$	12.3724	0.444142
$777$	4.68887i	0.168212i
$778$	− 2.38652i	− 0.0855609i
$779$	22.3123	0.799421
$780$	0	0
$781$	19.2851	0.690074
$782$	27.4610i	0.982005i
$783$	13.8098i	0.493523i
$784$	−19.1389	−0.683531
$785$	0	0
$786$	64.7085	2.30807
$787$	− 19.6660i	− 0.701018i	−0.936559	−	0.350509i	$-0.886009\pi$
$787$	− 19.6660i	− 0.701018i	0.936559	−	0.350509i	$-0.113991\pi$
$788$	28.6538i	1.02075i
$789$	2.18932	0.0779418
$790$	0	0
$791$	6.70568	0.238427
$792$	− 2.66404i	− 0.0946624i
$793$	− 9.34520i	− 0.331858i
$794$	−9.76984	−0.346719
$795$	0	0
$796$	24.4387	0.866207
$797$	10.9441i	0.387660i	0.981035	+	0.193830i	$0.0620910\pi$
$797$	10.9441i	0.387660i	−0.981035	+	0.193830i	$0.937909\pi$
$798$	11.7584i	0.416244i
$799$	−28.7318	−1.01646
$800$	0	0
$801$	26.5985	0.939812
$802$	50.2243i	1.77348i
$803$	− 1.55186i	− 0.0547639i
$804$	11.2851	0.397994
$805$	0	0
$806$	0.960663	0.0338379
$807$	7.22982i	0.254502i
$808$	− 1.85062i	− 0.0651046i
$809$	−16.4427	−0.578096	−0.289048	−	0.957315i	$-0.593339\pi$
$809$	−16.4427	−0.578096	−0.289048	+	0.957315i	$0.593339\pi$
$810$	0	0
$811$	22.6473	0.795253	0.397627	−	0.917547i	$-0.369834\pi$
$811$	22.6473	0.795253	0.397627	+	0.917547i	$0.369834\pi$
$812$	− 12.5484i	− 0.440364i
$813$	26.7118i	0.936823i
$814$	−8.96299	−0.314153
$815$	0	0
$816$	20.2474	0.708802
$817$	− 12.0023i	− 0.419906i
$818$	− 3.96067i	− 0.138482i
$819$	6.13887	0.214509
$820$	0	0
$821$	−39.7792	−1.38830	−0.694151	−	0.719829i	$-0.744220\pi$
$821$	−39.7792	−1.38830	−0.694151	+	0.719829i	$0.744220\pi$
$822$	− 3.16076i	− 0.110244i
$823$	− 16.5602i	− 0.577252i	−0.957442	−	0.288626i	$-0.906802\pi$
$823$	− 16.5602i	− 0.577252i	0.957442	−	0.288626i	$-0.0931984\pi$
$824$	7.38562	0.257290
$825$	0	0
$826$	−10.1770	−0.354102
$827$	− 26.0361i	− 0.905365i	−0.891672	−	0.452683i	$-0.850467\pi$
$827$	− 26.0361i	− 0.905365i	0.891672	−	0.452683i	$-0.149533\pi$
$828$	19.5885i	0.680746i
$829$	−5.14357	−0.178644	−0.0893218	−	0.996003i	$-0.528470\pi$
$829$	−5.14357	−0.178644	−0.0893218	+	0.996003i	$0.528470\pi$
$830$	0	0
$831$	26.0885	0.905002
$832$	35.4352i	1.22849i
$833$	17.4109i	0.603252i
$834$	75.9734	2.63074
$835$	0	0
$836$	−12.1389	−0.419832
$837$	0.350197i	0.0121046i
$838$	− 4.85467i	− 0.167702i
$839$	−38.5664	−1.33146	−0.665730	−	0.746193i	$-0.731880\pi$
$839$	−38.5664	−1.33146	−0.665730	+	0.746193i	$0.731880\pi$
$840$	0	0
$841$	−0.00936035	−0.000322771 0
$842$	49.9427i	1.72114i
$843$	− 54.1744i	− 1.86586i
$844$	20.3318	0.699850
$845$	0	0
$846$	−37.9491	−1.30472
$847$	− 6.94679i	− 0.238694i
$848$	24.0708i	0.826596i
$849$	−7.39447	−0.253777
$850$	0	0
$851$	9.77764	0.335173
$852$	− 49.7841i	− 1.70558i
$853$	− 9.14763i	− 0.313209i	−0.987661	−	0.156604i	$-0.949945\pi$
$853$	− 9.14763i	− 0.313209i	0.987661	−	0.156604i	$-0.0500548\pi$
$854$	−5.73175	−0.196136
$855$	0	0
$856$	3.49990	0.119624
$857$	− 13.6712i	− 0.466998i	−0.972357	−	0.233499i	$-0.924982\pi$
$857$	− 13.6712i	− 0.466998i	0.972357	−	0.233499i	$-0.0750176\pi$
$858$	30.9367i	1.05616i
$859$	35.6556	1.21655	0.608277	−	0.793725i	$-0.291861\pi$
$859$	35.6556	1.21655	0.608277	+	0.793725i	$0.291861\pi$
$860$	0	0
$861$	−18.8357	−0.641919
$862$	− 2.48621i	− 0.0846808i
$863$	− 33.9333i	− 1.15510i	−0.816354	−	0.577552i	$-0.804008\pi$
$863$	− 33.9333i	− 1.15510i	0.816354	−	0.577552i	$-0.195992\pi$
$864$	−20.7444	−0.705739
$865$	0	0
$866$	−53.4121	−1.81502
$867$	18.9550i	0.643744i
$868$	− 0.318210i	− 0.0108007i
$869$	−31.7017	−1.07541
$870$	0	0
$871$	−7.37494	−0.249890
$872$	− 11.8240i	− 0.400410i
$873$	31.2206i	1.05666i
$874$	24.5197	0.829391
$875$	0	0
$876$	−4.00610	−0.135354
$877$	28.6991i	0.969099i	0.874764	+	0.484550i	$0.161017\pi$
$877$	28.6991i	0.969099i	−0.874764	+	0.484550i	$0.838983\pi$
$878$	40.4006i	1.36345i
$879$	−19.7060	−0.664665
$880$	0	0
$881$	8.33039	0.280658	0.140329	−	0.990105i	$-0.455184\pi$
$881$	8.33039	0.280658	0.140329	+	0.990105i	$0.455184\pi$
$882$	22.9964i	0.774331i
$883$	50.3165i	1.69329i	0.532161	+	0.846643i	$0.321380\pi$
$883$	50.3165i	1.69329i	−0.532161	+	0.846643i	$0.678620\pi$
$884$	22.9352	0.771395
$885$	0	0
$886$	5.13529	0.172524
$887$	12.1186i	0.406903i	0.979085	+	0.203452i	$0.0652160\pi$
$887$	12.1186i	0.406903i	−0.979085	+	0.203452i	$0.934784\pi$
$888$	3.43276i	0.115196i
$889$	1.47948	0.0496202
$890$	0	0
$891$	−22.2777	−0.746332
$892$	66.5620i	2.22866i
$893$	25.6543i	0.858490i
$894$	−14.7233	−0.492422
$895$	0	0
$896$	5.68063	0.189777
$897$	− 33.7485i	− 1.12683i
$898$	− 29.9415i	− 0.999161i
$899$	0.735159	0.0245189
$900$	0	0
$901$	21.8976	0.729514
$902$	− 36.0053i	− 1.19884i
$903$	10.1321i	0.337176i
$904$	4.90928	0.163280
$905$	0	0
$906$	81.0827	2.69379
$907$	− 31.9105i	− 1.05957i	−0.848132	−	0.529786i	$-0.822272\pi$
$907$	− 31.9105i	− 1.05957i	0.848132	−	0.529786i	$-0.177728\pi$
$908$	− 52.2322i	− 1.73339i
$909$	4.66988	0.154890
$910$	0	0
$911$	−24.6880	−0.817949	−0.408975	−	0.912546i	$-0.634114\pi$
$911$	−24.6880	−0.817949	−0.408975	+	0.912546i	$0.634114\pi$
$912$	− 18.0788i	− 0.598647i
$913$	3.54220i	0.117230i
$914$	52.5416	1.73792
$915$	0	0
$916$	5.81371	0.192090
$917$	14.0074i	0.462565i
$918$	15.4810i	0.510951i
$919$	19.4850	0.642752	0.321376	−	0.946952i	$-0.395855\pi$
$919$	19.4850	0.642752	0.321376	+	0.946952i	$0.395855\pi$
$920$	0	0
$921$	−20.8446	−0.686854
$922$	− 60.1094i	− 1.97960i
$923$	32.5345i	1.07089i
$924$	10.2474	0.337116
$925$	0	0
$926$	66.4781	2.18461
$927$	18.6369i	0.612118i
$928$	43.5482i	1.42954i
$929$	−11.7642	−0.385972	−0.192986	−	0.981201i	$-0.561817\pi$
$929$	−11.7642	−0.385972	−0.192986	+	0.981201i	$0.561817\pi$
$930$	0	0
$931$	15.5460	0.509501
$932$	13.9873i	0.458168i
$933$	64.4862i	2.11118i
$934$	−89.8749	−2.94080
$935$	0	0
$936$	4.49431	0.146901
$937$	− 21.0683i	− 0.688271i	−0.938920	−	0.344136i	$-0.888172\pi$
$937$	− 21.0683i	− 0.688271i	0.938920	−	0.344136i	$-0.111828\pi$
$938$	4.52331i	0.147691i
$939$	41.5296	1.35527
$940$	0	0
$941$	−2.24706	−0.0732521	−0.0366261	−	0.999329i	$-0.511661\pi$
$941$	−2.24706	−0.0732521	−0.0366261	+	0.999329i	$0.511661\pi$
$942$	7.58787i	0.247226i
$943$	39.2778i	1.27906i
$944$	15.6472	0.509273
$945$	0	0
$946$	−19.3680	−0.629709
$947$	− 6.75625i	− 0.219549i	−0.993957	−	0.109774i	$-0.964987\pi$
$947$	− 6.75625i	− 0.219549i	0.993957	−	0.109774i	$-0.0350128\pi$
$948$	81.8374i	2.65795i
$949$	2.61803	0.0849850
$950$	0	0
$951$	−50.1021	−1.62467
$952$	− 2.08700i	− 0.0676400i
$953$	− 59.9534i	− 1.94208i	−0.238918	−	0.971040i	$-0.576793\pi$
$953$	− 59.9534i	− 1.94208i	0.238918	−	0.971040i	$-0.423207\pi$
$954$	28.9225	0.936400
$955$	0	0
$956$	−16.5150	−0.534135
$957$	23.6747i	0.765293i
$958$	43.8459i	1.41660i
$959$	0.684206	0.0220942
$960$	0	0
$961$	−30.9814	−0.999399
$962$	− 15.1208i	− 0.487516i
$963$	8.83169i	0.284597i
$964$	−2.77057	−0.0892342
$965$	0	0
$966$	−20.6992	−0.665984
$967$	− 9.05599i	− 0.291221i	−0.989342	−	0.145610i	$-0.953485\pi$
$967$	− 9.05599i	− 0.291221i	0.989342	−	0.145610i	$-0.0465146\pi$
$968$	− 5.08580i	− 0.163464i
$969$	−16.4465	−0.528338
$970$	0	0
$971$	47.3508	1.51956	0.759780	−	0.650180i	$-0.225307\pi$
$971$	47.3508	1.51956	0.759780	+	0.650180i	$0.225307\pi$
$972$	39.4397i	1.26503i
$973$	16.4459i	0.527231i
$974$	58.2191	1.86546
$975$	0	0
$976$	8.81263	0.282085
$977$	− 4.74467i	− 0.151795i	−0.997116	−	0.0758977i	$-0.975818\pi$
$977$	− 4.74467i	− 0.151795i	0.997116	−	0.0758977i	$-0.0241822\pi$
$978$	− 4.09364i	− 0.130900i
$979$	−29.0160	−0.927357
$980$	0	0
$981$	29.8367	0.952613
$982$	− 31.1981i	− 0.995570i
$983$	− 18.5656i	− 0.592150i	−0.955165	−	0.296075i	$-0.904322\pi$
$983$	− 18.5656i	− 0.592150i	0.955165	−	0.296075i	$-0.0956778\pi$
$984$	−13.7898	−0.439601
$985$	0	0
$986$	32.4990	1.03498
$987$	− 21.6570i	− 0.689350i
$988$	− 20.4786i	− 0.651513i
$989$	21.1284	0.671843
$990$	0	0
$991$	−39.7199	−1.26174	−0.630871	−	0.775887i	$-0.717302\pi$
$991$	−39.7199	−1.26174	−0.630871	+	0.775887i	$0.717302\pi$
$992$	1.10432i	0.0350621i
$993$	6.51411i	0.206719i
$994$	19.9546	0.632921
$995$	0	0
$996$	9.14414	0.289743
$997$	30.2914i	0.959338i	0.877449	+	0.479669i	$0.159243\pi$
$997$	30.2914i	0.959338i	−0.877449	+	0.479669i	$0.840757\pi$
$998$	− 92.4309i	− 2.92585i
$999$	5.51210	0.174395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.b.c.624.2		8
5.2	odd	4		625.2.a.f.1.7		8
5.3	odd	4		625.2.a.f.1.2		8
5.4	even	2	inner	625.2.b.c.624.7		8
15.2	even	4		5625.2.a.x.1.2		8
15.8	even	4		5625.2.a.x.1.7		8
20.3	even	4		10000.2.a.bj.1.7		8
20.7	even	4		10000.2.a.bj.1.2		8
25.2	odd	20		125.2.d.b.101.4		16
25.3	odd	20		625.2.d.o.376.4		16
25.4	even	10		625.2.e.i.249.2		8
25.6	even	5		625.2.e.i.374.2		8
25.8	odd	20		625.2.d.o.251.4		16
25.9	even	10		25.2.e.a.19.1	yes	8
25.11	even	5		25.2.e.a.4.1	✓	8
25.12	odd	20		125.2.d.b.26.4		16
25.13	odd	20		125.2.d.b.26.1		16
25.14	even	10		125.2.e.b.24.2		8
25.16	even	5		125.2.e.b.99.2		8
25.17	odd	20		625.2.d.o.251.1		16
25.19	even	10		625.2.e.a.374.1		8
25.21	even	5		625.2.e.a.249.1		8
25.22	odd	20		625.2.d.o.376.1		16
25.23	odd	20		125.2.d.b.101.1		16
75.11	odd	10		225.2.m.a.154.2		8
75.59	odd	10		225.2.m.a.19.2		8
100.11	odd	10		400.2.y.c.129.1		8
100.59	odd	10		400.2.y.c.369.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.4.1	✓	8	25.11	even	5
25.2.e.a.19.1	yes	8	25.9	even	10
125.2.d.b.26.1		16	25.13	odd	20
125.2.d.b.26.4		16	25.12	odd	20
125.2.d.b.101.1		16	25.23	odd	20
125.2.d.b.101.4		16	25.2	odd	20
125.2.e.b.24.2		8	25.14	even	10
125.2.e.b.99.2		8	25.16	even	5
225.2.m.a.19.2		8	75.59	odd	10
225.2.m.a.154.2		8	75.11	odd	10
400.2.y.c.129.1		8	100.11	odd	10
400.2.y.c.369.1		8	100.59	odd	10
625.2.a.f.1.2		8	5.3	odd	4
625.2.a.f.1.7		8	5.2	odd	4
625.2.b.c.624.2		8	1.1	even	1	trivial
625.2.b.c.624.7		8	5.4	even	2	inner
625.2.d.o.251.1		16	25.17	odd	20
625.2.d.o.251.4		16	25.8	odd	20
625.2.d.o.376.1		16	25.22	odd	20
625.2.d.o.376.4		16	25.3	odd	20
625.2.e.a.249.1		8	25.21	even	5
625.2.e.a.374.1		8	25.19	even	10
625.2.e.i.249.2		8	25.4	even	10
625.2.e.i.374.2		8	25.6	even	5
5625.2.a.x.1.2		8	15.2	even	4
5625.2.a.x.1.7		8	15.8	even	4
10000.2.a.bj.1.2		8	20.7	even	4
10000.2.a.bj.1.7		8	20.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.08529i q^{2} +2.19849i q^{3} -2.34841 q^{4} +4.58448 q^{6} +0.992398i q^{7} +0.726543i q^{8} -1.83337 q^{9} +O(q^{10})\)	\(q-2.08529i q^{2} +2.19849i q^{3} -2.34841 q^{4} +4.58448 q^{6} +0.992398i q^{7} +0.726543i q^{8} -1.83337 q^{9} +2.00000 q^{11} -5.16297i q^{12} +3.37406i q^{13} +2.06943 q^{14} -3.18178 q^{16} +2.89451i q^{17} +3.82309i q^{18} +2.58448 q^{19} -2.18178 q^{21} -4.17057i q^{22} +4.54963i q^{23} -1.59730 q^{24} +7.03588 q^{26} +2.56484i q^{27} -2.33056i q^{28} +5.38430 q^{29} +0.136538 q^{31} +8.08800i q^{32} +4.39698i q^{33} +6.03588 q^{34} +4.30550 q^{36} -2.14910i q^{37} -5.38938i q^{38} -7.41785 q^{39} +8.63318 q^{41} +4.54963i q^{42} -4.64398i q^{43} -4.69683 q^{44} +9.48728 q^{46} +9.92630i q^{47} -6.99512i q^{48} +6.01515 q^{49} -6.36356 q^{51} -7.92369i q^{52} -7.56521i q^{53} +5.34841 q^{54} -0.721020 q^{56} +5.68196i q^{57} -11.2278i q^{58} -4.91775 q^{59} -2.76972 q^{61} -0.284720i q^{62} -1.81943i q^{63} +10.5022 q^{64} +9.16896 q^{66} +2.18577i q^{67} -6.79751i q^{68} -10.0023 q^{69} +9.64254 q^{71} -1.33202i q^{72} -0.775929i q^{73} -4.48150 q^{74} -6.06943 q^{76} +1.98480i q^{77} +15.4683i q^{78} -15.8508 q^{79} -11.1389 q^{81} -18.0026i q^{82} +1.77110i q^{83} +5.12372 q^{84} -9.68401 q^{86} +11.8373i q^{87} +1.45309i q^{88} -14.5080 q^{89} -3.34841 q^{91} -10.6844i q^{92} +0.300177i q^{93} +20.6992 q^{94} -17.7814 q^{96} -17.0291i q^{97} -12.5433i q^{98} -3.66673 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10}) \) 8 * q - 6 * q^4 + 6 * q^6 - 4 * q^9	\( 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9} + 16 q^{11} - 12 q^{14} - 2 q^{16} - 10 q^{19} + 6 q^{21} - 20 q^{24} + 6 q^{26} - 20 q^{29} + 16 q^{31} - 2 q^{34} - 12 q^{36} - 18 q^{39} + 26 q^{41} - 12 q^{44} + 6 q^{46} + 14 q^{49} - 4 q^{51} + 30 q^{54} + 10 q^{56} - 30 q^{59} + 6 q^{61} + 44 q^{64} + 12 q^{66} - 8 q^{69} + 46 q^{71} - 12 q^{74} - 20 q^{76} - 10 q^{79} - 32 q^{81} + 18 q^{84} - 14 q^{86} - 30 q^{89} - 14 q^{91} + 68 q^{94} - 54 q^{96} - 8 q^{99}+O(q^{100}) \) 8 * q - 6 * q^4 + 6 * q^6 - 4 * q^9 + 16 * q^11 - 12 * q^14 - 2 * q^16 - 10 * q^19 + 6 * q^21 - 20 * q^24 + 6 * q^26 - 20 * q^29 + 16 * q^31 - 2 * q^34 - 12 * q^36 - 18 * q^39 + 26 * q^41 - 12 * q^44 + 6 * q^46 + 14 * q^49 - 4 * q^51 + 30 * q^54 + 10 * q^56 - 30 * q^59 + 6 * q^61 + 44 * q^64 + 12 * q^66 - 8 * q^69 + 46 * q^71 - 12 * q^74 - 20 * q^76 - 10 * q^79 - 32 * q^81 + 18 * q^84 - 14 * q^86 - 30 * q^89 - 14 * q^91 + 68 * q^94 - 54 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more