LMFDB - Embedded newform 31.3.b.b.30.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [31,3,Mod(30,31)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(31, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("31.30");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	31.b (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$0.844688819517$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			30.3
Root			$2.52892$ of defining polynomial
Character	$\chi$	$=$	31.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39543 q^{2} +1.73807 q^{4} -9.98218 q^{5} +10.2492 q^{7} -5.41830 q^{8} +9.00000 q^{9} +O(q^{10})$	$q+2.39543 q^{2} +1.73807 q^{4} -9.98218 q^{5} +10.2492 q^{7} -5.41830 q^{8} +9.00000 q^{9} -23.9116 q^{10} +24.5511 q^{14} -19.9314 q^{16} +21.5588 q^{18} -11.0501 q^{19} -17.3497 q^{20} +74.6439 q^{25} +17.8137 q^{28} -31.0000 q^{31} -26.0710 q^{32} -102.309 q^{35} +15.6426 q^{36} -26.4697 q^{38} +54.0864 q^{40} +12.3850 q^{41} -89.8396 q^{45} -30.0000 q^{47} +56.0454 q^{49} +178.804 q^{50} -55.5330 q^{56} +108.202 q^{59} -74.2582 q^{62} +92.2425 q^{63} +17.2744 q^{64} +10.0000 q^{67} -245.074 q^{70} -112.207 q^{71} -48.7647 q^{72} -19.2058 q^{76} +198.959 q^{80} +81.0000 q^{81} +29.6674 q^{82} -215.204 q^{90} -71.8628 q^{94} +110.304 q^{95} -104.731 q^{97} +134.253 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{4} - 45 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + 12 * q^4 - 45 * q^8 + 27 * q^9	$ 3 q + 12 q^{4} - 45 q^{8} + 27 q^{9} - 33 q^{10} - 9 q^{14} + 48 q^{16} + 27 q^{20} + 75 q^{25} + 75 q^{28} - 93 q^{31} - 180 q^{32} - 162 q^{35} + 108 q^{36} + 135 q^{38} - 132 q^{40} - 90 q^{47} + 147 q^{49} + 207 q^{50} - 36 q^{56} + 483 q^{64} + 30 q^{67} - 417 q^{70} - 405 q^{72} - 381 q^{76} + 495 q^{80} + 243 q^{81} - 345 q^{82} - 297 q^{90} + 198 q^{95} + 495 q^{98}+O(q^{100}) $ 3 * q + 12 * q^4 - 45 * q^8 + 27 * q^9 - 33 * q^10 - 9 * q^14 + 48 * q^16 + 27 * q^20 + 75 * q^25 + 75 * q^28 - 93 * q^31 - 180 * q^32 - 162 * q^35 + 108 * q^36 + 135 * q^38 - 132 * q^40 - 90 * q^47 + 147 * q^49 + 207 * q^50 - 36 * q^56 + 483 * q^64 + 30 * q^67 - 417 * q^70 - 405 * q^72 - 381 * q^76 + 495 * q^80 + 243 * q^81 - 345 * q^82 - 297 * q^90 + 198 * q^95 + 495 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	2.39543	1.19771	0.598857	−	0.800856i	$-0.295622\pi$
$2$	2.39543	1.19771	0.598857	+	0.800856i	$0.295622\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.73807	0.434516
$5$	−9.98218	−1.99644	−0.998218	−	0.0596728i	$-0.980994\pi$
$5$	−9.98218	−1.99644	−0.998218	+	0.0596728i	$0.980994\pi$
$6$	0	0
$7$	10.2492	1.46417	0.732083	−	0.681215i	$-0.238548\pi$
$7$	10.2492	1.46417	0.732083	+	0.681215i	$0.238548\pi$
$8$	−5.41830	−0.677287
$9$	9.00000	1.00000
$10$	−23.9116	−2.39116
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	24.5511	1.75365
$15$	0	0
$16$	−19.9314	−1.24571
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	21.5588	1.19771
$19$	−11.0501	−0.581585	−0.290793	−	0.956786i	$-0.593919\pi$
$19$	−11.0501	−0.581585	−0.290793	+	0.956786i	$0.593919\pi$
$20$	−17.3497	−0.867484
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	74.6439	2.98576
$26$	0	0
$27$	0	0
$28$	17.8137	0.636204
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−31.0000	−1.00000
$31$	−31.0000	−1.00000
$32$	−26.0710	−0.814718
$33$	0	0
$34$	0	0
$35$	−102.309	−2.92311
$36$	15.6426	0.434516
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−26.4697	−0.696572
$39$	0	0
$40$	54.0864	1.35216
$41$	12.3850	0.302074	0.151037	−	0.988528i	$-0.451739\pi$
$41$	12.3850	0.302074	0.151037	+	0.988528i	$0.451739\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−89.8396	−1.99644
$46$	0	0
$47$	−30.0000	−0.638298	−0.319149	−	0.947705i	$-0.603397\pi$
$47$	−30.0000	−0.638298	−0.319149	+	0.947705i	$0.603397\pi$
$48$	0	0
$49$	56.0454	1.14378
$50$	178.804	3.57608
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−55.5330	−0.991661
$57$	0	0
$58$	0	0
$59$	108.202	1.83393	0.916967	−	0.398964i	$-0.130630\pi$
$59$	108.202	1.83393	0.916967	+	0.398964i	$0.130630\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−74.2582	−1.19771
$63$	92.2425	1.46417
$64$	17.2744	0.269913
$65$	0	0
$66$	0	0
$67$	10.0000	0.149254	0.0746269	−	0.997212i	$-0.476223\pi$
$67$	10.0000	0.149254	0.0746269	+	0.997212i	$0.476223\pi$
$68$	0	0
$69$	0	0
$70$	−245.074	−3.50105
$71$	−112.207	−1.58038	−0.790189	−	0.612863i	$-0.790018\pi$
$71$	−112.207	−1.58038	−0.790189	+	0.612863i	$0.790018\pi$
$72$	−48.7647	−0.677287
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	−19.2058	−0.252708
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	198.959	2.48698
$81$	81.0000	1.00000
$82$	29.6674	0.361798
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−215.204	−2.39116
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−71.8628	−0.764498
$95$	110.304	1.16110
$96$	0	0
$97$	−104.731	−1.07970	−0.539852	−	0.841760i	$-0.681520\pi$
$97$	−104.731	−1.07970	−0.539852	+	0.841760i	$0.681520\pi$
$98$	134.253	1.36992
$99$	0	0
$100$	129.736	1.29736
$101$	191.263	1.89370	0.946848	−	0.321681i	$-0.104248\pi$
$101$	191.263	1.89370	0.946848	+	0.321681i	$0.104248\pi$
$102$	0	0
$103$	−186.725	−1.81286	−0.906430	−	0.422356i	$-0.861203\pi$
$103$	−186.725	−1.81286	−0.906430	+	0.422356i	$0.861203\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.57457	−0.0334072	−0.0167036	−	0.999860i	$-0.505317\pi$
$107$	−3.57457	−0.0334072	−0.0167036	+	0.999860i	$0.505317\pi$
$108$	0	0
$109$	185.924	1.70572	0.852861	−	0.522138i	$-0.174866\pi$
$109$	185.924	1.70572	0.852861	+	0.522138i	$0.174866\pi$
$110$	0	0
$111$	0	0
$112$	−204.280	−1.82393
$113$	−115.411	−1.02133	−0.510667	−	0.859779i	$-0.670601\pi$
$113$	−115.411	−1.02133	−0.510667	+	0.859779i	$0.670601\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	259.190	2.19653
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	−53.8800	−0.434516
$125$	−495.554	−3.96444
$126$	220.960	1.75365
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	145.664	1.13800
$129$	0	0
$130$	0	0
$131$	138.000	1.05344	0.526718	−	0.850040i	$-0.323423\pi$
$131$	138.000	1.05344	0.526718	+	0.850040i	$0.323423\pi$
$132$	0	0
$133$	−113.254	−0.851537
$134$	23.9543	0.178763
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−177.820	−1.27014
$141$	0	0
$142$	−268.783	−1.89284
$143$	0	0
$144$	−179.383	−1.24571
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−198.000	−1.32886	−0.664430	−	0.747351i	$-0.731325\pi$
$149$	−198.000	−1.32886	−0.664430	+	0.747351i	$0.731325\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	59.8728	0.393900
$153$	0	0
$154$	0	0
$155$	309.448	1.99644
$156$	0	0
$157$	100.727	0.641570	0.320785	−	0.947152i	$-0.396053\pi$
$157$	100.727	0.641570	0.320785	+	0.947152i	$0.396053\pi$
$158$	0	0
$159$	0	0
$160$	260.245	1.62653
$161$	0	0
$162$	194.030	1.19771
$163$	−293.221	−1.79890	−0.899451	−	0.437022i	$-0.856033\pi$
$163$	−293.221	−1.79890	−0.899451	+	0.437022i	$0.856033\pi$
$164$	21.5260	0.131256
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	−99.4510	−0.581585
$172$	0	0
$173$	−150.000	−0.867052	−0.433526	−	0.901141i	$-0.642731\pi$
$173$	−150.000	−0.867052	−0.433526	+	0.901141i	$0.642731\pi$
$174$	0	0
$175$	765.038	4.37164
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−156.147	−0.867484
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−52.1420	−0.277351
$189$	0	0
$190$	264.226	1.39066
$191$	−179.249	−0.938477	−0.469238	−	0.883072i	$-0.655471\pi$
$191$	−179.249	−0.938477	−0.469238	+	0.883072i	$0.655471\pi$
$192$	0	0
$193$	−301.705	−1.56324	−0.781619	−	0.623756i	$-0.785606\pi$
$193$	−301.705	−1.56324	−0.781619	+	0.623756i	$0.785606\pi$
$194$	−250.876	−1.29318
$195$	0	0
$196$	97.4105	0.496992
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−404.443	−2.02221
$201$	0	0
$202$	458.157	2.26810
$203$	0	0
$204$	0	0
$205$	−123.630	−0.603071
$206$	−447.285	−2.17129
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−404.998	−1.91942	−0.959710	−	0.280992i	$-0.909336\pi$
$211$	−404.998	−1.91942	−0.959710	+	0.280992i	$0.909336\pi$
$212$	0	0
$213$	0	0
$214$	−8.56261	−0.0400122
$215$	0	0
$216$	0	0
$217$	−317.724	−1.46417
$218$	445.366	2.04296
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−267.206	−1.19288
$225$	671.795	2.98576
$226$	−276.458	−1.22326
$227$	330.000	1.45374	0.726872	−	0.686773i	$-0.240973\pi$
$227$	330.000	1.45374	0.726872	+	0.686773i	$0.240973\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	414.876	1.78058	0.890292	−	0.455390i	$-0.150500\pi$
$233$	414.876	1.78058	0.890292	+	0.455390i	$0.150500\pi$
$234$	0	0
$235$	299.465	1.27432
$236$	188.062	0.796874
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	289.847	1.19771
$243$	0	0
$244$	0	0
$245$	−559.455	−2.28349
$246$	0	0
$247$	0	0
$248$	167.967	0.677287
$249$	0	0
$250$	−1187.06	−4.74826
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	160.323	0.636204
$253$	0	0
$254$	0	0
$255$	0	0
$256$	279.829	1.09308
$257$	398.857	1.55197	0.775986	−	0.630750i	$-0.217253\pi$
$257$	398.857	1.55197	0.775986	+	0.630750i	$0.217253\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	330.569	1.26171
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−271.293	−1.01990
$267$	0	0
$268$	17.3807	0.0648532
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−279.000	−1.00000
$280$	554.340	1.97979
$281$	303.099	1.07865	0.539323	−	0.842099i	$-0.318680\pi$
$281$	303.099	1.07865	0.539323	+	0.842099i	$0.318680\pi$
$282$	0	0
$283$	−550.000	−1.94346	−0.971731	−	0.236089i	$-0.924134\pi$
$283$	−550.000	−1.94346	−0.971731	+	0.236089i	$0.924134\pi$
$284$	−195.023	−0.686700
$285$	0	0
$286$	0	0
$287$	126.936	0.442287
$288$	−234.639	−0.814718
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	90.0000	0.307167	0.153584	−	0.988136i	$-0.450919\pi$
$293$	90.0000	0.307167	0.153584	+	0.988136i	$0.450919\pi$
$294$	0	0
$295$	−1080.09	−3.66133
$296$	0	0
$297$	0	0
$298$	−474.294	−1.59159
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	220.244	0.724487
$305$	0	0
$306$	0	0
$307$	592.686	1.93057	0.965287	−	0.261190i	$-0.0841151\pi$
$307$	592.686	1.93057	0.965287	+	0.261190i	$0.0841151\pi$
$308$	0	0
$309$	0	0
$310$	741.259	2.39116
$311$	603.306	1.93989	0.969946	−	0.243321i	$-0.0782368\pi$
$311$	603.306	1.93989	0.969946	+	0.243321i	$0.0782368\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	241.283	0.768417
$315$	−920.781	−2.92311
$316$	0	0
$317$	−271.862	−0.857610	−0.428805	−	0.903397i	$-0.641065\pi$
$317$	−271.862	−0.857610	−0.428805	+	0.903397i	$0.641065\pi$
$318$	0	0
$319$	0	0
$320$	−172.437	−0.538864
$321$	0	0
$322$	0	0
$323$	0	0
$324$	140.783	0.434516
$325$	0	0
$326$	−702.389	−2.15457
$327$	0	0
$328$	−67.1058	−0.204591
$329$	−307.475	−0.934574
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−99.8218	−0.297976
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	404.827	1.19771
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−238.228	−0.696572
$343$	72.2090	0.210522
$344$	0	0
$345$	0	0
$346$	−359.314	−1.03848
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	202.000	0.578797	0.289398	−	0.957209i	$-0.406545\pi$
$349$	202.000	0.578797	0.289398	+	0.957209i	$0.406545\pi$
$350$	1832.59	5.23598
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	1120.07	3.15512
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−703.128	−1.95857	−0.979287	−	0.202477i	$-0.935101\pi$
$359$	−703.128	−1.95857	−0.979287	+	0.202477i	$0.935101\pi$
$360$	486.778	1.35216
$361$	−238.895	−0.661759
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	111.465	0.302074
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−309.240	−0.829062	−0.414531	−	0.910035i	$-0.636054\pi$
$373$	−309.240	−0.829062	−0.414531	+	0.910035i	$0.636054\pi$
$374$	0	0
$375$	0	0
$376$	162.549	0.432311
$377$	0	0
$378$	0	0
$379$	−358.000	−0.944591	−0.472296	−	0.881440i	$-0.656574\pi$
$379$	−358.000	−0.944591	−0.472296	+	0.881440i	$0.656574\pi$
$380$	191.716	0.504516
$381$	0	0
$382$	−429.378	−1.12403
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−722.712	−1.87231
$387$	0	0
$388$	−182.030	−0.469149
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−303.670	−0.774669
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	789.660	1.98907	0.994534	−	0.104411i	$-0.0332959\pi$
$397$	789.660	1.98907	0.994534	+	0.104411i	$0.0332959\pi$
$398$	0	0
$399$	0	0
$400$	−1487.76	−3.71939
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	332.428	0.822842
$405$	−808.557	−1.99644
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−296.146	−0.722306
$411$	0	0
$412$	−324.540	−0.787718
$413$	1108.98	2.68518
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	331.874	0.792063	0.396031	−	0.918237i	$-0.370387\pi$
$419$	331.874	0.792063	0.396031	+	0.918237i	$0.370387\pi$
$420$	0	0
$421$	720.482	1.71136	0.855679	−	0.517506i	$-0.173140\pi$
$421$	720.482	1.71136	0.855679	+	0.517506i	$0.173140\pi$
$422$	−970.142	−2.29891
$423$	−270.000	−0.638298
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−6.21283	−0.0145160
$429$	0	0
$430$	0	0
$431$	738.000	1.71230	0.856148	−	0.516730i	$-0.172851\pi$
$431$	738.000	1.71230	0.856148	+	0.516730i	$0.172851\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−761.085	−1.75365
$435$	0	0
$436$	323.148	0.741164
$437$	0	0
$438$	0	0
$439$	384.974	0.876934	0.438467	−	0.898747i	$-0.355522\pi$
$439$	384.974	0.876934	0.438467	+	0.898747i	$0.355522\pi$
$440$	0	0
$441$	504.408	1.14378
$442$	0	0
$443$	−626.534	−1.41430	−0.707149	−	0.707065i	$-0.750019\pi$
$443$	−626.534	−1.41430	−0.707149	+	0.707065i	$0.750019\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	177.049	0.395198
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	1609.24	3.57608
$451$	0	0
$452$	−200.591	−0.443786
$453$	0	0
$454$	790.491	1.74117
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	993.805	2.13263
$467$	−894.822	−1.91611	−0.958053	−	0.286591i	$-0.907478\pi$
$467$	−894.822	−1.91611	−0.958053	+	0.286591i	$0.907478\pi$
$468$	0	0
$469$	102.492	0.218532
$470$	717.347	1.52627
$471$	0	0
$472$	−586.271	−1.24210
$473$	0	0
$474$	0	0
$475$	−824.824	−1.73647
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−770.170	−1.60787	−0.803936	−	0.594716i	$-0.797264\pi$
$479$	−770.170	−1.60787	−0.803936	+	0.594716i	$0.797264\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	210.306	0.434516
$485$	1045.45	2.15556
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−1340.13	−2.73496
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	617.873	1.24571
$497$	−1150.03	−2.31394
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−861.306	−1.72261
$501$	0	0
$502$	0	0
$503$	1005.80	1.99960	0.999799	−	0.0200726i	$-0.00638974\pi$
$503$	1005.80	1.99960	0.999799	+	0.0200726i	$0.00638974\pi$
$504$	−499.797	−0.991661
$505$	−1909.22	−3.78064
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	87.6544	0.171200
$513$	0	0
$514$	955.432	1.85882
$515$	1863.92	3.61926
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−942.000	−1.80806	−0.904031	−	0.427468i	$-0.859406\pi$
$521$	−942.000	−1.80806	−0.904031	+	0.427468i	$0.859406\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	239.853	0.457735
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	973.819	1.83393
$532$	−196.844	−0.370007
$533$	0	0
$534$	0	0
$535$	35.6820	0.0666953
$536$	−54.1830	−0.101088
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−634.009	−1.17192	−0.585961	−	0.810339i	$-0.699283\pi$
$541$	−634.009	−1.17192	−0.585961	+	0.810339i	$0.699283\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1855.92	−3.40536
$546$	0	0
$547$	−1081.12	−1.97645	−0.988223	−	0.153020i	$-0.951100\pi$
$547$	−1081.12	−1.97645	−0.988223	+	0.153020i	$0.951100\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−668.324	−1.19771
$559$	0	0
$560$	2039.16	3.64136
$561$	0	0
$562$	726.052	1.29191
$563$	156.021	0.277125	0.138563	−	0.990354i	$-0.455752\pi$
$563$	156.021	0.277125	0.138563	+	0.990354i	$0.455752\pi$
$564$	0	0
$565$	1152.05	2.03903
$566$	−1317.48	−2.32771
$567$	830.182	1.46417
$568$	607.970	1.07037
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	304.066	0.529732
$575$	0	0
$576$	155.470	0.269913
$577$	−830.000	−1.43847	−0.719237	−	0.694764i	$-0.755509\pi$
$577$	−830.000	−1.43847	−0.719237	+	0.694764i	$0.755509\pi$
$578$	692.278	1.19771
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	215.588	0.367898
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	342.554	0.581585
$590$	−2587.28	−4.38522
$591$	0	0
$592$	0	0
$593$	−224.102	−0.377913	−0.188956	−	0.981985i	$-0.560510\pi$
$593$	−224.102	−0.377913	−0.188956	+	0.981985i	$0.560510\pi$
$594$	0	0
$595$	0	0
$596$	−344.137	−0.577411
$597$	0	0
$598$	0	0
$599$	861.983	1.43904	0.719518	−	0.694474i	$-0.244363\pi$
$599$	861.983	1.43904	0.719518	+	0.694474i	$0.244363\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	90.0000	0.149254
$604$	0	0
$605$	−1207.84	−1.99644
$606$	0	0
$607$	1090.00	1.79572	0.897858	−	0.440284i	$-0.145122\pi$
$607$	1090.00	1.79572	0.897858	+	0.440284i	$0.145122\pi$
$608$	288.087	0.473828
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1419.74	2.31227
$615$	0	0
$616$	0	0
$617$	−750.000	−1.21556	−0.607780	−	0.794106i	$-0.707940\pi$
$617$	−750.000	−1.21556	−0.607780	+	0.794106i	$0.707940\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	537.840	0.867484
$621$	0	0
$622$	1445.18	2.32343
$623$	0	0
$624$	0	0
$625$	3080.62	4.92899
$626$	0	0
$627$	0	0
$628$	175.069	0.278773
$629$	0	0
$630$	−2205.66	−3.50105
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−651.226	−1.02717
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1009.86	−1.58038
$640$	−1454.04	−2.27194
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−438.882	−0.677287
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−509.637	−0.781652
$653$	810.000	1.24043	0.620214	−	0.784432i	$-0.287046\pi$
$653$	810.000	1.24043	0.620214	+	0.784432i	$0.287046\pi$
$654$	0	0
$655$	−1377.54	−2.10312
$656$	−246.851	−0.376297
$657$	0	0
$658$	−736.533	−1.11935
$659$	622.589	0.944747	0.472374	−	0.881398i	$-0.343397\pi$
$659$	622.589	0.944747	0.472374	+	0.881398i	$0.343397\pi$
$660$	0	0
$661$	−979.900	−1.48245	−0.741225	−	0.671256i	$-0.765755\pi$
$661$	−979.900	−1.48245	−0.741225	+	0.671256i	$0.765755\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1130.53	1.70004
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	−239.116	−0.356889
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	293.733	0.434516
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	−1073.41	−1.58087
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1185.42	−1.73560	−0.867802	−	0.496911i	$-0.834468\pi$
$683$	−1185.42	−1.73560	−0.867802	+	0.496911i	$0.834468\pi$
$684$	−172.852	−0.252708
$685$	0	0
$686$	172.971	0.252145
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−248.368	−0.359432	−0.179716	−	0.983719i	$-0.557518\pi$
$691$	−248.368	−0.359432	−0.179716	+	0.983719i	$0.557518\pi$
$692$	−260.710	−0.376748
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	483.876	0.693232
$699$	0	0
$700$	1329.69	1.89955
$701$	267.798	0.382023	0.191011	−	0.981588i	$-0.438823\pi$
$701$	267.798	0.382023	0.191011	+	0.981588i	$0.438823\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1960.29	2.77269
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	2683.04	3.77893
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−1684.29	−2.34581
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	1790.63	2.48698
$721$	−1913.77	−2.65433
$722$	−572.255	−0.792597
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−152.788	−0.210163	−0.105081	−	0.994464i	$-0.533510\pi$
$727$	−152.788	−0.210163	−0.105081	+	0.994464i	$0.533510\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.8032	0.0461162	0.0230581	−	0.999734i	$-0.492660\pi$
$733$	33.8032	0.0461162	0.0230581	+	0.999734i	$0.492660\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	267.007	0.361798
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	1976.47	2.65298
$746$	−740.762	−0.992978
$747$	0	0
$748$	0	0
$749$	−36.6363	−0.0489137
$750$	0	0
$751$	1490.28	1.98440	0.992198	−	0.124671i	$-0.0397875\pi$
$751$	1490.28	1.98440	0.992198	+	0.124671i	$0.0397875\pi$
$752$	597.942	0.795135
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−857.563	−1.13135
$759$	0	0
$760$	−597.661	−0.786396
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	1905.56	2.49746
$764$	−311.547	−0.407784
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1403.81	−1.82550	−0.912750	−	0.408519i	$-0.866045\pi$
$769$	−1403.81	−1.82550	−0.912750	+	0.408519i	$0.866045\pi$
$770$	0	0
$771$	0	0
$772$	−524.383	−0.679253
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−2313.96	−2.98576
$776$	567.465	0.731269
$777$	0	0
$778$	0	0
$779$	−136.856	−0.175682
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1117.06	−1.42482
$785$	−1005.47	−1.28085
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1182.86	−1.49540
$792$	0	0
$793$	0	0
$794$	1891.57	2.38233
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1946.04	−2.43255
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−1036.32	−1.28258
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−1936.84	−2.39116
$811$	−1478.00	−1.82244	−0.911221	−	0.411918i	$-0.864859\pi$
$811$	−1478.00	−1.82244	−0.911221	+	0.411918i	$0.864859\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2926.98	3.59139
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−214.876	−0.262044
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	1011.73	1.22783
$825$	0	0
$826$	2656.48	3.21608
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	794.980	0.948664
$839$	−1422.00	−1.69487	−0.847437	−	0.530895i	$-0.821856\pi$
$839$	−1422.00	−1.69487	−0.847437	+	0.530895i	$0.821856\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	1725.86	2.04972
$843$	0	0
$844$	−703.913	−0.834020
$845$	−1686.99	−1.99644
$846$	−646.765	−0.764498
$847$	1240.15	1.46417
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1210.00	1.41852	0.709261	−	0.704946i	$-0.249029\pi$
$853$	1210.00	1.41852	0.709261	+	0.704946i	$0.249029\pi$
$854$	0	0
$855$	992.738	1.16110
$856$	19.3681	0.0226262
$857$	−270.000	−0.315053	−0.157526	−	0.987515i	$-0.550352\pi$
$857$	−270.000	−0.315053	−0.157526	+	0.987515i	$0.550352\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	1767.82	2.05084
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	1497.33	1.73101
$866$	0	0
$867$	0	0
$868$	−552.225	−0.636204
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−1007.39	−1.15526
$873$	−942.582	−1.07970
$874$	0	0
$875$	−5079.02	−5.80459
$876$	0	0
$877$	−62.3110	−0.0710502	−0.0355251	−	0.999369i	$-0.511310\pi$
$877$	−62.3110	−0.0710502	−0.0355251	+	0.999369i	$0.511310\pi$
$878$	922.176	1.05031
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1208.27	1.36992
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−1500.82	−1.69392
$887$	−658.275	−0.742136	−0.371068	−	0.928606i	$-0.621008\pi$
$887$	−658.275	−0.742136	−0.371068	+	0.928606i	$0.621008\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	331.503	0.371224
$894$	0	0
$895$	0	0
$896$	1492.93	1.66622
$897$	0	0
$898$	0	0
$899$	0	0
$900$	1167.62	1.29736
$901$	0	0
$902$	0	0
$903$	0	0
$904$	625.329	0.691736
$905$	0	0
$906$	0	0
$907$	1266.55	1.39642	0.698208	−	0.715895i	$-0.253981\pi$
$907$	1266.55	1.39642	0.698208	+	0.715895i	$0.253981\pi$
$908$	573.562	0.631676
$909$	1721.37	1.89370
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1414.38	1.54240
$918$	0	0
$919$	−1262.00	−1.37323	−0.686616	−	0.727020i	$-0.740905\pi$
$919$	−1262.00	−1.37323	−0.686616	+	0.727020i	$0.740905\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1680.52	−1.81286
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−619.308	−0.665207
$932$	721.082	0.773693
$933$	0	0
$934$	−2143.48	−2.29495
$935$	0	0
$936$	0	0
$937$	−110.000	−0.117396	−0.0586980	−	0.998276i	$-0.518695\pi$
$937$	−110.000	−0.117396	−0.0586980	+	0.998276i	$0.518695\pi$
$938$	245.511	0.261739
$939$	0	0
$940$	520.491	0.553713
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−2156.62	−2.28455
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−1975.80	−2.07979
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	1789.30	1.87361
$956$	0	0
$957$	0	0
$958$	−1844.89	−1.92577
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	−32.1711	−0.0334072
$964$	0	0
$965$	3011.67	3.12091
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−655.614	−0.677287
$969$	0	0
$970$	2504.29	2.58174
$971$	−1158.00	−1.19258	−0.596292	−	0.802767i	$-0.703360\pi$
$971$	−1158.00	−1.19258	−0.596292	+	0.802767i	$0.703360\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1888.17	−1.93262	−0.966312	−	0.257372i	$-0.917143\pi$
$977$	−1888.17	−1.93262	−0.966312	+	0.257372i	$0.917143\pi$
$978$	0	0
$979$	0	0
$980$	−972.369	−0.992214
$981$	1673.31	1.70572
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	808.201	0.814718
$993$	0	0
$994$	−2754.80	−2.77143
$995$	0	0
$996$	0	0
$997$	−1523.12	−1.52770	−0.763852	−	0.645392i	$-0.776694\pi$
$997$	−1523.12	−1.52770	−0.763852	+	0.645392i	$0.776694\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.3.b.b.30.3	✓	3
3.2	odd	2		279.3.d.c.154.1		3
4.3	odd	2		496.3.e.c.433.1		3
5.2	odd	4		775.3.c.b.774.5		6
5.3	odd	4		775.3.c.b.774.2		6
5.4	even	2		775.3.d.d.526.1		3
31.30	odd	2	CM	31.3.b.b.30.3	✓	3
93.92	even	2		279.3.d.c.154.1		3
124.123	even	2		496.3.e.c.433.1		3
155.92	even	4		775.3.c.b.774.5		6
155.123	even	4		775.3.c.b.774.2		6
155.154	odd	2		775.3.d.d.526.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.3.b.b.30.3	✓	3	1.1	even	1	trivial
31.3.b.b.30.3	✓	3	31.30	odd	2	CM
279.3.d.c.154.1		3	3.2	odd	2
279.3.d.c.154.1		3	93.92	even	2
496.3.e.c.433.1		3	4.3	odd	2
496.3.e.c.433.1		3	124.123	even	2
775.3.c.b.774.2		6	5.3	odd	4
775.3.c.b.774.2		6	155.123	even	4
775.3.c.b.774.5		6	5.2	odd	4
775.3.c.b.774.5		6	155.92	even	4
775.3.d.d.526.1		3	5.4	even	2
775.3.d.d.526.1		3	155.154	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	31.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.39543 q^{2} +1.73807 q^{4} -9.98218 q^{5} +10.2492 q^{7} -5.41830 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.39543 q^{2} +1.73807 q^{4} -9.98218 q^{5} +10.2492 q^{7} -5.41830 q^{8} +9.00000 q^{9} -23.9116 q^{10} +24.5511 q^{14} -19.9314 q^{16} +21.5588 q^{18} -11.0501 q^{19} -17.3497 q^{20} +74.6439 q^{25} +17.8137 q^{28} -31.0000 q^{31} -26.0710 q^{32} -102.309 q^{35} +15.6426 q^{36} -26.4697 q^{38} +54.0864 q^{40} +12.3850 q^{41} -89.8396 q^{45} -30.0000 q^{47} +56.0454 q^{49} +178.804 q^{50} -55.5330 q^{56} +108.202 q^{59} -74.2582 q^{62} +92.2425 q^{63} +17.2744 q^{64} +10.0000 q^{67} -245.074 q^{70} -112.207 q^{71} -48.7647 q^{72} -19.2058 q^{76} +198.959 q^{80} +81.0000 q^{81} +29.6674 q^{82} -215.204 q^{90} -71.8628 q^{94} +110.304 q^{95} -104.731 q^{97} +134.253 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{4} - 45 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + 12 * q^4 - 45 * q^8 + 27 * q^9	\( 3 q + 12 q^{4} - 45 q^{8} + 27 q^{9} - 33 q^{10} - 9 q^{14} + 48 q^{16} + 27 q^{20} + 75 q^{25} + 75 q^{28} - 93 q^{31} - 180 q^{32} - 162 q^{35} + 108 q^{36} + 135 q^{38} - 132 q^{40} - 90 q^{47} + 147 q^{49} + 207 q^{50} - 36 q^{56} + 483 q^{64} + 30 q^{67} - 417 q^{70} - 405 q^{72} - 381 q^{76} + 495 q^{80} + 243 q^{81} - 345 q^{82} - 297 q^{90} + 198 q^{95} + 495 q^{98}+O(q^{100}) \) 3 * q + 12 * q^4 - 45 * q^8 + 27 * q^9 - 33 * q^10 - 9 * q^14 + 48 * q^16 + 27 * q^20 + 75 * q^25 + 75 * q^28 - 93 * q^31 - 180 * q^32 - 162 * q^35 + 108 * q^36 + 135 * q^38 - 132 * q^40 - 90 * q^47 + 147 * q^49 + 207 * q^50 - 36 * q^56 + 483 * q^64 + 30 * q^67 - 417 * q^70 - 405 * q^72 - 381 * q^76 + 495 * q^80 + 243 * q^81 - 345 * q^82 - 297 * q^90 + 198 * q^95 + 495 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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