LMFDB - Embedded newform 31.3.b.b.30.2

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			30.2
Root			$-2.36147$ 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)

$f(q)$	$=$	$q+1.57653 q^{2} -1.51454 q^{4} +5.50787 q^{5} -13.3839 q^{7} -8.69386 q^{8} +9.00000 q^{9} +O(q^{10})$	$q+1.57653 q^{2} -1.51454 q^{4} +5.50787 q^{5} -13.3839 q^{7} -8.69386 q^{8} +9.00000 q^{9} +8.68335 q^{10} -21.1001 q^{14} -7.64802 q^{16} +14.1888 q^{18} +37.0119 q^{19} -8.34188 q^{20} +5.33665 q^{25} +20.2704 q^{28} -31.0000 q^{31} +22.7181 q^{32} -73.7167 q^{35} -13.6308 q^{36} +58.3505 q^{38} -47.8847 q^{40} -76.3919 q^{41} +49.5708 q^{45} -30.0000 q^{47} +130.128 q^{49} +8.41341 q^{50} +116.358 q^{56} -13.3305 q^{59} -48.8726 q^{62} -120.455 q^{63} +66.4079 q^{64} +10.0000 q^{67} -116.217 q^{70} +131.471 q^{71} -78.2447 q^{72} -56.0559 q^{76} -42.1243 q^{80} +81.0000 q^{81} -120.435 q^{82} +78.1502 q^{90} -47.2960 q^{94} +203.857 q^{95} -89.0575 q^{97} +205.152 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.57653	0.788267	0.394134	−	0.919053i	$-0.371045\pi$
$2$	1.57653	0.788267	0.394134	+	0.919053i	$0.371045\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−1.51454	−0.378635
$5$	5.50787	1.10157	0.550787	−	0.834646i	$-0.314328\pi$
$5$	5.50787	1.10157	0.550787	+	0.834646i	$0.314328\pi$
$6$	0	0
$7$	−13.3839	−1.91198	−0.955991	−	0.293395i	$-0.905215\pi$
$7$	−13.3839	−1.91198	−0.955991	+	0.293395i	$0.905215\pi$
$8$	−8.69386	−1.08673
$9$	9.00000	1.00000
$10$	8.68335	0.868335
$11$	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			$\pi$
$14$	−21.1001	−1.50715
$15$	0	0
$16$	−7.64802	−0.478001
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	14.1888	0.788267
$19$	37.0119	1.94799	0.973997	−	0.226559i	$-0.0727477\pi$
$19$	37.0119	1.94799	0.973997	+	0.226559i	$0.0727477\pi$
$20$	−8.34188	−0.417094
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	5.33665	0.213466
$26$	0	0
$27$	0	0
$28$	20.2704	0.723943
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−31.0000	−1.00000
$31$	−31.0000	−1.00000
$32$	22.7181	0.709940
$33$	0	0
$34$	0	0
$35$	−73.7167	−2.10619
$36$	−13.6308	−0.378635
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	58.3505	1.53554
$39$	0	0
$40$	−47.8847	−1.19712
$41$	−76.3919	−1.86322	−0.931609	−	0.363462i	$-0.881595\pi$
$41$	−76.3919	−1.86322	−0.931609	+	0.363462i	$0.881595\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	49.5708	1.10157
$46$	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
$49$	130.128	2.65568
$50$	8.41341	0.168268
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	116.358	2.07781
$57$	0	0
$58$	0	0
$59$	−13.3305	−0.225941	−0.112971	−	0.993598i	$-0.536037\pi$
$59$	−13.3305	−0.225941	−0.112971	+	0.993598i	$0.536037\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−48.8726	−0.788267
$63$	−120.455	−1.91198
$64$	66.4079	1.03762
$65$	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
$70$	−116.217	−1.66024
$71$	131.471	1.85170	0.925850	−	0.377892i	$-0.123351\pi$
$71$	131.471	1.85170	0.925850	+	0.377892i	$0.123351\pi$
$72$	−78.2447	−1.08673
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	−56.0559	−0.737578
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−42.1243	−0.526554
$81$	81.0000	1.00000
$82$	−120.435	−1.46871
$83$	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			$\pi$
$90$	78.1502	0.868335
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−47.2960	−0.503149
$95$	203.857	2.14586
$96$	0	0
$97$	−89.0575	−0.918119	−0.459060	−	0.888406i	$-0.651814\pi$
$97$	−89.0575	−0.918119	−0.459060	+	0.888406i	$0.651814\pi$
$98$	205.152	2.09338
$99$	0	0
$100$	−8.08256	−0.0808256
$101$	−151.906	−1.50402	−0.752008	−	0.659154i	$-0.770915\pi$
$101$	−151.906	−1.50402	−0.752008	+	0.659154i	$0.770915\pi$
$102$	0	0
$103$	18.0135	0.174888	0.0874441	−	0.996169i	$-0.472130\pi$
$103$	18.0135	0.174888	0.0874441	+	0.996169i	$0.472130\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−183.516	−1.71511	−0.857553	−	0.514396i	$-0.828016\pi$
$107$	−183.516	−1.71511	−0.857553	+	0.514396i	$0.828016\pi$
$108$	0	0
$109$	5.61454	0.0515095	0.0257548	−	0.999668i	$-0.491801\pi$
$109$	5.61454	0.0515095	0.0257548	+	0.999668i	$0.491801\pi$
$110$	0	0
$111$	0	0
$112$	102.360	0.913930
$113$	225.983	1.99985	0.999924	−	0.0123608i	$-0.00393466\pi$
$113$	225.983	1.99985	0.999924	+	0.0123608i	$0.00393466\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−21.0161	−0.178102
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	46.9507	0.378635
$125$	−108.303	−0.866426
$126$	−189.901	−1.50715
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	13.8221	0.107985
$129$	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
$133$	−495.363	−3.72453
$134$	15.7653	0.117652
$135$	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			$\pi$
$140$	111.647	0.797477
$141$	0	0
$142$	207.268	1.45963
$143$	0	0
$144$	−68.8322	−0.478001
$145$	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			$\pi$
$152$	−321.776	−2.11695
$153$	0	0
$154$	0	0
$155$	−170.744	−1.10157
$156$	0	0
$157$	207.198	1.31973	0.659865	−	0.751384i	$-0.270613\pi$
$157$	207.198	1.31973	0.659865	+	0.751384i	$0.270613\pi$
$158$	0	0
$159$	0	0
$160$	125.128	0.782051
$161$	0	0
$162$	127.699	0.788267
$163$	269.992	1.65639	0.828197	−	0.560436i	$-0.189367\pi$
$163$	269.992	1.65639	0.828197	+	0.560436i	$0.189367\pi$
$164$	115.699	0.705479
$165$	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
$171$	333.107	1.94799
$172$	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
$175$	−71.4251	−0.408143
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−75.0769	−0.417094
$181$	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
$188$	45.4361	0.241682
$189$	0	0
$190$	321.387	1.69151
$191$	−202.515	−1.06029	−0.530143	−	0.847908i	$-0.677862\pi$
$191$	−202.515	−1.06029	−0.530143	+	0.847908i	$0.677862\pi$
$192$	0	0
$193$	−57.6602	−0.298757	−0.149379	−	0.988780i	$-0.547727\pi$
$193$	−57.6602	−0.298757	−0.149379	+	0.988780i	$0.547727\pi$
$194$	−140.402	−0.723723
$195$	0	0
$196$	−197.084	−1.00553
$197$	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			$\pi$
$200$	−46.3961	−0.231980
$201$	0	0
$202$	−239.484	−1.18557
$203$	0	0
$204$	0	0
$205$	−420.757	−2.05247
$206$	28.3989	0.137859
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	99.8066	0.473017	0.236509	−	0.971629i	$-0.423997\pi$
$211$	99.8066	0.473017	0.236509	+	0.971629i	$0.423997\pi$
$212$	0	0
$213$	0	0
$214$	−289.320	−1.35196
$215$	0	0
$216$	0	0
$217$	414.900	1.91198
$218$	8.85151	0.0406033
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−304.056	−1.35739
$225$	48.0298	0.213466
$226$	356.270	1.57641
$227$	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
$233$	−391.219	−1.67905	−0.839525	−	0.543320i	$-0.817167\pi$
$233$	−391.219	−1.67905	−0.839525	+	0.543320i	$0.817167\pi$
$234$	0	0
$235$	−165.236	−0.703133
$236$	20.1896	0.0855492
$237$	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			$\pi$
$242$	190.761	0.788267
$243$	0	0
$244$	0	0
$245$	716.729	2.92543
$246$	0	0
$247$	0	0
$248$	269.510	1.08673
$249$	0	0
$250$	−170.744	−0.682975
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	182.434	0.723943
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−243.841	−0.952503
$257$	81.3415	0.316504	0.158252	−	0.987399i	$-0.449414\pi$
$257$	81.3415	0.316504	0.158252	+	0.987399i	$0.449414\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	217.562	0.830389
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−780.957	−2.93593
$267$	0	0
$268$	−15.1454	−0.0565126
$269$	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
$280$	640.883	2.28887
$281$	−561.405	−1.99788	−0.998941	−	0.0460176i	$-0.985347\pi$
$281$	−561.405	−1.99788	−0.998941	+	0.0460176i	$0.985347\pi$
$282$	0	0
$283$	−550.000	−1.94346	−0.971731	−	0.236089i	$-0.924134\pi$
$283$	−550.000	−1.94346	−0.971731	+	0.236089i	$0.924134\pi$
$284$	−199.117	−0.701117
$285$	0	0
$286$	0	0
$287$	1022.42	3.56244
$288$	204.463	0.709940
$289$	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
$295$	−73.4229	−0.248891
$296$	0	0
$297$	0	0
$298$	−312.154	−1.04750
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−283.068	−0.931144
$305$	0	0
$306$	0	0
$307$	−435.229	−1.41768	−0.708841	−	0.705368i	$-0.750782\pi$
$307$	−435.229	−1.41768	−0.708841	+	0.705368i	$0.750782\pi$
$308$	0	0
$309$	0	0
$310$	−269.184	−0.868335
$311$	−170.584	−0.548502	−0.274251	−	0.961658i	$-0.588430\pi$
$311$	−170.584	−0.548502	−0.274251	+	0.961658i	$0.588430\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	326.654	1.04030
$315$	−663.450	−2.10619
$316$	0	0
$317$	−360.088	−1.13592	−0.567962	−	0.823054i	$-0.692268\pi$
$317$	−360.088	−1.13592	−0.567962	+	0.823054i	$0.692268\pi$
$318$	0	0
$319$	0	0
$320$	365.766	1.14302
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−122.678	−0.378635
$325$	0	0
$326$	425.652	1.30568
$327$	0	0
$328$	664.141	2.02482
$329$	401.516	1.22041
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	55.0787	0.164414
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	266.434	0.788267
$339$	0	0
$340$	0	0
$341$	0	0
$342$	525.155	1.53554
$343$	−1085.81	−3.16563
$344$	0	0
$345$	0	0
$346$	−236.480	−0.683469
$347$	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	$0.406545\pi$
$350$	−112.604	−0.321726
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	724.123	2.03978
$356$	0	0
$357$	0	0
$358$	0	0
$359$	225.663	0.628587	0.314294	−	0.949326i	$-0.398232\pi$
$359$	225.663	0.628587	0.314294	+	0.949326i	$0.398232\pi$
$360$	−430.962	−1.19712
$361$	1008.88	2.79468
$362$	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
$369$	−687.527	−1.86322
$370$	0	0
$371$	0	0
$372$	0	0
$373$	742.553	1.99076	0.995379	−	0.0960234i	$-0.0306124\pi$
$373$	742.553	1.99076	0.995379	+	0.0960234i	$0.0306124\pi$
$374$	0	0
$375$	0	0
$376$	260.816	0.693659
$377$	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	$0.656574\pi$
$380$	−308.749	−0.812497
$381$	0	0
$382$	−319.271	−0.835789
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−90.9033	−0.235501
$387$	0	0
$388$	134.881	0.347632
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1131.32	−2.88601
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−466.626	−1.17538	−0.587690	−	0.809086i	$-0.699963\pi$
$397$	−466.626	−1.17538	−0.587690	+	0.809086i	$0.699963\pi$
$398$	0	0
$399$	0	0
$400$	−40.8148	−0.102037
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	230.067	0.569472
$405$	446.138	1.10157
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−663.338	−1.61790
$411$	0	0
$412$	−27.2821	−0.0662187
$413$	178.414	0.431996
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−832.329	−1.98646	−0.993232	−	0.116145i	$-0.962946\pi$
$419$	−832.329	−1.98646	−0.993232	+	0.116145i	$0.962946\pi$
$420$	0	0
$421$	−737.603	−1.75203	−0.876013	−	0.482287i	$-0.839806\pi$
$421$	−737.603	−1.75203	−0.876013	+	0.482287i	$0.839806\pi$
$422$	157.349	0.372864
$423$	−270.000	−0.638298
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	277.942	0.649398
$429$	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			$\pi$
$434$	654.105	1.50715
$435$	0	0
$436$	−8.50343	−0.0195033
$437$	0	0
$438$	0	0
$439$	490.894	1.11821	0.559105	−	0.829097i	$-0.311145\pi$
$439$	490.894	1.11821	0.559105	+	0.829097i	$0.311145\pi$
$440$	0	0
$441$	1171.15	2.65568
$442$	0	0
$443$	855.797	1.93182	0.965910	−	0.258877i	$-0.0833523\pi$
$443$	855.797	1.93182	0.965910	+	0.258877i	$0.0833523\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−888.795	−1.98392
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	75.7207	0.168268
$451$	0	0
$452$	−342.259	−0.757211
$453$	0	0
$454$	520.256	1.14594
$455$	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
$466$	−616.770	−1.32354
$467$	679.225	1.45444	0.727221	−	0.686403i	$-0.240811\pi$
$467$	679.225	1.45444	0.727221	+	0.686403i	$0.240811\pi$
$468$	0	0
$469$	−133.839	−0.285371
$470$	−260.501	−0.554256
$471$	0	0
$472$	115.894	0.245538
$473$	0	0
$474$	0	0
$475$	197.520	0.415831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−108.323	−0.226143	−0.113072	−	0.993587i	$-0.536069\pi$
$479$	−108.323	−0.226143	−0.113072	+	0.993587i	$0.536069\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−183.259	−0.378635
$485$	−490.518	−1.01138
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	1129.95	2.30602
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	237.089	0.478001
$497$	−1759.59	−3.54042
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	164.029	0.328059
$501$	0	0
$502$	0	0
$503$	−485.411	−0.965032	−0.482516	−	0.875887i	$-0.660277\pi$
$503$	−485.411	−0.965032	−0.482516	+	0.875887i	$0.660277\pi$
$504$	1047.22	2.07781
$505$	−836.677	−1.65679
$506$	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
$512$	−439.712	−0.858812
$513$	0	0
$514$	128.238	0.249490
$515$	99.2159	0.192652
$516$	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			$\pi$
$524$	−209.006	−0.398867
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	−119.975	−0.225941
$532$	750.246	1.41024
$533$	0	0
$534$	0	0
$535$	−1010.78	−1.88932
$536$	−86.9386	−0.162199
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1076.32	1.98951	0.994755	−	0.102287i	$-0.0326160\pi$
$541$	1076.32	1.98951	0.994755	+	0.102287i	$0.0326160\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.9242	0.0567416
$546$	0	0
$547$	395.582	0.723184	0.361592	−	0.932336i	$-0.382233\pi$
$547$	395.582	0.723184	0.361592	+	0.932336i	$0.382233\pi$
$548$	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			$\pi$
$558$	−439.853	−0.788267
$559$	0	0
$560$	563.787	1.00676
$561$	0	0
$562$	−885.074	−1.57486
$563$	887.727	1.57678	0.788390	−	0.615176i	$-0.210915\pi$
$563$	887.727	1.57678	0.788390	+	0.615176i	$0.210915\pi$
$564$	0	0
$565$	1244.68	2.20298
$566$	−867.094	−1.53197
$567$	−1084.09	−1.91198
$568$	−1142.99	−2.01230
$569$	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
$574$	1611.88	2.80816
$575$	0	0
$576$	597.671	1.03762
$577$	−830.000	−1.43847	−0.719237	−	0.694764i	$-0.755509\pi$
$577$	−830.000	−1.43847	−0.719237	+	0.694764i	$0.755509\pi$
$578$	455.619	0.788267
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	141.888	0.242130
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−1147.37	−1.94799
$590$	−115.754	−0.196193
$591$	0	0
$592$	0	0
$593$	1120.65	1.88981	0.944903	−	0.327352i	$-0.106156\pi$
$593$	1120.65	1.88981	0.944903	+	0.327352i	$0.106156\pi$
$594$	0	0
$595$	0	0
$596$	299.879	0.503152
$597$	0	0
$598$	0	0
$599$	289.524	0.483346	0.241673	−	0.970358i	$-0.422304\pi$
$599$	289.524	0.483346	0.241673	+	0.970358i	$0.422304\pi$
$600$	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
$605$	666.452	1.10157
$606$	0	0
$607$	1090.00	1.79572	0.897858	−	0.440284i	$-0.145122\pi$
$607$	1090.00	1.79572	0.897858	+	0.440284i	$0.145122\pi$
$608$	840.839	1.38296
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−686.153	−1.11751
$615$	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			$\pi$
$620$	258.598	0.417094
$621$	0	0
$622$	−268.932	−0.432366
$623$	0	0
$624$	0	0
$625$	−729.936	−1.16790
$626$	0	0
$627$	0	0
$628$	−313.809	−0.499695
$629$	0	0
$630$	−1045.95	−1.66024
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−567.692	−0.895413
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1183.24	1.85170
$640$	76.1303	0.118954
$641$	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			$\pi$
$648$	−704.203	−1.08673
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−408.914	−0.627168
$653$	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
$655$	760.086	1.16044
$656$	584.247	0.890621
$657$	0	0
$658$	633.004	0.962013
$659$	−1317.34	−1.99900	−0.999500	−	0.0316115i	$-0.989936\pi$
$659$	−1317.34	−1.99900	−0.999500	+	0.0316115i	$0.989936\pi$
$660$	0	0
$661$	−278.562	−0.421425	−0.210712	−	0.977548i	$-0.567578\pi$
$661$	−278.562	−0.421425	−0.210712	+	0.977548i	$0.567578\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2728.39	−4.10285
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	86.8335	0.129602
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−255.957	−0.378635
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	1191.94	1.75543
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.86787	0.00712719	0.00356360	−	0.999994i	$-0.498866\pi$
$683$	4.86787	0.00712719	0.00356360	+	0.999994i	$0.498866\pi$
$684$	−504.503	−0.737578
$685$	0	0
$686$	−1711.82	−2.49536
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1053.18	−1.52413	−0.762067	−	0.647498i	$-0.775815\pi$
$691$	−1053.18	−1.52413	−0.762067	+	0.647498i	$0.775815\pi$
$692$	227.181	0.328296
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	318.460	0.456246
$699$	0	0
$700$	108.176	0.154537
$701$	1057.91	1.50915	0.754574	−	0.656215i	$-0.227843\pi$
$701$	1057.91	1.50915	0.754574	+	0.656215i	$0.227843\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2033.09	2.87565
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	1141.61	1.60790
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	355.765	0.495495
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−379.119	−0.526554
$721$	−241.090	−0.334383
$722$	1590.54	2.20296
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1328.62	1.82754	0.913771	−	0.406229i	$-0.133156\pi$
$727$	1328.62	1.82754	0.913771	+	0.406229i	$0.133156\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1286.16	−1.75465	−0.877324	−	0.479898i	$-0.840674\pi$
$733$	−1286.16	−1.75465	−0.877324	+	0.479898i	$0.840674\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	−1083.91	−1.46871
$739$	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
$745$	−1090.56	−1.46384
$746$	1170.66	1.56925
$747$	0	0
$748$	0	0
$749$	2456.16	3.27925
$750$	0	0
$751$	−907.309	−1.20813	−0.604067	−	0.796933i	$-0.706454\pi$
$751$	−907.309	−1.20813	−0.604067	+	0.796933i	$0.706454\pi$
$752$	229.441	0.305107
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−564.399	−0.744590
$759$	0	0
$760$	−1772.30	−2.33198
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	−75.1443	−0.0984853
$764$	306.716	0.401461
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1246.03	1.62033	0.810163	−	0.586205i	$-0.199379\pi$
$769$	1246.03	1.62033	0.810163	+	0.586205i	$0.199379\pi$
$770$	0	0
$771$	0	0
$772$	87.3286	0.113120
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−165.436	−0.213466
$776$	774.254	0.997750
$777$	0	0
$778$	0	0
$779$	−2827.41	−3.62954
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−995.223	−1.26942
$785$	1141.22	1.45378
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3024.53	−3.82367
$792$	0	0
$793$	0	0
$794$	−735.652	−0.926514
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	121.238	0.151548
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	1320.65	1.63446
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	703.351	0.868335
$811$	−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
$815$	1487.08	1.82464
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	637.253	0.777137
$821$	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			$\pi$
$824$	−156.607	−0.190057
$825$	0	0
$826$	281.276	0.340528
$827$	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
$838$	−1312.19	−1.56587
$839$	−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
$842$	−1162.86	−1.38107
$843$	0	0
$844$	−151.161	−0.179101
$845$	930.830	1.10157
$846$	−425.664	−0.503149
$847$	−1619.45	−1.91198
$848$	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
$855$	1834.71	2.14586
$856$	1595.47	1.86386
$857$	−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
$862$	1163.48	1.34975
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−826.181	−0.955122
$866$	0	0
$867$	0	0
$868$	−628.382	−0.723943
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−48.8120	−0.0559771
$873$	−801.518	−0.918119
$874$	0	0
$875$	1449.52	1.65659
$876$	0	0
$877$	1549.21	1.76648	0.883241	−	0.468919i	$-0.155356\pi$
$877$	1549.21	1.76648	0.883241	+	0.468919i	$0.155356\pi$
$878$	773.911	0.881448
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1846.36	2.09338
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	1349.19	1.52279
$887$	−1097.51	−1.23732	−0.618662	−	0.785657i	$-0.712325\pi$
$887$	−1097.51	−1.23732	−0.618662	+	0.785657i	$0.712325\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1110.36	−1.24340
$894$	0	0
$895$	0	0
$896$	−184.993	−0.206466
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−72.7430	−0.0808256
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−1964.66	−2.17330
$905$	0	0
$906$	0	0
$907$	491.374	0.541757	0.270879	−	0.962613i	$-0.412686\pi$
$907$	491.374	0.541757	0.270879	+	0.962613i	$0.412686\pi$
$908$	−499.798	−0.550438
$909$	−1367.15	−1.50402
$910$	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
$917$	−1846.98	−2.01415
$918$	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
$927$	162.121	0.174888
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	4816.29	5.17325
$932$	592.516	0.635747
$933$	0	0
$934$	1070.82	1.14649
$935$	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	$0.518695\pi$
$938$	−211.001	−0.224948
$939$	0	0
$940$	250.256	0.266230
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	101.952	0.108000
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	311.396	0.327786
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−1115.43	−1.16798
$956$	0	0
$957$	0	0
$958$	−170.774	−0.178261
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	−1651.65	−1.71511
$964$	0	0
$965$	−317.585	−0.329104
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−1051.96	−1.08673
$969$	0	0
$970$	−773.318	−0.797235
$971$	−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
$977$	508.559	0.520531	0.260266	−	0.965537i	$-0.416190\pi$
$977$	508.559	0.520531	0.260266	+	0.965537i	$0.416190\pi$
$978$	0	0
$979$	0	0
$980$	−1085.51	−1.10767
$981$	50.5308	0.0515095
$982$	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			$\pi$
$992$	−704.260	−0.709940
$993$	0	0
$994$	−2774.05	−2.79080
$995$	0	0
$996$	0	0
$997$	1876.06	1.88170	0.940851	−	0.338819i	$-0.110028\pi$
$997$	1876.06	1.88170	0.940851	+	0.338819i	$0.110028\pi$
$998$	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.30.2	✓	3
3.2	odd	2		279.3.d.c.154.2		3
4.3	odd	2		496.3.e.c.433.3		3
5.2	odd	4		775.3.c.b.774.4		6
5.3	odd	4		775.3.c.b.774.3		6
5.4	even	2		775.3.d.d.526.2		3
31.30	odd	2	CM	31.3.b.b.30.2	✓	3
93.92	even	2		279.3.d.c.154.2		3
124.123	even	2		496.3.e.c.433.3		3
155.92	even	4		775.3.c.b.774.4		6
155.123	even	4		775.3.c.b.774.3		6
155.154	odd	2		775.3.d.d.526.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.3.b.b.30.2	✓	3	1.1	even	1	trivial
31.3.b.b.30.2	✓	3	31.30	odd	2	CM
279.3.d.c.154.2		3	3.2	odd	2
279.3.d.c.154.2		3	93.92	even	2
496.3.e.c.433.3		3	4.3	odd	2
496.3.e.c.433.3		3	124.123	even	2
775.3.c.b.774.3		6	5.3	odd	4
775.3.c.b.774.3		6	155.123	even	4
775.3.c.b.774.4		6	5.2	odd	4
775.3.c.b.774.4		6	155.92	even	4
775.3.d.d.526.2		3	5.4	even	2
775.3.d.d.526.2		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)

\(f(q)\)	\(=\)	\(q+1.57653 q^{2} -1.51454 q^{4} +5.50787 q^{5} -13.3839 q^{7} -8.69386 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.57653 q^{2} -1.51454 q^{4} +5.50787 q^{5} -13.3839 q^{7} -8.69386 q^{8} +9.00000 q^{9} +8.68335 q^{10} -21.1001 q^{14} -7.64802 q^{16} +14.1888 q^{18} +37.0119 q^{19} -8.34188 q^{20} +5.33665 q^{25} +20.2704 q^{28} -31.0000 q^{31} +22.7181 q^{32} -73.7167 q^{35} -13.6308 q^{36} +58.3505 q^{38} -47.8847 q^{40} -76.3919 q^{41} +49.5708 q^{45} -30.0000 q^{47} +130.128 q^{49} +8.41341 q^{50} +116.358 q^{56} -13.3305 q^{59} -48.8726 q^{62} -120.455 q^{63} +66.4079 q^{64} +10.0000 q^{67} -116.217 q^{70} +131.471 q^{71} -78.2447 q^{72} -56.0559 q^{76} -42.1243 q^{80} +81.0000 q^{81} -120.435 q^{82} +78.1502 q^{90} -47.2960 q^{94} +203.857 q^{95} -89.0575 q^{97} +205.152 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{4} - 45 q^{8} + 27 q^{9}+O(q^{10}) \) 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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