LMFDB - Embedded newform 512.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [512,2,Mod(1,512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(512, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 512 = 2^{9} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	512.a (trivial)

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:	$4.08834058349$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{3} -2.65685 q^{9} +O(q^{10})$	$q+0.585786 q^{3} -2.65685 q^{9} +6.24264 q^{11} +5.65685 q^{17} +7.41421 q^{19} -5.00000 q^{25} -3.31371 q^{27} +3.65685 q^{33} -6.00000 q^{41} +13.0711 q^{43} -7.00000 q^{49} +3.31371 q^{51} +4.34315 q^{57} -14.2426 q^{59} -3.89949 q^{67} +16.9706 q^{73} -2.92893 q^{75} +6.02944 q^{81} -10.7279 q^{83} -5.65685 q^{89} -16.9706 q^{97} -16.5858 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 6 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 6 * q^9	$ 2 q + 4 q^{3} + 6 q^{9} + 4 q^{11} + 12 q^{19} - 10 q^{25} + 16 q^{27} - 4 q^{33} - 12 q^{41} + 12 q^{43} - 14 q^{49} - 16 q^{51} + 20 q^{57} - 20 q^{59} + 12 q^{67} - 20 q^{75} + 46 q^{81} + 4 q^{83} - 36 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 + 6 * q^9 + 4 * q^11 + 12 * q^19 - 10 * q^25 + 16 * q^27 - 4 * q^33 - 12 * q^41 + 12 * q^43 - 14 * q^49 - 16 * q^51 + 20 * q^57 - 20 * q^59 + 12 * q^67 - 20 * q^75 + 46 * q^81 + 4 * q^83 - 36 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	0.585786	0.338204	0.169102	−	0.985599i	$-0.445913\pi$
$3$	0.585786	0.338204	0.169102	+	0.985599i	$0.445913\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−2.65685	−0.885618
$10$	0	0
$11$	6.24264	1.88223	0.941113	−	0.338091i	$-0.109781\pi$
$11$	6.24264	1.88223	0.941113	+	0.338091i	$0.109781\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	0	0
$19$	7.41421	1.70094	0.850469	−	0.526026i	$-0.176318\pi$
$19$	7.41421	1.70094	0.850469	+	0.526026i	$0.176318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−3.31371	−0.637723
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	3.65685	0.636577
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	13.0711	1.99332	0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	13.0711	1.99332	0.996660	+	0.0816682i	$0.0260248\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	3.31371	0.464012
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.34315	0.575264
$58$	0	0
$59$	−14.2426	−1.85423	−0.927117	−	0.374772i	$-0.877721\pi$
$59$	−14.2426	−1.85423	−0.927117	+	0.374772i	$0.877721\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.89949	−0.476399	−0.238200	−	0.971216i	$-0.576557\pi$
$67$	−3.89949	−0.476399	−0.238200	+	0.971216i	$0.576557\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	16.9706	1.98625	0.993127	−	0.117041i	$-0.0373409\pi$
$73$	16.9706	1.98625	0.993127	+	0.117041i	$0.0373409\pi$
$74$	0	0
$75$	−2.92893	−0.338204
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	−10.7279	−1.17754	−0.588771	−	0.808300i	$-0.700388\pi$
$83$	−10.7279	−1.17754	−0.588771	+	0.808300i	$0.700388\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.65685	−0.599625	−0.299813	−	0.953998i	$-0.596924\pi$
$89$	−5.65685	−0.599625	−0.299813	+	0.953998i	$0.596924\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.9706	−1.72310	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	−16.9706	−1.72310	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	−16.5858	−1.66693
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.75736	−0.943280	−0.471640	−	0.881791i	$-0.656338\pi$
$107$	−9.75736	−0.943280	−0.471640	+	0.881791i	$0.656338\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	27.9706	2.54278
$122$	0	0
$123$	−3.51472	−0.316912
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	7.65685	0.674148
$130$	0	0
$131$	2.72792	0.238340	0.119170	−	0.992874i	$-0.461977\pi$
$131$	2.72792	0.238340	0.119170	+	0.992874i	$0.461977\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−9.55635	−0.810559	−0.405279	−	0.914193i	$-0.632826\pi$
$139$	−9.55635	−0.810559	−0.405279	+	0.914193i	$0.632826\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.10051	−0.338204
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−15.0294	−1.21506
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.5858	1.29910	0.649550	−	0.760319i	$-0.274958\pi$
$163$	16.5858	1.29910	0.649550	+	0.760319i	$0.274958\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−19.6985	−1.50638
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.34315	−0.627109
$178$	0	0
$179$	−26.7279	−1.99774	−0.998869	−	0.0475398i	$-0.984862\pi$
$179$	−26.7279	−1.99774	−0.998869	+	0.0475398i	$0.984862\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	35.3137	2.58239
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−16.9706	−1.22157	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	−16.9706	−1.22157	−0.610784	+	0.791797i	$0.709146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−2.28427	−0.161120
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	46.2843	3.20155
$210$	0	0
$211$	27.8995	1.92068	0.960340	−	0.278831i	$-0.0899469\pi$
$211$	27.8995	1.92068	0.960340	+	0.278831i	$0.0899469\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.94113	0.671759
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	13.2843	0.885618
$226$	0	0
$227$	23.2132	1.54071	0.770357	−	0.637613i	$-0.220078\pi$
$227$	23.2132	1.54071	0.770357	+	0.637613i	$0.220078\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.65685	−0.370593	−0.185296	−	0.982683i	$-0.559325\pi$
$233$	−5.65685	−0.370593	−0.185296	+	0.982683i	$0.559325\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−16.9706	−1.09317	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	−16.9706	−1.09317	−0.546585	+	0.837404i	$0.684072\pi$
$242$	0	0
$243$	13.4731	0.864299
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.28427	−0.398250
$250$	0	0
$251$	17.7574	1.12083	0.560417	−	0.828210i	$-0.310641\pi$
$251$	17.7574	1.12083	0.560417	+	0.828210i	$0.310641\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.31371	−0.202796
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−31.2132	−1.88223
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−28.2843	−1.68730	−0.843649	−	0.536895i	$-0.819597\pi$
$281$	−28.2843	−1.68730	−0.843649	+	0.536895i	$0.819597\pi$
$282$	0	0
$283$	33.5563	1.99472	0.997359	−	0.0726300i	$-0.0231392\pi$
$283$	33.5563	1.99472	0.997359	+	0.0726300i	$0.0231392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	−9.94113	−0.582759
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−20.6863	−1.20034
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	30.0416	1.71457	0.857283	−	0.514845i	$-0.172151\pi$
$307$	30.0416	1.71457	0.857283	+	0.514845i	$0.172151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−5.71573	−0.319021
$322$	0	0
$323$	41.9411	2.33367
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.384776	−0.0211492	−0.0105746	−	0.999944i	$-0.503366\pi$
$331$	−0.384776	−0.0211492	−0.0105746	+	0.999944i	$0.503366\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−10.5442	−0.572680
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.2426	−1.62351	−0.811755	−	0.583998i	$-0.801488\pi$
$347$	−30.2426	−1.62351	−0.811755	+	0.583998i	$0.801488\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	35.9706	1.89319
$362$	0	0
$363$	16.3848	0.859978
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	15.9411	0.829862
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−20.8701	−1.07202	−0.536011	−	0.844211i	$-0.680070\pi$
$379$	−20.8701	−1.07202	−0.536011	+	0.844211i	$0.680070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−34.7279	−1.76532
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.59798	0.0806074
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−39.5980	−1.97743	−0.988714	−	0.149813i	$-0.952133\pi$
$401$	−39.5980	−1.97743	−0.988714	+	0.149813i	$0.952133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	3.51472	0.173368
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.59798	−0.274134
$418$	0	0
$419$	−13.2721	−0.648383	−0.324192	−	0.945991i	$-0.605092\pi$
$419$	−13.2721	−0.648383	−0.324192	+	0.945991i	$0.605092\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−28.2843	−1.37199
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−16.9706	−0.815553	−0.407777	−	0.913082i	$-0.633696\pi$
$433$	−16.9706	−0.815553	−0.407777	+	0.913082i	$0.633696\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	18.5980	0.885618
$442$	0	0
$443$	−27.6985	−1.31599	−0.657997	−	0.753020i	$-0.728596\pi$
$443$	−27.6985	−1.31599	−0.657997	+	0.753020i	$0.728596\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.65685	0.266963	0.133482	−	0.991051i	$-0.457384\pi$
$449$	5.65685	0.266963	0.133482	+	0.991051i	$0.457384\pi$
$450$	0	0
$451$	−37.4558	−1.76373
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	0	0
$459$	−18.7452	−0.874949
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.786797	0.0364086	0.0182043	−	0.999834i	$-0.494205\pi$
$467$	0.786797	0.0364086	0.0182043	+	0.999834i	$0.494205\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	81.5980	3.75188
$474$	0	0
$475$	−37.0711	−1.70094
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	9.71573	0.439360
$490$	0	0
$491$	19.6985	0.888980	0.444490	−	0.895784i	$-0.353385\pi$
$491$	19.6985	0.888980	0.444490	+	0.895784i	$0.353385\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.1005	−0.899822	−0.449911	−	0.893073i	$-0.648544\pi$
$499$	−20.1005	−0.899822	−0.449911	+	0.893073i	$0.648544\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.61522	−0.338204
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−24.5685	−1.08473
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	44.8701	1.96203	0.981015	−	0.193930i	$-0.0621236\pi$
$523$	44.8701	1.96203	0.981015	+	0.193930i	$0.0621236\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	37.8406	1.64214
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−15.6569	−0.675643
$538$	0	0
$539$	−43.6985	−1.88223
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−26.5269	−1.13421	−0.567104	−	0.823646i	$-0.691936\pi$
$547$	−26.5269	−1.13421	−0.567104	+	0.823646i	$0.691936\pi$
$548$	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		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	20.6863	0.873376
$562$	0	0
$563$	−47.2132	−1.98980	−0.994900	−	0.100870i	$-0.967837\pi$
$563$	−47.2132	−1.98980	−0.994900	+	0.100870i	$0.967837\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−14.4437	−0.604448	−0.302224	−	0.953237i	$-0.597729\pi$
$571$	−14.4437	−0.604448	−0.302224	+	0.953237i	$0.597729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	−9.94113	−0.413139
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.2426	1.57844	0.789221	−	0.614109i	$-0.210484\pi$
$587$	38.2426	1.57844	0.789221	+	0.614109i	$0.210484\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	16.9706	0.692244	0.346122	−	0.938190i	$-0.387498\pi$
$601$	16.9706	0.692244	0.346122	+	0.938190i	$0.387498\pi$
$602$	0	0
$603$	10.3604	0.421908
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.5980	1.59415	0.797077	−	0.603877i	$-0.206378\pi$
$617$	39.5980	1.59415	0.797077	+	0.603877i	$0.206378\pi$
$618$	0	0
$619$	−48.3848	−1.94475	−0.972374	−	0.233428i	$-0.925006\pi$
$619$	−48.3848	−1.94475	−0.972374	+	0.233428i	$0.925006\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	27.1127	1.08278
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	16.3431	0.649582
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.2843	1.11716	0.558581	−	0.829450i	$-0.311346\pi$
$641$	28.2843	1.11716	0.558581	+	0.829450i	$0.311346\pi$
$642$	0	0
$643$	41.3553	1.63090	0.815448	−	0.578831i	$-0.196491\pi$
$643$	41.3553	1.63090	0.815448	+	0.578831i	$0.196491\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−88.9117	−3.49009
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−45.0883	−1.75906
$658$	0	0
$659$	21.2721	0.828643	0.414321	−	0.910131i	$-0.364019\pi$
$659$	21.2721	0.828643	0.414321	+	0.910131i	$0.364019\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	50.9117	1.96250	0.981251	−	0.192736i	$-0.0617360\pi$
$673$	50.9117	1.96250	0.981251	+	0.192736i	$0.0617360\pi$
$674$	0	0
$675$	16.5685	0.637723
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	13.5980	0.521076
$682$	0	0
$683$	51.6985	1.97819	0.989094	−	0.147287i	$-0.0470541\pi$
$683$	51.6985	1.97819	0.989094	+	0.147287i	$0.0470541\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	50.5269	1.92213	0.961067	−	0.276315i	$-0.0891133\pi$
$691$	50.5269	1.92213	0.961067	+	0.276315i	$0.0891133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−33.9411	−1.28561
$698$	0	0
$699$	−3.31371	−0.125336
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.94113	−0.369714
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−10.1960	−0.377628
$730$	0	0
$731$	73.9411	2.73481
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.3431	−0.896691
$738$	0	0
$739$	−54.0416	−1.98795	−0.993977	−	0.109591i	$-0.965046\pi$
$739$	−54.0416	−1.98795	−0.993977	+	0.109591i	$0.965046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	28.5025	1.04285
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	10.4020	0.379071
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	50.9117	1.83592	0.917961	−	0.396670i	$-0.129834\pi$
$769$	50.9117	1.83592	0.917961	+	0.396670i	$0.129834\pi$
$770$	0	0
$771$	17.5736	0.632897
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−44.4853	−1.59385
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−17.3553	−0.618651	−0.309326	−	0.950956i	$-0.600103\pi$
$787$	−17.3553	−0.618651	−0.309326	+	0.950956i	$0.600103\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	15.0294	0.531039
$802$	0	0
$803$	105.941	3.73858
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−3.12994	−0.109907	−0.0549536	−	0.998489i	$-0.517501\pi$
$811$	−3.12994	−0.109907	−0.0549536	+	0.998489i	$0.517501\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	96.9117	3.39051
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	−18.2843	−0.636577
$826$	0	0
$827$	24.1838	0.840952	0.420476	−	0.907304i	$-0.361863\pi$
$827$	24.1838	0.840952	0.420476	+	0.907304i	$0.361863\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−39.5980	−1.37199
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−16.5685	−0.570651
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	19.6569	0.674621
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	47.0122	1.60404	0.802018	−	0.597300i	$-0.203760\pi$
$859$	47.0122	1.60404	0.802018	+	0.597300i	$0.203760\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.78680	0.298415
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	45.0883	1.52601
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−40.5858	−1.36582	−0.682910	−	0.730502i	$-0.739286\pi$
$883$	−40.5858	−1.36582	−0.682910	+	0.730502i	$0.739286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	37.6396	1.26097
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−34.9289	−1.15980	−0.579898	−	0.814689i	$-0.696908\pi$
$907$	−34.9289	−1.15980	−0.579898	+	0.814689i	$0.696908\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−66.9706	−2.21640
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	17.5980	0.579873
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28.2843	0.927977	0.463988	−	0.885841i	$-0.346418\pi$
$929$	28.2843	0.927977	0.463988	+	0.885841i	$0.346418\pi$
$930$	0	0
$931$	−51.8995	−1.70094
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−50.9117	−1.66321	−0.831606	−	0.555366i	$-0.812578\pi$
$937$	−50.9117	−1.66321	−0.831606	+	0.555366i	$0.812578\pi$
$938$	0	0
$939$	5.85786	0.191164
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.7868	0.545498	0.272749	−	0.962085i	$-0.412067\pi$
$947$	16.7868	0.545498	0.272749	+	0.962085i	$0.412067\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	25.9239	0.835385
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	24.5685	0.789255
$970$	0	0
$971$	−16.1838	−0.519362	−0.259681	−	0.965694i	$-0.583617\pi$
$971$	−16.1838	−0.519362	−0.259681	+	0.965694i	$0.583617\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−62.2254	−1.99077	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	−62.2254	−1.99077	−0.995383	+	0.0959785i	$0.969402\pi$
$978$	0	0
$979$	−35.3137	−1.12863
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	−0.225397	−0.00715275
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	512.2.a.f.1.1	yes	2
3.2	odd	2		4608.2.a.i.1.1		2
4.3	odd	2		512.2.a.a.1.2	✓	2
8.3	odd	2	CM	512.2.a.f.1.1	yes	2
8.5	even	2		512.2.a.a.1.2	✓	2
12.11	even	2		4608.2.a.k.1.2		2
16.3	odd	4		512.2.b.c.257.2		4
16.5	even	4		512.2.b.c.257.2		4
16.11	odd	4		512.2.b.c.257.3		4
16.13	even	4		512.2.b.c.257.3		4
24.5	odd	2		4608.2.a.k.1.2		2
24.11	even	2		4608.2.a.i.1.1		2
32.3	odd	8		1024.2.e.o.257.1		4
32.5	even	8		1024.2.e.o.769.1		4
32.11	odd	8		1024.2.e.o.769.1		4
32.13	even	8		1024.2.e.o.257.1		4
32.19	odd	8		1024.2.e.g.257.2		4
32.21	even	8		1024.2.e.g.769.2		4
32.27	odd	8		1024.2.e.g.769.2		4
32.29	even	8		1024.2.e.g.257.2		4
48.5	odd	4		4608.2.d.k.2305.1		4
48.11	even	4		4608.2.d.k.2305.4		4
48.29	odd	4		4608.2.d.k.2305.4		4
48.35	even	4		4608.2.d.k.2305.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.a.1.2	✓	2	4.3	odd	2
512.2.a.a.1.2	✓	2	8.5	even	2
512.2.a.f.1.1	yes	2	1.1	even	1	trivial
512.2.a.f.1.1	yes	2	8.3	odd	2	CM
512.2.b.c.257.2		4	16.3	odd	4
512.2.b.c.257.2		4	16.5	even	4
512.2.b.c.257.3		4	16.11	odd	4
512.2.b.c.257.3		4	16.13	even	4
1024.2.e.g.257.2		4	32.19	odd	8
1024.2.e.g.257.2		4	32.29	even	8
1024.2.e.g.769.2		4	32.21	even	8
1024.2.e.g.769.2		4	32.27	odd	8
1024.2.e.o.257.1		4	32.3	odd	8
1024.2.e.o.257.1		4	32.13	even	8
1024.2.e.o.769.1		4	32.5	even	8
1024.2.e.o.769.1		4	32.11	odd	8
4608.2.a.i.1.1		2	3.2	odd	2
4608.2.a.i.1.1		2	24.11	even	2
4608.2.a.k.1.2		2	12.11	even	2
4608.2.a.k.1.2		2	24.5	odd	2
4608.2.d.k.2305.1		4	48.5	odd	4
4608.2.d.k.2305.1		4	48.35	even	4
4608.2.d.k.2305.4		4	48.11	even	4
4608.2.d.k.2305.4		4	48.29	odd	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:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.a (trivial)

\(f(q)\)	\(=\)	\(q+0.585786 q^{3} -2.65685 q^{9} +O(q^{10})\)	\(q+0.585786 q^{3} -2.65685 q^{9} +6.24264 q^{11} +5.65685 q^{17} +7.41421 q^{19} -5.00000 q^{25} -3.31371 q^{27} +3.65685 q^{33} -6.00000 q^{41} +13.0711 q^{43} -7.00000 q^{49} +3.31371 q^{51} +4.34315 q^{57} -14.2426 q^{59} -3.89949 q^{67} +16.9706 q^{73} -2.92893 q^{75} +6.02944 q^{81} -10.7279 q^{83} -5.65685 q^{89} -16.9706 q^{97} -16.5858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 6 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 6 * q^9	\( 2 q + 4 q^{3} + 6 q^{9} + 4 q^{11} + 12 q^{19} - 10 q^{25} + 16 q^{27} - 4 q^{33} - 12 q^{41} + 12 q^{43} - 14 q^{49} - 16 q^{51} + 20 q^{57} - 20 q^{59} + 12 q^{67} - 20 q^{75} + 46 q^{81} + 4 q^{83} - 36 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 + 6 * q^9 + 4 * q^11 + 12 * q^19 - 10 * q^25 + 16 * q^27 - 4 * q^33 - 12 * q^41 + 12 * q^43 - 14 * q^49 - 16 * q^51 + 20 * q^57 - 20 * q^59 + 12 * q^67 - 20 * q^75 + 46 * q^81 + 4 * q^83 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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