LMFDB - Embedded newform 688.2.g.d.687.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(687,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("688.687");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.g (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:	no
Analytic conductor:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-43})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} - 11x + 121 $ x^4 - x^3 - 10x^2 - 11x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{43}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			687.1
Root			$-2.58945 - 2.07237i$ of defining polynomial
Character	$\chi$	$=$	688.687
Dual form			688.2.g.d.687.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{9} -4.14474i q^{11} -7.17891 q^{13} -3.17891 q^{17} +2.78346i q^{23} +5.00000 q^{25} -11.0729i q^{31} -11.1789 q^{41} -6.55744i q^{43} +13.1149i q^{47} -7.00000 q^{49} +0.821092 q^{53} -13.1149i q^{59} +9.71166i q^{67} +13.1149i q^{79} +9.00000 q^{81} -18.0012i q^{83} +17.5367 q^{97} +12.4342i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 6 q^{13} + 10 q^{17} + 20 q^{25} - 22 q^{41} - 28 q^{49} + 26 q^{53} + 36 q^{81} + 2 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 - 6 * q^13 + 10 * q^17 + 20 * q^25 - 22 * q^41 - 28 * q^49 + 26 * q^53 + 36 * q^81 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	− 4.14474i	− 1.24969i	−0.780750	−	0.624844i	$-0.785163\pi$
$11$	− 4.14474i	− 1.24969i	0.780750	−	0.624844i	$-0.214837\pi$
$12$	0	0
$13$	−7.17891	−1.99107	−0.995535	−	0.0943882i	$-0.969911\pi$
$13$	−7.17891	−1.99107	−0.995535	+	0.0943882i	$0.969911\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.17891	−0.770999	−0.385499	−	0.922708i	$-0.625971\pi$
$17$	−3.17891	−0.770999	−0.385499	+	0.922708i	$0.625971\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.78346i	0.580391i	0.956967	+	0.290196i	$0.0937204\pi$
$23$	2.78346i	0.580391i	−0.956967	+	0.290196i	$0.906280\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 11.0729i	− 1.98876i	−0.105869	−	0.994380i	$-0.533762\pi$
$31$	− 11.0729i	− 1.98876i	0.105869	−	0.994380i	$-0.466238\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.1789	−1.74585	−0.872926	−	0.487852i	$-0.837780\pi$
$41$	−11.1789	−1.74585	−0.872926	+	0.487852i	$0.837780\pi$
$42$	0	0
$43$	− 6.55744i	− 1.00000i
$43$	− 6.55744i	− 1.00000i
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.1149i	1.91300i	0.291730	+	0.956501i	$0.405769\pi$
$47$	13.1149i	1.91300i	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.821092	0.112786	0.0563928	−	0.998409i	$-0.482040\pi$
$53$	0.821092	0.112786	0.0563928	+	0.998409i	$0.482040\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 13.1149i	− 1.70741i	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	− 13.1149i	− 1.70741i	0.520756	−	0.853706i	$-0.325650\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.71166i	1.18647i	0.805030	+	0.593234i	$0.202149\pi$
$67$	9.71166i	1.18647i	−0.805030	+	0.593234i	$0.797851\pi$
$68$	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
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.1149i	1.47554i	0.675053	+	0.737769i	$0.264121\pi$
$79$	13.1149i	1.47554i	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 18.0012i	− 1.97588i	−0.154828	−	0.987942i	$-0.549482\pi$
$83$	− 18.0012i	− 1.97588i	0.154828	−	0.987942i	$-0.450518\pi$
$84$	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			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.5367	1.78058	0.890292	−	0.455389i	$-0.150500\pi$
$97$	17.5367	1.78058	0.890292	+	0.455389i	$0.150500\pi$
$98$	0	0
$99$	12.4342i	1.24969i
$100$	0	0
$101$	−15.1789	−1.51036	−0.755179	−	0.655519i	$-0.772450\pi$
$101$	−15.1789	−1.51036	−0.755179	+	0.655519i	$0.772450\pi$
$102$	0	0
$103$	5.50603i	0.542525i	0.962505	+	0.271263i	$0.0874412\pi$
$103$	5.50603i	0.542525i	−0.962505	+	0.271263i	$0.912559\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.1149i	− 1.26786i	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	− 13.1149i	− 1.26786i	0.773389	−	0.633932i	$-0.218560\pi$
$108$	0	0
$109$	13.5367	1.29658	0.648292	−	0.761392i	$-0.275484\pi$
$109$	13.5367	1.29658	0.648292	+	0.761392i	$0.275484\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	21.5367	1.99107
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.17891	−0.561719
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.3624i	1.71814i	0.511859	+	0.859069i	$0.328957\pi$
$127$	19.3624i	1.71814i	−0.511859	+	0.859069i	$0.671043\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	− 1.42217i	− 0.120627i	−0.998179	−	0.0603135i	$-0.980790\pi$
$139$	− 1.42217i	− 0.120627i	0.998179	−	0.0603135i	$-0.0192101\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	29.7547i	2.48822i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	9.53673	0.770999
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 24.9294i	− 1.92909i	−0.263916	−	0.964546i	$-0.585014\pi$
$167$	− 24.9294i	− 1.92909i	0.263916	−	0.964546i	$-0.414986\pi$
$168$	0	0
$169$	38.5367	2.96436
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.1758i	0.963507i
$188$	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
$193$	−19.1789	−1.38053	−0.690264	−	0.723558i	$-0.742506\pi$
$193$	−19.1789	−1.38053	−0.690264	+	0.723558i	$0.742506\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.35038i	− 0.580391i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	22.8211	1.53511
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	5.53673	0.365877	0.182939	−	0.983124i	$-0.441439\pi$
$229$	5.53673	0.365877	0.182939	+	0.983124i	$0.441439\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.1149i	0.848331i	0.905585	+	0.424165i	$0.139432\pi$
$239$	13.1149i	0.848331i	−0.905585	+	0.424165i	$0.860568\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.5681i	1.48760i	0.668400	+	0.743802i	$0.266979\pi$
$251$	23.5681i	1.48760i	−0.668400	+	0.743802i	$0.733021\pi$
$252$	0	0
$253$	11.5367	0.725308
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	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
$267$	0	0
$268$	0	0
$269$	−27.8945	−1.70076	−0.850380	−	0.526169i	$-0.823628\pi$
$269$	−27.8945	−1.70076	−0.850380	+	0.526169i	$0.823628\pi$
$270$	0	0
$271$	− 13.7955i	− 0.838018i	−0.907982	−	0.419009i	$-0.862378\pi$
$271$	− 13.7955i	− 0.838018i	0.907982	−	0.419009i	$-0.137622\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 20.7237i	− 1.24969i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	33.2188i	1.98876i
$280$	0	0
$281$	−31.8945	−1.90267	−0.951334	−	0.308160i	$-0.900287\pi$
$281$	−31.8945	−1.90267	−0.951334	+	0.308160i	$0.900287\pi$
$282$	0	0
$283$	− 31.8576i	− 1.89374i	−0.321624	−	0.946868i	$-0.604229\pi$
$283$	− 31.8576i	− 1.89374i	0.321624	−	0.946868i	$-0.395771\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.89454	−0.405561
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 19.9822i	− 1.15560i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.2786i	− 0.871995i	−0.899948	−	0.435997i	$-0.856396\pi$
$307$	− 15.2786i	− 0.871995i	0.899948	−	0.435997i	$-0.143604\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 27.6519i	− 1.56800i	−0.620763	−	0.783998i	$-0.713177\pi$
$311$	− 27.6519i	− 1.56800i	0.620763	−	0.783998i	$-0.286823\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.1789	−1.30186	−0.650929	−	0.759139i	$-0.725620\pi$
$317$	−23.1789	−1.30186	−0.650929	+	0.759139i	$0.725620\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−35.8945	−1.99107
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.53673	0.0837107	0.0418554	−	0.999124i	$-0.486673\pi$
$337$	1.53673	0.0837107	0.0418554	+	0.999124i	$0.486673\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−45.8945	−2.48533
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.8211	0.682398	0.341199	−	0.939991i	$-0.389167\pi$
$353$	12.8211	0.682398	0.341199	+	0.939991i	$0.389167\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.0608864i	0.00321346i	0.999999	+	0.00160673i	$0.000511438\pi$
$359$	0.0608864i	0.00321346i	−0.999999	+	0.00160673i	$0.999489\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.1149i	0.684591i	0.939592	+	0.342296i	$0.111204\pi$
$367$	13.1149i	0.684591i	−0.939592	+	0.342296i	$0.888796\pi$
$368$	0	0
$369$	33.5367	1.74585
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 34.5801i	− 1.77626i	−0.459590	−	0.888131i	$-0.652004\pi$
$379$	− 34.5801i	− 1.77626i	0.459590	−	0.888131i	$-0.347996\pi$
$380$	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
$387$	19.6723i	1.00000i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	− 8.84836i	− 0.447481i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−39.8945	−1.99224	−0.996119	−	0.0880147i	$-0.971948\pi$
$401$	−39.8945	−1.99224	−0.996119	+	0.0880147i	$0.971948\pi$
$402$	0	0
$403$	79.4917i	3.95976i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	− 39.3446i	− 1.91300i
$424$	0	0
$425$	−15.8945	−0.770999
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 38.7858i	− 1.86824i	−0.356953	−	0.934122i	$-0.616185\pi$
$431$	− 38.7858i	− 1.86824i	0.356953	−	0.934122i	$-0.383815\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 22.2068i	− 1.05987i	−0.848038	−	0.529936i	$-0.822216\pi$
$439$	− 22.2068i	− 1.05987i	0.848038	−	0.529936i	$-0.177784\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	0	0
$443$	− 13.1149i	− 0.623107i	−0.950229	−	0.311553i	$-0.899151\pi$
$443$	− 13.1149i	− 0.623107i	0.950229	−	0.311553i	$-0.100849\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	46.3337i	2.18177i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	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
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	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	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−27.1789	−1.24969
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.46327	−0.112786
$478$	0	0
$479$	− 41.5083i	− 1.89656i	−0.317429	−	0.948282i	$-0.602819\pi$
$479$	− 41.5083i	− 1.89656i	0.317429	−	0.948282i	$-0.397181\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 39.3446i	− 1.78288i	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	− 39.3446i	− 1.78288i	0.453143	−	0.891438i	$-0.350303\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	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
$507$	0	0
$508$	0	0
$509$	−43.8945	−1.94559	−0.972796	−	0.231665i	$-0.925583\pi$
$509$	−43.8945	−1.94559	−0.972796	+	0.231665i	$0.925583\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	54.3578	2.39065
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	35.1999i	1.53333i
$528$	0	0
$529$	15.2524	0.663146
$530$	0	0
$531$	39.3446i	1.70741i
$532$	0	0
$533$	80.2524	3.47612
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	29.0132i	1.24969i
$540$	0	0
$541$	−11.8945	−0.511386	−0.255693	−	0.966758i	$-0.582304\pi$
$541$	−11.8945	−0.511386	−0.255693	+	0.966758i	$0.582304\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.1568i	0.648058i	0.946047	+	0.324029i	$0.105038\pi$
$547$	15.1568i	0.648058i	−0.946047	+	0.324029i	$0.894962\pi$
$548$	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
$557$	34.2524	1.45132	0.725660	−	0.688054i	$-0.241535\pi$
$557$	34.2524	1.45132	0.725660	+	0.688054i	$0.241535\pi$
$558$	0	0
$559$	47.0753i	1.99107i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.4245i	1.57725i	0.614872	+	0.788627i	$0.289208\pi$
$563$	37.4245i	1.57725i	−0.614872	+	0.788627i	$0.710792\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	46.2524	1.93900	0.969500	−	0.245092i	$-0.0788181\pi$
$569$	46.2524	1.93900	0.969500	+	0.245092i	$0.0788181\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.9173i	0.580391i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 3.40322i	− 0.140947i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.22860i	0.336212i	0.985769	+	0.168106i	$0.0537650\pi$
$599$	8.22860i	0.336212i	−0.985769	+	0.168106i	$0.946235\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	− 29.1350i	− 1.18647i
$604$	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
$611$	− 94.1505i	− 3.80892i
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.2524	1.21791	0.608957	−	0.793203i	$-0.291588\pi$
$617$	30.2524	1.21791	0.608957	+	0.793203i	$0.291588\pi$
$618$	0	0
$619$	− 13.1149i	− 0.527131i	−0.964641	−	0.263566i	$-0.915101\pi$
$619$	− 13.1149i	− 0.527131i	0.964641	−	0.263566i	$-0.0848986\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	50.2524	1.99107
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	39.3446i	1.55160i	0.630978	+	0.775800i	$0.282654\pi$
$643$	39.3446i	1.55160i	−0.630978	+	0.775800i	$0.717346\pi$
$644$	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
$649$	−54.3578	−2.13373
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.8696i	1.66996i	0.550278	+	0.834982i	$0.314522\pi$
$659$	42.8696i	1.66996i	−0.550278	+	0.834982i	$0.685478\pi$
$660$	0	0
$661$	−31.1789	−1.21272	−0.606359	−	0.795191i	$-0.707371\pi$
$661$	−31.1789	−1.21272	−0.606359	+	0.795191i	$0.707371\pi$
$662$	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
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.30040i	0.0497584i	0.999690	+	0.0248792i	$0.00792011\pi$
$683$	1.30040i	0.0497584i	−0.999690	+	0.0248792i	$0.992080\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.89454	−0.224564
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	35.5367	1.34605
$698$	0	0
$699$	0	0
$700$	0	0
$701$	46.0000	1.73740	0.868698	−	0.495342i	$-0.164957\pi$
$701$	46.0000	1.73740	0.868698	+	0.495342i	$0.164957\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.2524	0.985928	0.492964	−	0.870050i	$-0.335913\pi$
$709$	26.2524	0.985928	0.492964	+	0.870050i	$0.335913\pi$
$710$	0	0
$711$	− 39.3446i	− 1.47554i
$712$	0	0
$713$	30.8211	1.15426
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13.1149i	0.489102i	0.969636	+	0.244551i	$0.0786406\pi$
$719$	13.1149i	0.489102i	−0.969636	+	0.244551i	$0.921359\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	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
$729$	−27.0000	−1.00000
$730$	0	0
$731$	20.8455i	0.770999i
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.2524	1.48271
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	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
$747$	54.0035i	1.97588i
$748$	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
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	94.1505i	3.39958i
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 55.3647i	− 1.98876i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	39.3446i	1.40248i	0.712923	+	0.701242i	$0.247371\pi$
$787$	39.3446i	1.40248i	−0.712923	+	0.701242i	$0.752629\pi$
$788$	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
$797$	−50.0000	−1.77109	−0.885545	−	0.464553i	$-0.846215\pi$
$797$	−50.0000	−1.77109	−0.885545	+	0.464553i	$0.846215\pi$
$798$	0	0
$799$	− 41.6910i	− 1.47492i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.8945	−1.81113	−0.905566	−	0.424205i	$-0.860553\pi$
$821$	−51.8945	−1.81113	−0.905566	+	0.424205i	$0.860553\pi$
$822$	0	0
$823$	− 52.6422i	− 1.83499i	−0.397747	−	0.917495i	$-0.630208\pi$
$823$	− 52.6422i	− 1.83499i	0.397747	−	0.917495i	$-0.369792\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.1149i	− 0.456049i	−0.973655	−	0.228024i	$-0.926773\pi$
$827$	− 13.1149i	− 0.456049i	0.973655	−	0.228024i	$-0.0732266\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	22.2524	0.770999
$834$	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
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−10.4633	−0.358256	−0.179128	−	0.983826i	$-0.557328\pi$
$853$	−10.4633	−0.358256	−0.179128	+	0.983826i	$0.557328\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	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
$867$	0	0
$868$	0	0
$869$	54.3578	1.84396
$870$	0	0
$871$	− 69.7191i	− 2.36234i
$872$	0	0
$873$	−52.6102	−1.78058
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−48.6102	−1.64145	−0.820724	−	0.571324i	$-0.806430\pi$
$877$	−48.6102	−1.64145	−0.820724	+	0.571324i	$0.806430\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−35.1789	−1.18521	−0.592604	−	0.805494i	$-0.701900\pi$
$881$	−35.1789	−1.18521	−0.592604	+	0.805494i	$0.701900\pi$
$882$	0	0
$883$	− 23.4463i	− 0.789031i	−0.918889	−	0.394515i	$-0.870912\pi$
$883$	− 23.4463i	− 0.789031i	0.918889	−	0.394515i	$-0.129088\pi$
$884$	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
$891$	− 37.3027i	− 1.24969i
$892$	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
$901$	−2.61018	−0.0869575
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	56.7260i	1.88356i	0.336234	+	0.941778i	$0.390847\pi$
$907$	56.7260i	1.88356i	−0.336234	+	0.941778i	$0.609153\pi$
$908$	0	0
$909$	45.5367	1.51036
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−74.6102	−2.46924
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	27.7737i	0.916170i	0.888908	+	0.458085i	$0.151464\pi$
$919$	27.7737i	0.916170i	−0.888908	+	0.458085i	$0.848536\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 16.5181i	− 0.542525i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24.8211	0.809144	0.404572	−	0.914506i	$-0.367420\pi$
$941$	24.8211	0.809144	0.404572	+	0.914506i	$0.367420\pi$
$942$	0	0
$943$	− 31.1160i	− 1.01328i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.5560i	− 0.408015i	−0.978969	−	0.204008i	$-0.934603\pi$
$947$	− 12.5560i	− 0.408015i	0.978969	−	0.204008i	$-0.0653968\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−91.6102	−2.95517
$962$	0	0
$963$	39.3446i	1.26786i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	60.9317i	1.95943i	0.200396	+	0.979715i	$0.435777\pi$
$967$	60.9317i	1.95943i	−0.200396	+	0.979715i	$0.564223\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 62.2929i	− 1.99908i	−0.0303994	−	0.999538i	$-0.509678\pi$
$971$	− 62.2929i	− 1.99908i	0.0303994	−	0.999538i	$-0.490322\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−40.6102	−1.29658
$982$	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
$989$	18.2524	0.580391
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.g.d.687.1	✓	4
4.3	odd	2	inner	688.2.g.d.687.4	yes	4
8.3	odd	2		2752.2.g.d.2751.1		4
8.5	even	2		2752.2.g.d.2751.4		4
43.42	odd	2	CM	688.2.g.d.687.1	✓	4
172.171	even	2	inner	688.2.g.d.687.4	yes	4
344.85	odd	2		2752.2.g.d.2751.4		4
344.171	even	2		2752.2.g.d.2751.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.g.d.687.1	✓	4	1.1	even	1	trivial
688.2.g.d.687.1	✓	4	43.42	odd	2	CM
688.2.g.d.687.4	yes	4	4.3	odd	2	inner
688.2.g.d.687.4	yes	4	172.171	even	2	inner
2752.2.g.d.2751.1		4	8.3	odd	2
2752.2.g.d.2751.1		4	344.171	even	2
2752.2.g.d.2751.4		4	8.5	even	2
2752.2.g.d.2751.4		4	344.85	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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{9} -4.14474i q^{11} -7.17891 q^{13} -3.17891 q^{17} +2.78346i q^{23} +5.00000 q^{25} -11.0729i q^{31} -11.1789 q^{41} -6.55744i q^{43} +13.1149i q^{47} -7.00000 q^{49} +0.821092 q^{53} -13.1149i q^{59} +9.71166i q^{67} +13.1149i q^{79} +9.00000 q^{81} -18.0012i q^{83} +17.5367 q^{97} +12.4342i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 6 q^{13} + 10 q^{17} + 20 q^{25} - 22 q^{41} - 28 q^{49} + 26 q^{53} + 36 q^{81} + 2 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 - 6 * q^13 + 10 * q^17 + 20 * q^25 - 22 * q^41 - 28 * q^49 + 26 * q^53 + 36 * q^81 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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