LMFDB - Embedded newform 1760.2.c.b.879.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1760,2,Mod(879,1760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1760.879"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1760 = 2^{5} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1760.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$14.0536707557$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.599695360000.19
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{4} + 16 $ x^8 - 3*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			879.4
Root			$1.35246 + 0.413333i$ of defining polynomial
Character	$\chi$	$=$	1760.879
Dual form			1760.2.c.b.879.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1760\mathbb{Z}\right)^\times$.

$n$	$321$	$991$	$1057$	$1541$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	5.22173i	1.97363i	0.161853	+	0.986815i	$0.448253\pi$
$7$	5.22173i	1.97363i	−0.161853	+	0.986815i	$0.551747\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	3.31662	1.00000
$11$	3.31662	1.00000
$12$	0	0
$13$	3.56840i	0.989697i	0.868979	+	0.494848i	$0.164776\pi$
$13$	3.56840i	0.989697i	−0.868979	+	0.494848i	$0.835224\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.55341	−1.10436	−0.552182	−	0.833724i	$-0.686204\pi$
$17$	−4.55341	−1.10436	−0.552182	+	0.833724i	$0.686204\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	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
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.94427i	1.60644i	0.595683	+	0.803219i	$0.296881\pi$
$31$	8.94427i	1.60644i	−0.595683	+	0.803219i	$0.703119\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	11.6762	1.97363
$36$	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
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−9.96326	−1.51938	−0.759691	−	0.650284i	$-0.774650\pi$
$43$	−9.96326	−1.51938	−0.759691	+	0.650284i	$0.774650\pi$
$44$	0	0
$45$	− 6.70820i	− 1.00000i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−20.2665	−2.89521
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	− 7.41620i	− 1.00000i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	15.6652i	1.97363i
$64$	0	0
$65$	7.97919	0.989697
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.8324i	1.76028i	0.474713	+	0.880141i	$0.342552\pi$
$71$	14.8324i	1.76028i	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	17.0860	1.99977	0.999883	−	0.0153173i	$-0.00487585\pi$
$73$	17.0860	1.99977	0.999883	+	0.0153173i	$0.00487585\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.3185i	1.97363i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	13.3890	1.46964	0.734819	−	0.678263i	$-0.237267\pi$
$83$	13.3890	1.46964	0.734819	+	0.678263i	$0.237267\pi$
$84$	0	0
$85$	10.1817i	1.10436i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.2665	1.40625	0.703123	−	0.711068i	$-0.251788\pi$
$89$	13.2665	1.40625	0.703123	+	0.711068i	$0.251788\pi$
$90$	0	0
$91$	−18.6332	−1.95330
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	9.94987	1.00000
$100$	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
$107$	−8.25036	−0.797593	−0.398796	−	0.917039i	$-0.630572\pi$
$107$	−8.25036	−0.797593	−0.398796	+	0.917039i	$0.630572\pi$
$108$	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
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.7052i	0.989697i
$118$	0	0
$119$	− 23.7767i	− 2.17960i
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803i	1.00000i
$126$	0	0
$127$	− 1.39159i	− 0.123483i	−0.998092	−	0.0617417i	$-0.980334\pi$
$127$	− 1.39159i	− 0.123483i	0.998092	−	0.0617417i	$-0.0196655\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	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
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.8351i	0.989697i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	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
$153$	−13.6602	−1.10436
$154$	0	0
$155$	20.0000	1.60644
$156$	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
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.05188i	0.700455i	0.936665	+	0.350228i	$0.113896\pi$
$167$	9.05188i	0.700455i	−0.936665	+	0.350228i	$0.886104\pi$
$168$	0	0
$169$	0.266499	0.0204999
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.4553i	− 1.85930i	−0.368438	−	0.929652i	$-0.620107\pi$
$173$	− 24.4553i	− 1.85930i	0.368438	−	0.929652i	$-0.379893\pi$
$174$	0	0
$175$	− 26.1087i	− 1.97363i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.8997	−1.48738	−0.743689	−	0.668526i	$-0.766925\pi$
$179$	−19.8997	−1.48738	−0.743689	+	0.668526i	$0.766925\pi$
$180$	0	0
$181$	4.47214i	0.332411i	0.986091	+	0.166206i	$0.0531515\pi$
$181$	4.47214i	0.332411i	−0.986091	+	0.166206i	$0.946848\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−15.1019	−1.10436
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 26.8328i	− 1.94155i	−0.239983	−	0.970777i	$-0.577142\pi$
$191$	− 26.8328i	− 1.94155i	0.239983	−	0.970777i	$-0.422858\pi$
$192$	0	0
$193$	1.12762	0.0811678	0.0405839	−	0.999176i	$-0.487078\pi$
$193$	1.12762	0.0811678	0.0405839	+	0.999176i	$0.487078\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.04492i	− 0.216941i	−0.994100	−	0.108471i	$-0.965405\pi$
$197$	− 3.04492i	− 0.216941i	0.994100	−	0.108471i	$-0.0345954\pi$
$198$	0	0
$199$	14.8324i	1.05144i	0.850657	+	0.525720i	$0.176204\pi$
$199$	14.8324i	1.05144i	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	22.2785i	1.51938i
$216$	0	0
$217$	−46.7046	−3.17052
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 16.2484i	− 1.09298i
$222$	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
$227$	6.53747	0.433907	0.216954	−	0.976182i	$-0.430388\pi$
$227$	6.53747	0.433907	0.216954	+	0.976182i	$0.430388\pi$
$228$	0	0
$229$	29.6648i	1.96030i	0.198246	+	0.980152i	$0.436476\pi$
$229$	29.6648i	1.96030i	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	29.6186	1.94038	0.970190	−	0.242348i	$-0.0779174\pi$
$233$	29.6186	1.94038	0.970190	+	0.242348i	$0.0779174\pi$
$234$	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
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	45.3173i	2.89521i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	0	0
$254$	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
$263$	− 19.4953i	− 1.20213i	−0.799198	−	0.601067i	$-0.794742\pi$
$263$	− 19.4953i	− 1.20213i	0.799198	−	0.601067i	$-0.205258\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 29.6648i	− 1.80869i	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	− 29.6648i	− 1.80869i	0.426798	−	0.904347i	$-0.359642\pi$
$270$	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
$275$	−16.5831	−1.00000
$276$	0	0
$277$	− 31.0687i	− 1.86673i	−0.358923	−	0.933367i	$-0.616856\pi$
$277$	− 31.0687i	− 1.86673i	0.358923	−	0.933367i	$-0.383144\pi$
$278$	0	0
$279$	26.8328i	1.60644i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	33.3156	1.98040	0.990202	−	0.139641i	$-0.0445948\pi$
$283$	33.3156	1.98040	0.990202	+	0.139641i	$0.0445948\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.73350	0.219618
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.9319i	1.39811i	0.715066	+	0.699057i	$0.246397\pi$
$293$	23.9319i	1.39811i	−0.715066	+	0.699057i	$0.753603\pi$
$294$	0	0
$295$	− 8.94427i	− 0.520756i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 52.0255i	− 2.99870i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−31.6027	−1.80366	−0.901830	−	0.432092i	$-0.857776\pi$
$307$	−31.6027	−1.80366	−0.901830	+	0.432092i	$0.857776\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 14.8324i	− 0.841068i	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	− 14.8324i	− 0.841068i	0.907277	−	0.420534i	$-0.138157\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	35.0285	1.97363
$316$	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
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 17.8420i	− 0.989697i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.63325	0.364596	0.182298	−	0.983243i	$-0.441646\pi$
$331$	6.63325	0.364596	0.182298	+	0.983243i	$0.441646\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.4050	−0.621269	−0.310634	−	0.950529i	$-0.600541\pi$
$337$	−11.4050	−0.621269	−0.310634	+	0.950529i	$0.600541\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	29.6648i	1.60644i
$342$	0	0
$343$	− 69.2741i	− 3.74045i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.8148	−0.902667	−0.451334	−	0.892355i	$-0.649052\pi$
$347$	−16.8148	−0.902667	−0.451334	+	0.892355i	$0.649052\pi$
$348$	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
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	33.1662	1.76028
$356$	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
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 38.2055i	− 1.99977i
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 31.5921i	− 1.63578i	−0.575375	−	0.817890i	$-0.695144\pi$
$373$	− 31.5921i	− 1.63578i	0.575375	−	0.817890i	$-0.304856\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	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
$385$	38.7254	1.97363
$386$	0	0
$387$	−29.8898	−1.51938
$388$	0	0
$389$	29.6648i	1.50406i	0.659126	+	0.752032i	$0.270926\pi$
$389$	29.6648i	1.50406i	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	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
$401$	39.7995	1.98749	0.993746	−	0.111664i	$-0.0356180\pi$
$401$	39.7995	1.98749	0.993746	+	0.111664i	$0.0356180\pi$
$402$	0	0
$403$	−31.9168	−1.58989
$404$	0	0
$405$	− 20.1246i	− 1.00000i
$406$	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
$413$	20.8869i	1.02778i
$414$	0	0
$415$	− 29.9388i	− 1.46964i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.8997	0.972166	0.486083	−	0.873913i	$-0.338425\pi$
$419$	19.8997	0.972166	0.486083	+	0.873913i	$0.338425\pi$
$420$	0	0
$421$	− 31.3050i	− 1.52571i	−0.646570	−	0.762855i	$-0.723797\pi$
$421$	− 31.3050i	− 1.52571i	0.646570	−	0.762855i	$-0.276203\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	22.7670	1.10436
$426$	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
$433$	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
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−60.7995	−2.89521
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	− 29.6648i	− 1.40625i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.2665	−0.626085	−0.313042	−	0.949739i	$-0.601348\pi$
$449$	−13.2665	−0.626085	−0.313042	+	0.949739i	$0.601348\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	41.6652i	1.95330i
$456$	0	0
$457$	10.2344	0.478746	0.239373	−	0.970928i	$-0.423058\pi$
$457$	10.2344	0.478746	0.239373	+	0.970928i	$0.423058\pi$
$458$	0	0
$459$	0	0
$460$	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	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−33.0444	−1.51938
$474$	0	0
$475$	0	0
$476$	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
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 22.2486i	− 1.00000i
$496$	0	0
$497$	−77.4508	−3.47414
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 8.00491i	− 0.356921i	−0.983947	−	0.178461i	$-0.942888\pi$
$503$	− 8.00491i	− 0.356921i	0.983947	−	0.178461i	$-0.0571117\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 29.6648i	− 1.31487i	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	− 29.6648i	− 1.31487i	0.753512	−	0.657434i	$-0.228358\pi$
$510$	0	0
$511$	89.2186i	3.94680i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.7995	−1.74365	−0.871824	−	0.489820i	$-0.837063\pi$
$521$	−39.7995	−1.74365	−0.871824	+	0.489820i	$0.837063\pi$
$522$	0	0
$523$	28.1769	1.23209	0.616044	−	0.787711i	$-0.288734\pi$
$523$	28.1769	1.23209	0.616044	+	0.787711i	$0.288734\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 40.7269i	− 1.77409i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	18.4484i	0.797593i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−67.2164	−2.89521
$540$	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
$547$	−38.4542	−1.64418	−0.822092	−	0.569354i	$-0.807193\pi$
$547$	−38.4542	−1.64418	−0.822092	+	0.569354i	$0.807193\pi$
$548$	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
$557$	45.3423i	1.92121i	0.277911	+	0.960607i	$0.410358\pi$
$557$	45.3423i	1.92121i	−0.277911	+	0.960607i	$0.589642\pi$
$558$	0	0
$559$	− 35.5529i	− 1.50373i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.5277	0.780850	0.390425	−	0.920635i	$-0.372328\pi$
$563$	18.5277	0.780850	0.390425	+	0.920635i	$0.372328\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	46.9956i	1.97363i
$568$	0	0
$569$	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
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	69.9140i	2.90052i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	23.9376	0.989697
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.29817	0.0943744	0.0471872	−	0.998886i	$-0.484974\pi$
$593$	2.29817	0.0943744	0.0471872	+	0.998886i	$0.484974\pi$
$594$	0	0
$595$	−53.1662	−2.17960
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 44.7214i	− 1.82727i	−0.406541	−	0.913633i	$-0.633265\pi$
$599$	− 44.7214i	− 1.82727i	0.406541	−	0.913633i	$-0.366735\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 24.5967i	− 1.00000i
$606$	0	0
$607$	− 12.8820i	− 0.522865i	−0.965222	−	0.261433i	$-0.915805\pi$
$607$	− 12.8820i	− 0.522865i	0.965222	−	0.261433i	$-0.0841949\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 44.8188i	− 1.81021i	−0.425184	−	0.905107i	$-0.639791\pi$
$613$	− 44.8188i	− 1.81021i	0.425184	−	0.905107i	$-0.360209\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	46.4327	1.86629	0.933145	−	0.359501i	$-0.117053\pi$
$619$	46.4327	1.86629	0.933145	+	0.359501i	$0.117053\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	69.2741i	2.77541i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 14.8324i	− 0.590468i	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	− 14.8324i	− 0.590468i	0.955425	−	0.295234i	$-0.0953977\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.11168	−0.123483
$636$	0	0
$637$	− 72.3190i	− 2.86538i
$638$	0	0
$639$	44.4972i	1.76028i
$640$	0	0
$641$	39.7995	1.57199	0.785993	−	0.618236i	$-0.212152\pi$
$641$	39.7995	1.57199	0.785993	+	0.618236i	$0.212152\pi$
$642$	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		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	13.2665	0.520756
$650$	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
$657$	51.2580	1.99977
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	29.6648i	1.15383i	0.816805	+	0.576913i	$0.195743\pi$
$661$	29.6648i	1.15383i	−0.816805	+	0.576913i	$0.804257\pi$
$662$	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
$673$	−47.8322	−1.84380	−0.921898	−	0.387432i	$-0.873362\pi$
$673$	−47.8322	−1.84380	−0.921898	+	0.387432i	$0.873362\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.7289i	1.48847i	0.667915	+	0.744237i	$0.267187\pi$
$677$	38.7289i	1.48847i	−0.667915	+	0.744237i	$0.732813\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	51.9556i	1.97363i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	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
$709$	− 4.47214i	− 0.167955i	−0.996468	−	0.0839773i	$-0.973238\pi$
$709$	− 4.47214i	− 0.167955i	0.996468	−	0.0839773i	$-0.0267623\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	26.4640	0.989697
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 26.8328i	− 1.00070i	−0.865825	−	0.500348i	$-0.833206\pi$
$719$	− 26.8328i	− 1.00070i	0.865825	−	0.500348i	$-0.166794\pi$
$720$	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
$731$	45.3668	1.67795
$732$	0	0
$733$	− 11.2287i	− 0.414741i	−0.978262	−	0.207371i	$-0.933509\pi$
$733$	− 11.2287i	− 0.414741i	0.978262	−	0.207371i	$-0.0664906\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 23.3255i	− 0.855729i	−0.903843	−	0.427865i	$-0.859266\pi$
$743$	− 23.3255i	− 0.855729i	0.903843	−	0.427865i	$-0.140734\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	40.1671	1.46964
$748$	0	0
$749$	− 43.0812i	− 1.57415i
$750$	0	0
$751$	− 44.4972i	− 1.62373i	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	− 44.4972i	− 1.62373i	0.583848	−	0.811863i	$-0.301546\pi$
$752$	0	0
$753$	0	0
$754$	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
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	30.5452i	1.10436i
$766$	0	0
$767$	14.2736i	0.515390i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	− 44.7214i	− 1.60644i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	49.1935i	1.76028i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	54.9550	1.95893	0.979467	−	0.201607i	$-0.0646164\pi$
$787$	54.9550	1.95893	0.979467	+	0.201607i	$0.0646164\pi$
$788$	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
$801$	39.7995	1.40625
$802$	0	0
$803$	56.6679	1.99977
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−55.8997	−1.95330
$820$	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
$825$	0	0
$826$	0	0
$827$	−53.2421	−1.85141	−0.925704	−	0.378248i	$-0.876527\pi$
$827$	−53.2421	−1.85141	−0.925704	+	0.378248i	$0.876527\pi$
$828$	0	0
$829$	22.3607i	0.776619i	0.921529	+	0.388309i	$0.126941\pi$
$829$	22.3607i	0.776619i	−0.921529	+	0.388309i	$0.873059\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	92.2816	3.19737
$834$	0	0
$835$	20.2406	0.700455
$836$	0	0
$837$	0	0
$838$	0	0
$839$	14.8324i	0.512071i	0.966667	+	0.256036i	$0.0824164\pi$
$839$	14.8324i	0.512071i	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 0.595910i	− 0.0204999i
$846$	0	0
$847$	57.4391i	1.97363i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	52.4791i	1.79685i	0.439128	+	0.898425i	$0.355288\pi$
$853$	52.4791i	1.79685i	−0.439128	+	0.898425i	$0.644712\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.3412	−0.660684	−0.330342	−	0.943861i	$-0.607164\pi$
$857$	−19.3412	−0.660684	−0.330342	+	0.943861i	$0.607164\pi$
$858$	0	0
$859$	−46.4327	−1.58426	−0.792132	−	0.610349i	$-0.791029\pi$
$859$	−46.4327	−1.58426	−0.792132	+	0.610349i	$0.791029\pi$
$860$	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
$865$	−54.6838	−1.85930
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−58.3808	−1.97363
$876$	0	0
$877$	59.0924i	1.99541i	0.0677220	+	0.997704i	$0.478427\pi$
$877$	59.0924i	1.99541i	−0.0677220	+	0.997704i	$0.521573\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50.8257i	1.70656i	0.521452	+	0.853281i	$0.325391\pi$
$887$	50.8257i	1.70656i	−0.521452	+	0.853281i	$0.674609\pi$
$888$	0	0
$889$	7.26650	0.243711
$890$	0	0
$891$	29.8496	1.00000
$892$	0	0
$893$	0	0
$894$	0	0
$895$	44.4972i	1.48738i
$896$	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
$905$	10.0000	0.332411
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.94427i	0.296337i	0.988962	+	0.148168i	$0.0473378\pi$
$911$	8.94427i	0.296337i	−0.988962	+	0.148168i	$0.952662\pi$
$912$	0	0
$913$	44.4064	1.46964
$914$	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
$921$	0	0
$922$	0	0
$923$	−52.9280	−1.74215
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	33.7690i	1.10436i
$936$	0	0
$937$	36.4702	1.19143	0.595714	−	0.803196i	$-0.296869\pi$
$937$	36.4702	1.19143	0.595714	+	0.803196i	$0.296869\pi$
$938$	0	0
$939$	0	0
$940$	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
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	60.9697i	1.97916i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	60.3648	1.95541	0.977704	−	0.209987i	$-0.0673422\pi$
$953$	60.3648	1.95541	0.977704	+	0.209987i	$0.0673422\pi$
$954$	0	0
$955$	−60.0000	−1.94155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−49.0000	−1.58065
$962$	0	0
$963$	−24.7511	−0.797593
$964$	0	0
$965$	− 2.52143i	− 0.0811678i
$966$	0	0
$967$	− 49.7788i	− 1.60078i	−0.599481	−	0.800389i	$-0.704626\pi$
$967$	− 49.7788i	− 1.60078i	0.599481	−	0.800389i	$-0.295374\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−59.6992	−1.91584	−0.957920	−	0.287035i	$-0.907330\pi$
$971$	−59.6992	−1.91584	−0.957920	+	0.287035i	$0.907330\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	44.0000	1.40625
$980$	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
$985$	−6.80864	−0.216941
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 62.6099i	− 1.98887i	−0.105356	−	0.994435i	$-0.533598\pi$
$991$	− 62.6099i	− 1.98887i	0.105356	−	0.994435i	$-0.466402\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	33.1662	1.05144
$996$	0	0
$997$	23.4084i	0.741350i	0.928763	+	0.370675i	$0.120874\pi$
$997$	23.4084i	0.741350i	−0.928763	+	0.370675i	$0.879126\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1760.2.c.b.879.4		8
4.3	odd	2		440.2.c.b.219.1	✓	8
5.4	even	2	inner	1760.2.c.b.879.1		8
8.3	odd	2	inner	1760.2.c.b.879.5		8
8.5	even	2		440.2.c.b.219.2	yes	8
11.10	odd	2	inner	1760.2.c.b.879.1		8
20.19	odd	2		440.2.c.b.219.8	yes	8
40.19	odd	2	inner	1760.2.c.b.879.8		8
40.29	even	2		440.2.c.b.219.7	yes	8
44.43	even	2		440.2.c.b.219.8	yes	8
55.54	odd	2	CM	1760.2.c.b.879.4		8
88.21	odd	2		440.2.c.b.219.7	yes	8
88.43	even	2	inner	1760.2.c.b.879.8		8
220.219	even	2		440.2.c.b.219.1	✓	8
440.109	odd	2		440.2.c.b.219.2	yes	8
440.219	even	2	inner	1760.2.c.b.879.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.c.b.219.1	✓	8	4.3	odd	2
440.2.c.b.219.1	✓	8	220.219	even	2
440.2.c.b.219.2	yes	8	8.5	even	2
440.2.c.b.219.2	yes	8	440.109	odd	2
440.2.c.b.219.7	yes	8	40.29	even	2
440.2.c.b.219.7	yes	8	88.21	odd	2
440.2.c.b.219.8	yes	8	20.19	odd	2
440.2.c.b.219.8	yes	8	44.43	even	2
1760.2.c.b.879.1		8	5.4	even	2	inner
1760.2.c.b.879.1		8	11.10	odd	2	inner
1760.2.c.b.879.4		8	1.1	even	1	trivial
1760.2.c.b.879.4		8	55.54	odd	2	CM
1760.2.c.b.879.5		8	8.3	odd	2	inner
1760.2.c.b.879.5		8	440.219	even	2	inner
1760.2.c.b.879.8		8	40.19	odd	2	inner
1760.2.c.b.879.8		8	88.43	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1760 = 2^{5} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1760.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +5.22173i q^{7} +3.00000 q^{9} +3.31662 q^{11} +3.56840i q^{13} -4.55341 q^{17} -5.00000 q^{25} +8.94427i q^{31} +11.6762 q^{35} -9.96326 q^{43} -6.70820i q^{45} -20.2665 q^{49} -7.41620i q^{55} +4.00000 q^{59} +15.6652i q^{63} +7.97919 q^{65} +14.8324i q^{71} +17.0860 q^{73} +17.3185i q^{77} +9.00000 q^{81} +13.3890 q^{83} +10.1817i q^{85} +13.2665 q^{89} -18.6332 q^{91} +9.94987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9} - 40 q^{25} - 56 q^{49} + 32 q^{59} + 72 q^{81} - 96 q^{91}+O(q^{100}) \) 8 * q + 24 * q^9 - 40 * q^25 - 56 * q^49 + 32 * q^59 + 72 * q^81 - 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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