LMFDB - Embedded newform 1216.2.b.f.607.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,2,Mod(607,1216)] 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("1216.607"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-36,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 207x^{8} + 1014x^{6} + 1065x^{4} - 5508x^{2} + 8464 $ x^12 + 18x^10 + 207x^8 + 1014x^6 + 1065x^4 - 5508*x^2 + 8464
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			607.7
Root			$0.564101 + 2.36358i$ of defining polynomial
Character	$\chi$	$=$	1216.607
Dual form			1216.2.b.f.607.6

$q$-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 $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	1.12820i	0.651368i	0.945479	+	0.325684i	$0.105595\pi$
$3$	1.12820i	0.651368i	−0.945479	+	0.325684i	$0.894405\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	− 0.0952793i	− 0.0360122i	−0.999838	−	0.0180061i	$-0.994268\pi$
$7$	− 0.0952793i	− 0.0360122i	0.999838	−	0.0180061i	$-0.00573183\pi$
$8$	0	0
$9$	1.72716	0.575720
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	6.35390	1.76225	0.881127	−	0.472879i	$-0.156785\pi$
$13$	6.35390	1.76225	0.881127	+	0.472879i	$0.156785\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.53660	−1.10029	−0.550144	−	0.835070i	$-0.685427\pi$
$17$	−4.53660	−1.10029	−0.550144	+	0.835070i	$0.685427\pi$
$18$	0	0
$19$	4.35890i	1.00000i
$19$	4.35890i	1.00000i
$20$	0	0
$21$	0.107494	0.0234572
$22$	0	0
$23$	− 9.35904i	− 1.95149i	−0.218899	−	0.975747i	$-0.570247\pi$
$23$	− 9.35904i	− 1.95149i	0.218899	−	0.975747i	$-0.429753\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	5.33319i	1.02637i
$28$	0	0
$29$	−4.09750	−0.760886	−0.380443	−	0.924804i	$-0.624228\pi$
$29$	−4.09750	−0.760886	−0.380443	+	0.924804i	$0.624228\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.71780	1.43320	0.716599	−	0.697486i	$-0.245698\pi$
$37$	8.71780	1.43320	0.716599	+	0.697486i	$0.245698\pi$
$38$	0	0
$39$	7.16848i	1.14788i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	6.99092	0.998703
$50$	0	0
$51$	− 5.11820i	− 0.716692i
$52$	0	0
$53$	10.8667	1.49266	0.746329	−	0.665578i	$-0.231815\pi$
$53$	10.8667	1.49266	0.746329	+	0.665578i	$0.231815\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.91772	−0.651368
$58$	0	0
$59$	11.5796i	1.50754i	0.657141	+	0.753768i	$0.271766\pi$
$59$	11.5796i	1.50754i	−0.657141	+	0.753768i	$0.728234\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 0.164563i	− 0.0207329i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	16.0924i	1.96600i	0.183605	+	0.983000i	$0.441223\pi$
$67$	16.0924i	1.96600i	−0.183605	+	0.983000i	$0.558777\pi$
$68$	0	0
$69$	10.5589	1.27114
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−13.8004	−1.61521	−0.807605	−	0.589724i	$-0.799237\pi$
$73$	−13.8004	−1.61521	−0.807605	+	0.589724i	$0.799237\pi$
$74$	0	0
$75$	5.64101i	0.651368i
$76$	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
$81$	−0.835437	−0.0928264
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.62280i	− 0.495616i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 0.605395i	− 0.0634626i
$92$	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
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	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
$107$	− 18.3488i	− 1.77385i	−0.461917	−	0.886923i	$-0.652838\pi$
$107$	− 18.3488i	− 1.77385i	0.461917	−	0.886923i	$-0.347162\pi$
$108$	0	0
$109$	−13.1231	−1.25697	−0.628483	−	0.777823i	$-0.716324\pi$
$109$	−13.1231	−1.25697	−0.628483	+	0.777823i	$0.716324\pi$
$110$	0	0
$111$	9.83544i	0.933538i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.9742	1.01457
$118$	0	0
$119$	0.432244i	0.0396238i
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0.415313	0.0360122
$134$	0	0
$135$	0	0
$136$	0	0
$137$	23.2547	1.98678	0.993391	−	0.114781i	$-0.0366166\pi$
$137$	23.2547	1.98678	0.993391	+	0.114781i	$0.0366166\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−6.76921	−0.570071
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.88717i	0.650523i
$148$	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
$153$	−7.83544	−0.633458
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	12.2598i	0.972269i
$160$	0	0
$161$	−0.891723	−0.0702776
$162$	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
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	27.3720	2.10554
$170$	0	0
$171$	7.52852i	0.575720i
$172$	0	0
$173$	−26.1534	−1.98841	−0.994203	−	0.107521i	$-0.965709\pi$
$173$	−26.1534	−1.98841	−0.994203	+	0.107521i	$0.965709\pi$
$174$	0	0
$175$	− 0.476396i	− 0.0360122i
$176$	0	0
$177$	−13.0641	−0.981960
$178$	0	0
$179$	− 26.1534i	− 1.95480i	−0.211407	−	0.977398i	$-0.567804\pi$
$179$	− 26.1534i	− 1.95480i	0.211407	−	0.977398i	$-0.432196\pi$
$180$	0	0
$181$	8.71780	0.647989	0.323994	−	0.946059i	$-0.394974\pi$
$181$	8.71780	0.647989	0.323994	+	0.946059i	$0.394974\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0.508143	0.0369619
$190$	0	0
$191$	− 19.0039i	− 1.37508i	−0.726149	−	0.687538i	$-0.758692\pi$
$191$	− 19.0039i	− 1.37508i	0.726149	−	0.687538i	$-0.241308\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	− 27.8866i	− 1.97683i	−0.151788	−	0.988413i	$-0.548503\pi$
$199$	− 27.8866i	− 1.97683i	0.151788	−	0.988413i	$-0.451497\pi$
$200$	0	0
$201$	−18.1555	−1.28059
$202$	0	0
$203$	0.390406i	0.0274012i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 16.1646i	− 1.12351i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 28.8002i	− 1.98269i	−0.131291	−	0.991344i	$-0.541912\pi$
$211$	− 28.8002i	− 1.98269i	0.131291	−	0.991344i	$-0.458088\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 15.5696i	− 1.05210i
$220$	0	0
$221$	−28.8251	−1.93899
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	8.63580	0.575720
$226$	0	0
$227$	− 4.81038i	− 0.319276i	−0.987176	−	0.159638i	$-0.948967\pi$
$227$	− 4.81038i	− 0.319276i	0.987176	−	0.159638i	$-0.0510328\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.78737i	0.568407i	0.958764	+	0.284204i	$0.0917292\pi$
$239$	8.78737i	0.568407i	−0.958764	+	0.284204i	$0.908271\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	15.0570i	0.965909i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	27.6960i	1.76225i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	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
$259$	− 0.830626i	− 0.0516126i
$260$	0	0
$261$	−7.07703	−0.438057
$262$	0	0
$263$	30.0000i	1.84988i	0.380114	+	0.924940i	$0.375885\pi$
$263$	30.0000i	1.84988i	−0.380114	+	0.924940i	$0.624115\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.1534	−1.59460	−0.797300	−	0.603583i	$-0.793739\pi$
$269$	−26.1534	−1.59460	−0.797300	+	0.603583i	$0.793739\pi$
$270$	0	0
$271$	− 28.2677i	− 1.71714i	−0.512697	−	0.858570i	$-0.671354\pi$
$271$	− 28.2677i	− 1.71714i	0.512697	−	0.858570i	$-0.328646\pi$
$272$	0	0
$273$	0.683008	0.0413375
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.58075	0.210633
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.6359	−1.03030	−0.515151	−	0.857100i	$-0.672264\pi$
$293$	−17.6359	−1.03030	−0.515151	+	0.857100i	$0.672264\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 59.4664i	− 3.43903i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.71780i	0.497551i	0.968561	+	0.248776i	$0.0800281\pi$
$307$	8.71780i	0.497551i	−0.968561	+	0.248776i	$0.919972\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.93072i	0.563119i	0.959544	+	0.281560i	$0.0908517\pi$
$311$	9.93072i	0.563119i	−0.959544	+	0.281560i	$0.909148\pi$
$312$	0	0
$313$	−14.5626	−0.823127	−0.411563	−	0.911381i	$-0.635017\pi$
$313$	−14.5626	−0.823127	−0.411563	+	0.911381i	$0.635017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−34.0259	−1.91109	−0.955543	−	0.294853i	$-0.904729\pi$
$317$	−34.0259	−1.91109	−0.955543	+	0.294853i	$0.904729\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	20.7012	1.15543
$322$	0	0
$323$	− 19.7746i	− 1.10029i
$324$	0	0
$325$	31.7695	1.76225
$326$	0	0
$327$	− 14.8055i	− 0.818747i
$328$	0	0
$329$	0.571676	0.0315175
$330$	0	0
$331$	2.55398i	0.140379i	0.997534	+	0.0701897i	$0.0223605\pi$
$331$	2.55398i	0.140379i	−0.997534	+	0.0701897i	$0.977640\pi$
$332$	0	0
$333$	15.0570	0.825120
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.33305i	− 0.0719777i
$344$	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
$351$	33.8866i	1.80873i
$352$	0	0
$353$	−23.8264	−1.26815	−0.634075	−	0.773272i	$-0.718619\pi$
$353$	−23.8264	−1.26815	−0.634075	+	0.773272i	$0.718619\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.487659	−0.0258096
$358$	0	0
$359$	− 37.1503i	− 1.96072i	−0.197218	−	0.980360i	$-0.563191\pi$
$359$	− 37.1503i	− 1.96072i	0.197218	−	0.980360i	$-0.436809\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 12.4102i	− 0.651368i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.03537i	− 0.0537538i
$372$	0	0
$373$	−38.5387	−1.99546	−0.997729	−	0.0673505i	$-0.978545\pi$
$373$	−38.5387	−1.99546	−0.997729	+	0.0673505i	$0.978545\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−26.0351	−1.34087
$378$	0	0
$379$	35.5694i	1.82708i	0.406751	+	0.913539i	$0.366662\pi$
$379$	35.5694i	1.82708i	−0.406751	+	0.913539i	$0.633338\pi$
$380$	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
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	42.4582i	2.14721i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0.468557i	0.0234572i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	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
$411$	26.2360i	1.29413i
$412$	0	0
$413$	1.10330	0.0542896
$414$	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
$421$	19.8923	0.969493	0.484746	−	0.874655i	$-0.338912\pi$
$421$	19.8923	0.969493	0.484746	+	0.874655i	$0.338912\pi$
$422$	0	0
$423$	10.3630i	0.503864i
$424$	0	0
$425$	−22.6830	−1.10029
$426$	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
$437$	40.7951	1.95149
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	12.0744	0.574973
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	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
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	41.7822	1.95449	0.977245	−	0.212116i	$-0.0680353\pi$
$457$	41.7822	1.95449	0.977245	+	0.212116i	$0.0680353\pi$
$458$	0	0
$459$	− 24.1946i	− 1.12931i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	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
$469$	1.53327	0.0708000
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	21.7945i	1.00000i
$476$	0	0
$477$	18.7685	0.859353
$478$	0	0
$479$	− 42.0000i	− 1.91903i	−0.281659	−	0.959514i	$-0.590885\pi$
$479$	− 42.0000i	− 1.91903i	0.281659	−	0.959514i	$-0.409115\pi$
$480$	0	0
$481$	55.3920	2.52566
$482$	0	0
$483$	− 1.00604i	− 0.0457766i
$484$	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		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	18.5887	0.837193
$494$	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
$503$	37.7220i	1.68194i	0.541081	+	0.840970i	$0.318015\pi$
$503$	37.7220i	1.68194i	−0.541081	+	0.840970i	$0.681985\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	30.8812i	1.37148i
$508$	0	0
$509$	−26.1534	−1.15923	−0.579614	−	0.814891i	$-0.696797\pi$
$509$	−26.1534	−1.15923	−0.579614	+	0.814891i	$0.696797\pi$
$510$	0	0
$511$	1.31489i	0.0581673i
$512$	0	0
$513$	−23.2468	−1.02637
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	− 29.5063i	− 1.29518i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 15.2618i	− 0.667351i	−0.942688	−	0.333676i	$-0.891711\pi$
$523$	− 15.2618i	− 0.667351i	0.942688	−	0.333676i	$-0.108289\pi$
$524$	0	0
$525$	0.537471	0.0234572
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−64.5916	−2.80833
$530$	0	0
$531$	19.9998i	0.867918i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.5063	1.27329
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	9.83544i	0.422079i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.5890i	1.86373i	0.362804	+	0.931865i	$0.381819\pi$
$547$	43.5890i	1.86373i	−0.362804	+	0.931865i	$0.618181\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 17.8606i	− 0.760886i
$552$	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			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 26.1534i	− 1.10223i	−0.834428	−	0.551117i	$-0.814202\pi$
$563$	− 26.1534i	− 1.10223i	0.834428	−	0.551117i	$-0.185798\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.0795999i	0.00334288i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	21.4403	0.895680
$574$	0	0
$575$	− 46.7952i	− 1.95149i
$576$	0	0
$577$	−42.3539	−1.76322	−0.881608	−	0.471983i	$-0.843538\pi$
$577$	−42.3539	−1.76322	−0.881608	+	0.471983i	$0.843538\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	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
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.4617	1.28764
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	27.7942i	1.13187i
$604$	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
$609$	−0.440457	−0.0178482
$610$	0	0
$611$	38.1234i	1.54231i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	49.9136	2.00296
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−39.5492	−1.57693
$630$	0	0
$631$	− 34.0000i	− 1.35352i	−0.736204	−	0.676759i	$-0.763384\pi$
$631$	− 34.0000i	− 1.35352i	0.736204	−	0.676759i	$-0.236616\pi$
$632$	0	0
$633$	32.4924	1.29146
$634$	0	0
$635$	0	0
$636$	0	0
$637$	44.4196	1.75997
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.2235i	1.81723i	0.417630	+	0.908617i	$0.362861\pi$
$647$	46.2235i	1.81723i	−0.417630	+	0.908617i	$0.637139\pi$
$648$	0	0
$649$	0	0
$650$	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
$657$	−23.8354	−0.929909
$658$	0	0
$659$	13.0054i	0.506618i	0.967385	+	0.253309i	$0.0815189\pi$
$659$	13.0054i	0.506618i	−0.967385	+	0.253309i	$0.918481\pi$
$660$	0	0
$661$	51.2465	1.99326	0.996629	−	0.0820394i	$-0.0261433\pi$
$661$	51.2465	1.99326	0.996629	+	0.0820394i	$0.0261433\pi$
$662$	0	0
$663$	− 32.5205i	− 1.26299i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	38.3486i	1.48486i
$668$	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
$675$	26.6660i	1.02637i
$676$	0	0
$677$	42.2209	1.62268	0.811340	−	0.584574i	$-0.198738\pi$
$677$	42.2209	1.62268	0.811340	+	0.584574i	$0.198738\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.42709	0.207966
$682$	0	0
$683$	− 26.1534i	− 1.00073i	−0.865814	−	0.500366i	$-0.833199\pi$
$683$	− 26.1534i	− 1.00073i	0.865814	−	0.500366i	$-0.166801\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	69.0460	2.63044
$690$	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
$697$	0	0
$698$	0	0
$699$	20.3076i	0.768105i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	38.0000i	1.43320i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.91392	−0.370242
$718$	0	0
$719$	36.5787i	1.36415i	0.731281	+	0.682077i	$0.238923\pi$
$719$	36.5787i	1.36415i	−0.731281	+	0.682077i	$0.761077\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−20.4875	−0.760886
$726$	0	0
$727$	− 27.1243i	− 1.00599i	−0.864291	−	0.502993i	$-0.832232\pi$
$727$	− 27.1243i	− 1.00599i	0.864291	−	0.502993i	$-0.167768\pi$
$728$	0	0
$729$	−19.4937	−0.721988
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	−31.2467	−1.14788
$742$	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
$747$	0	0
$748$	0	0
$749$	−1.74826	−0.0638801
$750$	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
$761$	−51.6176	−1.87114	−0.935569	−	0.353144i	$-0.885113\pi$
$761$	−51.6176	−1.87114	−0.935569	+	0.353144i	$0.885113\pi$
$762$	0	0
$763$	1.25036i	0.0452661i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	73.5756i	2.65666i
$768$	0	0
$769$	−14.9437	−0.538884	−0.269442	−	0.963017i	$-0.586839\pi$
$769$	−14.9437	−0.538884	−0.269442	+	0.963017i	$0.586839\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−47.5643	−1.71077	−0.855385	−	0.517993i	$-0.826679\pi$
$773$	−47.5643	−1.71077	−0.855385	+	0.517993i	$0.826679\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.937113	0.0336188
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 21.8527i	− 0.780953i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.21523i	0.150257i	0.997174	+	0.0751284i	$0.0239367\pi$
$787$	4.21523i	0.150257i	−0.997174	+	0.0751284i	$0.976063\pi$
$788$	0	0
$789$	−33.8461	−1.20495
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	55.7593	1.97510	0.987548	−	0.157317i	$-0.0502844\pi$
$797$	55.7593	1.97510	0.987548	+	0.157317i	$0.0502844\pi$
$798$	0	0
$799$	− 27.2196i	− 0.962961i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 29.5063i	− 1.03867i
$808$	0	0
$809$	−5.67995	−0.199697	−0.0998483	−	0.995003i	$-0.531836\pi$
$809$	−5.67995	−0.199697	−0.0998483	+	0.995003i	$0.531836\pi$
$810$	0	0
$811$	− 54.2158i	− 1.90377i	−0.306449	−	0.951887i	$-0.599141\pi$
$811$	− 54.2158i	− 1.90377i	0.306449	−	0.951887i	$-0.400859\pi$
$812$	0	0
$813$	31.8917	1.11849
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 1.04561i	− 0.0365367i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 55.6778i	− 1.94081i	−0.241488	−	0.970404i	$-0.577635\pi$
$823$	− 55.6778i	− 1.94081i	0.241488	−	0.970404i	$-0.422365\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	56.4722i	1.96373i	0.189579	+	0.981866i	$0.439288\pi$
$827$	56.4722i	1.96373i	−0.189579	+	0.981866i	$0.560712\pi$
$828$	0	0
$829$	37.7081	1.30966	0.654828	−	0.755778i	$-0.272741\pi$
$829$	37.7081	1.30966	0.654828	+	0.755778i	$0.272741\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−31.7150	−1.09886
$834$	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
$841$	−12.2105	−0.421053
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.04807i	0.0360122i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 81.5902i	− 2.79688i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	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
$867$	4.03981i	0.137199i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	102.250i	3.46459i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.2311	0.615620	0.307810	−	0.951448i	$-0.400404\pi$
$877$	18.2311	0.615620	0.307810	+	0.951448i	$0.400404\pi$
$878$	0	0
$879$	− 19.8969i	− 0.671105i
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	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
$891$	0	0
$892$	0	0
$893$	−26.1534	−0.875190
$894$	0	0
$895$	0	0
$896$	0	0
$897$	67.0901	2.24007
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−49.2979	−1.64235
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	47.4466i	1.57544i	0.616035	+	0.787719i	$0.288738\pi$
$907$	47.4466i	1.57544i	−0.616035	+	0.787719i	$0.711262\pi$
$908$	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
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	55.4873i	1.83036i	0.403049	+	0.915178i	$0.367950\pi$
$919$	55.4873i	1.83036i	−0.403049	+	0.915178i	$0.632050\pi$
$920$	0	0
$921$	−9.83544	−0.324089
$922$	0	0
$923$	0	0
$924$	0	0
$925$	43.5890	1.43320
$926$	0	0
$927$	0	0
$928$	0	0
$929$	60.6908	1.99120	0.995601	−	0.0936938i	$-0.0298675\pi$
$929$	60.6908	1.99120	0.995601	+	0.0936938i	$0.0298675\pi$
$930$	0	0
$931$	30.4727i	0.998703i
$932$	0	0
$933$	−11.2039	−0.366798
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42.7350	−1.39609	−0.698046	−	0.716053i	$-0.745947\pi$
$937$	−42.7350	−1.39609	−0.698046	+	0.716053i	$0.745947\pi$
$938$	0	0
$939$	− 16.4296i	− 0.536158i
$940$	0	0
$941$	−35.4517	−1.15569	−0.577846	−	0.816146i	$-0.696107\pi$
$941$	−35.4517	−1.15569	−0.577846	+	0.816146i	$0.696107\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−87.6861	−2.84641
$950$	0	0
$951$	− 38.3881i	− 1.24482i
$952$	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
$959$	− 2.21569i	− 0.0715483i
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 31.6913i	− 1.02124i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 50.0000i	− 1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$	− 50.0000i	− 1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$	0	0
$969$	22.3097	0.716692
$970$	0	0
$971$	− 26.1534i	− 0.839302i	−0.907685	−	0.419651i	$-0.862152\pi$
$971$	− 26.1534i	− 0.839302i	0.907685	−	0.419651i	$-0.137848\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	35.8424i	1.14788i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−22.6657	−0.723661
$982$	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
$987$	0.644966i	0.0205295i
$988$	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
$993$	−2.88141	−0.0914387
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	46.4937i	1.47100i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.b.f.607.7	yes	12
4.3	odd	2	inner	1216.2.b.f.607.6	yes	12
8.3	odd	2	inner	1216.2.b.f.607.8	yes	12
8.5	even	2	inner	1216.2.b.f.607.5	✓	12
19.18	odd	2	inner	1216.2.b.f.607.5	✓	12
76.75	even	2	inner	1216.2.b.f.607.8	yes	12
152.37	odd	2	CM	1216.2.b.f.607.7	yes	12
152.75	even	2	inner	1216.2.b.f.607.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.b.f.607.5	✓	12	8.5	even	2	inner
1216.2.b.f.607.5	✓	12	19.18	odd	2	inner
1216.2.b.f.607.6	yes	12	4.3	odd	2	inner
1216.2.b.f.607.6	yes	12	152.75	even	2	inner
1216.2.b.f.607.7	yes	12	1.1	even	1	trivial
1216.2.b.f.607.7	yes	12	152.37	odd	2	CM
1216.2.b.f.607.8	yes	12	8.3	odd	2	inner
1216.2.b.f.607.8	yes	12	76.75	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.12820i q^{3} -0.0952793i q^{7} +1.72716 q^{9} +6.35390 q^{13} -4.53660 q^{17} +4.35890i q^{19} +0.107494 q^{21} -9.35904i q^{23} +5.00000 q^{25} +5.33319i q^{27} -4.09750 q^{29} +8.71780 q^{37} +7.16848i q^{39} +6.00000i q^{47} +6.99092 q^{49} -5.11820i q^{51} +10.8667 q^{53} -4.91772 q^{57} +11.5796i q^{59} -0.164563i q^{63} +16.0924i q^{67} +10.5589 q^{69} -13.8004 q^{73} +5.64101i q^{75} -0.835437 q^{81} -4.62280i q^{87} -0.605395i q^{91} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{9} + 60 q^{25} - 84 q^{49} + 108 q^{81}+O(q^{100}) \) 12 * q - 36 * q^9 + 60 * q^25 - 84 * q^49 + 108 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more