LMFDB - Embedded newform 4140.2.n.a.2069.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(2069,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.2069");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.n (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1352 x^{12} - 4836 x^{11} + 18782 x^{10} - 55300 x^{9} + \cdots + 11064600 $ x^16 - 8x^15 + 60x^14 - 280x^13 + 1352x^12 - 4836x^11 + 18782x^10 - 55300x^9 + 177369x^8 - 421148x^7 + 1135954x^6 - 2123100x^5 + 5223234x^4 - 7310160x^3 + 12160080x^2 - 8802000*x + 11064600
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2069.9
Root			$2.65988 + 2.13910i$ of defining polynomial
Character	$\chi$	$=$	4140.2069
Dual form			4140.2.n.a.2069.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} -5.21115 q^{7} +O(q^{10})$	$q+2.23607i q^{5} -5.21115 q^{7} +3.41156i q^{17} -4.79583i q^{23} -5.00000 q^{25} +1.80960i q^{29} +2.76383 q^{31} -11.6525i q^{35} -9.55813 q^{37} -8.54667i q^{41} -13.1069 q^{43} +20.1561 q^{49} +13.5780i q^{53} +13.4007i q^{59} +16.3604 q^{67} -9.78151i q^{71} -18.0501i q^{83} -7.62848 q^{85} +16.7269 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 80 q^{25} + 112 q^{49}+O(q^{100}) $ 16 * q - 80 * q^25 + 112 * q^49

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	0	0
$4$	0	0
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	−5.21115	−1.96963	−0.984816	−	0.173603i	$-0.944459\pi$
$7$	−5.21115	−1.96963	−0.984816	+	0.173603i	$0.944459\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.41156i	0.827425i	0.910408	+	0.413712i	$0.135768\pi$
$17$	3.41156i	0.827425i	−0.910408	+	0.413712i	$0.864232\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.79583i	− 1.00000i
$23$	− 4.79583i	− 1.00000i
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.80960i	0.336034i	0.985784	+	0.168017i	$0.0537364\pi$
$29$	1.80960i	0.336034i	−0.985784	+	0.168017i	$0.946264\pi$
$30$	0	0
$31$	2.76383	0.496398	0.248199	−	0.968709i	$-0.420161\pi$
$31$	2.76383	0.496398	0.248199	+	0.968709i	$0.420161\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 11.6525i	− 1.96963i
$36$	0	0
$37$	−9.55813	−1.57135	−0.785673	−	0.618642i	$-0.787683\pi$
$37$	−9.55813	−1.57135	−0.785673	+	0.618642i	$0.787683\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.54667i	− 1.33477i	−0.744715	−	0.667383i	$-0.767415\pi$
$41$	− 8.54667i	− 1.33477i	0.744715	−	0.667383i	$-0.232585\pi$
$42$	0	0
$43$	−13.1069	−1.99878	−0.999391	−	0.0349050i	$-0.988887\pi$
$43$	−13.1069	−1.99878	−0.999391	+	0.0349050i	$0.988887\pi$
$44$	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
$49$	20.1561	2.87945
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.5780i	1.86508i	0.361070	+	0.932539i	$0.382412\pi$
$53$	13.5780i	1.86508i	−0.361070	+	0.932539i	$0.617588\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.4007i	1.74462i	0.488950	+	0.872312i	$0.337380\pi$
$59$	13.4007i	1.74462i	−0.488950	+	0.872312i	$0.662620\pi$
$60$	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
$67$	16.3604	1.99874	0.999369	−	0.0355060i	$-0.0113043\pi$
$67$	16.3604	1.99874	0.999369	+	0.0355060i	$0.0113043\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 9.78151i	− 1.16085i	−0.814313	−	0.580426i	$-0.802886\pi$
$71$	− 9.78151i	− 1.16085i	0.814313	−	0.580426i	$-0.197114\pi$
$72$	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
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 18.0501i	− 1.98126i	−0.136586	−	0.990628i	$-0.543613\pi$
$83$	− 18.0501i	− 1.98126i	0.136586	−	0.990628i	$-0.456387\pi$
$84$	0	0
$85$	−7.62848	−0.827425
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.7269	1.69836	0.849182	−	0.528101i	$-0.177096\pi$
$97$	16.7269	1.69836	0.849182	+	0.528101i	$0.177096\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 14.7583i	− 1.46851i	−0.678875	−	0.734254i	$-0.737532\pi$
$101$	− 14.7583i	− 1.46851i	0.678875	−	0.734254i	$-0.262468\pi$
$102$	0	0
$103$	5.86678	0.578071	0.289036	−	0.957318i	$-0.406665\pi$
$103$	5.86678	0.578071	0.289036	+	0.957318i	$0.406665\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19.8263i	− 1.91668i	−0.285622	−	0.958342i	$-0.592200\pi$
$107$	− 19.8263i	− 1.91668i	0.285622	−	0.958342i	$-0.407800\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.63370i	− 0.435902i	−0.975960	−	0.217951i	$-0.930063\pi$
$113$	− 4.63370i	− 0.435902i	0.975960	−	0.217951i	$-0.0699372\pi$
$114$	0	0
$115$	10.7238	1.00000
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 17.7782i	− 1.62972i
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 20.8226i	− 1.81928i	−0.415398	−	0.909640i	$-0.636358\pi$
$131$	− 20.8226i	− 1.81928i	0.415398	−	0.909640i	$-0.363642\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.1833i	1.63894i	0.573121	+	0.819471i	$0.305733\pi$
$137$	19.1833i	1.63894i	−0.573121	+	0.819471i	$0.694267\pi$
$138$	0	0
$139$	23.5484	1.99735	0.998676	−	0.0514389i	$-0.0163808\pi$
$139$	23.5484	1.99735	0.998676	+	0.0514389i	$0.0163808\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.04639	−0.336034
$146$	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
$151$	21.4476	1.74538	0.872691	−	0.488273i	$-0.162373\pi$
$151$	21.4476	1.74538	0.872691	+	0.488273i	$0.162373\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.18010i	0.496398i
$156$	0	0
$157$	1.73369	0.138363	0.0691817	−	0.997604i	$-0.477961\pi$
$157$	1.73369	0.138363	0.0691817	+	0.997604i	$0.477961\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.9918i	1.96963i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	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
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	26.0558	1.96963
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3.85204i	− 0.287915i	−0.989584	−	0.143958i	$-0.954017\pi$
$179$	− 3.85204i	− 0.287915i	0.989584	−	0.143958i	$-0.0459829\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 21.3726i	− 1.57135i
$186$	0	0
$187$	0	0
$188$	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
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	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	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 9.43010i	− 0.661863i
$204$	0	0
$205$	19.1109	1.33477
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−18.0208	−1.24060	−0.620301	−	0.784364i	$-0.712990\pi$
$211$	−18.0208	−1.24060	−0.620301	+	0.784364i	$0.712990\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 29.3079i	− 1.99878i
$216$	0	0
$217$	−14.4027	−0.977721
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.94427i	− 0.593652i	−0.954932	−	0.296826i	$-0.904072\pi$
$227$	− 8.94427i	− 0.593652i	0.954932	−	0.296826i	$-0.0959282\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 17.8642i	− 1.15554i	−0.816201	−	0.577768i	$-0.803924\pi$
$239$	− 17.8642i	− 1.15554i	0.816201	−	0.577768i	$-0.196076\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	45.0705i	2.87945i
$246$	0	0
$247$	0	0
$248$	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		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	49.8089	3.09497
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.9944i	1.66454i	0.554367	+	0.832272i	$0.312960\pi$
$263$	26.9944i	1.66454i	−0.554367	+	0.832272i	$0.687040\pi$
$264$	0	0
$265$	−30.3613	−1.86508
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.33501i	− 0.142368i	−0.997463	−	0.0711840i	$-0.977322\pi$
$269$	− 2.33501i	− 0.142368i	0.997463	−	0.0711840i	$-0.0226778\pi$
$270$	0	0
$271$	−24.3613	−1.47984	−0.739921	−	0.672694i	$-0.765137\pi$
$271$	−24.3613	−1.47984	−0.739921	+	0.672694i	$0.765137\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	10.8486	0.644885	0.322442	−	0.946589i	$-0.395496\pi$
$283$	10.8486	0.644885	0.322442	+	0.946589i	$0.395496\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	44.5380i	2.62900i
$288$	0	0
$289$	5.36126	0.315368
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 22.5949i	− 1.32001i	−0.751262	−	0.660004i	$-0.770555\pi$
$293$	− 22.5949i	− 1.32001i	0.751262	−	0.660004i	$-0.229445\pi$
$294$	0	0
$295$	−29.9649	−1.74462
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	68.3020	3.93686
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.63324i	0.262727i	0.991334	+	0.131363i	$0.0419355\pi$
$311$	4.63324i	0.262727i	−0.991334	+	0.131363i	$0.958065\pi$
$312$	0	0
$313$	−29.5332	−1.66932	−0.834659	−	0.550768i	$-0.814335\pi$
$313$	−29.5332	−1.66932	−0.834659	+	0.550768i	$0.814335\pi$
$314$	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
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−36.3613	−1.99860	−0.999298	−	0.0374662i	$-0.988071\pi$
$331$	−36.3613	−1.99860	−0.999298	+	0.0374662i	$0.988071\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	36.5829i	1.99874i
$336$	0	0
$337$	35.7006	1.94474	0.972368	−	0.233454i	$-0.0750028\pi$
$337$	35.7006	1.94474	0.972368	+	0.233454i	$0.0750028\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−68.5586	−3.70182
$344$	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
$349$	−18.3613	−0.982856	−0.491428	−	0.870918i	$-0.663525\pi$
$349$	−18.3613	−0.982856	−0.491428	+	0.870918i	$0.663525\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	21.8721	1.16085
$356$	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
$365$	0	0
$366$	0	0
$367$	−35.4766	−1.85187	−0.925933	−	0.377689i	$-0.876719\pi$
$367$	−35.4766	−1.85187	−0.925933	+	0.377689i	$0.876719\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 70.7569i	− 3.67352i
$372$	0	0
$373$	−23.9670	−1.24097	−0.620483	−	0.784220i	$-0.713063\pi$
$373$	−23.9670	−1.24097	−0.620483	+	0.784220i	$0.713063\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.643015i	0.0328565i	0.999865	+	0.0164283i	$0.00522951\pi$
$383$	0.643015i	0.0328565i	−0.999865	+	0.0164283i	$0.994770\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	16.3613	0.827425
$392$	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	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−28.4131	−1.40494	−0.702468	−	0.711715i	$-0.747919\pi$
$409$	−28.4131	−1.40494	−0.702468	+	0.711715i	$0.747919\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 69.8332i	− 3.43627i
$414$	0	0
$415$	40.3613	1.98126
$416$	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
$423$	0	0
$424$	0	0
$425$	− 17.0578i	− 0.827425i
$426$	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
$433$	−15.0699	−0.724212	−0.362106	−	0.932137i	$-0.617942\pi$
$433$	−15.0699	−0.724212	−0.362106	+	0.932137i	$0.617942\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	21.4476	1.02364	0.511819	−	0.859093i	$-0.328972\pi$
$439$	21.4476	1.02364	0.511819	+	0.859093i	$0.328972\pi$
$440$	0	0
$441$	0	0
$442$	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
$449$	− 20.9700i	− 0.989635i	−0.868997	−	0.494817i	$-0.835235\pi$
$449$	− 20.9700i	− 0.989635i	0.868997	−	0.494817i	$-0.164765\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.2789	1.97772	0.988862	−	0.148838i	$-0.0475533\pi$
$457$	42.2789	1.97772	0.988862	+	0.148838i	$0.0475533\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.9173i	1.34681i	0.739273	+	0.673406i	$0.235169\pi$
$461$	28.9173i	1.34681i	−0.739273	+	0.673406i	$0.764831\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 36.2618i	− 1.67799i	−0.544135	−	0.838997i	$-0.683142\pi$
$467$	− 36.2618i	− 1.67799i	0.544135	−	0.838997i	$-0.316858\pi$
$468$	0	0
$469$	−85.2565	−3.93678
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	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
$485$	37.4026i	1.69836i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.770818i	0.0347865i	0.999849	+	0.0173933i	$0.00553673\pi$
$491$	0.770818i	0.0347865i	−0.999849	+	0.0173933i	$0.994463\pi$
$492$	0	0
$493$	−6.17355	−0.278043
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.9730i	2.28645i
$498$	0	0
$499$	−12.3613	−0.553366	−0.276683	−	0.960961i	$-0.589235\pi$
$499$	−12.3613	−0.553366	−0.276683	+	0.960961i	$0.589235\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.78271i	0.391602i	0.980644	+	0.195801i	$0.0627306\pi$
$503$	8.78271i	0.391602i	−0.980644	+	0.195801i	$0.937269\pi$
$504$	0	0
$505$	33.0006	1.46851
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.4320i	0.905633i	0.891604	+	0.452816i	$0.149581\pi$
$509$	20.4320i	0.905633i	−0.891604	+	0.452816i	$0.850419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.1185i	0.578071i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	24.8404	1.08620	0.543098	−	0.839669i	$-0.317251\pi$
$523$	24.8404	1.08620	0.543098	+	0.839669i	$0.317251\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.42896i	0.410732i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	44.3330	1.91668
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−42.8952	−1.84421	−0.922105	−	0.386940i	$-0.873532\pi$
$541$	−42.8952	−1.84421	−0.922105	+	0.386940i	$0.873532\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	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
$557$	− 43.0642i	− 1.82469i	−0.409422	−	0.912345i	$-0.634270\pi$
$557$	− 43.0642i	− 1.82469i	0.409422	−	0.912345i	$-0.365730\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 45.8328i	− 1.93162i	−0.259248	−	0.965811i	$-0.583475\pi$
$563$	− 45.8328i	− 1.93162i	0.259248	−	0.965811i	$-0.416525\pi$
$564$	0	0
$565$	10.3613	0.435902
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	23.9792i	1.00000i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	94.0619i	3.90234i
$582$	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
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	39.7532	1.62972
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 46.2784i	− 1.89089i	−0.325787	−	0.945443i	$-0.605629\pi$
$599$	− 46.2784i	− 1.89089i	0.325787	−	0.945443i	$-0.394371\pi$
$600$	0	0
$601$	46.3613	1.89112	0.945558	−	0.325455i	$-0.105517\pi$
$601$	46.3613	1.89112	0.945558	+	0.325455i	$0.105517\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 24.5967i	− 1.00000i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−4.99338	−0.201681	−0.100840	−	0.994903i	$-0.532153\pi$
$613$	−4.99338	−0.201681	−0.100840	+	0.994903i	$0.532153\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 49.6782i	− 1.99997i	−0.00563284	−	0.999984i	$-0.501793\pi$
$617$	− 49.6782i	− 1.99997i	0.00563284	−	0.999984i	$-0.498207\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	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
$629$	− 32.6081i	− 1.30017i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−22.5884	−0.890799	−0.445400	−	0.895332i	$-0.646938\pi$
$643$	−22.5884	−0.890799	−0.445400	+	0.895332i	$0.646938\pi$
$644$	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
$649$	0	0
$650$	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
$655$	46.5608	1.81928
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.67853	0.336034
$668$	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
$677$	− 22.8454i	− 0.878019i	−0.898482	−	0.439009i	$-0.855329\pi$
$677$	− 22.8454i	− 0.878019i	0.898482	−	0.439009i	$-0.144671\pi$
$678$	0	0
$679$	−87.1667	−3.34515
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	−42.8952	−1.63894
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	21.4476	0.815906	0.407953	−	0.913003i	$-0.366243\pi$
$691$	21.4476	0.815906	0.407953	+	0.913003i	$0.366243\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	52.6559i	1.99735i
$696$	0	0
$697$	29.1575	1.10442
$698$	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
$707$	76.9079i	2.89242i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 13.2548i	− 0.496398i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	52.0508i	1.94117i	0.240763	+	0.970584i	$0.422602\pi$
$719$	52.0508i	1.94117i	−0.240763	+	0.970584i	$0.577398\pi$
$720$	0	0
$721$	−30.5727	−1.13859
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 9.04800i	− 0.336034i
$726$	0	0
$727$	−24.4532	−0.906917	−0.453459	−	0.891277i	$-0.649810\pi$
$727$	−24.4532	−0.906917	−0.453459	+	0.891277i	$0.649810\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 44.7149i	− 1.65384i
$732$	0	0
$733$	−43.4229	−1.60386	−0.801931	−	0.597416i	$-0.796194\pi$
$733$	−43.4229	−1.60386	−0.801931	+	0.597416i	$0.796194\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−34.6037	−1.27292	−0.636460	−	0.771310i	$-0.719602\pi$
$739$	−34.6037	−1.27292	−0.636460	+	0.771310i	$0.719602\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9.59166i	− 0.351884i	−0.984401	−	0.175942i	$-0.943703\pi$
$743$	− 9.59166i	− 0.351884i	0.984401	−	0.175942i	$-0.0562971\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	103.318i	3.77516i
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	47.9583i	1.74538i
$756$	0	0
$757$	1.46535	0.0532592	0.0266296	−	0.999645i	$-0.491523\pi$
$757$	1.46535	0.0532592	0.0266296	+	0.999645i	$0.491523\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	55.1567i	1.99943i	0.0239357	+	0.999714i	$0.492380\pi$
$761$	55.1567i	1.99943i	−0.0239357	+	0.999714i	$0.507620\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38.3667i	− 1.37995i	−0.723832	−	0.689976i	$-0.757621\pi$
$773$	− 38.3667i	− 1.37995i	0.723832	−	0.689976i	$-0.242379\pi$
$774$	0	0
$775$	−13.8191	−0.496398
$776$	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
$785$	3.87665i	0.138363i
$786$	0	0
$787$	−19.1008	−0.680871	−0.340436	−	0.940268i	$-0.610575\pi$
$787$	−19.1008	−0.680871	−0.340436	+	0.940268i	$0.610575\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24.1469i	0.858565i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.94865i	0.316977i	0.987361	+	0.158489i	$0.0506621\pi$
$797$	8.94865i	0.316977i	−0.987361	+	0.158489i	$0.949338\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−55.8834	−1.96963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 32.2302i	− 1.13315i	−0.824009	−	0.566577i	$-0.808267\pi$
$809$	− 32.2302i	− 1.13315i	0.824009	−	0.566577i	$-0.191733\pi$
$810$	0	0
$811$	4.36126	0.153145	0.0765723	−	0.997064i	$-0.475602\pi$
$811$	4.36126	0.153145	0.0765723	+	0.997064i	$0.475602\pi$
$812$	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
$821$	3.46144i	0.120805i	0.998174	+	0.0604026i	$0.0192385\pi$
$821$	3.46144i	0.120805i	−0.998174	+	0.0604026i	$0.980762\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.3732i	0.638900i	0.947603	+	0.319450i	$0.103498\pi$
$827$	18.3732i	0.638900i	−0.947603	+	0.319450i	$0.896502\pi$
$828$	0	0
$829$	−44.9960	−1.56278	−0.781389	−	0.624045i	$-0.785488\pi$
$829$	−44.9960	−1.56278	−0.781389	+	0.624045i	$0.785488\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	68.7638i	2.38253i
$834$	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
$841$	25.7254	0.887081
$842$	0	0
$843$	0	0
$844$	0	0
$845$	29.0689i	1.00000i
$846$	0	0
$847$	57.3227	1.96963
$848$	0	0
$849$	0	0
$850$	0	0
$851$	45.8392i	1.57135i
$852$	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		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	52.3613	1.78654	0.893272	−	0.449517i	$-0.148404\pi$
$859$	52.3613	1.78654	0.893272	+	0.449517i	$0.148404\pi$
$860$	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
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	58.2625i	1.96963i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	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
$893$	0	0
$894$	0	0
$895$	8.61343	0.287915
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.00142i	0.166807i
$900$	0	0
$901$	−46.3220	−1.54321
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	43.4330	1.44217	0.721085	−	0.692847i	$-0.243644\pi$
$907$	43.4330	1.44217	0.721085	+	0.692847i	$0.243644\pi$
$908$	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
$917$	108.510i	3.58331i
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	47.7906	1.57135
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.1342i	0.463729i	0.972748	+	0.231864i	$0.0744826\pi$
$929$	14.1342i	0.463729i	−0.972748	+	0.231864i	$0.925517\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	54.6743	1.78613	0.893065	−	0.449927i	$-0.148550\pi$
$937$	54.6743	1.78613	0.893065	+	0.449927i	$0.148550\pi$
$938$	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
$943$	−40.9884	−1.33477
$944$	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
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 22.3607i	− 0.724333i	−0.932113	−	0.362167i	$-0.882037\pi$
$953$	− 22.3607i	− 0.724333i	0.932113	−	0.362167i	$-0.117963\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 99.9673i	− 3.22811i
$960$	0	0
$961$	−23.3613	−0.753589
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−122.715	−3.93405
$974$	0	0
$975$	0	0
$976$	0	0
$977$	55.4244i	1.77319i	0.462551	+	0.886593i	$0.346934\pi$
$977$	55.4244i	1.77319i	−0.462551	+	0.886593i	$0.653066\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 25.3634i	− 0.808968i	−0.914545	−	0.404484i	$-0.867451\pi$
$983$	− 25.3634i	− 0.808968i	0.914545	−	0.404484i	$-0.132549\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	62.8584i	1.99878i
$990$	0	0
$991$	−55.3884	−1.75947	−0.879734	−	0.475465i	$-0.842280\pi$
$991$	−55.3884	−1.75947	−0.879734	+	0.475465i	$0.842280\pi$
$992$	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	4140.2.n.a.2069.9	yes	16
3.2	odd	2	inner	4140.2.n.a.2069.1	✓	16
5.4	even	2	inner	4140.2.n.a.2069.8	yes	16
15.14	odd	2	inner	4140.2.n.a.2069.16	yes	16
23.22	odd	2	inner	4140.2.n.a.2069.8	yes	16
69.68	even	2	inner	4140.2.n.a.2069.16	yes	16
115.114	odd	2	CM	4140.2.n.a.2069.9	yes	16
345.344	even	2	inner	4140.2.n.a.2069.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.n.a.2069.1	✓	16	3.2	odd	2	inner
4140.2.n.a.2069.1	✓	16	345.344	even	2	inner
4140.2.n.a.2069.8	yes	16	5.4	even	2	inner
4140.2.n.a.2069.8	yes	16	23.22	odd	2	inner
4140.2.n.a.2069.9	yes	16	1.1	even	1	trivial
4140.2.n.a.2069.9	yes	16	115.114	odd	2	CM
4140.2.n.a.2069.16	yes	16	15.14	odd	2	inner
4140.2.n.a.2069.16	yes	16	69.68	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} -5.21115 q^{7} +O(q^{10})\)	\(q+2.23607i q^{5} -5.21115 q^{7} +3.41156i q^{17} -4.79583i q^{23} -5.00000 q^{25} +1.80960i q^{29} +2.76383 q^{31} -11.6525i q^{35} -9.55813 q^{37} -8.54667i q^{41} -13.1069 q^{43} +20.1561 q^{49} +13.5780i q^{53} +13.4007i q^{59} +16.3604 q^{67} -9.78151i q^{71} -18.0501i q^{83} -7.62848 q^{85} +16.7269 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 80 q^{25} + 112 q^{49}+O(q^{100}) \) 16 * q - 80 * q^25 + 112 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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