LMFDB - Embedded newform 4140.2.n.a.2069.5

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			2069.5
Root			$-0.0787370 + 1.43199i$ of defining polynomial
Character	$\chi$	$=$	4140.2069
Dual form			4140.2.n.a.2069.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +0.918623 q^{7} +O(q^{10})$	$q-2.23607i q^{5} +0.918623 q^{7} +3.41156i q^{17} -4.79583i q^{23} -5.00000 q^{25} -10.6172i q^{29} -2.76383 q^{31} -2.05410i q^{35} +7.52610 q^{37} +9.53701i q^{41} +0.457775 q^{43} -6.15613 q^{49} -13.5780i q^{53} -7.51139i q^{59} +0.581259 q^{67} -13.7230i q^{71} +18.0501i q^{83} +7.62848 q^{85} -10.4024 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	0.918623	0.347207	0.173603	−	0.984816i	$-0.444459\pi$
$7$	0.918623	0.347207	0.173603	+	0.984816i	$0.444459\pi$
$8$	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
$29$	− 10.6172i	− 1.97157i	−0.168017	−	0.985784i	$-0.553736\pi$
$29$	− 10.6172i	− 1.97157i	0.168017	−	0.985784i	$-0.446264\pi$
$30$	0	0
$31$	−2.76383	−0.496398	−0.248199	−	0.968709i	$-0.579839\pi$
$31$	−2.76383	−0.496398	−0.248199	+	0.968709i	$0.579839\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.05410i	− 0.347207i
$36$	0	0
$37$	7.52610	1.23728	0.618642	−	0.785673i	$-0.287683\pi$
$37$	7.52610	1.23728	0.618642	+	0.785673i	$0.287683\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.53701i	1.48943i	0.667383	+	0.744715i	$0.267415\pi$
$41$	9.53701i	1.48943i	−0.667383	+	0.744715i	$0.732585\pi$
$42$	0	0
$43$	0.457775	0.0698100	0.0349050	−	0.999391i	$-0.488887\pi$
$43$	0.457775	0.0698100	0.0349050	+	0.999391i	$0.488887\pi$
$44$	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
$49$	−6.15613	−0.879447
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 13.5780i	− 1.86508i	−0.361070	−	0.932539i	$-0.617588\pi$
$53$	− 13.5780i	− 1.86508i	0.361070	−	0.932539i	$-0.382412\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 7.51139i	− 0.977900i	−0.872312	−	0.488950i	$-0.837380\pi$
$59$	− 7.51139i	− 0.977900i	0.872312	−	0.488950i	$-0.162620\pi$
$60$	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
$67$	0.581259	0.0710120	0.0355060	−	0.999369i	$-0.488696\pi$
$67$	0.581259	0.0710120	0.0355060	+	0.999369i	$0.488696\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 13.7230i	− 1.62863i	−0.580426	−	0.814313i	$-0.697114\pi$
$71$	− 13.7230i	− 1.62863i	0.580426	−	0.814313i	$-0.302886\pi$
$72$	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	$0.456387\pi$
$83$	18.0501i	1.98126i	−0.136586	+	0.990628i	$0.543613\pi$
$84$	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
$97$	−10.4024	−1.05620	−0.528101	−	0.849182i	$-0.677096\pi$
$97$	−10.4024	−1.05620	−0.528101	+	0.849182i	$0.677096\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 13.6452i	− 1.35775i	−0.734254	−	0.678875i	$-0.762468\pi$
$101$	− 13.6452i	− 1.35775i	0.734254	−	0.678875i	$-0.237532\pi$
$102$	0	0
$103$	19.4314	1.91464	0.957318	−	0.289036i	$-0.0933346\pi$
$103$	19.4314	1.91464	0.957318	+	0.289036i	$0.0933346\pi$
$104$	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	$0.0699372\pi$
$113$	4.63370i	0.435902i	−0.975960	+	0.217951i	$0.930063\pi$
$114$	0	0
$115$	−10.7238	−1.00000
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.13394i	0.287288i
$120$	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
$131$	− 9.50890i	− 0.830796i	−0.909640	−	0.415398i	$-0.863642\pi$
$131$	− 9.50890i	− 0.830796i	0.909640	−	0.415398i	$-0.136358\pi$
$132$	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	$-0.983619\pi$
$139$	−23.5484	−1.99735	−0.998676	+	0.0514389i	$0.983619\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−23.7408	−1.97157
$146$	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	$-0.837627\pi$
$151$	−21.4476	−1.74538	−0.872691	+	0.488273i	$0.837627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.18010i	0.496398i
$156$	0	0
$157$	−24.9999	−1.99521	−0.997604	−	0.0691817i	$-0.977961\pi$
$157$	−24.9999	−1.99521	−0.997604	+	0.0691817i	$0.977961\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.40556i	− 0.347207i
$162$	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
$175$	−4.59312	−0.347207
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 26.4795i	− 1.97917i	−0.143958	−	0.989584i	$-0.545983\pi$
$179$	− 26.4795i	− 1.97917i	0.143958	−	0.989584i	$-0.454017\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 16.8289i	− 1.23728i
$186$	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
$203$	− 9.75322i	− 0.684542i
$204$	0	0
$205$	21.3254	1.48943
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	18.0208	1.24060	0.620301	−	0.784364i	$-0.287010\pi$
$211$	18.0208	1.24060	0.620301	+	0.784364i	$0.287010\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.02362i	− 0.0698100i
$216$	0	0
$217$	−2.53892	−0.172353
$218$	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	$0.0959282\pi$
$227$	8.94427i	0.593652i	−0.954932	+	0.296826i	$0.904072\pi$
$228$	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
$239$	− 25.2363i	− 1.63240i	−0.577768	−	0.816201i	$-0.696076\pi$
$239$	− 25.2363i	− 1.63240i	0.577768	−	0.816201i	$-0.303924\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	13.7655i	0.879447i
$246$	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
$259$	6.91365	0.429593
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 26.9944i	− 1.66454i	−0.554367	−	0.832272i	$-0.687040\pi$
$263$	− 26.9944i	− 1.66454i	0.554367	−	0.832272i	$-0.312960\pi$
$264$	0	0
$265$	−30.3613	−1.86508
$266$	0	0
$267$	0	0
$268$	0	0
$269$	32.7192i	1.99493i	0.0711840	+	0.997463i	$0.477322\pi$
$269$	32.7192i	1.99493i	−0.0711840	+	0.997463i	$0.522678\pi$
$270$	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
$283$	31.8482	1.89318	0.946589	−	0.322442i	$-0.104504\pi$
$283$	31.8482	1.89318	0.946589	+	0.322442i	$0.104504\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.76092i	0.517140i
$288$	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
$295$	−16.7960	−0.977900
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0.420522	0.0242385
$302$	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
$311$	− 34.9647i	− 1.98267i	−0.131363	−	0.991334i	$-0.541935\pi$
$311$	− 34.9647i	− 1.98267i	0.131363	−	0.991334i	$-0.458065\pi$
$312$	0	0
$313$	−19.4881	−1.10154	−0.550768	−	0.834659i	$-0.685665\pi$
$313$	−19.4881	−1.10154	−0.550768	+	0.834659i	$0.685665\pi$
$314$	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
$335$	− 1.29973i	− 0.0710120i
$336$	0	0
$337$	8.57128	0.466908	0.233454	−	0.972368i	$-0.424997\pi$
$337$	8.57128	0.466908	0.233454	+	0.972368i	$0.424997\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−12.0855	−0.652557
$344$	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
$355$	−30.6857	−1.62863
$356$	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
$367$	14.4709	0.755377	0.377689	−	0.925933i	$-0.376719\pi$
$367$	14.4709	0.755377	0.377689	+	0.925933i	$0.376719\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 12.4730i	− 0.647568i
$372$	0	0
$373$	30.2916	1.56844	0.784220	−	0.620483i	$-0.213063\pi$
$373$	30.2916	1.56844	0.784220	+	0.620483i	$0.213063\pi$
$374$	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	$-0.252081\pi$
$409$	28.4131	1.40494	0.702468	+	0.711715i	$0.252081\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.90014i	− 0.339534i
$414$	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
$433$	38.7930	1.86427	0.932137	−	0.362106i	$-0.117942\pi$
$433$	38.7930	1.86427	0.932137	+	0.362106i	$0.117942\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−21.4476	−1.02364	−0.511819	−	0.859093i	$-0.671028\pi$
$439$	−21.4476	−1.02364	−0.511819	+	0.859093i	$0.671028\pi$
$440$	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
$449$	− 36.8274i	− 1.73799i	−0.494817	−	0.868997i	$-0.664765\pi$
$449$	− 36.8274i	− 1.73799i	0.494817	−	0.868997i	$-0.335235\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.36359	−0.297676	−0.148838	−	0.988862i	$-0.547553\pi$
$457$	−6.36359	−0.297676	−0.148838	+	0.988862i	$0.547553\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.7457i	1.47855i	0.673406	+	0.739273i	$0.264831\pi$
$461$	31.7457i	1.47855i	−0.673406	+	0.739273i	$0.735169\pi$
$462$	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	$0.316858\pi$
$467$	36.2618i	1.67799i	−0.544135	+	0.838997i	$0.683142\pi$
$468$	0	0
$469$	0.533958	0.0246559
$470$	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
$485$	23.2604i	1.05620i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	44.3103i	1.99970i	0.0173933	+	0.999849i	$0.494463\pi$
$491$	44.3103i	1.99970i	−0.0173933	+	0.999849i	$0.505537\pi$
$492$	0	0
$493$	36.2213	1.63132
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.6063i	− 0.565470i
$498$	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	$-0.937269\pi$
$503$	− 8.78271i	− 0.391602i	0.980644	−	0.195801i	$-0.0627306\pi$
$504$	0	0
$505$	−30.5116	−1.35775
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.2310i	1.78321i	0.452816	+	0.891604i	$0.350419\pi$
$509$	40.2310i	1.78321i	−0.452816	+	0.891604i	$0.649581\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 43.4500i	− 1.91464i
$516$	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
$523$	38.4051	1.67934	0.839669	−	0.543098i	$-0.182749\pi$
$523$	38.4051	1.67934	0.839669	+	0.543098i	$0.182749\pi$
$524$	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	$-0.126468\pi$
$541$	42.8952	1.84421	0.922105	+	0.386940i	$0.126468\pi$
$542$	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
$581$	16.5812i	0.687906i
$582$	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
$595$	7.00770	0.287288
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15.9469i	0.651575i	0.945443	+	0.325787i	$0.105629\pi$
$599$	15.9469i	0.651575i	−0.945443	+	0.325787i	$0.894371\pi$
$600$	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
$613$	49.2653	1.98981	0.994903	−	0.100840i	$-0.0321531\pi$
$613$	49.2653	1.98981	0.994903	+	0.100840i	$0.0321531\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	49.6782i	1.99997i	0.00563284	+	0.999984i	$0.498207\pi$
$617$	49.6782i	1.99997i	−0.00563284	+	0.999984i	$0.501793\pi$
$618$	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
$629$	25.6757i	1.02376i
$630$	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
$643$	−45.4067	−1.79066	−0.895332	−	0.445400i	$-0.853062\pi$
$643$	−45.4067	−1.79066	−0.895332	+	0.445400i	$0.853062\pi$
$644$	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
$655$	−21.2625	−0.830796
$656$	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
$667$	−50.9184	−1.97157
$668$	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	$0.144671\pi$
$677$	22.8454i	0.878019i	−0.898482	+	0.439009i	$0.855329\pi$
$678$	0	0
$679$	−9.55587	−0.366721
$680$	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	$-0.633757\pi$
$691$	−21.4476	−0.815906	−0.407953	+	0.913003i	$0.633757\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	52.6559i	1.99735i
$696$	0	0
$697$	−32.5361	−1.23239
$698$	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
$707$	− 12.5348i	− 0.471420i
$708$	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
$719$	− 12.9117i	− 0.481525i	−0.970584	−	0.240763i	$-0.922602\pi$
$719$	− 12.9117i	− 0.481525i	0.970584	−	0.240763i	$-0.0773975\pi$
$720$	0	0
$721$	17.8502	0.664775
$722$	0	0
$723$	0	0
$724$	0	0
$725$	53.0861i	1.97157i
$726$	0	0
$727$	−48.0629	−1.78255	−0.891277	−	0.453459i	$-0.850190\pi$
$727$	−48.0629	−1.78255	−0.891277	+	0.453459i	$0.850190\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.56173i	0.0577625i
$732$	0	0
$733$	32.3489	1.19483	0.597416	−	0.801931i	$-0.296194\pi$
$733$	32.3489	1.19483	0.597416	+	0.801931i	$0.296194\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	34.6037	1.27292	0.636460	−	0.771310i	$-0.280398\pi$
$739$	34.6037	1.27292	0.636460	+	0.771310i	$0.280398\pi$
$740$	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
$749$	− 18.2129i	− 0.665486i
$750$	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
$757$	−55.0078	−1.99929	−0.999645	−	0.0266296i	$-0.991523\pi$
$757$	−55.0078	−1.99929	−0.999645	+	0.0266296i	$0.991523\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1.32059i	− 0.0478713i	−0.999714	−	0.0239357i	$-0.992380\pi$
$761$	− 1.32059i	− 0.0478713i	0.999714	−	0.0239357i	$-0.00761969\pi$
$762$	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
$785$	55.9014i	1.99521i
$786$	0	0
$787$	52.7556	1.88054	0.940268	−	0.340436i	$-0.110575\pi$
$787$	52.7556	1.88054	0.940268	+	0.340436i	$0.110575\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.25662i	0.151348i
$792$	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
$805$	−9.85114	−0.347207
$806$	0	0
$807$	0	0
$808$	0	0
$809$	46.8744i	1.64802i	0.566577	+	0.824009i	$0.308267\pi$
$809$	46.8744i	1.64802i	−0.566577	+	0.824009i	$0.691733\pi$
$810$	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
$821$	57.2016i	1.99635i	0.0604026	+	0.998174i	$0.480762\pi$
$821$	57.2016i	1.99635i	−0.0604026	+	0.998174i	$0.519238\pi$
$822$	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	$-0.896502\pi$
$827$	− 18.3732i	− 0.638900i	0.947603	−	0.319450i	$-0.103498\pi$
$828$	0	0
$829$	44.9960	1.56278	0.781389	−	0.624045i	$-0.214512\pi$
$829$	44.9960	1.56278	0.781389	+	0.624045i	$0.214512\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 21.0020i	− 0.727676i
$834$	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
$841$	−83.7254	−2.88708
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 29.0689i	− 1.00000i
$846$	0	0
$847$	−10.1049	−0.347207
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 36.0939i	− 1.23728i
$852$	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
$875$	10.2705i	0.347207i
$876$	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
$895$	−59.2099	−1.97917
$896$	0	0
$897$	0	0
$898$	0	0
$899$	29.3442i	0.978682i
$900$	0	0
$901$	46.3220	1.54321
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	41.7322	1.38569	0.692847	−	0.721085i	$-0.256356\pi$
$907$	41.7322	1.38569	0.692847	+	0.721085i	$0.256356\pi$
$908$	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
$917$	− 8.73509i	− 0.288458i
$918$	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
$925$	−37.6305	−1.23728
$926$	0	0
$927$	0	0
$928$	0	0
$929$	59.2978i	1.94550i	0.231864	+	0.972748i	$0.425517\pi$
$929$	59.2978i	1.94550i	−0.231864	+	0.972748i	$0.574483\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.5449	0.899854	0.449927	−	0.893065i	$-0.351450\pi$
$937$	27.5449	0.899854	0.449927	+	0.893065i	$0.351450\pi$
$938$	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
$943$	45.7379	1.48943
$944$	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	$0.117963\pi$
$953$	22.3607i	0.724333i	−0.932113	+	0.362167i	$0.882037\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.6222i	0.569052i
$960$	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
$973$	−21.6321	−0.693495
$974$	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
$989$	− 2.19541i	− 0.0698100i
$990$	0	0
$991$	55.3884	1.75947	0.879734	−	0.475465i	$-0.157720\pi$
$991$	55.3884	1.75947	0.879734	+	0.475465i	$0.157720\pi$
$992$	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.2069.5	yes	16
3.2	odd	2	inner	4140.2.n.a.2069.13	yes	16
5.4	even	2	inner	4140.2.n.a.2069.12	yes	16
15.14	odd	2	inner	4140.2.n.a.2069.4	✓	16
23.22	odd	2	inner	4140.2.n.a.2069.12	yes	16
69.68	even	2	inner	4140.2.n.a.2069.4	✓	16
115.114	odd	2	CM	4140.2.n.a.2069.5	yes	16
345.344	even	2	inner	4140.2.n.a.2069.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.n.a.2069.4	✓	16	15.14	odd	2	inner
4140.2.n.a.2069.4	✓	16	69.68	even	2	inner
4140.2.n.a.2069.5	yes	16	1.1	even	1	trivial
4140.2.n.a.2069.5	yes	16	115.114	odd	2	CM
4140.2.n.a.2069.12	yes	16	5.4	even	2	inner
4140.2.n.a.2069.12	yes	16	23.22	odd	2	inner
4140.2.n.a.2069.13	yes	16	3.2	odd	2	inner
4140.2.n.a.2069.13	yes	16	345.344	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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +0.918623 q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +0.918623 q^{7} +3.41156i q^{17} -4.79583i q^{23} -5.00000 q^{25} -10.6172i q^{29} -2.76383 q^{31} -2.05410i q^{35} +7.52610 q^{37} +9.53701i q^{41} +0.457775 q^{43} -6.15613 q^{49} -13.5780i q^{53} -7.51139i q^{59} +0.581259 q^{67} -13.7230i q^{71} +18.0501i q^{83} +7.62848 q^{85} -10.4024 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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