LMFDB - Embedded newform 5625.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

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:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.4
Root			$-0.209057$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} +3.19236 q^{7} +O(q^{10})$	$q-2.00000 q^{4} +3.19236 q^{7} -0.990863 q^{13} +4.00000 q^{16} -4.28135 q^{19} -6.38473 q^{28} -10.3995 q^{31} +12.1625 q^{37} -11.1716 q^{43} +3.19118 q^{49} +1.98173 q^{52} +4.21917 q^{61} -8.00000 q^{64} -13.2804 q^{67} +10.0881 q^{73} +8.56271 q^{76} -17.7089 q^{79} -3.16320 q^{91} +19.6651 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 5 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 - 5 * q^7	$ 4 q - 8 q^{4} - 5 q^{7} - 5 q^{13} + 16 q^{16} + q^{19} + 10 q^{28} + 7 q^{31} + 10 q^{37} - 5 q^{43} + 17 q^{49} + 10 q^{52} + 13 q^{61} - 32 q^{64} - 5 q^{67} + 10 q^{73} - 2 q^{76} + 4 q^{79} - 25 q^{91} - 5 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 5 * q^7 - 5 * q^13 + 16 * q^16 + q^19 + 10 * q^28 + 7 * q^31 + 10 * q^37 - 5 * q^43 + 17 * q^49 + 10 * q^52 + 13 * q^61 - 32 * q^64 - 5 * q^67 + 10 * q^73 - 2 * q^76 + 4 * q^79 - 25 * q^91 - 5 * q^97

Display 100 coefficients 10 coefficients

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.19236	1.20660	0.603300	−	0.797514i	$-0.293852\pi$
$7$	3.19236	1.20660	0.603300	+	0.797514i	$0.293852\pi$
$8$	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
$13$	−0.990863	−0.274816	−0.137408	−	0.990515i	$-0.543877\pi$
$13$	−0.990863	−0.274816	−0.137408	+	0.990515i	$0.543877\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.28135	−0.982210	−0.491105	−	0.871100i	$-0.663407\pi$
$19$	−4.28135	−0.982210	−0.491105	+	0.871100i	$0.663407\pi$
$20$	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
$25$	0	0
$26$	0	0
$27$	0	0
$28$	−6.38473	−1.20660
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−10.3995	−1.86781	−0.933904	−	0.357525i	$-0.883621\pi$
$31$	−10.3995	−1.86781	−0.933904	+	0.357525i	$0.883621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	12.1625	1.99950	0.999749	−	0.0224255i	$-0.00713887\pi$
$37$	12.1625	1.99950	0.999749	+	0.0224255i	$0.00713887\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−11.1716	−1.70365	−0.851827	−	0.523824i	$-0.824505\pi$
$43$	−11.1716	−1.70365	−0.851827	+	0.523824i	$0.824505\pi$
$44$	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
$49$	3.19118	0.455883
$50$	0	0
$51$	0	0
$52$	1.98173	0.274816
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	4.21917	0.540209	0.270105	−	0.962831i	$-0.412942\pi$
$61$	4.21917	0.540209	0.270105	+	0.962831i	$0.412942\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−13.2804	−1.62246	−0.811231	−	0.584726i	$-0.801202\pi$
$67$	−13.2804	−1.62246	−0.811231	+	0.584726i	$0.801202\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	10.0881	1.18072	0.590359	−	0.807141i	$-0.298986\pi$
$73$	10.0881	1.18072	0.590359	+	0.807141i	$0.298986\pi$
$74$	0	0
$75$	0	0
$76$	8.56271	0.982210
$77$	0	0
$78$	0	0
$79$	−17.7089	−1.99240	−0.996201	−	0.0870877i	$-0.972244\pi$
$79$	−17.7089	−1.99240	−0.996201	+	0.0870877i	$0.972244\pi$
$80$	0	0
$81$	0	0
$82$	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
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.16320	−0.331593
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	19.6651	1.99669	0.998346	−	0.0574829i	$-0.0183075\pi$
$97$	19.6651	1.99669	0.998346	+	0.0574829i	$0.0183075\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−14.1442	−1.39367	−0.696834	−	0.717232i	$-0.745409\pi$
$103$	−14.1442	−1.39367	−0.696834	+	0.717232i	$0.745409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	20.4617	1.95987	0.979937	−	0.199306i	$-0.0638687\pi$
$109$	20.4617	1.95987	0.979937	+	0.199306i	$0.0638687\pi$
$110$	0	0
$111$	0	0
$112$	12.7695	1.20660
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	20.7990	1.86781
$125$	0	0
$126$	0	0
$127$	−3.70333	−0.328617	−0.164309	−	0.986409i	$-0.552539\pi$
$127$	−3.70333	−0.328617	−0.164309	+	0.986409i	$0.552539\pi$
$128$	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
$133$	−13.6676	−1.18513
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	11.5285	0.977835	0.488918	−	0.872330i	$-0.337392\pi$
$139$	11.5285	0.977835	0.488918	+	0.872330i	$0.337392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−24.3249	−1.99950
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−1.12900	−0.0918767	−0.0459384	−	0.998944i	$-0.514628\pi$
$151$	−1.12900	−0.0918767	−0.0459384	+	0.998944i	$0.514628\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−25.0000	−1.99522	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−25.0000	−1.99522	−0.997609	+	0.0691164i	$0.977982\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−25.0000	−1.95815	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−25.0000	−1.95815	−0.979076	+	0.203497i	$0.934769\pi$
$164$	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
$169$	−12.0182	−0.924476
$170$	0	0
$171$	0	0
$172$	22.3432	1.70365
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	20.9342	1.55603	0.778014	−	0.628246i	$-0.216227\pi$
$181$	20.9342	1.55603	0.778014	+	0.628246i	$0.216227\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	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
$193$	−25.0000	−1.79954	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−25.0000	−1.79954	−0.899770	+	0.436365i	$0.856266\pi$
$194$	0	0
$195$	0	0
$196$	−6.38237	−0.455883
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	28.1084	1.99255	0.996274	−	0.0862398i	$-0.0274851\pi$
$199$	28.1084	1.99255	0.996274	+	0.0862398i	$0.0274851\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−3.96345	−0.274816
$209$	0	0
$210$	0	0
$211$	−4.75389	−0.327271	−0.163636	−	0.986521i	$-0.552322\pi$
$211$	−4.75389	−0.327271	−0.163636	+	0.986521i	$0.552322\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−33.1990	−2.25370
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−21.3523	−1.42986	−0.714929	−	0.699197i	$-0.753541\pi$
$223$	−21.3523	−1.42986	−0.714929	+	0.699197i	$0.753541\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−28.5519	−1.88676	−0.943380	−	0.331714i	$-0.892373\pi$
$229$	−28.5519	−1.88676	−0.943380	+	0.331714i	$0.892373\pi$
$230$	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
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−29.0244	−1.86963	−0.934813	−	0.355141i	$-0.884433\pi$
$241$	−29.0244	−1.86963	−0.934813	+	0.355141i	$0.884433\pi$
$242$	0	0
$243$	0	0
$244$	−8.43834	−0.540209
$245$	0	0
$246$	0	0
$247$	4.24223	0.269927
$248$	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
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	38.8270	2.41259
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	26.5608	1.62246
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	32.8332	1.99448	0.997238	−	0.0742738i	$-0.0236639\pi$
$271$	32.8332	1.99448	0.997238	+	0.0742738i	$0.0236639\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.7532	−1.78770	−0.893848	−	0.448370i	$-0.852005\pi$
$277$	−29.7532	−1.78770	−0.893848	+	0.448370i	$0.852005\pi$
$278$	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
$283$	−25.0000	−1.48610	−0.743048	−	0.669238i	$-0.766621\pi$
$283$	−25.0000	−1.48610	−0.743048	+	0.669238i	$0.766621\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	−20.1761	−1.18072
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−35.6638	−2.05563
$302$	0	0
$303$	0	0
$304$	−17.1254	−0.982210
$305$	0	0
$306$	0	0
$307$	−27.2975	−1.55795	−0.778976	−	0.627054i	$-0.784261\pi$
$307$	−27.2975	−1.55795	−0.778976	+	0.627054i	$0.784261\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	5.87376	0.332005	0.166002	−	0.986125i	$-0.446914\pi$
$313$	5.87376	0.332005	0.166002	+	0.986125i	$0.446914\pi$
$314$	0	0
$315$	0	0
$316$	35.4177	1.99240
$317$	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
$331$	−15.7078	−0.863379	−0.431690	−	0.902022i	$-0.642082\pi$
$331$	−15.7078	−0.863379	−0.431690	+	0.902022i	$0.642082\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	36.1379	1.96856	0.984279	−	0.176620i	$-0.0565163\pi$
$337$	36.1379	1.96856	0.984279	+	0.176620i	$0.0565163\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−12.1591	−0.656531
$344$	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
$349$	9.03524	0.483645	0.241823	−	0.970320i	$-0.422255\pi$
$349$	9.03524	0.483645	0.241823	+	0.970320i	$0.422255\pi$
$350$	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
$355$	0	0
$356$	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
$361$	−0.670020	−0.0352642
$362$	0	0
$363$	0	0
$364$	6.32639	0.331593
$365$	0	0
$366$	0	0
$367$	−37.4783	−1.95635	−0.978175	−	0.207785i	$-0.933375\pi$
$367$	−37.4783	−1.95635	−0.978175	+	0.207785i	$0.933375\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−15.7477	−0.808904	−0.404452	−	0.914559i	$-0.632538\pi$
$379$	−15.7477	−0.808904	−0.404452	+	0.914559i	$0.632538\pi$
$380$	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
$388$	−39.3303	−1.99669
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−17.1168	−0.859067	−0.429533	−	0.903051i	$-0.641322\pi$
$397$	−17.1168	−0.859067	−0.429533	+	0.903051i	$0.641322\pi$
$398$	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
$403$	10.3045	0.513303
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−34.2887	−1.69547	−0.847734	−	0.530422i	$-0.822033\pi$
$409$	−34.2887	−1.69547	−0.847734	+	0.530422i	$0.822033\pi$
$410$	0	0
$411$	0	0
$412$	28.2884	1.39367
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	26.1472	1.27434	0.637168	−	0.770725i	$-0.280106\pi$
$421$	26.1472	1.27434	0.637168	+	0.770725i	$0.280106\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.4691	0.651817
$428$	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
$433$	−10.5990	−0.509356	−0.254678	−	0.967026i	$-0.581970\pi$
$433$	−10.5990	−0.509356	−0.254678	+	0.967026i	$0.581970\pi$
$434$	0	0
$435$	0	0
$436$	−40.9234	−1.95987
$437$	0	0
$438$	0	0
$439$	−28.0793	−1.34015	−0.670077	−	0.742292i	$-0.733739\pi$
$439$	−28.0793	−1.34015	−0.670077	+	0.742292i	$0.733739\pi$
$440$	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
$448$	−25.5389	−1.20660
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.5239	1.52141	0.760703	−	0.649100i	$-0.224854\pi$
$457$	32.5239	1.52141	0.760703	+	0.649100i	$0.224854\pi$
$458$	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
$463$	6.21729	0.288942	0.144471	−	0.989509i	$-0.453852\pi$
$463$	6.21729	0.288942	0.144471	+	0.989509i	$0.453852\pi$
$464$	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
$469$	−42.3959	−1.95766
$470$	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
$481$	−12.0513	−0.549494
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	−25.0000	−1.13286	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−25.0000	−1.13286	−0.566429	+	0.824110i	$0.691675\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−41.5980	−1.86781
$497$	0	0
$498$	0	0
$499$	11.0000	0.492428	0.246214	−	0.969216i	$-0.420813\pi$
$499$	11.0000	0.492428	0.246214	+	0.969216i	$0.420813\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	7.40666	0.328617
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	32.2047	1.42465
$512$	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
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	38.8193	1.69745	0.848725	−	0.528834i	$-0.177371\pi$
$523$	38.8193	1.69745	0.848725	+	0.528834i	$0.177371\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	27.3353	1.18513
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.1076	0.774227	0.387113	−	0.922032i	$-0.373472\pi$
$547$	18.1076	0.774227	0.387113	+	0.922032i	$0.373472\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−56.5331	−2.40403
$554$	0	0
$555$	0	0
$556$	−23.0570	−0.977835
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	11.0695	0.468191
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	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
$571$	22.7602	0.952484	0.476242	−	0.879314i	$-0.341999\pi$
$571$	22.7602	0.952484	0.476242	+	0.879314i	$0.341999\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−47.6590	−1.98407	−0.992035	−	0.125962i	$-0.959798\pi$
$577$	−47.6590	−1.98407	−0.992035	+	0.125962i	$0.959798\pi$
$578$	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
$589$	44.5240	1.83458
$590$	0	0
$591$	0	0
$592$	48.6499	1.99950
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	23.0000	0.938190	0.469095	−	0.883148i	$-0.344580\pi$
$601$	23.0000	0.938190	0.469095	+	0.883148i	$0.344580\pi$
$602$	0	0
$603$	0	0
$604$	2.25800	0.0918767
$605$	0	0
$606$	0	0
$607$	−36.6489	−1.48753	−0.743766	−	0.668440i	$-0.766962\pi$
$607$	−36.6489	−1.48753	−0.743766	+	0.668440i	$0.766962\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	43.0336	1.73811	0.869056	−	0.494714i	$-0.164727\pi$
$613$	43.0336	1.73811	0.869056	+	0.494714i	$0.164727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	44.7322	1.79794	0.898970	−	0.438011i	$-0.144317\pi$
$619$	44.7322	1.79794	0.898970	+	0.438011i	$0.144317\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	50.0000	1.99522
$629$	0	0
$630$	0	0
$631$	19.5166	0.776944	0.388472	−	0.921460i	$-0.373003\pi$
$631$	19.5166	0.776944	0.388472	+	0.921460i	$0.373003\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.16203	−0.125284
$638$	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
$643$	−42.0117	−1.65678	−0.828390	−	0.560152i	$-0.810743\pi$
$643$	−42.0117	−1.65678	−0.828390	+	0.560152i	$0.810743\pi$
$644$	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
$652$	50.0000	1.95815
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	44.6882	1.73817	0.869085	−	0.494663i	$-0.164708\pi$
$661$	44.6882	1.73817	0.869085	+	0.494663i	$0.164708\pi$
$662$	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
$673$	50.0000	1.92736	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	50.0000	1.92736	0.963679	+	0.267063i	$0.0860531\pi$
$674$	0	0
$675$	0	0
$676$	24.0364	0.924476
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	62.7783	2.40921
$680$	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
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−44.6864	−1.70365
$689$	0	0
$690$	0	0
$691$	−46.9462	−1.78592	−0.892959	−	0.450138i	$-0.851375\pi$
$691$	−46.9462	−1.78592	−0.892959	+	0.450138i	$0.851375\pi$
$692$	0	0
$693$	0	0
$694$	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
$703$	−52.0718	−1.96393
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−45.1534	−1.68160
$722$	0	0
$723$	0	0
$724$	−41.8685	−1.55603
$725$	0	0
$726$	0	0
$727$	52.6107	1.95122	0.975612	−	0.219504i	$-0.0704439\pi$
$727$	52.6107	1.95122	0.975612	+	0.219504i	$0.0704439\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	50.0000	1.84679	0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000	1.84679	0.923396	+	0.383849i	$0.125402\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−17.5979	−0.647351	−0.323675	−	0.946168i	$-0.604919\pi$
$739$	−17.5979	−0.647351	−0.323675	+	0.946168i	$0.604919\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	31.8881	1.16361	0.581807	−	0.813327i	$-0.302346\pi$
$751$	31.8881	1.16361	0.581807	+	0.813327i	$0.302346\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18.6432	−0.677599	−0.338800	−	0.940859i	$-0.610021\pi$
$757$	−18.6432	−0.677599	−0.338800	+	0.940859i	$0.610021\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	65.3212	2.36478
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	9.56734	0.345007	0.172504	−	0.985009i	$-0.444814\pi$
$769$	9.56734	0.345007	0.172504	+	0.985009i	$0.444814\pi$
$770$	0	0
$771$	0	0
$772$	50.0000	1.79954
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	12.7647	0.455883
$785$	0	0
$786$	0	0
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.18062	−0.148458
$794$	0	0
$795$	0	0
$796$	−56.2167	−1.99255
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−37.0000	−1.29925	−0.649623	−	0.760257i	$-0.725073\pi$
$811$	−37.0000	−1.29925	−0.649623	+	0.760257i	$0.725073\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	47.8296	1.67334
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	29.5514	1.03010	0.515048	−	0.857161i	$-0.327774\pi$
$823$	29.5514	1.03010	0.515048	+	0.857161i	$0.327774\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	57.5763	1.99971	0.999853	−	0.0171408i	$-0.00545636\pi$
$829$	57.5763	1.99971	0.999853	+	0.0171408i	$0.00545636\pi$
$830$	0	0
$831$	0	0
$832$	7.92691	0.274816
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	9.50778	0.327271
$845$	0	0
$846$	0	0
$847$	−35.1160	−1.20660
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	55.2921	1.89317	0.946583	−	0.322461i	$-0.104510\pi$
$853$	55.2921	1.89317	0.946583	+	0.322461i	$0.104510\pi$
$854$	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
$859$	0.832170	0.0283933	0.0141966	−	0.999899i	$-0.495481\pi$
$859$	0.832170	0.0283933	0.0141966	+	0.999899i	$0.495481\pi$
$860$	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
$868$	66.3980	2.25370
$869$	0	0
$870$	0	0
$871$	13.1591	0.445878
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25.0000	−0.844190	−0.422095	−	0.906552i	$-0.638705\pi$
$877$	−25.0000	−0.844190	−0.422095	+	0.906552i	$0.638705\pi$
$878$	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
$883$	31.2610	1.05202	0.526008	−	0.850480i	$-0.323688\pi$
$883$	31.2610	1.05202	0.526008	+	0.850480i	$0.323688\pi$
$884$	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
$889$	−11.8224	−0.396510
$890$	0	0
$891$	0	0
$892$	42.7047	1.42986
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	58.8306	1.95344	0.976719	−	0.214523i	$-0.0688195\pi$
$907$	58.8306	1.95344	0.976719	+	0.214523i	$0.0688195\pi$
$908$	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
$916$	57.1037	1.88676
$917$	0	0
$918$	0	0
$919$	57.3457	1.89166	0.945830	−	0.324661i	$-0.105250\pi$
$919$	57.3457	1.89166	0.945830	+	0.324661i	$0.105250\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−13.6626	−0.447773
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−57.8397	−1.88954	−0.944771	−	0.327731i	$-0.893716\pi$
$937$	−57.8397	−1.88954	−0.944771	+	0.327731i	$0.893716\pi$
$938$	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		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−9.99588	−0.324480
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	77.1498	2.48870
$962$	0	0
$963$	0	0
$964$	58.0488	1.86963
$965$	0	0
$966$	0	0
$967$	62.1878	1.99982	0.999912	−	0.0132362i	$-0.00421334\pi$
$967$	62.1878	1.99982	0.999912	+	0.0132362i	$0.00421334\pi$
$968$	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
$973$	36.8032	1.17986
$974$	0	0
$975$	0	0
$976$	16.8767	0.540209
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	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
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−8.48447	−0.269927
$989$	0	0
$990$	0	0
$991$	21.8793	0.695019	0.347509	−	0.937676i	$-0.387027\pi$
$991$	21.8793	0.695019	0.347509	+	0.937676i	$0.387027\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−28.5605	−0.904520	−0.452260	−	0.891886i	$-0.649382\pi$
$997$	−28.5605	−0.904520	−0.452260	+	0.891886i	$0.649382\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.k.1.4	✓	4
3.2	odd	2	CM	5625.2.a.k.1.4	✓	4
5.4	even	2		5625.2.a.l.1.1	yes	4
15.14	odd	2		5625.2.a.l.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.k.1.4	✓	4	1.1	even	1	trivial
5625.2.a.k.1.4	✓	4	3.2	odd	2	CM
5625.2.a.l.1.1	yes	4	5.4	even	2
5625.2.a.l.1.1	yes	4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} +3.19236 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} +3.19236 q^{7} -0.990863 q^{13} +4.00000 q^{16} -4.28135 q^{19} -6.38473 q^{28} -10.3995 q^{31} +12.1625 q^{37} -11.1716 q^{43} +3.19118 q^{49} +1.98173 q^{52} +4.21917 q^{61} -8.00000 q^{64} -13.2804 q^{67} +10.0881 q^{73} +8.56271 q^{76} -17.7089 q^{79} -3.16320 q^{91} +19.6651 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 5 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 - 5 * q^7	\( 4 q - 8 q^{4} - 5 q^{7} - 5 q^{13} + 16 q^{16} + q^{19} + 10 q^{28} + 7 q^{31} + 10 q^{37} - 5 q^{43} + 17 q^{49} + 10 q^{52} + 13 q^{61} - 32 q^{64} - 5 q^{67} + 10 q^{73} - 2 q^{76} + 4 q^{79} - 25 q^{91} - 5 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 5 * q^7 - 5 * q^13 + 16 * q^16 + q^19 + 10 * q^28 + 7 * q^31 + 10 * q^37 - 5 * q^43 + 17 * q^49 + 10 * q^52 + 13 * q^61 - 32 * q^64 - 5 * q^67 + 10 * q^73 - 2 * q^76 + 4 * q^79 - 25 * q^91 - 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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