LMFDB - Embedded newform 7569.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, 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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$-1.95630$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} -5.25085 q^{7} +O(q^{10})$	$q-2.00000 q^{4} -5.25085 q^{7} +2.45426 q^{13} +4.00000 q^{16} -5.89939 q^{19} -5.00000 q^{25} +10.5017 q^{28} +11.1197 q^{31} +5.84497 q^{37} +9.98586 q^{43} +20.5714 q^{49} -4.90852 q^{52} +12.2536 q^{61} -8.00000 q^{64} -16.0257 q^{67} +0.0544214 q^{73} +11.7988 q^{76} -17.6437 q^{79} -12.8870 q^{91} +8.85199 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 + q^7	$ 4 q - 8 q^{4} + q^{7} - 2 q^{13} + 16 q^{16} - q^{19} - 20 q^{25} - 2 q^{28} - 4 q^{31} + 11 q^{37} + 5 q^{43} + 41 q^{49} + 4 q^{52} - 13 q^{61} - 32 q^{64} - 11 q^{67} - 10 q^{73} + 2 q^{76} - 13 q^{79} + 2 q^{91} + 14 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + q^7 - 2 * q^13 + 16 * q^16 - q^19 - 20 * q^25 - 2 * q^28 - 4 * q^31 + 11 * q^37 + 5 * q^43 + 41 * q^49 + 4 * q^52 - 13 * q^61 - 32 * q^64 - 11 * q^67 - 10 * q^73 + 2 * q^76 - 13 * q^79 + 2 * q^91 + 14 * 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		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−5.25085	−1.98464	−0.992318	−	0.123716i	$-0.960519\pi$
$7$	−5.25085	−1.98464	−0.992318	+	0.123716i	$0.960519\pi$
$8$	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
$13$	2.45426	0.680690	0.340345	−	0.940301i	$-0.389456\pi$
$13$	2.45426	0.680690	0.340345	+	0.940301i	$0.389456\pi$
$14$	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
$19$	−5.89939	−1.35341	−0.676706	−	0.736253i	$-0.736593\pi$
$19$	−5.89939	−1.35341	−0.676706	+	0.736253i	$0.736593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	10.5017	1.98464
$29$	0	0
$29$	0	0
$30$	0	0
$31$	11.1197	1.99716	0.998582	−	0.0532375i	$-0.0169540\pi$
$31$	11.1197	1.99716	0.998582	+	0.0532375i	$0.0169540\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.84497	0.960906	0.480453	−	0.877020i	$-0.340472\pi$
$37$	5.84497	0.960906	0.480453	+	0.877020i	$0.340472\pi$
$38$	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
$43$	9.98586	1.52283	0.761415	−	0.648265i	$-0.224505\pi$
$43$	9.98586	1.52283	0.761415	+	0.648265i	$0.224505\pi$
$44$	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
$49$	20.5714	2.93878
$50$	0	0
$51$	0	0
$52$	−4.90852	−0.680690
$53$	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
$61$	12.2536	1.56891	0.784457	−	0.620183i	$-0.212942\pi$
$61$	12.2536	1.56891	0.784457	+	0.620183i	$0.212942\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−16.0257	−1.95785	−0.978926	−	0.204217i	$-0.934535\pi$
$67$	−16.0257	−1.95785	−0.978926	+	0.204217i	$0.934535\pi$
$68$	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
$73$	0.0544214	0.00636955	0.00318477	−	0.999995i	$-0.498986\pi$
$73$	0.0544214	0.00636955	0.00318477	+	0.999995i	$0.498986\pi$
$74$	0	0
$75$	0	0
$76$	11.7988	1.35341
$77$	0	0
$78$	0	0
$79$	−17.6437	−1.98508	−0.992538	−	0.121937i	$-0.961089\pi$
$79$	−17.6437	−1.98508	−0.992538	+	0.121937i	$0.961089\pi$
$80$	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
$91$	−12.8870	−1.35092
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.85199	0.898783	0.449392	−	0.893335i	$-0.351641\pi$
$97$	8.85199	0.898783	0.449392	+	0.893335i	$0.351641\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	9.47490	0.933589	0.466795	−	0.884366i	$-0.345409\pi$
$103$	9.47490	0.933589	0.466795	+	0.884366i	$0.345409\pi$
$104$	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
$109$	−6.27766	−0.601291	−0.300645	−	0.953736i	$-0.597202\pi$
$109$	−6.27766	−0.601291	−0.300645	+	0.953736i	$0.597202\pi$
$110$	0	0
$111$	0	0
$112$	−21.0034	−1.98464
$113$	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
$124$	−22.2395	−1.99716
$125$	0	0
$126$	0	0
$127$	−21.1056	−1.87282	−0.936410	−	0.350909i	$-0.885873\pi$
$127$	−21.1056	−1.87282	−0.936410	+	0.350909i	$0.885873\pi$
$128$	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
$133$	30.9768	2.68603
$134$	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
$148$	−11.6899	−0.960906
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	6.20866	0.505253	0.252627	−	0.967564i	$-0.418706\pi$
$151$	6.20866	0.505253	0.252627	+	0.967564i	$0.418706\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	23.5431	1.87895	0.939473	−	0.342623i	$-0.111315\pi$
$157$	23.5431	1.87895	0.939473	+	0.342623i	$0.111315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−23.3733	−1.83074	−0.915371	−	0.402611i	$-0.868103\pi$
$163$	−23.3733	−1.83074	−0.915371	+	0.402611i	$0.868103\pi$
$164$	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
$169$	−6.97660	−0.536662
$170$	0	0
$171$	0	0
$172$	−19.9717	−1.52283
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	26.2543	1.98464
$176$	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
$193$	7.71811	0.555562	0.277781	−	0.960644i	$-0.410401\pi$
$193$	7.71811	0.555562	0.277781	+	0.960644i	$0.410401\pi$
$194$	0	0
$195$	0	0
$196$	−41.1429	−2.93878
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	24.6886	1.75013	0.875065	−	0.484005i	$-0.160818\pi$
$199$	24.6886	1.75013	0.875065	+	0.484005i	$0.160818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	9.81705	0.680690
$209$	0	0
$210$	0	0
$211$	−29.0000	−1.99644	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−29.0000	−1.99644	−0.998221	+	0.0596196i	$0.981011\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−58.3881	−3.96364
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	25.2275	1.68936	0.844678	−	0.535275i	$-0.179792\pi$
$223$	25.2275	1.68936	0.844678	+	0.535275i	$0.179792\pi$
$224$	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
$229$	−29.0000	−1.91637	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−29.0000	−1.91637	−0.958187	+	0.286143i	$0.907627\pi$
$230$	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
$241$	−22.0302	−1.41909	−0.709545	−	0.704660i	$-0.751100\pi$
$241$	−22.0302	−1.41909	−0.709545	+	0.704660i	$0.751100\pi$
$242$	0	0
$243$	0	0
$244$	−24.5072	−1.56891
$245$	0	0
$246$	0	0
$247$	−14.4786	−0.921254
$248$	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
$259$	−30.6910	−1.90705
$260$	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
$268$	32.0514	1.95785
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.3885	−1.40528	−0.702639	−	0.711546i	$-0.747995\pi$
$277$	−23.3885	−1.40528	−0.702639	+	0.711546i	$0.747995\pi$
$278$	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
$283$	−19.7801	−1.17580	−0.587902	−	0.808932i	$-0.700046\pi$
$283$	−19.7801	−1.17580	−0.587902	+	0.808932i	$0.700046\pi$
$284$	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
$292$	−0.108843	−0.00636955
$293$	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
$301$	−52.4343	−3.02226
$302$	0	0
$303$	0	0
$304$	−23.5975	−1.35341
$305$	0	0
$306$	0	0
$307$	29.3337	1.67416	0.837080	−	0.547080i	$-0.184261\pi$
$307$	29.3337	1.67416	0.837080	+	0.547080i	$0.184261\pi$
$308$	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
$313$	27.2811	1.54202	0.771008	−	0.636825i	$-0.219753\pi$
$313$	27.2811	1.54202	0.771008	+	0.636825i	$0.219753\pi$
$314$	0	0
$315$	0	0
$316$	35.2875	1.98508
$317$	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
$325$	−12.2713	−0.680690
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.58424	0.361902	0.180951	−	0.983492i	$-0.442082\pi$
$331$	6.58424	0.361902	0.180951	+	0.983492i	$0.442082\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−71.2616	−3.84777
$344$	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
$349$	−1.30013	−0.0695946	−0.0347973	−	0.999394i	$-0.511079\pi$
$349$	−1.30013	−0.0695946	−0.0347973	+	0.999394i	$0.511079\pi$
$350$	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
$361$	15.8028	0.831724
$362$	0	0
$363$	0	0
$364$	25.7739	1.35092
$365$	0	0
$366$	0	0
$367$	−25.6411	−1.33845	−0.669227	−	0.743058i	$-0.733375\pi$
$367$	−25.6411	−1.33845	−0.669227	+	0.743058i	$0.733375\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−35.7292	−1.84999	−0.924993	−	0.379985i	$-0.875929\pi$
$373$	−35.7292	−1.84999	−0.924993	+	0.379985i	$0.875929\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	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
$388$	−17.7040	−0.898783
$389$	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
$397$	−7.30447	−0.366601	−0.183300	−	0.983057i	$-0.558678\pi$
$397$	−7.30447	−0.366601	−0.183300	+	0.983057i	$0.558678\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	27.2907	1.35945
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	17.7526	0.877809	0.438904	−	0.898534i	$-0.355367\pi$
$409$	17.7526	0.877809	0.438904	+	0.898534i	$0.355367\pi$
$410$	0	0
$411$	0	0
$412$	−18.9498	−0.933589
$413$	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
$421$	6.00823	0.292823	0.146412	−	0.989224i	$-0.453228\pi$
$421$	6.00823	0.292823	0.146412	+	0.989224i	$0.453228\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−64.3419	−3.11372
$428$	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
$433$	−41.1324	−1.97670	−0.988350	−	0.152201i	$-0.951364\pi$
$433$	−41.1324	−1.97670	−0.988350	+	0.152201i	$0.951364\pi$
$434$	0	0
$435$	0	0
$436$	12.5553	0.601291
$437$	0	0
$438$	0	0
$439$	−38.2601	−1.82605	−0.913027	−	0.407900i	$-0.866261\pi$
$439$	−38.2601	−1.82605	−0.913027	+	0.407900i	$0.866261\pi$
$440$	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
$448$	42.0068	1.98464
$449$	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
$457$	24.2006	1.13206	0.566029	−	0.824385i	$-0.308479\pi$
$457$	24.2006	1.13206	0.566029	+	0.824385i	$0.308479\pi$
$458$	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
$463$	35.8058	1.66404	0.832018	−	0.554748i	$-0.187185\pi$
$463$	35.8058	1.66404	0.832018	+	0.554748i	$0.187185\pi$
$464$	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
$469$	84.1486	3.88562
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	29.4969	1.35341
$476$	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
$481$	14.3451	0.654079
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	−1.15413	−0.0522985	−0.0261493	−	0.999658i	$-0.508325\pi$
$487$	−1.15413	−0.0522985	−0.0261493	+	0.999658i	$0.508325\pi$
$488$	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
$496$	44.4789	1.99716
$497$	0	0
$498$	0	0
$499$	−37.7828	−1.69139	−0.845694	−	0.533667i	$-0.820813\pi$
$499$	−37.7828	−1.69139	−0.845694	+	0.533667i	$0.820813\pi$
$500$	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
$508$	42.2112	1.87282
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−0.285759	−0.0126412
$512$	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
$523$	−45.6229	−1.99495	−0.997474	−	0.0710325i	$-0.977371\pi$
$523$	−45.6229	−1.99495	−0.997474	+	0.0710325i	$0.977371\pi$
$524$	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
$532$	−61.9536	−2.68603
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−29.0000	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$541$	−29.0000	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.2970	1.20989	0.604946	−	0.796266i	$-0.293195\pi$
$547$	28.2970	1.20989	0.604946	+	0.796266i	$0.293195\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	92.6447	3.93965
$554$	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
$559$	24.5079	1.03657
$560$	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
$571$	39.4142	1.64943	0.824716	−	0.565547i	$-0.191335\pi$
$571$	39.4142	1.64943	0.824716	+	0.565547i	$0.191335\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	41.0773	1.71007	0.855036	−	0.518569i	$-0.173535\pi$
$577$	41.0773	1.71007	0.855036	+	0.518569i	$0.173535\pi$
$578$	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
$589$	−65.5996	−2.70299
$590$	0	0
$591$	0	0
$592$	23.3799	0.960906
$593$	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
$601$	16.7891	0.684842	0.342421	−	0.939547i	$-0.388753\pi$
$601$	16.7891	0.684842	0.342421	+	0.939547i	$0.388753\pi$
$602$	0	0
$603$	0	0
$604$	−12.4173	−0.505253
$605$	0	0
$606$	0	0
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	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
$619$	41.2413	1.65763	0.828814	−	0.559525i	$-0.189016\pi$
$619$	41.2413	1.65763	0.828814	+	0.559525i	$0.189016\pi$
$620$	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
$628$	−47.0862	−1.87895
$629$	0	0
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	50.4877	2.00040
$638$	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
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	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
$652$	46.7467	1.83074
$653$	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
$661$	11.0000	0.427850	0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000	0.427850	0.213925	+	0.976850i	$0.431375\pi$
$662$	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
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	0	0
$675$	0	0
$676$	13.9532	0.536662
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−46.4805	−1.78376
$680$	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
$688$	39.9434	1.52283
$689$	0	0
$690$	0	0
$691$	28.4430	1.08202	0.541012	−	0.841015i	$-0.318041\pi$
$691$	28.4430	1.08202	0.541012	+	0.841015i	$0.318041\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−52.5085	−1.98464
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−34.4817	−1.30050
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	50.5314	1.89775	0.948873	−	0.315659i	$-0.102225\pi$
$709$	50.5314	1.89775	0.948873	+	0.315659i	$0.102225\pi$
$710$	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
$721$	−49.7513	−1.85283
$722$	0	0
$723$	0	0
$724$	−41.8685	−1.55603
$725$	0	0
$726$	0	0
$727$	−52.8768	−1.96109	−0.980546	−	0.196290i	$-0.937111\pi$
$727$	−52.8768	−1.96109	−0.980546	+	0.196290i	$0.937111\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	4.31649	0.159433	0.0797166	−	0.996818i	$-0.474598\pi$
$733$	4.31649	0.159433	0.0797166	+	0.996818i	$0.474598\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−27.9088	−1.02664	−0.513322	−	0.858196i	$-0.671585\pi$
$739$	−27.9088	−1.02664	−0.513322	+	0.858196i	$0.671585\pi$
$740$	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
$751$	−54.4648	−1.98745	−0.993725	−	0.111854i	$-0.964321\pi$
$751$	−54.4648	−1.98745	−0.993725	+	0.111854i	$0.964321\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−29.0000	−1.05402	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−29.0000	−1.05402	−0.527011	+	0.849858i	$0.676688\pi$
$758$	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
$763$	32.9631	1.19334
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−53.3309	−1.92316	−0.961581	−	0.274520i	$-0.911481\pi$
$769$	−53.3309	−1.92316	−0.961581	+	0.274520i	$0.911481\pi$
$770$	0	0
$771$	0	0
$772$	−15.4362	−0.555562
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−55.5987	−1.99716
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	82.2858	2.93878
$785$	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
$793$	30.0736	1.06794
$794$	0	0
$795$	0	0
$796$	−49.3772	−1.75013
$797$	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
$811$	7.42128	0.260596	0.130298	−	0.991475i	$-0.458407\pi$
$811$	7.42128	0.260596	0.130298	+	0.991475i	$0.458407\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−58.9105	−2.06102
$818$	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	$-0.672226\pi$
$823$	−29.5514	−1.03010	−0.515048	+	0.857161i	$0.672226\pi$
$824$	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
$829$	−52.1971	−1.81288	−0.906439	−	0.422336i	$-0.861210\pi$
$829$	−52.1971	−1.81288	−0.906439	+	0.422336i	$0.861210\pi$
$830$	0	0
$831$	0	0
$832$	−19.6341	−0.680690
$833$	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
$841$	0	0
$842$	0	0
$843$	0	0
$844$	58.0000	1.99644
$845$	0	0
$846$	0	0
$847$	57.7594	1.98464
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	58.0000	1.98588	0.992941	−	0.118609i	$-0.0378434\pi$
$853$	58.0000	1.98588	0.992941	+	0.118609i	$0.0378434\pi$
$854$	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
$859$	35.1242	1.19842	0.599211	−	0.800591i	$-0.295481\pi$
$859$	35.1242	1.19842	0.599211	+	0.800591i	$0.295481\pi$
$860$	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
$868$	116.776	3.96364
$869$	0	0
$870$	0	0
$871$	−39.3313	−1.33269
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.1738	0.782525	0.391262	−	0.920279i	$-0.372038\pi$
$877$	23.1738	0.782525	0.391262	+	0.920279i	$0.372038\pi$
$878$	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
$883$	−53.5353	−1.80161	−0.900804	−	0.434227i	$-0.857022\pi$
$883$	−53.5353	−1.80161	−0.900804	+	0.434227i	$0.857022\pi$
$884$	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
$889$	110.822	3.71686
$890$	0	0
$891$	0	0
$892$	−50.4549	−1.68936
$893$	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	$-0.931180\pi$
$907$	−58.8306	−1.95344	−0.976719	+	0.214523i	$0.931180\pi$
$908$	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
$916$	58.0000	1.91637
$917$	0	0
$918$	0	0
$919$	13.5821	0.448033	0.224016	−	0.974585i	$-0.428083\pi$
$919$	13.5821	0.448033	0.224016	+	0.974585i	$0.428083\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−29.2248	−0.960906
$926$	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
$931$	−121.359	−3.97738
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.0000	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$938$	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
$949$	0.133564	0.00433568
$950$	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
$961$	92.6486	2.98866
$962$	0	0
$963$	0	0
$964$	44.0604	1.41909
$965$	0	0
$966$	0	0
$967$	5.68170	0.182711	0.0913556	−	0.995818i	$-0.470880\pi$
$967$	5.68170	0.182711	0.0913556	+	0.995818i	$0.470880\pi$
$968$	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
$973$	−60.5345	−1.94065
$974$	0	0
$975$	0	0
$976$	49.0144	1.56891
$977$	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
$988$	28.9573	0.921254
$989$	0	0
$990$	0	0
$991$	−33.6755	−1.06974	−0.534869	−	0.844935i	$-0.679639\pi$
$991$	−33.6755	−1.06974	−0.534869	+	0.844935i	$0.679639\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.18261	0.100794	0.0503972	−	0.998729i	$-0.483951\pi$
$997$	3.18261	0.100794	0.0503972	+	0.998729i	$0.483951\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.u.1.1	✓	4
3.2	odd	2	CM	7569.2.a.u.1.1	✓	4
29.28	even	2		7569.2.a.v.1.1	yes	4
87.86	odd	2		7569.2.a.v.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7569.2.a.u.1.1	✓	4	1.1	even	1	trivial
7569.2.a.u.1.1	✓	4	3.2	odd	2	CM
7569.2.a.v.1.1	yes	4	29.28	even	2
7569.2.a.v.1.1	yes	4	87.86	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -5.25085 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} -5.25085 q^{7} +2.45426 q^{13} +4.00000 q^{16} -5.89939 q^{19} -5.00000 q^{25} +10.5017 q^{28} +11.1197 q^{31} +5.84497 q^{37} +9.98586 q^{43} +20.5714 q^{49} -4.90852 q^{52} +12.2536 q^{61} -8.00000 q^{64} -16.0257 q^{67} +0.0544214 q^{73} +11.7988 q^{76} -17.6437 q^{79} -12.8870 q^{91} +8.85199 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 + q^7	\( 4 q - 8 q^{4} + q^{7} - 2 q^{13} + 16 q^{16} - q^{19} - 20 q^{25} - 2 q^{28} - 4 q^{31} + 11 q^{37} + 5 q^{43} + 41 q^{49} + 4 q^{52} - 13 q^{61} - 32 q^{64} - 11 q^{67} - 10 q^{73} + 2 q^{76} - 13 q^{79} + 2 q^{91} + 14 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + q^7 - 2 * q^13 + 16 * q^16 - q^19 - 20 * q^25 - 2 * q^28 - 4 * q^31 + 11 * q^37 + 5 * q^43 + 41 * q^49 + 4 * q^52 - 13 * q^61 - 32 * q^64 - 11 * q^67 - 10 * q^73 + 2 * q^76 - 13 * q^79 + 2 * q^91 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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