LMFDB - Embedded newform 7569.2.a.u.1.2

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			1.2
Root			$-0.209057$ 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)

$f(q)$	$=$	$q-2.00000 q^{4} -2.24520 q^{7} +O(q^{10})$	$q-2.00000 q^{4} -2.24520 q^{7} -5.97108 q^{13} +4.00000 q^{16} +8.54541 q^{19} -5.00000 q^{25} +4.49041 q^{28} +2.87238 q^{31} +1.54265 q^{37} +11.1716 q^{43} -1.95906 q^{49} +11.9422 q^{52} -5.42685 q^{61} -8.00000 q^{64} +14.9301 q^{67} -10.0881 q^{73} -17.0908 q^{76} +15.5482 q^{79} +13.4063 q^{91} +19.4708 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.24520	−0.848608	−0.424304	−	0.905520i	$-0.639481\pi$
$7$	−2.24520	−0.848608	−0.424304	+	0.905520i	$0.639481\pi$
$8$	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
$13$	−5.97108	−1.65608	−0.828040	−	0.560670i	$-0.810544\pi$
$13$	−5.97108	−1.65608	−0.828040	+	0.560670i	$0.810544\pi$
$14$	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
$19$	8.54541	1.96045	0.980226	−	0.197884i	$-0.0634068\pi$
$19$	8.54541	1.96045	0.980226	+	0.197884i	$0.0634068\pi$
$20$	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
$28$	4.49041	0.848608
$29$	0	0
$29$	0	0
$30$	0	0
$31$	2.87238	0.515894	0.257947	−	0.966159i	$-0.416954\pi$
$31$	2.87238	0.515894	0.257947	+	0.966159i	$0.416954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.54265	0.253610	0.126805	−	0.991928i	$-0.459528\pi$
$37$	1.54265	0.253610	0.126805	+	0.991928i	$0.459528\pi$
$38$	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	$-0.175495\pi$
$43$	11.1716	1.70365	0.851827	+	0.523824i	$0.175495\pi$
$44$	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
$49$	−1.95906	−0.279865
$50$	0	0
$51$	0	0
$52$	11.9422	1.65608
$53$	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
$61$	−5.42685	−0.694837	−0.347419	−	0.937710i	$-0.612942\pi$
$61$	−5.42685	−0.694837	−0.347419	+	0.937710i	$0.612942\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	14.9301	1.82401	0.912003	−	0.410183i	$-0.134535\pi$
$67$	14.9301	1.82401	0.912003	+	0.410183i	$0.134535\pi$
$68$	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	$-0.701014\pi$
$73$	−10.0881	−1.18072	−0.590359	+	0.807141i	$0.701014\pi$
$74$	0	0
$75$	0	0
$76$	−17.0908	−1.96045
$77$	0	0
$78$	0	0
$79$	15.5482	1.74931	0.874653	−	0.484750i	$-0.161089\pi$
$79$	15.5482	1.74931	0.874653	+	0.484750i	$0.161089\pi$
$80$	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
$91$	13.4063	1.40536
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	19.4708	1.97696	0.988482	−	0.151341i	$-0.0483592\pi$
$97$	19.4708	1.97696	0.988482	+	0.151341i	$0.0483592\pi$
$98$	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
$103$	−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
$109$	−20.8798	−1.99992	−0.999961	−	0.00878995i	$-0.997202\pi$
$109$	−20.8798	−1.99992	−0.999961	+	0.00878995i	$0.997202\pi$
$110$	0	0
$111$	0	0
$112$	−8.98082	−0.848608
$113$	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
$124$	−5.74475	−0.515894
$125$	0	0
$126$	0	0
$127$	−14.0440	−1.24620	−0.623101	−	0.782142i	$-0.714127\pi$
$127$	−14.0440	−1.24620	−0.623101	+	0.782142i	$0.714127\pi$
$128$	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
$133$	−19.1862	−1.66365
$134$	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
$139$	23.1250	1.96144	0.980719	−	0.195424i	$-0.0626082\pi$
$139$	23.1250	1.96144	0.980719	+	0.195424i	$0.0626082\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−3.08530	−0.253610
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.95418	0.728681	0.364340	−	0.931266i	$-0.381294\pi$
$151$	8.95418	0.728681	0.364340	+	0.931266i	$0.381294\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−24.0936	−1.92288	−0.961438	−	0.275020i	$-0.911315\pi$
$157$	−24.0936	−1.92288	−0.961438	+	0.275020i	$0.911315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.55448	0.200082	0.100041	−	0.994983i	$-0.468103\pi$
$163$	2.55448	0.200082	0.100041	+	0.994983i	$0.468103\pi$
$164$	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
$169$	22.6538	1.74260
$170$	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
$175$	11.2260	0.848608
$176$	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
$181$	−26.8723	−1.99740	−0.998701	−	0.0509566i	$-0.983773\pi$
$181$	−26.8723	−1.99740	−0.998701	+	0.0509566i	$0.983773\pi$
$182$	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
$193$	27.7701	1.99893	0.999466	−	0.0326701i	$-0.0104011\pi$
$193$	27.7701	1.99893	0.999466	+	0.0326701i	$0.0104011\pi$
$194$	0	0
$195$	0	0
$196$	3.91811	0.279865
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−11.9470	−0.846903	−0.423451	−	0.905919i	$-0.639182\pi$
$199$	−11.9470	−0.846903	−0.423451	+	0.905919i	$0.639182\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−23.8843	−1.65608
$209$	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
$217$	−6.44907	−0.437791
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−7.40858	−0.496115	−0.248058	−	0.968745i	$-0.579792\pi$
$223$	−7.40858	−0.496115	−0.248058	+	0.968745i	$0.579792\pi$
$224$	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
$241$	−27.6154	−1.77887	−0.889433	−	0.457066i	$-0.848900\pi$
$241$	−27.6154	−1.77887	−0.889433	+	0.457066i	$0.848900\pi$
$242$	0	0
$243$	0	0
$244$	10.8537	0.694837
$245$	0	0
$246$	0	0
$247$	−51.0253	−3.24666
$248$	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
$259$	−3.46356	−0.215215
$260$	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
$268$	−29.8603	−1.82401
$269$	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
$277$	32.8434	1.97337	0.986683	−	0.162652i	$-0.0520049\pi$
$277$	32.8434	1.97337	0.986683	+	0.162652i	$0.0520049\pi$
$278$	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
$283$	0.00487458	0.000289764 0	0.000144882	−	1.00000i	$-0.499954\pi$
$283$	0.00487458	0.000289764 0	0.000144882		1.00000i	$0.499954\pi$
$284$	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
$301$	−25.0825	−1.44573
$302$	0	0
$303$	0	0
$304$	34.1816	1.96045
$305$	0	0
$306$	0	0
$307$	−12.4629	−0.711294	−0.355647	−	0.934620i	$-0.615739\pi$
$307$	−12.4629	−0.711294	−0.355647	+	0.934620i	$0.615739\pi$
$308$	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
$313$	29.8606	1.68782	0.843911	−	0.536483i	$-0.180247\pi$
$313$	29.8606	1.68782	0.843911	+	0.536483i	$0.180247\pi$
$314$	0	0
$315$	0	0
$316$	−31.0963	−1.74931
$317$	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
$325$	29.8554	1.65608
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	36.0693	1.98255	0.991274	−	0.131821i	$-0.0420824\pi$
$331$	36.0693	1.98255	0.991274	+	0.131821i	$0.0420824\pi$
$332$	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
$343$	20.1149	1.08610
$344$	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
$349$	−20.8963	−1.11856	−0.559278	−	0.828980i	$-0.688921\pi$
$349$	−20.8963	−1.11856	−0.559278	+	0.828980i	$0.688921\pi$
$350$	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
$361$	54.0240	2.84337
$362$	0	0
$363$	0	0
$364$	−26.8126	−1.40536
$365$	0	0
$366$	0	0
$367$	19.1529	0.999775	0.499887	−	0.866090i	$-0.333375\pi$
$367$	19.1529	0.999775	0.499887	+	0.866090i	$0.333375\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.91817	0.151097	0.0755485	−	0.997142i	$-0.475929\pi$
$373$	2.91817	0.151097	0.0755485	+	0.997142i	$0.475929\pi$
$374$	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
$388$	−38.9417	−1.97696
$389$	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
$397$	−39.5144	−1.98317	−0.991586	−	0.129452i	$-0.958678\pi$
$397$	−39.5144	−1.98317	−0.991586	+	0.129452i	$0.958678\pi$
$398$	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
$403$	−17.1512	−0.854361
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−35.7243	−1.76645	−0.883226	−	0.468948i	$-0.844633\pi$
$409$	−35.7243	−1.76645	−0.883226	+	0.468948i	$0.844633\pi$
$410$	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
$421$	−28.7215	−1.39980	−0.699901	−	0.714240i	$-0.746772\pi$
$421$	−28.7215	−1.39980	−0.699901	+	0.714240i	$0.746772\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.1844	0.589644
$428$	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
$433$	29.5537	1.42026	0.710130	−	0.704071i	$-0.248636\pi$
$433$	29.5537	1.42026	0.710130	+	0.704071i	$0.248636\pi$
$434$	0	0
$435$	0	0
$436$	41.7596	1.99992
$437$	0	0
$438$	0	0
$439$	20.9061	0.997793	0.498897	−	0.866662i	$-0.333739\pi$
$439$	20.9061	0.997793	0.498897	+	0.866662i	$0.333739\pi$
$440$	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
$448$	17.9616	0.848608
$449$	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
$457$	−26.0432	−1.21825	−0.609124	−	0.793075i	$-0.708479\pi$
$457$	−26.0432	−1.21825	−0.609124	+	0.793075i	$0.708479\pi$
$458$	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
$463$	−14.9350	−0.694089	−0.347044	−	0.937849i	$-0.612815\pi$
$463$	−14.9350	−0.694089	−0.347044	+	0.937849i	$0.612815\pi$
$464$	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
$469$	−33.5212	−1.54787
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−42.7270	−1.96045
$476$	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
$481$	−9.21127	−0.419998
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	26.8674	1.21748	0.608739	−	0.793370i	$-0.291675\pi$
$487$	26.8674	1.21748	0.608739	+	0.793370i	$0.291675\pi$
$488$	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
$496$	11.4895	0.515894
$497$	0	0
$498$	0	0
$499$	−34.3510	−1.53776	−0.768882	−	0.639391i	$-0.779187\pi$
$499$	−34.3510	−1.53776	−0.768882	+	0.639391i	$0.779187\pi$
$500$	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
$508$	28.0880	1.24620
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	22.6498	1.00197
$512$	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
$532$	38.3724	1.66365
$533$	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
$547$	−44.7855	−1.91489	−0.957445	−	0.288614i	$-0.906805\pi$
$547$	−44.7855	−1.91489	−0.957445	+	0.288614i	$0.906805\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−34.9088	−1.48447
$554$	0	0
$555$	0	0
$556$	−46.2500	−1.96144
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−66.7065	−2.82139
$560$	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
$571$	−47.7735	−1.99926	−0.999629	−	0.0272193i	$-0.991335\pi$
$571$	−47.7735	−1.99926	−0.999629	+	0.0272193i	$0.991335\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	36.3872	1.51482	0.757409	−	0.652941i	$-0.226465\pi$
$577$	36.3872	1.51482	0.757409	+	0.652941i	$0.226465\pi$
$578$	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
$589$	24.5456	1.01138
$590$	0	0
$591$	0	0
$592$	6.17059	0.253610
$593$	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
$601$	−38.6238	−1.57550	−0.787748	−	0.615998i	$-0.788753\pi$
$601$	−38.6238	−1.57550	−0.787748	+	0.615998i	$0.788753\pi$
$602$	0	0
$603$	0	0
$604$	−17.9084	−0.728681
$605$	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
$613$	36.5962	1.47811	0.739054	−	0.673646i	$-0.235273\pi$
$613$	36.5962	1.47811	0.739054	+	0.673646i	$0.235273\pi$
$614$	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
$619$	−49.7298	−1.99881	−0.999405	−	0.0344993i	$-0.989016\pi$
$619$	−49.7298	−1.99881	−0.999405	+	0.0344993i	$0.989016\pi$
$620$	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
$628$	48.1872	1.92288
$629$	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
$637$	11.6977	0.463479
$638$	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
$652$	−5.10895	−0.200082
$653$	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
$676$	−45.3076	−1.74260
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−43.7160	−1.67767
$680$	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
$691$	2.97823	0.113297	0.0566486	−	0.998394i	$-0.481959\pi$
$691$	2.97823	0.113297	0.0566486	+	0.998394i	$0.481959\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−22.4520	−0.848608
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	13.1826	0.497189
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−50.7615	−1.90639	−0.953194	−	0.302360i	$-0.902225\pi$
$709$	−50.7615	−1.90639	−0.953194	+	0.302360i	$0.902225\pi$
$710$	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
$721$	31.7566	1.18268
$722$	0	0
$723$	0	0
$724$	53.7446	1.99740
$725$	0	0
$726$	0	0
$727$	36.5564	1.35580	0.677902	−	0.735152i	$-0.262889\pi$
$727$	36.5564	1.35580	0.677902	+	0.735152i	$0.262889\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	52.6677	1.94533	0.972664	−	0.232219i	$-0.0745985\pi$
$733$	52.6677	1.94533	0.972664	+	0.232219i	$0.0745985\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	35.7514	1.31514	0.657568	−	0.753395i	$-0.271585\pi$
$739$	35.7514	1.31514	0.657568	+	0.753395i	$0.271585\pi$
$740$	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
$751$	−22.6611	−0.826915	−0.413458	−	0.910523i	$-0.635679\pi$
$751$	−22.6611	−0.826915	−0.413458	+	0.910523i	$0.635679\pi$
$752$	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
$763$	46.8794	1.69715
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−30.9603	−1.11646	−0.558229	−	0.829687i	$-0.688519\pi$
$769$	−30.9603	−1.11646	−0.558229	+	0.829687i	$0.688519\pi$
$770$	0	0
$771$	0	0
$772$	−55.5401	−1.99893
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−14.3619	−0.515894
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−7.83622	−0.279865
$785$	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
$793$	32.4042	1.15071
$794$	0	0
$795$	0	0
$796$	23.8941	0.846903
$797$	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
$811$	−51.4134	−1.80537	−0.902684	−	0.430304i	$-0.858407\pi$
$811$	−51.4134	−1.80537	−0.902684	+	0.430304i	$0.858407\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	95.4659	3.33993
$818$	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
$823$	52.8151	1.84102	0.920509	−	0.390721i	$-0.127774\pi$
$823$	52.8151	1.84102	0.920509	+	0.390721i	$0.127774\pi$
$824$	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
$829$	−39.2596	−1.36354	−0.681770	−	0.731566i	$-0.738790\pi$
$829$	−39.2596	−1.36354	−0.681770	+	0.731566i	$0.738790\pi$
$830$	0	0
$831$	0	0
$832$	47.7686	1.65608
$833$	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
$847$	24.6973	0.848608
$848$	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
$859$	−0.832170	−0.0283933	−0.0141966	−	0.999899i	$-0.504519\pi$
$859$	−0.832170	−0.0283933	−0.0141966	+	0.999899i	$0.504519\pi$
$860$	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
$868$	12.8981	0.437791
$869$	0	0
$870$	0	0
$871$	−89.1490	−3.02070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44.6778	−1.50866	−0.754331	−	0.656495i	$-0.772038\pi$
$877$	−44.6778	−1.50866	−0.754331	+	0.656495i	$0.772038\pi$
$878$	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
$883$	−41.0866	−1.38268	−0.691338	−	0.722532i	$-0.742978\pi$
$883$	−41.0866	−1.38268	−0.691338	+	0.722532i	$0.742978\pi$
$884$	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
$889$	31.5316	1.05754
$890$	0	0
$891$	0	0
$892$	14.8172	0.496115
$893$	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
$907$	55.1899	1.83255	0.916275	−	0.400549i	$-0.131180\pi$
$907$	55.1899	1.83255	0.916275	+	0.400549i	$0.131180\pi$
$908$	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
$919$	60.3942	1.99222	0.996111	−	0.0881111i	$-0.0280831\pi$
$919$	60.3942	1.99222	0.996111	+	0.0881111i	$0.0280831\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−7.71324	−0.253610
$926$	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
$931$	−16.7409	−0.548662
$932$	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
$949$	60.2366	1.95536
$950$	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
$961$	−22.7495	−0.733854
$962$	0	0
$963$	0	0
$964$	55.2308	1.77887
$965$	0	0
$966$	0	0
$967$	31.8068	1.02284	0.511419	−	0.859331i	$-0.329120\pi$
$967$	31.8068	1.02284	0.511419	+	0.859331i	$0.329120\pi$
$968$	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
$973$	−51.9204	−1.66449
$974$	0	0
$975$	0	0
$976$	−21.7074	−0.694837
$977$	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
$988$	102.051	3.24666
$989$	0	0
$990$	0	0
$991$	40.1874	1.27659	0.638297	−	0.769790i	$-0.279639\pi$
$991$	40.1874	1.27659	0.638297	+	0.769790i	$0.279639\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	60.9670	1.93084	0.965422	−	0.260694i	$-0.0839513\pi$
$997$	60.9670	1.93084	0.965422	+	0.260694i	$0.0839513\pi$
$998$	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.1.2	✓	4
3.2	odd	2	CM	7569.2.a.u.1.2	✓	4
29.28	even	2		7569.2.a.v.1.2	yes	4
87.86	odd	2		7569.2.a.v.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7569.2.a.u.1.2	✓	4	1.1	even	1	trivial
7569.2.a.u.1.2	✓	4	3.2	odd	2	CM
7569.2.a.v.1.2	yes	4	29.28	even	2
7569.2.a.v.1.2	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -2.24520 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} -2.24520 q^{7} -5.97108 q^{13} +4.00000 q^{16} +8.54541 q^{19} -5.00000 q^{25} +4.49041 q^{28} +2.87238 q^{31} +1.54265 q^{37} +11.1716 q^{43} -1.95906 q^{49} +11.9422 q^{52} -5.42685 q^{61} -8.00000 q^{64} +14.9301 q^{67} -10.0881 q^{73} -17.0908 q^{76} +15.5482 q^{79} +13.4063 q^{91} +19.4708 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + q^{7}+O(q^{10}) \) 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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