LMFDB - Embedded newform 7569.2.a.u.1.3

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			1.3
Root			$1.82709$ 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)

$f(q)$	$=$	$q-2.00000 q^{4} +3.86324 q^{7} +O(q^{10})$	$q-2.00000 q^{4} +3.86324 q^{7} +7.20715 q^{13} +4.00000 q^{16} -7.92737 q^{19} -5.00000 q^{25} -7.72648 q^{28} -9.34451 q^{31} -8.34102 q^{37} -3.08143 q^{43} +7.92461 q^{49} -14.4143 q^{52} -15.6076 q^{61} -8.00000 q^{64} -8.13176 q^{67} +16.2684 q^{73} +15.8547 q^{76} -7.51373 q^{79} +27.8429 q^{91} +3.18165 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	3.86324	1.46017	0.730084	−	0.683358i	$-0.239481\pi$
$7$	3.86324	1.46017	0.730084	+	0.683358i	$0.239481\pi$
$8$	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
$13$	7.20715	1.99890	0.999451	−	0.0331183i	$-0.0105438\pi$
$13$	7.20715	1.99890	0.999451	+	0.0331183i	$0.0105438\pi$
$14$	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
$19$	−7.92737	−1.81866	−0.909332	−	0.416071i	$-0.863407\pi$
$19$	−7.92737	−1.81866	−0.909332	+	0.416071i	$0.863407\pi$
$20$	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
$28$	−7.72648	−1.46017
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−9.34451	−1.67832	−0.839162	−	0.543882i	$-0.816954\pi$
$31$	−9.34451	−1.67832	−0.839162	+	0.543882i	$0.816954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.34102	−1.37126	−0.685628	−	0.727952i	$-0.740472\pi$
$37$	−8.34102	−1.37126	−0.685628	+	0.727952i	$0.740472\pi$
$38$	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
$43$	−3.08143	−0.469914	−0.234957	−	0.972006i	$-0.575495\pi$
$43$	−3.08143	−0.469914	−0.234957	+	0.972006i	$0.575495\pi$
$44$	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
$49$	7.92461	1.13209
$50$	0	0
$51$	0	0
$52$	−14.4143	−1.99890
$53$	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
$61$	−15.6076	−1.99835	−0.999174	−	0.0406463i	$-0.987058\pi$
$61$	−15.6076	−1.99835	−0.999174	+	0.0406463i	$0.987058\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−8.13176	−0.993453	−0.496726	−	0.867907i	$-0.665465\pi$
$67$	−8.13176	−0.993453	−0.496726	+	0.867907i	$0.665465\pi$
$68$	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
$73$	16.2684	1.90407	0.952036	−	0.305987i	$-0.0989863\pi$
$73$	16.2684	1.90407	0.952036	+	0.305987i	$0.0989863\pi$
$74$	0	0
$75$	0	0
$76$	15.8547	1.81866
$77$	0	0
$78$	0	0
$79$	−7.51373	−0.845360	−0.422680	−	0.906279i	$-0.638911\pi$
$79$	−7.51373	−0.845360	−0.422680	+	0.906279i	$0.638911\pi$
$80$	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
$91$	27.8429	2.91873
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.18165	0.323047	0.161524	−	0.986869i	$-0.448359\pi$
$97$	3.18165	0.323047	0.161524	+	0.986869i	$0.448359\pi$
$98$	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
$103$	−18.2165	−1.79492	−0.897462	−	0.441092i	$-0.854591\pi$
$103$	−18.2165	−1.79492	−0.897462	+	0.441092i	$0.854591\pi$
$104$	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
$109$	−6.62677	−0.634730	−0.317365	−	0.948304i	$-0.602798\pi$
$109$	−6.62677	−0.634730	−0.317365	+	0.948304i	$0.602798\pi$
$110$	0	0
$111$	0	0
$112$	15.4530	1.46017
$113$	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
$124$	18.6890	1.67832
$125$	0	0
$126$	0	0
$127$	12.4259	1.10262	0.551312	−	0.834299i	$-0.314127\pi$
$127$	12.4259	1.10262	0.551312	+	0.834299i	$0.314127\pi$
$128$	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
$133$	−30.6253	−2.65555
$134$	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
$139$	2.76353	0.234400	0.117200	−	0.993108i	$-0.462608\pi$
$139$	2.76353	0.234400	0.117200	+	0.993108i	$0.462608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	16.6820	1.37126
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−20.6968	−1.68428	−0.842142	−	0.539256i	$-0.818706\pi$
$151$	−20.6968	−1.68428	−0.842142	+	0.539256i	$0.818706\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.4411	1.23233	0.616167	−	0.787616i	$-0.288685\pi$
$157$	15.4411	1.23233	0.616167	+	0.787616i	$0.288685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.9521	1.95440	0.977200	−	0.212322i	$-0.0681026\pi$
$163$	24.9521	1.95440	0.977200	+	0.212322i	$0.0681026\pi$
$164$	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
$169$	38.9430	2.99561
$170$	0	0
$171$	0	0
$172$	6.16286	0.469914
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−19.3162	−1.46017
$176$	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
$181$	22.5461	1.67584	0.837918	−	0.545797i	$-0.183773\pi$
$181$	22.5461	1.67584	0.837918	+	0.545797i	$0.183773\pi$
$182$	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
$193$	9.44473	0.679846	0.339923	−	0.940453i	$-0.389599\pi$
$193$	9.44473	0.679846	0.339923	+	0.940453i	$0.389599\pi$
$194$	0	0
$195$	0	0
$196$	−15.8492	−1.13209
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−5.35792	−0.379813	−0.189906	−	0.981802i	$-0.560818\pi$
$199$	−5.35792	−0.379813	−0.189906	+	0.981802i	$0.560818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	28.8286	1.99890
$209$	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
$217$	−36.1001	−2.45063
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−29.8062	−1.99597	−0.997986	−	0.0634420i	$-0.979792\pi$
$223$	−29.8062	−1.99597	−0.997986	+	0.0634420i	$0.979792\pi$
$224$	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
$241$	4.96294	0.319691	0.159846	−	0.987142i	$-0.448900\pi$
$241$	4.96294	0.319691	0.159846	+	0.987142i	$0.448900\pi$
$242$	0	0
$243$	0	0
$244$	31.2152	1.99835
$245$	0	0
$246$	0	0
$247$	−57.1337	−3.63533
$248$	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
$259$	−32.2234	−2.00226
$260$	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
$268$	16.2635	0.993453
$269$	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
$277$	−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
$283$	19.7722	1.17534	0.587668	−	0.809102i	$-0.300046\pi$
$283$	19.7722	1.17534	0.587668	+	0.809102i	$0.300046\pi$
$284$	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
$292$	−32.5368	−1.90407
$293$	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
$301$	−11.9043	−0.686153
$302$	0	0
$303$	0	0
$304$	−31.7095	−1.81866
$305$	0	0
$306$	0	0
$307$	−9.16832	−0.523263	−0.261632	−	0.965168i	$-0.584261\pi$
$307$	−9.16832	−0.523263	−0.261632	+	0.965168i	$0.584261\pi$
$308$	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
$313$	−8.82618	−0.498885	−0.249443	−	0.968390i	$-0.580247\pi$
$313$	−8.82618	−0.498885	−0.249443	+	0.968390i	$0.580247\pi$
$314$	0	0
$315$	0	0
$316$	15.0275	0.845360
$317$	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
$325$	−36.0357	−1.99890
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.7078	0.863379	0.431690	−	0.902022i	$-0.357918\pi$
$331$	15.7078	0.863379	0.431690	+	0.902022i	$0.357918\pi$
$332$	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
$343$	3.57200	0.192870
$344$	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
$349$	35.1111	1.87946	0.939728	−	0.341924i	$-0.111079\pi$
$349$	35.1111	1.87946	0.939728	+	0.341924i	$0.111079\pi$
$350$	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
$361$	43.8433	2.30754
$362$	0	0
$363$	0	0
$364$	−55.6859	−2.91873
$365$	0	0
$366$	0	0
$367$	37.4783	1.95635	0.978175	−	0.207785i	$-0.0666253\pi$
$367$	37.4783	1.95635	0.978175	+	0.207785i	$0.0666253\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	37.5327	1.94337	0.971684	−	0.236283i	$-0.0759292\pi$
$373$	37.5327	1.94337	0.971684	+	0.236283i	$0.0759292\pi$
$374$	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
$388$	−6.36329	−0.323047
$389$	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
$400$	−20.0000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−67.3473	−3.35481
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	40.0505	1.98037	0.990185	−	0.139761i	$-0.0446333\pi$
$409$	40.0505	1.98037	0.990185	+	0.139761i	$0.0446333\pi$
$410$	0	0
$411$	0	0
$412$	36.4330	1.79492
$413$	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
$421$	40.4642	1.97210	0.986051	−	0.166441i	$-0.0532276\pi$
$421$	40.4642	1.97210	0.986051	+	0.166441i	$0.0532276\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−60.2958	−2.91792
$428$	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
$433$	−6.68643	−0.321329	−0.160665	−	0.987009i	$-0.551364\pi$
$433$	−6.68643	−0.321329	−0.160665	+	0.987009i	$0.551364\pi$
$434$	0	0
$435$	0	0
$436$	13.2535	0.634730
$437$	0	0
$438$	0	0
$439$	4.43331	0.211590	0.105795	−	0.994388i	$-0.466261\pi$
$439$	4.43331	0.211590	0.105795	+	0.994388i	$0.466261\pi$
$440$	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
$448$	−30.9059	−1.46017
$449$	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
$457$	−40.2962	−1.88498	−0.942489	−	0.334238i	$-0.891521\pi$
$457$	−40.2962	−1.88498	−0.942489	+	0.334238i	$0.891521\pi$
$458$	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
$463$	−11.6405	−0.540978	−0.270489	−	0.962723i	$-0.587185\pi$
$463$	−11.6405	−0.540978	−0.270489	+	0.962723i	$0.587185\pi$
$464$	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
$469$	−31.4149	−1.45061
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	39.6369	1.81866
$476$	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
$481$	−60.1150	−2.74101
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	−42.3183	−1.91762	−0.958812	−	0.284042i	$-0.908325\pi$
$487$	−42.3183	−1.91762	−0.958812	+	0.284042i	$0.908325\pi$
$488$	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
$496$	−37.3780	−1.67832
$497$	0	0
$498$	0	0
$499$	16.5527	0.740999	0.370499	−	0.928833i	$-0.379187\pi$
$499$	16.5527	0.740999	0.370499	+	0.928833i	$0.379187\pi$
$500$	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
$508$	−24.8519	−1.10262
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	62.8487	2.78026
$512$	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
$523$	−17.1881	−0.751585	−0.375792	−	0.926704i	$-0.622629\pi$
$523$	−17.1881	−0.751585	−0.375792	+	0.926704i	$0.622629\pi$
$524$	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
$532$	61.2507	2.65555
$533$	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
$547$	44.1675	1.88847	0.944233	−	0.329278i	$-0.106805\pi$
$547$	44.1675	1.88847	0.944233	+	0.329278i	$0.106805\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−29.0273	−1.23437
$554$	0	0
$555$	0	0
$556$	−5.52707	−0.234400
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−22.2083	−0.939313
$560$	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
$571$	37.8850	1.58544	0.792718	−	0.609588i	$-0.208665\pi$
$571$	37.8850	1.58544	0.792718	+	0.609588i	$0.208665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−18.5888	−0.773862	−0.386931	−	0.922109i	$-0.626465\pi$
$577$	−18.5888	−0.773862	−0.386931	+	0.922109i	$0.626465\pi$
$578$	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
$589$	74.0774	3.05231
$590$	0	0
$591$	0	0
$592$	−33.3641	−1.37126
$593$	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
$601$	−40.6599	−1.65855	−0.829276	−	0.558839i	$-0.811247\pi$
$601$	−40.6599	−1.65855	−0.829276	+	0.558839i	$0.811247\pi$
$602$	0	0
$603$	0	0
$604$	41.3937	1.68428
$605$	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
$613$	−20.4159	−0.824590	−0.412295	−	0.911050i	$-0.635273\pi$
$613$	−20.4159	−0.824590	−0.412295	+	0.911050i	$0.635273\pi$
$614$	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
$619$	39.2232	1.57651	0.788257	−	0.615346i	$-0.210984\pi$
$619$	39.2232	1.57651	0.788257	+	0.615346i	$0.210984\pi$
$620$	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
$628$	−30.8822	−1.23233
$629$	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
$637$	57.1139	2.26293
$638$	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
$652$	−49.9042	−1.95440
$653$	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
$676$	−77.8859	−2.99561
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	12.2915	0.471703
$680$	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
$688$	−12.3257	−0.469914
$689$	0	0
$690$	0	0
$691$	−33.2619	−1.26534	−0.632671	−	0.774421i	$-0.718041\pi$
$691$	−33.2619	−1.26534	−0.632671	+	0.774421i	$0.718041\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	38.6324	1.46017
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	66.1224	2.49385
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	31.6024	1.18685	0.593427	−	0.804888i	$-0.297775\pi$
$709$	31.6024	1.18685	0.593427	+	0.804888i	$0.297775\pi$
$710$	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
$721$	−70.3746	−2.62089
$722$	0	0
$723$	0	0
$724$	−45.0921	−1.67584
$725$	0	0
$726$	0	0
$727$	−6.27278	−0.232645	−0.116322	−	0.993212i	$-0.537111\pi$
$727$	−6.27278	−0.232645	−0.116322	+	0.993212i	$0.537111\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	28.2340	1.04285	0.521423	−	0.853299i	$-0.325402\pi$
$733$	28.2340	1.04285	0.521423	+	0.853299i	$0.325402\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	50.0044	1.83944	0.919721	−	0.392572i	$-0.128415\pi$
$739$	50.0044	1.83944	0.919721	+	0.392572i	$0.128415\pi$
$740$	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
$751$	40.4595	1.47639	0.738194	−	0.674589i	$-0.235679\pi$
$751$	40.4595	1.47639	0.738194	+	0.674589i	$0.235679\pi$
$752$	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
$763$	−25.6008	−0.926811
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	34.1964	1.23315	0.616577	−	0.787295i	$-0.288519\pi$
$769$	34.1964	1.23315	0.616577	+	0.787295i	$0.288519\pi$
$770$	0	0
$771$	0	0
$772$	−18.8895	−0.679846
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	46.7226	1.67832
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	31.6985	1.13209
$785$	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
$793$	−112.486	−3.99450
$794$	0	0
$795$	0	0
$796$	10.7158	0.379813
$797$	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
$811$	−39.1965	−1.37638	−0.688188	−	0.725533i	$-0.741593\pi$
$811$	−39.1965	−1.37638	−0.688188	+	0.725533i	$0.741593\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	24.4277	0.854616
$818$	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
$823$	−55.9053	−1.94874	−0.974368	−	0.224962i	$-0.927774\pi$
$823$	−55.9053	−1.94874	−0.974368	+	0.224962i	$0.927774\pi$
$824$	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
$829$	27.9333	0.970164	0.485082	−	0.874469i	$-0.338790\pi$
$829$	27.9333	0.970164	0.485082	+	0.874469i	$0.338790\pi$
$830$	0	0
$831$	0	0
$832$	−57.6572	−1.99890
$833$	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
$847$	−42.4956	−1.46017
$848$	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
$859$	−33.7777	−1.15248	−0.576241	−	0.817280i	$-0.695481\pi$
$859$	−33.7777	−1.15248	−0.576241	+	0.817280i	$0.695481\pi$
$860$	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
$868$	72.2002	2.45063
$869$	0	0
$870$	0	0
$871$	−58.6068	−1.98582
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−50.7862	−1.71493	−0.857464	−	0.514543i	$-0.827962\pi$
$877$	−50.7862	−1.71493	−0.857464	+	0.514543i	$0.827962\pi$
$878$	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
$883$	28.1424	0.947066	0.473533	−	0.880776i	$-0.342978\pi$
$883$	28.1424	0.947066	0.473533	+	0.880776i	$0.342978\pi$
$884$	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
$889$	48.0044	1.61002
$890$	0	0
$891$	0	0
$892$	59.6124	1.99597
$893$	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
$907$	−30.4686	−1.01169	−0.505846	−	0.862624i	$-0.668820\pi$
$907$	−30.4686	−1.01169	−0.505846	+	0.862624i	$0.668820\pi$
$908$	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
$919$	23.7436	0.783228	0.391614	−	0.920130i	$-0.371917\pi$
$919$	23.7436	0.783228	0.391614	+	0.920130i	$0.371917\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	41.7051	1.37126
$926$	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
$931$	−62.8214	−2.05889
$932$	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
$949$	117.249	3.80605
$950$	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
$961$	56.3199	1.81677
$962$	0	0
$963$	0	0
$964$	−9.92589	−0.319691
$965$	0	0
$966$	0	0
$967$	−57.1462	−1.83770	−0.918849	−	0.394609i	$-0.870880\pi$
$967$	−57.1462	−1.83770	−0.918849	+	0.394609i	$0.870880\pi$
$968$	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
$973$	10.6762	0.342263
$974$	0	0
$975$	0	0
$976$	−62.4304	−1.99835
$977$	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
$988$	114.267	3.63533
$989$	0	0
$990$	0	0
$991$	58.5127	1.85872	0.929359	−	0.369178i	$-0.120361\pi$
$991$	58.5127	1.85872	0.929359	+	0.369178i	$0.120361\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34.4970	1.09253	0.546266	−	0.837612i	$-0.316049\pi$
$997$	34.4970	1.09253	0.546266	+	0.837612i	$0.316049\pi$
$998$	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.1.3	✓	4
3.2	odd	2	CM	7569.2.a.u.1.3	✓	4
29.28	even	2		7569.2.a.v.1.3	yes	4
87.86	odd	2		7569.2.a.v.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7569.2.a.u.1.3	✓	4	1.1	even	1	trivial
7569.2.a.u.1.3	✓	4	3.2	odd	2	CM
7569.2.a.v.1.3	yes	4	29.28	even	2
7569.2.a.v.1.3	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} +3.86324 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} +3.86324 q^{7} +7.20715 q^{13} +4.00000 q^{16} -7.92737 q^{19} -5.00000 q^{25} -7.72648 q^{28} -9.34451 q^{31} -8.34102 q^{37} -3.08143 q^{43} +7.92461 q^{49} -14.4143 q^{52} -15.6076 q^{61} -8.00000 q^{64} -8.13176 q^{67} +16.2684 q^{73} +15.8547 q^{76} -7.51373 q^{79} +27.8429 q^{91} +3.18165 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + q^{7}+O(q^{10}) \) 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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