LMFDB - Embedded newform 324.9.g.a.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,9,Mod(53,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("324.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	324.g (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$131.990669660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			269.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.269
Dual form			324.9.g.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2017.00 + 3493.55i) q^{7} +O(q^{10})$	$q+(-2017.00 + 3493.55i) q^{7} +(17903.0 + 31008.9i) q^{13} -258526. q^{19} +(-195312. + 338291. i) q^{25} +(904703. + 1.56699e6i) q^{31} +503522. q^{37} +(-1.74610e6 + 3.02433e6i) q^{43} +(-5.25418e6 - 9.10050e6i) q^{49} +(1.19133e7 - 2.06344e7i) q^{61} +(2.71070e6 + 4.69508e6i) q^{67} +1.61693e7 q^{73} +(9.44352e6 - 1.63567e7i) q^{79} -1.44441e8 q^{91} +(-8.84540e7 + 1.53207e8i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4034 q^{7}+O(q^{10}) $ 2 * q - 4034 * q^7	$ 2 q - 4034 q^{7} + 35806 q^{13} - 517052 q^{19} - 390625 q^{25} + 1809406 q^{31} + 1007044 q^{37} - 3492194 q^{43} - 10508355 q^{49} + 23826526 q^{61} + 5421406 q^{67} + 32338564 q^{73} + 18887038 q^{79} - 288882808 q^{91} - 176908034 q^{97}+O(q^{100}) $ 2 * q - 4034 * q^7 + 35806 * q^13 - 517052 * q^19 - 390625 * q^25 + 1809406 * q^31 + 1007044 * q^37 - 3492194 * q^43 - 10508355 * q^49 + 23826526 * q^61 + 5421406 * q^67 + 32338564 * q^73 + 18887038 * q^79 - 288882808 * q^91 - 176908034 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	−2017.00	+	3493.55i	−0.840067	+	1.45504i	0.0497709	+	0.998761i	$0.484151\pi$
$7$	−2017.00	+	3493.55i	−0.840067	+	1.45504i	−0.889838	+	0.456277i	$0.849182\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	17903.0	+	31008.9i	0.626834	+	1.08571i	0.988183	+	0.153277i	$0.0489828\pi$
$13$	17903.0	+	31008.9i	0.626834	+	1.08571i	−0.361349	+	0.932430i	$0.617684\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−258526.			−1.98376			−0.991882	−	0.127165i	$-0.959412\pi$
$19$	−258526.			−1.98376			−0.991882	+	0.127165i	$0.959412\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−195312.	+	338291.i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	904703.	+	1.56699e6i	0.979624	+	1.69676i	0.663746	+	0.747958i	$0.268966\pi$
$31$	904703.	+	1.56699e6i	0.979624	+	1.69676i	0.315877	+	0.948800i	$0.397701\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	503522.			0.268665			0.134333	−	0.990936i	$-0.457111\pi$
$37$	503522.			0.268665			0.134333	+	0.990936i	$0.457111\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	−1.74610e6	+	3.02433e6i	−0.510734	+	0.884617i	0.489189	+	0.872178i	$0.337293\pi$
$43$	−1.74610e6	+	3.02433e6i	−0.510734	+	0.884617i	−0.999923	+	0.0124389i	$0.996040\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	−5.25418e6	−	9.10050e6i	−0.911424	−	1.57863i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	1.19133e7	−	2.06344e7i	0.860422	−	1.49029i	−0.0111006	−	0.999938i	$-0.503533\pi$
$61$	1.19133e7	−	2.06344e7i	0.860422	−	1.49029i	0.871522	−	0.490356i	$-0.163133\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.71070e6	+	4.69508e6i	0.134519	+	0.232993i	0.925414	−	0.378959i	$-0.123718\pi$
$67$	2.71070e6	+	4.69508e6i	0.134519	+	0.232993i	−0.790895	+	0.611952i	$0.790384\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	1.61693e7			0.569376			0.284688	−	0.958620i	$-0.408110\pi$
$73$	1.61693e7			0.569376			0.284688	+	0.958620i	$0.408110\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	9.44352e6	−	1.63567e7i	0.242452	−	0.419939i	−0.718960	−	0.695051i	$-0.755382\pi$
$79$	9.44352e6	−	1.63567e7i	0.242452	−	0.419939i	0.961412	+	0.275112i	$0.0887150\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−1.44441e8			−2.10633
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.84540e7	+	1.53207e8i	−0.999150	+	1.73058i	−0.463872	+	0.885902i	$0.653540\pi$
$97$	−8.84540e7	+	1.53207e8i	−0.999150	+	1.73058i	−0.535278	+	0.844676i	$0.679793\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−2.22245e7	−	3.84939e7i	−0.197462	−	0.342014i	0.750243	−	0.661162i	$-0.229936\pi$
$103$	−2.22245e7	−	3.84939e7i	−0.197462	−	0.342014i	−0.947705	+	0.319149i	$0.896603\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	2.03181e8			1.43938			0.719692	−	0.694293i	$-0.244283\pi$
$109$	2.03181e8			1.43938			0.719692	+	0.694293i	$0.244283\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.07179e8	−	1.85640e8i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−4.00562e8			−1.53977			−0.769883	−	0.638185i	$-0.779686\pi$
$127$	−4.00562e8			−1.53977			−0.769883	+	0.638185i	$0.779686\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	5.21447e8	−	9.03173e8i	1.66649	−	2.88645i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−3.54715e8	−	6.14385e8i	−0.950213	−	1.64582i	−0.744961	−	0.667108i	$-0.767532\pi$
$139$	−3.54715e8	−	6.14385e8i	−0.950213	−	1.64582i	−0.205252	−	0.978709i	$-0.565801\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−1.35117e8	+	2.34030e8i	−0.259898	+	0.450157i	−0.966214	−	0.257740i	$-0.917022\pi$
$151$	−1.35117e8	+	2.34030e8i	−0.259898	+	0.450157i	0.706316	+	0.707896i	$0.250356\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.80867e7	−	3.13271e7i	−0.0297688	−	0.0515611i	0.850757	−	0.525559i	$-0.176144\pi$
$157$	−1.80867e7	−	3.13271e7i	−0.0297688	−	0.0515611i	−0.880526	+	0.473998i	$0.842810\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.17139e9			−1.65940			−0.829698	−	0.558212i	$-0.811488\pi$
$163$	−1.17139e9			−1.65940			−0.829698	+	0.558212i	$0.811488\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−2.33169e8	+	4.03861e8i	−0.285841	+	0.495092i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	−7.87891e8	−	1.36467e9i	−0.840067	−	1.45504i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−1.05268e9			−0.980801			−0.490400	−	0.871497i	$-0.663149\pi$
$181$	−1.05268e9			−0.980801			−0.490400	+	0.871497i	$0.663149\pi$
$182$		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		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−1.16335e9	−	2.01498e9i	−0.838457	−	1.45225i	−0.891185	−	0.453641i	$-0.850125\pi$
$193$	−1.16335e9	−	2.01498e9i	−0.838457	−	1.45225i	0.0527278	−	0.998609i	$-0.483208\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1.73472e9			1.10616			0.553080	−	0.833128i	$-0.313452\pi$
$199$	1.73472e9			1.10616			0.553080	+	0.833128i	$0.313452\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	9.18553e7	+	1.59098e8i	0.0463420	+	0.0802667i	0.888266	−	0.459329i	$-0.151910\pi$
$211$	9.18553e7	+	1.59098e8i	0.0463420	+	0.0802667i	−0.841924	+	0.539596i	$0.818577\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−7.29914e9			−3.29180
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	2.36366e9	−	4.09398e9i	0.955797	−	1.65549i	0.223263	−	0.974758i	$-0.428329\pi$
$223$	2.36366e9	−	4.09398e9i	0.955797	−	1.65549i	0.732534	−	0.680731i	$-0.238338\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−2.67939e9	−	4.64083e9i	−0.974302	−	1.68754i	−0.682220	−	0.731147i	$-0.738986\pi$
$229$	−2.67939e9	−	4.64083e9i	−0.974302	−	1.68754i	−0.292081	−	0.956393i	$-0.594348\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	2.78289e9	−	4.82011e9i	0.824951	−	1.42886i	−0.0770059	−	0.997031i	$-0.524536\pi$
$241$	2.78289e9	−	4.82011e9i	0.824951	−	1.42886i	0.901957	−	0.431826i	$-0.142131\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.62839e9	−	8.01661e9i	−1.24349	−	2.15379i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	−1.01560e9	+	1.75908e9i	−0.225697	+	0.390918i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−2.98709e9			−0.553824			−0.276912	−	0.960895i	$-0.589311\pi$
$271$	−2.98709e9			−0.553824			−0.276912	+	0.960895i	$0.589311\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.71404e9	+	6.43290e9i	−0.630852	+	1.09267i	0.356526	+	0.934285i	$0.383961\pi$
$277$	−3.71404e9	+	6.43290e9i	−0.630852	+	1.09267i	−0.987378	+	0.158382i	$0.949372\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	5.18486e9	+	8.98044e9i	0.808335	+	1.40008i	0.914016	+	0.405678i	$0.132964\pi$
$283$	5.18486e9	+	8.98044e9i	0.808335	+	1.40008i	−0.105681	+	0.994400i	$0.533702\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.97576e9			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−7.04376e9	−	1.22001e10i	−0.858101	−	1.48627i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.41256e10			−1.59020			−0.795101	−	0.606478i	$-0.792582\pi$
$307$	−1.41256e10			−1.59020			−0.795101	+	0.606478i	$0.792582\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−5.85032e9	+	1.01331e10i	−0.609541	+	1.05576i	0.381776	+	0.924255i	$0.375313\pi$
$313$	−5.85032e9	+	1.01331e10i	−0.609541	+	1.05576i	−0.991316	+	0.131500i	$0.958021\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.39867e10			−1.25367
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8.12475e9	−	1.40725e10i	0.676858	−	1.17235i	−0.299064	−	0.954233i	$-0.596674\pi$
$331$	8.12475e9	−	1.40725e10i	0.676858	−	1.17235i	0.975922	−	0.218120i	$-0.0699924\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.28845e10	+	2.23166e10i	0.998957	+	1.73024i	0.539014	+	0.842297i	$0.318797\pi$
$337$	1.28845e10	+	2.23166e10i	0.998957	+	1.73024i	0.459943	+	0.887948i	$0.347870\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.91355e10			1.38249
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	1.48002e10	−	2.56347e10i	0.997621	−	1.72793i	0.439106	−	0.898435i	$-0.355295\pi$
$349$	1.48002e10	−	2.56347e10i	0.997621	−	1.72793i	0.558514	−	0.829495i	$-0.311372\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	4.98521e10			2.93532
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.21528e10	+	2.10493e10i	−0.669903	+	1.16031i	0.308028	+	0.951377i	$0.400331\pi$
$367$	−1.21528e10	+	2.10493e10i	−0.669903	+	1.16031i	−0.977931	+	0.208929i	$0.933002\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.47234e9	−	4.28221e9i	−0.127724	−	0.221224i	0.795070	−	0.606517i	$-0.207434\pi$
$373$	−2.47234e9	−	4.28221e9i	−0.127724	−	0.221224i	−0.922794	+	0.385293i	$0.874100\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−3.49200e9			−0.169245			−0.0846227	−	0.996413i	$-0.526968\pi$
$379$	−3.49200e9			−0.169245			−0.0846227	+	0.996413i	$0.526968\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.57611e10			−0.634491			−0.317245	−	0.948343i	$-0.602758\pi$
$397$	−1.57611e10			−0.634491			−0.317245	+	0.948343i	$0.602758\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−3.23938e10	+	5.61077e10i	−1.22812	+	2.12717i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.63405e10	+	4.56231e10i	0.941305	+	1.63039i	0.762985	+	0.646416i	$0.223733\pi$
$409$	2.63405e10	+	4.56231e10i	0.941305	+	1.63039i	0.178320	+	0.983972i	$0.442934\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	5.80445e9	−	1.00536e10i	0.184771	−	0.320032i	−0.758729	−	0.651407i	$-0.774179\pi$
$421$	5.80445e9	−	1.00536e10i	0.184771	−	0.320032i	0.943499	+	0.331375i	$0.107512\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.80581e10	+	8.32391e10i	1.44562	+	2.50389i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	6.51683e10			1.85389			0.926947	−	0.375192i	$-0.122423\pi$
$433$	6.51683e10			1.85389			0.926947	+	0.375192i	$0.122423\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−3.37747e10	+	5.84994e10i	−0.909354	+	1.57505i	−0.0943897	+	0.995535i	$0.530090\pi$
$439$	−3.37747e10	+	5.84994e10i	−0.909354	+	1.57505i	−0.814964	+	0.579512i	$0.803243\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.36304e10	−	2.36085e10i	0.312495	−	0.541258i	−0.666407	−	0.745589i	$-0.732168\pi$
$457$	1.36304e10	−	2.36085e10i	0.312495	−	0.541258i	0.978902	+	0.204331i	$0.0655018\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	3.43926e10	+	5.95698e10i	0.748413	+	1.29629i	0.948583	+	0.316529i	$0.102517\pi$
$463$	3.43926e10	+	5.95698e10i	0.748413	+	1.29629i	−0.200169	+	0.979761i	$0.564149\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−2.18700e10			−0.452019
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	5.04934e10	−	8.74571e10i	0.991882	−	1.71799i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	9.01455e9	+	1.56137e10i	0.168408	+	0.291692i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	9.09984e10			1.61777			0.808887	−	0.587965i	$-0.200071\pi$
$487$	9.09984e10			1.61777			0.808887	+	0.587965i	$0.200071\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.16651e10	−	1.06807e11i	−0.994574	−	1.72265i	−0.587381	−	0.809311i	$-0.699841\pi$
$499$	−6.16651e10	−	1.06807e11i	−0.994574	−	1.72265i	−0.407193	−	0.913342i	$-0.633492\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−3.26134e10	+	5.64881e10i	−0.478314	+	0.828464i
$512$		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		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	2.44747e10			0.327122			0.163561	−	0.986533i	$-0.447702\pi$
$523$	2.44747e10			0.327122			0.163561	+	0.986533i	$0.447702\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.91555e10	+	6.78193e10i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	6.27323e10			0.732322			0.366161	−	0.930551i	$-0.380672\pi$
$541$	6.27323e10			0.732322			0.366161	+	0.930551i	$0.380672\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.09133e10	−	5.35434e10i	0.345300	−	0.598077i	−0.640108	−	0.768285i	$-0.721111\pi$
$547$	3.09133e10	−	5.35434e10i	0.345300	−	0.598077i	0.985408	+	0.170208i	$0.0544439\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	3.80952e10	+	6.59827e10i	0.407351	+	0.705553i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	−1.25041e11			−1.28058
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	9.74697e10	+	1.68822e11i	0.916907	+	1.58813i	0.804086	+	0.594513i	$0.202655\pi$
$571$	9.74697e10	+	1.68822e11i	0.916907	+	1.58813i	0.112821	+	0.993615i	$0.464011\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.54299e11			−1.39206			−0.696031	−	0.718012i	$-0.745052\pi$
$577$	−1.54299e11			−1.39206			−0.696031	+	0.718012i	$0.745052\pi$
$578$		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		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$	−2.33889e11	−	4.05108e11i	−1.94334	−	3.36597i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	3.11303e10	−	5.39193e10i	0.238608	−	0.413282i	−0.721707	−	0.692199i	$-0.756642\pi$
$601$	3.11303e10	−	5.39193e10i	0.238608	−	0.413282i	0.960315	+	0.278917i	$0.0899754\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.32994e11	+	2.30353e11i	0.979666	+	1.69683i	0.663587	+	0.748099i	$0.269033\pi$
$607$	1.32994e11	+	2.30353e11i	0.979666	+	1.69683i	0.316079	+	0.948733i	$0.397633\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.57998e10			−0.111894			−0.0559472	−	0.998434i	$-0.517818\pi$
$613$	−1.57998e10			−0.111894			−0.0559472	+	0.998434i	$0.517818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	−1.12767e11	+	1.95317e11i	−0.768100	+	1.33039i	0.170493	+	0.985359i	$0.445464\pi$
$619$	−1.12767e11	+	1.95317e11i	−0.768100	+	1.33039i	−0.938592	+	0.345029i	$0.887869\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.62939e10	−	1.32145e11i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−2.00069e11			−1.26201			−0.631004	−	0.775780i	$-0.717357\pi$
$631$	−2.00069e11			−1.26201			−0.631004	+	0.775780i	$0.717357\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.88131e11	−	3.25853e11i	1.14262	−	1.97908i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−9.42718e10	−	1.63284e11i	−0.551490	−	0.955210i	−0.998167	−	0.0605142i	$-0.980726\pi$
$643$	−9.42718e10	−	1.63284e11i	−0.551490	−	0.955210i	0.446677	−	0.894695i	$-0.352607\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−1.78004e11	−	3.08313e11i	−0.932449	−	1.61505i	−0.779121	−	0.626874i	$-0.784334\pi$
$661$	−1.78004e11	−	3.08313e11i	−0.932449	−	1.61505i	−0.153328	−	0.988175i	$-0.548999\pi$
$662$		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
$673$	1.14467e11	−	1.98263e11i	0.557983	−	0.966454i	−0.439682	−	0.898153i	$-0.644909\pi$
$673$	1.14467e11	−	1.98263e11i	0.557983	−	0.966454i	0.997665	−	0.0683008i	$-0.0217578\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−3.56824e11	−	6.18036e11i	−1.67870	−	2.90760i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−7.79006e10	+	1.34928e11i	−0.341687	+	0.591819i	−0.984746	−	0.173997i	$-0.944332\pi$
$691$	−7.79006e10	+	1.34928e11i	−0.341687	+	0.591819i	0.643059	+	0.765817i	$0.277665\pi$
$692$		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		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−1.30174e11			−0.532968
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.43843e11	−	4.22348e11i	0.964995	−	1.67142i	0.255367	−	0.966844i	$-0.417804\pi$
$709$	2.43843e11	−	4.22348e11i	0.964995	−	1.67142i	0.709628	−	0.704577i	$-0.248863\pi$
$710$		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		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	1.79307e11			0.663524
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−6.07502e10	+	1.05222e11i	−0.217475	+	0.376678i	−0.954035	−	0.299694i	$-0.903115\pi$
$727$	−6.07502e10	+	1.05222e11i	−0.217475	+	0.376678i	0.736560	+	0.676372i	$0.236449\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	2.88665e11	+	4.99982e11i	0.999949	+	1.73196i	0.508714	+	0.860936i	$0.330121\pi$
$733$	2.88665e11	+	4.99982e11i	0.999949	+	1.73196i	0.491235	+	0.871027i	$0.336546\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−4.35662e11			−1.46074			−0.730368	−	0.683054i	$-0.760651\pi$
$739$	−4.35662e11			−1.46074			−0.730368	+	0.683054i	$0.760651\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.67915e11	+	4.64043e11i	0.842244	+	1.45881i	0.887993	+	0.459856i	$0.152099\pi$
$751$	2.67915e11	+	4.64043e11i	0.842244	+	1.45881i	−0.0457494	+	0.998953i	$0.514568\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−4.59764e11			−1.40008			−0.700038	−	0.714106i	$-0.746833\pi$
$757$	−4.59764e11			−1.40008			−0.700038	+	0.714106i	$0.746833\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−4.09816e11	+	7.09822e11i	−1.20918	+	2.09436i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−3.35250e11	−	5.80670e11i	−0.958658	−	1.66044i	−0.725767	−	0.687941i	$-0.758515\pi$
$769$	−3.35250e11	−	5.80670e11i	−0.958658	−	1.66044i	−0.232890	−	0.972503i	$-0.574818\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−7.06799e11			−1.95925
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−3.37710e11	−	5.84931e11i	−0.880329	−	1.52477i	−0.850976	−	0.525205i	$-0.823989\pi$
$787$	−3.37710e11	−	5.84931e11i	−0.880329	−	1.52477i	−0.0293527	−	0.999569i	$-0.509345\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	8.53133e11			2.15737
$794$		0			0
$795$		0

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2017.00 + 3493.55i) q^{7} +O(q^{10})\)	\(q+(-2017.00 + 3493.55i) q^{7} +(17903.0 + 31008.9i) q^{13} -258526. q^{19} +(-195312. + 338291. i) q^{25} +(904703. + 1.56699e6i) q^{31} +503522. q^{37} +(-1.74610e6 + 3.02433e6i) q^{43} +(-5.25418e6 - 9.10050e6i) q^{49} +(1.19133e7 - 2.06344e7i) q^{61} +(2.71070e6 + 4.69508e6i) q^{67} +1.61693e7 q^{73} +(9.44352e6 - 1.63567e7i) q^{79} -1.44441e8 q^{91} +(-8.84540e7 + 1.53207e8i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4034 q^{7}+O(q^{10}) \) 2 * q - 4034 * q^7	\( 2 q - 4034 q^{7} + 35806 q^{13} - 517052 q^{19} - 390625 q^{25} + 1809406 q^{31} + 1007044 q^{37} - 3492194 q^{43} - 10508355 q^{49} + 23826526 q^{61} + 5421406 q^{67} + 32338564 q^{73} + 18887038 q^{79} - 288882808 q^{91} - 176908034 q^{97}+O(q^{100}) \) 2 * q - 4034 * q^7 + 35806 * q^13 - 517052 * q^19 - 390625 * q^25 + 1809406 * q^31 + 1007044 * q^37 - 3492194 * q^43 - 10508355 * q^49 + 23826526 * q^61 + 5421406 * q^67 + 32338564 * q^73 + 18887038 * q^79 - 288882808 * q^91 - 176908034 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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