LMFDB - Embedded newform 324.3.d.d.163.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(163,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("324.163");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.d (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			163.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	324.163

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +9.19615 q^{5} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} +9.19615 q^{5} +8.00000 q^{8} +18.3923 q^{10} -15.7846 q^{13} +16.0000 q^{16} -33.9808 q^{17} +36.7846 q^{20} +59.5692 q^{25} -31.5692 q^{26} +16.3731 q^{29} +32.0000 q^{32} -67.9615 q^{34} +14.2154 q^{37} +73.5692 q^{40} -80.0000 q^{41} +49.0000 q^{49} +119.138 q^{50} -63.1384 q^{52} -56.0000 q^{53} +32.7461 q^{58} +114.923 q^{61} +64.0000 q^{64} -145.158 q^{65} -135.923 q^{68} -138.138 q^{73} +28.4308 q^{74} +147.138 q^{80} -160.000 q^{82} -312.492 q^{85} -12.4500 q^{89} -130.000 q^{97} +98.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + 16 q^{10} + 10 q^{13} + 32 q^{16} - 16 q^{17} + 32 q^{20} + 36 q^{25} + 20 q^{26} - 40 q^{29} + 64 q^{32} - 32 q^{34} + 70 q^{37} + 64 q^{40} - 160 q^{41} + 98 q^{49} + 72 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 176 q^{65} - 64 q^{68} - 110 q^{73} + 140 q^{74} + 128 q^{80} - 320 q^{82} - 334 q^{85} - 160 q^{89} - 260 q^{97} + 196 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 16 * q^8 + 16 * q^10 + 10 * q^13 + 32 * q^16 - 16 * q^17 + 32 * q^20 + 36 * q^25 + 20 * q^26 - 40 * q^29 + 64 * q^32 - 32 * q^34 + 70 * q^37 + 64 * q^40 - 160 * q^41 + 98 * q^49 + 72 * q^50 + 40 * q^52 - 112 * q^53 - 80 * q^58 + 22 * q^61 + 128 * q^64 - 176 * q^65 - 64 * q^68 - 110 * q^73 + 140 * q^74 + 128 * q^80 - 320 * q^82 - 334 * q^85 - 160 * q^89 - 260 * q^97 + 196 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.00000	1.00000
$2$	2.00000	1.00000
$3$	0	0
$3$	0	0
$4$	4.00000	1.00000
$5$	9.19615	1.83923	0.919615	−	0.392820i	$-0.128501\pi$
$5$	9.19615	1.83923	0.919615	+	0.392820i	$0.128501\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	8.00000	1.00000
$9$	0	0
$10$	18.3923	1.83923
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−15.7846	−1.21420	−0.607100	−	0.794625i	$-0.707667\pi$
$13$	−15.7846	−1.21420	−0.607100	+	0.794625i	$0.707667\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	−33.9808	−1.99887	−0.999434	−	0.0336351i	$-0.989292\pi$
$17$	−33.9808	−1.99887	−0.999434	+	0.0336351i	$0.989292\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	36.7846	1.83923
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	59.5692	2.38277
$26$	−31.5692	−1.21420
$27$	0	0
$28$	0	0
$29$	16.3731	0.564589	0.282294	−	0.959328i	$-0.408905\pi$
$29$	16.3731	0.564589	0.282294	+	0.959328i	$0.408905\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.0000	1.00000
$33$	0	0
$34$	−67.9615	−1.99887
$35$	0	0
$36$	0	0
$37$	14.2154	0.384200	0.192100	−	0.981375i	$-0.438470\pi$
$37$	14.2154	0.384200	0.192100	+	0.981375i	$0.438470\pi$
$38$	0	0
$39$	0	0
$40$	73.5692	1.83923
$41$	−80.0000	−1.95122	−0.975610	−	0.219512i	$-0.929553\pi$
$41$	−80.0000	−1.95122	−0.975610	+	0.219512i	$0.929553\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	119.138	2.38277
$51$	0	0
$52$	−63.1384	−1.21420
$53$	−56.0000	−1.05660	−0.528302	−	0.849057i	$-0.677171\pi$
$53$	−56.0000	−1.05660	−0.528302	+	0.849057i	$0.677171\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	32.7461	0.564589
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	114.923	1.88398	0.941992	−	0.335635i	$-0.108951\pi$
$61$	114.923	1.88398	0.941992	+	0.335635i	$0.108951\pi$
$62$	0	0
$63$	0	0
$64$	64.0000	1.00000
$65$	−145.158	−2.23320
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−135.923	−1.99887
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−138.138	−1.89231	−0.946154	−	0.323718i	$-0.895067\pi$
$73$	−138.138	−1.89231	−0.946154	+	0.323718i	$0.895067\pi$
$74$	28.4308	0.384200
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	147.138	1.83923
$81$	0	0
$82$	−160.000	−1.95122
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−312.492	−3.67638
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.4500	−0.139888	−0.0699439	−	0.997551i	$-0.522282\pi$
$89$	−12.4500	−0.139888	−0.0699439	+	0.997551i	$0.522282\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−130.000	−1.34021	−0.670103	−	0.742268i	$-0.733750\pi$
$97$	−130.000	−1.34021	−0.670103	+	0.742268i	$0.733750\pi$
$98$	98.0000	1.00000
$99$	0	0
$100$	238.277	2.38277
$101$	40.0000	0.396040	0.198020	−	0.980198i	$-0.436549\pi$
$101$	40.0000	0.396040	0.198020	+	0.980198i	$0.436549\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−126.277	−1.21420
$105$	0	0
$106$	−112.000	−1.05660
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	12.9230	0.118560	0.0592800	−	0.998241i	$-0.481120\pi$
$109$	12.9230	0.118560	0.0592800	+	0.998241i	$0.481120\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	86.0192	0.761232	0.380616	−	0.924733i	$-0.375712\pi$
$113$	86.0192	0.761232	0.380616	+	0.924733i	$0.375712\pi$
$114$	0	0
$115$	0	0
$116$	65.4923	0.564589
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	229.846	1.88398
$123$	0	0
$124$	0	0
$125$	317.904	2.54323
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	128.000	1.00000
$129$	0	0
$130$	−290.315	−2.23320
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−271.846	−1.99887
$137$	−93.8653	−0.685148	−0.342574	−	0.939491i	$-0.611299\pi$
$137$	−93.8653	−0.685148	−0.342574	+	0.939491i	$0.611299\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	150.569	1.03841
$146$	−276.277	−1.89231
$147$	0	0
$148$	56.8616	0.384200
$149$	−228.335	−1.53245	−0.766223	−	0.642574i	$-0.777866\pi$
$149$	−228.335	−1.53245	−0.766223	+	0.642574i	$0.777866\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	143.631	0.914845	0.457423	−	0.889249i	$-0.348773\pi$
$157$	143.631	0.914845	0.457423	+	0.889249i	$0.348773\pi$
$158$	0	0
$159$	0	0
$160$	294.277	1.83923
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−320.000	−1.95122
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	80.1539	0.474283
$170$	−624.985	−3.67638
$171$	0	0
$172$	0	0
$173$	337.788	1.95253	0.976267	−	0.216570i	$-0.0694871\pi$
$173$	337.788	1.95253	0.976267	+	0.216570i	$0.0694871\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−24.9000	−0.139888
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	38.0000	0.209945	0.104972	−	0.994475i	$-0.466525\pi$
$181$	38.0000	0.209945	0.104972	+	0.994475i	$0.466525\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	130.727	0.706632
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	385.985	1.99992	0.999960	−	0.00895123i	$-0.00284930\pi$
$193$	385.985	1.99992	0.999960	+	0.00895123i	$0.00284930\pi$
$194$	−260.000	−1.34021
$195$	0	0
$196$	196.000	1.00000
$197$	−309.750	−1.57233	−0.786167	−	0.618014i	$-0.787938\pi$
$197$	−309.750	−1.57233	−0.786167	+	0.618014i	$0.787938\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	476.554	2.38277
$201$	0	0
$202$	80.0000	0.396040
$203$	0	0
$204$	0	0
$205$	−735.692	−3.58874
$206$	0	0
$207$	0	0
$208$	−252.554	−1.21420
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−224.000	−1.05660
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	25.8461	0.118560
$219$	0	0
$220$	0	0
$221$	536.373	2.42703
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	172.038	0.761232
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	324.923	1.41888	0.709439	−	0.704767i	$-0.248948\pi$
$229$	324.923	1.41888	0.709439	+	0.704767i	$0.248948\pi$
$230$	0	0
$231$	0	0
$232$	130.985	0.564589
$233$	26.1347	0.112166	0.0560830	−	0.998426i	$-0.482139\pi$
$233$	26.1347	0.112166	0.0560830	+	0.998426i	$0.482139\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	1.15390	0.00478798	0.00239399	−	0.999997i	$-0.499238\pi$
$241$	1.15390	0.00478798	0.00239399	+	0.999997i	$0.499238\pi$
$242$	242.000	1.00000
$243$	0	0
$244$	459.692	1.88398
$245$	450.611	1.83923
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	635.808	2.54323
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	409.673	1.59406	0.797029	−	0.603941i	$-0.206404\pi$
$257$	409.673	1.59406	0.797029	+	0.603941i	$0.206404\pi$
$258$	0	0
$259$	0	0
$260$	−580.631	−2.23320
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−514.985	−1.94334
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−379.512	−1.41082	−0.705412	−	0.708798i	$-0.749238\pi$
$269$	−379.512	−1.41082	−0.705412	+	0.708798i	$0.749238\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−543.692	−1.99887
$273$	0	0
$274$	−187.731	−0.685148
$275$	0	0
$276$	0	0
$277$	230.000	0.830325	0.415162	−	0.909747i	$-0.363725\pi$
$277$	230.000	0.830325	0.415162	+	0.909747i	$0.363725\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−240.104	−0.854462	−0.427231	−	0.904143i	$-0.640511\pi$
$281$	−240.104	−0.854462	−0.427231	+	0.904143i	$0.640511\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	865.692	2.99547
$290$	301.138	1.03841
$291$	0	0
$292$	−552.554	−1.89231
$293$	−561.634	−1.91684	−0.958421	−	0.285359i	$-0.907887\pi$
$293$	−561.634	−1.91684	−0.958421	+	0.285359i	$0.907887\pi$
$294$	0	0
$295$	0	0
$296$	113.723	0.384200
$297$	0	0
$298$	−456.669	−1.53245
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1056.85	3.46508
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	515.400	1.64664	0.823322	−	0.567574i	$-0.192118\pi$
$313$	515.400	1.64664	0.823322	+	0.567574i	$0.192118\pi$
$314$	287.261	0.914845
$315$	0	0
$316$	0	0
$317$	−178.096	−0.561818	−0.280909	−	0.959734i	$-0.590636\pi$
$317$	−178.096	−0.561818	−0.280909	+	0.959734i	$0.590636\pi$
$318$	0	0
$319$	0	0
$320$	588.554	1.83923
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−940.277	−2.89316
$326$	0	0
$327$	0	0
$328$	−640.000	−1.95122
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	350.000	1.03858	0.519288	−	0.854599i	$-0.326197\pi$
$337$	350.000	1.03858	0.519288	+	0.854599i	$0.326197\pi$
$338$	160.308	0.474283
$339$	0	0
$340$	−1249.97	−3.67638
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	675.577	1.95253
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−598.000	−1.71347	−0.856734	−	0.515759i	$-0.827510\pi$
$349$	−598.000	−1.71347	−0.856734	+	0.515759i	$0.827510\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	544.000	1.54108	0.770538	−	0.637394i	$-0.219988\pi$
$353$	544.000	1.54108	0.770538	+	0.637394i	$0.219988\pi$
$354$	0	0
$355$	0	0
$356$	−49.8001	−0.139888
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	76.0000	0.209945
$363$	0	0
$364$	0	0
$365$	−1270.34	−3.48039
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	261.454	0.706632
$371$	0	0
$372$	0	0
$373$	−550.000	−1.47453	−0.737265	−	0.675603i	$-0.763883\pi$
$373$	−550.000	−1.47453	−0.737265	+	0.675603i	$0.763883\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−258.442	−0.685524
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	771.969	1.99992
$387$	0	0
$388$	−520.000	−1.34021
$389$	−680.000	−1.74807	−0.874036	−	0.485861i	$-0.838506\pi$
$389$	−680.000	−1.74807	−0.874036	+	0.485861i	$0.838506\pi$
$390$	0	0
$391$	0	0
$392$	392.000	1.00000
$393$	0	0
$394$	−619.500	−1.57233
$395$	0	0
$396$	0	0
$397$	−719.908	−1.81337	−0.906685	−	0.421809i	$-0.861395\pi$
$397$	−719.908	−1.81337	−0.906685	+	0.421809i	$0.861395\pi$
$398$	0	0
$399$	0	0
$400$	953.108	2.38277
$401$	731.088	1.82316	0.911581	−	0.411120i	$-0.134862\pi$
$401$	731.088	1.82316	0.911581	+	0.411120i	$0.134862\pi$
$402$	0	0
$403$	0	0
$404$	160.000	0.396040
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−598.846	−1.46417	−0.732086	−	0.681213i	$-0.761453\pi$
$409$	−598.846	−1.46417	−0.732086	+	0.681213i	$0.761453\pi$
$410$	−1471.38	−3.58874
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−505.108	−1.21420
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	756.461	1.79682	0.898410	−	0.439157i	$-0.144723\pi$
$421$	756.461	1.79682	0.898410	+	0.439157i	$0.144723\pi$
$422$	0	0
$423$	0	0
$424$	−448.000	−1.05660
$425$	−2024.21	−4.76284
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−851.677	−1.96692	−0.983460	−	0.181123i	$-0.942027\pi$
$433$	−851.677	−1.96692	−0.983460	+	0.181123i	$0.942027\pi$
$434$	0	0
$435$	0	0
$436$	51.6922	0.118560
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	1072.75	2.42703
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−114.492	−0.257286
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−560.000	−1.24722	−0.623608	−	0.781737i	$-0.714334\pi$
$449$	−560.000	−1.24722	−0.623608	+	0.781737i	$0.714334\pi$
$450$	0	0
$451$	0	0
$452$	344.077	0.761232
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	715.985	1.56671	0.783353	−	0.621577i	$-0.213508\pi$
$457$	715.985	1.56671	0.783353	+	0.621577i	$0.213508\pi$
$458$	649.846	1.41888
$459$	0	0
$460$	0	0
$461$	760.000	1.64859	0.824295	−	0.566161i	$-0.191572\pi$
$461$	760.000	1.64859	0.824295	+	0.566161i	$0.191572\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	261.969	0.564589
$465$	0	0
$466$	52.2693	0.112166
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−224.384	−0.466496
$482$	2.30781	0.00478798
$483$	0	0
$484$	484.000	1.00000
$485$	−1195.50	−2.46495
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	919.384	1.88398
$489$	0	0
$490$	901.223	1.83923
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	−556.369	−1.12854
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	1271.62	2.54323
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	367.846	0.728408
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−440.000	−0.864440	−0.432220	−	0.901768i	$-0.642270\pi$
$509$	−440.000	−0.864440	−0.432220	+	0.901768i	$0.642270\pi$
$510$	0	0
$511$	0	0
$512$	512.000	1.00000
$513$	0	0
$514$	819.346	1.59406
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−1161.26	−2.23320
$521$	880.000	1.68906	0.844530	−	0.535509i	$-0.179880\pi$
$521$	880.000	1.68906	0.844530	+	0.535509i	$0.179880\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	−1029.97	−1.94334
$531$	0	0
$532$	0	0
$533$	1262.77	2.36917
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−759.023	−1.41082
$539$	0	0
$540$	0	0
$541$	1068.46	1.97497	0.987487	−	0.157698i	$-0.0504073\pi$
$541$	1068.46	1.97497	0.987487	+	0.157698i	$0.0504073\pi$
$542$	0	0
$543$	0	0
$544$	−1087.38	−1.99887
$545$	118.842	0.218059
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−375.461	−0.685148
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	460.000	0.830325
$555$	0	0
$556$	0	0
$557$	817.788	1.46820	0.734101	−	0.679040i	$-0.237604\pi$
$557$	817.788	1.46820	0.734101	+	0.679040i	$0.237604\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−480.207	−0.854462
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	791.046	1.40008
$566$	0	0
$567$	0	0
$568$	0	0
$569$	119.896	0.210714	0.105357	−	0.994434i	$-0.466401\pi$
$569$	119.896	0.210714	0.105357	+	0.994434i	$0.466401\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	491.862	0.852446	0.426223	−	0.904618i	$-0.359844\pi$
$577$	491.862	0.852446	0.426223	+	0.904618i	$0.359844\pi$
$578$	1731.38	2.99547
$579$	0	0
$580$	602.277	1.03841
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−1105.11	−1.89231
$585$	0	0
$586$	−1123.27	−1.91684
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	227.446	0.384200
$593$	−1173.40	−1.97876	−0.989379	−	0.145358i	$-0.953567\pi$
$593$	−1173.40	−1.97876	−0.989379	+	0.145358i	$0.953567\pi$
$594$	0	0
$595$	0	0
$596$	−913.338	−1.53245
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	966.692	1.60847	0.804236	−	0.594309i	$-0.202575\pi$
$601$	966.692	1.60847	0.804236	+	0.594309i	$0.202575\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1112.73	1.83923
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	2113.70	3.46508
$611$	0	0
$612$	0	0
$613$	−70.0000	−0.114192	−0.0570962	−	0.998369i	$-0.518184\pi$
$613$	−70.0000	−0.114192	−0.0570962	+	0.998369i	$0.518184\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−789.865	−1.28017	−0.640085	−	0.768304i	$-0.721101\pi$
$617$	−789.865	−1.28017	−0.640085	+	0.768304i	$0.721101\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1434.26	2.29482
$626$	1030.80	1.64664
$627$	0	0
$628$	574.523	0.914845
$629$	−483.050	−0.767965
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−356.192	−0.561818
$635$	0	0
$636$	0	0
$637$	−773.446	−1.21420
$638$	0	0
$639$	0	0
$640$	1177.11	1.83923
$641$	−1254.82	−1.95760	−0.978798	−	0.204828i	$-0.934336\pi$
$641$	−1254.82	−1.95760	−0.978798	+	0.204828i	$0.934336\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	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
$650$	−1880.55	−2.89316
$651$	0	0
$652$	0	0
$653$	1144.00	1.75191	0.875957	−	0.482389i	$-0.160231\pi$
$653$	1144.00	1.75191	0.875957	+	0.482389i	$0.160231\pi$
$654$	0	0
$655$	0	0
$656$	−1280.00	−1.95122
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−1108.62	−1.67718	−0.838589	−	0.544764i	$-0.816619\pi$
$661$	−1108.62	−1.67718	−0.838589	+	0.544764i	$0.816619\pi$
$662$	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
$673$	−1341.09	−1.99271	−0.996354	−	0.0853191i	$-0.972809\pi$
$673$	−1341.09	−1.99271	−0.996354	+	0.0853191i	$0.972809\pi$
$674$	700.000	1.03858
$675$	0	0
$676$	320.616	0.474283
$677$	−104.000	−0.153619	−0.0768095	−	0.997046i	$-0.524473\pi$
$677$	−104.000	−0.153619	−0.0768095	+	0.997046i	$0.524473\pi$
$678$	0	0
$679$	0	0
$680$	−2499.94	−3.67638
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−863.200	−1.26015
$686$	0	0
$687$	0	0
$688$	0	0
$689$	883.938	1.28293
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1351.15	1.95253
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2718.46	3.90023
$698$	−1196.00	−1.71347
$699$	0	0
$700$	0	0
$701$	867.565	1.23761	0.618805	−	0.785544i	$-0.287617\pi$
$701$	867.565	1.23761	0.618805	+	0.785544i	$0.287617\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	1088.00	1.54108
$707$	0	0
$708$	0	0
$709$	−1402.15	−1.97765	−0.988825	−	0.149082i	$-0.952368\pi$
$709$	−1402.15	−1.97765	−0.988825	+	0.149082i	$0.952368\pi$
$710$	0	0
$711$	0	0
$712$	−99.6001	−0.139888
$713$	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
$721$	0	0
$722$	722.000	1.00000
$723$	0	0
$724$	152.000	0.209945
$725$	975.331	1.34528
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	−2540.68	−3.48039
$731$	0	0
$732$	0	0
$733$	−1450.00	−1.97817	−0.989086	−	0.147340i	$-0.952929\pi$
$733$	−1450.00	−1.97817	−0.989086	+	0.147340i	$0.952929\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	522.908	0.706632
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−2099.80	−2.81852
$746$	−1100.00	−1.47453
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	−516.885	−0.685524
$755$	0	0
$756$	0	0
$757$	1190.00	1.57199	0.785997	−	0.618230i	$-0.212150\pi$
$757$	1190.00	1.57199	0.785997	+	0.618230i	$0.212150\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	827.550	1.08745	0.543725	−	0.839263i	$-0.317013\pi$
$761$	827.550	1.08745	0.543725	+	0.839263i	$0.317013\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	558.230	0.725917	0.362959	−	0.931805i	$-0.381767\pi$
$769$	558.230	0.725917	0.362959	+	0.931805i	$0.381767\pi$
$770$	0	0
$771$	0	0
$772$	1543.94	1.99992
$773$	410.250	0.530725	0.265362	−	0.964149i	$-0.414508\pi$
$773$	410.250	0.530725	0.265362	+	0.964149i	$0.414508\pi$
$774$	0	0
$775$	0	0
$776$	−1040.00	−1.34021
$777$	0	0
$778$	−1360.00	−1.74807
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	784.000	1.00000
$785$	1320.85	1.68261
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1239.00	−1.57233
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1814.02	−2.28754
$794$	−1439.82	−1.81337
$795$	0	0
$796$	0	0
$797$	−1533.29	−1.92382	−0.961912	−	0.273358i	$-0.911866\pi$
$797$	−1533.29	−1.92382	−0.961912	+	0.273358i	$0.911866\pi$
$798$	0	0
$799$	0	0
$800$	1906.22	2.38277
$801$	0	0
$802$	1462.18	1.82316
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	320.000	0.396040
$809$	1594.63	1.97111	0.985554	−	0.169361i	$-0.0541703\pi$
$809$	1594.63	1.97111	0.985554	+	0.169361i	$0.0541703\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−1197.69	−1.46417
$819$	0	0
$820$	−2942.77	−3.58874
$821$	−43.0498	−0.0524358	−0.0262179	−	0.999656i	$-0.508346\pi$
$821$	−43.0498	−0.0524358	−0.0262179	+	0.999656i	$0.508346\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1258.00	−1.51749	−0.758745	−	0.651387i	$-0.774187\pi$
$829$	−1258.00	−1.51749	−0.758745	+	0.651387i	$0.774187\pi$
$830$	0	0
$831$	0	0
$832$	−1010.22	−1.21420
$833$	−1665.06	−1.99887
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−572.923	−0.681240
$842$	1512.92	1.79682
$843$	0	0
$844$	0	0
$845$	737.108	0.872317
$846$	0	0
$847$	0	0
$848$	−896.000	−1.05660
$849$	0	0
$850$	−4048.41	−4.76284
$851$	0	0
$852$	0	0
$853$	410.000	0.480657	0.240328	−	0.970692i	$-0.422745\pi$
$853$	410.000	0.480657	0.240328	+	0.970692i	$0.422745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1196.94	−1.39667	−0.698333	−	0.715774i	$-0.746074\pi$
$857$	−1196.94	−1.39667	−0.698333	+	0.715774i	$0.746074\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	3106.35	3.59116
$866$	−1703.35	−1.96692
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	103.384	0.118560
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−202.246	−0.230612	−0.115306	−	0.993330i	$-0.536785\pi$
$877$	−202.246	−0.230612	−0.115306	+	0.993330i	$0.536785\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1600.00	1.81612	0.908059	−	0.418842i	$-0.137564\pi$
$881$	1600.00	1.81612	0.908059	+	0.418842i	$0.137564\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	2145.49	2.42703
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	−228.985	−0.257286
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−1120.00	−1.24722
$899$	0	0
$900$	0	0
$901$	1902.92	2.11201
$902$	0	0
$903$	0	0
$904$	688.154	0.761232
$905$	349.454	0.386137
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	1431.97	1.56671
$915$	0	0
$916$	1299.69	1.41888
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	1520.00	1.64859
$923$	0	0
$924$	0	0
$925$	846.800	0.915459
$926$	0	0
$927$	0	0
$928$	523.938	0.564589
$929$	−696.565	−0.749801	−0.374901	−	0.927065i	$-0.622323\pi$
$929$	−696.565	−0.749801	−0.374901	+	0.927065i	$0.622323\pi$
$930$	0	0
$931$	0	0
$932$	104.539	0.112166
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1364.63	−1.45638	−0.728191	−	0.685374i	$-0.759639\pi$
$937$	−1364.63	−1.45638	−0.728191	+	0.685374i	$0.759639\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1863.45	1.98029	0.990143	−	0.140058i	$-0.0447290\pi$
$941$	1863.45	1.98029	0.990143	+	0.140058i	$0.0447290\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	2180.46	2.29764
$950$	0	0
$951$	0	0
$952$	0	0
$953$	337.211	0.353842	0.176921	−	0.984225i	$-0.443386\pi$
$953$	337.211	0.353842	0.176921	+	0.984225i	$0.443386\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	−448.769	−0.466496
$963$	0	0
$964$	4.61561	0.00478798
$965$	3549.57	3.67831
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	968.000	1.00000
$969$	0	0
$970$	−2391.00	−2.46495
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	1838.77	1.88398
$977$	496.000	0.507677	0.253838	−	0.967247i	$-0.418307\pi$
$977$	496.000	0.507677	0.253838	+	0.967247i	$0.418307\pi$
$978$	0	0
$979$	0	0
$980$	1802.45	1.83923
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−2848.51	−2.89189
$986$	−1112.74	−1.12854
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1569.32	−1.57405	−0.787023	−	0.616924i	$-0.788378\pi$
$997$	−1569.32	−1.57405	−0.787023	+	0.616924i	$0.788378\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.d.d.163.2	yes	2
3.2	odd	2		324.3.d.a.163.1	✓	2
4.3	odd	2	CM	324.3.d.d.163.2	yes	2
9.2	odd	6		324.3.f.n.271.2		4
9.4	even	3		324.3.f.k.55.1		4
9.5	odd	6		324.3.f.n.55.2		4
9.7	even	3		324.3.f.k.271.1		4
12.11	even	2		324.3.d.a.163.1	✓	2
36.7	odd	6		324.3.f.k.271.1		4
36.11	even	6		324.3.f.n.271.2		4
36.23	even	6		324.3.f.n.55.2		4
36.31	odd	6		324.3.f.k.55.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.3.d.a.163.1	✓	2	3.2	odd	2
324.3.d.a.163.1	✓	2	12.11	even	2
324.3.d.d.163.2	yes	2	1.1	even	1	trivial
324.3.d.d.163.2	yes	2	4.3	odd	2	CM
324.3.f.k.55.1		4	9.4	even	3
324.3.f.k.55.1		4	36.31	odd	6
324.3.f.k.271.1		4	9.7	even	3
324.3.f.k.271.1		4	36.7	odd	6
324.3.f.n.55.2		4	9.5	odd	6
324.3.f.n.55.2		4	36.23	even	6
324.3.f.n.271.2		4	9.2	odd	6
324.3.f.n.271.2		4	36.11	even	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +9.19615 q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +9.19615 q^{5} +8.00000 q^{8} +18.3923 q^{10} -15.7846 q^{13} +16.0000 q^{16} -33.9808 q^{17} +36.7846 q^{20} +59.5692 q^{25} -31.5692 q^{26} +16.3731 q^{29} +32.0000 q^{32} -67.9615 q^{34} +14.2154 q^{37} +73.5692 q^{40} -80.0000 q^{41} +49.0000 q^{49} +119.138 q^{50} -63.1384 q^{52} -56.0000 q^{53} +32.7461 q^{58} +114.923 q^{61} +64.0000 q^{64} -145.158 q^{65} -135.923 q^{68} -138.138 q^{73} +28.4308 q^{74} +147.138 q^{80} -160.000 q^{82} -312.492 q^{85} -12.4500 q^{89} -130.000 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + 16 q^{10} + 10 q^{13} + 32 q^{16} - 16 q^{17} + 32 q^{20} + 36 q^{25} + 20 q^{26} - 40 q^{29} + 64 q^{32} - 32 q^{34} + 70 q^{37} + 64 q^{40} - 160 q^{41} + 98 q^{49} + 72 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 176 q^{65} - 64 q^{68} - 110 q^{73} + 140 q^{74} + 128 q^{80} - 320 q^{82} - 334 q^{85} - 160 q^{89} - 260 q^{97} + 196 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 16 * q^8 + 16 * q^10 + 10 * q^13 + 32 * q^16 - 16 * q^17 + 32 * q^20 + 36 * q^25 + 20 * q^26 - 40 * q^29 + 64 * q^32 - 32 * q^34 + 70 * q^37 + 64 * q^40 - 160 * q^41 + 98 * q^49 + 72 * q^50 + 40 * q^52 - 112 * q^53 - 80 * q^58 + 22 * q^61 + 128 * q^64 - 176 * q^65 - 64 * q^68 - 110 * q^73 + 140 * q^74 + 128 * q^80 - 320 * q^82 - 334 * q^85 - 160 * q^89 - 260 * q^97 + 196 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more