LMFDB - Embedded newform 648.2.f.a.323.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(323,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("648.323");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.f (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:	no
Analytic conductor:	$5.17430605098$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			323.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	648.323
Dual form			648.2.f.a.323.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} -3.78194i q^{11} +4.00000 q^{16} -8.02458i q^{17} -8.34847 q^{19} -5.34847 q^{22} -5.00000 q^{25} -5.65685i q^{32} -11.3485 q^{34} +11.8065i q^{38} +0.460702i q^{41} -2.34847 q^{43} +7.56388i q^{44} +7.00000 q^{49} +7.07107i q^{50} -12.2672i q^{59} -8.00000 q^{64} -14.3485 q^{67} +16.0492i q^{68} +13.6969 q^{73} +16.6969 q^{76} +0.651531 q^{82} -2.82843i q^{83} +3.32124i q^{86} +10.6969 q^{88} +5.65685i q^{89} +19.6969 q^{97} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} - 4 q^{19} + 8 q^{22} - 20 q^{25} - 16 q^{34} + 20 q^{43} + 28 q^{49} - 32 q^{64} - 28 q^{67} - 4 q^{73} + 8 q^{76} + 32 q^{82} - 16 q^{88} + 20 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 - 4 * q^19 + 8 * q^22 - 20 * q^25 - 16 * q^34 + 20 * q^43 + 28 * q^49 - 32 * q^64 - 28 * q^67 - 4 * q^73 + 8 * q^76 + 32 * q^82 - 16 * q^88 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	− 1.41421i	− 1.00000i
$2$	− 1.41421i	− 1.00000i
$3$	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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	2.82843i	1.00000i
$9$	0	0
$10$	0	0
$11$	− 3.78194i	− 1.14030i	−0.821541	−	0.570149i	$-0.806886\pi$
$11$	− 3.78194i	− 1.14030i	0.821541	−	0.570149i	$-0.193114\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 8.02458i	− 1.94625i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	− 8.02458i	− 1.94625i	0.230285	−	0.973123i	$-0.426034\pi$
$18$	0	0
$19$	−8.34847	−1.91527	−0.957635	−	0.287984i	$-0.907015\pi$
$19$	−8.34847	−1.91527	−0.957635	+	0.287984i	$0.907015\pi$
$20$	0	0
$21$	0	0
$22$	−5.34847	−1.14030
$23$	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
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	−11.3485	−1.94625
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	11.8065i	1.91527i
$39$	0	0
$40$	0	0
$41$	0.460702i	0.0719495i	0.999353	+	0.0359748i	$0.0114536\pi$
$41$	0.460702i	0.0719495i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$	0	0
$43$	−2.34847	−0.358138	−0.179069	−	0.983836i	$-0.557309\pi$
$43$	−2.34847	−0.358138	−0.179069	+	0.983836i	$0.557309\pi$
$44$	7.56388i	1.14030i
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	7.07107i	1.00000i
$51$	0	0
$52$	0	0
$53$	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
$59$	− 12.2672i	− 1.59706i	−0.601958	−	0.798528i	$-0.705612\pi$
$59$	− 12.2672i	− 1.59706i	0.601958	−	0.798528i	$-0.294388\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−14.3485	−1.75294	−0.876472	−	0.481452i	$-0.840109\pi$
$67$	−14.3485	−1.75294	−0.876472	+	0.481452i	$0.840109\pi$
$68$	16.0492i	1.94625i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	13.6969	1.60311	0.801553	−	0.597924i	$-0.204008\pi$
$73$	13.6969	1.60311	0.801553	+	0.597924i	$0.204008\pi$
$74$	0	0
$75$	0	0
$76$	16.6969	1.91527
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0.651531	0.0719495
$83$	− 2.82843i	− 0.310460i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	− 2.82843i	− 0.310460i	0.987878	−	0.155230i	$-0.0496119\pi$
$84$	0	0
$85$	0	0
$86$	3.32124i	0.358138i
$87$	0	0
$88$	10.6969	1.14030
$89$	5.65685i	0.599625i	0.953998	+	0.299813i	$0.0969242\pi$
$89$	5.65685i	0.599625i	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	19.6969	1.99992	0.999961	−	0.00888289i	$-0.00282755\pi$
$97$	19.6969	1.99992	0.999961	+	0.00888289i	$0.00282755\pi$
$98$	− 9.89949i	− 1.00000i
$99$	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
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.70334i	0.454689i	0.973814	+	0.227345i	$0.0730044\pi$
$107$	4.70334i	0.454689i	−0.973814	+	0.227345i	$0.926996\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 11.3137i	− 1.06430i	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	− 11.3137i	− 1.06430i	0.846649	−	0.532152i	$-0.178617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−17.3485	−1.59706
$119$	0	0
$120$	0	0
$121$	−3.30306	−0.300278
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	0	0
$131$	14.1421i	1.23560i	0.786334	+	0.617802i	$0.211977\pi$
$131$	14.1421i	1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	20.2918i	1.75294i
$135$	0	0
$136$	22.6969	1.94625
$137$	− 16.5099i	− 1.41053i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	− 16.5099i	− 1.41053i	0.708942	−	0.705266i	$-0.249173\pi$
$138$	0	0
$139$	3.65153	0.309719	0.154859	−	0.987937i	$-0.450508\pi$
$139$	3.65153	0.309719	0.154859	+	0.987937i	$0.450508\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	− 19.3704i	− 1.60311i
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	− 23.6130i	− 1.91527i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	− 0.921404i	− 0.0719495i
$165$	0	0
$166$	−4.00000	−0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	4.69694	0.358138
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	− 15.1278i	− 1.14030i
$177$	0	0
$178$	8.00000	0.599625
$179$	− 19.7990i	− 1.47985i	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	− 19.7990i	− 1.47985i	0.672692	−	0.739923i	$-0.265138\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−30.3485	−2.21930
$188$	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
$193$	25.6969	1.84971	0.924853	−	0.380325i	$-0.124188\pi$
$193$	25.6969	1.84971	0.924853	+	0.380325i	$0.124188\pi$
$194$	− 27.8557i	− 1.99992i
$195$	0	0
$196$	−14.0000	−1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 14.1421i	− 1.00000i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	31.5734i	2.18398i
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	0	0
$213$	0	0
$214$	6.65153	0.454689
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	−16.0000	−1.06430
$227$	27.3950i	1.81827i	0.416503	+	0.909134i	$0.363255\pi$
$227$	27.3950i	1.81827i	−0.416503	+	0.909134i	$0.636745\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.1523i	1.51676i	0.651813	+	0.758380i	$0.274009\pi$
$233$	23.1523i	1.51676i	−0.651813	+	0.758380i	$0.725991\pi$
$234$	0	0
$235$	0	0
$236$	24.5344i	1.59706i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	1.69694	0.109309	0.0546547	−	0.998505i	$-0.482594\pi$
$241$	1.69694	0.109309	0.0546547	+	0.998505i	$0.482594\pi$
$242$	4.67123i	0.300278i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 20.7525i	− 1.30989i	−0.755678	−	0.654943i	$-0.772693\pi$
$251$	− 20.7525i	− 1.30989i	0.755678	−	0.654943i	$-0.227307\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	31.6376i	1.97350i	0.162247	+	0.986750i	$0.448126\pi$
$257$	31.6376i	1.97350i	−0.162247	+	0.986750i	$0.551874\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	20.0000	1.23560
$263$	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
$268$	28.6969	1.75294
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	− 32.0983i	− 1.94625i
$273$	0	0
$274$	−23.3485	−1.41053
$275$	18.9097i	1.14030i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	− 5.16404i	− 0.309719i
$279$	0	0
$280$	0	0
$281$	− 28.2843i	− 1.68730i	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	− 28.2843i	− 1.68730i	0.536895	−	0.843649i	$-0.319597\pi$
$282$	0	0
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−47.3939	−2.78788
$290$	0	0
$291$	0	0
$292$	−27.3939	−1.60311
$293$	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
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−33.3939	−1.91527
$305$	0	0
$306$	0	0
$307$	9.65153	0.550842	0.275421	−	0.961324i	$-0.411183\pi$
$307$	9.65153	0.550842	0.275421	+	0.961324i	$0.411183\pi$
$308$	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
$313$	−24.3939	−1.37882	−0.689412	−	0.724370i	$-0.742131\pi$
$313$	−24.3939	−1.37882	−0.689412	+	0.724370i	$0.742131\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	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
$323$	66.9930i	3.72759i
$324$	0	0
$325$	0	0
$326$	− 2.82843i	− 0.156652i
$327$	0	0
$328$	−1.30306	−0.0719495
$329$	0	0
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	5.65685i	0.310460i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−36.3939	−1.98250	−0.991250	−	0.131995i	$-0.957862\pi$
$337$	−36.3939	−1.98250	−0.991250	+	0.131995i	$0.957862\pi$
$338$	− 18.3848i	− 1.00000i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	− 6.64247i	− 0.358138i
$345$	0	0
$346$	0	0
$347$	13.1886i	0.708002i	0.935245	+	0.354001i	$0.115179\pi$
$347$	13.1886i	0.708002i	−0.935245	+	0.354001i	$0.884821\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−21.3939	−1.14030
$353$	14.6671i	0.780648i	0.920677	+	0.390324i	$0.127637\pi$
$353$	14.6671i	0.780648i	−0.920677	+	0.390324i	$0.872363\pi$
$354$	0	0
$355$	0	0
$356$	− 11.3137i	− 0.599625i
$357$	0	0
$358$	−28.0000	−1.47985
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	50.6969	2.66826
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	42.9192i	2.21930i
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−26.3485	−1.35343	−0.676715	−	0.736245i	$-0.736597\pi$
$379$	−26.3485	−1.35343	−0.676715	+	0.736245i	$0.736597\pi$
$380$	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
$386$	− 36.3410i	− 1.84971i
$387$	0	0
$388$	−39.3939	−1.99992
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	19.7990i	1.00000i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	− 24.9951i	− 1.24820i	−0.781345	−	0.624099i	$-0.785466\pi$
$401$	− 24.9951i	− 1.24820i	0.781345	−	0.624099i	$-0.214534\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−18.3939	−0.909519	−0.454759	−	0.890614i	$-0.650275\pi$
$409$	−18.3939	−0.909519	−0.454759	+	0.890614i	$0.650275\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	44.6515	2.18398
$419$	− 36.7696i	− 1.79631i	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	− 36.7696i	− 1.79631i	0.439679	−	0.898155i	$-0.355092\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	− 19.7990i	− 0.963800i
$423$	0	0
$424$	0	0
$425$	40.1229i	1.94625i
$426$	0	0
$427$	0	0
$428$	− 9.40669i	− 0.454689i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−4.30306	−0.206792	−0.103396	−	0.994640i	$-0.532971\pi$
$433$	−4.30306	−0.206792	−0.103396	+	0.994640i	$0.532971\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 34.9589i	− 1.66095i	−0.557059	−	0.830473i	$-0.688070\pi$
$443$	− 34.9589i	− 1.66095i	0.557059	−	0.830473i	$-0.311930\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 39.2015i	− 1.85003i	−0.379927	−	0.925016i	$-0.624051\pi$
$449$	− 39.2015i	− 1.85003i	0.379927	−	0.925016i	$-0.375949\pi$
$450$	0	0
$451$	1.74235	0.0820439
$452$	22.6274i	1.06430i
$453$	0	0
$454$	38.7423	1.81827
$455$	0	0
$456$	0	0
$457$	−42.3939	−1.98310	−0.991551	−	0.129718i	$-0.958593\pi$
$457$	−42.3939	−1.98310	−0.991551	+	0.129718i	$0.958593\pi$
$458$	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
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	32.7423	1.51676
$467$	10.4244i	0.482384i	0.970477	+	0.241192i	$0.0775384\pi$
$467$	10.4244i	0.482384i	−0.970477	+	0.241192i	$0.922462\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	34.6969	1.59706
$473$	8.88177i	0.408384i
$474$	0	0
$475$	41.7423	1.91527
$476$	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
$481$	0	0
$482$	− 2.39983i	− 0.109309i
$483$	0	0
$484$	6.60612	0.300278
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 43.4441i	− 1.96061i	−0.197499	−	0.980303i	$-0.563282\pi$
$491$	− 43.4441i	− 1.96061i	0.197499	−	0.980303i	$-0.436718\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	29.7423	1.33145	0.665725	−	0.746197i	$-0.268122\pi$
$499$	29.7423	1.33145	0.665725	+	0.746197i	$0.268122\pi$
$500$	0	0
$501$	0	0
$502$	−29.3485	−1.30989
$503$	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
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	44.7423	1.97350
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.4313i	0.763678i	0.924229	+	0.381839i	$0.124709\pi$
$521$	17.4313i	0.763678i	−0.924229	+	0.381839i	$0.875291\pi$
$522$	0	0
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	− 28.2843i	− 1.23560i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	− 40.5836i	− 1.75294i
$537$	0	0
$538$	0	0
$539$	− 26.4736i	− 1.14030i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−45.3939	−1.94625
$545$	0	0
$546$	0	0
$547$	15.6515	0.669211	0.334606	−	0.942358i	$-0.391397\pi$
$547$	15.6515	0.669211	0.334606	+	0.942358i	$0.391397\pi$
$548$	33.0197i	1.41053i
$549$	0	0
$550$	26.7423	1.14030
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−7.30306	−0.309719
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−40.0000	−1.68730
$563$	44.3655i	1.86978i	0.354932	+	0.934892i	$0.384504\pi$
$563$	44.3655i	1.86978i	−0.354932	+	0.934892i	$0.615496\pi$
$564$	0	0
$565$	0	0
$566$	31.1127i	1.30776i
$567$	0	0
$568$	0	0
$569$	− 47.6868i	− 1.99913i	−0.0294311	−	0.999567i	$-0.509370\pi$
$569$	− 47.6868i	− 1.99913i	0.0294311	−	0.999567i	$-0.490630\pi$
$570$	0	0
$571$	47.7423	1.99796	0.998978	−	0.0452101i	$-0.0143957\pi$
$571$	47.7423	1.99796	0.998978	+	0.0452101i	$0.0143957\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−12.3939	−0.515964	−0.257982	−	0.966150i	$-0.583058\pi$
$577$	−12.3939	−0.515964	−0.257982	+	0.966150i	$0.583058\pi$
$578$	67.0251i	2.78788i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	38.7408i	1.60311i
$585$	0	0
$586$	0	0
$587$	− 29.2378i	− 1.20677i	−0.797449	−	0.603386i	$-0.793818\pi$
$587$	− 29.2378i	− 1.20677i	0.797449	−	0.603386i	$-0.206182\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 45.2548i	− 1.85839i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	− 45.2548i	− 1.85839i	0.369586	−	0.929197i	$-0.379500\pi$
$594$	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
$601$	37.6969	1.53769	0.768845	−	0.639435i	$-0.220832\pi$
$601$	37.6969	1.53769	0.768845	+	0.639435i	$0.220832\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	47.2261i	1.91527i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	− 13.6493i	− 0.550842i
$615$	0	0
$616$	0	0
$617$	6.18177i	0.248869i	0.992228	+	0.124434i	$0.0397116\pi$
$617$	6.18177i	0.248869i	−0.992228	+	0.124434i	$0.960288\pi$
$618$	0	0
$619$	23.7423	0.954285	0.477143	−	0.878826i	$-0.341672\pi$
$619$	23.7423	0.954285	0.477143	+	0.878826i	$0.341672\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	34.4982i	1.37882i
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 22.2309i	− 0.878069i	−0.898470	−	0.439034i	$-0.855321\pi$
$641$	− 22.2309i	− 0.878069i	0.898470	−	0.439034i	$-0.144679\pi$
$642$	0	0
$643$	−32.3485	−1.27570	−0.637850	−	0.770161i	$-0.720176\pi$
$643$	−32.3485	−1.27570	−0.637850	+	0.770161i	$0.720176\pi$
$644$	0	0
$645$	0	0
$646$	94.7423	3.72759
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−46.3939	−1.82112
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	1.84281i	0.0719495i
$657$	0	0
$658$	0	0
$659$	48.0833i	1.87306i	0.350590	+	0.936529i	$0.385981\pi$
$659$	48.0833i	1.87306i	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 36.7696i	− 1.42909i
$663$	0	0
$664$	8.00000	0.310460
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	51.4687i	1.98250i
$675$	0	0
$676$	−26.0000	−1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 51.9294i	− 1.98702i	−0.113728	−	0.993512i	$-0.536279\pi$
$683$	− 51.9294i	− 1.98702i	0.113728	−	0.993512i	$-0.463721\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−9.39388	−0.358138
$689$	0	0
$690$	0	0
$691$	−46.0000	−1.74992	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	0	0
$693$	0	0
$694$	18.6515	0.708002
$695$	0	0
$696$	0	0
$697$	3.69694	0.140032
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	30.2555i	1.14030i
$705$	0	0
$706$	20.7423	0.780648
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−16.0000	−0.599625
$713$	0	0
$714$	0	0
$715$	0	0
$716$	39.5980i	1.47985i
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 71.6963i	− 2.66826i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.8455i	0.697025i
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	54.2650i	1.99888i
$738$	0	0
$739$	53.7423	1.97694	0.988472	−	0.151403i	$-0.0483792\pi$
$739$	53.7423	1.97694	0.988472	+	0.151403i	$0.0483792\pi$
$740$	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
$748$	60.6969	2.21930
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	37.2624i	1.35343i
$759$	0	0
$760$	0	0
$761$	− 11.3137i	− 0.410122i	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	− 11.3137i	− 0.410122i	0.978749	−	0.205061i	$-0.0657392\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	−51.3939	−1.84971
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	55.7114i	1.99992i
$777$	0	0
$778$	0	0
$779$	− 3.84616i	− 0.137803i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	50.0000	1.78231	0.891154	−	0.453701i	$-0.149897\pi$
$787$	50.0000	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	28.2843i	1.00000i
$801$	0	0
$802$	−35.3485	−1.24820
$803$	− 51.8010i	− 1.82802i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 33.4804i	− 1.17711i	−0.808458	−	0.588555i	$-0.799697\pi$
$809$	− 33.4804i	− 1.17711i	0.808458	−	0.588555i	$-0.200303\pi$
$810$	0	0
$811$	17.7423	0.623018	0.311509	−	0.950243i	$-0.399166\pi$
$811$	17.7423	0.623018	0.311509	+	0.950243i	$0.399166\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	19.6061	0.685931
$818$	26.0129i	0.909519i
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 19.7990i	− 0.688478i	−0.938882	−	0.344239i	$-0.888137\pi$
$827$	− 19.7990i	− 0.688478i	0.938882	−	0.344239i	$-0.111863\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 56.1721i	− 1.94625i
$834$	0	0
$835$	0	0
$836$	− 63.1468i	− 2.18398i
$837$	0	0
$838$	−52.0000	−1.79631
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	−28.0000	−0.963800
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	56.7423	1.94625
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−13.3031	−0.454689
$857$	22.6274i	0.772938i	0.922302	+	0.386469i	$0.126305\pi$
$857$	22.6274i	0.772938i	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	21.6515	0.738741	0.369370	−	0.929282i	$-0.379573\pi$
$859$	21.6515	0.738741	0.369370	+	0.929282i	$0.379573\pi$
$860$	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
$866$	6.08545i	0.206792i
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	56.5685i	1.90584i	0.303218	+	0.952921i	$0.401939\pi$
$881$	56.5685i	1.90584i	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−52.4393	−1.76472	−0.882361	−	0.470573i	$-0.844047\pi$
$883$	−52.4393	−1.76472	−0.882361	+	0.470573i	$0.844047\pi$
$884$	0	0
$885$	0	0
$886$	−49.4393	−1.66095
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−55.4393	−1.85003
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 2.46405i	− 0.0820439i
$903$	0	0
$904$	32.0000	1.06430
$905$	0	0
$906$	0	0
$907$	−46.4393	−1.54199	−0.770996	−	0.636841i	$-0.780241\pi$
$907$	−46.4393	−1.54199	−0.770996	+	0.636841i	$0.780241\pi$
$908$	− 54.7900i	− 1.81827i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−10.6969	−0.354017
$914$	59.9540i	1.98310i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 28.2843i	− 0.927977i	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	− 28.2843i	− 0.927977i	0.885841	−	0.463988i	$-0.153582\pi$
$930$	0	0
$931$	−58.4393	−1.91527
$932$	− 46.3047i	− 1.51676i
$933$	0	0
$934$	14.7423	0.482384
$935$	0	0
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	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
$944$	− 49.0689i	− 1.59706i
$945$	0	0
$946$	12.5607	0.408384
$947$	52.8508i	1.71742i	0.512461	+	0.858710i	$0.328734\pi$
$947$	52.8508i	1.71742i	−0.512461	+	0.858710i	$0.671266\pi$
$948$	0	0
$949$	0	0
$950$	− 59.0326i	− 1.91527i
$951$	0	0
$952$	0	0
$953$	− 13.7456i	− 0.445265i	−0.974902	−	0.222633i	$-0.928535\pi$
$953$	− 13.7456i	− 0.445265i	0.974902	−	0.222633i	$-0.0714650\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	−3.39388	−0.109309
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	− 9.34247i	− 0.300278i
$969$	0	0
$970$	0	0
$971$	31.1127i	0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$	31.1127i	0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.9165i	0.829144i	0.910017	+	0.414572i	$0.136069\pi$
$977$	25.9165i	0.829144i	−0.910017	+	0.414572i	$0.863931\pi$
$978$	0	0
$979$	21.3939	0.683751
$980$	0	0
$981$	0	0
$982$	−61.4393	−1.96061
$983$	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
$988$	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
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 42.0620i	− 1.33145i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.f.a.323.1		4
3.2	odd	2	inner	648.2.f.a.323.4		4
4.3	odd	2		2592.2.f.a.1295.3		4
8.3	odd	2	CM	648.2.f.a.323.1		4
8.5	even	2		2592.2.f.a.1295.3		4
9.2	odd	6		216.2.l.a.179.2		4
9.4	even	3		216.2.l.a.35.2		4
9.5	odd	6		72.2.l.a.11.1	✓	4
9.7	even	3		72.2.l.a.59.1	yes	4
12.11	even	2		2592.2.f.a.1295.2		4
24.5	odd	2		2592.2.f.a.1295.2		4
24.11	even	2	inner	648.2.f.a.323.4		4
36.7	odd	6		288.2.p.a.239.1		4
36.11	even	6		864.2.p.a.719.2		4
36.23	even	6		288.2.p.a.47.1		4
36.31	odd	6		864.2.p.a.143.2		4
72.5	odd	6		288.2.p.a.47.1		4
72.11	even	6		216.2.l.a.179.2		4
72.13	even	6		864.2.p.a.143.2		4
72.29	odd	6		864.2.p.a.719.2		4
72.43	odd	6		72.2.l.a.59.1	yes	4
72.59	even	6		72.2.l.a.11.1	✓	4
72.61	even	6		288.2.p.a.239.1		4
72.67	odd	6		216.2.l.a.35.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.l.a.11.1	✓	4	9.5	odd	6
72.2.l.a.11.1	✓	4	72.59	even	6
72.2.l.a.59.1	yes	4	9.7	even	3
72.2.l.a.59.1	yes	4	72.43	odd	6
216.2.l.a.35.2		4	9.4	even	3
216.2.l.a.35.2		4	72.67	odd	6
216.2.l.a.179.2		4	9.2	odd	6
216.2.l.a.179.2		4	72.11	even	6
288.2.p.a.47.1		4	36.23	even	6
288.2.p.a.47.1		4	72.5	odd	6
288.2.p.a.239.1		4	36.7	odd	6
288.2.p.a.239.1		4	72.61	even	6
648.2.f.a.323.1		4	1.1	even	1	trivial
648.2.f.a.323.1		4	8.3	odd	2	CM
648.2.f.a.323.4		4	3.2	odd	2	inner
648.2.f.a.323.4		4	24.11	even	2	inner
864.2.p.a.143.2		4	36.31	odd	6
864.2.p.a.143.2		4	72.13	even	6
864.2.p.a.719.2		4	36.11	even	6
864.2.p.a.719.2		4	72.29	odd	6
2592.2.f.a.1295.2		4	12.11	even	2
2592.2.f.a.1295.2		4	24.5	odd	2
2592.2.f.a.1295.3		4	4.3	odd	2
2592.2.f.a.1295.3		4	8.5	even	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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} -3.78194i q^{11} +4.00000 q^{16} -8.02458i q^{17} -8.34847 q^{19} -5.34847 q^{22} -5.00000 q^{25} -5.65685i q^{32} -11.3485 q^{34} +11.8065i q^{38} +0.460702i q^{41} -2.34847 q^{43} +7.56388i q^{44} +7.00000 q^{49} +7.07107i q^{50} -12.2672i q^{59} -8.00000 q^{64} -14.3485 q^{67} +16.0492i q^{68} +13.6969 q^{73} +16.6969 q^{76} +0.651531 q^{82} -2.82843i q^{83} +3.32124i q^{86} +10.6969 q^{88} +5.65685i q^{89} +19.6969 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} - 4 q^{19} + 8 q^{22} - 20 q^{25} - 16 q^{34} + 20 q^{43} + 28 q^{49} - 32 q^{64} - 28 q^{67} - 4 q^{73} + 8 q^{76} + 32 q^{82} - 16 q^{88} + 20 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 - 4 * q^19 + 8 * q^22 - 20 * q^25 - 16 * q^34 + 20 * q^43 + 28 * q^49 - 32 * q^64 - 28 * q^67 - 4 * q^73 + 8 * q^76 + 32 * q^82 - 16 * q^88 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more