LMFDB - Embedded newform 648.2.f.a.323.3

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			323.3
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	648.323
Dual form			648.2.f.a.323.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} -6.61037i q^{11} +4.00000 q^{16} -2.36773i q^{17} +6.34847 q^{19} +9.34847 q^{22} -5.00000 q^{25} +5.65685i q^{32} +3.34847 q^{34} +8.97809i q^{38} -10.8530i q^{41} +12.3485 q^{43} +13.2207i q^{44} +7.00000 q^{49} -7.07107i q^{50} +1.87492i q^{59} -8.00000 q^{64} +0.348469 q^{67} +4.73545i q^{68} -15.6969 q^{73} -12.6969 q^{76} +15.3485 q^{82} +2.82843i q^{83} +17.4634i q^{86} -18.6969 q^{88} -5.65685i q^{89} -9.69694 q^{97} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	− 6.61037i	− 1.99310i	−0.0829925	−	0.996550i	$-0.526448\pi$
$11$	− 6.61037i	− 1.99310i	0.0829925	−	0.996550i	$-0.473552\pi$
$12$	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
$17$	− 2.36773i	− 0.574258i	−0.957892	−	0.287129i	$-0.907299\pi$
$17$	− 2.36773i	− 0.574258i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$	0	0
$19$	6.34847	1.45644	0.728219	−	0.685344i	$-0.240348\pi$
$19$	6.34847	1.45644	0.728219	+	0.685344i	$0.240348\pi$
$20$	0	0
$21$	0	0
$22$	9.34847	1.99310
$23$	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
$34$	3.34847	0.574258
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	8.97809i	1.45644i
$39$	0	0
$40$	0	0
$41$	− 10.8530i	− 1.69495i	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	− 10.8530i	− 1.69495i	0.530831	−	0.847477i	$-0.321880\pi$
$42$	0	0
$43$	12.3485	1.88312	0.941562	−	0.336840i	$-0.109358\pi$
$43$	12.3485	1.88312	0.941562	+	0.336840i	$0.109358\pi$
$44$	13.2207i	1.99310i
$45$	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
$59$	1.87492i	0.244093i	0.992524	+	0.122047i	$0.0389457\pi$
$59$	1.87492i	0.244093i	−0.992524	+	0.122047i	$0.961054\pi$
$60$	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
$67$	0.348469	0.0425723	0.0212861	−	0.999773i	$-0.493224\pi$
$67$	0.348469	0.0425723	0.0212861	+	0.999773i	$0.493224\pi$
$68$	4.73545i	0.574258i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−15.6969	−1.83719	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−15.6969	−1.83719	−0.918594	+	0.395203i	$0.870674\pi$
$74$	0	0
$75$	0	0
$76$	−12.6969	−1.45644
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	15.3485	1.69495
$83$	2.82843i	0.310460i	0.987878	+	0.155230i	$0.0496119\pi$
$83$	2.82843i	0.310460i	−0.987878	+	0.155230i	$0.950388\pi$
$84$	0	0
$85$	0	0
$86$	17.4634i	1.88312i
$87$	0	0
$88$	−18.6969	−1.99310
$89$	− 5.65685i	− 0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	− 5.65685i	− 0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.69694	−0.984575	−0.492287	−	0.870433i	$-0.663839\pi$
$97$	−9.69694	−0.984575	−0.492287	+	0.870433i	$0.663839\pi$
$98$	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
$107$	− 15.0956i	− 1.45935i	−0.683793	−	0.729676i	$-0.739671\pi$
$107$	− 15.0956i	− 1.45935i	0.683793	−	0.729676i	$-0.260329\pi$
$108$	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	$0.178617\pi$
$113$	11.3137i	1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−2.65153	−0.244093
$119$	0	0
$120$	0	0
$121$	−32.6969	−2.97245
$122$	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	$-0.788023\pi$
$131$	− 14.1421i	− 1.23560i	0.786334	−	0.617802i	$-0.211977\pi$
$132$	0	0
$133$	0	0
$134$	0.492810i	0.0425723i
$135$	0	0
$136$	−6.69694	−0.574258
$137$	6.11756i	0.522658i	0.965250	+	0.261329i	$0.0841608\pi$
$137$	6.11756i	0.522658i	−0.965250	+	0.261329i	$0.915839\pi$
$138$	0	0
$139$	18.3485	1.55630	0.778148	−	0.628080i	$-0.216159\pi$
$139$	18.3485	1.55630	0.778148	+	0.628080i	$0.216159\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	− 22.1988i	− 1.83719i
$147$	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			$\pi$
$152$	− 17.9562i	− 1.45644i
$153$	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	$0.475042\pi$
$164$	21.7060i	1.69495i
$165$	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
$172$	−24.6969	−1.88312
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	− 26.4415i	− 1.99310i
$177$	0	0
$178$	8.00000	0.599625
$179$	19.7990i	1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$	19.7990i	1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$	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
$187$	−15.6515	−1.14455
$188$	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
$193$	−3.69694	−0.266111	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	−3.69694	−0.266111	−0.133056	+	0.991109i	$0.542479\pi$
$194$	− 13.7135i	− 0.984575i
$195$	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
$209$	− 41.9657i	− 2.90283i
$210$	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
$214$	21.3485	1.45935
$215$	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
$227$	24.5665i	1.63054i	0.579082	+	0.815270i	$0.303411\pi$
$227$	24.5665i	1.63054i	−0.579082	+	0.815270i	$0.696589\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.8092i	1.88735i	0.330870	+	0.943676i	$0.392658\pi$
$233$	28.8092i	1.88735i	−0.330870	+	0.943676i	$0.607342\pi$
$234$	0	0
$235$	0	0
$236$	− 3.74983i	− 0.244093i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−27.6969	−1.78412	−0.892058	−	0.451920i	$-0.850739\pi$
$241$	−27.6969	−1.78412	−0.892058	+	0.451920i	$0.850739\pi$
$242$	− 46.2405i	− 2.97245i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.3602i	0.653930i	0.945036	+	0.326965i	$0.106026\pi$
$251$	10.3602i	0.653930i	−0.945036	+	0.326965i	$0.893974\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	20.3239i	1.26777i	0.773427	+	0.633885i	$0.218541\pi$
$257$	20.3239i	1.26777i	−0.773427	+	0.633885i	$0.781459\pi$
$258$	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
$268$	−0.696938	−0.0425723
$269$	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			$\pi$
$272$	− 9.47090i	− 0.574258i
$273$	0	0
$274$	−8.65153	−0.522658
$275$	33.0518i	1.99310i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	25.9487i	1.55630i
$279$	0	0
$280$	0	0
$281$	28.2843i	1.68730i	0.536895	+	0.843649i	$0.319597\pi$
$281$	28.2843i	1.68730i	−0.536895	+	0.843649i	$0.680403\pi$
$282$	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
$289$	11.3939	0.670228
$290$	0	0
$291$	0	0
$292$	31.3939	1.83719
$293$	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
$304$	25.3939	1.45644
$305$	0	0
$306$	0	0
$307$	24.3485	1.38964	0.694820	−	0.719183i	$-0.255484\pi$
$307$	24.3485	1.38964	0.694820	+	0.719183i	$0.255484\pi$
$308$	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
$313$	34.3939	1.94406	0.972028	−	0.234863i	$-0.0754642\pi$
$313$	34.3939	1.94406	0.972028	+	0.234863i	$0.0754642\pi$
$314$	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
$323$	− 15.0314i	− 0.836371i
$324$	0	0
$325$	0	0
$326$	2.82843i	0.156652i
$327$	0	0
$328$	−30.6969	−1.69495
$329$	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
$337$	22.3939	1.21987	0.609936	−	0.792451i	$-0.291195\pi$
$337$	22.3939	1.21987	0.609936	+	0.792451i	$0.291195\pi$
$338$	18.3848i	1.00000i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	− 34.9267i	− 1.88312i
$345$	0	0
$346$	0	0
$347$	− 23.5809i	− 1.26589i	−0.774197	−	0.632945i	$-0.781846\pi$
$347$	− 23.5809i	− 1.26589i	0.774197	−	0.632945i	$-0.218154\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	37.3939	1.99310
$353$	37.2945i	1.98498i	0.122308	+	0.992492i	$0.460970\pi$
$353$	37.2945i	1.98498i	−0.122308	+	0.992492i	$0.539030\pi$
$354$	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
$361$	21.3031	1.12121
$362$	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			$\pi$
$374$	− 22.1346i	− 1.14455i
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−11.6515	−0.598499	−0.299249	−	0.954175i	$-0.596736\pi$
$379$	−11.6515	−0.598499	−0.299249	+	0.954175i	$0.596736\pi$
$380$	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
$386$	− 5.22826i	− 0.266111i
$387$	0	0
$388$	19.3939	0.984575
$389$	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
$401$	14.6028i	0.729231i	0.931158	+	0.364615i	$0.118800\pi$
$401$	14.6028i	0.729231i	−0.931158	+	0.364615i	$0.881200\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	40.3939	1.99735	0.998674	−	0.0514740i	$-0.0163919\pi$
$409$	40.3939	1.99735	0.998674	+	0.0514740i	$0.0163919\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	59.3485	2.90283
$419$	36.7696i	1.79631i	0.439679	+	0.898155i	$0.355092\pi$
$419$	36.7696i	1.79631i	−0.439679	+	0.898155i	$0.644908\pi$
$420$	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
$425$	11.8386i	0.574258i
$426$	0	0
$427$	0	0
$428$	30.1913i	1.45935i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−33.6969	−1.61937	−0.809686	−	0.586864i	$-0.800362\pi$
$433$	−33.6969	−1.61937	−0.809686	+	0.586864i	$0.800362\pi$
$434$	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
$443$	− 37.7873i	− 1.79533i	−0.440681	−	0.897664i	$-0.645263\pi$
$443$	− 37.7873i	− 1.79533i	0.440681	−	0.897664i	$-0.354737\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 33.5446i	− 1.58307i	−0.611124	−	0.791535i	$-0.709282\pi$
$449$	− 33.5446i	− 1.58307i	0.611124	−	0.791535i	$-0.290718\pi$
$450$	0	0
$451$	−71.7423	−3.37822
$452$	− 22.6274i	− 1.06430i
$453$	0	0
$454$	−34.7423	−1.63054
$455$	0	0
$456$	0	0
$457$	16.3939	0.766873	0.383437	−	0.923567i	$-0.374740\pi$
$457$	16.3939	0.766873	0.383437	+	0.923567i	$0.374740\pi$
$458$	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
$466$	−40.7423	−1.88735
$467$	41.5371i	1.92211i	0.276360	+	0.961054i	$0.410872\pi$
$467$	41.5371i	1.92211i	−0.276360	+	0.961054i	$0.589128\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	5.30306	0.244093
$473$	− 81.6279i	− 3.75325i
$474$	0	0
$475$	−31.7423	−1.45644
$476$	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
$482$	− 39.1694i	− 1.78412i
$483$	0	0
$484$	65.3939	2.97245
$485$	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
$491$	− 29.3020i	− 1.32238i	−0.750218	−	0.661190i	$-0.770052\pi$
$491$	− 29.3020i	− 1.32238i	0.750218	−	0.661190i	$-0.229948\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−43.7423	−1.95818	−0.979088	−	0.203436i	$-0.934789\pi$
$499$	−43.7423	−1.95818	−0.979088	+	0.203436i	$0.934789\pi$
$500$	0	0
$501$	0	0
$502$	−14.6515	−0.653930
$503$	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
$514$	−28.7423	−1.26777
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 27.8236i	− 1.21897i	−0.792797	−	0.609486i	$-0.791376\pi$
$521$	− 27.8236i	− 1.21897i	0.792797	−	0.609486i	$-0.208624\pi$
$522$	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
$536$	− 0.985620i	− 0.0425723i
$537$	0	0
$538$	0	0
$539$	− 46.2726i	− 1.99310i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	13.3939	0.574258
$545$	0	0
$546$	0	0
$547$	30.3485	1.29761	0.648803	−	0.760956i	$-0.275270\pi$
$547$	30.3485	1.29761	0.648803	+	0.760956i	$0.275270\pi$
$548$	− 12.2351i	− 0.522658i
$549$	0	0
$550$	−46.7423	−1.99310
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−36.6969	−1.55630
$557$	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
$563$	7.59599i	0.320133i	0.987106	+	0.160066i	$0.0511708\pi$
$563$	7.59599i	0.320133i	−0.987106	+	0.160066i	$0.948829\pi$
$564$	0	0
$565$	0	0
$566$	− 31.1127i	− 1.30776i
$567$	0	0
$568$	0	0
$569$	− 25.0594i	− 1.05054i	−0.850935	−	0.525271i	$-0.823964\pi$
$569$	− 25.0594i	− 1.05054i	0.850935	−	0.525271i	$-0.176036\pi$
$570$	0	0
$571$	−25.7423	−1.07728	−0.538642	−	0.842535i	$-0.681062\pi$
$571$	−25.7423	−1.07728	−0.538642	+	0.842535i	$0.681062\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	46.3939	1.93140	0.965701	−	0.259656i	$-0.0836092\pi$
$577$	46.3939	1.93140	0.965701	+	0.259656i	$0.0836092\pi$
$578$	16.1134i	0.670228i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	44.3976i	1.83719i
$585$	0	0
$586$	0	0
$587$	18.8455i	0.777836i	0.921272	+	0.388918i	$0.127151\pi$
$587$	18.8455i	0.777836i	−0.921272	+	0.388918i	$0.872849\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	45.2548i	1.85839i	0.369586	+	0.929197i	$0.379500\pi$
$593$	45.2548i	1.85839i	−0.369586	+	0.929197i	$0.620500\pi$
$594$	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
$601$	8.30306	0.338689	0.169344	−	0.985557i	$-0.445835\pi$
$601$	8.30306	0.338689	0.169344	+	0.985557i	$0.445835\pi$
$602$	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			$\pi$
$608$	35.9124i	1.45644i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	34.4339i	1.38964i
$615$	0	0
$616$	0	0
$617$	45.7798i	1.84302i	0.388351	+	0.921512i	$0.373045\pi$
$617$	45.7798i	1.84302i	−0.388351	+	0.921512i	$0.626955\pi$
$618$	0	0
$619$	−49.7423	−1.99931	−0.999657	−	0.0261952i	$-0.991661\pi$
$619$	−49.7423	−1.99931	−0.999657	+	0.0261952i	$0.991661\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	48.6403i	1.94406i
$627$	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
$641$	− 50.5152i	− 1.99523i	−0.0690201	−	0.997615i	$-0.521987\pi$
$641$	− 50.5152i	− 1.99523i	0.0690201	−	0.997615i	$-0.478013\pi$
$642$	0	0
$643$	−17.6515	−0.696108	−0.348054	−	0.937474i	$-0.613157\pi$
$643$	−17.6515	−0.696108	−0.348054	+	0.937474i	$0.613157\pi$
$644$	0	0
$645$	0	0
$646$	21.2577	0.836371
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	12.3939	0.486502
$650$	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
$656$	− 43.4120i	− 1.69495i
$657$	0	0
$658$	0	0
$659$	− 48.0833i	− 1.87306i	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	− 48.0833i	− 1.87306i	0.350590	−	0.936529i	$-0.385981\pi$
$660$	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	$0.561736\pi$
$674$	31.6697i	1.21987i
$675$	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
$683$	− 20.8167i	− 0.796530i	−0.917270	−	0.398265i	$-0.869613\pi$
$683$	− 20.8167i	− 0.796530i	0.917270	−	0.398265i	$-0.130387\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	49.3939	1.88312
$689$	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
$694$	33.3485	1.26589
$695$	0	0
$696$	0	0
$697$	−25.6969	−0.973341
$698$	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
$704$	52.8829i	1.99310i
$705$	0	0
$706$	−52.7423	−1.98498
$707$	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
$722$	30.1271i	1.12121i
$723$	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
$731$	− 29.2378i	− 1.08140i
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.30351i	− 0.0848508i
$738$	0	0
$739$	−19.7423	−0.726234	−0.363117	−	0.931744i	$-0.618287\pi$
$739$	−19.7423	−0.726234	−0.363117	+	0.931744i	$0.618287\pi$
$740$	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
$748$	31.3031	1.14455
$749$	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			$\pi$
$758$	− 16.4778i	− 0.598499i
$759$	0	0
$760$	0	0
$761$	11.3137i	0.410122i	0.978749	+	0.205061i	$0.0657392\pi$
$761$	11.3137i	0.410122i	−0.978749	+	0.205061i	$0.934261\pi$
$762$	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
$772$	7.39388	0.266111
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	27.4271i	0.984575i
$777$	0	0
$778$	0	0
$779$	− 68.9000i	− 2.46860i
$780$	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
$802$	−20.6515	−0.729231
$803$	103.763i	3.66170i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.0881i	0.811735i	0.913932	+	0.405868i	$0.133031\pi$
$809$	23.0881i	0.811735i	−0.913932	+	0.405868i	$0.866969\pi$
$810$	0	0
$811$	−55.7423	−1.95738	−0.978689	−	0.205347i	$-0.934168\pi$
$811$	−55.7423	−1.95738	−0.978689	+	0.205347i	$0.934168\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	78.3939	2.74265
$818$	57.1256i	1.99735i
$819$	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	$0.111863\pi$
$827$	19.7990i	0.688478i	−0.938882	+	0.344239i	$0.888137\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 16.5741i	− 0.574258i
$834$	0	0
$835$	0	0
$836$	83.9314i	2.90283i
$837$	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
$850$	−16.7423	−0.574258
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−42.6969	−1.45935
$857$	− 22.6274i	− 0.772938i	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	− 22.6274i	− 0.772938i	0.922302	−	0.386469i	$-0.126305\pi$
$858$	0	0
$859$	36.3485	1.24019	0.620097	−	0.784525i	$-0.287093\pi$
$859$	36.3485	1.24019	0.620097	+	0.784525i	$0.287093\pi$
$860$	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
$866$	− 47.6547i	− 1.61937i
$867$	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	$-0.598061\pi$
$881$	− 56.5685i	− 1.90584i	0.303218	−	0.952921i	$-0.401939\pi$
$882$	0	0
$883$	50.4393	1.69742	0.848709	−	0.528861i	$-0.177381\pi$
$883$	50.4393	1.69742	0.848709	+	0.528861i	$0.177381\pi$
$884$	0	0
$885$	0	0
$886$	53.4393	1.79533
$887$	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
$898$	47.4393	1.58307
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 101.459i	− 3.37822i
$903$	0	0
$904$	32.0000	1.06430
$905$	0	0
$906$	0	0
$907$	56.4393	1.87404	0.937018	−	0.349281i	$-0.113574\pi$
$907$	56.4393	1.87404	0.937018	+	0.349281i	$0.113574\pi$
$908$	− 49.1331i	− 1.63054i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	18.6969	0.618778
$914$	23.1844i	0.766873i
$915$	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	$0.153582\pi$
$929$	28.2843i	0.927977i	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	44.4393	1.45644
$932$	− 57.6184i	− 1.88735i
$933$	0	0
$934$	−58.7423	−1.92211
$935$	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
$944$	7.49966i	0.244093i
$945$	0	0
$946$	115.439	3.75325
$947$	− 0.889296i	− 0.0288982i	−0.999896	−	0.0144491i	$-0.995401\pi$
$947$	− 0.889296i	− 0.0288982i	0.999896	−	0.0144491i	$-0.00459946\pi$
$948$	0	0
$949$	0	0
$950$	− 44.8905i	− 1.45644i
$951$	0	0
$952$	0	0
$953$	− 59.0005i	− 1.91121i	−0.294646	−	0.955607i	$-0.595202\pi$
$953$	− 59.0005i	− 1.91121i	0.294646	−	0.955607i	$-0.404798\pi$
$954$	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
$964$	55.3939	1.78412
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	92.4809i	2.97245i
$969$	0	0
$970$	0	0
$971$	− 31.1127i	− 0.998454i	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	− 31.1127i	− 0.998454i	0.866471	−	0.499227i	$-0.166383\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 36.3089i	− 1.16162i	−0.814038	−	0.580812i	$-0.802735\pi$
$977$	− 36.3089i	− 1.16162i	0.814038	−	0.580812i	$-0.197265\pi$
$978$	0	0
$979$	−37.3939	−1.19511
$980$	0	0
$981$	0	0
$982$	41.4393	1.32238
$983$	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			$\pi$
$998$	− 61.8610i	− 1.95818i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.f.a.323.3		4
3.2	odd	2	inner	648.2.f.a.323.2		4
4.3	odd	2		2592.2.f.a.1295.4		4
8.3	odd	2	CM	648.2.f.a.323.3		4
8.5	even	2		2592.2.f.a.1295.4		4
9.2	odd	6		216.2.l.a.179.1		4
9.4	even	3		216.2.l.a.35.1		4
9.5	odd	6		72.2.l.a.11.2	✓	4
9.7	even	3		72.2.l.a.59.2	yes	4
12.11	even	2		2592.2.f.a.1295.1		4
24.5	odd	2		2592.2.f.a.1295.1		4
24.11	even	2	inner	648.2.f.a.323.2		4
36.7	odd	6		288.2.p.a.239.2		4
36.11	even	6		864.2.p.a.719.1		4
36.23	even	6		288.2.p.a.47.2		4
36.31	odd	6		864.2.p.a.143.1		4
72.5	odd	6		288.2.p.a.47.2		4
72.11	even	6		216.2.l.a.179.1		4
72.13	even	6		864.2.p.a.143.1		4
72.29	odd	6		864.2.p.a.719.1		4
72.43	odd	6		72.2.l.a.59.2	yes	4
72.59	even	6		72.2.l.a.11.2	✓	4
72.61	even	6		288.2.p.a.239.2		4
72.67	odd	6		216.2.l.a.35.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.l.a.11.2	✓	4	9.5	odd	6
72.2.l.a.11.2	✓	4	72.59	even	6
72.2.l.a.59.2	yes	4	9.7	even	3
72.2.l.a.59.2	yes	4	72.43	odd	6
216.2.l.a.35.1		4	9.4	even	3
216.2.l.a.35.1		4	72.67	odd	6
216.2.l.a.179.1		4	9.2	odd	6
216.2.l.a.179.1		4	72.11	even	6
288.2.p.a.47.2		4	36.23	even	6
288.2.p.a.47.2		4	72.5	odd	6
288.2.p.a.239.2		4	36.7	odd	6
288.2.p.a.239.2		4	72.61	even	6
648.2.f.a.323.2		4	3.2	odd	2	inner
648.2.f.a.323.2		4	24.11	even	2	inner
648.2.f.a.323.3		4	1.1	even	1	trivial
648.2.f.a.323.3		4	8.3	odd	2	CM
864.2.p.a.143.1		4	36.31	odd	6
864.2.p.a.143.1		4	72.13	even	6
864.2.p.a.719.1		4	36.11	even	6
864.2.p.a.719.1		4	72.29	odd	6
2592.2.f.a.1295.1		4	12.11	even	2
2592.2.f.a.1295.1		4	24.5	odd	2
2592.2.f.a.1295.4		4	4.3	odd	2
2592.2.f.a.1295.4		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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} -6.61037i q^{11} +4.00000 q^{16} -2.36773i q^{17} +6.34847 q^{19} +9.34847 q^{22} -5.00000 q^{25} +5.65685i q^{32} +3.34847 q^{34} +8.97809i q^{38} -10.8530i q^{41} +12.3485 q^{43} +13.2207i q^{44} +7.00000 q^{49} -7.07107i q^{50} +1.87492i q^{59} -8.00000 q^{64} +0.348469 q^{67} +4.73545i q^{68} -15.6969 q^{73} -12.6969 q^{76} +15.3485 q^{82} +2.82843i q^{83} +17.4634i q^{86} -18.6969 q^{88} -5.65685i q^{89} -9.69694 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 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

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