LMFDB - Embedded newform 1200.3.c.c.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(449,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.c.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +2.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +2.00000i q^{7} -9.00000 q^{9} -22.0000i q^{13} +26.0000 q^{19} -6.00000 q^{21} -27.0000i q^{27} +46.0000 q^{31} -26.0000i q^{37} +66.0000 q^{39} +22.0000i q^{43} +45.0000 q^{49} +78.0000i q^{57} +74.0000 q^{61} -18.0000i q^{63} +122.000i q^{67} -46.0000i q^{73} -142.000 q^{79} +81.0000 q^{81} +44.0000 q^{91} +138.000i q^{93} -2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} + 52 q^{19} - 12 q^{21} + 92 q^{31} + 132 q^{39} + 90 q^{49} + 148 q^{61} - 284 q^{79} + 162 q^{81} + 88 q^{91}+O(q^{100}) $ 2 * q - 18 * q^9 + 52 * q^19 - 12 * q^21 + 92 * q^31 + 132 * q^39 + 90 * q^49 + 148 * q^61 - 284 * q^79 + 162 * q^81 + 88 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	3.00000i	1.00000i
$3$	3.00000i	1.00000i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.285714i	0.989743	+	0.142857i	$0.0456289\pi$
$7$	2.00000i	0.285714i	−0.989743	+	0.142857i	$0.954371\pi$
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 22.0000i	− 1.69231i	−0.532939	−	0.846154i	$-0.678912\pi$
$13$	− 22.0000i	− 1.69231i	0.532939	−	0.846154i	$-0.321088\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	26.0000	1.36842	0.684211	−	0.729285i	$-0.260147\pi$
$19$	26.0000	1.36842	0.684211	+	0.729285i	$0.260147\pi$
$20$	0	0
$21$	−6.00000	−0.285714
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	46.0000	1.48387	0.741935	−	0.670471i	$-0.233908\pi$
$31$	46.0000	1.48387	0.741935	+	0.670471i	$0.233908\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 26.0000i	− 0.702703i	−0.936244	−	0.351351i	$-0.885722\pi$
$37$	− 26.0000i	− 0.702703i	0.936244	−	0.351351i	$-0.114278\pi$
$38$	0	0
$39$	66.0000	1.69231
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	22.0000i	0.511628i	0.966726	+	0.255814i	$0.0823435\pi$
$43$	22.0000i	0.511628i	−0.966726	+	0.255814i	$0.917657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	45.0000	0.918367
$50$	0	0
$51$	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
$57$	78.0000i	1.36842i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	74.0000	1.21311	0.606557	−	0.795040i	$-0.292550\pi$
$61$	74.0000	1.21311	0.606557	+	0.795040i	$0.292550\pi$
$62$	0	0
$63$	− 18.0000i	− 0.285714i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	122.000i	1.82090i	0.413624	+	0.910448i	$0.364263\pi$
$67$	122.000i	1.82090i	−0.413624	+	0.910448i	$0.635737\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 46.0000i	− 0.630137i	−0.949069	−	0.315068i	$-0.897973\pi$
$73$	− 46.0000i	− 0.630137i	0.949069	−	0.315068i	$-0.102027\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−142.000	−1.79747	−0.898734	−	0.438494i	$-0.855512\pi$
$79$	−142.000	−1.79747	−0.898734	+	0.438494i	$0.855512\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	44.0000	0.483516
$92$	0	0
$93$	138.000i	1.48387i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.0206186i	−0.999947	−	0.0103093i	$-0.996718\pi$
$97$	− 2.00000i	− 0.0206186i	0.999947	−	0.0103093i	$-0.00328160\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 194.000i	− 1.88350i	−0.336321	−	0.941748i	$-0.609183\pi$
$103$	− 194.000i	− 1.88350i	0.336321	−	0.941748i	$-0.390817\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	214.000	1.96330	0.981651	−	0.190684i	$-0.0610707\pi$
$109$	214.000	1.96330	0.981651	+	0.190684i	$0.0610707\pi$
$110$	0	0
$111$	78.0000	0.702703
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	198.000i	1.69231i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	146.000i	1.14961i	0.818292	+	0.574803i	$0.194921\pi$
$127$	146.000i	1.14961i	−0.818292	+	0.574803i	$0.805079\pi$
$128$	0	0
$129$	−66.0000	−0.511628
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	52.0000i	0.390977i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−22.0000	−0.158273	−0.0791367	−	0.996864i	$-0.525216\pi$
$139$	−22.0000	−0.158273	−0.0791367	+	0.996864i	$0.525216\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	135.000i	0.918367i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	286.000	1.89404	0.947020	−	0.321175i	$-0.104078\pi$
$151$	286.000	1.89404	0.947020	+	0.321175i	$0.104078\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	118.000i	0.751592i	0.926702	+	0.375796i	$0.122631\pi$
$157$	118.000i	0.751592i	−0.926702	+	0.375796i	$0.877369\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	262.000i	1.60736i	0.595060	+	0.803681i	$0.297128\pi$
$163$	262.000i	1.60736i	−0.595060	+	0.803681i	$0.702872\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−315.000	−1.86391
$170$	0	0
$171$	−234.000	−1.36842
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	314.000	1.73481	0.867403	−	0.497606i	$-0.165787\pi$
$181$	314.000	1.73481	0.867403	+	0.497606i	$0.165787\pi$
$182$	0	0
$183$	222.000i	1.21311i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	54.0000	0.285714
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	− 382.000i	− 1.97927i	−0.143590	−	0.989637i	$-0.545865\pi$
$193$	− 382.000i	− 1.97927i	0.143590	−	0.989637i	$-0.454135\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	386.000	1.93970	0.969849	−	0.243706i	$-0.0783631\pi$
$199$	386.000	1.93970	0.969849	+	0.243706i	$0.0783631\pi$
$200$	0	0
$201$	−366.000	−1.82090
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	166.000	0.786730	0.393365	−	0.919382i	$-0.371311\pi$
$211$	166.000	0.786730	0.393365	+	0.919382i	$0.371311\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	92.0000i	0.423963i
$218$	0	0
$219$	138.000	0.630137
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 338.000i	− 1.51570i	−0.652432	−	0.757848i	$-0.726251\pi$
$223$	− 338.000i	− 1.51570i	0.652432	−	0.757848i	$-0.273749\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−26.0000	−0.113537	−0.0567686	−	0.998387i	$-0.518080\pi$
$229$	−26.0000	−0.113537	−0.0567686	+	0.998387i	$0.518080\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 426.000i	− 1.79747i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−286.000	−1.18672	−0.593361	−	0.804936i	$-0.702199\pi$
$241$	−286.000	−1.18672	−0.593361	+	0.804936i	$0.702199\pi$
$242$	0	0
$243$	243.000i	1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 572.000i	− 2.31579i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	52.0000	0.200772
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−242.000	−0.892989	−0.446494	−	0.894786i	$-0.647328\pi$
$271$	−242.000	−0.892989	−0.446494	+	0.894786i	$0.647328\pi$
$272$	0	0
$273$	132.000i	0.483516i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 122.000i	− 0.440433i	−0.975451	−	0.220217i	$-0.929324\pi$
$277$	− 122.000i	− 0.440433i	0.975451	−	0.220217i	$-0.0706764\pi$
$278$	0	0
$279$	−414.000	−1.48387
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 458.000i	− 1.61837i	−0.587551	−	0.809187i	$-0.699908\pi$
$283$	− 458.000i	− 1.61837i	0.587551	−	0.809187i	$-0.300092\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−289.000	−1.00000
$290$	0	0
$291$	6.00000	0.0206186
$292$	0	0
$293$	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
$301$	−44.0000	−0.146179
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 358.000i	− 1.16612i	−0.812428	−	0.583062i	$-0.801855\pi$
$307$	− 358.000i	− 1.16612i	0.812428	−	0.583062i	$-0.198145\pi$
$308$	0	0
$309$	582.000	1.88350
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 142.000i	− 0.453674i	−0.973933	−	0.226837i	$-0.927162\pi$
$313$	− 142.000i	− 0.453674i	0.973933	−	0.226837i	$-0.0728385\pi$
$314$	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
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	642.000i	1.96330i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−362.000	−1.09366	−0.546828	−	0.837245i	$-0.684165\pi$
$331$	−362.000	−1.09366	−0.546828	+	0.837245i	$0.684165\pi$
$332$	0	0
$333$	234.000i	0.702703i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 482.000i	− 1.43027i	−0.698988	−	0.715134i	$-0.746366\pi$
$337$	− 482.000i	− 1.43027i	0.698988	−	0.715134i	$-0.253634\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	188.000i	0.548105i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	502.000	1.43840	0.719198	−	0.694805i	$-0.244510\pi$
$349$	502.000	1.43840	0.719198	+	0.694805i	$0.244510\pi$
$350$	0	0
$351$	−594.000	−1.69231
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	315.000	0.872576
$362$	0	0
$363$	363.000i	1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 718.000i	− 1.95640i	−0.207657	−	0.978202i	$-0.566584\pi$
$367$	− 718.000i	− 1.95640i	0.207657	−	0.978202i	$-0.433416\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	698.000i	1.87131i	0.352911	+	0.935657i	$0.385192\pi$
$373$	698.000i	1.87131i	−0.352911	+	0.935657i	$0.614808\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−694.000	−1.83113	−0.915567	−	0.402165i	$-0.868258\pi$
$379$	−694.000	−1.83113	−0.915567	+	0.402165i	$0.868258\pi$
$380$	0	0
$381$	−438.000	−1.14961
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 198.000i	− 0.511628i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 362.000i	− 0.911839i	−0.890021	−	0.455919i	$-0.849311\pi$
$397$	− 362.000i	− 0.911839i	0.890021	−	0.455919i	$-0.150689\pi$
$398$	0	0
$399$	−156.000	−0.390977
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 1012.00i	− 2.51117i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−626.000	−1.53056	−0.765281	−	0.643696i	$-0.777400\pi$
$409$	−626.000	−1.53056	−0.765281	+	0.643696i	$0.777400\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 66.0000i	− 0.158273i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−358.000	−0.850356	−0.425178	−	0.905110i	$-0.639789\pi$
$421$	−358.000	−0.850356	−0.425178	+	0.905110i	$0.639789\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	148.000i	0.346604i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	− 862.000i	− 1.99076i	−0.0960028	−	0.995381i	$-0.530606\pi$
$433$	− 862.000i	− 1.99076i	0.0960028	−	0.995381i	$-0.469394\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−94.0000	−0.214123	−0.107062	−	0.994252i	$-0.534144\pi$
$439$	−94.0000	−0.214123	−0.107062	+	0.994252i	$0.534144\pi$
$440$	0	0
$441$	−405.000	−0.918367
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	858.000i	1.89404i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	814.000i	1.78118i	0.454805	+	0.890591i	$0.349709\pi$
$457$	814.000i	1.78118i	−0.454805	+	0.890591i	$0.650291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	526.000i	1.13607i	0.823005	+	0.568035i	$0.192296\pi$
$463$	526.000i	1.13607i	−0.823005	+	0.568035i	$0.807704\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−244.000	−0.520256
$470$	0	0
$471$	−354.000	−0.751592
$472$	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
$481$	−572.000	−1.18919
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	962.000i	1.97536i	0.156489	+	0.987680i	$0.449982\pi$
$487$	962.000i	1.97536i	−0.156489	+	0.987680i	$0.550018\pi$
$488$	0	0
$489$	−786.000	−1.60736
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.0000	0.0521042	0.0260521	−	0.999661i	$-0.491706\pi$
$499$	26.0000	0.0521042	0.0260521	+	0.999661i	$0.491706\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 945.000i	− 1.86391i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	92.0000	0.180039
$512$	0	0
$513$	− 702.000i	− 1.36842i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	982.000i	1.87763i	0.344423	+	0.938815i	$0.388075\pi$
$523$	982.000i	1.87763i	−0.344423	+	0.938815i	$0.611925\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1034.00	1.91128	0.955638	−	0.294545i	$-0.0951680\pi$
$541$	1034.00	1.91128	0.955638	+	0.294545i	$0.0951680\pi$
$542$	0	0
$543$	942.000i	1.73481i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	506.000i	0.925046i	0.886607	+	0.462523i	$0.153056\pi$
$547$	506.000i	0.925046i	−0.886607	+	0.462523i	$0.846944\pi$
$548$	0	0
$549$	−666.000	−1.21311
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 284.000i	− 0.513562i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	484.000	0.865832
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	162.000i	0.285714i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	886.000	1.55166	0.775832	−	0.630940i	$-0.217330\pi$
$571$	886.000	1.55166	0.775832	+	0.630940i	$0.217330\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 962.000i	− 1.66724i	−0.552335	−	0.833622i	$-0.686263\pi$
$577$	− 962.000i	− 1.66724i	0.552335	−	0.833622i	$-0.313737\pi$
$578$	0	0
$579$	1146.00	1.97927
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	1196.00	2.03056
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1158.00i	1.93970i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−526.000	−0.875208	−0.437604	−	0.899168i	$-0.644173\pi$
$601$	−526.000	−0.875208	−0.437604	+	0.899168i	$0.644173\pi$
$602$	0	0
$603$	− 1098.00i	− 1.82090i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 814.000i	− 1.34102i	−0.741900	−	0.670511i	$-0.766075\pi$
$607$	− 814.000i	− 1.34102i	0.741900	−	0.670511i	$-0.233925\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 1126.00i	− 1.83687i	−0.395574	−	0.918434i	$-0.629454\pi$
$613$	− 1126.00i	− 1.83687i	0.395574	−	0.918434i	$-0.370546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−214.000	−0.345719	−0.172859	−	0.984947i	$-0.555301\pi$
$619$	−214.000	−0.345719	−0.172859	+	0.984947i	$0.555301\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−674.000	−1.06815	−0.534073	−	0.845438i	$-0.679339\pi$
$631$	−674.000	−1.06815	−0.534073	+	0.845438i	$0.679339\pi$
$632$	0	0
$633$	498.000i	0.786730i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 990.000i	− 1.55416i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 314.000i	− 0.488336i	−0.969733	−	0.244168i	$-0.921485\pi$
$643$	− 314.000i	− 0.488336i	0.969733	−	0.244168i	$-0.0785148\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−276.000	−0.423963
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	414.000i	0.630137i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	122.000	0.184569	0.0922844	−	0.995733i	$-0.470583\pi$
$661$	122.000	0.184569	0.0922844	+	0.995733i	$0.470583\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1014.00	1.51570
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1154.00i	1.71471i	0.514725	+	0.857355i	$0.327894\pi$
$673$	1154.00i	1.71471i	−0.514725	+	0.857355i	$0.672106\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	4.00000	0.00589102
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 78.0000i	− 0.113537i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1318.00	1.90738	0.953690	−	0.300790i	$-0.0972504\pi$
$691$	1318.00	1.90738	0.953690	+	0.300790i	$0.0972504\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 676.000i	− 0.961593i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	934.000	1.31735	0.658674	−	0.752428i	$-0.271118\pi$
$709$	934.000	1.31735	0.658674	+	0.752428i	$0.271118\pi$
$710$	0	0
$711$	1278.00	1.79747
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	388.000	0.538141
$722$	0	0
$723$	− 858.000i	− 1.18672i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	482.000i	0.662999i	0.943455	+	0.331499i	$0.107554\pi$
$727$	482.000i	0.662999i	−0.943455	+	0.331499i	$0.892446\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1034.00i	1.41064i	0.708888	+	0.705321i	$0.249197\pi$
$733$	1034.00i	1.41064i	−0.708888	+	0.705321i	$0.750803\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−1222.00	−1.65359	−0.826793	−	0.562506i	$-0.809837\pi$
$739$	−1222.00	−1.65359	−0.826793	+	0.562506i	$0.809837\pi$
$740$	0	0
$741$	1716.00	2.31579
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1202.00	−1.60053	−0.800266	−	0.599645i	$-0.795309\pi$
$751$	−1202.00	−1.60053	−0.800266	+	0.599645i	$0.795309\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	838.000i	1.10700i	0.832849	+	0.553501i	$0.186708\pi$
$757$	838.000i	1.10700i	−0.832849	+	0.553501i	$0.813292\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	428.000i	0.560944i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1534.00	1.99480	0.997399	−	0.0720749i	$-0.0229621\pi$
$769$	1534.00	1.99480	0.997399	+	0.0720749i	$0.0229621\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	156.000i	0.200772i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1562.00i	1.98475i	0.123246	+	0.992376i	$0.460669\pi$
$787$	1562.00i	1.98475i	−0.123246	+	0.992376i	$0.539331\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 1628.00i	− 2.05296i
$794$	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
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−1514.00	−1.86683	−0.933416	−	0.358797i	$-0.883187\pi$
$811$	−1514.00	−1.86683	−0.933416	+	0.358797i	$0.883187\pi$
$812$	0	0
$813$	− 726.000i	− 0.892989i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	572.000i	0.700122i
$818$	0	0
$819$	−396.000	−0.483516
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 1058.00i	− 1.28554i	−0.766059	−	0.642770i	$-0.777785\pi$
$823$	− 1058.00i	− 1.28554i	0.766059	−	0.642770i	$-0.222215\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−458.000	−0.552473	−0.276236	−	0.961090i	$-0.589087\pi$
$829$	−458.000	−0.552473	−0.276236	+	0.961090i	$0.589087\pi$
$830$	0	0
$831$	366.000	0.440433
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1242.00i	− 1.48387i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	242.000i	0.285714i
$848$	0	0
$849$	1374.00	1.61837
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1658.00i	1.94373i	0.235543	+	0.971864i	$0.424313\pi$
$853$	1658.00i	1.94373i	−0.235543	+	0.971864i	$0.575687\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	1418.00	1.65076	0.825378	−	0.564580i	$-0.190962\pi$
$859$	1418.00	1.65076	0.825378	+	0.564580i	$0.190962\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 867.000i	− 1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2684.00	3.08152
$872$	0	0
$873$	18.0000i	0.0206186i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	598.000i	0.681870i	0.940087	+	0.340935i	$0.110744\pi$
$877$	598.000i	0.681870i	−0.940087	+	0.340935i	$0.889256\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	1702.00i	1.92752i	0.266771	+	0.963760i	$0.414043\pi$
$883$	1702.00i	1.92752i	−0.266771	+	0.963760i	$0.585957\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−292.000	−0.328459
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 132.000i	− 0.146179i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 214.000i	− 0.235943i	−0.993017	−	0.117971i	$-0.962361\pi$
$907$	− 214.000i	− 0.235943i	0.993017	−	0.117971i	$-0.0376391\pi$
$908$	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
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	866.000	0.942329	0.471164	−	0.882045i	$-0.343834\pi$
$919$	866.000	0.942329	0.471164	+	0.882045i	$0.343834\pi$
$920$	0	0
$921$	1074.00	1.16612
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1746.00i	1.88350i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1170.00	1.25671
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1198.00i	1.27855i	0.768979	+	0.639274i	$0.220765\pi$
$937$	1198.00i	1.27855i	−0.768979	+	0.639274i	$0.779235\pi$
$938$	0	0
$939$	426.000	0.453674
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−1012.00	−1.06639
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1155.00	1.20187
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1534.00i	− 1.58635i	−0.608994	−	0.793175i	$-0.708427\pi$
$967$	− 1534.00i	− 1.58635i	0.608994	−	0.793175i	$-0.291573\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	− 44.0000i	− 0.0452210i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−1926.00	−1.96330
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	46.0000	0.0464178	0.0232089	−	0.999731i	$-0.492612\pi$
$991$	46.0000	0.0464178	0.0232089	+	0.999731i	$0.492612\pi$
$992$	0	0
$993$	− 1086.00i	− 1.09366i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1894.00i	1.89970i	0.312707	+	0.949850i	$0.398764\pi$
$997$	1894.00i	1.89970i	−0.312707	+	0.949850i	$0.601236\pi$
$998$	0	0
$999$	−702.000	−0.702703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.c.449.2		2
3.2	odd	2	CM	1200.3.c.c.449.2		2
4.3	odd	2		300.3.b.a.149.1		2
5.2	odd	4		48.3.e.a.17.1		1
5.3	odd	4		1200.3.l.b.401.1		1
5.4	even	2	inner	1200.3.c.c.449.1		2
12.11	even	2		300.3.b.a.149.1		2
15.2	even	4		48.3.e.a.17.1		1
15.8	even	4		1200.3.l.b.401.1		1
15.14	odd	2	inner	1200.3.c.c.449.1		2
20.3	even	4		300.3.g.b.101.1		1
20.7	even	4		12.3.c.a.5.1	✓	1
20.19	odd	2		300.3.b.a.149.2		2
40.27	even	4		192.3.e.b.65.1		1
40.37	odd	4		192.3.e.a.65.1		1
45.2	even	12		1296.3.q.b.1025.1		2
45.7	odd	12		1296.3.q.b.1025.1		2
45.22	odd	12		1296.3.q.b.593.1		2
45.32	even	12		1296.3.q.b.593.1		2
60.23	odd	4		300.3.g.b.101.1		1
60.47	odd	4		12.3.c.a.5.1	✓	1
60.59	even	2		300.3.b.a.149.2		2
80.27	even	4		768.3.h.a.641.2		2
80.37	odd	4		768.3.h.b.641.1		2
80.67	even	4		768.3.h.a.641.1		2
80.77	odd	4		768.3.h.b.641.2		2
120.77	even	4		192.3.e.a.65.1		1
120.107	odd	4		192.3.e.b.65.1		1
140.27	odd	4		588.3.c.c.197.1		1
140.47	odd	12		588.3.p.b.557.1		2
140.67	even	12		588.3.p.c.569.1		2
140.87	odd	12		588.3.p.b.569.1		2
140.107	even	12		588.3.p.c.557.1		2
180.7	even	12		324.3.g.b.53.1		2
180.47	odd	12		324.3.g.b.53.1		2
180.67	even	12		324.3.g.b.269.1		2
180.167	odd	12		324.3.g.b.269.1		2
220.87	odd	4		1452.3.e.b.485.1		1
240.77	even	4		768.3.h.b.641.2		2
240.107	odd	4		768.3.h.a.641.2		2
240.197	even	4		768.3.h.b.641.1		2
240.227	odd	4		768.3.h.a.641.1		2
420.47	even	12		588.3.p.b.557.1		2
420.107	odd	12		588.3.p.c.557.1		2
420.167	even	4		588.3.c.c.197.1		1
420.227	even	12		588.3.p.b.569.1		2
420.347	odd	12		588.3.p.c.569.1		2
660.527	even	4		1452.3.e.b.485.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.3.c.a.5.1	✓	1	20.7	even	4
12.3.c.a.5.1	✓	1	60.47	odd	4
48.3.e.a.17.1		1	5.2	odd	4
48.3.e.a.17.1		1	15.2	even	4
192.3.e.a.65.1		1	40.37	odd	4
192.3.e.a.65.1		1	120.77	even	4
192.3.e.b.65.1		1	40.27	even	4
192.3.e.b.65.1		1	120.107	odd	4
300.3.b.a.149.1		2	4.3	odd	2
300.3.b.a.149.1		2	12.11	even	2
300.3.b.a.149.2		2	20.19	odd	2
300.3.b.a.149.2		2	60.59	even	2
300.3.g.b.101.1		1	20.3	even	4
300.3.g.b.101.1		1	60.23	odd	4
324.3.g.b.53.1		2	180.7	even	12
324.3.g.b.53.1		2	180.47	odd	12
324.3.g.b.269.1		2	180.67	even	12
324.3.g.b.269.1		2	180.167	odd	12
588.3.c.c.197.1		1	140.27	odd	4
588.3.c.c.197.1		1	420.167	even	4
588.3.p.b.557.1		2	140.47	odd	12
588.3.p.b.557.1		2	420.47	even	12
588.3.p.b.569.1		2	140.87	odd	12
588.3.p.b.569.1		2	420.227	even	12
588.3.p.c.557.1		2	140.107	even	12
588.3.p.c.557.1		2	420.107	odd	12
588.3.p.c.569.1		2	140.67	even	12
588.3.p.c.569.1		2	420.347	odd	12
768.3.h.a.641.1		2	80.67	even	4
768.3.h.a.641.1		2	240.227	odd	4
768.3.h.a.641.2		2	80.27	even	4
768.3.h.a.641.2		2	240.107	odd	4
768.3.h.b.641.1		2	80.37	odd	4
768.3.h.b.641.1		2	240.197	even	4
768.3.h.b.641.2		2	80.77	odd	4
768.3.h.b.641.2		2	240.77	even	4
1200.3.c.c.449.1		2	5.4	even	2	inner
1200.3.c.c.449.1		2	15.14	odd	2	inner
1200.3.c.c.449.2		2	1.1	even	1	trivial
1200.3.c.c.449.2		2	3.2	odd	2	CM
1200.3.l.b.401.1		1	5.3	odd	4
1200.3.l.b.401.1		1	15.8	even	4
1296.3.q.b.593.1		2	45.22	odd	12
1296.3.q.b.593.1		2	45.32	even	12
1296.3.q.b.1025.1		2	45.2	even	12
1296.3.q.b.1025.1		2	45.7	odd	12
1452.3.e.b.485.1		1	220.87	odd	4
1452.3.e.b.485.1		1	660.527	even	4

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +2.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +2.00000i q^{7} -9.00000 q^{9} -22.0000i q^{13} +26.0000 q^{19} -6.00000 q^{21} -27.0000i q^{27} +46.0000 q^{31} -26.0000i q^{37} +66.0000 q^{39} +22.0000i q^{43} +45.0000 q^{49} +78.0000i q^{57} +74.0000 q^{61} -18.0000i q^{63} +122.000i q^{67} -46.0000i q^{73} -142.000 q^{79} +81.0000 q^{81} +44.0000 q^{91} +138.000i q^{93} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} + 52 q^{19} - 12 q^{21} + 92 q^{31} + 132 q^{39} + 90 q^{49} + 148 q^{61} - 284 q^{79} + 162 q^{81} + 88 q^{91}+O(q^{100}) \) 2 * q - 18 * q^9 + 52 * q^19 - 12 * q^21 + 92 * q^31 + 132 * q^39 + 90 * q^49 + 148 * q^61 - 284 * q^79 + 162 * q^81 + 88 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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