LMFDB - Embedded newform 147.6.c.a.146.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(146,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("147.146");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	147.c (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:	$23.5764215125$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			146.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	147.146
Dual form			147.6.c.a.146.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.5885i q^{3} +32.0000 q^{4} -243.000 q^{9} +O(q^{10})$	$q+15.5885i q^{3} +32.0000 q^{4} -243.000 q^{9} +498.831i q^{12} +1141.42i q^{13} +1024.00 q^{16} +161.081i q^{19} -3125.00 q^{25} -3788.00i q^{27} +10349.0i q^{31} -7776.00 q^{36} +6661.00 q^{37} -17793.0 q^{39} -22475.0 q^{43} +15962.6i q^{48} +36525.5i q^{52} -2511.00 q^{57} +43432.9i q^{61} +32768.0 q^{64} -37939.0 q^{67} -46781.0i q^{73} -48713.9i q^{75} +5154.58i q^{76} +90857.0 q^{79} +59049.0 q^{81} -161325. q^{93} -127631. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 64 q^{4} - 486 q^{9}+O(q^{10}) $ 2 * q + 64 * q^4 - 486 * q^9	$ 2 q + 64 q^{4} - 486 q^{9} + 2048 q^{16} - 6250 q^{25} - 15552 q^{36} + 13322 q^{37} - 35586 q^{39} - 44950 q^{43} - 5022 q^{57} + 65536 q^{64} - 75878 q^{67} + 181714 q^{79} + 118098 q^{81} - 322650 q^{93}+O(q^{100}) $ 2 * q + 64 * q^4 - 486 * q^9 + 2048 * q^16 - 6250 * q^25 - 15552 * q^36 + 13322 * q^37 - 35586 * q^39 - 44950 * q^43 - 5022 * q^57 + 65536 * q^64 - 75878 * q^67 + 181714 * q^79 + 118098 * q^81 - 322650 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	15.5885i	1.00000i
$3$	15.5885i	1.00000i
$4$	32.0000	1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−243.000	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	498.831i	1.00000i
$13$	1141.42i	1.87322i	0.350380	+	0.936608i	$0.386052\pi$
$13$	1141.42i	1.87322i	−0.350380	+	0.936608i	$0.613948\pi$
$14$	0	0
$15$	0	0
$16$	1024.00	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	161.081i	0.102367i	0.998689	+	0.0511835i	$0.0162993\pi$
$19$	161.081i	0.102367i	−0.998689	+	0.0511835i	$0.983701\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−3125.00	−1.00000
$26$	0	0
$27$	− 3788.00i	− 1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	10349.0i	1.93417i	0.254456	+	0.967084i	$0.418103\pi$
$31$	10349.0i	1.93417i	−0.254456	+	0.967084i	$0.581897\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−7776.00	−1.00000
$37$	6661.00	0.799899	0.399949	−	0.916537i	$-0.369028\pi$
$37$	6661.00	0.799899	0.399949	+	0.916537i	$0.369028\pi$
$38$	0	0
$39$	−17793.0	−1.87322
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−22475.0	−1.85365	−0.926827	−	0.375489i	$-0.877475\pi$
$43$	−22475.0	−1.85365	−0.926827	+	0.375489i	$0.877475\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	15962.6i	1.00000i
$49$	0	0
$50$	0	0
$51$	0	0
$52$	36525.5i	1.87322i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2511.00	−0.102367
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	43432.9i	1.49449i	0.664546	+	0.747247i	$0.268625\pi$
$61$	43432.9i	1.49449i	−0.664546	+	0.747247i	$0.731375\pi$
$62$	0	0
$63$	0	0
$64$	32768.0	1.00000
$65$	0	0
$66$	0	0
$67$	−37939.0	−1.03252	−0.516260	−	0.856432i	$-0.672676\pi$
$67$	−37939.0	−1.03252	−0.516260	+	0.856432i	$0.672676\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 46781.0i	− 1.02745i	−0.857954	−	0.513727i	$-0.828265\pi$
$73$	− 46781.0i	− 1.02745i	0.857954	−	0.513727i	$-0.171735\pi$
$74$	0	0
$75$	− 48713.9i	− 1.00000i
$76$	5154.58i	0.102367i
$77$	0	0
$78$	0	0
$79$	90857.0	1.63791	0.818956	−	0.573856i	$-0.194553\pi$
$79$	90857.0	1.63791	0.818956	+	0.573856i	$0.194553\pi$
$80$	0	0
$81$	59049.0	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		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−161325.	−1.93417
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 127631.i	− 1.37730i	−0.725095	−	0.688649i	$-0.758204\pi$
$97$	− 127631.i	− 1.37730i	0.725095	−	0.688649i	$-0.241796\pi$
$98$	0	0
$99$	0	0
$100$	−100000.	−1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 40601.0i	− 0.377089i	−0.982065	−	0.188544i	$-0.939623\pi$
$103$	− 40601.0i	− 0.377089i	0.982065	−	0.188544i	$-0.0603769\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 121216.i	− 1.00000i
$109$	247843.	1.99807	0.999034	−	0.0439362i	$-0.0139898\pi$
$109$	247843.	1.99807	0.999034	+	0.0439362i	$0.0139898\pi$
$110$	0	0
$111$	103835.i	0.799899i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 277365.i	− 1.87322i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	161051.	1.00000
$122$	0	0
$123$	0	0
$124$	331168.i	1.93417i
$125$	0	0
$126$	0	0
$127$	347111.	1.90967	0.954837	−	0.297131i	$-0.0960299\pi$
$127$	347111.	1.90967	0.954837	+	0.297131i	$0.0960299\pi$
$128$	0	0
$129$	− 350351.i	− 1.85365i
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	− 29024.0i	− 0.127415i	−0.997969	−	0.0637074i	$-0.979708\pi$
$139$	− 29024.0i	− 0.127415i	0.997969	−	0.0637074i	$-0.0202924\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−248832.	−1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	213152.	0.799899
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	408724.	1.45877	0.729387	−	0.684102i	$-0.239806\pi$
$151$	408724.	1.45877	0.729387	+	0.684102i	$0.239806\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−569376.	−1.87322
$157$	607971.i	1.96849i	0.176807	+	0.984246i	$0.443423\pi$
$157$	607971.i	1.96849i	−0.176807	+	0.984246i	$0.556577\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	678248.	1.99949	0.999746	−	0.0225538i	$-0.00717969\pi$
$163$	678248.	1.99949	0.999746	+	0.0225538i	$0.00717969\pi$
$164$	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
$169$	−931550.	−2.50893
$170$	0	0
$171$	− 39142.6i	− 0.102367i
$172$	−719200.	−1.85365
$173$	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
$181$	222265.i	0.504284i	0.967690	+	0.252142i	$0.0811351\pi$
$181$	222265.i	0.504284i	−0.967690	+	0.252142i	$0.918865\pi$
$182$	0	0
$183$	−677052.	−1.49449
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	510803.i	1.00000i
$193$	−656375.	−1.26841	−0.634204	−	0.773166i	$-0.718672\pi$
$193$	−656375.	−1.26841	−0.634204	+	0.773166i	$0.718672\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	645047.i	1.15467i	0.816507	+	0.577336i	$0.195908\pi$
$199$	645047.i	1.15467i	−0.816507	+	0.577336i	$0.804092\pi$
$200$	0	0
$201$	− 591410.i	− 1.03252i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	1.16882e6i	1.87322i
$209$	0	0
$210$	0	0
$211$	288976.	0.446844	0.223422	−	0.974722i	$-0.428277\pi$
$211$	288976.	0.446844	0.223422	+	0.974722i	$0.428277\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	729243.	1.02745
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 1.45377e6i	− 1.95764i	−0.204718	−	0.978821i	$-0.565628\pi$
$223$	− 1.45377e6i	− 1.95764i	0.204718	−	0.978821i	$-0.434372\pi$
$224$	0	0
$225$	759375.	1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	−80352.0	−0.102367
$229$	− 623613.i	− 0.785826i	−0.919576	−	0.392913i	$-0.871467\pi$
$229$	− 623613.i	− 0.785826i	0.919576	−	0.392913i	$-0.128533\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.41632e6i	1.63791i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	− 1.25292e6i	− 1.38958i	−0.719215	−	0.694788i	$-0.755498\pi$
$241$	− 1.25292e6i	− 1.38958i	0.719215	−	0.694788i	$-0.244502\pi$
$242$	0	0
$243$	920483.i	1.00000i
$244$	1.38985e6i	1.49449i
$245$	0	0
$246$	0	0
$247$	−183861.	−0.191755
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.04858e6	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−1.21405e6	−1.03252
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	− 878236.i	− 0.726421i	−0.931707	−	0.363210i	$-0.881681\pi$
$271$	− 878236.i	− 0.726421i	0.931707	−	0.363210i	$-0.118319\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	738389.	0.578210	0.289105	−	0.957297i	$-0.406642\pi$
$277$	738389.	0.578210	0.289105	+	0.957297i	$0.406642\pi$
$278$	0	0
$279$	− 2.51481e6i	− 1.93417i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 1.34561e6i	− 0.998745i	−0.866387	−	0.499373i	$-0.833564\pi$
$283$	− 1.34561e6i	− 0.998745i	0.866387	−	0.499373i	$-0.166436\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.41986e6	−1.00000
$290$	0	0
$291$	1.98958e6	1.37730
$292$	− 1.49699e6i	− 1.02745i
$293$	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
$300$	− 1.55885e6i	− 1.00000i
$301$	0	0
$302$	0	0
$303$	0	0
$304$	164947.i	0.102367i
$305$	0	0
$306$	0	0
$307$	808260.i	0.489446i	0.969593	+	0.244723i	$0.0786971\pi$
$307$	808260.i	0.489446i	−0.969593	+	0.244723i	$0.921303\pi$
$308$	0	0
$309$	632907.	0.377089
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 2.32953e6i	− 1.34403i	−0.740539	−	0.672014i	$-0.765430\pi$
$313$	− 2.32953e6i	− 1.34403i	0.740539	−	0.672014i	$-0.234570\pi$
$314$	0	0
$315$	0	0
$316$	2.90742e6	1.63791
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	1.88957e6	1.00000
$325$	− 3.56694e6i	− 1.87322i
$326$	0	0
$327$	3.86349e6i	1.99807i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	546151.	0.273995	0.136998	−	0.990571i	$-0.456255\pi$
$331$	546151.	0.273995	0.136998	+	0.990571i	$0.456255\pi$
$332$	0	0
$333$	−1.61862e6	−0.799899
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.63172e6	−1.26231	−0.631155	−	0.775657i	$-0.717419\pi$
$337$	−2.63172e6	−1.26231	−0.631155	+	0.775657i	$0.717419\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	1.55867e6i	0.685001i	0.939518	+	0.342501i	$0.111274\pi$
$349$	1.55867e6i	0.685001i	−0.939518	+	0.342501i	$0.888726\pi$
$350$	0	0
$351$	4.32370e6	1.87322
$352$	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
$361$	2.45015e6	0.989521
$362$	0	0
$363$	2.51054e6i	1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.46748e6i	1.73140i	0.500563	+	0.865700i	$0.333126\pi$
$367$	4.46748e6i	1.73140i	−0.500563	+	0.865700i	$0.666874\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−5.16240e6	−1.93417
$373$	−5.31727e6	−1.97887	−0.989434	−	0.144983i	$-0.953687\pi$
$373$	−5.31727e6	−1.97887	−0.989434	+	0.144983i	$0.953687\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−69893.0	−0.0249940	−0.0124970	−	0.999922i	$-0.503978\pi$
$379$	−69893.0	−0.0249940	−0.0124970	+	0.999922i	$0.503978\pi$
$380$	0	0
$381$	5.41092e6i	1.90967i
$382$	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
$387$	5.46142e6	1.85365
$388$	− 4.08420e6i	− 1.37730i
$389$	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
$397$	3.79639e6i	1.20891i	0.796638	+	0.604456i	$0.206610\pi$
$397$	3.79639e6i	1.20891i	−0.796638	+	0.604456i	$0.793390\pi$
$398$	0	0
$399$	0	0
$400$	−3.20000e6	−1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−1.18126e7	−3.62311
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	4.36452e6i	1.29011i	0.764134	+	0.645057i	$0.223166\pi$
$409$	4.36452e6i	1.29011i	−0.764134	+	0.645057i	$0.776834\pi$
$410$	0	0
$411$	0	0
$412$	− 1.29923e6i	− 0.377089i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	452439.	0.127415
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−4.87580e6	−1.34073	−0.670364	−	0.742033i	$-0.733862\pi$
$421$	−4.87580e6	−1.34073	−0.670364	+	0.742033i	$0.733862\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 3.87891e6i	− 1.00000i
$433$	− 2.18049e6i	− 0.558901i	−0.960160	−	0.279450i	$-0.909848\pi$
$433$	− 2.18049e6i	− 0.558901i	0.960160	−	0.279450i	$-0.0901522\pi$
$434$	0	0
$435$	0	0
$436$	7.93098e6	1.99807
$437$	0	0
$438$	0	0
$439$	− 7.01951e6i	− 1.73838i	−0.494475	−	0.869192i	$-0.664640\pi$
$439$	− 7.01951e6i	− 1.73838i	0.494475	−	0.869192i	$-0.335360\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	3.32271e6i	0.799899i
$445$	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
$453$	6.37138e6i	1.45877i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.18576e6	−0.265587	−0.132793	−	0.991144i	$-0.542395\pi$
$457$	−1.18576e6	−0.265587	−0.132793	+	0.991144i	$0.542395\pi$
$458$	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
$463$	2.92217e6	0.633510	0.316755	−	0.948507i	$-0.397407\pi$
$463$	2.92217e6	0.633510	0.316755	+	0.948507i	$0.397407\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	− 8.87569e6i	− 1.87322i
$469$	0	0
$470$	0	0
$471$	−9.47732e6	−1.96849
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 503377.i	− 0.102367i
$476$	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
$481$	7.60301e6i	1.49838i
$482$	0	0
$483$	0	0
$484$	5.15363e6	1.00000
$485$	0	0
$486$	0	0
$487$	1.36487e6	0.260778	0.130389	−	0.991463i	$-0.458377\pi$
$487$	1.36487e6	0.260778	0.130389	+	0.991463i	$0.458377\pi$
$488$	0	0
$489$	1.05728e7i	1.99949i
$490$	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
$496$	1.05974e7i	1.93417i
$497$	0	0
$498$	0	0
$499$	2.17369e6	0.390793	0.195397	−	0.980724i	$-0.437401\pi$
$499$	2.17369e6	0.390793	0.195397	+	0.980724i	$0.437401\pi$
$500$	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
$507$	− 1.45214e7i	− 2.50893i
$508$	1.11076e7	1.90967
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	610173.	0.102367
$514$	0	0
$515$	0	0
$516$	− 1.12112e7i	− 1.85365i
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	4.35404e6i	0.696047i	0.937486	+	0.348023i	$0.113147\pi$
$523$	4.35404e6i	0.696047i	−0.937486	+	0.348023i	$0.886853\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	6.43634e6	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
$541$	−3.02235e6	−0.443968	−0.221984	−	0.975050i	$-0.571253\pi$
$541$	−3.02235e6	−0.443968	−0.221984	+	0.975050i	$0.571253\pi$
$542$	0	0
$543$	−3.46478e6	−0.504284
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.27982e7	1.82886	0.914430	−	0.404744i	$-0.132639\pi$
$547$	1.27982e7	1.82886	0.914430	+	0.404744i	$0.132639\pi$
$548$	0	0
$549$	− 1.05542e7i	− 1.49449i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	− 928767.i	− 0.127415i
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	− 2.56534e7i	− 3.47229i
$560$	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
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	5.88780e6	0.755723	0.377862	−	0.925862i	$-0.376660\pi$
$571$	5.88780e6	0.755723	0.377862	+	0.925862i	$0.376660\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−7.96262e6	−1.00000
$577$	9.44227e6i	1.18069i	0.807150	+	0.590347i	$0.201009\pi$
$577$	9.44227e6i	1.18069i	−0.807150	+	0.590347i	$0.798991\pi$
$578$	0	0
$579$	− 1.02319e7i	− 1.26841i
$580$	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
$589$	−1.66702e6	−0.197995
$590$	0	0
$591$	0	0
$592$	6.82086e6	0.799899
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.00553e7	−1.15467
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	3.11249e6i	0.351498i	0.984435	+	0.175749i	$0.0562346\pi$
$601$	3.11249e6i	0.351498i	−0.984435	+	0.175749i	$0.943765\pi$
$602$	0	0
$603$	9.21918e6	1.03252
$604$	1.30792e7	1.45877
$605$	0	0
$606$	0	0
$607$	− 9.15058e6i	− 1.00804i	−0.863693	−	0.504019i	$-0.831854\pi$
$607$	− 9.15058e6i	− 1.00804i	0.863693	−	0.504019i	$-0.168146\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.58234e7	−1.70079	−0.850394	−	0.526147i	$-0.823636\pi$
$613$	−1.58234e7	−1.70079	−0.850394	+	0.526147i	$0.823636\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 1.46298e7i	− 1.53466i	−0.641253	−	0.767330i	$-0.721585\pi$
$619$	− 1.46298e7i	− 1.53466i	0.641253	−	0.767330i	$-0.278415\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	−1.82200e7	−1.87322
$625$	9.76562e6	1.00000
$626$	0	0
$627$	0	0
$628$	1.94551e7i	1.96849i
$629$	0	0
$630$	0	0
$631$	−1.62439e7	−1.62411	−0.812057	−	0.583579i	$-0.801652\pi$
$631$	−1.62439e7	−1.62411	−0.812057	+	0.583579i	$0.801652\pi$
$632$	0	0
$633$	4.50469e6i	0.446844i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	1.61402e7i	1.53951i	0.638342	+	0.769753i	$0.279620\pi$
$643$	1.61402e7i	1.53951i	−0.638342	+	0.769753i	$0.720380\pi$
$644$	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
$651$	0	0
$652$	2.17039e7	1.99949
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.13678e7i	1.02745i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	2.24558e7i	1.99905i	0.0307548	+	0.999527i	$0.490209\pi$
$661$	2.24558e7i	1.99905i	−0.0307548	+	0.999527i	$0.509791\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.26620e7	1.95764
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.73115e7	−1.47332	−0.736661	−	0.676262i	$-0.763599\pi$
$673$	−1.73115e7	−1.47332	−0.736661	+	0.676262i	$0.763599\pi$
$674$	0	0
$675$	1.18375e7i	1.00000i
$676$	−2.98096e7	−2.50893
$677$	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
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	− 1.25256e6i	− 0.102367i
$685$	0	0
$686$	0	0
$687$	9.72116e6	0.785826
$688$	−2.30144e7	−1.85365
$689$	0	0
$690$	0	0
$691$	2.41017e7i	1.92022i	0.279616	+	0.960112i	$0.409793\pi$
$691$	2.41017e7i	1.92022i	−0.279616	+	0.960112i	$0.590207\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	1.07296e6i	0.0818832i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.27014e7	1.69604	0.848021	−	0.529962i	$-0.177794\pi$
$709$	2.27014e7	1.69604	0.848021	+	0.529962i	$0.177794\pi$
$710$	0	0
$711$	−2.20783e7	−1.63791
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.95312e7	1.38958
$724$	7.11249e6i	0.504284i
$725$	0	0
$726$	0	0
$727$	− 2.37945e7i	− 1.66971i	−0.550474	−	0.834853i	$-0.685553\pi$
$727$	− 2.37945e7i	− 1.66971i	0.550474	−	0.834853i	$-0.314447\pi$
$728$	0	0
$729$	−1.43489e7	−1.00000
$730$	0	0
$731$	0	0
$732$	−2.16657e7	−1.49449
$733$	− 2.31923e7i	− 1.59435i	−0.603746	−	0.797177i	$-0.706326\pi$
$733$	− 2.31923e7i	− 1.59435i	0.603746	−	0.797177i	$-0.293674\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.28170e7	0.863330	0.431665	−	0.902034i	$-0.357926\pi$
$739$	1.28170e7	0.863330	0.431665	+	0.902034i	$0.357926\pi$
$740$	0	0
$741$	− 2.86611e6i	− 0.191755i
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.73219e7	1.76771	0.883857	−	0.467758i	$-0.154938\pi$
$751$	2.73219e7	1.76771	0.883857	+	0.467758i	$0.154938\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.98526e7	−1.25915	−0.629575	−	0.776940i	$-0.716771\pi$
$757$	−1.98526e7	−1.25915	−0.629575	+	0.776940i	$0.716771\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.63457e7i	1.00000i
$769$	2.12277e7i	1.29446i	0.762296	+	0.647228i	$0.224072\pi$
$769$	2.12277e7i	1.29446i	−0.762296	+	0.647228i	$0.775928\pi$
$770$	0	0
$771$	0	0
$772$	−2.10040e7	−1.26841
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	− 3.23406e7i	− 1.93417i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.05647e7i	0.608024i	0.952668	+	0.304012i	$0.0983263\pi$
$787$	1.05647e7i	0.608024i	−0.952668	+	0.304012i	$0.901674\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.95753e7	−2.79951
$794$	0	0
$795$	0	0
$796$	2.06415e7i	1.15467i
$797$	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
$804$	− 1.89251e7i	− 1.03252i
$805$	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
$811$	− 2.97462e7i	− 1.58811i	−0.607849	−	0.794053i	$-0.707967\pi$
$811$	− 2.97462e7i	− 1.58811i	0.607849	−	0.794053i	$-0.292033\pi$
$812$	0	0
$813$	1.36904e7	0.726421
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.62029e6i	− 0.189753i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	2.22923e7	1.14725	0.573623	−	0.819120i	$-0.305538\pi$
$823$	2.22923e7	1.14725	0.573623	+	0.819120i	$0.305538\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	− 3.24529e7i	− 1.64009i	−0.572299	−	0.820045i	$-0.693948\pi$
$829$	− 3.24529e7i	− 1.64009i	0.572299	−	0.820045i	$-0.306052\pi$
$830$	0	0
$831$	1.15103e7i	0.578210i
$832$	3.74021e7i	1.87322i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.92020e7	1.93417
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	2.05111e7	1.00000
$842$	0	0
$843$	0	0
$844$	9.24723e6	0.446844
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.09761e7	0.998745
$850$	0	0
$851$	0	0
$852$	0	0
$853$	4.23947e7i	1.99498i	0.0707842	+	0.997492i	$0.477450\pi$
$853$	4.23947e7i	1.99498i	−0.0707842	+	0.997492i	$0.522550\pi$
$854$	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
$859$	− 4.31437e7i	− 1.99496i	−0.0709259	−	0.997482i	$-0.522595\pi$
$859$	− 4.31437e7i	− 1.99496i	0.0709259	−	0.997482i	$-0.477405\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.21334e7i	− 1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 4.33044e7i	− 1.93413i
$872$	0	0
$873$	3.10144e7i	1.37730i
$874$	0	0
$875$	0	0
$876$	2.33358e7	1.02745
$877$	−3.83931e6	−0.168560	−0.0842800	−	0.996442i	$-0.526859\pi$
$877$	−3.83931e6	−0.168560	−0.0842800	+	0.996442i	$0.526859\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	4.58066e7	1.97709	0.988545	−	0.150925i	$-0.0482253\pi$
$883$	4.58066e7	1.97709	0.988545	+	0.150925i	$0.0482253\pi$
$884$	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
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 4.65206e7i	− 1.95764i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	2.43000e7	1.00000
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.62120e7	−1.05799	−0.528995	−	0.848625i	$-0.677431\pi$
$907$	−2.62120e7	−1.05799	−0.528995	+	0.848625i	$0.677431\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−2.57126e6	−0.102367
$913$	0	0
$914$	0	0
$915$	0	0
$916$	− 1.99556e7i	− 0.785826i
$917$	0	0
$918$	0	0
$919$	4.21821e6	0.164755	0.0823777	−	0.996601i	$-0.473749\pi$
$919$	4.21821e6	0.164755	0.0823777	+	0.996601i	$0.473749\pi$
$920$	0	0
$921$	−1.25995e7	−0.489446
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−2.08156e7	−0.799899
$926$	0	0
$927$	9.86604e6i	0.377089i
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 5.34884e7i	− 1.99026i	−0.0985501	−	0.995132i	$-0.531420\pi$
$937$	− 5.34884e7i	− 1.99026i	0.0985501	−	0.995132i	$-0.468580\pi$
$938$	0	0
$939$	3.63138e7	1.34403
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	4.53223e7i	1.63791i
$949$	5.33968e7	1.92464
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7.84727e7	−2.74101
$962$	0	0
$963$	0	0
$964$	− 4.00936e7i	− 1.38958i
$965$	0	0
$966$	0	0
$967$	−3.23453e7	−1.11236	−0.556180	−	0.831062i	$-0.687733\pi$
$967$	−3.23453e7	−1.11236	−0.556180	+	0.831062i	$0.687733\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	2.94555e7i	1.00000i
$973$	0	0
$974$	0	0
$975$	5.56031e7	1.87322
$976$	4.44753e7i	1.49449i
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−6.02258e7	−1.99807
$982$	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
$988$	−5.88355e6	−0.191755
$989$	0	0
$990$	0	0
$991$	−6.05528e7	−1.95862	−0.979310	−	0.202365i	$-0.935137\pi$
$991$	−6.05528e7	−1.95862	−0.979310	+	0.202365i	$0.935137\pi$
$992$	0	0
$993$	8.51365e6i	0.273995i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.68344e7i	− 0.536365i	−0.963368	−	0.268183i	$-0.913577\pi$
$997$	− 1.68344e7i	− 0.536365i	0.963368	−	0.268183i	$-0.0864230\pi$
$998$	0	0
$999$	− 2.52318e7i	− 0.799899i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.6.c.a.146.2		2
3.2	odd	2	CM	147.6.c.a.146.2		2
7.4	even	3		21.6.g.a.5.1	✓	2
7.5	odd	6		21.6.g.a.17.1	yes	2
7.6	odd	2	inner	147.6.c.a.146.1		2
21.5	even	6		21.6.g.a.17.1	yes	2
21.11	odd	6		21.6.g.a.5.1	✓	2
21.20	even	2	inner	147.6.c.a.146.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.g.a.5.1	✓	2	7.4	even	3
21.6.g.a.5.1	✓	2	21.11	odd	6
21.6.g.a.17.1	yes	2	7.5	odd	6
21.6.g.a.17.1	yes	2	21.5	even	6
147.6.c.a.146.1		2	7.6	odd	2	inner
147.6.c.a.146.1		2	21.20	even	2	inner
147.6.c.a.146.2		2	1.1	even	1	trivial
147.6.c.a.146.2		2	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+15.5885i q^{3} +32.0000 q^{4} -243.000 q^{9} +O(q^{10})\)	\(q+15.5885i q^{3} +32.0000 q^{4} -243.000 q^{9} +498.831i q^{12} +1141.42i q^{13} +1024.00 q^{16} +161.081i q^{19} -3125.00 q^{25} -3788.00i q^{27} +10349.0i q^{31} -7776.00 q^{36} +6661.00 q^{37} -17793.0 q^{39} -22475.0 q^{43} +15962.6i q^{48} +36525.5i q^{52} -2511.00 q^{57} +43432.9i q^{61} +32768.0 q^{64} -37939.0 q^{67} -46781.0i q^{73} -48713.9i q^{75} +5154.58i q^{76} +90857.0 q^{79} +59049.0 q^{81} -161325. q^{93} -127631. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 64 q^{4} - 486 q^{9}+O(q^{10}) \) 2 * q + 64 * q^4 - 486 * q^9	\( 2 q + 64 q^{4} - 486 q^{9} + 2048 q^{16} - 6250 q^{25} - 15552 q^{36} + 13322 q^{37} - 35586 q^{39} - 44950 q^{43} - 5022 q^{57} + 65536 q^{64} - 75878 q^{67} + 181714 q^{79} + 118098 q^{81} - 322650 q^{93}+O(q^{100}) \) 2 * q + 64 * q^4 - 486 * q^9 + 2048 * q^16 - 6250 * q^25 - 15552 * q^36 + 13322 * q^37 - 35586 * q^39 - 44950 * q^43 - 5022 * q^57 + 65536 * q^64 - 75878 * q^67 + 181714 * q^79 + 118098 * q^81 - 322650 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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