LMFDB - Embedded newform 48.22.c.a.47.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(47,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.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:	$134.149125258$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			47.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.22.c.a.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+102276. i q^{3} -9.85315e8i q^{7} -1.04604e10 q^{9} +O(q^{10})$	$q+102276. i q^{3} -9.85315e8i q^{7} -1.04604e10 q^{9} +3.70077e11 q^{13} +3.99212e13i q^{19} +1.00774e14 q^{21} +4.76837e14 q^{25} -1.06984e15i q^{27} +1.25663e15i q^{31} -5.77763e16 q^{37} +3.78499e16i q^{39} -9.98580e16i q^{43} -4.12300e17 q^{49} -4.08297e18 q^{57} -1.08607e19 q^{61} +1.03067e19i q^{63} -2.90219e19i q^{67} +3.90986e19 q^{73} +4.87689e19i q^{75} -9.04974e18i q^{79} +1.09419e20 q^{81} -3.64642e20i q^{91} -1.28523e20 q^{93} +1.13207e21 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20920706406 q^{9}+O(q^{10}) $ 2 * q - 20920706406 * q^9	$ 2 q - 20920706406 q^{9} + 740153650460 q^{13} + 201547891726956 q^{21} + 953674316406250 q^{25} - 11\!\cdots\!80 q^{37}+ \cdots + 22\!\cdots\!60 q^{97}+O(q^{100}) $ 2 * q - 20920706406 * q^9 + 740153650460 * q^13 + 201547891726956 * q^21 + 953674316406250 * q^25 - 115552646878006580 * q^37 - 824599526003063242 * q^49 - 8165948250222754380 * q^57 - 21721383528929687876 * q^61 + 78197246686961002580 * q^73 + 218837978263024718418 * q^81 - 257046033705514549380 * q^93 + 2264136605088888703460 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	102276.i	1.00000i
$3$	102276.i	1.00000i
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	− 9.85315e8i	− 1.31840i	−0.751970	−	0.659198i	$-0.770896\pi$
$7$	− 9.85315e8i	− 1.31840i	0.751970	−	0.659198i	$-0.229104\pi$
$8$	0	0
$9$	−1.04604e10	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	3.70077e11	0.744538	0.372269	−	0.928125i	$-0.378580\pi$
$13$	3.70077e11	0.744538	0.372269	+	0.928125i	$0.378580\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	3.99212e13i	1.49379i	0.664940	+	0.746897i	$0.268457\pi$
$19$	3.99212e13i	1.49379i	−0.664940	+	0.746897i	$0.731543\pi$
$20$	0	0
$21$	1.00774e14	1.31840
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.76837e14	1.00000
$26$	0	0
$27$	− 1.06984e15i	− 1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	1.25663e15i	0.275366i	0.990476	+	0.137683i	$0.0439655\pi$
$31$	1.25663e15i	0.275366i	−0.990476	+	0.137683i	$0.956035\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.77763e16	−1.97529	−0.987647	−	0.156694i	$-0.949916\pi$
$37$	−5.77763e16	−1.97529	−0.987647	+	0.156694i	$0.949916\pi$
$38$	0	0
$39$	3.78499e16i	0.744538i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 9.98580e16i	− 0.704635i	−0.935881	−	0.352317i	$-0.885394\pi$
$43$	− 9.98580e16i	− 0.704635i	0.935881	−	0.352317i	$-0.114606\pi$
$44$	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
$49$	−4.12300e17	−0.738166
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.08297e18	−1.49379
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−1.08607e19	−1.94937	−0.974685	−	0.223583i	$-0.928225\pi$
$61$	−1.08607e19	−1.94937	−0.974685	+	0.223583i	$0.928225\pi$
$62$	0	0
$63$	1.03067e19i	1.31840i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.90219e19i	− 1.94509i	−0.232710	−	0.972546i	$-0.574759\pi$
$67$	− 2.90219e19i	− 1.94509i	0.232710	−	0.972546i	$-0.425241\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	3.90986e19	1.06481	0.532404	−	0.846490i	$-0.321289\pi$
$73$	3.90986e19	1.06481	0.532404	+	0.846490i	$0.321289\pi$
$74$	0	0
$75$	4.87689e19i	1.00000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 9.04974e18i	− 0.107535i	−0.998553	−	0.0537677i	$-0.982877\pi$
$79$	− 9.04974e18i	− 0.107535i	0.998553	−	0.0537677i	$-0.0171231\pi$
$80$	0	0
$81$	1.09419e20	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
$91$	− 3.64642e20i	− 0.981595i
$92$	0	0
$93$	−1.28523e20	−0.275366
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.13207e21	1.55872	0.779362	−	0.626573i	$-0.215543\pi$
$97$	1.13207e21	1.55872	0.779362	+	0.626573i	$0.215543\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	1.21098e21i	0.887864i	0.896060	+	0.443932i	$0.146417\pi$
$103$	1.21098e21i	0.887864i	−0.896060	+	0.443932i	$0.853583\pi$
$104$	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
$109$	−4.46449e21	−1.80632	−0.903158	−	0.429309i	$-0.858757\pi$
$109$	−4.46449e21	−1.80632	−0.903158	+	0.429309i	$0.858757\pi$
$110$	0	0
$111$	− 5.90912e21i	− 1.97529i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.87113e21	−0.744538
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.40025e21	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.46678e22i	− 1.19241i	−0.802832	−	0.596205i	$-0.796674\pi$
$127$	− 1.46678e22i	− 1.19241i	0.802832	−	0.596205i	$-0.203326\pi$
$128$	0	0
$129$	1.02131e22	0.704635
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	3.93349e22	1.96941
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	− 6.34187e22i	− 1.99785i	−0.0463902	−	0.998923i	$-0.514772\pi$
$139$	− 6.34187e22i	− 1.99785i	0.0463902	−	0.998923i	$-0.485228\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.21683e22i	− 0.738166i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	− 1.45063e23i	− 1.91557i	−0.287477	−	0.957787i	$-0.592817\pi$
$151$	− 1.45063e23i	− 1.91557i	0.287477	−	0.957787i	$-0.407183\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.68883e22	−0.235840	−0.117920	−	0.993023i	$-0.537623\pi$
$157$	−2.68883e22	−0.235840	−0.117920	+	0.993023i	$0.537623\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.10195e23i	− 1.83512i	−0.397594	−	0.917561i	$-0.630155\pi$
$163$	− 3.10195e23i	− 1.83512i	0.397594	−	0.917561i	$-0.369845\pi$
$164$	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
$169$	−1.10108e23	−0.445664
$170$	0	0
$171$	− 4.17590e23i	− 1.49379i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	− 4.69835e23i	− 1.31840i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−6.96900e23	−1.37260	−0.686302	−	0.727317i	$-0.740767\pi$
$181$	−6.96900e23	−1.37260	−0.686302	+	0.727317i	$0.740767\pi$
$182$	0	0
$183$	− 1.11079e24i	− 1.94937i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.05413e24	−1.31840
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	1.98909e24	1.99665	0.998326	−	0.0578425i	$-0.0184221\pi$
$193$	1.98909e24	1.99665	0.998326	+	0.0578425i	$0.0184221\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	− 1.45180e24i	− 1.05669i	−0.849029	−	0.528346i	$-0.822812\pi$
$199$	− 1.45180e24i	− 1.05669i	0.849029	−	0.528346i	$-0.177188\pi$
$200$	0	0
$201$	2.96824e24	1.94509
$202$	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
$211$	− 4.82670e24i	− 1.89970i	−0.312706	−	0.949850i	$-0.601236\pi$
$211$	− 4.82670e24i	− 1.89970i	0.312706	−	0.949850i	$-0.398764\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.23818e24	0.363041
$218$	0	0
$219$	3.99885e24i	1.06481i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.39856e24i	1.84929i	0.380836	+	0.924643i	$0.375636\pi$
$223$	8.39856e24i	1.84929i	−0.380836	+	0.924643i	$0.624364\pi$
$224$	0	0
$225$	−4.98789e24	−1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−1.17896e25	−1.96438	−0.982191	−	0.187884i	$-0.939837\pi$
$229$	−1.17896e25	−1.96438	−0.982191	+	0.187884i	$0.939837\pi$
$230$	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
$237$	9.25570e23	0.107535
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	7.96668e24	0.776423	0.388212	−	0.921570i	$-0.373093\pi$
$241$	7.96668e24	0.776423	0.388212	+	0.921570i	$0.373093\pi$
$242$	0	0
$243$	1.11909e25i	1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.47739e25i	1.11219i
$248$	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
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	5.69279e25i	2.60422i
$260$	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
$271$	5.65070e25i	1.60666i	0.595532	+	0.803332i	$0.296941\pi$
$271$	5.65070e25i	1.60666i	−0.595532	+	0.803332i	$0.703059\pi$
$272$	0	0
$273$	3.72941e25	0.981595
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.19618e24	0.0496169	0.0248085	−	0.999692i	$-0.492102\pi$
$277$	2.19618e24	0.0496169	0.0248085	+	0.999692i	$0.492102\pi$
$278$	0	0
$279$	− 1.31448e25i	− 0.275366i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 3.39377e25i	− 0.612244i	−0.951992	−	0.306122i	$-0.900968\pi$
$283$	− 3.39377e25i	− 0.612244i	0.951992	−	0.306122i	$-0.0990315\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	6.90919e25	1.00000
$290$	0	0
$291$	1.15783e26i	1.55872i
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−9.83916e25	−0.928987
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.24533e26i	− 1.72317i	−0.507614	−	0.861585i	$-0.669472\pi$
$307$	− 2.24533e26i	− 1.72317i	0.507614	−	0.861585i	$-0.330528\pi$
$308$	0	0
$309$	−1.23854e26	−0.887864
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−3.18989e26	−1.99784	−0.998919	−	0.0464917i	$-0.985196\pi$
$313$	−3.18989e26	−1.99784	−0.998919	+	0.0464917i	$0.985196\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	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
$324$	0	0
$325$	1.76466e26	0.744538
$326$	0	0
$327$	− 4.56610e26i	− 1.80632i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 4.81353e26i	− 1.67598i	−0.545684	−	0.837991i	$-0.683730\pi$
$331$	− 4.81353e26i	− 1.67598i	0.545684	−	0.837991i	$-0.316270\pi$
$332$	0	0
$333$	6.04361e26	1.97529
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.86905e26	−0.538898	−0.269449	−	0.963015i	$-0.586842\pi$
$337$	−1.86905e26	−0.538898	−0.269449	+	0.963015i	$0.586842\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.44098e26i	− 0.345200i
$344$	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
$349$	9.79269e26	1.95540	0.977699	−	0.210012i	$-0.0673503\pi$
$349$	9.79269e26	1.95540	0.977699	+	0.210012i	$0.0673503\pi$
$350$	0	0
$351$	− 3.95924e26i	− 0.744538i
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−8.79492e26	−1.23142
$362$	0	0
$363$	− 7.56867e26i	− 1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9.94502e26i	− 1.17115i	−0.810619	−	0.585574i	$-0.800869\pi$
$367$	− 9.94502e26i	− 1.17115i	0.810619	−	0.585574i	$-0.199131\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.60839e27	−1.59753	−0.798767	−	0.601641i	$-0.794514\pi$
$373$	−1.60839e27	−1.59753	−0.798767	+	0.601641i	$0.794514\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8.53526e26i	0.716977i	0.933534	+	0.358488i	$0.116708\pi$
$379$	8.53526e26i	0.716977i	−0.933534	+	0.358488i	$0.883292\pi$
$380$	0	0
$381$	1.50016e27	1.19241
$382$	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
$387$	1.04455e27i	0.704635i
$388$	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
$397$	−1.94928e27	−1.00595	−0.502973	−	0.864302i	$-0.667761\pi$
$397$	−1.94928e27	−1.00595	−0.502973	+	0.864302i	$0.667761\pi$
$398$	0	0
$399$	4.02302e27i	1.96941i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	4.65050e26i	0.205020i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.59171e27	−0.978343	−0.489171	−	0.872188i	$-0.662701\pi$
$409$	−2.59171e27	−0.978343	−0.489171	+	0.872188i	$0.662701\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.48621e27	1.99785
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−4.53662e27	−1.26407	−0.632034	−	0.774940i	$-0.717780\pi$
$421$	−4.53662e27	−1.26407	−0.632034	+	0.774940i	$0.717780\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.07012e28i	2.57004i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	8.16275e27	1.69322	0.846611	−	0.532213i	$-0.178639\pi$
$433$	8.16275e27	1.69322	0.846611	+	0.532213i	$0.178639\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.04995e28i	1.88490i	0.334341	+	0.942452i	$0.391486\pi$
$439$	1.04995e28i	1.88490i	−0.334341	+	0.942452i	$0.608514\pi$
$440$	0	0
$441$	4.31280e27	0.738166
$442$	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
$453$	1.48365e28	1.91557
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.64801e28	−1.94017	−0.970083	−	0.242775i	$-0.921942\pi$
$457$	−1.64801e28	−1.94017	−0.970083	+	0.242775i	$0.921942\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 1.45774e28i	− 1.49651i	−0.663411	−	0.748255i	$-0.730892\pi$
$463$	− 1.45774e28i	− 1.49651i	0.663411	−	0.748255i	$-0.269108\pi$
$464$	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
$469$	−2.85957e28	−2.56440
$470$	0	0
$471$	− 2.75002e27i	− 0.235840i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.90359e28i	1.49379i
$476$	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
$481$	−2.13817e28	−1.47068
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.30159e28i	− 1.99374i	−0.0790391	−	0.996872i	$-0.525185\pi$
$487$	− 3.30159e28i	− 1.99374i	0.0790391	−	0.996872i	$-0.474815\pi$
$488$	0	0
$489$	3.17255e28	1.83512
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 2.09447e28i	− 0.979530i	−0.871855	−	0.489765i	$-0.837083\pi$
$499$	− 2.09447e28i	− 0.979530i	0.871855	−	0.489765i	$-0.162917\pi$
$500$	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
$507$	− 1.12614e28i	− 0.445664i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	− 3.85245e28i	− 1.40384i
$512$	0	0
$513$	4.27094e28	1.49379
$514$	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
$523$	− 5.96873e28i	− 1.70457i	−0.523081	−	0.852283i	$-0.675217\pi$
$523$	− 5.96873e28i	− 1.70457i	0.523081	−	0.852283i	$-0.324783\pi$
$524$	0	0
$525$	4.80528e28	1.31840
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.94716e28	−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
$541$	−9.99080e28	−1.99999	−0.999997	−	0.00231282i	$-0.999264\pi$
$541$	−9.99080e28	−1.99999	−0.999997	+	0.00231282i	$0.999264\pi$
$542$	0	0
$543$	− 7.12760e28i	− 1.37260i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.01057e29i	1.80178i	0.434048	+	0.900890i	$0.357085\pi$
$547$	1.01057e29i	1.80178i	−0.434048	+	0.900890i	$0.642915\pi$
$548$	0	0
$549$	1.13607e29	1.94937
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.91685e27	−0.141774
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	− 3.69551e28i	− 0.524627i
$560$	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
$567$	− 1.07812e29i	− 1.31840i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	1.11396e29i	1.26529i	0.774443	+	0.632643i	$0.218030\pi$
$571$	1.11396e29i	1.26529i	−0.774443	+	0.632643i	$0.781970\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.94611e29	−1.98072	−0.990358	−	0.138529i	$-0.955763\pi$
$577$	−1.94611e29	−1.98072	−0.990358	+	0.138529i	$0.955763\pi$
$578$	0	0
$579$	2.03436e29i	1.99665i
$580$	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
$589$	−5.01662e28	−0.411340
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.48484e29	1.05669
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−2.22203e29	−1.47424	−0.737119	−	0.675762i	$-0.763815\pi$
$601$	−2.22203e29	−1.47424	−0.737119	+	0.675762i	$0.763815\pi$
$602$	0	0
$603$	3.03579e29i	1.94509i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.66798e29i	1.59478i	0.603461	+	0.797392i	$0.293788\pi$
$607$	2.66798e29i	1.59478i	−0.603461	+	0.797392i	$0.706212\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	3.35322e29	1.80770	0.903851	−	0.427848i	$-0.140728\pi$
$613$	3.35322e29	1.80770	0.903851	+	0.427848i	$0.140728\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	2.08457e29i	1.01453i	0.861790	+	0.507264i	$0.169343\pi$
$619$	2.08457e29i	1.01453i	−0.861790	+	0.507264i	$0.830657\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.27374e29	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 4.59445e29i	− 1.82778i	−0.405957	−	0.913892i	$-0.633062\pi$
$631$	− 4.59445e29i	− 1.82778i	0.405957	−	0.913892i	$-0.366938\pi$
$632$	0	0
$633$	4.93655e29	1.89970
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.52583e29	−0.549593
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	5.96033e29i	1.94561i	0.231629	+	0.972804i	$0.425594\pi$
$643$	5.96033e29i	1.94561i	−0.231629	+	0.972804i	$0.574406\pi$
$644$	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
$651$	1.26636e29i	0.363041i
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.08985e29	−1.06481
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−8.04773e29	−1.96588	−0.982942	−	0.183916i	$-0.941123\pi$
$661$	−8.04773e29	−1.96588	−0.982942	+	0.183916i	$0.941123\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−8.58970e29	−1.84929
$670$	0	0
$671$	0	0
$672$	0	0
$673$	8.12894e29	1.64390	0.821951	−	0.569559i	$-0.192886\pi$
$673$	8.12894e29	1.64390	0.821951	+	0.569559i	$0.192886\pi$
$674$	0	0
$675$	− 5.10140e29i	− 1.00000i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	− 1.11544e30i	− 2.05502i
$680$	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
$687$	− 1.20579e30i	− 1.96438i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.30194e30i	1.99560i	0.0663314	+	0.997798i	$0.478871\pi$
$691$	1.30194e30i	1.99560i	−0.0663314	+	0.997798i	$0.521129\pi$
$692$	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
$703$	− 2.30650e30i	− 2.95068i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.91584e29	0.926214	0.463107	−	0.886302i	$-0.346735\pi$
$709$	7.91584e29	0.926214	0.463107	+	0.886302i	$0.346735\pi$
$710$	0	0
$711$	9.46635e28i	0.107535i
$712$	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
$721$	1.19320e30	1.17056
$722$	0	0
$723$	8.14799e29i	0.776423i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.23125e29i	− 0.740209i	−0.928990	−	0.370105i	$-0.879322\pi$
$727$	− 8.23125e29i	− 0.740209i	0.928990	−	0.370105i	$-0.120678\pi$
$728$	0	0
$729$	−1.14456e30	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	9.86819e29	0.814040	0.407020	−	0.913419i	$-0.366568\pi$
$733$	9.86819e29	0.814040	0.407020	+	0.913419i	$0.366568\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 2.64103e30i	− 1.99989i	−0.0103371	−	0.999947i	$-0.503290\pi$
$739$	− 2.64103e30i	− 1.99989i	0.0103371	−	0.999947i	$-0.496710\pi$
$740$	0	0
$741$	−1.51101e30	−1.11219
$742$	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
$751$	− 1.41354e30i	− 0.903836i	−0.892059	−	0.451918i	$-0.850740\pi$
$751$	− 1.41354e30i	− 0.903836i	0.892059	−	0.451918i	$-0.149260\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.68402e30	1.57862	0.789311	−	0.613993i	$-0.210438\pi$
$757$	2.68402e30	1.57862	0.789311	+	0.613993i	$0.210438\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	4.39893e30i	2.38144i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.75585e30	1.37413	0.687066	−	0.726595i	$-0.258898\pi$
$769$	2.75585e30	1.37413	0.687066	+	0.726595i	$0.258898\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	5.99208e29i	0.275366i
$776$	0	0
$777$	−5.82235e30	−2.60422
$778$	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
$787$	4.92409e30i	1.92571i	0.270018	+	0.962855i	$0.412970\pi$
$787$	4.92409e30i	1.92571i	−0.270018	+	0.962855i	$0.587030\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.01929e30	−1.45138
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	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
$811$	4.57864e30i	1.30622i	0.757261	+	0.653112i	$0.226537\pi$
$811$	4.57864e30i	1.30622i	−0.757261	+	0.653112i	$0.773463\pi$
$812$	0	0
$813$	−5.77930e30	−1.60666
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.98645e30	1.05258
$818$	0	0
$819$	3.81429e30i	0.981595i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	2.11515e30i	0.517181i	0.965987	+	0.258591i	$0.0832580\pi$
$823$	2.11515e30i	0.517181i	−0.965987	+	0.258591i	$0.916742\pi$
$824$	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
$829$	4.85631e30	1.10023	0.550116	−	0.835088i	$-0.314584\pi$
$829$	4.85631e30	1.10023	0.550116	+	0.835088i	$0.314584\pi$
$830$	0	0
$831$	2.24616e29i	0.0496169i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.34440e30	0.275366
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	5.13284e30	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.29158e30i	1.31840i
$848$	0	0
$849$	3.47100e30	0.612244
$850$	0	0
$851$	0	0
$852$	0	0
$853$	9.51956e30	1.59828	0.799139	−	0.601147i	$-0.205289\pi$
$853$	9.51956e30	1.59828	0.799139	+	0.601147i	$0.205289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	2.05401e29i	0.0320387i	0.999872	+	0.0160193i	$0.00509933\pi$
$859$	2.05401e29i	0.0320387i	−0.999872	+	0.0160193i	$0.994901\pi$
$860$	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
$867$	7.06644e30i	1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 1.07403e31i	− 1.44819i
$872$	0	0
$873$	−1.18418e31	−1.55872
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.31229e30	−0.541019	−0.270509	−	0.962717i	$-0.587192\pi$
$877$	−4.31229e30	−0.541019	−0.270509	+	0.962717i	$0.587192\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 1.63282e31i	− 1.90700i	−0.301400	−	0.953498i	$-0.597454\pi$
$883$	− 1.63282e31i	− 1.90700i	0.301400	−	0.953498i	$-0.402546\pi$
$884$	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
$889$	−1.44524e31	−1.57207
$890$	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
$903$	− 1.00631e31i	− 0.928987i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 2.03730e31i	− 1.79547i	−0.440533	−	0.897736i	$-0.645210\pi$
$907$	− 2.03730e31i	− 1.79547i	0.440533	−	0.897736i	$-0.354790\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.45048e31i	1.88122i	0.339494	+	0.940608i	$0.389744\pi$
$919$	2.45048e31i	1.88122i	−0.339494	+	0.940608i	$0.610256\pi$
$920$	0	0
$921$	2.29643e31	1.72317
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−2.75499e31	−1.97529
$926$	0	0
$927$	− 1.26673e31i	− 0.887864i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	− 1.64595e31i	− 1.10267i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.77423e30	0.424223	0.212111	−	0.977245i	$-0.431966\pi$
$937$	6.77423e30	0.424223	0.212111	+	0.977245i	$0.431966\pi$
$938$	0	0
$939$	− 3.26249e31i	− 1.99784i
$940$	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
$949$	1.44695e31	0.792790
$950$	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
$961$	1.92464e31	0.924174
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3.68942e31i	− 1.65951i	−0.558128	−	0.829755i	$-0.688480\pi$
$967$	− 3.68942e31i	− 1.65951i	0.558128	−	0.829755i	$-0.311520\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−6.24874e31	−2.63395
$974$	0	0
$975$	1.80483e31i	0.744538i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.67001e31	1.80632
$982$	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
$991$	4.92838e31i	1.71368i	0.515579	+	0.856842i	$0.327577\pi$
$991$	4.92838e31i	1.71368i	−0.515579	+	0.856842i	$0.672423\pi$
$992$	0	0
$993$	4.92308e31	1.67598
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.20454e31	0.393118	0.196559	−	0.980492i	$-0.437023\pi$
$997$	1.20454e31	0.393118	0.196559	+	0.980492i	$0.437023\pi$
$998$	0	0
$999$	6.18115e31i	1.97529i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.c.a.47.2	yes	2
3.2	odd	2	CM	48.22.c.a.47.2	yes	2
4.3	odd	2	inner	48.22.c.a.47.1	✓	2
12.11	even	2	inner	48.22.c.a.47.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.22.c.a.47.1	✓	2	4.3	odd	2	inner
48.22.c.a.47.1	✓	2	12.11	even	2	inner
48.22.c.a.47.2	yes	2	1.1	even	1	trivial
48.22.c.a.47.2	yes	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+102276. i q^{3} -9.85315e8i q^{7} -1.04604e10 q^{9} +O(q^{10})\)	\(q+102276. i q^{3} -9.85315e8i q^{7} -1.04604e10 q^{9} +3.70077e11 q^{13} +3.99212e13i q^{19} +1.00774e14 q^{21} +4.76837e14 q^{25} -1.06984e15i q^{27} +1.25663e15i q^{31} -5.77763e16 q^{37} +3.78499e16i q^{39} -9.98580e16i q^{43} -4.12300e17 q^{49} -4.08297e18 q^{57} -1.08607e19 q^{61} +1.03067e19i q^{63} -2.90219e19i q^{67} +3.90986e19 q^{73} +4.87689e19i q^{75} -9.04974e18i q^{79} +1.09419e20 q^{81} -3.64642e20i q^{91} -1.28523e20 q^{93} +1.13207e21 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20920706406 q^{9}+O(q^{10}) \) 2 * q - 20920706406 * q^9	\( 2 q - 20920706406 q^{9} + 740153650460 q^{13} + 201547891726956 q^{21} + 953674316406250 q^{25} - 11\!\cdots\!80 q^{37}+ \cdots + 22\!\cdots\!60 q^{97}+O(q^{100}) \) 2 * q - 20920706406 * q^9 + 740153650460 * q^13 + 201547891726956 * q^21 + 953674316406250 * q^25 - 115552646878006580 * q^37 - 824599526003063242 * q^49 - 8165948250222754380 * q^57 - 21721383528929687876 * q^61 + 78197246686961002580 * q^73 + 218837978263024718418 * q^81 - 257046033705514549380 * q^93 + 2264136605088888703460 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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