LMFDB - Embedded newform 300.9.b.b.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,9,Mod(149,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	300.b (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:	$122.213583018$
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			149.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.9.b.b.149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000i q^{3} +4034.00i q^{7} -6561.00 q^{9} +O(q^{10})$	$q-81.0000i q^{3} +4034.00i q^{7} -6561.00 q^{9} +35806.0i q^{13} +258526. q^{19} +326754. q^{21} +531441. i q^{27} -1.80941e6 q^{31} +503522. i q^{37} +2.90029e6 q^{39} -3.49219e6i q^{43} -1.05084e7 q^{49} -2.09406e7i q^{57} -2.38265e7 q^{61} -2.64671e7i q^{63} -5.42141e6i q^{67} -1.61693e7i q^{73} +1.88870e7 q^{79} +4.30467e7 q^{81} -1.44441e8 q^{91} +1.46562e8i q^{93} +1.76908e8i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 13122 q^{9}+O(q^{10}) $ 2 * q - 13122 * q^9	$ 2 q - 13122 q^{9} + 517052 q^{19} + 653508 q^{21} - 3618812 q^{31} + 5800572 q^{39} - 21016710 q^{49} - 47653052 q^{61} + 37774076 q^{79} + 86093442 q^{81} - 288882808 q^{91}+O(q^{100}) $ 2 * q - 13122 * q^9 + 517052 * q^19 + 653508 * q^21 - 3618812 * q^31 + 5800572 * q^39 - 21016710 * q^49 - 47653052 * q^61 + 37774076 * q^79 + 86093442 * q^81 - 288882808 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	− 81.0000i	− 1.00000i
$3$	− 81.0000i	− 1.00000i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4034.00i	1.68013i	0.542483	+	0.840067i	$0.317484\pi$
$7$	4034.00i	1.68013i	−0.542483	+	0.840067i	$0.682516\pi$
$8$	0	0
$9$	−6561.00	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	35806.0i	1.25367i	0.779153	+	0.626834i	$0.215650\pi$
$13$	35806.0i	1.25367i	−0.779153	+	0.626834i	$0.784350\pi$
$14$	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
$19$	258526.	1.98376	0.991882	−	0.127165i	$-0.0405878\pi$
$19$	258526.	1.98376	0.991882	+	0.127165i	$0.0405878\pi$
$20$	0	0
$21$	326754.	1.68013
$22$	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
$27$	531441.i	1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.80941e6	−1.95925	−0.979624	−	0.200842i	$-0.935632\pi$
$31$	−1.80941e6	−1.95925	−0.979624	+	0.200842i	$0.935632\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	503522.i	0.268665i	0.990936	+	0.134333i	$0.0428891\pi$
$37$	503522.i	0.268665i	−0.990936	+	0.134333i	$0.957111\pi$
$38$	0	0
$39$	2.90029e6	1.25367
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 3.49219e6i	− 1.02147i	−0.859739	−	0.510734i	$-0.829374\pi$
$43$	− 3.49219e6i	− 1.02147i	0.859739	−	0.510734i	$-0.170626\pi$
$44$	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
$49$	−1.05084e7	−1.82285
$50$	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
$57$	− 2.09406e7i	− 1.98376i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−2.38265e7	−1.72084	−0.860422	−	0.509583i	$-0.829800\pi$
$61$	−2.38265e7	−1.72084	−0.860422	+	0.509583i	$0.829800\pi$
$62$	0	0
$63$	− 2.64671e7i	− 1.68013i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.42141e6i	− 0.269037i	−0.990911	−	0.134519i	$-0.957051\pi$
$67$	− 5.42141e6i	− 0.269037i	0.990911	−	0.134519i	$-0.0429488\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 1.61693e7i	− 0.569376i	−0.958620	−	0.284688i	$-0.908110\pi$
$73$	− 1.61693e7i	− 0.569376i	0.958620	−	0.284688i	$-0.0918900\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.88870e7	0.484904	0.242452	−	0.970163i	$-0.422048\pi$
$79$	1.88870e7	0.484904	0.242452	+	0.970163i	$0.422048\pi$
$80$	0	0
$81$	4.30467e7	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
$91$	−1.44441e8	−2.10633
$92$	0	0
$93$	1.46562e8i	1.95925i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.76908e8i	1.99830i	0.0412262	+	0.999150i	$0.486874\pi$
$97$	1.76908e8i	1.99830i	−0.0412262	+	0.999150i	$0.513126\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 4.44490e7i	− 0.394923i	−0.980311	−	0.197462i	$-0.936730\pi$
$103$	− 4.44490e7i	− 0.394923i	0.980311	−	0.197462i	$-0.0632698\pi$
$104$	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
$109$	−2.03181e8	−1.43938	−0.719692	−	0.694293i	$-0.755717\pi$
$109$	−2.03181e8	−1.43938	−0.719692	+	0.694293i	$0.755717\pi$
$110$	0	0
$111$	4.07853e7	0.268665
$112$	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
$117$	− 2.34923e8i	− 1.25367i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.14359e8	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.00562e8i	− 1.53977i	−0.638185	−	0.769883i	$-0.720314\pi$
$127$	− 4.00562e8i	− 1.53977i	0.638185	−	0.769883i	$-0.279686\pi$
$128$	0	0
$129$	−2.82868e8	−1.02147
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	1.04289e9i	3.33299i
$134$	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
$139$	−7.09431e8	−1.90043	−0.950213	−	0.311602i	$-0.899135\pi$
$139$	−7.09431e8	−1.90043	−0.950213	+	0.311602i	$0.899135\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	8.51177e8i	1.82285i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	2.70234e8	0.519796	0.259898	−	0.965636i	$-0.416311\pi$
$151$	2.70234e8	0.519796	0.259898	+	0.965636i	$0.416311\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.61735e7i	0.0595376i	0.999557	+	0.0297688i	$0.00947711\pi$
$157$	3.61735e7i	0.0595376i	−0.999557	+	0.0297688i	$0.990523\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.17139e9i	1.65940i	0.558212	+	0.829698i	$0.311488\pi$
$163$	1.17139e9i	1.65940i	−0.558212	+	0.829698i	$0.688512\pi$
$164$	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
$169$	−4.66339e8	−0.571682
$170$	0	0
$171$	−1.69619e9	−1.98376
$172$	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
$181$	−1.05268e9	−0.980801	−0.490400	−	0.871497i	$-0.663149\pi$
$181$	−1.05268e9	−0.980801	−0.490400	+	0.871497i	$0.663149\pi$
$182$	0	0
$183$	1.92995e9i	1.72084i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.14383e9	−1.68013
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	− 2.32670e9i	− 1.67691i	−0.544968	−	0.838457i	$-0.683458\pi$
$193$	− 2.32670e9i	− 1.67691i	0.544968	−	0.838457i	$-0.316542\pi$
$194$	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
$199$	−1.73472e9	−1.10616	−0.553080	−	0.833128i	$-0.686548\pi$
$199$	−1.73472e9	−1.10616	−0.553080	+	0.833128i	$0.686548\pi$
$200$	0	0
$201$	−4.39134e8	−0.269037
$202$	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
$211$	−1.83711e8	−0.0926840	−0.0463420	−	0.998926i	$-0.514756\pi$
$211$	−1.83711e8	−0.0926840	−0.0463420	+	0.998926i	$0.514756\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7.29914e9i	− 3.29180i
$218$	0	0
$219$	−1.30971e9	−0.569376
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.72732e9i	1.91159i	0.294027	+	0.955797i	$0.405004\pi$
$223$	4.72732e9i	1.91159i	−0.294027	+	0.955797i	$0.594996\pi$
$224$	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
$229$	−5.35877e9	−1.94860	−0.974302	−	0.225247i	$-0.927681\pi$
$229$	−5.35877e9	−1.94860	−0.974302	+	0.225247i	$0.927681\pi$
$230$	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
$237$	− 1.52985e9i	− 0.484904i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−5.56578e9	−1.64990	−0.824951	−	0.565204i	$-0.808797\pi$
$241$	−5.56578e9	−1.64990	−0.824951	+	0.565204i	$0.808797\pi$
$242$	0	0
$243$	− 3.48678e9i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.25678e9i	2.48698i
$248$	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
$259$	−2.03121e9	−0.451393
$260$	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
$271$	−2.98709e9	−0.553824	−0.276912	−	0.960895i	$-0.589311\pi$
$271$	−2.98709e9	−0.553824	−0.276912	+	0.960895i	$0.589311\pi$
$272$	0	0
$273$	1.16998e10i	2.10633i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.42807e9i	1.26170i	0.775904	+	0.630852i	$0.217294\pi$
$277$	7.42807e9i	1.26170i	−0.775904	+	0.630852i	$0.782706\pi$
$278$	0	0
$279$	1.18715e10	1.95925
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	1.03697e10i	1.61667i	0.588723	+	0.808335i	$0.299631\pi$
$283$	1.03697e10i	1.61667i	−0.588723	+	0.808335i	$0.700369\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.97576e9	−1.00000
$290$	0	0
$291$	1.43296e10	1.99830
$292$	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
$301$	1.40875e10	1.71620
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.41256e10i	− 1.59020i	−0.606478	−	0.795101i	$-0.707418\pi$
$307$	− 1.41256e10i	− 1.59020i	0.606478	−	0.795101i	$-0.292582\pi$
$308$	0	0
$309$	−3.60037e9	−0.394923
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 1.17006e10i	− 1.21908i	−0.792755	−	0.609541i	$-0.791354\pi$
$313$	− 1.17006e10i	− 1.21908i	0.792755	−	0.609541i	$-0.208646\pi$
$314$	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
$327$	1.64576e10i	1.43938i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.62495e10	−1.35372	−0.676858	−	0.736113i	$-0.736659\pi$
$331$	−1.62495e10	−1.35372	−0.676858	+	0.736113i	$0.736659\pi$
$332$	0	0
$333$	− 3.30361e9i	− 0.268665i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.57689e10i	− 1.99791i	−0.0456520	−	0.998957i	$-0.514537\pi$
$337$	− 2.57689e10i	− 1.99791i	0.0456520	−	0.998957i	$-0.485463\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.91355e10i	− 1.38249i
$344$	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
$349$	2.96004e10	1.99524	0.997621	−	0.0689403i	$-0.0219618\pi$
$349$	2.96004e10	1.99524	0.997621	+	0.0689403i	$0.0219618\pi$
$350$	0	0
$351$	−1.90288e10	−1.25367
$352$	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
$361$	4.98521e10	2.93532
$362$	0	0
$363$	− 1.73631e10i	− 1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.43056e10i	1.33981i	0.742448	+	0.669903i	$0.233664\pi$
$367$	2.43056e10i	1.33981i	−0.742448	+	0.669903i	$0.766336\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 4.94467e9i	− 0.255448i	−0.991810	−	0.127724i	$-0.959233\pi$
$373$	− 4.94467e9i	− 0.255448i	0.991810	−	0.127724i	$-0.0407672\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3.49200e9	0.169245	0.0846227	−	0.996413i	$-0.473032\pi$
$379$	3.49200e9	0.169245	0.0846227	+	0.996413i	$0.473032\pi$
$380$	0	0
$381$	−3.24455e10	−1.53977
$382$	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
$387$	2.29123e10i	1.02147i
$388$	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
$397$	− 1.57611e10i	− 0.634491i	−0.948343	−	0.317245i	$-0.897242\pi$
$397$	− 1.57611e10i	− 0.634491i	0.948343	−	0.317245i	$-0.102758\pi$
$398$	0	0
$399$	8.44744e10	3.33299
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 6.47876e10i	− 2.45624i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	5.26810e10	1.88261	0.941305	−	0.337556i	$-0.109600\pi$
$409$	5.26810e10	1.88261	0.941305	+	0.337556i	$0.109600\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.74639e10i	1.90043i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−1.16089e10	−0.369541	−0.184771	−	0.982782i	$-0.559154\pi$
$421$	−1.16089e10	−0.369541	−0.184771	+	0.982782i	$0.559154\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 9.61162e10i	− 2.89125i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	− 6.51683e10i	− 1.85389i	−0.375192	−	0.926947i	$-0.622423\pi$
$433$	− 6.51683e10i	− 1.85389i	0.375192	−	0.926947i	$-0.377577\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.75493e10	−1.81871	−0.909354	−	0.416024i	$-0.863423\pi$
$439$	−6.75493e10	−1.81871	−0.909354	+	0.416024i	$0.863423\pi$
$440$	0	0
$441$	6.89453e10	1.82285
$442$	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
$453$	− 2.18890e10i	− 0.519796i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.72608e10i	− 0.624991i	−0.949919	−	0.312495i	$-0.898835\pi$
$457$	− 2.72608e10i	− 0.624991i	0.949919	−	0.312495i	$-0.101165\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	6.87853e10i	1.49683i	0.663232	+	0.748413i	$0.269184\pi$
$463$	6.87853e10i	1.49683i	−0.663232	+	0.748413i	$0.730816\pi$
$464$	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
$469$	2.18700e10	0.452019
$470$	0	0
$471$	2.93005e9	0.0595376
$472$	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
$481$	−1.80291e10	−0.336817
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.09984e10i	1.61777i	0.587965	+	0.808887i	$0.299929\pi$
$487$	9.09984e10i	1.61777i	−0.587965	+	0.808887i	$0.700071\pi$
$488$	0	0
$489$	9.48824e10	1.65940
$490$	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
$499$	−1.23330e11	−1.98915	−0.994574	−	0.104032i	$-0.966826\pi$
$499$	−1.23330e11	−1.98915	−0.994574	+	0.104032i	$0.966826\pi$
$500$	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
$507$	3.77735e10i	0.571682i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	6.52269e10	0.956628
$512$	0	0
$513$	1.37391e11i	1.98376i
$514$	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
$523$	− 2.44747e10i	− 0.327122i	−0.986533	−	0.163561i	$-0.947702\pi$
$523$	− 2.44747e10i	− 0.327122i	0.986533	−	0.163561i	$-0.0522981\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.83110e10	−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
$541$	6.27323e10	0.732322	0.366161	−	0.930551i	$-0.380672\pi$
$541$	6.27323e10	0.732322	0.366161	+	0.930551i	$0.380672\pi$
$542$	0	0
$543$	8.52668e10i	0.980801i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 6.18266e10i	− 0.690599i	−0.938492	−	0.345300i	$-0.887777\pi$
$547$	− 6.18266e10i	− 0.690599i	0.938492	−	0.345300i	$-0.112223\pi$
$548$	0	0
$549$	1.56326e11	1.72084
$550$	0	0
$551$	0	0
$552$	0	0
$553$	7.61903e10i	0.814703i
$554$	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
$559$	1.25041e11	1.28058
$560$	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
$567$	1.73650e11i	1.68013i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−1.94939e11	−1.83381	−0.916907	−	0.399102i	$-0.869322\pi$
$571$	−1.94939e11	−1.83381	−0.916907	+	0.399102i	$0.869322\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.54299e11i	− 1.39206i	−0.718012	−	0.696031i	$-0.754948\pi$
$577$	− 1.54299e11i	− 1.39206i	0.718012	−	0.696031i	$-0.245052\pi$
$578$	0	0
$579$	−1.88462e11	−1.67691
$580$	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
$589$	−4.67778e11	−3.88668
$590$	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
$597$	1.40513e11i	1.10616i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−6.22607e10	−0.477217	−0.238608	−	0.971116i	$-0.576691\pi$
$601$	−6.22607e10	−0.477217	−0.238608	+	0.971116i	$0.576691\pi$
$602$	0	0
$603$	3.55698e10i	0.269037i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.65989e11i	− 1.95933i	−0.200634	−	0.979666i	$-0.564300\pi$
$607$	− 2.65989e11i	− 1.95933i	0.200634	−	0.979666i	$-0.435700\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.57998e10i	0.111894i	0.998434	+	0.0559472i	$0.0178179\pi$
$613$	1.57998e10i	0.111894i	−0.998434	+	0.0559472i	$0.982182\pi$
$614$	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
$619$	−2.25533e11	−1.53620	−0.768100	−	0.640330i	$-0.778797\pi$
$619$	−2.25533e11	−1.53620	−0.768100	+	0.640330i	$0.778797\pi$
$620$	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
$631$	−2.00069e11	−1.26201	−0.631004	−	0.775780i	$-0.717357\pi$
$631$	−2.00069e11	−1.26201	−0.631004	+	0.775780i	$0.717357\pi$
$632$	0	0
$633$	1.48806e10i	0.0926840i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.76262e11i	− 2.28525i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 1.88544e11i	− 1.10298i	−0.834181	−	0.551490i	$-0.814059\pi$
$643$	− 1.88544e11i	− 1.10298i	0.834181	−	0.551490i	$-0.185941\pi$
$644$	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
$651$	−5.91231e11	−3.29180
$652$	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
$657$	1.06087e11i	0.569376i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	3.56009e11	1.86490	0.932449	−	0.361302i	$-0.117668\pi$
$661$	3.56009e11	1.86490	0.932449	+	0.361302i	$0.117668\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	3.82913e11	1.91159
$670$	0	0
$671$	0	0
$672$	0	0
$673$	2.28934e11i	1.11597i	0.829853	+	0.557983i	$0.188424\pi$
$673$	2.28934e11i	1.11597i	−0.829853	+	0.557983i	$0.811576\pi$
$674$	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
$679$	−7.13647e11	−3.35741
$680$	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
$687$	4.34061e11i	1.94860i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.55801e11	0.683374	0.341687	−	0.939814i	$-0.389002\pi$
$691$	1.55801e11	0.683374	0.341687	+	0.939814i	$0.389002\pi$
$692$	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
$703$	1.30174e11i	0.532968i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.87686e11	1.92999	0.964995	−	0.262268i	$-0.0844703\pi$
$709$	4.87686e11	1.92999	0.964995	+	0.262268i	$0.0844703\pi$
$710$	0	0
$711$	−1.23918e11	−0.484904
$712$	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
$721$	1.79307e11	0.663524
$722$	0	0
$723$	4.50828e11i	1.64990i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.21500e11i	0.434951i	0.976066	+	0.217475i	$0.0697822\pi$
$727$	1.21500e11i	0.434951i	−0.976066	+	0.217475i	$0.930218\pi$
$728$	0	0
$729$	−2.82430e11	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	5.77330e11i	1.99990i	0.0100913	+	0.999949i	$0.496788\pi$
$733$	5.77330e11i	1.99990i	−0.0100913	+	0.999949i	$0.503212\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	4.35662e11	1.46074	0.730368	−	0.683054i	$-0.239349\pi$
$739$	4.35662e11	1.46074	0.730368	+	0.683054i	$0.239349\pi$
$740$	0	0
$741$	7.49799e11	2.48698
$742$	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
$751$	−5.35831e11	−1.68449	−0.842244	−	0.539097i	$-0.818766\pi$
$751$	−5.35831e11	−1.68449	−0.842244	+	0.539097i	$0.818766\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 4.59764e11i	− 1.40008i	−0.714106	−	0.700038i	$-0.753167\pi$
$757$	− 4.59764e11i	− 1.40008i	0.714106	−	0.700038i	$-0.246833\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 8.19631e11i	− 2.41836i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.70500e11	−1.91732	−0.958658	−	0.284562i	$-0.908152\pi$
$769$	−6.70500e11	−1.91732	−0.958658	+	0.284562i	$0.908152\pi$
$770$	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
$777$	1.64528e11i	0.451393i
$778$	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
$787$	6.75420e11i	1.76066i	0.474364	+	0.880329i	$0.342678\pi$
$787$	6.75420e11i	1.76066i	−0.474364	+	0.880329i	$0.657322\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 8.53133e11i	− 2.15737i
$794$	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
$811$	8.88545e10	0.205398	0.102699	−	0.994712i	$-0.467252\pi$
$811$	8.88545e10	0.205398	0.102699	+	0.994712i	$0.467252\pi$
$812$	0	0
$813$	2.41954e11i	0.553824i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.02823e11i	− 2.02635i
$818$	0	0
$819$	9.47680e11	2.10633
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	8.62186e11i	1.87932i	0.342105	+	0.939662i	$0.388860\pi$
$823$	8.62186e11i	1.87932i	−0.342105	+	0.939662i	$0.611140\pi$
$824$	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
$829$	−4.11968e11	−0.872258	−0.436129	−	0.899884i	$-0.643651\pi$
$829$	−4.11968e11	−0.872258	−0.436129	+	0.899884i	$0.643651\pi$
$830$	0	0
$831$	6.01674e11	1.26170
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 9.61593e11i	− 1.95925i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	5.00246e11	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.64724e11i	1.68013i
$848$	0	0
$849$	8.39948e11	1.61667
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 6.14949e11i	− 1.16156i	−0.814059	−	0.580782i	$-0.802747\pi$
$853$	− 6.14949e11i	− 1.16156i	0.814059	−	0.580782i	$-0.197253\pi$
$854$	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
$859$	8.02752e11	1.47438	0.737189	−	0.675686i	$-0.236153\pi$
$859$	8.02752e11	1.47438	0.737189	+	0.675686i	$0.236153\pi$
$860$	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
$867$	5.65036e11i	1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1.94119e11	0.337284
$872$	0	0
$873$	− 1.16069e12i	− 1.99830i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.10825e11i	0.356389i	0.983995	+	0.178194i	$0.0570256\pi$
$877$	2.10825e11i	0.356389i	−0.983995	+	0.178194i	$0.942974\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 5.72878e11i	− 0.942366i	−0.882035	−	0.471183i	$-0.843827\pi$
$883$	− 5.72878e11i	− 0.942366i	0.882035	−	0.471183i	$-0.156173\pi$
$884$	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
$889$	1.61587e12	2.58701
$890$	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
$903$	− 1.14109e12i	− 1.71620i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.20490e12i	1.78042i	0.455547	+	0.890212i	$0.349444\pi$
$907$	1.20490e12i	1.78042i	−0.455547	+	0.890212i	$0.650556\pi$
$908$	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
$919$	5.44534e11	0.763419	0.381709	−	0.924282i	$-0.375336\pi$
$919$	5.44534e11	0.763419	0.381709	+	0.924282i	$0.375336\pi$
$920$	0	0
$921$	−1.14417e12	−1.59020
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.91630e11i	0.394923i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−2.71668e12	−3.61610
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.43879e12i	− 1.86655i	−0.359168	−	0.933273i	$-0.616939\pi$
$937$	− 1.43879e12i	− 1.86655i	0.359168	−	0.933273i	$-0.383061\pi$
$938$	0	0
$939$	−9.47753e11	−1.21908
$940$	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
$949$	5.78957e11	0.713808
$950$	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
$961$	2.42106e12	2.83865
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.51552e12i	− 1.73322i	−0.498984	−	0.866611i	$-0.666293\pi$
$967$	− 1.51552e12i	− 1.73322i	0.498984	−	0.866611i	$-0.333707\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	− 2.86184e12i	− 3.19297i
$974$	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
$981$	1.33307e12	1.43938
$982$	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
$991$	1.92066e12	1.99139	0.995693	−	0.0927105i	$-0.0295531\pi$
$991$	1.92066e12	1.99139	0.995693	+	0.0927105i	$0.0295531\pi$
$992$	0	0
$993$	1.31621e12i	1.35372i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.81390e11i	0.588420i	0.955741	+	0.294210i	$0.0950565\pi$
$997$	5.81390e11i	0.588420i	−0.955741	+	0.294210i	$0.904944\pi$
$998$	0	0
$999$	−2.67592e11	−0.268665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.9.b.b.149.1		2
3.2	odd	2	CM	300.9.b.b.149.1		2
5.2	odd	4		300.9.g.a.101.1		1
5.3	odd	4		12.9.c.a.5.1	✓	1
5.4	even	2	inner	300.9.b.b.149.2		2
15.2	even	4		300.9.g.a.101.1		1
15.8	even	4		12.9.c.a.5.1	✓	1
15.14	odd	2	inner	300.9.b.b.149.2		2
20.3	even	4		48.9.e.a.17.1		1
40.3	even	4		192.9.e.b.65.1		1
40.13	odd	4		192.9.e.a.65.1		1
45.13	odd	12		324.9.g.a.269.1		2
45.23	even	12		324.9.g.a.269.1		2
45.38	even	12		324.9.g.a.53.1		2
45.43	odd	12		324.9.g.a.53.1		2
60.23	odd	4		48.9.e.a.17.1		1
120.53	even	4		192.9.e.a.65.1		1
120.83	odd	4		192.9.e.b.65.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.9.c.a.5.1	✓	1	5.3	odd	4
12.9.c.a.5.1	✓	1	15.8	even	4
48.9.e.a.17.1		1	20.3	even	4
48.9.e.a.17.1		1	60.23	odd	4
192.9.e.a.65.1		1	40.13	odd	4
192.9.e.a.65.1		1	120.53	even	4
192.9.e.b.65.1		1	40.3	even	4
192.9.e.b.65.1		1	120.83	odd	4
300.9.b.b.149.1		2	1.1	even	1	trivial
300.9.b.b.149.1		2	3.2	odd	2	CM
300.9.b.b.149.2		2	5.4	even	2	inner
300.9.b.b.149.2		2	15.14	odd	2	inner
300.9.g.a.101.1		1	5.2	odd	4
300.9.g.a.101.1		1	15.2	even	4
324.9.g.a.53.1		2	45.38	even	12
324.9.g.a.53.1		2	45.43	odd	12
324.9.g.a.269.1		2	45.13	odd	12
324.9.g.a.269.1		2	45.23	even	12

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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-81.0000i q^{3} +4034.00i q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q-81.0000i q^{3} +4034.00i q^{7} -6561.00 q^{9} +35806.0i q^{13} +258526. q^{19} +326754. q^{21} +531441. i q^{27} -1.80941e6 q^{31} +503522. i q^{37} +2.90029e6 q^{39} -3.49219e6i q^{43} -1.05084e7 q^{49} -2.09406e7i q^{57} -2.38265e7 q^{61} -2.64671e7i q^{63} -5.42141e6i q^{67} -1.61693e7i q^{73} +1.88870e7 q^{79} +4.30467e7 q^{81} -1.44441e8 q^{91} +1.46562e8i q^{93} +1.76908e8i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13122 q^{9}+O(q^{10}) \) 2 * q - 13122 * q^9	\( 2 q - 13122 q^{9} + 517052 q^{19} + 653508 q^{21} - 3618812 q^{31} + 5800572 q^{39} - 21016710 q^{49} - 47653052 q^{61} + 37774076 q^{79} + 86093442 q^{81} - 288882808 q^{91}+O(q^{100}) \) 2 * q - 13122 * q^9 + 517052 * q^19 + 653508 * q^21 - 3618812 * q^31 + 5800572 * q^39 - 21016710 * q^49 - 47653052 * q^61 + 37774076 * q^79 + 86093442 * q^81 - 288882808 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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