LMFDB - Embedded newform 384.6.f.a.191.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(191,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("384.191");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			191.2
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	384.191
Dual form			384.6.f.a.191.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5885i q^{3} +31.1127 q^{5} -93.0806i q^{7} -243.000 q^{9} +O(q^{10})$	$q-15.5885i q^{3} +31.1127 q^{5} -93.0806i q^{7} -243.000 q^{9} +308.305i q^{11} -484.999i q^{15} -1450.98 q^{21} -2157.00 q^{25} +3788.00i q^{27} -8793.58 q^{29} +8137.20i q^{31} +4806.00 q^{33} -2895.99i q^{35} -7560.39 q^{45} +8143.00 q^{49} -14778.5 q^{53} +9592.20i q^{55} -29004.9i q^{59} +22618.6i q^{63} -90706.0 q^{73} +33624.3i q^{75} +28697.2 q^{77} +109624. i q^{79} +59049.0 q^{81} -952.628i q^{83} +137078. i q^{87} +126846. q^{93} +90242.0 q^{97} -74918.1i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 972 q^{9}+O(q^{10}) $ 4 * q - 972 * q^9	$ 4 q - 972 q^{9} - 8628 q^{25} + 19224 q^{33} + 32572 q^{49} - 362824 q^{73} + 236196 q^{81} + 360968 q^{97}+O(q^{100}) $ 4 * q - 972 * q^9 - 8628 * q^25 + 19224 * q^33 + 32572 * q^49 - 362824 * q^73 + 236196 * q^81 + 360968 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	− 15.5885i	− 1.00000i
$3$	− 15.5885i	− 1.00000i
$4$	0	0
$5$	31.1127	0.556561	0.278280	−	0.960500i	$-0.410236\pi$
$5$	31.1127	0.556561	0.278280	+	0.960500i	$0.410236\pi$
$6$	0	0
$7$	− 93.0806i	− 0.717983i	−0.933341	−	0.358991i	$-0.883121\pi$
$7$	− 93.0806i	− 0.717983i	0.933341	−	0.358991i	$-0.116879\pi$
$8$	0	0
$9$	−243.000	−1.00000
$10$	0	0
$11$	308.305i	0.768244i	0.923282	+	0.384122i	$0.125496\pi$
$11$	308.305i	0.768244i	−0.923282	+	0.384122i	$0.874504\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	− 484.999i	− 0.556561i
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1450.98	−0.717983
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−2157.00	−0.690240
$26$	0	0
$27$	3788.00i	1.00000i
$28$	0	0
$29$	−8793.58	−1.94165	−0.970824	−	0.239791i	$-0.922921\pi$
$29$	−8793.58	−1.94165	−0.970824	+	0.239791i	$0.922921\pi$
$30$	0	0
$31$	8137.20i	1.52080i	0.649457	+	0.760398i	$0.274996\pi$
$31$	8137.20i	1.52080i	−0.649457	+	0.760398i	$0.725004\pi$
$32$	0	0
$33$	4806.00	0.768244
$34$	0	0
$35$	− 2895.99i	− 0.399601i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−7560.39	−0.556561
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	8143.00	0.484501
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−14778.5	−0.722673	−0.361336	−	0.932436i	$-0.617679\pi$
$53$	−14778.5	−0.722673	−0.361336	+	0.932436i	$0.617679\pi$
$54$	0	0
$55$	9592.20i	0.427574i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 29004.9i	− 1.08478i	−0.840127	−	0.542390i	$-0.817520\pi$
$59$	− 29004.9i	− 1.08478i	0.840127	−	0.542390i	$-0.182480\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	22618.6i	0.717983i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	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
$73$	−90706.0	−1.99218	−0.996091	−	0.0883368i	$-0.971845\pi$
$73$	−90706.0	−1.99218	−0.996091	+	0.0883368i	$0.971845\pi$
$74$	0	0
$75$	33624.3i	0.690240i
$76$	0	0
$77$	28697.2	0.551586
$78$	0	0
$79$	109624.i	1.97624i	0.153683	+	0.988120i	$0.450886\pi$
$79$	109624.i	1.97624i	−0.153683	+	0.988120i	$0.549114\pi$
$80$	0	0
$81$	59049.0	1.00000
$82$	0	0
$83$	− 952.628i	− 0.0151785i	−0.999971	−	0.00758924i	$-0.997584\pi$
$83$	− 952.628i	− 0.0151785i	0.999971	−	0.00758924i	$-0.00241575\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	137078.i	1.94165i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	126846.	1.52080
$94$	0	0
$95$	0	0
$96$	0	0
$97$	90242.0	0.973822	0.486911	−	0.873452i	$-0.338124\pi$
$97$	90242.0	0.973822	0.486911	+	0.873452i	$0.338124\pi$
$98$	0	0
$99$	− 74918.1i	− 0.768244i
$100$	0	0
$101$	132831.	1.29568	0.647839	−	0.761777i	$-0.275673\pi$
$101$	132831.	1.29568	0.647839	+	0.761777i	$0.275673\pi$
$102$	0	0
$103$	169588.i	1.57508i	0.616265	+	0.787539i	$0.288645\pi$
$103$	169588.i	1.57508i	−0.616265	+	0.787539i	$0.711355\pi$
$104$	0	0
$105$	−45144.0	−0.399601
$106$	0	0
$107$	229583.i	1.93857i	0.245945	+	0.969284i	$0.420902\pi$
$107$	229583.i	1.93857i	−0.245945	+	0.969284i	$0.579098\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	65999.0	0.409802
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−164337.	−0.940721
$126$	0	0
$127$	323240.i	1.77834i	0.457575	+	0.889171i	$0.348718\pi$
$127$	323240.i	1.77834i	−0.457575	+	0.889171i	$0.651282\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 270508.i	− 1.37722i	−0.725134	−	0.688608i	$-0.758222\pi$
$131$	− 270508.i	− 1.37722i	0.725134	−	0.688608i	$-0.241778\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	117855.i	0.556561i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−273592.	−1.08065
$146$	0	0
$147$	− 126937.i	− 0.484501i
$148$	0	0
$149$	−297293.	−1.09703	−0.548516	−	0.836140i	$-0.684807\pi$
$149$	−297293.	−1.09703	−0.548516	+	0.836140i	$0.684807\pi$
$150$	0	0
$151$	514515.i	1.83635i	0.396173	+	0.918176i	$0.370338\pi$
$151$	514515.i	1.83635i	−0.396173	+	0.918176i	$0.629662\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	253170.i	0.846416i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	230375.i	0.722673i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	149528.	0.427574
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	371293.	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−708230.	−1.79911	−0.899557	−	0.436803i	$-0.856111\pi$
$173$	−708230.	−1.79911	−0.899557	+	0.436803i	$0.856111\pi$
$174$	0	0
$175$	200775.i	0.495581i
$176$	0	0
$177$	−452142.	−1.08478
$178$	0	0
$179$	− 492516.i	− 1.14891i	−0.818535	−	0.574457i	$-0.805213\pi$
$179$	− 492516.i	− 1.14891i	0.818535	−	0.574457i	$-0.194787\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	352589.	0.717983
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	237094.	0.458171	0.229085	−	0.973406i	$-0.426427\pi$
$193$	237094.	0.458171	0.229085	+	0.973406i	$0.426427\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−523895.	−0.961787	−0.480894	−	0.876779i	$-0.659688\pi$
$197$	−523895.	−0.961787	−0.480894	+	0.876779i	$0.659688\pi$
$198$	0	0
$199$	− 955179.i	− 1.70983i	−0.518772	−	0.854913i	$-0.673610\pi$
$199$	− 955179.i	− 1.70983i	0.518772	−	0.854913i	$-0.326390\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	818512.i	1.39407i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	757416.	1.09191
$218$	0	0
$219$	1.41397e6i	1.99218i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 800410.i	− 1.07783i	−0.842360	−	0.538915i	$-0.818834\pi$
$223$	− 800410.i	− 1.07783i	0.842360	−	0.538915i	$-0.181166\pi$
$224$	0	0
$225$	524151.	0.690240
$226$	0	0
$227$	− 1.52485e6i	− 1.96410i	−0.188626	−	0.982049i	$-0.560403\pi$
$227$	− 1.52485e6i	− 1.96410i	0.188626	−	0.982049i	$-0.439597\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	− 447345.i	− 0.551586i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.70888e6	1.97624
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−1.79905e6	−1.99527	−0.997633	−	0.0687692i	$-0.978093\pi$
$241$	−1.79905e6	−1.99527	−0.997633	+	0.0687692i	$0.978093\pi$
$242$	0	0
$243$	− 920483.i	− 1.00000i
$244$	0	0
$245$	253351.	0.269654
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−14850.0	−0.0151785
$250$	0	0
$251$	1.24486e6i	1.24720i	0.781742	+	0.623602i	$0.214331\pi$
$251$	1.24486e6i	1.24720i	−0.781742	+	0.623602i	$0.785669\pi$
$252$	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
$259$	0	0
$260$	0	0
$261$	2.13684e6	1.94165
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−459800.	−0.402211
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−97413.9	−0.0820805	−0.0410403	−	0.999157i	$-0.513067\pi$
$269$	−97413.9	−0.0820805	−0.0410403	+	0.999157i	$0.513067\pi$
$270$	0	0
$271$	− 2.10154e6i	− 1.73826i	−0.494585	−	0.869129i	$-0.664680\pi$
$271$	− 2.10154e6i	− 1.73826i	0.494585	−	0.869129i	$-0.335320\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 665014.i	− 0.530272i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	− 1.97734e6i	− 1.52080i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.41986e6	1.00000
$290$	0	0
$291$	− 1.40673e6i	− 0.973822i
$292$	0	0
$293$	−2.49248e6	−1.69614	−0.848072	−	0.529881i	$-0.822237\pi$
$293$	−2.49248e6	−1.69614	−0.848072	+	0.529881i	$0.822237\pi$
$294$	0	0
$295$	− 902421.i	− 0.603746i
$296$	0	0
$297$	−1.16786e6	−0.768244
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 2.07064e6i	− 1.29568i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	2.64361e6	1.57508
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−579394.	−0.334282	−0.167141	−	0.985933i	$-0.553454\pi$
$313$	−579394.	−0.334282	−0.167141	+	0.985933i	$0.553454\pi$
$314$	0	0
$315$	703725.i	0.399601i
$316$	0	0
$317$	−2.94983e6	−1.64873	−0.824363	−	0.566062i	$-0.808466\pi$
$317$	−2.94983e6	−1.64873	−0.824363	+	0.566062i	$0.808466\pi$
$318$	0	0
$319$	− 2.71111e6i	− 1.49166i
$320$	0	0
$321$	3.57885e6	1.93857
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−295658.	−0.141813	−0.0709063	−	0.997483i	$-0.522589\pi$
$337$	−295658.	−0.141813	−0.0709063	+	0.997483i	$0.522589\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.50874e6	−1.16834
$342$	0	0
$343$	− 2.32236e6i	− 1.06585i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.75539e6i	0.782617i	0.920259	+	0.391309i	$0.127978\pi$
$347$	1.75539e6i	0.782617i	−0.920259	+	0.391309i	$0.872022\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	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
$361$	−2.47610e6	−1.00000
$362$	0	0
$363$	− 1.02882e6i	− 0.409802i
$364$	0	0
$365$	−2.82211e6	−1.10877
$366$	0	0
$367$	3.08626e6i	1.19610i	0.801458	+	0.598051i	$0.204058\pi$
$367$	3.08626e6i	1.19610i	−0.801458	+	0.598051i	$0.795942\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.37559e6i	0.518867i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	2.56176e6i	0.940721i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	5.03881e6	1.77834
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	892848.	0.306991
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.88438e6	−1.97163	−0.985817	−	0.167823i	$-0.946326\pi$
$389$	−5.88438e6	−1.97163	−0.985817	+	0.167823i	$0.946326\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4.21681e6	−1.37722
$394$	0	0
$395$	3.41071e6i	1.09990i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.83717e6	0.556561
$406$	0	0
$407$	0	0
$408$	0	0
$409$	6.41905e6	1.89742	0.948708	−	0.316154i	$-0.102392\pi$
$409$	6.41905e6	1.89742	0.948708	+	0.316154i	$0.102392\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.69980e6	−0.778854
$414$	0	0
$415$	− 29638.8i	− 0.00844774i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 6.89227e6i	− 1.91791i	−0.283566	−	0.958953i	$-0.591517\pi$
$419$	− 6.89227e6i	− 1.91791i	0.283566	−	0.958953i	$-0.408483\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	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		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−7.72129e6	−1.97911	−0.989556	−	0.144149i	$-0.953955\pi$
$433$	−7.72129e6	−1.97911	−0.989556	+	0.144149i	$0.953955\pi$
$434$	0	0
$435$	4.26488e6i	1.08065i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	161789.i	0.0400670i	0.999799	+	0.0200335i	$0.00637729\pi$
$439$	161789.i	0.0400670i	−0.999799	+	0.0200335i	$0.993623\pi$
$440$	0	0
$441$	−1.97875e6	−0.484501
$442$	0	0
$443$	− 7.07563e6i	− 1.71299i	−0.516152	−	0.856497i	$-0.672636\pi$
$443$	− 7.07563e6i	− 1.71299i	0.516152	−	0.856497i	$-0.327364\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.63434e6i	1.09703i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	8.02050e6	1.83635
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.46604e6	1.44826	0.724132	−	0.689661i	$-0.242240\pi$
$457$	6.46604e6	1.44826	0.724132	+	0.689661i	$0.242240\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.19132e6	1.35685	0.678424	−	0.734671i	$-0.262663\pi$
$461$	6.19132e6	1.35685	0.678424	+	0.734671i	$0.262663\pi$
$462$	0	0
$463$	9.17742e6i	1.98961i	0.101795	+	0.994805i	$0.467541\pi$
$463$	9.17742e6i	1.98961i	−0.101795	+	0.994805i	$0.532459\pi$
$464$	0	0
$465$	3.94654e6	0.846416
$466$	0	0
$467$	8.31289e6i	1.76384i	0.471396	+	0.881921i	$0.343750\pi$
$467$	8.31289e6i	1.76384i	−0.471396	+	0.881921i	$0.656250\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.59118e6	0.722673
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.80767e6	0.541991
$486$	0	0
$487$	1.02000e7i	1.94885i	0.224714	+	0.974425i	$0.427855\pi$
$487$	1.02000e7i	1.94885i	−0.224714	+	0.974425i	$0.572145\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 9.56252e6i	− 1.79006i	−0.446002	−	0.895032i	$-0.647153\pi$
$491$	− 9.56252e6i	− 1.79006i	0.446002	−	0.895032i	$-0.352847\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 2.33091e6i	− 0.427574i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	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
$505$	4.13274e6	0.721124
$506$	0	0
$507$	− 5.78789e6i	− 1.00000i
$508$	0	0
$509$	−3.60655e6	−0.617018	−0.308509	−	0.951221i	$-0.599830\pi$
$509$	−3.60655e6	−0.617018	−0.308509	+	0.951221i	$0.599830\pi$
$510$	0	0
$511$	8.44297e6i	1.43035i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.27634e6i	0.876627i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1.10402e7i	1.79911i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	3.12977e6	0.495581
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−6.43634e6	−1.00000
$530$	0	0
$531$	7.04820e6i	1.08478i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	7.14296e6i	1.07893i
$536$	0	0
$537$	−7.67756e6	−1.14891
$538$	0	0
$539$	2.51053e6i	0.372214i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.02039e7	1.41891
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.46264e7	−1.99756	−0.998781	−	0.0493530i	$-0.984284\pi$
$557$	−1.46264e7	−1.99756	−0.998781	+	0.0493530i	$0.984284\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.50222e7i	1.99739i	0.0510615	+	0.998696i	$0.483740\pi$
$563$	1.50222e7i	1.99739i	−0.0510615	+	0.998696i	$0.516260\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 5.49632e6i	− 0.717983i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.58140e7	−1.97744	−0.988720	−	0.149775i	$-0.952145\pi$
$577$	−1.58140e7	−1.97744	−0.988720	+	0.149775i	$0.952145\pi$
$578$	0	0
$579$	− 3.69593e6i	− 0.458171i
$580$	0	0
$581$	−88671.2	−0.0108979
$582$	0	0
$583$	− 4.55630e6i	− 0.555189i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.61487e7i	− 1.93438i	−0.254052	−	0.967191i	$-0.581764\pi$
$587$	− 1.61487e7i	− 1.93438i	0.254052	−	0.967191i	$-0.418236\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	8.16672e6i	0.961787i
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.48898e7	−1.70983
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−3.58800e6	−0.405197	−0.202599	−	0.979262i	$-0.564939\pi$
$601$	−3.58800e6	−0.405197	−0.202599	+	0.979262i	$0.564939\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.05341e6	0.228080
$606$	0	0
$607$	− 1.22863e7i	− 1.35347i	−0.736225	−	0.676736i	$-0.763394\pi$
$607$	− 1.22863e7i	− 1.35347i	0.736225	−	0.676736i	$-0.236606\pi$
$608$	0	0
$609$	1.27593e7	1.39407
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.62765e6	0.166671
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	9.38478e6i	0.938320i	0.883113	+	0.469160i	$0.155443\pi$
$631$	9.38478e6i	0.938320i	−0.883113	+	0.469160i	$0.844557\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.00569e7i	0.989755i
$636$	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
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	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
$649$	8.94236e6	0.833375
$650$	0	0
$651$	− 1.18069e7i	− 1.09191i
$652$	0	0
$653$	3.42790e6	0.314591	0.157295	−	0.987552i	$-0.449723\pi$
$653$	3.42790e6	0.314591	0.157295	+	0.987552i	$0.449723\pi$
$654$	0	0
$655$	− 8.41624e6i	− 0.766505i
$656$	0	0
$657$	2.20416e7	1.99218
$658$	0	0
$659$	1.40567e7i	1.26087i	0.776242	+	0.630435i	$0.217123\pi$
$659$	1.40567e7i	1.26087i	−0.776242	+	0.630435i	$0.782877\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−1.24772e7	−1.07783
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−9.82461e6	−0.836137	−0.418069	−	0.908415i	$-0.637293\pi$
$673$	−9.82461e6	−0.836137	−0.418069	+	0.908415i	$0.637293\pi$
$674$	0	0
$675$	− 8.17071e6i	− 0.690240i
$676$	0	0
$677$	−6.40534e6	−0.537119	−0.268559	−	0.963263i	$-0.586548\pi$
$677$	−6.40534e6	−0.537119	−0.268559	+	0.963263i	$0.586548\pi$
$678$	0	0
$679$	− 8.39978e6i	− 0.699187i
$680$	0	0
$681$	−2.37701e7	−1.96410
$682$	0	0
$683$	− 2.08857e7i	− 1.71316i	−0.516018	−	0.856578i	$-0.672586\pi$
$683$	− 2.08857e7i	− 1.71316i	0.516018	−	0.856578i	$-0.327414\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	−6.97342e6	−0.551586
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.66879e7	1.28265	0.641323	−	0.767271i	$-0.278386\pi$
$701$	1.66879e7	1.28265	0.641323	+	0.767271i	$0.278386\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.23640e7i	− 0.930275i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	− 2.66387e7i	− 1.97624i
$712$	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
$721$	1.57854e7	1.13088
$722$	0	0
$723$	2.80444e7i	1.99527i
$724$	0	0
$725$	1.89678e7	1.34020
$726$	0	0
$727$	− 2.80125e7i	− 1.96569i	−0.184421	−	0.982847i	$-0.559041\pi$
$727$	− 2.80125e7i	− 1.96569i	0.184421	−	0.982847i	$-0.440959\pi$
$728$	0	0
$729$	−1.43489e7	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	− 3.94935e6i	− 0.269654i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−9.24959e6	−0.610565
$746$	0	0
$747$	231489.i	0.0151785i
$748$	0	0
$749$	2.13698e7	1.39186
$750$	0	0
$751$	− 1.87911e7i	− 1.21577i	−0.794025	−	0.607885i	$-0.792018\pi$
$751$	− 1.87911e7i	− 1.21577i	0.794025	−	0.607885i	$-0.207982\pi$
$752$	0	0
$753$	1.94055e7	1.24720
$754$	0	0
$755$	1.60080e7i	1.02204i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−2.11786e7	−1.29146	−0.645730	−	0.763566i	$-0.723447\pi$
$769$	−2.11786e7	−1.29146	−0.645730	+	0.763566i	$0.723447\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.21977e7	1.93810	0.969050	−	0.246866i	$-0.0794008\pi$
$773$	3.21977e7	1.93810	0.969050	+	0.246866i	$0.0794008\pi$
$774$	0	0
$775$	− 1.75520e7i	− 1.04971i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 3.33100e7i	− 1.94165i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	7.16757e6i	0.402211i
$796$	0	0
$797$	425031.	0.0237014	0.0118507	−	0.999930i	$-0.496228\pi$
$797$	425031.	0.0237014	0.0118507	+	0.999930i	$0.496228\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2.79651e7i	− 1.53048i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.51853e6i	0.0820805i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	−3.27598e7	−1.73826
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.72747e7	1.92999	0.964997	−	0.262261i	$-0.0844681\pi$
$821$	3.72747e7	1.92999	0.964997	+	0.262261i	$0.0844681\pi$
$822$	0	0
$823$	2.38743e6i	0.122866i	0.998111	+	0.0614330i	$0.0195671\pi$
$823$	2.38743e6i	0.122866i	−0.998111	+	0.0614330i	$0.980433\pi$
$824$	0	0
$825$	−1.03665e7	−0.530272
$826$	0	0
$827$	3.10182e7i	1.57708i	0.614985	+	0.788539i	$0.289162\pi$
$827$	3.10182e7i	1.57708i	−0.614985	+	0.788539i	$0.710838\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.08237e7	−1.52080
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	5.68159e7	2.77000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.15519e7	0.556561
$846$	0	0
$847$	− 6.14323e6i	− 0.294231i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	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
$865$	−2.20349e7	−1.00132
$866$	0	0
$867$	− 2.21334e7i	− 1.00000i
$868$	0	0
$869$	−3.37978e7	−1.51823
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.19288e7	−0.973822
$874$	0	0
$875$	1.52966e7i	0.675422i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	3.88539e7i	1.69614i
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	−1.40674e7	−0.603746
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	3.00873e7	1.27682
$890$	0	0
$891$	1.82051e7i	0.768244i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 1.53235e7i	− 0.639440i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 7.15552e7i	− 2.95285i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−3.22780e7	−1.29568
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	293700.	0.0116608
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.51791e7	−0.988818
$918$	0	0
$919$	− 6.92925e6i	− 0.270643i	−0.990802	−	0.135322i	$-0.956793\pi$
$919$	− 6.92925e6i	− 0.270643i	0.990802	−	0.135322i	$-0.0432068\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.12099e7i	− 1.57508i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.13194e6	−0.116537	−0.0582686	−	0.998301i	$-0.518558\pi$
$937$	−3.13194e6	−0.116537	−0.0582686	+	0.998301i	$0.518558\pi$
$938$	0	0
$939$	9.03186e6i	0.334282i
$940$	0	0
$941$	5.31421e7	1.95643	0.978217	−	0.207586i	$-0.0665608\pi$
$941$	5.31421e7	1.95643	0.978217	+	0.207586i	$0.0665608\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.09700e7	0.399601
$946$	0	0
$947$	4.96034e7i	1.79737i	0.438600	+	0.898683i	$0.355475\pi$
$947$	4.96034e7i	1.79737i	−0.438600	+	0.898683i	$0.644525\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	4.59832e7i	1.64873i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.22619e7	−1.49166
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.75850e7	−1.31282
$962$	0	0
$963$	− 5.57888e7i	− 1.93857i
$964$	0	0
$965$	7.37663e6	0.255000
$966$	0	0
$967$	2.23393e7i	0.768250i	0.923281	+	0.384125i	$0.125497\pi$
$967$	2.23393e7i	0.768250i	−0.923281	+	0.384125i	$0.874503\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.61292e7i	0.548989i	0.961589	+	0.274495i	$0.0885105\pi$
$971$	1.61292e7i	0.548989i	−0.961589	+	0.274495i	$0.911489\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−1.62998e7	−0.535293
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	5.62747e7i	1.82024i	0.414343	+	0.910121i	$0.364011\pi$
$991$	5.62747e7i	1.82024i	−0.414343	+	0.910121i	$0.635989\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2.97182e7i	− 0.951622i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.f.a.191.2	yes	4
3.2	odd	2	inner	384.6.f.a.191.3	yes	4
4.3	odd	2	inner	384.6.f.a.191.4	yes	4
8.3	odd	2	inner	384.6.f.a.191.1	✓	4
8.5	even	2	inner	384.6.f.a.191.3	yes	4
12.11	even	2	inner	384.6.f.a.191.1	✓	4
24.5	odd	2	CM	384.6.f.a.191.2	yes	4
24.11	even	2	inner	384.6.f.a.191.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.f.a.191.1	✓	4	8.3	odd	2	inner
384.6.f.a.191.1	✓	4	12.11	even	2	inner
384.6.f.a.191.2	yes	4	1.1	even	1	trivial
384.6.f.a.191.2	yes	4	24.5	odd	2	CM
384.6.f.a.191.3	yes	4	3.2	odd	2	inner
384.6.f.a.191.3	yes	4	8.5	even	2	inner
384.6.f.a.191.4	yes	4	4.3	odd	2	inner
384.6.f.a.191.4	yes	4	24.11	even	2	inner

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.5885i q^{3} +31.1127 q^{5} -93.0806i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q-15.5885i q^{3} +31.1127 q^{5} -93.0806i q^{7} -243.000 q^{9} +308.305i q^{11} -484.999i q^{15} -1450.98 q^{21} -2157.00 q^{25} +3788.00i q^{27} -8793.58 q^{29} +8137.20i q^{31} +4806.00 q^{33} -2895.99i q^{35} -7560.39 q^{45} +8143.00 q^{49} -14778.5 q^{53} +9592.20i q^{55} -29004.9i q^{59} +22618.6i q^{63} -90706.0 q^{73} +33624.3i q^{75} +28697.2 q^{77} +109624. i q^{79} +59049.0 q^{81} -952.628i q^{83} +137078. i q^{87} +126846. q^{93} +90242.0 q^{97} -74918.1i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 972 q^{9}+O(q^{10}) \) 4 * q - 972 * q^9	\( 4 q - 972 q^{9} - 8628 q^{25} + 19224 q^{33} + 32572 q^{49} - 362824 q^{73} + 236196 q^{81} + 360968 q^{97}+O(q^{100}) \) 4 * q - 972 * q^9 - 8628 * q^25 + 19224 * q^33 + 32572 * q^49 - 362824 * q^73 + 236196 * q^81 + 360968 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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