LMFDB - Embedded newform 384.4.f.a.191.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,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, 4, names="a")

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

chi := DirichletCharacter("384.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
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:	$22.6567334422$
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}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			191.3
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	384.191
Dual form			384.4.f.a.191.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615i q^{3} -19.7990 q^{5} +14.6969i q^{7} -27.0000 q^{9} +O(q^{10})$	$q+5.19615i q^{3} -19.7990 q^{5} +14.6969i q^{7} -27.0000 q^{9} +72.7461i q^{11} -102.879i q^{15} -76.3675 q^{21} +267.000 q^{25} -140.296i q^{27} -223.446 q^{29} -338.030i q^{31} -378.000 q^{33} -290.985i q^{35} +534.573 q^{45} +127.000 q^{49} +579.828 q^{53} -1440.30i q^{55} +717.069i q^{59} -396.817i q^{63} -322.000 q^{73} +1387.37i q^{75} -1069.15 q^{77} +308.636i q^{79} +729.000 q^{81} -883.346i q^{83} -1161.06i q^{87} +1756.45 q^{93} -574.000 q^{97} -1964.15i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9}+O(q^{10}) $ 4 * q - 108 * q^9	$ 4 q - 108 q^{9} + 1068 q^{25} - 1512 q^{33} + 508 q^{49} - 1288 q^{73} + 2916 q^{81} - 2296 q^{97}+O(q^{100}) $ 4 * q - 108 * q^9 + 1068 * q^25 - 1512 * q^33 + 508 * q^49 - 1288 * q^73 + 2916 * q^81 - 2296 * 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
$3$	5.19615i	1.00000i
$3$	5.19615i	1.00000i
$4$	0	0
$5$	−19.7990	−1.77088	−0.885438	−	0.464758i	$-0.846141\pi$
$5$	−19.7990	−1.77088	−0.885438	+	0.464758i	$0.846141\pi$
$6$	0	0
$7$	14.6969i	0.793560i	0.917914	+	0.396780i	$0.129872\pi$
$7$	14.6969i	0.793560i	−0.917914	+	0.396780i	$0.870128\pi$
$8$	0	0
$9$	−27.0000	−1.00000
$10$	0	0
$11$	72.7461i	1.99398i	0.0775275	+	0.996990i	$0.475297\pi$
$11$	72.7461i	1.99398i	−0.0775275	+	0.996990i	$0.524703\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	− 102.879i	− 1.77088i
$16$	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
$21$	−76.3675	−0.793560
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	267.000	2.13600
$26$	0	0
$27$	− 140.296i	− 1.00000i
$28$	0	0
$29$	−223.446	−1.43079	−0.715394	−	0.698722i	$-0.753753\pi$
$29$	−223.446	−1.43079	−0.715394	+	0.698722i	$0.753753\pi$
$30$	0	0
$31$	− 338.030i	− 1.95845i	−0.202780	−	0.979224i	$-0.564998\pi$
$31$	− 338.030i	− 1.95845i	0.202780	−	0.979224i	$-0.435002\pi$
$32$	0	0
$33$	−378.000	−1.99398
$34$	0	0
$35$	− 290.985i	− 1.40530i
$36$	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
$45$	534.573	1.77088
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	127.000	0.370262
$50$	0	0
$51$	0	0
$52$	0	0
$53$	579.828	1.50274	0.751372	−	0.659879i	$-0.229392\pi$
$53$	579.828	1.50274	0.751372	+	0.659879i	$0.229392\pi$
$54$	0	0
$55$	− 1440.30i	− 3.53109i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	717.069i	1.58228i	0.611636	+	0.791139i	$0.290512\pi$
$59$	717.069i	1.58228i	−0.611636	+	0.791139i	$0.709488\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 396.817i	− 0.793560i
$64$	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
$73$	−322.000	−0.516264	−0.258132	−	0.966110i	$-0.583107\pi$
$73$	−322.000	−0.516264	−0.258132	+	0.966110i	$0.583107\pi$
$74$	0	0
$75$	1387.37i	2.13600i
$76$	0	0
$77$	−1069.15	−1.58234
$78$	0	0
$79$	308.636i	0.439547i	0.975551	+	0.219774i	$0.0705319\pi$
$79$	308.636i	0.439547i	−0.975551	+	0.219774i	$0.929468\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	− 883.346i	− 1.16819i	−0.811685	−	0.584095i	$-0.801449\pi$
$83$	− 883.346i	− 1.16819i	0.811685	−	0.584095i	$-0.198551\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1161.06i	− 1.43079i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1756.45	1.95845
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−574.000	−0.600834	−0.300417	−	0.953808i	$-0.597126\pi$
$97$	−574.000	−0.600834	−0.300417	+	0.953808i	$0.597126\pi$
$98$	0	0
$99$	− 1964.15i	− 1.99398i
$100$	0	0
$101$	−1762.11	−1.73601	−0.868003	−	0.496560i	$-0.834596\pi$
$101$	−1762.11	−1.73601	−0.868003	+	0.496560i	$0.834596\pi$
$102$	0	0
$103$	1366.82i	1.30754i	0.756695	+	0.653768i	$0.226813\pi$
$103$	1366.82i	1.30754i	−0.756695	+	0.653768i	$0.773187\pi$
$104$	0	0
$105$	1512.00	1.40530
$106$	0	0
$107$	− 363.731i	− 0.328628i	−0.986408	−	0.164314i	$-0.947459\pi$
$107$	− 363.731i	− 0.328628i	0.986408	−	0.164314i	$-0.0525410\pi$
$108$	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
$121$	−3961.00	−2.97596
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2811.46	−2.01171
$126$	0	0
$127$	− 1748.94i	− 1.22199i	−0.791634	−	0.610996i	$-0.790769\pi$
$127$	− 1748.94i	− 1.22199i	0.791634	−	0.610996i	$-0.209231\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1319.82i	− 0.880255i	−0.897935	−	0.440128i	$-0.854933\pi$
$131$	− 1319.82i	− 0.880255i	0.897935	−	0.440128i	$-0.145067\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2777.72i	1.77088i
$136$	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
$145$	4424.00	2.53375
$146$	0	0
$147$	659.911i	0.370262i
$148$	0	0
$149$	−1241.68	−0.682700	−0.341350	−	0.939936i	$-0.610884\pi$
$149$	−1241.68	−0.682700	−0.341350	+	0.939936i	$0.610884\pi$
$150$	0	0
$151$	3600.75i	1.94056i	0.241981	+	0.970281i	$0.422203\pi$
$151$	3600.75i	1.94056i	−0.241981	+	0.970281i	$0.577797\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6692.64i	3.46817i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	3012.87i	1.50274i
$160$	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
$165$	7484.02	3.53109
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2514.47	1.10504	0.552519	−	0.833500i	$-0.313666\pi$
$173$	2514.47	1.10504	0.552519	+	0.833500i	$0.313666\pi$
$174$	0	0
$175$	3924.08i	1.69504i
$176$	0	0
$177$	−3726.00	−1.58228
$178$	0	0
$179$	− 1236.68i	− 0.516392i	−0.966093	−	0.258196i	$-0.916872\pi$
$179$	− 1236.68i	− 0.516392i	0.966093	−	0.258196i	$-0.0831280\pi$
$180$	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
$189$	2061.92	0.793560
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−2522.00	−0.940609	−0.470304	−	0.882504i	$-0.655856\pi$
$193$	−2522.00	−0.940609	−0.470304	+	0.882504i	$0.655856\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5529.58	−1.99983	−0.999913	−	0.0131547i	$-0.995813\pi$
$197$	−5529.58	−1.99983	−0.999913	+	0.0131547i	$0.995813\pi$
$198$	0	0
$199$	73.4847i	0.0261768i	0.999914	+	0.0130884i	$0.00416629\pi$
$199$	73.4847i	0.0261768i	−0.999914	+	0.0130884i	$0.995834\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3283.97i	− 1.13542i
$204$	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
$217$	4968.00	1.55415
$218$	0	0
$219$	− 1673.16i	− 0.516264i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6657.71i	1.99925i	0.0273265	+	0.999627i	$0.491301\pi$
$223$	6657.71i	1.99925i	−0.0273265	+	0.999627i	$0.508699\pi$
$224$	0	0
$225$	−7209.00	−2.13600
$226$	0	0
$227$	− 5954.79i	− 1.74112i	−0.492066	−	0.870558i	$-0.663758\pi$
$227$	− 5954.79i	− 1.74112i	0.492066	−	0.870558i	$-0.336242\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	− 5555.44i	− 1.58234i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1603.72	−0.439547
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	6230.00	1.66518	0.832592	−	0.553886i	$-0.186856\pi$
$241$	6230.00	1.66518	0.832592	+	0.553886i	$0.186856\pi$
$242$	0	0
$243$	3788.00i	1.00000i
$244$	0	0
$245$	−2514.47	−0.655689
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4590.00	1.16819
$250$	0	0
$251$	− 6827.74i	− 1.71699i	−0.512826	−	0.858493i	$-0.671401\pi$
$251$	− 6827.74i	− 1.71699i	0.512826	−	0.858493i	$-0.328599\pi$
$252$	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
$261$	6033.04	1.43079
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−11480.0	−2.66117
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5009.14	−1.13536	−0.567682	−	0.823248i	$-0.692160\pi$
$269$	−5009.14	−1.13536	−0.567682	+	0.823248i	$0.692160\pi$
$270$	0	0
$271$	− 5217.41i	− 1.16950i	−0.811213	−	0.584751i	$-0.801192\pi$
$271$	− 5217.41i	− 1.16950i	0.811213	−	0.584751i	$-0.198808\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	19423.2i	4.25914i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	9126.80i	1.95845i
$280$	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
$289$	4913.00	1.00000
$290$	0	0
$291$	− 2982.59i	− 0.600834i
$292$	0	0
$293$	−9602.51	−1.91462	−0.957312	−	0.289058i	$-0.906658\pi$
$293$	−9602.51	−1.91462	−0.957312	+	0.289058i	$0.906658\pi$
$294$	0	0
$295$	− 14197.2i	− 2.80202i
$296$	0	0
$297$	10206.0	1.99398
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 9156.19i	− 1.73601i
$304$	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
$309$	−7102.18	−1.30754
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−7378.00	−1.33236	−0.666181	−	0.745790i	$-0.732072\pi$
$313$	−7378.00	−1.33236	−0.666181	+	0.745790i	$0.732072\pi$
$314$	0	0
$315$	7856.58i	1.40530i
$316$	0	0
$317$	−528.916	−0.0937125	−0.0468563	−	0.998902i	$-0.514920\pi$
$317$	−528.916	−0.0937125	−0.0468563	+	0.998902i	$0.514920\pi$
$318$	0	0
$319$	− 16254.8i	− 2.85296i
$320$	0	0
$321$	1890.00	0.328628
$322$	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
$337$	−11594.0	−1.87408	−0.937041	−	0.349220i	$-0.886447\pi$
$337$	−11594.0	−1.87408	−0.937041	+	0.349220i	$0.886447\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24590.3	3.90511
$342$	0	0
$343$	6907.56i	1.08739i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10984.7i	− 1.69939i	−0.527276	−	0.849694i	$-0.676787\pi$
$347$	− 10984.7i	− 1.69939i	0.527276	−	0.849694i	$-0.323213\pi$
$348$	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
$361$	−6859.00	−1.00000
$362$	0	0
$363$	− 20582.0i	− 2.97596i
$364$	0	0
$365$	6375.27	0.914239
$366$	0	0
$367$	− 5276.20i	− 0.750451i	−0.926934	−	0.375225i	$-0.877565\pi$
$367$	− 5276.20i	− 0.750451i	0.926934	−	0.375225i	$-0.122435\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8521.69i	1.19252i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	− 14608.8i	− 2.01171i
$376$	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
$381$	9087.74	1.22199
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	21168.0	2.80213
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6191.43	0.806987	0.403493	−	0.914983i	$-0.367796\pi$
$389$	6191.43	0.806987	0.403493	+	0.914983i	$0.367796\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	6858.00	0.880255
$394$	0	0
$395$	− 6110.68i	− 0.778383i
$396$	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
$405$	−14433.5	−1.77088
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11270.0	−1.36251	−0.681254	−	0.732047i	$-0.738565\pi$
$409$	−11270.0	−1.36251	−0.681254	+	0.732047i	$0.738565\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10538.7	−1.25563
$414$	0	0
$415$	17489.4i	2.06872i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 11940.8i	− 1.39223i	−0.717930	−	0.696115i	$-0.754910\pi$
$419$	− 11940.8i	− 1.39223i	0.717930	−	0.696115i	$-0.245090\pi$
$420$	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
$433$	15442.0	1.71385	0.856923	−	0.515445i	$-0.172373\pi$
$433$	15442.0	1.71385	0.856923	+	0.515445i	$0.172373\pi$
$434$	0	0
$435$	22987.8i	2.53375i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 17562.8i	− 1.90940i	−0.297562	−	0.954702i	$-0.596174\pi$
$439$	− 17562.8i	− 1.90940i	0.297562	−	0.954702i	$-0.403826\pi$
$440$	0	0
$441$	−3429.00	−0.370262
$442$	0	0
$443$	11130.2i	1.19370i	0.802352	+	0.596851i	$0.203582\pi$
$443$	11130.2i	1.19370i	−0.802352	+	0.596851i	$0.796418\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6451.96i	− 0.682700i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−18710.0	−1.94056
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2774.00	−0.283944	−0.141972	−	0.989871i	$-0.545344\pi$
$457$	−2774.00	−0.283944	−0.141972	+	0.989871i	$0.545344\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19620.8	1.98228	0.991141	−	0.132816i	$-0.0424020\pi$
$461$	19620.8	1.98228	0.991141	+	0.132816i	$0.0424020\pi$
$462$	0	0
$463$	7304.38i	0.733182i	0.930382	+	0.366591i	$0.119475\pi$
$463$	7304.38i	0.733182i	−0.930382	+	0.366591i	$0.880525\pi$
$464$	0	0
$465$	−34776.0	−3.46817
$466$	0	0
$467$	19069.9i	1.88961i	0.327630	+	0.944806i	$0.393750\pi$
$467$	19069.9i	1.88961i	−0.327630	+	0.944806i	$0.606250\pi$
$468$	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
$477$	−15655.3	−1.50274
$478$	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
$485$	11364.6	1.06400
$486$	0	0
$487$	21295.9i	1.98154i	0.135571	+	0.990768i	$0.456713\pi$
$487$	21295.9i	1.98154i	−0.135571	+	0.990768i	$0.543287\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	800.207i	0.0735496i	0.999324	+	0.0367748i	$0.0117084\pi$
$491$	800.207i	0.0735496i	−0.999324	+	0.0367748i	$0.988292\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	38888.1i	3.53109i
$496$	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
$505$	34888.0	3.07425
$506$	0	0
$507$	11415.9i	1.00000i
$508$	0	0
$509$	−4296.38	−0.374133	−0.187067	−	0.982347i	$-0.559898\pi$
$509$	−4296.38	−0.374133	−0.187067	+	0.982347i	$0.559898\pi$
$510$	0	0
$511$	− 4732.41i	− 0.409686i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 27061.6i	− 2.31549i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	13065.6i	1.10504i
$520$	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
$525$	−20390.1	−1.69504
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	− 19360.9i	− 1.58228i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	7201.50i	0.581959i
$536$	0	0
$537$	6426.00	0.516392
$538$	0	0
$539$	9238.76i	0.738296i
$540$	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
$553$	−4536.00	−0.348807
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20803.1	−1.58250	−0.791252	−	0.611490i	$-0.790570\pi$
$557$	−20803.1	−1.58250	−0.791252	+	0.611490i	$0.790570\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 9030.91i	− 0.676035i	−0.941140	−	0.338017i	$-0.890244\pi$
$563$	− 9030.91i	− 0.676035i	0.941140	−	0.338017i	$-0.109756\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10714.1i	0.793560i
$568$	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
$577$	−10906.0	−0.786868	−0.393434	−	0.919353i	$-0.628713\pi$
$577$	−10906.0	−0.786868	−0.393434	+	0.919353i	$0.628713\pi$
$578$	0	0
$579$	− 13104.7i	− 0.940609i
$580$	0	0
$581$	12982.5	0.927029
$582$	0	0
$583$	42180.2i	2.99644i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 25305.3i	− 1.77932i	−0.456625	−	0.889659i	$-0.650942\pi$
$587$	− 25305.3i	− 1.77932i	0.456625	−	0.889659i	$-0.349058\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	− 28732.5i	− 1.99983i
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−381.838	−0.0261768
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	3598.00	0.244202	0.122101	−	0.992518i	$-0.461037\pi$
$601$	3598.00	0.244202	0.122101	+	0.992518i	$0.461037\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	78423.8	5.27005
$606$	0	0
$607$	5423.17i	0.362635i	0.983425	+	0.181318i	$0.0580362\pi$
$607$	5423.17i	0.362635i	−0.983425	+	0.181318i	$0.941964\pi$
$608$	0	0
$609$	17064.0	1.13542
$610$	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
$625$	22289.0	1.42650
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 9156.19i	− 0.577658i	−0.957381	−	0.288829i	$-0.906734\pi$
$631$	− 9156.19i	− 0.577658i	0.957381	−	0.288829i	$-0.0932659\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	34627.2i	2.16400i
$636$	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
$649$	−52164.0	−3.15503
$650$	0	0
$651$	25814.5i	1.55415i
$652$	0	0
$653$	16973.4	1.01718	0.508591	−	0.861008i	$-0.330166\pi$
$653$	16973.4	1.01718	0.508591	+	0.861008i	$0.330166\pi$
$654$	0	0
$655$	26131.2i	1.55882i
$656$	0	0
$657$	8694.00	0.516264
$658$	0	0
$659$	33681.5i	1.99096i	0.0949654	+	0.995481i	$0.469726\pi$
$659$	33681.5i	1.99096i	−0.0949654	+	0.995481i	$0.530274\pi$
$660$	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
$669$	−34594.5	−1.99925
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−29342.0	−1.68061	−0.840305	−	0.542113i	$-0.817624\pi$
$673$	−29342.0	−1.68061	−0.840305	+	0.542113i	$0.817624\pi$
$674$	0	0
$675$	− 37459.1i	− 2.13600i
$676$	0	0
$677$	−5721.91	−0.324831	−0.162416	−	0.986722i	$-0.551929\pi$
$677$	−5721.91	−0.324831	−0.162416	+	0.986722i	$0.551929\pi$
$678$	0	0
$679$	− 8436.04i	− 0.476798i
$680$	0	0
$681$	30942.0	1.74112
$682$	0	0
$683$	33827.0i	1.89510i	0.319610	+	0.947549i	$0.396448\pi$
$683$	33827.0i	1.89510i	−0.319610	+	0.947549i	$0.603552\pi$
$684$	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
$693$	28866.9	1.58234
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27613.9	−1.48782	−0.743911	−	0.668278i	$-0.767032\pi$
$701$	−27613.9	−1.48782	−0.743911	+	0.668278i	$0.767032\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 25897.6i	− 1.37762i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	− 8333.16i	− 0.439547i
$712$	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
$721$	−20088.0	−1.03761
$722$	0	0
$723$	32372.0i	1.66518i
$724$	0	0
$725$	−59660.0	−3.05616
$726$	0	0
$727$	− 38961.6i	− 1.98763i	−0.111060	−	0.993814i	$-0.535425\pi$
$727$	− 38961.6i	− 1.98763i	0.111060	−	0.993814i	$-0.464575\pi$
$728$	0	0
$729$	−19683.0	−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
$735$	− 13065.6i	− 0.655689i
$736$	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
$745$	24584.0	1.20898
$746$	0	0
$747$	23850.3i	1.16819i
$748$	0	0
$749$	5345.73	0.260786
$750$	0	0
$751$	− 35081.6i	− 1.70459i	−0.523063	−	0.852294i	$-0.675211\pi$
$751$	− 35081.6i	− 1.70459i	0.523063	−	0.852294i	$-0.324789\pi$
$752$	0	0
$753$	35478.0	1.71699
$754$	0	0
$755$	− 71291.2i	− 3.43649i
$756$	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
$769$	42406.0	1.98856	0.994278	−	0.106824i	$-0.0340682\pi$
$769$	42406.0	1.98856	0.994278	+	0.106824i	$0.0340682\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38152.7	1.77523	0.887617	−	0.460583i	$-0.152360\pi$
$773$	38152.7	1.77523	0.887617	+	0.460583i	$0.152360\pi$
$774$	0	0
$775$	− 90253.9i	− 4.18325i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	31348.6i	1.43079i
$784$	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
$795$	− 59651.8i	− 2.66117i
$796$	0	0
$797$	−26708.8	−1.18705	−0.593523	−	0.804817i	$-0.702263\pi$
$797$	−26708.8	−1.18705	−0.593523	+	0.804817i	$0.702263\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 23424.3i	− 1.02942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 26028.3i	− 1.13536i
$808$	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
$813$	27110.5	1.16950
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7260.57	−0.308643	−0.154321	−	0.988021i	$-0.549319\pi$
$821$	−7260.57	−0.308643	−0.154321	+	0.988021i	$0.549319\pi$
$822$	0	0
$823$	44340.7i	1.87803i	0.343877	+	0.939015i	$0.388260\pi$
$823$	44340.7i	1.87803i	−0.343877	+	0.939015i	$0.611740\pi$
$824$	0	0
$825$	−100926.	−4.25914
$826$	0	0
$827$	24660.9i	1.03693i	0.855097	+	0.518467i	$0.173497\pi$
$827$	24660.9i	1.03693i	−0.855097	+	0.518467i	$0.826503\pi$
$828$	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
$837$	−47424.2	−1.95845
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	25539.0	1.04715
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−43498.4	−1.77088
$846$	0	0
$847$	− 58214.6i	− 2.36160i
$848$	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
$865$	−49784.0	−1.95689
$866$	0	0
$867$	25528.7i	1.00000i
$868$	0	0
$869$	−22452.1	−0.876449
$870$	0	0
$871$	0	0
$872$	0	0
$873$	15498.0	0.600834
$874$	0	0
$875$	− 41319.8i	− 1.59642i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	− 49896.1i	− 1.91462i
$880$	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
$885$	73771.0	2.80202
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	25704.0	0.969724
$890$	0	0
$891$	53031.9i	1.99398i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	24485.1i	0.914465i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	75531.3i	2.80212i
$900$	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
$909$	47577.0	1.73601
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	64260.0	2.32935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19397.4	0.698535
$918$	0	0
$919$	54217.0i	1.94609i	0.230622	+	0.973044i	$0.425924\pi$
$919$	54217.0i	1.94609i	−0.230622	+	0.973044i	$0.574076\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 36904.0i	− 1.30754i
$928$	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
$937$	32074.0	1.11826	0.559131	−	0.829079i	$-0.311135\pi$
$937$	32074.0	1.11826	0.559131	+	0.829079i	$0.311135\pi$
$938$	0	0
$939$	− 38337.2i	− 1.33236i
$940$	0	0
$941$	−24570.5	−0.851198	−0.425599	−	0.904912i	$-0.639937\pi$
$941$	−24570.5	−0.851198	−0.425599	+	0.904912i	$0.639937\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−40824.0	−1.40530
$946$	0	0
$947$	54632.3i	1.87467i	0.348429	+	0.937335i	$0.386715\pi$
$947$	54632.3i	1.87467i	−0.348429	+	0.937335i	$0.613285\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 2748.33i	− 0.0937125i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	84462.5	2.85296
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−84473.0	−2.83552
$962$	0	0
$963$	9820.73i	0.328628i
$964$	0	0
$965$	49933.1	1.66570
$966$	0	0
$967$	− 14094.4i	− 0.468712i	−0.972151	−	0.234356i	$-0.924702\pi$
$967$	− 14094.4i	− 0.468712i	0.972151	−	0.234356i	$-0.0752981\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10049.4i	0.332131i	0.986115	+	0.166066i	$0.0531063\pi$
$971$	10049.4i	0.332131i	−0.986115	+	0.166066i	$0.946894\pi$
$972$	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
$985$	109480.	3.54144
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 58126.4i	− 1.86321i	−0.363466	−	0.931607i	$-0.618407\pi$
$991$	− 58126.4i	− 1.86321i	0.363466	−	0.931607i	$-0.381593\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1454.92i	− 0.0463559i
$996$	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	384.4.f.a.191.3	yes	4
3.2	odd	2	inner	384.4.f.a.191.2	yes	4
4.3	odd	2	inner	384.4.f.a.191.1	✓	4
8.3	odd	2	inner	384.4.f.a.191.4	yes	4
8.5	even	2	inner	384.4.f.a.191.2	yes	4
12.11	even	2	inner	384.4.f.a.191.4	yes	4
16.3	odd	4		768.4.c.q.767.4		4
16.5	even	4		768.4.c.q.767.3		4
16.11	odd	4		768.4.c.q.767.1		4
16.13	even	4		768.4.c.q.767.2		4
24.5	odd	2	CM	384.4.f.a.191.3	yes	4
24.11	even	2	inner	384.4.f.a.191.1	✓	4
48.5	odd	4		768.4.c.q.767.2		4
48.11	even	4		768.4.c.q.767.4		4
48.29	odd	4		768.4.c.q.767.3		4
48.35	even	4		768.4.c.q.767.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.f.a.191.1	✓	4	4.3	odd	2	inner
384.4.f.a.191.1	✓	4	24.11	even	2	inner
384.4.f.a.191.2	yes	4	3.2	odd	2	inner
384.4.f.a.191.2	yes	4	8.5	even	2	inner
384.4.f.a.191.3	yes	4	1.1	even	1	trivial
384.4.f.a.191.3	yes	4	24.5	odd	2	CM
384.4.f.a.191.4	yes	4	8.3	odd	2	inner
384.4.f.a.191.4	yes	4	12.11	even	2	inner
768.4.c.q.767.1		4	16.11	odd	4
768.4.c.q.767.1		4	48.35	even	4
768.4.c.q.767.2		4	16.13	even	4
768.4.c.q.767.2		4	48.5	odd	4
768.4.c.q.767.3		4	16.5	even	4
768.4.c.q.767.3		4	48.29	odd	4
768.4.c.q.767.4		4	16.3	odd	4
768.4.c.q.767.4		4	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615i q^{3} -19.7990 q^{5} +14.6969i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q+5.19615i q^{3} -19.7990 q^{5} +14.6969i q^{7} -27.0000 q^{9} +72.7461i q^{11} -102.879i q^{15} -76.3675 q^{21} +267.000 q^{25} -140.296i q^{27} -223.446 q^{29} -338.030i q^{31} -378.000 q^{33} -290.985i q^{35} +534.573 q^{45} +127.000 q^{49} +579.828 q^{53} -1440.30i q^{55} +717.069i q^{59} -396.817i q^{63} -322.000 q^{73} +1387.37i q^{75} -1069.15 q^{77} +308.636i q^{79} +729.000 q^{81} -883.346i q^{83} -1161.06i q^{87} +1756.45 q^{93} -574.000 q^{97} -1964.15i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9}+O(q^{10}) \) 4 * q - 108 * q^9	\( 4 q - 108 q^{9} + 1068 q^{25} - 1512 q^{33} + 508 q^{49} - 1288 q^{73} + 2916 q^{81} - 2296 q^{97}+O(q^{100}) \) 4 * q - 108 * q^9 + 1068 * q^25 - 1512 * q^33 + 508 * q^49 - 1288 * q^73 + 2916 * q^81 - 2296 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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