LMFDB - Embedded newform 192.6.c.b.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,6,Mod(191,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	192.c (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:	$30.7936934041$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			191.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.6.c.b.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5885i q^{3} +107.387i q^{7} -243.000 q^{9} +O(q^{10})$	$q-15.5885i q^{3} +107.387i q^{7} -243.000 q^{9} +1202.00 q^{13} -2802.46i q^{19} +1674.00 q^{21} +3125.00 q^{25} +3788.00i q^{27} -2816.31i q^{31} -16550.0 q^{37} -18737.3i q^{39} -24016.6i q^{43} +5275.00 q^{49} -43686.0 q^{57} +38626.0 q^{61} -26095.1i q^{63} -64324.9i q^{67} +1450.00 q^{73} -48713.9i q^{75} -46852.0i q^{79} +59049.0 q^{81} +129079. i q^{91} -43902.0 q^{93} +134386. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 486 q^{9}+O(q^{10}) $ 2 * q - 486 * q^9	$ 2 q - 486 q^{9} + 2404 q^{13} + 3348 q^{21} + 6250 q^{25} - 33100 q^{37} + 10550 q^{49} - 87372 q^{57} + 77252 q^{61} + 2900 q^{73} + 118098 q^{81} - 87804 q^{93} + 268772 q^{97}+O(q^{100}) $ 2 * q - 486 * q^9 + 2404 * q^13 + 3348 * q^21 + 6250 * q^25 - 33100 * q^37 + 10550 * q^49 - 87372 * q^57 + 77252 * q^61 + 2900 * q^73 + 118098 * q^81 - 87804 * q^93 + 268772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

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
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	107.387i	0.828337i	0.910200	+	0.414169i	$0.135928\pi$
$7$	107.387i	0.828337i	−0.910200	+	0.414169i	$0.864072\pi$
$8$	0	0
$9$	−243.000	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1202.00	1.97263	0.986316	−	0.164866i	$-0.0527191\pi$
$13$	1202.00	1.97263	0.986316	+	0.164866i	$0.0527191\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	− 2802.46i	− 1.78096i	−0.455018	−	0.890482i	$-0.650367\pi$
$19$	− 2802.46i	− 1.78096i	0.455018	−	0.890482i	$-0.349633\pi$
$20$	0	0
$21$	1674.00	0.828337
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	3125.00	1.00000
$26$	0	0
$27$	3788.00i	1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 2816.31i	− 0.526353i	−0.964748	−	0.263176i	$-0.915230\pi$
$31$	− 2816.31i	− 0.526353i	0.964748	−	0.263176i	$-0.0847701\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−16550.0	−1.98744	−0.993719	−	0.111902i	$-0.964306\pi$
$37$	−16550.0	−1.98744	−0.993719	+	0.111902i	$0.964306\pi$
$38$	0	0
$39$	− 18737.3i	− 1.97263i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 24016.6i	− 1.98080i	−0.138230	−	0.990400i	$-0.544141\pi$
$43$	− 24016.6i	− 1.98080i	0.138230	−	0.990400i	$-0.455859\pi$
$44$	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
$49$	5275.00	0.313857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−43686.0	−1.78096
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	38626.0	1.32909	0.664546	−	0.747247i	$-0.268625\pi$
$61$	38626.0	1.32909	0.664546	+	0.747247i	$0.268625\pi$
$62$	0	0
$63$	− 26095.1i	− 0.828337i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 64324.9i	− 1.75062i	−0.483561	−	0.875310i	$-0.660657\pi$
$67$	− 64324.9i	− 1.75062i	0.483561	−	0.875310i	$-0.339343\pi$
$68$	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
$73$	1450.00	0.0318464	0.0159232	−	0.999873i	$-0.494931\pi$
$73$	1450.00	0.0318464	0.0159232	+	0.999873i	$0.494931\pi$
$74$	0	0
$75$	− 48713.9i	− 1.00000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 46852.0i	− 0.844618i	−0.906452	−	0.422309i	$-0.861220\pi$
$79$	− 46852.0i	− 0.844618i	0.906452	−	0.422309i	$-0.138780\pi$
$80$	0	0
$81$	59049.0	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	129079.i	1.63400i
$92$	0	0
$93$	−43902.0	−0.526353
$94$	0	0
$95$	0	0
$96$	0	0
$97$	134386.	1.45019	0.725095	−	0.688649i	$-0.241796\pi$
$97$	134386.	1.45019	0.725095	+	0.688649i	$0.241796\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 162844.i	− 1.51244i	−0.654317	−	0.756221i	$-0.727044\pi$
$103$	− 162844.i	− 1.51244i	0.654317	−	0.756221i	$-0.272956\pi$
$104$	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
$109$	114482.	0.922935	0.461467	−	0.887157i	$-0.347323\pi$
$109$	114482.	0.922935	0.461467	+	0.887157i	$0.347323\pi$
$110$	0	0
$111$	257989.i	1.98744i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−292086.	−1.97263
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−161051.	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	246599.i	1.35669i	0.734742	+	0.678347i	$0.237303\pi$
$127$	246599.i	1.35669i	−0.734742	+	0.678347i	$0.762697\pi$
$128$	0	0
$129$	−374382.	−1.98080
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	300948.	1.47524
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	379233.i	1.66482i	0.554157	+	0.832412i	$0.313041\pi$
$139$	379233.i	1.66482i	−0.554157	+	0.832412i	$0.686959\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 82229.1i	− 0.313857i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	383348.i	1.36820i	0.729387	+	0.684102i	$0.239806\pi$
$151$	383348.i	1.36820i	−0.729387	+	0.684102i	$0.760194\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−109214.	−0.353614	−0.176807	−	0.984246i	$-0.556577\pi$
$157$	−109214.	−0.353614	−0.176807	+	0.984246i	$0.556577\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 15300.9i	− 0.0451075i	−0.999746	−	0.0225538i	$-0.992820\pi$
$163$	− 15300.9i	− 0.0451075i	0.999746	−	0.0225538i	$-0.00717969\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.07351e6	2.89128
$170$	0	0
$171$	680997.i	1.78096i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	335585.i	0.828337i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	234026.	0.530967	0.265484	−	0.964115i	$-0.414468\pi$
$181$	234026.	0.530967	0.265484	+	0.964115i	$0.414468\pi$
$182$	0	0
$183$	− 602120.i	− 1.32909i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−406782.	−0.828337
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	364802.	0.704959	0.352480	−	0.935820i	$-0.385339\pi$
$193$	364802.	0.704959	0.352480	+	0.935820i	$0.385339\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	− 645047.i	− 1.15467i	−0.816507	−	0.577336i	$-0.804092\pi$
$199$	− 645047.i	− 1.15467i	0.816507	−	0.577336i	$-0.195908\pi$
$200$	0	0
$201$	−1.00273e6	−1.75062
$202$	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
$211$	− 1.26071e6i	− 1.94944i	−0.223422	−	0.974722i	$-0.571723\pi$
$211$	− 1.26071e6i	− 1.94944i	0.223422	−	0.974722i	$-0.428277\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	302436.	0.435998
$218$	0	0
$219$	− 22603.3i	− 0.0318464i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.45377e6i	1.95764i	0.204718	+	0.978821i	$0.434372\pi$
$223$	1.45377e6i	1.95764i	−0.204718	+	0.978821i	$0.565628\pi$
$224$	0	0
$225$	−759375.	−1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	1.26982e6	1.60012	0.800060	−	0.599919i	$-0.204801\pi$
$229$	1.26982e6	1.60012	0.800060	+	0.599919i	$0.204801\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−730350.	−0.844618
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−1.29697e6	−1.43843	−0.719215	−	0.694788i	$-0.755498\pi$
$241$	−1.29697e6	−1.43843	−0.719215	+	0.694788i	$0.755498\pi$
$242$	0	0
$243$	− 920483.i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.36855e6i	− 3.51319i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	− 1.77726e6i	− 1.64627i
$260$	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
$271$	− 878236.i	− 0.726421i	−0.931707	−	0.363210i	$-0.881681\pi$
$271$	− 878236.i	− 0.726421i	0.931707	−	0.363210i	$-0.118319\pi$
$272$	0	0
$273$	2.01215e6	1.63400
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.48661e6	−1.94719	−0.973596	−	0.228276i	$-0.926691\pi$
$277$	−2.48661e6	−1.94719	−0.973596	+	0.228276i	$0.926691\pi$
$278$	0	0
$279$	684364.i	0.526353i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	2.69461e6i	2.00000i	0.000724409		1.00000i	$0.499769\pi$
$283$	2.69461e6i	2.00000i	−0.000724409		1.00000i	$0.500231\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.41986e6	1.00000
$290$	0	0
$291$	− 2.09487e6i	− 1.45019i
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.57908e6	1.64077
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.36916e6i	1.43466i	0.696733	+	0.717331i	$0.254636\pi$
$307$	2.36916e6i	1.43466i	−0.696733	+	0.717331i	$0.745364\pi$
$308$	0	0
$309$	−2.53849e6	−1.51244
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	733898.	0.423423	0.211712	−	0.977332i	$-0.432096\pi$
$313$	733898.	0.423423	0.211712	+	0.977332i	$0.432096\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.75625e6	1.97263
$326$	0	0
$327$	− 1.78460e6i	− 0.922935i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 2.44747e6i	− 1.22786i	−0.789361	−	0.613929i	$-0.789588\pi$
$331$	− 2.44747e6i	− 1.22786i	0.789361	−	0.613929i	$-0.210412\pi$
$332$	0	0
$333$	4.02165e6	1.98744
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.48509e6	−0.712323	−0.356161	−	0.934424i	$-0.615915\pi$
$337$	−1.48509e6	−0.712323	−0.356161	+	0.934424i	$0.615915\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.37132e6i	1.08832i
$344$	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
$349$	−4.27561e6	−1.87904	−0.939518	−	0.342501i	$-0.888726\pi$
$349$	−4.27561e6	−1.87904	−0.939518	+	0.342501i	$0.888726\pi$
$350$	0	0
$351$	4.55317e6i	1.97263i
$352$	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
$361$	−5.37767e6	−2.17183
$362$	0	0
$363$	2.51054e6i	1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.47084e6i	1.73270i	0.499437	+	0.866350i	$0.333540\pi$
$367$	4.47084e6i	1.73270i	−0.499437	+	0.866350i	$0.666460\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.33340e6	−1.24055	−0.620276	−	0.784384i	$-0.712979\pi$
$373$	−3.33340e6	−1.24055	−0.620276	+	0.784384i	$0.712979\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.85670e6i	1.02157i	0.859709	+	0.510784i	$0.170645\pi$
$379$	2.85670e6i	1.02157i	−0.859709	+	0.510784i	$0.829355\pi$
$380$	0	0
$381$	3.84410e6	1.35669
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.83604e6i	1.98080i
$388$	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
$397$	−5.78949e6	−1.84359	−0.921794	−	0.387681i	$-0.873276\pi$
$397$	−5.78949e6	−1.84359	−0.921794	+	0.387681i	$0.873276\pi$
$398$	0	0
$399$	− 4.69132e6i	− 1.47524i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 3.38521e6i	− 1.03830i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	1.19468e6	0.353138	0.176569	−	0.984288i	$-0.443500\pi$
$409$	1.19468e6	0.353138	0.176569	+	0.984288i	$0.443500\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.91165e6	1.66482
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−2.23610e6	−0.614874	−0.307437	−	0.951568i	$-0.599471\pi$
$421$	−2.23610e6	−0.614874	−0.307437	+	0.951568i	$0.599471\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.14794e6i	1.10094i
$428$	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
$433$	1.85760e6	0.476138	0.238069	−	0.971248i	$-0.423486\pi$
$433$	1.85760e6	0.476138	0.238069	+	0.971248i	$0.423486\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	7.01951e6i	1.73838i	0.494475	+	0.869192i	$0.335360\pi$
$439$	7.01951e6i	1.73838i	−0.494475	+	0.869192i	$0.664640\pi$
$440$	0	0
$441$	−1.28182e6	−0.313857
$442$	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
$453$	5.97580e6	1.36820
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.25745e6	1.84950	0.924752	−	0.380569i	$-0.124272\pi$
$457$	8.25745e6	1.84950	0.924752	+	0.380569i	$0.124272\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 6.90583e6i	− 1.49714i	−0.663054	−	0.748572i	$-0.730740\pi$
$463$	− 6.90583e6i	− 1.49714i	0.663054	−	0.748572i	$-0.269260\pi$
$464$	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
$469$	6.90767e6	1.45010
$470$	0	0
$471$	1.70248e6i	0.353614i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 8.75768e6i	− 1.78096i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−1.98931e7	−3.92048
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.00717e6i	− 0.765623i	−0.923827	−	0.382811i	$-0.874956\pi$
$487$	− 4.00717e6i	− 0.765623i	0.923827	−	0.382811i	$-0.125044\pi$
$488$	0	0
$489$	−238518.	−0.0451075
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.33752e6i	1.31916i	0.751634	+	0.659581i	$0.229266\pi$
$499$	7.33752e6i	1.31916i	−0.751634	+	0.659581i	$0.770734\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.67344e7i	− 2.89128i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	155711.i	0.0263796i
$512$	0	0
$513$	1.06157e7	1.78096
$514$	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
$523$	− 7.98031e6i	− 1.27575i	−0.770140	−	0.637875i	$-0.779814\pi$
$523$	− 7.98031e6i	− 1.27575i	0.770140	−	0.637875i	$-0.220186\pi$
$524$	0	0
$525$	5.23125e6	0.828337
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−6.43634e6	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.98573e6	−1.46685	−0.733426	−	0.679769i	$-0.762080\pi$
$541$	−9.98573e6	−1.46685	−0.733426	+	0.679769i	$0.762080\pi$
$542$	0	0
$543$	− 3.64810e6i	− 0.530967i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.66472e6i	0.809487i	0.914430	+	0.404744i	$0.132639\pi$
$547$	5.66472e6i	0.809487i	−0.914430	+	0.404744i	$0.867361\pi$
$548$	0	0
$549$	−9.38612e6	−1.32909
$550$	0	0
$551$	0	0
$552$	0	0
$553$	5.03130e6	0.699628
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	− 2.88680e7i	− 3.90739i
$560$	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
$567$	6.34110e6i	0.828337i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	2.11436e6i	0.271387i	0.990751	+	0.135693i	$0.0433262\pi$
$571$	2.11436e6i	0.271387i	−0.990751	+	0.135693i	$0.956674\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.72229e6	−0.215360	−0.107680	−	0.994186i	$-0.534342\pi$
$577$	−1.72229e6	−0.215360	−0.107680	+	0.994186i	$0.534342\pi$
$578$	0	0
$579$	− 5.68670e6i	− 0.704959i
$580$	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
$589$	−7.89260e6	−0.937415
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.00553e7	−1.15467
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	6.02163e6	0.680029	0.340015	−	0.940420i	$-0.389568\pi$
$601$	6.02163e6	0.680029	0.340015	+	0.940420i	$0.389568\pi$
$602$	0	0
$603$	1.56310e7i	1.75062i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.00446e6i	0.991941i	0.868339	+	0.495970i	$0.165188\pi$
$607$	9.00446e6i	0.991941i	−0.868339	+	0.495970i	$0.834812\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.58234e7	1.70079	0.850394	−	0.526147i	$-0.176364\pi$
$613$	1.58234e7	1.70079	0.850394	+	0.526147i	$0.176364\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 1.79030e7i	− 1.87801i	−0.343901	−	0.939006i	$-0.611748\pi$
$619$	− 1.79030e7i	− 1.87801i	0.343901	−	0.939006i	$-0.388252\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9.76562e6	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.16735e7i	1.16716i	0.812057	+	0.583579i	$0.198348\pi$
$631$	1.16735e7i	1.16716i	−0.812057	+	0.583579i	$0.801652\pi$
$632$	0	0
$633$	−1.96526e7	−1.94944
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.34055e6	0.619125
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	3.52146e6i	0.335889i	0.985796	+	0.167944i	$0.0537129\pi$
$643$	3.52146e6i	0.335889i	−0.985796	+	0.167944i	$0.946287\pi$
$644$	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
$651$	− 4.71451e6i	− 0.435998i
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−352350.	−0.0318464
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	1.91018e7	1.70048	0.850238	−	0.526398i	$-0.176458\pi$
$661$	1.91018e7	1.70048	0.850238	+	0.526398i	$0.176458\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.26620e7	1.95764
$670$	0	0
$671$	0	0
$672$	0	0
$673$	5.10725e6	0.434660	0.217330	−	0.976098i	$-0.430265\pi$
$673$	5.10725e6	0.434660	0.217330	+	0.976098i	$0.430265\pi$
$674$	0	0
$675$	1.18375e7i	1.00000i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	1.44313e7i	1.20125i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.97945e7i	− 1.60012i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.81296e7i	1.44442i	0.691673	+	0.722211i	$0.256874\pi$
$691$	1.81296e7i	1.44442i	−0.691673	+	0.722211i	$0.743126\pi$
$692$	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
$703$	4.63807e7i	3.53956i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−2.27014e7	−1.69604	−0.848021	−	0.529962i	$-0.822206\pi$
$709$	−2.27014e7	−1.69604	−0.848021	+	0.529962i	$0.822206\pi$
$710$	0	0
$711$	1.13850e7i	0.844618i
$712$	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
$721$	1.74873e7	1.25281
$722$	0	0
$723$	2.02178e7i	1.43843i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.69007e6i	− 0.118596i	−0.998240	−	0.0592978i	$-0.981114\pi$
$727$	− 1.69007e6i	− 0.118596i	0.998240	−	0.0592978i	$-0.0188862\pi$
$728$	0	0
$729$	−1.43489e7	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.13028e7	−0.777006	−0.388503	−	0.921448i	$-0.627008\pi$
$733$	−1.13028e7	−0.777006	−0.388503	+	0.921448i	$0.627008\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	2.29176e6i	0.154368i	0.997017	+	0.0771842i	$0.0245930\pi$
$739$	2.29176e6i	0.154368i	−0.997017	+	0.0771842i	$0.975407\pi$
$740$	0	0
$741$	−5.25106e7	−3.51319
$742$	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
$751$	− 3.08912e7i	− 1.99864i	−0.0368381	−	0.999321i	$-0.511729\pi$
$751$	− 3.08912e7i	− 1.99864i	0.0368381	−	0.999321i	$-0.488271\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.98526e7	1.25915	0.629575	−	0.776940i	$-0.283229\pi$
$757$	1.98526e7	1.25915	0.629575	+	0.776940i	$0.283229\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	1.22939e7i	0.764501i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	5.88288e6	0.358735	0.179368	−	0.983782i	$-0.442595\pi$
$769$	5.88288e6	0.358735	0.179368	+	0.983782i	$0.442595\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 8.80098e6i	− 0.526353i
$776$	0	0
$777$	−2.77047e7	−1.64627
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.05647e7i	− 0.608024i	−0.952668	−	0.304012i	$-0.901674\pi$
$787$	− 1.05647e7i	− 0.608024i	0.952668	−	0.304012i	$-0.0983263\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.64285e7	2.62181
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	2.97462e7i	1.58811i	0.607849	+	0.794053i	$0.292033\pi$
$811$	2.97462e7i	1.58811i	−0.607849	+	0.794053i	$0.707967\pi$
$812$	0	0
$813$	−1.36904e7	−0.726421
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.73056e7	−3.52773
$818$	0	0
$819$	− 3.13663e7i	− 1.63400i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	3.18330e7i	1.63824i	0.573623	+	0.819120i	$0.305538\pi$
$823$	3.18330e7i	1.63824i	−0.573623	+	0.819120i	$0.694462\pi$
$824$	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
$829$	3.94293e7	1.99266	0.996329	−	0.0856034i	$-0.0272818\pi$
$829$	3.94293e7	1.99266	0.996329	+	0.0856034i	$0.0272818\pi$
$830$	0	0
$831$	3.87625e7i	1.94719i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.06682e7	0.526353
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	2.05111e7	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.72948e7i	− 0.828337i
$848$	0	0
$849$	4.20048e7	2.00000
$850$	0	0
$851$	0	0
$852$	0	0
$853$	3.52107e7	1.65692	0.828461	−	0.560047i	$-0.189217\pi$
$853$	3.52107e7	1.65692	0.828461	+	0.560047i	$0.189217\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	− 4.31437e7i	− 1.99496i	−0.0709259	−	0.997482i	$-0.522595\pi$
$859$	− 4.31437e7i	− 1.99496i	0.0709259	−	0.997482i	$-0.477405\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.21334e7i	− 1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 7.73185e7i	− 3.45333i
$872$	0	0
$873$	−3.26558e7	−1.45019
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.83931e6	0.168560	0.0842800	−	0.996442i	$-0.473141\pi$
$877$	3.83931e6	0.168560	0.0842800	+	0.996442i	$0.473141\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 3.61730e7i	− 1.56129i	−0.624978	−	0.780643i	$-0.714892\pi$
$883$	− 3.61730e7i	− 1.56129i	0.624978	−	0.780643i	$-0.285108\pi$
$884$	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
$889$	−2.64816e7	−1.12380
$890$	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
$903$	− 4.02038e7i	− 1.64077i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.37251e7i	1.76487i	0.470433	+	0.882436i	$0.344098\pi$
$907$	4.37251e7i	1.76487i	−0.470433	+	0.882436i	$0.655902\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 2.18628e7i	− 0.853919i	−0.904271	−	0.426959i	$-0.859585\pi$
$919$	− 2.18628e7i	− 0.853919i	0.904271	−	0.426959i	$-0.140415\pi$
$920$	0	0
$921$	3.69316e7	1.43466
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−5.17188e7	−1.98744
$926$	0	0
$927$	3.95711e7i	1.51244i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	− 1.47830e7i	− 0.558969i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.36738e7	1.62507	0.812535	−	0.582913i	$-0.198087\pi$
$937$	4.36738e7	1.62507	0.812535	+	0.582913i	$0.198087\pi$
$938$	0	0
$939$	− 1.14403e7i	− 0.423423i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	1.74290e6	0.0628213
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.06975e7	0.722953
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.84615e6i	0.132269i	0.997811	+	0.0661347i	$0.0210667\pi$
$967$	3.84615e6i	0.132269i	−0.997811	+	0.0661347i	$0.978933\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−4.07247e7	−1.37904
$974$	0	0
$975$	− 5.85541e7i	− 1.97263i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−2.78191e7	−0.922935
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	4.61840e7i	1.49385i	0.664908	+	0.746925i	$0.268471\pi$
$991$	4.61840e7i	1.49385i	−0.664908	+	0.746925i	$0.731529\pi$
$992$	0	0
$993$	−3.81524e7	−1.22786
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.48154e7	1.42787	0.713937	−	0.700210i	$-0.246910\pi$
$997$	4.48154e7	1.42787	0.713937	+	0.700210i	$0.246910\pi$
$998$	0	0
$999$	− 6.26913e7i	− 1.98744i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.6.c.b.191.1		2
3.2	odd	2	CM	192.6.c.b.191.1		2
4.3	odd	2	inner	192.6.c.b.191.2		2
8.3	odd	2		48.6.c.b.47.1	✓	2
8.5	even	2		48.6.c.b.47.2	yes	2
12.11	even	2	inner	192.6.c.b.191.2		2
24.5	odd	2		48.6.c.b.47.2	yes	2
24.11	even	2		48.6.c.b.47.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.6.c.b.47.1	✓	2	8.3	odd	2
48.6.c.b.47.1	✓	2	24.11	even	2
48.6.c.b.47.2	yes	2	8.5	even	2
48.6.c.b.47.2	yes	2	24.5	odd	2
192.6.c.b.191.1		2	1.1	even	1	trivial
192.6.c.b.191.1		2	3.2	odd	2	CM
192.6.c.b.191.2		2	4.3	odd	2	inner
192.6.c.b.191.2		2	12.11	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-15.5885i q^{3} +107.387i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q-15.5885i q^{3} +107.387i q^{7} -243.000 q^{9} +1202.00 q^{13} -2802.46i q^{19} +1674.00 q^{21} +3125.00 q^{25} +3788.00i q^{27} -2816.31i q^{31} -16550.0 q^{37} -18737.3i q^{39} -24016.6i q^{43} +5275.00 q^{49} -43686.0 q^{57} +38626.0 q^{61} -26095.1i q^{63} -64324.9i q^{67} +1450.00 q^{73} -48713.9i q^{75} -46852.0i q^{79} +59049.0 q^{81} +129079. i q^{91} -43902.0 q^{93} +134386. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 486 q^{9}+O(q^{10}) \) 2 * q - 486 * q^9	\( 2 q - 486 q^{9} + 2404 q^{13} + 3348 q^{21} + 6250 q^{25} - 33100 q^{37} + 10550 q^{49} - 87372 q^{57} + 77252 q^{61} + 2900 q^{73} + 118098 q^{81} - 87804 q^{93} + 268772 q^{97}+O(q^{100}) \) 2 * q - 486 * q^9 + 2404 * q^13 + 3348 * q^21 + 6250 * q^25 - 33100 * q^37 + 10550 * q^49 - 87372 * q^57 + 77252 * q^61 + 2900 * q^73 + 118098 * q^81 - 87804 * q^93 + 268772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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