LMFDB - Embedded newform 1536.2.c.l.1535.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(1535,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1536.1535");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1535.1
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1536.1535
Dual form			1536.2.c.l.1535.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -4.44949i q^{5} +2.04989i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} -4.44949i q^{5} +2.04989i q^{7} +3.00000 q^{9} -5.65685 q^{11} +7.70674i q^{15} -3.55051i q^{21} -14.7980 q^{25} -5.19615 q^{27} +5.34847i q^{29} -3.60697i q^{31} +9.79796 q^{33} +9.12096 q^{35} -13.3485i q^{45} +2.79796 q^{49} +12.4495i q^{53} +25.1701i q^{55} +10.3923 q^{59} +6.14966i q^{63} -9.79796 q^{73} +25.6308 q^{75} -11.5959i q^{77} +17.4634i q^{79} +9.00000 q^{81} +5.65685 q^{83} -9.26382i q^{87} +6.24745i q^{93} -2.00000 q^{97} -16.9706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} - 40 q^{25} - 56 q^{49} + 72 q^{81} - 16 q^{97}+O(q^{100}) $ 8 * q + 24 * q^9 - 40 * q^25 - 56 * q^49 + 72 * q^81 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$.

$n$	$511$	$517$	$1025$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−1.73205	−1.00000
$3$	−1.73205	−1.00000
$4$	0	0
$5$	− 4.44949i	− 1.98987i	−0.100509	−	0.994936i	$-0.532047\pi$
$5$	− 4.44949i	− 1.98987i	0.100509	−	0.994936i	$-0.467953\pi$
$6$	0	0
$7$	2.04989i	0.774785i	0.921915	+	0.387392i	$0.126624\pi$
$7$	2.04989i	0.774785i	−0.921915	+	0.387392i	$0.873376\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−5.65685	−1.70561	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	−5.65685	−1.70561	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	7.70674i	1.98987i
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	− 3.55051i	− 0.774785i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−14.7980	−2.95959
$26$	0	0
$27$	−5.19615	−1.00000
$28$	0	0
$29$	5.34847i	0.993186i	0.867984	+	0.496593i	$0.165416\pi$
$29$	5.34847i	0.993186i	−0.867984	+	0.496593i	$0.834584\pi$
$30$	0	0
$31$	− 3.60697i	− 0.647830i	−0.946086	−	0.323915i	$-0.895001\pi$
$31$	− 3.60697i	− 0.647830i	0.946086	−	0.323915i	$-0.104999\pi$
$32$	0	0
$33$	9.79796	1.70561
$34$	0	0
$35$	9.12096	1.54172
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	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	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 13.3485i	− 1.98987i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	2.79796	0.399708
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.4495i	1.71007i	0.518571	+	0.855034i	$0.326464\pi$
$53$	12.4495i	1.71007i	−0.518571	+	0.855034i	$0.673536\pi$
$54$	0	0
$55$	25.1701i	3.39394i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	6.14966i	0.774785i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	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
$73$	−9.79796	−1.14676	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	−9.79796	−1.14676	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	25.6308	2.95959
$76$	0	0
$77$	− 11.5959i	− 1.32148i
$78$	0	0
$79$	17.4634i	1.96478i	0.186834	+	0.982391i	$0.440177\pi$
$79$	17.4634i	1.96478i	−0.186834	+	0.982391i	$0.559823\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	5.65685	0.620920	0.310460	−	0.950586i	$-0.399517\pi$
$83$	5.65685	0.620920	0.310460	+	0.950586i	$0.399517\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 9.26382i	− 0.993186i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.24745i	0.647830i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−16.9706	−1.70561
$100$	0	0
$101$	11.5505i	1.14932i	0.818393	+	0.574659i	$0.194865\pi$
$101$	11.5505i	1.14932i	−0.818393	+	0.574659i	$0.805135\pi$
$102$	0	0
$103$	0.492810i	0.0485580i	0.999705	+	0.0242790i	$0.00772901\pi$
$103$	0.492810i	0.0485580i	−0.999705	+	0.0242790i	$0.992271\pi$
$104$	0	0
$105$	−15.7980	−1.54172
$106$	0	0
$107$	−17.3205	−1.67444	−0.837218	−	0.546869i	$-0.815820\pi$
$107$	−17.3205	−1.67444	−0.837218	+	0.546869i	$0.815820\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	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
$121$	21.0000	1.90909
$122$	0	0
$123$	0	0
$124$	0	0
$125$	43.5959i	3.89934i
$126$	0	0
$127$	19.0205i	1.68779i	0.536507	+	0.843896i	$0.319744\pi$
$127$	19.0205i	1.68779i	−0.536507	+	0.843896i	$0.680256\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	23.1202i	1.98987i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	23.7980	1.97631
$146$	0	0
$147$	−4.84621	−0.399708
$148$	0	0
$149$	15.1464i	1.24084i	0.784268	+	0.620422i	$0.213039\pi$
$149$	15.1464i	1.24084i	−0.784268	+	0.620422i	$0.786961\pi$
$150$	0	0
$151$	− 15.9063i	− 1.29444i	−0.762305	−	0.647218i	$-0.775932\pi$
$151$	− 15.9063i	− 1.29444i	0.762305	−	0.647218i	$-0.224068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.0492	−1.28910
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	− 21.5631i	− 1.71007i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	− 43.5959i	− 3.39394i
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.75255i	0.133244i	0.997778	+	0.0666220i	$0.0212222\pi$
$173$	1.75255i	0.133244i	−0.997778	+	0.0666220i	$0.978778\pi$
$174$	0	0
$175$	− 30.3342i	− 2.29305i
$176$	0	0
$177$	−18.0000	−1.35296
$178$	0	0
$179$	−24.2487	−1.81243	−0.906217	−	0.422813i	$-0.861043\pi$
$179$	−24.2487	−1.81243	−0.906217	+	0.422813i	$0.861043\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 10.6515i	− 0.774785i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	9.79796	0.705273	0.352636	−	0.935760i	$-0.385285\pi$
$193$	9.79796	0.705273	0.352636	+	0.935760i	$0.385285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.14643i	− 0.509162i	−0.967051	−	0.254581i	$-0.918062\pi$
$197$	− 7.14643i	− 0.509162i	0.967051	−	0.254581i	$-0.0819375\pi$
$198$	0	0
$199$	− 27.2200i	− 1.92957i	−0.263031	−	0.964787i	$-0.584722\pi$
$199$	− 27.2200i	− 1.92957i	0.263031	−	0.964787i	$-0.415278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.9638	−0.769505
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.39388	0.501929
$218$	0	0
$219$	16.9706	1.14676
$220$	0	0
$221$	0	0
$222$	0	0
$223$	28.7771i	1.92706i	0.267608	+	0.963528i	$0.413767\pi$
$223$	28.7771i	1.92706i	−0.267608	+	0.963528i	$0.586233\pi$
$224$	0	0
$225$	−44.3939	−2.95959
$226$	0	0
$227$	28.2843	1.87729	0.938647	−	0.344881i	$-0.112081\pi$
$227$	28.2843	1.87729	0.938647	+	0.344881i	$0.112081\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	20.0847i	1.32148i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 30.2474i	− 1.96478i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−29.3939	−1.89343	−0.946713	−	0.322078i	$-0.895619\pi$
$241$	−29.3939	−1.89343	−0.946713	+	0.322078i	$0.895619\pi$
$242$	0	0
$243$	−15.5885	−1.00000
$244$	0	0
$245$	− 12.4495i	− 0.795369i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−9.79796	−0.620920
$250$	0	0
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	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
$261$	16.0454i	0.993186i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	55.3939	3.40282
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 29.3485i	− 1.78941i	−0.446660	−	0.894704i	$-0.647387\pi$
$269$	− 29.3485i	− 1.78941i	0.446660	−	0.894704i	$-0.352613\pi$
$270$	0	0
$271$	− 32.8769i	− 1.99713i	−0.0535825	−	0.998563i	$-0.517064\pi$
$271$	− 32.8769i	− 1.99713i	0.0535825	−	0.998563i	$-0.482936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	83.7099	5.04790
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	− 10.8209i	− 0.647830i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	3.46410	0.203069
$292$	0	0
$293$	32.0454i	1.87211i	0.351850	+	0.936056i	$0.385553\pi$
$293$	32.0454i	1.87211i	−0.351850	+	0.936056i	$0.614447\pi$
$294$	0	0
$295$	− 46.2405i	− 2.69222i
$296$	0	0
$297$	29.3939	1.70561
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 20.0061i	− 1.14932i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	− 0.853572i	− 0.0485580i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−9.79796	−0.553813	−0.276907	−	0.960897i	$-0.589309\pi$
$313$	−9.79796	−0.553813	−0.276907	+	0.960897i	$0.589309\pi$
$314$	0	0
$315$	27.3629	1.54172
$316$	0	0
$317$	− 9.75255i	− 0.547758i	−0.961764	−	0.273879i	$-0.911693\pi$
$317$	− 9.75255i	− 0.547758i	0.961764	−	0.273879i	$-0.0883068\pi$
$318$	0	0
$319$	− 30.2555i	− 1.69398i
$320$	0	0
$321$	30.0000	1.67444
$322$	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	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.3939	−1.60119	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	−29.3939	−1.60119	−0.800593	+	0.599208i	$0.795482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.4041i	1.10494i
$342$	0	0
$343$	20.0847i	1.08447i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.2843	−1.51838	−0.759190	−	0.650870i	$-0.774404\pi$
$347$	−28.2843	−1.51838	−0.759190	+	0.650870i	$0.774404\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	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
$361$	19.0000	1.00000
$362$	0	0
$363$	−36.3731	−1.90909
$364$	0	0
$365$	43.5959i	2.28191i
$366$	0	0
$367$	30.3342i	1.58343i	0.610890	+	0.791715i	$0.290812\pi$
$367$	30.3342i	1.58343i	−0.610890	+	0.791715i	$0.709188\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−25.5201	−1.32494
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	− 75.5103i	− 3.89934i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	− 32.9444i	− 1.68779i
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−51.5959	−2.62957
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 39.1464i	− 1.98480i	−0.123043	−	0.992401i	$-0.539265\pi$
$389$	− 39.1464i	− 1.98480i	0.123043	−	0.992401i	$-0.460735\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	77.7031	3.90967
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	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
$405$	− 40.0454i	− 1.98987i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	21.3031i	1.04826i
$414$	0	0
$415$	− 25.1701i	− 1.23555i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−39.5980	−1.93449	−0.967244	−	0.253849i	$-0.918303\pi$
$419$	−39.5980	−1.93449	−0.967244	+	0.253849i	$0.918303\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	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
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−41.2193	−1.97631
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.72112i	0.320782i	0.987054	+	0.160391i	$0.0512755\pi$
$439$	6.72112i	0.320782i	−0.987054	+	0.160391i	$0.948725\pi$
$440$	0	0
$441$	8.39388	0.399708
$442$	0	0
$443$	−28.2843	−1.34383	−0.671913	−	0.740630i	$-0.734527\pi$
$443$	−28.2843	−1.34383	−0.671913	+	0.740630i	$0.734527\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 26.2344i	− 1.24084i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	27.5505i	1.29444i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	40.9444i	1.90697i	0.301440	+	0.953485i	$0.402533\pi$
$461$	40.9444i	1.90697i	−0.301440	+	0.953485i	$0.597467\pi$
$462$	0	0
$463$	− 42.6335i	− 1.98135i	−0.136260	−	0.990673i	$-0.543508\pi$
$463$	− 42.6335i	− 1.98135i	0.136260	−	0.990673i	$-0.456492\pi$
$464$	0	0
$465$	27.7980	1.28910
$466$	0	0
$467$	−39.5980	−1.83238	−0.916188	−	0.400749i	$-0.868750\pi$
$467$	−39.5980	−1.83238	−0.916188	+	0.400749i	$0.868750\pi$
$468$	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
$477$	37.3485i	1.71007i
$478$	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
$485$	8.89898i	0.404082i
$486$	0	0
$487$	29.7627i	1.34868i	0.738422	+	0.674338i	$0.235571\pi$
$487$	29.7627i	1.34868i	−0.738422	+	0.674338i	$0.764429\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.1051	−1.71966	−0.859830	−	0.510581i	$-0.829431\pi$
$491$	−38.1051	−1.71966	−0.859830	+	0.510581i	$0.829431\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	75.5103i	3.39394i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	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
$505$	51.3939	2.28700
$506$	0	0
$507$	22.5167	1.00000
$508$	0	0
$509$	− 33.8434i	− 1.50008i	−0.661392	−	0.750040i	$-0.730034\pi$
$509$	− 33.8434i	− 1.50008i	0.661392	−	0.750040i	$-0.269966\pi$
$510$	0	0
$511$	− 20.0847i	− 0.888496i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.19275	0.0966242
$516$	0	0
$517$	0	0
$518$	0	0
$519$	− 3.03551i	− 0.133244i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	52.5403i	2.29305i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	31.1769	1.35296
$532$	0	0
$533$	0	0
$534$	0	0
$535$	77.0674i	3.33192i
$536$	0	0
$537$	42.0000	1.81243
$538$	0	0
$539$	−15.8276	−0.681745
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−35.7980	−1.52228
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.8434i	1.77296i	0.462767	+	0.886480i	$0.346857\pi$
$557$	41.8434i	1.77296i	−0.462767	+	0.886480i	$0.653143\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.2843	1.19204	0.596020	−	0.802970i	$-0.296748\pi$
$563$	28.2843	1.19204	0.596020	+	0.802970i	$0.296748\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.4490i	0.774785i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−29.3939	−1.22368	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	−29.3939	−1.22368	−0.611842	+	0.790980i	$0.709571\pi$
$578$	0	0
$579$	−16.9706	−0.705273
$580$	0	0
$581$	11.5959i	0.481080i
$582$	0	0
$583$	− 70.4249i	− 2.91670i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.3205	0.714894	0.357447	−	0.933933i	$-0.383647\pi$
$587$	17.3205	0.714894	0.357447	+	0.933933i	$0.383647\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.3780i	0.509162i
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	47.1464i	1.92957i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−48.9898	−1.99834	−0.999168	−	0.0407909i	$-0.987012\pi$
$601$	−48.9898	−1.99834	−0.999168	+	0.0407909i	$0.987012\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 93.4393i	− 3.79885i
$606$	0	0
$607$	46.7333i	1.89684i	0.317010	+	0.948422i	$0.397321\pi$
$607$	46.7333i	1.89684i	−0.317010	+	0.948422i	$0.602679\pi$
$608$	0	0
$609$	18.9898	0.769505
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	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	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	119.990	4.79959
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 31.8912i	− 1.26957i	−0.772689	−	0.634785i	$-0.781089\pi$
$631$	− 31.8912i	− 1.26957i	0.772689	−	0.634785i	$-0.218911\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	84.6313	3.35849
$636$	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	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	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
$649$	−58.7878	−2.30762
$650$	0	0
$651$	−12.8066	−0.501929
$652$	0	0
$653$	− 46.2474i	− 1.80980i	−0.425622	−	0.904901i	$-0.639945\pi$
$653$	− 46.2474i	− 1.80980i	0.425622	−	0.904901i	$-0.360055\pi$
$654$	0	0
$655$	15.4135i	0.602255i
$656$	0	0
$657$	−29.3939	−1.14676
$658$	0	0
$659$	24.2487	0.944596	0.472298	−	0.881439i	$-0.343425\pi$
$659$	24.2487	0.944596	0.472298	+	0.881439i	$0.343425\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 49.8434i	− 1.92706i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	76.8925	2.95959
$676$	0	0
$677$	34.7423i	1.33526i	0.744495	+	0.667628i	$0.232690\pi$
$677$	34.7423i	1.33526i	−0.744495	+	0.667628i	$0.767310\pi$
$678$	0	0
$679$	− 4.09978i	− 0.157335i
$680$	0	0
$681$	−48.9898	−1.87729
$682$	0	0
$683$	−5.65685	−0.216454	−0.108227	−	0.994126i	$-0.534517\pi$
$683$	−5.65685	−0.216454	−0.108227	+	0.994126i	$0.534517\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	− 34.7878i	− 1.32148i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 0.944387i	− 0.0356690i	−0.999841	−	0.0178345i	$-0.994323\pi$
$701$	− 0.944387i	− 0.0356690i	0.999841	−	0.0178345i	$-0.00567720\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−23.6773	−0.890475
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	52.3901i	1.96478i
$712$	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
$721$	−1.01021	−0.0376220
$722$	0	0
$723$	50.9117	1.89343
$724$	0	0
$725$	− 79.1464i	− 2.93942i
$726$	0	0
$727$	39.5193i	1.46569i	0.680395	+	0.732845i	$0.261808\pi$
$727$	39.5193i	1.46569i	−0.680395	+	0.732845i	$0.738192\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	21.5631i	0.795369i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	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
$745$	67.3939	2.46912
$746$	0	0
$747$	16.9706	0.620920
$748$	0	0
$749$	− 35.5051i	− 1.29733i
$750$	0	0
$751$	45.1762i	1.64850i	0.566224	+	0.824251i	$0.308404\pi$
$751$	45.1762i	1.64850i	−0.566224	+	0.824251i	$0.691596\pi$
$752$	0	0
$753$	9.79796	0.357057
$754$	0	0
$755$	−70.7749	−2.57576
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	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
$769$	48.9898	1.76662	0.883309	−	0.468792i	$-0.155311\pi$
$769$	48.9898	1.76662	0.883309	+	0.468792i	$0.155311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	50.7423i	1.82508i	0.408993	+	0.912538i	$0.365880\pi$
$773$	50.7423i	1.82508i	−0.408993	+	0.912538i	$0.634120\pi$
$774$	0	0
$775$	53.3757i	1.91731i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 27.7915i	− 0.993186i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−95.9450	−3.40282
$796$	0	0
$797$	− 25.7526i	− 0.912202i	−0.889928	−	0.456101i	$-0.849246\pi$
$797$	− 25.7526i	− 0.912202i	0.889928	−	0.456101i	$-0.150754\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	55.4256	1.95593
$804$	0	0
$805$	0	0
$806$	0	0
$807$	50.8330i	1.78941i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	56.9444i	1.99713i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 56.0454i	− 1.95600i	−0.208609	−	0.977999i	$-0.566894\pi$
$821$	− 56.0454i	− 1.95600i	0.208609	−	0.977999i	$-0.433106\pi$
$822$	0	0
$823$	8.27820i	0.288560i	0.989537	+	0.144280i	$0.0460866\pi$
$823$	8.27820i	0.288560i	−0.989537	+	0.144280i	$0.953913\pi$
$824$	0	0
$825$	−144.990	−5.04790
$826$	0	0
$827$	10.3923	0.361376	0.180688	−	0.983540i	$-0.442168\pi$
$827$	10.3923	0.361376	0.180688	+	0.983540i	$0.442168\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	18.7423i	0.647830i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	0.393877	0.0135820
$842$	0	0
$843$	0	0
$844$	0	0
$845$	57.8434i	1.98987i
$846$	0	0
$847$	43.0476i	1.47913i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	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	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	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
$865$	7.79796	0.265139
$866$	0	0
$867$	−29.4449	−1.00000
$868$	0	0
$869$	− 98.7878i	− 3.35115i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	−89.3668	−3.02115
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	− 55.5043i	− 1.87211i
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	80.0908i	2.69222i
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−38.9898	−1.30768
$890$	0	0
$891$	−50.9117	−1.70561
$892$	0	0
$893$	0	0
$894$	0	0
$895$	107.894i	3.60651i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.2917	0.643416
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	34.6515i	1.14932i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.10102i	− 0.234496i
$918$	0	0
$919$	− 59.6041i	− 1.96616i	−0.183187	−	0.983078i	$-0.558641\pi$
$919$	− 59.6041i	− 1.96616i	0.183187	−	0.983078i	$-0.441359\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.47843i	0.0485580i
$928$	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
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	16.9706	0.553813
$940$	0	0
$941$	− 7.05561i	− 0.230006i	−0.993365	−	0.115003i	$-0.963312\pi$
$941$	− 7.05561i	− 0.230006i	0.993365	−	0.115003i	$-0.0366878\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−47.3939	−1.54172
$946$	0	0
$947$	−24.2487	−0.787977	−0.393989	−	0.919115i	$-0.628905\pi$
$947$	−24.2487	−0.787977	−0.393989	+	0.919115i	$0.628905\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	16.8919i	0.547758i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	52.4041i	1.69398i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	17.9898	0.580316
$962$	0	0
$963$	−51.9615	−1.67444
$964$	0	0
$965$	− 43.5959i	− 1.40340i
$966$	0	0
$967$	47.3047i	1.52122i	0.649211	+	0.760609i	$0.275099\pi$
$967$	47.3047i	1.52122i	−0.649211	+	0.760609i	$0.724901\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	62.2254	1.99691	0.998454	−	0.0555842i	$-0.0177021\pi$
$971$	62.2254	1.99691	0.998454	+	0.0555842i	$0.0177021\pi$
$972$	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
$985$	−31.7980	−1.01317
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	23.6917i	0.752591i	0.926500	+	0.376296i	$0.122802\pi$
$991$	23.6917i	0.752591i	−0.926500	+	0.376296i	$0.877198\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−121.115	−3.83961
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.c.l.1535.1	✓	8
3.2	odd	2	inner	1536.2.c.l.1535.8	yes	8
4.3	odd	2	inner	1536.2.c.l.1535.5	yes	8
8.3	odd	2	inner	1536.2.c.l.1535.4	yes	8
8.5	even	2	inner	1536.2.c.l.1535.8	yes	8
12.11	even	2	inner	1536.2.c.l.1535.4	yes	8
16.3	odd	4		1536.2.f.b.767.3		4
16.5	even	4		1536.2.f.h.767.4		4
16.11	odd	4		1536.2.f.h.767.2		4
16.13	even	4		1536.2.f.b.767.1		4
24.5	odd	2	CM	1536.2.c.l.1535.1	✓	8
24.11	even	2	inner	1536.2.c.l.1535.5	yes	8
48.5	odd	4		1536.2.f.b.767.1		4
48.11	even	4		1536.2.f.b.767.3		4
48.29	odd	4		1536.2.f.h.767.4		4
48.35	even	4		1536.2.f.h.767.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.c.l.1535.1	✓	8	1.1	even	1	trivial
1536.2.c.l.1535.1	✓	8	24.5	odd	2	CM
1536.2.c.l.1535.4	yes	8	8.3	odd	2	inner
1536.2.c.l.1535.4	yes	8	12.11	even	2	inner
1536.2.c.l.1535.5	yes	8	4.3	odd	2	inner
1536.2.c.l.1535.5	yes	8	24.11	even	2	inner
1536.2.c.l.1535.8	yes	8	3.2	odd	2	inner
1536.2.c.l.1535.8	yes	8	8.5	even	2	inner
1536.2.f.b.767.1		4	16.13	even	4
1536.2.f.b.767.1		4	48.5	odd	4
1536.2.f.b.767.3		4	16.3	odd	4
1536.2.f.b.767.3		4	48.11	even	4
1536.2.f.h.767.2		4	16.11	odd	4
1536.2.f.h.767.2		4	48.35	even	4
1536.2.f.h.767.4		4	16.5	even	4
1536.2.f.h.767.4		4	48.29	odd	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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -4.44949i q^{5} +2.04989i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} -4.44949i q^{5} +2.04989i q^{7} +3.00000 q^{9} -5.65685 q^{11} +7.70674i q^{15} -3.55051i q^{21} -14.7980 q^{25} -5.19615 q^{27} +5.34847i q^{29} -3.60697i q^{31} +9.79796 q^{33} +9.12096 q^{35} -13.3485i q^{45} +2.79796 q^{49} +12.4495i q^{53} +25.1701i q^{55} +10.3923 q^{59} +6.14966i q^{63} -9.79796 q^{73} +25.6308 q^{75} -11.5959i q^{77} +17.4634i q^{79} +9.00000 q^{81} +5.65685 q^{83} -9.26382i q^{87} +6.24745i q^{93} -2.00000 q^{97} -16.9706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} - 40 q^{25} - 56 q^{49} + 72 q^{81} - 16 q^{97}+O(q^{100}) \) 8 * q + 24 * q^9 - 40 * q^25 - 56 * q^49 + 72 * q^81 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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