LMFDB - Embedded newform 768.4.f.a.383.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(383,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.f (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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			383.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.4.f.a.383.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} -31.1769i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} -31.1769i q^{7} +27.0000 q^{9} -70.0000i q^{13} -155.885 q^{19} -162.000i q^{21} -125.000 q^{25} +140.296 q^{27} +155.885i q^{31} +110.000i q^{37} -363.731i q^{39} -218.238 q^{43} -629.000 q^{49} -810.000 q^{57} -182.000i q^{61} -841.777i q^{63} -654.715 q^{67} +1190.00 q^{73} -649.519 q^{75} -1091.19i q^{79} +729.000 q^{81} -2182.38 q^{91} +810.000i q^{93} +1330.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{9}+O(q^{10}) $ 4 * q + 108 * q^9	$ 4 q + 108 q^{9} - 500 q^{25} - 2516 q^{49} - 3240 q^{57} + 4760 q^{73} + 2916 q^{81} + 5320 q^{97}+O(q^{100}) $ 4 * q + 108 * q^9 - 500 * q^25 - 2516 * q^49 - 3240 * q^57 + 4760 * q^73 + 2916 * q^81 + 5320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	5.19615	1.00000
$3$	5.19615	1.00000
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	− 31.1769i	− 1.68340i	−0.539949	−	0.841698i	$-0.681557\pi$
$7$	− 31.1769i	− 1.68340i	0.539949	−	0.841698i	$-0.318443\pi$
$8$	0	0
$9$	27.0000	1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 70.0000i	− 1.49342i	−0.665148	−	0.746712i	$-0.731631\pi$
$13$	− 70.0000i	− 1.49342i	0.665148	−	0.746712i	$-0.268369\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−155.885	−1.88223	−0.941115	−	0.338086i	$-0.890220\pi$
$19$	−155.885	−1.88223	−0.941115	+	0.338086i	$0.890220\pi$
$20$	0	0
$21$	− 162.000i	− 1.68340i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	0	0
$27$	140.296	1.00000
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	155.885i	0.903151i	0.892233	+	0.451576i	$0.149138\pi$
$31$	155.885i	0.903151i	−0.892233	+	0.451576i	$0.850862\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	110.000i	0.488754i	0.969680	+	0.244377i	$0.0785834\pi$
$37$	110.000i	0.488754i	−0.969680	+	0.244377i	$0.921417\pi$
$38$	0	0
$39$	− 363.731i	− 1.49342i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−218.238	−0.773978	−0.386989	−	0.922084i	$-0.626485\pi$
$43$	−218.238	−0.773978	−0.386989	+	0.922084i	$0.626485\pi$
$44$	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
$49$	−629.000	−1.83382
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−810.000	−1.88223
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 182.000i	− 0.382012i	−0.981589	−	0.191006i	$-0.938825\pi$
$61$	− 182.000i	− 0.382012i	0.981589	−	0.191006i	$-0.0611749\pi$
$62$	0	0
$63$	− 841.777i	− 1.68340i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−654.715	−1.19382	−0.596912	−	0.802307i	$-0.703606\pi$
$67$	−654.715	−1.19382	−0.596912	+	0.802307i	$0.703606\pi$
$68$	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
$73$	1190.00	1.90793	0.953966	−	0.299916i	$-0.0969588\pi$
$73$	1190.00	1.90793	0.953966	+	0.299916i	$0.0969588\pi$
$74$	0	0
$75$	−649.519	−1.00000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 1091.19i	− 1.55403i	−0.629480	−	0.777017i	$-0.716732\pi$
$79$	− 1091.19i	− 1.55403i	0.629480	−	0.777017i	$-0.283268\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	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
$91$	−2182.38	−2.51402
$92$	0	0
$93$	810.000i	0.903151i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1330.00	1.39218	0.696088	−	0.717957i	$-0.254922\pi$
$97$	1330.00	1.39218	0.696088	+	0.717957i	$0.254922\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 1028.84i	− 0.984218i	−0.870534	−	0.492109i	$-0.836226\pi$
$103$	− 1028.84i	− 0.984218i	0.870534	−	0.492109i	$-0.163774\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 646.000i	− 0.567666i	−0.958874	−	0.283833i	$-0.908394\pi$
$109$	− 646.000i	− 0.567666i	0.958874	−	0.283833i	$-0.0916061\pi$
$110$	0	0
$111$	571.577i	0.488754i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1890.00i	− 1.49342i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1331.00	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2837.10i	− 1.98230i	−0.132754	−	0.991149i	$-0.542382\pi$
$127$	− 2837.10i	− 1.98230i	0.132754	−	0.991149i	$-0.457618\pi$
$128$	0	0
$129$	−1134.00	−0.773978
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	4860.00i	3.16854i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	2026.50	1.23659	0.618293	−	0.785948i	$-0.287825\pi$
$139$	2026.50	1.23659	0.618293	+	0.785948i	$0.287825\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3268.38	−1.83382
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	− 3273.58i	− 1.76424i	−0.471027	−	0.882119i	$-0.656117\pi$
$151$	− 3273.58i	− 1.76424i	0.471027	−	0.882119i	$-0.343883\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3850.00i	1.95709i	0.206028	+	0.978546i	$0.433946\pi$
$157$	3850.00i	1.95709i	−0.206028	+	0.978546i	$0.566054\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2400.62	−1.15357	−0.576783	−	0.816897i	$-0.695692\pi$
$163$	−2400.62	−1.15357	−0.576783	+	0.816897i	$0.695692\pi$
$164$	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
$169$	−2703.00	−1.23031
$170$	0	0
$171$	−4208.88	−1.88223
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	3897.11i	1.68340i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 3458.00i	− 1.42006i	−0.704171	−	0.710031i	$-0.748681\pi$
$181$	− 3458.00i	− 1.42006i	0.704171	−	0.710031i	$-0.251319\pi$
$182$	0	0
$183$	− 945.700i	− 0.382012i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 4374.00i	− 1.68340i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−1150.00	−0.428906	−0.214453	−	0.976734i	$-0.568797\pi$
$193$	−1150.00	−0.428906	−0.214453	+	0.976734i	$0.568797\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	− 2026.50i	− 0.721883i	−0.932588	−	0.360942i	$-0.882455\pi$
$199$	− 2026.50i	− 0.721883i	0.932588	−	0.360942i	$-0.117545\pi$
$200$	0	0
$201$	−3402.00	−1.19382
$202$	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
$211$	1091.19	0.356023	0.178011	−	0.984028i	$-0.443034\pi$
$211$	1091.19	0.356023	0.178011	+	0.984028i	$0.443034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4860.00	1.52036
$218$	0	0
$219$	6183.42	1.90793
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 5830.08i	− 1.75072i	−0.483469	−	0.875362i	$-0.660623\pi$
$223$	− 5830.08i	− 1.75072i	0.483469	−	0.875362i	$-0.339377\pi$
$224$	0	0
$225$	−3375.00	−1.00000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 4466.00i	− 1.28874i	−0.764714	−	0.644370i	$-0.777120\pi$
$229$	− 4466.00i	− 1.28874i	0.764714	−	0.644370i	$-0.222880\pi$
$230$	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
$237$	− 5670.00i	− 1.55403i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	7378.00	1.97203	0.986014	−	0.166662i	$-0.0532990\pi$
$241$	7378.00	1.97203	0.986014	+	0.166662i	$0.0532990\pi$
$242$	0	0
$243$	3788.00	1.00000
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10911.9i	2.81097i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	3429.46	0.822766
$260$	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		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	8885.42i	1.99170i	0.0910064	+	0.995850i	$0.470992\pi$
$271$	8885.42i	1.99170i	−0.0910064	+	0.995850i	$0.529008\pi$
$272$	0	0
$273$	−11340.0	−2.51402
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4030.00i	0.874149i	0.899425	+	0.437074i	$0.143985\pi$
$277$	4030.00i	0.874149i	−0.899425	+	0.437074i	$0.856015\pi$
$278$	0	0
$279$	4208.88i	0.903151i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−7700.70	−1.61752	−0.808761	−	0.588137i	$-0.799862\pi$
$283$	−7700.70	−1.61752	−0.808761	+	0.588137i	$0.799862\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4913.00	1.00000
$290$	0	0
$291$	6910.88	1.39218
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6804.00i	1.30291i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1590.02	0.295594	0.147797	−	0.989018i	$-0.452782\pi$
$307$	1590.02	0.295594	0.147797	+	0.989018i	$0.452782\pi$
$308$	0	0
$309$	− 5346.00i	− 0.984218i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−10010.0	−1.80766	−0.903832	−	0.427888i	$-0.859258\pi$
$313$	−10010.0	−1.80766	−0.903832	+	0.427888i	$0.859258\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	8750.00i	1.49342i
$326$	0	0
$327$	− 3356.71i	− 0.567666i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12003.1	1.99320	0.996602	−	0.0823644i	$-0.0262471\pi$
$331$	12003.1	1.99320	0.996602	+	0.0823644i	$0.0262471\pi$
$332$	0	0
$333$	2970.00i	0.488754i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4930.00	0.796897	0.398448	−	0.917191i	$-0.369549\pi$
$337$	4930.00	0.796897	0.398448	+	0.917191i	$0.369549\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8916.60i	1.40365i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	11914.0i	1.82734i	0.406456	+	0.913670i	$0.366764\pi$
$349$	11914.0i	1.82734i	−0.406456	+	0.913670i	$0.633236\pi$
$350$	0	0
$351$	− 9820.73i	− 1.49342i
$352$	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
$361$	17441.0	2.54279
$362$	0	0
$363$	6916.08	1.00000
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13374.9i	1.90235i	0.308646	+	0.951177i	$0.400124\pi$
$367$	13374.9i	1.90235i	−0.308646	+	0.951177i	$0.599876\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12350.0i	1.71437i	0.515011	+	0.857183i	$0.327788\pi$
$373$	12350.0i	1.71437i	−0.515011	+	0.857183i	$0.672212\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	12003.1	1.62680	0.813402	−	0.581702i	$-0.197613\pi$
$379$	12003.1	1.62680	0.813402	+	0.581702i	$0.197613\pi$
$380$	0	0
$381$	− 14742.0i	− 1.98230i
$382$	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
$387$	−5892.44	−0.773978
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1190.00i	− 0.150439i	−0.997167	−	0.0752196i	$-0.976034\pi$
$397$	− 1190.00i	− 0.150439i	0.997167	−	0.0752196i	$-0.0239658\pi$
$398$	0	0
$399$	25253.3i	3.16854i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	10911.9	1.34879
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	8246.00	0.996916	0.498458	−	0.866914i	$-0.333900\pi$
$409$	8246.00	0.996916	0.498458	+	0.866914i	$0.333900\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10530.0	1.23659
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	− 17138.0i	− 1.98398i	−0.126322	−	0.991989i	$-0.540317\pi$
$421$	− 17138.0i	− 1.98398i	0.126322	−	0.991989i	$-0.459683\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5674.20	−0.643077
$428$	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
$433$	−2590.00	−0.287454	−0.143727	−	0.989617i	$-0.545909\pi$
$433$	−2590.00	−0.287454	−0.143727	+	0.989617i	$0.545909\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 10756.0i	− 1.16938i	−0.811257	−	0.584690i	$-0.801216\pi$
$439$	− 10756.0i	− 1.16938i	0.811257	−	0.584690i	$-0.198784\pi$
$440$	0	0
$441$	−16983.0	−1.83382
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	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
$453$	− 17010.0i	− 1.76424i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12710.0	1.30098	0.650491	−	0.759514i	$-0.274563\pi$
$457$	12710.0	1.30098	0.650491	+	0.759514i	$0.274563\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	2400.62i	0.240964i	0.992716	+	0.120482i	$0.0384440\pi$
$463$	2400.62i	0.240964i	−0.992716	+	0.120482i	$0.961556\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	20412.0i	2.00968i
$470$	0	0
$471$	20005.2i	1.95709i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	19485.6	1.88223
$476$	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
$481$	7700.00	0.729916
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5019.48i	− 0.467052i	−0.972351	−	0.233526i	$-0.924974\pi$
$487$	− 5019.48i	− 0.467052i	0.972351	−	0.233526i	$-0.0750265\pi$
$488$	0	0
$489$	−12474.0	−1.15357
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−16367.9	−1.46839	−0.734195	−	0.678938i	$-0.762440\pi$
$499$	−16367.9	−1.46839	−0.734195	+	0.678938i	$0.762440\pi$
$500$	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
$507$	−14045.2	−1.23031
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	− 37100.5i	− 3.21180i
$512$	0	0
$513$	−21870.0	−1.88223
$514$	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
$523$	−20670.3	−1.72820	−0.864100	−	0.503320i	$-0.832112\pi$
$523$	−20670.3	−1.72820	−0.864100	+	0.503320i	$0.832112\pi$
$524$	0	0
$525$	20250.0i	1.68340i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−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
$541$	− 22678.0i	− 1.80222i	−0.433586	−	0.901112i	$-0.642752\pi$
$541$	− 22678.0i	− 1.80222i	0.433586	−	0.901112i	$-0.357248\pi$
$542$	0	0
$543$	− 17968.3i	− 1.42006i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25533.9	1.99589	0.997944	−	0.0640963i	$-0.0204165\pi$
$547$	25533.9	1.99589	0.997944	+	0.0640963i	$0.0204165\pi$
$548$	0	0
$549$	− 4914.00i	− 0.382012i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−34020.0	−2.61605
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	15276.7i	1.15588i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 22728.0i	− 1.68340i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−14185.5	−1.03966	−0.519829	−	0.854270i	$-0.674004\pi$
$571$	−14185.5	−1.03966	−0.519829	+	0.854270i	$0.674004\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−17710.0	−1.27778	−0.638888	−	0.769300i	$-0.720605\pi$
$577$	−17710.0	−1.27778	−0.638888	+	0.769300i	$0.720605\pi$
$578$	0	0
$579$	−5975.58	−0.428906
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	− 24300.0i	− 1.69994i
$590$	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
$597$	− 10530.0i	− 0.721883i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	29302.0	1.98877	0.994387	−	0.105801i	$-0.0337408\pi$
$601$	29302.0	1.98877	0.994387	+	0.105801i	$0.0337408\pi$
$602$	0	0
$603$	−17677.3	−1.19382
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 9321.90i	− 0.623335i	−0.950191	−	0.311667i	$-0.899113\pi$
$607$	− 9321.90i	− 0.623335i	0.950191	−	0.311667i	$-0.100887\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	17390.0i	1.14580i	0.819625	+	0.572900i	$0.194182\pi$
$613$	17390.0i	1.14580i	−0.819625	+	0.572900i	$0.805818\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−15432.6	−1.00208	−0.501040	−	0.865424i	$-0.667049\pi$
$619$	−15432.6	−1.00208	−0.501040	+	0.865424i	$0.667049\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	31644.6i	1.99643i	0.0596825	+	0.998217i	$0.480991\pi$
$631$	31644.6i	1.99643i	−0.0596825	+	0.998217i	$0.519009\pi$
$632$	0	0
$633$	5670.00	0.356023
$634$	0	0
$635$	0	0
$636$	0	0
$637$	44030.0i	2.73867i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−29836.3	−1.82991	−0.914953	−	0.403561i	$-0.867772\pi$
$643$	−29836.3	−1.82991	−0.914953	+	0.403561i	$0.867772\pi$
$644$	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
$651$	25253.3	1.52036
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32130.0	1.90793
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 20482.0i	− 1.20523i	−0.798032	−	0.602615i	$-0.794125\pi$
$661$	− 20482.0i	− 1.20523i	0.798032	−	0.602615i	$-0.205875\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 30294.0i	− 1.75072i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	24050.0	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$673$	24050.0	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$674$	0	0
$675$	−17537.0	−1.00000
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	− 41465.3i	− 2.34358i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 23206.0i	− 1.28874i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−32579.9	−1.79363	−0.896814	−	0.442408i	$-0.854124\pi$
$691$	−32579.9	−1.79363	−0.896814	+	0.442408i	$0.854124\pi$
$692$	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		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	− 17147.3i	− 0.919947i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 36146.0i	− 1.91466i	−0.289003	−	0.957328i	$-0.593324\pi$
$709$	− 36146.0i	− 1.91466i	0.289003	−	0.957328i	$-0.406676\pi$
$710$	0	0
$711$	− 29462.2i	− 1.55403i
$712$	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
$721$	−32076.0	−1.65683
$722$	0	0
$723$	38337.2	1.97203
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37692.9i	− 1.92290i	−0.274971	−	0.961452i	$-0.588668\pi$
$727$	− 37692.9i	− 1.92290i	0.274971	−	0.961452i	$-0.411332\pi$
$728$	0	0
$729$	19683.0	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	15050.0i	0.758369i	0.925321	+	0.379184i	$0.123795\pi$
$733$	15050.0i	0.758369i	−0.925321	+	0.379184i	$0.876205\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−25097.4	−1.24929	−0.624644	−	0.780910i	$-0.714756\pi$
$739$	−25097.4	−1.24929	−0.624644	+	0.780910i	$0.714756\pi$
$740$	0	0
$741$	56700.0i	2.81097i
$742$	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
$751$	33827.0i	1.64363i	0.569757	+	0.821813i	$0.307037\pi$
$751$	33827.0i	1.64363i	−0.569757	+	0.821813i	$0.692963\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41470.0i	1.99109i	0.0943039	+	0.995543i	$0.469937\pi$
$757$	41470.0i	1.99109i	−0.0943039	+	0.995543i	$0.530063\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	−20140.3	−0.955606
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−4606.00	−0.215990	−0.107995	−	0.994151i	$-0.534443\pi$
$769$	−4606.00	−0.215990	−0.107995	+	0.994151i	$0.534443\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	− 19485.6i	− 0.903151i
$776$	0	0
$777$	17820.0	0.822766
$778$	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
$787$	−8137.17	−0.368563	−0.184281	−	0.982874i	$-0.558996\pi$
$787$	−8137.17	−0.368563	−0.184281	+	0.982874i	$0.558996\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12740.0	−0.570505
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−24162.1	−1.04617	−0.523087	−	0.852280i	$-0.675220\pi$
$811$	−24162.1	−1.04617	−0.523087	+	0.852280i	$0.675220\pi$
$812$	0	0
$813$	46170.0i	1.99170i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	34020.0	1.45680
$818$	0	0
$819$	−58924.4	−2.51402
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	45611.8i	1.93187i	0.258786	+	0.965935i	$0.416677\pi$
$823$	45611.8i	1.93187i	−0.258786	+	0.965935i	$0.583323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	17066.0i	0.714990i	0.933915	+	0.357495i	$0.116369\pi$
$829$	17066.0i	0.714990i	−0.933915	+	0.357495i	$0.883631\pi$
$830$	0	0
$831$	20940.5i	0.874149i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	21870.0i	0.903151i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 41496.5i	− 1.68340i
$848$	0	0
$849$	−40014.0	−1.61752
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 46690.0i	− 1.87413i	−0.349151	−	0.937066i	$-0.613530\pi$
$853$	− 46690.0i	− 1.87413i	0.349151	−	0.937066i	$-0.386470\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	39438.8	1.56651	0.783256	−	0.621699i	$-0.213557\pi$
$859$	39438.8	1.56651	0.783256	+	0.621699i	$0.213557\pi$
$860$	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
$867$	25528.7	1.00000
$868$	0	0
$869$	0	0
$870$	0	0
$871$	45830.1i	1.78288i
$872$	0	0
$873$	35910.0	1.39218
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 50150.0i	− 1.93095i	−0.260491	−	0.965476i	$-0.583885\pi$
$877$	− 50150.0i	− 1.93095i	0.260491	−	0.965476i	$-0.416115\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	48230.7	1.83816	0.919078	−	0.394076i	$-0.128935\pi$
$883$	48230.7	1.83816	0.919078	+	0.394076i	$0.128935\pi$
$884$	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
$889$	−88452.0	−3.33699
$890$	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
$903$	35354.6i	1.30291i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	31208.1	1.14250	0.571250	−	0.820776i	$-0.306459\pi$
$907$	31208.1	1.14250	0.571250	+	0.820776i	$0.306459\pi$
$908$	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
$919$	− 55650.8i	− 1.99755i	−0.0494625	−	0.998776i	$-0.515751\pi$
$919$	− 55650.8i	− 1.99755i	0.0494625	−	0.998776i	$-0.484249\pi$
$920$	0	0
$921$	8262.00	0.295594
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 13750.0i	− 0.488754i
$926$	0	0
$927$	− 27778.6i	− 0.984218i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	98051.4	3.45167
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	55510.0	1.93536	0.967680	−	0.252181i	$-0.0811477\pi$
$937$	55510.0	1.93536	0.967680	+	0.252181i	$0.0811477\pi$
$938$	0	0
$939$	−52013.5	−1.80766
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	− 83300.0i	− 2.84935i
$950$	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
$961$	5491.00	0.184317
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	33390.5i	1.11041i	0.831714	+	0.555204i	$0.187360\pi$
$967$	33390.5i	1.11041i	−0.831714	+	0.555204i	$0.812640\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	− 63180.0i	− 2.08166i
$974$	0	0
$975$	45466.3i	1.49342i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 17442.0i	− 0.567666i
$982$	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
$991$	42556.5i	1.36413i	0.731292	+	0.682064i	$0.238918\pi$
$991$	42556.5i	1.36413i	−0.731292	+	0.682064i	$0.761082\pi$
$992$	0	0
$993$	62370.0	1.99320
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28910.0i	0.918344i	0.888347	+	0.459172i	$0.151854\pi$
$997$	28910.0i	0.918344i	−0.888347	+	0.459172i	$0.848146\pi$
$998$	0	0
$999$	15432.6i	0.488754i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.f.a.383.3		4
3.2	odd	2	CM	768.4.f.a.383.3		4
4.3	odd	2	inner	768.4.f.a.383.2		4
8.3	odd	2	inner	768.4.f.a.383.4		4
8.5	even	2	inner	768.4.f.a.383.1		4
12.11	even	2	inner	768.4.f.a.383.2		4
16.3	odd	4		48.4.c.a.47.1	✓	2
16.5	even	4		192.4.c.a.191.1		2
16.11	odd	4		192.4.c.a.191.2		2
16.13	even	4		48.4.c.a.47.2	yes	2
24.5	odd	2	inner	768.4.f.a.383.1		4
24.11	even	2	inner	768.4.f.a.383.4		4
48.5	odd	4		192.4.c.a.191.1		2
48.11	even	4		192.4.c.a.191.2		2
48.29	odd	4		48.4.c.a.47.2	yes	2
48.35	even	4		48.4.c.a.47.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.c.a.47.1	✓	2	16.3	odd	4
48.4.c.a.47.1	✓	2	48.35	even	4
48.4.c.a.47.2	yes	2	16.13	even	4
48.4.c.a.47.2	yes	2	48.29	odd	4
192.4.c.a.191.1		2	16.5	even	4
192.4.c.a.191.1		2	48.5	odd	4
192.4.c.a.191.2		2	16.11	odd	4
192.4.c.a.191.2		2	48.11	even	4
768.4.f.a.383.1		4	8.5	even	2	inner
768.4.f.a.383.1		4	24.5	odd	2	inner
768.4.f.a.383.2		4	4.3	odd	2	inner
768.4.f.a.383.2		4	12.11	even	2	inner
768.4.f.a.383.3		4	1.1	even	1	trivial
768.4.f.a.383.3		4	3.2	odd	2	CM
768.4.f.a.383.4		4	8.3	odd	2	inner
768.4.f.a.383.4		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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} -31.1769i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} -31.1769i q^{7} +27.0000 q^{9} -70.0000i q^{13} -155.885 q^{19} -162.000i q^{21} -125.000 q^{25} +140.296 q^{27} +155.885i q^{31} +110.000i q^{37} -363.731i q^{39} -218.238 q^{43} -629.000 q^{49} -810.000 q^{57} -182.000i q^{61} -841.777i q^{63} -654.715 q^{67} +1190.00 q^{73} -649.519 q^{75} -1091.19i q^{79} +729.000 q^{81} -2182.38 q^{91} +810.000i q^{93} +1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{9}+O(q^{10}) \) 4 * q + 108 * q^9	\( 4 q + 108 q^{9} - 500 q^{25} - 2516 q^{49} - 3240 q^{57} + 4760 q^{73} + 2916 q^{81} + 5320 q^{97}+O(q^{100}) \) 4 * q + 108 * q^9 - 500 * q^25 - 2516 * q^49 - 3240 * q^57 + 4760 * q^73 + 2916 * q^81 + 5320 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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