LMFDB - Embedded newform 768.2.c.g.767.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(767,768)]

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, 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("768.767");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.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:	$6.13251087523$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			767.4
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.767
Dual form			768.2.c.g.767.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +4.00000i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +4.00000i q^{7} -3.00000 q^{9} -6.92820 q^{13} -3.46410i q^{19} -6.92820 q^{21} +5.00000 q^{25} -5.19615i q^{27} +4.00000i q^{31} -6.92820 q^{37} -12.0000i q^{39} +10.3923i q^{43} -9.00000 q^{49} +6.00000 q^{57} -6.92820 q^{61} -12.0000i q^{63} +3.46410i q^{67} -10.0000 q^{73} +8.66025i q^{75} +4.00000i q^{79} +9.00000 q^{81} -27.7128i q^{91} -6.92820 q^{93} +14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 20 q^{25} - 36 q^{49} + 24 q^{57} - 40 q^{73} + 36 q^{81} + 56 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 + 20 * q^25 - 36 * q^49 + 24 * q^57 - 40 * q^73 + 36 * q^81 + 56 * 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
$3$	1.73205i	1.00000i
$3$	1.73205i	1.00000i
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−6.92820	−1.92154	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	−6.92820	−1.92154	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	0	0
$21$	−6.92820	−1.51186
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 5.19615i	− 1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.92820	−1.13899	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820	−1.13899	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	− 12.0000i	− 1.92154i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	10.3923i	1.58481i	0.609994	+	0.792406i	$0.291172\pi$
$43$	10.3923i	1.58481i	−0.609994	+	0.792406i	$0.708828\pi$
$44$	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
$49$	−9.00000	−1.28571
$50$	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
$57$	6.00000	0.794719
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−6.92820	−0.887066	−0.443533	−	0.896258i	$-0.646275\pi$
$61$	−6.92820	−0.887066	−0.443533	+	0.896258i	$0.646275\pi$
$62$	0	0
$63$	− 12.0000i	− 1.51186i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.46410i	0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	3.46410i	0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	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
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	8.66025i	1.00000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000i	0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$	4.00000i	0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$	0	0
$81$	9.00000	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
$91$	− 27.7128i	− 2.90509i
$92$	0	0
$93$	−6.92820	−0.718421
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	20.0000i	1.97066i	0.170664	+	0.985329i	$0.445409\pi$
$103$	20.0000i	1.97066i	−0.170664	+	0.985329i	$0.554591\pi$
$104$	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
$109$	20.7846	1.99080	0.995402	−	0.0957826i	$-0.0305354\pi$
$109$	20.7846	1.99080	0.995402	+	0.0957826i	$0.0305354\pi$
$110$	0	0
$111$	− 12.0000i	− 1.13899i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	20.7846	1.92154
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.0000i	1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$	20.0000i	1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$	0	0
$129$	−18.0000	−1.58481
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	13.8564	1.20150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	17.3205i	1.46911i	0.678551	+	0.734553i	$0.262608\pi$
$139$	17.3205i	1.46911i	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 15.5885i	− 1.28571i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	4.00000i	0.325515i	0.986666	+	0.162758i	$0.0520389\pi$
$151$	4.00000i	0.325515i	−0.986666	+	0.162758i	$0.947961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.7846	1.65879	0.829396	−	0.558661i	$-0.188685\pi$
$157$	20.7846	1.65879	0.829396	+	0.558661i	$0.188685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.2487i	1.89931i	0.313304	+	0.949653i	$0.398564\pi$
$163$	24.2487i	1.89931i	−0.313304	+	0.949653i	$0.601436\pi$
$164$	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
$169$	35.0000	2.69231
$170$	0	0
$171$	10.3923i	0.794719i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	20.0000i	1.51186i
$176$	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
$181$	−6.92820	−0.514969	−0.257485	−	0.966282i	$-0.582894\pi$
$181$	−6.92820	−0.514969	−0.257485	+	0.966282i	$0.582894\pi$
$182$	0	0
$183$	− 12.0000i	− 0.887066i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	20.7846	1.51186
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	− 28.0000i	− 1.98487i	−0.122782	−	0.992434i	$-0.539182\pi$
$199$	− 28.0000i	− 1.98487i	0.122782	−	0.992434i	$-0.460818\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	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
$211$	− 24.2487i	− 1.66935i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	− 24.2487i	− 1.66935i	0.550743	−	0.834675i	$-0.314345\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	− 17.3205i	− 1.17041i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 28.0000i	− 1.87502i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$	− 28.0000i	− 1.87502i	0.347960	−	0.937509i	$-0.386874\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	20.7846	1.37349	0.686743	−	0.726900i	$-0.259040\pi$
$229$	20.7846	1.37349	0.686743	+	0.726900i	$0.259040\pi$
$230$	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
$237$	−6.92820	−0.450035
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	15.5885i	1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	24.0000i	1.52708i
$248$	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
$259$	− 27.7128i	− 1.72199i
$260$	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
$271$	− 28.0000i	− 1.70088i	−0.526073	−	0.850439i	$-0.676336\pi$
$271$	− 28.0000i	− 1.70088i	0.526073	−	0.850439i	$-0.323664\pi$
$272$	0	0
$273$	48.0000	2.90509
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.7846	1.24883	0.624413	−	0.781094i	$-0.285338\pi$
$277$	20.7846	1.24883	0.624413	+	0.781094i	$0.285338\pi$
$278$	0	0
$279$	− 12.0000i	− 0.718421i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 10.3923i	− 0.617758i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	− 10.3923i	− 0.617758i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	24.2487i	1.42148i
$292$	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
$301$	−41.5692	−2.39601
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	31.1769i	1.77936i	0.456584	+	0.889680i	$0.349073\pi$
$307$	31.1769i	1.77936i	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	−34.6410	−1.97066
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	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
$325$	−34.6410	−1.92154
$326$	0	0
$327$	36.0000i	1.99080i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.3205i	0.952021i	0.879440	+	0.476011i	$0.157918\pi$
$331$	17.3205i	0.952021i	−0.879440	+	0.476011i	$0.842082\pi$
$332$	0	0
$333$	20.7846	1.13899
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	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
$349$	−34.6410	−1.85429	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	−34.6410	−1.85429	−0.927146	+	0.374701i	$0.877745\pi$
$350$	0	0
$351$	36.0000i	1.92154i
$352$	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
$361$	7.00000	0.368421
$362$	0	0
$363$	− 19.0526i	− 1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.92820	−0.358729	−0.179364	−	0.983783i	$-0.557404\pi$
$373$	−6.92820	−0.358729	−0.179364	+	0.983783i	$0.557404\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	38.1051i	1.95733i	0.205466	+	0.978664i	$0.434129\pi$
$379$	38.1051i	1.95733i	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−34.6410	−1.77471
$382$	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
$387$	− 31.1769i	− 1.58481i
$388$	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
$397$	20.7846	1.04315	0.521575	−	0.853206i	$-0.325345\pi$
$397$	20.7846	1.04315	0.521575	+	0.853206i	$0.325345\pi$
$398$	0	0
$399$	24.0000i	1.20150i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 27.7128i	− 1.38047i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−30.0000	−1.46911
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−34.6410	−1.68830	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	−34.6410	−1.68830	−0.844150	+	0.536107i	$0.819894\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 27.7128i	− 1.34112i
$428$	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
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 28.0000i	− 1.33637i	−0.743996	−	0.668184i	$-0.767072\pi$
$439$	− 28.0000i	− 1.33637i	0.743996	−	0.668184i	$-0.232928\pi$
$440$	0	0
$441$	27.0000	1.28571
$442$	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
$453$	−6.92820	−0.325515
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	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
$469$	−13.8564	−0.639829
$470$	0	0
$471$	36.0000i	1.65879i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 17.3205i	− 0.794719i
$476$	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
$481$	48.0000	2.18861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 44.0000i	− 1.99383i	−0.0784867	−	0.996915i	$-0.525009\pi$
$487$	− 44.0000i	− 1.99383i	0.0784867	−	0.996915i	$-0.474991\pi$
$488$	0	0
$489$	−42.0000	−1.89931
$490$	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
$499$	31.1769i	1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$	31.1769i	1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$	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
$507$	60.6218i	2.69231i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	− 40.0000i	− 1.76950i
$512$	0	0
$513$	−18.0000	−0.794719
$514$	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
$523$	− 45.0333i	− 1.96917i	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	− 45.0333i	− 1.96917i	0.174908	−	0.984585i	$-0.444037\pi$
$524$	0	0
$525$	−34.6410	−1.51186
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−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
$541$	−6.92820	−0.297867	−0.148933	−	0.988847i	$-0.547584\pi$
$541$	−6.92820	−0.297867	−0.148933	+	0.988847i	$0.547584\pi$
$542$	0	0
$543$	− 12.0000i	− 0.514969i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.2487i	1.03680i	0.855138	+	0.518400i	$0.173472\pi$
$547$	24.2487i	1.03680i	−0.855138	+	0.518400i	$0.826528\pi$
$548$	0	0
$549$	20.7846	0.887066
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	− 72.0000i	− 3.04528i
$560$	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
$567$	36.0000i	1.51186i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	45.0333i	1.88459i	0.334790	+	0.942293i	$0.391335\pi$
$571$	45.0333i	1.88459i	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	0	0
$579$	3.46410i	0.143963i
$580$	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
$589$	13.8564	0.570943
$590$	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
$597$	48.4974	1.98487
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	− 10.3923i	− 0.423207i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	48.4974	1.95879	0.979396	−	0.201948i	$-0.0647272\pi$
$613$	48.4974	1.95879	0.979396	+	0.201948i	$0.0647272\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 38.1051i	− 1.53157i	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	− 38.1051i	− 1.53157i	0.643094	−	0.765787i	$-0.277650\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 44.0000i	− 1.75161i	−0.482663	−	0.875806i	$-0.660330\pi$
$631$	− 44.0000i	− 1.75161i	0.482663	−	0.875806i	$-0.339670\pi$
$632$	0	0
$633$	42.0000	1.66935
$634$	0	0
$635$	0	0
$636$	0	0
$637$	62.3538	2.47055
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 31.1769i	− 1.22950i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	− 31.1769i	− 1.22950i	0.788723	−	0.614749i	$-0.210743\pi$
$644$	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
$651$	− 27.7128i	− 1.08615i
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.0000	1.17041
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−34.6410	−1.34738	−0.673690	−	0.739014i	$-0.735292\pi$
$661$	−34.6410	−1.34738	−0.673690	+	0.739014i	$0.735292\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	48.4974	1.87502
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	0	0
$675$	− 25.9808i	− 1.00000i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	56.0000i	2.14908i
$680$	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
$687$	36.0000i	1.37349i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	51.9615i	1.97671i	0.152167	+	0.988355i	$0.451375\pi$
$691$	51.9615i	1.97671i	−0.152167	+	0.988355i	$0.548625\pi$
$692$	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
$703$	24.0000i	0.905177i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	48.4974	1.82136	0.910679	−	0.413114i	$-0.135559\pi$
$709$	48.4974	1.82136	0.910679	+	0.413114i	$0.135559\pi$
$710$	0	0
$711$	− 12.0000i	− 0.450035i
$712$	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
$721$	−80.0000	−2.97936
$722$	0	0
$723$	− 24.2487i	− 0.901819i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 44.0000i	− 1.63187i	−0.578144	−	0.815935i	$-0.696223\pi$
$727$	− 44.0000i	− 1.63187i	0.578144	−	0.815935i	$-0.303777\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	20.7846	0.767697	0.383849	−	0.923396i	$-0.374598\pi$
$733$	20.7846	0.767697	0.383849	+	0.923396i	$0.374598\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 51.9615i	− 1.91144i	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	− 51.9615i	− 1.91144i	0.294285	−	0.955718i	$-0.404919\pi$
$740$	0	0
$741$	−41.5692	−1.52708
$742$	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
$751$	52.0000i	1.89751i	0.316017	+	0.948753i	$0.397654\pi$
$751$	52.0000i	1.89751i	−0.316017	+	0.948753i	$0.602346\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	48.4974	1.76267	0.881334	−	0.472493i	$-0.156646\pi$
$757$	48.4974	1.76267	0.881334	+	0.472493i	$0.156646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	83.1384i	3.00981i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	20.0000i	0.718421i
$776$	0	0
$777$	48.0000	1.72199
$778$	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
$787$	− 3.46410i	− 0.123482i	−0.998092	−	0.0617409i	$-0.980335\pi$
$787$	− 3.46410i	− 0.123482i	0.998092	−	0.0617409i	$-0.0196653\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	48.0000	1.70453
$794$	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
$811$	10.3923i	0.364923i	0.983213	+	0.182462i	$0.0584065\pi$
$811$	10.3923i	0.364923i	−0.983213	+	0.182462i	$0.941593\pi$
$812$	0	0
$813$	48.4974	1.70088
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.0000	1.25948
$818$	0	0
$819$	83.1384i	2.90509i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	52.0000i	1.81261i	0.422628	+	0.906303i	$0.361108\pi$
$823$	52.0000i	1.81261i	−0.422628	+	0.906303i	$0.638892\pi$
$824$	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
$829$	−34.6410	−1.20313	−0.601566	−	0.798823i	$-0.705456\pi$
$829$	−34.6410	−1.20313	−0.601566	+	0.798823i	$0.705456\pi$
$830$	0	0
$831$	36.0000i	1.24883i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.7846	0.718421
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	0	0
$849$	18.0000	0.617758
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−6.92820	−0.237217	−0.118609	−	0.992941i	$-0.537843\pi$
$853$	−6.92820	−0.237217	−0.118609	+	0.992941i	$0.537843\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	− 17.3205i	− 0.590968i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	− 17.3205i	− 0.590968i	0.955348	−	0.295484i	$-0.0954809\pi$
$860$	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
$867$	29.4449i	1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 24.0000i	− 0.813209i
$872$	0	0
$873$	−42.0000	−1.42148
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.4974	1.63764	0.818821	−	0.574049i	$-0.194628\pi$
$877$	48.4974	1.63764	0.818821	+	0.574049i	$0.194628\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 58.8897i	− 1.98180i	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	− 58.8897i	− 1.98180i	0.134611	−	0.990899i	$-0.457022\pi$
$884$	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
$889$	−80.0000	−2.68311
$890$	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
$903$	− 72.0000i	− 2.39601i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 45.0333i	− 1.49531i	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	− 45.0333i	− 1.49531i	0.664089	−	0.747653i	$-0.268820\pi$
$908$	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
$919$	52.0000i	1.71532i	0.514216	+	0.857661i	$0.328083\pi$
$919$	52.0000i	1.71532i	−0.514216	+	0.857661i	$0.671917\pi$
$920$	0	0
$921$	−54.0000	−1.77936
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−34.6410	−1.13899
$926$	0	0
$927$	− 60.0000i	− 1.97066i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	31.1769i	1.02178i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	− 38.1051i	− 1.24351i
$940$	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
$949$	69.2820	2.24899
$950$	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
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.0000i	0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$	20.0000i	0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$	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
$973$	−69.2820	−2.22108
$974$	0	0
$975$	− 60.0000i	− 1.92154i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−62.3538	−1.99080
$982$	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
$991$	− 44.0000i	− 1.39771i	−0.715265	−	0.698853i	$-0.753694\pi$
$991$	− 44.0000i	− 1.39771i	0.715265	−	0.698853i	$-0.246306\pi$
$992$	0	0
$993$	−30.0000	−0.952021
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−62.3538	−1.97477	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	−62.3538	−1.97477	−0.987383	+	0.158352i	$0.949382\pi$
$998$	0	0
$999$	36.0000i	1.13899i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.c.g.767.4		4
3.2	odd	2	CM	768.2.c.g.767.4		4
4.3	odd	2	inner	768.2.c.g.767.1		4
8.3	odd	2	inner	768.2.c.g.767.3		4
8.5	even	2	inner	768.2.c.g.767.2		4
12.11	even	2	inner	768.2.c.g.767.1		4
16.3	odd	4		192.2.f.b.95.4	yes	4
16.5	even	4		192.2.f.b.95.3	yes	4
16.11	odd	4		192.2.f.b.95.2	yes	4
16.13	even	4		192.2.f.b.95.1	✓	4
24.5	odd	2	inner	768.2.c.g.767.2		4
24.11	even	2	inner	768.2.c.g.767.3		4
48.5	odd	4		192.2.f.b.95.3	yes	4
48.11	even	4		192.2.f.b.95.2	yes	4
48.29	odd	4		192.2.f.b.95.1	✓	4
48.35	even	4		192.2.f.b.95.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.2.f.b.95.1	✓	4	16.13	even	4
192.2.f.b.95.1	✓	4	48.29	odd	4
192.2.f.b.95.2	yes	4	16.11	odd	4
192.2.f.b.95.2	yes	4	48.11	even	4
192.2.f.b.95.3	yes	4	16.5	even	4
192.2.f.b.95.3	yes	4	48.5	odd	4
192.2.f.b.95.4	yes	4	16.3	odd	4
192.2.f.b.95.4	yes	4	48.35	even	4
768.2.c.g.767.1		4	4.3	odd	2	inner
768.2.c.g.767.1		4	12.11	even	2	inner
768.2.c.g.767.2		4	8.5	even	2	inner
768.2.c.g.767.2		4	24.5	odd	2	inner
768.2.c.g.767.3		4	8.3	odd	2	inner
768.2.c.g.767.3		4	24.11	even	2	inner
768.2.c.g.767.4		4	1.1	even	1	trivial
768.2.c.g.767.4		4	3.2	odd	2	CM

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +4.00000i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +4.00000i q^{7} -3.00000 q^{9} -6.92820 q^{13} -3.46410i q^{19} -6.92820 q^{21} +5.00000 q^{25} -5.19615i q^{27} +4.00000i q^{31} -6.92820 q^{37} -12.0000i q^{39} +10.3923i q^{43} -9.00000 q^{49} +6.00000 q^{57} -6.92820 q^{61} -12.0000i q^{63} +3.46410i q^{67} -10.0000 q^{73} +8.66025i q^{75} +4.00000i q^{79} +9.00000 q^{81} -27.7128i q^{91} -6.92820 q^{93} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 20 q^{25} - 36 q^{49} + 24 q^{57} - 40 q^{73} + 36 q^{81} + 56 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 + 20 * q^25 - 36 * q^49 + 24 * q^57 - 40 * q^73 + 36 * q^81 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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