LMFDB - Embedded newform 1152.2.f.a.575.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,2,Mod(575,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.575"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.575
Dual form			1152.2.f.a.575.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{5} -2.82843i q^{7} +4.00000i q^{11} -2.00000i q^{13} -1.41421i q^{17} -5.65685 q^{19} -4.00000 q^{23} -3.00000 q^{25} +7.07107 q^{29} -8.48528i q^{31} +4.00000i q^{35} +8.00000i q^{37} +4.24264i q^{41} -11.3137 q^{43} -12.0000 q^{47} -1.00000 q^{49} -12.7279 q^{53} -5.65685i q^{55} -8.00000i q^{61} +2.82843i q^{65} -5.65685 q^{67} -4.00000 q^{71} +8.00000 q^{73} +11.3137 q^{77} -2.82843i q^{79} +12.0000i q^{83} +2.00000i q^{85} -15.5563i q^{89} -5.65685 q^{91} +8.00000 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{23} - 12 q^{25} - 48 q^{47} - 4 q^{49} - 16 q^{71} + 32 q^{73} + 32 q^{95}+O(q^{100}) $ 4 * q - 16 * q^23 - 12 * q^25 - 48 * q^47 - 4 * q^49 - 16 * q^71 + 32 * q^73 + 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.41421i	− 0.342997i	−0.985184	−	0.171499i	$-0.945139\pi$
$17$	− 1.41421i	− 0.342997i	0.985184	−	0.171499i	$-0.0548609\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.07107	1.31306	0.656532	−	0.754298i	$-0.272023\pi$
$29$	7.07107	1.31306	0.656532	+	0.754298i	$0.272023\pi$
$30$	0	0
$31$	− 8.48528i	− 1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$	− 8.48528i	− 1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000i	0.676123i
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.24264i	0.662589i	0.943527	+	0.331295i	$0.107485\pi$
$41$	4.24264i	0.662589i	−0.943527	+	0.331295i	$0.892515\pi$
$42$	0	0
$43$	−11.3137	−1.72532	−0.862662	−	0.505781i	$-0.831205\pi$
$43$	−11.3137	−1.72532	−0.862662	+	0.505781i	$0.831205\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.7279	−1.74831	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	−12.7279	−1.74831	−0.874157	+	0.485643i	$0.838586\pi$
$54$	0	0
$55$	− 5.65685i	− 0.762770i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.82843i	0.350823i
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.3137	1.28932
$78$	0	0
$79$	− 2.82843i	− 0.318223i	−0.987261	−	0.159111i	$-0.949137\pi$
$79$	− 2.82843i	− 0.318223i	0.987261	−	0.159111i	$-0.0508629\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	2.00000i	0.216930i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 15.5563i	− 1.64897i	−0.565884	−	0.824485i	$-0.691465\pi$
$89$	− 15.5563i	− 1.64897i	0.565884	−	0.824485i	$-0.308535\pi$
$90$	0	0
$91$	−5.65685	−0.592999
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.89949	−0.985037	−0.492518	−	0.870302i	$-0.663924\pi$
$101$	−9.89949	−0.985037	−0.492518	+	0.870302i	$0.663924\pi$
$102$	0	0
$103$	− 8.48528i	− 0.836080i	−0.908429	−	0.418040i	$-0.862717\pi$
$103$	− 8.48528i	− 0.836080i	0.908429	−	0.418040i	$-0.137283\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.41421i	− 0.133038i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	− 1.41421i	− 0.133038i	0.997785	−	0.0665190i	$-0.0211893\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137	1.01193
$126$	0	0
$127$	2.82843i	0.250982i	0.992095	+	0.125491i	$0.0400507\pi$
$127$	2.82843i	0.250982i	−0.992095	+	0.125491i	$0.959949\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 8.00000i	− 0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$	− 8.00000i	− 0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.3848i	1.57072i	0.619041	+	0.785359i	$0.287521\pi$
$137$	18.3848i	1.57072i	−0.619041	+	0.785359i	$0.712479\pi$
$138$	0	0
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.89949	−0.810998	−0.405499	−	0.914095i	$-0.632902\pi$
$149$	−9.89949	−0.810998	−0.405499	+	0.914095i	$0.632902\pi$
$150$	0	0
$151$	8.48528i	0.690522i	0.938507	+	0.345261i	$0.112210\pi$
$151$	8.48528i	0.690522i	−0.938507	+	0.345261i	$0.887790\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.0000i	0.963863i
$156$	0	0
$157$	8.00000i	0.638470i	0.947676	+	0.319235i	$0.103426\pi$
$157$	8.00000i	0.638470i	−0.947676	+	0.319235i	$0.896574\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	11.3137i	0.891645i
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.07107	0.537603	0.268802	−	0.963196i	$-0.413372\pi$
$173$	7.07107	0.537603	0.268802	+	0.963196i	$0.413372\pi$
$174$	0	0
$175$	8.48528i	0.641427i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 24.0000i	− 1.79384i	−0.442189	−	0.896922i	$-0.645798\pi$
$179$	− 24.0000i	− 1.79384i	0.442189	−	0.896922i	$-0.354202\pi$
$180$	0	0
$181$	6.00000i	0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	6.00000i	0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 11.3137i	− 0.831800i
$186$	0	0
$187$	5.65685	0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.41421	0.100759	0.0503793	−	0.998730i	$-0.483957\pi$
$197$	1.41421	0.100759	0.0503793	+	0.998730i	$0.483957\pi$
$198$	0	0
$199$	25.4558i	1.80452i	0.431196	+	0.902258i	$0.358092\pi$
$199$	25.4558i	1.80452i	−0.431196	+	0.902258i	$0.641908\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 20.0000i	− 1.40372i
$204$	0	0
$205$	− 6.00000i	− 0.419058i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 22.6274i	− 1.56517i
$210$	0	0
$211$	5.65685	0.389434	0.194717	−	0.980859i	$-0.437621\pi$
$211$	5.65685	0.389434	0.194717	+	0.980859i	$0.437621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.82843	−0.190261
$222$	0	0
$223$	14.1421i	0.947027i	0.880786	+	0.473514i	$0.157015\pi$
$223$	14.1421i	0.947027i	−0.880786	+	0.473514i	$0.842985\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.5563i	1.01913i	0.860432	+	0.509565i	$0.170194\pi$
$233$	15.5563i	1.01913i	−0.860432	+	0.509565i	$0.829806\pi$
$234$	0	0
$235$	16.9706	1.10704
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.41421	0.0903508
$246$	0	0
$247$	11.3137i	0.719874i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 20.0000i	− 1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$	− 20.0000i	− 1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.41421i	0.0882162i	0.999027	+	0.0441081i	$0.0140446\pi$
$257$	1.41421i	0.0882162i	−0.999027	+	0.0441081i	$0.985955\pi$
$258$	0	0
$259$	22.6274	1.40600
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.07107	−0.431131	−0.215565	−	0.976489i	$-0.569159\pi$
$269$	−7.07107	−0.431131	−0.215565	+	0.976489i	$0.569159\pi$
$270$	0	0
$271$	2.82843i	0.171815i	0.996303	+	0.0859074i	$0.0273789\pi$
$271$	2.82843i	0.171815i	−0.996303	+	0.0859074i	$0.972621\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 12.0000i	− 0.723627i
$276$	0	0
$277$	6.00000i	0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$	6.00000i	0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 26.8701i	− 1.60293i	−0.598040	−	0.801467i	$-0.704053\pi$
$281$	− 26.8701i	− 1.60293i	0.598040	−	0.801467i	$-0.295947\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.41421	−0.0826192	−0.0413096	−	0.999146i	$-0.513153\pi$
$293$	−1.41421	−0.0826192	−0.0413096	+	0.999146i	$0.513153\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	32.0000i	1.84445i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.3137i	0.647821i
$306$	0	0
$307$	16.9706	0.968561	0.484281	−	0.874913i	$-0.339081\pi$
$307$	16.9706	0.968561	0.484281	+	0.874913i	$0.339081\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.3848	−1.03259	−0.516296	−	0.856410i	$-0.672690\pi$
$317$	−18.3848	−1.03259	−0.516296	+	0.856410i	$0.672690\pi$
$318$	0	0
$319$	28.2843i	1.58362i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	33.9411i	1.87123i
$330$	0	0
$331$	−5.65685	−0.310929	−0.155464	−	0.987841i	$-0.549687\pi$
$331$	−5.65685	−0.310929	−0.155464	+	0.987841i	$0.549687\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−24.0000	−1.30736	−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000	−1.30736	−0.653682	+	0.756770i	$0.726776\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.9411	1.83801
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	0	0
$349$	8.00000i	0.428230i	0.976808	+	0.214115i	$0.0686868\pi$
$349$	8.00000i	0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.89949i	0.526897i	0.964673	+	0.263448i	$0.0848599\pi$
$353$	9.89949i	0.526897i	−0.964673	+	0.263448i	$0.915140\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11.3137	−0.592187
$366$	0	0
$367$	− 8.48528i	− 0.442928i	−0.975169	−	0.221464i	$-0.928916\pi$
$367$	− 8.48528i	− 0.442928i	0.975169	−	0.221464i	$-0.0710835\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	36.0000i	1.86903i
$372$	0	0
$373$	− 24.0000i	− 1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$	− 24.0000i	− 1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.1421i	− 0.728357i
$378$	0	0
$379$	16.9706	0.871719	0.435860	−	0.900015i	$-0.356444\pi$
$379$	16.9706	0.871719	0.435860	+	0.900015i	$0.356444\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.7279	0.645331	0.322666	−	0.946513i	$-0.395421\pi$
$389$	12.7279	0.645331	0.322666	+	0.946513i	$0.395421\pi$
$390$	0	0
$391$	5.65685i	0.286079i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.00000i	0.201262i
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 24.0416i	− 1.20058i	−0.799782	−	0.600291i	$-0.795051\pi$
$401$	− 24.0416i	− 1.20058i	0.799782	−	0.600291i	$-0.204949\pi$
$402$	0	0
$403$	−16.9706	−0.845364
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−32.0000	−1.58618
$408$	0	0
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 16.9706i	− 0.833052i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 36.0000i	− 1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$	− 36.0000i	− 1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	0	0
$421$	− 26.0000i	− 1.26716i	−0.773676	−	0.633581i	$-0.781584\pi$
$421$	− 26.0000i	− 1.26716i	0.773676	−	0.633581i	$-0.218416\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.24264i	0.205798i
$426$	0	0
$427$	−22.6274	−1.09502
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.6274	1.08242
$438$	0	0
$439$	8.48528i	0.404980i	0.979284	+	0.202490i	$0.0649034\pi$
$439$	8.48528i	0.404980i	−0.979284	+	0.202490i	$0.935097\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	22.0000i	1.04290i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.7279i	0.600668i	0.953834	+	0.300334i	$0.0970981\pi$
$449$	12.7279i	0.600668i	−0.953834	+	0.300334i	$0.902902\pi$
$450$	0	0
$451$	−16.9706	−0.799113
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.6985	−1.38320	−0.691598	−	0.722282i	$-0.743093\pi$
$461$	−29.6985	−1.38320	−0.691598	+	0.722282i	$0.743093\pi$
$462$	0	0
$463$	− 25.4558i	− 1.18303i	−0.806293	−	0.591517i	$-0.798529\pi$
$463$	− 25.4558i	− 1.18303i	0.806293	−	0.591517i	$-0.201471\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	16.0000i	0.738811i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 45.2548i	− 2.08082i
$474$	0	0
$475$	16.9706	0.778663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.48528i	− 0.384505i	−0.981346	−	0.192252i	$-0.938421\pi$
$487$	− 8.48528i	− 0.384505i	0.981346	−	0.192252i	$-0.0615792\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.00000i	0.361035i	0.983572	+	0.180517i	$0.0577772\pi$
$491$	8.00000i	0.361035i	−0.983572	+	0.180517i	$0.942223\pi$
$492$	0	0
$493$	− 10.0000i	− 0.450377i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.3137i	0.507489i
$498$	0	0
$499$	−22.6274	−1.01294	−0.506471	−	0.862257i	$-0.669050\pi$
$499$	−22.6274	−1.01294	−0.506471	+	0.862257i	$0.669050\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.1838	−1.69247	−0.846233	−	0.532813i	$-0.821135\pi$
$509$	−38.1838	−1.69247	−0.846233	+	0.532813i	$0.821135\pi$
$510$	0	0
$511$	− 22.6274i	− 1.00098i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.0000i	0.528783i
$516$	0	0
$517$	− 48.0000i	− 2.11104i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.6985i	1.30111i	0.759457	+	0.650557i	$0.225465\pi$
$521$	29.6985i	1.30111i	−0.759457	+	0.650557i	$0.774535\pi$
$522$	0	0
$523$	−39.5980	−1.73150	−0.865749	−	0.500478i	$-0.833158\pi$
$523$	−39.5980	−1.73150	−0.865749	+	0.500478i	$0.833158\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.48528	0.367538
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.00000i	− 0.172292i
$540$	0	0
$541$	2.00000i	0.0859867i	0.999075	+	0.0429934i	$0.0136894\pi$
$541$	2.00000i	0.0859867i	−0.999075	+	0.0429934i	$0.986311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19.7990i	0.848096i
$546$	0	0
$547$	−33.9411	−1.45122	−0.725609	−	0.688107i	$-0.758442\pi$
$547$	−33.9411	−1.45122	−0.725609	+	0.688107i	$0.758442\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−40.0000	−1.70406
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.3848	−0.778988	−0.389494	−	0.921029i	$-0.627350\pi$
$557$	−18.3848	−0.778988	−0.389494	+	0.921029i	$0.627350\pi$
$558$	0	0
$559$	22.6274i	0.957038i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	2.00000i	0.0841406i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.07107i	0.296435i	0.988955	+	0.148217i	$0.0473535\pi$
$569$	7.07107i	0.296435i	−0.988955	+	0.148217i	$0.952646\pi$
$570$	0	0
$571$	−33.9411	−1.42039	−0.710196	−	0.704004i	$-0.751394\pi$
$571$	−33.9411	−1.42039	−0.710196	+	0.704004i	$0.751394\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	33.9411	1.40812
$582$	0	0
$583$	− 50.9117i	− 2.10855i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.0000i	1.32078i	0.750922	+	0.660391i	$0.229609\pi$
$587$	32.0000i	1.32078i	−0.750922	+	0.660391i	$0.770391\pi$
$588$	0	0
$589$	48.0000i	1.97781i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 43.8406i	− 1.80032i	−0.435561	−	0.900159i	$-0.643450\pi$
$593$	− 43.8406i	− 1.80032i	0.435561	−	0.900159i	$-0.356550\pi$
$594$	0	0
$595$	5.65685	0.231908
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.07107	0.287480
$606$	0	0
$607$	− 19.7990i	− 0.803616i	−0.915724	−	0.401808i	$-0.868382\pi$
$607$	− 19.7990i	− 0.803616i	0.915724	−	0.401808i	$-0.131618\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.0000i	0.970936i
$612$	0	0
$613$	24.0000i	0.969351i	0.874694	+	0.484675i	$0.161062\pi$
$613$	24.0000i	0.969351i	−0.874694	+	0.484675i	$0.838938\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.07107i	0.284670i	0.989819	+	0.142335i	$0.0454611\pi$
$617$	7.07107i	0.284670i	−0.989819	+	0.142335i	$0.954539\pi$
$618$	0	0
$619$	33.9411	1.36421	0.682105	−	0.731255i	$-0.261065\pi$
$619$	33.9411	1.36421	0.682105	+	0.731255i	$0.261065\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−44.0000	−1.76282
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.3137	0.451107
$630$	0	0
$631$	− 42.4264i	− 1.68897i	−0.535580	−	0.844484i	$-0.679907\pi$
$631$	− 42.4264i	− 1.68897i	0.535580	−	0.844484i	$-0.320093\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 4.00000i	− 0.158735i
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.41421i	− 0.0558581i	−0.999610	−	0.0279290i	$-0.991109\pi$
$641$	− 1.41421i	− 0.0558581i	0.999610	−	0.0279290i	$-0.00889125\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.6985	1.16219	0.581096	−	0.813835i	$-0.302624\pi$
$653$	29.6985	1.16219	0.581096	+	0.813835i	$0.302624\pi$
$654$	0	0
$655$	11.3137i	0.442063i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.0000i	0.623272i	0.950202	+	0.311636i	$0.100877\pi$
$659$	16.0000i	0.623272i	−0.950202	+	0.311636i	$0.899123\pi$
$660$	0	0
$661$	− 40.0000i	− 1.55582i	−0.628376	−	0.777910i	$-0.716280\pi$
$661$	− 40.0000i	− 1.55582i	0.628376	−	0.777910i	$-0.283720\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 22.6274i	− 0.877454i
$666$	0	0
$667$	−28.2843	−1.09517
$668$	0	0
$669$	0	0
$670$	0	0
$671$	32.0000	1.23535
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.2132	0.815290	0.407645	−	0.913141i	$-0.366350\pi$
$677$	21.2132	0.815290	0.407645	+	0.913141i	$0.366350\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	− 26.0000i	− 0.993409i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	25.4558i	0.969790i
$690$	0	0
$691$	−22.6274	−0.860788	−0.430394	−	0.902641i	$-0.641625\pi$
$691$	−22.6274	−0.860788	−0.430394	+	0.902641i	$0.641625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−26.8701	−1.01487	−0.507434	−	0.861691i	$-0.669406\pi$
$701$	−26.8701	−1.01487	−0.507434	+	0.861691i	$0.669406\pi$
$702$	0	0
$703$	− 45.2548i	− 1.70682i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.0000i	1.05305i
$708$	0	0
$709$	26.0000i	0.976450i	0.872718	+	0.488225i	$0.162356\pi$
$709$	26.0000i	0.976450i	−0.872718	+	0.488225i	$0.837644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	33.9411i	1.27111i
$714$	0	0
$715$	−11.3137	−0.423109
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21.2132	−0.787839
$726$	0	0
$727$	14.1421i	0.524503i	0.965000	+	0.262251i	$0.0844650\pi$
$727$	14.1421i	0.524503i	−0.965000	+	0.262251i	$0.915535\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.0000i	0.591781i
$732$	0	0
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 22.6274i	− 0.833492i
$738$	0	0
$739$	33.9411	1.24854	0.624272	−	0.781207i	$-0.285396\pi$
$739$	33.9411	1.24854	0.624272	+	0.781207i	$0.285396\pi$
$740$	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
$745$	14.0000	0.512920
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 19.7990i	− 0.722475i	−0.932474	−	0.361238i	$-0.882354\pi$
$751$	− 19.7990i	− 0.722475i	0.932474	−	0.361238i	$-0.117646\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 12.0000i	− 0.436725i
$756$	0	0
$757$	− 6.00000i	− 0.218074i	−0.994038	−	0.109037i	$-0.965223\pi$
$757$	− 6.00000i	− 0.218074i	0.994038	−	0.109037i	$-0.0347767\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26.8701i	0.974039i	0.873391	+	0.487019i	$0.161916\pi$
$761$	26.8701i	0.974039i	−0.873391	+	0.487019i	$0.838084\pi$
$762$	0	0
$763$	−39.5980	−1.43354
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−35.3553	−1.27164	−0.635822	−	0.771836i	$-0.719339\pi$
$773$	−35.3553	−1.27164	−0.635822	+	0.771836i	$0.719339\pi$
$774$	0	0
$775$	25.4558i	0.914401i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	− 16.0000i	− 0.572525i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 11.3137i	− 0.403804i
$786$	0	0
$787$	28.2843	1.00823	0.504113	−	0.863638i	$-0.331820\pi$
$787$	28.2843	1.00823	0.504113	+	0.863638i	$0.331820\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	−16.0000	−0.568177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.5563	−0.551034	−0.275517	−	0.961296i	$-0.588849\pi$
$797$	−15.5563	−0.551034	−0.275517	+	0.961296i	$0.588849\pi$
$798$	0	0
$799$	16.9706i	0.600375i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	32.0000i	1.12926i
$804$	0	0
$805$	− 16.0000i	− 0.563926i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 18.3848i	− 0.646374i	−0.946335	−	0.323187i	$-0.895246\pi$
$809$	− 18.3848i	− 0.646374i	0.946335	−	0.323187i	$-0.104754\pi$
$810$	0	0
$811$	16.9706	0.595917	0.297959	−	0.954579i	$-0.403694\pi$
$811$	16.9706	0.595917	0.297959	+	0.954579i	$0.403694\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	64.0000	2.23908
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.41421	0.0493564	0.0246782	−	0.999695i	$-0.492144\pi$
$821$	1.41421	0.0493564	0.0246782	+	0.999695i	$0.492144\pi$
$822$	0	0
$823$	8.48528i	0.295778i	0.989004	+	0.147889i	$0.0472479\pi$
$823$	8.48528i	0.295778i	−0.989004	+	0.147889i	$0.952752\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.00000i	0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$	8.00000i	0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$	0	0
$829$	− 30.0000i	− 1.04194i	−0.853574	−	0.520972i	$-0.825570\pi$
$829$	− 30.0000i	− 1.04194i	0.853574	−	0.520972i	$-0.174430\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.41421i	0.0489996i
$834$	0	0
$835$	−22.6274	−0.783054
$836$	0	0
$837$	0	0
$838$	0	0
$839$	52.0000	1.79524	0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000	1.79524	0.897620	+	0.440771i	$0.145295\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.7279	−0.437854
$846$	0	0
$847$	14.1421i	0.485930i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 32.0000i	− 1.09695i
$852$	0	0
$853$	24.0000i	0.821744i	0.911693	+	0.410872i	$0.134776\pi$
$853$	24.0000i	0.821744i	−0.911693	+	0.410872i	$0.865224\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 52.3259i	− 1.78742i	−0.448646	−	0.893709i	$-0.648094\pi$
$857$	− 52.3259i	− 1.78742i	0.448646	−	0.893709i	$-0.351906\pi$
$858$	0	0
$859$	33.9411	1.15806	0.579028	−	0.815308i	$-0.303432\pi$
$859$	33.9411	1.15806	0.579028	+	0.815308i	$0.303432\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.3137	0.383791
$870$	0	0
$871$	11.3137i	0.383350i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 32.0000i	− 1.08180i
$876$	0	0
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.7279i	0.428815i	0.976744	+	0.214407i	$0.0687820\pi$
$881$	12.7279i	0.428815i	−0.976744	+	0.214407i	$0.931218\pi$
$882$	0	0
$883$	−33.9411	−1.14221	−0.571105	−	0.820877i	$-0.693485\pi$
$883$	−33.9411	−1.14221	−0.571105	+	0.820877i	$0.693485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	67.8823	2.27159
$894$	0	0
$895$	33.9411i	1.13453i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 60.0000i	− 2.00111i
$900$	0	0
$901$	18.0000i	0.599667i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 8.48528i	− 0.282060i
$906$	0	0
$907$	33.9411	1.12700	0.563498	−	0.826117i	$-0.309455\pi$
$907$	33.9411	1.12700	0.563498	+	0.826117i	$0.309455\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−22.6274	−0.747223
$918$	0	0
$919$	31.1127i	1.02631i	0.858295	+	0.513157i	$0.171524\pi$
$919$	31.1127i	1.02631i	−0.858295	+	0.513157i	$0.828476\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.00000i	0.263323i
$924$	0	0
$925$	− 24.0000i	− 0.789115i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.5269i	1.06717i	0.845745	+	0.533587i	$0.179156\pi$
$929$	32.5269i	1.06717i	−0.845745	+	0.533587i	$0.820844\pi$
$930$	0	0
$931$	5.65685	0.185396
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	15.5563	0.507122	0.253561	−	0.967319i	$-0.418398\pi$
$941$	15.5563	0.507122	0.253561	+	0.967319i	$0.418398\pi$
$942$	0	0
$943$	− 16.9706i	− 0.552638i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.0000i	− 0.519930i	−0.965618	−	0.259965i	$-0.916289\pi$
$947$	− 16.0000i	− 0.519930i	0.965618	−	0.259965i	$-0.0837111\pi$
$948$	0	0
$949$	− 16.0000i	− 0.519382i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	41.0122i	1.32852i	0.747504	+	0.664258i	$0.231252\pi$
$953$	41.0122i	1.32852i	−0.747504	+	0.664258i	$0.768748\pi$
$954$	0	0
$955$	22.6274	0.732206
$956$	0	0
$957$	0	0
$958$	0	0
$959$	52.0000	1.67917
$960$	0	0
$961$	−41.0000	−1.32258
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14.1421	−0.455251
$966$	0	0
$967$	− 31.1127i	− 1.00052i	−0.865876	−	0.500258i	$-0.833238\pi$
$967$	− 31.1127i	− 1.00052i	0.865876	−	0.500258i	$-0.166762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.0000i	0.641831i	0.947108	+	0.320915i	$0.103990\pi$
$971$	20.0000i	0.641831i	−0.947108	+	0.320915i	$0.896010\pi$
$972$	0	0
$973$	− 48.0000i	− 1.53881i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 21.2132i	− 0.678671i	−0.940666	−	0.339335i	$-0.889798\pi$
$977$	− 21.2132i	− 0.678671i	0.940666	−	0.339335i	$-0.110202\pi$
$978$	0	0
$979$	62.2254	1.98873
$980$	0	0
$981$	0	0
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.2548	1.43902
$990$	0	0
$991$	48.0833i	1.52742i	0.645562	+	0.763708i	$0.276623\pi$
$991$	48.0833i	1.52742i	−0.645562	+	0.763708i	$0.723377\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 36.0000i	− 1.14128i
$996$	0	0
$997$	40.0000i	1.26681i	0.773819	+	0.633406i	$0.218344\pi$
$997$	40.0000i	1.26681i	−0.773819	+	0.633406i	$0.781656\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.f.a.575.1	✓	4
3.2	odd	2		1152.2.f.d.575.3	yes	4
4.3	odd	2		1152.2.f.d.575.2	yes	4
8.3	odd	2		1152.2.f.d.575.4	yes	4
8.5	even	2	inner	1152.2.f.a.575.3	yes	4
12.11	even	2	inner	1152.2.f.a.575.4	yes	4
16.3	odd	4		2304.2.c.b.2303.2		2
16.5	even	4		2304.2.c.a.2303.1		2
16.11	odd	4		2304.2.c.g.2303.1		2
16.13	even	4		2304.2.c.h.2303.2		2
24.5	odd	2		1152.2.f.d.575.1	yes	4
24.11	even	2	inner	1152.2.f.a.575.2	yes	4
48.5	odd	4		2304.2.c.g.2303.2		2
48.11	even	4		2304.2.c.a.2303.2		2
48.29	odd	4		2304.2.c.b.2303.1		2
48.35	even	4		2304.2.c.h.2303.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.f.a.575.1	✓	4	1.1	even	1	trivial
1152.2.f.a.575.2	yes	4	24.11	even	2	inner
1152.2.f.a.575.3	yes	4	8.5	even	2	inner
1152.2.f.a.575.4	yes	4	12.11	even	2	inner
1152.2.f.d.575.1	yes	4	24.5	odd	2
1152.2.f.d.575.2	yes	4	4.3	odd	2
1152.2.f.d.575.3	yes	4	3.2	odd	2
1152.2.f.d.575.4	yes	4	8.3	odd	2
2304.2.c.a.2303.1		2	16.5	even	4
2304.2.c.a.2303.2		2	48.11	even	4
2304.2.c.b.2303.1		2	48.29	odd	4
2304.2.c.b.2303.2		2	16.3	odd	4
2304.2.c.g.2303.1		2	16.11	odd	4
2304.2.c.g.2303.2		2	48.5	odd	4
2304.2.c.h.2303.1		2	48.35	even	4
2304.2.c.h.2303.2		2	16.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{5} -2.82843i q^{7} +4.00000i q^{11} -2.00000i q^{13} -1.41421i q^{17} -5.65685 q^{19} -4.00000 q^{23} -3.00000 q^{25} +7.07107 q^{29} -8.48528i q^{31} +4.00000i q^{35} +8.00000i q^{37} +4.24264i q^{41} -11.3137 q^{43} -12.0000 q^{47} -1.00000 q^{49} -12.7279 q^{53} -5.65685i q^{55} -8.00000i q^{61} +2.82843i q^{65} -5.65685 q^{67} -4.00000 q^{71} +8.00000 q^{73} +11.3137 q^{77} -2.82843i q^{79} +12.0000i q^{83} +2.00000i q^{85} -15.5563i q^{89} -5.65685 q^{91} +8.00000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{23} - 12 q^{25} - 48 q^{47} - 4 q^{49} - 16 q^{71} + 32 q^{73} + 32 q^{95}+O(q^{100}) \) 4 * q - 16 * q^23 - 12 * q^25 - 48 * q^47 - 4 * q^49 - 16 * q^71 + 32 * q^73 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more