LMFDB - Embedded newform 792.2.m.b.197.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [792,2,Mod(197,792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("792.197"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 792 = 2^{3} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	792.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-8,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$6.32415184009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 11x^{2} - 10x + 3 $ x^4 - 2x^3 + 11x^2 - 10*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			197.3
Root			$0.500000 + 3.07253i$ of defining polynomial
Character	$\chi$	$=$	792.197
Dual form			792.2.m.b.197.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} -3.31662i q^{11} -2.69042 q^{13} +4.00000 q^{16} -8.69042 q^{19} +4.69042 q^{22} -8.04746i q^{23} -5.00000 q^{25} -3.80482i q^{26} -6.63325i q^{29} +9.38083 q^{31} +5.65685i q^{32} -12.2901i q^{38} +3.30958 q^{43} +6.63325i q^{44} +11.3808 q^{46} +0.437818i q^{47} +7.00000 q^{49} -7.07107i q^{50} +5.38083 q^{52} +9.38083 q^{58} -14.6904 q^{61} +13.2665i q^{62} -8.00000 q^{64} -16.5327i q^{71} +17.3808 q^{76} -6.63325i q^{83} +4.68046i q^{86} -9.38083 q^{88} +11.8523i q^{89} +16.0949i q^{92} -0.619168 q^{94} -18.7617 q^{97} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 8 q^{13} + 16 q^{16} - 16 q^{19} - 20 q^{25} + 32 q^{43} + 8 q^{46} + 28 q^{49} - 16 q^{52} - 40 q^{61} - 32 q^{64} + 32 q^{76} - 40 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 + 8 * q^13 + 16 * q^16 - 16 * q^19 - 20 * q^25 + 32 * q^43 + 8 * q^46 + 28 * q^49 - 16 * q^52 - 40 * q^61 - 32 * q^64 + 32 * q^76 - 40 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/792\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$	$397$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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 $

$2$	1.41421i	1.00000i
$2$	1.41421i	1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 2.82843i	− 1.00000i
$9$	0	0
$10$	0	0
$11$	− 3.31662i	− 1.00000i
$11$	− 3.31662i	− 1.00000i
$12$	0	0
$13$	−2.69042	−0.746187	−0.373094	−	0.927794i	$-0.621703\pi$
$13$	−2.69042	−0.746187	−0.373094	+	0.927794i	$0.621703\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−8.69042	−1.99372	−0.996859	−	0.0791961i	$-0.974765\pi$
$19$	−8.69042	−1.99372	−0.996859	+	0.0791961i	$0.974765\pi$
$20$	0	0
$21$	0	0
$22$	4.69042	1.00000
$23$	− 8.04746i	− 1.67801i	−0.544122	−	0.839006i	$-0.683137\pi$
$23$	− 8.04746i	− 1.67801i	0.544122	−	0.839006i	$-0.316863\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	− 3.80482i	− 0.746187i
$27$	0	0
$28$	0	0
$29$	− 6.63325i	− 1.23176i	−0.787839	−	0.615882i	$-0.788800\pi$
$29$	− 6.63325i	− 1.23176i	0.787839	−	0.615882i	$-0.211200\pi$
$30$	0	0
$31$	9.38083	1.68485	0.842424	−	0.538816i	$-0.181128\pi$
$31$	9.38083	1.68485	0.842424	+	0.538816i	$0.181128\pi$
$32$	5.65685i	1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	− 12.2901i	− 1.99372i
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	3.30958	0.504707	0.252353	−	0.967635i	$-0.418795\pi$
$43$	3.30958	0.504707	0.252353	+	0.967635i	$0.418795\pi$
$44$	6.63325i	1.00000i
$45$	0	0
$46$	11.3808	1.67801
$47$	0.437818i	0.0638624i	0.999490	+	0.0319312i	$0.0101657\pi$
$47$	0.437818i	0.0638624i	−0.999490	+	0.0319312i	$0.989834\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	− 7.07107i	− 1.00000i
$51$	0	0
$52$	5.38083	0.746187
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	9.38083	1.23176
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−14.6904	−1.88091	−0.940457	−	0.339911i	$-0.889603\pi$
$61$	−14.6904	−1.88091	−0.940457	+	0.339911i	$0.889603\pi$
$62$	13.2665i	1.68485i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 16.5327i	− 1.96208i	−0.193816	−	0.981038i	$-0.562086\pi$
$71$	− 16.5327i	− 1.96208i	0.193816	−	0.981038i	$-0.437914\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	17.3808	1.99372
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.63325i	− 0.728094i	−0.931381	−	0.364047i	$-0.881395\pi$
$83$	− 6.63325i	− 0.728094i	0.931381	−	0.364047i	$-0.118605\pi$
$84$	0	0
$85$	0	0
$86$	4.68046i	0.504707i
$87$	0	0
$88$	−9.38083	−1.00000
$89$	11.8523i	1.25634i	0.778076	+	0.628170i	$0.216196\pi$
$89$	11.8523i	1.25634i	−0.778076	+	0.628170i	$0.783804\pi$
$90$	0	0
$91$	0	0
$92$	16.0949i	1.67801i
$93$	0	0
$94$	−0.619168	−0.0638624
$95$	0	0
$96$	0	0
$97$	−18.7617	−1.90496	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	−18.7617	−1.90496	−0.952479	+	0.304604i	$0.901476\pi$
$98$	9.89949i	1.00000i
$99$	0	0
$100$	10.0000	1.00000
$101$	2.82843i	0.281439i	0.990050	+	0.140720i	$0.0449416\pi$
$101$	2.82843i	0.281439i	−0.990050	+	0.140720i	$0.955058\pi$
$102$	0	0
$103$	9.38083	0.924321	0.462160	−	0.886796i	$-0.347074\pi$
$103$	9.38083	0.924321	0.462160	+	0.886796i	$0.347074\pi$
$104$	7.60964i	0.746187i
$105$	0	0
$106$	0	0
$107$	− 5.65685i	− 0.546869i	−0.961891	−	0.273434i	$-0.911840\pi$
$107$	− 5.65685i	− 0.546869i	0.961891	−	0.273434i	$-0.0881596\pi$
$108$	0	0
$109$	9.30958	0.891696	0.445848	−	0.895109i	$-0.352902\pi$
$109$	9.30958	0.891696	0.445848	+	0.895109i	$0.352902\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.3376i	1.91320i	0.291409	+	0.956599i	$0.405876\pi$
$113$	20.3376i	1.91320i	−0.291409	+	0.956599i	$0.594124\pi$
$114$	0	0
$115$	0	0
$116$	13.2665i	1.23176i
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 20.7754i	− 1.88091i
$123$	0	0
$124$	−18.7617	−1.68485
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 11.3137i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	11.3137i	0.988483i	0.869325	+	0.494242i	$0.164554\pi$
$131$	11.3137i	0.988483i	−0.869325	+	0.494242i	$0.835446\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.36700i	0.287663i	0.989602	+	0.143831i	$0.0459423\pi$
$137$	3.36700i	0.287663i	−0.989602	+	0.143831i	$0.954058\pi$
$138$	0	0
$139$	−20.6904	−1.75494	−0.877469	−	0.479633i	$-0.840770\pi$
$139$	−20.6904	−1.75494	−0.877469	+	0.479633i	$0.840770\pi$
$140$	0	0
$141$	0	0
$142$	23.3808	1.96208
$143$	8.92310i	0.746187i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 14.1421i	− 1.15857i	−0.815125	−	0.579284i	$-0.803332\pi$
$149$	− 14.1421i	− 1.15857i	0.815125	−	0.579284i	$-0.196668\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	24.5802i	1.99372i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	9.38083	0.728094
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−5.76166	−0.443205
$170$	0	0
$171$	0	0
$172$	−6.61917	−0.504707
$173$	− 6.63325i	− 0.504317i	−0.967686	−	0.252158i	$-0.918860\pi$
$173$	− 6.63325i	− 0.504317i	0.967686	−	0.252158i	$-0.0811404\pi$
$174$	0	0
$175$	0	0
$176$	− 13.2665i	− 1.00000i
$177$	0	0
$178$	−16.7617	−1.25634
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−22.7617	−1.67801
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 0.875636i	− 0.0638624i
$189$	0	0
$190$	0	0
$191$	− 25.0180i	− 1.81024i	−0.425155	−	0.905120i	$-0.639781\pi$
$191$	− 25.0180i	− 1.81024i	0.425155	−	0.905120i	$-0.360219\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	− 26.5330i	− 1.90496i
$195$	0	0
$196$	−14.0000	−1.00000
$197$	19.7990i	1.41062i	0.708899	+	0.705310i	$0.249192\pi$
$197$	19.7990i	1.41062i	−0.708899	+	0.705310i	$0.750808\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	14.1421i	1.00000i
$201$	0	0
$202$	−4.00000	−0.281439
$203$	0	0
$204$	0	0
$205$	0	0
$206$	13.2665i	0.924321i
$207$	0	0
$208$	−10.7617	−0.746187
$209$	28.8228i	1.99372i
$210$	0	0
$211$	15.3096	1.05396	0.526978	−	0.849879i	$-0.323325\pi$
$211$	15.3096	1.05396	0.526978	+	0.849879i	$0.323325\pi$
$212$	0	0
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	0	0
$218$	13.1657i	0.891696i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	0	0
$226$	−28.7617	−1.91320
$227$	− 22.6274i	− 1.50183i	−0.660396	−	0.750917i	$-0.729612\pi$
$227$	− 22.6274i	− 1.50183i	0.660396	−	0.750917i	$-0.270388\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−18.7617	−1.23176
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	− 15.5563i	− 1.00000i
$243$	0	0
$244$	29.3808	1.88091
$245$	0	0
$246$	0	0
$247$	23.3808	1.48769
$248$	− 26.5330i	− 1.68485i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−26.6904	−1.67801
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 5.11828i	− 0.319269i	−0.987176	−	0.159635i	$-0.948968\pi$
$257$	− 5.11828i	− 0.319269i	0.987176	−	0.159635i	$-0.0510316\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−16.0000	−0.988483
$263$	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
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	−4.76166	−0.287663
$275$	16.5831i	1.00000i
$276$	0	0
$277$	25.4521	1.52927	0.764634	−	0.644465i	$-0.222920\pi$
$277$	25.4521	1.52927	0.764634	+	0.644465i	$0.222920\pi$
$278$	− 29.2607i	− 1.75494i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	19.4521	1.15631	0.578153	−	0.815928i	$-0.303774\pi$
$283$	19.4521	1.15631	0.578153	+	0.815928i	$0.303774\pi$
$284$	33.0655i	1.96208i
$285$	0	0
$286$	−12.6192	−0.746187
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	33.1662i	1.93759i	0.247858	+	0.968796i	$0.420273\pi$
$293$	33.1662i	1.93759i	−0.247858	+	0.968796i	$0.579727\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	20.0000	1.15857
$299$	21.6510i	1.25211i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−34.7617	−1.99372
$305$	0	0
$306$	0	0
$307$	31.4521	1.79506	0.897532	−	0.440948i	$-0.145358\pi$
$307$	31.4521	1.79506	0.897532	+	0.440948i	$0.145358\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.4084i	0.987139i	0.869706	+	0.493569i	$0.164308\pi$
$311$	17.4084i	0.987139i	−0.869706	+	0.493569i	$0.835692\pi$
$312$	0	0
$313$	−18.7617	−1.06047	−0.530236	−	0.847850i	$-0.677897\pi$
$313$	−18.7617	−1.06047	−0.530236	+	0.847850i	$0.677897\pi$
$314$	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
$319$	−22.0000	−1.23176
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	13.4521	0.746187
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	13.2665i	0.728094i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	− 8.14822i	− 0.443205i
$339$	0	0
$340$	0	0
$341$	− 31.1127i	− 1.68485i
$342$	0	0
$343$	0	0
$344$	− 9.36092i	− 0.504707i
$345$	0	0
$346$	9.38083	0.504317
$347$	33.1662i	1.78046i	0.455514	+	0.890229i	$0.349456\pi$
$347$	33.1662i	1.78046i	−0.455514	+	0.890229i	$0.650544\pi$
$348$	0	0
$349$	21.3096	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$349$	21.3096	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$350$	0	0
$351$	0	0
$352$	18.7617	1.00000
$353$	− 27.9472i	− 1.48748i	−0.668469	−	0.743740i	$-0.733050\pi$
$353$	− 27.9472i	− 1.48748i	0.668469	−	0.743740i	$-0.266950\pi$
$354$	0	0
$355$	0	0
$356$	− 23.7046i	− 1.25634i
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	56.5233	2.97491
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	− 32.1899i	− 1.67801i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	37.4521	1.93920	0.969598	−	0.244705i	$-0.0786911\pi$
$373$	37.4521	1.93920	0.969598	+	0.244705i	$0.0786911\pi$
$374$	0	0
$375$	0	0
$376$	1.23834	0.0638624
$377$	17.8462i	0.919126i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	35.3808	1.81024
$383$	− 33.5033i	− 1.71194i	−0.517026	−	0.855970i	$-0.672961\pi$
$383$	− 33.5033i	− 1.71194i	0.517026	−	0.855970i	$-0.327039\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	37.5233	1.90496
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	− 19.7990i	− 1.00000i
$393$	0	0
$394$	−28.0000	−1.41062
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	2.82843i	0.141776i
$399$	0	0
$400$	−20.0000	−1.00000
$401$	− 36.4325i	− 1.81935i	−0.415319	−	0.909676i	$-0.636330\pi$
$401$	− 36.4325i	− 1.81935i	0.415319	−	0.909676i	$-0.363670\pi$
$402$	0	0
$403$	−25.2383	−1.25721
$404$	− 5.65685i	− 0.281439i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−18.7617	−0.924321
$413$	0	0
$414$	0	0
$415$	0	0
$416$	− 15.2193i	− 0.746187i
$417$	0	0
$418$	−40.7617	−1.99372
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	21.6510i	1.05396i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	11.3137i	0.546869i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	37.5233	1.80326	0.901628	−	0.432512i	$-0.142373\pi$
$433$	37.5233	1.80326	0.901628	+	0.432512i	$0.142373\pi$
$434$	0	0
$435$	0	0
$436$	−18.6192	−0.891696
$437$	69.9358i	3.34548i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	− 14.1421i	− 0.669650i
$447$	0	0
$448$	0	0
$449$	− 13.6036i	− 0.641992i	−0.947081	−	0.320996i	$-0.895982\pi$
$449$	− 13.6036i	− 0.641992i	0.947081	−	0.320996i	$-0.104018\pi$
$450$	0	0
$451$	0	0
$452$	− 40.6751i	− 1.91320i
$453$	0	0
$454$	32.0000	1.50183
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 6.63325i	− 0.308941i	−0.987997	−	0.154471i	$-0.950633\pi$
$461$	− 6.63325i	− 0.308941i	0.987997	−	0.154471i	$-0.0493672\pi$
$462$	0	0
$463$	9.38083	0.435964	0.217982	−	0.975953i	$-0.430053\pi$
$463$	9.38083	0.435964	0.217982	+	0.975953i	$0.430053\pi$
$464$	− 26.5330i	− 1.23176i
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 10.9766i	− 0.504707i
$474$	0	0
$475$	43.4521	1.99372
$476$	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
$481$	0	0
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	41.5508i	1.88091i
$489$	0	0
$490$	0	0
$491$	− 39.5980i	− 1.78703i	−0.449032	−	0.893516i	$-0.648231\pi$
$491$	− 39.5980i	− 1.78703i	0.449032	−	0.893516i	$-0.351769\pi$
$492$	0	0
$493$	0	0
$494$	33.0655i	1.48769i
$495$	0	0
$496$	37.5233	1.68485
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	− 37.7459i	− 1.67801i
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	1.00000i
$513$	0	0
$514$	7.23834	0.319269
$515$	0	0
$516$	0	0
$517$	1.45208	0.0638624
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 44.9178i	− 1.96788i	−0.178490	−	0.983942i	$-0.557121\pi$
$521$	− 44.9178i	− 1.96788i	0.178490	−	0.983942i	$-0.442879\pi$
$522$	0	0
$523$	27.3096	1.19416	0.597082	−	0.802180i	$-0.296327\pi$
$523$	27.3096	1.19416	0.597082	+	0.802180i	$0.296327\pi$
$524$	− 22.6274i	− 0.988483i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−41.7617	−1.81572
$530$	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
$539$	− 23.2164i	− 1.00000i
$540$	0	0
$541$	−30.8329	−1.32561	−0.662805	−	0.748792i	$-0.730634\pi$
$541$	−30.8329	−1.32561	−0.662805	+	0.748792i	$0.730634\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−36.8329	−1.57486	−0.787431	−	0.616403i	$-0.788589\pi$
$547$	−36.8329	−1.57486	−0.787431	+	0.616403i	$0.788589\pi$
$548$	− 6.73401i	− 0.287663i
$549$	0	0
$550$	−23.4521	−1.00000
$551$	57.6457i	2.45579i
$552$	0	0
$553$	0	0
$554$	35.9947i	1.52927i
$555$	0	0
$556$	41.3808	1.75494
$557$	− 46.4327i	− 1.96742i	−0.179766	−	0.983709i	$-0.557534\pi$
$557$	− 46.4327i	− 1.96742i	0.179766	−	0.983709i	$-0.442466\pi$
$558$	0	0
$559$	−8.90416	−0.376606
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.1662i	1.39779i	0.715224	+	0.698895i	$0.246325\pi$
$563$	33.1662i	1.39779i	−0.715224	+	0.698895i	$0.753675\pi$
$564$	0	0
$565$	0	0
$566$	27.5094i	1.15631i
$567$	0	0
$568$	−46.7617	−1.96208
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−24.8329	−1.03922	−0.519612	−	0.854402i	$-0.673924\pi$
$571$	−24.8329	−1.03922	−0.519612	+	0.854402i	$0.673924\pi$
$572$	− 17.8462i	− 0.746187i
$573$	0	0
$574$	0	0
$575$	40.2373i	1.67801i
$576$	0	0
$577$	37.5233	1.56212	0.781058	−	0.624458i	$-0.214680\pi$
$577$	37.5233	1.56212	0.781058	+	0.624458i	$0.214680\pi$
$578$	− 24.0416i	− 1.00000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−46.9042	−1.93759
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−81.5233	−3.35911
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	28.2843i	1.15857i
$597$	0	0
$598$	−30.6192	−1.25211
$599$	23.2668i	0.950654i	0.879809	+	0.475327i	$0.157670\pi$
$599$	23.2668i	0.950654i	−0.879809	+	0.475327i	$0.842330\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	− 49.1604i	− 1.99372i
$609$	0	0
$610$	0	0
$611$	− 1.17791i	− 0.0476533i
$612$	0	0
$613$	49.4521	1.99735	0.998675	−	0.0514548i	$-0.0163858\pi$
$613$	49.4521	1.99735	0.998675	+	0.0514548i	$0.0163858\pi$
$614$	44.4800i	1.79506i
$615$	0	0
$616$	0	0
$617$	45.7934i	1.84357i	0.387698	+	0.921787i	$0.373271\pi$
$617$	45.7934i	1.84357i	−0.387698	+	0.921787i	$0.626729\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	−24.6192	−0.987139
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	− 26.5330i	− 1.06047i
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−46.9042	−1.86723	−0.933613	−	0.358284i	$-0.883362\pi$
$631$	−46.9042	−1.86723	−0.933613	+	0.358284i	$0.883362\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−18.8329	−0.746187
$638$	− 31.1127i	− 1.23176i
$639$	0	0
$640$	0	0
$641$	− 2.49137i	− 0.0984031i	−0.998789	−	0.0492016i	$-0.984332\pi$
$641$	− 2.49137i	− 0.0984031i	0.998789	−	0.0492016i	$-0.0156677\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 41.9886i	− 1.65074i	−0.564591	−	0.825371i	$-0.690966\pi$
$647$	− 41.9886i	− 1.65074i	0.564591	−	0.825371i	$-0.309034\pi$
$648$	0	0
$649$	0	0
$650$	19.0241i	0.746187i
$651$	0	0
$652$	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
$657$	0	0
$658$	0	0
$659$	− 6.63325i	− 0.258395i	−0.991619	−	0.129197i	$-0.958760\pi$
$659$	− 6.63325i	− 0.258395i	0.991619	−	0.129197i	$-0.0412401\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	−18.7617	−0.728094
$665$	0	0
$666$	0	0
$667$	−53.3808	−2.06691
$668$	0	0
$669$	0	0
$670$	0	0
$671$	48.7226i	1.88091i
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	11.5233	0.443205
$677$	− 48.0833i	− 1.84799i	−0.382405	−	0.923995i	$-0.624904\pi$
$677$	− 48.0833i	− 1.84799i	0.382405	−	0.923995i	$-0.375096\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	44.0000	1.68485
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	13.2383	0.504707
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	13.2665i	0.504317i
$693$	0	0
$694$	−46.9042	−1.78046
$695$	0	0
$696$	0	0
$697$	0	0
$698$	30.1363i	1.14068i
$699$	0	0
$700$	0	0
$701$	− 46.4327i	− 1.75374i	−0.480727	−	0.876870i	$-0.659627\pi$
$701$	− 46.4327i	− 1.75374i	0.480727	−	0.876870i	$-0.340373\pi$
$702$	0	0
$703$	0	0
$704$	26.5330i	1.00000i
$705$	0	0
$706$	39.5233	1.48748
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	33.5233	1.25634
$713$	− 75.4919i	− 2.82719i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14.7815i	0.551256i	0.961264	+	0.275628i	$0.0888858\pi$
$719$	14.7815i	0.551256i	−0.961264	+	0.275628i	$0.911114\pi$
$720$	0	0
$721$	0	0
$722$	79.9361i	2.97491i
$723$	0	0
$724$	0	0
$725$	33.1662i	1.23176i
$726$	0	0
$727$	−46.0000	−1.70605	−0.853023	−	0.521874i	$-0.825233\pi$
$727$	−46.0000	−1.70605	−0.853023	+	0.521874i	$0.825233\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.3096	1.23032	0.615159	−	0.788403i	$-0.289092\pi$
$733$	33.3096	1.23032	0.615159	+	0.788403i	$0.289092\pi$
$734$	36.7696i	1.35719i
$735$	0	0
$736$	45.5233	1.67801
$737$	0	0
$738$	0	0
$739$	−12.8329	−0.472066	−0.236033	−	0.971745i	$-0.575847\pi$
$739$	−12.8329	−0.472066	−0.236033	+	0.971745i	$0.575847\pi$
$740$	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
$745$	0	0
$746$	52.9652i	1.93920i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.38083	0.342311	0.171156	−	0.985244i	$-0.445250\pi$
$751$	9.38083	0.342311	0.171156	+	0.985244i	$0.445250\pi$
$752$	1.75127i	0.0638624i
$753$	0	0
$754$	−25.2383	−0.919126
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	50.0361i	1.81024i
$765$	0	0
$766$	47.3808	1.71194
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	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
$775$	−46.9042	−1.68485
$776$	53.0660i	1.90496i
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−54.8329	−1.96208
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	55.4521	1.97665	0.988327	−	0.152350i	$-0.0486842\pi$
$787$	55.4521	1.97665	0.988327	+	0.152350i	$0.0486842\pi$
$788$	− 39.5980i	− 1.41062i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	39.5233	1.40351
$794$	0	0
$795$	0	0
$796$	−4.00000	−0.141776
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	− 28.2843i	− 1.00000i
$801$	0	0
$802$	51.5233	1.81935
$803$	0	0
$804$	0	0
$805$	0	0
$806$	− 35.6924i	− 1.25721i
$807$	0	0
$808$	8.00000	0.281439
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−44.6904	−1.56929	−0.784646	−	0.619944i	$-0.787155\pi$
$811$	−44.6904	−1.56929	−0.784646	+	0.619944i	$0.787155\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−28.7617	−1.00624
$818$	0	0
$819$	0	0
$820$	0	0
$821$	53.7401i	1.87554i	0.347253	+	0.937771i	$0.387115\pi$
$821$	53.7401i	1.87554i	−0.347253	+	0.937771i	$0.612885\pi$
$822$	0	0
$823$	50.0000	1.74289	0.871445	−	0.490493i	$-0.163183\pi$
$823$	50.0000	1.74289	0.871445	+	0.490493i	$0.163183\pi$
$824$	− 26.5330i	− 0.924321i
$825$	0	0
$826$	0	0
$827$	− 46.4327i	− 1.61462i	−0.590124	−	0.807312i	$-0.700921\pi$
$827$	− 46.4327i	− 1.61462i	0.590124	−	0.807312i	$-0.299079\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	21.5233	0.746187
$833$	0	0
$834$	0	0
$835$	0	0
$836$	− 57.6457i	− 1.99372i
$837$	0	0
$838$	0	0
$839$	57.2079i	1.97504i	0.157508	+	0.987518i	$0.449654\pi$
$839$	57.2079i	1.97504i	−0.157508	+	0.987518i	$0.550346\pi$
$840$	0	0
$841$	−15.0000	−0.517241
$842$	0	0
$843$	0	0
$844$	−30.6192	−1.05396
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−10.5479	−0.361154	−0.180577	−	0.983561i	$-0.557796\pi$
$853$	−10.5479	−0.361154	−0.180577	+	0.983561i	$0.557796\pi$
$854$	0	0
$855$	0	0
$856$	−16.0000	−0.546869
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.3789i	1.17027i	0.810935	+	0.585136i	$0.198959\pi$
$863$	34.3789i	1.17027i	−0.810935	+	0.585136i	$0.801041\pi$
$864$	0	0
$865$	0	0
$866$	53.0660i	1.80326i
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 26.3315i	− 0.891696i
$873$	0	0
$874$	−98.9042	−3.34548
$875$	0	0
$876$	0	0
$877$	−6.83291	−0.230731	−0.115365	−	0.993323i	$-0.536804\pi$
$877$	−6.83291	−0.230731	−0.115365	+	0.993323i	$0.536804\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.99391i	0.201940i	0.994889	+	0.100970i	$0.0321946\pi$
$881$	5.99391i	0.201940i	−0.994889	+	0.100970i	$0.967805\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	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
$889$	0	0
$890$	0	0
$891$	0	0
$892$	20.0000	0.669650
$893$	− 3.80482i	− 0.127324i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	19.2383	0.641992
$899$	− 62.2254i	− 2.07533i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	57.5233	1.91320
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	45.2548i	1.50183i
$909$	0	0
$910$	0	0
$911$	6.29619i	0.208602i	0.994546	+	0.104301i	$0.0332605\pi$
$911$	6.29619i	0.208602i	−0.994546	+	0.104301i	$0.966739\pi$
$912$	0	0
$913$	−22.0000	−0.728094
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	9.38083	0.308941
$923$	44.4800i	1.46408i
$924$	0	0
$925$	0	0
$926$	13.2665i	0.435964i
$927$	0	0
$928$	37.5233	1.23176
$929$	54.2787i	1.78083i	0.455153	+	0.890413i	$0.349584\pi$
$929$	54.2787i	1.78083i	−0.455153	+	0.890413i	$0.650416\pi$
$930$	0	0
$931$	−60.8329	−1.99372
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 14.1421i	− 0.461020i	−0.973070	−	0.230510i	$-0.925960\pi$
$941$	− 14.1421i	− 0.461020i	0.973070	−	0.230510i	$-0.0740395\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	15.5233	0.504707
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	61.4505i	1.99372i
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	57.0000	1.83871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	31.1127i	1.00000i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	− 48.0833i	− 1.54069i
$975$	0	0
$976$	−58.7617	−1.88091
$977$	− 61.8883i	− 1.97998i	−0.141126	−	0.989992i	$-0.545072\pi$
$977$	− 61.8883i	− 1.97998i	0.141126	−	0.989992i	$-0.454928\pi$
$978$	0	0
$979$	39.3096	1.25634
$980$	0	0
$981$	0	0
$982$	56.0000	1.78703
$983$	− 50.4739i	− 1.60987i	−0.593365	−	0.804933i	$-0.702201\pi$
$983$	− 50.4739i	− 1.60987i	0.593365	−	0.804933i	$-0.297799\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−46.7617	−1.48769
$989$	− 26.6338i	− 0.846904i
$990$	0	0
$991$	−46.9042	−1.48996	−0.744980	−	0.667087i	$-0.767541\pi$
$991$	−46.9042	−1.48996	−0.744980	+	0.667087i	$0.767541\pi$
$992$	53.0660i	1.68485i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	61.4521	1.94621	0.973103	−	0.230369i	$-0.0739932\pi$
$997$	61.4521	1.94621	0.973103	+	0.230369i	$0.0739932\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	792.2.m.b.197.3	yes	4
3.2	odd	2	inner	792.2.m.b.197.2	yes	4
4.3	odd	2		3168.2.m.b.593.3		4
8.3	odd	2		3168.2.m.a.593.2		4
8.5	even	2		792.2.m.a.197.2	✓	4
11.10	odd	2		792.2.m.a.197.2	✓	4
12.11	even	2		3168.2.m.b.593.1		4
24.5	odd	2		792.2.m.a.197.3	yes	4
24.11	even	2		3168.2.m.a.593.4		4
33.32	even	2		792.2.m.a.197.3	yes	4
44.43	even	2		3168.2.m.a.593.2		4
88.21	odd	2	CM	792.2.m.b.197.3	yes	4
88.43	even	2		3168.2.m.b.593.3		4
132.131	odd	2		3168.2.m.a.593.4		4
264.131	odd	2		3168.2.m.b.593.1		4
264.197	even	2	inner	792.2.m.b.197.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.m.a.197.2	✓	4	8.5	even	2
792.2.m.a.197.2	✓	4	11.10	odd	2
792.2.m.a.197.3	yes	4	24.5	odd	2
792.2.m.a.197.3	yes	4	33.32	even	2
792.2.m.b.197.2	yes	4	3.2	odd	2	inner
792.2.m.b.197.2	yes	4	264.197	even	2	inner
792.2.m.b.197.3	yes	4	1.1	even	1	trivial
792.2.m.b.197.3	yes	4	88.21	odd	2	CM
3168.2.m.a.593.2		4	8.3	odd	2
3168.2.m.a.593.2		4	44.43	even	2
3168.2.m.a.593.4		4	24.11	even	2
3168.2.m.a.593.4		4	132.131	odd	2
3168.2.m.b.593.1		4	12.11	even	2
3168.2.m.b.593.1		4	264.131	odd	2
3168.2.m.b.593.3		4	4.3	odd	2
3168.2.m.b.593.3		4	88.43	even	2

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:	\( N \)	\(=\)	\( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	792.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} -3.31662i q^{11} -2.69042 q^{13} +4.00000 q^{16} -8.69042 q^{19} +4.69042 q^{22} -8.04746i q^{23} -5.00000 q^{25} -3.80482i q^{26} -6.63325i q^{29} +9.38083 q^{31} +5.65685i q^{32} -12.2901i q^{38} +3.30958 q^{43} +6.63325i q^{44} +11.3808 q^{46} +0.437818i q^{47} +7.00000 q^{49} -7.07107i q^{50} +5.38083 q^{52} +9.38083 q^{58} -14.6904 q^{61} +13.2665i q^{62} -8.00000 q^{64} -16.5327i q^{71} +17.3808 q^{76} -6.63325i q^{83} +4.68046i q^{86} -9.38083 q^{88} +11.8523i q^{89} +16.0949i q^{92} -0.619168 q^{94} -18.7617 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 8 q^{13} + 16 q^{16} - 16 q^{19} - 20 q^{25} + 32 q^{43} + 8 q^{46} + 28 q^{49} - 16 q^{52} - 40 q^{61} - 32 q^{64} + 32 q^{76} - 40 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 + 8 * q^13 + 16 * q^16 - 16 * q^19 - 20 * q^25 + 32 * q^43 + 8 * q^46 + 28 * q^49 - 16 * q^52 - 40 * q^61 - 32 * q^64 + 32 * q^76 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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