LMFDB - Embedded newform 792.2.m.a.197.4

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			197.4
Root			$0.500000 + 0.244099i$ of defining polynomial
Character	$\chi$	$=$	792.197
Dual form			792.2.m.a.197.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} +3.31662i q^{11} -6.69042 q^{13} +4.00000 q^{16} -0.690416 q^{19} -4.69042 q^{22} -5.21904i q^{23} -5.00000 q^{25} -9.46168i q^{26} +6.63325i q^{29} -9.38083 q^{31} +5.65685i q^{32} -0.976395i q^{38} -12.6904 q^{43} -6.63325i q^{44} +7.38083 q^{46} -13.7043i q^{47} +7.00000 q^{49} -7.07107i q^{50} +13.3808 q^{52} -9.38083 q^{58} +5.30958 q^{61} -13.2665i q^{62} -8.00000 q^{64} +3.26625i q^{71} +1.38083 q^{76} +6.63325i q^{83} -17.9470i q^{86} +9.38083 q^{88} +14.6807i q^{89} +10.4381i q^{92} +19.3808 q^{94} +18.7617 q^{97} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−6.69042	−1.85559	−0.927794	−	0.373094i	$-0.878297\pi$
$13$	−6.69042	−1.85559	−0.927794	+	0.373094i	$0.878297\pi$
$14$	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
$19$	−0.690416	−0.158392	−0.0791961	−	0.996859i	$-0.525235\pi$
$19$	−0.690416	−0.158392	−0.0791961	+	0.996859i	$0.525235\pi$
$20$	0	0
$21$	0	0
$22$	−4.69042	−1.00000
$23$	− 5.21904i	− 1.08824i	−0.839006	−	0.544122i	$-0.816863\pi$
$23$	− 5.21904i	− 1.08824i	0.839006	−	0.544122i	$-0.183137\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	− 9.46168i	− 1.85559i
$27$	0	0
$28$	0	0
$29$	6.63325i	1.23176i	0.787839	+	0.615882i	$0.211200\pi$
$29$	6.63325i	1.23176i	−0.787839	+	0.615882i	$0.788800\pi$
$30$	0	0
$31$	−9.38083	−1.68485	−0.842424	−	0.538816i	$-0.818872\pi$
$31$	−9.38083	−1.68485	−0.842424	+	0.538816i	$0.818872\pi$
$32$	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			$\pi$
$38$	− 0.976395i	− 0.158392i
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−12.6904	−1.93527	−0.967635	−	0.252353i	$-0.918795\pi$
$43$	−12.6904	−1.93527	−0.967635	+	0.252353i	$0.918795\pi$
$44$	− 6.63325i	− 1.00000i
$45$	0	0
$46$	7.38083	1.08824
$47$	− 13.7043i	− 1.99898i	−0.0319312	−	0.999490i	$-0.510166\pi$
$47$	− 13.7043i	− 1.99898i	0.0319312	−	0.999490i	$-0.489834\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	− 7.07107i	− 1.00000i
$51$	0	0
$52$	13.3808	1.85559
$53$	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
$61$	5.30958	0.679823	0.339911	−	0.940457i	$-0.389603\pi$
$61$	5.30958	0.679823	0.339911	+	0.940457i	$0.389603\pi$
$62$	− 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
$71$	3.26625i	0.387632i	0.981038	+	0.193816i	$0.0620865\pi$
$71$	3.26625i	0.387632i	−0.981038	+	0.193816i	$0.937914\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	1.38083	0.158392
$77$	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	$0.118605\pi$
$83$	6.63325i	0.728094i	−0.931381	+	0.364047i	$0.881395\pi$
$84$	0	0
$85$	0	0
$86$	− 17.9470i	− 1.93527i
$87$	0	0
$88$	9.38083	1.00000
$89$	14.6807i	1.55615i	0.628170	+	0.778076i	$0.283804\pi$
$89$	14.6807i	1.55615i	−0.628170	+	0.778076i	$0.716196\pi$
$90$	0	0
$91$	0	0
$92$	10.4381i	1.08824i
$93$	0	0
$94$	19.3808	1.99898
$95$	0	0
$96$	0	0
$97$	18.7617	1.90496	0.952479	−	0.304604i	$-0.0985241\pi$
$97$	18.7617	1.90496	0.952479	+	0.304604i	$0.0985241\pi$
$98$	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	$-0.652926\pi$
$103$	−9.38083	−0.924321	−0.462160	+	0.886796i	$0.652926\pi$
$104$	18.9234i	1.85559i
$105$	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
$109$	−18.6904	−1.79022	−0.895109	−	0.445848i	$-0.852902\pi$
$109$	−18.6904	−1.79022	−0.895109	+	0.445848i	$0.852902\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.19543i	0.582817i	0.956599	+	0.291409i	$0.0941239\pi$
$113$	6.19543i	0.582817i	−0.956599	+	0.291409i	$0.905876\pi$
$114$	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
$122$	7.50889i	0.679823i
$123$	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
$137$	23.1660i	1.97920i	0.143831	+	0.989602i	$0.454058\pi$
$137$	23.1660i	1.97920i	−0.143831	+	0.989602i	$0.545942\pi$
$138$	0	0
$139$	11.3096	0.959266	0.479633	−	0.877469i	$-0.340770\pi$
$139$	11.3096	0.959266	0.479633	+	0.877469i	$0.340770\pi$
$140$	0	0
$141$	0	0
$142$	−4.61917	−0.387632
$143$	− 22.1896i	− 1.85559i
$144$	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			$\pi$
$152$	1.95279i	0.158392i
$153$	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
$169$	31.7617	2.44320
$170$	0	0
$171$	0	0
$172$	25.3808	1.93527
$173$	6.63325i	0.504317i	0.967686	+	0.252158i	$0.0811404\pi$
$173$	6.63325i	0.504317i	−0.967686	+	0.252158i	$0.918860\pi$
$174$	0	0
$175$	0	0
$176$	13.2665i	1.00000i
$177$	0	0
$178$	−20.7617	−1.55615
$179$	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
$184$	−14.7617	−1.08824
$185$	0	0
$186$	0	0
$187$	0	0
$188$	27.4086i	1.99898i
$189$	0	0
$190$	0	0
$191$	11.7515i	0.850311i	0.905120	+	0.425155i	$0.139781\pi$
$191$	11.7515i	0.850311i	−0.905120	+	0.425155i	$0.860219\pi$
$192$	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
$208$	−26.7617	−1.85559
$209$	− 2.28985i	− 0.158392i
$210$	0	0
$211$	−24.6904	−1.69976	−0.849879	−	0.526978i	$-0.823325\pi$
$211$	−24.6904	−1.69976	−0.849879	+	0.526978i	$0.823325\pi$
$212$	0	0
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 26.4322i	− 1.79022i
$219$	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
$226$	−8.76166	−0.582817
$227$	− 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
$244$	−10.6192	−0.679823
$245$	0	0
$246$	0	0
$247$	4.61917	0.293911
$248$	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
$253$	17.3096	1.08824
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	31.6513i	1.97435i	0.159635	+	0.987176i	$0.448968\pi$
$257$	31.6513i	1.97435i	−0.159635	+	0.987176i	$0.551032\pi$
$258$	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
$274$	−32.7617	−1.97920
$275$	− 16.5831i	− 1.00000i
$276$	0	0
$277$	21.4521	1.28893	0.644465	−	0.764634i	$-0.277080\pi$
$277$	21.4521	1.28893	0.644465	+	0.764634i	$0.277080\pi$
$278$	15.9942i	0.959266i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	27.4521	1.63186	0.815928	−	0.578153i	$-0.196226\pi$
$283$	27.4521	1.63186	0.815928	+	0.578153i	$0.196226\pi$
$284$	− 6.53249i	− 0.387632i
$285$	0	0
$286$	31.3808	1.85559
$287$	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	$-0.579727\pi$
$293$	− 33.1662i	− 1.93759i	0.247858	−	0.968796i	$-0.420273\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	20.0000	1.15857
$299$	34.9175i	2.01933i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−2.76166	−0.158392
$305$	0	0
$306$	0	0
$307$	15.4521	0.881897	0.440948	−	0.897532i	$-0.354642\pi$
$307$	15.4521	0.881897	0.440948	+	0.897532i	$0.354642\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 30.6749i	− 1.73941i	−0.493569	−	0.869706i	$-0.664308\pi$
$311$	− 30.6749i	− 1.73941i	0.493569	−	0.869706i	$-0.335692\pi$
$312$	0	0
$313$	18.7617	1.06047	0.530236	−	0.847850i	$-0.322103\pi$
$313$	18.7617	1.06047	0.530236	+	0.847850i	$0.322103\pi$
$314$	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
$325$	33.4521	1.85559
$326$	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			$\pi$
$338$	44.9178i	2.44320i
$339$	0	0
$340$	0	0
$341$	− 31.1127i	− 1.68485i
$342$	0	0
$343$	0	0
$344$	35.8939i	1.93527i
$345$	0	0
$346$	−9.38083	−0.504317
$347$	− 33.1662i	− 1.78046i	−0.455514	−	0.890229i	$-0.650544\pi$
$347$	− 33.1662i	− 1.78046i	0.455514	−	0.890229i	$-0.349456\pi$
$348$	0	0
$349$	−30.6904	−1.64282	−0.821410	−	0.570338i	$-0.806812\pi$
$349$	−30.6904	−1.64282	−0.821410	+	0.570338i	$0.806812\pi$
$350$	0	0
$351$	0	0
$352$	−18.7617	−1.00000
$353$	− 25.1188i	− 1.33694i	−0.743740	−	0.668469i	$-0.766950\pi$
$353$	− 25.1188i	− 1.33694i	0.743740	−	0.668469i	$-0.233050\pi$
$354$	0	0
$355$	0	0
$356$	− 29.3614i	− 1.55615i
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−18.5233	−0.974912
$362$	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	$0.262589\pi$
$368$	− 20.8761i	− 1.08824i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	9.45208	0.489410	0.244705	−	0.969598i	$-0.421309\pi$
$373$	9.45208	0.489410	0.244705	+	0.969598i	$0.421309\pi$
$374$	0	0
$375$	0	0
$376$	−38.7617	−1.99898
$377$	− 44.3792i	− 2.28564i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	−16.6192	−0.850311
$383$	20.2368i	1.03405i	0.855970	+	0.517026i	$0.172961\pi$
$383$	20.2368i	1.03405i	−0.855970	+	0.517026i	$0.827039\pi$
$384$	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
$401$	− 16.6335i	− 0.830638i	−0.909676	−	0.415319i	$-0.863670\pi$
$401$	− 16.6335i	− 0.830638i	0.909676	−	0.415319i	$-0.136330\pi$
$402$	0	0
$403$	62.7617	3.12638
$404$	− 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
$416$	− 37.8467i	− 1.85559i
$417$	0	0
$418$	3.23834	0.158392
$419$	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			$\pi$
$422$	− 34.9175i	− 1.69976i
$423$	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	$-0.857627\pi$
$433$	−37.5233	−1.80326	−0.901628	+	0.432512i	$0.857627\pi$
$434$	0	0
$435$	0	0
$436$	37.3808	1.79022
$437$	3.60330i	0.172369i
$438$	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
$449$	40.1366i	1.89416i	0.320996	+	0.947081i	$0.395982\pi$
$449$	40.1366i	1.89416i	−0.320996	+	0.947081i	$0.604018\pi$
$450$	0	0
$451$	0	0
$452$	− 12.3909i	− 0.582817i
$453$	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	$0.0493672\pi$
$461$	6.63325i	0.308941i	−0.987997	+	0.154471i	$0.950633\pi$
$462$	0	0
$463$	−9.38083	−0.435964	−0.217982	−	0.975953i	$-0.569947\pi$
$463$	−9.38083	−0.435964	−0.217982	+	0.975953i	$0.569947\pi$
$464$	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
$473$	− 42.0893i	− 1.93527i
$474$	0	0
$475$	3.45208	0.158392
$476$	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	$0.779915\pi$
$488$	− 15.0178i	− 0.679823i
$489$	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
$494$	6.53249i	0.293911i
$495$	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
$506$	24.4794i	1.08824i
$507$	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
$514$	−44.7617	−1.97435
$515$	0	0
$516$	0	0
$517$	45.4521	1.99898
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8.14822i	− 0.356980i	−0.983942	−	0.178490i	$-0.942879\pi$
$521$	− 8.14822i	− 0.356980i	0.983942	−	0.178490i	$-0.0571212\pi$
$522$	0	0
$523$	−36.6904	−1.60436	−0.802180	−	0.597082i	$-0.796327\pi$
$523$	−36.6904	−1.60436	−0.802180	+	0.597082i	$0.796327\pi$
$524$	− 22.6274i	− 0.988483i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−4.23834	−0.184276
$530$	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
$541$	−34.8329	−1.49758	−0.748792	−	0.662805i	$-0.769366\pi$
$541$	−34.8329	−1.49758	−0.748792	+	0.662805i	$0.769366\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−28.8329	−1.23281	−0.616403	−	0.787431i	$-0.711411\pi$
$547$	−28.8329	−1.23281	−0.616403	+	0.787431i	$0.711411\pi$
$548$	− 46.3320i	− 1.97920i
$549$	0	0
$550$	23.4521	1.00000
$551$	− 4.57970i	− 0.195102i
$552$	0	0
$553$	0	0
$554$	30.3378i	1.28893i
$555$	0	0
$556$	−22.6192	−0.959266
$557$	46.4327i	1.96742i	0.179766	+	0.983709i	$0.442466\pi$
$557$	46.4327i	1.96742i	−0.179766	+	0.983709i	$0.557534\pi$
$558$	0	0
$559$	84.9042	3.59106
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 33.1662i	− 1.39779i	−0.715224	−	0.698895i	$-0.753675\pi$
$563$	− 33.1662i	− 1.39779i	0.715224	−	0.698895i	$-0.246325\pi$
$564$	0	0
$565$	0	0
$566$	38.8231i	1.63186i
$567$	0	0
$568$	9.23834	0.387632
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−40.8329	−1.70880	−0.854402	−	0.519612i	$-0.826076\pi$
$571$	−40.8329	−1.70880	−0.854402	+	0.519612i	$0.826076\pi$
$572$	44.3792i	1.85559i
$573$	0	0
$574$	0	0
$575$	26.0952i	1.08824i
$576$	0	0
$577$	−37.5233	−1.56212	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	−37.5233	−1.56212	−0.781058	+	0.624458i	$0.785320\pi$
$578$	− 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
$589$	6.47667	0.266867
$590$	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
$598$	−49.3808	−2.01933
$599$	43.0657i	1.75962i	0.475327	+	0.879809i	$0.342330\pi$
$599$	43.0657i	1.75962i	−0.475327	+	0.879809i	$0.657670\pi$
$600$	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			$\pi$
$608$	− 3.90558i	− 0.158392i
$609$	0	0
$610$	0	0
$611$	91.6876i	3.70928i
$612$	0	0
$613$	−2.54792	−0.102910	−0.0514548	−	0.998675i	$-0.516386\pi$
$613$	−2.54792	−0.102910	−0.0514548	+	0.998675i	$0.516386\pi$
$614$	21.8525i	0.881897i
$615$	0	0
$616$	0	0
$617$	− 19.2604i	− 0.775395i	−0.921787	−	0.387698i	$-0.873271\pi$
$617$	− 19.2604i	− 0.775395i	0.921787	−	0.387698i	$-0.126729\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	43.3808	1.73941
$623$	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	$-0.116638\pi$
$631$	46.9042	1.86723	0.933613	+	0.358284i	$0.116638\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−46.8329	−1.85559
$638$	− 31.1127i	− 1.23176i
$639$	0	0
$640$	0	0
$641$	− 50.5746i	− 1.99758i	−0.0492016	−	0.998789i	$-0.515668\pi$
$641$	− 50.5746i	− 1.99758i	0.0492016	−	0.998789i	$-0.484332\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.7221i	1.12918i	0.825371	+	0.564591i	$0.190966\pi$
$647$	28.7221i	1.12918i	−0.825371	+	0.564591i	$0.809034\pi$
$648$	0	0
$649$	0	0
$650$	47.3084i	1.85559i
$651$	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	$0.0412401\pi$
$659$	6.63325i	0.258395i	−0.991619	+	0.129197i	$0.958760\pi$
$660$	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
$667$	34.6192	1.34046
$668$	0	0
$669$	0	0
$670$	0	0
$671$	17.6099i	0.679823i
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−63.5233	−2.44320
$677$	− 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
$688$	−50.7617	−1.93527
$689$	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
$698$	− 43.4028i	− 1.64282i
$699$	0	0
$700$	0	0
$701$	46.4327i	1.75374i	0.480727	+	0.876870i	$0.340373\pi$
$701$	46.4327i	1.75374i	−0.480727	+	0.876870i	$0.659627\pi$
$702$	0	0
$703$	0	0
$704$	− 26.5330i	− 1.00000i
$705$	0	0
$706$	35.5233	1.33694
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	41.5233	1.55615
$713$	48.9589i	1.83353i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	51.5510i	1.92253i	0.275628	+	0.961264i	$0.411114\pi$
$719$	51.5510i	1.92253i	−0.275628	+	0.961264i	$0.588886\pi$
$720$	0	0
$721$	0	0
$722$	− 26.1959i	− 0.974912i
$723$	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
$733$	−42.6904	−1.57681	−0.788403	−	0.615159i	$-0.789092\pi$
$733$	−42.6904	−1.57681	−0.788403	+	0.615159i	$0.789092\pi$
$734$	36.7696i	1.35719i
$735$	0	0
$736$	29.5233	1.08824
$737$	0	0
$738$	0	0
$739$	−52.8329	−1.94349	−0.971745	−	0.236033i	$-0.924153\pi$
$739$	−52.8329	−1.94349	−0.971745	+	0.236033i	$0.924153\pi$
$740$	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
$746$	13.3673i	0.489410i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−9.38083	−0.342311	−0.171156	−	0.985244i	$-0.554750\pi$
$751$	−9.38083	−0.342311	−0.171156	+	0.985244i	$0.554750\pi$
$752$	− 54.8173i	− 1.99898i
$753$	0	0
$754$	62.7617	2.28564
$755$	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
$764$	− 23.5031i	− 0.850311i
$765$	0	0
$766$	−28.6192	−1.03405
$767$	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
$781$	−10.8329	−0.387632
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	−8.54792	−0.304700	−0.152350	−	0.988327i	$-0.548684\pi$
$787$	−8.54792	−0.304700	−0.152350	+	0.988327i	$0.548684\pi$
$788$	− 39.5980i	− 1.41062i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−35.5233	−1.26147
$794$	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
$802$	23.5233	0.830638
$803$	0	0
$804$	0	0
$805$	0	0
$806$	88.7584i	3.12638i
$807$	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
$811$	35.3096	1.23989	0.619944	−	0.784646i	$-0.287155\pi$
$811$	35.3096	1.23989	0.619944	+	0.784646i	$0.287155\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.76166	0.306532
$818$	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	$0.299079\pi$
$827$	46.4327i	1.61462i	−0.590124	+	0.807312i	$0.700921\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	53.5233	1.85559
$833$	0	0
$834$	0	0
$835$	0	0
$836$	4.57970i	0.158392i
$837$	0	0
$838$	0	0
$839$	9.12462i	0.315017i	0.987518	+	0.157508i	$0.0503461\pi$
$839$	9.12462i	0.315017i	−0.987518	+	0.157508i	$0.949654\pi$
$840$	0	0
$841$	−15.0000	−0.517241
$842$	0	0
$843$	0	0
$844$	49.3808	1.69976
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	57.4521	1.96712	0.983561	−	0.180577i	$-0.0577965\pi$
$853$	57.4521	1.96712	0.983561	+	0.180577i	$0.0577965\pi$
$854$	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
$863$	− 47.6454i	− 1.62187i	−0.585136	−	0.810935i	$-0.698959\pi$
$863$	− 47.6454i	− 1.62187i	0.585136	−	0.810935i	$-0.301041\pi$
$864$	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
$872$	52.8645i	1.79022i
$873$	0	0
$874$	−5.09584	−0.172369
$875$	0	0
$876$	0	0
$877$	−58.8329	−1.98665	−0.993323	−	0.115365i	$-0.963196\pi$
$877$	−58.8329	−1.98665	−0.993323	+	0.115365i	$0.963196\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 59.0599i	− 1.98978i	−0.100970	−	0.994889i	$-0.532195\pi$
$881$	− 59.0599i	− 1.98978i	0.100970	−	0.994889i	$-0.467805\pi$
$882$	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
$893$	9.46168i	0.316623i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−56.7617	−1.89416
$899$	− 62.2254i	− 2.07533i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	17.5233	0.582817
$905$	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
$911$	60.0363i	1.98909i	0.104301	+	0.994546i	$0.466739\pi$
$911$	60.0363i	1.98909i	−0.104301	+	0.994546i	$0.533261\pi$
$912$	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
$923$	− 21.8525i	− 0.719285i
$924$	0	0
$925$	0	0
$926$	− 13.2665i	− 0.435964i
$927$	0	0
$928$	−37.5233	−1.23176
$929$	− 27.7457i	− 0.910307i	−0.890413	−	0.455153i	$-0.849584\pi$
$929$	− 27.7457i	− 0.910307i	0.890413	−	0.455153i	$-0.150416\pi$
$930$	0	0
$931$	−4.83291	−0.158392
$932$	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
$946$	59.5233	1.93527
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	4.88198i	0.158392i
$951$	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
$976$	21.2383	0.679823
$977$	8.82234i	0.282252i	0.989992	+	0.141126i	$0.0450722\pi$
$977$	8.82234i	0.282252i	−0.989992	+	0.141126i	$0.954928\pi$
$978$	0	0
$979$	−48.6904	−1.55615
$980$	0	0
$981$	0	0
$982$	56.0000	1.78703
$983$	37.2074i	1.18673i	0.804933	+	0.593365i	$0.202201\pi$
$983$	37.2074i	1.18673i	−0.804933	+	0.593365i	$0.797799\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−9.23834	−0.293911
$989$	66.2317i	2.10605i
$990$	0	0
$991$	46.9042	1.48996	0.744980	−	0.667087i	$-0.232459\pi$
$991$	46.9042	1.48996	0.744980	+	0.667087i	$0.232459\pi$
$992$	− 53.0660i	− 1.68485i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.5479	−0.460737	−0.230369	−	0.973103i	$-0.573993\pi$
$997$	−14.5479	−0.460737	−0.230369	+	0.973103i	$0.573993\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	792.2.m.a.197.4	yes	4
3.2	odd	2	inner	792.2.m.a.197.1	✓	4
4.3	odd	2		3168.2.m.a.593.1		4
8.3	odd	2		3168.2.m.b.593.4		4
8.5	even	2		792.2.m.b.197.1	yes	4
11.10	odd	2		792.2.m.b.197.1	yes	4
12.11	even	2		3168.2.m.a.593.3		4
24.5	odd	2		792.2.m.b.197.4	yes	4
24.11	even	2		3168.2.m.b.593.2		4
33.32	even	2		792.2.m.b.197.4	yes	4
44.43	even	2		3168.2.m.b.593.4		4
88.21	odd	2	CM	792.2.m.a.197.4	yes	4
88.43	even	2		3168.2.m.a.593.1		4
132.131	odd	2		3168.2.m.b.593.2		4
264.131	odd	2		3168.2.m.a.593.3		4
264.197	even	2	inner	792.2.m.a.197.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.m.a.197.1	✓	4	3.2	odd	2	inner
792.2.m.a.197.1	✓	4	264.197	even	2	inner
792.2.m.a.197.4	yes	4	1.1	even	1	trivial
792.2.m.a.197.4	yes	4	88.21	odd	2	CM
792.2.m.b.197.1	yes	4	8.5	even	2
792.2.m.b.197.1	yes	4	11.10	odd	2
792.2.m.b.197.4	yes	4	24.5	odd	2
792.2.m.b.197.4	yes	4	33.32	even	2
3168.2.m.a.593.1		4	4.3	odd	2
3168.2.m.a.593.1		4	88.43	even	2
3168.2.m.a.593.3		4	12.11	even	2
3168.2.m.a.593.3		4	264.131	odd	2
3168.2.m.b.593.2		4	24.11	even	2
3168.2.m.b.593.2		4	132.131	odd	2
3168.2.m.b.593.4		4	8.3	odd	2
3168.2.m.b.593.4		4	44.43	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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} +3.31662i q^{11} -6.69042 q^{13} +4.00000 q^{16} -0.690416 q^{19} -4.69042 q^{22} -5.21904i q^{23} -5.00000 q^{25} -9.46168i q^{26} +6.63325i q^{29} -9.38083 q^{31} +5.65685i q^{32} -0.976395i q^{38} -12.6904 q^{43} -6.63325i q^{44} +7.38083 q^{46} -13.7043i q^{47} +7.00000 q^{49} -7.07107i q^{50} +13.3808 q^{52} -9.38083 q^{58} +5.30958 q^{61} -13.2665i q^{62} -8.00000 q^{64} +3.26625i q^{71} +1.38083 q^{76} +6.63325i q^{83} -17.9470i q^{86} +9.38083 q^{88} +14.6807i q^{89} +10.4381i q^{92} +19.3808 q^{94} +18.7617 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 8 q^{13} + 16 q^{16} + 16 q^{19} - 20 q^{25} - 32 q^{43} - 8 q^{46} + 28 q^{49} + 16 q^{52} + 40 q^{61} - 32 q^{64} - 32 q^{76} + 40 q^{94}+O(q^{100}) \) 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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