LMFDB - Embedded newform 3168.2.f.h.1585.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3168,2,Mod(1585,3168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3168.1585"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	20.0.74831334220841134637329678336.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2x^{18} + 5x^{16} - 8x^{14} + 28x^{12} - 64x^{10} + 112x^{8} - 128x^{6} + 320x^{4} - 512x^{2} + 1024 $ x^20 - 2x^18 + 5x^16 - 8x^14 + 28x^12 - 64x^10 + 112x^8 - 128x^6 + 320x^4 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{29} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.4
Root			$1.25295 + 0.655834i$ of defining polynomial
Character	$\chi$	$=$	3168.1585
Dual form			3168.2.f.h.1585.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54164i q^{5} -4.84329 q^{7} +1.00000i q^{11} +4.85572i q^{13} +7.15011 q^{17} -1.80543i q^{19} +3.29371 q^{23} -1.45995 q^{25} +4.82588i q^{29} +0.623337 q^{31} +12.3099i q^{35} -10.3565i q^{37} -6.34400 q^{41} +6.34400i q^{43} -6.08486 q^{47} +16.4575 q^{49} -4.47736i q^{53} +2.54164 q^{55} -7.75087i q^{59} -2.14112i q^{61} +12.3415 q^{65} -9.21748i q^{67} -0.911979 q^{71} +0.0991030 q^{73} -4.84329i q^{77} +7.99086 q^{79} +1.41148i q^{83} -18.1730i q^{85} +14.5622 q^{89} -23.5177i q^{91} -4.58877 q^{95} -2.04425 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{25} - 40 q^{31} + 36 q^{49} + 16 q^{55} - 56 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) $ 20 * q - 12 * q^25 - 40 * q^31 + 36 * q^49 + 16 * q^55 - 56 * q^73 + 16 * q^79 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 2.54164i	− 1.13666i	−0.822802	−	0.568329i	$-0.807590\pi$
$5$	− 2.54164i	− 1.13666i	0.822802	−	0.568329i	$-0.192410\pi$
$6$	0	0
$7$	−4.84329	−1.83059	−0.915296	−	0.402781i	$-0.868044\pi$
$7$	−4.84329	−1.83059	−0.915296	+	0.402781i	$0.868044\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	4.85572i	1.34674i	0.739308	+	0.673368i	$0.235153\pi$
$13$	4.85572i	1.34674i	−0.739308	+	0.673368i	$0.764847\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.15011	1.73416	0.867078	−	0.498172i	$-0.165995\pi$
$17$	7.15011	1.73416	0.867078	+	0.498172i	$0.165995\pi$
$18$	0	0
$19$	− 1.80543i	− 0.414195i	−0.978320	−	0.207097i	$-0.933598\pi$
$19$	− 1.80543i	− 0.414195i	0.978320	−	0.207097i	$-0.0664017\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.29371	0.686785	0.343393	−	0.939192i	$-0.388424\pi$
$23$	3.29371	0.686785	0.343393	+	0.939192i	$0.388424\pi$
$24$	0	0
$25$	−1.45995	−0.291990
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.82588i	0.896144i	0.893997	+	0.448072i	$0.147889\pi$
$29$	4.82588i	0.896144i	−0.893997	+	0.448072i	$0.852111\pi$
$30$	0	0
$31$	0.623337	0.111955	0.0559773	−	0.998432i	$-0.482173\pi$
$31$	0.623337	0.111955	0.0559773	+	0.998432i	$0.482173\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	12.3099i	2.08076i
$36$	0	0
$37$	− 10.3565i	− 1.70259i	−0.524685	−	0.851297i	$-0.675817\pi$
$37$	− 10.3565i	− 1.70259i	0.524685	−	0.851297i	$-0.324183\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.34400	−0.990766	−0.495383	−	0.868675i	$-0.664972\pi$
$41$	−6.34400	−0.990766	−0.495383	+	0.868675i	$0.664972\pi$
$42$	0	0
$43$	6.34400i	0.967450i	0.875220	+	0.483725i	$0.160717\pi$
$43$	6.34400i	0.967450i	−0.875220	+	0.483725i	$0.839283\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.08486	−0.887569	−0.443784	−	0.896134i	$-0.646364\pi$
$47$	−6.08486	−0.887569	−0.443784	+	0.896134i	$0.646364\pi$
$48$	0	0
$49$	16.4575	2.35107
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4.47736i	− 0.615013i	−0.951546	−	0.307506i	$-0.900505\pi$
$53$	− 4.47736i	− 0.615013i	0.951546	−	0.307506i	$-0.0994945\pi$
$54$	0	0
$55$	2.54164	0.342715
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 7.75087i	− 1.00908i	−0.863389	−	0.504539i	$-0.831663\pi$
$59$	− 7.75087i	− 1.00908i	0.863389	−	0.504539i	$-0.168337\pi$
$60$	0	0
$61$	− 2.14112i	− 0.274142i	−0.990561	−	0.137071i	$-0.956231\pi$
$61$	− 2.14112i	− 0.274142i	0.990561	−	0.137071i	$-0.0437689\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.3415	1.53078
$66$	0	0
$67$	− 9.21748i	− 1.12609i	−0.826425	−	0.563047i	$-0.809629\pi$
$67$	− 9.21748i	− 1.12609i	0.826425	−	0.563047i	$-0.190371\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.911979	−0.108232	−0.0541160	−	0.998535i	$-0.517234\pi$
$71$	−0.911979	−0.108232	−0.0541160	+	0.998535i	$0.517234\pi$
$72$	0	0
$73$	0.0991030	0.0115991	0.00579956	−	0.999983i	$-0.498154\pi$
$73$	0.0991030	0.0115991	0.00579956	+	0.999983i	$0.498154\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.84329i	− 0.551945i
$78$	0	0
$79$	7.99086	0.899042	0.449521	−	0.893270i	$-0.351595\pi$
$79$	7.99086	0.899042	0.449521	+	0.893270i	$0.351595\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.41148i	0.154930i	0.996995	+	0.0774652i	$0.0246827\pi$
$83$	1.41148i	0.154930i	−0.996995	+	0.0774652i	$0.975317\pi$
$84$	0	0
$85$	− 18.1730i	− 1.97114i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.5622	1.54359	0.771793	−	0.635874i	$-0.219360\pi$
$89$	14.5622	1.54359	0.771793	+	0.635874i	$0.219360\pi$
$90$	0	0
$91$	− 23.5177i	− 2.46532i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.58877	−0.470798
$96$	0	0
$97$	−2.04425	−0.207562	−0.103781	−	0.994600i	$-0.533094\pi$
$97$	−2.04425	−0.207562	−0.103781	+	0.994600i	$0.533094\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.90917i	0.189970i	0.995479	+	0.0949848i	$0.0302802\pi$
$101$	1.90917i	0.189970i	−0.995479	+	0.0949848i	$0.969720\pi$
$102$	0	0
$103$	19.5566	1.92697	0.963484	−	0.267764i	$-0.0862849\pi$
$103$	19.5566	1.92697	0.963484	+	0.267764i	$0.0862849\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.52320i	− 0.920643i	−0.887752	−	0.460321i	$-0.847734\pi$
$107$	− 9.52320i	− 0.920643i	0.887752	−	0.460321i	$-0.152266\pi$
$108$	0	0
$109$	− 17.8559i	− 1.71028i	−0.518396	−	0.855141i	$-0.673471\pi$
$109$	− 17.8559i	− 1.71028i	0.518396	−	0.855141i	$-0.326529\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.584117	0.0549491	0.0274746	−	0.999623i	$-0.491253\pi$
$113$	0.584117	0.0549491	0.0274746	+	0.999623i	$0.491253\pi$
$114$	0	0
$115$	− 8.37142i	− 0.780639i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−34.6301	−3.17453
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 8.99754i	− 0.804765i
$126$	0	0
$127$	−12.5593	−1.11446	−0.557231	−	0.830358i	$-0.688136\pi$
$127$	−12.5593	−1.11446	−0.557231	+	0.830358i	$0.688136\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.2298i	1.15590i	0.816074	+	0.577948i	$0.196146\pi$
$131$	13.2298i	1.15590i	−0.816074	+	0.577948i	$0.803854\pi$
$132$	0	0
$133$	8.74425i	0.758222i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.14943	−0.696253	−0.348126	−	0.937448i	$-0.613182\pi$
$137$	−8.14943	−0.696253	−0.348126	+	0.937448i	$0.613182\pi$
$138$	0	0
$139$	− 15.2150i	− 1.29052i	−0.763963	−	0.645260i	$-0.776749\pi$
$139$	− 15.2150i	− 1.29052i	0.763963	−	0.645260i	$-0.223251\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.85572	−0.406056
$144$	0	0
$145$	12.2657	1.01861
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 18.6759i	− 1.52999i	−0.644039	−	0.764993i	$-0.722742\pi$
$149$	− 18.6759i	− 1.52999i	0.644039	−	0.764993i	$-0.277258\pi$
$150$	0	0
$151$	3.92339	0.319281	0.159641	−	0.987175i	$-0.448966\pi$
$151$	3.92339	0.319281	0.159641	+	0.987175i	$0.448966\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.58430i	− 0.127254i
$156$	0	0
$157$	− 8.19964i	− 0.654402i	−0.944955	−	0.327201i	$-0.893895\pi$
$157$	− 8.19964i	− 0.654402i	0.944955	−	0.327201i	$-0.106105\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.9524	−1.25722
$162$	0	0
$163$	1.15259i	0.0902776i	0.998981	+	0.0451388i	$0.0143730\pi$
$163$	1.15259i	0.0902776i	−0.998981	+	0.0451388i	$0.985627\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.9481	1.38887	0.694434	−	0.719556i	$-0.255655\pi$
$167$	17.9481	1.38887	0.694434	+	0.719556i	$0.255655\pi$
$168$	0	0
$169$	−10.5781	−0.813697
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 0.697316i	− 0.0530160i	−0.999649	−	0.0265080i	$-0.991561\pi$
$173$	− 0.697316i	− 0.0530160i	0.999649	−	0.0265080i	$-0.00843874\pi$
$174$	0	0
$175$	7.07096	0.534515
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.2551i	1.36445i	0.731143	+	0.682224i	$0.238987\pi$
$179$	18.2551i	1.36445i	−0.731143	+	0.682224i	$0.761013\pi$
$180$	0	0
$181$	− 10.0236i	− 0.745048i	−0.928023	−	0.372524i	$-0.878492\pi$
$181$	− 10.0236i	− 0.745048i	0.928023	−	0.372524i	$-0.121508\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−26.3225	−1.93527
$186$	0	0
$187$	7.15011i	0.522868i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.3380	0.965106	0.482553	−	0.875867i	$-0.339710\pi$
$191$	13.3380	0.965106	0.482553	+	0.875867i	$0.339710\pi$
$192$	0	0
$193$	−12.0590	−0.868028	−0.434014	−	0.900906i	$-0.642903\pi$
$193$	−12.0590	−0.868028	−0.434014	+	0.900906i	$0.642903\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.4806i	1.81541i	0.419604	+	0.907707i	$0.362169\pi$
$197$	25.4806i	1.81541i	−0.419604	+	0.907707i	$0.637831\pi$
$198$	0	0
$199$	10.7899	0.764876	0.382438	−	0.923981i	$-0.375084\pi$
$199$	10.7899	0.764876	0.382438	+	0.923981i	$0.375084\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 23.3732i	− 1.64048i
$204$	0	0
$205$	16.1242i	1.12616i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.80543	0.124884
$210$	0	0
$211$	− 21.3658i	− 1.47088i	−0.677589	−	0.735441i	$-0.736975\pi$
$211$	− 21.3658i	− 1.47088i	0.677589	−	0.735441i	$-0.263025\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.1242	1.09966
$216$	0	0
$217$	−3.01900	−0.204943
$218$	0	0
$219$	0	0
$220$	0	0
$221$	34.7190i	2.33545i
$222$	0	0
$223$	4.27192	0.286069	0.143034	−	0.989718i	$-0.454314\pi$
$223$	4.27192	0.286069	0.143034	+	0.989718i	$0.454314\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	18.2440i	1.20559i	0.797894	+	0.602797i	$0.205947\pi$
$229$	18.2440i	1.20559i	−0.797894	+	0.602797i	$0.794053\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.5657	0.888721	0.444360	−	0.895848i	$-0.353431\pi$
$233$	13.5657	0.888721	0.444360	+	0.895848i	$0.353431\pi$
$234$	0	0
$235$	15.4656i	1.00886i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.66440	−0.172346	−0.0861728	−	0.996280i	$-0.527464\pi$
$239$	−2.66440	−0.172346	−0.0861728	+	0.996280i	$0.527464\pi$
$240$	0	0
$241$	10.2657	0.661270	0.330635	−	0.943759i	$-0.392737\pi$
$241$	10.2657	0.661270	0.330635	+	0.943759i	$0.392737\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 41.8291i	− 2.67236i
$246$	0	0
$247$	8.76669	0.557811
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 21.9871i	− 1.38781i	−0.720065	−	0.693906i	$-0.755888\pi$
$251$	− 21.9871i	− 1.38781i	0.720065	−	0.693906i	$-0.244112\pi$
$252$	0	0
$253$	3.29371i	0.207074i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.0204	−1.06171	−0.530853	−	0.847464i	$-0.678128\pi$
$257$	−17.0204	−1.06171	−0.530853	+	0.847464i	$0.678128\pi$
$258$	0	0
$259$	50.1594i	3.11676i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.8876	1.28799	0.643993	−	0.765031i	$-0.277277\pi$
$263$	20.8876	1.28799	0.643993	+	0.765031i	$0.277277\pi$
$264$	0	0
$265$	−11.3798	−0.699059
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.3916i	1.48718i	0.668634	+	0.743592i	$0.266879\pi$
$269$	24.3916i	1.48718i	−0.668634	+	0.743592i	$0.733121\pi$
$270$	0	0
$271$	12.0583	0.732492	0.366246	−	0.930518i	$-0.380643\pi$
$271$	12.0583	0.732492	0.366246	+	0.930518i	$0.380643\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.45995i	− 0.0880383i
$276$	0	0
$277$	0.142476i	0.00856054i	0.999991	+	0.00428027i	$0.00136246\pi$
$277$	0.142476i	0.00856054i	−0.999991	+	0.00428027i	$0.998638\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.14875	0.545769	0.272884	−	0.962047i	$-0.412022\pi$
$281$	9.14875	0.545769	0.272884	+	0.962047i	$0.412022\pi$
$282$	0	0
$283$	− 7.37210i	− 0.438226i	−0.975699	−	0.219113i	$-0.929684\pi$
$283$	− 7.37210i	− 0.438226i	0.975699	−	0.219113i	$-0.0703163\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.7258	1.81369
$288$	0	0
$289$	34.1240	2.00730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 4.38597i	− 0.256231i	−0.991759	−	0.128116i	$-0.959107\pi$
$293$	− 4.38597i	− 0.256231i	0.991759	−	0.128116i	$-0.0408928\pi$
$294$	0	0
$295$	−19.6999	−1.14698
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.9933i	0.924918i
$300$	0	0
$301$	− 30.7258i	− 1.77101i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.44196	−0.311606
$306$	0	0
$307$	− 27.3791i	− 1.56261i	−0.624150	−	0.781304i	$-0.714555\pi$
$307$	− 27.3791i	− 1.56261i	0.624150	−	0.781304i	$-0.285445\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.0540	−0.853634	−0.426817	−	0.904338i	$-0.640365\pi$
$311$	−15.0540	−0.853634	−0.426817	+	0.904338i	$0.640365\pi$
$312$	0	0
$313$	−21.0355	−1.18900	−0.594500	−	0.804096i	$-0.702650\pi$
$313$	−21.0355	−1.18900	−0.594500	+	0.804096i	$0.702650\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.29178i	0.184885i	0.995718	+	0.0924425i	$0.0294674\pi$
$317$	3.29178i	0.184885i	−0.995718	+	0.0924425i	$0.970533\pi$
$318$	0	0
$319$	−4.82588	−0.270198
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 12.9090i	− 0.718279i
$324$	0	0
$325$	− 7.08911i	− 0.393233i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	29.4708	1.62478
$330$	0	0
$331$	− 12.6880i	− 0.697395i	−0.937235	−	0.348698i	$-0.886624\pi$
$331$	− 12.6880i	− 0.697395i	0.937235	−	0.348698i	$-0.113376\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−23.4275	−1.27998
$336$	0	0
$337$	−6.48274	−0.353137	−0.176569	−	0.984288i	$-0.556500\pi$
$337$	−6.48274	−0.353137	−0.176569	+	0.984288i	$0.556500\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.623337i	0.0337556i
$342$	0	0
$343$	−45.8054	−2.47326
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.0102i	1.44998i	0.688759	+	0.724991i	$0.258156\pi$
$347$	27.0102i	1.44998i	−0.688759	+	0.724991i	$0.741844\pi$
$348$	0	0
$349$	− 2.22276i	− 0.118981i	−0.998229	−	0.0594907i	$-0.981052\pi$
$349$	− 2.22276i	− 0.118981i	0.998229	−	0.0594907i	$-0.0189477\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.9731	1.38241	0.691204	−	0.722660i	$-0.257081\pi$
$353$	25.9731	1.38241	0.691204	+	0.722660i	$0.257081\pi$
$354$	0	0
$355$	2.31792i	0.123023i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.6003	0.823352	0.411676	−	0.911330i	$-0.364944\pi$
$359$	15.6003	0.823352	0.411676	+	0.911330i	$0.364944\pi$
$360$	0	0
$361$	15.7404	0.828443
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.251884i	− 0.0131842i
$366$	0	0
$367$	−2.56706	−0.133999	−0.0669997	−	0.997753i	$-0.521343\pi$
$367$	−2.56706	−0.133999	−0.0669997	+	0.997753i	$0.521343\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	21.6852i	1.12584i
$372$	0	0
$373$	20.9799i	1.08630i	0.839636	+	0.543149i	$0.182768\pi$
$373$	20.9799i	1.08630i	−0.839636	+	0.543149i	$0.817232\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23.4332	−1.20687
$378$	0	0
$379$	− 2.32082i	− 0.119213i	−0.998222	−	0.0596063i	$-0.981015\pi$
$379$	− 2.32082i	− 0.119213i	0.998222	−	0.0596063i	$-0.0189845\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.30839	0.373441	0.186721	−	0.982413i	$-0.440214\pi$
$383$	7.30839	0.373441	0.186721	+	0.982413i	$0.440214\pi$
$384$	0	0
$385$	−12.3099	−0.627372
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.4190i	0.629669i	0.949147	+	0.314835i	$0.101949\pi$
$389$	12.4190i	0.629669i	−0.949147	+	0.314835i	$0.898051\pi$
$390$	0	0
$391$	23.5504	1.19099
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 20.3099i	− 1.02190i
$396$	0	0
$397$	− 12.8462i	− 0.644731i	−0.946615	−	0.322366i	$-0.895522\pi$
$397$	− 12.8462i	− 0.644731i	0.946615	−	0.322366i	$-0.104478\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.73490	−0.336325	−0.168163	−	0.985759i	$-0.553783\pi$
$401$	−6.73490	−0.336325	−0.168163	+	0.985759i	$0.553783\pi$
$402$	0	0
$403$	3.02675i	0.150773i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.3565	0.513351
$408$	0	0
$409$	6.06964	0.300124	0.150062	−	0.988677i	$-0.452053\pi$
$409$	6.06964	0.300124	0.150062	+	0.988677i	$0.452053\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	37.5397i	1.84721i
$414$	0	0
$415$	3.58748	0.176103
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 18.3756i	− 0.897708i	−0.893605	−	0.448854i	$-0.851832\pi$
$419$	− 18.3756i	− 0.897708i	0.893605	−	0.448854i	$-0.148168\pi$
$420$	0	0
$421$	− 14.8185i	− 0.722208i	−0.932525	−	0.361104i	$-0.882400\pi$
$421$	− 14.8185i	− 0.722208i	0.932525	−	0.361104i	$-0.117600\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.4388	−0.506356
$426$	0	0
$427$	10.3701i	0.501843i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.7144	1.52763	0.763815	−	0.645435i	$-0.223324\pi$
$431$	31.7144	1.52763	0.763815	+	0.645435i	$0.223324\pi$
$432$	0	0
$433$	14.8878	0.715464	0.357732	−	0.933824i	$-0.383550\pi$
$433$	14.8878	0.715464	0.357732	+	0.933824i	$0.383550\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.94657i	− 0.284463i
$438$	0	0
$439$	−29.4357	−1.40489	−0.702446	−	0.711737i	$-0.747909\pi$
$439$	−29.4357	−1.40489	−0.702446	+	0.711737i	$0.747909\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.34361i	− 0.158860i	−0.996840	−	0.0794298i	$-0.974690\pi$
$443$	− 3.34361i	− 0.158860i	0.996840	−	0.0794298i	$-0.0253100\pi$
$444$	0	0
$445$	− 37.0118i	− 1.75453i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−19.2984	−0.910749	−0.455374	−	0.890300i	$-0.650495\pi$
$449$	−19.2984	−0.910749	−0.455374	+	0.890300i	$0.650495\pi$
$450$	0	0
$451$	− 6.34400i	− 0.298727i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−59.7736	−2.80223
$456$	0	0
$457$	2.68162	0.125441	0.0627205	−	0.998031i	$-0.480022\pi$
$457$	2.68162	0.125441	0.0627205	+	0.998031i	$0.480022\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.240855i	0.0112177i	0.999984	+	0.00560887i	$0.00178537\pi$
$461$	0.240855i	0.0112177i	−0.999984	+	0.00560887i	$0.998215\pi$
$462$	0	0
$463$	2.80818	0.130507	0.0652536	−	0.997869i	$-0.479214\pi$
$463$	2.80818	0.130507	0.0652536	+	0.997869i	$0.479214\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 0.198206i	− 0.00917188i	−0.999989	−	0.00458594i	$-0.998540\pi$
$467$	− 0.198206i	− 0.00917188i	0.999989	−	0.00458594i	$-0.00145975\pi$
$468$	0	0
$469$	44.6430i	2.06142i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.34400	−0.291697
$474$	0	0
$475$	2.63584i	0.120941i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−0.634324	−0.0289830	−0.0144915	−	0.999895i	$-0.504613\pi$
$479$	−0.634324	−0.0289830	−0.0144915	+	0.999895i	$0.504613\pi$
$480$	0	0
$481$	50.2882	2.29294
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.19575i	0.235927i
$486$	0	0
$487$	24.7182	1.12009	0.560044	−	0.828463i	$-0.310784\pi$
$487$	24.7182	1.12009	0.560044	+	0.828463i	$0.310784\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 30.5099i	− 1.37689i	−0.725288	−	0.688446i	$-0.758293\pi$
$491$	− 30.5099i	− 1.37689i	0.725288	−	0.688446i	$-0.241707\pi$
$492$	0	0
$493$	34.5056i	1.55405i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.41698	0.198129
$498$	0	0
$499$	− 30.1394i	− 1.34923i	−0.738171	−	0.674613i	$-0.764310\pi$
$499$	− 30.1394i	− 1.34923i	0.738171	−	0.674613i	$-0.235690\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.4586	1.26890	0.634452	−	0.772962i	$-0.281226\pi$
$503$	28.4586	1.26890	0.634452	+	0.772962i	$0.281226\pi$
$504$	0	0
$505$	4.85243	0.215930
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.27915i	0.366967i	0.983023	+	0.183484i	$0.0587374\pi$
$509$	8.27915i	0.366967i	−0.983023	+	0.183484i	$0.941263\pi$
$510$	0	0
$511$	−0.479985	−0.0212333
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 49.7059i	− 2.19030i
$516$	0	0
$517$	− 6.08486i	− 0.267612i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.9245	−0.653855	−0.326927	−	0.945050i	$-0.606013\pi$
$521$	−14.9245	−0.653855	−0.326927	+	0.945050i	$0.606013\pi$
$522$	0	0
$523$	28.0034i	1.22450i	0.790663	+	0.612252i	$0.209736\pi$
$523$	28.0034i	1.22450i	−0.790663	+	0.612252i	$0.790264\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.45692	0.194147
$528$	0	0
$529$	−12.1515	−0.528326
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 30.8047i	− 1.33430i
$534$	0	0
$535$	−24.2046	−1.04646
$536$	0	0
$537$	0	0
$538$	0	0
$539$	16.4575i	0.708874i
$540$	0	0
$541$	42.3272i	1.81979i	0.414842	+	0.909893i	$0.363837\pi$
$541$	42.3272i	1.81979i	−0.414842	+	0.909893i	$0.636163\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−45.3832	−1.94400
$546$	0	0
$547$	5.55376i	0.237462i	0.992926	+	0.118731i	$0.0378826\pi$
$547$	5.55376i	0.237462i	−0.992926	+	0.118731i	$0.962117\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.71281	0.371178
$552$	0	0
$553$	−38.7021	−1.64578
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.05429i	− 0.256528i	−0.991740	−	0.128264i	$-0.959059\pi$
$557$	− 6.05429i	− 0.256528i	0.991740	−	0.128264i	$-0.0409405\pi$
$558$	0	0
$559$	−30.8047	−1.30290
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.5314i	1.53961i	0.638277	+	0.769806i	$0.279647\pi$
$563$	36.5314i	1.53961i	−0.638277	+	0.769806i	$0.720353\pi$
$564$	0	0
$565$	− 1.48462i	− 0.0624583i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	39.1923	1.64303	0.821514	−	0.570188i	$-0.193130\pi$
$569$	39.1923	1.64303	0.821514	+	0.570188i	$0.193130\pi$
$570$	0	0
$571$	0.253449i	0.0106065i	0.999986	+	0.00530325i	$0.00168808\pi$
$571$	0.253449i	0.0106065i	−0.999986	+	0.00530325i	$0.998312\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.80864	−0.200534
$576$	0	0
$577$	−25.2484	−1.05111	−0.525553	−	0.850761i	$-0.676142\pi$
$577$	−25.2484	−1.05111	−0.525553	+	0.850761i	$0.676142\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.83622i	− 0.283614i
$582$	0	0
$583$	4.47736	0.185433
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.1304i	1.61509i	0.589809	+	0.807543i	$0.299203\pi$
$587$	39.1304i	1.61509i	−0.589809	+	0.807543i	$0.700797\pi$
$588$	0	0
$589$	− 1.12539i	− 0.0463710i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.4476	−1.04501	−0.522504	−	0.852637i	$-0.675002\pi$
$593$	−25.4476	−1.04501	−0.522504	+	0.852637i	$0.675002\pi$
$594$	0	0
$595$	88.0173i	3.60836i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.0670	1.51452	0.757259	−	0.653115i	$-0.226538\pi$
$599$	37.0670	1.51452	0.757259	+	0.653115i	$0.226538\pi$
$600$	0	0
$601$	30.5587	1.24652	0.623259	−	0.782016i	$-0.285808\pi$
$601$	30.5587	1.24652	0.623259	+	0.782016i	$0.285808\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.54164i	0.103332i
$606$	0	0
$607$	−15.4063	−0.625322	−0.312661	−	0.949865i	$-0.601220\pi$
$607$	−15.4063	−0.625322	−0.312661	+	0.949865i	$0.601220\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 29.5464i	− 1.19532i
$612$	0	0
$613$	24.0437i	0.971116i	0.874205	+	0.485558i	$0.161383\pi$
$613$	24.0437i	0.971116i	−0.874205	+	0.485558i	$0.838617\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.77372	0.353217	0.176608	−	0.984281i	$-0.443487\pi$
$617$	8.77372	0.353217	0.176608	+	0.984281i	$0.443487\pi$
$618$	0	0
$619$	1.81682i	0.0730242i	0.999333	+	0.0365121i	$0.0116248\pi$
$619$	1.81682i	0.0730242i	−0.999333	+	0.0365121i	$0.988375\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−70.5288	−2.82568
$624$	0	0
$625$	−30.1683	−1.20673
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 74.0499i	− 2.95256i
$630$	0	0
$631$	−2.72027	−0.108292	−0.0541461	−	0.998533i	$-0.517244\pi$
$631$	−2.72027	−0.108292	−0.0541461	+	0.998533i	$0.517244\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	31.9214i	1.26676i
$636$	0	0
$637$	79.9130i	3.16627i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.10619	−0.241180	−0.120590	−	0.992702i	$-0.538479\pi$
$641$	−6.10619	−0.241180	−0.120590	+	0.992702i	$0.538479\pi$
$642$	0	0
$643$	32.9170i	1.29812i	0.760737	+	0.649060i	$0.224838\pi$
$643$	32.9170i	1.29812i	−0.760737	+	0.649060i	$0.775162\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.0567	−0.434684	−0.217342	−	0.976095i	$-0.569739\pi$
$647$	−11.0567	−0.434684	−0.217342	+	0.976095i	$0.569739\pi$
$648$	0	0
$649$	7.75087	0.304248
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18.6278i	− 0.728963i	−0.931211	−	0.364482i	$-0.881246\pi$
$653$	− 18.6278i	− 0.728963i	0.931211	−	0.364482i	$-0.118754\pi$
$654$	0	0
$655$	33.6255	1.31386
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 26.2315i	− 1.02184i	−0.859629	−	0.510918i	$-0.829306\pi$
$659$	− 26.2315i	− 1.02184i	0.859629	−	0.510918i	$-0.170694\pi$
$660$	0	0
$661$	− 38.4922i	− 1.49717i	−0.663038	−	0.748586i	$-0.730733\pi$
$661$	− 38.4922i	− 1.49717i	0.663038	−	0.748586i	$-0.269267\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	22.2248	0.861839
$666$	0	0
$667$	15.8950i	0.615459i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.14112	0.0826570
$672$	0	0
$673$	−29.0436	−1.11955	−0.559773	−	0.828646i	$-0.689112\pi$
$673$	−29.0436	−1.11955	−0.559773	+	0.828646i	$0.689112\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14.6056i	− 0.561340i	−0.959804	−	0.280670i	$-0.909443\pi$
$677$	− 14.6056i	− 0.561340i	0.959804	−	0.280670i	$-0.0905567\pi$
$678$	0	0
$679$	9.90090	0.379962
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.6575i	1.13481i	0.823438	+	0.567406i	$0.192053\pi$
$683$	29.6575i	1.13481i	−0.823438	+	0.567406i	$0.807947\pi$
$684$	0	0
$685$	20.7129i	0.791401i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21.7408	0.828259
$690$	0	0
$691$	− 16.6181i	− 0.632184i	−0.948729	−	0.316092i	$-0.897629\pi$
$691$	− 16.6181i	− 0.632184i	0.948729	−	0.316092i	$-0.102371\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−38.6711	−1.46688
$696$	0	0
$697$	−45.3603	−1.71814
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.9173i	1.39435i	0.716902	+	0.697174i	$0.245559\pi$
$701$	36.9173i	1.39435i	−0.716902	+	0.697174i	$0.754441\pi$
$702$	0	0
$703$	−18.6979	−0.705206
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.24667i	− 0.347757i
$708$	0	0
$709$	1.80322i	0.0677215i	0.999427	+	0.0338608i	$0.0107803\pi$
$709$	1.80322i	0.0677215i	−0.999427	+	0.0338608i	$0.989220\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.05309	0.0768887
$714$	0	0
$715$	12.3415i	0.461547i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30.0049	−1.11899	−0.559496	−	0.828833i	$-0.689005\pi$
$719$	−30.0049	−1.11899	−0.559496	+	0.828833i	$0.689005\pi$
$720$	0	0
$721$	−94.7183	−3.52750
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.04555i	− 0.261665i
$726$	0	0
$727$	−5.71187	−0.211842	−0.105921	−	0.994375i	$-0.533779\pi$
$727$	−5.71187	−0.211842	−0.105921	+	0.994375i	$0.533779\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	45.3603i	1.67771i
$732$	0	0
$733$	− 13.4418i	− 0.496483i	−0.968698	−	0.248242i	$-0.920147\pi$
$733$	− 13.4418i	− 0.496483i	0.968698	−	0.248242i	$-0.0798527\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.21748	0.339530
$738$	0	0
$739$	− 15.6188i	− 0.574548i	−0.957849	−	0.287274i	$-0.907251\pi$
$739$	− 15.6188i	− 0.574548i	0.957849	−	0.287274i	$-0.0927489\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.2940	−1.51493	−0.757466	−	0.652875i	$-0.773563\pi$
$743$	−41.2940	−1.51493	−0.757466	+	0.652875i	$0.773563\pi$
$744$	0	0
$745$	−47.4674	−1.73907
$746$	0	0
$747$	0	0
$748$	0	0
$749$	46.1237i	1.68532i
$750$	0	0
$751$	−5.34503	−0.195043	−0.0975215	−	0.995233i	$-0.531091\pi$
$751$	−5.34503	−0.195043	−0.0975215	+	0.995233i	$0.531091\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 9.97187i	− 0.362913i
$756$	0	0
$757$	13.6401i	0.495757i	0.968791	+	0.247878i	$0.0797334\pi$
$757$	13.6401i	0.495757i	−0.968791	+	0.247878i	$0.920267\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.1959	1.05835	0.529176	−	0.848512i	$-0.322501\pi$
$761$	29.1959	1.05835	0.529176	+	0.848512i	$0.322501\pi$
$762$	0	0
$763$	86.4812i	3.13083i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	37.6361	1.35896
$768$	0	0
$769$	23.4723	0.846432	0.423216	−	0.906029i	$-0.360901\pi$
$769$	23.4723	0.846432	0.423216	+	0.906029i	$0.360901\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4.20825i	− 0.151360i	−0.997132	−	0.0756802i	$-0.975887\pi$
$773$	− 4.20825i	− 0.151360i	0.997132	−	0.0756802i	$-0.0241128\pi$
$774$	0	0
$775$	−0.910040	−0.0326896
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.4537i	0.410370i
$780$	0	0
$781$	− 0.911979i	− 0.0326332i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−20.8405	−0.743831
$786$	0	0
$787$	17.8152i	0.635042i	0.948251	+	0.317521i	$0.102850\pi$
$787$	17.8152i	0.635042i	−0.948251	+	0.317521i	$0.897150\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.82905	−0.100590
$792$	0	0
$793$	10.3967	0.369197
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.5339i	1.25868i	0.777131	+	0.629338i	$0.216674\pi$
$797$	35.5339i	1.25868i	−0.777131	+	0.629338i	$0.783326\pi$
$798$	0	0
$799$	−43.5074	−1.53918
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.0991030i	0.00349727i
$804$	0	0
$805$	40.5453i	1.42903i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.57918	−0.125837	−0.0629187	−	0.998019i	$-0.520041\pi$
$809$	−3.57918	−0.125837	−0.0629187	+	0.998019i	$0.520041\pi$
$810$	0	0
$811$	− 14.0896i	− 0.494753i	−0.968919	−	0.247377i	$-0.920432\pi$
$811$	− 14.0896i	− 0.494753i	0.968919	−	0.247377i	$-0.0795685\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.92946	0.102615
$816$	0	0
$817$	11.4537	0.400713
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 47.8288i	− 1.66924i	−0.550829	−	0.834618i	$-0.685688\pi$
$821$	− 47.8288i	− 1.66924i	0.550829	−	0.834618i	$-0.314312\pi$
$822$	0	0
$823$	31.6451	1.10308	0.551539	−	0.834149i	$-0.314041\pi$
$823$	31.6451	1.10308	0.551539	+	0.834149i	$0.314041\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 25.5630i	− 0.888911i	−0.895801	−	0.444456i	$-0.853397\pi$
$827$	− 25.5630i	− 0.888911i	0.895801	−	0.444456i	$-0.146603\pi$
$828$	0	0
$829$	− 14.1148i	− 0.490228i	−0.969494	−	0.245114i	$-0.921175\pi$
$829$	− 14.1148i	− 0.490228i	0.969494	−	0.245114i	$-0.0788254\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	117.673	4.07712
$834$	0	0
$835$	− 45.6177i	− 1.57867i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.13279	0.211727	0.105864	−	0.994381i	$-0.466239\pi$
$839$	6.13279	0.211727	0.105864	+	0.994381i	$0.466239\pi$
$840$	0	0
$841$	5.71084	0.196926
$842$	0	0
$843$	0	0
$844$	0	0
$845$	26.8856i	0.924894i
$846$	0	0
$847$	4.84329	0.166418
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 34.1112i	− 1.16932i
$852$	0	0
$853$	− 3.74773i	− 0.128320i	−0.997940	−	0.0641599i	$-0.979563\pi$
$853$	− 3.74773i	− 0.128320i	0.997940	−	0.0641599i	$-0.0204368\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.2567	−1.40930	−0.704652	−	0.709553i	$-0.748897\pi$
$857$	−41.2567	−1.40930	−0.704652	+	0.709553i	$0.748897\pi$
$858$	0	0
$859$	9.21458i	0.314398i	0.987567	+	0.157199i	$0.0502463\pi$
$859$	9.21458i	0.314398i	−0.987567	+	0.157199i	$0.949754\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−39.3004	−1.33780	−0.668901	−	0.743352i	$-0.733235\pi$
$863$	−39.3004	−1.33780	−0.668901	+	0.743352i	$0.733235\pi$
$864$	0	0
$865$	−1.77233	−0.0602610
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.99086i	0.271071i
$870$	0	0
$871$	44.7575	1.51655
$872$	0	0
$873$	0	0
$874$	0	0
$875$	43.5778i	1.47320i
$876$	0	0
$877$	19.2061i	0.648545i	0.945964	+	0.324273i	$0.105120\pi$
$877$	19.2061i	0.648545i	−0.945964	+	0.324273i	$0.894880\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.78843	−0.195017	−0.0975087	−	0.995235i	$-0.531087\pi$
$881$	−5.78843	−0.195017	−0.0975087	+	0.995235i	$0.531087\pi$
$882$	0	0
$883$	27.1286i	0.912949i	0.889737	+	0.456474i	$0.150888\pi$
$883$	27.1286i	0.912949i	−0.889737	+	0.456474i	$0.849112\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.55834	0.320938	0.160469	−	0.987041i	$-0.448699\pi$
$887$	9.55834	0.320938	0.160469	+	0.987041i	$0.448699\pi$
$888$	0	0
$889$	60.8286	2.04013
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.9858i	0.367626i
$894$	0	0
$895$	46.3979	1.55091
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.00815i	0.100327i
$900$	0	0
$901$	− 32.0136i	− 1.06653i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−25.4764	−0.846864
$906$	0	0
$907$	− 23.2370i	− 0.771571i	−0.922588	−	0.385786i	$-0.873930\pi$
$907$	− 23.2370i	− 0.771571i	0.922588	−	0.385786i	$-0.126070\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	30.9370	1.02499	0.512494	−	0.858691i	$-0.328722\pi$
$911$	30.9370	1.02499	0.512494	+	0.858691i	$0.328722\pi$
$912$	0	0
$913$	−1.41148	−0.0467132
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 64.0759i	− 2.11597i
$918$	0	0
$919$	−44.8067	−1.47804	−0.739019	−	0.673684i	$-0.764711\pi$
$919$	−44.8067	−1.47804	−0.739019	+	0.673684i	$0.764711\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 4.42832i	− 0.145760i
$924$	0	0
$925$	15.1199i	0.497140i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39.2081	1.28638	0.643189	−	0.765708i	$-0.277611\pi$
$929$	39.2081	1.28638	0.643189	+	0.765708i	$0.277611\pi$
$930$	0	0
$931$	− 29.7129i	− 0.973802i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18.1730	0.594321
$936$	0	0
$937$	55.8729	1.82529	0.912644	−	0.408756i	$-0.134037\pi$
$937$	55.8729	1.82529	0.912644	+	0.408756i	$0.134037\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 43.3369i	− 1.41274i	−0.707842	−	0.706371i	$-0.750331\pi$
$941$	− 43.3369i	− 1.41274i	0.707842	−	0.706371i	$-0.249669\pi$
$942$	0	0
$943$	−20.8953	−0.680443
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.07170i	0.197304i	0.995122	+	0.0986519i	$0.0314530\pi$
$947$	6.07170i	0.197304i	−0.995122	+	0.0986519i	$0.968547\pi$
$948$	0	0
$949$	0.481217i	0.0156210i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.0254	−0.454326	−0.227163	−	0.973857i	$-0.572945\pi$
$953$	−14.0254	−0.454326	−0.227163	+	0.973857i	$0.572945\pi$
$954$	0	0
$955$	− 33.9005i	− 1.09699i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	39.4701	1.27456
$960$	0	0
$961$	−30.6115	−0.987466
$962$	0	0
$963$	0	0
$964$	0	0
$965$	30.6498i	0.986651i
$966$	0	0
$967$	7.68589	0.247162	0.123581	−	0.992335i	$-0.460562\pi$
$967$	7.68589	0.247162	0.123581	+	0.992335i	$0.460562\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 26.3648i	− 0.846086i	−0.906110	−	0.423043i	$-0.860962\pi$
$971$	− 26.3648i	− 0.846086i	0.906110	−	0.423043i	$-0.139038\pi$
$972$	0	0
$973$	73.6907i	2.36242i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.2530	1.86368	0.931840	−	0.362870i	$-0.118203\pi$
$977$	58.2530	1.86368	0.931840	+	0.362870i	$0.118203\pi$
$978$	0	0
$979$	14.5622i	0.465409i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.7987	−0.918537	−0.459268	−	0.888298i	$-0.651888\pi$
$983$	−28.7987	−0.918537	−0.459268	+	0.888298i	$0.651888\pi$
$984$	0	0
$985$	64.7625	2.06350
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.8953i	0.664430i
$990$	0	0
$991$	32.1947	1.02270	0.511349	−	0.859373i	$-0.329146\pi$
$991$	32.1947	1.02270	0.511349	+	0.859373i	$0.329146\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 27.4241i	− 0.869402i
$996$	0	0
$997$	− 32.6398i	− 1.03371i	−0.856072	−	0.516856i	$-0.827102\pi$
$997$	− 32.6398i	− 1.03371i	0.856072	−	0.516856i	$-0.172898\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.f.h.1585.4		20
3.2	odd	2	inner	3168.2.f.h.1585.18		20
4.3	odd	2		792.2.f.h.397.4	yes	20
8.3	odd	2		792.2.f.h.397.3	✓	20
8.5	even	2	inner	3168.2.f.h.1585.17		20
12.11	even	2		792.2.f.h.397.17	yes	20
24.5	odd	2	inner	3168.2.f.h.1585.3		20
24.11	even	2		792.2.f.h.397.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.f.h.397.3	✓	20	8.3	odd	2
792.2.f.h.397.4	yes	20	4.3	odd	2
792.2.f.h.397.17	yes	20	12.11	even	2
792.2.f.h.397.18	yes	20	24.11	even	2
3168.2.f.h.1585.3		20	24.5	odd	2	inner
3168.2.f.h.1585.4		20	1.1	even	1	trivial
3168.2.f.h.1585.17		20	8.5	even	2	inner
3168.2.f.h.1585.18		20	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.54164i q^{5} -4.84329 q^{7} +1.00000i q^{11} +4.85572i q^{13} +7.15011 q^{17} -1.80543i q^{19} +3.29371 q^{23} -1.45995 q^{25} +4.82588i q^{29} +0.623337 q^{31} +12.3099i q^{35} -10.3565i q^{37} -6.34400 q^{41} +6.34400i q^{43} -6.08486 q^{47} +16.4575 q^{49} -4.47736i q^{53} +2.54164 q^{55} -7.75087i q^{59} -2.14112i q^{61} +12.3415 q^{65} -9.21748i q^{67} -0.911979 q^{71} +0.0991030 q^{73} -4.84329i q^{77} +7.99086 q^{79} +1.41148i q^{83} -18.1730i q^{85} +14.5622 q^{89} -23.5177i q^{91} -4.58877 q^{95} -2.04425 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{25} - 40 q^{31} + 36 q^{49} + 16 q^{55} - 56 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) 20 * q - 12 * q^25 - 40 * q^31 + 36 * q^49 + 16 * q^55 - 56 * q^73 + 16 * q^79 + 8 * q^97

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