LMFDB - Embedded newform 3168.2.f.h.1585.13

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			1585.13
Root			$1.38682 + 0.277013i$ of defining polynomial
Character	$\chi$	$=$	3168.1585
Dual form			3168.2.f.h.1585.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.972802i q^{5} +0.372730 q^{7} +1.00000i q^{11} -4.87017i q^{13} -3.93517 q^{17} +1.92228i q^{19} -2.39623 q^{23} +4.05366 q^{25} -6.73164i q^{29} -0.891949 q^{31} +0.362592i q^{35} -3.53439i q^{37} +5.34412 q^{41} -5.34412i q^{43} -0.216045 q^{47} -6.86107 q^{49} -5.30526i q^{53} -0.972802 q^{55} +7.02352i q^{59} -11.6208i q^{61} +4.73771 q^{65} -9.68560i q^{67} -6.53455 q^{71} +8.43977 q^{73} +0.372730i q^{77} -8.59639 q^{79} +15.6098i q^{83} -3.82814i q^{85} +7.67271 q^{89} -1.81526i q^{91} -1.87000 q^{95} +4.18597 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.972802i	0.435050i	0.976055	+	0.217525i	$0.0697984\pi$
$5$	0.972802i	0.435050i	−0.976055	+	0.217525i	$0.930202\pi$
$6$	0	0
$7$	0.372730	0.140879	0.0704393	−	0.997516i	$-0.477560\pi$
$7$	0.372730	0.140879	0.0704393	+	0.997516i	$0.477560\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	− 4.87017i	− 1.35074i	−0.737478	−	0.675371i	$-0.763983\pi$
$13$	− 4.87017i	− 1.35074i	0.737478	−	0.675371i	$-0.236017\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.93517	−0.954419	−0.477209	−	0.878790i	$-0.658352\pi$
$17$	−3.93517	−0.954419	−0.477209	+	0.878790i	$0.658352\pi$
$18$	0	0
$19$	1.92228i	0.441002i	0.975387	+	0.220501i	$0.0707693\pi$
$19$	1.92228i	0.441002i	−0.975387	+	0.220501i	$0.929231\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.39623	−0.499648	−0.249824	−	0.968291i	$-0.580373\pi$
$23$	−2.39623	−0.499648	−0.249824	+	0.968291i	$0.580373\pi$
$24$	0	0
$25$	4.05366	0.810731
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.73164i	− 1.25003i	−0.780611	−	0.625017i	$-0.785092\pi$
$29$	− 6.73164i	− 1.25003i	0.780611	−	0.625017i	$-0.214908\pi$
$30$	0	0
$31$	−0.891949	−0.160199	−0.0800994	−	0.996787i	$-0.525524\pi$
$31$	−0.891949	−0.160199	−0.0800994	+	0.996787i	$0.525524\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.362592i	0.0612893i
$36$	0	0
$37$	− 3.53439i	− 0.581049i	−0.956867	−	0.290525i	$-0.906170\pi$
$37$	− 3.53439i	− 0.581049i	0.956867	−	0.290525i	$-0.0938298\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.34412	0.834611	0.417305	−	0.908766i	$-0.362975\pi$
$41$	5.34412	0.834611	0.417305	+	0.908766i	$0.362975\pi$
$42$	0	0
$43$	− 5.34412i	− 0.814970i	−0.913212	−	0.407485i	$-0.866406\pi$
$43$	− 5.34412i	− 0.814970i	0.913212	−	0.407485i	$-0.133594\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.216045	−0.0315134	−0.0157567	−	0.999876i	$-0.505016\pi$
$47$	−0.216045	−0.0315134	−0.0157567	+	0.999876i	$0.505016\pi$
$48$	0	0
$49$	−6.86107	−0.980153
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5.30526i	− 0.728733i	−0.931256	−	0.364367i	$-0.881286\pi$
$53$	− 5.30526i	− 0.728733i	0.931256	−	0.364367i	$-0.118714\pi$
$54$	0	0
$55$	−0.972802	−0.131173
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.02352i	0.914384i	0.889368	+	0.457192i	$0.151145\pi$
$59$	7.02352i	0.914384i	−0.889368	+	0.457192i	$0.848855\pi$
$60$	0	0
$61$	− 11.6208i	− 1.48789i	−0.668242	−	0.743944i	$-0.732953\pi$
$61$	− 11.6208i	− 1.48789i	0.668242	−	0.743944i	$-0.267047\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.73771	0.587641
$66$	0	0
$67$	− 9.68560i	− 1.18328i	−0.806201	−	0.591642i	$-0.798480\pi$
$67$	− 9.68560i	− 1.18328i	0.806201	−	0.591642i	$-0.201520\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.53455	−0.775509	−0.387754	−	0.921763i	$-0.626749\pi$
$71$	−6.53455	−0.775509	−0.387754	+	0.921763i	$0.626749\pi$
$72$	0	0
$73$	8.43977	0.987800	0.493900	−	0.869519i	$-0.335571\pi$
$73$	8.43977	0.987800	0.493900	+	0.869519i	$0.335571\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.372730i	0.0424765i
$78$	0	0
$79$	−8.59639	−0.967170	−0.483585	−	0.875297i	$-0.660666\pi$
$79$	−8.59639	−0.967170	−0.483585	+	0.875297i	$0.660666\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.6098i	1.71340i	0.515819	+	0.856698i	$0.327488\pi$
$83$	15.6098i	1.71340i	−0.515819	+	0.856698i	$0.672512\pi$
$84$	0	0
$85$	− 3.82814i	− 0.415220i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.67271	0.813306	0.406653	−	0.913583i	$-0.366696\pi$
$89$	7.67271	0.813306	0.406653	+	0.913583i	$0.366696\pi$
$90$	0	0
$91$	− 1.81526i	− 0.190291i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.87000	−0.191858
$96$	0	0
$97$	4.18597	0.425020	0.212510	−	0.977159i	$-0.431836\pi$
$97$	4.18597	0.425020	0.212510	+	0.977159i	$0.431836\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 16.6772i	− 1.65945i	−0.558174	−	0.829724i	$-0.688498\pi$
$101$	− 16.6772i	− 1.65945i	0.558174	−	0.829724i	$-0.311502\pi$
$102$	0	0
$103$	4.57869	0.451152	0.225576	−	0.974226i	$-0.427574\pi$
$103$	4.57869	0.451152	0.225576	+	0.974226i	$0.427574\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.90717i	0.474394i	0.971462	+	0.237197i	$0.0762286\pi$
$107$	4.90717i	0.474394i	−0.971462	+	0.237197i	$0.923771\pi$
$108$	0	0
$109$	− 5.27648i	− 0.505395i	−0.967545	−	0.252698i	$-0.918682\pi$
$109$	− 5.27648i	− 0.505395i	0.967545	−	0.252698i	$-0.0813178\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.18851	−0.770310	−0.385155	−	0.922852i	$-0.625852\pi$
$113$	−8.18851	−0.770310	−0.385155	+	0.922852i	$0.625852\pi$
$114$	0	0
$115$	− 2.33106i	− 0.217372i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.46675	−0.134457
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.80742i	0.787759i
$126$	0	0
$127$	20.1141	1.78484	0.892418	−	0.451210i	$-0.149007\pi$
$127$	20.1141	1.78484	0.892418	+	0.451210i	$0.149007\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 9.74472i	− 0.851400i	−0.904864	−	0.425700i	$-0.860028\pi$
$131$	− 9.74472i	− 0.851400i	0.904864	−	0.425700i	$-0.139972\pi$
$132$	0	0
$133$	0.716492i	0.0621277i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.26640	0.620811	0.310405	−	0.950604i	$-0.399535\pi$
$137$	7.26640	0.620811	0.310405	+	0.950604i	$0.399535\pi$
$138$	0	0
$139$	− 19.7674i	− 1.67665i	−0.545170	−	0.838326i	$-0.683535\pi$
$139$	− 19.7674i	− 1.67665i	0.545170	−	0.838326i	$-0.316465\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.87017	0.407264
$144$	0	0
$145$	6.54856	0.543828
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 0.684605i	− 0.0560850i	−0.999607	−	0.0280425i	$-0.991073\pi$
$149$	− 0.684605i	− 0.0560850i	0.999607	−	0.0280425i	$-0.00892738\pi$
$150$	0	0
$151$	9.73458	0.792189	0.396095	−	0.918210i	$-0.370365\pi$
$151$	9.73458	0.792189	0.396095	+	0.918210i	$0.370365\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.867690i	− 0.0696945i
$156$	0	0
$157$	1.97457i	0.157588i	0.996891	+	0.0787938i	$0.0251069\pi$
$157$	1.97457i	0.157588i	−0.996891	+	0.0787938i	$0.974893\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.893146	−0.0703897
$162$	0	0
$163$	− 14.0170i	− 1.09790i	−0.835856	−	0.548948i	$-0.815028\pi$
$163$	− 14.0170i	− 1.09790i	0.835856	−	0.548948i	$-0.184972\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.2679	1.41361	0.706805	−	0.707408i	$-0.250136\pi$
$167$	18.2679	1.41361	0.706805	+	0.707408i	$0.250136\pi$
$168$	0	0
$169$	−10.7186	−0.824505
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.17552i	0.165402i	0.996574	+	0.0827010i	$0.0263546\pi$
$173$	2.17552i	0.165402i	−0.996574	+	0.0827010i	$0.973645\pi$
$174$	0	0
$175$	1.51092	0.114215
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.26314i	− 0.617616i	−0.951124	−	0.308808i	$-0.900070\pi$
$179$	− 8.26314i	− 0.617616i	0.951124	−	0.308808i	$-0.0999301\pi$
$180$	0	0
$181$	− 11.0945i	− 0.824651i	−0.911037	−	0.412325i	$-0.864717\pi$
$181$	− 11.0945i	− 0.824651i	0.911037	−	0.412325i	$-0.135283\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.43826	0.252786
$186$	0	0
$187$	− 3.93517i	− 0.287768i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.6967	−1.42521	−0.712603	−	0.701568i	$-0.752484\pi$
$191$	−19.6967	−1.42521	−0.712603	+	0.701568i	$0.752484\pi$
$192$	0	0
$193$	−16.6240	−1.19662	−0.598309	−	0.801265i	$-0.704161\pi$
$193$	−16.6240	−1.19662	−0.598309	+	0.801265i	$0.704161\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.71137i	0.193177i	0.995324	+	0.0965885i	$0.0307931\pi$
$197$	2.71137i	0.193177i	−0.995324	+	0.0965885i	$0.969207\pi$
$198$	0	0
$199$	−4.78316	−0.339069	−0.169535	−	0.985524i	$-0.554226\pi$
$199$	−4.78316	−0.339069	−0.169535	+	0.985524i	$0.554226\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.50908i	− 0.176103i
$204$	0	0
$205$	5.19877i	0.363098i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.92228	−0.132967
$210$	0	0
$211$	− 19.1635i	− 1.31927i	−0.751587	−	0.659634i	$-0.770711\pi$
$211$	− 19.1635i	− 1.31927i	0.751587	−	0.659634i	$-0.229289\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.19877	0.354553
$216$	0	0
$217$	−0.332456	−0.0225686
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.1650i	1.28917i
$222$	0	0
$223$	−2.30232	−0.154175	−0.0770873	−	0.997024i	$-0.524562\pi$
$223$	−2.30232	−0.154175	−0.0770873	+	0.997024i	$0.524562\pi$
$224$	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
$229$	− 19.2751i	− 1.27373i	−0.770974	−	0.636866i	$-0.780230\pi$
$229$	− 19.2751i	− 1.27373i	0.770974	−	0.636866i	$-0.219770\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.0333	−0.853837	−0.426918	−	0.904290i	$-0.640401\pi$
$233$	−13.0333	−0.853837	−0.426918	+	0.904290i	$0.640401\pi$
$234$	0	0
$235$	− 0.210169i	− 0.0137099i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.7828	1.40901	0.704505	−	0.709699i	$-0.251169\pi$
$239$	21.7828	1.40901	0.704505	+	0.709699i	$0.251169\pi$
$240$	0	0
$241$	4.54856	0.292998	0.146499	−	0.989211i	$-0.453199\pi$
$241$	4.54856	0.292998	0.146499	+	0.989211i	$0.453199\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.67447i	− 0.426416i
$246$	0	0
$247$	9.36185	0.595680
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 18.5209i	− 1.16903i	−0.811383	−	0.584515i	$-0.801285\pi$
$251$	− 18.5209i	− 1.16903i	0.811383	−	0.584515i	$-0.198715\pi$
$252$	0	0
$253$	− 2.39623i	− 0.150650i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.8451	−1.11315	−0.556575	−	0.830798i	$-0.687885\pi$
$257$	−17.8451	−1.11315	−0.556575	+	0.830798i	$0.687885\pi$
$258$	0	0
$259$	− 1.31737i	− 0.0818574i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.6628	−0.780822	−0.390411	−	0.920641i	$-0.627667\pi$
$263$	−12.6628	−0.780822	−0.390411	+	0.920641i	$0.627667\pi$
$264$	0	0
$265$	5.16097	0.317036
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.4434i	0.880632i	0.897843	+	0.440316i	$0.145134\pi$
$269$	14.4434i	0.880632i	−0.897843	+	0.440316i	$0.854866\pi$
$270$	0	0
$271$	−26.9274	−1.63572	−0.817861	−	0.575416i	$-0.804840\pi$
$271$	−26.9274	−1.63572	−0.817861	+	0.575416i	$0.804840\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.05366i	0.244445i
$276$	0	0
$277$	18.2832i	1.09853i	0.835647	+	0.549266i	$0.185093\pi$
$277$	18.2832i	1.09853i	−0.835647	+	0.549266i	$0.814907\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.5976	−0.632202	−0.316101	−	0.948726i	$-0.602374\pi$
$281$	−10.5976	−0.632202	−0.316101	+	0.948726i	$0.602374\pi$
$282$	0	0
$283$	− 5.66229i	− 0.336588i	−0.985737	−	0.168294i	$-0.946174\pi$
$283$	− 5.66229i	− 0.336588i	0.985737	−	0.168294i	$-0.0538258\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.99191	0.117579
$288$	0	0
$289$	−1.51444	−0.0890845
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 0.229918i	− 0.0134320i	−0.999977	−	0.00671598i	$-0.997862\pi$
$293$	− 0.229918i	− 0.0134320i	0.999977	−	0.00671598i	$-0.00213778\pi$
$294$	0	0
$295$	−6.83250	−0.397803
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.6700i	0.674896i
$300$	0	0
$301$	− 1.99191i	− 0.114812i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.3047	0.647306
$306$	0	0
$307$	22.0069i	1.25600i	0.778213	+	0.628000i	$0.216126\pi$
$307$	22.0069i	1.25600i	−0.778213	+	0.628000i	$0.783874\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.5072	0.765923	0.382961	−	0.923764i	$-0.374904\pi$
$311$	13.5072	0.765923	0.382961	+	0.923764i	$0.374904\pi$
$312$	0	0
$313$	2.14251	0.121102	0.0605508	−	0.998165i	$-0.480714\pi$
$313$	2.14251	0.121102	0.0605508	+	0.998165i	$0.480714\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.8640i	1.17184i	0.810369	+	0.585920i	$0.199267\pi$
$317$	20.8640i	1.17184i	−0.810369	+	0.585920i	$0.800733\pi$
$318$	0	0
$319$	6.73164	0.376900
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 7.56451i	− 0.420901i
$324$	0	0
$325$	− 19.7420i	− 1.09509i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.0805264	−0.00443956
$330$	0	0
$331$	10.6882i	0.587478i	0.955886	+	0.293739i	$0.0948997\pi$
$331$	10.6882i	0.587478i	−0.955886	+	0.293739i	$0.905100\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.42217	0.514788
$336$	0	0
$337$	20.3795	1.11014	0.555071	−	0.831803i	$-0.312691\pi$
$337$	20.3795	1.11014	0.555071	+	0.831803i	$0.312691\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 0.891949i	− 0.0483017i
$342$	0	0
$343$	−5.16643	−0.278961
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 27.7641i	− 1.49046i	−0.666810	−	0.745228i	$-0.732341\pi$
$347$	− 27.7641i	− 1.49046i	0.666810	−	0.745228i	$-0.267659\pi$
$348$	0	0
$349$	− 4.68897i	− 0.250995i	−0.992094	−	0.125497i	$-0.959947\pi$
$349$	− 4.68897i	− 0.250995i	0.992094	−	0.125497i	$-0.0400527\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	36.0249	1.91741	0.958706	−	0.284401i	$-0.0917946\pi$
$353$	36.0249	1.91741	0.958706	+	0.284401i	$0.0917946\pi$
$354$	0	0
$355$	− 6.35683i	− 0.337385i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.8873	−1.36628	−0.683141	−	0.730287i	$-0.739386\pi$
$359$	−25.8873	−1.36628	−0.683141	+	0.730287i	$0.739386\pi$
$360$	0	0
$361$	15.3048	0.805517
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.21022i	0.429743i
$366$	0	0
$367$	26.7496	1.39632	0.698160	−	0.715942i	$-0.254002\pi$
$367$	26.7496	1.39632	0.698160	+	0.715942i	$0.254002\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.97743i	− 0.102663i
$372$	0	0
$373$	0.328598i	0.0170142i	0.999964	+	0.00850708i	$0.00270792\pi$
$373$	0.328598i	0.0170142i	−0.999964	+	0.00850708i	$0.997292\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−32.7843	−1.68847
$378$	0	0
$379$	30.3940i	1.56124i	0.625008	+	0.780618i	$0.285096\pi$
$379$	30.3940i	1.56124i	−0.625008	+	0.780618i	$0.714904\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−36.9042	−1.88571	−0.942857	−	0.333197i	$-0.891873\pi$
$383$	−36.9042	−1.88571	−0.942857	+	0.333197i	$0.891873\pi$
$384$	0	0
$385$	−0.362592	−0.0184794
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 23.7033i	− 1.20181i	−0.799322	−	0.600904i	$-0.794808\pi$
$389$	− 23.7033i	− 1.20181i	0.799322	−	0.600904i	$-0.205192\pi$
$390$	0	0
$391$	9.42957	0.476874
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 8.36259i	− 0.420768i
$396$	0	0
$397$	− 1.48318i	− 0.0744387i	−0.999307	−	0.0372193i	$-0.988150\pi$
$397$	− 1.48318i	− 0.0744387i	0.999307	−	0.0372193i	$-0.0118500\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.79245	0.439074	0.219537	−	0.975604i	$-0.429545\pi$
$401$	8.79245	0.439074	0.219537	+	0.975604i	$0.429545\pi$
$402$	0	0
$403$	4.34394i	0.216387i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.53439	0.175193
$408$	0	0
$409$	31.4357	1.55439	0.777197	−	0.629258i	$-0.216641\pi$
$409$	31.4357	1.55439	0.777197	+	0.629258i	$0.216641\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.61787i	0.128817i
$414$	0	0
$415$	−15.1852	−0.745413
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 15.3165i	− 0.748260i	−0.927376	−	0.374130i	$-0.877941\pi$
$419$	− 15.3165i	− 0.748260i	0.927376	−	0.374130i	$-0.122059\pi$
$420$	0	0
$421$	18.9907i	0.925548i	0.886476	+	0.462774i	$0.153146\pi$
$421$	18.9907i	0.925548i	−0.886476	+	0.462774i	$0.846854\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.9518	−0.773777
$426$	0	0
$427$	− 4.33140i	− 0.209611i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	23.8779	1.15016	0.575079	−	0.818098i	$-0.304971\pi$
$431$	23.8779	1.15016	0.575079	+	0.818098i	$0.304971\pi$
$432$	0	0
$433$	−32.0589	−1.54065	−0.770326	−	0.637651i	$-0.779906\pi$
$433$	−32.0589	−1.54065	−0.770326	+	0.637651i	$0.779906\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.60623i	− 0.220346i
$438$	0	0
$439$	−26.4992	−1.26474	−0.632370	−	0.774667i	$-0.717918\pi$
$439$	−26.4992	−1.26474	−0.632370	+	0.774667i	$0.717918\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	39.1108i	1.85821i	0.369818	+	0.929104i	$0.379420\pi$
$443$	39.1108i	1.85821i	−0.369818	+	0.929104i	$0.620580\pi$
$444$	0	0
$445$	7.46403i	0.353829i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.54272	−0.261577	−0.130789	−	0.991410i	$-0.541751\pi$
$449$	−5.54272	−0.261577	−0.130789	+	0.991410i	$0.541751\pi$
$450$	0	0
$451$	5.34412i	0.251645i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.76589	0.0827860
$456$	0	0
$457$	−28.8027	−1.34733	−0.673666	−	0.739036i	$-0.735281\pi$
$457$	−28.8027	−1.34733	−0.673666	+	0.739036i	$0.735281\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.2610i	1.17652i	0.808671	+	0.588261i	$0.200187\pi$
$461$	25.2610i	1.17652i	−0.808671	+	0.588261i	$0.799813\pi$
$462$	0	0
$463$	20.4096	0.948516	0.474258	−	0.880386i	$-0.342716\pi$
$463$	20.4096	0.948516	0.474258	+	0.880386i	$0.342716\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16.8795i	− 0.781092i	−0.920583	−	0.390546i	$-0.872286\pi$
$467$	− 16.8795i	− 0.781092i	0.920583	−	0.390546i	$-0.127714\pi$
$468$	0	0
$469$	− 3.61011i	− 0.166699i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.34412	0.245723
$474$	0	0
$475$	7.79228i	0.357534i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.89668	0.132353	0.0661763	−	0.997808i	$-0.478920\pi$
$479$	2.89668	0.132353	0.0661763	+	0.997808i	$0.478920\pi$
$480$	0	0
$481$	−17.2131	−0.784848
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.07212i	0.184905i
$486$	0	0
$487$	−25.3697	−1.14961	−0.574806	−	0.818290i	$-0.694922\pi$
$487$	−25.3697	−1.14961	−0.574806	+	0.818290i	$0.694922\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 3.95724i	− 0.178588i	−0.996005	−	0.0892939i	$-0.971539\pi$
$491$	− 3.95724i	− 0.178588i	0.996005	−	0.0892939i	$-0.0284611\pi$
$492$	0	0
$493$	26.4902i	1.19306i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.43562	−0.109253
$498$	0	0
$499$	39.2380i	1.75654i	0.478169	+	0.878268i	$0.341301\pi$
$499$	39.2380i	1.75654i	−0.478169	+	0.878268i	$0.658699\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.4657	1.35840	0.679200	−	0.733953i	$-0.262327\pi$
$503$	30.4657	1.35840	0.679200	+	0.733953i	$0.262327\pi$
$504$	0	0
$505$	16.2237	0.721944
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 7.57428i	− 0.335724i	−0.985811	−	0.167862i	$-0.946314\pi$
$509$	− 7.57428i	− 0.335724i	0.985811	−	0.167862i	$-0.0536863\pi$
$510$	0	0
$511$	3.14575	0.139160
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.45416i	0.196274i
$516$	0	0
$517$	− 0.216045i	− 0.00950165i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−33.7994	−1.48078	−0.740390	−	0.672178i	$-0.765359\pi$
$521$	−33.7994	−1.48078	−0.740390	+	0.672178i	$0.765359\pi$
$522$	0	0
$523$	19.6629i	0.859797i	0.902877	+	0.429898i	$0.141451\pi$
$523$	19.6629i	0.859797i	−0.902877	+	0.429898i	$0.858549\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.50997	0.152897
$528$	0	0
$529$	−17.2581	−0.750351
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 26.0268i	− 1.12734i
$534$	0	0
$535$	−4.77370	−0.206385
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6.86107i	− 0.295527i
$540$	0	0
$541$	4.50069i	0.193500i	0.995309	+	0.0967499i	$0.0308447\pi$
$541$	4.50069i	0.193500i	−0.995309	+	0.0967499i	$0.969155\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.13297	0.219872
$546$	0	0
$547$	34.7996i	1.48792i	0.668222	+	0.743962i	$0.267056\pi$
$547$	34.7996i	1.48792i	−0.668222	+	0.743962i	$0.732944\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.9401	0.551268
$552$	0	0
$553$	−3.20413	−0.136253
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.7083i	1.76724i	0.468206	+	0.883619i	$0.344901\pi$
$557$	41.7083i	1.76724i	−0.468206	+	0.883619i	$0.655099\pi$
$558$	0	0
$559$	−26.0268	−1.10081
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.0971i	1.05772i	0.848710	+	0.528859i	$0.177380\pi$
$563$	25.0971i	1.05772i	−0.848710	+	0.528859i	$0.822620\pi$
$564$	0	0
$565$	− 7.96580i	− 0.335124i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.4176	1.61055	0.805274	−	0.592902i	$-0.202018\pi$
$569$	38.4176	1.61055	0.805274	+	0.592902i	$0.202018\pi$
$570$	0	0
$571$	− 44.0148i	− 1.84196i	−0.389608	−	0.920981i	$-0.627389\pi$
$571$	− 44.0148i	− 1.84196i	0.389608	−	0.920981i	$-0.372611\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.71349	−0.405081
$576$	0	0
$577$	9.06878	0.377538	0.188769	−	0.982021i	$-0.439550\pi$
$577$	9.06878	0.377538	0.188769	+	0.982021i	$0.439550\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.81822i	0.241381i
$582$	0	0
$583$	5.30526	0.219721
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 32.6203i	− 1.34638i	−0.739468	−	0.673191i	$-0.764923\pi$
$587$	− 32.6203i	− 1.34638i	0.739468	−	0.673191i	$-0.235077\pi$
$588$	0	0
$589$	− 1.71458i	− 0.0706480i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.1304	1.03198	0.515992	−	0.856594i	$-0.327424\pi$
$593$	25.1304	1.03198	0.515992	+	0.856594i	$0.327424\pi$
$594$	0	0
$595$	− 1.42686i	− 0.0584956i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.4554	−0.999221	−0.499611	−	0.866250i	$-0.666524\pi$
$599$	−24.4554	−0.999221	−0.499611	+	0.866250i	$0.666524\pi$
$600$	0	0
$601$	−7.04967	−0.287562	−0.143781	−	0.989610i	$-0.545926\pi$
$601$	−7.04967	−0.287562	−0.143781	+	0.989610i	$0.545926\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 0.972802i	− 0.0395500i
$606$	0	0
$607$	−29.4951	−1.19717	−0.598585	−	0.801059i	$-0.704270\pi$
$607$	−29.4951	−1.19717	−0.598585	+	0.801059i	$0.704270\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.05218i	0.0425665i
$612$	0	0
$613$	34.6553i	1.39972i	0.714282	+	0.699858i	$0.246753\pi$
$613$	34.6553i	1.39972i	−0.714282	+	0.699858i	$0.753247\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.4034	1.38503	0.692514	−	0.721405i	$-0.256503\pi$
$617$	34.4034	1.38503	0.692514	+	0.721405i	$0.256503\pi$
$618$	0	0
$619$	33.5984i	1.35043i	0.737620	+	0.675216i	$0.235950\pi$
$619$	33.5984i	1.35043i	−0.737620	+	0.675216i	$0.764050\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.85985	0.114577
$624$	0	0
$625$	11.7004	0.468016
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.9084i	0.554565i
$630$	0	0
$631$	38.2188	1.52147	0.760733	−	0.649064i	$-0.224839\pi$
$631$	38.2188	1.52147	0.760733	+	0.649064i	$0.224839\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19.5670i	0.776493i
$636$	0	0
$637$	33.4146i	1.32393i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−36.3107	−1.43418	−0.717092	−	0.696978i	$-0.754527\pi$
$641$	−36.3107	−1.43418	−0.717092	+	0.696978i	$0.754527\pi$
$642$	0	0
$643$	− 28.7600i	− 1.13418i	−0.823655	−	0.567091i	$-0.808069\pi$
$643$	− 28.7600i	− 1.13418i	0.823655	−	0.567091i	$-0.191931\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.182275	0.00716597	0.00358299	−	0.999994i	$-0.498859\pi$
$647$	0.182275	0.00716597	0.00358299	+	0.999994i	$0.498859\pi$
$648$	0	0
$649$	−7.02352	−0.275697
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 19.9879i	− 0.782187i	−0.920351	−	0.391093i	$-0.872097\pi$
$653$	− 19.9879i	− 0.782187i	0.920351	−	0.391093i	$-0.127903\pi$
$654$	0	0
$655$	9.47969	0.370402
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.5539i	0.956485i	0.878228	+	0.478242i	$0.158726\pi$
$659$	24.5539i	0.956485i	−0.878228	+	0.478242i	$0.841274\pi$
$660$	0	0
$661$	3.00617i	0.116926i	0.998290	+	0.0584632i	$0.0186200\pi$
$661$	3.00617i	0.116926i	−0.998290	+	0.0584632i	$0.981380\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.697005	−0.0270287
$666$	0	0
$667$	16.1306i	0.624578i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.6208	0.448615
$672$	0	0
$673$	26.2783	1.01295	0.506476	−	0.862254i	$-0.330948\pi$
$673$	26.2783	1.01295	0.506476	+	0.862254i	$0.330948\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 42.2493i	− 1.62377i	−0.583815	−	0.811887i	$-0.698441\pi$
$677$	− 42.2493i	− 1.62377i	0.583815	−	0.811887i	$-0.301559\pi$
$678$	0	0
$679$	1.56023	0.0598762
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 8.26642i	− 0.316306i	−0.987415	−	0.158153i	$-0.949446\pi$
$683$	− 8.26642i	− 0.316306i	0.987415	−	0.158153i	$-0.0505539\pi$
$684$	0	0
$685$	7.06877i	0.270084i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−25.8375	−0.984331
$690$	0	0
$691$	14.2272i	0.541227i	0.962688	+	0.270614i	$0.0872265\pi$
$691$	14.2272i	0.541227i	−0.962688	+	0.270614i	$0.912773\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19.2298	0.729428
$696$	0	0
$697$	−21.0300	−0.796568
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.3270i	0.805511i	0.915308	+	0.402755i	$0.131947\pi$
$701$	21.3270i	0.805511i	−0.915308	+	0.402755i	$0.868053\pi$
$702$	0	0
$703$	6.79409	0.256244
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.21610i	− 0.233781i
$708$	0	0
$709$	41.4642i	1.55722i	0.627509	+	0.778609i	$0.284075\pi$
$709$	41.4642i	1.55722i	−0.627509	+	0.778609i	$0.715925\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.13731	0.0800431
$714$	0	0
$715$	4.73771i	0.177180i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.1036	−1.27185	−0.635925	−	0.771751i	$-0.719381\pi$
$719$	−34.1036	−1.27185	−0.635925	+	0.771751i	$0.719381\pi$
$720$	0	0
$721$	1.70661	0.0635576
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 27.2878i	− 1.01344i
$726$	0	0
$727$	11.7396	0.435397	0.217698	−	0.976016i	$-0.430145\pi$
$727$	11.7396	0.435397	0.217698	+	0.976016i	$0.430145\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	21.0300i	0.777823i
$732$	0	0
$733$	16.3251i	0.602981i	0.953469	+	0.301491i	$0.0974842\pi$
$733$	16.3251i	0.602981i	−0.953469	+	0.301491i	$0.902516\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.68560	0.356774
$738$	0	0
$739$	10.8959i	0.400814i	0.979713	+	0.200407i	$0.0642264\pi$
$739$	10.8959i	0.400814i	−0.979713	+	0.200407i	$0.935774\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.7775	−0.578821	−0.289410	−	0.957205i	$-0.593459\pi$
$743$	−15.7775	−0.578821	−0.289410	+	0.957205i	$0.593459\pi$
$744$	0	0
$745$	0.665985	0.0243998
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.82905i	0.0668319i
$750$	0	0
$751$	23.8788	0.871350	0.435675	−	0.900104i	$-0.356510\pi$
$751$	23.8788	0.871350	0.435675	+	0.900104i	$0.356510\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9.46982i	0.344642i
$756$	0	0
$757$	49.4564i	1.79752i	0.438437	+	0.898762i	$0.355532\pi$
$757$	49.4564i	1.79752i	−0.438437	+	0.898762i	$0.644468\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.5915	0.420190	0.210095	−	0.977681i	$-0.432623\pi$
$761$	11.5915	0.420190	0.210095	+	0.977681i	$0.432623\pi$
$762$	0	0
$763$	− 1.96670i	− 0.0711993i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	34.2057	1.23510
$768$	0	0
$769$	10.9488	0.394826	0.197413	−	0.980320i	$-0.436746\pi$
$769$	10.9488	0.394826	0.197413	+	0.980320i	$0.436746\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 55.2931i	− 1.98876i	−0.105888	−	0.994378i	$-0.533769\pi$
$773$	− 55.2931i	− 1.98876i	0.105888	−	0.994378i	$-0.466231\pi$
$774$	0	0
$775$	−3.61565	−0.129878
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.2729i	0.368065i
$780$	0	0
$781$	− 6.53455i	− 0.233825i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.92086	−0.0685585
$786$	0	0
$787$	− 16.2666i	− 0.579840i	−0.957051	−	0.289920i	$-0.906371\pi$
$787$	− 16.2666i	− 0.579840i	0.957051	−	0.289920i	$-0.0936287\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.05210	−0.108520
$792$	0	0
$793$	−56.5951	−2.00975
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.1218i	0.819016i	0.912307	+	0.409508i	$0.134300\pi$
$797$	23.1218i	0.819016i	−0.912307	+	0.409508i	$0.865700\pi$
$798$	0	0
$799$	0.850174	0.0300770
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.43977i	0.297833i
$804$	0	0
$805$	− 0.868854i	− 0.0306231i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.85499	−0.205850	−0.102925	−	0.994689i	$-0.532820\pi$
$809$	−5.85499	−0.205850	−0.102925	+	0.994689i	$0.532820\pi$
$810$	0	0
$811$	− 18.0528i	− 0.633921i	−0.948439	−	0.316961i	$-0.897338\pi$
$811$	− 18.0528i	− 0.633921i	0.948439	−	0.316961i	$-0.102662\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.6358	0.477640
$816$	0	0
$817$	10.2729	0.359404
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 48.1747i	− 1.68131i	−0.541573	−	0.840654i	$-0.682171\pi$
$821$	− 48.1747i	− 1.68131i	0.541573	−	0.840654i	$-0.317829\pi$
$822$	0	0
$823$	4.20676	0.146639	0.0733193	−	0.997309i	$-0.476641\pi$
$823$	4.20676	0.146639	0.0733193	+	0.997309i	$0.476641\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.5271i	0.887666i	0.896110	+	0.443833i	$0.146382\pi$
$827$	25.5271i	0.887666i	−0.896110	+	0.443833i	$0.853618\pi$
$828$	0	0
$829$	4.31017i	0.149699i	0.997195	+	0.0748493i	$0.0238476\pi$
$829$	4.31017i	0.149699i	−0.997195	+	0.0748493i	$0.976152\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.9995	0.935477
$834$	0	0
$835$	17.7710i	0.614992i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43.9106	−1.51596	−0.757981	−	0.652277i	$-0.773814\pi$
$839$	−43.9106	−1.51596	−0.757981	+	0.652277i	$0.773814\pi$
$840$	0	0
$841$	−16.3150	−0.562587
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 10.4270i	− 0.358701i
$846$	0	0
$847$	−0.372730	−0.0128071
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.46920i	0.290320i
$852$	0	0
$853$	27.7678i	0.950750i	0.879783	+	0.475375i	$0.157688\pi$
$853$	27.7678i	0.950750i	−0.879783	+	0.475375i	$0.842312\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.7294	0.571464	0.285732	−	0.958310i	$-0.407763\pi$
$857$	16.7294	0.571464	0.285732	+	0.958310i	$0.407763\pi$
$858$	0	0
$859$	33.7228i	1.15061i	0.817940	+	0.575303i	$0.195116\pi$
$859$	33.7228i	1.15061i	−0.817940	+	0.575303i	$0.804884\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.156749	−0.00533580	−0.00266790	−	0.999996i	$-0.500849\pi$
$863$	−0.156749	−0.00533580	−0.00266790	+	0.999996i	$0.500849\pi$
$864$	0	0
$865$	−2.11635	−0.0719582
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8.59639i	− 0.291613i
$870$	0	0
$871$	−47.1705	−1.59831
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.28278i	0.110978i
$876$	0	0
$877$	− 7.44868i	− 0.251524i	−0.992060	−	0.125762i	$-0.959862\pi$
$877$	− 7.44868i	− 0.251524i	0.992060	−	0.125762i	$-0.0401376\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.7307	0.900579	0.450289	−	0.892883i	$-0.351321\pi$
$881$	26.7307	0.900579	0.450289	+	0.892883i	$0.351321\pi$
$882$	0	0
$883$	− 2.02931i	− 0.0682918i	−0.999417	−	0.0341459i	$-0.989129\pi$
$883$	− 2.02931i	− 0.0682918i	0.999417	−	0.0341459i	$-0.0108711\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.9468	−1.60990	−0.804948	−	0.593346i	$-0.797807\pi$
$887$	−47.9468	−1.60990	−0.804948	+	0.593346i	$0.797807\pi$
$888$	0	0
$889$	7.49711	0.251445
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 0.415300i	− 0.0138975i
$894$	0	0
$895$	8.03840	0.268694
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.00428i	0.200254i
$900$	0	0
$901$	20.8771i	0.695517i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.7928	0.358765
$906$	0	0
$907$	31.2433i	1.03742i	0.854952	+	0.518708i	$0.173587\pi$
$907$	31.2433i	1.03742i	−0.854952	+	0.518708i	$0.826413\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−52.2465	−1.73100	−0.865502	−	0.500905i	$-0.833001\pi$
$911$	−52.2465	−1.73100	−0.865502	+	0.500905i	$0.833001\pi$
$912$	0	0
$913$	−15.6098	−0.516608
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.63214i	− 0.119944i
$918$	0	0
$919$	26.7583	0.882675	0.441338	−	0.897341i	$-0.354504\pi$
$919$	26.7583	0.882675	0.441338	+	0.897341i	$0.354504\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	31.8244i	1.04751i
$924$	0	0
$925$	− 14.3272i	− 0.471075i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.8346	−0.421091	−0.210546	−	0.977584i	$-0.567524\pi$
$929$	−12.8346	−0.421091	−0.210546	+	0.977584i	$0.567524\pi$
$930$	0	0
$931$	− 13.1889i	− 0.432250i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.82814	0.125194
$936$	0	0
$937$	−7.16455	−0.234055	−0.117028	−	0.993129i	$-0.537337\pi$
$937$	−7.16455	−0.234055	−0.117028	+	0.993129i	$0.537337\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.9782i	0.716469i	0.933632	+	0.358235i	$0.116621\pi$
$941$	21.9782i	0.716469i	−0.933632	+	0.358235i	$0.883379\pi$
$942$	0	0
$943$	−12.8057	−0.417012
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.5866i	1.31889i	0.751755	+	0.659443i	$0.229208\pi$
$947$	40.5866i	1.31889i	−0.751755	+	0.659443i	$0.770792\pi$
$948$	0	0
$949$	− 41.1031i	− 1.33426i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.80164	−0.123147	−0.0615736	−	0.998103i	$-0.519612\pi$
$953$	−3.80164	−0.123147	−0.0615736	+	0.998103i	$0.519612\pi$
$954$	0	0
$955$	− 19.1610i	− 0.620036i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.70840	0.0874589
$960$	0	0
$961$	−30.2044	−0.974336
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 16.1718i	− 0.520589i
$966$	0	0
$967$	−46.2968	−1.48880	−0.744402	−	0.667732i	$-0.767265\pi$
$967$	−46.2968	−1.48880	−0.744402	+	0.667732i	$0.767265\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 28.9883i	− 0.930280i	−0.885237	−	0.465140i	$-0.846004\pi$
$971$	− 28.9883i	− 0.930280i	0.885237	−	0.465140i	$-0.153996\pi$
$972$	0	0
$973$	− 7.36790i	− 0.236204i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−56.0820	−1.79422	−0.897112	−	0.441804i	$-0.854339\pi$
$977$	−56.0820	−1.79422	−0.897112	+	0.441804i	$0.854339\pi$
$978$	0	0
$979$	7.67271i	0.245221i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.0408	−1.59605	−0.798027	−	0.602622i	$-0.794123\pi$
$983$	−50.0408	−1.59605	−0.798027	+	0.602622i	$0.794123\pi$
$984$	0	0
$985$	−2.63763	−0.0840417
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.8057i	0.407199i
$990$	0	0
$991$	26.4967	0.841694	0.420847	−	0.907132i	$-0.361733\pi$
$991$	26.4967	0.841694	0.420847	+	0.907132i	$0.361733\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 4.65307i	− 0.147512i
$996$	0	0
$997$	21.3660i	0.676669i	0.941026	+	0.338335i	$0.109864\pi$
$997$	21.3660i	0.676669i	−0.941026	+	0.338335i	$0.890136\pi$
$998$	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.1585.13		20
3.2	odd	2	inner	3168.2.f.h.1585.7		20
4.3	odd	2		792.2.f.h.397.2	yes	20
8.3	odd	2		792.2.f.h.397.1	✓	20
8.5	even	2	inner	3168.2.f.h.1585.8		20
12.11	even	2		792.2.f.h.397.19	yes	20
24.5	odd	2	inner	3168.2.f.h.1585.14		20
24.11	even	2		792.2.f.h.397.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.f.h.397.1	✓	20	8.3	odd	2
792.2.f.h.397.2	yes	20	4.3	odd	2
792.2.f.h.397.19	yes	20	12.11	even	2
792.2.f.h.397.20	yes	20	24.11	even	2
3168.2.f.h.1585.7		20	3.2	odd	2	inner
3168.2.f.h.1585.8		20	8.5	even	2	inner
3168.2.f.h.1585.13		20	1.1	even	1	trivial
3168.2.f.h.1585.14		20	24.5	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)

\(f(q)\)	\(=\)	\(q+0.972802i q^{5} +0.372730 q^{7} +1.00000i q^{11} -4.87017i q^{13} -3.93517 q^{17} +1.92228i q^{19} -2.39623 q^{23} +4.05366 q^{25} -6.73164i q^{29} -0.891949 q^{31} +0.362592i q^{35} -3.53439i q^{37} +5.34412 q^{41} -5.34412i q^{43} -0.216045 q^{47} -6.86107 q^{49} -5.30526i q^{53} -0.972802 q^{55} +7.02352i q^{59} -11.6208i q^{61} +4.73771 q^{65} -9.68560i q^{67} -6.53455 q^{71} +8.43977 q^{73} +0.372730i q^{77} -8.59639 q^{79} +15.6098i q^{83} -3.82814i q^{85} +7.67271 q^{89} -1.81526i q^{91} -1.87000 q^{95} +4.18597 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{25} - 40 q^{31} + 36 q^{49} + 16 q^{55} - 56 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) 20 * q - 12 * q^25 - 40 * q^31 + 36 * q^49 + 16 * q^55 - 56 * q^73 + 16 * q^79 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more