LMFDB - Embedded newform 3168.2.f.h.1585.6

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)

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);

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")

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			1585.6
Root			$-0.484785 + 1.32853i$ of defining polynomial
Character	$\chi$	$=$	3168.1585
Dual form			3168.2.f.h.1585.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21156i q^{5} -1.26179 q^{7} -1.00000i q^{11} +0.807008i q^{13} +5.69004 q^{17} -5.87553i q^{19} -7.89855 q^{23} +0.109010 q^{25} -7.17010i q^{29} -7.31411 q^{31} +2.79052i q^{35} +9.62643i q^{37} +2.83003 q^{41} +2.83003i q^{43} +7.22017 q^{47} -5.40788 q^{49} -3.01732i q^{53} -2.21156 q^{55} +3.32934i q^{59} -8.97382i q^{61} +1.78474 q^{65} -12.3983i q^{67} +2.56065 q^{71} -9.01085 q^{73} +1.26179i q^{77} -2.35557 q^{79} +2.50251i q^{83} -12.5838i q^{85} -0.832776 q^{89} -1.01828i q^{91} -12.9941 q^{95} -15.0666 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 2.21156i	− 0.989039i	−0.869167	−	0.494519i	$-0.835344\pi$
$5$	− 2.21156i	− 0.989039i	0.869167	−	0.494519i	$-0.164656\pi$
$6$	0	0
$7$	−1.26179	−0.476912	−0.238456	−	0.971153i	$-0.576641\pi$
$7$	−1.26179	−0.476912	−0.238456	+	0.971153i	$0.576641\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	0.807008i	0.223824i	0.993718	+	0.111912i	$0.0356974\pi$
$13$	0.807008i	0.223824i	−0.993718	+	0.111912i	$0.964303\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.69004	1.38004	0.690018	−	0.723792i	$-0.257603\pi$
$17$	5.69004	1.38004	0.690018	+	0.723792i	$0.257603\pi$
$18$	0	0
$19$	− 5.87553i	− 1.34794i	−0.738759	−	0.673970i	$-0.764588\pi$
$19$	− 5.87553i	− 1.34794i	0.738759	−	0.673970i	$-0.235412\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.89855	−1.64696	−0.823481	−	0.567344i	$-0.807971\pi$
$23$	−7.89855	−1.64696	−0.823481	+	0.567344i	$0.807971\pi$
$24$	0	0
$25$	0.109010	0.0218020
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.17010i	− 1.33145i	−0.746196	−	0.665727i	$-0.768122\pi$
$29$	− 7.17010i	− 1.33145i	0.746196	−	0.665727i	$-0.231878\pi$
$30$	0	0
$31$	−7.31411	−1.31365	−0.656826	−	0.754042i	$-0.728101\pi$
$31$	−7.31411	−1.31365	−0.656826	+	0.754042i	$0.728101\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.79052i	0.471685i
$36$	0	0
$37$	9.62643i	1.58257i	0.611444	+	0.791287i	$0.290589\pi$
$37$	9.62643i	1.58257i	−0.611444	+	0.791287i	$0.709411\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.83003	0.441976	0.220988	−	0.975277i	$-0.429072\pi$
$41$	2.83003	0.441976	0.220988	+	0.975277i	$0.429072\pi$
$42$	0	0
$43$	2.83003i	0.431575i	0.976440	+	0.215788i	$0.0692319\pi$
$43$	2.83003i	0.431575i	−0.976440	+	0.215788i	$0.930768\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.22017	1.05317	0.526585	−	0.850122i	$-0.323472\pi$
$47$	7.22017	1.05317	0.526585	+	0.850122i	$0.323472\pi$
$48$	0	0
$49$	−5.40788	−0.772555
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.01732i	− 0.414460i	−0.978292	−	0.207230i	$-0.933555\pi$
$53$	− 3.01732i	− 0.414460i	0.978292	−	0.207230i	$-0.0664449\pi$
$54$	0	0
$55$	−2.21156	−0.298206
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.32934i	0.433443i	0.976233	+	0.216722i	$0.0695364\pi$
$59$	3.32934i	0.433443i	−0.976233	+	0.216722i	$0.930464\pi$
$60$	0	0
$61$	− 8.97382i	− 1.14898i	−0.818512	−	0.574490i	$-0.805201\pi$
$61$	− 8.97382i	− 1.14898i	0.818512	−	0.574490i	$-0.194799\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.78474	0.221370
$66$	0	0
$67$	− 12.3983i	− 1.51470i	−0.653010	−	0.757350i	$-0.726494\pi$
$67$	− 12.3983i	− 1.51470i	0.653010	−	0.757350i	$-0.273506\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.56065	0.303893	0.151947	−	0.988389i	$-0.451446\pi$
$71$	2.56065	0.303893	0.151947	+	0.988389i	$0.451446\pi$
$72$	0	0
$73$	−9.01085	−1.05464	−0.527320	−	0.849667i	$-0.676803\pi$
$73$	−9.01085	−1.05464	−0.527320	+	0.849667i	$0.676803\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.26179i	0.143794i
$78$	0	0
$79$	−2.35557	−0.265022	−0.132511	−	0.991182i	$-0.542304\pi$
$79$	−2.35557	−0.265022	−0.132511	+	0.991182i	$0.542304\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.50251i	0.274686i	0.990524	+	0.137343i	$0.0438562\pi$
$83$	2.50251i	0.274686i	−0.990524	+	0.137343i	$0.956144\pi$
$84$	0	0
$85$	− 12.5838i	− 1.36491i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.832776	−0.0882741	−0.0441371	−	0.999025i	$-0.514054\pi$
$89$	−0.832776	−0.0882741	−0.0441371	+	0.999025i	$0.514054\pi$
$90$	0	0
$91$	− 1.01828i	− 0.106744i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12.9941	−1.33317
$96$	0	0
$97$	−15.0666	−1.52978	−0.764889	−	0.644162i	$-0.777206\pi$
$97$	−15.0666	−1.52978	−0.764889	+	0.644162i	$0.777206\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.25302i	0.522695i	0.965245	+	0.261347i	$0.0841668\pi$
$101$	5.25302i	0.522695i	−0.965245	+	0.261347i	$0.915833\pi$
$102$	0	0
$103$	−11.4187	−1.12512	−0.562561	−	0.826756i	$-0.690184\pi$
$103$	−11.4187	−1.12512	−0.562561	+	0.826756i	$0.690184\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.72868i	0.843833i	0.906635	+	0.421917i	$0.138643\pi$
$107$	8.72868i	0.843833i	−0.906635	+	0.421917i	$0.861357\pi$
$108$	0	0
$109$	− 8.73133i	− 0.836310i	−0.908376	−	0.418155i	$-0.862677\pi$
$109$	− 8.73133i	− 0.836310i	0.908376	−	0.418155i	$-0.137323\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.6536	−1.66071	−0.830356	−	0.557233i	$-0.811863\pi$
$113$	−17.6536	−1.66071	−0.830356	+	0.557233i	$0.811863\pi$
$114$	0	0
$115$	17.4681i	1.62891i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.17964	−0.658156
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.2989i	− 1.01060i
$126$	0	0
$127$	−16.4077	−1.45595	−0.727976	−	0.685603i	$-0.759539\pi$
$127$	−16.4077	−1.45595	−0.727976	+	0.685603i	$0.759539\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.00854i	0.437599i	0.975770	+	0.218799i	$0.0702140\pi$
$131$	5.00854i	0.437599i	−0.975770	+	0.218799i	$0.929786\pi$
$132$	0	0
$133$	7.41370i	0.642849i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.70556	0.743766	0.371883	−	0.928280i	$-0.378712\pi$
$137$	8.70556	0.743766	0.371883	+	0.928280i	$0.378712\pi$
$138$	0	0
$139$	− 7.78357i	− 0.660194i	−0.943947	−	0.330097i	$-0.892919\pi$
$139$	− 7.78357i	− 0.660194i	0.943947	−	0.330097i	$-0.107081\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.807008	0.0674854
$144$	0	0
$145$	−15.8571	−1.31686
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.48858i	0.613489i	0.951792	+	0.306744i	$0.0992396\pi$
$149$	7.48858i	0.613489i	−0.951792	+	0.306744i	$0.900760\pi$
$150$	0	0
$151$	3.47981	0.283183	0.141592	−	0.989925i	$-0.454778\pi$
$151$	3.47981	0.283183	0.141592	+	0.989925i	$0.454778\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.1756i	1.29925i
$156$	0	0
$157$	1.24302i	0.0992036i	0.998769	+	0.0496018i	$0.0157952\pi$
$157$	1.24302i	0.0992036i	−0.998769	+	0.0496018i	$0.984205\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.96632	0.785456
$162$	0	0
$163$	− 1.07526i	− 0.0842212i	−0.999113	−	0.0421106i	$-0.986592\pi$
$163$	− 1.07526i	− 0.0842212i	0.999113	−	0.0421106i	$-0.0134082\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.13747	0.0880199	0.0440099	−	0.999031i	$-0.485987\pi$
$167$	1.13747	0.0880199	0.0440099	+	0.999031i	$0.485987\pi$
$168$	0	0
$169$	12.3487	0.949903
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.44142i	− 0.185618i	−0.995684	−	0.0928088i	$-0.970415\pi$
$173$	− 2.44142i	− 0.185618i	0.995684	−	0.0928088i	$-0.0295845\pi$
$174$	0	0
$175$	−0.137548	−0.0103976
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 25.2869i	− 1.89003i	−0.327025	−	0.945016i	$-0.606046\pi$
$179$	− 25.2869i	− 1.89003i	0.327025	−	0.945016i	$-0.393954\pi$
$180$	0	0
$181$	− 3.87828i	− 0.288271i	−0.989558	−	0.144135i	$-0.953960\pi$
$181$	− 3.87828i	− 0.288271i	0.989558	−	0.144135i	$-0.0460400\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	21.2894	1.56523
$186$	0	0
$187$	− 5.69004i	− 0.416097i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.99214	0.288861	0.144431	−	0.989515i	$-0.453865\pi$
$191$	3.99214	0.288861	0.144431	+	0.989515i	$0.453865\pi$
$192$	0	0
$193$	−23.5108	−1.69234	−0.846171	−	0.532912i	$-0.821098\pi$
$193$	−23.5108	−1.69234	−0.846171	+	0.532912i	$0.821098\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.2276i	1.01367i	0.862043	+	0.506836i	$0.169185\pi$
$197$	14.2276i	1.01367i	−0.862043	+	0.506836i	$0.830815\pi$
$198$	0	0
$199$	−16.1603	−1.14558	−0.572788	−	0.819704i	$-0.694138\pi$
$199$	−16.1603	−1.14558	−0.572788	+	0.819704i	$0.694138\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.04716i	0.634986i
$204$	0	0
$205$	− 6.25877i	− 0.437131i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.87553	−0.406419
$210$	0	0
$211$	12.3021i	0.846909i	0.905917	+	0.423454i	$0.139183\pi$
$211$	12.3021i	0.846909i	−0.905917	+	0.423454i	$0.860817\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.25877	0.426845
$216$	0	0
$217$	9.22887	0.626497
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.59190i	0.308885i
$222$	0	0
$223$	13.6672	0.915225	0.457613	−	0.889152i	$-0.348705\pi$
$223$	13.6672	0.915225	0.457613	+	0.889152i	$0.348705\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	− 13.1337i	− 0.867900i	−0.900937	−	0.433950i	$-0.857119\pi$
$229$	− 13.1337i	− 0.867900i	0.900937	−	0.433950i	$-0.142881\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.3322	−1.72508	−0.862539	−	0.505991i	$-0.831127\pi$
$233$	−26.3322	−1.72508	−0.862539	+	0.505991i	$0.831127\pi$
$234$	0	0
$235$	− 15.9678i	− 1.04163i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.78177	0.115253	0.0576265	−	0.998338i	$-0.481647\pi$
$239$	1.78177	0.115253	0.0576265	+	0.998338i	$0.481647\pi$
$240$	0	0
$241$	−17.8571	−1.15028	−0.575138	−	0.818056i	$-0.695052\pi$
$241$	−17.8571	−1.15028	−0.575138	+	0.818056i	$0.695052\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11.9598i	0.764087i
$246$	0	0
$247$	4.74160	0.301701
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 25.1501i	− 1.58746i	−0.608269	−	0.793731i	$-0.708136\pi$
$251$	− 25.1501i	− 1.58746i	0.608269	−	0.793731i	$-0.291864\pi$
$252$	0	0
$253$	7.89855i	0.496578i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.6591	0.852032	0.426016	−	0.904716i	$-0.359917\pi$
$257$	13.6591	0.852032	0.426016	+	0.904716i	$0.359917\pi$
$258$	0	0
$259$	− 12.1465i	− 0.754749i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.41704	−0.272366	−0.136183	−	0.990684i	$-0.543484\pi$
$263$	−4.41704	−0.272366	−0.136183	+	0.990684i	$0.543484\pi$
$264$	0	0
$265$	−6.67297	−0.409917
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6.10693i	− 0.372346i	−0.982517	−	0.186173i	$-0.940392\pi$
$269$	− 6.10693i	− 0.372346i	0.982517	−	0.186173i	$-0.0596085\pi$
$270$	0	0
$271$	−8.19095	−0.497565	−0.248782	−	0.968559i	$-0.580030\pi$
$271$	−8.19095	−0.497565	−0.248782	+	0.968559i	$0.580030\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 0.109010i	− 0.00657355i
$276$	0	0
$277$	− 19.8174i	− 1.19071i	−0.803463	−	0.595355i	$-0.797011\pi$
$277$	− 19.8174i	− 1.19071i	0.803463	−	0.595355i	$-0.202989\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−23.1012	−1.37810	−0.689050	−	0.724714i	$-0.741972\pi$
$281$	−23.1012	−1.37810	−0.689050	+	0.724714i	$0.741972\pi$
$282$	0	0
$283$	31.8637i	1.89410i	0.321085	+	0.947050i	$0.395953\pi$
$283$	31.8637i	1.89410i	−0.321085	+	0.947050i	$0.604047\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.57090	−0.210784
$288$	0	0
$289$	15.3765	0.904500
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.98170i	− 0.115772i	−0.998323	−	0.0578860i	$-0.981564\pi$
$293$	− 1.98170i	− 0.115772i	0.998323	−	0.0578860i	$-0.0184360\pi$
$294$	0	0
$295$	7.36303	0.428692
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6.37419i	− 0.368629i
$300$	0	0
$301$	− 3.57090i	− 0.205823i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−19.8461	−1.13639
$306$	0	0
$307$	10.9821i	0.626780i	0.949625	+	0.313390i	$0.101465\pi$
$307$	10.9821i	0.626780i	−0.949625	+	0.313390i	$0.898535\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.3552	1.60787	0.803937	−	0.594714i	$-0.202735\pi$
$311$	28.3552	1.60787	0.803937	+	0.594714i	$0.202735\pi$
$312$	0	0
$313$	23.7566	1.34280	0.671402	−	0.741094i	$-0.265692\pi$
$313$	23.7566	1.34280	0.671402	+	0.741094i	$0.265692\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 27.0578i	− 1.51972i	−0.650089	−	0.759858i	$-0.725268\pi$
$317$	− 27.0578i	− 1.51972i	0.650089	−	0.759858i	$-0.274732\pi$
$318$	0	0
$319$	−7.17010	−0.401448
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 33.4320i	− 1.86021i
$324$	0	0
$325$	0.0879719i	0.00487980i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.11035	−0.502270
$330$	0	0
$331$	− 5.66005i	− 0.311105i	−0.987828	−	0.155552i	$-0.950284\pi$
$331$	− 5.66005i	− 0.311105i	0.987828	−	0.155552i	$-0.0497157\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−27.4197	−1.49810
$336$	0	0
$337$	−9.88755	−0.538609	−0.269305	−	0.963055i	$-0.586794\pi$
$337$	−9.88755	−0.538609	−0.269305	+	0.963055i	$0.586794\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.31411i	0.396081i
$342$	0	0
$343$	15.6562	0.845353
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.9432i	− 1.01692i	−0.861085	−	0.508461i	$-0.830215\pi$
$347$	− 18.9432i	− 1.01692i	0.861085	−	0.508461i	$-0.169785\pi$
$348$	0	0
$349$	35.6892i	1.91040i	0.295961	+	0.955200i	$0.404360\pi$
$349$	35.6892i	1.91040i	−0.295961	+	0.955200i	$0.595640\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.0504	1.38652	0.693261	−	0.720687i	$-0.256173\pi$
$353$	26.0504	1.38652	0.693261	+	0.720687i	$0.256173\pi$
$354$	0	0
$355$	− 5.66303i	− 0.300562i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.6845	1.61946	0.809732	−	0.586800i	$-0.199612\pi$
$359$	30.6845	1.61946	0.809732	+	0.586800i	$0.199612\pi$
$360$	0	0
$361$	−15.5219	−0.816942
$362$	0	0
$363$	0	0
$364$	0	0
$365$	19.9280i	1.04308i
$366$	0	0
$367$	1.56749	0.0818225	0.0409113	−	0.999163i	$-0.486974\pi$
$367$	1.56749	0.0818225	0.0409113	+	0.999163i	$0.486974\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.80722i	0.197661i
$372$	0	0
$373$	− 5.45176i	− 0.282282i	−0.989990	−	0.141141i	$-0.954923\pi$
$373$	− 5.45176i	− 0.282282i	0.989990	−	0.141141i	$-0.0450770\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.78632	0.298011
$378$	0	0
$379$	− 34.2320i	− 1.75838i	−0.476474	−	0.879189i	$-0.658085\pi$
$379$	− 34.2320i	− 1.75838i	0.476474	−	0.879189i	$-0.341915\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.5726	1.00012	0.500058	−	0.865992i	$-0.333312\pi$
$383$	19.5726	1.00012	0.500058	+	0.865992i	$0.333312\pi$
$384$	0	0
$385$	2.79052	0.142218
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 4.79316i	− 0.243023i	−0.992590	−	0.121511i	$-0.961226\pi$
$389$	− 4.79316i	− 0.243023i	0.992590	−	0.121511i	$-0.0387741\pi$
$390$	0	0
$391$	−44.9430	−2.27287
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.20948i	0.262117i
$396$	0	0
$397$	31.5145i	1.58167i	0.612029	+	0.790835i	$0.290353\pi$
$397$	31.5145i	1.58167i	−0.612029	+	0.790835i	$0.709647\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.43202	−0.121449	−0.0607247	−	0.998155i	$-0.519341\pi$
$401$	−2.43202	−0.121449	−0.0607247	+	0.998155i	$0.519341\pi$
$402$	0	0
$403$	− 5.90254i	− 0.294027i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.62643	0.477164
$408$	0	0
$409$	−17.6332	−0.871907	−0.435953	−	0.899969i	$-0.643589\pi$
$409$	−17.6332	−0.871907	−0.435953	+	0.899969i	$0.643589\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 4.20093i	− 0.206714i
$414$	0	0
$415$	5.53444	0.271675
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.3460i	1.18938i	0.803954	+	0.594691i	$0.202726\pi$
$419$	24.3460i	1.18938i	−0.803954	+	0.594691i	$0.797274\pi$
$420$	0	0
$421$	20.1604i	0.982556i	0.871003	+	0.491278i	$0.163470\pi$
$421$	20.1604i	0.982556i	−0.871003	+	0.491278i	$0.836530\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.620271	0.0300876
$426$	0	0
$427$	11.3231i	0.547962i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	40.1197	1.93250	0.966250	−	0.257608i	$-0.0829342\pi$
$431$	40.1197	1.93250	0.966250	+	0.257608i	$0.0829342\pi$
$432$	0	0
$433$	38.8560	1.86730	0.933649	−	0.358189i	$-0.116606\pi$
$433$	38.8560	1.86730	0.933649	+	0.358189i	$0.116606\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	46.4082i	2.22001i
$438$	0	0
$439$	15.0055	0.716173	0.358086	−	0.933688i	$-0.383429\pi$
$439$	15.0055	0.716173	0.358086	+	0.933688i	$0.383429\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 7.84122i	− 0.372548i	−0.982498	−	0.186274i	$-0.940359\pi$
$443$	− 7.84122i	− 0.372548i	0.982498	−	0.186274i	$-0.0596412\pi$
$444$	0	0
$445$	1.84173i	0.0873065i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−30.3904	−1.43421	−0.717107	−	0.696963i	$-0.754534\pi$
$449$	−30.3904	−1.43421	−0.717107	+	0.696963i	$0.754534\pi$
$450$	0	0
$451$	− 2.83003i	− 0.133261i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.25198	−0.105574
$456$	0	0
$457$	38.3253	1.79278	0.896391	−	0.443264i	$-0.146180\pi$
$457$	38.3253	1.79278	0.896391	+	0.443264i	$0.146180\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 20.9345i	− 0.975018i	−0.873118	−	0.487509i	$-0.837906\pi$
$461$	− 20.9345i	− 0.975018i	0.873118	−	0.487509i	$-0.162094\pi$
$462$	0	0
$463$	−3.44920	−0.160298	−0.0801491	−	0.996783i	$-0.525540\pi$
$463$	−3.44920	−0.160298	−0.0801491	+	0.996783i	$0.525540\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.0217i	− 0.833945i	−0.908919	−	0.416973i	$-0.863091\pi$
$467$	− 18.0217i	− 0.833945i	0.908919	−	0.416973i	$-0.136909\pi$
$468$	0	0
$469$	15.6441i	0.722378i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.83003	0.130125
$474$	0	0
$475$	− 0.640492i	− 0.0293878i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.70503	0.352052	0.176026	−	0.984386i	$-0.443676\pi$
$479$	7.70503	0.352052	0.176026	+	0.984386i	$0.443676\pi$
$480$	0	0
$481$	−7.76860	−0.354218
$482$	0	0
$483$	0	0
$484$	0	0
$485$	33.3206i	1.51301i
$486$	0	0
$487$	17.5367	0.794666	0.397333	−	0.917675i	$-0.369936\pi$
$487$	17.5367	0.794666	0.397333	+	0.917675i	$0.369936\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 33.7842i	− 1.52466i	−0.647189	−	0.762329i	$-0.724056\pi$
$491$	− 33.7842i	− 1.52466i	0.647189	−	0.762329i	$-0.275944\pi$
$492$	0	0
$493$	− 40.7981i	− 1.83745i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.23101	−0.144930
$498$	0	0
$499$	− 21.9959i	− 0.984672i	−0.870405	−	0.492336i	$-0.836143\pi$
$499$	− 21.9959i	− 0.984672i	0.870405	−	0.492336i	$-0.163857\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.6750	−1.27855	−0.639277	−	0.768977i	$-0.720766\pi$
$503$	−28.6750	−1.27855	−0.639277	+	0.768977i	$0.720766\pi$
$504$	0	0
$505$	11.6174	0.516966
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 19.0044i	− 0.842355i	−0.906978	−	0.421177i	$-0.861617\pi$
$509$	− 19.0044i	− 0.842355i	0.906978	−	0.421177i	$-0.138383\pi$
$510$	0	0
$511$	11.3698	0.502971
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25.2532i	1.11279i
$516$	0	0
$517$	− 7.22017i	− 0.317543i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.85153	−0.300171	−0.150086	−	0.988673i	$-0.547955\pi$
$521$	−6.85153	−0.300171	−0.150086	+	0.988673i	$0.547955\pi$
$522$	0	0
$523$	− 6.45353i	− 0.282193i	−0.989996	−	0.141097i	$-0.954937\pi$
$523$	− 6.45353i	− 0.282193i	0.989996	−	0.141097i	$-0.0450628\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−41.6175	−1.81289
$528$	0	0
$529$	39.3871	1.71248
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.28385i	0.0989247i
$534$	0	0
$535$	19.3040	0.834584
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.40788i	0.232934i
$540$	0	0
$541$	− 16.9996i	− 0.730869i	−0.930837	−	0.365435i	$-0.880920\pi$
$541$	− 16.9996i	− 0.730869i	0.930837	−	0.365435i	$-0.119080\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19.3098	−0.827143
$546$	0	0
$547$	− 3.77902i	− 0.161579i	−0.996731	−	0.0807896i	$-0.974256\pi$
$547$	− 3.77902i	− 0.161579i	0.996731	−	0.0807896i	$-0.0257442\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−42.1282	−1.79472
$552$	0	0
$553$	2.97223	0.126392
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.1693i	− 1.19357i	−0.802401	−	0.596785i	$-0.796445\pi$
$557$	− 28.1693i	− 1.19357i	0.802401	−	0.596785i	$-0.203555\pi$
$558$	0	0
$559$	−2.28385	−0.0965967
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.7142i	0.830853i	0.909627	+	0.415427i	$0.136368\pi$
$563$	19.7142i	0.830853i	−0.909627	+	0.415427i	$0.863632\pi$
$564$	0	0
$565$	39.0420i	1.64251i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.16301	0.300289	0.150145	−	0.988664i	$-0.452026\pi$
$569$	7.16301	0.300289	0.150145	+	0.988664i	$0.452026\pi$
$570$	0	0
$571$	16.1436i	0.675588i	0.941220	+	0.337794i	$0.109681\pi$
$571$	16.1436i	0.675588i	−0.941220	+	0.337794i	$0.890319\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.861021	−0.0359071
$576$	0	0
$577$	21.6002	0.899226	0.449613	−	0.893223i	$-0.351562\pi$
$577$	21.6002	0.899226	0.449613	+	0.893223i	$0.351562\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 3.15764i	− 0.131001i
$582$	0	0
$583$	−3.01732	−0.124964
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.90577i	0.202483i	0.994862	+	0.101241i	$0.0322814\pi$
$587$	4.90577i	0.202483i	−0.994862	+	0.101241i	$0.967719\pi$
$588$	0	0
$589$	42.9743i	1.77072i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	40.5123	1.66364	0.831820	−	0.555046i	$-0.187299\pi$
$593$	40.5123	1.66364	0.831820	+	0.555046i	$0.187299\pi$
$594$	0	0
$595$	15.8782i	0.650942i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.2021	0.416845	0.208423	−	0.978039i	$-0.433167\pi$
$599$	10.2021	0.416845	0.208423	+	0.978039i	$0.433167\pi$
$600$	0	0
$601$	−15.0279	−0.613003	−0.306501	−	0.951870i	$-0.599158\pi$
$601$	−15.0279	−0.613003	−0.306501	+	0.951870i	$0.599158\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.21156i	0.0899126i
$606$	0	0
$607$	43.6279	1.77080	0.885400	−	0.464830i	$-0.153884\pi$
$607$	43.6279	1.77080	0.885400	+	0.464830i	$0.153884\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.82674i	0.235725i
$612$	0	0
$613$	− 12.8628i	− 0.519522i	−0.965673	−	0.259761i	$-0.916356\pi$
$613$	− 12.8628i	− 0.519522i	0.965673	−	0.259761i	$-0.0836438\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.2341	−0.532785	−0.266392	−	0.963865i	$-0.585832\pi$
$617$	−13.2341	−0.532785	−0.266392	+	0.963865i	$0.585832\pi$
$618$	0	0
$619$	41.8398i	1.68168i	0.541281	+	0.840842i	$0.317939\pi$
$619$	41.8398i	1.68168i	−0.541281	+	0.840842i	$0.682061\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.05079	0.0420990
$624$	0	0
$625$	−24.4431	−0.977723
$626$	0	0
$627$	0	0
$628$	0	0
$629$	54.7747i	2.18401i
$630$	0	0
$631$	0.527116	0.0209842	0.0104921	−	0.999945i	$-0.496660\pi$
$631$	0.527116	0.0209842	0.0104921	+	0.999945i	$0.496660\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	36.2867i	1.43999i
$636$	0	0
$637$	− 4.36420i	− 0.172916i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.0689	0.911168	0.455584	−	0.890193i	$-0.349430\pi$
$641$	23.0689	0.911168	0.455584	+	0.890193i	$0.349430\pi$
$642$	0	0
$643$	0.368023i	0.0145134i	0.999974	+	0.00725670i	$0.00230990\pi$
$643$	0.368023i	0.0145134i	−0.999974	+	0.00725670i	$0.997690\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.2272	−1.14904	−0.574520	−	0.818491i	$-0.694811\pi$
$647$	−29.2272	−1.14904	−0.574520	+	0.818491i	$0.694811\pi$
$648$	0	0
$649$	3.32934	0.130688
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 42.3725i	− 1.65817i	−0.559126	−	0.829083i	$-0.688863\pi$
$653$	− 42.3725i	− 1.65817i	0.559126	−	0.829083i	$-0.311137\pi$
$654$	0	0
$655$	11.0767	0.432802
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.2808i	0.828981i	0.910054	+	0.414490i	$0.136040\pi$
$659$	21.2808i	0.828981i	−0.910054	+	0.414490i	$0.863960\pi$
$660$	0	0
$661$	− 35.7298i	− 1.38973i	−0.719142	−	0.694863i	$-0.755465\pi$
$661$	− 35.7298i	− 1.38973i	0.719142	−	0.694863i	$-0.244535\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.3958	0.635803
$666$	0	0
$667$	56.6334i	2.19285i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.97382	−0.346430
$672$	0	0
$673$	9.20425	0.354798	0.177399	−	0.984139i	$-0.443232\pi$
$673$	9.20425	0.354798	0.177399	+	0.984139i	$0.443232\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.93341i	− 0.0743069i	−0.999310	−	0.0371534i	$-0.988171\pi$
$677$	− 1.93341i	− 0.0743069i	0.999310	−	0.0371534i	$-0.0118290\pi$
$678$	0	0
$679$	19.0109	0.729570
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.4015i	1.58419i	0.610401	+	0.792093i	$0.291008\pi$
$683$	41.4015i	1.58419i	−0.610401	+	0.792093i	$0.708992\pi$
$684$	0	0
$685$	− 19.2529i	− 0.735614i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.43500	0.0927660
$690$	0	0
$691$	17.0431i	0.648350i	0.945997	+	0.324175i	$0.105087\pi$
$691$	17.0431i	0.648350i	−0.945997	+	0.324175i	$0.894913\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.2138	−0.652958
$696$	0	0
$697$	16.1030	0.609943
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25.6959i	0.970520i	0.874370	+	0.485260i	$0.161275\pi$
$701$	25.6959i	0.970520i	−0.874370	+	0.485260i	$0.838725\pi$
$702$	0	0
$703$	56.5604	2.13322
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.62821i	− 0.249280i
$708$	0	0
$709$	20.8903i	0.784551i	0.919848	+	0.392276i	$0.128312\pi$
$709$	20.8903i	0.784551i	−0.919848	+	0.392276i	$0.871688\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	57.7709	2.16354
$714$	0	0
$715$	− 1.78474i	− 0.0667457i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	38.9407	1.45224	0.726121	−	0.687567i	$-0.241321\pi$
$719$	38.9407	1.45224	0.726121	+	0.687567i	$0.241321\pi$
$720$	0	0
$721$	14.4081	0.536584
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 0.781612i	− 0.0290284i
$726$	0	0
$727$	20.4422	0.758160	0.379080	−	0.925364i	$-0.376241\pi$
$727$	20.4422	0.758160	0.379080	+	0.925364i	$0.376241\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.1030i	0.595589i
$732$	0	0
$733$	− 45.3953i	− 1.67671i	−0.545122	−	0.838357i	$-0.683517\pi$
$733$	− 45.3953i	− 1.67671i	0.545122	−	0.838357i	$-0.316483\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.3983	−0.456699
$738$	0	0
$739$	31.4387i	1.15649i	0.815863	+	0.578246i	$0.196262\pi$
$739$	31.4387i	1.15649i	−0.815863	+	0.578246i	$0.803738\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.1059	0.590868	0.295434	−	0.955363i	$-0.404536\pi$
$743$	16.1059	0.590868	0.295434	+	0.955363i	$0.404536\pi$
$744$	0	0
$745$	16.5614	0.606764
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 11.0138i	− 0.402434i
$750$	0	0
$751$	−12.4896	−0.455751	−0.227876	−	0.973690i	$-0.573178\pi$
$751$	−12.4896	−0.455751	−0.227876	+	0.973690i	$0.573178\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 7.69580i	− 0.280079i
$756$	0	0
$757$	48.8353i	1.77495i	0.460855	+	0.887475i	$0.347543\pi$
$757$	48.8353i	1.77495i	−0.460855	+	0.887475i	$0.652457\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.8577	−1.11859	−0.559296	−	0.828968i	$-0.688928\pi$
$761$	−30.8577	−1.11859	−0.559296	+	0.828968i	$0.688928\pi$
$762$	0	0
$763$	11.0171i	0.398846i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.68680	−0.0970148
$768$	0	0
$769$	0.0363100	0.00130937	0.000654686	−	1.00000i	$-0.499792\pi$
$769$	0.0363100	0.00130937	0.000654686		1.00000i	$0.499792\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 40.8771i	− 1.47025i	−0.677934	−	0.735123i	$-0.737125\pi$
$773$	− 40.8771i	− 1.47025i	0.677934	−	0.735123i	$-0.262875\pi$
$774$	0	0
$775$	−0.797311	−0.0286402
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 16.6279i	− 0.595757i
$780$	0	0
$781$	− 2.56065i	− 0.0916273i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.74901	0.0981163
$786$	0	0
$787$	− 30.8252i	− 1.09880i	−0.835559	−	0.549400i	$-0.814856\pi$
$787$	− 30.8252i	− 1.09880i	0.835559	−	0.549400i	$-0.185144\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22.2752	0.792014
$792$	0	0
$793$	7.24194	0.257169
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.1946i	− 0.679906i	−0.940442	−	0.339953i	$-0.889589\pi$
$797$	− 19.1946i	− 0.679906i	0.940442	−	0.339953i	$-0.110411\pi$
$798$	0	0
$799$	41.0830	1.45341
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.01085i	0.317986i
$804$	0	0
$805$	− 22.0411i	− 0.776847i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.9454	0.736402	0.368201	−	0.929746i	$-0.379974\pi$
$809$	20.9454	0.736402	0.368201	+	0.929746i	$0.379974\pi$
$810$	0	0
$811$	− 50.7579i	− 1.78235i	−0.453660	−	0.891175i	$-0.649882\pi$
$811$	− 50.7579i	− 1.78235i	0.453660	−	0.891175i	$-0.350118\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.37801	−0.0832980
$816$	0	0
$817$	16.6279	0.581737
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.43224i	0.119786i	0.998205	+	0.0598930i	$0.0190760\pi$
$821$	3.43224i	0.119786i	−0.998205	+	0.0598930i	$0.980924\pi$
$822$	0	0
$823$	26.7144	0.931205	0.465603	−	0.884994i	$-0.345838\pi$
$823$	26.7144	0.931205	0.465603	+	0.884994i	$0.345838\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.7010i	− 1.06758i	−0.845617	−	0.533789i	$-0.820768\pi$
$827$	− 30.7010i	− 1.06758i	0.845617	−	0.533789i	$-0.179232\pi$
$828$	0	0
$829$	16.1045i	0.559332i	0.960097	+	0.279666i	$0.0902237\pi$
$829$	16.1045i	0.559332i	−0.960097	+	0.279666i	$0.909776\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−30.7710	−1.06615
$834$	0	0
$835$	− 2.51558i	− 0.0870551i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−33.3502	−1.15138	−0.575689	−	0.817669i	$-0.695266\pi$
$839$	−33.3502	−1.15138	−0.575689	+	0.817669i	$0.695266\pi$
$840$	0	0
$841$	−22.4103	−0.772769
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 27.3100i	− 0.939491i
$846$	0	0
$847$	1.26179	0.0433557
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 76.0349i	− 2.60644i
$852$	0	0
$853$	41.2723i	1.41314i	0.707645	+	0.706568i	$0.249758\pi$
$853$	41.2723i	1.41314i	−0.707645	+	0.706568i	$0.750242\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.90574	−0.270055	−0.135027	−	0.990842i	$-0.543112\pi$
$857$	−7.90574	−0.270055	−0.135027	+	0.990842i	$0.543112\pi$
$858$	0	0
$859$	− 27.4966i	− 0.938173i	−0.883152	−	0.469087i	$-0.844583\pi$
$859$	− 27.4966i	− 0.938173i	0.883152	−	0.469087i	$-0.155417\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.0970	−0.377748	−0.188874	−	0.982001i	$-0.560484\pi$
$863$	−11.0970	−0.377748	−0.188874	+	0.982001i	$0.560484\pi$
$864$	0	0
$865$	−5.39934	−0.183583
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.35557i	0.0799072i
$870$	0	0
$871$	10.0056	0.339026
$872$	0	0
$873$	0	0
$874$	0	0
$875$	14.2568i	0.481968i
$876$	0	0
$877$	− 20.5216i	− 0.692967i	−0.938056	−	0.346483i	$-0.887376\pi$
$877$	− 20.5216i	− 0.692967i	0.938056	−	0.346483i	$-0.112624\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.4013	−0.417811	−0.208906	−	0.977936i	$-0.566990\pi$
$881$	−12.4013	−0.417811	−0.208906	+	0.977936i	$0.566990\pi$
$882$	0	0
$883$	12.7693i	0.429723i	0.976645	+	0.214861i	$0.0689300\pi$
$883$	12.7693i	0.429723i	−0.976645	+	0.214861i	$0.931070\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.3628	0.448680	0.224340	−	0.974511i	$-0.427977\pi$
$887$	13.3628	0.448680	0.224340	+	0.974511i	$0.427977\pi$
$888$	0	0
$889$	20.7031	0.694361
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 42.4224i	− 1.41961i
$894$	0	0
$895$	−55.9234	−1.86931
$896$	0	0
$897$	0	0
$898$	0	0
$899$	52.4429i	1.74907i
$900$	0	0
$901$	− 17.1686i	− 0.571970i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.57705	−0.285111
$906$	0	0
$907$	35.9604i	1.19405i	0.802224	+	0.597024i	$0.203650\pi$
$907$	35.9604i	1.19405i	−0.802224	+	0.597024i	$0.796350\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.88541	0.194992	0.0974962	−	0.995236i	$-0.468917\pi$
$911$	5.88541	0.194992	0.0974962	+	0.995236i	$0.468917\pi$
$912$	0	0
$913$	2.50251	0.0828208
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.31974i	− 0.208696i
$918$	0	0
$919$	0.160480	0.00529376	0.00264688	−	0.999996i	$-0.499157\pi$
$919$	0.160480	0.00529376	0.00264688	+	0.999996i	$0.499157\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.06647i	0.0680185i
$924$	0	0
$925$	1.04938i	0.0345033i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.22825	0.0402975	0.0201488	−	0.999797i	$-0.493586\pi$
$929$	1.22825	0.0402975	0.0201488	+	0.999797i	$0.493586\pi$
$930$	0	0
$931$	31.7742i	1.04136i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.5838	−0.411536
$936$	0	0
$937$	−15.4915	−0.506086	−0.253043	−	0.967455i	$-0.581431\pi$
$937$	−15.4915	−0.506086	−0.253043	+	0.967455i	$0.581431\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 35.1914i	− 1.14721i	−0.819134	−	0.573603i	$-0.805545\pi$
$941$	− 35.1914i	− 1.14721i	0.819134	−	0.573603i	$-0.194455\pi$
$942$	0	0
$943$	−22.3531	−0.727918
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.6971i	0.445095i	0.974922	+	0.222548i	$0.0714373\pi$
$947$	13.6971i	0.445095i	−0.974922	+	0.222548i	$0.928563\pi$
$948$	0	0
$949$	− 7.27183i	− 0.236054i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.3673	0.335830	0.167915	−	0.985801i	$-0.446297\pi$
$953$	10.3673	0.335830	0.167915	+	0.985801i	$0.446297\pi$
$954$	0	0
$955$	− 8.82886i	− 0.285695i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.9846	−0.354711
$960$	0	0
$961$	22.4962	0.725682
$962$	0	0
$963$	0	0
$964$	0	0
$965$	51.9954i	1.67379i
$966$	0	0
$967$	−31.1781	−1.00262	−0.501310	−	0.865268i	$-0.667149\pi$
$967$	−31.1781	−1.00262	−0.501310	+	0.865268i	$0.667149\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 10.8679i	− 0.348769i	−0.984678	−	0.174384i	$-0.944206\pi$
$971$	− 10.8679i	− 0.348769i	0.984678	−	0.174384i	$-0.0557935\pi$
$972$	0	0
$973$	9.82124i	0.314855i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.6984	0.854157	0.427079	−	0.904215i	$-0.359543\pi$
$977$	26.6984	0.854157	0.427079	+	0.904215i	$0.359543\pi$
$978$	0	0
$979$	0.832776i	0.0266156i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.3201	−0.616216	−0.308108	−	0.951351i	$-0.599696\pi$
$983$	−19.3201	−0.616216	−0.308108	+	0.951351i	$0.599696\pi$
$984$	0	0
$985$	31.4651	1.00256
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 22.3531i	− 0.710788i
$990$	0	0
$991$	−8.28865	−0.263298	−0.131649	−	0.991296i	$-0.542027\pi$
$991$	−8.28865	−0.263298	−0.131649	+	0.991296i	$0.542027\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	35.7395i	1.13302i
$996$	0	0
$997$	− 34.9020i	− 1.10536i	−0.833394	−	0.552679i	$-0.813606\pi$
$997$	− 34.9020i	− 1.10536i	0.833394	−	0.552679i	$-0.186394\pi$
$998$	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.1585.6		20
3.2	odd	2	inner	3168.2.f.h.1585.16		20
4.3	odd	2		792.2.f.h.397.12	yes	20
8.3	odd	2		792.2.f.h.397.11	yes	20
8.5	even	2	inner	3168.2.f.h.1585.15		20
12.11	even	2		792.2.f.h.397.9	✓	20
24.5	odd	2	inner	3168.2.f.h.1585.5		20
24.11	even	2		792.2.f.h.397.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.f.h.397.9	✓	20	12.11	even	2
792.2.f.h.397.10	yes	20	24.11	even	2
792.2.f.h.397.11	yes	20	8.3	odd	2
792.2.f.h.397.12	yes	20	4.3	odd	2
3168.2.f.h.1585.5		20	24.5	odd	2	inner
3168.2.f.h.1585.6		20	1.1	even	1	trivial
3168.2.f.h.1585.15		20	8.5	even	2	inner
3168.2.f.h.1585.16		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)

\(f(q)\)	\(=\)	\(q-2.21156i q^{5} -1.26179 q^{7} -1.00000i q^{11} +0.807008i q^{13} +5.69004 q^{17} -5.87553i q^{19} -7.89855 q^{23} +0.109010 q^{25} -7.17010i q^{29} -7.31411 q^{31} +2.79052i q^{35} +9.62643i q^{37} +2.83003 q^{41} +2.83003i q^{43} +7.22017 q^{47} -5.40788 q^{49} -3.01732i q^{53} -2.21156 q^{55} +3.32934i q^{59} -8.97382i q^{61} +1.78474 q^{65} -12.3983i q^{67} +2.56065 q^{71} -9.01085 q^{73} +1.26179i q^{77} -2.35557 q^{79} +2.50251i q^{83} -12.5838i q^{85} -0.832776 q^{89} -1.01828i q^{91} -12.9941 q^{95} -15.0666 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{25} - 40 q^{31} + 36 q^{49} + 16 q^{55} - 56 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) 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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