LMFDB - Embedded newform 6336.2.d.g.3455.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(3455,6336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 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("6336.3455"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,0,0,-10,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.5236158660608.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{8} - 4x^{7} + 3x^{6} + 8x^{5} + 6x^{4} - 16x^{3} - 16x^{2} + 32 $ x^10 - 2x^8 - 4x^7 + 3x^6 + 8x^5 + 6x^4 - 16x^3 - 16*x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 396)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3455.8
Root			$-0.382373 - 1.36154i$ of defining polynomial
Character	$\chi$	$=$	6336.3455
Dual form			6336.2.d.g.3455.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05483i q^{5} +4.18915i q^{7} -1.00000 q^{11} -3.27394 q^{13} +0.332700i q^{17} -8.47431i q^{19} +3.52212 q^{23} +0.777673 q^{25} -9.52712i q^{29} -3.95895i q^{31} -8.60800 q^{35} +1.37648 q^{37} -5.42788i q^{41} +1.53622i q^{43} -6.93286 q^{47} -10.5490 q^{49} +8.83749i q^{53} -2.05483i q^{55} -3.70203 q^{59} -14.8572 q^{61} -6.72739i q^{65} -6.32795i q^{67} +5.77871 q^{71} -8.72042 q^{73} -4.18915i q^{77} -13.2903i q^{79} +0.984127 q^{83} -0.683642 q^{85} +1.46267i q^{89} -13.7150i q^{91} +17.4133 q^{95} +5.99367 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{11} - 8 q^{13} + 8 q^{23} - 10 q^{25} + 16 q^{35} + 4 q^{37} - 34 q^{49} + 40 q^{59} - 20 q^{61} + 16 q^{71} + 24 q^{73} + 8 q^{83} + 36 q^{85} + 96 q^{95} + 24 q^{97}+O(q^{100}) $ 10 * q - 10 * q^11 - 8 * q^13 + 8 * q^23 - 10 * q^25 + 16 * q^35 + 4 * q^37 - 34 * q^49 + 40 * q^59 - 20 * q^61 + 16 * q^71 + 24 * q^73 + 8 * q^83 + 36 * q^85 + 96 * q^95 + 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times$.

$n$	$1729$	$3521$	$4159$	$4357$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	2.05483i	0.918948i	0.888191	+	0.459474i	$0.151962\pi$
$5$	2.05483i	0.918948i	−0.888191	+	0.459474i	$0.848038\pi$
$6$	0	0
$7$	4.18915i	1.58335i	0.610941	+	0.791676i	$0.290791\pi$
$7$	4.18915i	1.58335i	−0.610941	+	0.791676i	$0.709209\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.27394	−0.908028	−0.454014	−	0.890995i	$-0.650008\pi$
$13$	−3.27394	−0.908028	−0.454014	+	0.890995i	$0.650008\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.332700i	0.0806916i	0.999186	+	0.0403458i	$0.0128460\pi$
$17$	0.332700i	0.0806916i	−0.999186	+	0.0403458i	$0.987154\pi$
$18$	0	0
$19$	− 8.47431i	− 1.94414i	−0.234693	−	0.972070i	$-0.575408\pi$
$19$	− 8.47431i	− 1.94414i	0.234693	−	0.972070i	$-0.424592\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.52212	0.734413	0.367207	−	0.930139i	$-0.380314\pi$
$23$	3.52212	0.734413	0.367207	+	0.930139i	$0.380314\pi$
$24$	0	0
$25$	0.777673	0.155535
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.52712i	− 1.76914i	−0.466406	−	0.884571i	$-0.654451\pi$
$29$	− 9.52712i	− 1.76914i	0.466406	−	0.884571i	$-0.345549\pi$
$30$	0	0
$31$	− 3.95895i	− 0.711049i	−0.934667	−	0.355524i	$-0.884302\pi$
$31$	− 3.95895i	− 0.711049i	0.934667	−	0.355524i	$-0.115698\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.60800	−1.45502
$36$	0	0
$37$	1.37648	0.226291	0.113146	−	0.993578i	$-0.463907\pi$
$37$	1.37648	0.226291	0.113146	+	0.993578i	$0.463907\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.42788i	− 0.847692i	−0.905734	−	0.423846i	$-0.860680\pi$
$41$	− 5.42788i	− 0.847692i	0.905734	−	0.423846i	$-0.139320\pi$
$42$	0	0
$43$	1.53622i	0.234271i	0.993116	+	0.117136i	$0.0373712\pi$
$43$	1.53622i	0.234271i	−0.993116	+	0.117136i	$0.962629\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.93286	−1.01126	−0.505631	−	0.862750i	$-0.668740\pi$
$47$	−6.93286	−1.01126	−0.505631	+	0.862750i	$0.668740\pi$
$48$	0	0
$49$	−10.5490	−1.50700
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.83749i	1.21392i	0.794731	+	0.606961i	$0.207612\pi$
$53$	8.83749i	1.21392i	−0.794731	+	0.606961i	$0.792388\pi$
$54$	0	0
$55$	− 2.05483i	− 0.277073i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.70203	−0.481964	−0.240982	−	0.970530i	$-0.577469\pi$
$59$	−3.70203	−0.481964	−0.240982	+	0.970530i	$0.577469\pi$
$60$	0	0
$61$	−14.8572	−1.90227	−0.951136	−	0.308771i	$-0.900082\pi$
$61$	−14.8572	−1.90227	−0.951136	+	0.308771i	$0.900082\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.72739i	− 0.834430i
$66$	0	0
$67$	− 6.32795i	− 0.773083i	−0.922272	−	0.386541i	$-0.873670\pi$
$67$	− 6.32795i	− 0.773083i	0.922272	−	0.386541i	$-0.126330\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.77871	0.685807	0.342904	−	0.939371i	$-0.388590\pi$
$71$	5.77871	0.685807	0.342904	+	0.939371i	$0.388590\pi$
$72$	0	0
$73$	−8.72042	−1.02065	−0.510324	−	0.859982i	$-0.670475\pi$
$73$	−8.72042	−1.02065	−0.510324	+	0.859982i	$0.670475\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.18915i	− 0.477399i
$78$	0	0
$79$	− 13.2903i	− 1.49527i	−0.664109	−	0.747636i	$-0.731189\pi$
$79$	− 13.2903i	− 1.49527i	0.664109	−	0.747636i	$-0.268811\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.984127	0.108022	0.0540110	−	0.998540i	$-0.482799\pi$
$83$	0.984127	0.108022	0.0540110	+	0.998540i	$0.482799\pi$
$84$	0	0
$85$	−0.683642	−0.0741514
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.46267i	0.155043i	0.996991	+	0.0775214i	$0.0247006\pi$
$89$	1.46267i	0.155043i	−0.996991	+	0.0775214i	$0.975299\pi$
$90$	0	0
$91$	− 13.7150i	− 1.43773i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	17.4133	1.78656
$96$	0	0
$97$	5.99367	0.608565	0.304283	−	0.952582i	$-0.401583\pi$
$97$	5.99367	0.608565	0.304283	+	0.952582i	$0.401583\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5.11346i	− 0.508808i	−0.967098	−	0.254404i	$-0.918121\pi$
$101$	− 5.11346i	− 0.508808i	0.967098	−	0.254404i	$-0.0818793\pi$
$102$	0	0
$103$	0.816108i	0.0804135i	0.999191	+	0.0402067i	$0.0128017\pi$
$103$	0.816108i	0.0804135i	−0.999191	+	0.0402067i	$0.987198\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1725	1.37011	0.685056	−	0.728491i	$-0.259778\pi$
$107$	14.1725	1.37011	0.685056	+	0.728491i	$0.259778\pi$
$108$	0	0
$109$	2.61615	0.250582	0.125291	−	0.992120i	$-0.460014\pi$
$109$	2.61615	0.250582	0.125291	+	0.992120i	$0.460014\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.81271i	0.829030i	0.910043	+	0.414515i	$0.136049\pi$
$113$	8.81271i	0.829030i	−0.910043	+	0.414515i	$0.863951\pi$
$114$	0	0
$115$	7.23736i	0.674888i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.39373	−0.127763
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.8721i	1.06188i
$126$	0	0
$127$	− 18.0169i	− 1.59874i	−0.600840	−	0.799369i	$-0.705167\pi$
$127$	− 18.0169i	− 1.59874i	0.600840	−	0.799369i	$-0.294833\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.0276	1.40034	0.700168	−	0.713978i	$-0.253108\pi$
$131$	16.0276	1.40034	0.700168	+	0.713978i	$0.253108\pi$
$132$	0	0
$133$	35.5002	3.07826
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.1920i	− 0.870758i	−0.900247	−	0.435379i	$-0.856614\pi$
$137$	− 10.1920i	− 0.870758i	0.900247	−	0.435379i	$-0.143386\pi$
$138$	0	0
$139$	1.58468i	0.134410i	0.997739	+	0.0672052i	$0.0214082\pi$
$139$	1.58468i	0.134410i	−0.997739	+	0.0672052i	$0.978592\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.27394	0.273781
$144$	0	0
$145$	19.5766	1.62575
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.05740i	− 0.250472i	−0.992127	−	0.125236i	$-0.960031\pi$
$149$	− 3.05740i	− 0.250472i	0.992127	−	0.125236i	$-0.0399689\pi$
$150$	0	0
$151$	− 16.3143i	− 1.32763i	−0.747895	−	0.663817i	$-0.768935\pi$
$151$	− 16.3143i	− 1.32763i	0.747895	−	0.663817i	$-0.231065\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.13497	0.653417
$156$	0	0
$157$	−2.29129	−0.182865	−0.0914325	−	0.995811i	$-0.529145\pi$
$157$	−2.29129	−0.182865	−0.0914325	+	0.995811i	$0.529145\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	14.7547i	1.16283i
$162$	0	0
$163$	− 0.530812i	− 0.0415764i	−0.999784	−	0.0207882i	$-0.993382\pi$
$163$	− 0.530812i	− 0.0415764i	0.999784	−	0.0207882i	$-0.00661756\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.0224	−1.31723	−0.658616	−	0.752479i	$-0.728858\pi$
$167$	−17.0224	−1.31723	−0.658616	+	0.752479i	$0.728858\pi$
$168$	0	0
$169$	−2.28131	−0.175485
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 5.38984i	− 0.409782i	−0.978785	−	0.204891i	$-0.934316\pi$
$173$	− 5.38984i	− 0.409782i	0.978785	−	0.204891i	$-0.0656840\pi$
$174$	0	0
$175$	3.25779i	0.246266i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.0619	−1.20052	−0.600260	−	0.799805i	$-0.704936\pi$
$179$	−16.0619	−1.20052	−0.600260	+	0.799805i	$0.704936\pi$
$180$	0	0
$181$	−7.65892	−0.569283	−0.284642	−	0.958634i	$-0.591875\pi$
$181$	−7.65892	−0.569283	−0.284642	+	0.958634i	$0.591875\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.82843i	0.207950i
$186$	0	0
$187$	− 0.332700i	− 0.0243294i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.1828	−0.953872	−0.476936	−	0.878938i	$-0.658253\pi$
$191$	−13.1828	−0.953872	−0.476936	+	0.878938i	$0.658253\pi$
$192$	0	0
$193$	17.5508	1.26333	0.631665	−	0.775241i	$-0.282372\pi$
$193$	17.5508	1.26333	0.631665	+	0.775241i	$0.282372\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.6432i	0.829541i	0.909926	+	0.414770i	$0.136138\pi$
$197$	11.6432i	0.829541i	−0.909926	+	0.414770i	$0.863862\pi$
$198$	0	0
$199$	5.34814i	0.379119i	0.981869	+	0.189560i	$0.0607060\pi$
$199$	5.34814i	0.379119i	−0.981869	+	0.189560i	$0.939294\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	39.9106	2.80117
$204$	0	0
$205$	11.1534	0.778985
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.47431i	0.586180i
$210$	0	0
$211$	− 17.3016i	− 1.19109i	−0.803321	−	0.595546i	$-0.796936\pi$
$211$	− 17.3016i	− 1.19109i	0.803321	−	0.595546i	$-0.203064\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.15667	−0.215283
$216$	0	0
$217$	16.5847	1.12584
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.08924i	− 0.0732702i
$222$	0	0
$223$	− 22.3672i	− 1.49782i	−0.662674	−	0.748908i	$-0.730579\pi$
$223$	− 22.3672i	− 1.49782i	0.662674	−	0.748908i	$-0.269421\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.0898	1.73164	0.865820	−	0.500355i	$-0.166797\pi$
$227$	26.0898	1.73164	0.865820	+	0.500355i	$0.166797\pi$
$228$	0	0
$229$	9.87457	0.652530	0.326265	−	0.945278i	$-0.394210\pi$
$229$	9.87457	0.652530	0.326265	+	0.945278i	$0.394210\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.6093i	1.54669i	0.633983	+	0.773347i	$0.281419\pi$
$233$	23.6093i	1.54669i	−0.633983	+	0.773347i	$0.718581\pi$
$234$	0	0
$235$	− 14.2459i	− 0.929297i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.88758	0.251466	0.125733	−	0.992064i	$-0.459872\pi$
$239$	3.88758	0.251466	0.125733	+	0.992064i	$0.459872\pi$
$240$	0	0
$241$	−22.7455	−1.46517	−0.732583	−	0.680678i	$-0.761685\pi$
$241$	−22.7455	−1.46517	−0.732583	+	0.680678i	$0.761685\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 21.6764i	− 1.38486i
$246$	0	0
$247$	27.7444i	1.76533i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.93642	−0.311584	−0.155792	−	0.987790i	$-0.549793\pi$
$251$	−4.93642	−0.311584	−0.155792	+	0.987790i	$0.549793\pi$
$252$	0	0
$253$	−3.52212	−0.221434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.6561i	1.66276i	0.555702	+	0.831381i	$0.312449\pi$
$257$	26.6561i	1.66276i	−0.555702	+	0.831381i	$0.687551\pi$
$258$	0	0
$259$	5.76628i	0.358299i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.7901	1.22031	0.610155	−	0.792282i	$-0.291107\pi$
$263$	19.7901	1.22031	0.610155	+	0.792282i	$0.291107\pi$
$264$	0	0
$265$	−18.1595	−1.11553
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.1510i	0.984741i	0.870386	+	0.492371i	$0.163870\pi$
$269$	16.1510i	0.984741i	−0.870386	+	0.492371i	$0.836130\pi$
$270$	0	0
$271$	0.854317i	0.0518961i	0.999663	+	0.0259480i	$0.00826045\pi$
$271$	0.854317i	0.0518961i	−0.999663	+	0.0259480i	$0.991740\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.777673	−0.0468955
$276$	0	0
$277$	−22.4384	−1.34819	−0.674097	−	0.738643i	$-0.735467\pi$
$277$	−22.4384	−1.34819	−0.674097	+	0.738643i	$0.735467\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.7617i	− 1.65612i	−0.560637	−	0.828062i	$-0.689444\pi$
$281$	− 27.7617i	− 1.65612i	0.560637	−	0.828062i	$-0.310556\pi$
$282$	0	0
$283$	13.7582i	0.817842i	0.912570	+	0.408921i	$0.134095\pi$
$283$	13.7582i	0.817842i	−0.912570	+	0.408921i	$0.865905\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22.7382	1.34219
$288$	0	0
$289$	16.8893	0.993489
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.5988i	1.37866i	0.724448	+	0.689330i	$0.242095\pi$
$293$	23.5988i	1.37866i	−0.724448	+	0.689330i	$0.757905\pi$
$294$	0	0
$295$	− 7.60705i	− 0.442900i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.5312	−0.666868
$300$	0	0
$301$	−6.43546	−0.370934
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 30.5291i	− 1.74809i
$306$	0	0
$307$	− 4.42205i	− 0.252380i	−0.992006	−	0.126190i	$-0.959725\pi$
$307$	− 4.42205i	− 0.252380i	0.992006	−	0.126190i	$-0.0402749\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.2418	1.60144	0.800722	−	0.599037i	$-0.204450\pi$
$311$	28.2418	1.60144	0.800722	+	0.599037i	$0.204450\pi$
$312$	0	0
$313$	−5.28764	−0.298875	−0.149437	−	0.988771i	$-0.547746\pi$
$313$	−5.28764	−0.298875	−0.149437	+	0.988771i	$0.547746\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.21786i	0.236898i	0.992960	+	0.118449i	$0.0377923\pi$
$317$	4.21786i	0.236898i	−0.992960	+	0.118449i	$0.962208\pi$
$318$	0	0
$319$	9.52712i	0.533416i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.81940	0.156876
$324$	0	0
$325$	−2.54606	−0.141230
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 29.0428i	− 1.60118i
$330$	0	0
$331$	− 21.1046i	− 1.16002i	−0.814611	−	0.580008i	$-0.803050\pi$
$331$	− 21.1046i	− 1.16002i	0.814611	−	0.580008i	$-0.196950\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.0029	0.710423
$336$	0	0
$337$	14.3857	0.783638	0.391819	−	0.920042i	$-0.371846\pi$
$337$	14.3857	0.783638	0.391819	+	0.920042i	$0.371846\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.95895i	0.214389i
$342$	0	0
$343$	− 14.8674i	− 0.802764i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.1450	0.544610	0.272305	−	0.962211i	$-0.412214\pi$
$347$	10.1450	0.544610	0.272305	+	0.962211i	$0.412214\pi$
$348$	0	0
$349$	3.67592	0.196768	0.0983838	−	0.995149i	$-0.468633\pi$
$349$	3.67592	0.196768	0.0983838	+	0.995149i	$0.468633\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 0.508364i	− 0.0270575i	−0.999908	−	0.0135288i	$-0.995694\pi$
$353$	− 0.508364i	− 0.0270575i	0.999908	−	0.0135288i	$-0.00430647\pi$
$354$	0	0
$355$	11.8743i	0.630221i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.3560	1.54935	0.774676	−	0.632359i	$-0.217913\pi$
$359$	29.3560	1.54935	0.774676	+	0.632359i	$0.217913\pi$
$360$	0	0
$361$	−52.8139	−2.77968
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 17.9190i	− 0.937923i
$366$	0	0
$367$	− 31.1945i	− 1.62834i	−0.580628	−	0.814169i	$-0.697193\pi$
$367$	− 31.1945i	− 1.62834i	0.580628	−	0.814169i	$-0.302807\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−37.0216	−1.92207
$372$	0	0
$373$	−10.8720	−0.562929	−0.281464	−	0.959572i	$-0.590820\pi$
$373$	−10.8720	−0.562929	−0.281464	+	0.959572i	$0.590820\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.1912i	1.60643i
$378$	0	0
$379$	0.305469i	0.0156909i	0.999969	+	0.00784545i	$0.00249731\pi$
$379$	0.305469i	0.0156909i	−0.999969	+	0.00784545i	$0.997503\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−18.0390	−0.921748	−0.460874	−	0.887466i	$-0.652464\pi$
$383$	−18.0390	−0.921748	−0.460874	+	0.887466i	$0.652464\pi$
$384$	0	0
$385$	8.60800	0.438704
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 3.06226i	− 0.155263i	−0.996982	−	0.0776315i	$-0.975264\pi$
$389$	− 3.06226i	− 0.155263i	0.996982	−	0.0776315i	$-0.0247358\pi$
$390$	0	0
$391$	1.17181i	0.0592610i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	27.3092	1.37408
$396$	0	0
$397$	19.1036	0.958781	0.479391	−	0.877602i	$-0.340858\pi$
$397$	19.1036	0.958781	0.479391	+	0.877602i	$0.340858\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 21.1452i	− 1.05594i	−0.849262	−	0.527972i	$-0.822953\pi$
$401$	− 21.1452i	− 1.05594i	0.849262	−	0.527972i	$-0.177047\pi$
$402$	0	0
$403$	12.9614i	0.645652i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.37648	−0.0682295
$408$	0	0
$409$	16.6503	0.823306	0.411653	−	0.911341i	$-0.364952\pi$
$409$	16.6503	0.823306	0.411653	+	0.911341i	$0.364952\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 15.5084i	− 0.763118i
$414$	0	0
$415$	2.02221i	0.0992666i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.0057	−1.27046	−0.635232	−	0.772322i	$-0.719095\pi$
$419$	−26.0057	−1.27046	−0.635232	+	0.772322i	$0.719095\pi$
$420$	0	0
$421$	−28.5887	−1.39333	−0.696663	−	0.717398i	$-0.745333\pi$
$421$	−28.5887	−1.39333	−0.696663	+	0.717398i	$0.745333\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.258732i	0.0125503i
$426$	0	0
$427$	− 62.2392i	− 3.01197i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.9908	0.673913	0.336957	−	0.941520i	$-0.390602\pi$
$431$	13.9908	0.673913	0.336957	+	0.941520i	$0.390602\pi$
$432$	0	0
$433$	−10.7986	−0.518947	−0.259473	−	0.965750i	$-0.583549\pi$
$433$	−10.7986	−0.518947	−0.259473	+	0.965750i	$0.583549\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 29.8476i	− 1.42780i
$438$	0	0
$439$	4.67151i	0.222959i	0.993767	+	0.111480i	$0.0355589\pi$
$439$	4.67151i	0.222959i	−0.993767	+	0.111480i	$0.964441\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.0906	0.859509	0.429755	−	0.902946i	$-0.358600\pi$
$443$	18.0906	0.859509	0.429755	+	0.902946i	$0.358600\pi$
$444$	0	0
$445$	−3.00554	−0.142476
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 24.6355i	− 1.16262i	−0.813682	−	0.581311i	$-0.802540\pi$
$449$	− 24.6355i	− 1.16262i	0.813682	−	0.581311i	$-0.197460\pi$
$450$	0	0
$451$	5.42788i	0.255589i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	28.1821	1.32120
$456$	0	0
$457$	−19.4637	−0.910476	−0.455238	−	0.890370i	$-0.650446\pi$
$457$	−19.4637	−0.910476	−0.455238	+	0.890370i	$0.650446\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1.07568i	− 0.0500993i	−0.999686	−	0.0250496i	$-0.992026\pi$
$461$	− 1.07568i	− 0.0500993i	0.999686	−	0.0250496i	$-0.00797439\pi$
$462$	0	0
$463$	− 4.81991i	− 0.224000i	−0.993708	−	0.112000i	$-0.964274\pi$
$463$	− 4.81991i	− 0.224000i	0.993708	−	0.112000i	$-0.0357257\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.41821	−0.389548	−0.194774	−	0.980848i	$-0.562397\pi$
$467$	−8.41821	−0.389548	−0.194774	+	0.980848i	$0.562397\pi$
$468$	0	0
$469$	26.5088	1.22406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.53622i	− 0.0706354i
$474$	0	0
$475$	− 6.59024i	− 0.302381i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.1234	0.965151	0.482575	−	0.875854i	$-0.339701\pi$
$479$	21.1234	0.965151	0.482575	+	0.875854i	$0.339701\pi$
$480$	0	0
$481$	−4.50651	−0.205479
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.3160i	0.559240i
$486$	0	0
$487$	8.21718i	0.372356i	0.982516	+	0.186178i	$0.0596101\pi$
$487$	8.21718i	0.372356i	−0.982516	+	0.186178i	$0.940390\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.17027	0.368719	0.184360	−	0.982859i	$-0.440979\pi$
$491$	8.17027	0.368719	0.184360	+	0.982859i	$0.440979\pi$
$492$	0	0
$493$	3.16967	0.142755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.2079i	1.08587i
$498$	0	0
$499$	− 21.9921i	− 0.984501i	−0.870454	−	0.492250i	$-0.836174\pi$
$499$	− 21.9921i	− 0.984501i	0.870454	−	0.492250i	$-0.163826\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.0446	−0.804569	−0.402284	−	0.915515i	$-0.631784\pi$
$503$	−18.0446	−0.804569	−0.402284	+	0.915515i	$0.631784\pi$
$504$	0	0
$505$	10.5073	0.467568
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.74310i	0.254558i	0.991867	+	0.127279i	$0.0406244\pi$
$509$	5.74310i	0.254558i	−0.991867	+	0.127279i	$0.959376\pi$
$510$	0	0
$511$	− 36.5312i	− 1.61605i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.67696	−0.0738958
$516$	0	0
$517$	6.93286	0.304907
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 13.3348i	− 0.584208i	−0.956386	−	0.292104i	$-0.905645\pi$
$521$	− 13.3348i	− 0.584208i	0.956386	−	0.292104i	$-0.0943554\pi$
$522$	0	0
$523$	− 31.7086i	− 1.38652i	−0.720687	−	0.693260i	$-0.756174\pi$
$523$	− 31.7086i	− 1.38652i	0.720687	−	0.693260i	$-0.243826\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.31714	0.0573757
$528$	0	0
$529$	−10.5946	−0.460637
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.7706i	0.769728i
$534$	0	0
$535$	29.1222i	1.25906i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.5490	0.454378
$540$	0	0
$541$	−31.1160	−1.33778	−0.668890	−	0.743361i	$-0.733230\pi$
$541$	−31.1160	−1.33778	−0.668890	+	0.743361i	$0.733230\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.37575i	0.230272i
$546$	0	0
$547$	− 13.1543i	− 0.562437i	−0.959644	−	0.281218i	$-0.909261\pi$
$547$	− 13.1543i	− 0.562437i	0.959644	−	0.281218i	$-0.0907385\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−80.7357	−3.43946
$552$	0	0
$553$	55.6750	2.36754
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.1015i	0.512756i	0.966577	+	0.256378i	$0.0825292\pi$
$557$	12.1015i	0.512756i	−0.966577	+	0.256378i	$0.917471\pi$
$558$	0	0
$559$	− 5.02949i	− 0.212725i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.2233	−0.852309	−0.426155	−	0.904650i	$-0.640132\pi$
$563$	−20.2233	−0.852309	−0.426155	+	0.904650i	$0.640132\pi$
$564$	0	0
$565$	−18.1086	−0.761835
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.66888i	0.405340i	0.979247	+	0.202670i	$0.0649619\pi$
$569$	9.66888i	0.405340i	−0.979247	+	0.202670i	$0.935038\pi$
$570$	0	0
$571$	17.7620i	0.743318i	0.928369	+	0.371659i	$0.121211\pi$
$571$	17.7620i	0.743318i	−0.928369	+	0.371659i	$0.878789\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.73906	0.114227
$576$	0	0
$577$	−22.1545	−0.922304	−0.461152	−	0.887321i	$-0.652564\pi$
$577$	−22.1545	−0.922304	−0.461152	+	0.887321i	$0.652564\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.12266i	0.171037i
$582$	0	0
$583$	− 8.83749i	− 0.366011i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.9785	−1.36117	−0.680584	−	0.732670i	$-0.738274\pi$
$587$	−32.9785	−1.36117	−0.680584	+	0.732670i	$0.738274\pi$
$588$	0	0
$589$	−33.5494	−1.38238
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.23373i	0.297054i	0.988908	+	0.148527i	$0.0474531\pi$
$593$	7.23373i	0.297054i	−0.988908	+	0.148527i	$0.952547\pi$
$594$	0	0
$595$	− 2.86388i	− 0.117408i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.9917	1.10285	0.551425	−	0.834224i	$-0.314084\pi$
$599$	26.9917	1.10285	0.551425	+	0.834224i	$0.314084\pi$
$600$	0	0
$601$	33.8244	1.37972	0.689862	−	0.723941i	$-0.257671\pi$
$601$	33.8244	1.37972	0.689862	+	0.723941i	$0.257671\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.05483i	0.0835407i
$606$	0	0
$607$	− 18.4992i	− 0.750861i	−0.926851	−	0.375430i	$-0.877495\pi$
$607$	− 18.4992i	− 0.750861i	0.926851	−	0.375430i	$-0.122505\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	22.6978	0.918254
$612$	0	0
$613$	−18.8425	−0.761041	−0.380520	−	0.924772i	$-0.624255\pi$
$613$	−18.8425	−0.761041	−0.380520	+	0.924772i	$0.624255\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19.5867i	− 0.788529i	−0.918997	−	0.394265i	$-0.870999\pi$
$617$	− 19.5867i	− 0.788529i	0.918997	−	0.394265i	$-0.129001\pi$
$618$	0	0
$619$	20.9215i	0.840908i	0.907314	+	0.420454i	$0.138129\pi$
$619$	20.9215i	0.840908i	−0.907314	+	0.420454i	$0.861871\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.12735	−0.245487
$624$	0	0
$625$	−20.5069	−0.820274
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.457954i	0.0182598i
$630$	0	0
$631$	− 1.27651i	− 0.0508172i	−0.999677	−	0.0254086i	$-0.991911\pi$
$631$	− 1.27651i	− 0.0508172i	0.999677	−	0.0254086i	$-0.00808868\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	37.0216	1.46916
$636$	0	0
$637$	34.5369	1.36840
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 20.0324i	− 0.791234i	−0.918416	−	0.395617i	$-0.870531\pi$
$641$	− 20.0324i	− 0.791234i	0.918416	−	0.395617i	$-0.129469\pi$
$642$	0	0
$643$	9.71125i	0.382974i	0.981495	+	0.191487i	$0.0613310\pi$
$643$	9.71125i	0.382974i	−0.981495	+	0.191487i	$0.938669\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31.8921	−1.25381	−0.626904	−	0.779097i	$-0.715678\pi$
$647$	−31.8921	−1.25381	−0.626904	+	0.779097i	$0.715678\pi$
$648$	0	0
$649$	3.70203	0.145318
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 20.6601i	− 0.808491i	−0.914651	−	0.404245i	$-0.867534\pi$
$653$	− 20.6601i	− 0.808491i	0.914651	−	0.404245i	$-0.132466\pi$
$654$	0	0
$655$	32.9340i	1.28684i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28.5649	−1.11273	−0.556365	−	0.830938i	$-0.687804\pi$
$659$	−28.5649	−1.11273	−0.556365	+	0.830938i	$0.687804\pi$
$660$	0	0
$661$	20.3230	0.790475	0.395237	−	0.918579i	$-0.370662\pi$
$661$	20.3230	0.790475	0.395237	+	0.918579i	$0.370662\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	72.9468i	2.82876i
$666$	0	0
$667$	− 33.5557i	− 1.29928i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.8572	0.573557
$672$	0	0
$673$	−40.4387	−1.55880	−0.779398	−	0.626529i	$-0.784475\pi$
$673$	−40.4387	−1.55880	−0.779398	+	0.626529i	$0.784475\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 25.2867i	− 0.971848i	−0.874001	−	0.485924i	$-0.838483\pi$
$677$	− 25.2867i	− 0.971848i	0.874001	−	0.485924i	$-0.161517\pi$
$678$	0	0
$679$	25.1084i	0.963573i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.12760	0.0814102	0.0407051	−	0.999171i	$-0.487040\pi$
$683$	2.12760	0.0814102	0.0407051	+	0.999171i	$0.487040\pi$
$684$	0	0
$685$	20.9428	0.800181
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 28.9334i	− 1.10228i
$690$	0	0
$691$	33.9780i	1.29258i	0.763090	+	0.646292i	$0.223681\pi$
$691$	33.9780i	1.29258i	−0.763090	+	0.646292i	$0.776319\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.25624	−0.123516
$696$	0	0
$697$	1.80585	0.0684016
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 43.1893i	− 1.63124i	−0.578590	−	0.815618i	$-0.696397\pi$
$701$	− 43.1893i	− 1.63124i	0.578590	−	0.815618i	$-0.303603\pi$
$702$	0	0
$703$	− 11.6647i	− 0.439942i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.4211	0.805622
$708$	0	0
$709$	−2.99634	−0.112530	−0.0562650	−	0.998416i	$-0.517919\pi$
$709$	−2.99634	−0.112530	−0.0562650	+	0.998416i	$0.517919\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 13.9439i	− 0.522204i
$714$	0	0
$715$	6.72739i	0.251590i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−44.6554	−1.66536	−0.832682	−	0.553751i	$-0.813196\pi$
$719$	−44.6554	−1.66536	−0.832682	+	0.553751i	$0.813196\pi$
$720$	0	0
$721$	−3.41880	−0.127323
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.40899i	− 0.275163i
$726$	0	0
$727$	− 22.6022i	− 0.838270i	−0.907924	−	0.419135i	$-0.862333\pi$
$727$	− 22.6022i	− 0.838270i	0.907924	−	0.419135i	$-0.137667\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.511100	−0.0189037
$732$	0	0
$733$	28.3580	1.04743	0.523713	−	0.851895i	$-0.324546\pi$
$733$	28.3580	1.04743	0.523713	+	0.851895i	$0.324546\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.32795i	0.233093i
$738$	0	0
$739$	43.9526i	1.61682i	0.588619	+	0.808411i	$0.299672\pi$
$739$	43.9526i	1.61682i	−0.588619	+	0.808411i	$0.700328\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.8324	−1.16782	−0.583909	−	0.811819i	$-0.698478\pi$
$743$	−31.8324	−1.16782	−0.583909	+	0.811819i	$0.698478\pi$
$744$	0	0
$745$	6.28245	0.230171
$746$	0	0
$747$	0	0
$748$	0	0
$749$	59.3710i	2.16937i
$750$	0	0
$751$	− 11.1655i	− 0.407433i	−0.979030	−	0.203717i	$-0.934698\pi$
$751$	− 11.1655i	− 0.407433i	0.979030	−	0.203717i	$-0.0653021\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	33.5230	1.22003
$756$	0	0
$757$	−11.7438	−0.426834	−0.213417	−	0.976961i	$-0.568459\pi$
$757$	−11.7438	−0.426834	−0.213417	+	0.976961i	$0.568459\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.11196i	0.294058i	0.989132	+	0.147029i	$0.0469711\pi$
$761$	8.11196i	0.294058i	−0.989132	+	0.147029i	$0.953029\pi$
$762$	0	0
$763$	10.9595i	0.396760i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.1202	0.437636
$768$	0	0
$769$	−36.1570	−1.30386	−0.651928	−	0.758281i	$-0.726039\pi$
$769$	−36.1570	−1.30386	−0.651928	+	0.758281i	$0.726039\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 23.8614i	− 0.858236i	−0.903249	−	0.429118i	$-0.858825\pi$
$773$	− 23.8614i	− 0.858236i	0.903249	−	0.429118i	$-0.141175\pi$
$774$	0	0
$775$	− 3.07877i	− 0.110593i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−45.9975	−1.64803
$780$	0	0
$781$	−5.77871	−0.206779
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.70822i	− 0.168043i
$786$	0	0
$787$	43.8822i	1.56423i	0.623134	+	0.782115i	$0.285859\pi$
$787$	43.8822i	1.56423i	−0.623134	+	0.782115i	$0.714141\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−36.9178	−1.31265
$792$	0	0
$793$	48.6417	1.72732
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.3351i	1.00368i	0.864961	+	0.501840i	$0.167343\pi$
$797$	28.3351i	1.00368i	−0.864961	+	0.501840i	$0.832657\pi$
$798$	0	0
$799$	− 2.30656i	− 0.0816004i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.72042	0.307737
$804$	0	0
$805$	−30.3184	−1.06858
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.5019i	0.650492i	0.945629	+	0.325246i	$0.105447\pi$
$809$	18.5019i	0.650492i	−0.945629	+	0.325246i	$0.894553\pi$
$810$	0	0
$811$	− 0.0475401i	− 0.00166936i	−1.00000		0.000834679i	$-0.999734\pi$
$811$	− 0.0475401i	− 0.00166936i	1.00000		0.000834679i	$-0.000265687\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.09073	0.0382065
$816$	0	0
$817$	13.0184	0.455456
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 39.2412i	− 1.36953i	−0.728765	−	0.684764i	$-0.759905\pi$
$821$	− 39.2412i	− 1.36953i	0.728765	−	0.684764i	$-0.240095\pi$
$822$	0	0
$823$	− 37.4363i	− 1.30495i	−0.757812	−	0.652473i	$-0.773732\pi$
$823$	− 37.4363i	− 1.30495i	0.757812	−	0.652473i	$-0.226268\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.8240	1.31527	0.657635	−	0.753337i	$-0.271557\pi$
$827$	37.8240	1.31527	0.657635	+	0.753337i	$0.271557\pi$
$828$	0	0
$829$	14.3390	0.498014	0.249007	−	0.968502i	$-0.419896\pi$
$829$	14.3390	0.498014	0.249007	+	0.968502i	$0.419896\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.50966i	− 0.121602i
$834$	0	0
$835$	− 34.9781i	− 1.21047i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.56063	−0.295546	−0.147773	−	0.989021i	$-0.547210\pi$
$839$	−8.56063	−0.295546	−0.147773	+	0.989021i	$0.547210\pi$
$840$	0	0
$841$	−61.7660	−2.12986
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4.68770i	− 0.161262i
$846$	0	0
$847$	4.18915i	0.143941i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.84812	0.166192
$852$	0	0
$853$	54.1181	1.85297	0.926484	−	0.376335i	$-0.122816\pi$
$853$	54.1181	1.85297	0.926484	+	0.376335i	$0.122816\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.8279i	0.779788i	0.920860	+	0.389894i	$0.127488\pi$
$857$	22.8279i	0.779788i	−0.920860	+	0.389894i	$0.872512\pi$
$858$	0	0
$859$	− 0.288906i	− 0.00985734i	−0.999988	−	0.00492867i	$-0.998431\pi$
$859$	− 0.288906i	− 0.00985734i	0.999988	−	0.00492867i	$-0.00156885\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.722052	0.0245789	0.0122895	−	0.999924i	$-0.496088\pi$
$863$	0.722052	0.0245789	0.0122895	+	0.999924i	$0.496088\pi$
$864$	0	0
$865$	11.0752	0.376568
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.2903i	0.450842i
$870$	0	0
$871$	20.7173i	0.701981i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−49.7342	−1.68132
$876$	0	0
$877$	25.4051	0.857869	0.428935	−	0.903335i	$-0.358889\pi$
$877$	25.4051	0.857869	0.428935	+	0.903335i	$0.358889\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 18.2594i	− 0.615175i	−0.951520	−	0.307588i	$-0.900478\pi$
$881$	− 18.2594i	− 0.615175i	0.951520	−	0.307588i	$-0.0995217\pi$
$882$	0	0
$883$	10.2183i	0.343872i	0.985108	+	0.171936i	$0.0550023\pi$
$883$	10.2183i	0.343872i	−0.985108	+	0.171936i	$0.944998\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.4055	−1.18880	−0.594399	−	0.804170i	$-0.702610\pi$
$887$	−35.4055	−1.18880	−0.594399	+	0.804170i	$0.702610\pi$
$888$	0	0
$889$	75.4755	2.53137
$890$	0	0
$891$	0	0
$892$	0	0
$893$	58.7512i	1.96603i
$894$	0	0
$895$	− 33.0044i	− 1.10321i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−37.7174	−1.25795
$900$	0	0
$901$	−2.94023	−0.0979533
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 15.7378i	− 0.523142i
$906$	0	0
$907$	− 23.9880i	− 0.796508i	−0.917275	−	0.398254i	$-0.869616\pi$
$907$	− 23.9880i	− 0.796508i	0.917275	−	0.398254i	$-0.130384\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0.671680	0.0222537	0.0111269	−	0.999938i	$-0.496458\pi$
$911$	0.671680	0.0222537	0.0111269	+	0.999938i	$0.496458\pi$
$912$	0	0
$913$	−0.984127	−0.0325699
$914$	0	0
$915$	0	0
$916$	0	0
$917$	67.1421i	2.21723i
$918$	0	0
$919$	12.6625i	0.417699i	0.977948	+	0.208849i	$0.0669719\pi$
$919$	12.6625i	0.417699i	−0.977948	+	0.208849i	$0.933028\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.9192	−0.622732
$924$	0	0
$925$	1.07045	0.0351962
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.2495i	0.565939i	0.959129	+	0.282969i	$0.0913195\pi$
$929$	17.2495i	0.565939i	−0.959129	+	0.282969i	$0.908681\pi$
$930$	0	0
$931$	89.3956i	2.92982i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.683642	0.0223575
$936$	0	0
$937$	25.9299	0.847091	0.423546	−	0.905875i	$-0.360785\pi$
$937$	25.9299	0.847091	0.423546	+	0.905875i	$0.360785\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 38.5622i	− 1.25709i	−0.777773	−	0.628545i	$-0.783651\pi$
$941$	− 38.5622i	− 1.25709i	0.777773	−	0.628545i	$-0.216349\pi$
$942$	0	0
$943$	− 19.1177i	− 0.622556i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.0835	−0.327668	−0.163834	−	0.986488i	$-0.552386\pi$
$947$	−10.0835	−0.327668	−0.163834	+	0.986488i	$0.552386\pi$
$948$	0	0
$949$	28.5502	0.926777
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.8961i	0.385353i	0.981262	+	0.192677i	$0.0617168\pi$
$953$	11.8961i	0.385353i	−0.981262	+	0.192677i	$0.938283\pi$
$954$	0	0
$955$	− 27.0884i	− 0.876559i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	42.6957	1.37872
$960$	0	0
$961$	15.3267	0.494410
$962$	0	0
$963$	0	0
$964$	0	0
$965$	36.0638i	1.16094i
$966$	0	0
$967$	40.7204i	1.30948i	0.755855	+	0.654739i	$0.227222\pi$
$967$	40.7204i	1.30948i	−0.755855	+	0.654739i	$0.772778\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.171053	0.00548936	0.00274468	−	0.999996i	$-0.499126\pi$
$971$	0.171053	0.00548936	0.00274468	+	0.999996i	$0.499126\pi$
$972$	0	0
$973$	−6.63845	−0.212819
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.4361i	1.35765i	0.734299	+	0.678826i	$0.237511\pi$
$977$	42.4361i	1.35765i	−0.734299	+	0.678826i	$0.762489\pi$
$978$	0	0
$979$	− 1.46267i	− 0.0467471i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.0332	−0.511380	−0.255690	−	0.966759i	$-0.582303\pi$
$983$	−16.0332	−0.511380	−0.255690	+	0.966759i	$0.582303\pi$
$984$	0	0
$985$	−23.9247	−0.762305
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.41075i	0.172052i
$990$	0	0
$991$	− 19.7092i	− 0.626085i	−0.949739	−	0.313042i	$-0.898652\pi$
$991$	− 19.7092i	− 0.626085i	0.949739	−	0.313042i	$-0.101348\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.9895	−0.348391
$996$	0	0
$997$	−14.9855	−0.474596	−0.237298	−	0.971437i	$-0.576262\pi$
$997$	−14.9855	−0.474596	−0.237298	+	0.971437i	$0.576262\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.d.g.3455.8		10
3.2	odd	2		6336.2.d.h.3455.3		10
4.3	odd	2		6336.2.d.h.3455.8		10
8.3	odd	2		396.2.c.a.287.6	yes	10
8.5	even	2		396.2.c.b.287.6	yes	10
12.11	even	2	inner	6336.2.d.g.3455.3		10
24.5	odd	2		396.2.c.a.287.5	✓	10
24.11	even	2		396.2.c.b.287.5	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
396.2.c.a.287.5	✓	10	24.5	odd	2
396.2.c.a.287.6	yes	10	8.3	odd	2
396.2.c.b.287.5	yes	10	24.11	even	2
396.2.c.b.287.6	yes	10	8.5	even	2
6336.2.d.g.3455.3		10	12.11	even	2	inner
6336.2.d.g.3455.8		10	1.1	even	1	trivial
6336.2.d.h.3455.3		10	3.2	odd	2
6336.2.d.h.3455.8		10	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.05483i q^{5} +4.18915i q^{7} -1.00000 q^{11} -3.27394 q^{13} +0.332700i q^{17} -8.47431i q^{19} +3.52212 q^{23} +0.777673 q^{25} -9.52712i q^{29} -3.95895i q^{31} -8.60800 q^{35} +1.37648 q^{37} -5.42788i q^{41} +1.53622i q^{43} -6.93286 q^{47} -10.5490 q^{49} +8.83749i q^{53} -2.05483i q^{55} -3.70203 q^{59} -14.8572 q^{61} -6.72739i q^{65} -6.32795i q^{67} +5.77871 q^{71} -8.72042 q^{73} -4.18915i q^{77} -13.2903i q^{79} +0.984127 q^{83} -0.683642 q^{85} +1.46267i q^{89} -13.7150i q^{91} +17.4133 q^{95} +5.99367 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{11} - 8 q^{13} + 8 q^{23} - 10 q^{25} + 16 q^{35} + 4 q^{37} - 34 q^{49} + 40 q^{59} - 20 q^{61} + 16 q^{71} + 24 q^{73} + 8 q^{83} + 36 q^{85} + 96 q^{95} + 24 q^{97}+O(q^{100}) \) 10 * q - 10 * q^11 - 8 * q^13 + 8 * q^23 - 10 * q^25 + 16 * q^35 + 4 * q^37 - 34 * q^49 + 40 * q^59 - 20 * q^61 + 16 * q^71 + 24 * q^73 + 8 * q^83 + 36 * q^85 + 96 * q^95 + 24 * q^97

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