LMFDB - Embedded newform 6336.2.d.e.3455.6

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 = [8,0,0,0,0,0,0,0,0,0,8,0,8] 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:	$8$
Coefficient field:	8.0.1768034304.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 9x^{6} - 2x^{5} + 34x^{4} - 18x^{3} + 51x^{2} + 18x + 9 $ x^8 - 2x^7 + 9x^6 - 2x^5 + 34x^4 - 18x^3 + 51x^2 + 18*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1584)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3455.6
Root			$-0.197356 - 0.341831i$ of defining polynomial
Character	$\chi$	$=$	6336.3455
Dual form			6336.2.d.e.3455.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.97242i q^{5} -1.97242i q^{7} +1.00000 q^{11} -6.65525 q^{13} +5.31217i q^{17} -4.53363i q^{19} +6.54569 q^{23} +1.10955 q^{25} +3.46410i q^{29} -8.77627i q^{31} +3.89045 q^{35} +2.72311 q^{37} -8.77627i q^{41} +9.07407i q^{43} -13.3784 q^{47} +3.10955 q^{49} +4.95578i q^{53} +1.97242i q^{55} +10.4778 q^{59} -10.6552 q^{61} -13.1270i q^{65} +5.02118i q^{67} +6.54569 q^{71} +8.47783 q^{73} -1.97242i q^{77} +8.65191i q^{79} +8.10955 q^{83} -10.4778 q^{85} -6.63040i q^{89} +13.1270i q^{91} +8.94222 q^{95} +0.109555 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 16 q^{25} + 56 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{59} - 24 q^{61} + 16 q^{71} + 40 q^{83} - 16 q^{85} - 8 q^{95} - 24 q^{97}+O(q^{100}) $ 8 * q + 8 * q^11 + 8 * q^13 + 16 * q^23 - 16 * q^25 + 56 * q^35 - 8 * q^37 - 16 * q^47 + 16 * q^59 - 24 * q^61 + 16 * q^71 + 40 * q^83 - 16 * q^85 - 8 * 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
$5$	1.97242i	0.882094i	0.897484	+	0.441047i	$0.145393\pi$
$5$	1.97242i	0.882094i	−0.897484	+	0.441047i	$0.854607\pi$
$6$	0	0
$7$	− 1.97242i	− 0.745505i	−0.927931	−	0.372753i	$-0.878414\pi$
$7$	− 1.97242i	− 0.745505i	0.927931	−	0.372753i	$-0.121586\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.65525	−1.84583	−0.922917	−	0.384999i	$-0.874202\pi$
$13$	−6.65525	−1.84583	−0.922917	+	0.384999i	$0.874202\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.31217i	1.28839i	0.764861	+	0.644195i	$0.222807\pi$
$17$	5.31217i	1.28839i	−0.764861	+	0.644195i	$0.777193\pi$
$18$	0	0
$19$	− 4.53363i	− 1.04009i	−0.854140	−	0.520043i	$-0.825916\pi$
$19$	− 4.53363i	− 1.04009i	0.854140	−	0.520043i	$-0.174084\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.54569	1.36487	0.682436	−	0.730946i	$-0.260921\pi$
$23$	6.54569	1.36487	0.682436	+	0.730946i	$0.260921\pi$
$24$	0	0
$25$	1.10955	0.221911
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.46410i	0.643268i	0.946864	+	0.321634i	$0.104232\pi$
$29$	3.46410i	0.643268i	−0.946864	+	0.321634i	$0.895768\pi$
$30$	0	0
$31$	− 8.77627i	− 1.57626i	−0.615506	−	0.788132i	$-0.711048\pi$
$31$	− 8.77627i	− 1.57626i	0.615506	−	0.788132i	$-0.288952\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.89045	0.657605
$36$	0	0
$37$	2.72311	0.447677	0.223838	−	0.974626i	$-0.428141\pi$
$37$	2.72311	0.447677	0.223838	+	0.974626i	$0.428141\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.77627i	− 1.37062i	−0.728250	−	0.685311i	$-0.759666\pi$
$41$	− 8.77627i	− 1.37062i	0.728250	−	0.685311i	$-0.240334\pi$
$42$	0	0
$43$	9.07407i	1.38378i	0.722002	+	0.691891i	$0.243222\pi$
$43$	9.07407i	1.38378i	−0.722002	+	0.691891i	$0.756778\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13.3784	−1.95143	−0.975717	−	0.219034i	$-0.929709\pi$
$47$	−13.3784	−1.95143	−0.975717	+	0.219034i	$0.929709\pi$
$48$	0	0
$49$	3.10955	0.444222
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.95578i	0.680729i	0.940294	+	0.340365i	$0.110550\pi$
$53$	4.95578i	0.680729i	−0.940294	+	0.340365i	$0.889450\pi$
$54$	0	0
$55$	1.97242i	0.265961i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.4778	1.36410	0.682049	−	0.731307i	$-0.261089\pi$
$59$	10.4778	1.36410	0.682049	+	0.731307i	$0.261089\pi$
$60$	0	0
$61$	−10.6552	−1.36426	−0.682132	−	0.731229i	$-0.738947\pi$
$61$	−10.6552	−1.36426	−0.682132	+	0.731229i	$0.738947\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 13.1270i	− 1.62820i
$66$	0	0
$67$	5.02118i	0.613435i	0.951801	+	0.306717i	$0.0992306\pi$
$67$	5.02118i	0.613435i	−0.951801	+	0.306717i	$0.900769\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.54569	0.776831	0.388415	−	0.921484i	$-0.373023\pi$
$71$	6.54569	0.776831	0.388415	+	0.921484i	$0.373023\pi$
$72$	0	0
$73$	8.47783	0.992255	0.496127	−	0.868250i	$-0.334755\pi$
$73$	8.47783	0.992255	0.496127	+	0.868250i	$0.334755\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.97242i	− 0.224778i
$78$	0	0
$79$	8.65191i	0.973416i	0.873565	+	0.486708i	$0.161802\pi$
$79$	8.65191i	0.973416i	−0.873565	+	0.486708i	$0.838198\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.10955	0.890139	0.445070	−	0.895496i	$-0.353179\pi$
$83$	8.10955	0.890139	0.445070	+	0.895496i	$0.353179\pi$
$84$	0	0
$85$	−10.4778	−1.13648
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6.63040i	− 0.702822i	−0.936221	−	0.351411i	$-0.885702\pi$
$89$	− 6.63040i	− 0.702822i	0.936221	−	0.351411i	$-0.114298\pi$
$90$	0	0
$91$	13.1270i	1.37608i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.94222	0.917453
$96$	0	0
$97$	0.109555	0.0111236	0.00556180	−	0.999985i	$-0.498230\pi$
$97$	0.109555	0.0111236	0.00556180	+	0.999985i	$0.498230\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.65386i	0.363572i	0.983338	+	0.181786i	$0.0581879\pi$
$101$	3.65386i	0.363572i	−0.983338	+	0.181786i	$0.941812\pi$
$102$	0	0
$103$	11.7596i	1.15871i	0.815075	+	0.579355i	$0.196696\pi$
$103$	11.7596i	1.15871i	−0.815075	+	0.579355i	$0.803304\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.832667	−0.0804970	−0.0402485	−	0.999190i	$-0.512815\pi$
$107$	−0.832667	−0.0804970	−0.0402485	+	0.999190i	$0.512815\pi$
$108$	0	0
$109$	13.2688	1.27092	0.635461	−	0.772133i	$-0.280810\pi$
$109$	13.2688	1.27092	0.635461	+	0.772133i	$0.280810\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.3099i	1.25209i	0.779787	+	0.626045i	$0.215327\pi$
$113$	13.3099i	1.25209i	−0.779787	+	0.626045i	$0.784673\pi$
$114$	0	0
$115$	12.9109i	1.20394i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.4778	0.960501
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0506i	1.07784i
$126$	0	0
$127$	4.95578i	0.439755i	0.975528	+	0.219877i	$0.0705657\pi$
$127$	4.95578i	0.439755i	−0.975528	+	0.219877i	$0.929434\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.72311	0.412660	0.206330	−	0.978482i	$-0.433848\pi$
$131$	4.72311	0.412660	0.206330	+	0.978482i	$0.433848\pi$
$132$	0	0
$133$	−8.94222	−0.775389
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9221i	0.933141i	0.884484	+	0.466570i	$0.154511\pi$
$137$	10.9221i	0.933141i	−0.884484	+	0.466570i	$0.845489\pi$
$138$	0	0
$139$	− 7.26827i	− 0.616487i	−0.951307	−	0.308244i	$-0.900259\pi$
$139$	− 7.26827i	− 0.616487i	0.951307	−	0.308244i	$-0.0997412\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.65525	−0.556540
$144$	0	0
$145$	−6.83267	−0.567422
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.25247i	0.102606i	0.998683	+	0.0513031i	$0.0163375\pi$
$149$	1.25247i	0.102606i	−0.998683	+	0.0513031i	$0.983663\pi$
$150$	0	0
$151$	2.56802i	0.208982i	0.994526	+	0.104491i	$0.0333214\pi$
$151$	2.56802i	0.208982i	−0.994526	+	0.104491i	$0.966679\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.3105	1.39041
$156$	0	0
$157$	1.64516	0.131298	0.0656491	−	0.997843i	$-0.479088\pi$
$157$	1.64516	0.131298	0.0656491	+	0.997843i	$0.479088\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 12.9109i	− 1.01752i
$162$	0	0
$163$	7.52380i	0.589309i	0.955604	+	0.294655i	$0.0952046\pi$
$163$	7.52380i	0.589309i	−0.955604	+	0.294655i	$0.904795\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.5874	1.90263	0.951314	−	0.308222i	$-0.0997340\pi$
$167$	24.5874	1.90263	0.951314	+	0.308222i	$0.0997340\pi$
$168$	0	0
$169$	31.2923	2.40710
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 14.0884i	− 1.07112i	−0.844496	−	0.535562i	$-0.820100\pi$
$173$	− 14.0884i	− 1.07112i	0.844496	−	0.535562i	$-0.179900\pi$
$174$	0	0
$175$	− 2.18851i	− 0.165436i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.5692	−0.864724	−0.432362	−	0.901700i	$-0.642320\pi$
$179$	−11.5692	−0.864724	−0.432362	+	0.901700i	$0.642320\pi$
$180$	0	0
$181$	−15.8145	−1.17548	−0.587741	−	0.809049i	$-0.699983\pi$
$181$	−15.8145	−1.17548	−0.587741	+	0.809049i	$0.699983\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.37113i	0.394893i
$186$	0	0
$187$	5.31217i	0.388464i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.31858	−0.0954089	−0.0477045	−	0.998861i	$-0.515191\pi$
$191$	−1.31858	−0.0954089	−0.0477045	+	0.998861i	$0.515191\pi$
$192$	0	0
$193$	−4.25872	−0.306549	−0.153275	−	0.988184i	$-0.548982\pi$
$193$	−4.25872	−0.306549	−0.153275	+	0.988184i	$0.548982\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.23583i	− 0.301790i	−0.988550	−	0.150895i	$-0.951784\pi$
$197$	− 4.23583i	− 0.301790i	0.988550	−	0.150895i	$-0.0482156\pi$
$198$	0	0
$199$	− 1.06271i	− 0.0753338i	−0.999290	−	0.0376669i	$-0.988007\pi$
$199$	− 1.06271i	− 0.0753338i	0.999290	−	0.0376669i	$-0.0119926\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.83267	0.479559
$204$	0	0
$205$	17.3105	1.20902
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.53363i	− 0.313598i
$210$	0	0
$211$	27.8368i	1.91636i	0.286159	+	0.958182i	$0.407621\pi$
$211$	27.8368i	1.91636i	−0.286159	+	0.958182i	$0.592379\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−17.8979	−1.22063
$216$	0	0
$217$	−17.3105	−1.17511
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 35.3538i	− 2.37815i
$222$	0	0
$223$	− 6.19875i	− 0.415099i	−0.978225	−	0.207549i	$-0.933451\pi$
$223$	− 6.19875i	− 0.415099i	0.978225	−	0.207549i	$-0.0665488\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.7883	−1.44614	−0.723071	−	0.690774i	$-0.757270\pi$
$227$	−21.7883	−1.44614	−0.723071	+	0.690774i	$0.757270\pi$
$228$	0	0
$229$	19.3105	1.27607	0.638037	−	0.770006i	$-0.279747\pi$
$229$	19.3105	1.27607	0.638037	+	0.770006i	$0.279747\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 15.7045i	− 1.02883i	−0.857540	−	0.514417i	$-0.828008\pi$
$233$	− 15.7045i	− 1.02883i	0.857540	−	0.514417i	$-0.171992\pi$
$234$	0	0
$235$	− 26.3878i	− 1.72135i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.63173	0.234917	0.117458	−	0.993078i	$-0.462525\pi$
$239$	3.63173	0.234917	0.117458	+	0.993078i	$0.462525\pi$
$240$	0	0
$241$	−9.86427	−0.635414	−0.317707	−	0.948189i	$-0.602913\pi$
$241$	−9.86427	−0.635414	−0.317707	+	0.948189i	$0.602913\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.13335i	0.391845i
$246$	0	0
$247$	30.1724i	1.91982i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.7049	0.991287	0.495644	−	0.868526i	$-0.334932\pi$
$251$	15.7049	0.991287	0.495644	+	0.868526i	$0.334932\pi$
$252$	0	0
$253$	6.54569	0.411524
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.5843i	1.65828i	0.559037	+	0.829142i	$0.311171\pi$
$257$	26.5843i	1.65828i	−0.559037	+	0.829142i	$0.688829\pi$
$258$	0	0
$259$	− 5.37113i	− 0.333745i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.504003	−0.0310782	−0.0155391	−	0.999879i	$-0.504946\pi$
$263$	−0.504003	−0.0310782	−0.0155391	+	0.999879i	$0.504946\pi$
$264$	0	0
$265$	−9.77489	−0.600467
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.52950i	0.215197i	0.994194	+	0.107599i	$0.0343162\pi$
$269$	3.52950i	0.215197i	−0.994194	+	0.107599i	$0.965684\pi$
$270$	0	0
$271$	21.0820i	1.28064i	0.768107	+	0.640321i	$0.221199\pi$
$271$	21.0820i	1.28064i	−0.768107	+	0.640321i	$0.778801\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.10955	0.0669087
$276$	0	0
$277$	22.6552	1.36122	0.680611	−	0.732645i	$-0.261714\pi$
$277$	22.6552	1.36122	0.680611	+	0.732645i	$0.261714\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.0628i	0.659951i	0.943989	+	0.329976i	$0.107041\pi$
$281$	11.0628i	0.659951i	−0.943989	+	0.329976i	$0.892959\pi$
$282$	0	0
$283$	8.47847i	0.503993i	0.967728	+	0.251996i	$0.0810871\pi$
$283$	8.47847i	0.503993i	−0.967728	+	0.251996i	$0.918913\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−17.3105	−1.02181
$288$	0	0
$289$	−11.2191	−0.659948
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.21539i	0.187845i	0.995580	+	0.0939225i	$0.0299406\pi$
$293$	3.21539i	0.187845i	−0.995580	+	0.0939225i	$0.970059\pi$
$294$	0	0
$295$	20.6667i	1.20326i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−43.5632	−2.51933
$300$	0	0
$301$	17.8979	1.03162
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 21.0166i	− 1.20341i
$306$	0	0
$307$	13.2676i	0.757223i	0.925556	+	0.378612i	$0.123598\pi$
$307$	13.2676i	0.757223i	−0.925556	+	0.378612i	$0.876402\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.5457	1.73209	0.866044	−	0.499968i	$-0.166655\pi$
$311$	30.5457	1.73209	0.866044	+	0.499968i	$0.166655\pi$
$312$	0	0
$313$	−22.4019	−1.26623	−0.633115	−	0.774058i	$-0.718224\pi$
$313$	−22.4019	−1.26623	−0.633115	+	0.774058i	$0.718224\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.0820i	1.18409i	0.805907	+	0.592043i	$0.201678\pi$
$317$	21.0820i	1.18409i	−0.805907	+	0.592043i	$0.798322\pi$
$318$	0	0
$319$	3.46410i	0.193952i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0834	1.34004
$324$	0	0
$325$	−7.38436	−0.409611
$326$	0	0
$327$	0	0
$328$	0	0
$329$	26.3878i	1.45480i
$330$	0	0
$331$	− 33.9275i	− 1.86482i	−0.361397	−	0.932412i	$-0.617700\pi$
$331$	− 33.9275i	− 1.86482i	0.361397	−	0.932412i	$-0.382300\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.90388	−0.541107
$336$	0	0
$337$	12.1828	0.663638	0.331819	−	0.943343i	$-0.392338\pi$
$337$	12.1828	0.663638	0.331819	+	0.943343i	$0.392338\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 8.77627i	− 0.475262i
$342$	0	0
$343$	− 19.9403i	− 1.07667i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.11756	−0.489456	−0.244728	−	0.969592i	$-0.578699\pi$
$347$	−9.11756	−0.489456	−0.244728	+	0.969592i	$0.578699\pi$
$348$	0	0
$349$	−6.43614	−0.344519	−0.172259	−	0.985052i	$-0.555107\pi$
$349$	−6.43614	−0.344519	−0.172259	+	0.985052i	$0.555107\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.297798i	0.0158502i	0.999969	+	0.00792510i	$0.00252267\pi$
$353$	0.297798i	0.0158502i	−0.999969	+	0.00792510i	$0.997477\pi$
$354$	0	0
$355$	12.9109i	0.685238i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.12772	0.165075	0.0825374	−	0.996588i	$-0.473698\pi$
$359$	3.12772	0.165075	0.0825374	+	0.996588i	$0.473698\pi$
$360$	0	0
$361$	−1.55377	−0.0817776
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16.7218i	0.875262i
$366$	0	0
$367$	27.3629i	1.42833i	0.699977	+	0.714165i	$0.253193\pi$
$367$	27.3629i	1.42833i	−0.699977	+	0.714165i	$0.746807\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.77489	0.507487
$372$	0	0
$373$	−11.7466	−0.608218	−0.304109	−	0.952637i	$-0.598359\pi$
$373$	−11.7466	−0.608218	−0.304109	+	0.952637i	$0.598359\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.0545i	− 1.18736i
$378$	0	0
$379$	− 8.48528i	− 0.435860i	−0.975964	−	0.217930i	$-0.930070\pi$
$379$	− 8.48528i	− 0.435860i	0.975964	−	0.217930i	$-0.0699304\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.7205	1.62084	0.810420	−	0.585849i	$-0.199239\pi$
$383$	31.7205	1.62084	0.810420	+	0.585849i	$0.199239\pi$
$384$	0	0
$385$	3.89045	0.198275
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 29.7834i	− 1.51008i	−0.655680	−	0.755039i	$-0.727618\pi$
$389$	− 29.7834i	− 1.51008i	0.655680	−	0.755039i	$-0.272382\pi$
$390$	0	0
$391$	34.7718i	1.75849i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−17.0652	−0.858644
$396$	0	0
$397$	−3.07795	−0.154478	−0.0772390	−	0.997013i	$-0.524610\pi$
$397$	−3.07795	−0.154478	−0.0772390	+	0.997013i	$0.524610\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.7394i	1.58499i	0.609880	+	0.792494i	$0.291218\pi$
$401$	31.7394i	1.58499i	−0.609880	+	0.792494i	$0.708782\pi$
$402$	0	0
$403$	58.4082i	2.90952i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.72311	0.134980
$408$	0	0
$409$	−3.03160	−0.149903	−0.0749516	−	0.997187i	$-0.523880\pi$
$409$	−3.03160	−0.149903	−0.0749516	+	0.997187i	$0.523880\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 20.6667i	− 1.01694i
$414$	0	0
$415$	15.9955i	0.785186i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.9557	1.02375	0.511875	−	0.859060i	$-0.328951\pi$
$419$	20.9557	1.02375	0.511875	+	0.859060i	$0.328951\pi$
$420$	0	0
$421$	29.3441	1.43014	0.715072	−	0.699051i	$-0.246394\pi$
$421$	29.3441	1.43014	0.715072	+	0.699051i	$0.246394\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.89414i	0.285908i
$426$	0	0
$427$	21.0166i	1.01707i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.4517	0.985122	0.492561	−	0.870278i	$-0.336061\pi$
$431$	20.4517	0.985122	0.492561	+	0.870278i	$0.336061\pi$
$432$	0	0
$433$	−6.18277	−0.297125	−0.148563	−	0.988903i	$-0.547465\pi$
$433$	−6.18277	−0.297125	−0.148563	+	0.988903i	$0.547465\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 29.6757i	− 1.41958i
$438$	0	0
$439$	− 5.32167i	− 0.253989i	−0.991903	−	0.126995i	$-0.959467\pi$
$439$	− 5.32167i	− 0.253989i	0.991903	−	0.126995i	$-0.0405331\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.4933	1.49629	0.748145	−	0.663535i	$-0.230945\pi$
$443$	31.4933	1.49629	0.748145	+	0.663535i	$0.230945\pi$
$444$	0	0
$445$	13.0780	0.619954
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.0227i	0.661770i	0.943671	+	0.330885i	$0.107347\pi$
$449$	14.0227i	0.661770i	−0.943671	+	0.330885i	$0.892653\pi$
$450$	0	0
$451$	− 8.77627i	− 0.413258i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−25.8919	−1.21383
$456$	0	0
$457$	−24.6210	−1.15172	−0.575861	−	0.817548i	$-0.695333\pi$
$457$	−24.6210	−1.15172	−0.575861	+	0.817548i	$0.695333\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.1883i	1.07999i	0.841669	+	0.539994i	$0.181573\pi$
$461$	23.1883i	1.07999i	−0.841669	+	0.539994i	$0.818427\pi$
$462$	0	0
$463$	27.1595i	1.26221i	0.775698	+	0.631104i	$0.217398\pi$
$463$	27.1595i	1.26221i	−0.775698	+	0.631104i	$0.782602\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.74128	0.265675	0.132837	−	0.991138i	$-0.457591\pi$
$467$	5.74128	0.265675	0.132837	+	0.991138i	$0.457591\pi$
$468$	0	0
$469$	9.90388	0.457319
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.07407i	0.417226i
$474$	0	0
$475$	− 5.03031i	− 0.230806i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.7151	−0.718042	−0.359021	−	0.933330i	$-0.616889\pi$
$479$	−15.7151	−0.718042	−0.359021	+	0.933330i	$0.616889\pi$
$480$	0	0
$481$	−18.1230	−0.826337
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.216088i	0.00981205i
$486$	0	0
$487$	13.2577i	0.600766i	0.953819	+	0.300383i	$0.0971145\pi$
$487$	13.2577i	0.600766i	−0.953819	+	0.300383i	$0.902886\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−24.3421	−1.09854	−0.549272	−	0.835644i	$-0.685095\pi$
$491$	−24.3421	−1.09854	−0.549272	+	0.835644i	$0.685095\pi$
$492$	0	0
$493$	−18.4019	−0.828779
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.9109i	− 0.579131i
$498$	0	0
$499$	14.3205i	0.641072i	0.947236	+	0.320536i	$0.103863\pi$
$499$	14.3205i	0.641072i	−0.947236	+	0.320536i	$0.896137\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.7702	1.19362	0.596811	−	0.802382i	$-0.296434\pi$
$503$	26.7702	1.19362	0.596811	+	0.802382i	$0.296434\pi$
$504$	0	0
$505$	−7.20694	−0.320705
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 14.8673i	− 0.658984i	−0.944158	−	0.329492i	$-0.893123\pi$
$509$	− 14.8673i	− 0.658984i	0.944158	−	0.329492i	$-0.106877\pi$
$510$	0	0
$511$	− 16.7218i	− 0.739731i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−23.1949	−1.02209
$516$	0	0
$517$	−13.3784	−0.588380
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.8761i	1.35270i	0.736578	+	0.676352i	$0.236440\pi$
$521$	30.8761i	1.35270i	−0.736578	+	0.676352i	$0.763560\pi$
$522$	0	0
$523$	− 9.32278i	− 0.407657i	−0.979007	−	0.203828i	$-0.934662\pi$
$523$	− 9.32278i	− 0.407657i	0.979007	−	0.203828i	$-0.0653384\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	46.6210	2.03084
$528$	0	0
$529$	19.8461	0.862874
$530$	0	0
$531$	0	0
$532$	0	0
$533$	58.4082i	2.52994i
$534$	0	0
$535$	− 1.64237i	− 0.0710059i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.10955	0.133938
$540$	0	0
$541$	16.3965	0.704942	0.352471	−	0.935823i	$-0.385342\pi$
$541$	16.3965	0.704942	0.352471	+	0.935823i	$0.385342\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	26.1717i	1.12107i
$546$	0	0
$547$	− 0.954670i	− 0.0408187i	−0.999792	−	0.0204094i	$-0.993503\pi$
$547$	− 0.954670i	− 0.0408187i	0.999792	−	0.0204094i	$-0.00649695\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15.7049	0.669053
$552$	0	0
$553$	17.0652	0.725687
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.13835i	− 0.217719i	−0.994057	−	0.108859i	$-0.965280\pi$
$557$	− 5.13835i	− 0.217719i	0.994057	−	0.108859i	$-0.0347198\pi$
$558$	0	0
$559$	− 60.3902i	− 2.55423i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.6733	0.618408	0.309204	−	0.950996i	$-0.399937\pi$
$563$	14.6733	0.618408	0.309204	+	0.950996i	$0.399937\pi$
$564$	0	0
$565$	−26.2527	−1.10446
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.12097i	0.172760i	0.996262	+	0.0863801i	$0.0275299\pi$
$569$	4.12097i	0.172760i	−0.996262	+	0.0863801i	$0.972470\pi$
$570$	0	0
$571$	− 11.8087i	− 0.494179i	−0.968993	−	0.247090i	$-0.920526\pi$
$571$	− 11.8087i	− 0.494179i	0.968993	−	0.247090i	$-0.0794742\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.26281	0.302880
$576$	0	0
$577$	21.3367	0.888257	0.444129	−	0.895963i	$-0.353513\pi$
$577$	21.3367	0.888257	0.444129	+	0.895963i	$0.353513\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 15.9955i	− 0.663603i
$582$	0	0
$583$	4.95578i	0.205248i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.7958	1.97274	0.986371	−	0.164536i	$-0.0526127\pi$
$587$	47.7958	1.97274	0.986371	+	0.164536i	$0.0526127\pi$
$588$	0	0
$589$	−39.7883	−1.63945
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 32.8089i	− 1.34730i	−0.739051	−	0.673650i	$-0.764726\pi$
$593$	− 32.8089i	− 1.34730i	0.739051	−	0.673650i	$-0.235274\pi$
$594$	0	0
$595$	20.6667i	0.847252i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.9994	−0.980587	−0.490294	−	0.871557i	$-0.663110\pi$
$599$	−23.9994	−0.980587	−0.490294	+	0.871557i	$0.663110\pi$
$600$	0	0
$601$	−47.2346	−1.92674	−0.963369	−	0.268180i	$-0.913578\pi$
$601$	−47.2346	−1.92674	−0.963369	+	0.268180i	$0.913578\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.97242i	0.0801903i
$606$	0	0
$607$	− 35.1872i	− 1.42820i	−0.700042	−	0.714101i	$-0.746836\pi$
$607$	− 35.1872i	− 1.42820i	0.700042	−	0.714101i	$-0.253164\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	89.0363	3.60202
$612$	0	0
$613$	−44.1849	−1.78461	−0.892305	−	0.451434i	$-0.850913\pi$
$613$	−44.1849	−1.78461	−0.892305	+	0.451434i	$0.850913\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 37.9078i	− 1.52611i	−0.646333	−	0.763055i	$-0.723698\pi$
$617$	− 37.9078i	− 1.52611i	0.646333	−	0.763055i	$-0.276302\pi$
$618$	0	0
$619$	− 48.2320i	− 1.93861i	−0.245864	−	0.969304i	$-0.579072\pi$
$619$	− 48.2320i	− 1.93861i	0.245864	−	0.969304i	$-0.420928\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.0780	−0.523957
$624$	0	0
$625$	−18.2211	−0.728845
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.4656i	0.576782i
$630$	0	0
$631$	0.422528i	0.0168206i	0.999965	+	0.00841029i	$0.00267711\pi$
$631$	0.422528i	0.0168206i	−0.999965	+	0.00841029i	$0.997323\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.77489	−0.387905
$636$	0	0
$637$	−20.6949	−0.819960
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 19.6561i	− 0.776370i	−0.921581	−	0.388185i	$-0.873102\pi$
$641$	− 19.6561i	− 0.776370i	0.921581	−	0.388185i	$-0.126898\pi$
$642$	0	0
$643$	− 5.00756i	− 0.197479i	−0.995113	−	0.0987394i	$-0.968519\pi$
$643$	− 5.00756i	− 0.197479i	0.995113	−	0.0987394i	$-0.0314810\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.9321	1.09813	0.549063	−	0.835781i	$-0.314985\pi$
$647$	27.9321	1.09813	0.549063	+	0.835781i	$0.314985\pi$
$648$	0	0
$649$	10.4778	0.411291
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 13.2250i	− 0.517533i	−0.965940	−	0.258767i	$-0.916684\pi$
$653$	− 13.2250i	− 0.517533i	0.965940	−	0.258767i	$-0.0833161\pi$
$654$	0	0
$655$	9.31597i	0.364005i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−34.6210	−1.34864	−0.674321	−	0.738438i	$-0.735564\pi$
$659$	−34.6210	−1.34864	−0.674321	+	0.738438i	$0.735564\pi$
$660$	0	0
$661$	47.3777	1.84278	0.921390	−	0.388640i	$-0.127055\pi$
$661$	47.3777	1.84278	0.921390	+	0.388640i	$0.127055\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 17.6378i	− 0.683966i
$666$	0	0
$667$	22.6749i	0.877977i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.6552	−0.411341
$672$	0	0
$673$	12.5612	0.484199	0.242100	−	0.970251i	$-0.422164\pi$
$673$	12.5612	0.484199	0.242100	+	0.970251i	$0.422164\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.7460i	0.451436i	0.974193	+	0.225718i	$0.0724727\pi$
$677$	11.7460i	0.451436i	−0.974193	+	0.225718i	$0.927527\pi$
$678$	0	0
$679$	− 0.216088i	− 0.00829270i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13.1512	−0.503216	−0.251608	−	0.967829i	$-0.580959\pi$
$683$	−13.1512	−0.503216	−0.251608	+	0.967829i	$0.580959\pi$
$684$	0	0
$685$	−21.5430	−0.823117
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 32.9820i	− 1.25651i
$690$	0	0
$691$	− 25.6742i	− 0.976694i	−0.872649	−	0.488347i	$-0.837600\pi$
$691$	− 25.6742i	− 0.976694i	0.872649	−	0.488347i	$-0.162400\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.3361	0.543799
$696$	0	0
$697$	46.6210	1.76590
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.8088i	0.861477i	0.902477	+	0.430738i	$0.141747\pi$
$701$	22.8088i	0.861477i	−0.902477	+	0.430738i	$0.858253\pi$
$702$	0	0
$703$	− 12.3456i	− 0.465622i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.20694	0.271045
$708$	0	0
$709$	28.7567	1.07998	0.539991	−	0.841671i	$-0.318428\pi$
$709$	28.7567	1.07998	0.539991	+	0.841671i	$0.318428\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 57.4468i	− 2.15140i
$714$	0	0
$715$	− 13.1270i	− 0.490920i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−43.2030	−1.61120	−0.805600	−	0.592459i	$-0.798157\pi$
$719$	−43.2030	−1.61120	−0.805600	+	0.592459i	$0.798157\pi$
$720$	0	0
$721$	23.1949	0.863825
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.84361i	0.142748i
$726$	0	0
$727$	− 1.44222i	− 0.0534891i	−0.999642	−	0.0267445i	$-0.991486\pi$
$727$	− 1.44222i	− 0.0534891i	0.999642	−	0.0267445i	$-0.00851406\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−48.2029	−1.78285
$732$	0	0
$733$	4.83059	0.178422	0.0892109	−	0.996013i	$-0.471565\pi$
$733$	4.83059	0.178422	0.0892109	+	0.996013i	$0.471565\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.02118i	0.184958i
$738$	0	0
$739$	− 14.9282i	− 0.549144i	−0.961566	−	0.274572i	$-0.911464\pi$
$739$	− 14.9282i	− 0.549144i	0.961566	−	0.274572i	$-0.0885362\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.6008	−0.902517	−0.451258	−	0.892393i	$-0.649025\pi$
$743$	−24.6008	−0.902517	−0.451258	+	0.892393i	$0.649025\pi$
$744$	0	0
$745$	−2.47039	−0.0905082
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.64237i	0.0600109i
$750$	0	0
$751$	10.5494i	0.384954i	0.981301	+	0.192477i	$0.0616521\pi$
$751$	10.5494i	0.384954i	−0.981301	+	0.192477i	$0.938348\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.06521	−0.184342
$756$	0	0
$757$	32.9059	1.19598	0.597992	−	0.801502i	$-0.295965\pi$
$757$	32.9059	1.19598	0.597992	+	0.801502i	$0.295965\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	52.5005i	1.90314i	0.307431	+	0.951570i	$0.400531\pi$
$761$	52.5005i	1.90314i	−0.307431	+	0.951570i	$0.599469\pi$
$762$	0	0
$763$	− 26.1717i	− 0.947478i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−69.7326	−2.51790
$768$	0	0
$769$	−21.1116	−0.761302	−0.380651	−	0.924719i	$-0.624300\pi$
$769$	−21.1116	−0.761302	−0.380651	+	0.924719i	$0.624300\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 0.297425i	− 0.0106976i	−0.999986	−	0.00534882i	$-0.998297\pi$
$773$	− 0.297425i	− 0.0106976i	0.999986	−	0.00534882i	$-0.00170259\pi$
$774$	0	0
$775$	− 9.73775i	− 0.349790i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−39.7883	−1.42556
$780$	0	0
$781$	6.54569	0.234223
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.24495i	0.115817i
$786$	0	0
$787$	− 12.8881i	− 0.459412i	−0.973260	−	0.229706i	$-0.926224\pi$
$787$	− 12.8881i	− 0.459412i	0.973260	−	0.229706i	$-0.0737764\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	26.2527	0.933439
$792$	0	0
$793$	70.9133	2.51821
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.1357i	− 1.06746i	−0.845655	−	0.533730i	$-0.820790\pi$
$797$	− 30.1357i	− 1.06746i	0.845655	−	0.533730i	$-0.179210\pi$
$798$	0	0
$799$	− 71.0681i	− 2.51421i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.47783	0.299176
$804$	0	0
$805$	25.4657	0.897547
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 32.6614i	− 1.14831i	−0.818745	−	0.574157i	$-0.805330\pi$
$809$	− 32.6614i	− 1.14831i	0.818745	−	0.574157i	$-0.194670\pi$
$810$	0	0
$811$	27.9213i	0.980451i	0.871596	+	0.490225i	$0.163086\pi$
$811$	27.9213i	0.980451i	−0.871596	+	0.490225i	$0.836914\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.8401	−0.519826
$816$	0	0
$817$	41.1384	1.43925
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 16.3191i	− 0.569539i	−0.958596	−	0.284770i	$-0.908083\pi$
$821$	− 16.3191i	− 0.569539i	0.958596	−	0.284770i	$-0.0919171\pi$
$822$	0	0
$823$	5.58953i	0.194839i	0.995243	+	0.0974194i	$0.0310588\pi$
$823$	5.58953i	0.194839i	−0.995243	+	0.0974194i	$0.968941\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.7829	0.374958	0.187479	−	0.982269i	$-0.439968\pi$
$827$	10.7829	0.374958	0.187479	+	0.982269i	$0.439968\pi$
$828$	0	0
$829$	−33.1116	−1.15001	−0.575006	−	0.818149i	$-0.695000\pi$
$829$	−33.1116	−1.15001	−0.575006	+	0.818149i	$0.695000\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16.5185i	0.572331i
$834$	0	0
$835$	48.4967i	1.67830i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.50135	0.120880	0.0604400	−	0.998172i	$-0.480750\pi$
$839$	3.50135	0.120880	0.0604400	+	0.998172i	$0.480750\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	61.7217i	2.12329i
$846$	0	0
$847$	− 1.97242i	− 0.0677732i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	17.8247	0.611022
$852$	0	0
$853$	−31.5511	−1.08029	−0.540145	−	0.841572i	$-0.681631\pi$
$853$	−31.5511	−1.08029	−0.540145	+	0.841572i	$0.681631\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.3575i	1.24195i	0.783830	+	0.620975i	$0.213263\pi$
$857$	36.3575i	1.24195i	−0.783830	+	0.620975i	$0.786737\pi$
$858$	0	0
$859$	13.3454i	0.455338i	0.973739	+	0.227669i	$0.0731104\pi$
$859$	13.3454i	0.455338i	−0.973739	+	0.227669i	$0.926890\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.3784	−1.27237	−0.636187	−	0.771535i	$-0.719489\pi$
$863$	−37.3784	−1.27237	−0.636187	+	0.771535i	$0.719489\pi$
$864$	0	0
$865$	27.7883	0.944831
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.65191i	0.293496i
$870$	0	0
$871$	− 33.4172i	− 1.13230i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	23.7689	0.803535
$876$	0	0
$877$	11.9657	0.404054	0.202027	−	0.979380i	$-0.435247\pi$
$877$	11.9657	0.404054	0.202027	+	0.979380i	$0.435247\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	17.2221i	0.580228i	0.956992	+	0.290114i	$0.0936932\pi$
$881$	17.2221i	0.580228i	−0.956992	+	0.290114i	$0.906307\pi$
$882$	0	0
$883$	23.0711i	0.776406i	0.921574	+	0.388203i	$0.126904\pi$
$883$	23.0711i	0.776406i	−0.921574	+	0.388203i	$0.873096\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.5296	1.12581	0.562907	−	0.826520i	$-0.309683\pi$
$887$	33.5296	1.12581	0.562907	+	0.826520i	$0.309683\pi$
$888$	0	0
$889$	9.77489	0.327839
$890$	0	0
$891$	0	0
$892$	0	0
$893$	60.6525i	2.02966i
$894$	0	0
$895$	− 22.8194i	− 0.762767i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.4019	1.01396
$900$	0	0
$901$	−26.3259	−0.877044
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 31.1929i	− 1.03689i
$906$	0	0
$907$	14.6704i	0.487123i	0.969885	+	0.243561i	$0.0783157\pi$
$907$	14.6704i	0.487123i	−0.969885	+	0.243561i	$0.921684\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.37093	−0.177947	−0.0889734	−	0.996034i	$-0.528359\pi$
$911$	−5.37093	−0.177947	−0.0889734	+	0.996034i	$0.528359\pi$
$912$	0	0
$913$	8.10955	0.268387
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9.31597i	− 0.307640i
$918$	0	0
$919$	32.7858i	1.08150i	0.841183	+	0.540751i	$0.181860\pi$
$919$	32.7858i	1.08150i	−0.841183	+	0.540751i	$0.818140\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−43.5632	−1.43390
$924$	0	0
$925$	3.02144	0.0993444
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25.9697i	0.852039i	0.904714	+	0.426020i	$0.140085\pi$
$929$	25.9697i	0.852039i	−0.904714	+	0.426020i	$0.859915\pi$
$930$	0	0
$931$	− 14.0976i	− 0.462029i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.4778	−0.342662
$936$	0	0
$937$	−5.92405	−0.193530	−0.0967652	−	0.995307i	$-0.530850\pi$
$937$	−5.92405	−0.193530	−0.0967652	+	0.995307i	$0.530850\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 50.1104i	− 1.63355i	−0.576955	−	0.816776i	$-0.695759\pi$
$941$	− 50.1104i	− 1.63355i	0.576955	−	0.816776i	$-0.304241\pi$
$942$	0	0
$943$	− 57.4468i	− 1.87072i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.4019	0.987928	0.493964	−	0.869482i	$-0.335547\pi$
$947$	30.4019	0.987928	0.493964	+	0.869482i	$0.335547\pi$
$948$	0	0
$949$	−56.4221	−1.83154
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 54.6372i	− 1.76987i	−0.465712	−	0.884936i	$-0.654202\pi$
$953$	− 54.6372i	− 1.76987i	0.465712	−	0.884936i	$-0.345798\pi$
$954$	0	0
$955$	− 2.60079i	− 0.0841596i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	21.5430	0.695661
$960$	0	0
$961$	−46.0229	−1.48461
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 8.39999i	− 0.270405i
$966$	0	0
$967$	19.5250i	0.627880i	0.949443	+	0.313940i	$0.101649\pi$
$967$	19.5250i	0.627880i	−0.949443	+	0.313940i	$0.898351\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−50.7567	−1.62886	−0.814430	−	0.580261i	$-0.802950\pi$
$971$	−50.7567	−1.62886	−0.814430	+	0.580261i	$0.802950\pi$
$972$	0	0
$973$	−14.3361	−0.459594
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 30.0181i	− 0.960365i	−0.877169	−	0.480182i	$-0.840570\pi$
$977$	− 30.0181i	− 0.960365i	0.877169	−	0.480182i	$-0.159430\pi$
$978$	0	0
$979$	− 6.63040i	− 0.211909i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.9637	−0.923800	−0.461900	−	0.886932i	$-0.652832\pi$
$983$	−28.9637	−0.923800	−0.461900	+	0.886932i	$0.652832\pi$
$984$	0	0
$985$	8.35484	0.266207
$986$	0	0
$987$	0	0
$988$	0	0
$989$	59.3961i	1.88868i
$990$	0	0
$991$	− 5.45965i	− 0.173431i	−0.996233	−	0.0867157i	$-0.972363\pi$
$991$	− 5.45965i	− 0.173431i	0.996233	−	0.0867157i	$-0.0276372\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.09612	0.0664514
$996$	0	0
$997$	−7.06987	−0.223905	−0.111953	−	0.993714i	$-0.535710\pi$
$997$	−7.06987	−0.223905	−0.111953	+	0.993714i	$0.535710\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.d.e.3455.6		8
3.2	odd	2		6336.2.d.c.3455.3		8
4.3	odd	2		6336.2.d.c.3455.6		8
8.3	odd	2		1584.2.d.d.287.3	yes	8
8.5	even	2		1584.2.d.c.287.3	✓	8
12.11	even	2	inner	6336.2.d.e.3455.3		8
24.5	odd	2		1584.2.d.d.287.6	yes	8
24.11	even	2		1584.2.d.c.287.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1584.2.d.c.287.3	✓	8	8.5	even	2
1584.2.d.c.287.6	yes	8	24.11	even	2
1584.2.d.d.287.3	yes	8	8.3	odd	2
1584.2.d.d.287.6	yes	8	24.5	odd	2
6336.2.d.c.3455.3		8	3.2	odd	2
6336.2.d.c.3455.6		8	4.3	odd	2
6336.2.d.e.3455.3		8	12.11	even	2	inner
6336.2.d.e.3455.6		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.97242i q^{5} -1.97242i q^{7} +1.00000 q^{11} -6.65525 q^{13} +5.31217i q^{17} -4.53363i q^{19} +6.54569 q^{23} +1.10955 q^{25} +3.46410i q^{29} -8.77627i q^{31} +3.89045 q^{35} +2.72311 q^{37} -8.77627i q^{41} +9.07407i q^{43} -13.3784 q^{47} +3.10955 q^{49} +4.95578i q^{53} +1.97242i q^{55} +10.4778 q^{59} -10.6552 q^{61} -13.1270i q^{65} +5.02118i q^{67} +6.54569 q^{71} +8.47783 q^{73} -1.97242i q^{77} +8.65191i q^{79} +8.10955 q^{83} -10.4778 q^{85} -6.63040i q^{89} +13.1270i q^{91} +8.94222 q^{95} +0.109555 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 16 q^{25} + 56 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{59} - 24 q^{61} + 16 q^{71} + 40 q^{83} - 16 q^{85} - 8 q^{95} - 24 q^{97}+O(q^{100}) \) 8 * q + 8 * q^11 + 8 * q^13 + 16 * q^23 - 16 * q^25 + 56 * q^35 - 8 * q^37 - 16 * q^47 + 16 * q^59 - 24 * q^61 + 16 * q^71 + 40 * q^83 - 16 * q^85 - 8 * q^95 - 24 * q^97

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