LMFDB - Embedded newform 5850.2.e.bk.5149.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5850,2,Mod(5149,5850)] 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("5850.5149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.3
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.bk.5149.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -2.82843i q^{7} -1.00000i q^{8} +5.65685 q^{11} -1.00000i q^{13} +2.82843 q^{14} +1.00000 q^{16} -4.82843i q^{17} +2.82843 q^{19} +5.65685i q^{22} -8.48528i q^{23} +1.00000 q^{26} +2.82843i q^{28} -3.17157 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.82843 q^{34} +0.343146i q^{37} +2.82843i q^{38} -3.65685 q^{41} -1.65685i q^{43} -5.65685 q^{44} +8.48528 q^{46} -8.00000i q^{47} -1.00000 q^{49} +1.00000i q^{52} +9.31371i q^{53} -2.82843 q^{56} -3.17157i q^{58} -13.6569 q^{59} +6.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +5.65685i q^{67} +4.82843i q^{68} -5.65685 q^{71} +2.48528i q^{73} -0.343146 q^{74} -2.82843 q^{76} -16.0000i q^{77} -13.6569 q^{79} -3.65685i q^{82} -17.6569i q^{83} +1.65685 q^{86} -5.65685i q^{88} -4.34315 q^{89} -2.82843 q^{91} +8.48528i q^{92} +8.00000 q^{94} +8.82843i q^{97} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{16} + 4 q^{26} - 24 q^{29} + 16 q^{31} + 8 q^{34} + 8 q^{41} - 4 q^{49} - 32 q^{59} + 24 q^{61} - 4 q^{64} - 24 q^{74} - 32 q^{79} - 16 q^{86} - 40 q^{89} + 32 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^16 + 4 * q^26 - 24 * q^29 + 16 * q^31 + 8 * q^34 + 8 * q^41 - 4 * q^49 - 32 * q^59 + 24 * q^61 - 4 * q^64 - 24 * q^74 - 32 * q^79 - 16 * q^86 - 40 * q^89 + 32 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	2.82843	0.755929
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.82843i	− 1.17107i	−0.810649	−	0.585533i	$-0.800885\pi$
$17$	− 4.82843i	− 1.17107i	0.810649	−	0.585533i	$-0.199115\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	5.65685i	1.20605i
$23$	− 8.48528i	− 1.76930i	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	− 8.48528i	− 1.76930i	0.466252	−	0.884652i	$-0.345604\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	2.82843i	0.534522i
$29$	−3.17157	−0.588946	−0.294473	−	0.955660i	$-0.595144\pi$
$29$	−3.17157	−0.588946	−0.294473	+	0.955660i	$0.595144\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	4.82843	0.828068
$35$	0	0
$36$	0	0
$37$	0.343146i	0.0564128i	0.999602	+	0.0282064i	$0.00897957\pi$
$37$	0.343146i	0.0564128i	−0.999602	+	0.0282064i	$0.991020\pi$
$38$	2.82843i	0.458831i
$39$	0	0
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	− 1.65685i	− 0.252668i	−0.991988	−	0.126334i	$-0.959679\pi$
$43$	− 1.65685i	− 0.252668i	0.991988	−	0.126334i	$-0.0403211\pi$
$44$	−5.65685	−0.852803
$45$	0	0
$46$	8.48528	1.25109
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	9.31371i	1.27934i	0.768651	+	0.639668i	$0.220928\pi$
$53$	9.31371i	1.27934i	−0.768651	+	0.639668i	$0.779072\pi$
$54$	0	0
$55$	0	0
$56$	−2.82843	−0.377964
$57$	0	0
$58$	− 3.17157i	− 0.416448i
$59$	−13.6569	−1.77797	−0.888985	−	0.457935i	$-0.848589\pi$
$59$	−13.6569	−1.77797	−0.888985	+	0.457935i	$0.848589\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	5.65685i	0.691095i	0.938401	+	0.345547i	$0.112307\pi$
$67$	5.65685i	0.691095i	−0.938401	+	0.345547i	$0.887693\pi$
$68$	4.82843i	0.585533i
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	2.48528i	0.290880i	0.989367	+	0.145440i	$0.0464598\pi$
$73$	2.48528i	0.290880i	−0.989367	+	0.145440i	$0.953540\pi$
$74$	−0.343146	−0.0398899
$75$	0	0
$76$	−2.82843	−0.324443
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	−13.6569	−1.53652	−0.768258	−	0.640140i	$-0.778876\pi$
$79$	−13.6569	−1.53652	−0.768258	+	0.640140i	$0.778876\pi$
$80$	0	0
$81$	0	0
$82$	− 3.65685i	− 0.403832i
$83$	− 17.6569i	− 1.93809i	−0.246881	−	0.969046i	$-0.579406\pi$
$83$	− 17.6569i	− 1.93809i	0.246881	−	0.969046i	$-0.420594\pi$
$84$	0	0
$85$	0	0
$86$	1.65685	0.178663
$87$	0	0
$88$	− 5.65685i	− 0.603023i
$89$	−4.34315	−0.460373	−0.230186	−	0.973147i	$-0.573934\pi$
$89$	−4.34315	−0.460373	−0.230186	+	0.973147i	$0.573934\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	8.48528i	0.884652i
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	0	0
$97$	8.82843i	0.896391i	0.893936	+	0.448195i	$0.147933\pi$
$97$	8.82843i	0.896391i	−0.893936	+	0.448195i	$0.852067\pi$
$98$	− 1.00000i	− 0.101015i
$99$	0	0
$100$	0	0
$101$	12.1421	1.20819	0.604094	−	0.796913i	$-0.293535\pi$
$101$	12.1421	1.20819	0.604094	+	0.796913i	$0.293535\pi$
$102$	0	0
$103$	9.65685i	0.951518i	0.879576	+	0.475759i	$0.157827\pi$
$103$	9.65685i	0.951518i	−0.879576	+	0.475759i	$0.842173\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−9.31371	−0.904627
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	−3.17157	−0.303782	−0.151891	−	0.988397i	$-0.548536\pi$
$109$	−3.17157	−0.303782	−0.151891	+	0.988397i	$0.548536\pi$
$110$	0	0
$111$	0	0
$112$	− 2.82843i	− 0.267261i
$113$	10.4853i	0.986372i	0.869924	+	0.493186i	$0.164168\pi$
$113$	10.4853i	0.986372i	−0.869924	+	0.493186i	$0.835832\pi$
$114$	0	0
$115$	0	0
$116$	3.17157	0.294473
$117$	0	0
$118$	− 13.6569i	− 1.25722i
$119$	−13.6569	−1.25192
$120$	0	0
$121$	21.0000	1.90909
$122$	6.00000i	0.543214i
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	1.65685i	0.147022i	0.997294	+	0.0735110i	$0.0234204\pi$
$127$	1.65685i	0.147022i	−0.997294	+	0.0735110i	$0.976580\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−22.1421	−1.93457	−0.967284	−	0.253697i	$-0.918353\pi$
$131$	−22.1421	−1.93457	−0.967284	+	0.253697i	$0.918353\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	−5.65685	−0.488678
$135$	0	0
$136$	−4.82843	−0.414034
$137$	5.31371i	0.453981i	0.973897	+	0.226990i	$0.0728886\pi$
$137$	5.31371i	0.453981i	−0.973897	+	0.226990i	$0.927111\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	0	0
$141$	0	0
$142$	− 5.65685i	− 0.474713i
$143$	− 5.65685i	− 0.473050i
$144$	0	0
$145$	0	0
$146$	−2.48528	−0.205683
$147$	0	0
$148$	− 0.343146i	− 0.0282064i
$149$	−7.65685	−0.627274	−0.313637	−	0.949543i	$-0.601547\pi$
$149$	−7.65685	−0.627274	−0.313637	+	0.949543i	$0.601547\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	− 2.82843i	− 0.229416i
$153$	0	0
$154$	16.0000	1.28932
$155$	0	0
$156$	0	0
$157$	17.3137i	1.38178i	0.722958	+	0.690892i	$0.242782\pi$
$157$	17.3137i	1.38178i	−0.722958	+	0.690892i	$0.757218\pi$
$158$	− 13.6569i	− 1.08648i
$159$	0	0
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	− 11.3137i	− 0.886158i	−0.896483	−	0.443079i	$-0.853886\pi$
$163$	− 11.3137i	− 0.886158i	0.896483	−	0.443079i	$-0.146114\pi$
$164$	3.65685	0.285552
$165$	0	0
$166$	17.6569	1.37044
$167$	− 24.9706i	− 1.93228i	−0.258018	−	0.966140i	$-0.583069\pi$
$167$	− 24.9706i	− 1.93228i	0.258018	−	0.966140i	$-0.416931\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	1.65685i	0.126334i
$173$	− 13.3137i	− 1.01222i	−0.862468	−	0.506111i	$-0.831083\pi$
$173$	− 13.3137i	− 1.01222i	0.862468	−	0.506111i	$-0.168917\pi$
$174$	0	0
$175$	0	0
$176$	5.65685	0.426401
$177$	0	0
$178$	− 4.34315i	− 0.325533i
$179$	24.4853	1.83012	0.915058	−	0.403322i	$-0.132145\pi$
$179$	24.4853	1.83012	0.915058	+	0.403322i	$0.132145\pi$
$180$	0	0
$181$	3.65685	0.271812	0.135906	−	0.990722i	$-0.456606\pi$
$181$	3.65685	0.271812	0.135906	+	0.990722i	$0.456606\pi$
$182$	− 2.82843i	− 0.209657i
$183$	0	0
$184$	−8.48528	−0.625543
$185$	0	0
$186$	0	0
$187$	− 27.3137i	− 1.99738i
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	− 14.4853i	− 1.04267i	−0.853351	−	0.521337i	$-0.825434\pi$
$193$	− 14.4853i	− 1.04267i	0.853351	−	0.521337i	$-0.174566\pi$
$194$	−8.82843	−0.633844
$195$	0	0
$196$	1.00000	0.0714286
$197$	9.31371i	0.663574i	0.943354	+	0.331787i	$0.107652\pi$
$197$	9.31371i	0.663574i	−0.943354	+	0.331787i	$0.892348\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	0	0
$201$	0	0
$202$	12.1421i	0.854318i
$203$	8.97056i	0.629610i
$204$	0	0
$205$	0	0
$206$	−9.65685	−0.672825
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	16.0000	1.10674
$210$	0	0
$211$	23.3137	1.60498	0.802491	−	0.596664i	$-0.203508\pi$
$211$	23.3137	1.60498	0.802491	+	0.596664i	$0.203508\pi$
$212$	− 9.31371i	− 0.639668i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	− 11.3137i	− 0.768025i
$218$	− 3.17157i	− 0.214806i
$219$	0	0
$220$	0	0
$221$	−4.82843	−0.324795
$222$	0	0
$223$	5.17157i	0.346314i	0.984894	+	0.173157i	$0.0553968\pi$
$223$	5.17157i	0.346314i	−0.984894	+	0.173157i	$0.944603\pi$
$224$	2.82843	0.188982
$225$	0	0
$226$	−10.4853	−0.697471
$227$	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
$229$	24.1421	1.59536	0.797679	−	0.603083i	$-0.206061\pi$
$229$	24.1421	1.59536	0.797679	+	0.603083i	$0.206061\pi$
$230$	0	0
$231$	0	0
$232$	3.17157i	0.208224i
$233$	− 22.4853i	− 1.47306i	−0.676405	−	0.736530i	$-0.736463\pi$
$233$	− 22.4853i	− 1.47306i	0.676405	−	0.736530i	$-0.263537\pi$
$234$	0	0
$235$	0	0
$236$	13.6569	0.888985
$237$	0	0
$238$	− 13.6569i	− 0.885242i
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−17.3137	−1.11527	−0.557637	−	0.830085i	$-0.688292\pi$
$241$	−17.3137	−1.11527	−0.557637	+	0.830085i	$0.688292\pi$
$242$	21.0000i	1.34993i
$243$	0	0
$244$	−6.00000	−0.384111
$245$	0	0
$246$	0	0
$247$	− 2.82843i	− 0.179969i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−5.17157	−0.326427	−0.163213	−	0.986591i	$-0.552186\pi$
$251$	−5.17157	−0.326427	−0.163213	+	0.986591i	$0.552186\pi$
$252$	0	0
$253$	− 48.0000i	− 3.01773i
$254$	−1.65685	−0.103960
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.828427i	0.0516759i	0.999666	+	0.0258379i	$0.00822538\pi$
$257$	0.828427i	0.0516759i	−0.999666	+	0.0258379i	$0.991775\pi$
$258$	0	0
$259$	0.970563	0.0603078
$260$	0	0
$261$	0	0
$262$	− 22.1421i	− 1.36795i
$263$	0.485281i	0.0299237i	0.999888	+	0.0149619i	$0.00476269\pi$
$263$	0.485281i	0.0299237i	−0.999888	+	0.0149619i	$0.995237\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	− 5.65685i	− 0.345547i
$269$	2.48528	0.151530	0.0757651	−	0.997126i	$-0.475860\pi$
$269$	2.48528	0.151530	0.0757651	+	0.997126i	$0.475860\pi$
$270$	0	0
$271$	−15.3137	−0.930242	−0.465121	−	0.885247i	$-0.653989\pi$
$271$	−15.3137	−0.930242	−0.465121	+	0.885247i	$0.653989\pi$
$272$	− 4.82843i	− 0.292766i
$273$	0	0
$274$	−5.31371	−0.321013
$275$	0	0
$276$	0	0
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	− 17.6569i	− 1.05899i
$279$	0	0
$280$	0	0
$281$	−19.6569	−1.17263	−0.586315	−	0.810083i	$-0.699422\pi$
$281$	−19.6569	−1.17263	−0.586315	+	0.810083i	$0.699422\pi$
$282$	0	0
$283$	− 6.34315i	− 0.377061i	−0.982067	−	0.188530i	$-0.939628\pi$
$283$	− 6.34315i	− 0.377061i	0.982067	−	0.188530i	$-0.0603724\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	5.65685	0.334497
$287$	10.3431i	0.610537i
$288$	0	0
$289$	−6.31371	−0.371395
$290$	0	0
$291$	0	0
$292$	− 2.48528i	− 0.145440i
$293$	− 28.6274i	− 1.67243i	−0.548401	−	0.836216i	$-0.684763\pi$
$293$	− 28.6274i	− 1.67243i	0.548401	−	0.836216i	$-0.315237\pi$
$294$	0	0
$295$	0	0
$296$	0.343146	0.0199449
$297$	0	0
$298$	− 7.65685i	− 0.443550i
$299$	−8.48528	−0.490716
$300$	0	0
$301$	−4.68629	−0.270113
$302$	12.0000i	0.690522i
$303$	0	0
$304$	2.82843	0.162221
$305$	0	0
$306$	0	0
$307$	10.3431i	0.590315i	0.955449	+	0.295157i	$0.0953720\pi$
$307$	10.3431i	0.590315i	−0.955449	+	0.295157i	$0.904628\pi$
$308$	16.0000i	0.911685i
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	2.97056i	0.167906i	0.996470	+	0.0839531i	$0.0267546\pi$
$313$	2.97056i	0.167906i	−0.996470	+	0.0839531i	$0.973245\pi$
$314$	−17.3137	−0.977069
$315$	0	0
$316$	13.6569	0.768258
$317$	2.68629i	0.150877i	0.997150	+	0.0754386i	$0.0240357\pi$
$317$	2.68629i	0.150877i	−0.997150	+	0.0754386i	$0.975964\pi$
$318$	0	0
$319$	−17.9411	−1.00451
$320$	0	0
$321$	0	0
$322$	− 24.0000i	− 1.33747i
$323$	− 13.6569i	− 0.759888i
$324$	0	0
$325$	0	0
$326$	11.3137	0.626608
$327$	0	0
$328$	3.65685i	0.201916i
$329$	−22.6274	−1.24749
$330$	0	0
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	17.6569i	0.969046i
$333$	0	0
$334$	24.9706	1.36633
$335$	0	0
$336$	0	0
$337$	22.9706i	1.25129i	0.780109	+	0.625643i	$0.215163\pi$
$337$	22.9706i	1.25129i	−0.780109	+	0.625643i	$0.784837\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	22.6274	1.22534
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	−1.65685	−0.0893316
$345$	0	0
$346$	13.3137	0.715749
$347$	1.65685i	0.0889446i	0.999011	+	0.0444723i	$0.0141606\pi$
$347$	1.65685i	0.0889446i	−0.999011	+	0.0444723i	$0.985839\pi$
$348$	0	0
$349$	16.1421	0.864069	0.432034	−	0.901857i	$-0.357796\pi$
$349$	16.1421	0.864069	0.432034	+	0.901857i	$0.357796\pi$
$350$	0	0
$351$	0	0
$352$	5.65685i	0.301511i
$353$	17.3137i	0.921516i	0.887526	+	0.460758i	$0.152422\pi$
$353$	17.3137i	0.921516i	−0.887526	+	0.460758i	$0.847578\pi$
$354$	0	0
$355$	0	0
$356$	4.34315	0.230186
$357$	0	0
$358$	24.4853i	1.29409i
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	3.65685i	0.192200i
$363$	0	0
$364$	2.82843	0.148250
$365$	0	0
$366$	0	0
$367$	− 14.3431i	− 0.748706i	−0.927286	−	0.374353i	$-0.877865\pi$
$367$	− 14.3431i	− 0.748706i	0.927286	−	0.374353i	$-0.122135\pi$
$368$	− 8.48528i	− 0.442326i
$369$	0	0
$370$	0	0
$371$	26.3431	1.36767
$372$	0	0
$373$	− 25.3137i	− 1.31069i	−0.755328	−	0.655347i	$-0.772522\pi$
$373$	− 25.3137i	− 1.31069i	0.755328	−	0.655347i	$-0.227478\pi$
$374$	27.3137	1.41236
$375$	0	0
$376$	−8.00000	−0.412568
$377$	3.17157i	0.163344i
$378$	0	0
$379$	24.4853	1.25772	0.628862	−	0.777517i	$-0.283521\pi$
$379$	24.4853	1.25772	0.628862	+	0.777517i	$0.283521\pi$
$380$	0	0
$381$	0	0
$382$	11.3137i	0.578860i
$383$	18.3431i	0.937291i	0.883386	+	0.468645i	$0.155258\pi$
$383$	18.3431i	0.937291i	−0.883386	+	0.468645i	$0.844742\pi$
$384$	0	0
$385$	0	0
$386$	14.4853	0.737281
$387$	0	0
$388$	− 8.82843i	− 0.448195i
$389$	10.4853	0.531625	0.265812	−	0.964025i	$-0.414360\pi$
$389$	10.4853	0.531625	0.265812	+	0.964025i	$0.414360\pi$
$390$	0	0
$391$	−40.9706	−2.07197
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	−9.31371	−0.469218
$395$	0	0
$396$	0	0
$397$	26.2843i	1.31917i	0.751630	+	0.659585i	$0.229268\pi$
$397$	26.2843i	1.31917i	−0.751630	+	0.659585i	$0.770732\pi$
$398$	− 21.6569i	− 1.08556i
$399$	0	0
$400$	0	0
$401$	−6.97056	−0.348093	−0.174047	−	0.984737i	$-0.555684\pi$
$401$	−6.97056	−0.348093	−0.174047	+	0.984737i	$0.555684\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	−12.1421	−0.604094
$405$	0	0
$406$	−8.97056	−0.445202
$407$	1.94113i	0.0962180i
$408$	0	0
$409$	−7.65685	−0.378607	−0.189304	−	0.981919i	$-0.560623\pi$
$409$	−7.65685	−0.378607	−0.189304	+	0.981919i	$0.560623\pi$
$410$	0	0
$411$	0	0
$412$	− 9.65685i	− 0.475759i
$413$	38.6274i	1.90073i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	16.0000i	0.782586i
$419$	−5.17157	−0.252648	−0.126324	−	0.991989i	$-0.540318\pi$
$419$	−5.17157	−0.252648	−0.126324	+	0.991989i	$0.540318\pi$
$420$	0	0
$421$	4.14214	0.201875	0.100938	−	0.994893i	$-0.467816\pi$
$421$	4.14214	0.201875	0.100938	+	0.994893i	$0.467816\pi$
$422$	23.3137i	1.13489i
$423$	0	0
$424$	9.31371	0.452314
$425$	0	0
$426$	0	0
$427$	− 16.9706i	− 0.821263i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	10.9706i	0.527212i	0.964630	+	0.263606i	$0.0849118\pi$
$433$	10.9706i	0.527212i	−0.964630	+	0.263606i	$0.915088\pi$
$434$	11.3137	0.543075
$435$	0	0
$436$	3.17157	0.151891
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	22.6274	1.07995	0.539974	−	0.841682i	$-0.318434\pi$
$439$	22.6274	1.07995	0.539974	+	0.841682i	$0.318434\pi$
$440$	0	0
$441$	0	0
$442$	− 4.82843i	− 0.229665i
$443$	− 41.6569i	− 1.97918i	−0.143926	−	0.989588i	$-0.545973\pi$
$443$	− 41.6569i	− 1.97918i	0.143926	−	0.989588i	$-0.454027\pi$
$444$	0	0
$445$	0	0
$446$	−5.17157	−0.244881
$447$	0	0
$448$	2.82843i	0.133631i
$449$	−30.2843	−1.42920	−0.714602	−	0.699532i	$-0.753392\pi$
$449$	−30.2843	−1.42920	−0.714602	+	0.699532i	$0.753392\pi$
$450$	0	0
$451$	−20.6863	−0.974079
$452$	− 10.4853i	− 0.493186i
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	− 15.1716i	− 0.709696i	−0.934924	−	0.354848i	$-0.884533\pi$
$457$	− 15.1716i	− 0.709696i	0.934924	−	0.354848i	$-0.115467\pi$
$458$	24.1421i	1.12809i
$459$	0	0
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	35.7990i	1.66372i	0.554985	+	0.831860i	$0.312724\pi$
$463$	35.7990i	1.66372i	−0.554985	+	0.831860i	$0.687276\pi$
$464$	−3.17157	−0.147237
$465$	0	0
$466$	22.4853	1.04161
$467$	− 15.3137i	− 0.708634i	−0.935125	−	0.354317i	$-0.884713\pi$
$467$	− 15.3137i	− 0.708634i	0.935125	−	0.354317i	$-0.115287\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	13.6569i	0.628608i
$473$	− 9.37258i	− 0.430952i
$474$	0	0
$475$	0	0
$476$	13.6569	0.625961
$477$	0	0
$478$	16.0000i	0.731823i
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	0.343146	0.0156461
$482$	− 17.3137i	− 0.788618i
$483$	0	0
$484$	−21.0000	−0.954545
$485$	0	0
$486$	0	0
$487$	0.485281i	0.0219902i	0.999940	+	0.0109951i	$0.00349992\pi$
$487$	0.485281i	0.0219902i	−0.999940	+	0.0109951i	$0.996500\pi$
$488$	− 6.00000i	− 0.271607i
$489$	0	0
$490$	0	0
$491$	9.85786	0.444879	0.222440	−	0.974946i	$-0.428598\pi$
$491$	9.85786	0.444879	0.222440	+	0.974946i	$0.428598\pi$
$492$	0	0
$493$	15.3137i	0.689695i
$494$	2.82843	0.127257
$495$	0	0
$496$	4.00000	0.179605
$497$	16.0000i	0.717698i
$498$	0	0
$499$	16.4853	0.737983	0.368991	−	0.929433i	$-0.379703\pi$
$499$	16.4853	0.737983	0.368991	+	0.929433i	$0.379703\pi$
$500$	0	0
$501$	0	0
$502$	− 5.17157i	− 0.230819i
$503$	40.4853i	1.80515i	0.430533	+	0.902575i	$0.358326\pi$
$503$	40.4853i	1.80515i	−0.430533	+	0.902575i	$0.641674\pi$
$504$	0	0
$505$	0	0
$506$	48.0000	2.13386
$507$	0	0
$508$	− 1.65685i	− 0.0735110i
$509$	−14.6863	−0.650958	−0.325479	−	0.945549i	$-0.605526\pi$
$509$	−14.6863	−0.650958	−0.325479	+	0.945549i	$0.605526\pi$
$510$	0	0
$511$	7.02944	0.310964
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−0.828427	−0.0365404
$515$	0	0
$516$	0	0
$517$	− 45.2548i	− 1.99031i
$518$	0.970563i	0.0426441i
$519$	0	0
$520$	0	0
$521$	6.97056	0.305386	0.152693	−	0.988274i	$-0.451205\pi$
$521$	6.97056	0.305386	0.152693	+	0.988274i	$0.451205\pi$
$522$	0	0
$523$	34.6274i	1.51415i	0.653327	+	0.757076i	$0.273373\pi$
$523$	34.6274i	1.51415i	−0.653327	+	0.757076i	$0.726627\pi$
$524$	22.1421	0.967284
$525$	0	0
$526$	−0.485281	−0.0211593
$527$	− 19.3137i	− 0.841318i
$528$	0	0
$529$	−49.0000	−2.13043
$530$	0	0
$531$	0	0
$532$	8.00000i	0.346844i
$533$	3.65685i	0.158396i
$534$	0	0
$535$	0	0
$536$	5.65685	0.244339
$537$	0	0
$538$	2.48528i	0.107148i
$539$	−5.65685	−0.243658
$540$	0	0
$541$	−2.48528	−0.106851	−0.0534253	−	0.998572i	$-0.517014\pi$
$541$	−2.48528	−0.106851	−0.0534253	+	0.998572i	$0.517014\pi$
$542$	− 15.3137i	− 0.657780i
$543$	0	0
$544$	4.82843	0.207017
$545$	0	0
$546$	0	0
$547$	− 23.3137i	− 0.996822i	−0.866941	−	0.498411i	$-0.833917\pi$
$547$	− 23.3137i	− 0.996822i	0.866941	−	0.498411i	$-0.166083\pi$
$548$	− 5.31371i	− 0.226990i
$549$	0	0
$550$	0	0
$551$	−8.97056	−0.382159
$552$	0	0
$553$	38.6274i	1.64260i
$554$	26.0000	1.10463
$555$	0	0
$556$	17.6569	0.748817
$557$	33.3137i	1.41155i	0.708437	+	0.705774i	$0.249400\pi$
$557$	33.3137i	1.41155i	−0.708437	+	0.705774i	$0.750600\pi$
$558$	0	0
$559$	−1.65685	−0.0700775
$560$	0	0
$561$	0	0
$562$	− 19.6569i	− 0.829174i
$563$	41.6569i	1.75563i	0.479002	+	0.877814i	$0.340998\pi$
$563$	41.6569i	1.75563i	−0.479002	+	0.877814i	$0.659002\pi$
$564$	0	0
$565$	0	0
$566$	6.34315	0.266622
$567$	0	0
$568$	5.65685i	0.237356i
$569$	20.3431	0.852829	0.426415	−	0.904528i	$-0.359776\pi$
$569$	20.3431	0.852829	0.426415	+	0.904528i	$0.359776\pi$
$570$	0	0
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	5.65685i	0.236525i
$573$	0	0
$574$	−10.3431	−0.431715
$575$	0	0
$576$	0	0
$577$	− 27.4558i	− 1.14300i	−0.820601	−	0.571501i	$-0.806361\pi$
$577$	− 27.4558i	− 1.14300i	0.820601	−	0.571501i	$-0.193639\pi$
$578$	− 6.31371i	− 0.262616i
$579$	0	0
$580$	0	0
$581$	−49.9411	−2.07191
$582$	0	0
$583$	52.6863i	2.18204i
$584$	2.48528	0.102842
$585$	0	0
$586$	28.6274	1.18259
$587$	− 42.6274i	− 1.75942i	−0.475509	−	0.879711i	$-0.657736\pi$
$587$	− 42.6274i	− 1.75942i	0.475509	−	0.879711i	$-0.342264\pi$
$588$	0	0
$589$	11.3137	0.466173
$590$	0	0
$591$	0	0
$592$	0.343146i	0.0141032i
$593$	11.6569i	0.478690i	0.970935	+	0.239345i	$0.0769326\pi$
$593$	11.6569i	0.478690i	−0.970935	+	0.239345i	$0.923067\pi$
$594$	0	0
$595$	0	0
$596$	7.65685	0.313637
$597$	0	0
$598$	− 8.48528i	− 0.346989i
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	6.68629	0.272740	0.136370	−	0.990658i	$-0.456456\pi$
$601$	6.68629	0.272740	0.136370	+	0.990658i	$0.456456\pi$
$602$	− 4.68629i	− 0.190999i
$603$	0	0
$604$	−12.0000	−0.488273
$605$	0	0
$606$	0	0
$607$	4.97056i	0.201749i	0.994899	+	0.100874i	$0.0321640\pi$
$607$	4.97056i	0.201749i	−0.994899	+	0.100874i	$0.967836\pi$
$608$	2.82843i	0.114708i
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	− 34.2843i	− 1.38473i	−0.721548	−	0.692364i	$-0.756569\pi$
$613$	− 34.2843i	− 1.38473i	0.721548	−	0.692364i	$-0.243431\pi$
$614$	−10.3431	−0.417415
$615$	0	0
$616$	−16.0000	−0.644658
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	29.1716	1.17250	0.586252	−	0.810129i	$-0.300603\pi$
$619$	29.1716	1.17250	0.586252	+	0.810129i	$0.300603\pi$
$620$	0	0
$621$	0	0
$622$	24.0000i	0.962312i
$623$	12.2843i	0.492159i
$624$	0	0
$625$	0	0
$626$	−2.97056	−0.118728
$627$	0	0
$628$	− 17.3137i	− 0.690892i
$629$	1.65685	0.0660631
$630$	0	0
$631$	−22.3431	−0.889467	−0.444733	−	0.895663i	$-0.646702\pi$
$631$	−22.3431	−0.889467	−0.444733	+	0.895663i	$0.646702\pi$
$632$	13.6569i	0.543240i
$633$	0	0
$634$	−2.68629	−0.106686
$635$	0	0
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	− 17.9411i	− 0.710296i
$639$	0	0
$640$	0	0
$641$	−40.6274	−1.60469	−0.802343	−	0.596863i	$-0.796414\pi$
$641$	−40.6274	−1.60469	−0.802343	+	0.596863i	$0.796414\pi$
$642$	0	0
$643$	− 39.5980i	− 1.56159i	−0.624786	−	0.780796i	$-0.714814\pi$
$643$	− 39.5980i	− 1.56159i	0.624786	−	0.780796i	$-0.285186\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	13.6569	0.537322
$647$	− 8.48528i	− 0.333591i	−0.985992	−	0.166795i	$-0.946658\pi$
$647$	− 8.48528i	− 0.333591i	0.985992	−	0.166795i	$-0.0533419\pi$
$648$	0	0
$649$	−77.2548	−3.03252
$650$	0	0
$651$	0	0
$652$	11.3137i	0.443079i
$653$	− 14.2843i	− 0.558987i	−0.960148	−	0.279493i	$-0.909834\pi$
$653$	− 14.2843i	− 0.558987i	0.960148	−	0.279493i	$-0.0901665\pi$
$654$	0	0
$655$	0	0
$656$	−3.65685	−0.142776
$657$	0	0
$658$	− 22.6274i	− 0.882109i
$659$	−24.4853	−0.953811	−0.476906	−	0.878955i	$-0.658242\pi$
$659$	−24.4853	−0.953811	−0.476906	+	0.878955i	$0.658242\pi$
$660$	0	0
$661$	20.1421	0.783438	0.391719	−	0.920085i	$-0.371880\pi$
$661$	20.1421	0.783438	0.391719	+	0.920085i	$0.371880\pi$
$662$	8.48528i	0.329790i
$663$	0	0
$664$	−17.6569	−0.685219
$665$	0	0
$666$	0	0
$667$	26.9117i	1.04202i
$668$	24.9706i	0.966140i
$669$	0	0
$670$	0	0
$671$	33.9411	1.31028
$672$	0	0
$673$	− 12.6274i	− 0.486751i	−0.969932	−	0.243376i	$-0.921745\pi$
$673$	− 12.6274i	− 0.486751i	0.969932	−	0.243376i	$-0.0782548\pi$
$674$	−22.9706	−0.884793
$675$	0	0
$676$	1.00000	0.0384615
$677$	23.6569i	0.909207i	0.890694	+	0.454603i	$0.150219\pi$
$677$	23.6569i	0.909207i	−0.890694	+	0.454603i	$0.849781\pi$
$678$	0	0
$679$	24.9706	0.958282
$680$	0	0
$681$	0	0
$682$	22.6274i	0.866449i
$683$	− 22.3431i	− 0.854937i	−0.904030	−	0.427468i	$-0.859406\pi$
$683$	− 22.3431i	− 0.854937i	0.904030	−	0.427468i	$-0.140594\pi$
$684$	0	0
$685$	0	0
$686$	16.9706	0.647939
$687$	0	0
$688$	− 1.65685i	− 0.0631670i
$689$	9.31371	0.354824
$690$	0	0
$691$	11.7990	0.448855	0.224427	−	0.974491i	$-0.427949\pi$
$691$	11.7990	0.448855	0.224427	+	0.974491i	$0.427949\pi$
$692$	13.3137i	0.506111i
$693$	0	0
$694$	−1.65685	−0.0628933
$695$	0	0
$696$	0	0
$697$	17.6569i	0.668801i
$698$	16.1421i	0.610989i
$699$	0	0
$700$	0	0
$701$	28.1421	1.06291	0.531457	−	0.847085i	$-0.321645\pi$
$701$	28.1421	1.06291	0.531457	+	0.847085i	$0.321645\pi$
$702$	0	0
$703$	0.970563i	0.0366055i
$704$	−5.65685	−0.213201
$705$	0	0
$706$	−17.3137	−0.651610
$707$	− 34.3431i	− 1.29161i
$708$	0	0
$709$	12.8284	0.481782	0.240891	−	0.970552i	$-0.422560\pi$
$709$	12.8284	0.481782	0.240891	+	0.970552i	$0.422560\pi$
$710$	0	0
$711$	0	0
$712$	4.34315i	0.162766i
$713$	− 33.9411i	− 1.27111i
$714$	0	0
$715$	0	0
$716$	−24.4853	−0.915058
$717$	0	0
$718$	28.2843i	1.05556i
$719$	−18.3431	−0.684084	−0.342042	−	0.939685i	$-0.611118\pi$
$719$	−18.3431	−0.684084	−0.342042	+	0.939685i	$0.611118\pi$
$720$	0	0
$721$	27.3137	1.01722
$722$	− 11.0000i	− 0.409378i
$723$	0	0
$724$	−3.65685	−0.135906
$725$	0	0
$726$	0	0
$727$	− 21.9411i	− 0.813751i	−0.913484	−	0.406876i	$-0.866618\pi$
$727$	− 21.9411i	− 0.813751i	0.913484	−	0.406876i	$-0.133382\pi$
$728$	2.82843i	0.104828i
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	− 11.6569i	− 0.430556i	−0.976553	−	0.215278i	$-0.930934\pi$
$733$	− 11.6569i	− 0.430556i	0.976553	−	0.215278i	$-0.0690657\pi$
$734$	14.3431	0.529415
$735$	0	0
$736$	8.48528	0.312772
$737$	32.0000i	1.17874i
$738$	0	0
$739$	−14.1421	−0.520227	−0.260113	−	0.965578i	$-0.583760\pi$
$739$	−14.1421	−0.520227	−0.260113	+	0.965578i	$0.583760\pi$
$740$	0	0
$741$	0	0
$742$	26.3431i	0.967087i
$743$	− 20.2843i	− 0.744158i	−0.928201	−	0.372079i	$-0.878645\pi$
$743$	− 20.2843i	− 0.744158i	0.928201	−	0.372079i	$-0.121355\pi$
$744$	0	0
$745$	0	0
$746$	25.3137	0.926801
$747$	0	0
$748$	27.3137i	0.998688i
$749$	−11.3137	−0.413394
$750$	0	0
$751$	−11.3137	−0.412843	−0.206422	−	0.978463i	$-0.566182\pi$
$751$	−11.3137	−0.412843	−0.206422	+	0.978463i	$0.566182\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	−3.17157	−0.115502
$755$	0	0
$756$	0	0
$757$	47.9411i	1.74245i	0.490884	+	0.871225i	$0.336674\pi$
$757$	47.9411i	1.74245i	−0.490884	+	0.871225i	$0.663326\pi$
$758$	24.4853i	0.889345i
$759$	0	0
$760$	0	0
$761$	−16.3431	−0.592439	−0.296219	−	0.955120i	$-0.595726\pi$
$761$	−16.3431	−0.592439	−0.296219	+	0.955120i	$0.595726\pi$
$762$	0	0
$763$	8.97056i	0.324756i
$764$	−11.3137	−0.409316
$765$	0	0
$766$	−18.3431	−0.662765
$767$	13.6569i	0.493120i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	14.4853i	0.521337i
$773$	30.6863i	1.10371i	0.833940	+	0.551855i	$0.186080\pi$
$773$	30.6863i	1.10371i	−0.833940	+	0.551855i	$0.813920\pi$
$774$	0	0
$775$	0	0
$776$	8.82843	0.316922
$777$	0	0
$778$	10.4853i	0.375916i
$779$	−10.3431	−0.370582
$780$	0	0
$781$	−32.0000	−1.14505
$782$	− 40.9706i	− 1.46510i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	24.0000i	0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$	24.0000i	0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$	− 9.31371i	− 0.331787i
$789$	0	0
$790$	0	0
$791$	29.6569	1.05448
$792$	0	0
$793$	− 6.00000i	− 0.213066i
$794$	−26.2843	−0.932794
$795$	0	0
$796$	21.6569	0.767607
$797$	− 28.6274i	− 1.01404i	−0.861936	−	0.507018i	$-0.830748\pi$
$797$	− 28.6274i	− 1.01404i	0.861936	−	0.507018i	$-0.169252\pi$
$798$	0	0
$799$	−38.6274	−1.36654
$800$	0	0
$801$	0	0
$802$	− 6.97056i	− 0.246139i
$803$	14.0589i	0.496127i
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	− 12.1421i	− 0.427159i
$809$	−9.31371	−0.327453	−0.163726	−	0.986506i	$-0.552351\pi$
$809$	−9.31371	−0.327453	−0.163726	+	0.986506i	$0.552351\pi$
$810$	0	0
$811$	−30.1421	−1.05843	−0.529217	−	0.848487i	$-0.677514\pi$
$811$	−30.1421	−1.05843	−0.529217	+	0.848487i	$0.677514\pi$
$812$	− 8.97056i	− 0.314805i
$813$	0	0
$814$	−1.94113	−0.0680364
$815$	0	0
$816$	0	0
$817$	− 4.68629i	− 0.163953i
$818$	− 7.65685i	− 0.267716i
$819$	0	0
$820$	0	0
$821$	22.2843	0.777726	0.388863	−	0.921296i	$-0.372868\pi$
$821$	22.2843	0.777726	0.388863	+	0.921296i	$0.372868\pi$
$822$	0	0
$823$	− 19.0294i	− 0.663324i	−0.943398	−	0.331662i	$-0.892391\pi$
$823$	− 19.0294i	− 0.663324i	0.943398	−	0.331662i	$-0.107609\pi$
$824$	9.65685	0.336412
$825$	0	0
$826$	−38.6274	−1.34402
$827$	− 1.65685i	− 0.0576145i	−0.999585	−	0.0288072i	$-0.990829\pi$
$827$	− 1.65685i	− 0.0576145i	0.999585	−	0.0288072i	$-0.00917090\pi$
$828$	0	0
$829$	30.6863	1.06578	0.532889	−	0.846185i	$-0.321106\pi$
$829$	30.6863	1.06578	0.532889	+	0.846185i	$0.321106\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	4.82843i	0.167295i
$834$	0	0
$835$	0	0
$836$	−16.0000	−0.553372
$837$	0	0
$838$	− 5.17157i	− 0.178649i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	4.14214i	0.142747i
$843$	0	0
$844$	−23.3137	−0.802491
$845$	0	0
$846$	0	0
$847$	− 59.3970i	− 2.04090i
$848$	9.31371i	0.319834i
$849$	0	0
$850$	0	0
$851$	2.91169	0.0998114
$852$	0	0
$853$	− 18.2843i	− 0.626042i	−0.949746	−	0.313021i	$-0.898659\pi$
$853$	− 18.2843i	− 0.626042i	0.949746	−	0.313021i	$-0.101341\pi$
$854$	16.9706	0.580721
$855$	0	0
$856$	−4.00000	−0.136717
$857$	− 15.1716i	− 0.518251i	−0.965844	−	0.259126i	$-0.916566\pi$
$857$	− 15.1716i	− 0.518251i	0.965844	−	0.259126i	$-0.0834343\pi$
$858$	0	0
$859$	29.9411	1.02158	0.510789	−	0.859706i	$-0.329353\pi$
$859$	29.9411	1.02158	0.510789	+	0.859706i	$0.329353\pi$
$860$	0	0
$861$	0	0
$862$	16.0000i	0.544962i
$863$	− 28.2843i	− 0.962808i	−0.876499	−	0.481404i	$-0.840127\pi$
$863$	− 28.2843i	− 0.962808i	0.876499	−	0.481404i	$-0.159873\pi$
$864$	0	0
$865$	0	0
$866$	−10.9706	−0.372795
$867$	0	0
$868$	11.3137i	0.384012i
$869$	−77.2548	−2.62069
$870$	0	0
$871$	5.65685	0.191675
$872$	3.17157i	0.107403i
$873$	0	0
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	− 39.2548i	− 1.32554i	−0.748822	−	0.662771i	$-0.769380\pi$
$877$	− 39.2548i	− 1.32554i	0.748822	−	0.662771i	$-0.230620\pi$
$878$	22.6274i	0.763638i
$879$	0	0
$880$	0	0
$881$	−46.2843	−1.55936	−0.779678	−	0.626180i	$-0.784617\pi$
$881$	−46.2843	−1.55936	−0.779678	+	0.626180i	$0.784617\pi$
$882$	0	0
$883$	8.68629i	0.292317i	0.989261	+	0.146158i	$0.0466909\pi$
$883$	8.68629i	0.292317i	−0.989261	+	0.146158i	$0.953309\pi$
$884$	4.82843	0.162398
$885$	0	0
$886$	41.6569	1.39949
$887$	− 23.5147i	− 0.789547i	−0.918778	−	0.394773i	$-0.870823\pi$
$887$	− 23.5147i	− 0.789547i	0.918778	−	0.394773i	$-0.129177\pi$
$888$	0	0
$889$	4.68629	0.157173
$890$	0	0
$891$	0	0
$892$	− 5.17157i	− 0.173157i
$893$	− 22.6274i	− 0.757198i
$894$	0	0
$895$	0	0
$896$	−2.82843	−0.0944911
$897$	0	0
$898$	− 30.2843i	− 1.01060i
$899$	−12.6863	−0.423112
$900$	0	0
$901$	44.9706	1.49819
$902$	− 20.6863i	− 0.688778i
$903$	0	0
$904$	10.4853	0.348735
$905$	0	0
$906$	0	0
$907$	48.2843i	1.60325i	0.597825	+	0.801626i	$0.296032\pi$
$907$	48.2843i	1.60325i	−0.597825	+	0.801626i	$0.703968\pi$
$908$	− 4.00000i	− 0.132745i
$909$	0	0
$910$	0	0
$911$	8.97056	0.297208	0.148604	−	0.988897i	$-0.452522\pi$
$911$	8.97056	0.297208	0.148604	+	0.988897i	$0.452522\pi$
$912$	0	0
$913$	− 99.8823i	− 3.30562i
$914$	15.1716	0.501831
$915$	0	0
$916$	−24.1421	−0.797679
$917$	62.6274i	2.06814i
$918$	0	0
$919$	25.9411	0.855719	0.427859	−	0.903845i	$-0.359268\pi$
$919$	25.9411	0.855719	0.427859	+	0.903845i	$0.359268\pi$
$920$	0	0
$921$	0	0
$922$	− 14.0000i	− 0.461065i
$923$	5.65685i	0.186198i
$924$	0	0
$925$	0	0
$926$	−35.7990	−1.17643
$927$	0	0
$928$	− 3.17157i	− 0.104112i
$929$	45.5980	1.49602	0.748011	−	0.663687i	$-0.231009\pi$
$929$	45.5980	1.49602	0.748011	+	0.663687i	$0.231009\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	22.4853i	0.736530i
$933$	0	0
$934$	15.3137	0.501080
$935$	0	0
$936$	0	0
$937$	28.6274i	0.935217i	0.883936	+	0.467608i	$0.154884\pi$
$937$	28.6274i	0.935217i	−0.883936	+	0.467608i	$0.845116\pi$
$938$	16.0000i	0.522419i
$939$	0	0
$940$	0	0
$941$	−21.0294	−0.685540	−0.342770	−	0.939419i	$-0.611365\pi$
$941$	−21.0294	−0.685540	−0.342770	+	0.939419i	$0.611365\pi$
$942$	0	0
$943$	31.0294i	1.01046i
$944$	−13.6569	−0.444493
$945$	0	0
$946$	9.37258	0.304729
$947$	41.6569i	1.35367i	0.736137	+	0.676833i	$0.236648\pi$
$947$	41.6569i	1.35367i	−0.736137	+	0.676833i	$0.763352\pi$
$948$	0	0
$949$	2.48528	0.0806756
$950$	0	0
$951$	0	0
$952$	13.6569i	0.442621i
$953$	56.1421i	1.81862i	0.416117	+	0.909311i	$0.363391\pi$
$953$	56.1421i	1.81862i	−0.416117	+	0.909311i	$0.636609\pi$
$954$	0	0
$955$	0	0
$956$	−16.0000	−0.517477
$957$	0	0
$958$	11.3137i	0.365529i
$959$	15.0294	0.485326
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0.343146i	0.0110635i
$963$	0	0
$964$	17.3137	0.557637
$965$	0	0
$966$	0	0
$967$	24.4853i	0.787394i	0.919240	+	0.393697i	$0.128804\pi$
$967$	24.4853i	0.787394i	−0.919240	+	0.393697i	$0.871196\pi$
$968$	− 21.0000i	− 0.674966i
$969$	0	0
$970$	0	0
$971$	−32.4853	−1.04250	−0.521251	−	0.853403i	$-0.674535\pi$
$971$	−32.4853	−1.04250	−0.521251	+	0.853403i	$0.674535\pi$
$972$	0	0
$973$	49.9411i	1.60104i
$974$	−0.485281	−0.0155494
$975$	0	0
$976$	6.00000	0.192055
$977$	− 19.6569i	− 0.628878i	−0.949278	−	0.314439i	$-0.898184\pi$
$977$	− 19.6569i	− 0.628878i	0.949278	−	0.314439i	$-0.101816\pi$
$978$	0	0
$979$	−24.5685	−0.785214
$980$	0	0
$981$	0	0
$982$	9.85786i	0.314577i
$983$	13.6569i	0.435586i	0.975995	+	0.217793i	$0.0698858\pi$
$983$	13.6569i	0.435586i	−0.975995	+	0.217793i	$0.930114\pi$
$984$	0	0
$985$	0	0
$986$	−15.3137	−0.487688
$987$	0	0
$988$	2.82843i	0.0899843i
$989$	−14.0589	−0.447046
$990$	0	0
$991$	58.9117	1.87139	0.935696	−	0.352808i	$-0.114773\pi$
$991$	58.9117	1.87139	0.935696	+	0.352808i	$0.114773\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	−16.0000	−0.507489
$995$	0	0
$996$	0	0
$997$	− 38.6863i	− 1.22521i	−0.790390	−	0.612604i	$-0.790122\pi$
$997$	− 38.6863i	− 1.22521i	0.790390	−	0.612604i	$-0.209878\pi$
$998$	16.4853i	0.521832i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.bk.5149.3		4
3.2	odd	2		1950.2.e.o.1249.1		4
5.2	odd	4		1170.2.a.o.1.2		2
5.3	odd	4		5850.2.a.cl.1.1		2
5.4	even	2	inner	5850.2.e.bk.5149.2		4
15.2	even	4		390.2.a.h.1.2	✓	2
15.8	even	4		1950.2.a.bd.1.1		2
15.14	odd	2		1950.2.e.o.1249.4		4
20.7	even	4		9360.2.a.ch.1.1		2
60.47	odd	4		3120.2.a.bc.1.1		2
195.47	odd	4		5070.2.b.q.1351.3		4
195.77	even	4		5070.2.a.bc.1.1		2
195.122	odd	4		5070.2.b.q.1351.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.2	✓	2	15.2	even	4
1170.2.a.o.1.2		2	5.2	odd	4
1950.2.a.bd.1.1		2	15.8	even	4
1950.2.e.o.1249.1		4	3.2	odd	2
1950.2.e.o.1249.4		4	15.14	odd	2
3120.2.a.bc.1.1		2	60.47	odd	4
5070.2.a.bc.1.1		2	195.77	even	4
5070.2.b.q.1351.2		4	195.122	odd	4
5070.2.b.q.1351.3		4	195.47	odd	4
5850.2.a.cl.1.1		2	5.3	odd	4
5850.2.e.bk.5149.2		4	5.4	even	2	inner
5850.2.e.bk.5149.3		4	1.1	even	1	trivial
9360.2.a.ch.1.1		2	20.7	even	4

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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.82843i q^{7} -1.00000i q^{8} +5.65685 q^{11} -1.00000i q^{13} +2.82843 q^{14} +1.00000 q^{16} -4.82843i q^{17} +2.82843 q^{19} +5.65685i q^{22} -8.48528i q^{23} +1.00000 q^{26} +2.82843i q^{28} -3.17157 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.82843 q^{34} +0.343146i q^{37} +2.82843i q^{38} -3.65685 q^{41} -1.65685i q^{43} -5.65685 q^{44} +8.48528 q^{46} -8.00000i q^{47} -1.00000 q^{49} +1.00000i q^{52} +9.31371i q^{53} -2.82843 q^{56} -3.17157i q^{58} -13.6569 q^{59} +6.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +5.65685i q^{67} +4.82843i q^{68} -5.65685 q^{71} +2.48528i q^{73} -0.343146 q^{74} -2.82843 q^{76} -16.0000i q^{77} -13.6569 q^{79} -3.65685i q^{82} -17.6569i q^{83} +1.65685 q^{86} -5.65685i q^{88} -4.34315 q^{89} -2.82843 q^{91} +8.48528i q^{92} +8.00000 q^{94} +8.82843i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{16} + 4 q^{26} - 24 q^{29} + 16 q^{31} + 8 q^{34} + 8 q^{41} - 4 q^{49} - 32 q^{59} + 24 q^{61} - 4 q^{64} - 24 q^{74} - 32 q^{79} - 16 q^{86} - 40 q^{89} + 32 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^16 + 4 * q^26 - 24 * q^29 + 16 * q^31 + 8 * q^34 + 8 * q^41 - 4 * q^49 - 32 * q^59 + 24 * q^61 - 4 * q^64 - 24 * q^74 - 32 * q^79 - 16 * q^86 - 40 * q^89 + 32 * q^94

Introduction

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