LMFDB - Embedded newform 4356.3.f.g.1693.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4356,3,Mod(1693,4356)] 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("4356.1693"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4356, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4356.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-100,0,-34,0,0, 0,0,0,98,0,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,110,0,22,0,0,0,-250,0,0,0,0, 0,-112,0,0,0,0,0,0,0,10,0,0,0,198,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-18, 0,440] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	no
Analytic conductor:	$118.692403155$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 19x^{6} - 37x^{5} + 229x^{4} + 196x^{3} + 1496x^{2} + 2952x + 26896 $ x^8 - x^7 + 19x^6 - 37x^5 + 229x^4 + 196x^3 + 1496x^2 + 2952x + 26896
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 11^{3} $
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1693.8
Root			$3.08941 + 2.24459i$ of defining polynomial
Character	$\chi$	$=$	4356.1693
Dual form			4356.3.f.g.1693.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.17883 q^{5} +1.10828i q^{7} -12.3893i q^{13} +11.8588i q^{17} -26.4404i q^{19} -19.9563 q^{23} -7.53741 q^{25} +48.4719i q^{29} +17.2974 q^{31} +4.63130i q^{35} -72.6211 q^{37} +41.2570i q^{41} +57.9276i q^{43} +8.75419 q^{47} +47.7717 q^{49} -0.259379 q^{53} +43.5555 q^{59} +81.8700i q^{61} -51.7725i q^{65} +52.4087 q^{67} +66.3063 q^{71} -57.9034i q^{73} +8.59534i q^{79} +87.2900i q^{83} +49.5560i q^{85} +3.65450 q^{89} +13.7307 q^{91} -110.490i q^{95} +6.85977 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{5} - 100 q^{23} - 34 q^{25} + 98 q^{31} - 210 q^{37} + 110 q^{47} + 22 q^{49} - 250 q^{53} - 112 q^{59} + 10 q^{67} + 198 q^{71} - 18 q^{89} + 440 q^{91} - 220 q^{97}+O(q^{100}) $ 8 * q - 14 * q^5 - 100 * q^23 - 34 * q^25 + 98 * q^31 - 210 * q^37 + 110 * q^47 + 22 * q^49 - 250 * q^53 - 112 * q^59 + 10 * q^67 + 198 * q^71 - 18 * q^89 + 440 * q^91 - 220 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4356\mathbb{Z}\right)^\times$.

$n$	$1333$	$1937$	$2179$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.17883	0.835765	0.417883	−	0.908501i	$-0.362772\pi$
$5$	4.17883	0.835765	0.417883	+	0.908501i	$0.362772\pi$
$6$	0	0
$7$	1.10828i	0.158325i	0.996862	+	0.0791627i	$0.0252247\pi$
$7$	1.10828i	0.158325i	−0.996862	+	0.0791627i	$0.974775\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 12.3893i	− 0.953019i	−0.879169	−	0.476510i	$-0.841902\pi$
$13$	− 12.3893i	− 0.953019i	0.879169	−	0.476510i	$-0.158098\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	11.8588i	0.697578i	0.937201	+	0.348789i	$0.113407\pi$
$17$	11.8588i	0.697578i	−0.937201	+	0.348789i	$0.886593\pi$
$18$	0	0
$19$	− 26.4404i	− 1.39160i	−0.718235	−	0.695800i	$-0.755050\pi$
$19$	− 26.4404i	− 1.39160i	0.718235	−	0.695800i	$-0.244950\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−19.9563	−0.867664	−0.433832	−	0.900994i	$-0.642839\pi$
$23$	−19.9563	−0.867664	−0.433832	+	0.900994i	$0.642839\pi$
$24$	0	0
$25$	−7.53741	−0.301497
$26$	0	0
$27$	0	0
$28$	0	0
$29$	48.4719i	1.67145i	0.549151	+	0.835723i	$0.314951\pi$
$29$	48.4719i	1.67145i	−0.549151	+	0.835723i	$0.685049\pi$
$30$	0	0
$31$	17.2974	0.557980	0.278990	−	0.960294i	$-0.410000\pi$
$31$	17.2974	0.557980	0.278990	+	0.960294i	$0.410000\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.63130i	0.132323i
$36$	0	0
$37$	−72.6211	−1.96273	−0.981367	−	0.192145i	$-0.938456\pi$
$37$	−72.6211	−1.96273	−0.981367	+	0.192145i	$0.938456\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	41.2570i	1.00627i	0.864208	+	0.503135i	$0.167820\pi$
$41$	41.2570i	1.00627i	−0.864208	+	0.503135i	$0.832180\pi$
$42$	0	0
$43$	57.9276i	1.34715i	0.739117	+	0.673577i	$0.235243\pi$
$43$	57.9276i	1.34715i	−0.739117	+	0.673577i	$0.764757\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.75419	0.186259	0.0931297	−	0.995654i	$-0.470313\pi$
$47$	8.75419	0.186259	0.0931297	+	0.995654i	$0.470313\pi$
$48$	0	0
$49$	47.7717	0.974933
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.259379	−0.00489395	−0.00244697	−	0.999997i	$-0.500779\pi$
$53$	−0.259379	−0.00489395	−0.00244697	+	0.999997i	$0.500779\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	43.5555	0.738229	0.369115	−	0.929384i	$-0.379661\pi$
$59$	43.5555	0.738229	0.369115	+	0.929384i	$0.379661\pi$
$60$	0	0
$61$	81.8700i	1.34213i	0.741398	+	0.671066i	$0.234163\pi$
$61$	81.8700i	1.34213i	−0.741398	+	0.671066i	$0.765837\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 51.7725i	− 0.796500i
$66$	0	0
$67$	52.4087	0.782220	0.391110	−	0.920344i	$-0.372091\pi$
$67$	52.4087	0.782220	0.391110	+	0.920344i	$0.372091\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	66.3063	0.933891	0.466946	−	0.884286i	$-0.345354\pi$
$71$	66.3063	0.933891	0.466946	+	0.884286i	$0.345354\pi$
$72$	0	0
$73$	− 57.9034i	− 0.793197i	−0.917992	−	0.396598i	$-0.870191\pi$
$73$	− 57.9034i	− 0.793197i	0.917992	−	0.396598i	$-0.129809\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.59534i	0.108802i	0.998519	+	0.0544009i	$0.0173249\pi$
$79$	8.59534i	0.108802i	−0.998519	+	0.0544009i	$0.982675\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	87.2900i	1.05169i	0.850581	+	0.525843i	$0.176250\pi$
$83$	87.2900i	1.05169i	−0.850581	+	0.525843i	$0.823750\pi$
$84$	0	0
$85$	49.5560i	0.583011i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.65450	0.0410618	0.0205309	−	0.999789i	$-0.493464\pi$
$89$	3.65450	0.0410618	0.0205309	+	0.999789i	$0.493464\pi$
$90$	0	0
$91$	13.7307	0.150887
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 110.490i	− 1.16305i
$96$	0	0
$97$	6.85977	0.0707192	0.0353596	−	0.999375i	$-0.488742\pi$
$97$	6.85977	0.0707192	0.0353596	+	0.999375i	$0.488742\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 85.7335i	− 0.848847i	−0.905464	−	0.424423i	$-0.860477\pi$
$101$	− 85.7335i	− 0.848847i	0.905464	−	0.424423i	$-0.139523\pi$
$102$	0	0
$103$	140.889	1.36786	0.683929	−	0.729548i	$-0.260270\pi$
$103$	140.889	1.36786	0.683929	+	0.729548i	$0.260270\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	118.566i	1.10810i	0.832484	+	0.554048i	$0.186918\pi$
$107$	118.566i	1.10810i	−0.832484	+	0.554048i	$0.813082\pi$
$108$	0	0
$109$	39.5835i	0.363152i	0.983377	+	0.181576i	$0.0581198\pi$
$109$	39.5835i	0.363152i	−0.983377	+	0.181576i	$0.941880\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	168.331	1.48966	0.744828	−	0.667256i	$-0.232531\pi$
$113$	168.331	1.48966	0.744828	+	0.667256i	$0.232531\pi$
$114$	0	0
$115$	−83.3938	−0.725163
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−13.1429	−0.110444
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−135.968	−1.08775
$126$	0	0
$127$	− 38.2286i	− 0.301013i	−0.988609	−	0.150506i	$-0.951910\pi$
$127$	− 38.2286i	− 0.301013i	0.988609	−	0.150506i	$-0.0480904\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	241.214i	1.84133i	0.390354	+	0.920665i	$0.372353\pi$
$131$	241.214i	1.84133i	−0.390354	+	0.920665i	$0.627647\pi$
$132$	0	0
$133$	29.3033	0.220326
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−112.255	−0.819382	−0.409691	−	0.912224i	$-0.634363\pi$
$137$	−112.255	−0.819382	−0.409691	+	0.912224i	$0.634363\pi$
$138$	0	0
$139$	100.051i	0.719792i	0.932992	+	0.359896i	$0.117188\pi$
$139$	100.051i	0.719792i	−0.932992	+	0.359896i	$0.882812\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	202.556i	1.39694i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 126.123i	− 0.846461i	−0.906022	−	0.423231i	$-0.860896\pi$
$149$	− 126.123i	− 0.846461i	0.906022	−	0.423231i	$-0.139104\pi$
$150$	0	0
$151$	216.075i	1.43096i	0.698632	+	0.715482i	$0.253793\pi$
$151$	216.075i	1.43096i	−0.698632	+	0.715482i	$0.746207\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	72.2827	0.466340
$156$	0	0
$157$	118.805	0.756718	0.378359	−	0.925659i	$-0.376488\pi$
$157$	118.805	0.756718	0.378359	+	0.925659i	$0.376488\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 22.1171i	− 0.137373i
$162$	0	0
$163$	−189.108	−1.16017	−0.580085	−	0.814556i	$-0.696981\pi$
$163$	−189.108	−1.16017	−0.580085	+	0.814556i	$0.696981\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 98.6256i	− 0.590572i	−0.955409	−	0.295286i	$-0.904585\pi$
$167$	− 98.6256i	− 0.590572i	0.955409	−	0.295286i	$-0.0954150\pi$
$168$	0	0
$169$	15.5064	0.0917540
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 147.804i	− 0.854356i	−0.904168	−	0.427178i	$-0.859508\pi$
$173$	− 147.804i	− 0.854356i	0.904168	−	0.427178i	$-0.140492\pi$
$174$	0	0
$175$	− 8.35355i	− 0.0477346i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	151.074	0.843988	0.421994	−	0.906599i	$-0.361330\pi$
$179$	151.074	0.843988	0.421994	+	0.906599i	$0.361330\pi$
$180$	0	0
$181$	252.503	1.39504	0.697521	−	0.716564i	$-0.254286\pi$
$181$	252.503	1.39504	0.697521	+	0.716564i	$0.254286\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−303.471	−1.64038
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−207.500	−1.08638	−0.543192	−	0.839608i	$-0.682785\pi$
$191$	−207.500	−1.08638	−0.543192	+	0.839608i	$0.682785\pi$
$192$	0	0
$193$	241.474i	1.25116i	0.780160	+	0.625580i	$0.215137\pi$
$193$	241.474i	1.25116i	−0.780160	+	0.625580i	$0.784863\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 172.018i	− 0.873188i	−0.899659	−	0.436594i	$-0.856185\pi$
$197$	− 172.018i	− 0.873188i	0.899659	−	0.436594i	$-0.143815\pi$
$198$	0	0
$199$	33.4659	0.168171	0.0840853	−	0.996459i	$-0.473203\pi$
$199$	33.4659	0.168171	0.0840853	+	0.996459i	$0.473203\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−53.7204	−0.264632
$204$	0	0
$205$	172.406i	0.841005i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 169.306i	− 0.802399i	−0.915991	−	0.401199i	$-0.868593\pi$
$211$	− 169.306i	− 0.802399i	0.915991	−	0.401199i	$-0.131407\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	242.069i	1.12590i
$216$	0	0
$217$	19.1703i	0.0883425i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	146.922	0.664805
$222$	0	0
$223$	−19.7426	−0.0885319	−0.0442660	−	0.999020i	$-0.514095\pi$
$223$	−19.7426	−0.0885319	−0.0442660	+	0.999020i	$0.514095\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	421.037i	1.85479i	0.374087	+	0.927393i	$0.377956\pi$
$227$	421.037i	1.85479i	−0.374087	+	0.927393i	$0.622044\pi$
$228$	0	0
$229$	−26.6503	−0.116377	−0.0581883	−	0.998306i	$-0.518532\pi$
$229$	−26.6503	−0.116377	−0.0581883	+	0.998306i	$0.518532\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 311.001i	− 1.33477i	−0.744714	−	0.667383i	$-0.767414\pi$
$233$	− 311.001i	− 1.33477i	0.744714	−	0.667383i	$-0.232586\pi$
$234$	0	0
$235$	36.5822	0.155669
$236$	0	0
$237$	0	0
$238$	0	0
$239$	412.161i	1.72452i	0.506463	+	0.862262i	$0.330953\pi$
$239$	412.161i	1.72452i	−0.506463	+	0.862262i	$0.669047\pi$
$240$	0	0
$241$	− 112.453i	− 0.466608i	−0.972404	−	0.233304i	$-0.925046\pi$
$241$	− 112.453i	− 0.466608i	0.972404	−	0.233304i	$-0.0749537\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	199.630	0.814815
$246$	0	0
$247$	−327.577	−1.32622
$248$	0	0
$249$	0	0
$250$	0	0
$251$	75.5126	0.300847	0.150424	−	0.988622i	$-0.451936\pi$
$251$	75.5126	0.300847	0.150424	+	0.988622i	$0.451936\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	202.357	0.787383	0.393692	−	0.919243i	$-0.371198\pi$
$257$	202.357	0.787383	0.393692	+	0.919243i	$0.371198\pi$
$258$	0	0
$259$	− 80.4844i	− 0.310751i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 13.9761i	− 0.0531412i	−0.999647	−	0.0265706i	$-0.991541\pi$
$263$	− 13.9761i	− 0.0531412i	0.999647	−	0.0265706i	$-0.00845867\pi$
$264$	0	0
$265$	−1.08390	−0.00409019
$266$	0	0
$267$	0	0
$268$	0	0
$269$	101.905	0.378829	0.189414	−	0.981897i	$-0.439341\pi$
$269$	101.905	0.378829	0.189414	+	0.981897i	$0.439341\pi$
$270$	0	0
$271$	310.833i	1.14698i	0.819211	+	0.573492i	$0.194412\pi$
$271$	310.833i	1.14698i	−0.819211	+	0.573492i	$0.805588\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 477.335i	− 1.72323i	−0.507562	−	0.861615i	$-0.669453\pi$
$277$	− 477.335i	− 1.72323i	0.507562	−	0.861615i	$-0.330547\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	474.292i	1.68787i	0.536445	+	0.843935i	$0.319767\pi$
$281$	474.292i	1.68787i	−0.536445	+	0.843935i	$0.680233\pi$
$282$	0	0
$283$	− 441.956i	− 1.56168i	−0.624729	−	0.780841i	$-0.714791\pi$
$283$	− 441.956i	− 1.56168i	0.624729	−	0.780841i	$-0.285209\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−45.7243	−0.159318
$288$	0	0
$289$	148.368	0.513385
$290$	0	0
$291$	0	0
$292$	0	0
$293$	189.144i	0.645542i	0.946477	+	0.322771i	$0.104614\pi$
$293$	189.144i	0.645542i	−0.946477	+	0.322771i	$0.895386\pi$
$294$	0	0
$295$	182.011	0.616986
$296$	0	0
$297$	0	0
$298$	0	0
$299$	247.243i	0.826901i
$300$	0	0
$301$	−64.1999	−0.213289
$302$	0	0
$303$	0	0
$304$	0	0
$305$	342.121i	1.12171i
$306$	0	0
$307$	− 26.0108i	− 0.0847257i	−0.999102	−	0.0423628i	$-0.986511\pi$
$307$	− 26.0108i	− 0.0847257i	0.999102	−	0.0423628i	$-0.0134885\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	103.585	0.333071	0.166535	−	0.986035i	$-0.446742\pi$
$311$	103.585	0.333071	0.166535	+	0.986035i	$0.446742\pi$
$312$	0	0
$313$	−235.138	−0.751240	−0.375620	−	0.926774i	$-0.622570\pi$
$313$	−235.138	−0.751240	−0.375620	+	0.926774i	$0.622570\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.4466	−0.0676549	−0.0338274	−	0.999428i	$-0.510770\pi$
$317$	−21.4466	−0.0676549	−0.0338274	+	0.999428i	$0.510770\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	313.552	0.970750
$324$	0	0
$325$	93.3829i	0.287332i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.70208i	0.0294896i
$330$	0	0
$331$	41.5345	0.125482	0.0627410	−	0.998030i	$-0.480016\pi$
$331$	41.5345	0.125482	0.0627410	+	0.998030i	$0.480016\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	219.007	0.653752
$336$	0	0
$337$	− 332.113i	− 0.985499i	−0.870171	−	0.492749i	$-0.835992\pi$
$337$	− 332.113i	− 0.985499i	0.870171	−	0.492749i	$-0.164008\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	107.250i	0.312682i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	115.865i	0.333905i	0.985965	+	0.166953i	$0.0533927\pi$
$347$	115.865i	0.333905i	−0.985965	+	0.166953i	$0.946607\pi$
$348$	0	0
$349$	34.2319i	0.0980855i	0.998797	+	0.0490428i	$0.0156171\pi$
$349$	34.2319i	0.0980855i	−0.998797	+	0.0490428i	$0.984383\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−198.775	−0.563102	−0.281551	−	0.959546i	$-0.590849\pi$
$353$	−198.775	−0.563102	−0.281551	+	0.959546i	$0.590849\pi$
$354$	0	0
$355$	277.082	0.780514
$356$	0	0
$357$	0	0
$358$	0	0
$359$	351.303i	0.978561i	0.872127	+	0.489280i	$0.162741\pi$
$359$	351.303i	0.978561i	−0.872127	+	0.489280i	$0.837259\pi$
$360$	0	0
$361$	−338.095	−0.936551
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 241.968i	− 0.662926i
$366$	0	0
$367$	−548.053	−1.49333	−0.746666	−	0.665199i	$-0.768347\pi$
$367$	−548.053	−1.49333	−0.746666	+	0.665199i	$0.768347\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 0.287464i	0 0.000774837i
$372$	0	0
$373$	142.065i	0.380872i	0.981700	+	0.190436i	$0.0609901\pi$
$373$	142.065i	0.380872i	−0.981700	+	0.190436i	$0.939010\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	600.531	1.59292
$378$	0	0
$379$	128.747	0.339701	0.169850	−	0.985470i	$-0.445672\pi$
$379$	128.747	0.339701	0.169850	+	0.985470i	$0.445672\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−738.153	−1.92729	−0.963646	−	0.267181i	$-0.913908\pi$
$383$	−738.153	−1.92729	−0.963646	+	0.267181i	$0.913908\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	219.312	0.563785	0.281893	−	0.959446i	$-0.409038\pi$
$389$	219.312	0.563785	0.281893	+	0.959446i	$0.409038\pi$
$390$	0	0
$391$	− 236.658i	− 0.605263i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	35.9184i	0.0909328i
$396$	0	0
$397$	−4.97040	−0.0125199	−0.00625995	−	0.999980i	$-0.501993\pi$
$397$	−4.97040	−0.0125199	−0.00625995	+	0.999980i	$0.501993\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−292.908	−0.730444	−0.365222	−	0.930921i	$-0.619007\pi$
$401$	−292.908	−0.730444	−0.365222	+	0.930921i	$0.619007\pi$
$402$	0	0
$403$	− 214.302i	− 0.531766i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 279.292i	− 0.682866i	−0.939906	−	0.341433i	$-0.889088\pi$
$409$	− 279.292i	− 0.682866i	0.939906	−	0.341433i	$-0.110912\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	48.2717i	0.116881i
$414$	0	0
$415$	364.770i	0.878963i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	582.160	1.38940	0.694701	−	0.719298i	$-0.255537\pi$
$419$	582.160	1.38940	0.694701	+	0.719298i	$0.255537\pi$
$420$	0	0
$421$	148.957	0.353817	0.176909	−	0.984227i	$-0.443390\pi$
$421$	148.957	0.353817	0.176909	+	0.984227i	$0.443390\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 89.3849i	− 0.210317i
$426$	0	0
$427$	−90.7348	−0.212494
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 175.230i	− 0.406567i	−0.979120	−	0.203283i	$-0.934839\pi$
$431$	− 175.230i	− 0.406567i	0.979120	−	0.203283i	$-0.0651613\pi$
$432$	0	0
$433$	289.806	0.669298	0.334649	−	0.942343i	$-0.391382\pi$
$433$	289.806	0.669298	0.334649	+	0.942343i	$0.391382\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	527.652i	1.20744i
$438$	0	0
$439$	439.213i	1.00049i	0.865885	+	0.500243i	$0.166756\pi$
$439$	439.213i	1.00049i	−0.865885	+	0.500243i	$0.833244\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	588.827	1.32918	0.664590	−	0.747208i	$-0.268606\pi$
$443$	588.827	1.32918	0.664590	+	0.747208i	$0.268606\pi$
$444$	0	0
$445$	15.2715	0.0343180
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−206.664	−0.460277	−0.230139	−	0.973158i	$-0.573918\pi$
$449$	−206.664	−0.460277	−0.230139	+	0.973158i	$0.573918\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	57.3784	0.126106
$456$	0	0
$457$	− 5.40423i	− 0.0118254i	−0.999983	−	0.00591272i	$-0.998118\pi$
$457$	− 5.40423i	− 0.0118254i	0.999983	−	0.00591272i	$-0.00188209\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	858.622i	1.86252i	0.364355	+	0.931260i	$0.381289\pi$
$461$	858.622i	1.86252i	−0.364355	+	0.931260i	$0.618711\pi$
$462$	0	0
$463$	382.107	0.825285	0.412643	−	0.910893i	$-0.364606\pi$
$463$	382.107	0.825285	0.412643	+	0.910893i	$0.364606\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−358.586	−0.767850	−0.383925	−	0.923364i	$-0.625428\pi$
$467$	−358.586	−0.767850	−0.383925	+	0.923364i	$0.625428\pi$
$468$	0	0
$469$	58.0835i	0.123845i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	199.292i	0.419563i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	683.067i	1.42603i	0.701150	+	0.713014i	$0.252670\pi$
$479$	683.067i	1.42603i	−0.701150	+	0.713014i	$0.747330\pi$
$480$	0	0
$481$	899.721i	1.87052i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.6658	0.0591047
$486$	0	0
$487$	798.119	1.63885	0.819424	−	0.573188i	$-0.194294\pi$
$487$	798.119	1.63885	0.819424	+	0.573188i	$0.194294\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	263.493i	0.536645i	0.963329	+	0.268323i	$0.0864693\pi$
$491$	263.493i	0.536645i	−0.963329	+	0.268323i	$0.913531\pi$
$492$	0	0
$493$	−574.820	−1.16596
$494$	0	0
$495$	0	0
$496$	0	0
$497$	73.4858i	0.147859i
$498$	0	0
$499$	177.263	0.355237	0.177618	−	0.984099i	$-0.443161\pi$
$499$	177.263	0.355237	0.177618	+	0.984099i	$0.443161\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 26.9594i	− 0.0535973i	−0.999641	−	0.0267986i	$-0.991469\pi$
$503$	− 26.9594i	− 0.0535973i	0.999641	−	0.0267986i	$-0.00853129\pi$
$504$	0	0
$505$	− 358.265i	− 0.709436i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−52.5671	−0.103275	−0.0516376	−	0.998666i	$-0.516444\pi$
$509$	−52.5671	−0.103275	−0.0516376	+	0.998666i	$0.516444\pi$
$510$	0	0
$511$	64.1731	0.125583
$512$	0	0
$513$	0	0
$514$	0	0
$515$	588.752	1.14321
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−878.205	−1.68561	−0.842807	−	0.538216i	$-0.819099\pi$
$521$	−878.205	−1.68561	−0.842807	+	0.538216i	$0.819099\pi$
$522$	0	0
$523$	974.006i	1.86234i	0.364579	+	0.931172i	$0.381213\pi$
$523$	974.006i	1.86234i	−0.364579	+	0.931172i	$0.618787\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	205.127i	0.389235i
$528$	0	0
$529$	−130.747	−0.247159
$530$	0	0
$531$	0	0
$532$	0	0
$533$	511.144	0.958994
$534$	0	0
$535$	495.468i	0.926109i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 552.954i	− 1.02210i	−0.859552	−	0.511048i	$-0.829257\pi$
$541$	− 552.954i	− 1.02210i	0.859552	−	0.511048i	$-0.170743\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	165.413i	0.303510i
$546$	0	0
$547$	415.428i	0.759466i	0.925096	+	0.379733i	$0.123984\pi$
$547$	415.428i	0.759466i	−0.925096	+	0.379733i	$0.876016\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1281.62	2.32598
$552$	0	0
$553$	−9.52603	−0.0172261
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 448.918i	− 0.805958i	−0.915209	−	0.402979i	$-0.867975\pi$
$557$	− 448.918i	− 0.805958i	0.915209	−	0.402979i	$-0.132025\pi$
$558$	0	0
$559$	717.680	1.28386
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 82.5651i	− 0.146652i	−0.997308	−	0.0733260i	$-0.976639\pi$
$563$	− 82.5651i	− 0.146652i	0.997308	−	0.0733260i	$-0.0233614\pi$
$564$	0	0
$565$	703.427	1.24500
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.5753i	0.0748248i	0.999300	+	0.0374124i	$0.0119115\pi$
$569$	42.5753i	0.0748248i	−0.999300	+	0.0374124i	$0.988088\pi$
$570$	0	0
$571$	− 460.190i	− 0.805936i	−0.915214	−	0.402968i	$-0.867979\pi$
$571$	− 460.190i	− 0.805936i	0.915214	−	0.402968i	$-0.132021\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	150.419	0.261598
$576$	0	0
$577$	161.005	0.279038	0.139519	−	0.990219i	$-0.455444\pi$
$577$	161.005	0.279038	0.139519	+	0.990219i	$0.455444\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−96.7416	−0.166509
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−141.810	−0.241584	−0.120792	−	0.992678i	$-0.538543\pi$
$587$	−141.810	−0.241584	−0.120792	+	0.992678i	$0.538543\pi$
$588$	0	0
$589$	− 457.350i	− 0.776485i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 279.117i	− 0.470686i	−0.971912	−	0.235343i	$-0.924379\pi$
$593$	− 279.117i	− 0.470686i	0.971912	−	0.235343i	$-0.0756213\pi$
$594$	0	0
$595$	−54.9218	−0.0923056
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.5052	0.0676214	0.0338107	−	0.999428i	$-0.489236\pi$
$599$	40.5052	0.0676214	0.0338107	+	0.999428i	$0.489236\pi$
$600$	0	0
$601$	425.882i	0.708623i	0.935127	+	0.354311i	$0.115285\pi$
$601$	425.882i	0.708623i	−0.935127	+	0.354311i	$0.884715\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	747.551i	1.23155i	0.787922	+	0.615775i	$0.211157\pi$
$607$	747.551i	1.23155i	−0.787922	+	0.615775i	$0.788843\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 108.458i	− 0.177509i
$612$	0	0
$613$	83.9925i	0.137019i	0.997650	+	0.0685094i	$0.0218243\pi$
$613$	83.9925i	0.137019i	−0.997650	+	0.0685094i	$0.978176\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	247.279	0.400776	0.200388	−	0.979717i	$-0.435780\pi$
$617$	247.279	0.400776	0.200388	+	0.979717i	$0.435780\pi$
$618$	0	0
$619$	−658.298	−1.06349	−0.531743	−	0.846906i	$-0.678463\pi$
$619$	−658.298	−1.06349	−0.531743	+	0.846906i	$0.678463\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.05020i	0.00650113i
$624$	0	0
$625$	−379.752	−0.607603
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 861.201i	− 1.36916i
$630$	0	0
$631$	−688.820	−1.09163	−0.545816	−	0.837905i	$-0.683780\pi$
$631$	−688.820	−1.09163	−0.545816	+	0.837905i	$0.683780\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 159.751i	− 0.251576i
$636$	0	0
$637$	− 591.856i	− 0.929130i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−605.576	−0.944736	−0.472368	−	0.881401i	$-0.656601\pi$
$641$	−605.576	−0.944736	−0.472368	+	0.881401i	$0.656601\pi$
$642$	0	0
$643$	924.883	1.43839	0.719194	−	0.694810i	$-0.244511\pi$
$643$	924.883	1.43839	0.719194	+	0.694810i	$0.244511\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−672.280	−1.03907	−0.519536	−	0.854448i	$-0.673895\pi$
$647$	−672.280	−1.03907	−0.519536	+	0.854448i	$0.673895\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−822.691	−1.25986	−0.629932	−	0.776650i	$-0.716917\pi$
$653$	−822.691	−1.25986	−0.629932	+	0.776650i	$0.716917\pi$
$654$	0	0
$655$	1007.99i	1.53892i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	378.008i	0.573608i	0.957989	+	0.286804i	$0.0925928\pi$
$659$	378.008i	0.573608i	−0.957989	+	0.286804i	$0.907407\pi$
$660$	0	0
$661$	−1273.25	−1.92625	−0.963127	−	0.269046i	$-0.913292\pi$
$661$	−1273.25	−1.92625	−0.963127	+	0.269046i	$0.913292\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	122.454	0.184141
$666$	0	0
$667$	− 967.319i	− 1.45025i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	507.430i	0.753982i	0.926217	+	0.376991i	$0.123041\pi$
$673$	507.430i	0.753982i	−0.926217	+	0.376991i	$0.876959\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	755.138i	1.11542i	0.830036	+	0.557709i	$0.188320\pi$
$677$	755.138i	1.11542i	−0.830036	+	0.557709i	$0.811680\pi$
$678$	0	0
$679$	7.60253i	0.0111967i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1145.46	−1.67710	−0.838549	−	0.544826i	$-0.816596\pi$
$683$	−1145.46	−1.67710	−0.838549	+	0.544826i	$0.816596\pi$
$684$	0	0
$685$	−469.095	−0.684811
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.21352i	0.00466403i
$690$	0	0
$691$	20.7784	0.0300701	0.0150351	−	0.999887i	$-0.495214\pi$
$691$	20.7784	0.0300701	0.0150351	+	0.999887i	$0.495214\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	418.096i	0.601577i
$696$	0	0
$697$	−489.260	−0.701951
$698$	0	0
$699$	0	0
$700$	0	0
$701$	184.332i	0.262956i	0.991319	+	0.131478i	$0.0419722\pi$
$701$	184.332i	0.262956i	−0.991319	+	0.131478i	$0.958028\pi$
$702$	0	0
$703$	1920.13i	2.73134i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	95.0166	0.134394
$708$	0	0
$709$	420.388	0.592931	0.296466	−	0.955044i	$-0.404192\pi$
$709$	420.388	0.592931	0.296466	+	0.955044i	$0.404192\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−345.191	−0.484139
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	491.056	0.682971	0.341486	−	0.939887i	$-0.389070\pi$
$719$	491.056	0.682971	0.341486	+	0.939887i	$0.389070\pi$
$720$	0	0
$721$	156.145i	0.216567i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 365.353i	− 0.503935i
$726$	0	0
$727$	−29.1173	−0.0400514	−0.0200257	−	0.999799i	$-0.506375\pi$
$727$	−29.1173	−0.0400514	−0.0200257	+	0.999799i	$0.506375\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−686.953	−0.939745
$732$	0	0
$733$	1030.07i	1.40528i	0.711544	+	0.702641i	$0.247996\pi$
$733$	1030.07i	1.40528i	−0.711544	+	0.702641i	$0.752004\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 949.857i	− 1.28533i	−0.766148	−	0.642664i	$-0.777829\pi$
$739$	− 949.857i	− 1.28533i	0.766148	−	0.642664i	$-0.222171\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 23.3401i	− 0.0314133i	−0.999877	−	0.0157067i	$-0.995000\pi$
$743$	− 23.3401i	− 0.0314133i	0.999877	−	0.0157067i	$-0.00499979\pi$
$744$	0	0
$745$	− 527.045i	− 0.707443i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−131.405	−0.175440
$750$	0	0
$751$	438.132	0.583398	0.291699	−	0.956510i	$-0.405779\pi$
$751$	438.132	0.583398	0.291699	+	0.956510i	$0.405779\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	902.942i	1.19595i
$756$	0	0
$757$	987.970	1.30511	0.652556	−	0.757741i	$-0.273697\pi$
$757$	987.970	1.30511	0.652556	+	0.757741i	$0.273697\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	49.9181i	0.0655954i	0.999462	+	0.0327977i	$0.0104417\pi$
$761$	49.9181i	0.0655954i	−0.999462	+	0.0327977i	$0.989558\pi$
$762$	0	0
$763$	−43.8696	−0.0574962
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 539.621i	− 0.703547i
$768$	0	0
$769$	197.596i	0.256952i	0.991713	+	0.128476i	$0.0410086\pi$
$769$	197.596i	0.256952i	−0.991713	+	0.128476i	$0.958991\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1155.34	1.49462	0.747309	−	0.664477i	$-0.231345\pi$
$773$	1155.34	1.49462	0.747309	+	0.664477i	$0.231345\pi$
$774$	0	0
$775$	−130.378	−0.168229
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1090.85	1.40032
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	496.464	0.632439
$786$	0	0
$787$	− 55.3936i	− 0.0703857i	−0.999381	−	0.0351929i	$-0.988795\pi$
$787$	− 55.3936i	− 0.0703857i	0.999381	−	0.0351929i	$-0.0112046\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	186.558i	0.235851i
$792$	0	0
$793$	1014.31	1.27908
$794$	0	0
$795$	0	0
$796$	0	0
$797$	896.713	1.12511	0.562555	−	0.826760i	$-0.309818\pi$
$797$	896.713	1.12511	0.562555	+	0.826760i	$0.309818\pi$
$798$	0	0
$799$	103.814i	0.129930i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 92.4235i	− 0.114812i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 747.195i	− 0.923603i	−0.886983	−	0.461801i	$-0.847203\pi$
$809$	− 747.195i	− 0.923603i	0.886983	−	0.461801i	$-0.152797\pi$
$810$	0	0
$811$	− 622.640i	− 0.767744i	−0.923386	−	0.383872i	$-0.874590\pi$
$811$	− 622.640i	− 0.767744i	0.923386	−	0.383872i	$-0.125410\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−790.248	−0.969629
$816$	0	0
$817$	1531.63	1.87470
$818$	0	0
$819$	0	0
$820$	0	0
$821$	499.842i	0.608821i	0.952541	+	0.304410i	$0.0984594\pi$
$821$	499.842i	0.608821i	−0.952541	+	0.304410i	$0.901541\pi$
$822$	0	0
$823$	−351.874	−0.427551	−0.213775	−	0.976883i	$-0.568576\pi$
$823$	−351.874	−0.427551	−0.213775	+	0.976883i	$0.568576\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	785.766i	0.950140i	0.879948	+	0.475070i	$0.157577\pi$
$827$	785.766i	0.950140i	−0.879948	+	0.475070i	$0.842423\pi$
$828$	0	0
$829$	−319.544	−0.385457	−0.192729	−	0.981252i	$-0.561734\pi$
$829$	−319.544	−0.385457	−0.192729	+	0.981252i	$0.561734\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	566.517i	0.680092i
$834$	0	0
$835$	− 412.139i	− 0.493580i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−709.047	−0.845110	−0.422555	−	0.906337i	$-0.638867\pi$
$839$	−709.047	−0.845110	−0.422555	+	0.906337i	$0.638867\pi$
$840$	0	0
$841$	−1508.53	−1.79373
$842$	0	0
$843$	0	0
$844$	0	0
$845$	64.7987	0.0766848
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1449.25	1.70299
$852$	0	0
$853$	− 1000.17i	− 1.17253i	−0.810118	−	0.586266i	$-0.800597\pi$
$853$	− 1000.17i	− 1.17253i	0.810118	−	0.586266i	$-0.199403\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	118.189i	0.137910i	0.997620	+	0.0689552i	$0.0219665\pi$
$857$	118.189i	0.137910i	−0.997620	+	0.0689552i	$0.978033\pi$
$858$	0	0
$859$	709.931	0.826463	0.413231	−	0.910626i	$-0.364400\pi$
$859$	709.931	0.826463	0.413231	+	0.910626i	$0.364400\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	374.324	0.433747	0.216873	−	0.976200i	$-0.430414\pi$
$863$	374.324	0.433747	0.216873	+	0.976200i	$0.430414\pi$
$864$	0	0
$865$	− 617.645i	− 0.714041i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 649.305i	− 0.745471i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 150.691i	− 0.172218i
$876$	0	0
$877$	− 813.286i	− 0.927351i	−0.886005	−	0.463675i	$-0.846530\pi$
$877$	− 813.286i	− 0.927351i	0.886005	−	0.463675i	$-0.153470\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−782.139	−0.887786	−0.443893	−	0.896080i	$-0.646403\pi$
$881$	−782.139	−0.887786	−0.443893	+	0.896080i	$0.646403\pi$
$882$	0	0
$883$	−1282.01	−1.45188	−0.725939	−	0.687759i	$-0.758595\pi$
$883$	−1282.01	−1.45188	−0.725939	+	0.687759i	$0.758595\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 939.076i	− 1.05871i	−0.848400	−	0.529355i	$-0.822434\pi$
$887$	− 939.076i	− 1.05871i	0.848400	−	0.529355i	$-0.177566\pi$
$888$	0	0
$889$	42.3679	0.0476580
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 231.464i	− 0.259198i
$894$	0	0
$895$	631.311	0.705376
$896$	0	0
$897$	0	0
$898$	0	0
$899$	838.437i	0.932633i
$900$	0	0
$901$	− 3.07593i	− 0.00341391i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1055.17	1.16593
$906$	0	0
$907$	−285.210	−0.314454	−0.157227	−	0.987562i	$-0.550255\pi$
$907$	−285.210	−0.314454	−0.157227	+	0.987562i	$0.550255\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1238.24	−1.35920	−0.679602	−	0.733581i	$-0.737848\pi$
$911$	−1238.24	−1.35920	−0.679602	+	0.733581i	$0.737848\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−267.333	−0.291529
$918$	0	0
$919$	− 626.565i	− 0.681790i	−0.940101	−	0.340895i	$-0.889270\pi$
$919$	− 626.565i	− 0.681790i	0.940101	−	0.340895i	$-0.110730\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 821.485i	− 0.890016i
$924$	0	0
$925$	547.375	0.591757
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−62.0672	−0.0668107	−0.0334054	−	0.999442i	$-0.510635\pi$
$929$	−62.0672	−0.0668107	−0.0334054	+	0.999442i	$0.510635\pi$
$930$	0	0
$931$	− 1263.10i	− 1.35672i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	484.663i	0.517250i	0.965978	+	0.258625i	$0.0832694\pi$
$937$	484.663i	0.517250i	−0.965978	+	0.258625i	$0.916731\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 127.978i	− 0.136002i	−0.997685	−	0.0680009i	$-0.978338\pi$
$941$	− 127.978i	− 0.136002i	0.997685	−	0.0680009i	$-0.0216621\pi$
$942$	0	0
$943$	− 823.337i	− 0.873104i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1560.24	1.64757	0.823783	−	0.566906i	$-0.191860\pi$
$947$	1560.24	1.64757	0.823783	+	0.566906i	$0.191860\pi$
$948$	0	0
$949$	−717.379	−0.755932
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 827.137i	− 0.867929i	−0.900930	−	0.433965i	$-0.857114\pi$
$953$	− 827.137i	− 0.867929i	0.900930	−	0.433965i	$-0.142886\pi$
$954$	0	0
$955$	−867.104	−0.907963
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 124.410i	− 0.129729i
$960$	0	0
$961$	−661.801	−0.688658
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1009.08i	1.04568i
$966$	0	0
$967$	− 867.442i	− 0.897044i	−0.893772	−	0.448522i	$-0.851951\pi$
$967$	− 867.442i	− 0.897044i	0.893772	−	0.448522i	$-0.148049\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1584.40	−1.63172	−0.815860	−	0.578249i	$-0.803736\pi$
$971$	−1584.40	−1.63172	−0.815860	+	0.578249i	$0.803736\pi$
$972$	0	0
$973$	−110.885	−0.113961
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1534.99	1.57112	0.785562	−	0.618783i	$-0.212374\pi$
$977$	1534.99	1.57112	0.785562	+	0.618783i	$0.212374\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	143.526	0.146008	0.0730042	−	0.997332i	$-0.476741\pi$
$983$	143.526	0.146008	0.0730042	+	0.997332i	$0.476741\pi$
$984$	0	0
$985$	− 718.834i	− 0.729780i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1156.02i	− 1.16888i
$990$	0	0
$991$	169.830	0.171372	0.0856860	−	0.996322i	$-0.472692\pi$
$991$	169.830	0.171372	0.0856860	+	0.996322i	$0.472692\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	139.848	0.140551
$996$	0	0
$997$	− 669.850i	− 0.671865i	−0.941886	−	0.335933i	$-0.890949\pi$
$997$	− 669.850i	− 0.671865i	0.941886	−	0.335933i	$-0.109051\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4356.3.f.g.1693.8		8
3.2	odd	2		484.3.d.c.241.6		8
11.6	odd	10		396.3.t.a.217.1		8
11.9	even	5		396.3.t.a.73.1		8
11.10	odd	2	inner	4356.3.f.g.1693.7		8
33.2	even	10		484.3.f.a.161.2		8
33.5	odd	10		484.3.f.a.481.2		8
33.8	even	10		484.3.f.e.233.1		8
33.14	odd	10		484.3.f.d.233.1		8
33.17	even	10		44.3.f.a.41.2	yes	8
33.20	odd	10		44.3.f.a.29.2	✓	8
33.26	odd	10		484.3.f.e.457.1		8
33.29	even	10		484.3.f.d.457.1		8
33.32	even	2		484.3.d.c.241.5		8
132.83	odd	10		176.3.n.c.129.1		8
132.119	even	10		176.3.n.c.161.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.3.f.a.29.2	✓	8	33.20	odd	10
44.3.f.a.41.2	yes	8	33.17	even	10
176.3.n.c.129.1		8	132.83	odd	10
176.3.n.c.161.1		8	132.119	even	10
396.3.t.a.73.1		8	11.9	even	5
396.3.t.a.217.1		8	11.6	odd	10
484.3.d.c.241.5		8	33.32	even	2
484.3.d.c.241.6		8	3.2	odd	2
484.3.f.a.161.2		8	33.2	even	10
484.3.f.a.481.2		8	33.5	odd	10
484.3.f.d.233.1		8	33.14	odd	10
484.3.f.d.457.1		8	33.29	even	10
484.3.f.e.233.1		8	33.8	even	10
484.3.f.e.457.1		8	33.26	odd	10
4356.3.f.g.1693.7		8	11.10	odd	2	inner
4356.3.f.g.1693.8		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:	\( N \)	\(=\)	\( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4356.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.17883 q^{5} +1.10828i q^{7} -12.3893i q^{13} +11.8588i q^{17} -26.4404i q^{19} -19.9563 q^{23} -7.53741 q^{25} +48.4719i q^{29} +17.2974 q^{31} +4.63130i q^{35} -72.6211 q^{37} +41.2570i q^{41} +57.9276i q^{43} +8.75419 q^{47} +47.7717 q^{49} -0.259379 q^{53} +43.5555 q^{59} +81.8700i q^{61} -51.7725i q^{65} +52.4087 q^{67} +66.3063 q^{71} -57.9034i q^{73} +8.59534i q^{79} +87.2900i q^{83} +49.5560i q^{85} +3.65450 q^{89} +13.7307 q^{91} -110.490i q^{95} +6.85977 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{5} - 100 q^{23} - 34 q^{25} + 98 q^{31} - 210 q^{37} + 110 q^{47} + 22 q^{49} - 250 q^{53} - 112 q^{59} + 10 q^{67} + 198 q^{71} - 18 q^{89} + 440 q^{91} - 220 q^{97}+O(q^{100}) \) 8 * q - 14 * q^5 - 100 * q^23 - 34 * q^25 + 98 * q^31 - 210 * q^37 + 110 * q^47 + 22 * q^49 - 250 * q^53 - 112 * q^59 + 10 * q^67 + 198 * q^71 - 18 * q^89 + 440 * q^91 - 220 * q^97

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