LMFDB - Embedded newform 558.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [558,2,Mod(1,558)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(558, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("558.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 558 = 2 \cdot 3^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	558.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-3,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.45565243279$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	558.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -3.00000 q^{10} -3.00000 q^{11} +5.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -7.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -6.00000 q^{23} +4.00000 q^{25} +5.00000 q^{26} -4.00000 q^{28} -6.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +12.0000 q^{35} +2.00000 q^{37} -7.00000 q^{38} -3.00000 q^{40} +12.0000 q^{41} +8.00000 q^{43} -3.00000 q^{44} -6.00000 q^{46} +3.00000 q^{47} +9.00000 q^{49} +4.00000 q^{50} +5.00000 q^{52} -12.0000 q^{53} +9.00000 q^{55} -4.00000 q^{56} -6.00000 q^{58} -6.00000 q^{59} +5.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -15.0000 q^{65} +11.0000 q^{67} -3.00000 q^{68} +12.0000 q^{70} +3.00000 q^{71} -4.00000 q^{73} +2.00000 q^{74} -7.00000 q^{76} +12.0000 q^{77} -13.0000 q^{79} -3.00000 q^{80} +12.0000 q^{82} +3.00000 q^{83} +9.00000 q^{85} +8.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} -20.0000 q^{91} -6.00000 q^{92} +3.00000 q^{94} +21.0000 q^{95} -7.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	0	0
$28$	−4.00000	−0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	12.0000	2.02837
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−7.00000	−1.13555
$39$	0	0
$40$	−3.00000	−0.474342
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	4.00000	0.565685
$51$	0	0
$52$	5.00000	0.693375
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	−4.00000	−0.534522
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−15.0000	−1.86052
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	12.0000	1.43427
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−7.00000	−0.802955
$77$	12.0000	1.36753
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	12.0000	1.32518
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	9.00000	0.976187
$86$	8.00000	0.862662
$87$	0	0
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−20.0000	−2.09657
$92$	−6.00000	−0.625543
$93$	0	0
$94$	3.00000	0.309426
$95$	21.0000	2.15455
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	4.00000	0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	9.00000	0.858116
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−6.00000	−0.552345
$119$	12.0000	1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	5.00000	0.452679
$123$	0	0
$124$	1.00000	0.0898027
$125$	3.00000	0.268328
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−15.0000	−1.31559
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	28.0000	2.42791
$134$	11.0000	0.950255
$135$	0	0
$136$	−3.00000	−0.257248
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	12.0000	1.01419
$141$	0	0
$142$	3.00000	0.251754
$143$	−15.0000	−1.25436
$144$	0	0
$145$	18.0000	1.49482
$146$	−4.00000	−0.331042
$147$	0	0
$148$	2.00000	0.164399
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	−7.00000	−0.567775
$153$	0	0
$154$	12.0000	0.966988
$155$	−3.00000	−0.240966
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−13.0000	−1.03422
$159$	0	0
$160$	−3.00000	−0.237171
$161$	24.0000	1.89146
$162$	0	0
$163$	−7.00000	−0.548282	−0.274141	−	0.961689i	$-0.588394\pi$
$163$	−7.00000	−0.548282	−0.274141	+	0.961689i	$0.588394\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	3.00000	0.232845
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	9.00000	0.690268
$171$	0	0
$172$	8.00000	0.609994
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	0	0
$175$	−16.0000	−1.20949
$176$	−3.00000	−0.226134
$177$	0	0
$178$	6.00000	0.449719
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−20.0000	−1.48250
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−6.00000	−0.441129
$186$	0	0
$187$	9.00000	0.658145
$188$	3.00000	0.218797
$189$	0	0
$190$	21.0000	1.52350
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	9.00000	0.642857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	6.00000	0.422159
$203$	24.0000	1.68447
$204$	0	0
$205$	−36.0000	−2.51435
$206$	−4.00000	−0.278693
$207$	0	0
$208$	5.00000	0.346688
$209$	21.0000	1.45260
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−18.0000	−1.23045
$215$	−24.0000	−1.63679
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−16.0000	−1.08366
$219$	0	0
$220$	9.00000	0.606780
$221$	−15.0000	−1.00901
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	−6.00000	−0.393919
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	−6.00000	−0.390567
$237$	0	0
$238$	12.0000	0.777844
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	5.00000	0.320092
$245$	−27.0000	−1.72497
$246$	0	0
$247$	−35.0000	−2.22700
$248$	1.00000	0.0635001
$249$	0	0
$250$	3.00000	0.189737
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	−15.0000	−0.930261
$261$	0	0
$262$	6.00000	0.370681
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	28.0000	1.71679
$267$	0	0
$268$	11.0000	0.671932
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	18.0000	1.08742
$275$	−12.0000	−0.723627
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	2.00000	0.119952
$279$	0	0
$280$	12.0000	0.717137
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	−15.0000	−0.886969
$287$	−48.0000	−2.83335
$288$	0	0
$289$	−8.00000	−0.470588
$290$	18.0000	1.05700
$291$	0	0
$292$	−4.00000	−0.234082
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	2.00000	0.116248
$297$	0	0
$298$	9.00000	0.521356
$299$	−30.0000	−1.73494
$300$	0	0
$301$	−32.0000	−1.84445
$302$	−7.00000	−0.402805
$303$	0	0
$304$	−7.00000	−0.401478
$305$	−15.0000	−0.858898
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	12.0000	0.683763
$309$	0	0
$310$	−3.00000	−0.170389
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−13.0000	−0.731307
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	−3.00000	−0.167705
$321$	0	0
$322$	24.0000	1.33747
$323$	21.0000	1.16847
$324$	0	0
$325$	20.0000	1.10940
$326$	−7.00000	−0.387694
$327$	0	0
$328$	12.0000	0.662589
$329$	−12.0000	−0.661581
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	3.00000	0.164646
$333$	0	0
$334$	6.00000	0.328305
$335$	−33.0000	−1.80298
$336$	0	0
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	9.00000	0.488094
$341$	−3.00000	−0.162459
$342$	0	0
$343$	−8.00000	−0.431959
$344$	8.00000	0.431331
$345$	0	0
$346$	−21.0000	−1.12897
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	−16.0000	−0.855236
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	−9.00000	−0.477670
$356$	6.00000	0.317999
$357$	0	0
$358$	−21.0000	−1.10988
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	2.00000	0.105118
$363$	0	0
$364$	−20.0000	−1.04828
$365$	12.0000	0.628109
$366$	0	0
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	−6.00000	−0.311925
$371$	48.0000	2.49204
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	3.00000	0.154713
$377$	−30.0000	−1.54508
$378$	0	0
$379$	−13.0000	−0.667765	−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000	−0.667765	−0.333883	+	0.942615i	$0.608359\pi$
$380$	21.0000	1.07728
$381$	0	0
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	−36.0000	−1.83473
$386$	−1.00000	−0.0508987
$387$	0	0
$388$	−7.00000	−0.355371
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	9.00000	0.454569
$393$	0	0
$394$	−6.00000	−0.302276
$395$	39.0000	1.96230
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−1.00000	−0.0501255
$399$	0	0
$400$	4.00000	0.200000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	5.00000	0.249068
$404$	6.00000	0.298511
$405$	0	0
$406$	24.0000	1.19110
$407$	−6.00000	−0.297409
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−36.0000	−1.77791
$411$	0	0
$412$	−4.00000	−0.197066
$413$	24.0000	1.18096
$414$	0	0
$415$	−9.00000	−0.441793
$416$	5.00000	0.245145
$417$	0	0
$418$	21.0000	1.02714
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	−12.0000	−0.582772
$425$	−12.0000	−0.582086
$426$	0	0
$427$	−20.0000	−0.967868
$428$	−18.0000	−0.870063
$429$	0	0
$430$	−24.0000	−1.15738
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−16.0000	−0.766261
$437$	42.0000	2.00913
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	9.00000	0.429058
$441$	0	0
$442$	−15.0000	−0.713477
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	−19.0000	−0.899676
$447$	0	0
$448$	−4.00000	−0.188982
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−12.0000	−0.563188
$455$	60.0000	2.81284
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	5.00000	0.233635
$459$	0	0
$460$	18.0000	0.839254
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	24.0000	1.11178
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−44.0000	−2.03173
$470$	−9.00000	−0.415139
$471$	0	0
$472$	−6.00000	−0.276172
$473$	−24.0000	−1.10352
$474$	0	0
$475$	−28.0000	−1.28473
$476$	12.0000	0.550019
$477$	0	0
$478$	6.00000	0.274434
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	20.0000	0.910975
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	21.0000	0.953561
$486$	0	0
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	5.00000	0.226339
$489$	0	0
$490$	−27.0000	−1.21974
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	−35.0000	−1.57472
$495$	0	0
$496$	1.00000	0.0449013
$497$	−12.0000	−0.538274
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	12.0000	0.535586
$503$	39.0000	1.73892	0.869462	−	0.494000i	$-0.164466\pi$
$503$	39.0000	1.73892	0.869462	+	0.494000i	$0.164466\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	18.0000	0.800198
$507$	0	0
$508$	−16.0000	−0.709885
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	0	0
$514$	0	0
$515$	12.0000	0.528783
$516$	0	0
$517$	−9.00000	−0.395820
$518$	−8.00000	−0.351500
$519$	0	0
$520$	−15.0000	−0.657794
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	0	0
$527$	−3.00000	−0.130682
$528$	0	0
$529$	13.0000	0.565217
$530$	36.0000	1.56374
$531$	0	0
$532$	28.0000	1.21395
$533$	60.0000	2.59889
$534$	0	0
$535$	54.0000	2.33462
$536$	11.0000	0.475128
$537$	0	0
$538$	24.0000	1.03471
$539$	−27.0000	−1.16297
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	11.0000	0.472490
$543$	0	0
$544$	−3.00000	−0.128624
$545$	48.0000	2.05609
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	−12.0000	−0.511682
$551$	42.0000	1.78926
$552$	0	0
$553$	52.0000	2.21126
$554$	−19.0000	−0.807233
$555$	0	0
$556$	2.00000	0.0848189
$557$	36.0000	1.52537	0.762684	−	0.646771i	$-0.223881\pi$
$557$	36.0000	1.52537	0.762684	+	0.646771i	$0.223881\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	12.0000	0.507093
$561$	0	0
$562$	12.0000	0.506189
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	−13.0000	−0.546431
$567$	0	0
$568$	3.00000	0.125877
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−15.0000	−0.627182
$573$	0	0
$574$	−48.0000	−2.00348
$575$	−24.0000	−1.00087
$576$	0	0
$577$	35.0000	1.45707	0.728535	−	0.685009i	$-0.240202\pi$
$577$	35.0000	1.45707	0.728535	+	0.685009i	$0.240202\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	18.0000	0.747409
$581$	−12.0000	−0.497844
$582$	0	0
$583$	36.0000	1.49097
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−6.00000	−0.247858
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	−7.00000	−0.288430
$590$	18.0000	0.741048
$591$	0	0
$592$	2.00000	0.0821995
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	−36.0000	−1.47586
$596$	9.00000	0.368654
$597$	0	0
$598$	−30.0000	−1.22679
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	−32.0000	−1.30422
$603$	0	0
$604$	−7.00000	−0.284826
$605$	6.00000	0.243935
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	−15.0000	−0.607332
$611$	15.0000	0.606835
$612$	0	0
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	12.0000	0.483494
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	−3.00000	−0.120483
$621$	0	0
$622$	−3.00000	−0.120289
$623$	−24.0000	−0.961540
$624$	0	0
$625$	−29.0000	−1.16000
$626$	20.0000	0.799361
$627$	0	0
$628$	−22.0000	−0.877896
$629$	−6.00000	−0.239236
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−13.0000	−0.517112
$633$	0	0
$634$	−21.0000	−0.834017
$635$	48.0000	1.90482
$636$	0	0
$637$	45.0000	1.78296
$638$	18.0000	0.712627
$639$	0	0
$640$	−3.00000	−0.118585
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	0	0
$643$	50.0000	1.97181	0.985904	−	0.167313i	$-0.0535092\pi$
$643$	50.0000	1.97181	0.985904	+	0.167313i	$0.0535092\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	21.0000	0.826234
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	20.0000	0.784465
$651$	0	0
$652$	−7.00000	−0.274141
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	12.0000	0.468521
$657$	0	0
$658$	−12.0000	−0.467809
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	2.00000	0.0777322
$663$	0	0
$664$	3.00000	0.116423
$665$	−84.0000	−3.25738
$666$	0	0
$667$	36.0000	1.39393
$668$	6.00000	0.232147
$669$	0	0
$670$	−33.0000	−1.27490
$671$	−15.0000	−0.579069
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	20.0000	0.770371
$675$	0	0
$676$	12.0000	0.461538
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	0	0
$679$	28.0000	1.07454
$680$	9.00000	0.345134
$681$	0	0
$682$	−3.00000	−0.114876
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	−54.0000	−2.06323
$686$	−8.00000	−0.305441
$687$	0	0
$688$	8.00000	0.304997
$689$	−60.0000	−2.28582
$690$	0	0
$691$	−37.0000	−1.40755	−0.703773	−	0.710425i	$-0.748503\pi$
$691$	−37.0000	−1.40755	−0.703773	+	0.710425i	$0.748503\pi$
$692$	−21.0000	−0.798300
$693$	0	0
$694$	−3.00000	−0.113878
$695$	−6.00000	−0.227593
$696$	0	0
$697$	−36.0000	−1.36360
$698$	8.00000	0.302804
$699$	0	0
$700$	−16.0000	−0.604743
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−21.0000	−0.790345
$707$	−24.0000	−0.902613
$708$	0	0
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	−9.00000	−0.337764
$711$	0	0
$712$	6.00000	0.224860
$713$	−6.00000	−0.224702
$714$	0	0
$715$	45.0000	1.68290
$716$	−21.0000	−0.784807
$717$	0	0
$718$	3.00000	0.111959
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	30.0000	1.11648
$723$	0	0
$724$	2.00000	0.0743294
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	−20.0000	−0.741249
$729$	0	0
$730$	12.0000	0.444140
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−25.0000	−0.922767
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−33.0000	−1.21557
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	48.0000	1.76214
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−27.0000	−0.989203
$746$	−4.00000	−0.146450
$747$	0	0
$748$	9.00000	0.329073
$749$	72.0000	2.63082
$750$	0	0
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	−30.0000	−1.09254
$755$	21.0000	0.764268
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−13.0000	−0.472181
$759$	0	0
$760$	21.0000	0.761750
$761$	−9.00000	−0.326250	−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000	−0.326250	−0.163125	+	0.986605i	$0.552157\pi$
$762$	0	0
$763$	64.0000	2.31696
$764$	0	0
$765$	0	0
$766$	36.0000	1.30073
$767$	−30.0000	−1.08324
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	−36.0000	−1.29735
$771$	0	0
$772$	−1.00000	−0.0359908
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	−7.00000	−0.251285
$777$	0	0
$778$	−30.0000	−1.07555
$779$	−84.0000	−3.00961
$780$	0	0
$781$	−9.00000	−0.322045
$782$	18.0000	0.643679
$783$	0	0
$784$	9.00000	0.321429
$785$	66.0000	2.35564
$786$	0	0
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	39.0000	1.38756
$791$	24.0000	0.853342
$792$	0	0
$793$	25.0000	0.887776
$794$	2.00000	0.0709773
$795$	0	0
$796$	−1.00000	−0.0354441
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	4.00000	0.141421
$801$	0	0
$802$	3.00000	0.105934
$803$	12.0000	0.423471
$804$	0	0
$805$	−72.0000	−2.53767
$806$	5.00000	0.176117
$807$	0	0
$808$	6.00000	0.211079
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	−6.00000	−0.210300
$815$	21.0000	0.735598
$816$	0	0
$817$	−56.0000	−1.95919
$818$	32.0000	1.11885
$819$	0	0
$820$	−36.0000	−1.25717
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	0	0
$823$	−7.00000	−0.244005	−0.122002	−	0.992530i	$-0.538932\pi$
$823$	−7.00000	−0.244005	−0.122002	+	0.992530i	$0.538932\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	15.0000	0.521601	0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000	0.521601	0.260801	+	0.965393i	$0.416014\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−9.00000	−0.312395
$831$	0	0
$832$	5.00000	0.173344
$833$	−27.0000	−0.935495
$834$	0	0
$835$	−18.0000	−0.622916
$836$	21.0000	0.726300
$837$	0	0
$838$	30.0000	1.03633
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−22.0000	−0.758170
$843$	0	0
$844$	8.00000	0.275371
$845$	−36.0000	−1.23844
$846$	0	0
$847$	8.00000	0.274883
$848$	−12.0000	−0.412082
$849$	0	0
$850$	−12.0000	−0.411597
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	−20.0000	−0.684386
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	0	0
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	0	0
$865$	63.0000	2.14206
$866$	−16.0000	−0.543702
$867$	0	0
$868$	−4.00000	−0.135769
$869$	39.0000	1.32298
$870$	0	0
$871$	55.0000	1.86360
$872$	−16.0000	−0.541828
$873$	0	0
$874$	42.0000	1.42067
$875$	−12.0000	−0.405674
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	26.0000	0.877457
$879$	0	0
$880$	9.00000	0.303390
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	−15.0000	−0.504505
$885$	0	0
$886$	−18.0000	−0.604722
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	64.0000	2.14649
$890$	−18.0000	−0.603361
$891$	0	0
$892$	−19.0000	−0.636167
$893$	−21.0000	−0.702738
$894$	0	0
$895$	63.0000	2.10586
$896$	−4.00000	−0.133631
$897$	0	0
$898$	−21.0000	−0.700779
$899$	−6.00000	−0.200111
$900$	0	0
$901$	36.0000	1.19933
$902$	−36.0000	−1.19867
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−6.00000	−0.199447
$906$	0	0
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	60.0000	1.98898
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	26.0000	0.860004
$915$	0	0
$916$	5.00000	0.165205
$917$	−24.0000	−0.792550
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	18.0000	0.593442
$921$	0	0
$922$	−30.0000	−0.987997
$923$	15.0000	0.493731
$924$	0	0
$925$	8.00000	0.263038
$926$	−1.00000	−0.0328620
$927$	0	0
$928$	−6.00000	−0.196960
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−63.0000	−2.06474
$932$	24.0000	0.786146
$933$	0	0
$934$	0	0
$935$	−27.0000	−0.882994
$936$	0	0
$937$	−31.0000	−1.01273	−0.506363	−	0.862320i	$-0.669010\pi$
$937$	−31.0000	−1.01273	−0.506363	+	0.862320i	$0.669010\pi$
$938$	−44.0000	−1.43665
$939$	0	0
$940$	−9.00000	−0.293548
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	−72.0000	−2.34464
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−24.0000	−0.780307
$947$	15.0000	0.487435	0.243717	−	0.969846i	$-0.421633\pi$
$947$	15.0000	0.487435	0.243717	+	0.969846i	$0.421633\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	−28.0000	−0.908440
$951$	0	0
$952$	12.0000	0.388922
$953$	33.0000	1.06897	0.534487	−	0.845176i	$-0.320505\pi$
$953$	33.0000	1.06897	0.534487	+	0.845176i	$0.320505\pi$
$954$	0	0
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	−15.0000	−0.484628
$959$	−72.0000	−2.32500
$960$	0	0
$961$	1.00000	0.0322581
$962$	10.0000	0.322413
$963$	0	0
$964$	20.0000	0.644157
$965$	3.00000	0.0965734
$966$	0	0
$967$	11.0000	0.353736	0.176868	−	0.984235i	$-0.443403\pi$
$967$	11.0000	0.353736	0.176868	+	0.984235i	$0.443403\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	21.0000	0.674269
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	23.0000	0.736968
$975$	0	0
$976$	5.00000	0.160046
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	−27.0000	−0.862483
$981$	0	0
$982$	−36.0000	−1.14881
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	18.0000	0.573237
$987$	0	0
$988$	−35.0000	−1.11350
$989$	−48.0000	−1.52631
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	−12.0000	−0.380617
$995$	3.00000	0.0951064
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	558.2.a.e.1.1	yes	1
3.2	odd	2		558.2.a.d.1.1	✓	1
4.3	odd	2		4464.2.a.d.1.1		1
12.11	even	2		4464.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
558.2.a.d.1.1	✓	1	3.2	odd	2
558.2.a.e.1.1	yes	1	1.1	even	1	trivial
4464.2.a.d.1.1		1	4.3	odd	2
4464.2.a.y.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 558 = 2 \cdot 3^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	558.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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