LMFDB - Embedded newform 118.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [118,2,Mod(1,118)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(118, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("118.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 118 = 2 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	118.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$0.942234743851$
Analytic rank:	$0$
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	$\chi$	$=$	118.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} +3.00000 q^{13} +3.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +2.00000 q^{20} -6.00000 q^{21} -1.00000 q^{22} +8.00000 q^{23} -2.00000 q^{24} -1.00000 q^{25} -3.00000 q^{26} -4.00000 q^{27} -3.00000 q^{28} -4.00000 q^{29} -4.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} -6.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +8.00000 q^{38} +6.00000 q^{39} -2.00000 q^{40} +5.00000 q^{41} +6.00000 q^{42} -9.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -8.00000 q^{46} +2.00000 q^{47} +2.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +3.00000 q^{52} +12.0000 q^{53} +4.00000 q^{54} +2.00000 q^{55} +3.00000 q^{56} -16.0000 q^{57} +4.00000 q^{58} +1.00000 q^{59} +4.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -2.00000 q^{66} +4.00000 q^{67} -1.00000 q^{68} +16.0000 q^{69} +6.00000 q^{70} -15.0000 q^{71} -1.00000 q^{72} +10.0000 q^{73} +1.00000 q^{74} -2.00000 q^{75} -8.00000 q^{76} -3.00000 q^{77} -6.00000 q^{78} +11.0000 q^{79} +2.00000 q^{80} -11.0000 q^{81} -5.00000 q^{82} -11.0000 q^{83} -6.00000 q^{84} -2.00000 q^{85} +9.00000 q^{86} -8.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -9.00000 q^{91} +8.00000 q^{92} -8.00000 q^{93} -2.00000 q^{94} -16.0000 q^{95} -2.00000 q^{96} +14.0000 q^{97} -2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−2.00000	−0.816497
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	2.00000	0.577350
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	3.00000	0.801784
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	−1.00000	−0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	2.00000	0.447214
$21$	−6.00000	−1.30931
$22$	−1.00000	−0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−2.00000	−0.408248
$25$	−1.00000	−0.200000
$26$	−3.00000	−0.588348
$27$	−4.00000	−0.769800
$28$	−3.00000	−0.566947
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−4.00000	−0.730297
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	1.00000	0.171499
$35$	−6.00000	−1.01419
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	8.00000	1.29777
$39$	6.00000	0.960769
$40$	−2.00000	−0.316228
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	6.00000	0.925820
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	1.00000	0.150756
$45$	2.00000	0.298142
$46$	−8.00000	−1.17954
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	2.00000	0.288675
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	3.00000	0.416025
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	4.00000	0.544331
$55$	2.00000	0.269680
$56$	3.00000	0.400892
$57$	−16.0000	−2.11925
$58$	4.00000	0.525226
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	4.00000	0.516398
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	4.00000	0.508001
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−2.00000	−0.246183
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−1.00000	−0.121268
$69$	16.0000	1.92617
$70$	6.00000	0.717137
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	1.00000	0.116248
$75$	−2.00000	−0.230940
$76$	−8.00000	−0.917663
$77$	−3.00000	−0.341882
$78$	−6.00000	−0.679366
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	2.00000	0.223607
$81$	−11.0000	−1.22222
$82$	−5.00000	−0.552158
$83$	−11.0000	−1.20741	−0.603703	−	0.797209i	$-0.706309\pi$
$83$	−11.0000	−1.20741	−0.603703	+	0.797209i	$0.706309\pi$
$84$	−6.00000	−0.654654
$85$	−2.00000	−0.216930
$86$	9.00000	0.970495
$87$	−8.00000	−0.857690
$88$	−1.00000	−0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−2.00000	−0.210819
$91$	−9.00000	−0.943456
$92$	8.00000	0.834058
$93$	−8.00000	−0.829561
$94$	−2.00000	−0.206284
$95$	−16.0000	−1.64157
$96$	−2.00000	−0.204124
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−2.00000	−0.202031
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	2.00000	0.198030
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	−3.00000	−0.294174
$105$	−12.0000	−1.17108
$106$	−12.0000	−1.16554
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	−4.00000	−0.384900
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−2.00000	−0.190693
$111$	−2.00000	−0.189832
$112$	−3.00000	−0.283473
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	16.0000	1.49854
$115$	16.0000	1.49201
$116$	−4.00000	−0.371391
$117$	3.00000	0.277350
$118$	−1.00000	−0.0920575
$119$	3.00000	0.275010
$120$	−4.00000	−0.365148
$121$	−10.0000	−0.909091
$122$	−10.0000	−0.905357
$123$	10.0000	0.901670
$124$	−4.00000	−0.359211
$125$	−12.0000	−1.07331
$126$	3.00000	0.267261
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	−18.0000	−1.58481
$130$	−6.00000	−0.526235
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	2.00000	0.174078
$133$	24.0000	2.08106
$134$	−4.00000	−0.345547
$135$	−8.00000	−0.688530
$136$	1.00000	0.0857493
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−16.0000	−1.36201
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	−6.00000	−0.507093
$141$	4.00000	0.336861
$142$	15.0000	1.25877
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	−8.00000	−0.664364
$146$	−10.0000	−0.827606
$147$	4.00000	0.329914
$148$	−1.00000	−0.0821995
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	2.00000	0.163299
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	8.00000	0.648886
$153$	−1.00000	−0.0808452
$154$	3.00000	0.241747
$155$	−8.00000	−0.642575
$156$	6.00000	0.480384
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−11.0000	−0.875113
$159$	24.0000	1.90332
$160$	−2.00000	−0.158114
$161$	−24.0000	−1.89146
$162$	11.0000	0.864242
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	5.00000	0.390434
$165$	4.00000	0.311400
$166$	11.0000	0.853766
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	6.00000	0.462910
$169$	−4.00000	−0.307692
$170$	2.00000	0.153393
$171$	−8.00000	−0.611775
$172$	−9.00000	−0.686244
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	8.00000	0.606478
$175$	3.00000	0.226779
$176$	1.00000	0.0753778
$177$	2.00000	0.150329
$178$	6.00000	0.449719
$179$	−17.0000	−1.27064	−0.635320	−	0.772249i	$-0.719132\pi$
$179$	−17.0000	−1.27064	−0.635320	+	0.772249i	$0.719132\pi$
$180$	2.00000	0.149071
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	9.00000	0.667124
$183$	20.0000	1.47844
$184$	−8.00000	−0.589768
$185$	−2.00000	−0.147043
$186$	8.00000	0.586588
$187$	−1.00000	−0.0731272
$188$	2.00000	0.145865
$189$	12.0000	0.872872
$190$	16.0000	1.16076
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	2.00000	0.144338
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	−14.0000	−1.00514
$195$	12.0000	0.859338
$196$	2.00000	0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−1.00000	−0.0710669
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	−9.00000	−0.633238
$203$	12.0000	0.842235
$204$	−2.00000	−0.140028
$205$	10.0000	0.698430
$206$	−2.00000	−0.139347
$207$	8.00000	0.556038
$208$	3.00000	0.208013
$209$	−8.00000	−0.553372
$210$	12.0000	0.828079
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	12.0000	0.824163
$213$	−30.0000	−2.05557
$214$	−18.0000	−1.23045
$215$	−18.0000	−1.22759
$216$	4.00000	0.272166
$217$	12.0000	0.814613
$218$	2.00000	0.135457
$219$	20.0000	1.35147
$220$	2.00000	0.134840
$221$	−3.00000	−0.201802
$222$	2.00000	0.134231
$223$	−3.00000	−0.200895	−0.100447	−	0.994942i	$-0.532027\pi$
$223$	−3.00000	−0.200895	−0.100447	+	0.994942i	$0.532027\pi$
$224$	3.00000	0.200446
$225$	−1.00000	−0.0666667
$226$	−4.00000	−0.266076
$227$	1.00000	0.0663723	0.0331862	−	0.999449i	$-0.489435\pi$
$227$	1.00000	0.0663723	0.0331862	+	0.999449i	$0.489435\pi$
$228$	−16.0000	−1.05963
$229$	29.0000	1.91637	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000	1.91637	0.958187	+	0.286143i	$0.0923732\pi$
$230$	−16.0000	−1.05501
$231$	−6.00000	−0.394771
$232$	4.00000	0.262613
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−3.00000	−0.196116
$235$	4.00000	0.260931
$236$	1.00000	0.0650945
$237$	22.0000	1.42905
$238$	−3.00000	−0.194461
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	4.00000	0.258199
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	10.0000	0.642824
$243$	−10.0000	−0.641500
$244$	10.0000	0.640184
$245$	4.00000	0.255551
$246$	−10.0000	−0.637577
$247$	−24.0000	−1.52708
$248$	4.00000	0.254000
$249$	−22.0000	−1.39419
$250$	12.0000	0.758947
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	−3.00000	−0.188982
$253$	8.00000	0.502956
$254$	−16.0000	−1.00393
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	18.0000	1.12063
$259$	3.00000	0.186411
$260$	6.00000	0.372104
$261$	−4.00000	−0.247594
$262$	−12.0000	−0.741362
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	−2.00000	−0.123091
$265$	24.0000	1.47431
$266$	−24.0000	−1.47153
$267$	−12.0000	−0.734388
$268$	4.00000	0.244339
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	8.00000	0.486864
$271$	−1.00000	−0.0607457	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000	−0.0607457	−0.0303728	+	0.999539i	$0.509669\pi$
$272$	−1.00000	−0.0606339
$273$	−18.0000	−1.08941
$274$	−9.00000	−0.543710
$275$	−1.00000	−0.0603023
$276$	16.0000	0.963087
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−2.00000	−0.119952
$279$	−4.00000	−0.239474
$280$	6.00000	0.358569
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	−4.00000	−0.238197
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	−15.0000	−0.890086
$285$	−32.0000	−1.89552
$286$	−3.00000	−0.177394
$287$	−15.0000	−0.885422
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	8.00000	0.469776
$291$	28.0000	1.64139
$292$	10.0000	0.585206
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	−4.00000	−0.233285
$295$	2.00000	0.116445
$296$	1.00000	0.0581238
$297$	−4.00000	−0.232104
$298$	−5.00000	−0.289642
$299$	24.0000	1.38796
$300$	−2.00000	−0.115470
$301$	27.0000	1.55625
$302$	22.0000	1.26596
$303$	18.0000	1.03407
$304$	−8.00000	−0.458831
$305$	20.0000	1.14520
$306$	1.00000	0.0571662
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−3.00000	−0.170941
$309$	4.00000	0.227552
$310$	8.00000	0.454369
$311$	−13.0000	−0.737162	−0.368581	−	0.929596i	$-0.620156\pi$
$311$	−13.0000	−0.737162	−0.368581	+	0.929596i	$0.620156\pi$
$312$	−6.00000	−0.339683
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	18.0000	1.01580
$315$	−6.00000	−0.338062
$316$	11.0000	0.618798
$317$	−4.00000	−0.224662	−0.112331	−	0.993671i	$-0.535832\pi$
$317$	−4.00000	−0.224662	−0.112331	+	0.993671i	$0.535832\pi$
$318$	−24.0000	−1.34585
$319$	−4.00000	−0.223957
$320$	2.00000	0.111803
$321$	36.0000	2.00932
$322$	24.0000	1.33747
$323$	8.00000	0.445132
$324$	−11.0000	−0.611111
$325$	−3.00000	−0.166410
$326$	−10.0000	−0.553849
$327$	−4.00000	−0.221201
$328$	−5.00000	−0.276079
$329$	−6.00000	−0.330791
$330$	−4.00000	−0.220193
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−11.0000	−0.603703
$333$	−1.00000	−0.0547997
$334$	−12.0000	−0.656611
$335$	8.00000	0.437087
$336$	−6.00000	−0.327327
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	4.00000	0.217571
$339$	8.00000	0.434500
$340$	−2.00000	−0.108465
$341$	−4.00000	−0.216612
$342$	8.00000	0.432590
$343$	15.0000	0.809924
$344$	9.00000	0.485247
$345$	32.0000	1.72282
$346$	13.0000	0.698884
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−8.00000	−0.428845
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	−3.00000	−0.160357
$351$	−12.0000	−0.640513
$352$	−1.00000	−0.0533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−2.00000	−0.106299
$355$	−30.0000	−1.59223
$356$	−6.00000	−0.317999
$357$	6.00000	0.317554
$358$	17.0000	0.898478
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	−2.00000	−0.105409
$361$	45.0000	2.36842
$362$	−8.00000	−0.420471
$363$	−20.0000	−1.04973
$364$	−9.00000	−0.471728
$365$	20.0000	1.04685
$366$	−20.0000	−1.04542
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	8.00000	0.417029
$369$	5.00000	0.260290
$370$	2.00000	0.103975
$371$	−36.0000	−1.86903
$372$	−8.00000	−0.414781
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	1.00000	0.0517088
$375$	−24.0000	−1.23935
$376$	−2.00000	−0.103142
$377$	−12.0000	−0.618031
$378$	−12.0000	−0.617213
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	−16.0000	−0.820783
$381$	32.0000	1.63941
$382$	−18.0000	−0.920960
$383$	7.00000	0.357683	0.178842	−	0.983878i	$-0.442765\pi$
$383$	7.00000	0.357683	0.178842	+	0.983878i	$0.442765\pi$
$384$	−2.00000	−0.102062
$385$	−6.00000	−0.305788
$386$	23.0000	1.17067
$387$	−9.00000	−0.457496
$388$	14.0000	0.710742
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	−12.0000	−0.607644
$391$	−8.00000	−0.404577
$392$	−2.00000	−0.101015
$393$	24.0000	1.21064
$394$	12.0000	0.604551
$395$	22.0000	1.10694
$396$	1.00000	0.0502519
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	8.00000	0.401004
$399$	48.0000	2.40301
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−8.00000	−0.399004
$403$	−12.0000	−0.597763
$404$	9.00000	0.447767
$405$	−22.0000	−1.09319
$406$	−12.0000	−0.595550
$407$	−1.00000	−0.0495682
$408$	2.00000	0.0990148
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	−10.0000	−0.493865
$411$	18.0000	0.887875
$412$	2.00000	0.0985329
$413$	−3.00000	−0.147620
$414$	−8.00000	−0.393179
$415$	−22.0000	−1.07994
$416$	−3.00000	−0.147087
$417$	4.00000	0.195881
$418$	8.00000	0.391293
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	−12.0000	−0.585540
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	11.0000	0.535472
$423$	2.00000	0.0972433
$424$	−12.0000	−0.582772
$425$	1.00000	0.0485071
$426$	30.0000	1.45350
$427$	−30.0000	−1.45180
$428$	18.0000	0.870063
$429$	6.00000	0.289683
$430$	18.0000	0.868037
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−4.00000	−0.192450
$433$	−35.0000	−1.68199	−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−35.0000	−1.68199	−0.840996	+	0.541041i	$0.818030\pi$
$434$	−12.0000	−0.576018
$435$	−16.0000	−0.767141
$436$	−2.00000	−0.0957826
$437$	−64.0000	−3.06154
$438$	−20.0000	−0.955637
$439$	11.0000	0.525001	0.262501	−	0.964932i	$-0.415453\pi$
$439$	11.0000	0.525001	0.262501	+	0.964932i	$0.415453\pi$
$440$	−2.00000	−0.0953463
$441$	2.00000	0.0952381
$442$	3.00000	0.142695
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	−2.00000	−0.0949158
$445$	−12.0000	−0.568855
$446$	3.00000	0.142054
$447$	10.0000	0.472984
$448$	−3.00000	−0.141737
$449$	−17.0000	−0.802280	−0.401140	−	0.916017i	$-0.631386\pi$
$449$	−17.0000	−0.802280	−0.401140	+	0.916017i	$0.631386\pi$
$450$	1.00000	0.0471405
$451$	5.00000	0.235441
$452$	4.00000	0.188144
$453$	−44.0000	−2.06730
$454$	−1.00000	−0.0469323
$455$	−18.0000	−0.843853
$456$	16.0000	0.749269
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−29.0000	−1.35508
$459$	4.00000	0.186704
$460$	16.0000	0.746004
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	6.00000	0.279145
$463$	−30.0000	−1.39422	−0.697109	−	0.716965i	$-0.745531\pi$
$463$	−30.0000	−1.39422	−0.697109	+	0.716965i	$0.745531\pi$
$464$	−4.00000	−0.185695
$465$	−16.0000	−0.741982
$466$	0	0
$467$	−1.00000	−0.0462745	−0.0231372	−	0.999732i	$-0.507365\pi$
$467$	−1.00000	−0.0462745	−0.0231372	+	0.999732i	$0.507365\pi$
$468$	3.00000	0.138675
$469$	−12.0000	−0.554109
$470$	−4.00000	−0.184506
$471$	−36.0000	−1.65879
$472$	−1.00000	−0.0460287
$473$	−9.00000	−0.413820
$474$	−22.0000	−1.01049
$475$	8.00000	0.367065
$476$	3.00000	0.137505
$477$	12.0000	0.549442
$478$	8.00000	0.365911
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	−4.00000	−0.182574
$481$	−3.00000	−0.136788
$482$	25.0000	1.13872
$483$	−48.0000	−2.18408
$484$	−10.0000	−0.454545
$485$	28.0000	1.27141
$486$	10.0000	0.453609
$487$	−5.00000	−0.226572	−0.113286	−	0.993562i	$-0.536138\pi$
$487$	−5.00000	−0.226572	−0.113286	+	0.993562i	$0.536138\pi$
$488$	−10.0000	−0.452679
$489$	20.0000	0.904431
$490$	−4.00000	−0.180702
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	10.0000	0.450835
$493$	4.00000	0.180151
$494$	24.0000	1.07981
$495$	2.00000	0.0898933
$496$	−4.00000	−0.179605
$497$	45.0000	2.01853
$498$	22.0000	0.985844
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	−12.0000	−0.536656
$501$	24.0000	1.07224
$502$	10.0000	0.446322
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	3.00000	0.133631
$505$	18.0000	0.800989
$506$	−8.00000	−0.355643
$507$	−8.00000	−0.355292
$508$	16.0000	0.709885
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	4.00000	0.177123
$511$	−30.0000	−1.32712
$512$	−1.00000	−0.0441942
$513$	32.0000	1.41283
$514$	15.0000	0.661622
$515$	4.00000	0.176261
$516$	−18.0000	−0.792406
$517$	2.00000	0.0879599
$518$	−3.00000	−0.131812
$519$	−26.0000	−1.14127
$520$	−6.00000	−0.263117
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	4.00000	0.175075
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	12.0000	0.524222
$525$	6.00000	0.261861
$526$	−3.00000	−0.130806
$527$	4.00000	0.174243
$528$	2.00000	0.0870388
$529$	41.0000	1.78261
$530$	−24.0000	−1.04249
$531$	1.00000	0.0433963
$532$	24.0000	1.04053
$533$	15.0000	0.649722
$534$	12.0000	0.519291
$535$	36.0000	1.55642
$536$	−4.00000	−0.172774
$537$	−34.0000	−1.46721
$538$	5.00000	0.215565
$539$	2.00000	0.0861461
$540$	−8.00000	−0.344265
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	1.00000	0.0429537
$543$	16.0000	0.686626
$544$	1.00000	0.0428746
$545$	−4.00000	−0.171341
$546$	18.0000	0.770329
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	9.00000	0.384461
$549$	10.0000	0.426790
$550$	1.00000	0.0426401
$551$	32.0000	1.36325
$552$	−16.0000	−0.681005
$553$	−33.0000	−1.40330
$554$	−18.0000	−0.764747
$555$	−4.00000	−0.169791
$556$	2.00000	0.0848189
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	4.00000	0.169334
$559$	−27.0000	−1.14198
$560$	−6.00000	−0.253546
$561$	−2.00000	−0.0844401
$562$	−7.00000	−0.295277
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	4.00000	0.168430
$565$	8.00000	0.336563
$566$	13.0000	0.546431
$567$	33.0000	1.38587
$568$	15.0000	0.629386
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	32.0000	1.34033
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	3.00000	0.125436
$573$	36.0000	1.50392
$574$	15.0000	0.626088
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	16.0000	0.665512
$579$	−46.0000	−1.91169
$580$	−8.00000	−0.332182
$581$	33.0000	1.36907
$582$	−28.0000	−1.16064
$583$	12.0000	0.496989
$584$	−10.0000	−0.413803
$585$	6.00000	0.248069
$586$	4.00000	0.165238
$587$	−33.0000	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$588$	4.00000	0.164957
$589$	32.0000	1.31854
$590$	−2.00000	−0.0823387
$591$	−24.0000	−0.987228
$592$	−1.00000	−0.0410997
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	4.00000	0.164122
$595$	6.00000	0.245976
$596$	5.00000	0.204808
$597$	−16.0000	−0.654836
$598$	−24.0000	−0.981433
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	2.00000	0.0816497
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	−27.0000	−1.10044
$603$	4.00000	0.162893
$604$	−22.0000	−0.895167
$605$	−20.0000	−0.813116
$606$	−18.0000	−0.731200
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	8.00000	0.324443
$609$	24.0000	0.972529
$610$	−20.0000	−0.809776
$611$	6.00000	0.242734
$612$	−1.00000	−0.0404226
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	8.00000	0.322854
$615$	20.0000	0.806478
$616$	3.00000	0.120873
$617$	−23.0000	−0.925945	−0.462973	−	0.886373i	$-0.653217\pi$
$617$	−23.0000	−0.925945	−0.462973	+	0.886373i	$0.653217\pi$
$618$	−4.00000	−0.160904
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	−8.00000	−0.321288
$621$	−32.0000	−1.28412
$622$	13.0000	0.521253
$623$	18.0000	0.721155
$624$	6.00000	0.240192
$625$	−19.0000	−0.760000
$626$	18.0000	0.719425
$627$	−16.0000	−0.638978
$628$	−18.0000	−0.718278
$629$	1.00000	0.0398726
$630$	6.00000	0.239046
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	−11.0000	−0.437557
$633$	−22.0000	−0.874421
$634$	4.00000	0.158860
$635$	32.0000	1.26988
$636$	24.0000	0.951662
$637$	6.00000	0.237729
$638$	4.00000	0.158362
$639$	−15.0000	−0.593391
$640$	−2.00000	−0.0790569
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−36.0000	−1.42081
$643$	−38.0000	−1.49857	−0.749287	−	0.662246i	$-0.769604\pi$
$643$	−38.0000	−1.49857	−0.749287	+	0.662246i	$0.769604\pi$
$644$	−24.0000	−0.945732
$645$	−36.0000	−1.41750
$646$	−8.00000	−0.314756
$647$	−44.0000	−1.72982	−0.864909	−	0.501928i	$-0.832624\pi$
$647$	−44.0000	−1.72982	−0.864909	+	0.501928i	$0.832624\pi$
$648$	11.0000	0.432121
$649$	1.00000	0.0392534
$650$	3.00000	0.117670
$651$	24.0000	0.940634
$652$	10.0000	0.391630
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	4.00000	0.156412
$655$	24.0000	0.937758
$656$	5.00000	0.195217
$657$	10.0000	0.390137
$658$	6.00000	0.233904
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	4.00000	0.155700
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−12.0000	−0.466393
$663$	−6.00000	−0.233021
$664$	11.0000	0.426883
$665$	48.0000	1.86136
$666$	1.00000	0.0387492
$667$	−32.0000	−1.23904
$668$	12.0000	0.464294
$669$	−6.00000	−0.231973
$670$	−8.00000	−0.309067
$671$	10.0000	0.386046
$672$	6.00000	0.231455
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	2.00000	0.0770371
$675$	4.00000	0.153960
$676$	−4.00000	−0.153846
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	−8.00000	−0.307238
$679$	−42.0000	−1.61181
$680$	2.00000	0.0766965
$681$	2.00000	0.0766402
$682$	4.00000	0.153168
$683$	35.0000	1.33924	0.669619	−	0.742705i	$-0.266457\pi$
$683$	35.0000	1.33924	0.669619	+	0.742705i	$0.266457\pi$
$684$	−8.00000	−0.305888
$685$	18.0000	0.687745
$686$	−15.0000	−0.572703
$687$	58.0000	2.21284
$688$	−9.00000	−0.343122
$689$	36.0000	1.37149
$690$	−32.0000	−1.21822
$691$	−11.0000	−0.418460	−0.209230	−	0.977866i	$-0.567096\pi$
$691$	−11.0000	−0.418460	−0.209230	+	0.977866i	$0.567096\pi$
$692$	−13.0000	−0.494186
$693$	−3.00000	−0.113961
$694$	−28.0000	−1.06287
$695$	4.00000	0.151729
$696$	8.00000	0.303239
$697$	−5.00000	−0.189389
$698$	17.0000	0.643459
$699$	0	0
$700$	3.00000	0.113389
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	12.0000	0.452911
$703$	8.00000	0.301726
$704$	1.00000	0.0376889
$705$	8.00000	0.301297
$706$	−14.0000	−0.526897
$707$	−27.0000	−1.01544
$708$	2.00000	0.0751646
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	30.0000	1.12588
$711$	11.0000	0.412532
$712$	6.00000	0.224860
$713$	−32.0000	−1.19841
$714$	−6.00000	−0.224544
$715$	6.00000	0.224387
$716$	−17.0000	−0.635320
$717$	−16.0000	−0.597531
$718$	−3.00000	−0.111959
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	2.00000	0.0745356
$721$	−6.00000	−0.223452
$722$	−45.0000	−1.67473
$723$	−50.0000	−1.85952
$724$	8.00000	0.297318
$725$	4.00000	0.148556
$726$	20.0000	0.742270
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	9.00000	0.333562
$729$	13.0000	0.481481
$730$	−20.0000	−0.740233
$731$	9.00000	0.332877
$732$	20.0000	0.739221
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	−4.00000	−0.147643
$735$	8.00000	0.295084
$736$	−8.00000	−0.294884
$737$	4.00000	0.147342
$738$	−5.00000	−0.184053
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	−2.00000	−0.0735215
$741$	−48.0000	−1.76332
$742$	36.0000	1.32160
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	8.00000	0.293294
$745$	10.0000	0.366372
$746$	14.0000	0.512576
$747$	−11.0000	−0.402469
$748$	−1.00000	−0.0365636
$749$	−54.0000	−1.97312
$750$	24.0000	0.876356
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	2.00000	0.0729325
$753$	−20.0000	−0.728841
$754$	12.0000	0.437014
$755$	−44.0000	−1.60132
$756$	12.0000	0.436436
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−26.0000	−0.944363
$759$	16.0000	0.580763
$760$	16.0000	0.580381
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	−32.0000	−1.15924
$763$	6.00000	0.217215
$764$	18.0000	0.651217
$765$	−2.00000	−0.0723102
$766$	−7.00000	−0.252920
$767$	3.00000	0.108324
$768$	2.00000	0.0721688
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	6.00000	0.216225
$771$	−30.0000	−1.08042
$772$	−23.0000	−0.827788
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	9.00000	0.323498
$775$	4.00000	0.143684
$776$	−14.0000	−0.502571
$777$	6.00000	0.215249
$778$	−16.0000	−0.573628
$779$	−40.0000	−1.43315
$780$	12.0000	0.429669
$781$	−15.0000	−0.536742
$782$	8.00000	0.286079
$783$	16.0000	0.571793
$784$	2.00000	0.0714286
$785$	−36.0000	−1.28490
$786$	−24.0000	−0.856052
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	−12.0000	−0.427482
$789$	6.00000	0.213606
$790$	−22.0000	−0.782725
$791$	−12.0000	−0.426671
$792$	−1.00000	−0.0355335
$793$	30.0000	1.06533
$794$	2.00000	0.0709773
$795$	48.0000	1.70238
$796$	−8.00000	−0.283552
$797$	49.0000	1.73567	0.867835	−	0.496853i	$-0.165511\pi$
$797$	49.0000	1.73567	0.867835	+	0.496853i	$0.165511\pi$
$798$	−48.0000	−1.69918
$799$	−2.00000	−0.0707549
$800$	1.00000	0.0353553
$801$	−6.00000	−0.212000
$802$	−18.0000	−0.635602
$803$	10.0000	0.352892
$804$	8.00000	0.282138
$805$	−48.0000	−1.69178
$806$	12.0000	0.422682
$807$	−10.0000	−0.352017
$808$	−9.00000	−0.316619
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	22.0000	0.773001
$811$	39.0000	1.36948	0.684738	−	0.728790i	$-0.259917\pi$
$811$	39.0000	1.36948	0.684738	+	0.728790i	$0.259917\pi$
$812$	12.0000	0.421117
$813$	−2.00000	−0.0701431
$814$	1.00000	0.0350500
$815$	20.0000	0.700569
$816$	−2.00000	−0.0700140
$817$	72.0000	2.51896
$818$	2.00000	0.0699284
$819$	−9.00000	−0.314485
$820$	10.0000	0.349215
$821$	23.0000	0.802706	0.401353	−	0.915924i	$-0.368540\pi$
$821$	23.0000	0.802706	0.401353	+	0.915924i	$0.368540\pi$
$822$	−18.0000	−0.627822
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−2.00000	−0.0696733
$825$	−2.00000	−0.0696311
$826$	3.00000	0.104383
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	8.00000	0.278019
$829$	−32.0000	−1.11141	−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000	−1.11141	−0.555703	+	0.831381i	$0.687551\pi$
$830$	22.0000	0.763631
$831$	36.0000	1.24883
$832$	3.00000	0.104006
$833$	−2.00000	−0.0692959
$834$	−4.00000	−0.138509
$835$	24.0000	0.830554
$836$	−8.00000	−0.276686
$837$	16.0000	0.553041
$838$	−3.00000	−0.103633
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	12.0000	0.414039
$841$	−13.0000	−0.448276
$842$	−19.0000	−0.654783
$843$	14.0000	0.482186
$844$	−11.0000	−0.378636
$845$	−8.00000	−0.275208
$846$	−2.00000	−0.0687614
$847$	30.0000	1.03081
$848$	12.0000	0.412082
$849$	−26.0000	−0.892318
$850$	−1.00000	−0.0342997
$851$	−8.00000	−0.274236
$852$	−30.0000	−1.02778
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	30.0000	1.02658
$855$	−16.0000	−0.547188
$856$	−18.0000	−0.615227
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	−6.00000	−0.204837
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−18.0000	−0.613795
$861$	−30.0000	−1.02240
$862$	−12.0000	−0.408722
$863$	26.0000	0.885050	0.442525	−	0.896756i	$-0.354083\pi$
$863$	26.0000	0.885050	0.442525	+	0.896756i	$0.354083\pi$
$864$	4.00000	0.136083
$865$	−26.0000	−0.884027
$866$	35.0000	1.18935
$867$	−32.0000	−1.08678
$868$	12.0000	0.407307
$869$	11.0000	0.373149
$870$	16.0000	0.542451
$871$	12.0000	0.406604
$872$	2.00000	0.0677285
$873$	14.0000	0.473828
$874$	64.0000	2.16483
$875$	36.0000	1.21702
$876$	20.0000	0.675737
$877$	44.0000	1.48577	0.742887	−	0.669417i	$-0.233456\pi$
$877$	44.0000	1.48577	0.742887	+	0.669417i	$0.233456\pi$
$878$	−11.0000	−0.371232
$879$	−8.00000	−0.269833
$880$	2.00000	0.0674200
$881$	−20.0000	−0.673817	−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000	−0.673817	−0.336909	+	0.941537i	$0.609381\pi$
$882$	−2.00000	−0.0673435
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	−3.00000	−0.100901
$885$	4.00000	0.134459
$886$	−39.0000	−1.31023
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	2.00000	0.0671156
$889$	−48.0000	−1.60987
$890$	12.0000	0.402241
$891$	−11.0000	−0.368514
$892$	−3.00000	−0.100447
$893$	−16.0000	−0.535420
$894$	−10.0000	−0.334450
$895$	−34.0000	−1.13649
$896$	3.00000	0.100223
$897$	48.0000	1.60267
$898$	17.0000	0.567297
$899$	16.0000	0.533630
$900$	−1.00000	−0.0333333
$901$	−12.0000	−0.399778
$902$	−5.00000	−0.166482
$903$	54.0000	1.79701
$904$	−4.00000	−0.133038
$905$	16.0000	0.531858
$906$	44.0000	1.46180
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	1.00000	0.0331862
$909$	9.00000	0.298511
$910$	18.0000	0.596694
$911$	25.0000	0.828287	0.414143	−	0.910212i	$-0.364081\pi$
$911$	25.0000	0.828287	0.414143	+	0.910212i	$0.364081\pi$
$912$	−16.0000	−0.529813
$913$	−11.0000	−0.364047
$914$	22.0000	0.727695
$915$	40.0000	1.32236
$916$	29.0000	0.958187
$917$	−36.0000	−1.18882
$918$	−4.00000	−0.132020
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	−16.0000	−0.527504
$921$	−16.0000	−0.527218
$922$	42.0000	1.38320
$923$	−45.0000	−1.48119
$924$	−6.00000	−0.197386
$925$	1.00000	0.0328798
$926$	30.0000	0.985861
$927$	2.00000	0.0656886
$928$	4.00000	0.131306
$929$	16.0000	0.524943	0.262471	−	0.964940i	$-0.415462\pi$
$929$	16.0000	0.524943	0.262471	+	0.964940i	$0.415462\pi$
$930$	16.0000	0.524661
$931$	−16.0000	−0.524379
$932$	0	0
$933$	−26.0000	−0.851202
$934$	1.00000	0.0327210
$935$	−2.00000	−0.0654070
$936$	−3.00000	−0.0980581
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	12.0000	0.391814
$939$	−36.0000	−1.17482
$940$	4.00000	0.130466
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	36.0000	1.17294
$943$	40.0000	1.30258
$944$	1.00000	0.0325472
$945$	24.0000	0.780720
$946$	9.00000	0.292615
$947$	14.0000	0.454939	0.227469	−	0.973785i	$-0.426955\pi$
$947$	14.0000	0.454939	0.227469	+	0.973785i	$0.426955\pi$
$948$	22.0000	0.714527
$949$	30.0000	0.973841
$950$	−8.00000	−0.259554
$951$	−8.00000	−0.259418
$952$	−3.00000	−0.0972306
$953$	23.0000	0.745043	0.372522	−	0.928024i	$-0.378493\pi$
$953$	23.0000	0.745043	0.372522	+	0.928024i	$0.378493\pi$
$954$	−12.0000	−0.388514
$955$	36.0000	1.16493
$956$	−8.00000	−0.258738
$957$	−8.00000	−0.258603
$958$	16.0000	0.516937
$959$	−27.0000	−0.871875
$960$	4.00000	0.129099
$961$	−15.0000	−0.483871
$962$	3.00000	0.0967239
$963$	18.0000	0.580042
$964$	−25.0000	−0.805196
$965$	−46.0000	−1.48079
$966$	48.0000	1.54437
$967$	30.0000	0.964735	0.482367	−	0.875969i	$-0.339777\pi$
$967$	30.0000	0.964735	0.482367	+	0.875969i	$0.339777\pi$
$968$	10.0000	0.321412
$969$	16.0000	0.513994
$970$	−28.0000	−0.899026
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	−10.0000	−0.320750
$973$	−6.00000	−0.192351
$974$	5.00000	0.160210
$975$	−6.00000	−0.192154
$976$	10.0000	0.320092
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	−20.0000	−0.639529
$979$	−6.00000	−0.191761
$980$	4.00000	0.127775
$981$	−2.00000	−0.0638551
$982$	−2.00000	−0.0638226
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	−10.0000	−0.318788
$985$	−24.0000	−0.764704
$986$	−4.00000	−0.127386
$987$	−12.0000	−0.381964
$988$	−24.0000	−0.763542
$989$	−72.0000	−2.28947
$990$	−2.00000	−0.0635642
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	4.00000	0.127000
$993$	24.0000	0.761617
$994$	−45.0000	−1.42731
$995$	−16.0000	−0.507234
$996$	−22.0000	−0.697097
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	6.00000	0.189927
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	118.2.a.b.1.1	✓	1
3.2	odd	2		1062.2.a.g.1.1		1
4.3	odd	2		944.2.a.d.1.1		1
5.2	odd	4		2950.2.c.c.1299.1		2
5.3	odd	4		2950.2.c.c.1299.2		2
5.4	even	2		2950.2.a.l.1.1		1
7.6	odd	2		5782.2.a.b.1.1		1
8.3	odd	2		3776.2.a.v.1.1		1
8.5	even	2		3776.2.a.d.1.1		1
12.11	even	2		8496.2.a.g.1.1		1
59.58	odd	2		6962.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
118.2.a.b.1.1	✓	1	1.1	even	1	trivial
944.2.a.d.1.1		1	4.3	odd	2
1062.2.a.g.1.1		1	3.2	odd	2
2950.2.a.l.1.1		1	5.4	even	2
2950.2.c.c.1299.1		2	5.2	odd	4
2950.2.c.c.1299.2		2	5.3	odd	4
3776.2.a.d.1.1		1	8.5	even	2
3776.2.a.v.1.1		1	8.3	odd	2
5782.2.a.b.1.1		1	7.6	odd	2
6962.2.a.m.1.1		1	59.58	odd	2
8496.2.a.g.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 118 = 2 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	118.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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