LMFDB - Embedded newform 418.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [418,2,Mod(1,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("418.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 418 = 2 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	418.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:	$3.33774680449$
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	$\chi$	$=$	418.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -3.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} -2.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} +3.00000 q^{21} -1.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} +5.00000 q^{27} -3.00000 q^{28} +1.00000 q^{29} +2.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -7.00000 q^{34} +6.00000 q^{35} -2.00000 q^{36} -6.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +4.00000 q^{45} -5.00000 q^{46} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +7.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} +5.00000 q^{54} +2.00000 q^{55} -3.00000 q^{56} -1.00000 q^{57} +1.00000 q^{58} +3.00000 q^{59} +2.00000 q^{60} -12.0000 q^{61} +10.0000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} +3.00000 q^{67} -7.00000 q^{68} +5.00000 q^{69} +6.00000 q^{70} -10.0000 q^{71} -2.00000 q^{72} +3.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +8.00000 q^{83} +3.00000 q^{84} +14.0000 q^{85} -4.00000 q^{86} -1.00000 q^{87} -1.00000 q^{88} -8.00000 q^{89} +4.00000 q^{90} -3.00000 q^{91} -5.00000 q^{92} -10.0000 q^{93} -2.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} +2.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	−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
$9$	−2.00000	−0.666667
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−3.00000	−0.801784
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	−2.00000	−0.471405
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.00000	−0.447214
$21$	3.00000	0.654654
$22$	−1.00000	−0.213201
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	5.00000	0.962250
$28$	−3.00000	−0.566947
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	2.00000	0.365148
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−7.00000	−1.20049
$35$	6.00000	1.01419
$36$	−2.00000	−0.333333
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	1.00000	0.162221
$39$	−1.00000	−0.160128
$40$	−2.00000	−0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	3.00000	0.462910
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	4.00000	0.596285
$46$	−5.00000	−0.737210
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	7.00000	0.980196
$52$	1.00000	0.138675
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	5.00000	0.680414
$55$	2.00000	0.269680
$56$	−3.00000	−0.400892
$57$	−1.00000	−0.132453
$58$	1.00000	0.131306
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	2.00000	0.258199
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	10.0000	1.27000
$63$	6.00000	0.755929
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	1.00000	0.123091
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	−7.00000	−0.848875
$69$	5.00000	0.601929
$70$	6.00000	0.717137
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	−2.00000	−0.235702
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	1.00000	0.114708
$77$	3.00000	0.341882
$78$	−1.00000	−0.113228
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	3.00000	0.327327
$85$	14.0000	1.51851
$86$	−4.00000	−0.431331
$87$	−1.00000	−0.107211
$88$	−1.00000	−0.106600
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	4.00000	0.421637
$91$	−3.00000	−0.314485
$92$	−5.00000	−0.521286
$93$	−10.0000	−1.03695
$94$	0	0
$95$	−2.00000	−0.205196
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	2.00000	0.202031
$99$	2.00000	0.201008
$100$	−1.00000	−0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	7.00000	0.693103
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	1.00000	0.0980581
$105$	−6.00000	−0.585540
$106$	−1.00000	−0.0971286
$107$	−13.0000	−1.25676	−0.628379	−	0.777908i	$-0.716281\pi$
$107$	−13.0000	−1.25676	−0.628379	+	0.777908i	$0.716281\pi$
$108$	5.00000	0.481125
$109$	−13.0000	−1.24517	−0.622587	−	0.782551i	$-0.713918\pi$
$109$	−13.0000	−1.24517	−0.622587	+	0.782551i	$0.713918\pi$
$110$	2.00000	0.190693
$111$	6.00000	0.569495
$112$	−3.00000	−0.283473
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	−1.00000	−0.0936586
$115$	10.0000	0.932505
$116$	1.00000	0.0928477
$117$	−2.00000	−0.184900
$118$	3.00000	0.276172
$119$	21.0000	1.92507
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	−12.0000	−1.08643
$123$	−6.00000	−0.541002
$124$	10.0000	0.898027
$125$	12.0000	1.07331
$126$	6.00000	0.534522
$127$	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
$129$	4.00000	0.352180
$130$	−2.00000	−0.175412
$131$	−22.0000	−1.92215	−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000	−1.92215	−0.961074	+	0.276289i	$0.910895\pi$
$132$	1.00000	0.0870388
$133$	−3.00000	−0.260133
$134$	3.00000	0.259161
$135$	−10.0000	−0.860663
$136$	−7.00000	−0.600245
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	5.00000	0.425628
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	−10.0000	−0.839181
$143$	−1.00000	−0.0836242
$144$	−2.00000	−0.166667
$145$	−2.00000	−0.166091
$146$	3.00000	0.248282
$147$	−2.00000	−0.164957
$148$	−6.00000	−0.493197
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	1.00000	0.0816497
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	1.00000	0.0811107
$153$	14.0000	1.13183
$154$	3.00000	0.241747
$155$	−20.0000	−1.60644
$156$	−1.00000	−0.0800641
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	8.00000	0.636446
$159$	1.00000	0.0793052
$160$	−2.00000	−0.158114
$161$	15.0000	1.18217
$162$	1.00000	0.0785674
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	6.00000	0.468521
$165$	−2.00000	−0.155700
$166$	8.00000	0.620920
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	3.00000	0.231455
$169$	−12.0000	−0.923077
$170$	14.0000	1.07375
$171$	−2.00000	−0.152944
$172$	−4.00000	−0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−1.00000	−0.0758098
$175$	3.00000	0.226779
$176$	−1.00000	−0.0753778
$177$	−3.00000	−0.225494
$178$	−8.00000	−0.599625
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	4.00000	0.298142
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−3.00000	−0.222375
$183$	12.0000	0.887066
$184$	−5.00000	−0.368605
$185$	12.0000	0.882258
$186$	−10.0000	−0.733236
$187$	7.00000	0.511891
$188$	0	0
$189$	−15.0000	−1.09109
$190$	−2.00000	−0.145095
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	−1.00000	−0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	8.00000	0.574367
$195$	2.00000	0.143223
$196$	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	$0.640599\pi$
$198$	2.00000	0.142134
$199$	23.0000	1.63043	0.815213	−	0.579161i	$-0.196620\pi$
$199$	23.0000	1.63043	0.815213	+	0.579161i	$0.196620\pi$
$200$	−1.00000	−0.0707107
$201$	−3.00000	−0.211604
$202$	6.00000	0.422159
$203$	−3.00000	−0.210559
$204$	7.00000	0.490098
$205$	−12.0000	−0.838116
$206$	12.0000	0.836080
$207$	10.0000	0.695048
$208$	1.00000	0.0693375
$209$	−1.00000	−0.0691714
$210$	−6.00000	−0.414039
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	−1.00000	−0.0686803
$213$	10.0000	0.685189
$214$	−13.0000	−0.888662
$215$	8.00000	0.545595
$216$	5.00000	0.340207
$217$	−30.0000	−2.03653
$218$	−13.0000	−0.880471
$219$	−3.00000	−0.202721
$220$	2.00000	0.134840
$221$	−7.00000	−0.470871
$222$	6.00000	0.402694
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	−3.00000	−0.200446
$225$	2.00000	0.133333
$226$	−12.0000	−0.798228
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	−1.00000	−0.0662266
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	10.0000	0.659380
$231$	−3.00000	−0.197386
$232$	1.00000	0.0656532
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	3.00000	0.195283
$237$	−8.00000	−0.519656
$238$	21.0000	1.36123
$239$	25.0000	1.61712	0.808558	−	0.588417i	$-0.200249\pi$
$239$	25.0000	1.61712	0.808558	+	0.588417i	$0.200249\pi$
$240$	2.00000	0.129099
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	1.00000	0.0642824
$243$	−16.0000	−1.02640
$244$	−12.0000	−0.768221
$245$	−4.00000	−0.255551
$246$	−6.00000	−0.382546
$247$	1.00000	0.0636285
$248$	10.0000	0.635001
$249$	−8.00000	−0.506979
$250$	12.0000	0.758947
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	6.00000	0.377964
$253$	5.00000	0.314347
$254$	16.0000	1.00393
$255$	−14.0000	−0.876714
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	4.00000	0.249029
$259$	18.0000	1.11847
$260$	−2.00000	−0.124035
$261$	−2.00000	−0.123797
$262$	−22.0000	−1.35916
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	1.00000	0.0615457
$265$	2.00000	0.122859
$266$	−3.00000	−0.183942
$267$	8.00000	0.489592
$268$	3.00000	0.183254
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−10.0000	−0.608581
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	−7.00000	−0.424437
$273$	3.00000	0.181568
$274$	−17.0000	−1.02701
$275$	1.00000	0.0603023
$276$	5.00000	0.300965
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	−14.0000	−0.839664
$279$	−20.0000	−1.19737
$280$	6.00000	0.358569
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−10.0000	−0.593391
$285$	2.00000	0.118470
$286$	−1.00000	−0.0591312
$287$	−18.0000	−1.06251
$288$	−2.00000	−0.117851
$289$	32.0000	1.88235
$290$	−2.00000	−0.117444
$291$	−8.00000	−0.468968
$292$	3.00000	0.175562
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	−2.00000	−0.116642
$295$	−6.00000	−0.349334
$296$	−6.00000	−0.348743
$297$	−5.00000	−0.290129
$298$	−18.0000	−1.04271
$299$	−5.00000	−0.289157
$300$	1.00000	0.0577350
$301$	12.0000	0.691669
$302$	12.0000	0.690522
$303$	−6.00000	−0.344691
$304$	1.00000	0.0573539
$305$	24.0000	1.37424
$306$	14.0000	0.800327
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	3.00000	0.170941
$309$	−12.0000	−0.682656
$310$	−20.0000	−1.13592
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	−1.00000	−0.0566139
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	−22.0000	−1.24153
$315$	−12.0000	−0.676123
$316$	8.00000	0.450035
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	1.00000	0.0560772
$319$	−1.00000	−0.0559893
$320$	−2.00000	−0.111803
$321$	13.0000	0.725589
$322$	15.0000	0.835917
$323$	−7.00000	−0.389490
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	−2.00000	−0.110770
$327$	13.0000	0.718902
$328$	6.00000	0.331295
$329$	0	0
$330$	−2.00000	−0.110096
$331$	29.0000	1.59398	0.796992	−	0.603990i	$-0.206423\pi$
$331$	29.0000	1.59398	0.796992	+	0.603990i	$0.206423\pi$
$332$	8.00000	0.439057
$333$	12.0000	0.657596
$334$	−22.0000	−1.20379
$335$	−6.00000	−0.327815
$336$	3.00000	0.163663
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	−12.0000	−0.652714
$339$	12.0000	0.651751
$340$	14.0000	0.759257
$341$	−10.0000	−0.541530
$342$	−2.00000	−0.108148
$343$	15.0000	0.809924
$344$	−4.00000	−0.215666
$345$	−10.0000	−0.538382
$346$	14.0000	0.752645
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	−1.00000	−0.0536056
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	3.00000	0.160357
$351$	5.00000	0.266880
$352$	−1.00000	−0.0533002
$353$	1.00000	0.0532246	0.0266123	−	0.999646i	$-0.491528\pi$
$353$	1.00000	0.0532246	0.0266123	+	0.999646i	$0.491528\pi$
$354$	−3.00000	−0.159448
$355$	20.0000	1.06149
$356$	−8.00000	−0.423999
$357$	−21.0000	−1.11144
$358$	4.00000	0.211407
$359$	−17.0000	−0.897226	−0.448613	−	0.893726i	$-0.648082\pi$
$359$	−17.0000	−0.897226	−0.448613	+	0.893726i	$0.648082\pi$
$360$	4.00000	0.210819
$361$	1.00000	0.0526316
$362$	−2.00000	−0.105118
$363$	−1.00000	−0.0524864
$364$	−3.00000	−0.157243
$365$	−6.00000	−0.314054
$366$	12.0000	0.627250
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−5.00000	−0.260643
$369$	−12.0000	−0.624695
$370$	12.0000	0.623850
$371$	3.00000	0.155752
$372$	−10.0000	−0.518476
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	7.00000	0.361961
$375$	−12.0000	−0.619677
$376$	0	0
$377$	1.00000	0.0515026
$378$	−15.0000	−0.771517
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	−2.00000	−0.102598
$381$	−16.0000	−0.819705
$382$	3.00000	0.153493
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	−1.00000	−0.0510310
$385$	−6.00000	−0.305788
$386$	−4.00000	−0.203595
$387$	8.00000	0.406663
$388$	8.00000	0.406138
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	2.00000	0.101274
$391$	35.0000	1.77003
$392$	2.00000	0.101015
$393$	22.0000	1.10975
$394$	−12.0000	−0.604551
$395$	−16.0000	−0.805047
$396$	2.00000	0.100504
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	23.0000	1.15289
$399$	3.00000	0.150188
$400$	−1.00000	−0.0500000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	−3.00000	−0.149626
$403$	10.0000	0.498135
$404$	6.00000	0.298511
$405$	−2.00000	−0.0993808
$406$	−3.00000	−0.148888
$407$	6.00000	0.297409
$408$	7.00000	0.346552
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	−12.0000	−0.592638
$411$	17.0000	0.838548
$412$	12.0000	0.591198
$413$	−9.00000	−0.442861
$414$	10.0000	0.491473
$415$	−16.0000	−0.785409
$416$	1.00000	0.0490290
$417$	14.0000	0.685583
$418$	−1.00000	−0.0489116
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	−6.00000	−0.292770
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	1.00000	0.0486792
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	7.00000	0.339550
$426$	10.0000	0.484502
$427$	36.0000	1.74216
$428$	−13.0000	−0.628379
$429$	1.00000	0.0482805
$430$	8.00000	0.385794
$431$	34.0000	1.63772	0.818861	−	0.573992i	$-0.194606\pi$
$431$	34.0000	1.63772	0.818861	+	0.573992i	$0.194606\pi$
$432$	5.00000	0.240563
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	−30.0000	−1.44005
$435$	2.00000	0.0958927
$436$	−13.0000	−0.622587
$437$	−5.00000	−0.239182
$438$	−3.00000	−0.143346
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	2.00000	0.0953463
$441$	−4.00000	−0.190476
$442$	−7.00000	−0.332956
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	6.00000	0.284747
$445$	16.0000	0.758473
$446$	10.0000	0.473514
$447$	18.0000	0.851371
$448$	−3.00000	−0.141737
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	2.00000	0.0942809
$451$	−6.00000	−0.282529
$452$	−12.0000	−0.564433
$453$	−12.0000	−0.563809
$454$	−15.0000	−0.703985
$455$	6.00000	0.281284
$456$	−1.00000	−0.0468293
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	20.0000	0.934539
$459$	−35.0000	−1.63366
$460$	10.0000	0.466252
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	−3.00000	−0.139573
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	1.00000	0.0464238
$465$	20.0000	0.927478
$466$	−10.0000	−0.463241
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	−2.00000	−0.0924500
$469$	−9.00000	−0.415581
$470$	0	0
$471$	22.0000	1.01371
$472$	3.00000	0.138086
$473$	4.00000	0.183920
$474$	−8.00000	−0.367452
$475$	−1.00000	−0.0458831
$476$	21.0000	0.962533
$477$	2.00000	0.0915737
$478$	25.0000	1.14347
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	2.00000	0.0912871
$481$	−6.00000	−0.273576
$482$	−4.00000	−0.182195
$483$	−15.0000	−0.682524
$484$	1.00000	0.0454545
$485$	−16.0000	−0.726523
$486$	−16.0000	−0.725775
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−12.0000	−0.543214
$489$	2.00000	0.0904431
$490$	−4.00000	−0.180702
$491$	10.0000	0.451294	0.225647	−	0.974209i	$-0.427550\pi$
$491$	10.0000	0.451294	0.225647	+	0.974209i	$0.427550\pi$
$492$	−6.00000	−0.270501
$493$	−7.00000	−0.315264
$494$	1.00000	0.0449921
$495$	−4.00000	−0.179787
$496$	10.0000	0.449013
$497$	30.0000	1.34568
$498$	−8.00000	−0.358489
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	12.0000	0.536656
$501$	22.0000	0.982888
$502$	−6.00000	−0.267793
$503$	−19.0000	−0.847168	−0.423584	−	0.905857i	$-0.639228\pi$
$503$	−19.0000	−0.847168	−0.423584	+	0.905857i	$0.639228\pi$
$504$	6.00000	0.267261
$505$	−12.0000	−0.533993
$506$	5.00000	0.222277
$507$	12.0000	0.532939
$508$	16.0000	0.709885
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	−14.0000	−0.619930
$511$	−9.00000	−0.398137
$512$	1.00000	0.0441942
$513$	5.00000	0.220755
$514$	−22.0000	−0.970378
$515$	−24.0000	−1.05757
$516$	4.00000	0.176090
$517$	0	0
$518$	18.0000	0.790875
$519$	−14.0000	−0.614532
$520$	−2.00000	−0.0877058
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	−2.00000	−0.0875376
$523$	−21.0000	−0.918266	−0.459133	−	0.888368i	$-0.651840\pi$
$523$	−21.0000	−0.918266	−0.459133	+	0.888368i	$0.651840\pi$
$524$	−22.0000	−0.961074
$525$	−3.00000	−0.130931
$526$	4.00000	0.174408
$527$	−70.0000	−3.04925
$528$	1.00000	0.0435194
$529$	2.00000	0.0869565
$530$	2.00000	0.0868744
$531$	−6.00000	−0.260378
$532$	−3.00000	−0.130066
$533$	6.00000	0.259889
$534$	8.00000	0.346194
$535$	26.0000	1.12408
$536$	3.00000	0.129580
$537$	−4.00000	−0.172613
$538$	18.0000	0.776035
$539$	−2.00000	−0.0861461
$540$	−10.0000	−0.430331
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	−15.0000	−0.644305
$543$	2.00000	0.0858282
$544$	−7.00000	−0.300123
$545$	26.0000	1.11372
$546$	3.00000	0.128388
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−17.0000	−0.726204
$549$	24.0000	1.02430
$550$	1.00000	0.0426401
$551$	1.00000	0.0426014
$552$	5.00000	0.212814
$553$	−24.0000	−1.02058
$554$	−16.0000	−0.679775
$555$	−12.0000	−0.509372
$556$	−14.0000	−0.593732
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−20.0000	−0.846668
$559$	−4.00000	−0.169182
$560$	6.00000	0.253546
$561$	−7.00000	−0.295540
$562$	16.0000	0.674919
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	−14.0000	−0.588464
$567$	−3.00000	−0.125988
$568$	−10.0000	−0.419591
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	2.00000	0.0837708
$571$	−14.0000	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$572$	−1.00000	−0.0418121
$573$	−3.00000	−0.125327
$574$	−18.0000	−0.751305
$575$	5.00000	0.208514
$576$	−2.00000	−0.0833333
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	32.0000	1.33102
$579$	4.00000	0.166234
$580$	−2.00000	−0.0830455
$581$	−24.0000	−0.995688
$582$	−8.00000	−0.331611
$583$	1.00000	0.0414158
$584$	3.00000	0.124141
$585$	4.00000	0.165380
$586$	−9.00000	−0.371787
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−2.00000	−0.0824786
$589$	10.0000	0.412043
$590$	−6.00000	−0.247016
$591$	12.0000	0.493614
$592$	−6.00000	−0.246598
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	−5.00000	−0.205152
$595$	−42.0000	−1.72183
$596$	−18.0000	−0.737309
$597$	−23.0000	−0.941327
$598$	−5.00000	−0.204465
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	1.00000	0.0408248
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	12.0000	0.489083
$603$	−6.00000	−0.244339
$604$	12.0000	0.488273
$605$	−2.00000	−0.0813116
$606$	−6.00000	−0.243733
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	1.00000	0.0405554
$609$	3.00000	0.121566
$610$	24.0000	0.971732
$611$	0	0
$612$	14.0000	0.565916
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	−16.0000	−0.645707
$615$	12.0000	0.483887
$616$	3.00000	0.120873
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	−12.0000	−0.482711
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−20.0000	−0.803219
$621$	−25.0000	−1.00322
$622$	3.00000	0.120289
$623$	24.0000	0.961540
$624$	−1.00000	−0.0400320
$625$	−19.0000	−0.760000
$626$	21.0000	0.839329
$627$	1.00000	0.0399362
$628$	−22.0000	−0.877896
$629$	42.0000	1.67465
$630$	−12.0000	−0.478091
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	8.00000	0.318223
$633$	−1.00000	−0.0397464
$634$	21.0000	0.834017
$635$	−32.0000	−1.26988
$636$	1.00000	0.0396526
$637$	2.00000	0.0792429
$638$	−1.00000	−0.0395904
$639$	20.0000	0.791188
$640$	−2.00000	−0.0790569
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	13.0000	0.513069
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	15.0000	0.591083
$645$	−8.00000	−0.315000
$646$	−7.00000	−0.275411
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	1.00000	0.0392837
$649$	−3.00000	−0.117760
$650$	−1.00000	−0.0392232
$651$	30.0000	1.17579
$652$	−2.00000	−0.0783260
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	13.0000	0.508340
$655$	44.0000	1.71922
$656$	6.00000	0.234261
$657$	−6.00000	−0.234082
$658$	0	0
$659$	11.0000	0.428499	0.214250	−	0.976779i	$-0.431269\pi$
$659$	11.0000	0.428499	0.214250	+	0.976779i	$0.431269\pi$
$660$	−2.00000	−0.0778499
$661$	29.0000	1.12797	0.563985	−	0.825785i	$-0.309268\pi$
$661$	29.0000	1.12797	0.563985	+	0.825785i	$0.309268\pi$
$662$	29.0000	1.12712
$663$	7.00000	0.271857
$664$	8.00000	0.310460
$665$	6.00000	0.232670
$666$	12.0000	0.464991
$667$	−5.00000	−0.193601
$668$	−22.0000	−0.851206
$669$	−10.0000	−0.386622
$670$	−6.00000	−0.231800
$671$	12.0000	0.463255
$672$	3.00000	0.115728
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−20.0000	−0.770371
$675$	−5.00000	−0.192450
$676$	−12.0000	−0.461538
$677$	−29.0000	−1.11456	−0.557280	−	0.830324i	$-0.688155\pi$
$677$	−29.0000	−1.11456	−0.557280	+	0.830324i	$0.688155\pi$
$678$	12.0000	0.460857
$679$	−24.0000	−0.921035
$680$	14.0000	0.536875
$681$	15.0000	0.574801
$682$	−10.0000	−0.382920
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	−2.00000	−0.0764719
$685$	34.0000	1.29907
$686$	15.0000	0.572703
$687$	−20.0000	−0.763048
$688$	−4.00000	−0.152499
$689$	−1.00000	−0.0380970
$690$	−10.0000	−0.380693
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	14.0000	0.532200
$693$	−6.00000	−0.227921
$694$	−26.0000	−0.986947
$695$	28.0000	1.06210
$696$	−1.00000	−0.0379049
$697$	−42.0000	−1.59086
$698$	10.0000	0.378506
$699$	10.0000	0.378235
$700$	3.00000	0.113389
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	5.00000	0.188713
$703$	−6.00000	−0.226294
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	1.00000	0.0376355
$707$	−18.0000	−0.676960
$708$	−3.00000	−0.112747
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	20.0000	0.750587
$711$	−16.0000	−0.600047
$712$	−8.00000	−0.299813
$713$	−50.0000	−1.87251
$714$	−21.0000	−0.785905
$715$	2.00000	0.0747958
$716$	4.00000	0.149487
$717$	−25.0000	−0.933642
$718$	−17.0000	−0.634434
$719$	−33.0000	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$719$	−33.0000	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$720$	4.00000	0.149071
$721$	−36.0000	−1.34071
$722$	1.00000	0.0372161
$723$	4.00000	0.148762
$724$	−2.00000	−0.0743294
$725$	−1.00000	−0.0371391
$726$	−1.00000	−0.0371135
$727$	47.0000	1.74313	0.871567	−	0.490277i	$-0.163104\pi$
$727$	47.0000	1.74313	0.871567	+	0.490277i	$0.163104\pi$
$728$	−3.00000	−0.111187
$729$	13.0000	0.481481
$730$	−6.00000	−0.222070
$731$	28.0000	1.03562
$732$	12.0000	0.443533
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	16.0000	0.590571
$735$	4.00000	0.147542
$736$	−5.00000	−0.184302
$737$	−3.00000	−0.110506
$738$	−12.0000	−0.441726
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	12.0000	0.441129
$741$	−1.00000	−0.0367359
$742$	3.00000	0.110133
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−10.0000	−0.366618
$745$	36.0000	1.31894
$746$	11.0000	0.402739
$747$	−16.0000	−0.585409
$748$	7.00000	0.255945
$749$	39.0000	1.42503
$750$	−12.0000	−0.438178
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	1.00000	0.0364179
$755$	−24.0000	−0.873449
$756$	−15.0000	−0.545545
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−5.00000	−0.181608
$759$	−5.00000	−0.181489
$760$	−2.00000	−0.0725476
$761$	−11.0000	−0.398750	−0.199375	−	0.979923i	$-0.563891\pi$
$761$	−11.0000	−0.398750	−0.199375	+	0.979923i	$0.563891\pi$
$762$	−16.0000	−0.579619
$763$	39.0000	1.41189
$764$	3.00000	0.108536
$765$	−28.0000	−1.01234
$766$	2.00000	0.0722629
$767$	3.00000	0.108324
$768$	−1.00000	−0.0360844
$769$	−49.0000	−1.76699	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−49.0000	−1.76699	−0.883493	+	0.468445i	$0.844814\pi$
$770$	−6.00000	−0.216225
$771$	22.0000	0.792311
$772$	−4.00000	−0.143963
$773$	−23.0000	−0.827253	−0.413626	−	0.910447i	$-0.635738\pi$
$773$	−23.0000	−0.827253	−0.413626	+	0.910447i	$0.635738\pi$
$774$	8.00000	0.287554
$775$	−10.0000	−0.359211
$776$	8.00000	0.287183
$777$	−18.0000	−0.645746
$778$	18.0000	0.645331
$779$	6.00000	0.214972
$780$	2.00000	0.0716115
$781$	10.0000	0.357828
$782$	35.0000	1.25160
$783$	5.00000	0.178685
$784$	2.00000	0.0714286
$785$	44.0000	1.57043
$786$	22.0000	0.784714
$787$	49.0000	1.74666	0.873331	−	0.487128i	$-0.161955\pi$
$787$	49.0000	1.74666	0.873331	+	0.487128i	$0.161955\pi$
$788$	−12.0000	−0.427482
$789$	−4.00000	−0.142404
$790$	−16.0000	−0.569254
$791$	36.0000	1.28001
$792$	2.00000	0.0710669
$793$	−12.0000	−0.426132
$794$	34.0000	1.20661
$795$	−2.00000	−0.0709327
$796$	23.0000	0.815213
$797$	19.0000	0.673015	0.336507	−	0.941681i	$-0.390754\pi$
$797$	19.0000	0.673015	0.336507	+	0.941681i	$0.390754\pi$
$798$	3.00000	0.106199
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	16.0000	0.565332
$802$	−2.00000	−0.0706225
$803$	−3.00000	−0.105868
$804$	−3.00000	−0.105802
$805$	−30.0000	−1.05736
$806$	10.0000	0.352235
$807$	−18.0000	−0.633630
$808$	6.00000	0.211079
$809$	−13.0000	−0.457056	−0.228528	−	0.973537i	$-0.573391\pi$
$809$	−13.0000	−0.457056	−0.228528	+	0.973537i	$0.573391\pi$
$810$	−2.00000	−0.0702728
$811$	−25.0000	−0.877869	−0.438934	−	0.898519i	$-0.644644\pi$
$811$	−25.0000	−0.877869	−0.438934	+	0.898519i	$0.644644\pi$
$812$	−3.00000	−0.105279
$813$	15.0000	0.526073
$814$	6.00000	0.210300
$815$	4.00000	0.140114
$816$	7.00000	0.245049
$817$	−4.00000	−0.139942
$818$	34.0000	1.18878
$819$	6.00000	0.209657
$820$	−12.0000	−0.419058
$821$	−16.0000	−0.558404	−0.279202	−	0.960232i	$-0.590070\pi$
$821$	−16.0000	−0.558404	−0.279202	+	0.960232i	$0.590070\pi$
$822$	17.0000	0.592943
$823$	−43.0000	−1.49889	−0.749443	−	0.662069i	$-0.769679\pi$
$823$	−43.0000	−1.49889	−0.749443	+	0.662069i	$0.769679\pi$
$824$	12.0000	0.418040
$825$	−1.00000	−0.0348155
$826$	−9.00000	−0.313150
$827$	37.0000	1.28662	0.643308	−	0.765607i	$-0.277561\pi$
$827$	37.0000	1.28662	0.643308	+	0.765607i	$0.277561\pi$
$828$	10.0000	0.347524
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	−16.0000	−0.555368
$831$	16.0000	0.555034
$832$	1.00000	0.0346688
$833$	−14.0000	−0.485071
$834$	14.0000	0.484780
$835$	44.0000	1.52268
$836$	−1.00000	−0.0345857
$837$	50.0000	1.72825
$838$	10.0000	0.345444
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	−6.00000	−0.207020
$841$	−28.0000	−0.965517
$842$	15.0000	0.516934
$843$	−16.0000	−0.551069
$844$	1.00000	0.0344214
$845$	24.0000	0.825625
$846$	0	0
$847$	−3.00000	−0.103081
$848$	−1.00000	−0.0343401
$849$	14.0000	0.480479
$850$	7.00000	0.240098
$851$	30.0000	1.02839
$852$	10.0000	0.342594
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	36.0000	1.23189
$855$	4.00000	0.136797
$856$	−13.0000	−0.444331
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	1.00000	0.0341394
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	8.00000	0.272798
$861$	18.0000	0.613438
$862$	34.0000	1.15804
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	5.00000	0.170103
$865$	−28.0000	−0.952029
$866$	24.0000	0.815553
$867$	−32.0000	−1.08678
$868$	−30.0000	−1.01827
$869$	−8.00000	−0.271381
$870$	2.00000	0.0678064
$871$	3.00000	0.101651
$872$	−13.0000	−0.440236
$873$	−16.0000	−0.541518
$874$	−5.00000	−0.169128
$875$	−36.0000	−1.21702
$876$	−3.00000	−0.101361
$877$	27.0000	0.911725	0.455863	−	0.890050i	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	−14.0000	−0.472477
$879$	9.00000	0.303562
$880$	2.00000	0.0674200
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	−4.00000	−0.134687
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−7.00000	−0.235435
$885$	6.00000	0.201688
$886$	18.0000	0.604722
$887$	−4.00000	−0.134307	−0.0671534	−	0.997743i	$-0.521392\pi$
$887$	−4.00000	−0.134307	−0.0671534	+	0.997743i	$0.521392\pi$
$888$	6.00000	0.201347
$889$	−48.0000	−1.60987
$890$	16.0000	0.536321
$891$	−1.00000	−0.0335013
$892$	10.0000	0.334825
$893$	0	0
$894$	18.0000	0.602010
$895$	−8.00000	−0.267411
$896$	−3.00000	−0.100223
$897$	5.00000	0.166945
$898$	24.0000	0.800890
$899$	10.0000	0.333519
$900$	2.00000	0.0666667
$901$	7.00000	0.233204
$902$	−6.00000	−0.199778
$903$	−12.0000	−0.399335
$904$	−12.0000	−0.399114
$905$	4.00000	0.132964
$906$	−12.0000	−0.398673
$907$	13.0000	0.431658	0.215829	−	0.976431i	$-0.430755\pi$
$907$	13.0000	0.431658	0.215829	+	0.976431i	$0.430755\pi$
$908$	−15.0000	−0.497792
$909$	−12.0000	−0.398015
$910$	6.00000	0.198898
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	−1.00000	−0.0331133
$913$	−8.00000	−0.264761
$914$	−37.0000	−1.22385
$915$	−24.0000	−0.793416
$916$	20.0000	0.660819
$917$	66.0000	2.17951
$918$	−35.0000	−1.15517
$919$	3.00000	0.0989609	0.0494804	−	0.998775i	$-0.484243\pi$
$919$	3.00000	0.0989609	0.0494804	+	0.998775i	$0.484243\pi$
$920$	10.0000	0.329690
$921$	16.0000	0.527218
$922$	−8.00000	−0.263466
$923$	−10.0000	−0.329154
$924$	−3.00000	−0.0986928
$925$	6.00000	0.197279
$926$	16.0000	0.525793
$927$	−24.0000	−0.788263
$928$	1.00000	0.0328266
$929$	29.0000	0.951459	0.475730	−	0.879592i	$-0.342184\pi$
$929$	29.0000	0.951459	0.475730	+	0.879592i	$0.342184\pi$
$930$	20.0000	0.655826
$931$	2.00000	0.0655474
$932$	−10.0000	−0.327561
$933$	−3.00000	−0.0982156
$934$	20.0000	0.654420
$935$	−14.0000	−0.457849
$936$	−2.00000	−0.0653720
$937$	−5.00000	−0.163343	−0.0816714	−	0.996659i	$-0.526026\pi$
$937$	−5.00000	−0.163343	−0.0816714	+	0.996659i	$0.526026\pi$
$938$	−9.00000	−0.293860
$939$	−21.0000	−0.685309
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	22.0000	0.716799
$943$	−30.0000	−0.976934
$944$	3.00000	0.0976417
$945$	30.0000	0.975900
$946$	4.00000	0.130051
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	−8.00000	−0.259828
$949$	3.00000	0.0973841
$950$	−1.00000	−0.0324443
$951$	−21.0000	−0.680972
$952$	21.0000	0.680614
$953$	60.0000	1.94359	0.971795	−	0.235826i	$-0.0757795\pi$
$953$	60.0000	1.94359	0.971795	+	0.235826i	$0.0757795\pi$
$954$	2.00000	0.0647524
$955$	−6.00000	−0.194155
$956$	25.0000	0.808558
$957$	1.00000	0.0323254
$958$	−12.0000	−0.387702
$959$	51.0000	1.64688
$960$	2.00000	0.0645497
$961$	69.0000	2.22581
$962$	−6.00000	−0.193448
$963$	26.0000	0.837838
$964$	−4.00000	−0.128831
$965$	8.00000	0.257529
$966$	−15.0000	−0.482617
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	1.00000	0.0321412
$969$	7.00000	0.224872
$970$	−16.0000	−0.513729
$971$	−44.0000	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$971$	−44.0000	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$972$	−16.0000	−0.513200
$973$	42.0000	1.34646
$974$	−12.0000	−0.384505
$975$	1.00000	0.0320256
$976$	−12.0000	−0.384111
$977$	−60.0000	−1.91957	−0.959785	−	0.280736i	$-0.909421\pi$
$977$	−60.0000	−1.91957	−0.959785	+	0.280736i	$0.909421\pi$
$978$	2.00000	0.0639529
$979$	8.00000	0.255681
$980$	−4.00000	−0.127775
$981$	26.0000	0.830116
$982$	10.0000	0.319113
$983$	−46.0000	−1.46717	−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000	−1.46717	−0.733586	+	0.679597i	$0.762155\pi$
$984$	−6.00000	−0.191273
$985$	24.0000	0.764704
$986$	−7.00000	−0.222925
$987$	0	0
$988$	1.00000	0.0318142
$989$	20.0000	0.635963
$990$	−4.00000	−0.127128
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	10.0000	0.317500
$993$	−29.0000	−0.920287
$994$	30.0000	0.951542
$995$	−46.0000	−1.45830
$996$	−8.00000	−0.253490
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−28.0000	−0.886325
$999$	−30.0000	−0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	418.2.a.a.1.1	✓	1
3.2	odd	2		3762.2.a.g.1.1		1
4.3	odd	2		3344.2.a.h.1.1		1
11.10	odd	2		4598.2.a.b.1.1		1
19.18	odd	2		7942.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.a.1.1	✓	1	1.1	even	1	trivial
3344.2.a.h.1.1		1	4.3	odd	2
3762.2.a.g.1.1		1	3.2	odd	2
4598.2.a.b.1.1		1	11.10	odd	2
7942.2.a.i.1.1		1	19.18	odd	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:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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