LMFDB - Embedded newform 6422.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	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	$0.618522\pi$
$18$	−2.00000	−0.471405
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	1.00000	0.188982
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	1.00000	0.182574
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	−1.00000	−0.169031
$36$	−2.00000	−0.333333
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−1.00000	−0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	6.00000	0.884652
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	−4.00000	−0.565685
$51$	3.00000	0.420084
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	1.00000	0.133631
$57$	−1.00000	−0.132453
$58$	−8.00000	−1.05045
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	1.00000	0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000	1.01600
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−3.00000	−0.363803
$69$	−6.00000	−0.722315
$70$	−1.00000	−0.119523
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	−2.00000	−0.235702
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	5.00000	0.581238
$75$	4.00000	0.461880
$76$	1.00000	0.114708
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	−1.00000	−0.109109
$85$	3.00000	0.325396
$86$	−1.00000	−0.107833
$87$	8.00000	0.857690
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	6.00000	0.625543
$93$	−8.00000	−0.829561
$94$	−3.00000	−0.309426
$95$	−1.00000	−0.102598
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−6.00000	−0.606092
$99$	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	$0.403510\pi$
$102$	3.00000	0.297044
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	−2.00000	−0.194257
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	5.00000	0.481125
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	1.00000	0.0944911
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	−1.00000	−0.0936586
$115$	−6.00000	−0.559503
$116$	−8.00000	−0.742781
$117$	0	0
$118$	10.0000	0.920575
$119$	−3.00000	−0.275010
$120$	1.00000	0.0912871
$121$	−11.0000	−1.00000
$122$	−14.0000	−1.26750
$123$	−2.00000	−0.180334
$124$	8.00000	0.718421
$125$	9.00000	0.804984
$126$	−2.00000	−0.178174
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	1.00000	0.0883883
$129$	1.00000	0.0880451
$130$	0	0
$131$	19.0000	1.66004	0.830019	−	0.557735i	$-0.188330\pi$
$131$	19.0000	1.66004	0.830019	+	0.557735i	$0.188330\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	4.00000	0.345547
$135$	−5.00000	−0.430331
$136$	−3.00000	−0.257248
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	−6.00000	−0.510754
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	−1.00000	−0.0845154
$141$	3.00000	0.252646
$142$	−3.00000	−0.251754
$143$	0	0
$144$	−2.00000	−0.166667
$145$	8.00000	0.664364
$146$	−16.0000	−1.32417
$147$	6.00000	0.494872
$148$	5.00000	0.410997
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	4.00000	0.326599
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	1.00000	0.0811107
$153$	6.00000	0.485071
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	4.00000	0.318223
$159$	2.00000	0.158610
$160$	−1.00000	−0.0790569
$161$	6.00000	0.472866
$162$	1.00000	0.0785674
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−16.0000	−1.24184
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	3.00000	0.230089
$171$	−2.00000	−0.152944
$172$	−1.00000	−0.0762493
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	8.00000	0.606478
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−10.0000	−0.751646
$178$	−8.00000	−0.599625
$179$	1.00000	0.0747435	0.0373718	−	0.999301i	$-0.488101\pi$
$179$	1.00000	0.0747435	0.0373718	+	0.999301i	$0.488101\pi$
$180$	2.00000	0.149071
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	6.00000	0.442326
$185$	−5.00000	−0.367607
$186$	−8.00000	−0.586588
$187$	0	0
$188$	−3.00000	−0.218797
$189$	5.00000	0.363696
$190$	−1.00000	−0.0725476
$191$	14.0000	1.01300	0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000	1.01300	0.506502	+	0.862239i	$0.330938\pi$
$192$	−1.00000	−0.0721688
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	0	0
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	−4.00000	−0.282843
$201$	−4.00000	−0.282138
$202$	6.00000	0.422159
$203$	−8.00000	−0.561490
$204$	3.00000	0.210042
$205$	−2.00000	−0.139686
$206$	−10.0000	−0.696733
$207$	−12.0000	−0.834058
$208$	0	0
$209$	0	0
$210$	1.00000	0.0690066
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	−2.00000	−0.137361
$213$	3.00000	0.205557
$214$	8.00000	0.546869
$215$	1.00000	0.0681994
$216$	5.00000	0.340207
$217$	8.00000	0.543075
$218$	−19.0000	−1.28684
$219$	16.0000	1.08118
$220$	0	0
$221$	0	0
$222$	−5.00000	−0.335578
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	1.00000	0.0668153
$225$	8.00000	0.533333
$226$	−10.0000	−0.665190
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	−1.00000	−0.0662266
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	10.0000	0.650945
$237$	−4.00000	−0.259828
$238$	−3.00000	−0.194461
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	1.00000	0.0645497
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−11.0000	−0.707107
$243$	−16.0000	−1.02640
$244$	−14.0000	−0.896258
$245$	6.00000	0.383326
$246$	−2.00000	−0.127515
$247$	0	0
$248$	8.00000	0.508001
$249$	16.0000	1.01396
$250$	9.00000	0.569210
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−2.00000	−0.125491
$255$	−3.00000	−0.187867
$256$	1.00000	0.0625000
$257$	−17.0000	−1.06043	−0.530215	−	0.847863i	$-0.677889\pi$
$257$	−17.0000	−1.06043	−0.530215	+	0.847863i	$0.677889\pi$
$258$	1.00000	0.0622573
$259$	5.00000	0.310685
$260$	0	0
$261$	16.0000	0.990375
$262$	19.0000	1.17382
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	1.00000	0.0613139
$267$	8.00000	0.489592
$268$	4.00000	0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−5.00000	−0.304290
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−4.00000	−0.241649
$275$	0	0
$276$	−6.00000	−0.361158
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	−13.0000	−0.779688
$279$	−16.0000	−0.957895
$280$	−1.00000	−0.0597614
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	3.00000	0.178647
$283$	32.0000	1.90220	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	32.0000	1.90220	0.951101	+	0.308879i	$0.0999539\pi$
$284$	−3.00000	−0.178017
$285$	1.00000	0.0592349
$286$	0	0
$287$	2.00000	0.118056
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	8.00000	0.469776
$291$	−10.0000	−0.586210
$292$	−16.0000	−0.936329
$293$	−11.0000	−0.642627	−0.321313	−	0.946973i	$-0.604124\pi$
$293$	−11.0000	−0.642627	−0.321313	+	0.946973i	$0.604124\pi$
$294$	6.00000	0.349927
$295$	−10.0000	−0.582223
$296$	5.00000	0.290619
$297$	0	0
$298$	−10.0000	−0.579284
$299$	0	0
$300$	4.00000	0.230940
$301$	−1.00000	−0.0576390
$302$	5.00000	0.287718
$303$	−6.00000	−0.344691
$304$	1.00000	0.0573539
$305$	14.0000	0.801638
$306$	6.00000	0.342997
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	−8.00000	−0.454369
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	0	0
$313$	−9.00000	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$314$	10.0000	0.564333
$315$	2.00000	0.112687
$316$	4.00000	0.225018
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	2.00000	0.112154
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−8.00000	−0.446516
$322$	6.00000	0.334367
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	0	0
$326$	−18.0000	−0.996928
$327$	19.0000	1.05070
$328$	2.00000	0.110432
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	−16.0000	−0.878114
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−4.00000	−0.218543
$336$	−1.00000	−0.0545545
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	10.0000	0.543125
$340$	3.00000	0.162698
$341$	0	0
$342$	−2.00000	−0.108148
$343$	−13.0000	−0.701934
$344$	−1.00000	−0.0539164
$345$	6.00000	0.323029
$346$	2.00000	0.107521
$347$	−17.0000	−0.912608	−0.456304	−	0.889824i	$-0.650827\pi$
$347$	−17.0000	−0.912608	−0.456304	+	0.889824i	$0.650827\pi$
$348$	8.00000	0.428845
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	−10.0000	−0.531494
$355$	3.00000	0.159223
$356$	−8.00000	−0.423999
$357$	3.00000	0.158777
$358$	1.00000	0.0528516
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	2.00000	0.105409
$361$	1.00000	0.0526316
$362$	−16.0000	−0.840941
$363$	11.0000	0.577350
$364$	0	0
$365$	16.0000	0.837478
$366$	14.0000	0.731792
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	6.00000	0.312772
$369$	−4.00000	−0.208232
$370$	−5.00000	−0.259938
$371$	−2.00000	−0.103835
$372$	−8.00000	−0.414781
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	−3.00000	−0.154713
$377$	0	0
$378$	5.00000	0.257172
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	−1.00000	−0.0512989
$381$	2.00000	0.102463
$382$	14.0000	0.716302
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	16.0000	0.814379
$387$	2.00000	0.101666
$388$	10.0000	0.507673
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	−6.00000	−0.303046
$393$	−19.0000	−0.958423
$394$	−7.00000	−0.352655
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−26.0000	−1.30326
$399$	−1.00000	−0.0500626
$400$	−4.00000	−0.200000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	−8.00000	−0.397033
$407$	0	0
$408$	3.00000	0.148522
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	−2.00000	−0.0987730
$411$	4.00000	0.197305
$412$	−10.0000	−0.492665
$413$	10.0000	0.492068
$414$	−12.0000	−0.589768
$415$	16.0000	0.785409
$416$	0	0
$417$	13.0000	0.636613
$418$	0	0
$419$	17.0000	0.830504	0.415252	−	0.909706i	$-0.363693\pi$
$419$	17.0000	0.830504	0.415252	+	0.909706i	$0.363693\pi$
$420$	1.00000	0.0487950
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	1.00000	0.0486792
$423$	6.00000	0.291730
$424$	−2.00000	−0.0971286
$425$	12.0000	0.582086
$426$	3.00000	0.145350
$427$	−14.0000	−0.677507
$428$	8.00000	0.386695
$429$	0	0
$430$	1.00000	0.0482243
$431$	−5.00000	−0.240842	−0.120421	−	0.992723i	$-0.538424\pi$
$431$	−5.00000	−0.240842	−0.120421	+	0.992723i	$0.538424\pi$
$432$	5.00000	0.240563
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	8.00000	0.384012
$435$	−8.00000	−0.383571
$436$	−19.0000	−0.909935
$437$	6.00000	0.287019
$438$	16.0000	0.764510
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	17.0000	0.807694	0.403847	−	0.914826i	$-0.367673\pi$
$443$	17.0000	0.807694	0.403847	+	0.914826i	$0.367673\pi$
$444$	−5.00000	−0.237289
$445$	8.00000	0.379236
$446$	−7.00000	−0.331460
$447$	10.0000	0.472984
$448$	1.00000	0.0472456
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	8.00000	0.377124
$451$	0	0
$452$	−10.0000	−0.470360
$453$	−5.00000	−0.234920
$454$	14.0000	0.657053
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−11.0000	−0.513996
$459$	−15.0000	−0.700140
$460$	−6.00000	−0.279751
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−8.00000	−0.371391
$465$	8.00000	0.370991
$466$	−3.00000	−0.138972
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	3.00000	0.138380
$471$	−10.0000	−0.460776
$472$	10.0000	0.460287
$473$	0	0
$474$	−4.00000	−0.183726
$475$	−4.00000	−0.183533
$476$	−3.00000	−0.137505
$477$	4.00000	0.183147
$478$	1.00000	0.0457389
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−18.0000	−0.819878
$483$	−6.00000	−0.273009
$484$	−11.0000	−0.500000
$485$	−10.0000	−0.454077
$486$	−16.0000	−0.725775
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−14.0000	−0.633750
$489$	18.0000	0.813988
$490$	6.00000	0.271052
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	−2.00000	−0.0901670
$493$	24.0000	1.08091
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	−3.00000	−0.134568
$498$	16.0000	0.716977
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	−28.0000	−1.24970
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	−2.00000	−0.0890871
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	−3.00000	−0.132842
$511$	−16.0000	−0.707798
$512$	1.00000	0.0441942
$513$	5.00000	0.220755
$514$	−17.0000	−0.749838
$515$	10.0000	0.440653
$516$	1.00000	0.0440225
$517$	0	0
$518$	5.00000	0.219687
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	16.0000	0.700301
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	19.0000	0.830019
$525$	4.00000	0.174574
$526$	6.00000	0.261612
$527$	−24.0000	−1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	2.00000	0.0868744
$531$	−20.0000	−0.867926
$532$	1.00000	0.0433555
$533$	0	0
$534$	8.00000	0.346194
$535$	−8.00000	−0.345870
$536$	4.00000	0.172774
$537$	−1.00000	−0.0431532
$538$	18.0000	0.776035
$539$	0	0
$540$	−5.00000	−0.215166
$541$	−35.0000	−1.50477	−0.752384	−	0.658725i	$-0.771096\pi$
$541$	−35.0000	−1.50477	−0.752384	+	0.658725i	$0.771096\pi$
$542$	5.00000	0.214768
$543$	16.0000	0.686626
$544$	−3.00000	−0.128624
$545$	19.0000	0.813871
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	−4.00000	−0.170872
$549$	28.0000	1.19501
$550$	0	0
$551$	−8.00000	−0.340811
$552$	−6.00000	−0.255377
$553$	4.00000	0.170097
$554$	−30.0000	−1.27458
$555$	5.00000	0.212238
$556$	−13.0000	−0.551323
$557$	21.0000	0.889799	0.444899	−	0.895581i	$-0.353239\pi$
$557$	21.0000	0.889799	0.444899	+	0.895581i	$0.353239\pi$
$558$	−16.0000	−0.677334
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	22.0000	0.928014
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	3.00000	0.126323
$565$	10.0000	0.420703
$566$	32.0000	1.34506
$567$	1.00000	0.0419961
$568$	−3.00000	−0.125877
$569$	−23.0000	−0.964210	−0.482105	−	0.876113i	$-0.660128\pi$
$569$	−23.0000	−0.964210	−0.482105	+	0.876113i	$0.660128\pi$
$570$	1.00000	0.0418854
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	0	0
$573$	−14.0000	−0.584858
$574$	2.00000	0.0834784
$575$	−24.0000	−1.00087
$576$	−2.00000	−0.0833333
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	−16.0000	−0.664937
$580$	8.00000	0.332182
$581$	−16.0000	−0.663792
$582$	−10.0000	−0.414513
$583$	0	0
$584$	−16.0000	−0.662085
$585$	0	0
$586$	−11.0000	−0.454406
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	6.00000	0.247436
$589$	8.00000	0.329634
$590$	−10.0000	−0.411693
$591$	7.00000	0.287942
$592$	5.00000	0.205499
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	−10.0000	−0.409616
$597$	26.0000	1.06411
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	4.00000	0.163299
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	−1.00000	−0.0407570
$603$	−8.00000	−0.325785
$604$	5.00000	0.203447
$605$	11.0000	0.447214
$606$	−6.00000	−0.243733
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	1.00000	0.0405554
$609$	8.00000	0.324176
$610$	14.0000	0.566843
$611$	0	0
$612$	6.00000	0.242536
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	4.00000	0.161427
$615$	2.00000	0.0806478
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	10.0000	0.402259
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	−8.00000	−0.321288
$621$	30.0000	1.20386
$622$	−14.0000	−0.561349
$623$	−8.00000	−0.320513
$624$	0	0
$625$	11.0000	0.440000
$626$	−9.00000	−0.359712
$627$	0	0
$628$	10.0000	0.399043
$629$	−15.0000	−0.598089
$630$	2.00000	0.0796819
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	4.00000	0.159111
$633$	−1.00000	−0.0397464
$634$	−6.00000	−0.238290
$635$	2.00000	0.0793676
$636$	2.00000	0.0793052
$637$	0	0
$638$	0	0
$639$	6.00000	0.237356
$640$	−1.00000	−0.0395285
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	−8.00000	−0.315735
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	6.00000	0.236433
$645$	−1.00000	−0.0393750
$646$	−3.00000	−0.118033
$647$	−34.0000	−1.33668	−0.668339	−	0.743857i	$-0.732994\pi$
$647$	−34.0000	−1.33668	−0.668339	+	0.743857i	$0.732994\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	−18.0000	−0.704934
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	19.0000	0.742959
$655$	−19.0000	−0.742391
$656$	2.00000	0.0780869
$657$	32.0000	1.24844
$658$	−3.00000	−0.116952
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−32.0000	−1.24372
$663$	0	0
$664$	−16.0000	−0.620920
$665$	−1.00000	−0.0387783
$666$	−10.0000	−0.387492
$667$	−48.0000	−1.85857
$668$	0	0
$669$	7.00000	0.270636
$670$	−4.00000	−0.154533
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	−23.0000	−0.885927
$675$	−20.0000	−0.769800
$676$	0	0
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	10.0000	0.384048
$679$	10.0000	0.383765
$680$	3.00000	0.115045
$681$	−14.0000	−0.536481
$682$	0	0
$683$	−14.0000	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$684$	−2.00000	−0.0764719
$685$	4.00000	0.152832
$686$	−13.0000	−0.496342
$687$	11.0000	0.419676
$688$	−1.00000	−0.0381246
$689$	0	0
$690$	6.00000	0.228416
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−17.0000	−0.645311
$695$	13.0000	0.493118
$696$	8.00000	0.303239
$697$	−6.00000	−0.227266
$698$	11.0000	0.416356
$699$	3.00000	0.113470
$700$	−4.00000	−0.151186
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	0	0
$705$	−3.00000	−0.112987
$706$	8.00000	0.301084
$707$	6.00000	0.225653
$708$	−10.0000	−0.375823
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	3.00000	0.112588
$711$	−8.00000	−0.300023
$712$	−8.00000	−0.299813
$713$	48.0000	1.79761
$714$	3.00000	0.112272
$715$	0	0
$716$	1.00000	0.0373718
$717$	−1.00000	−0.0373457
$718$	−8.00000	−0.298557
$719$	34.0000	1.26799	0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000	1.26799	0.633993	+	0.773339i	$0.281415\pi$
$720$	2.00000	0.0745356
$721$	−10.0000	−0.372419
$722$	1.00000	0.0372161
$723$	18.0000	0.669427
$724$	−16.0000	−0.594635
$725$	32.0000	1.18845
$726$	11.0000	0.408248
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	16.0000	0.592187
$731$	3.00000	0.110959
$732$	14.0000	0.517455
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	16.0000	0.590571
$735$	−6.00000	−0.221313
$736$	6.00000	0.221163
$737$	0	0
$738$	−4.00000	−0.147242
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	−5.00000	−0.183804
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−25.0000	−0.917161	−0.458581	−	0.888653i	$-0.651642\pi$
$743$	−25.0000	−0.917161	−0.458581	+	0.888653i	$0.651642\pi$
$744$	−8.00000	−0.293294
$745$	10.0000	0.366372
$746$	26.0000	0.951928
$747$	32.0000	1.17082
$748$	0	0
$749$	8.00000	0.292314
$750$	−9.00000	−0.328634
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−3.00000	−0.109399
$753$	28.0000	1.02038
$754$	0	0
$755$	−5.00000	−0.181969
$756$	5.00000	0.181848
$757$	14.0000	0.508839	0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000	0.508839	0.254419	+	0.967094i	$0.418116\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	−1.00000	−0.0362738
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	2.00000	0.0724524
$763$	−19.0000	−0.687846
$764$	14.0000	0.506502
$765$	−6.00000	−0.216930
$766$	1.00000	0.0361315
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	17.0000	0.612240
$772$	16.0000	0.575853
$773$	−11.0000	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$773$	−11.0000	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$774$	2.00000	0.0718885
$775$	−32.0000	−1.14947
$776$	10.0000	0.358979
$777$	−5.00000	−0.179374
$778$	−36.0000	−1.29066
$779$	2.00000	0.0716574
$780$	0	0
$781$	0	0
$782$	−18.0000	−0.643679
$783$	−40.0000	−1.42948
$784$	−6.00000	−0.214286
$785$	−10.0000	−0.356915
$786$	−19.0000	−0.677708
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	−7.00000	−0.249365
$789$	−6.00000	−0.213606
$790$	−4.00000	−0.142314
$791$	−10.0000	−0.355559
$792$	0	0
$793$	0	0
$794$	−18.0000	−0.638796
$795$	−2.00000	−0.0709327
$796$	−26.0000	−0.921546
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	−1.00000	−0.0353996
$799$	9.00000	0.318397
$800$	−4.00000	−0.141421
$801$	16.0000	0.565332
$802$	10.0000	0.353112
$803$	0	0
$804$	−4.00000	−0.141069
$805$	−6.00000	−0.211472
$806$	0	0
$807$	−18.0000	−0.633630
$808$	6.00000	0.211079
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	−1.00000	−0.0351364
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	−8.00000	−0.280745
$813$	−5.00000	−0.175358
$814$	0	0
$815$	18.0000	0.630512
$816$	3.00000	0.105021
$817$	−1.00000	−0.0349856
$818$	8.00000	0.279713
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−25.0000	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$821$	−25.0000	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$822$	4.00000	0.139516
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−10.0000	−0.348367
$825$	0	0
$826$	10.0000	0.347945
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	−12.0000	−0.417029
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	16.0000	0.555368
$831$	30.0000	1.04069
$832$	0	0
$833$	18.0000	0.623663
$834$	13.0000	0.450153
$835$	0	0
$836$	0	0
$837$	40.0000	1.38260
$838$	17.0000	0.587255
$839$	32.0000	1.10476	0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000	1.10476	0.552381	+	0.833592i	$0.313719\pi$
$840$	1.00000	0.0345033
$841$	35.0000	1.20690
$842$	13.0000	0.448010
$843$	−22.0000	−0.757720
$844$	1.00000	0.0344214
$845$	0	0
$846$	6.00000	0.206284
$847$	−11.0000	−0.377964
$848$	−2.00000	−0.0686803
$849$	−32.0000	−1.09824
$850$	12.0000	0.411597
$851$	30.0000	1.02839
$852$	3.00000	0.102778
$853$	5.00000	0.171197	0.0855984	−	0.996330i	$-0.472720\pi$
$853$	5.00000	0.171197	0.0855984	+	0.996330i	$0.472720\pi$
$854$	−14.0000	−0.479070
$855$	2.00000	0.0683986
$856$	8.00000	0.273434
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	1.00000	0.0340997
$861$	−2.00000	−0.0681598
$862$	−5.00000	−0.170301
$863$	−29.0000	−0.987171	−0.493586	−	0.869697i	$-0.664314\pi$
$863$	−29.0000	−0.987171	−0.493586	+	0.869697i	$0.664314\pi$
$864$	5.00000	0.170103
$865$	−2.00000	−0.0680020
$866$	29.0000	0.985460
$867$	8.00000	0.271694
$868$	8.00000	0.271538
$869$	0	0
$870$	−8.00000	−0.271225
$871$	0	0
$872$	−19.0000	−0.643421
$873$	−20.0000	−0.676897
$874$	6.00000	0.202953
$875$	9.00000	0.304256
$876$	16.0000	0.540590
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	−26.0000	−0.877457
$879$	11.0000	0.371021
$880$	0	0
$881$	−31.0000	−1.04442	−0.522208	−	0.852818i	$-0.674892\pi$
$881$	−31.0000	−1.04442	−0.522208	+	0.852818i	$0.674892\pi$
$882$	12.0000	0.404061
$883$	25.0000	0.841317	0.420658	−	0.907219i	$-0.361799\pi$
$883$	25.0000	0.841317	0.420658	+	0.907219i	$0.361799\pi$
$884$	0	0
$885$	10.0000	0.336146
$886$	17.0000	0.571126
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	−5.00000	−0.167789
$889$	−2.00000	−0.0670778
$890$	8.00000	0.268161
$891$	0	0
$892$	−7.00000	−0.234377
$893$	−3.00000	−0.100391
$894$	10.0000	0.334450
$895$	−1.00000	−0.0334263
$896$	1.00000	0.0334077
$897$	0	0
$898$	−26.0000	−0.867631
$899$	−64.0000	−2.13452
$900$	8.00000	0.266667
$901$	6.00000	0.199889
$902$	0	0
$903$	1.00000	0.0332779
$904$	−10.0000	−0.332595
$905$	16.0000	0.531858
$906$	−5.00000	−0.166114
$907$	−35.0000	−1.16216	−0.581078	−	0.813848i	$-0.697369\pi$
$907$	−35.0000	−1.16216	−0.581078	+	0.813848i	$0.697369\pi$
$908$	14.0000	0.464606
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	−2.00000	−0.0661541
$915$	−14.0000	−0.462826
$916$	−11.0000	−0.363450
$917$	19.0000	0.627435
$918$	−15.0000	−0.495074
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−6.00000	−0.197814
$921$	−4.00000	−0.131804
$922$	3.00000	0.0987997
$923$	0	0
$924$	0	0
$925$	−20.0000	−0.657596
$926$	16.0000	0.525793
$927$	20.0000	0.656886
$928$	−8.00000	−0.262613
$929$	60.0000	1.96854	0.984268	−	0.176682i	$-0.0565363\pi$
$929$	60.0000	1.96854	0.984268	+	0.176682i	$0.0565363\pi$
$930$	8.00000	0.262330
$931$	−6.00000	−0.196642
$932$	−3.00000	−0.0982683
$933$	14.0000	0.458339
$934$	−8.00000	−0.261768
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	4.00000	0.130605
$939$	9.00000	0.293704
$940$	3.00000	0.0978492
$941$	−37.0000	−1.20617	−0.603083	−	0.797679i	$-0.706061\pi$
$941$	−37.0000	−1.20617	−0.603083	+	0.797679i	$0.706061\pi$
$942$	−10.0000	−0.325818
$943$	12.0000	0.390774
$944$	10.0000	0.325472
$945$	−5.00000	−0.162650
$946$	0	0
$947$	58.0000	1.88475	0.942373	−	0.334563i	$-0.108589\pi$
$947$	58.0000	1.88475	0.942373	+	0.334563i	$0.108589\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	−4.00000	−0.129777
$951$	6.00000	0.194563
$952$	−3.00000	−0.0972306
$953$	−3.00000	−0.0971795	−0.0485898	−	0.998819i	$-0.515473\pi$
$953$	−3.00000	−0.0971795	−0.0485898	+	0.998819i	$0.515473\pi$
$954$	4.00000	0.129505
$955$	−14.0000	−0.453029
$956$	1.00000	0.0323423
$957$	0	0
$958$	21.0000	0.678479
$959$	−4.00000	−0.129167
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	0	0
$963$	−16.0000	−0.515593
$964$	−18.0000	−0.579741
$965$	−16.0000	−0.515058
$966$	−6.00000	−0.193047
$967$	−33.0000	−1.06121	−0.530604	−	0.847620i	$-0.678035\pi$
$967$	−33.0000	−1.06121	−0.530604	+	0.847620i	$0.678035\pi$
$968$	−11.0000	−0.353553
$969$	3.00000	0.0963739
$970$	−10.0000	−0.321081
$971$	39.0000	1.25157	0.625785	−	0.779996i	$-0.284779\pi$
$971$	39.0000	1.25157	0.625785	+	0.779996i	$0.284779\pi$
$972$	−16.0000	−0.513200
$973$	−13.0000	−0.416761
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−14.0000	−0.448129
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	18.0000	0.575577
$979$	0	0
$980$	6.00000	0.191663
$981$	38.0000	1.21325
$982$	15.0000	0.478669
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	−2.00000	−0.0637577
$985$	7.00000	0.223039
$986$	24.0000	0.764316
$987$	3.00000	0.0954911
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	62.0000	1.96949	0.984747	−	0.173990i	$-0.0556660\pi$
$991$	62.0000	1.96949	0.984747	+	0.173990i	$0.0556660\pi$
$992$	8.00000	0.254000
$993$	32.0000	1.01549
$994$	−3.00000	−0.0951542
$995$	26.0000	0.824255
$996$	16.0000	0.506979
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	22.0000	0.696398
$999$	25.0000	0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.e.1.1		1
13.12	even	2		494.2.a.a.1.1	✓	1
39.38	odd	2		4446.2.a.n.1.1		1
52.51	odd	2		3952.2.a.i.1.1		1
247.246	odd	2		9386.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.a.a.1.1	✓	1	13.12	even	2
3952.2.a.i.1.1		1	52.51	odd	2
4446.2.a.n.1.1		1	39.38	odd	2
6422.2.a.e.1.1		1	1.1	even	1	trivial
9386.2.a.l.1.1		1	247.246	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more