LMFDB - Embedded newform 1014.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,2,Mod(1,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, 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("1014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1014 = 2 \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1014.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:	$8.09683076496$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +3.00000 q^{20} +2.00000 q^{21} -6.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +3.00000 q^{29} -3.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +3.00000 q^{34} +6.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} -2.00000 q^{38} -3.00000 q^{40} -3.00000 q^{41} -2.00000 q^{42} -10.0000 q^{43} +6.00000 q^{44} +3.00000 q^{45} +6.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +3.00000 q^{53} -1.00000 q^{54} +18.0000 q^{55} -2.00000 q^{56} +2.00000 q^{57} -3.00000 q^{58} +3.00000 q^{60} -7.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} -10.0000 q^{67} -3.00000 q^{68} -6.00000 q^{69} -6.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} -13.0000 q^{73} +7.00000 q^{74} +4.00000 q^{75} +2.00000 q^{76} +12.0000 q^{77} -4.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +3.00000 q^{82} -6.00000 q^{83} +2.00000 q^{84} -9.00000 q^{85} +10.0000 q^{86} +3.00000 q^{87} -6.00000 q^{88} +18.0000 q^{89} -3.00000 q^{90} -6.00000 q^{92} -4.00000 q^{93} -6.00000 q^{94} +6.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	3.00000	0.774597
$16$	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	$0.618522\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	3.00000	0.670820
$21$	2.00000	0.436436
$22$	−6.00000	−1.27920
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−3.00000	−0.547723
$31$	−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
$33$	6.00000	1.04447
$34$	3.00000	0.514496
$35$	6.00000	1.01419
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	−2.00000	−0.308607
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	6.00000	0.904534
$45$	3.00000	0.447214
$46$	6.00000	0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	−3.00000	−0.420084
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−1.00000	−0.136083
$55$	18.0000	2.42712
$56$	−2.00000	−0.267261
$57$	2.00000	0.264906
$58$	−3.00000	−0.393919
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	3.00000	0.387298
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	4.00000	0.508001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−3.00000	−0.363803
$69$	−6.00000	−0.722315
$70$	−6.00000	−0.717137
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−1.00000	−0.117851
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	7.00000	0.813733
$75$	4.00000	0.461880
$76$	2.00000	0.229416
$77$	12.0000	1.36753
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	3.00000	0.331295
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	2.00000	0.218218
$85$	−9.00000	−0.976187
$86$	10.0000	1.07833
$87$	3.00000	0.321634
$88$	−6.00000	−0.639602
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−4.00000	−0.414781
$94$	−6.00000	−0.618853
$95$	6.00000	0.615587
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	3.00000	0.303046
$99$	6.00000	0.603023
$100$	4.00000	0.400000
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	3.00000	0.297044
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	−3.00000	−0.291386
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−18.0000	−1.71623
$111$	−7.00000	−0.664411
$112$	2.00000	0.188982
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	−2.00000	−0.187317
$115$	−18.0000	−1.67851
$116$	3.00000	0.278543
$117$	0	0
$118$	0	0
$119$	−6.00000	−0.550019
$120$	−3.00000	−0.273861
$121$	25.0000	2.27273
$122$	7.00000	0.633750
$123$	−3.00000	−0.270501
$124$	−4.00000	−0.359211
$125$	−3.00000	−0.268328
$126$	−2.00000	−0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	6.00000	0.522233
$133$	4.00000	0.346844
$134$	10.0000	0.863868
$135$	3.00000	0.258199
$136$	3.00000	0.257248
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	6.00000	0.510754
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	6.00000	0.507093
$141$	6.00000	0.505291
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	9.00000	0.747409
$146$	13.0000	1.07589
$147$	−3.00000	−0.247436
$148$	−7.00000	−0.575396
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	−4.00000	−0.326599
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−2.00000	−0.162221
$153$	−3.00000	−0.242536
$154$	−12.0000	−0.966988
$155$	−12.0000	−0.963863
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	4.00000	0.318223
$159$	3.00000	0.237915
$160$	−3.00000	−0.237171
$161$	−12.0000	−0.945732
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−3.00000	−0.234261
$165$	18.0000	1.40130
$166$	6.00000	0.465690
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	9.00000	0.690268
$171$	2.00000	0.152944
$172$	−10.0000	−0.762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−3.00000	−0.227429
$175$	8.00000	0.604743
$176$	6.00000	0.452267
$177$	0	0
$178$	−18.0000	−1.34916
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	3.00000	0.223607
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	−7.00000	−0.517455
$184$	6.00000	0.442326
$185$	−21.0000	−1.54395
$186$	4.00000	0.293294
$187$	−18.0000	−1.31629
$188$	6.00000	0.437595
$189$	2.00000	0.145479
$190$	−6.00000	−0.435286
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	23.0000	1.65558	0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000	1.65558	0.827788	+	0.561041i	$0.189599\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−6.00000	−0.426401
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	−4.00000	−0.282843
$201$	−10.0000	−0.705346
$202$	−15.0000	−1.05540
$203$	6.00000	0.421117
$204$	−3.00000	−0.210042
$205$	−9.00000	−0.628587
$206$	−14.0000	−0.975426
$207$	−6.00000	−0.417029
$208$	0	0
$209$	12.0000	0.830057
$210$	−6.00000	−0.414039
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	3.00000	0.206041
$213$	6.00000	0.411113
$214$	6.00000	0.410152
$215$	−30.0000	−2.04598
$216$	−1.00000	−0.0680414
$217$	−8.00000	−0.543075
$218$	−14.0000	−0.948200
$219$	−13.0000	−0.878459
$220$	18.0000	1.21356
$221$	0	0
$222$	7.00000	0.469809
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−2.00000	−0.133631
$225$	4.00000	0.266667
$226$	3.00000	0.199557
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	2.00000	0.132453
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	18.0000	1.18688
$231$	12.0000	0.789542
$232$	−3.00000	−0.196960
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	0	0
$237$	−4.00000	−0.259828
$238$	6.00000	0.388922
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	3.00000	0.193649
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	−25.0000	−1.60706
$243$	1.00000	0.0641500
$244$	−7.00000	−0.448129
$245$	−9.00000	−0.574989
$246$	3.00000	0.191273
$247$	0	0
$248$	4.00000	0.254000
$249$	−6.00000	−0.380235
$250$	3.00000	0.189737
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.00000	0.125988
$253$	−36.0000	−2.26330
$254$	4.00000	0.250982
$255$	−9.00000	−0.563602
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	10.0000	0.622573
$259$	−14.0000	−0.869918
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	−6.00000	−0.369274
$265$	9.00000	0.552866
$266$	−4.00000	−0.245256
$267$	18.0000	1.10158
$268$	−10.0000	−0.610847
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−3.00000	−0.182574
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−9.00000	−0.543710
$275$	24.0000	1.44725
$276$	−6.00000	−0.361158
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	4.00000	0.239904
$279$	−4.00000	−0.239474
$280$	−6.00000	−0.358569
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	−6.00000	−0.357295
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	6.00000	0.356034
$285$	6.00000	0.355409
$286$	0	0
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	−9.00000	−0.528498
$291$	14.0000	0.820695
$292$	−13.0000	−0.760767
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	7.00000	0.406867
$297$	6.00000	0.348155
$298$	9.00000	0.521356
$299$	0	0
$300$	4.00000	0.230940
$301$	−20.0000	−1.15278
$302$	10.0000	0.575435
$303$	15.0000	0.861727
$304$	2.00000	0.114708
$305$	−21.0000	−1.20246
$306$	3.00000	0.171499
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	12.0000	0.683763
$309$	14.0000	0.796432
$310$	12.0000	0.681554
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−5.00000	−0.282166
$315$	6.00000	0.338062
$316$	−4.00000	−0.225018
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	−3.00000	−0.168232
$319$	18.0000	1.00781
$320$	3.00000	0.167705
$321$	−6.00000	−0.334887
$322$	12.0000	0.668734
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	14.0000	0.774202
$328$	3.00000	0.165647
$329$	12.0000	0.661581
$330$	−18.0000	−0.990867
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−6.00000	−0.329293
$333$	−7.00000	−0.383598
$334$	0	0
$335$	−30.0000	−1.63908
$336$	2.00000	0.109109
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	−3.00000	−0.162938
$340$	−9.00000	−0.488094
$341$	−24.0000	−1.29967
$342$	−2.00000	−0.108148
$343$	−20.0000	−1.07990
$344$	10.0000	0.539164
$345$	−18.0000	−0.969087
$346$	6.00000	0.322562
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	3.00000	0.160817
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−8.00000	−0.427618
$351$	0	0
$352$	−6.00000	−0.319801
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	18.0000	0.953998
$357$	−6.00000	−0.317554
$358$	−6.00000	−0.317110
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	−3.00000	−0.158114
$361$	−15.0000	−0.789474
$362$	7.00000	0.367912
$363$	25.0000	1.31216
$364$	0	0
$365$	−39.0000	−2.04135
$366$	7.00000	0.365896
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	−6.00000	−0.312772
$369$	−3.00000	−0.156174
$370$	21.0000	1.09174
$371$	6.00000	0.311504
$372$	−4.00000	−0.207390
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	18.0000	0.930758
$375$	−3.00000	−0.154919
$376$	−6.00000	−0.309426
$377$	0	0
$378$	−2.00000	−0.102869
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	6.00000	0.307794
$381$	−4.00000	−0.204926
$382$	12.0000	0.613973
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	36.0000	1.83473
$386$	−23.0000	−1.17067
$387$	−10.0000	−0.508329
$388$	14.0000	0.710742
$389$	39.0000	1.97738	0.988689	−	0.149979i	$-0.0479205\pi$
$389$	39.0000	1.97738	0.988689	+	0.149979i	$0.0479205\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	3.00000	0.151523
$393$	0	0
$394$	−6.00000	−0.302276
$395$	−12.0000	−0.603786
$396$	6.00000	0.301511
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	10.0000	0.501255
$399$	4.00000	0.200250
$400$	4.00000	0.200000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	10.0000	0.498755
$403$	0	0
$404$	15.0000	0.746278
$405$	3.00000	0.149071
$406$	−6.00000	−0.297775
$407$	−42.0000	−2.08186
$408$	3.00000	0.148522
$409$	−1.00000	−0.0494468	−0.0247234	−	0.999694i	$-0.507871\pi$
$409$	−1.00000	−0.0494468	−0.0247234	+	0.999694i	$0.507871\pi$
$410$	9.00000	0.444478
$411$	9.00000	0.443937
$412$	14.0000	0.689730
$413$	0	0
$414$	6.00000	0.294884
$415$	−18.0000	−0.883585
$416$	0	0
$417$	−4.00000	−0.195881
$418$	−12.0000	−0.586939
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	6.00000	0.292770
$421$	29.0000	1.41337	0.706687	−	0.707527i	$-0.250189\pi$
$421$	29.0000	1.41337	0.706687	+	0.707527i	$0.250189\pi$
$422$	16.0000	0.778868
$423$	6.00000	0.291730
$424$	−3.00000	−0.145693
$425$	−12.0000	−0.582086
$426$	−6.00000	−0.290701
$427$	−14.0000	−0.677507
$428$	−6.00000	−0.290021
$429$	0	0
$430$	30.0000	1.44673
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	1.00000	0.0481125
$433$	−13.0000	−0.624740	−0.312370	−	0.949960i	$-0.601123\pi$
$433$	−13.0000	−0.624740	−0.312370	+	0.949960i	$0.601123\pi$
$434$	8.00000	0.384012
$435$	9.00000	0.431517
$436$	14.0000	0.670478
$437$	−12.0000	−0.574038
$438$	13.0000	0.621164
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	−18.0000	−0.858116
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−7.00000	−0.332205
$445$	54.0000	2.55985
$446$	−8.00000	−0.378811
$447$	−9.00000	−0.425685
$448$	2.00000	0.0944911
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	−4.00000	−0.188562
$451$	−18.0000	−0.847587
$452$	−3.00000	−0.141108
$453$	−10.0000	−0.469841
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	22.0000	1.02799
$459$	−3.00000	−0.140028
$460$	−18.0000	−0.839254
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	−12.0000	−0.558291
$463$	38.0000	1.76601	0.883005	−	0.469364i	$-0.155517\pi$
$463$	38.0000	1.76601	0.883005	+	0.469364i	$0.155517\pi$
$464$	3.00000	0.139272
$465$	−12.0000	−0.556487
$466$	6.00000	0.277945
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	−18.0000	−0.830278
$471$	5.00000	0.230388
$472$	0	0
$473$	−60.0000	−2.75880
$474$	4.00000	0.183726
$475$	8.00000	0.367065
$476$	−6.00000	−0.275010
$477$	3.00000	0.137361
$478$	−6.00000	−0.274434
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−3.00000	−0.136931
$481$	0	0
$482$	1.00000	0.0455488
$483$	−12.0000	−0.546019
$484$	25.0000	1.13636
$485$	42.0000	1.90712
$486$	−1.00000	−0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	7.00000	0.316875
$489$	−4.00000	−0.180886
$490$	9.00000	0.406579
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	−3.00000	−0.135250
$493$	−9.00000	−0.405340
$494$	0	0
$495$	18.0000	0.809040
$496$	−4.00000	−0.179605
$497$	12.0000	0.538274
$498$	6.00000	0.268866
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	12.0000	0.535586
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	−2.00000	−0.0890871
$505$	45.0000	2.00247
$506$	36.0000	1.60040
$507$	0	0
$508$	−4.00000	−0.177471
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	9.00000	0.398527
$511$	−26.0000	−1.15017
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	3.00000	0.132324
$515$	42.0000	1.85074
$516$	−10.0000	−0.440225
$517$	36.0000	1.58328
$518$	14.0000	0.615125
$519$	−6.00000	−0.263371
$520$	0	0
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	−3.00000	−0.131306
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	0	0
$525$	8.00000	0.349149
$526$	6.00000	0.261612
$527$	12.0000	0.522728
$528$	6.00000	0.261116
$529$	13.0000	0.565217
$530$	−9.00000	−0.390935
$531$	0	0
$532$	4.00000	0.173422
$533$	0	0
$534$	−18.0000	−0.778936
$535$	−18.0000	−0.778208
$536$	10.0000	0.431934
$537$	6.00000	0.258919
$538$	−18.0000	−0.776035
$539$	−18.0000	−0.775315
$540$	3.00000	0.129099
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	16.0000	0.687259
$543$	−7.00000	−0.300399
$544$	3.00000	0.128624
$545$	42.0000	1.79908
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	9.00000	0.384461
$549$	−7.00000	−0.298753
$550$	−24.0000	−1.02336
$551$	6.00000	0.255609
$552$	6.00000	0.255377
$553$	−8.00000	−0.340195
$554$	−17.0000	−0.722261
$555$	−21.0000	−0.891400
$556$	−4.00000	−0.169638
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	6.00000	0.253546
$561$	−18.0000	−0.759961
$562$	−9.00000	−0.379642
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	6.00000	0.252646
$565$	−9.00000	−0.378633
$566$	−14.0000	−0.588464
$567$	2.00000	0.0839921
$568$	−6.00000	−0.251754
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	−6.00000	−0.251312
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	6.00000	0.250435
$575$	−24.0000	−1.00087
$576$	1.00000	0.0416667
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	8.00000	0.332756
$579$	23.0000	0.955847
$580$	9.00000	0.373705
$581$	−12.0000	−0.497844
$582$	−14.0000	−0.580319
$583$	18.0000	0.745484
$584$	13.0000	0.537944
$585$	0	0
$586$	21.0000	0.867502
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−3.00000	−0.123718
$589$	−8.00000	−0.329634
$590$	0	0
$591$	6.00000	0.246807
$592$	−7.00000	−0.287698
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	−6.00000	−0.246183
$595$	−18.0000	−0.737928
$596$	−9.00000	−0.368654
$597$	−10.0000	−0.409273
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−4.00000	−0.163299
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	20.0000	0.815139
$603$	−10.0000	−0.407231
$604$	−10.0000	−0.406894
$605$	75.0000	3.04918
$606$	−15.0000	−0.609333
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	−2.00000	−0.0811107
$609$	6.00000	0.243132
$610$	21.0000	0.850265
$611$	0	0
$612$	−3.00000	−0.121268
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	10.0000	0.403567
$615$	−9.00000	−0.362915
$616$	−12.0000	−0.483494
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	−14.0000	−0.563163
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	−12.0000	−0.481932
$621$	−6.00000	−0.240772
$622$	30.0000	1.20289
$623$	36.0000	1.44231
$624$	0	0
$625$	−29.0000	−1.16000
$626$	10.0000	0.399680
$627$	12.0000	0.479234
$628$	5.00000	0.199522
$629$	21.0000	0.837325
$630$	−6.00000	−0.239046
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	4.00000	0.159111
$633$	−16.0000	−0.635943
$634$	−3.00000	−0.119145
$635$	−12.0000	−0.476205
$636$	3.00000	0.118958
$637$	0	0
$638$	−18.0000	−0.712627
$639$	6.00000	0.237356
$640$	−3.00000	−0.118585
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	6.00000	0.236801
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−12.0000	−0.472866
$645$	−30.0000	−1.18125
$646$	6.00000	0.236067
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	−4.00000	−0.156652
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	−3.00000	−0.117130
$657$	−13.0000	−0.507178
$658$	−12.0000	−0.467809
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	18.0000	0.700649
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	6.00000	0.232845
$665$	12.0000	0.465340
$666$	7.00000	0.271244
$667$	−18.0000	−0.696963
$668$	0	0
$669$	8.00000	0.309298
$670$	30.0000	1.15900
$671$	−42.0000	−1.62139
$672$	−2.00000	−0.0771517
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	−23.0000	−0.885927
$675$	4.00000	0.153960
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	3.00000	0.115214
$679$	28.0000	1.07454
$680$	9.00000	0.345134
$681$	18.0000	0.689761
$682$	24.0000	0.919007
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	2.00000	0.0764719
$685$	27.0000	1.03162
$686$	20.0000	0.763604
$687$	−22.0000	−0.839352
$688$	−10.0000	−0.381246
$689$	0	0
$690$	18.0000	0.685248
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	−6.00000	−0.228086
$693$	12.0000	0.455842
$694$	30.0000	1.13878
$695$	−12.0000	−0.455186
$696$	−3.00000	−0.113715
$697$	9.00000	0.340899
$698$	10.0000	0.378506
$699$	−6.00000	−0.226941
$700$	8.00000	0.302372
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	6.00000	0.226134
$705$	18.0000	0.677919
$706$	15.0000	0.564532
$707$	30.0000	1.12827
$708$	0	0
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	−18.0000	−0.675528
$711$	−4.00000	−0.150012
$712$	−18.0000	−0.674579
$713$	24.0000	0.898807
$714$	6.00000	0.224544
$715$	0	0
$716$	6.00000	0.224231
$717$	6.00000	0.224074
$718$	−6.00000	−0.223918
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	3.00000	0.111803
$721$	28.0000	1.04277
$722$	15.0000	0.558242
$723$	−1.00000	−0.0371904
$724$	−7.00000	−0.260153
$725$	12.0000	0.445669
$726$	−25.0000	−0.927837
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	39.0000	1.44345
$731$	30.0000	1.10959
$732$	−7.00000	−0.258727
$733$	−31.0000	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$734$	−2.00000	−0.0738213
$735$	−9.00000	−0.331970
$736$	6.00000	0.221163
$737$	−60.0000	−2.21013
$738$	3.00000	0.110432
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−21.0000	−0.771975
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	4.00000	0.146647
$745$	−27.0000	−0.989203
$746$	−29.0000	−1.06177
$747$	−6.00000	−0.219529
$748$	−18.0000	−0.658145
$749$	−12.0000	−0.438470
$750$	3.00000	0.109545
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	6.00000	0.218797
$753$	−12.0000	−0.437304
$754$	0	0
$755$	−30.0000	−1.09181
$756$	2.00000	0.0727393
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−20.0000	−0.726433
$759$	−36.0000	−1.30672
$760$	−6.00000	−0.217643
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	4.00000	0.144905
$763$	28.0000	1.01367
$764$	−12.0000	−0.434145
$765$	−9.00000	−0.325396
$766$	−24.0000	−0.867155
$767$	0	0
$768$	1.00000	0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	−36.0000	−1.29735
$771$	−3.00000	−0.108042
$772$	23.0000	0.827788
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	10.0000	0.359443
$775$	−16.0000	−0.574737
$776$	−14.0000	−0.502571
$777$	−14.0000	−0.502247
$778$	−39.0000	−1.39822
$779$	−6.00000	−0.214972
$780$	0	0
$781$	36.0000	1.28818
$782$	−18.0000	−0.643679
$783$	3.00000	0.107211
$784$	−3.00000	−0.107143
$785$	15.0000	0.535373
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	6.00000	0.213741
$789$	−6.00000	−0.213606
$790$	12.0000	0.426941
$791$	−6.00000	−0.213335
$792$	−6.00000	−0.213201
$793$	0	0
$794$	−14.0000	−0.496841
$795$	9.00000	0.319197
$796$	−10.0000	−0.354441
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	−4.00000	−0.141598
$799$	−18.0000	−0.636794
$800$	−4.00000	−0.141421
$801$	18.0000	0.635999
$802$	3.00000	0.105934
$803$	−78.0000	−2.75256
$804$	−10.0000	−0.352673
$805$	−36.0000	−1.26883
$806$	0	0
$807$	18.0000	0.633630
$808$	−15.0000	−0.527698
$809$	−51.0000	−1.79306	−0.896532	−	0.442978i	$-0.853922\pi$
$809$	−51.0000	−1.79306	−0.896532	+	0.442978i	$0.853922\pi$
$810$	−3.00000	−0.105409
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	6.00000	0.210559
$813$	−16.0000	−0.561144
$814$	42.0000	1.47210
$815$	−12.0000	−0.420342
$816$	−3.00000	−0.105021
$817$	−20.0000	−0.699711
$818$	1.00000	0.0349642
$819$	0	0
$820$	−9.00000	−0.314294
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	−9.00000	−0.313911
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−14.0000	−0.487713
$825$	24.0000	0.835573
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	−6.00000	−0.208514
$829$	17.0000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$829$	17.0000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$830$	18.0000	0.624789
$831$	17.0000	0.589723
$832$	0	0
$833$	9.00000	0.311832
$834$	4.00000	0.138509
$835$	0	0
$836$	12.0000	0.415029
$837$	−4.00000	−0.138260
$838$	−24.0000	−0.829066
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	−6.00000	−0.207020
$841$	−20.0000	−0.689655
$842$	−29.0000	−0.999406
$843$	9.00000	0.309976
$844$	−16.0000	−0.550743
$845$	0	0
$846$	−6.00000	−0.206284
$847$	50.0000	1.71802
$848$	3.00000	0.103020
$849$	14.0000	0.480479
$850$	12.0000	0.411597
$851$	42.0000	1.43974
$852$	6.00000	0.205557
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	14.0000	0.479070
$855$	6.00000	0.205196
$856$	6.00000	0.205076
$857$	21.0000	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	−30.0000	−1.02299
$861$	−6.00000	−0.204479
$862$	6.00000	0.204361
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	−1.00000	−0.0340207
$865$	−18.0000	−0.612018
$866$	13.0000	0.441758
$867$	−8.00000	−0.271694
$868$	−8.00000	−0.271538
$869$	−24.0000	−0.814144
$870$	−9.00000	−0.305129
$871$	0	0
$872$	−14.0000	−0.474100
$873$	14.0000	0.473828
$874$	12.0000	0.405906
$875$	−6.00000	−0.202837
$876$	−13.0000	−0.439229
$877$	41.0000	1.38447	0.692236	−	0.721671i	$-0.256626\pi$
$877$	41.0000	1.38447	0.692236	+	0.721671i	$0.256626\pi$
$878$	−14.0000	−0.472477
$879$	−21.0000	−0.708312
$880$	18.0000	0.606780
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	3.00000	0.101015
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	0	0
$886$	36.0000	1.20944
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	7.00000	0.234905
$889$	−8.00000	−0.268311
$890$	−54.0000	−1.81008
$891$	6.00000	0.201008
$892$	8.00000	0.267860
$893$	12.0000	0.401565
$894$	9.00000	0.301005
$895$	18.0000	0.601674
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−12.0000	−0.400222
$900$	4.00000	0.133333
$901$	−9.00000	−0.299833
$902$	18.0000	0.599334
$903$	−20.0000	−0.665558
$904$	3.00000	0.0997785
$905$	−21.0000	−0.698064
$906$	10.0000	0.332228
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	18.0000	0.597351
$909$	15.0000	0.497519
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	2.00000	0.0662266
$913$	−36.0000	−1.19143
$914$	−11.0000	−0.363848
$915$	−21.0000	−0.694239
$916$	−22.0000	−0.726900
$917$	0	0
$918$	3.00000	0.0990148
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	18.0000	0.593442
$921$	−10.0000	−0.329511
$922$	−15.0000	−0.493999
$923$	0	0
$924$	12.0000	0.394771
$925$	−28.0000	−0.920634
$926$	−38.0000	−1.24876
$927$	14.0000	0.459820
$928$	−3.00000	−0.0984798
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	12.0000	0.393496
$931$	−6.00000	−0.196642
$932$	−6.00000	−0.196537
$933$	−30.0000	−0.982156
$934$	18.0000	0.588978
$935$	−54.0000	−1.76599
$936$	0	0
$937$	47.0000	1.53542	0.767712	−	0.640796i	$-0.221395\pi$
$937$	47.0000	1.53542	0.767712	+	0.640796i	$0.221395\pi$
$938$	20.0000	0.653023
$939$	−10.0000	−0.326338
$940$	18.0000	0.587095
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	−5.00000	−0.162909
$943$	18.0000	0.586161
$944$	0	0
$945$	6.00000	0.195180
$946$	60.0000	1.95077
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	−8.00000	−0.259554
$951$	3.00000	0.0972817
$952$	6.00000	0.194461
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	−3.00000	−0.0971286
$955$	−36.0000	−1.16493
$956$	6.00000	0.194054
$957$	18.0000	0.581857
$958$	0	0
$959$	18.0000	0.581250
$960$	3.00000	0.0968246
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−6.00000	−0.193347
$964$	−1.00000	−0.0322078
$965$	69.0000	2.22119
$966$	12.0000	0.386094
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	−25.0000	−0.803530
$969$	−6.00000	−0.192748
$970$	−42.0000	−1.34854
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	1.00000	0.0320750
$973$	−8.00000	−0.256468
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−7.00000	−0.224065
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	4.00000	0.127906
$979$	108.000	3.45169
$980$	−9.00000	−0.287494
$981$	14.0000	0.446986
$982$	18.0000	0.574403
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	3.00000	0.0956365
$985$	18.0000	0.573528
$986$	9.00000	0.286618
$987$	12.0000	0.381964
$988$	0	0
$989$	60.0000	1.90789
$990$	−18.0000	−0.572078
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	4.00000	0.127000
$993$	−4.00000	−0.126936
$994$	−12.0000	−0.380617
$995$	−30.0000	−0.951064
$996$	−6.00000	−0.190117
$997$	5.00000	0.158352	0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000	0.158352	0.0791758	+	0.996861i	$0.474771\pi$
$998$	−32.0000	−1.01294
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1014.2.a.c.1.1		1
3.2	odd	2		3042.2.a.i.1.1		1
4.3	odd	2		8112.2.a.m.1.1		1
13.2	odd	12		1014.2.i.b.823.1		4
13.3	even	3		78.2.e.a.61.1	yes	2
13.4	even	6		1014.2.e.a.991.1		2
13.5	odd	4		1014.2.b.c.337.2		2
13.6	odd	12		1014.2.i.b.361.2		4
13.7	odd	12		1014.2.i.b.361.1		4
13.8	odd	4		1014.2.b.c.337.1		2
13.9	even	3		78.2.e.a.55.1	✓	2
13.10	even	6		1014.2.e.a.529.1		2
13.11	odd	12		1014.2.i.b.823.2		4
13.12	even	2		1014.2.a.f.1.1		1
39.5	even	4		3042.2.b.h.1351.1		2
39.8	even	4		3042.2.b.h.1351.2		2
39.29	odd	6		234.2.h.a.217.1		2
39.35	odd	6		234.2.h.a.55.1		2
39.38	odd	2		3042.2.a.h.1.1		1
52.3	odd	6		624.2.q.g.529.1		2
52.35	odd	6		624.2.q.g.289.1		2
52.51	odd	2		8112.2.a.c.1.1		1
65.3	odd	12		1950.2.z.g.1699.1		4
65.9	even	6		1950.2.i.m.601.1		2
65.22	odd	12		1950.2.z.g.1849.1		4
65.29	even	6		1950.2.i.m.451.1		2
65.42	odd	12		1950.2.z.g.1699.2		4
65.48	odd	12		1950.2.z.g.1849.2		4
156.35	even	6		1872.2.t.c.289.1		2
156.107	even	6		1872.2.t.c.1153.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.e.a.55.1	✓	2	13.9	even	3
78.2.e.a.61.1	yes	2	13.3	even	3
234.2.h.a.55.1		2	39.35	odd	6
234.2.h.a.217.1		2	39.29	odd	6
624.2.q.g.289.1		2	52.35	odd	6
624.2.q.g.529.1		2	52.3	odd	6
1014.2.a.c.1.1		1	1.1	even	1	trivial
1014.2.a.f.1.1		1	13.12	even	2
1014.2.b.c.337.1		2	13.8	odd	4
1014.2.b.c.337.2		2	13.5	odd	4
1014.2.e.a.529.1		2	13.10	even	6
1014.2.e.a.991.1		2	13.4	even	6
1014.2.i.b.361.1		4	13.7	odd	12
1014.2.i.b.361.2		4	13.6	odd	12
1014.2.i.b.823.1		4	13.2	odd	12
1014.2.i.b.823.2		4	13.11	odd	12
1872.2.t.c.289.1		2	156.35	even	6
1872.2.t.c.1153.1		2	156.107	even	6
1950.2.i.m.451.1		2	65.29	even	6
1950.2.i.m.601.1		2	65.9	even	6
1950.2.z.g.1699.1		4	65.3	odd	12
1950.2.z.g.1699.2		4	65.42	odd	12
1950.2.z.g.1849.1		4	65.22	odd	12
1950.2.z.g.1849.2		4	65.48	odd	12
3042.2.a.h.1.1		1	39.38	odd	2
3042.2.a.i.1.1		1	3.2	odd	2
3042.2.b.h.1351.1		2	39.5	even	4
3042.2.b.h.1351.2		2	39.8	even	4
8112.2.a.c.1.1		1	52.51	odd	2
8112.2.a.m.1.1		1	4.3	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:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1014.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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