LMFDB - Embedded newform 5070.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−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
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	1.00000	0.223607
$21$	3.00000	0.654654
$22$	3.00000	0.639602
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−1.00000	−0.182574
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	0	0
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	−3.00000	−0.486664
$39$	0	0
$40$	1.00000	0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	3.00000	0.462910
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	3.00000	0.452267
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	0	0
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−1.00000	−0.136083
$55$	3.00000	0.404520
$56$	−3.00000	−0.400892
$57$	3.00000	0.397360
$58$	−4.00000	−0.525226
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−1.00000	−0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−6.00000	−0.762001
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	−3.00000	−0.358569
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	1.00000	0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−9.00000	−1.04623
$75$	−1.00000	−0.115470
$76$	−3.00000	−0.344124
$77$	−9.00000	−1.02565
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−10.0000	−1.07833
$87$	4.00000	0.428845
$88$	3.00000	0.319801
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	6.00000	0.622171
$94$	3.00000	0.309426
$95$	−3.00000	−0.307794
$96$	−1.00000	−0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	2.00000	0.202031
$99$	3.00000	0.301511
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−15.0000	−1.47799	−0.738997	−	0.673709i	$-0.764700\pi$
$103$	−15.0000	−1.47799	−0.738997	+	0.673709i	$0.764700\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	9.00000	0.874157
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	3.00000	0.286039
$111$	9.00000	0.854242
$112$	−3.00000	−0.283473
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	3.00000	0.280976
$115$	−4.00000	−0.373002
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	−6.00000	−0.543214
$123$	−10.0000	−0.901670
$124$	−6.00000	−0.538816
$125$	1.00000	0.0894427
$126$	−3.00000	−0.267261
$127$	3.00000	0.266207	0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000	0.266207	0.133103	+	0.991102i	$0.457506\pi$
$128$	1.00000	0.0883883
$129$	10.0000	0.880451
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	−3.00000	−0.261116
$133$	9.00000	0.780399
$134$	8.00000	0.691095
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	4.00000	0.340503
$139$	−17.0000	−1.44192	−0.720961	−	0.692976i	$-0.756299\pi$
$139$	−17.0000	−1.44192	−0.720961	+	0.692976i	$0.756299\pi$
$140$	−3.00000	−0.253546
$141$	−3.00000	−0.252646
$142$	14.0000	1.17485
$143$	0	0
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	8.00000	0.662085
$147$	−2.00000	−0.164957
$148$	−9.00000	−0.739795
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−1.00000	−0.0816497
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	−9.00000	−0.725241
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	6.00000	0.477334
$159$	−9.00000	−0.713746
$160$	1.00000	0.0790569
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	10.0000	0.780869
$165$	−3.00000	−0.233550
$166$	−16.0000	−1.24184
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	3.00000	0.231455
$169$	0	0
$170$	0	0
$171$	−3.00000	−0.229416
$172$	−10.0000	−0.762493
$173$	13.0000	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	4.00000	0.303239
$175$	−3.00000	−0.226779
$176$	3.00000	0.226134
$177$	12.0000	0.901975
$178$	3.00000	0.224860
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	1.00000	0.0745356
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	−4.00000	−0.294884
$185$	−9.00000	−0.661693
$186$	6.00000	0.439941
$187$	0	0
$188$	3.00000	0.218797
$189$	3.00000	0.218218
$190$	−3.00000	−0.217643
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	2.00000	0.142857
$197$	−11.0000	−0.783718	−0.391859	−	0.920025i	$-0.628168\pi$
$197$	−11.0000	−0.783718	−0.391859	+	0.920025i	$0.628168\pi$
$198$	3.00000	0.213201
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	1.00000	0.0707107
$201$	−8.00000	−0.564276
$202$	0	0
$203$	12.0000	0.842235
$204$	0	0
$205$	10.0000	0.698430
$206$	−15.0000	−1.04510
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−9.00000	−0.622543
$210$	3.00000	0.207020
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	9.00000	0.618123
$213$	−14.0000	−0.959264
$214$	2.00000	0.136717
$215$	−10.0000	−0.681994
$216$	−1.00000	−0.0680414
$217$	18.0000	1.22192
$218$	2.00000	0.135457
$219$	−8.00000	−0.540590
$220$	3.00000	0.202260
$221$	0	0
$222$	9.00000	0.604040
$223$	11.0000	0.736614	0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000	0.736614	0.368307	+	0.929704i	$0.379937\pi$
$224$	−3.00000	−0.200446
$225$	1.00000	0.0666667
$226$	−8.00000	−0.532152
$227$	16.0000	1.06196	0.530979	−	0.847385i	$-0.321824\pi$
$227$	16.0000	1.06196	0.530979	+	0.847385i	$0.321824\pi$
$228$	3.00000	0.198680
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−4.00000	−0.263752
$231$	9.00000	0.592157
$232$	−4.00000	−0.262613
$233$	−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
$235$	3.00000	0.195698
$236$	−12.0000	−0.781133
$237$	−6.00000	−0.389742
$238$	0	0
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	−1.00000	−0.0645497
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	−2.00000	−0.128565
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	2.00000	0.127775
$246$	−10.0000	−0.637577
$247$	0	0
$248$	−6.00000	−0.381000
$249$	16.0000	1.01396
$250$	1.00000	0.0632456
$251$	−1.00000	−0.0631194	−0.0315597	−	0.999502i	$-0.510047\pi$
$251$	−1.00000	−0.0631194	−0.0315597	+	0.999502i	$0.510047\pi$
$252$	−3.00000	−0.188982
$253$	−12.0000	−0.754434
$254$	3.00000	0.188237
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	10.0000	0.622573
$259$	27.0000	1.67770
$260$	0	0
$261$	−4.00000	−0.247594
$262$	−3.00000	−0.185341
$263$	−31.0000	−1.91154	−0.955771	−	0.294112i	$-0.904976\pi$
$263$	−31.0000	−1.91154	−0.955771	+	0.294112i	$0.904976\pi$
$264$	−3.00000	−0.184637
$265$	9.00000	0.552866
$266$	9.00000	0.551825
$267$	−3.00000	−0.183597
$268$	8.00000	0.488678
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	−1.00000	−0.0608581
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	0	0
$274$	−12.0000	−0.724947
$275$	3.00000	0.180907
$276$	4.00000	0.240772
$277$	−31.0000	−1.86261	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−31.0000	−1.86261	−0.931305	+	0.364241i	$0.881328\pi$
$278$	−17.0000	−1.01959
$279$	−6.00000	−0.359211
$280$	−3.00000	−0.179284
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−3.00000	−0.178647
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	14.0000	0.830747
$285$	3.00000	0.177705
$286$	0	0
$287$	−30.0000	−1.77084
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	−4.00000	−0.234888
$291$	8.00000	0.468968
$292$	8.00000	0.468165
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	−2.00000	−0.116642
$295$	−12.0000	−0.698667
$296$	−9.00000	−0.523114
$297$	−3.00000	−0.174078
$298$	2.00000	0.115857
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	30.0000	1.72917
$302$	−14.0000	−0.805609
$303$	0	0
$304$	−3.00000	−0.172062
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	−9.00000	−0.512823
$309$	15.0000	0.853320
$310$	−6.00000	−0.340777
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−17.0000	−0.959366
$315$	−3.00000	−0.169031
$316$	6.00000	0.337526
$317$	−17.0000	−0.954815	−0.477408	−	0.878682i	$-0.658423\pi$
$317$	−17.0000	−0.954815	−0.477408	+	0.878682i	$0.658423\pi$
$318$	−9.00000	−0.504695
$319$	−12.0000	−0.671871
$320$	1.00000	0.0559017
$321$	−2.00000	−0.111629
$322$	12.0000	0.668734
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	−2.00000	−0.110600
$328$	10.0000	0.552158
$329$	−9.00000	−0.496186
$330$	−3.00000	−0.165145
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−16.0000	−0.878114
$333$	−9.00000	−0.493197
$334$	−9.00000	−0.492458
$335$	8.00000	0.437087
$336$	3.00000	0.163663
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	8.00000	0.434500
$340$	0	0
$341$	−18.0000	−0.974755
$342$	−3.00000	−0.162221
$343$	15.0000	0.809924
$344$	−10.0000	−0.539164
$345$	4.00000	0.215353
$346$	13.0000	0.698884
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	4.00000	0.214423
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	3.00000	0.159901
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	12.0000	0.637793
$355$	14.0000	0.743043
$356$	3.00000	0.159000
$357$	0	0
$358$	4.00000	0.211407
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	1.00000	0.0527046
$361$	−10.0000	−0.526316
$362$	−10.0000	−0.525588
$363$	2.00000	0.104973
$364$	0	0
$365$	8.00000	0.418739
$366$	6.00000	0.313625
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000	−0.208514
$369$	10.0000	0.520579
$370$	−9.00000	−0.467888
$371$	−27.0000	−1.40177
$372$	6.00000	0.311086
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	3.00000	0.154713
$377$	0	0
$378$	3.00000	0.154303
$379$	33.0000	1.69510	0.847548	−	0.530719i	$-0.178078\pi$
$379$	33.0000	1.69510	0.847548	+	0.530719i	$0.178078\pi$
$380$	−3.00000	−0.153897
$381$	−3.00000	−0.153695
$382$	−6.00000	−0.306987
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	−1.00000	−0.0510310
$385$	−9.00000	−0.458682
$386$	−8.00000	−0.407189
$387$	−10.0000	−0.508329
$388$	−8.00000	−0.406138
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	0	0
$392$	2.00000	0.101015
$393$	3.00000	0.151330
$394$	−11.0000	−0.554172
$395$	6.00000	0.301893
$396$	3.00000	0.150756
$397$	23.0000	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$398$	−10.0000	−0.501255
$399$	−9.00000	−0.450564
$400$	1.00000	0.0500000
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	−8.00000	−0.399004
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	12.0000	0.595550
$407$	−27.0000	−1.33834
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	10.0000	0.493865
$411$	12.0000	0.591916
$412$	−15.0000	−0.738997
$413$	36.0000	1.77144
$414$	−4.00000	−0.196589
$415$	−16.0000	−0.785409
$416$	0	0
$417$	17.0000	0.832494
$418$	−9.00000	−0.440204
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	3.00000	0.146385
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	9.00000	0.438113
$423$	3.00000	0.145865
$424$	9.00000	0.437079
$425$	0	0
$426$	−14.0000	−0.678302
$427$	18.0000	0.871081
$428$	2.00000	0.0966736
$429$	0	0
$430$	−10.0000	−0.482243
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	18.0000	0.864028
$435$	4.00000	0.191785
$436$	2.00000	0.0957826
$437$	12.0000	0.574038
$438$	−8.00000	−0.382255
$439$	−30.0000	−1.43182	−0.715911	−	0.698192i	$-0.753988\pi$
$439$	−30.0000	−1.43182	−0.715911	+	0.698192i	$0.753988\pi$
$440$	3.00000	0.143019
$441$	2.00000	0.0952381
$442$	0	0
$443$	−10.0000	−0.475114	−0.237557	−	0.971374i	$-0.576347\pi$
$443$	−10.0000	−0.475114	−0.237557	+	0.971374i	$0.576347\pi$
$444$	9.00000	0.427121
$445$	3.00000	0.142214
$446$	11.0000	0.520865
$447$	−2.00000	−0.0945968
$448$	−3.00000	−0.141737
$449$	35.0000	1.65175	0.825876	−	0.563852i	$-0.190681\pi$
$449$	35.0000	1.65175	0.825876	+	0.563852i	$0.190681\pi$
$450$	1.00000	0.0471405
$451$	30.0000	1.41264
$452$	−8.00000	−0.376288
$453$	14.0000	0.657777
$454$	16.0000	0.750917
$455$	0	0
$456$	3.00000	0.140488
$457$	42.0000	1.96468	0.982339	−	0.187112i	$-0.0599128\pi$
$457$	42.0000	1.96468	0.982339	+	0.187112i	$0.0599128\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	−4.00000	−0.186501
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	9.00000	0.418718
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−4.00000	−0.185695
$465$	6.00000	0.278243
$466$	−6.00000	−0.277945
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	3.00000	0.138380
$471$	17.0000	0.783319
$472$	−12.0000	−0.552345
$473$	−30.0000	−1.37940
$474$	−6.00000	−0.275589
$475$	−3.00000	−0.137649
$476$	0	0
$477$	9.00000	0.412082
$478$	26.0000	1.18921
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−7.00000	−0.318841
$483$	−12.0000	−0.546019
$484$	−2.00000	−0.0909091
$485$	−8.00000	−0.363261
$486$	−1.00000	−0.0453609
$487$	−29.0000	−1.31412	−0.657058	−	0.753840i	$-0.728199\pi$
$487$	−29.0000	−1.31412	−0.657058	+	0.753840i	$0.728199\pi$
$488$	−6.00000	−0.271607
$489$	20.0000	0.904431
$490$	2.00000	0.0903508
$491$	5.00000	0.225647	0.112823	−	0.993615i	$-0.464011\pi$
$491$	5.00000	0.225647	0.112823	+	0.993615i	$0.464011\pi$
$492$	−10.0000	−0.450835
$493$	0	0
$494$	0	0
$495$	3.00000	0.134840
$496$	−6.00000	−0.269408
$497$	−42.0000	−1.88396
$498$	16.0000	0.716977
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	1.00000	0.0447214
$501$	9.00000	0.402090
$502$	−1.00000	−0.0446322
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	3.00000	0.133103
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	1.00000	0.0441942
$513$	3.00000	0.132453
$514$	12.0000	0.529297
$515$	−15.0000	−0.660979
$516$	10.0000	0.440225
$517$	9.00000	0.395820
$518$	27.0000	1.18631
$519$	−13.0000	−0.570637
$520$	0	0
$521$	−35.0000	−1.53338	−0.766689	−	0.642019i	$-0.778097\pi$
$521$	−35.0000	−1.53338	−0.766689	+	0.642019i	$0.778097\pi$
$522$	−4.00000	−0.175075
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	−3.00000	−0.131056
$525$	3.00000	0.130931
$526$	−31.0000	−1.35166
$527$	0	0
$528$	−3.00000	−0.130558
$529$	−7.00000	−0.304348
$530$	9.00000	0.390935
$531$	−12.0000	−0.520756
$532$	9.00000	0.390199
$533$	0	0
$534$	−3.00000	−0.129823
$535$	2.00000	0.0864675
$536$	8.00000	0.345547
$537$	−4.00000	−0.172613
$538$	−4.00000	−0.172452
$539$	6.00000	0.258438
$540$	−1.00000	−0.0430331
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	12.0000	0.515444
$543$	10.0000	0.429141
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−12.0000	−0.512615
$549$	−6.00000	−0.256074
$550$	3.00000	0.127920
$551$	12.0000	0.511217
$552$	4.00000	0.170251
$553$	−18.0000	−0.765438
$554$	−31.0000	−1.31706
$555$	9.00000	0.382029
$556$	−17.0000	−0.720961
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	−6.00000	−0.254000
$559$	0	0
$560$	−3.00000	−0.126773
$561$	0	0
$562$	30.0000	1.26547
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	−3.00000	−0.126323
$565$	−8.00000	−0.336563
$566$	6.00000	0.252199
$567$	−3.00000	−0.125988
$568$	14.0000	0.587427
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	3.00000	0.125656
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	−30.0000	−1.25218
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−17.0000	−0.707107
$579$	8.00000	0.332469
$580$	−4.00000	−0.166091
$581$	48.0000	1.99138
$582$	8.00000	0.331611
$583$	27.0000	1.11823
$584$	8.00000	0.331042
$585$	0	0
$586$	1.00000	0.0413096
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	−2.00000	−0.0824786
$589$	18.0000	0.741677
$590$	−12.0000	−0.494032
$591$	11.0000	0.452480
$592$	−9.00000	−0.369898
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	2.00000	0.0819232
$597$	10.0000	0.409273
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−1.00000	−0.0408248
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	30.0000	1.22271
$603$	8.00000	0.325785
$604$	−14.0000	−0.569652
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	−3.00000	−0.121666
$609$	−12.0000	−0.486265
$610$	−6.00000	−0.242933
$611$	0	0
$612$	0	0
$613$	7.00000	0.282727	0.141364	−	0.989958i	$-0.454851\pi$
$613$	7.00000	0.282727	0.141364	+	0.989958i	$0.454851\pi$
$614$	−26.0000	−1.04927
$615$	−10.0000	−0.403239
$616$	−9.00000	−0.362620
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	15.0000	0.603388
$619$	23.0000	0.924448	0.462224	−	0.886763i	$-0.347052\pi$
$619$	23.0000	0.924448	0.462224	+	0.886763i	$0.347052\pi$
$620$	−6.00000	−0.240966
$621$	4.00000	0.160514
$622$	−4.00000	−0.160385
$623$	−9.00000	−0.360577
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	9.00000	0.359425
$628$	−17.0000	−0.678374
$629$	0	0
$630$	−3.00000	−0.119523
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	6.00000	0.238667
$633$	−9.00000	−0.357718
$634$	−17.0000	−0.675156
$635$	3.00000	0.119051
$636$	−9.00000	−0.356873
$637$	0	0
$638$	−12.0000	−0.475085
$639$	14.0000	0.553831
$640$	1.00000	0.0395285
$641$	35.0000	1.38242	0.691208	−	0.722655i	$-0.257079\pi$
$641$	35.0000	1.38242	0.691208	+	0.722655i	$0.257079\pi$
$642$	−2.00000	−0.0789337
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	12.0000	0.472866
$645$	10.0000	0.393750
$646$	0	0
$647$	−45.0000	−1.76913	−0.884566	−	0.466415i	$-0.845546\pi$
$647$	−45.0000	−1.76913	−0.884566	+	0.466415i	$0.845546\pi$
$648$	1.00000	0.0392837
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−18.0000	−0.705476
$652$	−20.0000	−0.783260
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	−2.00000	−0.0782062
$655$	−3.00000	−0.117220
$656$	10.0000	0.390434
$657$	8.00000	0.312110
$658$	−9.00000	−0.350857
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	−3.00000	−0.116775
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	−16.0000	−0.620920
$665$	9.00000	0.349005
$666$	−9.00000	−0.348743
$667$	16.0000	0.619522
$668$	−9.00000	−0.348220
$669$	−11.0000	−0.425285
$670$	8.00000	0.309067
$671$	−18.0000	−0.694882
$672$	3.00000	0.115728
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	6.00000	0.231111
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	8.00000	0.307238
$679$	24.0000	0.921035
$680$	0	0
$681$	−16.0000	−0.613121
$682$	−18.0000	−0.689256
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	−3.00000	−0.114708
$685$	−12.0000	−0.458496
$686$	15.0000	0.572703
$687$	−10.0000	−0.381524
$688$	−10.0000	−0.381246
$689$	0	0
$690$	4.00000	0.152277
$691$	7.00000	0.266293	0.133146	−	0.991096i	$-0.457492\pi$
$691$	7.00000	0.266293	0.133146	+	0.991096i	$0.457492\pi$
$692$	13.0000	0.494186
$693$	−9.00000	−0.341882
$694$	−24.0000	−0.911028
$695$	−17.0000	−0.644847
$696$	4.00000	0.151620
$697$	0	0
$698$	−16.0000	−0.605609
$699$	6.00000	0.226941
$700$	−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	$0.350270\pi$
$702$	0	0
$703$	27.0000	1.01832
$704$	3.00000	0.113067
$705$	−3.00000	−0.112987
$706$	8.00000	0.301084
$707$	0	0
$708$	12.0000	0.450988
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	14.0000	0.525411
$711$	6.00000	0.225018
$712$	3.00000	0.112430
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	−26.0000	−0.970988
$718$	−30.0000	−1.11959
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	1.00000	0.0372678
$721$	45.0000	1.67589
$722$	−10.0000	−0.372161
$723$	7.00000	0.260333
$724$	−10.0000	−0.371647
$725$	−4.00000	−0.148556
$726$	2.00000	0.0742270
$727$	−3.00000	−0.111264	−0.0556319	−	0.998451i	$-0.517717\pi$
$727$	−3.00000	−0.111264	−0.0556319	+	0.998451i	$0.517717\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	8.00000	0.296093
$731$	0	0
$732$	6.00000	0.221766
$733$	−53.0000	−1.95760	−0.978800	−	0.204819i	$-0.934339\pi$
$733$	−53.0000	−1.95760	−0.978800	+	0.204819i	$0.934339\pi$
$734$	8.00000	0.295285
$735$	−2.00000	−0.0737711
$736$	−4.00000	−0.147442
$737$	24.0000	0.884051
$738$	10.0000	0.368105
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	−9.00000	−0.330847
$741$	0	0
$742$	−27.0000	−0.991201
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	6.00000	0.219971
$745$	2.00000	0.0732743
$746$	−26.0000	−0.951928
$747$	−16.0000	−0.585409
$748$	0	0
$749$	−6.00000	−0.219235
$750$	−1.00000	−0.0365148
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	3.00000	0.109399
$753$	1.00000	0.0364420
$754$	0	0
$755$	−14.0000	−0.509512
$756$	3.00000	0.109109
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	33.0000	1.19861
$759$	12.0000	0.435572
$760$	−3.00000	−0.108821
$761$	31.0000	1.12375	0.561875	−	0.827222i	$-0.310080\pi$
$761$	31.0000	1.12375	0.561875	+	0.827222i	$0.310080\pi$
$762$	−3.00000	−0.108679
$763$	−6.00000	−0.217215
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−4.00000	−0.144526
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	−9.00000	−0.324337
$771$	−12.0000	−0.432169
$772$	−8.00000	−0.287926
$773$	7.00000	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$773$	7.00000	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$774$	−10.0000	−0.359443
$775$	−6.00000	−0.215526
$776$	−8.00000	−0.287183
$777$	−27.0000	−0.968620
$778$	24.0000	0.860442
$779$	−30.0000	−1.07486
$780$	0	0
$781$	42.0000	1.50288
$782$	0	0
$783$	4.00000	0.142948
$784$	2.00000	0.0714286
$785$	−17.0000	−0.606756
$786$	3.00000	0.107006
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	−11.0000	−0.391859
$789$	31.0000	1.10363
$790$	6.00000	0.213470
$791$	24.0000	0.853342
$792$	3.00000	0.106600
$793$	0	0
$794$	23.0000	0.816239
$795$	−9.00000	−0.319197
$796$	−10.0000	−0.354441
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−9.00000	−0.318597
$799$	0	0
$800$	1.00000	0.0353553
$801$	3.00000	0.106000
$802$	27.0000	0.953403
$803$	24.0000	0.846942
$804$	−8.00000	−0.282138
$805$	12.0000	0.422944
$806$	0	0
$807$	4.00000	0.140807
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	1.00000	0.0351364
$811$	31.0000	1.08856	0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000	1.08856	0.544279	+	0.838905i	$0.316803\pi$
$812$	12.0000	0.421117
$813$	−12.0000	−0.420858
$814$	−27.0000	−0.946350
$815$	−20.0000	−0.700569
$816$	0	0
$817$	30.0000	1.04957
$818$	7.00000	0.244749
$819$	0	0
$820$	10.0000	0.349215
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	12.0000	0.418548
$823$	−11.0000	−0.383436	−0.191718	−	0.981450i	$-0.561406\pi$
$823$	−11.0000	−0.383436	−0.191718	+	0.981450i	$0.561406\pi$
$824$	−15.0000	−0.522550
$825$	−3.00000	−0.104447
$826$	36.0000	1.25260
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	−4.00000	−0.139010
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	−16.0000	−0.555368
$831$	31.0000	1.07538
$832$	0	0
$833$	0	0
$834$	17.0000	0.588662
$835$	−9.00000	−0.311458
$836$	−9.00000	−0.311272
$837$	6.00000	0.207390
$838$	−12.0000	−0.414533
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	3.00000	0.103510
$841$	−13.0000	−0.448276
$842$	28.0000	0.964944
$843$	−30.0000	−1.03325
$844$	9.00000	0.309793
$845$	0	0
$846$	3.00000	0.103142
$847$	6.00000	0.206162
$848$	9.00000	0.309061
$849$	−6.00000	−0.205919
$850$	0	0
$851$	36.0000	1.23406
$852$	−14.0000	−0.479632
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	18.0000	0.615947
$855$	−3.00000	−0.102598
$856$	2.00000	0.0683586
$857$	−50.0000	−1.70797	−0.853984	−	0.520300i	$-0.825820\pi$
$857$	−50.0000	−1.70797	−0.853984	+	0.520300i	$0.825820\pi$
$858$	0	0
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	−10.0000	−0.340997
$861$	30.0000	1.02240
$862$	12.0000	0.408722
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−1.00000	−0.0340207
$865$	13.0000	0.442013
$866$	−16.0000	−0.543702
$867$	17.0000	0.577350
$868$	18.0000	0.610960
$869$	18.0000	0.610608
$870$	4.00000	0.135613
$871$	0	0
$872$	2.00000	0.0677285
$873$	−8.00000	−0.270759
$874$	12.0000	0.405906
$875$	−3.00000	−0.101419
$876$	−8.00000	−0.270295
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−30.0000	−1.01245
$879$	−1.00000	−0.0337292
$880$	3.00000	0.101130
$881$	−23.0000	−0.774890	−0.387445	−	0.921893i	$-0.626642\pi$
$881$	−23.0000	−0.774890	−0.387445	+	0.921893i	$0.626642\pi$
$882$	2.00000	0.0673435
$883$	22.0000	0.740359	0.370179	−	0.928960i	$-0.379296\pi$
$883$	22.0000	0.740359	0.370179	+	0.928960i	$0.379296\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	−10.0000	−0.335957
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	9.00000	0.302020
$889$	−9.00000	−0.301850
$890$	3.00000	0.100560
$891$	3.00000	0.100504
$892$	11.0000	0.368307
$893$	−9.00000	−0.301174
$894$	−2.00000	−0.0668900
$895$	4.00000	0.133705
$896$	−3.00000	−0.100223
$897$	0	0
$898$	35.0000	1.16797
$899$	24.0000	0.800445
$900$	1.00000	0.0333333
$901$	0	0
$902$	30.0000	0.998891
$903$	−30.0000	−0.998337
$904$	−8.00000	−0.266076
$905$	−10.0000	−0.332411
$906$	14.0000	0.465119
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	16.0000	0.530979
$909$	0	0
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	3.00000	0.0993399
$913$	−48.0000	−1.58857
$914$	42.0000	1.38924
$915$	6.00000	0.198354
$916$	10.0000	0.330409
$917$	9.00000	0.297206
$918$	0	0
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	−4.00000	−0.131876
$921$	26.0000	0.856729
$922$	32.0000	1.05386
$923$	0	0
$924$	9.00000	0.296078
$925$	−9.00000	−0.295918
$926$	24.0000	0.788689
$927$	−15.0000	−0.492665
$928$	−4.00000	−0.131306
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	6.00000	0.196748
$931$	−6.00000	−0.196642
$932$	−6.00000	−0.196537
$933$	4.00000	0.130954
$934$	8.00000	0.261768
$935$	0	0
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	−24.0000	−0.783628
$939$	−26.0000	−0.848478
$940$	3.00000	0.0978492
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	17.0000	0.553890
$943$	−40.0000	−1.30258
$944$	−12.0000	−0.390567
$945$	3.00000	0.0975900
$946$	−30.0000	−0.975384
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	−6.00000	−0.194871
$949$	0	0
$950$	−3.00000	−0.0973329
$951$	17.0000	0.551263
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	9.00000	0.291386
$955$	−6.00000	−0.194155
$956$	26.0000	0.840900
$957$	12.0000	0.387905
$958$	−12.0000	−0.387702
$959$	36.0000	1.16250
$960$	−1.00000	−0.0322749
$961$	5.00000	0.161290
$962$	0	0
$963$	2.00000	0.0644491
$964$	−7.00000	−0.225455
$965$	−8.00000	−0.257529
$966$	−12.0000	−0.386094
$967$	−25.0000	−0.803946	−0.401973	−	0.915652i	$-0.631675\pi$
$967$	−25.0000	−0.803946	−0.401973	+	0.915652i	$0.631675\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	−8.00000	−0.256865
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	−1.00000	−0.0320750
$973$	51.0000	1.63498
$974$	−29.0000	−0.929220
$975$	0	0
$976$	−6.00000	−0.192055
$977$	44.0000	1.40768	0.703842	−	0.710356i	$-0.251466\pi$
$977$	44.0000	1.40768	0.703842	+	0.710356i	$0.251466\pi$
$978$	20.0000	0.639529
$979$	9.00000	0.287641
$980$	2.00000	0.0638877
$981$	2.00000	0.0638551
$982$	5.00000	0.159556
$983$	−55.0000	−1.75423	−0.877114	−	0.480283i	$-0.840534\pi$
$983$	−55.0000	−1.75423	−0.877114	+	0.480283i	$0.840534\pi$
$984$	−10.0000	−0.318788
$985$	−11.0000	−0.350489
$986$	0	0
$987$	9.00000	0.286473
$988$	0	0
$989$	40.0000	1.27193
$990$	3.00000	0.0953463
$991$	58.0000	1.84243	0.921215	−	0.389053i	$-0.127198\pi$
$991$	58.0000	1.84243	0.921215	+	0.389053i	$0.127198\pi$
$992$	−6.00000	−0.190500
$993$	−28.0000	−0.888553
$994$	−42.0000	−1.33216
$995$	−10.0000	−0.317021
$996$	16.0000	0.506979
$997$	47.0000	1.48850	0.744252	−	0.667898i	$-0.232806\pi$
$997$	47.0000	1.48850	0.744252	+	0.667898i	$0.232806\pi$
$998$	−20.0000	−0.633089
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.r.1.1		1
13.4	even	6		390.2.i.e.211.1	yes	2
13.5	odd	4		5070.2.b.b.1351.1		2
13.8	odd	4		5070.2.b.b.1351.2		2
13.10	even	6		390.2.i.e.61.1	✓	2
13.12	even	2		5070.2.a.b.1.1		1
39.17	odd	6		1170.2.i.c.991.1		2
39.23	odd	6		1170.2.i.c.451.1		2
65.4	even	6		1950.2.i.f.601.1		2
65.17	odd	12		1950.2.z.d.1849.1		4
65.23	odd	12		1950.2.z.d.1699.1		4
65.43	odd	12		1950.2.z.d.1849.2		4
65.49	even	6		1950.2.i.f.451.1		2
65.62	odd	12		1950.2.z.d.1699.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.e.61.1	✓	2	13.10	even	6
390.2.i.e.211.1	yes	2	13.4	even	6
1170.2.i.c.451.1		2	39.23	odd	6
1170.2.i.c.991.1		2	39.17	odd	6
1950.2.i.f.451.1		2	65.49	even	6
1950.2.i.f.601.1		2	65.4	even	6
1950.2.z.d.1699.1		4	65.23	odd	12
1950.2.z.d.1699.2		4	65.62	odd	12
1950.2.z.d.1849.1		4	65.17	odd	12
1950.2.z.d.1849.2		4	65.43	odd	12
5070.2.a.b.1.1		1	13.12	even	2
5070.2.a.r.1.1		1	1.1	even	1	trivial
5070.2.b.b.1351.1		2	13.5	odd	4
5070.2.b.b.1351.2		2	13.8	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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