LMFDB - Embedded newform 6270.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6270,2,Mod(1,6270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6270, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6270.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6270 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6270.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:	$50.0662020673$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6270.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−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
$7$	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
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−1.00000	−0.213201
$23$	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
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	−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
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	2.00000	0.342997
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	1.00000	0.162221
$39$	2.00000	0.320256
$40$	−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	$0.450084\pi$
$42$	2.00000	0.308607
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	−2.00000	−0.267261
$57$	1.00000	0.132453
$58$	4.00000	0.525226
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	−1.00000	−0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000	1.01600
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	1.00000	0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−2.00000	−0.242536
$69$	−6.00000	−0.722315
$70$	−2.00000	−0.239046
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	−1.00000	−0.114708
$77$	2.00000	0.227921
$78$	−2.00000	−0.226455
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	−2.00000	−0.218218
$85$	−2.00000	−0.216930
$86$	0	0
$87$	4.00000	0.428845
$88$	−1.00000	−0.106600
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	−1.00000	−0.105409
$91$	−4.00000	−0.419314
$92$	6.00000	0.625543
$93$	8.00000	0.829561
$94$	6.00000	0.618853
$95$	−1.00000	−0.102598
$96$	1.00000	0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	3.00000	0.303046
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	−2.00000	−0.198030
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	2.00000	0.196116
$105$	−2.00000	−0.195180
$106$	10.0000	0.971286
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−1.00000	−0.0953463
$111$	−2.00000	−0.189832
$112$	2.00000	0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−1.00000	−0.0936586
$115$	6.00000	0.559503
$116$	−4.00000	−0.371391
$117$	−2.00000	−0.184900
$118$	−8.00000	−0.736460
$119$	−4.00000	−0.366679
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	−2.00000	−0.180334
$124$	−8.00000	−0.718421
$125$	1.00000	0.0894427
$126$	−2.00000	−0.178174
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−1.00000	−0.0870388
$133$	−2.00000	−0.173422
$134$	4.00000	0.345547
$135$	−1.00000	−0.0860663
$136$	2.00000	0.171499
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	6.00000	0.510754
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	2.00000	0.169031
$141$	6.00000	0.505291
$142$	10.0000	0.839181
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−4.00000	−0.331042
$147$	3.00000	0.247436
$148$	2.00000	0.164399
$149$	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
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	1.00000	0.0811107
$153$	−2.00000	−0.161690
$154$	−2.00000	−0.161165
$155$	−8.00000	−0.642575
$156$	2.00000	0.160128
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	6.00000	0.477334
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	12.0000	0.945732
$162$	−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
$165$	−1.00000	−0.0778499
$166$	14.0000	1.08661
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	2.00000	0.154303
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−4.00000	−0.303239
$175$	2.00000	0.151186
$176$	1.00000	0.0753778
$177$	−8.00000	−0.601317
$178$	−12.0000	−0.899438
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	1.00000	0.0745356
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	4.00000	0.296500
$183$	−2.00000	−0.147844
$184$	−6.00000	−0.442326
$185$	2.00000	0.147043
$186$	−8.00000	−0.586588
$187$	−2.00000	−0.146254
$188$	−6.00000	−0.437595
$189$	−2.00000	−0.145479
$190$	1.00000	0.0725476
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	14.0000	1.00514
$195$	2.00000	0.143223
$196$	−3.00000	−0.214286
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	−1.00000	−0.0710669
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	−18.0000	−1.26648
$203$	−8.00000	−0.561490
$204$	2.00000	0.140028
$205$	2.00000	0.139686
$206$	16.0000	1.11477
$207$	6.00000	0.417029
$208$	−2.00000	−0.138675
$209$	−1.00000	−0.0691714
$210$	2.00000	0.138013
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−10.0000	−0.686803
$213$	10.0000	0.685189
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	−16.0000	−1.08615
$218$	10.0000	0.677285
$219$	−4.00000	−0.270295
$220$	1.00000	0.0674200
$221$	4.00000	0.269069
$222$	2.00000	0.134231
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	1.00000	0.0662266
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−6.00000	−0.395628
$231$	−2.00000	−0.131590
$232$	4.00000	0.262613
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	2.00000	0.130744
$235$	−6.00000	−0.391397
$236$	8.00000	0.520756
$237$	6.00000	0.389742
$238$	4.00000	0.259281
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	−1.00000	−0.0645497
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	−3.00000	−0.191663
$246$	2.00000	0.127515
$247$	2.00000	0.127257
$248$	8.00000	0.508001
$249$	14.0000	0.887214
$250$	−1.00000	−0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.00000	0.125988
$253$	6.00000	0.377217
$254$	0	0
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	−2.00000	−0.124035
$261$	−4.00000	−0.247594
$262$	0	0
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	1.00000	0.0615457
$265$	−10.0000	−0.614295
$266$	2.00000	0.122628
$267$	−12.0000	−0.734388
$268$	−4.00000	−0.244339
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	1.00000	0.0608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−2.00000	−0.121268
$273$	4.00000	0.242091
$274$	0	0
$275$	1.00000	0.0603023
$276$	−6.00000	−0.361158
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	12.0000	0.719712
$279$	−8.00000	−0.478947
$280$	−2.00000	−0.119523
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−6.00000	−0.357295
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−10.0000	−0.593391
$285$	1.00000	0.0592349
$286$	2.00000	0.118262
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	4.00000	0.234888
$291$	14.0000	0.820695
$292$	4.00000	0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−3.00000	−0.174964
$295$	8.00000	0.465778
$296$	−2.00000	−0.116248
$297$	−1.00000	−0.0580259
$298$	−2.00000	−0.115857
$299$	−12.0000	−0.693978
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−22.0000	−1.26596
$303$	−18.0000	−1.03407
$304$	−1.00000	−0.0573539
$305$	2.00000	0.114520
$306$	2.00000	0.114332
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	2.00000	0.113961
$309$	16.0000	0.910208
$310$	8.00000	0.454369
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	−2.00000	−0.113228
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	−4.00000	−0.225733
$315$	2.00000	0.112687
$316$	−6.00000	−0.337526
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	−10.0000	−0.560772
$319$	−4.00000	−0.223957
$320$	1.00000	0.0559017
$321$	4.00000	0.223258
$322$	−12.0000	−0.668734
$323$	2.00000	0.111283
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	18.0000	0.996928
$327$	10.0000	0.553001
$328$	−2.00000	−0.110432
$329$	−12.0000	−0.661581
$330$	1.00000	0.0550482
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	−14.0000	−0.768350
$333$	2.00000	0.109599
$334$	0	0
$335$	−4.00000	−0.218543
$336$	−2.00000	−0.109109
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	9.00000	0.489535
$339$	6.00000	0.325875
$340$	−2.00000	−0.108465
$341$	−8.00000	−0.433224
$342$	1.00000	0.0540738
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−6.00000	−0.323029
$346$	−14.0000	−0.752645
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	4.00000	0.214423
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−2.00000	−0.106904
$351$	2.00000	0.106752
$352$	−1.00000	−0.0533002
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	8.00000	0.425195
$355$	−10.0000	−0.530745
$356$	12.0000	0.635999
$357$	4.00000	0.211702
$358$	−24.0000	−1.26844
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−1.00000	−0.0527046
$361$	1.00000	0.0526316
$362$	12.0000	0.630706
$363$	−1.00000	−0.0524864
$364$	−4.00000	−0.209657
$365$	4.00000	0.209370
$366$	2.00000	0.104542
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	6.00000	0.312772
$369$	2.00000	0.104116
$370$	−2.00000	−0.103975
$371$	−20.0000	−1.03835
$372$	8.00000	0.414781
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	2.00000	0.103418
$375$	−1.00000	−0.0516398
$376$	6.00000	0.309426
$377$	8.00000	0.412021
$378$	2.00000	0.102869
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	−1.00000	−0.0512989
$381$	0	0
$382$	4.00000	0.204658
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	2.00000	0.101929
$386$	14.0000	0.712581
$387$	0	0
$388$	−14.0000	−0.710742
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	−2.00000	−0.101274
$391$	−12.0000	−0.606866
$392$	3.00000	0.151523
$393$	0	0
$394$	−8.00000	−0.403034
$395$	−6.00000	−0.301893
$396$	1.00000	0.0502519
$397$	−24.0000	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$397$	−24.0000	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	1.00000	0.0500000
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	−4.00000	−0.199502
$403$	16.0000	0.797017
$404$	18.0000	0.895533
$405$	1.00000	0.0496904
$406$	8.00000	0.397033
$407$	2.00000	0.0991363
$408$	−2.00000	−0.0990148
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	−16.0000	−0.788263
$413$	16.0000	0.787309
$414$	−6.00000	−0.294884
$415$	−14.0000	−0.687233
$416$	2.00000	0.0980581
$417$	12.0000	0.587643
$418$	1.00000	0.0489116
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	−2.00000	−0.0975900
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	−12.0000	−0.584151
$423$	−6.00000	−0.291730
$424$	10.0000	0.485643
$425$	−2.00000	−0.0970143
$426$	−10.0000	−0.484502
$427$	4.00000	0.193574
$428$	−4.00000	−0.193347
$429$	2.00000	0.0965609
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−1.00000	−0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	16.0000	0.768025
$435$	4.00000	0.191785
$436$	−10.0000	−0.478913
$437$	−6.00000	−0.287019
$438$	4.00000	0.191127
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	−1.00000	−0.0476731
$441$	−3.00000	−0.142857
$442$	−4.00000	−0.190261
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−2.00000	−0.0949158
$445$	12.0000	0.568855
$446$	−16.0000	−0.757622
$447$	−2.00000	−0.0945968
$448$	2.00000	0.0944911
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	−1.00000	−0.0471405
$451$	2.00000	0.0941763
$452$	−6.00000	−0.282216
$453$	−22.0000	−1.03365
$454$	8.00000	0.375459
$455$	−4.00000	−0.187523
$456$	−1.00000	−0.0468293
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	10.0000	0.467269
$459$	2.00000	0.0933520
$460$	6.00000	0.279751
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	2.00000	0.0930484
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−4.00000	−0.185695
$465$	8.00000	0.370991
$466$	2.00000	0.0926482
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	−2.00000	−0.0924500
$469$	−8.00000	−0.369406
$470$	6.00000	0.276759
$471$	−4.00000	−0.184310
$472$	−8.00000	−0.368230
$473$	0	0
$474$	−6.00000	−0.275589
$475$	−1.00000	−0.0458831
$476$	−4.00000	−0.183340
$477$	−10.0000	−0.457869
$478$	−24.0000	−1.09773
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	1.00000	0.0456435
$481$	−4.00000	−0.182384
$482$	−8.00000	−0.364390
$483$	−12.0000	−0.546019
$484$	1.00000	0.0454545
$485$	−14.0000	−0.635707
$486$	1.00000	0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−2.00000	−0.0905357
$489$	18.0000	0.813988
$490$	3.00000	0.135526
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−2.00000	−0.0901670
$493$	8.00000	0.360302
$494$	−2.00000	−0.0899843
$495$	1.00000	0.0449467
$496$	−8.00000	−0.359211
$497$	−20.0000	−0.897123
$498$	−14.0000	−0.627355
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−2.00000	−0.0890871
$505$	18.0000	0.800989
$506$	−6.00000	−0.266733
$507$	9.00000	0.399704
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	−2.00000	−0.0885615
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	−6.00000	−0.264649
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−6.00000	−0.263880
$518$	−4.00000	−0.175750
$519$	−14.0000	−0.614532
$520$	2.00000	0.0877058
$521$	−40.0000	−1.75243	−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000	−1.75243	−0.876216	+	0.481919i	$0.839940\pi$
$522$	4.00000	0.175075
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	4.00000	0.174408
$527$	16.0000	0.696971
$528$	−1.00000	−0.0435194
$529$	13.0000	0.565217
$530$	10.0000	0.434372
$531$	8.00000	0.347170
$532$	−2.00000	−0.0867110
$533$	−4.00000	−0.173259
$534$	12.0000	0.519291
$535$	−4.00000	−0.172935
$536$	4.00000	0.172774
$537$	−24.0000	−1.03568
$538$	10.0000	0.431131
$539$	−3.00000	−0.129219
$540$	−1.00000	−0.0430331
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−20.0000	−0.859074
$543$	12.0000	0.514969
$544$	2.00000	0.0857493
$545$	−10.0000	−0.428353
$546$	−4.00000	−0.171184
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	−1.00000	−0.0426401
$551$	4.00000	0.170406
$552$	6.00000	0.255377
$553$	−12.0000	−0.510292
$554$	−2.00000	−0.0849719
$555$	−2.00000	−0.0848953
$556$	−12.0000	−0.508913
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	2.00000	0.0845154
$561$	2.00000	0.0844401
$562$	18.0000	0.759284
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	6.00000	0.252646
$565$	−6.00000	−0.252422
$566$	4.00000	0.168133
$567$	2.00000	0.0839921
$568$	10.0000	0.419591
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	−1.00000	−0.0418854
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−2.00000	−0.0836242
$573$	4.00000	0.167102
$574$	−4.00000	−0.166957
$575$	6.00000	0.250217
$576$	1.00000	0.0416667
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	13.0000	0.540729
$579$	14.0000	0.581820
$580$	−4.00000	−0.166091
$581$	−28.0000	−1.16164
$582$	−14.0000	−0.580319
$583$	−10.0000	−0.414158
$584$	−4.00000	−0.165521
$585$	−2.00000	−0.0826898
$586$	−6.00000	−0.247858
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	3.00000	0.123718
$589$	8.00000	0.329634
$590$	−8.00000	−0.329355
$591$	−8.00000	−0.329076
$592$	2.00000	0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	1.00000	0.0410305
$595$	−4.00000	−0.163984
$596$	2.00000	0.0819232
$597$	0	0
$598$	12.0000	0.490716
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	1.00000	0.0408248
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	22.0000	0.895167
$605$	1.00000	0.0406558
$606$	18.0000	0.731200
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	1.00000	0.0405554
$609$	8.00000	0.324176
$610$	−2.00000	−0.0809776
$611$	12.0000	0.485468
$612$	−2.00000	−0.0808452
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	−12.0000	−0.484281
$615$	−2.00000	−0.0806478
$616$	−2.00000	−0.0805823
$617$	48.0000	1.93241	0.966204	−	0.257780i	$-0.0829910\pi$
$617$	48.0000	1.93241	0.966204	+	0.257780i	$0.0829910\pi$
$618$	−16.0000	−0.643614
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	−8.00000	−0.321288
$621$	−6.00000	−0.240772
$622$	4.00000	0.160385
$623$	24.0000	0.961540
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	34.0000	1.35891
$627$	1.00000	0.0399362
$628$	4.00000	0.159617
$629$	−4.00000	−0.159490
$630$	−2.00000	−0.0796819
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	6.00000	0.238667
$633$	−12.0000	−0.476957
$634$	30.0000	1.19145
$635$	0	0
$636$	10.0000	0.396526
$637$	6.00000	0.237729
$638$	4.00000	0.158362
$639$	−10.0000	−0.395594
$640$	−1.00000	−0.0395285
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	−4.00000	−0.157867
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	22.0000	0.864909	0.432455	−	0.901656i	$-0.357648\pi$
$647$	22.0000	0.864909	0.432455	+	0.901656i	$0.357648\pi$
$648$	−1.00000	−0.0392837
$649$	8.00000	0.314027
$650$	2.00000	0.0784465
$651$	16.0000	0.627089
$652$	−18.0000	−0.704934
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	2.00000	0.0780869
$657$	4.00000	0.156055
$658$	12.0000	0.467809
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	−1.00000	−0.0389249
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	−18.0000	−0.699590
$663$	−4.00000	−0.155347
$664$	14.0000	0.543305
$665$	−2.00000	−0.0775567
$666$	−2.00000	−0.0774984
$667$	−24.0000	−0.929284
$668$	0	0
$669$	−16.0000	−0.618596
$670$	4.00000	0.154533
$671$	2.00000	0.0772091
$672$	2.00000	0.0771517
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−2.00000	−0.0770371
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	−6.00000	−0.230429
$679$	−28.0000	−1.07454
$680$	2.00000	0.0766965
$681$	8.00000	0.306561
$682$	8.00000	0.306336
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	20.0000	0.763604
$687$	10.0000	0.381524
$688$	0	0
$689$	20.0000	0.761939
$690$	6.00000	0.228416
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	14.0000	0.532200
$693$	2.00000	0.0759737
$694$	10.0000	0.379595
$695$	−12.0000	−0.455186
$696$	−4.00000	−0.151620
$697$	−4.00000	−0.151511
$698$	2.00000	0.0757011
$699$	2.00000	0.0756469
$700$	2.00000	0.0755929
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−2.00000	−0.0754851
$703$	−2.00000	−0.0754314
$704$	1.00000	0.0376889
$705$	6.00000	0.225973
$706$	−4.00000	−0.150542
$707$	36.0000	1.35392
$708$	−8.00000	−0.300658
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	10.0000	0.375293
$711$	−6.00000	−0.225018
$712$	−12.0000	−0.449719
$713$	−48.0000	−1.79761
$714$	−4.00000	−0.149696
$715$	−2.00000	−0.0747958
$716$	24.0000	0.896922
$717$	−24.0000	−0.896296
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	1.00000	0.0372678
$721$	−32.0000	−1.19174
$722$	−1.00000	−0.0372161
$723$	−8.00000	−0.297523
$724$	−12.0000	−0.445976
$725$	−4.00000	−0.148556
$726$	1.00000	0.0371135
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	4.00000	0.148250
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	0	0
$732$	−2.00000	−0.0739221
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	−12.0000	−0.442928
$735$	3.00000	0.110657
$736$	−6.00000	−0.221163
$737$	−4.00000	−0.147342
$738$	−2.00000	−0.0736210
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	2.00000	0.0735215
$741$	−2.00000	−0.0734718
$742$	20.0000	0.734223
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	−8.00000	−0.293294
$745$	2.00000	0.0732743
$746$	−2.00000	−0.0732252
$747$	−14.0000	−0.512233
$748$	−2.00000	−0.0731272
$749$	−8.00000	−0.292314
$750$	1.00000	0.0365148
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−6.00000	−0.218797
$753$	−12.0000	−0.437304
$754$	−8.00000	−0.291343
$755$	22.0000	0.800662
$756$	−2.00000	−0.0727393
$757$	−24.0000	−0.872295	−0.436147	−	0.899875i	$-0.643657\pi$
$757$	−24.0000	−0.872295	−0.436147	+	0.899875i	$0.643657\pi$
$758$	2.00000	0.0726433
$759$	−6.00000	−0.217786
$760$	1.00000	0.0362738
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	−4.00000	−0.144715
$765$	−2.00000	−0.0723102
$766$	−12.0000	−0.433578
$767$	−16.0000	−0.577727
$768$	−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	$0.511481\pi$
$770$	−2.00000	−0.0720750
$771$	−6.00000	−0.216085
$772$	−14.0000	−0.503871
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	14.0000	0.502571
$777$	−4.00000	−0.143499
$778$	14.0000	0.501924
$779$	−2.00000	−0.0716574
$780$	2.00000	0.0716115
$781$	−10.0000	−0.357828
$782$	12.0000	0.429119
$783$	4.00000	0.142948
$784$	−3.00000	−0.107143
$785$	4.00000	0.142766
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	8.00000	0.284988
$789$	4.00000	0.142404
$790$	6.00000	0.213470
$791$	−12.0000	−0.426671
$792$	−1.00000	−0.0355335
$793$	−4.00000	−0.142044
$794$	24.0000	0.851728
$795$	10.0000	0.354663
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	−2.00000	−0.0707992
$799$	12.0000	0.424529
$800$	−1.00000	−0.0353553
$801$	12.0000	0.423999
$802$	8.00000	0.282490
$803$	4.00000	0.141157
$804$	4.00000	0.141069
$805$	12.0000	0.422944
$806$	−16.0000	−0.563576
$807$	10.0000	0.352017
$808$	−18.0000	−0.633238
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	−1.00000	−0.0351364
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	−8.00000	−0.280745
$813$	−20.0000	−0.701431
$814$	−2.00000	−0.0701000
$815$	−18.0000	−0.630512
$816$	2.00000	0.0700140
$817$	0	0
$818$	20.0000	0.699284
$819$	−4.00000	−0.139771
$820$	2.00000	0.0698430
$821$	14.0000	0.488603	0.244302	−	0.969699i	$-0.421441\pi$
$821$	14.0000	0.488603	0.244302	+	0.969699i	$0.421441\pi$
$822$	0	0
$823$	−36.0000	−1.25488	−0.627441	−	0.778664i	$-0.715897\pi$
$823$	−36.0000	−1.25488	−0.627441	+	0.778664i	$0.715897\pi$
$824$	16.0000	0.557386
$825$	−1.00000	−0.0348155
$826$	−16.0000	−0.556711
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	6.00000	0.208514
$829$	−32.0000	−1.11141	−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000	−1.11141	−0.555703	+	0.831381i	$0.687551\pi$
$830$	14.0000	0.485947
$831$	−2.00000	−0.0693792
$832$	−2.00000	−0.0693375
$833$	6.00000	0.207888
$834$	−12.0000	−0.415526
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	8.00000	0.276520
$838$	−36.0000	−1.24360
$839$	38.0000	1.31191	0.655953	−	0.754802i	$-0.272267\pi$
$839$	38.0000	1.31191	0.655953	+	0.754802i	$0.272267\pi$
$840$	2.00000	0.0690066
$841$	−13.0000	−0.448276
$842$	8.00000	0.275698
$843$	18.0000	0.619953
$844$	12.0000	0.413057
$845$	−9.00000	−0.309609
$846$	6.00000	0.206284
$847$	2.00000	0.0687208
$848$	−10.0000	−0.343401
$849$	4.00000	0.137280
$850$	2.00000	0.0685994
$851$	12.0000	0.411355
$852$	10.0000	0.342594
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	−4.00000	−0.136877
$855$	−1.00000	−0.0341993
$856$	4.00000	0.136717
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	−2.00000	−0.0682789
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	12.0000	0.408722
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	1.00000	0.0340207
$865$	14.0000	0.476014
$866$	26.0000	0.883516
$867$	13.0000	0.441503
$868$	−16.0000	−0.543075
$869$	−6.00000	−0.203536
$870$	−4.00000	−0.135613
$871$	8.00000	0.271070
$872$	10.0000	0.338643
$873$	−14.0000	−0.473828
$874$	6.00000	0.202953
$875$	2.00000	0.0676123
$876$	−4.00000	−0.135147
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	10.0000	0.337484
$879$	−6.00000	−0.202375
$880$	1.00000	0.0337100
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	3.00000	0.101015
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	4.00000	0.134535
$885$	−8.00000	−0.268917
$886$	28.0000	0.940678
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	−12.0000	−0.402241
$891$	1.00000	0.0335013
$892$	16.0000	0.535720
$893$	6.00000	0.200782
$894$	2.00000	0.0668900
$895$	24.0000	0.802232
$896$	−2.00000	−0.0668153
$897$	12.0000	0.400668
$898$	8.00000	0.266963
$899$	32.0000	1.06726
$900$	1.00000	0.0333333
$901$	20.0000	0.666297
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	6.00000	0.199557
$905$	−12.0000	−0.398893
$906$	22.0000	0.730901
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	−8.00000	−0.265489
$909$	18.0000	0.597022
$910$	4.00000	0.132599
$911$	−10.0000	−0.331315	−0.165657	−	0.986183i	$-0.552975\pi$
$911$	−10.0000	−0.331315	−0.165657	+	0.986183i	$0.552975\pi$
$912$	1.00000	0.0331133
$913$	−14.0000	−0.463332
$914$	32.0000	1.05847
$915$	−2.00000	−0.0661180
$916$	−10.0000	−0.330409
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	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
$921$	−12.0000	−0.395413
$922$	−26.0000	−0.856264
$923$	20.0000	0.658308
$924$	−2.00000	−0.0657952
$925$	2.00000	0.0657596
$926$	40.0000	1.31448
$927$	−16.0000	−0.525509
$928$	4.00000	0.131306
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	−8.00000	−0.262330
$931$	3.00000	0.0983210
$932$	−2.00000	−0.0655122
$933$	4.00000	0.130954
$934$	4.00000	0.130884
$935$	−2.00000	−0.0654070
$936$	2.00000	0.0653720
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	8.00000	0.261209
$939$	34.0000	1.10955
$940$	−6.00000	−0.195698
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	4.00000	0.130327
$943$	12.0000	0.390774
$944$	8.00000	0.260378
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	6.00000	0.194871
$949$	−8.00000	−0.259691
$950$	1.00000	0.0324443
$951$	30.0000	0.972817
$952$	4.00000	0.129641
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	10.0000	0.323762
$955$	−4.00000	−0.129437
$956$	24.0000	0.776215
$957$	4.00000	0.129302
$958$	32.0000	1.03387
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	4.00000	0.128965
$963$	−4.00000	−0.128898
$964$	8.00000	0.257663
$965$	−14.0000	−0.450676
$966$	12.0000	0.386094
$967$	−42.0000	−1.35063	−0.675314	−	0.737530i	$-0.735992\pi$
$967$	−42.0000	−1.35063	−0.675314	+	0.737530i	$0.735992\pi$
$968$	−1.00000	−0.0321412
$969$	−2.00000	−0.0642493
$970$	14.0000	0.449513
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	−1.00000	−0.0320750
$973$	−24.0000	−0.769405
$974$	8.00000	0.256337
$975$	2.00000	0.0640513
$976$	2.00000	0.0640184
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−18.0000	−0.575577
$979$	12.0000	0.383522
$980$	−3.00000	−0.0958315
$981$	−10.0000	−0.319275
$982$	−20.0000	−0.638226
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	2.00000	0.0637577
$985$	8.00000	0.254901
$986$	−8.00000	−0.254772
$987$	12.0000	0.381964
$988$	2.00000	0.0636285
$989$	0	0
$990$	−1.00000	−0.0317821
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	8.00000	0.254000
$993$	−18.0000	−0.571213
$994$	20.0000	0.634361
$995$	0	0
$996$	14.0000	0.443607
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	4.00000	0.126618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6270.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6270.2.a.e.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6270 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6270.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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