LMFDB - Embedded newform 390.8.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,8,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("390.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	390.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:	$121.830159939$
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	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} -1154.00 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} -1154.00 q^{7} +512.000 q^{8} +729.000 q^{9} +1000.00 q^{10} +1692.00 q^{11} +1728.00 q^{12} -2197.00 q^{13} -9232.00 q^{14} +3375.00 q^{15} +4096.00 q^{16} +5116.00 q^{17} +5832.00 q^{18} -39146.0 q^{19} +8000.00 q^{20} -31158.0 q^{21} +13536.0 q^{22} -40030.0 q^{23} +13824.0 q^{24} +15625.0 q^{25} -17576.0 q^{26} +19683.0 q^{27} -73856.0 q^{28} -32224.0 q^{29} +27000.0 q^{30} -12572.0 q^{31} +32768.0 q^{32} +45684.0 q^{33} +40928.0 q^{34} -144250. q^{35} +46656.0 q^{36} +261998. q^{37} -313168. q^{38} -59319.0 q^{39} +64000.0 q^{40} +176154. q^{41} -249264. q^{42} -733212. q^{43} +108288. q^{44} +91125.0 q^{45} -320240. q^{46} -645864. q^{47} +110592. q^{48} +508173. q^{49} +125000. q^{50} +138132. q^{51} -140608. q^{52} -794306. q^{53} +157464. q^{54} +211500. q^{55} -590848. q^{56} -1.05694e6 q^{57} -257792. q^{58} -1.20330e6 q^{59} +216000. q^{60} -1.11399e6 q^{61} -100576. q^{62} -841266. q^{63} +262144. q^{64} -274625. q^{65} +365472. q^{66} -950728. q^{67} +327424. q^{68} -1.08081e6 q^{69} -1.15400e6 q^{70} -4.66638e6 q^{71} +373248. q^{72} -1.51298e6 q^{73} +2.09598e6 q^{74} +421875. q^{75} -2.50534e6 q^{76} -1.95257e6 q^{77} -474552. q^{78} +2.91345e6 q^{79} +512000. q^{80} +531441. q^{81} +1.40923e6 q^{82} -199996. q^{83} -1.99411e6 q^{84} +639500. q^{85} -5.86570e6 q^{86} -870048. q^{87} +866304. q^{88} -4.23651e6 q^{89} +729000. q^{90} +2.53534e6 q^{91} -2.56192e6 q^{92} -339444. q^{93} -5.16691e6 q^{94} -4.89325e6 q^{95} +884736. q^{96} +6.84538e6 q^{97} +4.06538e6 q^{98} +1.23347e6 q^{99} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	216.000	0.408248
$7$	−1154.00	−1.27164	−0.635818	−	0.771839i	$-0.719337\pi$
$7$	−1154.00	−1.27164	−0.635818	+	0.771839i	$0.719337\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	1000.00	0.316228
$11$	1692.00	0.383289	0.191644	−	0.981464i	$-0.438618\pi$
$11$	1692.00	0.383289	0.191644	+	0.981464i	$0.438618\pi$
$12$	1728.00	0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	−9232.00	−0.899182
$15$	3375.00	0.258199
$16$	4096.00	0.250000
$17$	5116.00	0.252557	0.126278	−	0.991995i	$-0.459697\pi$
$17$	5116.00	0.252557	0.126278	+	0.991995i	$0.459697\pi$
$18$	5832.00	0.235702
$19$	−39146.0	−1.30933	−0.654666	−	0.755918i	$-0.727191\pi$
$19$	−39146.0	−1.30933	−0.654666	+	0.755918i	$0.727191\pi$
$20$	8000.00	0.223607
$21$	−31158.0	−0.734179
$22$	13536.0	0.271026
$23$	−40030.0	−0.686022	−0.343011	−	0.939331i	$-0.611447\pi$
$23$	−40030.0	−0.686022	−0.343011	+	0.939331i	$0.611447\pi$
$24$	13824.0	0.204124
$25$	15625.0	0.200000
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	−73856.0	−0.635818
$29$	−32224.0	−0.245350	−0.122675	−	0.992447i	$-0.539147\pi$
$29$	−32224.0	−0.245350	−0.122675	+	0.992447i	$0.539147\pi$
$30$	27000.0	0.182574
$31$	−12572.0	−0.0757946	−0.0378973	−	0.999282i	$-0.512066\pi$
$31$	−12572.0	−0.0757946	−0.0378973	+	0.999282i	$0.512066\pi$
$32$	32768.0	0.176777
$33$	45684.0	0.221292
$34$	40928.0	0.178585
$35$	−144250.	−0.568693
$36$	46656.0	0.166667
$37$	261998.	0.850339	0.425169	−	0.905114i	$-0.360215\pi$
$37$	261998.	0.850339	0.425169	+	0.905114i	$0.360215\pi$
$38$	−313168.	−0.925838
$39$	−59319.0	−0.160128
$40$	64000.0	0.158114
$41$	176154.	0.399162	0.199581	−	0.979881i	$-0.436042\pi$
$41$	176154.	0.399162	0.199581	+	0.979881i	$0.436042\pi$
$42$	−249264.	−0.519143
$43$	−733212.	−1.40634	−0.703169	−	0.711022i	$-0.748232\pi$
$43$	−733212.	−1.40634	−0.703169	+	0.711022i	$0.748232\pi$
$44$	108288.	0.191644
$45$	91125.0	0.149071
$46$	−320240.	−0.485091
$47$	−645864.	−0.907400	−0.453700	−	0.891155i	$-0.649896\pi$
$47$	−645864.	−0.907400	−0.453700	+	0.891155i	$0.649896\pi$
$48$	110592.	0.144338
$49$	508173.	0.617057
$50$	125000.	0.141421
$51$	138132.	0.145814
$52$	−140608.	−0.138675
$53$	−794306.	−0.732862	−0.366431	−	0.930445i	$-0.619420\pi$
$53$	−794306.	−0.732862	−0.366431	+	0.930445i	$0.619420\pi$
$54$	157464.	0.136083
$55$	211500.	0.171412
$56$	−590848.	−0.449591
$57$	−1.05694e6	−0.755943
$58$	−257792.	−0.173489
$59$	−1.20330e6	−0.762767	−0.381383	−	0.924417i	$-0.624552\pi$
$59$	−1.20330e6	−0.762767	−0.381383	+	0.924417i	$0.624552\pi$
$60$	216000.	0.129099
$61$	−1.11399e6	−0.628389	−0.314194	−	0.949359i	$-0.601734\pi$
$61$	−1.11399e6	−0.628389	−0.314194	+	0.949359i	$0.601734\pi$
$62$	−100576.	−0.0535949
$63$	−841266.	−0.423879
$64$	262144.	0.125000
$65$	−274625.	−0.124035
$66$	365472.	0.156477
$67$	−950728.	−0.386184	−0.193092	−	0.981181i	$-0.561852\pi$
$67$	−950728.	−0.386184	−0.193092	+	0.981181i	$0.561852\pi$
$68$	327424.	0.126278
$69$	−1.08081e6	−0.396075
$70$	−1.15400e6	−0.402126
$71$	−4.66638e6	−1.54731	−0.773653	−	0.633609i	$-0.781573\pi$
$71$	−4.66638e6	−1.54731	−0.773653	+	0.633609i	$0.781573\pi$
$72$	373248.	0.117851
$73$	−1.51298e6	−0.455202	−0.227601	−	0.973754i	$-0.573088\pi$
$73$	−1.51298e6	−0.455202	−0.227601	+	0.973754i	$0.573088\pi$
$74$	2.09598e6	0.601280
$75$	421875.	0.115470
$76$	−2.50534e6	−0.654666
$77$	−1.95257e6	−0.487403
$78$	−474552.	−0.113228
$79$	2.91345e6	0.664833	0.332416	−	0.943133i	$-0.392136\pi$
$79$	2.91345e6	0.664833	0.332416	+	0.943133i	$0.392136\pi$
$80$	512000.	0.111803
$81$	531441.	0.111111
$82$	1.40923e6	0.282250
$83$	−199996.	−0.0383926	−0.0191963	−	0.999816i	$-0.506111\pi$
$83$	−199996.	−0.0383926	−0.0191963	+	0.999816i	$0.506111\pi$
$84$	−1.99411e6	−0.367090
$85$	639500.	0.112947
$86$	−5.86570e6	−0.994432
$87$	−870048.	−0.141653
$88$	866304.	0.135513
$89$	−4.23651e6	−0.637005	−0.318502	−	0.947922i	$-0.603180\pi$
$89$	−4.23651e6	−0.637005	−0.318502	+	0.947922i	$0.603180\pi$
$90$	729000.	0.105409
$91$	2.53534e6	0.352688
$92$	−2.56192e6	−0.343011
$93$	−339444.	−0.0437601
$94$	−5.16691e6	−0.641628
$95$	−4.89325e6	−0.585551
$96$	884736.	0.102062
$97$	6.84538e6	0.761546	0.380773	−	0.924669i	$-0.375658\pi$
$97$	6.84538e6	0.761546	0.380773	+	0.924669i	$0.375658\pi$
$98$	4.06538e6	0.436325
$99$	1.23347e6	0.127763
$100$	1.00000e6	0.100000
$101$	−2.05596e6	−0.198559	−0.0992794	−	0.995060i	$-0.531654\pi$
$101$	−2.05596e6	−0.198559	−0.0992794	+	0.995060i	$0.531654\pi$
$102$	1.10506e6	0.103106
$103$	1.16106e7	1.04694	0.523472	−	0.852043i	$-0.324637\pi$
$103$	1.16106e7	1.04694	0.523472	+	0.852043i	$0.324637\pi$
$104$	−1.12486e6	−0.0980581
$105$	−3.89475e6	−0.328335
$106$	−6.35445e6	−0.518212
$107$	3.37295e6	0.266174	0.133087	−	0.991104i	$-0.457511\pi$
$107$	3.37295e6	0.266174	0.133087	+	0.991104i	$0.457511\pi$
$108$	1.25971e6	0.0962250
$109$	2.23835e7	1.65553	0.827763	−	0.561077i	$-0.189613\pi$
$109$	2.23835e7	1.65553	0.827763	+	0.561077i	$0.189613\pi$
$110$	1.69200e6	0.121207
$111$	7.07395e6	0.490943
$112$	−4.72678e6	−0.317909
$113$	945752.	0.0616599	0.0308299	−	0.999525i	$-0.490185\pi$
$113$	945752.	0.0616599	0.0308299	+	0.999525i	$0.490185\pi$
$114$	−8.45554e6	−0.534533
$115$	−5.00375e6	−0.306798
$116$	−2.06234e6	−0.122675
$117$	−1.60161e6	−0.0924500
$118$	−9.62640e6	−0.539358
$119$	−5.90386e6	−0.321160
$120$	1.72800e6	0.0912871
$121$	−1.66243e7	−0.853090
$122$	−8.91195e6	−0.444338
$123$	4.75616e6	0.230456
$124$	−804608.	−0.0378973
$125$	1.95312e6	0.0894427
$126$	−6.73013e6	−0.299727
$127$	−2.25645e7	−0.977489	−0.488744	−	0.872427i	$-0.662545\pi$
$127$	−2.25645e7	−0.977489	−0.488744	+	0.872427i	$0.662545\pi$
$128$	2.09715e6	0.0883883
$129$	−1.97967e7	−0.811950
$130$	−2.19700e6	−0.0877058
$131$	−2.25443e7	−0.876166	−0.438083	−	0.898934i	$-0.644342\pi$
$131$	−2.25443e7	−0.876166	−0.438083	+	0.898934i	$0.644342\pi$
$132$	2.92378e6	0.110646
$133$	4.51745e7	1.66499
$134$	−7.60582e6	−0.273073
$135$	2.46038e6	0.0860663
$136$	2.61939e6	0.0892924
$137$	−6.60208e6	−0.219361	−0.109680	−	0.993967i	$-0.534983\pi$
$137$	−6.60208e6	−0.219361	−0.109680	+	0.993967i	$0.534983\pi$
$138$	−8.64648e6	−0.280067
$139$	−5.74086e7	−1.81311	−0.906557	−	0.422084i	$-0.861299\pi$
$139$	−5.74086e7	−1.81311	−0.906557	+	0.422084i	$0.861299\pi$
$140$	−9.23200e6	−0.284346
$141$	−1.74383e7	−0.523887
$142$	−3.73311e7	−1.09411
$143$	−3.71732e6	−0.106305
$144$	2.98598e6	0.0833333
$145$	−4.02800e6	−0.109724
$146$	−1.21039e7	−0.321877
$147$	1.37207e7	0.356258
$148$	1.67679e7	0.425169
$149$	−3.58726e7	−0.888405	−0.444203	−	0.895926i	$-0.646513\pi$
$149$	−3.58726e7	−0.888405	−0.444203	+	0.895926i	$0.646513\pi$
$150$	3.37500e6	0.0816497
$151$	−2.12620e7	−0.502555	−0.251278	−	0.967915i	$-0.580851\pi$
$151$	−2.12620e7	−0.502555	−0.251278	+	0.967915i	$0.580851\pi$
$152$	−2.00428e7	−0.462919
$153$	3.72956e6	0.0841856
$154$	−1.56205e7	−0.344646
$155$	−1.57150e6	−0.0338964
$156$	−3.79642e6	−0.0800641
$157$	−1.42389e6	−0.0293648	−0.0146824	−	0.999892i	$-0.504674\pi$
$157$	−1.42389e6	−0.0293648	−0.0146824	+	0.999892i	$0.504674\pi$
$158$	2.33076e7	0.470108
$159$	−2.14463e7	−0.423118
$160$	4.09600e6	0.0790569
$161$	4.61946e7	0.872370
$162$	4.25153e6	0.0785674
$163$	3.47166e7	0.627886	0.313943	−	0.949442i	$-0.398350\pi$
$163$	3.47166e7	0.627886	0.313943	+	0.949442i	$0.398350\pi$
$164$	1.12739e7	0.199581
$165$	5.71050e6	0.0989647
$166$	−1.59997e6	−0.0271477
$167$	9.73064e6	0.161672	0.0808358	−	0.996727i	$-0.474241\pi$
$167$	9.73064e6	0.161672	0.0808358	+	0.996727i	$0.474241\pi$
$168$	−1.59529e7	−0.259572
$169$	4.82681e6	0.0769231
$170$	5.11600e6	0.0798655
$171$	−2.85374e7	−0.436444
$172$	−4.69256e7	−0.703169
$173$	3.69461e7	0.542510	0.271255	−	0.962508i	$-0.412561\pi$
$173$	3.69461e7	0.542510	0.271255	+	0.962508i	$0.412561\pi$
$174$	−6.96038e6	−0.100164
$175$	−1.80312e7	−0.254327
$176$	6.93043e6	0.0958222
$177$	−3.24891e7	−0.440384
$178$	−3.38920e7	−0.450431
$179$	1.40882e8	1.83599	0.917994	−	0.396594i	$-0.129808\pi$
$179$	1.40882e8	1.83599	0.917994	+	0.396594i	$0.129808\pi$
$180$	5.83200e6	0.0745356
$181$	−4.52594e7	−0.567328	−0.283664	−	0.958924i	$-0.591550\pi$
$181$	−4.52594e7	−0.567328	−0.283664	+	0.958924i	$0.591550\pi$
$182$	2.02827e7	0.249388
$183$	−3.00778e7	−0.362800
$184$	−2.04954e7	−0.242545
$185$	3.27498e7	0.380283
$186$	−2.71555e6	−0.0309430
$187$	8.65627e6	0.0968022
$188$	−4.13353e7	−0.453700
$189$	−2.27142e7	−0.244726
$190$	−3.91460e7	−0.414047
$191$	−1.24812e8	−1.29611	−0.648053	−	0.761595i	$-0.724416\pi$
$191$	−1.24812e8	−1.29611	−0.648053	+	0.761595i	$0.724416\pi$
$192$	7.07789e6	0.0721688
$193$	−1.16885e7	−0.117033	−0.0585167	−	0.998286i	$-0.518637\pi$
$193$	−1.16885e7	−0.117033	−0.0585167	+	0.998286i	$0.518637\pi$
$194$	5.47630e7	0.538494
$195$	−7.41488e6	−0.0716115
$196$	3.25231e7	0.308529
$197$	−5.87031e7	−0.547053	−0.273526	−	0.961865i	$-0.588190\pi$
$197$	−5.87031e7	−0.547053	−0.273526	+	0.961865i	$0.588190\pi$
$198$	9.86774e6	0.0903420
$199$	−2.50861e7	−0.225656	−0.112828	−	0.993615i	$-0.535991\pi$
$199$	−2.50861e7	−0.225656	−0.112828	+	0.993615i	$0.535991\pi$
$200$	8.00000e6	0.0707107
$201$	−2.56697e7	−0.222964
$202$	−1.64476e7	−0.140402
$203$	3.71865e7	0.311996
$204$	8.84045e6	0.0729069
$205$	2.20192e7	0.178511
$206$	9.28846e7	0.740301
$207$	−2.91819e7	−0.228674
$208$	−8.99891e6	−0.0693375
$209$	−6.62350e7	−0.501852
$210$	−3.11580e7	−0.232168
$211$	−5.69564e6	−0.0417402	−0.0208701	−	0.999782i	$-0.506644\pi$
$211$	−5.69564e6	−0.0417402	−0.0208701	+	0.999782i	$0.506644\pi$
$212$	−5.08356e7	−0.366431
$213$	−1.25992e8	−0.893338
$214$	2.69836e7	0.188214
$215$	−9.16515e7	−0.628934
$216$	1.00777e7	0.0680414
$217$	1.45081e7	0.0963831
$218$	1.79068e8	1.17063
$219$	−4.08506e7	−0.262811
$220$	1.35360e7	0.0857059
$221$	−1.12399e7	−0.0700467
$222$	5.65916e7	0.347149
$223$	1.63339e8	0.986330	0.493165	−	0.869936i	$-0.335840\pi$
$223$	1.63339e8	0.986330	0.493165	+	0.869936i	$0.335840\pi$
$224$	−3.78143e7	−0.224796
$225$	1.13906e7	0.0666667
$226$	7.56602e6	0.0436001
$227$	−2.33872e8	−1.32705	−0.663525	−	0.748154i	$-0.730941\pi$
$227$	−2.33872e8	−1.32705	−0.663525	+	0.748154i	$0.730941\pi$
$228$	−6.76443e7	−0.377972
$229$	2.42915e8	1.33669	0.668345	−	0.743852i	$-0.267003\pi$
$229$	2.42915e8	1.33669	0.668345	+	0.743852i	$0.267003\pi$
$230$	−4.00300e7	−0.216939
$231$	−5.27193e7	−0.281403
$232$	−1.64987e7	−0.0867444
$233$	−1.66136e8	−0.860435	−0.430218	−	0.902725i	$-0.641563\pi$
$233$	−1.66136e8	−0.860435	−0.430218	+	0.902725i	$0.641563\pi$
$234$	−1.28129e7	−0.0653720
$235$	−8.07330e7	−0.405801
$236$	−7.70112e7	−0.381383
$237$	7.86631e7	0.383841
$238$	−4.72309e7	−0.227095
$239$	2.63535e7	0.124866	0.0624332	−	0.998049i	$-0.480114\pi$
$239$	2.63535e7	0.124866	0.0624332	+	0.998049i	$0.480114\pi$
$240$	1.38240e7	0.0645497
$241$	5.88896e7	0.271006	0.135503	−	0.990777i	$-0.456735\pi$
$241$	5.88896e7	0.271006	0.135503	+	0.990777i	$0.456735\pi$
$242$	−1.32994e8	−0.603226
$243$	1.43489e7	0.0641500
$244$	−7.12956e7	−0.314194
$245$	6.35216e7	0.275956
$246$	3.80493e7	0.162957
$247$	8.60038e7	0.363143
$248$	−6.43686e6	−0.0267974
$249$	−5.39989e6	−0.0221660
$250$	1.56250e7	0.0632456
$251$	−2.35282e8	−0.939141	−0.469570	−	0.882895i	$-0.655591\pi$
$251$	−2.35282e8	−0.939141	−0.469570	+	0.882895i	$0.655591\pi$
$252$	−5.38410e7	−0.211939
$253$	−6.77308e7	−0.262945
$254$	−1.80516e8	−0.691189
$255$	1.72665e7	0.0652099
$256$	1.67772e7	0.0625000
$257$	3.97302e8	1.46000	0.730002	−	0.683445i	$-0.239519\pi$
$257$	3.97302e8	1.46000	0.730002	+	0.683445i	$0.239519\pi$
$258$	−1.58374e8	−0.574135
$259$	−3.02346e8	−1.08132
$260$	−1.75760e7	−0.0620174
$261$	−2.34913e7	−0.0817834
$262$	−1.80354e8	−0.619543
$263$	−7.25396e7	−0.245884	−0.122942	−	0.992414i	$-0.539233\pi$
$263$	−7.25396e7	−0.245884	−0.122942	+	0.992414i	$0.539233\pi$
$264$	2.33902e7	0.0782385
$265$	−9.92882e7	−0.327746
$266$	3.61396e8	1.17733
$267$	−1.14386e8	−0.367775
$268$	−6.08466e7	−0.193092
$269$	1.85589e8	0.581325	0.290663	−	0.956826i	$-0.406124\pi$
$269$	1.85589e8	0.581325	0.290663	+	0.956826i	$0.406124\pi$
$270$	1.96830e7	0.0608581
$271$	−4.30701e8	−1.31457	−0.657285	−	0.753642i	$-0.728295\pi$
$271$	−4.30701e8	−1.31457	−0.657285	+	0.753642i	$0.728295\pi$
$272$	2.09551e7	0.0631392
$273$	6.84541e7	0.203625
$274$	−5.28167e7	−0.155111
$275$	2.64375e7	0.0766577
$276$	−6.91718e7	−0.198038
$277$	5.73836e8	1.62221	0.811107	−	0.584897i	$-0.198865\pi$
$277$	5.73836e8	1.62221	0.811107	+	0.584897i	$0.198865\pi$
$278$	−4.59269e8	−1.28206
$279$	−9.16499e6	−0.0252649
$280$	−7.38560e7	−0.201063
$281$	4.98001e7	0.133893	0.0669465	−	0.997757i	$-0.478674\pi$
$281$	4.98001e7	0.133893	0.0669465	+	0.997757i	$0.478674\pi$
$282$	−1.39507e8	−0.370444
$283$	−1.97105e8	−0.516945	−0.258473	−	0.966019i	$-0.583219\pi$
$283$	−1.97105e8	−0.516945	−0.258473	+	0.966019i	$0.583219\pi$
$284$	−2.98649e8	−0.773653
$285$	−1.32118e8	−0.338068
$286$	−2.97386e7	−0.0751691
$287$	−2.03282e8	−0.507588
$288$	2.38879e7	0.0589256
$289$	−3.84165e8	−0.936215
$290$	−3.22240e7	−0.0775866
$291$	1.84825e8	0.439679
$292$	−9.68310e7	−0.227601
$293$	2.13074e8	0.494873	0.247436	−	0.968904i	$-0.420412\pi$
$293$	2.13074e8	0.494873	0.247436	+	0.968904i	$0.420412\pi$
$294$	1.09765e8	0.251912
$295$	−1.50412e8	−0.341120
$296$	1.34143e8	0.300640
$297$	3.33036e7	0.0737639
$298$	−2.86981e8	−0.628197
$299$	8.79459e7	0.190268
$300$	2.70000e7	0.0577350
$301$	8.46127e8	1.78835
$302$	−1.70096e8	−0.355360
$303$	−5.55108e7	−0.114638
$304$	−1.60342e8	−0.327333
$305$	−1.39249e8	−0.281024
$306$	2.98365e7	0.0595282
$307$	8.32444e8	1.64199	0.820995	−	0.570935i	$-0.193419\pi$
$307$	8.32444e8	1.64199	0.820995	+	0.570935i	$0.193419\pi$
$308$	−1.24964e8	−0.243702
$309$	3.13485e8	0.604453
$310$	−1.25720e7	−0.0239684
$311$	1.27543e8	0.240434	0.120217	−	0.992748i	$-0.461641\pi$
$311$	1.27543e8	0.240434	0.120217	+	0.992748i	$0.461641\pi$
$312$	−3.03713e7	−0.0566139
$313$	9.57712e7	0.176535	0.0882673	−	0.996097i	$-0.471867\pi$
$313$	9.57712e7	0.176535	0.0882673	+	0.996097i	$0.471867\pi$
$314$	−1.13911e7	−0.0207641
$315$	−1.05158e8	−0.189564
$316$	1.86461e8	0.332416
$317$	−7.93037e8	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$317$	−7.93037e8	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$318$	−1.71570e8	−0.299190
$319$	−5.45230e7	−0.0940400
$320$	3.27680e7	0.0559017
$321$	9.10696e7	0.153676
$322$	3.69557e8	0.616859
$323$	−2.00271e8	−0.330681
$324$	3.40122e7	0.0555556
$325$	−3.43281e7	−0.0554700
$326$	2.77733e8	0.443982
$327$	6.04356e8	0.955819
$328$	9.01908e7	0.141125
$329$	7.45327e8	1.15388
$330$	4.56840e7	0.0699786
$331$	7.95291e8	1.20539	0.602696	−	0.797971i	$-0.294093\pi$
$331$	7.95291e8	1.20539	0.602696	+	0.797971i	$0.294093\pi$
$332$	−1.27997e7	−0.0191963
$333$	1.90997e8	0.283446
$334$	7.78451e7	0.114319
$335$	−1.18841e8	−0.172707
$336$	−1.27623e8	−0.183545
$337$	−1.01492e8	−0.144453	−0.0722264	−	0.997388i	$-0.523010\pi$
$337$	−1.01492e8	−0.144453	−0.0722264	+	0.997388i	$0.523010\pi$
$338$	3.86145e7	0.0543928
$339$	2.55353e7	0.0355994
$340$	4.09280e7	0.0564734
$341$	−2.12718e7	−0.0290512
$342$	−2.28299e8	−0.308613
$343$	3.63937e8	0.486964
$344$	−3.75405e8	−0.497216
$345$	−1.35101e8	−0.177130
$346$	2.95569e8	0.383612
$347$	−3.10728e8	−0.399234	−0.199617	−	0.979874i	$-0.563970\pi$
$347$	−3.10728e8	−0.399234	−0.199617	+	0.979874i	$0.563970\pi$
$348$	−5.56831e7	−0.0708265
$349$	3.00192e8	0.378017	0.189008	−	0.981975i	$-0.439473\pi$
$349$	3.00192e8	0.378017	0.189008	+	0.981975i	$0.439473\pi$
$350$	−1.44250e8	−0.179836
$351$	−4.32436e7	−0.0533761
$352$	5.54435e7	0.0677565
$353$	−7.30035e8	−0.883349	−0.441675	−	0.897175i	$-0.645615\pi$
$353$	−7.30035e8	−0.883349	−0.441675	+	0.897175i	$0.645615\pi$
$354$	−2.59913e8	−0.311398
$355$	−5.83298e8	−0.691976
$356$	−2.71136e8	−0.318502
$357$	−1.59404e8	−0.185422
$358$	1.12706e9	1.29824
$359$	1.20666e9	1.37644	0.688218	−	0.725504i	$-0.258393\pi$
$359$	1.20666e9	1.37644	0.688218	+	0.725504i	$0.258393\pi$
$360$	4.66560e7	0.0527046
$361$	6.38538e8	0.714350
$362$	−3.62075e8	−0.401161
$363$	−4.48856e8	−0.492532
$364$	1.62262e8	0.176344
$365$	−1.89123e8	−0.203573
$366$	−2.40623e8	−0.256539
$367$	−8.61069e8	−0.909299	−0.454650	−	0.890670i	$-0.650235\pi$
$367$	−8.61069e8	−0.909299	−0.454650	+	0.890670i	$0.650235\pi$
$368$	−1.63963e8	−0.171506
$369$	1.28416e8	0.133054
$370$	2.61998e8	0.268901
$371$	9.16629e8	0.931934
$372$	−2.17244e7	−0.0218800
$373$	−5.18347e8	−0.517178	−0.258589	−	0.965987i	$-0.583257\pi$
$373$	−5.18347e8	−0.517178	−0.258589	+	0.965987i	$0.583257\pi$
$374$	6.92502e7	0.0684495
$375$	5.27344e7	0.0516398
$376$	−3.30682e8	−0.320814
$377$	7.07961e7	0.0680479
$378$	−1.81713e8	−0.173048
$379$	−7.76635e8	−0.732790	−0.366395	−	0.930459i	$-0.619408\pi$
$379$	−7.76635e8	−0.732790	−0.366395	+	0.930459i	$0.619408\pi$
$380$	−3.13168e8	−0.292776
$381$	−6.09240e8	−0.564353
$382$	−9.98498e8	−0.916485
$383$	−3.32393e6	−0.00302313	−0.00151156	−	0.999999i	$-0.500481\pi$
$383$	−3.32393e6	−0.00302313	−0.00151156	+	0.999999i	$0.500481\pi$
$384$	5.66231e7	0.0510310
$385$	−2.44071e8	−0.217973
$386$	−9.35083e7	−0.0827551
$387$	−5.34512e8	−0.468780
$388$	4.38104e8	0.380773
$389$	−2.83020e8	−0.243777	−0.121889	−	0.992544i	$-0.538895\pi$
$389$	−2.83020e8	−0.243777	−0.121889	+	0.992544i	$0.538895\pi$
$390$	−5.93190e7	−0.0506370
$391$	−2.04793e8	−0.173260
$392$	2.60185e8	0.218163
$393$	−6.08695e8	−0.505855
$394$	−4.69625e8	−0.386825
$395$	3.64181e8	0.297322
$396$	7.89420e7	0.0638814
$397$	2.07870e9	1.66735	0.833673	−	0.552258i	$-0.186234\pi$
$397$	2.07870e9	1.66735	0.833673	+	0.552258i	$0.186234\pi$
$398$	−2.00688e8	−0.159563
$399$	1.21971e9	0.961284
$400$	6.40000e7	0.0500000
$401$	5.67213e8	0.439280	0.219640	−	0.975581i	$-0.429512\pi$
$401$	5.67213e8	0.439280	0.219640	+	0.975581i	$0.429512\pi$
$402$	−2.05357e8	−0.157659
$403$	2.76207e7	0.0210216
$404$	−1.31581e8	−0.0992794
$405$	6.64301e7	0.0496904
$406$	2.97492e8	0.220615
$407$	4.43301e8	0.325925
$408$	7.07236e7	0.0515530
$409$	−3.83935e8	−0.277477	−0.138738	−	0.990329i	$-0.544305\pi$
$409$	−3.83935e8	−0.277477	−0.138738	+	0.990329i	$0.544305\pi$
$410$	1.76154e8	0.126226
$411$	−1.78256e8	−0.126648
$412$	7.43077e8	0.523472
$413$	1.38861e9	0.969962
$414$	−2.33455e8	−0.161697
$415$	−2.49995e7	−0.0171697
$416$	−7.19913e7	−0.0490290
$417$	−1.55003e9	−1.04680
$418$	−5.29880e8	−0.354863
$419$	2.80880e9	1.86540	0.932700	−	0.360652i	$-0.117446\pi$
$419$	2.80880e9	1.86540	0.932700	+	0.360652i	$0.117446\pi$
$420$	−2.49264e8	−0.164167
$421$	−5.61104e8	−0.366485	−0.183242	−	0.983068i	$-0.558659\pi$
$421$	−5.61104e8	−0.366485	−0.183242	+	0.983068i	$0.558659\pi$
$422$	−4.55652e7	−0.0295148
$423$	−4.70835e8	−0.302467
$424$	−4.06685e8	−0.259106
$425$	7.99375e7	0.0505114
$426$	−1.00794e9	−0.631685
$427$	1.28555e9	0.799082
$428$	2.15869e8	0.133087
$429$	−1.00368e8	−0.0613753
$430$	−7.33212e8	−0.444723
$431$	5.21635e8	0.313831	0.156916	−	0.987612i	$-0.449845\pi$
$431$	5.21635e8	0.313831	0.156916	+	0.987612i	$0.449845\pi$
$432$	8.06216e7	0.0481125
$433$	−7.23382e8	−0.428213	−0.214107	−	0.976810i	$-0.568684\pi$
$433$	−7.23382e8	−0.428213	−0.214107	+	0.976810i	$0.568684\pi$
$434$	1.16065e8	0.0681532
$435$	−1.08756e8	−0.0633492
$436$	1.43255e9	0.827763
$437$	1.56701e9	0.898231
$438$	−3.26805e8	−0.185836
$439$	1.55754e9	0.878642	0.439321	−	0.898330i	$-0.355219\pi$
$439$	1.55754e9	0.878642	0.439321	+	0.898330i	$0.355219\pi$
$440$	1.08288e8	0.0606033
$441$	3.70458e8	0.205686
$442$	−8.99188e7	−0.0495305
$443$	1.39034e9	0.759814	0.379907	−	0.925025i	$-0.375956\pi$
$443$	1.39034e9	0.759814	0.379907	+	0.925025i	$0.375956\pi$
$444$	4.52733e8	0.245472
$445$	−5.29563e8	−0.284877
$446$	1.30671e9	0.697441
$447$	−9.68561e8	−0.512921
$448$	−3.02514e8	−0.158954
$449$	7.87555e8	0.410600	0.205300	−	0.978699i	$-0.434183\pi$
$449$	7.87555e8	0.410600	0.205300	+	0.978699i	$0.434183\pi$
$450$	9.11250e7	0.0471405
$451$	2.98053e8	0.152994
$452$	6.05281e7	0.0308299
$453$	−5.74073e8	−0.290150
$454$	−1.87098e9	−0.938367
$455$	3.16917e8	0.157727
$456$	−5.41154e8	−0.267266
$457$	1.63150e8	0.0799615	0.0399808	−	0.999200i	$-0.487270\pi$
$457$	1.63150e8	0.0799615	0.0399808	+	0.999200i	$0.487270\pi$
$458$	1.94332e9	0.945182
$459$	1.00698e8	0.0486046
$460$	−3.20240e8	−0.153399
$461$	2.06609e9	0.982192	0.491096	−	0.871105i	$-0.336596\pi$
$461$	2.06609e9	0.982192	0.491096	+	0.871105i	$0.336596\pi$
$462$	−4.21755e8	−0.198982
$463$	3.66420e9	1.71572	0.857858	−	0.513887i	$-0.171795\pi$
$463$	3.66420e9	1.71572	0.857858	+	0.513887i	$0.171795\pi$
$464$	−1.31990e8	−0.0613376
$465$	−4.24305e7	−0.0195701
$466$	−1.32909e9	−0.608420
$467$	2.89511e9	1.31539	0.657697	−	0.753283i	$-0.271531\pi$
$467$	2.89511e9	1.31539	0.657697	+	0.753283i	$0.271531\pi$
$468$	−1.02503e8	−0.0462250
$469$	1.09714e9	0.491086
$470$	−6.45864e8	−0.286945
$471$	−3.84450e7	−0.0169538
$472$	−6.16090e8	−0.269679
$473$	−1.24059e9	−0.539034
$474$	6.29305e8	0.271417
$475$	−6.11656e8	−0.261866
$476$	−3.77847e8	−0.160580
$477$	−5.79049e8	−0.244287
$478$	2.10828e8	0.0882939
$479$	2.83793e9	1.17985	0.589925	−	0.807458i	$-0.299157\pi$
$479$	2.83793e9	1.17985	0.589925	+	0.807458i	$0.299157\pi$
$480$	1.10592e8	0.0456435
$481$	−5.75610e8	−0.235842
$482$	4.71117e8	0.191630
$483$	1.24725e9	0.503663
$484$	−1.06396e9	−0.426545
$485$	8.55672e8	0.340574
$486$	1.14791e8	0.0453609
$487$	−1.36365e9	−0.534997	−0.267499	−	0.963558i	$-0.586197\pi$
$487$	−1.36365e9	−0.534997	−0.267499	+	0.963558i	$0.586197\pi$
$488$	−5.70365e8	−0.222169
$489$	9.37348e8	0.362510
$490$	5.08173e8	0.195131
$491$	2.62097e9	0.999256	0.499628	−	0.866240i	$-0.333470\pi$
$491$	2.62097e9	0.999256	0.499628	+	0.866240i	$0.333470\pi$
$492$	3.04394e8	0.115228
$493$	−1.64858e8	−0.0619649
$494$	6.88030e8	0.256781
$495$	1.54184e8	0.0571373
$496$	−5.14949e7	−0.0189487
$497$	5.38501e9	1.96761
$498$	−4.31991e7	−0.0156737
$499$	7.19179e8	0.259111	0.129555	−	0.991572i	$-0.458645\pi$
$499$	7.19179e8	0.259111	0.129555	+	0.991572i	$0.458645\pi$
$500$	1.25000e8	0.0447214
$501$	2.62727e8	0.0933411
$502$	−1.88226e9	−0.664073
$503$	2.31116e9	0.809732	0.404866	−	0.914376i	$-0.367318\pi$
$503$	2.31116e9	0.809732	0.404866	+	0.914376i	$0.367318\pi$
$504$	−4.30728e8	−0.149864
$505$	−2.56995e8	−0.0887982
$506$	−5.41846e8	−0.185930
$507$	1.30324e8	0.0444116
$508$	−1.44412e9	−0.488744
$509$	3.25759e7	0.0109492	0.00547462	−	0.999985i	$-0.498257\pi$
$509$	3.25759e7	0.0109492	0.00547462	+	0.999985i	$0.498257\pi$
$510$	1.38132e8	0.0461104
$511$	1.74598e9	0.578851
$512$	1.34218e8	0.0441942
$513$	−7.70511e8	−0.251981
$514$	3.17841e9	1.03238
$515$	1.45132e9	0.468207
$516$	−1.26699e9	−0.405975
$517$	−1.09280e9	−0.347796
$518$	−2.41877e9	−0.764609
$519$	9.97545e8	0.313218
$520$	−1.40608e8	−0.0438529
$521$	−4.93257e9	−1.52806	−0.764031	−	0.645179i	$-0.776783\pi$
$521$	−4.93257e9	−1.52806	−0.764031	+	0.645179i	$0.776783\pi$
$522$	−1.87930e8	−0.0578296
$523$	2.73884e9	0.837166	0.418583	−	0.908179i	$-0.362527\pi$
$523$	2.73884e9	0.837166	0.418583	+	0.908179i	$0.362527\pi$
$524$	−1.44283e9	−0.438083
$525$	−4.86844e8	−0.146836
$526$	−5.80317e8	−0.173866
$527$	−6.43184e7	−0.0191425
$528$	1.87122e8	0.0553230
$529$	−1.80242e9	−0.529374
$530$	−7.94306e8	−0.231751
$531$	−8.77206e8	−0.254256
$532$	2.89117e9	0.832497
$533$	−3.87010e8	−0.110708
$534$	−9.15085e8	−0.260056
$535$	4.21618e8	0.119037
$536$	−4.86773e8	−0.136537
$537$	3.80381e9	1.06001
$538$	1.48471e9	0.411059
$539$	8.59829e8	0.236511
$540$	1.57464e8	0.0430331
$541$	2.99821e9	0.814088	0.407044	−	0.913409i	$-0.366560\pi$
$541$	2.99821e9	0.814088	0.407044	+	0.913409i	$0.366560\pi$
$542$	−3.44561e9	−0.929541
$543$	−1.22200e9	−0.327547
$544$	1.67641e8	0.0446462
$545$	2.79794e9	0.740374
$546$	5.47633e8	0.143984
$547$	−4.78749e9	−1.25070	−0.625348	−	0.780346i	$-0.715043\pi$
$547$	−4.78749e9	−1.25070	−0.625348	+	0.780346i	$0.715043\pi$
$548$	−4.22533e8	−0.109680
$549$	−8.12102e8	−0.209463
$550$	2.11500e8	0.0542052
$551$	1.26144e9	0.321245
$552$	−5.53375e8	−0.140034
$553$	−3.36212e9	−0.845425
$554$	4.59068e9	1.14708
$555$	8.84243e8	0.219557
$556$	−3.67415e9	−0.906557
$557$	2.41166e9	0.591320	0.295660	−	0.955293i	$-0.404460\pi$
$557$	2.41166e9	0.591320	0.295660	+	0.955293i	$0.404460\pi$
$558$	−7.33199e7	−0.0178650
$559$	1.61087e9	0.390048
$560$	−5.90848e8	−0.142173
$561$	2.33719e8	0.0558888
$562$	3.98401e8	0.0946767
$563$	−3.00711e9	−0.710182	−0.355091	−	0.934832i	$-0.615550\pi$
$563$	−3.00711e9	−0.710182	−0.355091	+	0.934832i	$0.615550\pi$
$564$	−1.11605e9	−0.261944
$565$	1.18219e8	0.0275751
$566$	−1.57684e9	−0.365535
$567$	−6.13283e8	−0.141293
$568$	−2.38919e9	−0.547055
$569$	3.64659e9	0.829839	0.414919	−	0.909858i	$-0.363810\pi$
$569$	3.64659e9	0.829839	0.414919	+	0.909858i	$0.363810\pi$
$570$	−1.05694e9	−0.239050
$571$	−3.98117e9	−0.894921	−0.447460	−	0.894304i	$-0.647671\pi$
$571$	−3.98117e9	−0.894921	−0.447460	+	0.894304i	$0.647671\pi$
$572$	−2.37909e8	−0.0531526
$573$	−3.36993e9	−0.748307
$574$	−1.62625e9	−0.358919
$575$	−6.25469e8	−0.137204
$576$	1.91103e8	0.0416667
$577$	5.97365e9	1.29457	0.647283	−	0.762250i	$-0.275905\pi$
$577$	5.97365e9	1.29457	0.647283	+	0.762250i	$0.275905\pi$
$578$	−3.07332e9	−0.662004
$579$	−3.15591e8	−0.0675692
$580$	−2.57792e8	−0.0548620
$581$	2.30795e8	0.0488214
$582$	1.47860e9	0.310900
$583$	−1.34397e9	−0.280898
$584$	−7.74648e8	−0.160938
$585$	−2.00202e8	−0.0413449
$586$	1.70459e9	0.349928
$587$	7.66305e9	1.56375	0.781877	−	0.623433i	$-0.214263\pi$
$587$	7.66305e9	1.56375	0.781877	+	0.623433i	$0.214263\pi$
$588$	8.78123e8	0.178129
$589$	4.92144e8	0.0992403
$590$	−1.20330e9	−0.241208
$591$	−1.58498e9	−0.315841
$592$	1.07314e9	0.212585
$593$	3.63255e8	0.0715353	0.0357676	−	0.999360i	$-0.488612\pi$
$593$	3.63255e8	0.0715353	0.0357676	+	0.999360i	$0.488612\pi$
$594$	2.66429e8	0.0521590
$595$	−7.37983e8	−0.143627
$596$	−2.29585e9	−0.444203
$597$	−6.77324e8	−0.130282
$598$	7.03567e8	0.134540
$599$	5.13534e9	0.976283	0.488141	−	0.872765i	$-0.337675\pi$
$599$	5.13534e9	0.976283	0.488141	+	0.872765i	$0.337675\pi$
$600$	2.16000e8	0.0408248
$601$	−7.11295e9	−1.33656	−0.668281	−	0.743909i	$-0.732970\pi$
$601$	−7.11295e9	−1.33656	−0.668281	+	0.743909i	$0.732970\pi$
$602$	6.76901e9	1.26455
$603$	−6.93081e8	−0.128728
$604$	−1.36077e9	−0.251278
$605$	−2.07804e9	−0.381513
$606$	−4.44086e8	−0.0810613
$607$	−1.63161e9	−0.296113	−0.148056	−	0.988979i	$-0.547302\pi$
$607$	−1.63161e9	−0.296113	−0.148056	+	0.988979i	$0.547302\pi$
$608$	−1.28274e9	−0.231459
$609$	1.00404e9	0.180131
$610$	−1.11399e9	−0.198714
$611$	1.41896e9	0.251667
$612$	2.38692e8	0.0420928
$613$	−9.62839e9	−1.68827	−0.844135	−	0.536131i	$-0.819885\pi$
$613$	−9.62839e9	−1.68827	−0.844135	+	0.536131i	$0.819885\pi$
$614$	6.65955e9	1.16106
$615$	5.94520e8	0.103063
$616$	−9.99715e8	−0.172323
$617$	−1.46592e9	−0.251253	−0.125626	−	0.992078i	$-0.540094\pi$
$617$	−1.46592e9	−0.251253	−0.125626	+	0.992078i	$0.540094\pi$
$618$	2.50788e9	0.427413
$619$	−5.44771e9	−0.923202	−0.461601	−	0.887088i	$-0.652725\pi$
$619$	−5.44771e9	−0.923202	−0.461601	+	0.887088i	$0.652725\pi$
$620$	−1.00576e8	−0.0169482
$621$	−7.87910e8	−0.132025
$622$	1.02034e9	0.170012
$623$	4.88893e9	0.810038
$624$	−2.42971e8	−0.0400320
$625$	2.44141e8	0.0400000
$626$	7.66170e8	0.124829
$627$	−1.78835e9	−0.289744
$628$	−9.11290e7	−0.0146824
$629$	1.34038e9	0.214759
$630$	−8.41266e8	−0.134042
$631$	2.74825e9	0.435465	0.217732	−	0.976008i	$-0.430134\pi$
$631$	2.74825e9	0.435465	0.217732	+	0.976008i	$0.430134\pi$
$632$	1.49169e9	0.235054
$633$	−1.53782e8	−0.0240987
$634$	−6.34430e9	−0.988715
$635$	−2.82056e9	−0.437146
$636$	−1.37256e9	−0.211559
$637$	−1.11646e9	−0.171141
$638$	−4.36184e8	−0.0664963
$639$	−3.40179e9	−0.515769
$640$	2.62144e8	0.0395285
$641$	1.68058e8	0.0252033	0.0126016	−	0.999921i	$-0.495989\pi$
$641$	1.68058e8	0.0252033	0.0126016	+	0.999921i	$0.495989\pi$
$642$	7.28557e8	0.108665
$643$	−3.79289e9	−0.562642	−0.281321	−	0.959614i	$-0.590773\pi$
$643$	−3.79289e9	−0.562642	−0.281321	+	0.959614i	$0.590773\pi$
$644$	2.95646e9	0.436185
$645$	−2.47459e9	−0.363115
$646$	−1.60217e9	−0.233827
$647$	1.37045e9	0.198929	0.0994643	−	0.995041i	$-0.468287\pi$
$647$	1.37045e9	0.198929	0.0994643	+	0.995041i	$0.468287\pi$
$648$	2.72098e8	0.0392837
$649$	−2.03598e9	−0.292360
$650$	−2.74625e8	−0.0392232
$651$	3.91718e8	0.0556468
$652$	2.22186e9	0.313943
$653$	−6.37896e9	−0.896507	−0.448253	−	0.893907i	$-0.647954\pi$
$653$	−6.37896e9	−0.896507	−0.448253	+	0.893907i	$0.647954\pi$
$654$	4.83485e9	0.675866
$655$	−2.81803e9	−0.391833
$656$	7.21527e8	0.0997905
$657$	−1.10297e9	−0.151734
$658$	5.96262e9	0.815917
$659$	−4.13817e9	−0.563261	−0.281630	−	0.959523i	$-0.590875\pi$
$659$	−4.13817e9	−0.563261	−0.281630	+	0.959523i	$0.590875\pi$
$660$	3.65472e8	0.0494824
$661$	1.02508e9	0.138055	0.0690276	−	0.997615i	$-0.478010\pi$
$661$	1.02508e9	0.138055	0.0690276	+	0.997615i	$0.478010\pi$
$662$	6.36233e9	0.852341
$663$	−3.03476e8	−0.0404415
$664$	−1.02398e8	−0.0135738
$665$	5.64681e9	0.744608
$666$	1.52797e9	0.200427
$667$	1.28993e9	0.168316
$668$	6.22761e8	0.0808358
$669$	4.41015e9	0.569458
$670$	−9.50728e8	−0.122122
$671$	−1.88488e9	−0.240854
$672$	−1.02099e9	−0.129786
$673$	−1.59762e9	−0.202033	−0.101016	−	0.994885i	$-0.532209\pi$
$673$	−1.59762e9	−0.202033	−0.101016	+	0.994885i	$0.532209\pi$
$674$	−8.11934e8	−0.102144
$675$	3.07547e8	0.0384900
$676$	3.08916e8	0.0384615
$677$	−5.21173e9	−0.645537	−0.322769	−	0.946478i	$-0.604614\pi$
$677$	−5.21173e9	−0.645537	−0.322769	+	0.946478i	$0.604614\pi$
$678$	2.04282e8	0.0251725
$679$	−7.89956e9	−0.968409
$680$	3.27424e8	0.0399328
$681$	−6.31454e9	−0.766173
$682$	−1.70175e8	−0.0205423
$683$	−8.60236e9	−1.03311	−0.516553	−	0.856255i	$-0.672785\pi$
$683$	−8.60236e9	−1.03311	−0.516553	+	0.856255i	$0.672785\pi$
$684$	−1.82640e9	−0.218222
$685$	−8.25260e8	−0.0981011
$686$	2.91150e9	0.344335
$687$	6.55871e9	0.771738
$688$	−3.00324e9	−0.351585
$689$	1.74509e9	0.203259
$690$	−1.08081e9	−0.125250
$691$	−1.01919e10	−1.17512	−0.587558	−	0.809182i	$-0.699911\pi$
$691$	−1.01919e10	−1.17512	−0.587558	+	0.809182i	$0.699911\pi$
$692$	2.36455e9	0.271255
$693$	−1.42342e9	−0.162468
$694$	−2.48582e9	−0.282301
$695$	−7.17607e9	−0.810849
$696$	−4.45465e8	−0.0500819
$697$	9.01204e8	0.100811
$698$	2.40154e9	0.267298
$699$	−4.48567e9	−0.496773
$700$	−1.15400e9	−0.127164
$701$	2.23493e9	0.245048	0.122524	−	0.992466i	$-0.460901\pi$
$701$	2.23493e9	0.245048	0.122524	+	0.992466i	$0.460901\pi$
$702$	−3.45948e8	−0.0377426
$703$	−1.02562e10	−1.11338
$704$	4.43548e8	0.0479111
$705$	−2.17979e9	−0.234290
$706$	−5.84028e9	−0.624622
$707$	2.37257e9	0.252494
$708$	−2.07930e9	−0.220192
$709$	−3.15615e9	−0.332580	−0.166290	−	0.986077i	$-0.553179\pi$
$709$	−3.15615e9	−0.332580	−0.166290	+	0.986077i	$0.553179\pi$
$710$	−4.66638e9	−0.489301
$711$	2.12390e9	0.221611
$712$	−2.16909e9	−0.225215
$713$	5.03257e8	0.0519968
$714$	−1.27523e9	−0.131113
$715$	−4.64666e8	−0.0475411
$716$	9.01645e9	0.917994
$717$	7.11545e8	0.0720917
$718$	9.65331e9	0.973287
$719$	4.49367e9	0.450869	0.225435	−	0.974258i	$-0.427620\pi$
$719$	4.49367e9	0.450869	0.225435	+	0.974258i	$0.427620\pi$
$720$	3.73248e8	0.0372678
$721$	−1.33986e10	−1.33133
$722$	5.10830e9	0.505122
$723$	1.59002e9	0.156466
$724$	−2.89660e9	−0.283664
$725$	−5.03500e8	−0.0490700
$726$	−3.59085e9	−0.348272
$727$	1.23922e10	1.19612	0.598062	−	0.801450i	$-0.295938\pi$
$727$	1.23922e10	1.19612	0.598062	+	0.801450i	$0.295938\pi$
$728$	1.29809e9	0.124694
$729$	3.87420e8	0.0370370
$730$	−1.51298e9	−0.143948
$731$	−3.75111e9	−0.355181
$732$	−1.92498e9	−0.181400
$733$	−1.14944e10	−1.07801	−0.539005	−	0.842303i	$-0.681200\pi$
$733$	−1.14944e10	−1.07801	−0.539005	+	0.842303i	$0.681200\pi$
$734$	−6.88855e9	−0.642972
$735$	1.71508e9	0.159323
$736$	−1.31170e9	−0.121273
$737$	−1.60863e9	−0.148020
$738$	1.02733e9	0.0940833
$739$	−1.71896e10	−1.56679	−0.783393	−	0.621526i	$-0.786513\pi$
$739$	−1.71896e10	−1.56679	−0.783393	+	0.621526i	$0.786513\pi$
$740$	2.09598e9	0.190142
$741$	2.32210e9	0.209661
$742$	7.33303e9	0.658977
$743$	5.29333e9	0.473443	0.236722	−	0.971577i	$-0.423927\pi$
$743$	5.29333e9	0.473443	0.236722	+	0.971577i	$0.423927\pi$
$744$	−1.73795e8	−0.0154715
$745$	−4.48408e9	−0.397307
$746$	−4.14678e9	−0.365700
$747$	−1.45797e8	−0.0127975
$748$	5.54001e8	0.0484011
$749$	−3.89238e9	−0.338477
$750$	4.21875e8	0.0365148
$751$	−1.88532e10	−1.62422	−0.812109	−	0.583505i	$-0.801681\pi$
$751$	−1.88532e10	−1.62422	−0.812109	+	0.583505i	$0.801681\pi$
$752$	−2.64546e9	−0.226850
$753$	−6.35262e9	−0.542213
$754$	5.66369e8	0.0481171
$755$	−2.65774e9	−0.224750
$756$	−1.45371e9	−0.122363
$757$	1.54479e10	1.29430	0.647148	−	0.762364i	$-0.275961\pi$
$757$	1.54479e10	1.29430	0.647148	+	0.762364i	$0.275961\pi$
$758$	−6.21308e9	−0.518161
$759$	−1.82873e9	−0.151811
$760$	−2.50534e9	−0.207024
$761$	−1.49030e9	−0.122583	−0.0612913	−	0.998120i	$-0.519522\pi$
$761$	−1.49030e9	−0.122583	−0.0612913	+	0.998120i	$0.519522\pi$
$762$	−4.87392e9	−0.399058
$763$	−2.58306e10	−2.10523
$764$	−7.98798e9	−0.648053
$765$	4.66196e8	0.0376490
$766$	−2.65915e7	−0.00213767
$767$	2.64365e9	0.211553
$768$	4.52985e8	0.0360844
$769$	−9.75325e9	−0.773405	−0.386703	−	0.922205i	$-0.626386\pi$
$769$	−9.75325e9	−0.773405	−0.386703	+	0.922205i	$0.626386\pi$
$770$	−1.95257e9	−0.154131
$771$	1.07271e10	0.842934
$772$	−7.48067e8	−0.0585167
$773$	4.96454e9	0.386590	0.193295	−	0.981141i	$-0.438083\pi$
$773$	4.96454e9	0.386590	0.193295	+	0.981141i	$0.438083\pi$
$774$	−4.27609e9	−0.331477
$775$	−1.96438e8	−0.0151589
$776$	3.50483e9	0.269247
$777$	−8.16333e9	−0.624301
$778$	−2.26416e9	−0.172376
$779$	−6.89572e9	−0.522635
$780$	−4.74552e8	−0.0358057
$781$	−7.89552e9	−0.593065
$782$	−1.63835e9	−0.122513
$783$	−6.34265e8	−0.0472177
$784$	2.08148e9	0.154264
$785$	−1.77986e8	−0.0131324
$786$	−4.86956e9	−0.357693
$787$	−1.78064e10	−1.30216	−0.651081	−	0.759008i	$-0.725684\pi$
$787$	−1.78064e10	−1.30216	−0.651081	+	0.759008i	$0.725684\pi$
$788$	−3.75700e9	−0.273526
$789$	−1.95857e9	−0.141961
$790$	2.91345e9	0.210239
$791$	−1.09140e9	−0.0784089
$792$	6.31536e8	0.0451710
$793$	2.44744e9	0.174284
$794$	1.66296e10	1.17899
$795$	−2.68078e9	−0.189224
$796$	−1.60551e9	−0.112828
$797$	−1.74433e10	−1.22046	−0.610232	−	0.792223i	$-0.708924\pi$
$797$	−1.74433e10	−1.22046	−0.610232	+	0.792223i	$0.708924\pi$
$798$	9.75769e9	0.679731
$799$	−3.30424e9	−0.229170
$800$	5.12000e8	0.0353553
$801$	−3.08841e9	−0.212335
$802$	4.53770e9	0.310618
$803$	−2.55997e9	−0.174474
$804$	−1.64286e9	−0.111482
$805$	5.77433e9	0.390136
$806$	2.20965e8	0.0148646
$807$	5.01090e9	0.335628
$808$	−1.05265e9	−0.0702011
$809$	2.44878e10	1.62603	0.813016	−	0.582241i	$-0.197824\pi$
$809$	2.44878e10	1.62603	0.813016	+	0.582241i	$0.197824\pi$
$810$	5.31441e8	0.0351364
$811$	9.54270e9	0.628201	0.314100	−	0.949390i	$-0.398297\pi$
$811$	9.54270e9	0.628201	0.314100	+	0.949390i	$0.398297\pi$
$812$	2.37994e9	0.155998
$813$	−1.16289e10	−0.758967
$814$	3.54640e9	0.230464
$815$	4.33958e9	0.280799
$816$	5.65789e8	0.0364535
$817$	2.87023e10	1.84136
$818$	−3.07148e9	−0.196206
$819$	1.84826e9	0.117563
$820$	1.40923e9	0.0892553
$821$	8.38191e9	0.528618	0.264309	−	0.964438i	$-0.414856\pi$
$821$	8.38191e9	0.528618	0.264309	+	0.964438i	$0.414856\pi$
$822$	−1.42605e9	−0.0895537
$823$	−1.56364e10	−0.977769	−0.488884	−	0.872349i	$-0.662596\pi$
$823$	−1.56364e10	−0.977769	−0.488884	+	0.872349i	$0.662596\pi$
$824$	5.94461e9	0.370150
$825$	7.13812e8	0.0442584
$826$	1.11089e10	0.685866
$827$	−1.18246e10	−0.726973	−0.363486	−	0.931599i	$-0.618414\pi$
$827$	−1.18246e10	−0.726973	−0.363486	+	0.931599i	$0.618414\pi$
$828$	−1.86764e9	−0.114337
$829$	1.60752e10	0.979976	0.489988	−	0.871729i	$-0.337001\pi$
$829$	1.60752e10	0.979976	0.489988	+	0.871729i	$0.337001\pi$
$830$	−1.99996e8	−0.0121408
$831$	1.54936e10	0.936586
$832$	−5.75930e8	−0.0346688
$833$	2.59981e9	0.155842
$834$	−1.24003e10	−0.740201
$835$	1.21633e9	0.0723017
$836$	−4.23904e9	−0.250926
$837$	−2.47455e8	−0.0145867
$838$	2.24704e10	1.31904
$839$	−6.48664e8	−0.0379186	−0.0189593	−	0.999820i	$-0.506035\pi$
$839$	−6.48664e8	−0.0379186	−0.0189593	+	0.999820i	$0.506035\pi$
$840$	−1.99411e9	−0.116084
$841$	−1.62115e10	−0.939803
$842$	−4.48883e9	−0.259144
$843$	1.34460e9	0.0773032
$844$	−3.64521e8	−0.0208701
$845$	6.03351e8	0.0344010
$846$	−3.76668e9	−0.213876
$847$	1.91845e10	1.08482
$848$	−3.25348e9	−0.183216
$849$	−5.32183e9	−0.298458
$850$	6.39500e8	0.0357169
$851$	−1.04878e10	−0.583351
$852$	−8.06351e9	−0.446669
$853$	3.41580e8	0.0188439	0.00942196	−	0.999956i	$-0.497001\pi$
$853$	3.41580e8	0.0188439	0.00942196	+	0.999956i	$0.497001\pi$
$854$	1.02844e10	0.565036
$855$	−3.56718e9	−0.195184
$856$	1.72695e9	0.0941069
$857$	3.16259e10	1.71637	0.858184	−	0.513342i	$-0.171593\pi$
$857$	3.16259e10	1.71637	0.858184	+	0.513342i	$0.171593\pi$
$858$	−8.02942e8	−0.0433989
$859$	−9.46857e9	−0.509693	−0.254846	−	0.966982i	$-0.582025\pi$
$859$	−9.46857e9	−0.509693	−0.254846	+	0.966982i	$0.582025\pi$
$860$	−5.86570e9	−0.314467
$861$	−5.48861e9	−0.293056
$862$	4.17308e9	0.221912
$863$	3.63329e10	1.92425	0.962127	−	0.272603i	$-0.0878844\pi$
$863$	3.63329e10	1.92425	0.962127	+	0.272603i	$0.0878844\pi$
$864$	6.44973e8	0.0340207
$865$	4.61826e9	0.242618
$866$	−5.78705e9	−0.302792
$867$	−1.03725e10	−0.540524
$868$	9.28518e8	0.0481916
$869$	4.92955e9	0.254823
$870$	−8.70048e8	−0.0447946
$871$	2.08875e9	0.107108
$872$	1.14604e10	0.585317
$873$	4.99028e9	0.253849
$874$	1.25361e10	0.635145
$875$	−2.25391e9	−0.113739
$876$	−2.61444e9	−0.131406
$877$	−1.05049e10	−0.525888	−0.262944	−	0.964811i	$-0.584693\pi$
$877$	−1.05049e10	−0.525888	−0.262944	+	0.964811i	$0.584693\pi$
$878$	1.24603e10	0.621294
$879$	5.75299e9	0.285715
$880$	8.66304e8	0.0428530
$881$	−3.36643e10	−1.65864	−0.829322	−	0.558770i	$-0.811273\pi$
$881$	−3.36643e10	−1.65864	−0.829322	+	0.558770i	$0.811273\pi$
$882$	2.96366e9	0.145442
$883$	2.68521e9	0.131255	0.0656275	−	0.997844i	$-0.479095\pi$
$883$	2.68521e9	0.131255	0.0656275	+	0.997844i	$0.479095\pi$
$884$	−7.19351e8	−0.0350233
$885$	−4.06114e9	−0.196946
$886$	1.11227e10	0.537269
$887$	1.27311e10	0.612537	0.306269	−	0.951945i	$-0.400919\pi$
$887$	1.27311e10	0.612537	0.306269	+	0.951945i	$0.400919\pi$
$888$	3.62186e9	0.173575
$889$	2.60394e10	1.24301
$890$	−4.23651e9	−0.201439
$891$	8.99198e8	0.0425876
$892$	1.04537e10	0.493165
$893$	2.52830e10	1.18809
$894$	−7.74849e9	−0.362690
$895$	1.76102e10	0.821079
$896$	−2.42011e9	−0.112398
$897$	2.37454e9	0.109851
$898$	6.30044e9	0.290338
$899$	4.05120e8	0.0185962
$900$	7.29000e8	0.0333333
$901$	−4.06367e9	−0.185089
$902$	2.38442e9	0.108183
$903$	2.28454e10	1.03250
$904$	4.84225e8	0.0218001
$905$	−5.65743e9	−0.253717
$906$	−4.59258e9	−0.205167
$907$	1.45606e9	0.0647967	0.0323983	−	0.999475i	$-0.489685\pi$
$907$	1.45606e9	0.0647967	0.0323983	+	0.999475i	$0.489685\pi$
$908$	−1.49678e10	−0.663525
$909$	−1.49879e9	−0.0661862
$910$	2.53534e9	0.111530
$911$	−3.34993e10	−1.46799	−0.733993	−	0.679158i	$-0.762345\pi$
$911$	−3.34993e10	−1.46799	−0.733993	+	0.679158i	$0.762345\pi$
$912$	−4.32923e9	−0.188986
$913$	−3.38393e8	−0.0147155
$914$	1.30520e9	0.0565413
$915$	−3.75973e9	−0.162249
$916$	1.55466e10	0.668345
$917$	2.60161e10	1.11416
$918$	8.05586e8	0.0343686
$919$	1.62741e9	0.0691660	0.0345830	−	0.999402i	$-0.488990\pi$
$919$	1.62741e9	0.0691660	0.0345830	+	0.999402i	$0.488990\pi$
$920$	−2.56192e9	−0.108470
$921$	2.24760e10	0.948004
$922$	1.65287e10	0.694515
$923$	1.02520e10	0.429146
$924$	−3.37404e9	−0.140701
$925$	4.09372e9	0.170068
$926$	2.93136e10	1.21319
$927$	8.46411e9	0.348981
$928$	−1.05592e9	−0.0433722
$929$	−7.15616e9	−0.292836	−0.146418	−	0.989223i	$-0.546775\pi$
$929$	−7.15616e9	−0.292836	−0.146418	+	0.989223i	$0.546775\pi$
$930$	−3.39444e8	−0.0138381
$931$	−1.98929e10	−0.807933
$932$	−1.06327e10	−0.430218
$933$	3.44366e9	0.138814
$934$	2.31609e10	0.930124
$935$	1.08203e9	0.0432913
$936$	−8.20026e8	−0.0326860
$937$	−1.75119e10	−0.695418	−0.347709	−	0.937603i	$-0.613040\pi$
$937$	−1.75119e10	−0.695418	−0.347709	+	0.937603i	$0.613040\pi$
$938$	8.77712e9	0.347250
$939$	2.58582e9	0.101922
$940$	−5.16691e9	−0.202901
$941$	−2.15821e10	−0.844366	−0.422183	−	0.906511i	$-0.638736\pi$
$941$	−2.15821e10	−0.844366	−0.422183	+	0.906511i	$0.638736\pi$
$942$	−3.07560e8	−0.0119881
$943$	−7.05144e9	−0.273834
$944$	−4.92872e9	−0.190692
$945$	−2.83927e9	−0.109445
$946$	−9.92476e9	−0.381154
$947$	−8.30435e9	−0.317747	−0.158873	−	0.987299i	$-0.550786\pi$
$947$	−8.30435e9	−0.317747	−0.158873	+	0.987299i	$0.550786\pi$
$948$	5.03444e9	0.191921
$949$	3.32403e9	0.126250
$950$	−4.89325e9	−0.185168
$951$	−2.14120e10	−0.807283
$952$	−3.02278e9	−0.113547
$953$	−1.02597e10	−0.383981	−0.191990	−	0.981397i	$-0.561494\pi$
$953$	−1.02597e10	−0.383981	−0.191990	+	0.981397i	$0.561494\pi$
$954$	−4.63239e9	−0.172737
$955$	−1.56015e10	−0.579636
$956$	1.68662e9	0.0624332
$957$	−1.47212e9	−0.0542940
$958$	2.27034e10	0.834280
$959$	7.61880e9	0.278947
$960$	8.84736e8	0.0322749
$961$	−2.73546e10	−0.994255
$962$	−4.60488e9	−0.166765
$963$	2.45888e9	0.0887248
$964$	3.76894e9	0.135503
$965$	−1.46107e9	−0.0523389
$966$	9.97804e9	0.356144
$967$	−9.47422e9	−0.336939	−0.168469	−	0.985707i	$-0.553882\pi$
$967$	−9.47422e9	−0.336939	−0.168469	+	0.985707i	$0.553882\pi$
$968$	−8.51165e9	−0.301613
$969$	−5.40732e9	−0.190919
$970$	6.84538e9	0.240822
$971$	−3.84338e10	−1.34724	−0.673621	−	0.739077i	$-0.735262\pi$
$971$	−3.84338e10	−1.34724	−0.673621	+	0.739077i	$0.735262\pi$
$972$	9.18330e8	0.0320750
$973$	6.62495e10	2.30562
$974$	−1.09092e10	−0.378300
$975$	−9.26859e8	−0.0320256
$976$	−4.56292e9	−0.157097
$977$	2.78345e10	0.954887	0.477444	−	0.878662i	$-0.341563\pi$
$977$	2.78345e10	0.954887	0.477444	+	0.878662i	$0.341563\pi$
$978$	7.49879e9	0.256333
$979$	−7.16817e9	−0.244157
$980$	4.06538e9	0.137978
$981$	1.63176e10	0.551842
$982$	2.09677e10	0.706580
$983$	3.37142e10	1.13208	0.566038	−	0.824379i	$-0.308475\pi$
$983$	3.37142e10	1.13208	0.566038	+	0.824379i	$0.308475\pi$
$984$	2.43515e9	0.0814786
$985$	−7.33788e9	−0.244649
$986$	−1.31886e9	−0.0438158
$987$	2.01238e10	0.666194
$988$	5.50424e9	0.181572
$989$	2.93505e10	0.964780
$990$	1.23347e9	0.0404022
$991$	4.40008e9	0.143616	0.0718081	−	0.997418i	$-0.477123\pi$
$991$	4.40008e9	0.143616	0.0718081	+	0.997418i	$0.477123\pi$
$992$	−4.11959e8	−0.0133987
$993$	2.14729e10	0.695933
$994$	4.30801e10	1.39131
$995$	−3.13576e9	−0.100916
$996$	−3.45593e8	−0.0110830
$997$	2.54127e10	0.812115	0.406058	−	0.913847i	$-0.366903\pi$
$997$	2.54127e10	0.812115	0.406058	+	0.913847i	$0.366903\pi$
$998$	5.75343e9	0.183219
$999$	5.15691e9	0.163648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.8.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.8.a.f.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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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