LMFDB - Embedded newform 390.8.a.g.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)

$f(q)$

$=$

$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} +363.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} +363.000 q^{7} +512.000 q^{8} +729.000 q^{9} +1000.00 q^{10} -8409.00 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2904.00 q^{14} +3375.00 q^{15} +4096.00 q^{16} -3801.00 q^{17} +5832.00 q^{18} -11322.0 q^{19} +8000.00 q^{20} +9801.00 q^{21} -67272.0 q^{22} -28375.0 q^{23} +13824.0 q^{24} +15625.0 q^{25} -17576.0 q^{26} +19683.0 q^{27} +23232.0 q^{28} -114586. q^{29} +27000.0 q^{30} -131934. q^{31} +32768.0 q^{32} -227043. q^{33} -30408.0 q^{34} +45375.0 q^{35} +46656.0 q^{36} -280089. q^{37} -90576.0 q^{38} -59319.0 q^{39} +64000.0 q^{40} -352465. q^{41} +78408.0 q^{42} -341308. q^{43} -538176. q^{44} +91125.0 q^{45} -227000. q^{46} -400554. q^{47} +110592. q^{48} -691774. q^{49} +125000. q^{50} -102627. q^{51} -140608. q^{52} +1.39436e6 q^{53} +157464. q^{54} -1.05112e6 q^{55} +185856. q^{56} -305694. q^{57} -916688. q^{58} +1.34482e6 q^{59} +216000. q^{60} +390093. q^{61} -1.05547e6 q^{62} +264627. q^{63} +262144. q^{64} -274625. q^{65} -1.81634e6 q^{66} -144572. q^{67} -243264. q^{68} -766125. q^{69} +363000. q^{70} +5.59975e6 q^{71} +373248. q^{72} +2.63435e6 q^{73} -2.24071e6 q^{74} +421875. q^{75} -724608. q^{76} -3.05247e6 q^{77} -474552. q^{78} -6.28338e6 q^{79} +512000. q^{80} +531441. q^{81} -2.81972e6 q^{82} -7.35520e6 q^{83} +627264. q^{84} -475125. q^{85} -2.73046e6 q^{86} -3.09382e6 q^{87} -4.30541e6 q^{88} +3.09434e6 q^{89} +729000. q^{90} -797511. q^{91} -1.81600e6 q^{92} -3.56222e6 q^{93} -3.20443e6 q^{94} -1.41525e6 q^{95} +884736. q^{96} -1.66991e6 q^{97} -5.53419e6 q^{98} -6.13016e6 q^{99} +O(q^{100})\)

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
$7$	363.000	0.400003	0.200002	−	0.979796i	$-0.435905\pi$
$7$	363.000	0.400003	0.200002	+	0.979796i	$0.435905\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	1000.00	0.316228
$11$	−8409.00	−1.90489	−0.952445	−	0.304710i	$-0.901440\pi$
$11$	−8409.00	−1.90489	−0.952445	+	0.304710i	$0.901440\pi$
$12$	1728.00	0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	2904.00	0.282845
$15$	3375.00	0.258199
$16$	4096.00	0.250000
$17$	−3801.00	−0.187641	−0.0938203	−	0.995589i	$-0.529908\pi$
$17$	−3801.00	−0.187641	−0.0938203	+	0.995589i	$0.529908\pi$
$18$	5832.00	0.235702
$19$	−11322.0	−0.378691	−0.189346	−	0.981910i	$-0.560637\pi$
$19$	−11322.0	−0.378691	−0.189346	+	0.981910i	$0.560637\pi$
$20$	8000.00	0.223607
$21$	9801.00	0.230942
$22$	−67272.0	−1.34696
$23$	−28375.0	−0.486282	−0.243141	−	0.969991i	$-0.578178\pi$
$23$	−28375.0	−0.486282	−0.243141	+	0.969991i	$0.578178\pi$
$24$	13824.0	0.204124
$25$	15625.0	0.200000
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	23232.0	0.200002
$29$	−114586.	−0.872446	−0.436223	−	0.899839i	$-0.643684\pi$
$29$	−114586.	−0.872446	−0.436223	+	0.899839i	$0.643684\pi$
$30$	27000.0	0.182574
$31$	−131934.	−0.795410	−0.397705	−	0.917513i	$-0.630193\pi$
$31$	−131934.	−0.795410	−0.397705	+	0.917513i	$0.630193\pi$
$32$	32768.0	0.176777
$33$	−227043.	−1.09979
$34$	−30408.0	−0.132682
$35$	45375.0	0.178887
$36$	46656.0	0.166667
$37$	−280089.	−0.909055	−0.454527	−	0.890733i	$-0.650192\pi$
$37$	−280089.	−0.909055	−0.454527	+	0.890733i	$0.650192\pi$
$38$	−90576.0	−0.267775
$39$	−59319.0	−0.160128
$40$	64000.0	0.158114
$41$	−352465.	−0.798679	−0.399340	−	0.916803i	$-0.630761\pi$
$41$	−352465.	−0.798679	−0.399340	+	0.916803i	$0.630761\pi$
$42$	78408.0	0.163301
$43$	−341308.	−0.654647	−0.327323	−	0.944912i	$-0.606147\pi$
$43$	−341308.	−0.654647	−0.327323	+	0.944912i	$0.606147\pi$
$44$	−538176.	−0.952445
$45$	91125.0	0.149071
$46$	−227000.	−0.343853
$47$	−400554.	−0.562754	−0.281377	−	0.959597i	$-0.590791\pi$
$47$	−400554.	−0.562754	−0.281377	+	0.959597i	$0.590791\pi$
$48$	110592.	0.144338
$49$	−691774.	−0.839997
$50$	125000.	0.141421
$51$	−102627.	−0.108334
$52$	−140608.	−0.138675
$53$	1.39436e6	1.28649	0.643247	−	0.765659i	$-0.277587\pi$
$53$	1.39436e6	1.28649	0.643247	+	0.765659i	$0.277587\pi$
$54$	157464.	0.136083
$55$	−1.05112e6	−0.851893
$56$	185856.	0.141422
$57$	−305694.	−0.218638
$58$	−916688.	−0.616912
$59$	1.34482e6	0.852473	0.426237	−	0.904612i	$-0.359839\pi$
$59$	1.34482e6	0.852473	0.426237	+	0.904612i	$0.359839\pi$
$60$	216000.	0.129099
$61$	390093.	0.220046	0.110023	−	0.993929i	$-0.464908\pi$
$61$	390093.	0.220046	0.110023	+	0.993929i	$0.464908\pi$
$62$	−1.05547e6	−0.562439
$63$	264627.	0.133334
$64$	262144.	0.125000
$65$	−274625.	−0.124035
$66$	−1.81634e6	−0.777668
$67$	−144572.	−0.0587249	−0.0293625	−	0.999569i	$-0.509348\pi$
$67$	−144572.	−0.0587249	−0.0293625	+	0.999569i	$0.509348\pi$
$68$	−243264.	−0.0938203
$69$	−766125.	−0.280755
$70$	363000.	0.126492
$71$	5.59975e6	1.85680	0.928398	−	0.371587i	$-0.121186\pi$
$71$	5.59975e6	1.85680	0.928398	+	0.371587i	$0.121186\pi$
$72$	373248.	0.117851
$73$	2.63435e6	0.792579	0.396290	−	0.918126i	$-0.370298\pi$
$73$	2.63435e6	0.792579	0.396290	+	0.918126i	$0.370298\pi$
$74$	−2.24071e6	−0.642799
$75$	421875.	0.115470
$76$	−724608.	−0.189346
$77$	−3.05247e6	−0.761962
$78$	−474552.	−0.113228
$79$	−6.28338e6	−1.43383	−0.716917	−	0.697159i	$-0.754447\pi$
$79$	−6.28338e6	−1.43383	−0.716917	+	0.697159i	$0.754447\pi$
$80$	512000.	0.111803
$81$	531441.	0.111111
$82$	−2.81972e6	−0.564752
$83$	−7.35520e6	−1.41196	−0.705978	−	0.708233i	$-0.749492\pi$
$83$	−7.35520e6	−1.41196	−0.705978	+	0.708233i	$0.749492\pi$
$84$	627264.	0.115471
$85$	−475125.	−0.0839154
$86$	−2.73046e6	−0.462905
$87$	−3.09382e6	−0.503707
$88$	−4.30541e6	−0.673480
$89$	3.09434e6	0.465268	0.232634	−	0.972564i	$-0.425266\pi$
$89$	3.09434e6	0.465268	0.232634	+	0.972564i	$0.425266\pi$
$90$	729000.	0.105409
$91$	−797511.	−0.110941
$92$	−1.81600e6	−0.243141
$93$	−3.56222e6	−0.459230
$94$	−3.20443e6	−0.397927
$95$	−1.41525e6	−0.169356
$96$	884736.	0.102062
$97$	−1.66991e6	−0.185778	−0.0928888	−	0.995676i	$-0.529610\pi$
$97$	−1.66991e6	−0.185778	−0.0928888	+	0.995676i	$0.529610\pi$
$98$	−5.53419e6	−0.593968
$99$	−6.13016e6	−0.634963
$100$	1.00000e6	0.100000
$101$	1.06725e7	1.03072	0.515360	−	0.856974i	$-0.327658\pi$
$101$	1.06725e7	1.03072	0.515360	+	0.856974i	$0.327658\pi$
$102$	−821016.	−0.0766039
$103$	5.08037e6	0.458105	0.229052	−	0.973414i	$-0.426437\pi$
$103$	5.08037e6	0.458105	0.229052	+	0.973414i	$0.426437\pi$
$104$	−1.12486e6	−0.0980581
$105$	1.22512e6	0.103280
$106$	1.11548e7	0.909689
$107$	−1.74441e6	−0.137659	−0.0688297	−	0.997628i	$-0.521926\pi$
$107$	−1.74441e6	−0.137659	−0.0688297	+	0.997628i	$0.521926\pi$
$108$	1.25971e6	0.0962250
$109$	−9.22585e6	−0.682360	−0.341180	−	0.939998i	$-0.610827\pi$
$109$	−9.22585e6	−0.682360	−0.341180	+	0.939998i	$0.610827\pi$
$110$	−8.40900e6	−0.602379
$111$	−7.56240e6	−0.524843
$112$	1.48685e6	0.100001
$113$	−1.08961e7	−0.710389	−0.355195	−	0.934792i	$-0.615585\pi$
$113$	−1.08961e7	−0.710389	−0.355195	+	0.934792i	$0.615585\pi$
$114$	−2.44555e6	−0.154600
$115$	−3.54688e6	−0.217472
$116$	−7.33350e6	−0.436223
$117$	−1.60161e6	−0.0924500
$118$	1.07585e7	0.602790
$119$	−1.37976e6	−0.0750568
$120$	1.72800e6	0.0912871
$121$	5.12241e7	2.62861
$122$	3.12074e6	0.155596
$123$	−9.51656e6	−0.461118
$124$	−8.44378e6	−0.397705
$125$	1.95312e6	0.0894427
$126$	2.11702e6	0.0942817
$127$	−2.21469e7	−0.959399	−0.479700	−	0.877433i	$-0.659254\pi$
$127$	−2.21469e7	−0.959399	−0.479700	+	0.877433i	$0.659254\pi$
$128$	2.09715e6	0.0883883
$129$	−9.21532e6	−0.377960
$130$	−2.19700e6	−0.0877058
$131$	−2.77664e7	−1.07912	−0.539561	−	0.841946i	$-0.681410\pi$
$131$	−2.77664e7	−1.07912	−0.539561	+	0.841946i	$0.681410\pi$
$132$	−1.45308e7	−0.549894
$133$	−4.10989e6	−0.151478
$134$	−1.15658e6	−0.0415248
$135$	2.46038e6	0.0860663
$136$	−1.94611e6	−0.0663409
$137$	−1.20131e7	−0.399148	−0.199574	−	0.979883i	$-0.563956\pi$
$137$	−1.20131e7	−0.399148	−0.199574	+	0.979883i	$0.563956\pi$
$138$	−6.12900e6	−0.198524
$139$	1.97716e7	0.624439	0.312220	−	0.950010i	$-0.398928\pi$
$139$	1.97716e7	0.624439	0.312220	+	0.950010i	$0.398928\pi$
$140$	2.90400e6	0.0894434
$141$	−1.08150e7	−0.324906
$142$	4.47980e7	1.31295
$143$	1.84746e7	0.528321
$144$	2.98598e6	0.0833333
$145$	−1.43232e7	−0.390170
$146$	2.10748e7	0.560438
$147$	−1.86779e7	−0.484973
$148$	−1.79257e7	−0.454527
$149$	3.37325e7	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$149$	3.37325e7	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$150$	3.37500e6	0.0816497
$151$	−5.96190e7	−1.40918	−0.704588	−	0.709617i	$-0.748868\pi$
$151$	−5.96190e7	−1.40918	−0.704588	+	0.709617i	$0.748868\pi$
$152$	−5.79686e6	−0.133888
$153$	−2.77093e6	−0.0625468
$154$	−2.44197e7	−0.538789
$155$	−1.64918e7	−0.355718
$156$	−3.79642e6	−0.0800641
$157$	−8.25758e7	−1.70296	−0.851480	−	0.524388i	$-0.824294\pi$
$157$	−8.25758e7	−1.70296	−0.851480	+	0.524388i	$0.824294\pi$
$158$	−5.02671e7	−1.01387
$159$	3.76476e7	0.742758
$160$	4.09600e6	0.0790569
$161$	−1.03001e7	−0.194514
$162$	4.25153e6	0.0785674
$163$	2.42833e7	0.439189	0.219594	−	0.975591i	$-0.429527\pi$
$163$	2.42833e7	0.439189	0.219594	+	0.975591i	$0.429527\pi$
$164$	−2.25578e7	−0.399340
$165$	−2.83804e7	−0.491841
$166$	−5.88416e7	−0.998404
$167$	5.93488e7	0.986062	0.493031	−	0.870012i	$-0.335889\pi$
$167$	5.93488e7	0.986062	0.493031	+	0.870012i	$0.335889\pi$
$168$	5.01811e6	0.0816503
$169$	4.82681e6	0.0769231
$170$	−3.80100e6	−0.0593371
$171$	−8.25374e6	−0.126230
$172$	−2.18437e7	−0.327323
$173$	9.80869e6	0.144029	0.0720145	−	0.997404i	$-0.477057\pi$
$173$	9.80869e6	0.144029	0.0720145	+	0.997404i	$0.477057\pi$
$174$	−2.47506e7	−0.356175
$175$	5.67188e6	0.0800006
$176$	−3.44433e7	−0.476223
$177$	3.63100e7	0.492176
$178$	2.47547e7	0.328994
$179$	−9.30501e7	−1.21264	−0.606319	−	0.795222i	$-0.707355\pi$
$179$	−9.30501e7	−1.21264	−0.606319	+	0.795222i	$0.707355\pi$
$180$	5.83200e6	0.0745356
$181$	−6.85145e7	−0.858831	−0.429415	−	0.903107i	$-0.641280\pi$
$181$	−6.85145e7	−0.858831	−0.429415	+	0.903107i	$0.641280\pi$
$182$	−6.38009e6	−0.0784471
$183$	1.05325e7	0.127044
$184$	−1.45280e7	−0.171927
$185$	−3.50111e7	−0.406542
$186$	−2.84977e7	−0.324725
$187$	3.19626e7	0.357435
$188$	−2.56355e7	−0.281377
$189$	7.14493e6	0.0769807
$190$	−1.13220e7	−0.119753
$191$	3.41306e7	0.354427	0.177214	−	0.984172i	$-0.443292\pi$
$191$	3.41306e7	0.354427	0.177214	+	0.984172i	$0.443292\pi$
$192$	7.07789e6	0.0721688
$193$	1.71985e8	1.72203	0.861015	−	0.508579i	$-0.169829\pi$
$193$	1.71985e8	1.72203	0.861015	+	0.508579i	$0.169829\pi$
$194$	−1.33593e7	−0.131365
$195$	−7.41488e6	−0.0716115
$196$	−4.42735e7	−0.419999
$197$	3.33928e7	0.311187	0.155593	−	0.987821i	$-0.450271\pi$
$197$	3.33928e7	0.311187	0.155593	+	0.987821i	$0.450271\pi$
$198$	−4.90413e7	−0.448987
$199$	3.62056e7	0.325679	0.162840	−	0.986653i	$-0.447935\pi$
$199$	3.62056e7	0.325679	0.162840	+	0.986653i	$0.447935\pi$
$200$	8.00000e6	0.0707107
$201$	−3.90344e6	−0.0339048
$202$	8.53799e7	0.728829
$203$	−4.15947e7	−0.348981
$204$	−6.56813e6	−0.0541672
$205$	−4.40581e7	−0.357180
$206$	4.06429e7	0.323929
$207$	−2.06854e7	−0.162094
$208$	−8.99891e6	−0.0693375
$209$	9.52067e7	0.721366
$210$	9.80100e6	0.0730303
$211$	−1.13608e8	−0.832570	−0.416285	−	0.909234i	$-0.636668\pi$
$211$	−1.13608e8	−0.832570	−0.416285	+	0.909234i	$0.636668\pi$
$212$	8.92387e7	0.643247
$213$	1.51193e8	1.07202
$214$	−1.39553e7	−0.0973398
$215$	−4.26635e7	−0.292767
$216$	1.00777e7	0.0680414
$217$	−4.78920e7	−0.318166
$218$	−7.38068e7	−0.482501
$219$	7.11273e7	0.457596
$220$	−6.72720e7	−0.425946
$221$	8.35080e6	0.0520421
$222$	−6.04992e7	−0.371120
$223$	−2.69575e8	−1.62784	−0.813921	−	0.580976i	$-0.802671\pi$
$223$	−2.69575e8	−1.62784	−0.813921	+	0.580976i	$0.802671\pi$
$224$	1.18948e7	0.0707112
$225$	1.13906e7	0.0666667
$226$	−8.71688e7	−0.502321
$227$	−1.79406e8	−1.01800	−0.509000	−	0.860767i	$-0.669985\pi$
$227$	−1.79406e8	−1.01800	−0.509000	+	0.860767i	$0.669985\pi$
$228$	−1.95644e7	−0.109319
$229$	2.87536e8	1.58222	0.791111	−	0.611672i	$-0.209503\pi$
$229$	2.87536e8	1.58222	0.791111	+	0.611672i	$0.209503\pi$
$230$	−2.83750e7	−0.153776
$231$	−8.24166e7	−0.439919
$232$	−5.86680e7	−0.308456
$233$	−1.90464e8	−0.986430	−0.493215	−	0.869907i	$-0.664178\pi$
$233$	−1.90464e8	−0.986430	−0.493215	+	0.869907i	$0.664178\pi$
$234$	−1.28129e7	−0.0653720
$235$	−5.00692e7	−0.251671
$236$	8.60682e7	0.426237
$237$	−1.69651e8	−0.827824
$238$	−1.10381e7	−0.0530732
$239$	−9.67026e7	−0.458190	−0.229095	−	0.973404i	$-0.573577\pi$
$239$	−9.67026e7	−0.458190	−0.229095	+	0.973404i	$0.573577\pi$
$240$	1.38240e7	0.0645497
$241$	−2.52632e8	−1.16260	−0.581299	−	0.813690i	$-0.697455\pi$
$241$	−2.52632e8	−1.16260	−0.581299	+	0.813690i	$0.697455\pi$
$242$	4.09793e8	1.85871
$243$	1.43489e7	0.0641500
$244$	2.49660e7	0.110023
$245$	−8.64718e7	−0.375658
$246$	−7.61324e7	−0.326060
$247$	2.48744e7	0.105030
$248$	−6.75502e7	−0.281220
$249$	−1.98591e8	−0.815194
$250$	1.56250e7	0.0632456
$251$	4.65365e8	1.85753	0.928764	−	0.370672i	$-0.120872\pi$
$251$	4.65365e8	1.85753	0.928764	+	0.370672i	$0.120872\pi$
$252$	1.69361e7	0.0666672
$253$	2.38605e8	0.926314
$254$	−1.77175e8	−0.678398
$255$	−1.28284e7	−0.0484486
$256$	1.67772e7	0.0625000
$257$	−2.34907e8	−0.863237	−0.431618	−	0.902056i	$-0.642057\pi$
$257$	−2.34907e8	−0.863237	−0.431618	+	0.902056i	$0.642057\pi$
$258$	−7.37225e7	−0.267258
$259$	−1.01672e8	−0.363625
$260$	−1.75760e7	−0.0620174
$261$	−8.35332e7	−0.290815
$262$	−2.22132e8	−0.763055
$263$	−3.32948e8	−1.12858	−0.564289	−	0.825577i	$-0.690850\pi$
$263$	−3.32948e8	−1.12858	−0.564289	+	0.825577i	$0.690850\pi$
$264$	−1.16246e8	−0.388834
$265$	1.74294e8	0.575338
$266$	−3.28791e7	−0.107111
$267$	8.35472e7	0.268623
$268$	−9.25261e6	−0.0293625
$269$	−1.80299e8	−0.564756	−0.282378	−	0.959303i	$-0.591123\pi$
$269$	−1.80299e8	−0.564756	−0.282378	+	0.959303i	$0.591123\pi$
$270$	1.96830e7	0.0608581
$271$	4.66718e8	1.42450	0.712249	−	0.701927i	$-0.247677\pi$
$271$	4.66718e8	1.42450	0.712249	+	0.701927i	$0.247677\pi$
$272$	−1.55689e7	−0.0469101
$273$	−2.15328e7	−0.0640518
$274$	−9.61049e7	−0.282240
$275$	−1.31391e8	−0.380978
$276$	−4.90320e7	−0.140378
$277$	2.04937e8	0.579349	0.289675	−	0.957125i	$-0.406453\pi$
$277$	2.04937e8	0.579349	0.289675	+	0.957125i	$0.406453\pi$
$278$	1.58173e8	0.441545
$279$	−9.61799e7	−0.265137
$280$	2.32320e7	0.0632461
$281$	2.43543e8	0.654792	0.327396	−	0.944887i	$-0.393829\pi$
$281$	2.43543e8	0.654792	0.327396	+	0.944887i	$0.393829\pi$
$282$	−8.65197e7	−0.229743
$283$	2.42427e8	0.635813	0.317906	−	0.948122i	$-0.397020\pi$
$283$	2.42427e8	0.635813	0.317906	+	0.948122i	$0.397020\pi$
$284$	3.58384e8	0.928398
$285$	−3.82118e7	−0.0977777
$286$	1.47797e8	0.373580
$287$	−1.27945e8	−0.319474
$288$	2.38879e7	0.0589256
$289$	−3.95891e8	−0.964791
$290$	−1.14586e8	−0.275892
$291$	−4.50877e7	−0.107259
$292$	1.68598e8	0.396290
$293$	−2.81056e7	−0.0652763	−0.0326381	−	0.999467i	$-0.510391\pi$
$293$	−2.81056e7	−0.0652763	−0.0326381	+	0.999467i	$0.510391\pi$
$294$	−1.49423e8	−0.342928
$295$	1.68102e8	0.381238
$296$	−1.43406e8	−0.321399
$297$	−1.65514e8	−0.366596
$298$	2.69860e8	0.590720
$299$	6.23399e7	0.134870
$300$	2.70000e7	0.0577350
$301$	−1.23895e8	−0.261861
$302$	−4.76952e8	−0.996438
$303$	2.88157e8	0.595087
$304$	−4.63749e7	−0.0946729
$305$	4.87616e7	0.0984076
$306$	−2.21674e7	−0.0442273
$307$	−5.76296e7	−0.113674	−0.0568370	−	0.998383i	$-0.518102\pi$
$307$	−5.76296e7	−0.113674	−0.0568370	+	0.998383i	$0.518102\pi$
$308$	−1.95358e8	−0.380981
$309$	1.37170e8	0.264487
$310$	−1.31934e8	−0.251531
$311$	−1.92130e8	−0.362187	−0.181094	−	0.983466i	$-0.557964\pi$
$311$	−1.92130e8	−0.362187	−0.181094	+	0.983466i	$0.557964\pi$
$312$	−3.03713e7	−0.0566139
$313$	1.94306e8	0.358163	0.179081	−	0.983834i	$-0.442687\pi$
$313$	1.94306e8	0.358163	0.179081	+	0.983834i	$0.442687\pi$
$314$	−6.60607e8	−1.20417
$315$	3.30784e7	0.0596290
$316$	−4.02137e8	−0.716917
$317$	4.54134e8	0.800712	0.400356	−	0.916360i	$-0.368886\pi$
$317$	4.54134e8	0.800712	0.400356	+	0.916360i	$0.368886\pi$
$318$	3.01181e8	0.525209
$319$	9.63554e8	1.66191
$320$	3.27680e7	0.0559017
$321$	−4.70991e7	−0.0794776
$322$	−8.24010e7	−0.137543
$323$	4.30349e7	0.0710579
$324$	3.40122e7	0.0555556
$325$	−3.43281e7	−0.0554700
$326$	1.94266e8	0.310553
$327$	−2.49098e8	−0.393961
$328$	−1.80462e8	−0.282376
$329$	−1.45401e8	−0.225103
$330$	−2.27043e8	−0.347784
$331$	1.11255e9	1.68625	0.843126	−	0.537717i	$-0.180713\pi$
$331$	1.11255e9	1.68625	0.843126	+	0.537717i	$0.180713\pi$
$332$	−4.70733e8	−0.705978
$333$	−2.04185e8	−0.303018
$334$	4.74791e8	0.697251
$335$	−1.80715e7	−0.0262626
$336$	4.01449e7	0.0577355
$337$	−1.88145e7	−0.0267786	−0.0133893	−	0.999910i	$-0.504262\pi$
$337$	−1.88145e7	−0.0267786	−0.0133893	+	0.999910i	$0.504262\pi$
$338$	3.86145e7	0.0543928
$339$	−2.94195e8	−0.410143
$340$	−3.04080e7	−0.0419577
$341$	1.10943e9	1.51517
$342$	−6.60299e7	−0.0892584
$343$	−5.50060e8	−0.736005
$344$	−1.74750e8	−0.231453
$345$	−9.57656e7	−0.125558
$346$	7.84695e7	0.101844
$347$	−3.32179e8	−0.426794	−0.213397	−	0.976966i	$-0.568453\pi$
$347$	−3.32179e8	−0.426794	−0.213397	+	0.976966i	$0.568453\pi$
$348$	−1.98005e8	−0.251853
$349$	−1.30837e9	−1.64757	−0.823783	−	0.566905i	$-0.808140\pi$
$349$	−1.30837e9	−1.64757	−0.823783	+	0.566905i	$0.808140\pi$
$350$	4.53750e7	0.0565690
$351$	−4.32436e7	−0.0533761
$352$	−2.75546e8	−0.336740
$353$	2.66341e8	0.322274	0.161137	−	0.986932i	$-0.448484\pi$
$353$	2.66341e8	0.322274	0.161137	+	0.986932i	$0.448484\pi$
$354$	2.90480e8	0.348021
$355$	6.99968e8	0.830384
$356$	1.98038e8	0.232634
$357$	−3.72536e7	−0.0433341
$358$	−7.44401e8	−0.857464
$359$	4.17742e8	0.476516	0.238258	−	0.971202i	$-0.423424\pi$
$359$	4.17742e8	0.476516	0.238258	+	0.971202i	$0.423424\pi$
$360$	4.66560e7	0.0527046
$361$	−7.65684e8	−0.856593
$362$	−5.48116e8	−0.607285
$363$	1.38305e9	1.51763
$364$	−5.10407e7	−0.0554705
$365$	3.29293e8	0.354452
$366$	8.42601e7	0.0898335
$367$	1.28865e8	0.136083	0.0680415	−	0.997682i	$-0.478325\pi$
$367$	1.28865e8	0.136083	0.0680415	+	0.997682i	$0.478325\pi$
$368$	−1.16224e8	−0.121571
$369$	−2.56947e8	−0.266226
$370$	−2.80089e8	−0.287468
$371$	5.06151e8	0.514602
$372$	−2.27982e8	−0.229615
$373$	−5.04194e8	−0.503056	−0.251528	−	0.967850i	$-0.580933\pi$
$373$	−5.04194e8	−0.503056	−0.251528	+	0.967850i	$0.580933\pi$
$374$	2.55701e8	0.252744
$375$	5.27344e7	0.0516398
$376$	−2.05084e8	−0.198964
$377$	2.51745e8	0.241973
$378$	5.71594e7	0.0544335
$379$	1.13143e9	1.06756	0.533778	−	0.845625i	$-0.320772\pi$
$379$	1.13143e9	1.06756	0.533778	+	0.845625i	$0.320772\pi$
$380$	−9.05760e7	−0.0846780
$381$	−5.97965e8	−0.553909
$382$	2.73045e8	0.250618
$383$	−1.48826e9	−1.35357	−0.676787	−	0.736179i	$-0.736628\pi$
$383$	−1.48826e9	−1.35357	−0.676787	+	0.736179i	$0.736628\pi$
$384$	5.66231e7	0.0510310
$385$	−3.81558e8	−0.340760
$386$	1.37588e9	1.21766
$387$	−2.48814e8	−0.218216
$388$	−1.06875e8	−0.0928888
$389$	1.30729e9	1.12603	0.563013	−	0.826448i	$-0.309642\pi$
$389$	1.30729e9	1.12603	0.563013	+	0.826448i	$0.309642\pi$
$390$	−5.93190e7	−0.0506370
$391$	1.07853e8	0.0912463
$392$	−3.54188e8	−0.296984
$393$	−7.49694e8	−0.623032
$394$	2.67143e8	0.220042
$395$	−7.85423e8	−0.641230
$396$	−3.92330e8	−0.317482
$397$	−7.27954e8	−0.583898	−0.291949	−	0.956434i	$-0.594304\pi$
$397$	−7.27954e8	−0.583898	−0.291949	+	0.956434i	$0.594304\pi$
$398$	2.89645e8	0.230290
$399$	−1.10967e8	−0.0874558
$400$	6.40000e7	0.0500000
$401$	1.38421e9	1.07200	0.536001	−	0.844217i	$-0.319934\pi$
$401$	1.38421e9	1.07200	0.536001	+	0.844217i	$0.319934\pi$
$402$	−3.12276e7	−0.0239743
$403$	2.89859e8	0.220607
$404$	6.83039e8	0.515360
$405$	6.64301e7	0.0496904
$406$	−3.32758e8	−0.246767
$407$	2.35527e9	1.73165
$408$	−5.25450e7	−0.0383020
$409$	5.31524e8	0.384142	0.192071	−	0.981381i	$-0.438480\pi$
$409$	5.31524e8	0.384142	0.192071	+	0.981381i	$0.438480\pi$
$410$	−3.52465e8	−0.252565
$411$	−3.24354e8	−0.230448
$412$	3.25144e8	0.229052
$413$	4.88168e8	0.340992
$414$	−1.65483e8	−0.114618
$415$	−9.19400e8	−0.631446
$416$	−7.19913e7	−0.0490290
$417$	5.33833e8	0.360520
$418$	7.61654e8	0.510083
$419$	1.56185e9	1.03727	0.518634	−	0.854996i	$-0.326441\pi$
$419$	1.56185e9	1.03727	0.518634	+	0.854996i	$0.326441\pi$
$420$	7.84080e7	0.0516402
$421$	1.08115e9	0.706154	0.353077	−	0.935594i	$-0.385135\pi$
$421$	1.08115e9	0.706154	0.353077	+	0.935594i	$0.385135\pi$
$422$	−9.08865e8	−0.588716
$423$	−2.92004e8	−0.187585
$424$	7.13910e8	0.454844
$425$	−5.93906e7	−0.0375281
$426$	1.20955e9	0.758034
$427$	1.41604e8	0.0880192
$428$	−1.11642e8	−0.0688297
$429$	4.98813e8	0.305027
$430$	−3.41308e8	−0.207017
$431$	1.45876e9	0.877637	0.438818	−	0.898576i	$-0.355397\pi$
$431$	1.45876e9	0.877637	0.438818	+	0.898576i	$0.355397\pi$
$432$	8.06216e7	0.0481125
$433$	2.08284e9	1.23296	0.616478	−	0.787372i	$-0.288559\pi$
$433$	2.08284e9	1.23296	0.616478	+	0.787372i	$0.288559\pi$
$434$	−3.83136e8	−0.224978
$435$	−3.86728e8	−0.225265
$436$	−5.90454e8	−0.341180
$437$	3.21262e8	0.184151
$438$	5.69019e8	0.323569
$439$	1.05796e9	0.596823	0.298411	−	0.954437i	$-0.403543\pi$
$439$	1.05796e9	0.596823	0.298411	+	0.954437i	$0.403543\pi$
$440$	−5.38176e8	−0.301190
$441$	−5.04303e8	−0.279999
$442$	6.68064e7	0.0367993
$443$	6.13167e8	0.335093	0.167547	−	0.985864i	$-0.446416\pi$
$443$	6.13167e8	0.335093	0.167547	+	0.985864i	$0.446416\pi$
$444$	−4.83994e8	−0.262421
$445$	3.86793e8	0.208074
$446$	−2.15660e9	−1.15106
$447$	9.10777e8	0.482320
$448$	9.51583e7	0.0500004
$449$	3.38166e9	1.76306	0.881531	−	0.472127i	$-0.156514\pi$
$449$	3.38166e9	1.76306	0.881531	+	0.472127i	$0.156514\pi$
$450$	9.11250e7	0.0471405
$451$	2.96388e9	1.52140
$452$	−6.97350e8	−0.355195
$453$	−1.60971e9	−0.813588
$454$	−1.43525e9	−0.719834
$455$	−9.96889e7	−0.0496143
$456$	−1.56515e8	−0.0773001
$457$	1.30500e9	0.639595	0.319797	−	0.947486i	$-0.396385\pi$
$457$	1.30500e9	0.639595	0.319797	+	0.947486i	$0.396385\pi$
$458$	2.30029e9	1.11880
$459$	−7.48151e7	−0.0361114
$460$	−2.27000e8	−0.108736
$461$	−3.09818e9	−1.47283	−0.736417	−	0.676528i	$-0.763484\pi$
$461$	−3.09818e9	−1.47283	−0.736417	+	0.676528i	$0.763484\pi$
$462$	−6.59333e8	−0.311070
$463$	−3.81797e9	−1.78772	−0.893859	−	0.448347i	$-0.852013\pi$
$463$	−3.81797e9	−1.78772	−0.893859	+	0.448347i	$0.852013\pi$
$464$	−4.69344e8	−0.218112
$465$	−4.45277e8	−0.205374
$466$	−1.52371e9	−0.697511
$467$	3.74910e9	1.70340	0.851702	−	0.524026i	$-0.175571\pi$
$467$	3.74910e9	1.70340	0.851702	+	0.524026i	$0.175571\pi$
$468$	−1.02503e8	−0.0462250
$469$	−5.24796e7	−0.0234902
$470$	−4.00554e8	−0.177958
$471$	−2.22955e9	−0.983204
$472$	6.88546e8	0.301395
$473$	2.87006e9	1.24703
$474$	−1.35721e9	−0.585360
$475$	−1.76906e8	−0.0757383
$476$	−8.83048e7	−0.0375284
$477$	1.01648e9	0.428831
$478$	−7.73621e8	−0.323989
$479$	−1.53001e9	−0.636094	−0.318047	−	0.948075i	$-0.603027\pi$
$479$	−1.53001e9	−0.636094	−0.318047	+	0.948075i	$0.603027\pi$
$480$	1.10592e8	0.0456435
$481$	6.15356e8	0.252126
$482$	−2.02106e9	−0.822081
$483$	−2.78103e8	−0.112303
$484$	3.27834e9	1.31430
$485$	−2.08739e8	−0.0830823
$486$	1.14791e8	0.0453609
$487$	2.24285e9	0.879932	0.439966	−	0.898014i	$-0.354991\pi$
$487$	2.24285e9	0.879932	0.439966	+	0.898014i	$0.354991\pi$
$488$	1.99728e8	0.0777981
$489$	6.55649e8	0.253566
$490$	−6.91774e8	−0.265631
$491$	−1.86808e9	−0.712215	−0.356108	−	0.934445i	$-0.615896\pi$
$491$	−1.86808e9	−0.712215	−0.356108	+	0.934445i	$0.615896\pi$
$492$	−6.09060e8	−0.230559
$493$	4.35541e8	0.163706
$494$	1.98995e8	0.0742675
$495$	−7.66270e8	−0.283964
$496$	−5.40402e8	−0.198852
$497$	2.03271e9	0.742724
$498$	−1.58872e9	−0.576429
$499$	9.25745e6	0.00333534	0.00166767	−	0.999999i	$-0.499469\pi$
$499$	9.25745e6	0.00333534	0.00166767	+	0.999999i	$0.499469\pi$
$500$	1.25000e8	0.0447214
$501$	1.60242e9	0.569303
$502$	3.72292e9	1.31347
$503$	−9.71660e8	−0.340429	−0.170214	−	0.985407i	$-0.554446\pi$
$503$	−9.71660e8	−0.340429	−0.170214	+	0.985407i	$0.554446\pi$
$504$	1.35489e8	0.0471408
$505$	1.33406e9	0.460952
$506$	1.90884e9	0.655003
$507$	1.30324e8	0.0444116
$508$	−1.41740e9	−0.479700
$509$	3.35880e9	1.12894	0.564472	−	0.825453i	$-0.309080\pi$
$509$	3.35880e9	1.12894	0.564472	+	0.825453i	$0.309080\pi$
$510$	−1.02627e8	−0.0342583
$511$	9.56268e8	0.317034
$512$	1.34218e8	0.0441942
$513$	−2.22851e8	−0.0728792
$514$	−1.87926e9	−0.610400
$515$	6.35046e8	0.204871
$516$	−5.89780e8	−0.188980
$517$	3.36826e9	1.07198
$518$	−8.13378e8	−0.257122
$519$	2.64835e8	0.0831551
$520$	−1.40608e8	−0.0438529
$521$	2.85907e9	0.885713	0.442857	−	0.896592i	$-0.353965\pi$
$521$	2.85907e9	0.885713	0.442857	+	0.896592i	$0.353965\pi$
$522$	−6.68266e8	−0.205637
$523$	3.24501e9	0.991882	0.495941	−	0.868356i	$-0.334823\pi$
$523$	3.24501e9	0.991882	0.495941	+	0.868356i	$0.334823\pi$
$524$	−1.77705e9	−0.539561
$525$	1.53141e8	0.0461884
$526$	−2.66359e9	−0.798025
$527$	5.01481e8	0.149251
$528$	−9.29968e8	−0.274947
$529$	−2.59968e9	−0.763530
$530$	1.39436e9	0.406825
$531$	9.80371e8	0.284158
$532$	−2.63033e8	−0.0757389
$533$	7.74366e8	0.221514
$534$	6.68378e8	0.189945
$535$	−2.18051e8	−0.0615631
$536$	−7.40209e7	−0.0207624
$537$	−2.51235e9	−0.700117
$538$	−1.44239e9	−0.399343
$539$	5.81713e9	1.60010
$540$	1.57464e8	0.0430331
$541$	−1.20424e9	−0.326981	−0.163491	−	0.986545i	$-0.552275\pi$
$541$	−1.20424e9	−0.326981	−0.163491	+	0.986545i	$0.552275\pi$
$542$	3.73374e9	1.00727
$543$	−1.84989e9	−0.495846
$544$	−1.24551e8	−0.0331705
$545$	−1.15323e9	−0.305161
$546$	−1.72262e8	−0.0452914
$547$	1.06584e8	0.0278442	0.0139221	−	0.999903i	$-0.495568\pi$
$547$	1.06584e8	0.0278442	0.0139221	+	0.999903i	$0.495568\pi$
$548$	−7.68839e8	−0.199574
$549$	2.84378e8	0.0733487
$550$	−1.05112e9	−0.269392
$551$	1.29734e9	0.330388
$552$	−3.92256e8	−0.0992620
$553$	−2.28087e9	−0.573538
$554$	1.63949e9	0.409662
$555$	−9.45300e8	−0.234717
$556$	1.26538e9	0.312220
$557$	7.85683e9	1.92643	0.963217	−	0.268723i	$-0.0866017\pi$
$557$	7.85683e9	1.92643	0.963217	+	0.268723i	$0.0866017\pi$
$558$	−7.69439e8	−0.187480
$559$	7.49854e8	0.181566
$560$	1.85856e8	0.0447217
$561$	8.62990e8	0.206365
$562$	1.94834e9	0.463008
$563$	−1.42157e9	−0.335729	−0.167864	−	0.985810i	$-0.553687\pi$
$563$	−1.42157e9	−0.335729	−0.167864	+	0.985810i	$0.553687\pi$
$564$	−6.92157e8	−0.162453
$565$	−1.36201e9	−0.317696
$566$	1.93942e9	0.449588
$567$	1.92913e8	0.0444448
$568$	2.86707e9	0.656477
$569$	6.95006e9	1.58160	0.790799	−	0.612076i	$-0.209666\pi$
$569$	6.95006e9	1.58160	0.790799	+	0.612076i	$0.209666\pi$
$570$	−3.05694e8	−0.0691393
$571$	2.21091e9	0.496987	0.248494	−	0.968634i	$-0.420065\pi$
$571$	2.21091e9	0.496987	0.248494	+	0.968634i	$0.420065\pi$
$572$	1.18237e9	0.264161
$573$	9.21526e8	0.204629
$574$	−1.02356e9	−0.225902
$575$	−4.43359e8	−0.0972565
$576$	1.91103e8	0.0416667
$577$	−7.97083e9	−1.72738	−0.863691	−	0.504022i	$-0.831853\pi$
$577$	−7.97083e9	−1.72738	−0.863691	+	0.504022i	$0.831853\pi$
$578$	−3.16713e9	−0.682210
$579$	4.64360e9	0.994215
$580$	−9.16688e8	−0.195085
$581$	−2.66994e9	−0.564787
$582$	−3.60702e8	−0.0758434
$583$	−1.17251e10	−2.45063
$584$	1.34879e9	0.280219
$585$	−2.00202e8	−0.0413449
$586$	−2.24844e8	−0.0461573
$587$	3.78425e8	0.0772230	0.0386115	−	0.999254i	$-0.487707\pi$
$587$	3.78425e8	0.0772230	0.0386115	+	0.999254i	$0.487707\pi$
$588$	−1.19539e9	−0.242486
$589$	1.49376e9	0.301215
$590$	1.34482e9	0.269576
$591$	9.01606e8	0.179664
$592$	−1.14724e9	−0.227264
$593$	3.71415e9	0.731421	0.365711	−	0.930729i	$-0.380826\pi$
$593$	3.71415e9	0.731421	0.365711	+	0.930729i	$0.380826\pi$
$594$	−1.32411e9	−0.259223
$595$	−1.72470e8	−0.0335664
$596$	2.15888e9	0.417702
$597$	9.77552e8	0.188031
$598$	4.98719e8	0.0953678
$599$	−4.11459e9	−0.782226	−0.391113	−	0.920343i	$-0.627910\pi$
$599$	−4.11459e9	−0.782226	−0.391113	+	0.920343i	$0.627910\pi$
$600$	2.16000e8	0.0408248
$601$	−4.65545e9	−0.874785	−0.437392	−	0.899271i	$-0.644098\pi$
$601$	−4.65545e9	−0.874785	−0.437392	+	0.899271i	$0.644098\pi$
$602$	−9.91158e8	−0.185163
$603$	−1.05393e8	−0.0195750
$604$	−3.81561e9	−0.704588
$605$	6.40301e9	1.17555
$606$	2.30526e9	0.420790
$607$	−6.84700e9	−1.24263	−0.621313	−	0.783563i	$-0.713400\pi$
$607$	−6.84700e9	−1.24263	−0.621313	+	0.783563i	$0.713400\pi$
$608$	−3.70999e8	−0.0669438
$609$	−1.12306e9	−0.201484
$610$	3.90093e8	0.0695847
$611$	8.80017e8	0.156080
$612$	−1.77339e8	−0.0312734
$613$	−6.18600e9	−1.08467	−0.542336	−	0.840162i	$-0.682460\pi$
$613$	−6.18600e9	−1.08467	−0.542336	+	0.840162i	$0.682460\pi$
$614$	−4.61037e8	−0.0803797
$615$	−1.18957e9	−0.206218
$616$	−1.56286e9	−0.269394
$617$	4.83285e8	0.0828333	0.0414167	−	0.999142i	$-0.486813\pi$
$617$	4.83285e8	0.0828333	0.0414167	+	0.999142i	$0.486813\pi$
$618$	1.09736e9	0.187021
$619$	2.92844e9	0.496271	0.248136	−	0.968725i	$-0.420182\pi$
$619$	2.92844e9	0.496271	0.248136	+	0.968725i	$0.420182\pi$
$620$	−1.05547e9	−0.177859
$621$	−5.58505e8	−0.0935851
$622$	−1.53704e9	−0.256105
$623$	1.12325e9	0.186109
$624$	−2.42971e8	−0.0400320
$625$	2.44141e8	0.0400000
$626$	1.55445e9	0.253259
$627$	2.57058e9	0.416481
$628$	−5.28485e9	−0.851480
$629$	1.06462e9	0.170576
$630$	2.64627e8	0.0421640
$631$	−6.85180e9	−1.08568	−0.542840	−	0.839836i	$-0.682651\pi$
$631$	−6.85180e9	−1.08568	−0.542840	+	0.839836i	$0.682651\pi$
$632$	−3.21709e9	−0.506937
$633$	−3.06742e9	−0.480684
$634$	3.63307e9	0.566189
$635$	−2.76836e9	−0.429056
$636$	2.40945e9	0.371379
$637$	1.51983e9	0.232973
$638$	7.70843e9	1.17515
$639$	4.08222e9	0.618932
$640$	2.62144e8	0.0395285
$641$	−1.79522e9	−0.269224	−0.134612	−	0.990898i	$-0.542979\pi$
$641$	−1.79522e9	−0.269224	−0.134612	+	0.990898i	$0.542979\pi$
$642$	−3.76793e8	−0.0561992
$643$	−1.04918e10	−1.55636	−0.778180	−	0.628041i	$-0.783857\pi$
$643$	−1.04918e10	−1.55636	−0.778180	+	0.628041i	$0.783857\pi$
$644$	−6.59208e8	−0.0972572
$645$	−1.15191e9	−0.169029
$646$	3.44279e8	0.0502455
$647$	−7.96757e7	−0.0115654	−0.00578270	−	0.999983i	$-0.501841\pi$
$647$	−7.96757e7	−0.0115654	−0.00578270	+	0.999983i	$0.501841\pi$
$648$	2.72098e8	0.0392837
$649$	−1.13086e10	−1.62387
$650$	−2.74625e8	−0.0392232
$651$	−1.29309e9	−0.183693
$652$	1.55413e9	0.219594
$653$	1.31974e9	0.185478	0.0927392	−	0.995690i	$-0.470438\pi$
$653$	1.31974e9	0.185478	0.0927392	+	0.995690i	$0.470438\pi$
$654$	−1.99278e9	−0.278572
$655$	−3.47081e9	−0.482598
$656$	−1.44370e9	−0.199670
$657$	1.92044e9	0.264193
$658$	−1.16321e9	−0.159172
$659$	1.48737e9	0.202451	0.101226	−	0.994864i	$-0.467724\pi$
$659$	1.48737e9	0.202451	0.101226	+	0.994864i	$0.467724\pi$
$660$	−1.81634e9	−0.245920
$661$	−6.88443e9	−0.927178	−0.463589	−	0.886050i	$-0.653439\pi$
$661$	−6.88443e9	−0.927178	−0.463589	+	0.886050i	$0.653439\pi$
$662$	8.90042e9	1.19236
$663$	2.25472e8	0.0300465
$664$	−3.76586e9	−0.499202
$665$	−5.13736e8	−0.0677429
$666$	−1.63348e9	−0.214266
$667$	3.25138e9	0.424255
$668$	3.79832e9	0.493031
$669$	−7.27852e9	−0.939834
$670$	−1.44572e8	−0.0185704
$671$	−3.28029e9	−0.419164
$672$	3.21159e8	0.0408252
$673$	−1.44346e10	−1.82537	−0.912684	−	0.408665i	$-0.865994\pi$
$673$	−1.44346e10	−1.82537	−0.912684	+	0.408665i	$0.865994\pi$
$674$	−1.50516e8	−0.0189354
$675$	3.07547e8	0.0384900
$676$	3.08916e8	0.0384615
$677$	−6.04739e9	−0.749044	−0.374522	−	0.927218i	$-0.622193\pi$
$677$	−6.04739e9	−0.749044	−0.374522	+	0.927218i	$0.622193\pi$
$678$	−2.35356e9	−0.290015
$679$	−6.06179e8	−0.0743116
$680$	−2.43264e8	−0.0296686
$681$	−4.84397e9	−0.587742
$682$	8.87546e9	1.07139
$683$	1.35409e10	1.62620	0.813100	−	0.582124i	$-0.197778\pi$
$683$	1.35409e10	1.62620	0.813100	+	0.582124i	$0.197778\pi$
$684$	−5.28239e8	−0.0631152
$685$	−1.50164e9	−0.178504
$686$	−4.40048e9	−0.520434
$687$	7.76346e9	0.913497
$688$	−1.39800e9	−0.163662
$689$	−3.06340e9	−0.356809
$690$	−7.66125e8	−0.0887826
$691$	3.61852e9	0.417213	0.208606	−	0.978000i	$-0.433107\pi$
$691$	3.61852e9	0.417213	0.208606	+	0.978000i	$0.433107\pi$
$692$	6.27756e8	0.0720145
$693$	−2.22525e9	−0.253987
$694$	−2.65743e9	−0.301789
$695$	2.47145e9	0.279258
$696$	−1.58404e9	−0.178087
$697$	1.33972e9	0.149865
$698$	−1.04670e10	−1.16500
$699$	−5.14252e9	−0.569516
$700$	3.63000e8	0.0400003
$701$	3.52349e7	0.00386332	0.00193166	−	0.999998i	$-0.499385\pi$
$701$	3.52349e7	0.00386332	0.00193166	+	0.999998i	$0.499385\pi$
$702$	−3.45948e8	−0.0377426
$703$	3.17117e9	0.344251
$704$	−2.20437e9	−0.238111
$705$	−1.35187e9	−0.145302
$706$	2.13072e9	0.227882
$707$	3.87411e9	0.412291
$708$	2.32384e9	0.246088
$709$	−2.65033e9	−0.279279	−0.139640	−	0.990202i	$-0.544594\pi$
$709$	−2.65033e9	−0.279279	−0.139640	+	0.990202i	$0.544594\pi$
$710$	5.59975e9	0.587170
$711$	−4.58059e9	−0.477944
$712$	1.58430e9	0.164497
$713$	3.74363e9	0.386794
$714$	−2.98029e8	−0.0306418
$715$	2.30932e9	0.236273
$716$	−5.95520e9	−0.606319
$717$	−2.61097e9	−0.264536
$718$	3.34194e9	0.336948
$719$	3.92078e8	0.0393389	0.0196694	−	0.999807i	$-0.493739\pi$
$719$	3.92078e8	0.0393389	0.0196694	+	0.999807i	$0.493739\pi$
$720$	3.73248e8	0.0372678
$721$	1.84417e9	0.183243
$722$	−6.12547e9	−0.605703
$723$	−6.82108e9	−0.671226
$724$	−4.38493e9	−0.429415
$725$	−1.79041e9	−0.174489
$726$	1.10644e10	1.07312
$727$	−7.35649e9	−0.710068	−0.355034	−	0.934853i	$-0.615531\pi$
$727$	−7.35649e9	−0.710068	−0.355034	+	0.934853i	$0.615531\pi$
$728$	−4.08326e8	−0.0392235
$729$	3.87420e8	0.0370370
$730$	2.63435e9	0.250636
$731$	1.29731e9	0.122838
$732$	6.74081e8	0.0635218
$733$	−4.22284e9	−0.396042	−0.198021	−	0.980198i	$-0.563451\pi$
$733$	−4.22284e9	−0.396042	−0.198021	+	0.980198i	$0.563451\pi$
$734$	1.03092e9	0.0962253
$735$	−2.33474e9	−0.216886
$736$	−9.29792e8	−0.0859634
$737$	1.21571e9	0.111865
$738$	−2.05558e9	−0.188251
$739$	−1.25928e9	−0.114780	−0.0573899	−	0.998352i	$-0.518278\pi$
$739$	−1.25928e9	−0.114780	−0.0573899	+	0.998352i	$0.518278\pi$
$740$	−2.24071e9	−0.203271
$741$	6.71610e8	0.0606392
$742$	4.04921e9	0.363878
$743$	−3.90911e9	−0.349637	−0.174819	−	0.984601i	$-0.555934\pi$
$743$	−3.90911e9	−0.349637	−0.174819	+	0.984601i	$0.555934\pi$
$744$	−1.82386e9	−0.162362
$745$	4.21656e9	0.373604
$746$	−4.03355e9	−0.355714
$747$	−5.36194e9	−0.470652
$748$	2.04561e9	0.178717
$749$	−6.33221e8	−0.0550642
$750$	4.21875e8	0.0365148
$751$	−5.18730e9	−0.446891	−0.223446	−	0.974716i	$-0.571731\pi$
$751$	−5.18730e9	−0.446891	−0.223446	+	0.974716i	$0.571731\pi$
$752$	−1.64067e9	−0.140688
$753$	1.25648e10	1.07244
$754$	2.01396e9	0.171101
$755$	−7.45237e9	−0.630203
$756$	4.57275e8	0.0384903
$757$	9.32420e9	0.781225	0.390612	−	0.920555i	$-0.372263\pi$
$757$	9.32420e9	0.781225	0.390612	+	0.920555i	$0.372263\pi$
$758$	9.05145e9	0.754876
$759$	6.44235e9	0.534808
$760$	−7.24608e8	−0.0598764
$761$	−1.97932e10	−1.62806	−0.814028	−	0.580826i	$-0.802730\pi$
$761$	−1.97932e10	−1.62806	−0.814028	+	0.580826i	$0.802730\pi$
$762$	−4.78372e9	−0.391673
$763$	−3.34898e9	−0.272946
$764$	2.18436e9	0.177214
$765$	−3.46366e8	−0.0279718
$766$	−1.19060e10	−0.957121
$767$	−2.95456e9	−0.236434
$768$	4.52985e8	0.0360844
$769$	−1.41492e10	−1.12199	−0.560997	−	0.827818i	$-0.689582\pi$
$769$	−1.41492e10	−1.12199	−0.560997	+	0.827818i	$0.689582\pi$
$770$	−3.05247e9	−0.240954
$771$	−6.34249e9	−0.498390
$772$	1.10071e10	0.861015
$773$	−5.39199e9	−0.419876	−0.209938	−	0.977715i	$-0.567326\pi$
$773$	−5.39199e9	−0.419876	−0.209938	+	0.977715i	$0.567326\pi$
$774$	−1.99051e9	−0.154302
$775$	−2.06147e9	−0.159082
$776$	−8.54996e8	−0.0656823
$777$	−2.74515e9	−0.209939
$778$	1.04583e10	0.796221
$779$	3.99061e9	0.302453
$780$	−4.74552e8	−0.0358057
$781$	−4.70883e10	−3.53699
$782$	8.62827e8	0.0645209
$783$	−2.25540e9	−0.167902
$784$	−2.83351e9	−0.209999
$785$	−1.03220e10	−0.761586
$786$	−5.99755e9	−0.440550
$787$	−4.91535e9	−0.359454	−0.179727	−	0.983717i	$-0.557521\pi$
$787$	−4.91535e9	−0.359454	−0.179727	+	0.983717i	$0.557521\pi$
$788$	2.13714e9	0.155593
$789$	−8.98960e9	−0.651585
$790$	−6.28338e9	−0.453418
$791$	−3.95528e9	−0.284158
$792$	−3.13864e9	−0.224493
$793$	−8.57034e8	−0.0610298
$794$	−5.82363e9	−0.412879
$795$	4.70595e9	0.332171
$796$	2.31716e9	0.162840
$797$	9.53876e9	0.667403	0.333701	−	0.942679i	$-0.391702\pi$
$797$	9.53876e9	0.667403	0.333701	+	0.942679i	$0.391702\pi$
$798$	−8.87735e8	−0.0618406
$799$	1.52251e9	0.105595
$800$	5.12000e8	0.0353553
$801$	2.25577e9	0.155089
$802$	1.10736e10	0.758020
$803$	−2.21522e10	−1.50978
$804$	−2.49820e8	−0.0169524
$805$	−1.28752e9	−0.0869895
$806$	2.31887e9	0.155993
$807$	−4.86808e9	−0.326062
$808$	5.46431e9	0.364415
$809$	−1.16398e10	−0.772906	−0.386453	−	0.922309i	$-0.626300\pi$
$809$	−1.16398e10	−0.772906	−0.386453	+	0.922309i	$0.626300\pi$
$810$	5.31441e8	0.0351364
$811$	−9.64450e9	−0.634902	−0.317451	−	0.948275i	$-0.602827\pi$
$811$	−9.64450e9	−0.634902	−0.317451	+	0.948275i	$0.602827\pi$
$812$	−2.66206e9	−0.174491
$813$	1.26014e10	0.822434
$814$	1.88421e10	1.22446
$815$	3.03541e9	0.196411
$816$	−4.20360e8	−0.0270836
$817$	3.86429e9	0.247909
$818$	4.25220e9	0.271629
$819$	−5.81386e8	−0.0369803
$820$	−2.81972e9	−0.178590
$821$	−2.68552e10	−1.69366	−0.846831	−	0.531862i	$-0.821493\pi$
$821$	−2.68552e10	−1.69366	−0.846831	+	0.531862i	$0.821493\pi$
$822$	−2.59483e9	−0.162951
$823$	1.27394e10	0.796614	0.398307	−	0.917252i	$-0.369598\pi$
$823$	1.27394e10	0.796614	0.398307	+	0.917252i	$0.369598\pi$
$824$	2.60115e9	0.161965
$825$	−3.54755e9	−0.219958
$826$	3.90535e9	0.241118
$827$	2.59724e10	1.59677	0.798387	−	0.602145i	$-0.205687\pi$
$827$	2.59724e10	1.59677	0.798387	+	0.602145i	$0.205687\pi$
$828$	−1.32386e9	−0.0810470
$829$	3.67664e9	0.224135	0.112068	−	0.993701i	$-0.464253\pi$
$829$	3.67664e9	0.224135	0.112068	+	0.993701i	$0.464253\pi$
$830$	−7.35520e9	−0.446500
$831$	5.53329e9	0.334488
$832$	−5.75930e8	−0.0346688
$833$	2.62943e9	0.157618
$834$	4.27067e9	0.254926
$835$	7.41860e9	0.440981
$836$	6.09323e9	0.360683
$837$	−2.59686e9	−0.153077
$838$	1.24948e10	0.733460
$839$	1.56891e10	0.917128	0.458564	−	0.888661i	$-0.348364\pi$
$839$	1.56891e10	0.917128	0.458564	+	0.888661i	$0.348364\pi$
$840$	6.27264e8	0.0365151
$841$	−4.11992e9	−0.238838
$842$	8.64921e9	0.499326
$843$	6.57566e9	0.378045
$844$	−7.27092e9	−0.416285
$845$	6.03351e8	0.0344010
$846$	−2.33603e9	−0.132642
$847$	1.85944e10	1.05145
$848$	5.71128e9	0.321624
$849$	6.54554e9	0.367087
$850$	−4.75125e8	−0.0265364
$851$	7.94753e9	0.442057
$852$	9.67636e9	0.536011
$853$	1.14117e10	0.629548	0.314774	−	0.949167i	$-0.398071\pi$
$853$	1.14117e10	0.629548	0.314774	+	0.949167i	$0.398071\pi$
$854$	1.13283e9	0.0622389
$855$	−1.03172e9	−0.0564520
$856$	−8.93138e8	−0.0486699
$857$	2.91501e10	1.58201	0.791003	−	0.611813i	$-0.209559\pi$
$857$	2.91501e10	1.58201	0.791003	+	0.611813i	$0.209559\pi$
$858$	3.99051e9	0.215686
$859$	−2.50977e10	−1.35101	−0.675503	−	0.737357i	$-0.736073\pi$
$859$	−2.50977e10	−1.35101	−0.675503	+	0.737357i	$0.736073\pi$
$860$	−2.73046e9	−0.146383
$861$	−3.45451e9	−0.184449
$862$	1.16701e10	0.620583
$863$	−3.67744e10	−1.94764	−0.973818	−	0.227329i	$-0.927001\pi$
$863$	−3.67744e10	−1.94764	−0.973818	+	0.227329i	$0.927001\pi$
$864$	6.44973e8	0.0340207
$865$	1.22609e9	0.0644117
$866$	1.66627e10	0.871831
$867$	−1.06891e10	−0.557022
$868$	−3.06509e9	−0.159083
$869$	5.28370e10	2.73130
$870$	−3.09382e9	−0.159286
$871$	3.17625e8	0.0162874
$872$	−4.72363e9	−0.241251
$873$	−1.21737e9	−0.0619259
$874$	2.57009e9	0.130214
$875$	7.08984e8	0.0357774
$876$	4.55215e9	0.228798
$877$	1.38127e10	0.691482	0.345741	−	0.938330i	$-0.387628\pi$
$877$	1.38127e10	0.691482	0.345741	+	0.938330i	$0.387628\pi$
$878$	8.46372e9	0.422017
$879$	−7.58850e8	−0.0376873
$880$	−4.30541e9	−0.212973
$881$	−8.00225e9	−0.394272	−0.197136	−	0.980376i	$-0.563164\pi$
$881$	−8.00225e9	−0.394272	−0.197136	+	0.980376i	$0.563164\pi$
$882$	−4.03443e9	−0.197989
$883$	−1.82272e10	−0.890957	−0.445479	−	0.895293i	$-0.646967\pi$
$883$	−1.82272e10	−0.890957	−0.445479	+	0.895293i	$0.646967\pi$
$884$	5.34451e8	0.0260211
$885$	4.53875e9	0.220108
$886$	4.90533e9	0.236947
$887$	6.70660e9	0.322678	0.161339	−	0.986899i	$-0.448419\pi$
$887$	6.70660e9	0.322678	0.161339	+	0.986899i	$0.448419\pi$
$888$	−3.87195e9	−0.185560
$889$	−8.03931e9	−0.383763
$890$	3.09434e9	0.147131
$891$	−4.46889e9	−0.211654
$892$	−1.72528e10	−0.813921
$893$	4.53507e9	0.213110
$894$	7.28622e9	0.341052
$895$	−1.16313e10	−0.542308
$896$	7.61266e8	0.0353556
$897$	1.68318e9	0.0778675
$898$	2.70533e10	1.24667
$899$	1.51178e10	0.693952
$900$	7.29000e8	0.0333333
$901$	−5.29994e9	−0.241398
$902$	2.37110e10	1.07579
$903$	−3.34516e9	−0.151185
$904$	−5.57880e9	−0.251161
$905$	−8.56432e9	−0.384081
$906$	−1.28777e10	−0.575294
$907$	6.88929e9	0.306584	0.153292	−	0.988181i	$-0.451013\pi$
$907$	6.88929e9	0.306584	0.153292	+	0.988181i	$0.451013\pi$
$908$	−1.14820e10	−0.509000
$909$	7.78024e9	0.343573
$910$	−7.97511e8	−0.0350826
$911$	2.01962e10	0.885023	0.442511	−	0.896763i	$-0.354088\pi$
$911$	2.01962e10	0.885023	0.442511	+	0.896763i	$0.354088\pi$
$912$	−1.25212e9	−0.0546594
$913$	6.18499e10	2.68962
$914$	1.04400e10	0.452262
$915$	1.31656e9	0.0568157
$916$	1.84023e10	0.791111
$917$	−1.00792e10	−0.431652
$918$	−5.98521e8	−0.0255346
$919$	−2.79955e10	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$919$	−2.79955e10	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$920$	−1.81600e9	−0.0768880
$921$	−1.55600e9	−0.0656297
$922$	−2.47855e10	−1.04145
$923$	−1.23026e10	−0.514983
$924$	−5.27466e9	−0.219960
$925$	−4.37639e9	−0.181811
$926$	−3.05438e10	−1.26411
$927$	3.70359e9	0.152702
$928$	−3.75475e9	−0.154228
$929$	−2.27908e10	−0.932618	−0.466309	−	0.884622i	$-0.654417\pi$
$929$	−2.27908e10	−0.932618	−0.466309	+	0.884622i	$0.654417\pi$
$930$	−3.56222e9	−0.145221
$931$	7.83227e9	0.318100
$932$	−1.21897e10	−0.493215
$933$	−5.18750e9	−0.209109
$934$	2.99928e10	1.20449
$935$	3.99533e9	0.159850
$936$	−8.20026e8	−0.0326860
$937$	2.35907e10	0.936813	0.468407	−	0.883513i	$-0.344828\pi$
$937$	2.35907e10	0.936813	0.468407	+	0.883513i	$0.344828\pi$
$938$	−4.19837e8	−0.0166100
$939$	5.24626e9	0.206785
$940$	−3.20443e9	−0.125836
$941$	4.31905e10	1.68976	0.844880	−	0.534956i	$-0.179672\pi$
$941$	4.31905e10	1.68976	0.844880	+	0.534956i	$0.179672\pi$
$942$	−1.78364e10	−0.695230
$943$	1.00012e10	0.388384
$944$	5.50837e9	0.213118
$945$	8.93116e8	0.0344268
$946$	2.29605e10	0.881783
$947$	3.12877e10	1.19715	0.598576	−	0.801066i	$-0.295734\pi$
$947$	3.12877e10	1.19715	0.598576	+	0.801066i	$0.295734\pi$
$948$	−1.08577e10	−0.413912
$949$	−5.78766e9	−0.219822
$950$	−1.41525e9	−0.0535551
$951$	1.22616e10	0.462291
$952$	−7.06439e8	−0.0265366
$953$	−6.61629e9	−0.247622	−0.123811	−	0.992306i	$-0.539512\pi$
$953$	−6.61629e9	−0.247622	−0.123811	+	0.992306i	$0.539512\pi$
$954$	8.13188e9	0.303230
$955$	4.26632e9	0.158505
$956$	−6.18897e9	−0.229095
$957$	2.60159e10	0.959506
$958$	−1.22401e10	−0.449786
$959$	−4.36076e9	−0.159660
$960$	8.84736e8	0.0322749
$961$	−1.01060e10	−0.367324
$962$	4.92284e9	0.178280
$963$	−1.27168e9	−0.0458864
$964$	−1.61685e10	−0.581299
$965$	2.14982e10	0.770115
$966$	−2.22483e9	−0.0794102
$967$	1.87170e10	0.665646	0.332823	−	0.942989i	$-0.391999\pi$
$967$	1.87170e10	0.665646	0.332823	+	0.942989i	$0.391999\pi$
$968$	2.62267e10	0.929353
$969$	1.16194e9	0.0410253
$970$	−1.66992e9	−0.0587480
$971$	−2.49665e10	−0.875167	−0.437583	−	0.899178i	$-0.644165\pi$
$971$	−2.49665e10	−0.875167	−0.437583	+	0.899178i	$0.644165\pi$
$972$	9.18330e8	0.0320750
$973$	7.17709e9	0.249778
$974$	1.79428e10	0.622206
$975$	−9.26859e8	−0.0320256
$976$	1.59782e9	0.0550115
$977$	9.98809e9	0.342650	0.171325	−	0.985215i	$-0.445195\pi$
$977$	9.98809e9	0.342650	0.171325	+	0.985215i	$0.445195\pi$
$978$	5.24519e9	0.179298
$979$	−2.60203e10	−0.886284
$980$	−5.53419e9	−0.187829
$981$	−6.72564e9	−0.227453
$982$	−1.49447e10	−0.503612
$983$	4.62374e9	0.155259	0.0776294	−	0.996982i	$-0.475265\pi$
$983$	4.62374e9	0.155259	0.0776294	+	0.996982i	$0.475265\pi$
$984$	−4.87248e9	−0.163030
$985$	4.17410e9	0.139167
$986$	3.48433e9	0.115758
$987$	−3.92583e9	−0.129964
$988$	1.59196e9	0.0525151
$989$	9.68461e9	0.318343
$990$	−6.13016e9	−0.200793
$991$	−1.63617e10	−0.534035	−0.267017	−	0.963692i	$-0.586038\pi$
$991$	−1.63617e10	−0.534035	−0.267017	+	0.963692i	$0.586038\pi$
$992$	−4.32321e9	−0.140610
$993$	3.00389e10	0.973558
$994$	1.62617e10	0.525185
$995$	4.52570e9	0.145648
$996$	−1.27098e10	−0.407597
$997$	9.41349e9	0.300828	0.150414	−	0.988623i	$-0.451939\pi$
$997$	9.41349e9	0.300828	0.150414	+	0.988623i	$0.451939\pi$
$998$	7.40596e7	0.00235844
$999$	−5.51299e9	−0.174948

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.8.a.g.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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