LMFDB - Embedded newform 390.8.a.b.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)

$f(q)$

$=$

$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} -216.000 q^{6} -832.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} -216.000 q^{6} -832.000 q^{7} -512.000 q^{8} +729.000 q^{9} -1000.00 q^{10} -1620.00 q^{11} +1728.00 q^{12} +2197.00 q^{13} +6656.00 q^{14} +3375.00 q^{15} +4096.00 q^{16} -6270.00 q^{17} -5832.00 q^{18} -15340.0 q^{19} +8000.00 q^{20} -22464.0 q^{21} +12960.0 q^{22} +56760.0 q^{23} -13824.0 q^{24} +15625.0 q^{25} -17576.0 q^{26} +19683.0 q^{27} -53248.0 q^{28} +36822.0 q^{29} -27000.0 q^{30} +269576. q^{31} -32768.0 q^{32} -43740.0 q^{33} +50160.0 q^{34} -104000. q^{35} +46656.0 q^{36} +179054. q^{37} +122720. q^{38} +59319.0 q^{39} -64000.0 q^{40} -381462. q^{41} +179712. q^{42} -535756. q^{43} -103680. q^{44} +91125.0 q^{45} -454080. q^{46} -751128. q^{47} +110592. q^{48} -131319. q^{49} -125000. q^{50} -169290. q^{51} +140608. q^{52} +631758. q^{53} -157464. q^{54} -202500. q^{55} +425984. q^{56} -414180. q^{57} -294576. q^{58} -1.64788e6 q^{59} +216000. q^{60} -2.07476e6 q^{61} -2.15661e6 q^{62} -606528. q^{63} +262144. q^{64} +274625. q^{65} +349920. q^{66} -2.62750e6 q^{67} -401280. q^{68} +1.53252e6 q^{69} +832000. q^{70} +5.03539e6 q^{71} -373248. q^{72} +3.72852e6 q^{73} -1.43243e6 q^{74} +421875. q^{75} -981760. q^{76} +1.34784e6 q^{77} -474552. q^{78} +772784. q^{79} +512000. q^{80} +531441. q^{81} +3.05170e6 q^{82} -5.04337e6 q^{83} -1.43770e6 q^{84} -783750. q^{85} +4.28605e6 q^{86} +994194. q^{87} +829440. q^{88} -1.11422e7 q^{89} -729000. q^{90} -1.82790e6 q^{91} +3.63264e6 q^{92} +7.27855e6 q^{93} +6.00902e6 q^{94} -1.91750e6 q^{95} -884736. q^{96} +1.55659e6 q^{97} +1.05055e6 q^{98} -1.18098e6 q^{99} +O(q^{100})\)

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
$7$	−832.000	−0.916812	−0.458406	−	0.888743i	$-0.651579\pi$
$7$	−832.000	−0.916812	−0.458406	+	0.888743i	$0.651579\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	−1000.00	−0.316228
$11$	−1620.00	−0.366978	−0.183489	−	0.983022i	$-0.558739\pi$
$11$	−1620.00	−0.366978	−0.183489	+	0.983022i	$0.558739\pi$
$12$	1728.00	0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	6656.00	0.648284
$15$	3375.00	0.258199
$16$	4096.00	0.250000
$17$	−6270.00	−0.309525	−0.154763	−	0.987952i	$-0.549461\pi$
$17$	−6270.00	−0.309525	−0.154763	+	0.987952i	$0.549461\pi$
$18$	−5832.00	−0.235702
$19$	−15340.0	−0.513083	−0.256542	−	0.966533i	$-0.582583\pi$
$19$	−15340.0	−0.513083	−0.256542	+	0.966533i	$0.582583\pi$
$20$	8000.00	0.223607
$21$	−22464.0	−0.529322
$22$	12960.0	0.259493
$23$	56760.0	0.972736	0.486368	−	0.873754i	$-0.338321\pi$
$23$	56760.0	0.972736	0.486368	+	0.873754i	$0.338321\pi$
$24$	−13824.0	−0.204124
$25$	15625.0	0.200000
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	−53248.0	−0.458406
$29$	36822.0	0.280359	0.140179	−	0.990126i	$-0.455232\pi$
$29$	36822.0	0.280359	0.140179	+	0.990126i	$0.455232\pi$
$30$	−27000.0	−0.182574
$31$	269576.	1.62523	0.812616	−	0.582800i	$-0.198043\pi$
$31$	269576.	1.62523	0.812616	+	0.582800i	$0.198043\pi$
$32$	−32768.0	−0.176777
$33$	−43740.0	−0.211875
$34$	50160.0	0.218868
$35$	−104000.	−0.410011
$36$	46656.0	0.166667
$37$	179054.	0.581136	0.290568	−	0.956854i	$-0.406156\pi$
$37$	179054.	0.581136	0.290568	+	0.956854i	$0.406156\pi$
$38$	122720.	0.362805
$39$	59319.0	0.160128
$40$	−64000.0	−0.158114
$41$	−381462.	−0.864386	−0.432193	−	0.901781i	$-0.642260\pi$
$41$	−381462.	−0.864386	−0.432193	+	0.901781i	$0.642260\pi$
$42$	179712.	0.374287
$43$	−535756.	−1.02761	−0.513804	−	0.857908i	$-0.671764\pi$
$43$	−535756.	−1.02761	−0.513804	+	0.857908i	$0.671764\pi$
$44$	−103680.	−0.183489
$45$	91125.0	0.149071
$46$	−454080.	−0.687828
$47$	−751128.	−1.05529	−0.527645	−	0.849465i	$-0.676925\pi$
$47$	−751128.	−1.05529	−0.527645	+	0.849465i	$0.676925\pi$
$48$	110592.	0.144338
$49$	−131319.	−0.159456
$50$	−125000.	−0.141421
$51$	−169290.	−0.178705
$52$	140608.	0.138675
$53$	631758.	0.582888	0.291444	−	0.956588i	$-0.405864\pi$
$53$	631758.	0.582888	0.291444	+	0.956588i	$0.405864\pi$
$54$	−157464.	−0.136083
$55$	−202500.	−0.164118
$56$	425984.	0.324142
$57$	−414180.	−0.296229
$58$	−294576.	−0.198244
$59$	−1.64788e6	−1.04458	−0.522291	−	0.852767i	$-0.674922\pi$
$59$	−1.64788e6	−1.04458	−0.522291	+	0.852767i	$0.674922\pi$
$60$	216000.	0.129099
$61$	−2.07476e6	−1.17034	−0.585172	−	0.810909i	$-0.698973\pi$
$61$	−2.07476e6	−1.17034	−0.585172	+	0.810909i	$0.698973\pi$
$62$	−2.15661e6	−1.14921
$63$	−606528.	−0.305604
$64$	262144.	0.125000
$65$	274625.	0.124035
$66$	349920.	0.149818
$67$	−2.62750e6	−1.06729	−0.533643	−	0.845710i	$-0.679177\pi$
$67$	−2.62750e6	−1.06729	−0.533643	+	0.845710i	$0.679177\pi$
$68$	−401280.	−0.154763
$69$	1.53252e6	0.561609
$70$	832000.	0.289921
$71$	5.03539e6	1.66966	0.834832	−	0.550505i	$-0.185565\pi$
$71$	5.03539e6	1.66966	0.834832	+	0.550505i	$0.185565\pi$
$72$	−373248.	−0.117851
$73$	3.72852e6	1.12178	0.560889	−	0.827891i	$-0.310460\pi$
$73$	3.72852e6	1.12178	0.560889	+	0.827891i	$0.310460\pi$
$74$	−1.43243e6	−0.410925
$75$	421875.	0.115470
$76$	−981760.	−0.256542
$77$	1.34784e6	0.336450
$78$	−474552.	−0.113228
$79$	772784.	0.176345	0.0881725	−	0.996105i	$-0.471897\pi$
$79$	772784.	0.176345	0.0881725	+	0.996105i	$0.471897\pi$
$80$	512000.	0.111803
$81$	531441.	0.111111
$82$	3.05170e6	0.611213
$83$	−5.04337e6	−0.968161	−0.484081	−	0.875023i	$-0.660846\pi$
$83$	−5.04337e6	−0.968161	−0.484081	+	0.875023i	$0.660846\pi$
$84$	−1.43770e6	−0.264661
$85$	−783750.	−0.138424
$86$	4.28605e6	0.726629
$87$	994194.	0.161865
$88$	829440.	0.129746
$89$	−1.11422e7	−1.67536	−0.837679	−	0.546163i	$-0.816088\pi$
$89$	−1.11422e7	−1.67536	−0.837679	+	0.546163i	$0.816088\pi$
$90$	−729000.	−0.105409
$91$	−1.82790e6	−0.254278
$92$	3.63264e6	0.486368
$93$	7.27855e6	0.938328
$94$	6.00902e6	0.746202
$95$	−1.91750e6	−0.229458
$96$	−884736.	−0.102062
$97$	1.55659e6	0.173171	0.0865853	−	0.996244i	$-0.472404\pi$
$97$	1.55659e6	0.173171	0.0865853	+	0.996244i	$0.472404\pi$
$98$	1.05055e6	0.112753
$99$	−1.18098e6	−0.122326
$100$	1.00000e6	0.100000
$101$	−823602.	−0.0795413	−0.0397706	−	0.999209i	$-0.512663\pi$
$101$	−823602.	−0.0795413	−0.0397706	+	0.999209i	$0.512663\pi$
$102$	1.35432e6	0.126363
$103$	1.41509e7	1.27601	0.638004	−	0.770033i	$-0.279760\pi$
$103$	1.41509e7	1.27601	0.638004	+	0.770033i	$0.279760\pi$
$104$	−1.12486e6	−0.0980581
$105$	−2.80800e6	−0.236720
$106$	−5.05406e6	−0.412164
$107$	−1.40291e6	−0.110710	−0.0553549	−	0.998467i	$-0.517629\pi$
$107$	−1.40291e6	−0.110710	−0.0553549	+	0.998467i	$0.517629\pi$
$108$	1.25971e6	0.0962250
$109$	−1.02553e7	−0.758501	−0.379250	−	0.925294i	$-0.623818\pi$
$109$	−1.02553e7	−0.758501	−0.379250	+	0.925294i	$0.623818\pi$
$110$	1.62000e6	0.116049
$111$	4.83446e6	0.335519
$112$	−3.40787e6	−0.229203
$113$	6.68455e6	0.435810	0.217905	−	0.975970i	$-0.430078\pi$
$113$	6.68455e6	0.435810	0.217905	+	0.975970i	$0.430078\pi$
$114$	3.31344e6	0.209465
$115$	7.09500e6	0.435021
$116$	2.35661e6	0.140179
$117$	1.60161e6	0.0924500
$118$	1.31830e7	0.738631
$119$	5.21664e6	0.283777
$120$	−1.72800e6	−0.0912871
$121$	−1.68628e7	−0.865327
$122$	1.65981e7	0.827559
$123$	−1.02995e7	−0.499054
$124$	1.72529e7	0.812616
$125$	1.95312e6	0.0894427
$126$	4.85222e6	0.216095
$127$	1.03593e7	0.448765	0.224383	−	0.974501i	$-0.427963\pi$
$127$	1.03593e7	0.448765	0.224383	+	0.974501i	$0.427963\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.44654e7	−0.593290
$130$	−2.19700e6	−0.0877058
$131$	2.62587e7	1.02053	0.510263	−	0.860018i	$-0.329548\pi$
$131$	2.62587e7	1.02053	0.510263	+	0.860018i	$0.329548\pi$
$132$	−2.79936e6	−0.105938
$133$	1.27629e7	0.470401
$134$	2.10200e7	0.754685
$135$	2.46038e6	0.0860663
$136$	3.21024e6	0.109434
$137$	−2.03201e7	−0.675155	−0.337578	−	0.941298i	$-0.609608\pi$
$137$	−2.03201e7	−0.675155	−0.337578	+	0.941298i	$0.609608\pi$
$138$	−1.22602e7	−0.397118
$139$	−4.05898e7	−1.28193	−0.640966	−	0.767569i	$-0.721466\pi$
$139$	−4.05898e7	−1.28193	−0.640966	+	0.767569i	$0.721466\pi$
$140$	−6.65600e6	−0.205005
$141$	−2.02805e7	−0.609271
$142$	−4.02831e7	−1.18063
$143$	−3.55914e6	−0.101782
$144$	2.98598e6	0.0833333
$145$	4.60275e6	0.125380
$146$	−2.98282e7	−0.793216
$147$	−3.54561e6	−0.0920621
$148$	1.14595e7	0.290568
$149$	−4.26166e7	−1.05542	−0.527712	−	0.849424i	$-0.676950\pi$
$149$	−4.26166e7	−1.05542	−0.527712	+	0.849424i	$0.676950\pi$
$150$	−3.37500e6	−0.0816497
$151$	1.27591e7	0.301579	0.150789	−	0.988566i	$-0.451818\pi$
$151$	1.27591e7	0.301579	0.150789	+	0.988566i	$0.451818\pi$
$152$	7.85408e6	0.181402
$153$	−4.57083e6	−0.103175
$154$	−1.07827e7	−0.237906
$155$	3.36970e7	0.726826
$156$	3.79642e6	0.0800641
$157$	−7.97107e7	−1.64387	−0.821936	−	0.569580i	$-0.807106\pi$
$157$	−7.97107e7	−1.64387	−0.821936	+	0.569580i	$0.807106\pi$
$158$	−6.18227e6	−0.124695
$159$	1.70575e7	0.336531
$160$	−4.09600e6	−0.0790569
$161$	−4.72243e7	−0.891816
$162$	−4.25153e6	−0.0785674
$163$	−7.91284e7	−1.43112	−0.715560	−	0.698552i	$-0.753828\pi$
$163$	−7.91284e7	−1.43112	−0.715560	+	0.698552i	$0.753828\pi$
$164$	−2.44136e7	−0.432193
$165$	−5.46750e6	−0.0947534
$166$	4.03470e7	0.684593
$167$	−4.77466e7	−0.793295	−0.396648	−	0.917971i	$-0.629827\pi$
$167$	−4.77466e7	−0.793295	−0.396648	+	0.917971i	$0.629827\pi$
$168$	1.15016e7	0.187143
$169$	4.82681e6	0.0769231
$170$	6.27000e6	0.0978805
$171$	−1.11829e7	−0.171028
$172$	−3.42884e7	−0.513804
$173$	1.00377e8	1.47392	0.736958	−	0.675938i	$-0.236261\pi$
$173$	1.00377e8	1.47392	0.736958	+	0.675938i	$0.236261\pi$
$174$	−7.95355e6	−0.114456
$175$	−1.30000e7	−0.183362
$176$	−6.63552e6	−0.0917446
$177$	−4.44927e7	−0.603090
$178$	8.91380e7	1.18466
$179$	−3.01422e7	−0.392816	−0.196408	−	0.980522i	$-0.562928\pi$
$179$	−3.01422e7	−0.392816	−0.196408	+	0.980522i	$0.562928\pi$
$180$	5.83200e6	0.0745356
$181$	−4.91714e7	−0.616365	−0.308182	−	0.951327i	$-0.599721\pi$
$181$	−4.91714e7	−0.616365	−0.308182	+	0.951327i	$0.599721\pi$
$182$	1.46232e7	0.179802
$183$	−5.60186e7	−0.675699
$184$	−2.90611e7	−0.343914
$185$	2.23818e7	0.259892
$186$	−5.82284e7	−0.663498
$187$	1.01574e7	0.113589
$188$	−4.80722e7	−0.527645
$189$	−1.63763e7	−0.176441
$190$	1.53400e7	0.162251
$191$	1.19608e8	1.24207	0.621033	−	0.783784i	$-0.286713\pi$
$191$	1.19608e8	1.24207	0.621033	+	0.783784i	$0.286713\pi$
$192$	7.07789e6	0.0721688
$193$	−1.61741e8	−1.61946	−0.809730	−	0.586803i	$-0.800386\pi$
$193$	−1.61741e8	−1.61946	−0.809730	+	0.586803i	$0.800386\pi$
$194$	−1.24528e7	−0.122450
$195$	7.41488e6	0.0716115
$196$	−8.40442e6	−0.0797281
$197$	−1.13280e7	−0.105566	−0.0527828	−	0.998606i	$-0.516809\pi$
$197$	−1.13280e7	−0.105566	−0.0527828	+	0.998606i	$0.516809\pi$
$198$	9.44784e6	0.0864977
$199$	−2.61145e7	−0.234907	−0.117453	−	0.993078i	$-0.537473\pi$
$199$	−2.61145e7	−0.234907	−0.117453	+	0.993078i	$0.537473\pi$
$200$	−8.00000e6	−0.0707107
$201$	−7.09425e7	−0.616198
$202$	6.58882e6	0.0562442
$203$	−3.06359e7	−0.257036
$204$	−1.08346e7	−0.0893523
$205$	−4.76828e7	−0.386565
$206$	−1.13207e8	−0.902274
$207$	4.13780e7	0.324245
$208$	8.99891e6	0.0693375
$209$	2.48508e7	0.188290
$210$	2.24640e7	0.167386
$211$	−2.04160e8	−1.49618	−0.748088	−	0.663599i	$-0.769028\pi$
$211$	−2.04160e8	−1.49618	−0.748088	+	0.663599i	$0.769028\pi$
$212$	4.04325e7	0.291444
$213$	1.35956e8	0.963981
$214$	1.12233e7	0.0782836
$215$	−6.69695e7	−0.459560
$216$	−1.00777e7	−0.0680414
$217$	−2.24287e8	−1.49003
$218$	8.20424e7	0.536341
$219$	1.00670e8	0.647659
$220$	−1.29600e7	−0.0820589
$221$	−1.37752e7	−0.0858469
$222$	−3.86757e7	−0.237248
$223$	1.10406e8	0.666690	0.333345	−	0.942805i	$-0.391823\pi$
$223$	1.10406e8	0.666690	0.333345	+	0.942805i	$0.391823\pi$
$224$	2.72630e7	0.162071
$225$	1.13906e7	0.0666667
$226$	−5.34764e7	−0.308164
$227$	1.79085e7	0.101618	0.0508089	−	0.998708i	$-0.483820\pi$
$227$	1.79085e7	0.101618	0.0508089	+	0.998708i	$0.483820\pi$
$228$	−2.65075e7	−0.148114
$229$	−2.41964e8	−1.33145	−0.665727	−	0.746196i	$-0.731878\pi$
$229$	−2.41964e8	−1.33145	−0.665727	+	0.746196i	$0.731878\pi$
$230$	−5.67600e7	−0.307606
$231$	3.63917e7	0.194250
$232$	−1.88529e7	−0.0991218
$233$	6.14886e7	0.318456	0.159228	−	0.987242i	$-0.449100\pi$
$233$	6.14886e7	0.318456	0.159228	+	0.987242i	$0.449100\pi$
$234$	−1.28129e7	−0.0653720
$235$	−9.38910e7	−0.471940
$236$	−1.05464e8	−0.522291
$237$	2.08652e7	0.101813
$238$	−4.17331e7	−0.200660
$239$	−1.33235e8	−0.631288	−0.315644	−	0.948878i	$-0.602220\pi$
$239$	−1.33235e8	−0.631288	−0.315644	+	0.948878i	$0.602220\pi$
$240$	1.38240e7	0.0645497
$241$	−1.92189e8	−0.884440	−0.442220	−	0.896907i	$-0.645809\pi$
$241$	−1.92189e8	−0.884440	−0.442220	+	0.896907i	$0.645809\pi$
$242$	1.34902e8	0.611878
$243$	1.43489e7	0.0641500
$244$	−1.32785e8	−0.585172
$245$	−1.64149e7	−0.0713110
$246$	8.23958e7	0.352884
$247$	−3.37020e7	−0.142304
$248$	−1.38023e8	−0.574606
$249$	−1.36171e8	−0.558968
$250$	−1.56250e7	−0.0632456
$251$	−2.15085e8	−0.858522	−0.429261	−	0.903181i	$-0.641226\pi$
$251$	−2.15085e8	−0.858522	−0.429261	+	0.903181i	$0.641226\pi$
$252$	−3.88178e7	−0.152802
$253$	−9.19512e7	−0.356973
$254$	−8.28748e7	−0.317325
$255$	−2.11612e7	−0.0799191
$256$	1.67772e7	0.0625000
$257$	−2.40526e8	−0.883886	−0.441943	−	0.897043i	$-0.645711\pi$
$257$	−2.40526e8	−0.883886	−0.441943	+	0.897043i	$0.645711\pi$
$258$	1.15723e8	0.419519
$259$	−1.48973e8	−0.532793
$260$	1.75760e7	0.0620174
$261$	2.68432e7	0.0934530
$262$	−2.10070e8	−0.721622
$263$	−3.56463e7	−0.120829	−0.0604143	−	0.998173i	$-0.519242\pi$
$263$	−3.56463e7	−0.120829	−0.0604143	+	0.998173i	$0.519242\pi$
$264$	2.23949e7	0.0749092
$265$	7.89698e7	0.260675
$266$	−1.02103e8	−0.332624
$267$	−3.00841e8	−0.967269
$268$	−1.68160e8	−0.533643
$269$	−1.10084e8	−0.344820	−0.172410	−	0.985025i	$-0.555155\pi$
$269$	−1.10084e8	−0.344820	−0.172410	+	0.985025i	$0.555155\pi$
$270$	−1.96830e7	−0.0608581
$271$	3.07412e8	0.938270	0.469135	−	0.883127i	$-0.344566\pi$
$271$	3.07412e8	0.938270	0.469135	+	0.883127i	$0.344566\pi$
$272$	−2.56819e7	−0.0773814
$273$	−4.93534e7	−0.146807
$274$	1.62561e8	0.477407
$275$	−2.53125e7	−0.0733957
$276$	9.80813e7	0.280805
$277$	−5.08360e8	−1.43712	−0.718559	−	0.695466i	$-0.755198\pi$
$277$	−5.08360e8	−1.43712	−0.718559	+	0.695466i	$0.755198\pi$
$278$	3.24718e8	0.906463
$279$	1.96521e8	0.541744
$280$	5.32480e7	0.144961
$281$	4.64714e8	1.24944	0.624718	−	0.780850i	$-0.285214\pi$
$281$	4.64714e8	1.24944	0.624718	+	0.780850i	$0.285214\pi$
$282$	1.62244e8	0.430820
$283$	−1.98450e8	−0.520472	−0.260236	−	0.965545i	$-0.583800\pi$
$283$	−1.98450e8	−0.520472	−0.260236	+	0.965545i	$0.583800\pi$
$284$	3.22265e8	0.834832
$285$	−5.17725e7	−0.132478
$286$	2.84731e7	0.0719704
$287$	3.17376e8	0.792479
$288$	−2.38879e7	−0.0589256
$289$	−3.71026e8	−0.904194
$290$	−3.68220e7	−0.0886573
$291$	4.20280e7	0.0999801
$292$	2.38625e8	0.560889
$293$	−6.36117e8	−1.47741	−0.738703	−	0.674031i	$-0.764562\pi$
$293$	−6.36117e8	−1.47741	−0.738703	+	0.674031i	$0.764562\pi$
$294$	2.83649e7	0.0650977
$295$	−2.05984e8	−0.467151
$296$	−9.16756e7	−0.205463
$297$	−3.18865e7	−0.0706250
$298$	3.40933e8	0.746297
$299$	1.24702e8	0.269788
$300$	2.70000e7	0.0577350
$301$	4.45749e8	0.942123
$302$	−1.02073e8	−0.213248
$303$	−2.22373e7	−0.0459232
$304$	−6.28326e7	−0.128271
$305$	−2.59345e8	−0.523394
$306$	3.65666e7	0.0729558
$307$	−8.22351e8	−1.62208	−0.811041	−	0.584989i	$-0.801099\pi$
$307$	−8.22351e8	−1.62208	−0.811041	+	0.584989i	$0.801099\pi$
$308$	8.62618e7	0.168225
$309$	3.82074e8	0.736704
$310$	−2.69576e8	−0.513943
$311$	1.18109e8	0.222650	0.111325	−	0.993784i	$-0.464491\pi$
$311$	1.18109e8	0.222650	0.111325	+	0.993784i	$0.464491\pi$
$312$	−3.03713e7	−0.0566139
$313$	6.79774e8	1.25302	0.626512	−	0.779412i	$-0.284482\pi$
$313$	6.79774e8	1.25302	0.626512	+	0.779412i	$0.284482\pi$
$314$	6.37686e8	1.16239
$315$	−7.58160e7	−0.136670
$316$	4.94582e7	0.0881725
$317$	−9.09493e7	−0.160358	−0.0801792	−	0.996780i	$-0.525549\pi$
$317$	−9.09493e7	−0.160358	−0.0801792	+	0.996780i	$0.525549\pi$
$318$	−1.36460e8	−0.237963
$319$	−5.96516e7	−0.102886
$320$	3.27680e7	0.0559017
$321$	−3.78785e7	−0.0639183
$322$	3.77795e8	0.630609
$323$	9.61818e7	0.158812
$324$	3.40122e7	0.0555556
$325$	3.43281e7	0.0554700
$326$	6.33027e8	1.01195
$327$	−2.76893e8	−0.437920
$328$	1.95309e8	0.305607
$329$	6.24938e8	0.967501
$330$	4.37400e7	0.0670008
$331$	2.02419e8	0.306799	0.153399	−	0.988164i	$-0.450978\pi$
$331$	2.02419e8	0.306799	0.153399	+	0.988164i	$0.450978\pi$
$332$	−3.22776e8	−0.484081
$333$	1.30530e8	0.193712
$334$	3.81973e8	0.560944
$335$	−3.28437e8	−0.477305
$336$	−9.20125e7	−0.132330
$337$	1.87581e8	0.266983	0.133492	−	0.991050i	$-0.457381\pi$
$337$	1.87581e8	0.266983	0.133492	+	0.991050i	$0.457381\pi$
$338$	−3.86145e7	−0.0543928
$339$	1.80483e8	0.251615
$340$	−5.01600e7	−0.0692120
$341$	−4.36713e8	−0.596425
$342$	8.94629e7	0.120935
$343$	7.94445e8	1.06300
$344$	2.74307e8	0.363314
$345$	1.91565e8	0.251159
$346$	−8.03016e8	−1.04222
$347$	8.47541e8	1.08895	0.544474	−	0.838777i	$-0.316729\pi$
$347$	8.47541e8	1.08895	0.544474	+	0.838777i	$0.316729\pi$
$348$	6.36284e7	0.0809326
$349$	−4.29257e8	−0.540541	−0.270270	−	0.962784i	$-0.587113\pi$
$349$	−4.29257e8	−0.540541	−0.270270	+	0.962784i	$0.587113\pi$
$350$	1.04000e8	0.129657
$351$	4.32436e7	0.0533761
$352$	5.30842e7	0.0648732
$353$	9.30820e8	1.12630	0.563150	−	0.826355i	$-0.309589\pi$
$353$	9.30820e8	1.12630	0.563150	+	0.826355i	$0.309589\pi$
$354$	3.55941e8	0.426449
$355$	6.29424e8	0.746696
$356$	−7.13104e8	−0.837679
$357$	1.40849e8	0.163838
$358$	2.41138e8	0.277763
$359$	−1.21647e8	−0.138762	−0.0693812	−	0.997590i	$-0.522102\pi$
$359$	−1.21647e8	−0.138762	−0.0693812	+	0.997590i	$0.522102\pi$
$360$	−4.66560e7	−0.0527046
$361$	−6.58556e8	−0.736746
$362$	3.93371e8	0.435836
$363$	−4.55295e8	−0.499597
$364$	−1.16986e8	−0.127139
$365$	4.66065e8	0.501674
$366$	4.48149e8	0.477791
$367$	1.39120e9	1.46913	0.734564	−	0.678539i	$-0.237387\pi$
$367$	1.39120e9	1.46913	0.734564	+	0.678539i	$0.237387\pi$
$368$	2.32489e8	0.243184
$369$	−2.78086e8	−0.288129
$370$	−1.79054e8	−0.183771
$371$	−5.25623e8	−0.534399
$372$	4.65827e8	0.469164
$373$	−8.77102e8	−0.875123	−0.437562	−	0.899188i	$-0.644158\pi$
$373$	−8.77102e8	−0.875123	−0.437562	+	0.899188i	$0.644158\pi$
$374$	−8.12592e7	−0.0803197
$375$	5.27344e7	0.0516398
$376$	3.84578e8	0.373101
$377$	8.08979e7	0.0777576
$378$	1.31010e8	0.124762
$379$	−1.14084e9	−1.07644	−0.538219	−	0.842805i	$-0.680903\pi$
$379$	−1.14084e9	−1.07644	−0.538219	+	0.842805i	$0.680903\pi$
$380$	−1.22720e8	−0.114729
$381$	2.79702e8	0.259095
$382$	−9.56867e8	−0.878274
$383$	1.40382e8	0.127678	0.0638388	−	0.997960i	$-0.479666\pi$
$383$	1.40382e8	0.127678	0.0638388	+	0.997960i	$0.479666\pi$
$384$	−5.66231e7	−0.0510310
$385$	1.68480e8	0.150465
$386$	1.29393e9	1.14513
$387$	−3.90566e8	−0.342536
$388$	9.96220e7	0.0865853
$389$	1.43243e9	1.23382	0.616908	−	0.787035i	$-0.288385\pi$
$389$	1.43243e9	1.23382	0.616908	+	0.787035i	$0.288385\pi$
$390$	−5.93190e7	−0.0506370
$391$	−3.55885e8	−0.301086
$392$	6.72353e7	0.0563763
$393$	7.08986e8	0.589202
$394$	9.06241e7	0.0746461
$395$	9.65980e7	0.0788639
$396$	−7.55827e7	−0.0611631
$397$	1.77561e8	0.142423	0.0712116	−	0.997461i	$-0.477313\pi$
$397$	1.77561e8	0.142423	0.0712116	+	0.997461i	$0.477313\pi$
$398$	2.08916e8	0.166104
$399$	3.44598e8	0.271586
$400$	6.40000e7	0.0500000
$401$	7.89211e7	0.0611206	0.0305603	−	0.999533i	$-0.490271\pi$
$401$	7.89211e7	0.0611206	0.0305603	+	0.999533i	$0.490271\pi$
$402$	5.67540e8	0.435718
$403$	5.92258e8	0.450758
$404$	−5.27105e7	−0.0397706
$405$	6.64301e7	0.0496904
$406$	2.45087e8	0.181752
$407$	−2.90067e8	−0.213265
$408$	8.66765e7	0.0631816
$409$	7.09705e8	0.512916	0.256458	−	0.966555i	$-0.417445\pi$
$409$	7.09705e8	0.512916	0.256458	+	0.966555i	$0.417445\pi$
$410$	3.81462e8	0.273343
$411$	−5.48642e8	−0.389801
$412$	9.05657e8	0.638004
$413$	1.37103e9	0.957685
$414$	−3.31024e8	−0.229276
$415$	−6.30422e8	−0.432975
$416$	−7.19913e7	−0.0490290
$417$	−1.09592e9	−0.740124
$418$	−1.98806e8	−0.133141
$419$	9.46309e8	0.628469	0.314235	−	0.949345i	$-0.398252\pi$
$419$	9.46309e8	0.628469	0.314235	+	0.949345i	$0.398252\pi$
$420$	−1.79712e8	−0.118360
$421$	−1.75088e9	−1.14359	−0.571795	−	0.820397i	$-0.693753\pi$
$421$	−1.75088e9	−1.14359	−0.571795	+	0.820397i	$0.693753\pi$
$422$	1.63328e9	1.05796
$423$	−5.47572e8	−0.351763
$424$	−3.23460e8	−0.206082
$425$	−9.79688e7	−0.0619051
$426$	−1.08764e9	−0.681637
$427$	1.72620e9	1.07299
$428$	−8.97861e7	−0.0553549
$429$	−9.60968e7	−0.0587636
$430$	5.35756e8	0.324958
$431$	−5.22066e8	−0.314091	−0.157045	−	0.987591i	$-0.550197\pi$
$431$	−5.22066e8	−0.314091	−0.157045	+	0.987591i	$0.550197\pi$
$432$	8.06216e7	0.0481125
$433$	2.42047e9	1.43282	0.716411	−	0.697678i	$-0.245784\pi$
$433$	2.42047e9	1.43282	0.716411	+	0.697678i	$0.245784\pi$
$434$	1.79430e9	1.05361
$435$	1.24274e8	0.0723884
$436$	−6.56340e8	−0.379250
$437$	−8.70698e8	−0.499094
$438$	−8.05361e8	−0.457964
$439$	−1.19202e9	−0.672446	−0.336223	−	0.941782i	$-0.609150\pi$
$439$	−1.19202e9	−0.672446	−0.336223	+	0.941782i	$0.609150\pi$
$440$	1.03680e8	0.0580244
$441$	−9.57316e7	−0.0531521
$442$	1.10202e8	0.0607029
$443$	1.38125e9	0.754849	0.377425	−	0.926040i	$-0.376810\pi$
$443$	1.38125e9	0.754849	0.377425	+	0.926040i	$0.376810\pi$
$444$	3.09405e8	0.167760
$445$	−1.39278e9	−0.749243
$446$	−8.83244e8	−0.471421
$447$	−1.15065e9	−0.609349
$448$	−2.18104e8	−0.114601
$449$	8.41320e8	0.438631	0.219315	−	0.975654i	$-0.429618\pi$
$449$	8.41320e8	0.438631	0.219315	+	0.975654i	$0.429618\pi$
$450$	−9.11250e7	−0.0471405
$451$	6.17968e8	0.317211
$452$	4.27811e8	0.217905
$453$	3.44496e8	0.174117
$454$	−1.43268e8	−0.0718546
$455$	−2.28488e8	−0.113717
$456$	2.12060e8	0.104733
$457$	−2.44873e9	−1.20015	−0.600073	−	0.799945i	$-0.704862\pi$
$457$	−2.44873e9	−1.20015	−0.600073	+	0.799945i	$0.704862\pi$
$458$	1.93571e9	0.941480
$459$	−1.23412e8	−0.0595682
$460$	4.54080e8	0.217510
$461$	1.76100e9	0.837154	0.418577	−	0.908181i	$-0.362529\pi$
$461$	1.76100e9	0.837154	0.418577	+	0.908181i	$0.362529\pi$
$462$	−2.91133e8	−0.137355
$463$	1.44158e9	0.675000	0.337500	−	0.941326i	$-0.390419\pi$
$463$	1.44158e9	0.675000	0.337500	+	0.941326i	$0.390419\pi$
$464$	1.50823e8	0.0700897
$465$	9.09819e8	0.419633
$466$	−4.91909e8	−0.225182
$467$	8.90140e8	0.404435	0.202218	−	0.979341i	$-0.435185\pi$
$467$	8.90140e8	0.404435	0.202218	+	0.979341i	$0.435185\pi$
$468$	1.02503e8	0.0462250
$469$	2.18608e9	0.978501
$470$	7.51128e8	0.333712
$471$	−2.15219e9	−0.949090
$472$	8.43713e8	0.369315
$473$	8.67925e8	0.377110
$474$	−1.66921e8	−0.0719926
$475$	−2.39687e8	−0.102617
$476$	3.33865e8	0.141888
$477$	4.60552e8	0.194296
$478$	1.06588e9	0.446388
$479$	−5.28886e8	−0.219881	−0.109941	−	0.993938i	$-0.535066\pi$
$479$	−5.28886e8	−0.219881	−0.109941	+	0.993938i	$0.535066\pi$
$480$	−1.10592e8	−0.0456435
$481$	3.93382e8	0.161178
$482$	1.53751e9	0.625394
$483$	−1.27506e9	−0.514890
$484$	−1.07922e9	−0.432663
$485$	1.94574e8	0.0774443
$486$	−1.14791e8	−0.0453609
$487$	−3.63779e9	−1.42721	−0.713603	−	0.700551i	$-0.752938\pi$
$487$	−3.63779e9	−1.42721	−0.713603	+	0.700551i	$0.752938\pi$
$488$	1.06228e9	0.413779
$489$	−2.13647e9	−0.826257
$490$	1.31319e8	0.0504245
$491$	−3.40558e9	−1.29839	−0.649196	−	0.760621i	$-0.724894\pi$
$491$	−3.40558e9	−1.29839	−0.649196	+	0.760621i	$0.724894\pi$
$492$	−6.59166e8	−0.249527
$493$	−2.30874e8	−0.0867782
$494$	2.69616e8	0.100624
$495$	−1.47622e8	−0.0547059
$496$	1.10418e9	0.406308
$497$	−4.18945e9	−1.53077
$498$	1.08937e9	0.395250
$499$	−2.97186e9	−1.07072	−0.535361	−	0.844624i	$-0.679824\pi$
$499$	−2.97186e9	−1.07072	−0.535361	+	0.844624i	$0.679824\pi$
$500$	1.25000e8	0.0447214
$501$	−1.28916e9	−0.458009
$502$	1.72068e9	0.607066
$503$	2.51360e9	0.880658	0.440329	−	0.897836i	$-0.354862\pi$
$503$	2.51360e9	0.880658	0.440329	+	0.897836i	$0.354862\pi$
$504$	3.10542e8	0.108047
$505$	−1.02950e8	−0.0355719
$506$	7.35610e8	0.252418
$507$	1.30324e8	0.0444116
$508$	6.62998e8	0.224383
$509$	1.04588e9	0.351534	0.175767	−	0.984432i	$-0.443759\pi$
$509$	1.04588e9	0.351534	0.175767	+	0.984432i	$0.443759\pi$
$510$	1.69290e8	0.0565114
$511$	−3.10213e9	−1.02846
$512$	−1.34218e8	−0.0441942
$513$	−3.01937e8	−0.0987429
$514$	1.92421e9	0.625001
$515$	1.76886e9	0.570648
$516$	−9.25786e8	−0.296645
$517$	1.21683e9	0.387268
$518$	1.19178e9	0.376741
$519$	2.71018e9	0.850966
$520$	−1.40608e8	−0.0438529
$521$	−3.10468e9	−0.961800	−0.480900	−	0.876776i	$-0.659690\pi$
$521$	−3.10468e9	−0.961800	−0.480900	+	0.876776i	$0.659690\pi$
$522$	−2.14746e8	−0.0660812
$523$	−2.77553e9	−0.848380	−0.424190	−	0.905573i	$-0.639441\pi$
$523$	−2.77553e9	−0.848380	−0.424190	+	0.905573i	$0.639441\pi$
$524$	1.68056e9	0.510263
$525$	−3.51000e8	−0.105864
$526$	2.85170e8	0.0854387
$527$	−1.69024e9	−0.503051
$528$	−1.79159e8	−0.0529688
$529$	−1.83128e8	−0.0537848
$530$	−6.31758e8	−0.184325
$531$	−1.20130e9	−0.348194
$532$	8.16824e8	0.235200
$533$	−8.38072e8	−0.239738
$534$	2.40673e9	0.683962
$535$	−1.75364e8	−0.0495109
$536$	1.34528e9	0.377343
$537$	−8.13840e8	−0.226793
$538$	8.80673e8	0.243824
$539$	2.12737e8	0.0585170
$540$	1.57464e8	0.0430331
$541$	−3.75396e9	−1.01929	−0.509647	−	0.860384i	$-0.670224\pi$
$541$	−3.75396e9	−1.01929	−0.509647	+	0.860384i	$0.670224\pi$
$542$	−2.45929e9	−0.663457
$543$	−1.32763e9	−0.355858
$544$	2.05455e8	0.0547169
$545$	−1.28191e9	−0.339212
$546$	3.94827e8	0.103808
$547$	4.67599e9	1.22157	0.610785	−	0.791797i	$-0.290854\pi$
$547$	4.67599e9	1.22157	0.610785	+	0.791797i	$0.290854\pi$
$548$	−1.30049e9	−0.337578
$549$	−1.51250e9	−0.390115
$550$	2.02500e8	0.0518986
$551$	−5.64849e8	−0.143847
$552$	−7.84650e8	−0.198559
$553$	−6.42956e8	−0.161675
$554$	4.06688e9	1.01620
$555$	6.04307e8	0.150049
$556$	−2.59775e9	−0.640966
$557$	8.14035e9	1.99595	0.997976	−	0.0635904i	$-0.0202551\pi$
$557$	8.14035e9	1.99595	0.997976	+	0.0635904i	$0.0202551\pi$
$558$	−1.57217e9	−0.383071
$559$	−1.17706e9	−0.285007
$560$	−4.25984e8	−0.102503
$561$	2.74250e8	0.0655807
$562$	−3.71772e9	−0.883485
$563$	6.97672e9	1.64768	0.823838	−	0.566826i	$-0.191829\pi$
$563$	6.97672e9	1.64768	0.823838	+	0.566826i	$0.191829\pi$
$564$	−1.29795e9	−0.304636
$565$	8.35568e8	0.194900
$566$	1.58760e9	0.368030
$567$	−4.42159e8	−0.101868
$568$	−2.57812e9	−0.590315
$569$	3.66809e9	0.834732	0.417366	−	0.908738i	$-0.362953\pi$
$569$	3.66809e9	0.834732	0.417366	+	0.908738i	$0.362953\pi$
$570$	4.14180e8	0.0936757
$571$	−5.17645e9	−1.16361	−0.581803	−	0.813330i	$-0.697652\pi$
$571$	−5.17645e9	−1.16361	−0.581803	+	0.813330i	$0.697652\pi$
$572$	−2.27785e8	−0.0508908
$573$	3.22943e9	0.717107
$574$	−2.53901e9	−0.560368
$575$	8.86875e8	0.194547
$576$	1.91103e8	0.0416667
$577$	5.84683e9	1.26708	0.633542	−	0.773708i	$-0.281600\pi$
$577$	5.84683e9	1.26708	0.633542	+	0.773708i	$0.281600\pi$
$578$	2.96821e9	0.639362
$579$	−4.36701e9	−0.934996
$580$	2.94576e8	0.0626902
$581$	4.19609e9	0.887622
$582$	−3.36224e8	−0.0706966
$583$	−1.02345e9	−0.213907
$584$	−1.90900e9	−0.396608
$585$	2.00202e8	0.0413449
$586$	5.08893e9	1.04468
$587$	−1.30442e9	−0.266184	−0.133092	−	0.991104i	$-0.542491\pi$
$587$	−1.30442e9	−0.266184	−0.133092	+	0.991104i	$0.542491\pi$
$588$	−2.26919e8	−0.0460310
$589$	−4.13530e9	−0.833879
$590$	1.64788e9	0.330326
$591$	−3.05856e8	−0.0609483
$592$	7.33405e8	0.145284
$593$	−8.28240e9	−1.63104	−0.815520	−	0.578728i	$-0.803549\pi$
$593$	−8.28240e9	−1.63104	−0.815520	+	0.578728i	$0.803549\pi$
$594$	2.55092e8	0.0499394
$595$	6.52080e8	0.126909
$596$	−2.72746e9	−0.527712
$597$	−7.05091e8	−0.135624
$598$	−9.97614e8	−0.190769
$599$	−1.23192e9	−0.234200	−0.117100	−	0.993120i	$-0.537360\pi$
$599$	−1.23192e9	−0.234200	−0.117100	+	0.993120i	$0.537360\pi$
$600$	−2.16000e8	−0.0408248
$601$	−1.21346e9	−0.228016	−0.114008	−	0.993480i	$-0.536369\pi$
$601$	−1.21346e9	−0.228016	−0.114008	+	0.993480i	$0.536369\pi$
$602$	−3.56599e9	−0.666182
$603$	−1.91545e9	−0.355762
$604$	8.16583e8	0.150789
$605$	−2.10785e9	−0.386986
$606$	1.77898e8	0.0324726
$607$	−9.04510e9	−1.64155	−0.820773	−	0.571254i	$-0.806457\pi$
$607$	−9.04510e9	−1.64155	−0.820773	+	0.571254i	$0.806457\pi$
$608$	5.02661e8	0.0907011
$609$	−8.27169e8	−0.148400
$610$	2.07476e9	0.370096
$611$	−1.65023e9	−0.292685
$612$	−2.92533e8	−0.0515876
$613$	7.26329e9	1.27357	0.636784	−	0.771042i	$-0.280264\pi$
$613$	7.26329e9	1.27357	0.636784	+	0.771042i	$0.280264\pi$
$614$	6.57881e9	1.14699
$615$	−1.28743e9	−0.223184
$616$	−6.90094e8	−0.118953
$617$	4.46369e9	0.765060	0.382530	−	0.923943i	$-0.375053\pi$
$617$	4.46369e9	0.765060	0.382530	+	0.923943i	$0.375053\pi$
$618$	−3.05659e9	−0.520928
$619$	5.10307e9	0.864796	0.432398	−	0.901683i	$-0.357667\pi$
$619$	5.10307e9	0.864796	0.432398	+	0.901683i	$0.357667\pi$
$620$	2.15661e9	0.363413
$621$	1.11721e9	0.187203
$622$	−9.44875e8	−0.157437
$623$	9.27035e9	1.53599
$624$	2.42971e8	0.0400320
$625$	2.44141e8	0.0400000
$626$	−5.43819e9	−0.886022
$627$	6.70972e8	0.108710
$628$	−5.10149e9	−0.821936
$629$	−1.12267e9	−0.179876
$630$	6.06528e8	0.0966404
$631$	−1.22287e8	−0.0193766	−0.00968830	−	0.999953i	$-0.503084\pi$
$631$	−1.22287e8	−0.0193766	−0.00968830	+	0.999953i	$0.503084\pi$
$632$	−3.95665e8	−0.0623474
$633$	−5.51233e9	−0.863818
$634$	7.27594e8	0.113391
$635$	1.29492e9	0.200694
$636$	1.09168e9	0.168265
$637$	−2.88508e8	−0.0442252
$638$	4.77213e8	0.0727512
$639$	3.67080e9	0.556555
$640$	−2.62144e8	−0.0395285
$641$	−8.18040e8	−0.122679	−0.0613397	−	0.998117i	$-0.519537\pi$
$641$	−8.18040e8	−0.122679	−0.0613397	+	0.998117i	$0.519537\pi$
$642$	3.03028e8	0.0451971
$643$	3.47596e9	0.515627	0.257814	−	0.966195i	$-0.416998\pi$
$643$	3.47596e9	0.515627	0.257814	+	0.966195i	$0.416998\pi$
$644$	−3.02236e9	−0.445908
$645$	−1.80818e9	−0.265327
$646$	−7.69454e8	−0.112297
$647$	6.02782e9	0.874975	0.437487	−	0.899225i	$-0.355868\pi$
$647$	6.02782e9	0.874975	0.437487	+	0.899225i	$0.355868\pi$
$648$	−2.72098e8	−0.0392837
$649$	2.66956e9	0.383339
$650$	−2.74625e8	−0.0392232
$651$	−6.05576e9	−0.860270
$652$	−5.06422e9	−0.715560
$653$	−9.18685e9	−1.29113	−0.645566	−	0.763705i	$-0.723378\pi$
$653$	−9.18685e9	−1.29113	−0.645566	+	0.763705i	$0.723378\pi$
$654$	2.21515e9	0.309657
$655$	3.28234e9	0.456394
$656$	−1.56247e9	−0.216097
$657$	2.71809e9	0.373926
$658$	−4.99951e9	−0.684127
$659$	−5.12328e9	−0.697348	−0.348674	−	0.937244i	$-0.613368\pi$
$659$	−5.12328e9	−0.697348	−0.348674	+	0.937244i	$0.613368\pi$
$660$	−3.49920e8	−0.0473767
$661$	−2.72882e9	−0.367510	−0.183755	−	0.982972i	$-0.558825\pi$
$661$	−2.72882e9	−0.367510	−0.183755	+	0.982972i	$0.558825\pi$
$662$	−1.61935e9	−0.216940
$663$	−3.71930e8	−0.0495637
$664$	2.58221e9	0.342297
$665$	1.59536e9	0.210370
$666$	−1.04424e9	−0.136975
$667$	2.09002e9	0.272715
$668$	−3.05578e9	−0.396648
$669$	2.98095e9	0.384914
$670$	2.62750e9	0.337506
$671$	3.36111e9	0.429491
$672$	7.36100e8	0.0935717
$673$	1.00840e10	1.27520	0.637600	−	0.770368i	$-0.279927\pi$
$673$	1.00840e10	1.27520	0.637600	+	0.770368i	$0.279927\pi$
$674$	−1.50065e9	−0.188786
$675$	3.07547e8	0.0384900
$676$	3.08916e8	0.0384615
$677$	7.96445e9	0.986497	0.493248	−	0.869889i	$-0.335809\pi$
$677$	7.96445e9	0.986497	0.493248	+	0.869889i	$0.335809\pi$
$678$	−1.44386e9	−0.177919
$679$	−1.29509e9	−0.158765
$680$	4.01280e8	0.0489403
$681$	4.83531e8	0.0586691
$682$	3.49370e9	0.421736
$683$	−6.48415e8	−0.0778718	−0.0389359	−	0.999242i	$-0.512397\pi$
$683$	−6.48415e8	−0.0778718	−0.0389359	+	0.999242i	$0.512397\pi$
$684$	−7.15703e8	−0.0855139
$685$	−2.54001e9	−0.301939
$686$	−6.35556e9	−0.751657
$687$	−6.53302e9	−0.768715
$688$	−2.19446e9	−0.256902
$689$	1.38797e9	0.161664
$690$	−1.53252e9	−0.177596
$691$	−4.18756e9	−0.482823	−0.241411	−	0.970423i	$-0.577610\pi$
$691$	−4.18756e9	−0.482823	−0.241411	+	0.970423i	$0.577610\pi$
$692$	6.42413e9	0.736958
$693$	9.82575e8	0.112150
$694$	−6.78033e9	−0.770003
$695$	−5.07372e9	−0.573297
$696$	−5.09027e8	−0.0572280
$697$	2.39177e9	0.267549
$698$	3.43406e9	0.382220
$699$	1.66019e9	0.183860
$700$	−8.32000e8	−0.0916812
$701$	−1.25173e10	−1.37246	−0.686228	−	0.727387i	$-0.740735\pi$
$701$	−1.25173e10	−1.37246	−0.686228	+	0.727387i	$0.740735\pi$
$702$	−3.45948e8	−0.0377426
$703$	−2.74669e9	−0.298171
$704$	−4.24673e8	−0.0458723
$705$	−2.53506e9	−0.272474
$706$	−7.44656e9	−0.796415
$707$	6.85237e8	0.0729244
$708$	−2.84753e9	−0.301545
$709$	−6.35338e9	−0.669488	−0.334744	−	0.942309i	$-0.608650\pi$
$709$	−6.35338e9	−0.669488	−0.334744	+	0.942309i	$0.608650\pi$
$710$	−5.03539e9	−0.527994
$711$	5.63360e8	0.0587817
$712$	5.70483e9	0.592329
$713$	1.53011e10	1.58092
$714$	−1.12679e9	−0.115851
$715$	−4.44893e8	−0.0455181
$716$	−1.92910e9	−0.196408
$717$	−3.59736e9	−0.364474
$718$	9.73178e8	0.0981199
$719$	1.79725e9	0.180326	0.0901630	−	0.995927i	$-0.471261\pi$
$719$	1.79725e9	0.180326	0.0901630	+	0.995927i	$0.471261\pi$
$720$	3.73248e8	0.0372678
$721$	−1.17735e10	−1.16986
$722$	5.26845e9	0.520958
$723$	−5.18910e9	−0.510632
$724$	−3.14697e9	−0.308182
$725$	5.75344e8	0.0560718
$726$	3.64236e9	0.353268
$727$	1.23272e10	1.18985	0.594926	−	0.803781i	$-0.297181\pi$
$727$	1.23272e10	1.18985	0.594926	+	0.803781i	$0.297181\pi$
$728$	9.35887e8	0.0899008
$729$	3.87420e8	0.0370370
$730$	−3.72852e9	−0.354737
$731$	3.35919e9	0.318071
$732$	−3.58519e9	−0.337849
$733$	−2.09796e9	−0.196758	−0.0983790	−	0.995149i	$-0.531366\pi$
$733$	−2.09796e9	−0.196758	−0.0983790	+	0.995149i	$0.531366\pi$
$734$	−1.11296e10	−1.03883
$735$	−4.43202e8	−0.0411714
$736$	−1.85991e9	−0.171957
$737$	4.25655e9	0.391671
$738$	2.22469e9	0.203738
$739$	9.39528e9	0.856355	0.428178	−	0.903695i	$-0.359156\pi$
$739$	9.39528e9	0.856355	0.428178	+	0.903695i	$0.359156\pi$
$740$	1.43243e9	0.129946
$741$	−9.09953e8	−0.0821591
$742$	4.20498e9	0.377877
$743$	−1.98025e10	−1.77117	−0.885584	−	0.464479i	$-0.846242\pi$
$743$	−1.98025e10	−1.77117	−0.885584	+	0.464479i	$0.846242\pi$
$744$	−3.72662e9	−0.331749
$745$	−5.32707e9	−0.472000
$746$	7.01682e9	0.618806
$747$	−3.67662e9	−0.322720
$748$	6.50074e8	0.0567946
$749$	1.16722e9	0.101500
$750$	−4.21875e8	−0.0365148
$751$	1.28042e10	1.10309	0.551546	−	0.834145i	$-0.314038\pi$
$751$	1.28042e10	1.10309	0.551546	+	0.834145i	$0.314038\pi$
$752$	−3.07662e9	−0.263822
$753$	−5.80728e9	−0.495668
$754$	−6.47183e8	−0.0549829
$755$	1.59489e9	0.134870
$756$	−1.04808e9	−0.0882203
$757$	1.26151e10	1.05695	0.528474	−	0.848949i	$-0.322764\pi$
$757$	1.26151e10	1.05695	0.528474	+	0.848949i	$0.322764\pi$
$758$	9.12675e9	0.761156
$759$	−2.48268e9	−0.206099
$760$	9.81760e8	0.0811256
$761$	5.58105e9	0.459060	0.229530	−	0.973302i	$-0.426281\pi$
$761$	5.58105e9	0.459060	0.229530	+	0.973302i	$0.426281\pi$
$762$	−2.23762e9	−0.183208
$763$	8.53241e9	0.695402
$764$	7.65494e9	0.621033
$765$	−5.71354e8	−0.0461413
$766$	−1.12305e9	−0.0902817
$767$	−3.62038e9	−0.289715
$768$	4.52985e8	0.0360844
$769$	−1.33276e10	−1.05684	−0.528421	−	0.848982i	$-0.677216\pi$
$769$	−1.33276e10	−1.05684	−0.528421	+	0.848982i	$0.677216\pi$
$770$	−1.34784e9	−0.106395
$771$	−6.49420e9	−0.510312
$772$	−1.03514e10	−0.809730
$773$	1.47538e10	1.14888	0.574441	−	0.818546i	$-0.305219\pi$
$773$	1.47538e10	1.14888	0.574441	+	0.818546i	$0.305219\pi$
$774$	3.12453e9	0.242210
$775$	4.21212e9	0.325046
$776$	−7.96976e8	−0.0612251
$777$	−4.02227e9	−0.307608
$778$	−1.14595e10	−0.872440
$779$	5.85163e9	0.443502
$780$	4.74552e8	0.0358057
$781$	−8.15734e9	−0.612731
$782$	2.84708e9	0.212900
$783$	7.24767e8	0.0539551
$784$	−5.37883e8	−0.0398640
$785$	−9.96384e9	−0.735162
$786$	−5.67189e9	−0.416628
$787$	1.05157e10	0.768999	0.384499	−	0.923125i	$-0.374374\pi$
$787$	1.05157e10	0.768999	0.384499	+	0.923125i	$0.374374\pi$
$788$	−7.24993e8	−0.0527828
$789$	−9.62450e8	−0.0697604
$790$	−7.72784e8	−0.0557652
$791$	−5.56154e9	−0.399556
$792$	6.04662e8	0.0432488
$793$	−4.55825e9	−0.324595
$794$	−1.42049e9	−0.100708
$795$	2.13218e9	0.150501
$796$	−1.67133e9	−0.117453
$797$	−9.68385e8	−0.0677554	−0.0338777	−	0.999426i	$-0.510786\pi$
$797$	−9.68385e8	−0.0677554	−0.0338777	+	0.999426i	$0.510786\pi$
$798$	−2.75678e9	−0.192040
$799$	4.70957e9	0.326639
$800$	−5.12000e8	−0.0353553
$801$	−8.12270e9	−0.558453
$802$	−6.31369e8	−0.0432188
$803$	−6.04021e9	−0.411668
$804$	−4.54032e9	−0.308099
$805$	−5.90304e9	−0.398832
$806$	−4.73807e9	−0.318734
$807$	−2.97227e9	−0.199082
$808$	4.21684e8	0.0281221
$809$	−3.87523e9	−0.257322	−0.128661	−	0.991689i	$-0.541068\pi$
$809$	−3.87523e9	−0.257322	−0.128661	+	0.991689i	$0.541068\pi$
$810$	−5.31441e8	−0.0351364
$811$	−9.66509e9	−0.636258	−0.318129	−	0.948048i	$-0.603054\pi$
$811$	−9.66509e9	−0.636258	−0.318129	+	0.948048i	$0.603054\pi$
$812$	−1.96070e9	−0.128518
$813$	8.30011e9	0.541710
$814$	2.32054e9	0.150801
$815$	−9.89105e9	−0.640016
$816$	−6.93412e8	−0.0446761
$817$	8.21850e9	0.527248
$818$	−5.67764e9	−0.362686
$819$	−1.33254e9	−0.0847593
$820$	−3.05170e9	−0.193283
$821$	6.39815e9	0.403509	0.201755	−	0.979436i	$-0.435336\pi$
$821$	6.39815e9	0.403509	0.201755	+	0.979436i	$0.435336\pi$
$822$	4.38914e9	0.275631
$823$	−5.75631e9	−0.359952	−0.179976	−	0.983671i	$-0.557602\pi$
$823$	−5.75631e9	−0.359952	−0.179976	+	0.983671i	$0.557602\pi$
$824$	−7.24525e9	−0.451137
$825$	−6.83438e8	−0.0423750
$826$	−1.09683e10	−0.677186
$827$	2.10020e10	1.29120	0.645598	−	0.763677i	$-0.276608\pi$
$827$	2.10020e10	1.29120	0.645598	+	0.763677i	$0.276608\pi$
$828$	2.64819e9	0.162123
$829$	1.55396e9	0.0947327	0.0473663	−	0.998878i	$-0.484917\pi$
$829$	1.55396e9	0.0947327	0.0473663	+	0.998878i	$0.484917\pi$
$830$	5.04337e9	0.306159
$831$	−1.37257e10	−0.829720
$832$	5.75930e8	0.0346688
$833$	8.23370e8	0.0493557
$834$	8.76739e9	0.523347
$835$	−5.96833e9	−0.354772
$836$	1.59045e9	0.0941452
$837$	5.30606e9	0.312776
$838$	−7.57047e9	−0.444395
$839$	−2.37036e10	−1.38563	−0.692816	−	0.721115i	$-0.743630\pi$
$839$	−2.37036e10	−1.38563	−0.692816	+	0.721115i	$0.743630\pi$
$840$	1.43770e9	0.0836931
$841$	−1.58940e10	−0.921399
$842$	1.40071e10	0.808640
$843$	1.25473e10	0.721362
$844$	−1.30663e10	−0.748088
$845$	6.03351e8	0.0344010
$846$	4.38058e9	0.248734
$847$	1.40298e10	0.793342
$848$	2.58768e9	0.145722
$849$	−5.35814e9	−0.300495
$850$	7.83750e8	0.0437735
$851$	1.01631e10	0.565292
$852$	8.70116e9	0.481990
$853$	−1.38978e10	−0.766699	−0.383350	−	0.923603i	$-0.625230\pi$
$853$	−1.38978e10	−0.766699	−0.383350	+	0.923603i	$0.625230\pi$
$854$	−1.38096e10	−0.758716
$855$	−1.39786e9	−0.0764859
$856$	7.18289e8	0.0391418
$857$	1.14905e10	0.623601	0.311800	−	0.950148i	$-0.399068\pi$
$857$	1.14905e10	0.623601	0.311800	+	0.950148i	$0.399068\pi$
$858$	7.68774e8	0.0415521
$859$	9.64022e9	0.518933	0.259466	−	0.965752i	$-0.416453\pi$
$859$	9.64022e9	0.518933	0.259466	+	0.965752i	$0.416453\pi$
$860$	−4.28605e9	−0.229780
$861$	8.56916e9	0.457538
$862$	4.17653e9	0.222096
$863$	−1.62057e9	−0.0858283	−0.0429142	−	0.999079i	$-0.513664\pi$
$863$	−1.62057e9	−0.0858283	−0.0429142	+	0.999079i	$0.513664\pi$
$864$	−6.44973e8	−0.0340207
$865$	1.25471e10	0.659155
$866$	−1.93638e10	−1.01316
$867$	−1.00177e10	−0.522037
$868$	−1.43544e10	−0.745016
$869$	−1.25191e9	−0.0647148
$870$	−9.94194e8	−0.0511863
$871$	−5.77262e9	−0.296012
$872$	5.25072e9	0.268170
$873$	1.13476e9	0.0577235
$874$	6.96559e9	0.352913
$875$	−1.62500e9	−0.0820021
$876$	6.44289e9	0.323829
$877$	−3.73615e9	−0.187036	−0.0935181	−	0.995618i	$-0.529811\pi$
$877$	−3.73615e9	−0.187036	−0.0935181	+	0.995618i	$0.529811\pi$
$878$	9.53616e9	0.475491
$879$	−1.71751e10	−0.852981
$880$	−8.29440e8	−0.0410294
$881$	2.76995e10	1.36476	0.682379	−	0.730999i	$-0.260946\pi$
$881$	2.76995e10	1.36476	0.682379	+	0.730999i	$0.260946\pi$
$882$	7.65852e8	0.0375842
$883$	−4.92373e9	−0.240676	−0.120338	−	0.992733i	$-0.538398\pi$
$883$	−4.92373e9	−0.240676	−0.120338	+	0.992733i	$0.538398\pi$
$884$	−8.81612e8	−0.0429235
$885$	−5.56158e9	−0.269710
$886$	−1.10500e10	−0.533759
$887$	−3.10772e9	−0.149523	−0.0747616	−	0.997201i	$-0.523820\pi$
$887$	−3.10772e9	−0.149523	−0.0747616	+	0.997201i	$0.523820\pi$
$888$	−2.47524e9	−0.118624
$889$	−8.61897e9	−0.411433
$890$	1.11422e10	0.529795
$891$	−8.60934e8	−0.0407754
$892$	7.06595e9	0.333345
$893$	1.15223e10	0.541451
$894$	9.20518e9	0.430875
$895$	−3.76778e9	−0.175673
$896$	1.74483e9	0.0810355
$897$	3.36695e9	0.155762
$898$	−6.73056e9	−0.310159
$899$	9.92633e9	0.455648
$900$	7.29000e8	0.0333333
$901$	−3.96112e9	−0.180419
$902$	−4.94375e9	−0.224302
$903$	1.20352e10	0.543935
$904$	−3.42249e9	−0.154082
$905$	−6.14643e9	−0.275647
$906$	−2.75597e9	−0.123119
$907$	3.77483e10	1.67986	0.839928	−	0.542698i	$-0.182597\pi$
$907$	3.77483e10	1.67986	0.839928	+	0.542698i	$0.182597\pi$
$908$	1.14615e9	0.0508089
$909$	−6.00406e8	−0.0265138
$910$	1.82790e9	0.0804097
$911$	1.18830e10	0.520729	0.260364	−	0.965510i	$-0.416157\pi$
$911$	1.18830e10	0.520729	0.260364	+	0.965510i	$0.416157\pi$
$912$	−1.69648e9	−0.0740572
$913$	8.17026e9	0.355294
$914$	1.95898e10	0.848631
$915$	−7.00232e9	−0.302182
$916$	−1.54857e10	−0.665727
$917$	−2.18473e10	−0.935631
$918$	9.87299e8	0.0421211
$919$	3.17859e10	1.35092	0.675462	−	0.737395i	$-0.263944\pi$
$919$	3.17859e10	1.35092	0.675462	+	0.737395i	$0.263944\pi$
$920$	−3.63264e9	−0.153803
$921$	−2.22035e10	−0.936510
$922$	−1.40880e10	−0.591958
$923$	1.10628e10	0.463081
$924$	2.32907e9	0.0971248
$925$	2.79772e9	0.116227
$926$	−1.15326e10	−0.477297
$927$	1.03160e10	0.425336
$928$	−1.20658e9	−0.0495609
$929$	1.69826e10	0.694944	0.347472	−	0.937690i	$-0.387040\pi$
$929$	1.69826e10	0.694944	0.347472	+	0.937690i	$0.387040\pi$
$930$	−7.27855e9	−0.296725
$931$	2.01443e9	0.0818143
$932$	3.93527e9	0.159228
$933$	3.18895e9	0.128547
$934$	−7.12112e9	−0.285979
$935$	1.26968e9	0.0507986
$936$	−8.20026e8	−0.0326860
$937$	−4.90314e9	−0.194709	−0.0973544	−	0.995250i	$-0.531038\pi$
$937$	−4.90314e9	−0.194709	−0.0973544	+	0.995250i	$0.531038\pi$
$938$	−1.74886e10	−0.691904
$939$	1.83539e10	0.723434
$940$	−6.00902e9	−0.235970
$941$	−3.61621e10	−1.41478	−0.707392	−	0.706821i	$-0.750128\pi$
$941$	−3.61621e10	−1.41478	−0.707392	+	0.706821i	$0.750128\pi$
$942$	1.72175e10	0.671108
$943$	−2.16518e10	−0.840819
$944$	−6.74970e9	−0.261145
$945$	−2.04703e9	−0.0789066
$946$	−6.94340e9	−0.266657
$947$	3.80511e10	1.45594	0.727968	−	0.685611i	$-0.240465\pi$
$947$	3.80511e10	1.45594	0.727968	+	0.685611i	$0.240465\pi$
$948$	1.33537e9	0.0509064
$949$	8.19156e9	0.311125
$950$	1.91750e9	0.0725609
$951$	−2.45563e9	−0.0925830
$952$	−2.67092e9	−0.100330
$953$	−3.14982e10	−1.17886	−0.589428	−	0.807821i	$-0.700647\pi$
$953$	−3.14982e10	−1.17886	−0.589428	+	0.807821i	$0.700647\pi$
$954$	−3.68441e9	−0.137388
$955$	1.49510e10	0.555469
$956$	−8.52707e9	−0.315644
$957$	−1.61059e9	−0.0594011
$958$	4.23109e9	0.155479
$959$	1.69063e10	0.618990
$960$	8.84736e8	0.0322749
$961$	4.51586e10	1.64138
$962$	−3.14705e9	−0.113970
$963$	−1.02272e9	−0.0369033
$964$	−1.23001e10	−0.442220
$965$	−2.02177e10	−0.724244
$966$	1.02005e10	0.364082
$967$	2.56598e10	0.912558	0.456279	−	0.889837i	$-0.349182\pi$
$967$	2.56598e10	0.912558	0.456279	+	0.889837i	$0.349182\pi$
$968$	8.63374e9	0.305939
$969$	2.59691e9	0.0916903
$970$	−1.55659e9	−0.0547614
$971$	−2.98277e10	−1.04557	−0.522783	−	0.852466i	$-0.675106\pi$
$971$	−2.98277e10	−1.04557	−0.522783	+	0.852466i	$0.675106\pi$
$972$	9.18330e8	0.0320750
$973$	3.37707e10	1.17529
$974$	2.91023e10	1.00919
$975$	9.26859e8	0.0320256
$976$	−8.49823e9	−0.292586
$977$	−1.30148e10	−0.446486	−0.223243	−	0.974763i	$-0.571664\pi$
$977$	−1.30148e10	−0.446486	−0.223243	+	0.974763i	$0.571664\pi$
$978$	1.70917e10	0.584252
$979$	1.80504e10	0.614820
$980$	−1.05055e9	−0.0356555
$981$	−7.47612e9	−0.252834
$982$	2.72446e10	0.918102
$983$	5.29303e10	1.77733	0.888663	−	0.458560i	$-0.151635\pi$
$983$	5.29303e10	1.77733	0.888663	+	0.458560i	$0.151635\pi$
$984$	5.27333e9	0.176442
$985$	−1.41600e9	−0.0472103
$986$	1.84699e9	0.0613615
$987$	1.68733e10	0.558587
$988$	−2.15693e9	−0.0711518
$989$	−3.04095e10	−0.999591
$990$	1.18098e9	0.0386829
$991$	−3.17231e10	−1.03542	−0.517711	−	0.855555i	$-0.673216\pi$
$991$	−3.17231e10	−1.03542	−0.517711	+	0.855555i	$0.673216\pi$
$992$	−8.83347e9	−0.287303
$993$	5.46532e9	0.177130
$994$	3.35156e10	1.08242
$995$	−3.26431e9	−0.105054
$996$	−8.71495e9	−0.279484
$997$	−9.25426e9	−0.295739	−0.147869	−	0.989007i	$-0.547242\pi$
$997$	−9.25426e9	−0.295739	−0.147869	+	0.989007i	$0.547242\pi$
$998$	2.37749e10	0.757114
$999$	3.52432e9	0.111840

Twists

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

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