LMFDB - Embedded newform 390.8.a.c.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)

$f(q)$

$=$

$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} -216.000 q^{6} +203.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} -216.000 q^{6} +203.000 q^{7} -512.000 q^{8} +729.000 q^{9} -1000.00 q^{10} -7455.00 q^{11} +1728.00 q^{12} +2197.00 q^{13} -1624.00 q^{14} +3375.00 q^{15} +4096.00 q^{16} +15765.0 q^{17} -5832.00 q^{18} +1190.00 q^{19} +8000.00 q^{20} +5481.00 q^{21} +59640.0 q^{22} +2145.00 q^{23} -13824.0 q^{24} +15625.0 q^{25} -17576.0 q^{26} +19683.0 q^{27} +12992.0 q^{28} +87882.0 q^{29} -27000.0 q^{30} -164374. q^{31} -32768.0 q^{32} -201285. q^{33} -126120. q^{34} +25375.0 q^{35} +46656.0 q^{36} -540511. q^{37} -9520.00 q^{38} +59319.0 q^{39} -64000.0 q^{40} +758823. q^{41} -43848.0 q^{42} -37636.0 q^{43} -477120. q^{44} +91125.0 q^{45} -17160.0 q^{46} -100338. q^{47} +110592. q^{48} -782334. q^{49} -125000. q^{50} +425655. q^{51} +140608. q^{52} -2.02275e6 q^{53} -157464. q^{54} -931875. q^{55} -103936. q^{56} +32130.0 q^{57} -703056. q^{58} +192864. q^{59} +216000. q^{60} +2.63663e6 q^{61} +1.31499e6 q^{62} +147987. q^{63} +262144. q^{64} +274625. q^{65} +1.61028e6 q^{66} +216500. q^{67} +1.00896e6 q^{68} +57915.0 q^{69} -203000. q^{70} -2.58282e6 q^{71} -373248. q^{72} -6.10014e6 q^{73} +4.32409e6 q^{74} +421875. q^{75} +76160.0 q^{76} -1.51336e6 q^{77} -474552. q^{78} -358141. q^{79} +512000. q^{80} +531441. q^{81} -6.07058e6 q^{82} +6.94577e6 q^{83} +350784. q^{84} +1.97062e6 q^{85} +301088. q^{86} +2.37281e6 q^{87} +3.81696e6 q^{88} -8.87681e6 q^{89} -729000. q^{90} +445991. q^{91} +137280. q^{92} -4.43810e6 q^{93} +802704. q^{94} +148750. q^{95} -884736. q^{96} -5.71776e6 q^{97} +6.25867e6 q^{98} -5.43470e6 q^{99} +O(q^{100})\)

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
$7$	203.000	0.223693	0.111847	−	0.993725i	$-0.464323\pi$
$7$	203.000	0.223693	0.111847	+	0.993725i	$0.464323\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	−1000.00	−0.316228
$11$	−7455.00	−1.68878	−0.844390	−	0.535728i	$-0.820037\pi$
$11$	−7455.00	−1.68878	−0.844390	+	0.535728i	$0.820037\pi$
$12$	1728.00	0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	−1624.00	−0.158175
$15$	3375.00	0.258199
$16$	4096.00	0.250000
$17$	15765.0	0.778256	0.389128	−	0.921184i	$-0.372776\pi$
$17$	15765.0	0.778256	0.389128	+	0.921184i	$0.372776\pi$
$18$	−5832.00	−0.235702
$19$	1190.00	0.0398024	0.0199012	−	0.999802i	$-0.493665\pi$
$19$	1190.00	0.0398024	0.0199012	+	0.999802i	$0.493665\pi$
$20$	8000.00	0.223607
$21$	5481.00	0.129149
$22$	59640.0	1.19415
$23$	2145.00	0.0367604	0.0183802	−	0.999831i	$-0.494149\pi$
$23$	2145.00	0.0367604	0.0183802	+	0.999831i	$0.494149\pi$
$24$	−13824.0	−0.204124
$25$	15625.0	0.200000
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	12992.0	0.111847
$29$	87882.0	0.669125	0.334562	−	0.942374i	$-0.391412\pi$
$29$	87882.0	0.669125	0.334562	+	0.942374i	$0.391412\pi$
$30$	−27000.0	−0.182574
$31$	−164374.	−0.990985	−0.495493	−	0.868612i	$-0.665012\pi$
$31$	−164374.	−0.990985	−0.495493	+	0.868612i	$0.665012\pi$
$32$	−32768.0	−0.176777
$33$	−201285.	−0.975018
$34$	−126120.	−0.550310
$35$	25375.0	0.100039
$36$	46656.0	0.166667
$37$	−540511.	−1.75428	−0.877139	−	0.480236i	$-0.840551\pi$
$37$	−540511.	−1.75428	−0.877139	+	0.480236i	$0.840551\pi$
$38$	−9520.00	−0.0281446
$39$	59319.0	0.160128
$40$	−64000.0	−0.158114
$41$	758823.	1.71948	0.859740	−	0.510732i	$-0.170626\pi$
$41$	758823.	1.71948	0.859740	+	0.510732i	$0.170626\pi$
$42$	−43848.0	−0.0913224
$43$	−37636.0	−0.0721878	−0.0360939	−	0.999348i	$-0.511492\pi$
$43$	−37636.0	−0.0721878	−0.0360939	+	0.999348i	$0.511492\pi$
$44$	−477120.	−0.844390
$45$	91125.0	0.149071
$46$	−17160.0	−0.0259935
$47$	−100338.	−0.140969	−0.0704844	−	0.997513i	$-0.522454\pi$
$47$	−100338.	−0.140969	−0.0704844	+	0.997513i	$0.522454\pi$
$48$	110592.	0.144338
$49$	−782334.	−0.949961
$50$	−125000.	−0.141421
$51$	425655.	0.449327
$52$	140608.	0.138675
$53$	−2.02275e6	−1.86628	−0.933138	−	0.359518i	$-0.882941\pi$
$53$	−2.02275e6	−1.86628	−0.933138	+	0.359518i	$0.882941\pi$
$54$	−157464.	−0.136083
$55$	−931875.	−0.755246
$56$	−103936.	−0.0790875
$57$	32130.0	0.0229799
$58$	−703056.	−0.473142
$59$	192864.	0.122256	0.0611279	−	0.998130i	$-0.480530\pi$
$59$	192864.	0.122256	0.0611279	+	0.998130i	$0.480530\pi$
$60$	216000.	0.129099
$61$	2.63663e6	1.48729	0.743644	−	0.668575i	$-0.233096\pi$
$61$	2.63663e6	1.48729	0.743644	+	0.668575i	$0.233096\pi$
$62$	1.31499e6	0.700732
$63$	147987.	0.0745644
$64$	262144.	0.125000
$65$	274625.	0.124035
$66$	1.61028e6	0.689442
$67$	216500.	0.0879419	0.0439710	−	0.999033i	$-0.485999\pi$
$67$	216500.	0.0879419	0.0439710	+	0.999033i	$0.485999\pi$
$68$	1.00896e6	0.389128
$69$	57915.0	0.0212236
$70$	−203000.	−0.0707380
$71$	−2.58282e6	−0.856427	−0.428214	−	0.903678i	$-0.640857\pi$
$71$	−2.58282e6	−0.856427	−0.428214	+	0.903678i	$0.640857\pi$
$72$	−373248.	−0.117851
$73$	−6.10014e6	−1.83531	−0.917655	−	0.397377i	$-0.869921\pi$
$73$	−6.10014e6	−1.83531	−0.917655	+	0.397377i	$0.869921\pi$
$74$	4.32409e6	1.24046
$75$	421875.	0.115470
$76$	76160.0	0.0199012
$77$	−1.51336e6	−0.377769
$78$	−474552.	−0.113228
$79$	−358141.	−0.0817258	−0.0408629	−	0.999165i	$-0.513011\pi$
$79$	−358141.	−0.0817258	−0.0408629	+	0.999165i	$0.513011\pi$
$80$	512000.	0.111803
$81$	531441.	0.111111
$82$	−6.07058e6	−1.21586
$83$	6.94577e6	1.33336	0.666679	−	0.745345i	$-0.267715\pi$
$83$	6.94577e6	1.33336	0.666679	+	0.745345i	$0.267715\pi$
$84$	350784.	0.0645747
$85$	1.97062e6	0.348047
$86$	301088.	0.0510445
$87$	2.37281e6	0.386319
$88$	3.81696e6	0.597074
$89$	−8.87681e6	−1.33473	−0.667363	−	0.744733i	$-0.732577\pi$
$89$	−8.87681e6	−1.33473	−0.667363	+	0.744733i	$0.732577\pi$
$90$	−729000.	−0.105409
$91$	445991.	0.0620413
$92$	137280.	0.0183802
$93$	−4.43810e6	−0.572146
$94$	802704.	0.0996800
$95$	148750.	0.0178002
$96$	−884736.	−0.102062
$97$	−5.71776e6	−0.636099	−0.318050	−	0.948074i	$-0.603028\pi$
$97$	−5.71776e6	−0.636099	−0.318050	+	0.948074i	$0.603028\pi$
$98$	6.25867e6	0.671724
$99$	−5.43470e6	−0.562927
$100$	1.00000e6	0.100000
$101$	5.90063e6	0.569867	0.284933	−	0.958547i	$-0.408028\pi$
$101$	5.90063e6	0.569867	0.284933	+	0.958547i	$0.408028\pi$
$102$	−3.40524e6	−0.317722
$103$	1.11091e7	1.00172	0.500861	−	0.865528i	$-0.333017\pi$
$103$	1.11091e7	1.00172	0.500861	+	0.865528i	$0.333017\pi$
$104$	−1.12486e6	−0.0980581
$105$	685125.	0.0577574
$106$	1.61820e7	1.31966
$107$	2.48754e6	0.196303	0.0981514	−	0.995171i	$-0.468707\pi$
$107$	2.48754e6	0.196303	0.0981514	+	0.995171i	$0.468707\pi$
$108$	1.25971e6	0.0962250
$109$	−6.93482e6	−0.512911	−0.256456	−	0.966556i	$-0.582555\pi$
$109$	−6.93482e6	−0.512911	−0.256456	+	0.966556i	$0.582555\pi$
$110$	7.45500e6	0.534039
$111$	−1.45938e7	−1.01283
$112$	831488.	0.0559233
$113$	−1.22942e7	−0.801539	−0.400770	−	0.916179i	$-0.631257\pi$
$113$	−1.22942e7	−0.801539	−0.400770	+	0.916179i	$0.631257\pi$
$114$	−257040.	−0.0162493
$115$	268125.	0.0164397
$116$	5.62445e6	0.334562
$117$	1.60161e6	0.0924500
$118$	−1.54291e6	−0.0864478
$119$	3.20030e6	0.174091
$120$	−1.72800e6	−0.0912871
$121$	3.60899e7	1.85198
$122$	−2.10931e7	−1.05167
$123$	2.04882e7	0.992742
$124$	−1.05199e7	−0.495493
$125$	1.95312e6	0.0894427
$126$	−1.18390e6	−0.0527250
$127$	2.04438e7	0.885620	0.442810	−	0.896615i	$-0.353982\pi$
$127$	2.04438e7	0.885620	0.442810	+	0.896615i	$0.353982\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.01617e6	−0.0416777
$130$	−2.19700e6	−0.0877058
$131$	2.62737e7	1.02111	0.510553	−	0.859846i	$-0.329441\pi$
$131$	2.62737e7	1.02111	0.510553	+	0.859846i	$0.329441\pi$
$132$	−1.28822e7	−0.487509
$133$	241570.	0.00890353
$134$	−1.73200e6	−0.0621843
$135$	2.46038e6	0.0860663
$136$	−8.07168e6	−0.275155
$137$	−3.76387e6	−0.125058	−0.0625291	−	0.998043i	$-0.519917\pi$
$137$	−3.76387e6	−0.125058	−0.0625291	+	0.998043i	$0.519917\pi$
$138$	−463320.	−0.0150074
$139$	1.56103e7	0.493013	0.246507	−	0.969141i	$-0.420717\pi$
$139$	1.56103e7	0.493013	0.246507	+	0.969141i	$0.420717\pi$
$140$	1.62400e6	0.0500193
$141$	−2.70913e6	−0.0813884
$142$	2.06626e7	0.605585
$143$	−1.63786e7	−0.468383
$144$	2.98598e6	0.0833333
$145$	1.09852e7	0.299242
$146$	4.88011e7	1.29776
$147$	−2.11230e7	−0.548460
$148$	−3.45927e7	−0.877139
$149$	−7.12006e7	−1.76332	−0.881661	−	0.471884i	$-0.843574\pi$
$149$	−7.12006e7	−1.76332	−0.881661	+	0.471884i	$0.843574\pi$
$150$	−3.37500e6	−0.0816497
$151$	−7.60305e7	−1.79709	−0.898543	−	0.438887i	$-0.855373\pi$
$151$	−7.60305e7	−1.79709	−0.898543	+	0.438887i	$0.855373\pi$
$152$	−609280.	−0.0140723
$153$	1.14927e7	0.259419
$154$	1.21069e7	0.267123
$155$	−2.05468e7	−0.443182
$156$	3.79642e6	0.0800641
$157$	−7.69756e7	−1.58746	−0.793732	−	0.608267i	$-0.791865\pi$
$157$	−7.69756e7	−1.58746	−0.793732	+	0.608267i	$0.791865\pi$
$158$	2.86513e6	0.0577889
$159$	−5.46142e7	−1.07750
$160$	−4.09600e6	−0.0790569
$161$	435435.	0.00822305
$162$	−4.25153e6	−0.0785674
$163$	3.15602e6	0.0570799	0.0285400	−	0.999593i	$-0.490914\pi$
$163$	3.15602e6	0.0570799	0.0285400	+	0.999593i	$0.490914\pi$
$164$	4.85647e7	0.859740
$165$	−2.51606e7	−0.436041
$166$	−5.55661e7	−0.942827
$167$	9.26782e7	1.53982	0.769910	−	0.638152i	$-0.220301\pi$
$167$	9.26782e7	1.53982	0.769910	+	0.638152i	$0.220301\pi$
$168$	−2.80627e6	−0.0456612
$169$	4.82681e6	0.0769231
$170$	−1.57650e7	−0.246106
$171$	867510.	0.0132675
$172$	−2.40870e6	−0.0360939
$173$	−1.27790e8	−1.87644	−0.938219	−	0.346043i	$-0.887525\pi$
$173$	−1.27790e8	−1.87644	−0.938219	+	0.346043i	$0.887525\pi$
$174$	−1.89825e7	−0.273169
$175$	3.17187e6	0.0447387
$176$	−3.05357e7	−0.422195
$177$	5.20733e6	0.0705844
$178$	7.10145e7	0.943793
$179$	1.92226e7	0.250511	0.125256	−	0.992124i	$-0.460025\pi$
$179$	1.92226e7	0.250511	0.125256	+	0.992124i	$0.460025\pi$
$180$	5.83200e6	0.0745356
$181$	−1.29165e8	−1.61908	−0.809541	−	0.587063i	$-0.800284\pi$
$181$	−1.29165e8	−1.61908	−0.809541	+	0.587063i	$0.800284\pi$
$182$	−3.56793e6	−0.0438699
$183$	7.11891e7	0.858686
$184$	−1.09824e6	−0.0129968
$185$	−6.75639e7	−0.784537
$186$	3.55048e7	0.404568
$187$	−1.17528e8	−1.31430
$188$	−6.42163e6	−0.0704844
$189$	3.99565e6	0.0430498
$190$	−1.19000e6	−0.0125866
$191$	−6.40331e7	−0.664948	−0.332474	−	0.943112i	$-0.607883\pi$
$191$	−6.40331e7	−0.664948	−0.332474	+	0.943112i	$0.607883\pi$
$192$	7.07789e6	0.0721688
$193$	−1.16484e8	−1.16632	−0.583158	−	0.812358i	$-0.698183\pi$
$193$	−1.16484e8	−1.16632	−0.583158	+	0.812358i	$0.698183\pi$
$194$	4.57421e7	0.449790
$195$	7.41488e6	0.0716115
$196$	−5.00694e7	−0.474981
$197$	1.52434e8	1.42053	0.710265	−	0.703935i	$-0.248575\pi$
$197$	1.52434e8	1.42053	0.710265	+	0.703935i	$0.248575\pi$
$198$	4.34776e7	0.398049
$199$	−2.16184e8	−1.94463	−0.972315	−	0.233676i	$-0.924925\pi$
$199$	−2.16184e8	−1.94463	−0.972315	+	0.233676i	$0.924925\pi$
$200$	−8.00000e6	−0.0707107
$201$	5.84550e6	0.0507733
$202$	−4.72050e7	−0.402957
$203$	1.78400e7	0.149679
$204$	2.72419e7	0.224663
$205$	9.48529e7	0.768975
$206$	−8.88725e7	−0.708325
$207$	1.56371e6	0.0122535
$208$	8.99891e6	0.0693375
$209$	−8.87145e6	−0.0672175
$210$	−5.48100e6	−0.0408406
$211$	−1.00768e7	−0.0738469	−0.0369235	−	0.999318i	$-0.511756\pi$
$211$	−1.00768e7	−0.0738469	−0.0369235	+	0.999318i	$0.511756\pi$
$212$	−1.29456e8	−0.933138
$213$	−6.97362e7	−0.494458
$214$	−1.99003e7	−0.138807
$215$	−4.70450e6	−0.0322834
$216$	−1.00777e7	−0.0680414
$217$	−3.33679e7	−0.221677
$218$	5.54785e7	0.362683
$219$	−1.64704e8	−1.05962
$220$	−5.96400e7	−0.377623
$221$	3.46357e7	0.215850
$222$	1.16750e8	0.716181
$223$	6.21109e6	0.0375060	0.0187530	−	0.999824i	$-0.494030\pi$
$223$	6.21109e6	0.0375060	0.0187530	+	0.999824i	$0.494030\pi$
$224$	−6.65190e6	−0.0395438
$225$	1.13906e7	0.0666667
$226$	9.83534e7	0.566774
$227$	−1.33127e8	−0.755398	−0.377699	−	0.925928i	$-0.623285\pi$
$227$	−1.33127e8	−0.755398	−0.377699	+	0.925928i	$0.623285\pi$
$228$	2.05632e6	0.0114900
$229$	−2.78714e8	−1.53368	−0.766839	−	0.641840i	$-0.778171\pi$
$229$	−2.78714e8	−1.53368	−0.766839	+	0.641840i	$0.778171\pi$
$230$	−2.14500e6	−0.0116246
$231$	−4.08609e7	−0.218105
$232$	−4.49956e7	−0.236571
$233$	−1.12173e8	−0.580955	−0.290478	−	0.956882i	$-0.593814\pi$
$233$	−1.12173e8	−0.580955	−0.290478	+	0.956882i	$0.593814\pi$
$234$	−1.28129e7	−0.0653720
$235$	−1.25422e7	−0.0630432
$236$	1.23433e7	0.0611279
$237$	−9.66981e6	−0.0471844
$238$	−2.56024e7	−0.123101
$239$	2.42337e8	1.14823	0.574114	−	0.818776i	$-0.305347\pi$
$239$	2.42337e8	1.14823	0.574114	+	0.818776i	$0.305347\pi$
$240$	1.38240e7	0.0645497
$241$	−8.28600e7	−0.381316	−0.190658	−	0.981656i	$-0.561062\pi$
$241$	−8.28600e7	−0.381316	−0.190658	+	0.981656i	$0.561062\pi$
$242$	−2.88719e8	−1.30955
$243$	1.43489e7	0.0641500
$244$	1.68745e8	0.743644
$245$	−9.77918e7	−0.424836
$246$	−1.63906e8	−0.701975
$247$	2.61443e6	0.0110392
$248$	8.41595e7	0.350366
$249$	1.87536e8	0.769815
$250$	−1.56250e7	−0.0632456
$251$	−1.65509e7	−0.0660639	−0.0330319	−	0.999454i	$-0.510516\pi$
$251$	−1.65509e7	−0.0660639	−0.0330319	+	0.999454i	$0.510516\pi$
$252$	9.47117e6	0.0372822
$253$	−1.59910e7	−0.0620802
$254$	−1.63550e8	−0.626228
$255$	5.32069e7	0.200945
$256$	1.67772e7	0.0625000
$257$	−2.20973e8	−0.812032	−0.406016	−	0.913866i	$-0.633082\pi$
$257$	−2.20973e8	−0.812032	−0.406016	+	0.913866i	$0.633082\pi$
$258$	8.12938e6	0.0294705
$259$	−1.09724e8	−0.392420
$260$	1.75760e7	0.0620174
$261$	6.40660e7	0.223042
$262$	−2.10189e8	−0.722031
$263$	−8.39805e7	−0.284665	−0.142332	−	0.989819i	$-0.545460\pi$
$263$	−8.39805e7	−0.284665	−0.142332	+	0.989819i	$0.545460\pi$
$264$	1.03058e8	0.344721
$265$	−2.52843e8	−0.834624
$266$	−1.93256e6	−0.00629575
$267$	−2.39674e8	−0.770604
$268$	1.38560e7	0.0439710
$269$	2.20704e8	0.691316	0.345658	−	0.938361i	$-0.387656\pi$
$269$	2.20704e8	0.691316	0.345658	+	0.938361i	$0.387656\pi$
$270$	−1.96830e7	−0.0608581
$271$	−6.07973e8	−1.85563	−0.927816	−	0.373038i	$-0.878316\pi$
$271$	−6.07973e8	−1.85563	−0.927816	+	0.373038i	$0.878316\pi$
$272$	6.45734e7	0.194564
$273$	1.20418e7	0.0358196
$274$	3.01109e7	0.0884295
$275$	−1.16484e8	−0.337756
$276$	3.70656e6	0.0106118
$277$	−2.99074e8	−0.845474	−0.422737	−	0.906252i	$-0.638931\pi$
$277$	−2.99074e8	−0.845474	−0.422737	+	0.906252i	$0.638931\pi$
$278$	−1.24882e8	−0.348613
$279$	−1.19829e8	−0.330328
$280$	−1.29920e7	−0.0353690
$281$	−337770.	−0.000908132 0	−0.000454066	−	1.00000i	$-0.500145\pi$
$281$	−337770.	−0.000908132 0	−0.000454066		1.00000i	$0.500145\pi$
$282$	2.16730e7	0.0575503
$283$	2.52534e8	0.662319	0.331159	−	0.943575i	$-0.392560\pi$
$283$	2.52534e8	0.662319	0.331159	+	0.943575i	$0.392560\pi$
$284$	−1.65301e8	−0.428214
$285$	4.01625e6	0.0102769
$286$	1.31029e8	0.331197
$287$	1.54041e8	0.384636
$288$	−2.38879e7	−0.0589256
$289$	−1.61803e8	−0.394317
$290$	−8.78820e7	−0.211596
$291$	−1.54380e8	−0.367252
$292$	−3.90409e8	−0.917655
$293$	3.54268e8	0.822802	0.411401	−	0.911454i	$-0.365040\pi$
$293$	3.54268e8	0.822802	0.411401	+	0.911454i	$0.365040\pi$
$294$	1.68984e8	0.387820
$295$	2.41080e7	0.0546744
$296$	2.76742e8	0.620231
$297$	−1.46737e8	−0.325006
$298$	5.69605e8	1.24686
$299$	4.71256e6	0.0101955
$300$	2.70000e7	0.0577350
$301$	−7.64011e6	−0.0161479
$302$	6.08244e8	1.27073
$303$	1.59317e8	0.329013
$304$	4.87424e6	0.00995060
$305$	3.29579e8	0.665136
$306$	−9.19415e7	−0.183437
$307$	6.40555e7	0.126349	0.0631746	−	0.998002i	$-0.479878\pi$
$307$	6.40555e7	0.126349	0.0631746	+	0.998002i	$0.479878\pi$
$308$	−9.68554e7	−0.188884
$309$	2.99945e8	0.578345
$310$	1.64374e8	0.313377
$311$	4.29351e8	0.809378	0.404689	−	0.914454i	$-0.367380\pi$
$311$	4.29351e8	0.809378	0.404689	+	0.914454i	$0.367380\pi$
$312$	−3.03713e7	−0.0566139
$313$	5.33615e8	0.983610	0.491805	−	0.870705i	$-0.336337\pi$
$313$	5.33615e8	0.983610	0.491805	+	0.870705i	$0.336337\pi$
$314$	6.15805e8	1.12251
$315$	1.84984e7	0.0333462
$316$	−2.29210e7	−0.0408629
$317$	−2.54348e8	−0.448457	−0.224229	−	0.974537i	$-0.571986\pi$
$317$	−2.54348e8	−0.448457	−0.224229	+	0.974537i	$0.571986\pi$
$318$	4.36913e8	0.761904
$319$	−6.55160e8	−1.13000
$320$	3.27680e7	0.0559017
$321$	6.71635e7	0.113335
$322$	−3.48348e6	−0.00581457
$323$	1.87603e7	0.0309765
$324$	3.40122e7	0.0555556
$325$	3.43281e7	0.0554700
$326$	−2.52482e7	−0.0403616
$327$	−1.87240e8	−0.296129
$328$	−3.88517e8	−0.607928
$329$	−2.03686e7	−0.0315338
$330$	2.01285e8	0.308328
$331$	−6.28985e7	−0.0953328	−0.0476664	−	0.998863i	$-0.515178\pi$
$331$	−6.28985e7	−0.0953328	−0.0476664	+	0.998863i	$0.515178\pi$
$332$	4.44529e8	0.666679
$333$	−3.94033e8	−0.584759
$334$	−7.41426e8	−1.08882
$335$	2.70625e7	0.0393288
$336$	2.24502e7	0.0322873
$337$	−2.79906e8	−0.398389	−0.199195	−	0.979960i	$-0.563833\pi$
$337$	−2.79906e8	−0.398389	−0.199195	+	0.979960i	$0.563833\pi$
$338$	−3.86145e7	−0.0543928
$339$	−3.31943e8	−0.462769
$340$	1.26120e8	0.174023
$341$	1.22541e9	1.67356
$342$	−6.94008e6	−0.00938152
$343$	−3.25993e8	−0.436193
$344$	1.92696e7	0.0255222
$345$	7.23937e6	0.00949149
$346$	1.02232e9	1.32684
$347$	−1.10742e9	−1.42285	−0.711423	−	0.702764i	$-0.751949\pi$
$347$	−1.10742e9	−1.42285	−0.711423	+	0.702764i	$0.751949\pi$
$348$	1.51860e8	0.193160
$349$	1.16157e9	1.46270	0.731349	−	0.682003i	$-0.238891\pi$
$349$	1.16157e9	1.46270	0.731349	+	0.682003i	$0.238891\pi$
$350$	−2.53750e7	−0.0316350
$351$	4.32436e7	0.0533761
$352$	2.44285e8	0.298537
$353$	7.32456e7	0.0886278	0.0443139	−	0.999018i	$-0.485890\pi$
$353$	7.32456e7	0.0886278	0.0443139	+	0.999018i	$0.485890\pi$
$354$	−4.16586e7	−0.0499107
$355$	−3.22853e8	−0.383006
$356$	−5.68116e8	−0.667363
$357$	8.64080e7	0.100511
$358$	−1.53781e8	−0.177138
$359$	3.65483e8	0.416904	0.208452	−	0.978033i	$-0.433157\pi$
$359$	3.65483e8	0.416904	0.208452	+	0.978033i	$0.433157\pi$
$360$	−4.66560e7	−0.0527046
$361$	−8.92456e8	−0.998416
$362$	1.03332e9	1.14486
$363$	9.74426e8	1.06924
$364$	2.85434e7	0.0310207
$365$	−7.62517e8	−0.820776
$366$	−5.69513e8	−0.607183
$367$	−1.52630e9	−1.61179	−0.805895	−	0.592058i	$-0.798316\pi$
$367$	−1.52630e9	−1.61179	−0.805895	+	0.592058i	$0.798316\pi$
$368$	8.78592e6	0.00919009
$369$	5.53182e8	0.573160
$370$	5.40511e8	0.554752
$371$	−4.10618e8	−0.417474
$372$	−2.84038e8	−0.286073
$373$	1.32101e9	1.31803	0.659016	−	0.752129i	$-0.270973\pi$
$373$	1.32101e9	1.31803	0.659016	+	0.752129i	$0.270973\pi$
$374$	9.40225e8	0.929354
$375$	5.27344e7	0.0516398
$376$	5.13731e7	0.0498400
$377$	1.93077e8	0.185582
$378$	−3.19652e7	−0.0304408
$379$	1.23523e9	1.16550	0.582748	−	0.812653i	$-0.301977\pi$
$379$	1.23523e9	1.16550	0.582748	+	0.812653i	$0.301977\pi$
$380$	9.52000e6	0.00890009
$381$	5.51981e8	0.511313
$382$	5.12265e8	0.470189
$383$	1.54632e9	1.40638	0.703190	−	0.711002i	$-0.251758\pi$
$383$	1.54632e9	1.40638	0.703190	+	0.711002i	$0.251758\pi$
$384$	−5.66231e7	−0.0510310
$385$	−1.89171e8	−0.168943
$386$	9.31874e8	0.824711
$387$	−2.74366e7	−0.0240626
$388$	−3.65937e8	−0.318050
$389$	−1.42530e9	−1.22767	−0.613835	−	0.789434i	$-0.710374\pi$
$389$	−1.42530e9	−1.22767	−0.613835	+	0.789434i	$0.710374\pi$
$390$	−5.93190e7	−0.0506370
$391$	3.38159e7	0.0286090
$392$	4.00555e8	0.335862
$393$	7.09389e8	0.589536
$394$	−1.21947e9	−1.00447
$395$	−4.47676e7	−0.0365489
$396$	−3.47820e8	−0.281463
$397$	2.92795e8	0.234853	0.117427	−	0.993082i	$-0.462535\pi$
$397$	2.92795e8	0.234853	0.117427	+	0.993082i	$0.462535\pi$
$398$	1.72947e9	1.37506
$399$	6.52239e6	0.00514046
$400$	6.40000e7	0.0500000
$401$	−1.04546e9	−0.809661	−0.404830	−	0.914392i	$-0.632669\pi$
$401$	−1.04546e9	−0.809661	−0.404830	+	0.914392i	$0.632669\pi$
$402$	−4.67640e7	−0.0359022
$403$	−3.61130e8	−0.274850
$404$	3.77640e8	0.284933
$405$	6.64301e7	0.0496904
$406$	−1.42720e8	−0.105839
$407$	4.02951e9	2.96259
$408$	−2.17935e8	−0.158861
$409$	−1.80726e8	−0.130614	−0.0653068	−	0.997865i	$-0.520803\pi$
$409$	−1.80726e8	−0.130614	−0.0653068	+	0.997865i	$0.520803\pi$
$410$	−7.58823e8	−0.543747
$411$	−1.01624e8	−0.0722024
$412$	7.10980e8	0.500861
$413$	3.91514e7	0.0273478
$414$	−1.25096e7	−0.00866450
$415$	8.68221e8	0.596296
$416$	−7.19913e7	−0.0490290
$417$	4.21477e8	0.284641
$418$	7.09716e7	0.0475300
$419$	−7.27105e8	−0.482890	−0.241445	−	0.970414i	$-0.577621\pi$
$419$	−7.27105e8	−0.482890	−0.241445	+	0.970414i	$0.577621\pi$
$420$	4.38480e7	0.0288787
$421$	−2.44362e8	−0.159605	−0.0798025	−	0.996811i	$-0.525429\pi$
$421$	−2.44362e8	−0.159605	−0.0798025	+	0.996811i	$0.525429\pi$
$422$	8.06141e7	0.0522177
$423$	−7.31464e7	−0.0469896
$424$	1.03565e9	0.659828
$425$	2.46328e8	0.155651
$426$	5.57890e8	0.349635
$427$	5.35236e8	0.332696
$428$	1.59202e8	0.0981514
$429$	−4.42223e8	−0.270421
$430$	3.76360e7	0.0228278
$431$	1.42522e9	0.857457	0.428728	−	0.903433i	$-0.358962\pi$
$431$	1.42522e9	0.857457	0.428728	+	0.903433i	$0.358962\pi$
$432$	8.06216e7	0.0481125
$433$	2.17994e9	1.29044	0.645219	−	0.763998i	$-0.276766\pi$
$433$	2.17994e9	1.29044	0.645219	+	0.763998i	$0.276766\pi$
$434$	2.66943e8	0.156749
$435$	2.96602e8	0.172767
$436$	−4.43828e8	−0.256456
$437$	2.55255e6	0.00146315
$438$	1.31763e9	0.749263
$439$	3.10549e9	1.75188	0.875939	−	0.482422i	$-0.160243\pi$
$439$	3.10549e9	1.75188	0.875939	+	0.482422i	$0.160243\pi$
$440$	4.77120e8	0.267020
$441$	−5.70321e8	−0.316654
$442$	−2.77086e8	−0.152629
$443$	1.75892e9	0.961243	0.480621	−	0.876928i	$-0.340411\pi$
$443$	1.75892e9	0.961243	0.480621	+	0.876928i	$0.340411\pi$
$444$	−9.34003e8	−0.506417
$445$	−1.10960e9	−0.596907
$446$	−4.96887e7	−0.0265207
$447$	−1.92242e9	−1.01805
$448$	5.32152e7	0.0279617
$449$	1.21270e9	0.632255	0.316127	−	0.948717i	$-0.397617\pi$
$449$	1.21270e9	0.632255	0.316127	+	0.948717i	$0.397617\pi$
$450$	−9.11250e7	−0.0471405
$451$	−5.65703e9	−2.90382
$452$	−7.86827e8	−0.400770
$453$	−2.05282e9	−1.03755
$454$	1.06502e9	0.534147
$455$	5.57489e7	0.0277457
$456$	−1.64506e7	−0.00812463
$457$	8.62186e8	0.422566	0.211283	−	0.977425i	$-0.432236\pi$
$457$	8.62186e8	0.422566	0.211283	+	0.977425i	$0.432236\pi$
$458$	2.22971e9	1.08447
$459$	3.10302e8	0.149776
$460$	1.71600e7	0.00821987
$461$	−3.64339e9	−1.73202	−0.866010	−	0.500027i	$-0.833323\pi$
$461$	−3.64339e9	−1.73202	−0.866010	+	0.500027i	$0.833323\pi$
$462$	3.26887e8	0.154223
$463$	−3.27625e9	−1.53406	−0.767032	−	0.641609i	$-0.778267\pi$
$463$	−3.27625e9	−1.53406	−0.767032	+	0.641609i	$0.778267\pi$
$464$	3.59965e8	0.167281
$465$	−5.54762e8	−0.255871
$466$	8.97384e8	0.410798
$467$	−1.64317e9	−0.746575	−0.373288	−	0.927716i	$-0.621770\pi$
$467$	−1.64317e9	−0.746575	−0.373288	+	0.927716i	$0.621770\pi$
$468$	1.02503e8	0.0462250
$469$	4.39495e7	0.0196720
$470$	1.00338e8	0.0445782
$471$	−2.07834e9	−0.916523
$472$	−9.87464e7	−0.0432239
$473$	2.80576e8	0.121909
$474$	7.73585e7	0.0333644
$475$	1.85937e7	0.00796048
$476$	2.04819e8	0.0870454
$477$	−1.47458e9	−0.622092
$478$	−1.93870e9	−0.811919
$479$	2.41116e9	1.00242	0.501211	−	0.865325i	$-0.332888\pi$
$479$	2.41116e9	1.00242	0.501211	+	0.865325i	$0.332888\pi$
$480$	−1.10592e8	−0.0456435
$481$	−1.18750e9	−0.486549
$482$	6.62880e8	0.269631
$483$	1.17567e7	0.00474758
$484$	2.30975e9	0.925990
$485$	−7.14720e8	−0.284472
$486$	−1.14791e8	−0.0453609
$487$	1.95224e9	0.765916	0.382958	−	0.923766i	$-0.374905\pi$
$487$	1.95224e9	0.765916	0.382958	+	0.923766i	$0.374905\pi$
$488$	−1.34996e9	−0.525836
$489$	8.52126e7	0.0329551
$490$	7.82334e8	0.300404
$491$	−3.51945e9	−1.34180	−0.670902	−	0.741546i	$-0.734093\pi$
$491$	−3.51945e9	−1.34180	−0.670902	+	0.741546i	$0.734093\pi$
$492$	1.31125e9	0.496371
$493$	1.38546e9	0.520750
$494$	−2.09154e7	−0.00780589
$495$	−6.79337e8	−0.251749
$496$	−6.73276e8	−0.247746
$497$	−5.24313e8	−0.191577
$498$	−1.50029e9	−0.544341
$499$	−3.23476e9	−1.16544	−0.582721	−	0.812673i	$-0.698012\pi$
$499$	−3.23476e9	−1.16544	−0.582721	+	0.812673i	$0.698012\pi$
$500$	1.25000e8	0.0447214
$501$	2.50231e9	0.889016
$502$	1.32407e8	0.0467142
$503$	3.18448e9	1.11571	0.557854	−	0.829939i	$-0.311625\pi$
$503$	3.18448e9	1.11571	0.557854	+	0.829939i	$0.311625\pi$
$504$	−7.57693e7	−0.0263625
$505$	7.37578e8	0.254852
$506$	1.27928e8	0.0438973
$507$	1.30324e8	0.0444116
$508$	1.30840e9	0.442810
$509$	−2.17599e9	−0.731382	−0.365691	−	0.930736i	$-0.619167\pi$
$509$	−2.17599e9	−0.731382	−0.365691	+	0.930736i	$0.619167\pi$
$510$	−4.25655e8	−0.142090
$511$	−1.23833e9	−0.410547
$512$	−1.34218e8	−0.0441942
$513$	2.34228e7	0.00765998
$514$	1.76778e9	0.574193
$515$	1.38863e9	0.447984
$516$	−6.50350e7	−0.0208388
$517$	7.48020e8	0.238065
$518$	8.77790e8	0.277483
$519$	−3.45032e9	−1.08336
$520$	−1.40608e8	−0.0438529
$521$	4.00793e9	1.24162	0.620810	−	0.783961i	$-0.286804\pi$
$521$	4.00793e9	1.24162	0.620810	+	0.783961i	$0.286804\pi$
$522$	−5.12528e8	−0.157714
$523$	−5.21552e9	−1.59420	−0.797099	−	0.603849i	$-0.793633\pi$
$523$	−5.21552e9	−1.59420	−0.797099	+	0.603849i	$0.793633\pi$
$524$	1.68151e9	0.510553
$525$	8.56406e7	0.0258299
$526$	6.71844e8	0.201288
$527$	−2.59136e9	−0.771241
$528$	−8.24463e8	−0.243754
$529$	−3.40022e9	−0.998649
$530$	2.02275e9	0.590168
$531$	1.40598e8	0.0407519
$532$	1.54605e7	0.00445177
$533$	1.66713e9	0.476898
$534$	1.91739e9	0.544899
$535$	3.10942e8	0.0877892
$536$	−1.10848e8	−0.0310922
$537$	5.19011e8	0.144633
$538$	−1.76563e9	−0.488834
$539$	5.83230e9	1.60428
$540$	1.57464e8	0.0430331
$541$	−1.90069e9	−0.516083	−0.258042	−	0.966134i	$-0.583077\pi$
$541$	−1.90069e9	−0.516083	−0.258042	+	0.966134i	$0.583077\pi$
$542$	4.86379e9	1.31213
$543$	−3.48745e9	−0.934778
$544$	−5.16588e8	−0.137578
$545$	−8.66852e8	−0.229381
$546$	−9.63341e7	−0.0253283
$547$	−4.35102e8	−0.113667	−0.0568337	−	0.998384i	$-0.518100\pi$
$547$	−4.35102e8	−0.113667	−0.0568337	+	0.998384i	$0.518100\pi$
$548$	−2.40887e8	−0.0625291
$549$	1.92211e9	0.495763
$550$	9.31875e8	0.238830
$551$	1.04580e8	0.0266328
$552$	−2.96525e7	−0.00750368
$553$	−7.27026e7	−0.0182815
$554$	2.39260e9	0.597840
$555$	−1.82422e9	−0.452953
$556$	9.99058e8	0.246507
$557$	9.25917e8	0.227028	0.113514	−	0.993536i	$-0.463789\pi$
$557$	9.25917e8	0.227028	0.113514	+	0.993536i	$0.463789\pi$
$558$	9.58629e8	0.233577
$559$	−8.26863e7	−0.0200213
$560$	1.03936e8	0.0250097
$561$	−3.17326e9	−0.758814
$562$	2.70216e6	0.000642146 0
$563$	4.65874e9	1.10024	0.550122	−	0.835084i	$-0.314581\pi$
$563$	4.65874e9	1.10024	0.550122	+	0.835084i	$0.314581\pi$
$564$	−1.73384e8	−0.0406942
$565$	−1.53677e9	−0.358459
$566$	−2.02027e9	−0.468330
$567$	1.07883e8	0.0248548
$568$	1.32241e9	0.302793
$569$	−6.07493e8	−0.138245	−0.0691224	−	0.997608i	$-0.522020\pi$
$569$	−6.07493e8	−0.138245	−0.0691224	+	0.997608i	$0.522020\pi$
$570$	−3.21300e7	−0.00726689
$571$	4.30311e9	0.967289	0.483644	−	0.875265i	$-0.339313\pi$
$571$	4.30311e9	0.967289	0.483644	+	0.875265i	$0.339313\pi$
$572$	−1.04823e9	−0.234192
$573$	−1.72889e9	−0.383908
$574$	−1.23233e9	−0.271979
$575$	3.35156e7	0.00735207
$576$	1.91103e8	0.0416667
$577$	3.42341e9	0.741898	0.370949	−	0.928653i	$-0.379032\pi$
$577$	3.42341e9	0.741898	0.370949	+	0.928653i	$0.379032\pi$
$578$	1.29443e9	0.278824
$579$	−3.14507e9	−0.673373
$580$	7.03056e8	0.149621
$581$	1.40999e9	0.298263
$582$	1.23504e9	0.259686
$583$	1.50796e10	3.15173
$584$	3.12327e9	0.648880
$585$	2.00202e8	0.0413449
$586$	−2.83415e9	−0.581809
$587$	−7.56037e9	−1.54280	−0.771400	−	0.636350i	$-0.780443\pi$
$587$	−7.56037e9	−1.54280	−0.771400	+	0.636350i	$0.780443\pi$
$588$	−1.35187e9	−0.274230
$589$	−1.95605e8	−0.0394436
$590$	−1.92864e8	−0.0386606
$591$	4.11572e9	0.820143
$592$	−2.21393e9	−0.438570
$593$	4.63891e9	0.913533	0.456767	−	0.889587i	$-0.349007\pi$
$593$	4.63891e9	0.913533	0.456767	+	0.889587i	$0.349007\pi$
$594$	1.17389e9	0.229814
$595$	4.00037e8	0.0778557
$596$	−4.55684e9	−0.881661
$597$	−5.83695e9	−1.12273
$598$	−3.77005e7	−0.00720930
$599$	4.52708e9	0.860645	0.430323	−	0.902675i	$-0.358400\pi$
$599$	4.52708e9	0.860645	0.430323	+	0.902675i	$0.358400\pi$
$600$	−2.16000e8	−0.0408248
$601$	−8.82087e9	−1.65749	−0.828745	−	0.559627i	$-0.810945\pi$
$601$	−8.82087e9	−1.65749	−0.828745	+	0.559627i	$0.810945\pi$
$602$	6.11209e7	0.0114183
$603$	1.57828e8	0.0293140
$604$	−4.86595e9	−0.898543
$605$	4.51123e9	0.828231
$606$	−1.27454e9	−0.232647
$607$	−1.13410e9	−0.205822	−0.102911	−	0.994691i	$-0.532816\pi$
$607$	−1.13410e9	−0.205822	−0.102911	+	0.994691i	$0.532816\pi$
$608$	−3.89939e7	−0.00703614
$609$	4.81681e8	0.0864170
$610$	−2.63663e9	−0.470322
$611$	−2.20443e8	−0.0390977
$612$	7.35532e8	0.129709
$613$	−8.86069e9	−1.55366	−0.776830	−	0.629711i	$-0.783173\pi$
$613$	−8.86069e9	−1.55366	−0.776830	+	0.629711i	$0.783173\pi$
$614$	−5.12444e8	−0.0893423
$615$	2.56103e9	0.443968
$616$	7.74843e8	0.133561
$617$	5.56921e9	0.954543	0.477272	−	0.878756i	$-0.341626\pi$
$617$	5.56921e9	0.954543	0.477272	+	0.878756i	$0.341626\pi$
$618$	−2.39956e9	−0.408951
$619$	−3.12441e9	−0.529481	−0.264741	−	0.964320i	$-0.585286\pi$
$619$	−3.12441e9	−0.529481	−0.264741	+	0.964320i	$0.585286\pi$
$620$	−1.31499e9	−0.221591
$621$	4.22200e7	0.00707454
$622$	−3.43481e9	−0.572317
$623$	−1.80199e9	−0.298569
$624$	2.42971e8	0.0400320
$625$	2.44141e8	0.0400000
$626$	−4.26892e9	−0.695518
$627$	−2.39529e8	−0.0388081
$628$	−4.92644e9	−0.793732
$629$	−8.52116e9	−1.36528
$630$	−1.47987e8	−0.0235793
$631$	5.25840e9	0.833202	0.416601	−	0.909089i	$-0.363221\pi$
$631$	5.25840e9	0.833202	0.416601	+	0.909089i	$0.363221\pi$
$632$	1.83368e8	0.0288944
$633$	−2.72073e8	−0.0426355
$634$	2.03478e9	0.317107
$635$	2.55547e9	0.396061
$636$	−3.49531e9	−0.538748
$637$	−1.71879e9	−0.263472
$638$	5.24128e9	0.799034
$639$	−1.88288e9	−0.285476
$640$	−2.62144e8	−0.0395285
$641$	2.02244e9	0.303300	0.151650	−	0.988434i	$-0.451541\pi$
$641$	2.02244e9	0.303300	0.151650	+	0.988434i	$0.451541\pi$
$642$	−5.37308e8	−0.0801403
$643$	−7.02124e9	−1.04154	−0.520769	−	0.853697i	$-0.674355\pi$
$643$	−7.02124e9	−1.04154	−0.520769	+	0.853697i	$0.674355\pi$
$644$	2.78678e7	0.00411152
$645$	−1.27021e8	−0.0186388
$646$	−1.50083e8	−0.0219037
$647$	5.68604e9	0.825362	0.412681	−	0.910876i	$-0.364592\pi$
$647$	5.68604e9	0.825362	0.412681	+	0.910876i	$0.364592\pi$
$648$	−2.72098e8	−0.0392837
$649$	−1.43780e9	−0.206463
$650$	−2.74625e8	−0.0392232
$651$	−9.00934e8	−0.127985
$652$	2.01985e8	0.0285400
$653$	−1.57054e9	−0.220726	−0.110363	−	0.993891i	$-0.535201\pi$
$653$	−1.57054e9	−0.220726	−0.110363	+	0.993891i	$0.535201\pi$
$654$	1.49792e9	0.209395
$655$	3.28421e9	0.456653
$656$	3.10814e9	0.429870
$657$	−4.44700e9	−0.611770
$658$	1.62949e8	0.0222977
$659$	4.92979e9	0.671011	0.335505	−	0.942038i	$-0.391093\pi$
$659$	4.92979e9	0.671011	0.335505	+	0.942038i	$0.391093\pi$
$660$	−1.61028e9	−0.218021
$661$	6.35782e8	0.0856255	0.0428128	−	0.999083i	$-0.486368\pi$
$661$	6.35782e8	0.0856255	0.0428128	+	0.999083i	$0.486368\pi$
$662$	5.03188e8	0.0674105
$663$	9.35164e8	0.124621
$664$	−3.55623e9	−0.471413
$665$	3.01962e7	0.00398178
$666$	3.15226e9	0.413487
$667$	1.88507e8	0.0245973
$668$	5.93141e9	0.769910
$669$	1.67699e8	0.0216541
$670$	−2.16500e8	−0.0278097
$671$	−1.96561e10	−2.51170
$672$	−1.79601e8	−0.0228306
$673$	3.07107e9	0.388362	0.194181	−	0.980966i	$-0.437795\pi$
$673$	3.07107e9	0.388362	0.194181	+	0.980966i	$0.437795\pi$
$674$	2.23925e9	0.281704
$675$	3.07547e8	0.0384900
$676$	3.08916e8	0.0384615
$677$	9.50500e9	1.17731	0.588656	−	0.808384i	$-0.299657\pi$
$677$	9.50500e9	1.17731	0.588656	+	0.808384i	$0.299657\pi$
$678$	2.65554e9	0.327227
$679$	−1.16071e9	−0.142291
$680$	−1.00896e9	−0.123053
$681$	−3.59443e9	−0.436130
$682$	−9.80327e9	−1.18338
$683$	−2.89103e9	−0.347200	−0.173600	−	0.984816i	$-0.555540\pi$
$683$	−2.89103e9	−0.347200	−0.173600	+	0.984816i	$0.555540\pi$
$684$	5.55206e7	0.00663373
$685$	−4.70483e8	−0.0559277
$686$	2.60794e9	0.308435
$687$	−7.52527e9	−0.885469
$688$	−1.54157e8	−0.0180470
$689$	−4.44398e9	−0.517612
$690$	−5.79150e7	−0.00671149
$691$	−8.36892e9	−0.964931	−0.482465	−	0.875915i	$-0.660259\pi$
$691$	−8.36892e9	−0.964931	−0.482465	+	0.875915i	$0.660259\pi$
$692$	−8.17853e9	−0.938219
$693$	−1.10324e9	−0.125923
$694$	8.85934e9	1.00610
$695$	1.95128e9	0.220482
$696$	−1.21488e9	−0.136584
$697$	1.19628e10	1.33820
$698$	−9.29253e9	−1.03428
$699$	−3.02867e9	−0.335415
$700$	2.03000e8	0.0223693
$701$	1.33761e10	1.46661	0.733306	−	0.679899i	$-0.237976\pi$
$701$	1.33761e10	1.46661	0.733306	+	0.679899i	$0.237976\pi$
$702$	−3.45948e8	−0.0377426
$703$	−6.43208e8	−0.0698245
$704$	−1.95428e9	−0.211098
$705$	−3.38641e8	−0.0363980
$706$	−5.85965e8	−0.0626693
$707$	1.19783e9	0.127475
$708$	3.33269e8	0.0352922
$709$	1.39470e10	1.46966	0.734832	−	0.678249i	$-0.237261\pi$
$709$	1.39470e10	1.46966	0.734832	+	0.678249i	$0.237261\pi$
$710$	2.58282e9	0.270826
$711$	−2.61085e8	−0.0272419
$712$	4.54493e9	0.471897
$713$	−3.52582e8	−0.0364290
$714$	−6.91264e8	−0.0710722
$715$	−2.04733e9	−0.209467
$716$	1.23025e9	0.125256
$717$	6.54311e9	0.662929
$718$	−2.92386e9	−0.294796
$719$	−1.89770e10	−1.90405	−0.952023	−	0.306026i	$-0.901001\pi$
$719$	−1.89770e10	−1.90405	−0.952023	+	0.306026i	$0.901001\pi$
$720$	3.73248e8	0.0372678
$721$	2.25514e9	0.224079
$722$	7.13965e9	0.705987
$723$	−2.23722e9	−0.220153
$724$	−8.26654e9	−0.809541
$725$	1.37316e9	0.133825
$726$	−7.79541e9	−0.756068
$727$	1.69583e10	1.63686	0.818431	−	0.574605i	$-0.194844\pi$
$727$	1.69583e10	1.63686	0.818431	+	0.574605i	$0.194844\pi$
$728$	−2.28347e8	−0.0219349
$729$	3.87420e8	0.0370370
$730$	6.10014e9	0.580376
$731$	−5.93332e8	−0.0561806
$732$	4.55610e9	0.429343
$733$	8.04926e9	0.754904	0.377452	−	0.926029i	$-0.376800\pi$
$733$	8.04926e9	0.754904	0.377452	+	0.926029i	$0.376800\pi$
$734$	1.22104e10	1.13971
$735$	−2.64038e9	−0.245279
$736$	−7.02874e7	−0.00649838
$737$	−1.61401e9	−0.148515
$738$	−4.42546e9	−0.405285
$739$	−1.24829e10	−1.13778	−0.568891	−	0.822413i	$-0.692627\pi$
$739$	−1.24829e10	−1.13778	−0.568891	+	0.822413i	$0.692627\pi$
$740$	−4.32409e9	−0.392269
$741$	7.05896e7	0.00637349
$742$	3.28494e9	0.295198
$743$	1.69724e10	1.51804	0.759018	−	0.651069i	$-0.225679\pi$
$743$	1.69724e10	1.51804	0.759018	+	0.651069i	$0.225679\pi$
$744$	2.27231e9	0.202284
$745$	−8.90007e9	−0.788581
$746$	−1.05681e10	−0.931990
$747$	5.06346e9	0.444453
$748$	−7.52180e9	−0.657152
$749$	5.04970e8	0.0439116
$750$	−4.21875e8	−0.0365148
$751$	−1.50087e10	−1.29301	−0.646506	−	0.762909i	$-0.723771\pi$
$751$	−1.50087e10	−1.29301	−0.646506	+	0.762909i	$0.723771\pi$
$752$	−4.10984e8	−0.0352422
$753$	−4.46875e8	−0.0381420
$754$	−1.54461e9	−0.131226
$755$	−9.50381e9	−0.803681
$756$	2.55722e8	0.0215249
$757$	2.16395e10	1.81306	0.906528	−	0.422145i	$-0.138723\pi$
$757$	2.16395e10	1.81306	0.906528	+	0.422145i	$0.138723\pi$
$758$	−9.88185e9	−0.824131
$759$	−4.31756e8	−0.0358420
$760$	−7.61600e7	−0.00629331
$761$	−1.86911e10	−1.53741	−0.768704	−	0.639605i	$-0.779098\pi$
$761$	−1.86911e10	−1.53741	−0.768704	+	0.639605i	$0.779098\pi$
$762$	−4.41585e9	−0.361553
$763$	−1.40777e9	−0.114735
$764$	−4.09812e9	−0.332474
$765$	1.43659e9	0.116016
$766$	−1.23705e10	−0.994461
$767$	4.23722e8	0.0339076
$768$	4.52985e8	0.0360844
$769$	1.17344e10	0.930502	0.465251	−	0.885179i	$-0.345964\pi$
$769$	1.17344e10	0.930502	0.465251	+	0.885179i	$0.345964\pi$
$770$	1.51336e9	0.119461
$771$	−5.96627e9	−0.468827
$772$	−7.45499e9	−0.583158
$773$	−1.39918e10	−1.08954	−0.544771	−	0.838585i	$-0.683384\pi$
$773$	−1.39918e10	−1.08954	−0.544771	+	0.838585i	$0.683384\pi$
$774$	2.19493e8	0.0170148
$775$	−2.56834e9	−0.198197
$776$	2.92749e9	0.224895
$777$	−2.96254e9	−0.226564
$778$	1.14024e10	0.868094
$779$	9.02999e8	0.0684394
$780$	4.74552e8	0.0358057
$781$	1.92549e10	1.44632
$782$	−2.70527e8	−0.0202296
$783$	1.72978e9	0.128773
$784$	−3.20444e9	−0.237490
$785$	−9.62195e9	−0.709936
$786$	−5.67511e9	−0.416865
$787$	7.60623e9	0.556234	0.278117	−	0.960547i	$-0.410290\pi$
$787$	7.60623e9	0.556234	0.278117	+	0.960547i	$0.410290\pi$
$788$	9.75578e9	0.710265
$789$	−2.26747e9	−0.164351
$790$	3.58141e8	0.0258440
$791$	−2.49572e9	−0.179299
$792$	2.78256e9	0.199025
$793$	5.79268e9	0.412500
$794$	−2.34236e9	−0.166066
$795$	−6.82677e9	−0.481871
$796$	−1.38357e10	−0.972315
$797$	−2.94649e9	−0.206158	−0.103079	−	0.994673i	$-0.532869\pi$
$797$	−2.94649e9	−0.206158	−0.103079	+	0.994673i	$0.532869\pi$
$798$	−5.21791e7	−0.00363485
$799$	−1.58183e9	−0.109710
$800$	−5.12000e8	−0.0353553
$801$	−6.47120e9	−0.444908
$802$	8.36370e9	0.572516
$803$	4.54765e10	3.09944
$804$	3.74112e8	0.0253867
$805$	5.44294e7	0.00367746
$806$	2.88904e9	0.194348
$807$	5.95900e9	0.399131
$808$	−3.02112e9	−0.201478
$809$	1.92509e10	1.27830	0.639149	−	0.769083i	$-0.279287\pi$
$809$	1.92509e10	1.27830	0.639149	+	0.769083i	$0.279287\pi$
$810$	−5.31441e8	−0.0351364
$811$	−2.83546e10	−1.86659	−0.933297	−	0.359106i	$-0.883082\pi$
$811$	−2.83546e10	−1.86659	−0.933297	+	0.359106i	$0.883082\pi$
$812$	1.14176e9	0.0748393
$813$	−1.64153e10	−1.07135
$814$	−3.22361e10	−2.09487
$815$	3.94503e8	0.0255269
$816$	1.74348e9	0.112332
$817$	−4.47868e7	−0.00287325
$818$	1.44581e9	0.0923578
$819$	3.25127e8	0.0206804
$820$	6.07058e9	0.384487
$821$	−6.18006e9	−0.389755	−0.194877	−	0.980828i	$-0.562431\pi$
$821$	−6.18006e9	−0.389755	−0.194877	+	0.980828i	$0.562431\pi$
$822$	8.12995e8	0.0510548
$823$	−3.06897e9	−0.191908	−0.0959542	−	0.995386i	$-0.530590\pi$
$823$	−3.06897e9	−0.191908	−0.0959542	+	0.995386i	$0.530590\pi$
$824$	−5.68784e9	−0.354162
$825$	−3.14508e9	−0.195004
$826$	−3.13211e8	−0.0193378
$827$	−1.70610e10	−1.04890	−0.524450	−	0.851441i	$-0.675729\pi$
$827$	−1.70610e10	−1.04890	−0.524450	+	0.851441i	$0.675729\pi$
$828$	1.00077e8	0.00612673
$829$	4.04328e9	0.246487	0.123243	−	0.992376i	$-0.460670\pi$
$829$	4.04328e9	0.246487	0.123243	+	0.992376i	$0.460670\pi$
$830$	−6.94577e9	−0.421645
$831$	−8.07501e9	−0.488134
$832$	5.75930e8	0.0346688
$833$	−1.23335e10	−0.739314
$834$	−3.37182e9	−0.201272
$835$	1.15848e10	0.688629
$836$	−5.67773e8	−0.0336088
$837$	−3.23537e9	−0.190715
$838$	5.81684e9	0.341455
$839$	2.37953e9	0.139099	0.0695497	−	0.997578i	$-0.477844\pi$
$839$	2.37953e9	0.139099	0.0695497	+	0.997578i	$0.477844\pi$
$840$	−3.50784e8	−0.0204203
$841$	−9.52663e9	−0.552272
$842$	1.95490e9	0.112858
$843$	−9.11979e6	−0.000524310 0
$844$	−6.44913e8	−0.0369235
$845$	6.03351e8	0.0344010
$846$	5.85171e8	0.0332267
$847$	7.32624e9	0.414275
$848$	−8.28517e9	−0.466569
$849$	6.81841e9	0.382390
$850$	−1.97062e9	−0.110062
$851$	−1.15940e9	−0.0644879
$852$	−4.46312e9	−0.247229
$853$	2.99702e9	0.165336	0.0826682	−	0.996577i	$-0.473656\pi$
$853$	2.99702e9	0.165336	0.0826682	+	0.996577i	$0.473656\pi$
$854$	−4.28189e9	−0.235252
$855$	1.08439e8	0.00593339
$856$	−1.27362e9	−0.0694035
$857$	2.25136e10	1.22183	0.610917	−	0.791695i	$-0.290801\pi$
$857$	2.25136e10	1.22183	0.610917	+	0.791695i	$0.290801\pi$
$858$	3.53779e9	0.191217
$859$	2.37679e9	0.127942	0.0639712	−	0.997952i	$-0.479623\pi$
$859$	2.37679e9	0.127942	0.0639712	+	0.997952i	$0.479623\pi$
$860$	−3.01088e8	−0.0161417
$861$	4.15911e9	0.222070
$862$	−1.14018e10	−0.606313
$863$	6.04220e9	0.320005	0.160003	−	0.987117i	$-0.448850\pi$
$863$	6.04220e9	0.320005	0.160003	+	0.987117i	$0.448850\pi$
$864$	−6.44973e8	−0.0340207
$865$	−1.59737e10	−0.839168
$866$	−1.74395e10	−0.912477
$867$	−4.36869e9	−0.227659
$868$	−2.13555e9	−0.110838
$869$	2.66994e9	0.138017
$870$	−2.37281e9	−0.122165
$871$	4.75650e8	0.0243907
$872$	3.55063e9	0.181342
$873$	−4.16825e9	−0.212033
$874$	−2.04204e7	−0.00103460
$875$	3.96484e8	0.0200077
$876$	−1.05410e10	−0.529809
$877$	1.72325e8	0.00862678	0.00431339	−	0.999991i	$-0.498627\pi$
$877$	1.72325e8	0.00862678	0.00431339	+	0.999991i	$0.498627\pi$
$878$	−2.48439e10	−1.23877
$879$	9.56524e9	0.475045
$880$	−3.81696e9	−0.188811
$881$	4.46757e9	0.220118	0.110059	−	0.993925i	$-0.464896\pi$
$881$	4.46757e9	0.220118	0.110059	+	0.993925i	$0.464896\pi$
$882$	4.56257e9	0.223908
$883$	3.38049e10	1.65241	0.826203	−	0.563373i	$-0.190497\pi$
$883$	3.38049e10	1.65241	0.826203	+	0.563373i	$0.190497\pi$
$884$	2.21669e9	0.107925
$885$	6.50916e8	0.0315663
$886$	−1.40714e10	−0.679701
$887$	3.25703e10	1.56707	0.783537	−	0.621345i	$-0.213413\pi$
$887$	3.25703e10	1.56707	0.783537	+	0.621345i	$0.213413\pi$
$888$	7.47202e9	0.358091
$889$	4.15008e9	0.198107
$890$	8.87681e9	0.422077
$891$	−3.96189e9	−0.187642
$892$	3.97510e8	0.0187530
$893$	−1.19402e8	−0.00561090
$894$	1.53793e10	0.719873
$895$	2.40283e9	0.112032
$896$	−4.25722e8	−0.0197719
$897$	1.27239e8	0.00588637
$898$	−9.70162e9	−0.447072
$899$	−1.44455e10	−0.663093
$900$	7.29000e8	0.0333333
$901$	−3.18886e10	−1.45244
$902$	4.52562e10	2.05331
$903$	−2.06283e8	−0.00932301
$904$	6.29462e9	0.283387
$905$	−1.61456e10	−0.724076
$906$	1.64226e10	0.733657
$907$	2.84191e10	1.26469	0.632345	−	0.774687i	$-0.282092\pi$
$907$	2.84191e10	1.26469	0.632345	+	0.774687i	$0.282092\pi$
$908$	−8.52014e9	−0.377699
$909$	4.30156e9	0.189956
$910$	−4.45991e8	−0.0196192
$911$	−2.28802e9	−0.100264	−0.0501321	−	0.998743i	$-0.515964\pi$
$911$	−2.28802e9	−0.100264	−0.0501321	+	0.998743i	$0.515964\pi$
$912$	1.31604e8	0.00574498
$913$	−5.17807e10	−2.25175
$914$	−6.89749e9	−0.298799
$915$	8.89864e9	0.384016
$916$	−1.78377e10	−0.766839
$917$	5.33355e9	0.228415
$918$	−2.48242e9	−0.105907
$919$	−2.42847e10	−1.03211	−0.516057	−	0.856554i	$-0.672601\pi$
$919$	−2.42847e10	−1.03211	−0.516057	+	0.856554i	$0.672601\pi$
$920$	−1.37280e8	−0.00581232
$921$	1.72950e9	0.0729477
$922$	2.91471e10	1.22472
$923$	−5.67446e9	−0.237530
$924$	−2.61509e9	−0.109052
$925$	−8.44548e9	−0.350856
$926$	2.62100e10	1.08475
$927$	8.09851e9	0.333907
$928$	−2.87972e9	−0.118286
$929$	1.21568e10	0.497467	0.248734	−	0.968572i	$-0.419986\pi$
$929$	1.21568e10	0.497467	0.248734	+	0.968572i	$0.419986\pi$
$930$	4.43810e9	0.180928
$931$	−9.30977e8	−0.0378107
$932$	−7.17908e9	−0.290478
$933$	1.15925e10	0.467295
$934$	1.31454e10	0.527908
$935$	−1.46910e10	−0.587775
$936$	−8.20026e8	−0.0326860
$937$	−7.94555e9	−0.315526	−0.157763	−	0.987477i	$-0.550428\pi$
$937$	−7.94555e9	−0.315526	−0.157763	+	0.987477i	$0.550428\pi$
$938$	−3.51596e8	−0.0139102
$939$	1.44076e10	0.567888
$940$	−8.02704e8	−0.0315216
$941$	1.94938e10	0.762662	0.381331	−	0.924439i	$-0.375466\pi$
$941$	1.94938e10	0.762662	0.381331	+	0.924439i	$0.375466\pi$
$942$	1.66267e10	0.648080
$943$	1.62768e9	0.0632087
$944$	7.89971e8	0.0305639
$945$	4.99456e8	0.0192525
$946$	−2.24461e9	−0.0862029
$947$	1.94572e10	0.744486	0.372243	−	0.928135i	$-0.378589\pi$
$947$	1.94572e10	0.744486	0.372243	+	0.928135i	$0.378589\pi$
$948$	−6.18868e8	−0.0235922
$949$	−1.34020e10	−0.509024
$950$	−1.48750e8	−0.00562891
$951$	−6.86740e9	−0.258917
$952$	−1.63855e9	−0.0615504
$953$	3.37430e10	1.26287	0.631434	−	0.775429i	$-0.282467\pi$
$953$	3.37430e10	1.26287	0.631434	+	0.775429i	$0.282467\pi$
$954$	1.17967e10	0.439886
$955$	−8.00414e9	−0.297374
$956$	1.55096e10	0.574114
$957$	−1.76893e10	−0.652408
$958$	−1.92892e10	−0.708820
$959$	−7.64065e8	−0.0279747
$960$	8.84736e8	0.0322749
$961$	−4.93802e8	−0.0179482
$962$	9.50002e9	0.344042
$963$	1.81341e9	0.0654342
$964$	−5.30304e9	−0.190658
$965$	−1.45605e10	−0.521593
$966$	−9.40540e7	−0.00335704
$967$	−2.03331e10	−0.723121	−0.361561	−	0.932349i	$-0.617756\pi$
$967$	−2.03331e10	−0.723121	−0.361561	+	0.932349i	$0.617756\pi$
$968$	−1.84780e10	−0.654774
$969$	5.06529e8	0.0178843
$970$	5.71776e9	0.201152
$971$	3.37284e10	1.18230	0.591152	−	0.806560i	$-0.298673\pi$
$971$	3.37284e10	1.18230	0.591152	+	0.806560i	$0.298673\pi$
$972$	9.18330e8	0.0320750
$973$	3.16889e9	0.110284
$974$	−1.56179e10	−0.541584
$975$	9.26859e8	0.0320256
$976$	1.07996e10	0.371822
$977$	−1.61617e10	−0.554441	−0.277220	−	0.960806i	$-0.589413\pi$
$977$	−1.61617e10	−0.554441	−0.277220	+	0.960806i	$0.589413\pi$
$978$	−6.81701e8	−0.0233028
$979$	6.61766e10	2.25406
$980$	−6.25867e9	−0.212418
$981$	−5.05548e9	−0.170970
$982$	2.81556e10	0.948799
$983$	−1.64008e10	−0.550715	−0.275358	−	0.961342i	$-0.588796\pi$
$983$	−1.64008e10	−0.550715	−0.275358	+	0.961342i	$0.588796\pi$
$984$	−1.04900e10	−0.350987
$985$	1.90543e10	0.635280
$986$	−1.10837e10	−0.368226
$987$	−5.49953e8	−0.0182060
$988$	1.67324e8	0.00551960
$989$	−8.07292e7	−0.00265365
$990$	5.43469e9	0.178013
$991$	4.58157e10	1.49540	0.747698	−	0.664039i	$-0.231159\pi$
$991$	4.58157e10	1.49540	0.747698	+	0.664039i	$0.231159\pi$
$992$	5.38621e9	0.175183
$993$	−1.69826e9	−0.0550404
$994$	4.19450e9	0.135465
$995$	−2.70229e10	−0.869665
$996$	1.20023e10	0.384907
$997$	2.86766e9	0.0916421	0.0458211	−	0.998950i	$-0.485410\pi$
$997$	2.86766e9	0.0916421	0.0458211	+	0.998950i	$0.485410\pi$
$998$	2.58781e10	0.824091
$999$	−1.06389e10	−0.337611

Twists

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

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