LMFDB - Embedded newform 240.8.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("240.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.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:	$74.9724061162$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+27.0000 q^{3} -125.000 q^{5} -1380.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q+27.0000 q^{3} -125.000 q^{5} -1380.00 q^{7} +729.000 q^{9} +3304.00 q^{11} +8506.00 q^{13} -3375.00 q^{15} -9994.00 q^{17} -41236.0 q^{19} -37260.0 q^{21} -84120.0 q^{23} +15625.0 q^{25} +19683.0 q^{27} +132802. q^{29} +55800.0 q^{31} +89208.0 q^{33} +172500. q^{35} +228170. q^{37} +229662. q^{39} -139670. q^{41} +755492. q^{43} -91125.0 q^{45} -836984. q^{47} +1.08086e6 q^{49} -269838. q^{51} +1.64165e6 q^{53} -413000. q^{55} -1.11337e6 q^{57} +989656. q^{59} -1.65816e6 q^{61} -1.00602e6 q^{63} -1.06325e6 q^{65} +4.52384e6 q^{67} -2.27124e6 q^{69} +389408. q^{71} +5.61733e6 q^{73} +421875. q^{75} -4.55952e6 q^{77} -3.90108e6 q^{79} +531441. q^{81} +9.39412e6 q^{83} +1.24925e6 q^{85} +3.58565e6 q^{87} +2.80375e6 q^{89} -1.17383e7 q^{91} +1.50660e6 q^{93} +5.15450e6 q^{95} +5.09943e6 q^{97} +2.40862e6 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−1380.00	−1.52067	−0.760337	−	0.649529i	$-0.774966\pi$
$7$	−1380.00	−1.52067	−0.760337	+	0.649529i	$0.774966\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	3304.00	0.748455	0.374227	−	0.927337i	$-0.377908\pi$
$11$	3304.00	0.748455	0.374227	+	0.927337i	$0.377908\pi$
$12$	0	0
$13$	8506.00	1.07380	0.536900	−	0.843646i	$-0.319595\pi$
$13$	8506.00	1.07380	0.536900	+	0.843646i	$0.319595\pi$
$14$	0	0
$15$	−3375.00	−0.258199
$16$	0	0
$17$	−9994.00	−0.493365	−0.246682	−	0.969096i	$-0.579340\pi$
$17$	−9994.00	−0.493365	−0.246682	+	0.969096i	$0.579340\pi$
$18$	0	0
$19$	−41236.0	−1.37924	−0.689619	−	0.724173i	$-0.742222\pi$
$19$	−41236.0	−1.37924	−0.689619	+	0.724173i	$0.742222\pi$
$20$	0	0
$21$	−37260.0	−0.877961
$22$	0	0
$23$	−84120.0	−1.44162	−0.720812	−	0.693131i	$-0.756231\pi$
$23$	−84120.0	−1.44162	−0.720812	+	0.693131i	$0.756231\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	132802.	1.01114	0.505570	−	0.862785i	$-0.331282\pi$
$29$	132802.	1.01114	0.505570	+	0.862785i	$0.331282\pi$
$30$	0	0
$31$	55800.0	0.336410	0.168205	−	0.985752i	$-0.446203\pi$
$31$	55800.0	0.336410	0.168205	+	0.985752i	$0.446203\pi$
$32$	0	0
$33$	89208.0	0.432121
$34$	0	0
$35$	172500.	0.680066
$36$	0	0
$37$	228170.	0.740547	0.370273	−	0.928923i	$-0.379264\pi$
$37$	228170.	0.740547	0.370273	+	0.928923i	$0.379264\pi$
$38$	0	0
$39$	229662.	0.619959
$40$	0	0
$41$	−139670.	−0.316490	−0.158245	−	0.987400i	$-0.550584\pi$
$41$	−139670.	−0.316490	−0.158245	+	0.987400i	$0.550584\pi$
$42$	0	0
$43$	755492.	1.44907	0.724537	−	0.689236i	$-0.242054\pi$
$43$	755492.	1.44907	0.724537	+	0.689236i	$0.242054\pi$
$44$	0	0
$45$	−91125.0	−0.149071
$46$	0	0
$47$	−836984.	−1.17591	−0.587956	−	0.808893i	$-0.700067\pi$
$47$	−836984.	−1.17591	−0.587956	+	0.808893i	$0.700067\pi$
$48$	0	0
$49$	1.08086e6	1.31245
$50$	0	0
$51$	−269838.	−0.284844
$52$	0	0
$53$	1.64165e6	1.51466	0.757330	−	0.653033i	$-0.226504\pi$
$53$	1.64165e6	1.51466	0.757330	+	0.653033i	$0.226504\pi$
$54$	0	0
$55$	−413000.	−0.334719
$56$	0	0
$57$	−1.11337e6	−0.796303
$58$	0	0
$59$	989656.	0.627339	0.313669	−	0.949532i	$-0.398442\pi$
$59$	989656.	0.627339	0.313669	+	0.949532i	$0.398442\pi$
$60$	0	0
$61$	−1.65816e6	−0.935347	−0.467673	−	0.883901i	$-0.654908\pi$
$61$	−1.65816e6	−0.935347	−0.467673	+	0.883901i	$0.654908\pi$
$62$	0	0
$63$	−1.00602e6	−0.506891
$64$	0	0
$65$	−1.06325e6	−0.480218
$66$	0	0
$67$	4.52384e6	1.83758	0.918789	−	0.394749i	$-0.129168\pi$
$67$	4.52384e6	1.83758	0.918789	+	0.394749i	$0.129168\pi$
$68$	0	0
$69$	−2.27124e6	−0.832322
$70$	0	0
$71$	389408.	0.129122	0.0645611	−	0.997914i	$-0.479435\pi$
$71$	389408.	0.129122	0.0645611	+	0.997914i	$0.479435\pi$
$72$	0	0
$73$	5.61733e6	1.69005	0.845026	−	0.534726i	$-0.179585\pi$
$73$	5.61733e6	1.69005	0.845026	+	0.534726i	$0.179585\pi$
$74$	0	0
$75$	421875.	0.115470
$76$	0	0
$77$	−4.55952e6	−1.13816
$78$	0	0
$79$	−3.90108e6	−0.890205	−0.445103	−	0.895480i	$-0.646833\pi$
$79$	−3.90108e6	−0.890205	−0.445103	+	0.895480i	$0.646833\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	9.39412e6	1.80336	0.901680	−	0.432403i	$-0.142334\pi$
$83$	9.39412e6	1.80336	0.901680	+	0.432403i	$0.142334\pi$
$84$	0	0
$85$	1.24925e6	0.220639
$86$	0	0
$87$	3.58565e6	0.583782
$88$	0	0
$89$	2.80375e6	0.421574	0.210787	−	0.977532i	$-0.432397\pi$
$89$	2.80375e6	0.421574	0.210787	+	0.977532i	$0.432397\pi$
$90$	0	0
$91$	−1.17383e7	−1.63290
$92$	0	0
$93$	1.50660e6	0.194226
$94$	0	0
$95$	5.15450e6	0.616814
$96$	0	0
$97$	5.09943e6	0.567310	0.283655	−	0.958926i	$-0.408453\pi$
$97$	5.09943e6	0.567310	0.283655	+	0.958926i	$0.408453\pi$
$98$	0	0
$99$	2.40862e6	0.249485
$100$	0	0
$101$	1.51723e7	1.46530	0.732648	−	0.680607i	$-0.238284\pi$
$101$	1.51723e7	1.46530	0.732648	+	0.680607i	$0.238284\pi$
$102$	0	0
$103$	−4.70527e6	−0.424281	−0.212141	−	0.977239i	$-0.568044\pi$
$103$	−4.70527e6	−0.424281	−0.212141	+	0.977239i	$0.568044\pi$
$104$	0	0
$105$	4.65750e6	0.392636
$106$	0	0
$107$	−2.63120e6	−0.207640	−0.103820	−	0.994596i	$-0.533107\pi$
$107$	−2.63120e6	−0.207640	−0.103820	+	0.994596i	$0.533107\pi$
$108$	0	0
$109$	−4.30059e6	−0.318080	−0.159040	−	0.987272i	$-0.550840\pi$
$109$	−4.30059e6	−0.318080	−0.159040	+	0.987272i	$0.550840\pi$
$110$	0	0
$111$	6.16059e6	0.427555
$112$	0	0
$113$	3.98233e6	0.259635	0.129817	−	0.991538i	$-0.458561\pi$
$113$	3.98233e6	0.259635	0.129817	+	0.991538i	$0.458561\pi$
$114$	0	0
$115$	1.05150e7	0.644714
$116$	0	0
$117$	6.20087e6	0.357934
$118$	0	0
$119$	1.37917e7	0.750247
$120$	0	0
$121$	−8.57076e6	−0.439815
$122$	0	0
$123$	−3.77109e6	−0.182725
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−2.80177e7	−1.21372	−0.606861	−	0.794808i	$-0.707571\pi$
$127$	−2.80177e7	−1.21372	−0.606861	+	0.794808i	$0.707571\pi$
$128$	0	0
$129$	2.03983e7	0.836623
$130$	0	0
$131$	8.19919e6	0.318656	0.159328	−	0.987226i	$-0.449067\pi$
$131$	8.19919e6	0.318656	0.159328	+	0.987226i	$0.449067\pi$
$132$	0	0
$133$	5.69057e7	2.09737
$134$	0	0
$135$	−2.46038e6	−0.0860663
$136$	0	0
$137$	−1.66646e6	−0.0553697	−0.0276849	−	0.999617i	$-0.508813\pi$
$137$	−1.66646e6	−0.0553697	−0.0276849	+	0.999617i	$0.508813\pi$
$138$	0	0
$139$	5.87456e7	1.85534	0.927670	−	0.373401i	$-0.121808\pi$
$139$	5.87456e7	1.85534	0.927670	+	0.373401i	$0.121808\pi$
$140$	0	0
$141$	−2.25986e7	−0.678913
$142$	0	0
$143$	2.81038e7	0.803691
$144$	0	0
$145$	−1.66002e7	−0.452196
$146$	0	0
$147$	2.91831e7	0.757742
$148$	0	0
$149$	−1.93697e7	−0.479702	−0.239851	−	0.970810i	$-0.577099\pi$
$149$	−1.93697e7	−0.479702	−0.239851	+	0.970810i	$0.577099\pi$
$150$	0	0
$151$	5.33952e7	1.26207	0.631034	−	0.775755i	$-0.282631\pi$
$151$	5.33952e7	1.26207	0.631034	+	0.775755i	$0.282631\pi$
$152$	0	0
$153$	−7.28563e6	−0.164455
$154$	0	0
$155$	−6.97500e6	−0.150447
$156$	0	0
$157$	2.04529e7	0.421800	0.210900	−	0.977508i	$-0.432361\pi$
$157$	2.04529e7	0.421800	0.210900	+	0.977508i	$0.432361\pi$
$158$	0	0
$159$	4.43246e7	0.874489
$160$	0	0
$161$	1.16086e8	2.19224
$162$	0	0
$163$	733588.	0.0132677	0.00663385	−	0.999978i	$-0.497888\pi$
$163$	733588.	0.0132677	0.00663385	+	0.999978i	$0.497888\pi$
$164$	0	0
$165$	−1.11510e7	−0.193250
$166$	0	0
$167$	−1.68925e7	−0.280664	−0.140332	−	0.990105i	$-0.544817\pi$
$167$	−1.68925e7	−0.280664	−0.140332	+	0.990105i	$0.544817\pi$
$168$	0	0
$169$	9.60352e6	0.153048
$170$	0	0
$171$	−3.00610e7	−0.459746
$172$	0	0
$173$	1.18186e7	0.173541	0.0867707	−	0.996228i	$-0.472345\pi$
$173$	1.18186e7	0.173541	0.0867707	+	0.996228i	$0.472345\pi$
$174$	0	0
$175$	−2.15625e7	−0.304135
$176$	0	0
$177$	2.67207e7	0.362194
$178$	0	0
$179$	3.13746e7	0.408877	0.204439	−	0.978879i	$-0.434463\pi$
$179$	3.13746e7	0.408877	0.204439	+	0.978879i	$0.434463\pi$
$180$	0	0
$181$	−5.83555e7	−0.731488	−0.365744	−	0.930716i	$-0.619185\pi$
$181$	−5.83555e7	−0.731488	−0.365744	+	0.930716i	$0.619185\pi$
$182$	0	0
$183$	−4.47704e7	−0.540023
$184$	0	0
$185$	−2.85213e7	−0.331183
$186$	0	0
$187$	−3.30202e7	−0.369261
$188$	0	0
$189$	−2.71625e7	−0.292654
$190$	0	0
$191$	−4.06166e6	−0.0421780	−0.0210890	−	0.999778i	$-0.506713\pi$
$191$	−4.06166e6	−0.0421780	−0.0210890	+	0.999778i	$0.506713\pi$
$192$	0	0
$193$	−1.33221e8	−1.33389	−0.666946	−	0.745106i	$-0.732399\pi$
$193$	−1.33221e8	−1.33389	−0.666946	+	0.745106i	$0.732399\pi$
$194$	0	0
$195$	−2.87078e7	−0.277254
$196$	0	0
$197$	1.30771e7	0.121866	0.0609328	−	0.998142i	$-0.480592\pi$
$197$	1.30771e7	0.121866	0.0609328	+	0.998142i	$0.480592\pi$
$198$	0	0
$199$	6.98502e7	0.628322	0.314161	−	0.949370i	$-0.398277\pi$
$199$	6.98502e7	0.628322	0.314161	+	0.949370i	$0.398277\pi$
$200$	0	0
$201$	1.22144e8	1.06093
$202$	0	0
$203$	−1.83267e8	−1.53761
$204$	0	0
$205$	1.74588e7	0.141539
$206$	0	0
$207$	−6.13235e7	−0.480541
$208$	0	0
$209$	−1.36244e8	−1.03230
$210$	0	0
$211$	−3.28535e7	−0.240765	−0.120382	−	0.992728i	$-0.538412\pi$
$211$	−3.28535e7	−0.240765	−0.120382	+	0.992728i	$0.538412\pi$
$212$	0	0
$213$	1.05140e7	0.0745487
$214$	0	0
$215$	−9.44365e7	−0.648045
$216$	0	0
$217$	−7.70040e7	−0.511569
$218$	0	0
$219$	1.51668e8	0.975752
$220$	0	0
$221$	−8.50090e7	−0.529775
$222$	0	0
$223$	−6.95194e7	−0.419796	−0.209898	−	0.977723i	$-0.567313\pi$
$223$	−6.95194e7	−0.419796	−0.209898	+	0.977723i	$0.567313\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	2.30779e8	1.30950	0.654750	−	0.755845i	$-0.272774\pi$
$227$	2.30779e8	1.30950	0.654750	+	0.755845i	$0.272774\pi$
$228$	0	0
$229$	1.46157e8	0.804258	0.402129	−	0.915583i	$-0.368270\pi$
$229$	1.46157e8	0.804258	0.402129	+	0.915583i	$0.368270\pi$
$230$	0	0
$231$	−1.23107e8	−0.657114
$232$	0	0
$233$	3.11907e8	1.61540	0.807700	−	0.589594i	$-0.200712\pi$
$233$	3.11907e8	1.61540	0.807700	+	0.589594i	$0.200712\pi$
$234$	0	0
$235$	1.04623e8	0.525884
$236$	0	0
$237$	−1.05329e8	−0.513960
$238$	0	0
$239$	−2.27310e8	−1.07703	−0.538513	−	0.842617i	$-0.681014\pi$
$239$	−2.27310e8	−1.07703	−0.538513	+	0.842617i	$0.681014\pi$
$240$	0	0
$241$	−1.98483e8	−0.913404	−0.456702	−	0.889620i	$-0.650969\pi$
$241$	−1.98483e8	−0.913404	−0.456702	+	0.889620i	$0.650969\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−1.35107e8	−0.586944
$246$	0	0
$247$	−3.50753e8	−1.48103
$248$	0	0
$249$	2.53641e8	1.04117
$250$	0	0
$251$	1.32536e8	0.529024	0.264512	−	0.964382i	$-0.414789\pi$
$251$	1.32536e8	0.529024	0.264512	+	0.964382i	$0.414789\pi$
$252$	0	0
$253$	−2.77932e8	−1.07899
$254$	0	0
$255$	3.37298e7	0.127386
$256$	0	0
$257$	−3.58642e7	−0.131794	−0.0658970	−	0.997826i	$-0.520991\pi$
$257$	−3.58642e7	−0.131794	−0.0658970	+	0.997826i	$0.520991\pi$
$258$	0	0
$259$	−3.14875e8	−1.12613
$260$	0	0
$261$	9.68127e7	0.337047
$262$	0	0
$263$	4.79640e8	1.62581	0.812907	−	0.582394i	$-0.197884\pi$
$263$	4.79640e8	1.62581	0.812907	+	0.582394i	$0.197884\pi$
$264$	0	0
$265$	−2.05206e8	−0.677376
$266$	0	0
$267$	7.57011e7	0.243396
$268$	0	0
$269$	−7.08764e7	−0.222008	−0.111004	−	0.993820i	$-0.535407\pi$
$269$	−7.08764e7	−0.222008	−0.111004	+	0.993820i	$0.535407\pi$
$270$	0	0
$271$	−4.07490e8	−1.24373	−0.621863	−	0.783126i	$-0.713624\pi$
$271$	−4.07490e8	−1.24373	−0.621863	+	0.783126i	$0.713624\pi$
$272$	0	0
$273$	−3.16934e8	−0.942755
$274$	0	0
$275$	5.16250e7	0.149691
$276$	0	0
$277$	−7.93477e7	−0.224313	−0.112157	−	0.993691i	$-0.535776\pi$
$277$	−7.93477e7	−0.224313	−0.112157	+	0.993691i	$0.535776\pi$
$278$	0	0
$279$	4.06782e7	0.112137
$280$	0	0
$281$	−8.70068e7	−0.233928	−0.116964	−	0.993136i	$-0.537316\pi$
$281$	−8.70068e7	−0.233928	−0.116964	+	0.993136i	$0.537316\pi$
$282$	0	0
$283$	−2.77612e8	−0.728091	−0.364045	−	0.931381i	$-0.618605\pi$
$283$	−2.77612e8	−0.728091	−0.364045	+	0.931381i	$0.618605\pi$
$284$	0	0
$285$	1.39172e8	0.356117
$286$	0	0
$287$	1.92745e8	0.481278
$288$	0	0
$289$	−3.10459e8	−0.756591
$290$	0	0
$291$	1.37685e8	0.327536
$292$	0	0
$293$	2.46490e8	0.572484	0.286242	−	0.958157i	$-0.407594\pi$
$293$	2.46490e8	0.572484	0.286242	+	0.958157i	$0.407594\pi$
$294$	0	0
$295$	−1.23707e8	−0.280554
$296$	0	0
$297$	6.50326e7	0.144040
$298$	0	0
$299$	−7.15525e8	−1.54802
$300$	0	0
$301$	−1.04258e9	−2.20357
$302$	0	0
$303$	4.09651e8	0.845990
$304$	0	0
$305$	2.07270e8	0.418300
$306$	0	0
$307$	−3.84965e8	−0.759340	−0.379670	−	0.925122i	$-0.623963\pi$
$307$	−3.84965e8	−0.759340	−0.379670	+	0.925122i	$0.623963\pi$
$308$	0	0
$309$	−1.27042e8	−0.244959
$310$	0	0
$311$	−4.64435e7	−0.0875514	−0.0437757	−	0.999041i	$-0.513939\pi$
$311$	−4.64435e7	−0.0875514	−0.0437757	+	0.999041i	$0.513939\pi$
$312$	0	0
$313$	2.10558e8	0.388120	0.194060	−	0.980990i	$-0.437834\pi$
$313$	2.10558e8	0.388120	0.194060	+	0.980990i	$0.437834\pi$
$314$	0	0
$315$	1.25752e8	0.226689
$316$	0	0
$317$	−9.60971e8	−1.69435	−0.847175	−	0.531314i	$-0.821698\pi$
$317$	−9.60971e8	−1.69435	−0.847175	+	0.531314i	$0.821698\pi$
$318$	0	0
$319$	4.38778e8	0.756793
$320$	0	0
$321$	−7.10425e7	−0.119881
$322$	0	0
$323$	4.12113e8	0.680467
$324$	0	0
$325$	1.32906e8	0.214760
$326$	0	0
$327$	−1.16116e8	−0.183643
$328$	0	0
$329$	1.15504e9	1.78818
$330$	0	0
$331$	−3.99923e8	−0.606147	−0.303074	−	0.952967i	$-0.598013\pi$
$331$	−3.99923e8	−0.606147	−0.303074	+	0.952967i	$0.598013\pi$
$332$	0	0
$333$	1.66336e8	0.246849
$334$	0	0
$335$	−5.65480e8	−0.821790
$336$	0	0
$337$	2.69185e8	0.383129	0.191565	−	0.981480i	$-0.438644\pi$
$337$	2.69185e8	0.383129	0.191565	+	0.981480i	$0.438644\pi$
$338$	0	0
$339$	1.07523e8	0.149900
$340$	0	0
$341$	1.84363e8	0.251787
$342$	0	0
$343$	−3.55093e8	−0.475131
$344$	0	0
$345$	2.83905e8	0.372226
$346$	0	0
$347$	8.21868e8	1.05596	0.527982	−	0.849256i	$-0.322949\pi$
$347$	8.21868e8	1.05596	0.527982	+	0.849256i	$0.322949\pi$
$348$	0	0
$349$	6.48354e8	0.816438	0.408219	−	0.912884i	$-0.366150\pi$
$349$	6.48354e8	0.816438	0.408219	+	0.912884i	$0.366150\pi$
$350$	0	0
$351$	1.67424e8	0.206653
$352$	0	0
$353$	6.52666e8	0.789732	0.394866	−	0.918739i	$-0.370791\pi$
$353$	6.52666e8	0.789732	0.394866	+	0.918739i	$0.370791\pi$
$354$	0	0
$355$	−4.86760e7	−0.0577452
$356$	0	0
$357$	3.72376e8	0.433155
$358$	0	0
$359$	8.77431e8	1.00088	0.500440	−	0.865771i	$-0.333171\pi$
$359$	8.77431e8	1.00088	0.500440	+	0.865771i	$0.333171\pi$
$360$	0	0
$361$	8.06536e8	0.902295
$362$	0	0
$363$	−2.31410e8	−0.253927
$364$	0	0
$365$	−7.02166e8	−0.755814
$366$	0	0
$367$	−2.86989e8	−0.303064	−0.151532	−	0.988452i	$-0.548421\pi$
$367$	−2.86989e8	−0.303064	−0.151532	+	0.988452i	$0.548421\pi$
$368$	0	0
$369$	−1.01819e8	−0.105497
$370$	0	0
$371$	−2.26548e9	−2.30330
$372$	0	0
$373$	−1.77013e9	−1.76614	−0.883069	−	0.469242i	$-0.844527\pi$
$373$	−1.77013e9	−1.76614	−0.883069	+	0.469242i	$0.844527\pi$
$374$	0	0
$375$	−5.27344e7	−0.0516398
$376$	0	0
$377$	1.12961e9	1.08576
$378$	0	0
$379$	−1.46311e9	−1.38051	−0.690257	−	0.723565i	$-0.742502\pi$
$379$	−1.46311e9	−1.38051	−0.690257	+	0.723565i	$0.742502\pi$
$380$	0	0
$381$	−7.56477e8	−0.700742
$382$	0	0
$383$	9.90456e8	0.900823	0.450412	−	0.892821i	$-0.351277\pi$
$383$	9.90456e8	0.900823	0.450412	+	0.892821i	$0.351277\pi$
$384$	0	0
$385$	5.69940e8	0.508999
$386$	0	0
$387$	5.50754e8	0.483024
$388$	0	0
$389$	−7.96901e8	−0.686406	−0.343203	−	0.939261i	$-0.611512\pi$
$389$	−7.96901e8	−0.686406	−0.343203	+	0.939261i	$0.611512\pi$
$390$	0	0
$391$	8.40695e8	0.711246
$392$	0	0
$393$	2.21378e8	0.183976
$394$	0	0
$395$	4.87635e8	0.398112
$396$	0	0
$397$	7.95584e8	0.638145	0.319073	−	0.947730i	$-0.396629\pi$
$397$	7.95584e8	0.638145	0.319073	+	0.947730i	$0.396629\pi$
$398$	0	0
$399$	1.53645e9	1.21092
$400$	0	0
$401$	−2.01632e9	−1.56154	−0.780771	−	0.624817i	$-0.785174\pi$
$401$	−2.01632e9	−1.56154	−0.780771	+	0.624817i	$0.785174\pi$
$402$	0	0
$403$	4.74635e8	0.361237
$404$	0	0
$405$	−6.64301e7	−0.0496904
$406$	0	0
$407$	7.53874e8	0.554266
$408$	0	0
$409$	−5.48030e7	−0.0396070	−0.0198035	−	0.999804i	$-0.506304\pi$
$409$	−5.48030e7	−0.0396070	−0.0198035	+	0.999804i	$0.506304\pi$
$410$	0	0
$411$	−4.49944e7	−0.0319677
$412$	0	0
$413$	−1.36573e9	−0.953978
$414$	0	0
$415$	−1.17426e9	−0.806487
$416$	0	0
$417$	1.58613e9	1.07118
$418$	0	0
$419$	−1.10925e8	−0.0736679	−0.0368340	−	0.999321i	$-0.511727\pi$
$419$	−1.10925e8	−0.0736679	−0.0368340	+	0.999321i	$0.511727\pi$
$420$	0	0
$421$	−2.12064e9	−1.38509	−0.692547	−	0.721373i	$-0.743511\pi$
$421$	−2.12064e9	−1.38509	−0.692547	+	0.721373i	$0.743511\pi$
$422$	0	0
$423$	−6.10161e8	−0.391971
$424$	0	0
$425$	−1.56156e8	−0.0986730
$426$	0	0
$427$	2.28826e9	1.42236
$428$	0	0
$429$	7.58803e8	0.464011
$430$	0	0
$431$	2.48533e9	1.49525	0.747624	−	0.664123i	$-0.231195\pi$
$431$	2.48533e9	1.49525	0.747624	+	0.664123i	$0.231195\pi$
$432$	0	0
$433$	2.99956e7	0.0177562	0.00887811	−	0.999961i	$-0.497174\pi$
$433$	2.99956e7	0.0177562	0.00887811	+	0.999961i	$0.497174\pi$
$434$	0	0
$435$	−4.48207e8	−0.261075
$436$	0	0
$437$	3.46877e9	1.98834
$438$	0	0
$439$	1.04873e9	0.591616	0.295808	−	0.955247i	$-0.404411\pi$
$439$	1.04873e9	0.591616	0.295808	+	0.955247i	$0.404411\pi$
$440$	0	0
$441$	7.87945e8	0.437483
$442$	0	0
$443$	−2.04679e8	−0.111856	−0.0559282	−	0.998435i	$-0.517812\pi$
$443$	−2.04679e8	−0.111856	−0.0559282	+	0.998435i	$0.517812\pi$
$444$	0	0
$445$	−3.50468e8	−0.188534
$446$	0	0
$447$	−5.22983e8	−0.276956
$448$	0	0
$449$	2.63962e9	1.37619	0.688097	−	0.725619i	$-0.258446\pi$
$449$	2.63962e9	1.37619	0.688097	+	0.725619i	$0.258446\pi$
$450$	0	0
$451$	−4.61470e8	−0.236878
$452$	0	0
$453$	1.44167e9	0.728656
$454$	0	0
$455$	1.46728e9	0.730255
$456$	0	0
$457$	1.83738e9	0.900519	0.450260	−	0.892898i	$-0.351331\pi$
$457$	1.83738e9	0.900519	0.450260	+	0.892898i	$0.351331\pi$
$458$	0	0
$459$	−1.96712e8	−0.0949481
$460$	0	0
$461$	3.57678e9	1.70035	0.850176	−	0.526499i	$-0.176495\pi$
$461$	3.57678e9	1.70035	0.850176	+	0.526499i	$0.176495\pi$
$462$	0	0
$463$	−1.28068e9	−0.599662	−0.299831	−	0.953992i	$-0.596930\pi$
$463$	−1.28068e9	−0.599662	−0.299831	+	0.953992i	$0.596930\pi$
$464$	0	0
$465$	−1.88325e8	−0.0868606
$466$	0	0
$467$	4.19984e9	1.90820	0.954100	−	0.299490i	$-0.0968164\pi$
$467$	4.19984e9	1.90820	0.954100	+	0.299490i	$0.0968164\pi$
$468$	0	0
$469$	−6.24290e9	−2.79436
$470$	0	0
$471$	5.52229e8	0.243526
$472$	0	0
$473$	2.49615e9	1.08457
$474$	0	0
$475$	−6.44312e8	−0.275847
$476$	0	0
$477$	1.19676e9	0.504887
$478$	0	0
$479$	−2.48202e9	−1.03188	−0.515942	−	0.856623i	$-0.672558\pi$
$479$	−2.48202e9	−1.03188	−0.515942	+	0.856623i	$0.672558\pi$
$480$	0	0
$481$	1.94081e9	0.795200
$482$	0	0
$483$	3.13431e9	1.26569
$484$	0	0
$485$	−6.37428e8	−0.253709
$486$	0	0
$487$	1.08065e9	0.423970	0.211985	−	0.977273i	$-0.432007\pi$
$487$	1.08065e9	0.423970	0.211985	+	0.977273i	$0.432007\pi$
$488$	0	0
$489$	1.98069e7	0.00766011
$490$	0	0
$491$	9.47787e8	0.361348	0.180674	−	0.983543i	$-0.442172\pi$
$491$	9.47787e8	0.361348	0.180674	+	0.983543i	$0.442172\pi$
$492$	0	0
$493$	−1.32722e9	−0.498861
$494$	0	0
$495$	−3.01077e8	−0.111573
$496$	0	0
$497$	−5.37383e8	−0.196353
$498$	0	0
$499$	2.07692e9	0.748286	0.374143	−	0.927371i	$-0.377937\pi$
$499$	2.07692e9	0.748286	0.374143	+	0.927371i	$0.377937\pi$
$500$	0	0
$501$	−4.56098e8	−0.162041
$502$	0	0
$503$	−2.73265e9	−0.957408	−0.478704	−	0.877976i	$-0.658893\pi$
$503$	−2.73265e9	−0.957408	−0.478704	+	0.877976i	$0.658893\pi$
$504$	0	0
$505$	−1.89653e9	−0.655301
$506$	0	0
$507$	2.59295e8	0.0883622
$508$	0	0
$509$	4.52470e9	1.52082	0.760409	−	0.649444i	$-0.224998\pi$
$509$	4.52470e9	1.52082	0.760409	+	0.649444i	$0.224998\pi$
$510$	0	0
$511$	−7.75192e9	−2.57002
$512$	0	0
$513$	−8.11648e8	−0.265434
$514$	0	0
$515$	5.88158e8	0.189744
$516$	0	0
$517$	−2.76540e9	−0.880117
$518$	0	0
$519$	3.19101e8	0.100194
$520$	0	0
$521$	−2.79533e8	−0.0865965	−0.0432983	−	0.999062i	$-0.513787\pi$
$521$	−2.79533e8	−0.0865965	−0.0432983	+	0.999062i	$0.513787\pi$
$522$	0	0
$523$	−1.46376e9	−0.447419	−0.223710	−	0.974656i	$-0.571817\pi$
$523$	−1.46376e9	−0.447419	−0.223710	+	0.974656i	$0.571817\pi$
$524$	0	0
$525$	−5.82188e8	−0.175592
$526$	0	0
$527$	−5.57665e8	−0.165973
$528$	0	0
$529$	3.67135e9	1.07828
$530$	0	0
$531$	7.21459e8	0.209113
$532$	0	0
$533$	−1.18803e9	−0.339847
$534$	0	0
$535$	3.28900e8	0.0928595
$536$	0	0
$537$	8.47115e8	0.236065
$538$	0	0
$539$	3.57115e9	0.982308
$540$	0	0
$541$	−2.72194e9	−0.739073	−0.369537	−	0.929216i	$-0.620484\pi$
$541$	−2.72194e9	−0.739073	−0.369537	+	0.929216i	$0.620484\pi$
$542$	0	0
$543$	−1.57560e9	−0.422325
$544$	0	0
$545$	5.37574e8	0.142249
$546$	0	0
$547$	6.73048e8	0.175829	0.0879145	−	0.996128i	$-0.471980\pi$
$547$	6.73048e8	0.175829	0.0879145	+	0.996128i	$0.471980\pi$
$548$	0	0
$549$	−1.20880e9	−0.311782
$550$	0	0
$551$	−5.47622e9	−1.39460
$552$	0	0
$553$	5.38349e9	1.35371
$554$	0	0
$555$	−7.70074e8	−0.191208
$556$	0	0
$557$	−7.19994e9	−1.76537	−0.882685	−	0.469964i	$-0.844267\pi$
$557$	−7.19994e9	−1.76537	−0.882685	+	0.469964i	$0.844267\pi$
$558$	0	0
$559$	6.42621e9	1.55602
$560$	0	0
$561$	−8.91545e8	−0.213193
$562$	0	0
$563$	−5.80114e9	−1.37004	−0.685021	−	0.728524i	$-0.740207\pi$
$563$	−5.80114e9	−1.37004	−0.685021	+	0.728524i	$0.740207\pi$
$564$	0	0
$565$	−4.97792e8	−0.116112
$566$	0	0
$567$	−7.33389e8	−0.168964
$568$	0	0
$569$	−1.67069e9	−0.380192	−0.190096	−	0.981766i	$-0.560880\pi$
$569$	−1.67069e9	−0.380192	−0.190096	+	0.981766i	$0.560880\pi$
$570$	0	0
$571$	−5.70139e9	−1.28160	−0.640802	−	0.767706i	$-0.721398\pi$
$571$	−5.70139e9	−1.28160	−0.640802	+	0.767706i	$0.721398\pi$
$572$	0	0
$573$	−1.09665e8	−0.0243515
$574$	0	0
$575$	−1.31438e9	−0.288325
$576$	0	0
$577$	2.43063e9	0.526750	0.263375	−	0.964694i	$-0.415164\pi$
$577$	2.43063e9	0.526750	0.263375	+	0.964694i	$0.415164\pi$
$578$	0	0
$579$	−3.59696e9	−0.770123
$580$	0	0
$581$	−1.29639e10	−2.74232
$582$	0	0
$583$	5.42401e9	1.13365
$584$	0	0
$585$	−7.75109e8	−0.160073
$586$	0	0
$587$	3.33378e9	0.680305	0.340153	−	0.940370i	$-0.389521\pi$
$587$	3.33378e9	0.680305	0.340153	+	0.940370i	$0.389521\pi$
$588$	0	0
$589$	−2.30097e9	−0.463988
$590$	0	0
$591$	3.53083e8	0.0703591
$592$	0	0
$593$	3.42280e9	0.674048	0.337024	−	0.941496i	$-0.390580\pi$
$593$	3.42280e9	0.674048	0.337024	+	0.941496i	$0.390580\pi$
$594$	0	0
$595$	−1.72396e9	−0.335521
$596$	0	0
$597$	1.88596e9	0.362762
$598$	0	0
$599$	9.28661e8	0.176548	0.0882741	−	0.996096i	$-0.471865\pi$
$599$	9.28661e8	0.176548	0.0882741	+	0.996096i	$0.471865\pi$
$600$	0	0
$601$	−6.56974e9	−1.23449	−0.617245	−	0.786771i	$-0.711751\pi$
$601$	−6.56974e9	−1.23449	−0.617245	+	0.786771i	$0.711751\pi$
$602$	0	0
$603$	3.29788e9	0.612526
$604$	0	0
$605$	1.07134e9	0.196691
$606$	0	0
$607$	3.50790e9	0.636629	0.318315	−	0.947985i	$-0.396883\pi$
$607$	3.50790e9	0.636629	0.318315	+	0.947985i	$0.396883\pi$
$608$	0	0
$609$	−4.94820e9	−0.887742
$610$	0	0
$611$	−7.11939e9	−1.26269
$612$	0	0
$613$	3.63970e9	0.638196	0.319098	−	0.947722i	$-0.396620\pi$
$613$	3.63970e9	0.638196	0.319098	+	0.947722i	$0.396620\pi$
$614$	0	0
$615$	4.71386e8	0.0817173
$616$	0	0
$617$	7.82559e9	1.34128	0.670639	−	0.741784i	$-0.266020\pi$
$617$	7.82559e9	1.34128	0.670639	+	0.741784i	$0.266020\pi$
$618$	0	0
$619$	−8.35173e9	−1.41533	−0.707667	−	0.706546i	$-0.750252\pi$
$619$	−8.35173e9	−1.41533	−0.707667	+	0.706546i	$0.750252\pi$
$620$	0	0
$621$	−1.65573e9	−0.277441
$622$	0	0
$623$	−3.86917e9	−0.641076
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	−3.67858e9	−0.595997
$628$	0	0
$629$	−2.28033e9	−0.365360
$630$	0	0
$631$	2.16820e7	0.00343555	0.00171777	−	0.999999i	$-0.499453\pi$
$631$	2.16820e7	0.00343555	0.00171777	+	0.999999i	$0.499453\pi$
$632$	0	0
$633$	−8.87044e8	−0.139006
$634$	0	0
$635$	3.50221e9	0.542793
$636$	0	0
$637$	9.19377e9	1.40931
$638$	0	0
$639$	2.83878e8	0.0430407
$640$	0	0
$641$	−7.85361e9	−1.17779	−0.588893	−	0.808211i	$-0.700436\pi$
$641$	−7.85361e9	−1.17779	−0.588893	+	0.808211i	$0.700436\pi$
$642$	0	0
$643$	8.81820e9	1.30810	0.654051	−	0.756451i	$-0.273068\pi$
$643$	8.81820e9	1.30810	0.654051	+	0.756451i	$0.273068\pi$
$644$	0	0
$645$	−2.54979e9	−0.374149
$646$	0	0
$647$	8.71997e9	1.26576	0.632878	−	0.774252i	$-0.281873\pi$
$647$	8.71997e9	1.26576	0.632878	+	0.774252i	$0.281873\pi$
$648$	0	0
$649$	3.26982e9	0.469535
$650$	0	0
$651$	−2.07911e9	−0.295354
$652$	0	0
$653$	−6.65755e8	−0.0935661	−0.0467830	−	0.998905i	$-0.514897\pi$
$653$	−6.65755e8	−0.0935661	−0.0467830	+	0.998905i	$0.514897\pi$
$654$	0	0
$655$	−1.02490e9	−0.142507
$656$	0	0
$657$	4.09503e9	0.563350
$658$	0	0
$659$	9.99513e9	1.36047	0.680236	−	0.732994i	$-0.261877\pi$
$659$	9.99513e9	1.36047	0.680236	+	0.732994i	$0.261877\pi$
$660$	0	0
$661$	−2.89160e9	−0.389434	−0.194717	−	0.980859i	$-0.562379\pi$
$661$	−2.89160e9	−0.389434	−0.194717	+	0.980859i	$0.562379\pi$
$662$	0	0
$663$	−2.29524e9	−0.305866
$664$	0	0
$665$	−7.11321e9	−0.937972
$666$	0	0
$667$	−1.11713e10	−1.45768
$668$	0	0
$669$	−1.87702e9	−0.242370
$670$	0	0
$671$	−5.47857e9	−0.700065
$672$	0	0
$673$	−5.83236e9	−0.737550	−0.368775	−	0.929519i	$-0.620223\pi$
$673$	−5.83236e9	−0.737550	−0.368775	+	0.929519i	$0.620223\pi$
$674$	0	0
$675$	3.07547e8	0.0384900
$676$	0	0
$677$	−1.22524e10	−1.51762	−0.758808	−	0.651315i	$-0.774218\pi$
$677$	−1.22524e10	−1.51762	−0.758808	+	0.651315i	$0.774218\pi$
$678$	0	0
$679$	−7.03721e9	−0.862693
$680$	0	0
$681$	6.23103e9	0.756040
$682$	0	0
$683$	9.69293e9	1.16408	0.582040	−	0.813160i	$-0.302255\pi$
$683$	9.69293e9	1.16408	0.582040	+	0.813160i	$0.302255\pi$
$684$	0	0
$685$	2.08307e8	0.0247621
$686$	0	0
$687$	3.94624e9	0.464339
$688$	0	0
$689$	1.39639e10	1.62644
$690$	0	0
$691$	−3.33285e9	−0.384276	−0.192138	−	0.981368i	$-0.561542\pi$
$691$	−3.33285e9	−0.384276	−0.192138	+	0.981368i	$0.561542\pi$
$692$	0	0
$693$	−3.32389e9	−0.379385
$694$	0	0
$695$	−7.34320e9	−0.829733
$696$	0	0
$697$	1.39586e9	0.156145
$698$	0	0
$699$	8.42150e9	0.932651
$700$	0	0
$701$	−2.73199e9	−0.299548	−0.149774	−	0.988720i	$-0.547855\pi$
$701$	−2.73199e9	−0.299548	−0.149774	+	0.988720i	$0.547855\pi$
$702$	0	0
$703$	−9.40882e9	−1.02139
$704$	0	0
$705$	2.82482e9	0.303619
$706$	0	0
$707$	−2.09377e10	−2.22824
$708$	0	0
$709$	3.44620e8	0.0363144	0.0181572	−	0.999835i	$-0.494220\pi$
$709$	3.44620e8	0.0363144	0.0181572	+	0.999835i	$0.494220\pi$
$710$	0	0
$711$	−2.84389e9	−0.296735
$712$	0	0
$713$	−4.69390e9	−0.484976
$714$	0	0
$715$	−3.51298e9	−0.359422
$716$	0	0
$717$	−6.13738e9	−0.621822
$718$	0	0
$719$	−3.35749e9	−0.336871	−0.168436	−	0.985713i	$-0.553872\pi$
$719$	−3.35749e9	−0.336871	−0.168436	+	0.985713i	$0.553872\pi$
$720$	0	0
$721$	6.49327e9	0.645194
$722$	0	0
$723$	−5.35903e9	−0.527354
$724$	0	0
$725$	2.07503e9	0.202228
$726$	0	0
$727$	1.32017e10	1.27426	0.637132	−	0.770755i	$-0.280121\pi$
$727$	1.32017e10	1.27426	0.637132	+	0.770755i	$0.280121\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−7.55039e9	−0.714922
$732$	0	0
$733$	4.30179e9	0.403446	0.201723	−	0.979443i	$-0.435346\pi$
$733$	4.30179e9	0.403446	0.201723	+	0.979443i	$0.435346\pi$
$734$	0	0
$735$	−3.64789e9	−0.338873
$736$	0	0
$737$	1.49468e10	1.37534
$738$	0	0
$739$	1.82414e10	1.66266	0.831330	−	0.555780i	$-0.187580\pi$
$739$	1.82414e10	1.66266	0.831330	+	0.555780i	$0.187580\pi$
$740$	0	0
$741$	−9.47034e9	−0.855071
$742$	0	0
$743$	2.36847e8	0.0211840	0.0105920	−	0.999944i	$-0.496628\pi$
$743$	2.36847e8	0.0211840	0.0105920	+	0.999944i	$0.496628\pi$
$744$	0	0
$745$	2.42122e9	0.214529
$746$	0	0
$747$	6.84831e9	0.601120
$748$	0	0
$749$	3.63106e9	0.315753
$750$	0	0
$751$	1.44326e10	1.24338	0.621690	−	0.783264i	$-0.286447\pi$
$751$	1.44326e10	1.24338	0.621690	+	0.783264i	$0.286447\pi$
$752$	0	0
$753$	3.57847e9	0.305432
$754$	0	0
$755$	−6.67440e9	−0.564414
$756$	0	0
$757$	1.64934e10	1.38189	0.690945	−	0.722907i	$-0.257195\pi$
$757$	1.64934e10	1.38189	0.690945	+	0.722907i	$0.257195\pi$
$758$	0	0
$759$	−7.50418e9	−0.622955
$760$	0	0
$761$	−2.96845e9	−0.244165	−0.122083	−	0.992520i	$-0.538957\pi$
$761$	−2.96845e9	−0.244165	−0.122083	+	0.992520i	$0.538957\pi$
$762$	0	0
$763$	5.93482e9	0.483695
$764$	0	0
$765$	9.10703e8	0.0735465
$766$	0	0
$767$	8.41801e9	0.673637
$768$	0	0
$769$	8.15400e9	0.646590	0.323295	−	0.946298i	$-0.395209\pi$
$769$	8.15400e9	0.646590	0.323295	+	0.946298i	$0.395209\pi$
$770$	0	0
$771$	−9.68334e8	−0.0760913
$772$	0	0
$773$	−2.78059e9	−0.216525	−0.108263	−	0.994122i	$-0.534529\pi$
$773$	−2.78059e9	−0.216525	−0.108263	+	0.994122i	$0.534529\pi$
$774$	0	0
$775$	8.71875e8	0.0672819
$776$	0	0
$777$	−8.50161e9	−0.650171
$778$	0	0
$779$	5.75943e9	0.436514
$780$	0	0
$781$	1.28660e9	0.0966421
$782$	0	0
$783$	2.61394e9	0.194594
$784$	0	0
$785$	−2.55662e9	−0.188635
$786$	0	0
$787$	8.20982e9	0.600374	0.300187	−	0.953880i	$-0.402951\pi$
$787$	8.20982e9	0.600374	0.300187	+	0.953880i	$0.402951\pi$
$788$	0	0
$789$	1.29503e10	0.938664
$790$	0	0
$791$	−5.49562e9	−0.394820
$792$	0	0
$793$	−1.41043e10	−1.00438
$794$	0	0
$795$	−5.54057e9	−0.391083
$796$	0	0
$797$	−5.47916e7	−0.00383363	−0.00191681	−	0.999998i	$-0.500610\pi$
$797$	−5.47916e7	−0.00383363	−0.00191681	+	0.999998i	$0.500610\pi$
$798$	0	0
$799$	8.36482e9	0.580153
$800$	0	0
$801$	2.04393e9	0.140525
$802$	0	0
$803$	1.85597e10	1.26493
$804$	0	0
$805$	−1.45107e10	−0.980399
$806$	0	0
$807$	−1.91366e9	−0.128176
$808$	0	0
$809$	2.04973e9	0.136106	0.0680529	−	0.997682i	$-0.478321\pi$
$809$	2.04973e9	0.136106	0.0680529	+	0.997682i	$0.478321\pi$
$810$	0	0
$811$	1.59471e10	1.04981	0.524903	−	0.851162i	$-0.324101\pi$
$811$	1.59471e10	1.04981	0.524903	+	0.851162i	$0.324101\pi$
$812$	0	0
$813$	−1.10022e10	−0.718066
$814$	0	0
$815$	−9.16985e7	−0.00593350
$816$	0	0
$817$	−3.11535e10	−1.99862
$818$	0	0
$819$	−8.55721e9	−0.544300
$820$	0	0
$821$	9.65923e9	0.609174	0.304587	−	0.952484i	$-0.401482\pi$
$821$	9.65923e9	0.609174	0.304587	+	0.952484i	$0.401482\pi$
$822$	0	0
$823$	−3.90398e9	−0.244123	−0.122061	−	0.992523i	$-0.538950\pi$
$823$	−3.90398e9	−0.244123	−0.122061	+	0.992523i	$0.538950\pi$
$824$	0	0
$825$	1.39388e9	0.0864241
$826$	0	0
$827$	1.32224e10	0.812910	0.406455	−	0.913671i	$-0.366765\pi$
$827$	1.32224e10	0.812910	0.406455	+	0.913671i	$0.366765\pi$
$828$	0	0
$829$	1.20482e10	0.734484	0.367242	−	0.930125i	$-0.380302\pi$
$829$	1.20482e10	0.734484	0.367242	+	0.930125i	$0.380302\pi$
$830$	0	0
$831$	−2.14239e9	−0.129507
$832$	0	0
$833$	−1.08021e10	−0.647515
$834$	0	0
$835$	2.11156e9	0.125517
$836$	0	0
$837$	1.09831e9	0.0647420
$838$	0	0
$839$	−3.10603e10	−1.81568	−0.907839	−	0.419319i	$-0.862269\pi$
$839$	−3.10603e10	−1.81568	−0.907839	+	0.419319i	$0.862269\pi$
$840$	0	0
$841$	3.86495e8	0.0224057
$842$	0	0
$843$	−2.34918e9	−0.135058
$844$	0	0
$845$	−1.20044e9	−0.0684450
$846$	0	0
$847$	1.18276e10	0.668815
$848$	0	0
$849$	−7.49552e9	−0.420363
$850$	0	0
$851$	−1.91937e10	−1.06759
$852$	0	0
$853$	2.52157e10	1.39107	0.695536	−	0.718491i	$-0.255167\pi$
$853$	2.52157e10	1.39107	0.695536	+	0.718491i	$0.255167\pi$
$854$	0	0
$855$	3.75763e9	0.205605
$856$	0	0
$857$	2.25314e10	1.22280	0.611401	−	0.791321i	$-0.290606\pi$
$857$	2.25314e10	1.22280	0.611401	+	0.791321i	$0.290606\pi$
$858$	0	0
$859$	−4.07863e9	−0.219552	−0.109776	−	0.993956i	$-0.535013\pi$
$859$	−4.07863e9	−0.219552	−0.109776	+	0.993956i	$0.535013\pi$
$860$	0	0
$861$	5.20410e9	0.277866
$862$	0	0
$863$	1.10643e10	0.585986	0.292993	−	0.956115i	$-0.405349\pi$
$863$	1.10643e10	0.585986	0.292993	+	0.956115i	$0.405349\pi$
$864$	0	0
$865$	−1.47732e9	−0.0776101
$866$	0	0
$867$	−8.38238e9	−0.436818
$868$	0	0
$869$	−1.28892e10	−0.666278
$870$	0	0
$871$	3.84798e10	1.97319
$872$	0	0
$873$	3.71748e9	0.189103
$874$	0	0
$875$	2.69531e9	0.136013
$876$	0	0
$877$	3.60595e10	1.80518	0.902590	−	0.430501i	$-0.141663\pi$
$877$	3.60595e10	1.80518	0.902590	+	0.430501i	$0.141663\pi$
$878$	0	0
$879$	6.65524e9	0.330524
$880$	0	0
$881$	−2.62954e10	−1.29558	−0.647789	−	0.761820i	$-0.724306\pi$
$881$	−2.62954e10	−1.29558	−0.647789	+	0.761820i	$0.724306\pi$
$882$	0	0
$883$	−2.78873e9	−0.136315	−0.0681575	−	0.997675i	$-0.521712\pi$
$883$	−2.78873e9	−0.136315	−0.0681575	+	0.997675i	$0.521712\pi$
$884$	0	0
$885$	−3.34009e9	−0.161978
$886$	0	0
$887$	1.82970e10	0.880332	0.440166	−	0.897916i	$-0.354920\pi$
$887$	1.82970e10	0.880332	0.440166	+	0.897916i	$0.354920\pi$
$888$	0	0
$889$	3.86644e10	1.84567
$890$	0	0
$891$	1.75588e9	0.0831617
$892$	0	0
$893$	3.45139e10	1.62186
$894$	0	0
$895$	−3.92183e9	−0.182856
$896$	0	0
$897$	−1.93192e10	−0.893748
$898$	0	0
$899$	7.41035e9	0.340157
$900$	0	0
$901$	−1.64067e10	−0.747280
$902$	0	0
$903$	−2.81496e10	−1.27223
$904$	0	0
$905$	7.29444e9	0.327131
$906$	0	0
$907$	1.21245e9	0.0539559	0.0269780	−	0.999636i	$-0.491412\pi$
$907$	1.21245e9	0.0539559	0.0269780	+	0.999636i	$0.491412\pi$
$908$	0	0
$909$	1.10606e10	0.488432
$910$	0	0
$911$	2.09181e10	0.916659	0.458329	−	0.888782i	$-0.348448\pi$
$911$	2.09181e10	0.916659	0.458329	+	0.888782i	$0.348448\pi$
$912$	0	0
$913$	3.10382e10	1.34973
$914$	0	0
$915$	5.59630e9	0.241505
$916$	0	0
$917$	−1.13149e10	−0.484571
$918$	0	0
$919$	1.81545e10	0.771577	0.385789	−	0.922587i	$-0.373929\pi$
$919$	1.81545e10	0.771577	0.385789	+	0.922587i	$0.373929\pi$
$920$	0	0
$921$	−1.03940e10	−0.438405
$922$	0	0
$923$	3.31230e9	0.138651
$924$	0	0
$925$	3.56516e9	0.148109
$926$	0	0
$927$	−3.43014e9	−0.141427
$928$	0	0
$929$	−3.39225e10	−1.38814	−0.694069	−	0.719908i	$-0.744184\pi$
$929$	−3.39225e10	−1.38814	−0.694069	+	0.719908i	$0.744184\pi$
$930$	0	0
$931$	−4.45702e10	−1.81018
$932$	0	0
$933$	−1.25397e9	−0.0505478
$934$	0	0
$935$	4.12752e9	0.165139
$936$	0	0
$937$	3.64023e10	1.44557	0.722786	−	0.691072i	$-0.242861\pi$
$937$	3.64023e10	1.44557	0.722786	+	0.691072i	$0.242861\pi$
$938$	0	0
$939$	5.68507e9	0.224081
$940$	0	0
$941$	5.07117e10	1.98401	0.992007	−	0.126181i	$-0.0402719\pi$
$941$	5.07117e10	1.98401	0.992007	+	0.126181i	$0.0402719\pi$
$942$	0	0
$943$	1.17490e10	0.456259
$944$	0	0
$945$	3.39532e9	0.130879
$946$	0	0
$947$	−2.30994e10	−0.883845	−0.441922	−	0.897053i	$-0.645703\pi$
$947$	−2.30994e10	−0.883845	−0.441922	+	0.897053i	$0.645703\pi$
$948$	0	0
$949$	4.77810e10	1.81478
$950$	0	0
$951$	−2.59462e10	−0.978233
$952$	0	0
$953$	−4.16368e9	−0.155830	−0.0779151	−	0.996960i	$-0.524826\pi$
$953$	−4.16368e9	−0.155830	−0.0779151	+	0.996960i	$0.524826\pi$
$954$	0	0
$955$	5.07707e8	0.0188626
$956$	0	0
$957$	1.18470e10	0.436935
$958$	0	0
$959$	2.29971e9	0.0841993
$960$	0	0
$961$	−2.43990e10	−0.886829
$962$	0	0
$963$	−1.91815e9	−0.0692134
$964$	0	0
$965$	1.66526e10	0.596535
$966$	0	0
$967$	5.20468e10	1.85098	0.925489	−	0.378775i	$-0.123655\pi$
$967$	5.20468e10	1.85098	0.925489	+	0.378775i	$0.123655\pi$
$968$	0	0
$969$	1.11270e10	0.392868
$970$	0	0
$971$	3.64573e8	0.0127796	0.00638979	−	0.999980i	$-0.497966\pi$
$971$	3.64573e8	0.0127796	0.00638979	+	0.999980i	$0.497966\pi$
$972$	0	0
$973$	−8.10689e10	−2.82137
$974$	0	0
$975$	3.58847e9	0.123992
$976$	0	0
$977$	−1.81785e10	−0.623631	−0.311816	−	0.950143i	$-0.600937\pi$
$977$	−1.81785e10	−0.623631	−0.311816	+	0.950143i	$0.600937\pi$
$978$	0	0
$979$	9.26358e9	0.315529
$980$	0	0
$981$	−3.13513e9	−0.106027
$982$	0	0
$983$	1.85778e10	0.623816	0.311908	−	0.950112i	$-0.399032\pi$
$983$	1.85778e10	0.623816	0.311908	+	0.950112i	$0.399032\pi$
$984$	0	0
$985$	−1.63464e9	−0.0544999
$986$	0	0
$987$	3.11860e10	1.03240
$988$	0	0
$989$	−6.35520e10	−2.08902
$990$	0	0
$991$	−1.82520e10	−0.595735	−0.297867	−	0.954607i	$-0.596275\pi$
$991$	−1.82520e10	−0.595735	−0.297867	+	0.954607i	$0.596275\pi$
$992$	0	0
$993$	−1.07979e10	−0.349959
$994$	0	0
$995$	−8.73128e9	−0.280994
$996$	0	0
$997$	−1.24534e9	−0.0397975	−0.0198987	−	0.999802i	$-0.506334\pi$
$997$	−1.24534e9	−0.0397975	−0.0198987	+	0.999802i	$0.506334\pi$
$998$	0	0
$999$	4.49107e9	0.142518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.h.1.1		1
4.3	odd	2		15.8.a.b.1.1	✓	1
12.11	even	2		45.8.a.e.1.1		1
20.3	even	4		75.8.b.b.49.2		2
20.7	even	4		75.8.b.b.49.1		2
20.19	odd	2		75.8.a.b.1.1		1
60.23	odd	4		225.8.b.c.199.1		2
60.47	odd	4		225.8.b.c.199.2		2
60.59	even	2		225.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.a.b.1.1	✓	1	4.3	odd	2
45.8.a.e.1.1		1	12.11	even	2
75.8.a.b.1.1		1	20.19	odd	2
75.8.b.b.49.1		2	20.7	even	4
75.8.b.b.49.2		2	20.3	even	4
225.8.a.c.1.1		1	60.59	even	2
225.8.b.c.199.1		2	60.23	odd	4
225.8.b.c.199.2		2	60.47	odd	4
240.8.a.h.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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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