LMFDB - Embedded newform 342.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,8,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("342.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	342.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:	$106.835678716$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +64.0000 q^{4} -440.000 q^{5} +951.000 q^{7} +512.000 q^{8} +O(q^{10})$

\(q+8.00000 q^{2} +64.0000 q^{4} -440.000 q^{5} +951.000 q^{7} +512.000 q^{8} -3520.00 q^{10} +8398.00 q^{11} -6223.00 q^{13} +7608.00 q^{14} +4096.00 q^{16} -26211.0 q^{17} -6859.00 q^{19} -28160.0 q^{20} +67184.0 q^{22} +64213.0 q^{23} +115475. q^{25} -49784.0 q^{26} +60864.0 q^{28} -65845.0 q^{29} -32708.0 q^{31} +32768.0 q^{32} -209688. q^{34} -418440. q^{35} -436694. q^{37} -54872.0 q^{38} -225280. q^{40} +28808.0 q^{41} +650272. q^{43} +537472. q^{44} +513704. q^{46} -58736.0 q^{47} +80858.0 q^{49} +923800. q^{50} -398272. q^{52} +918703. q^{53} -3.69512e6 q^{55} +486912. q^{56} -526760. q^{58} +787635. q^{59} +3.10686e6 q^{61} -261664. q^{62} +262144. q^{64} +2.73812e6 q^{65} +2.72600e6 q^{67} -1.67750e6 q^{68} -3.34752e6 q^{70} +1.80096e6 q^{71} -1.43622e6 q^{73} -3.49355e6 q^{74} -438976. q^{76} +7.98650e6 q^{77} +3.40211e6 q^{79} -1.80224e6 q^{80} +230464. q^{82} +9.45404e6 q^{83} +1.15328e7 q^{85} +5.20218e6 q^{86} +4.29978e6 q^{88} +40980.0 q^{89} -5.91807e6 q^{91} +4.10963e6 q^{92} -469888. q^{94} +3.01796e6 q^{95} +4.28165e6 q^{97} +646864. q^{98} +O(q^{100})\)

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
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−440.000	−1.57419	−0.787096	−	0.616831i	$-0.788416\pi$
$5$	−440.000	−1.57419	−0.787096	+	0.616831i	$0.788416\pi$
$6$	0	0
$7$	951.000	1.04794	0.523971	−	0.851736i	$-0.324450\pi$
$7$	951.000	1.04794	0.523971	+	0.851736i	$0.324450\pi$
$8$	512.000	0.353553
$9$	0	0
$10$	−3520.00	−1.11312
$11$	8398.00	1.90240	0.951199	−	0.308578i	$-0.0998529\pi$
$11$	8398.00	1.90240	0.951199	+	0.308578i	$0.0998529\pi$
$12$	0	0
$13$	−6223.00	−0.785594	−0.392797	−	0.919625i	$-0.628492\pi$
$13$	−6223.00	−0.785594	−0.392797	+	0.919625i	$0.628492\pi$
$14$	7608.00	0.741007
$15$	0	0
$16$	4096.00	0.250000
$17$	−26211.0	−1.29393	−0.646967	−	0.762518i	$-0.723963\pi$
$17$	−26211.0	−1.29393	−0.646967	+	0.762518i	$0.723963\pi$
$18$	0	0
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	−28160.0	−0.787096
$21$	0	0
$22$	67184.0	1.34520
$23$	64213.0	1.10046	0.550232	−	0.835012i	$-0.314539\pi$
$23$	64213.0	1.10046	0.550232	+	0.835012i	$0.314539\pi$
$24$	0	0
$25$	115475.	1.47808
$26$	−49784.0	−0.555499
$27$	0	0
$28$	60864.0	0.523971
$29$	−65845.0	−0.501337	−0.250669	−	0.968073i	$-0.580650\pi$
$29$	−65845.0	−0.501337	−0.250669	+	0.968073i	$0.580650\pi$
$30$	0	0
$31$	−32708.0	−0.197191	−0.0985957	−	0.995128i	$-0.531435\pi$
$31$	−32708.0	−0.197191	−0.0985957	+	0.995128i	$0.531435\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	−209688.	−0.914950
$35$	−418440.	−1.64966
$36$	0	0
$37$	−436694.	−1.41733	−0.708665	−	0.705545i	$-0.750702\pi$
$37$	−436694.	−1.41733	−0.708665	+	0.705545i	$0.750702\pi$
$38$	−54872.0	−0.162221
$39$	0	0
$40$	−225280.	−0.556561
$41$	28808.0	0.0652784	0.0326392	−	0.999467i	$-0.489609\pi$
$41$	28808.0	0.0652784	0.0326392	+	0.999467i	$0.489609\pi$
$42$	0	0
$43$	650272.	1.24726	0.623628	−	0.781721i	$-0.285658\pi$
$43$	650272.	1.24726	0.623628	+	0.781721i	$0.285658\pi$
$44$	537472.	0.951199
$45$	0	0
$46$	513704.	0.778145
$47$	−58736.0	−0.0825205	−0.0412603	−	0.999148i	$-0.513137\pi$
$47$	−58736.0	−0.0825205	−0.0412603	+	0.999148i	$0.513137\pi$
$48$	0	0
$49$	80858.0	0.0981831
$50$	923800.	1.04516
$51$	0	0
$52$	−398272.	−0.392797
$53$	918703.	0.847636	0.423818	−	0.905747i	$-0.360690\pi$
$53$	918703.	0.847636	0.423818	+	0.905747i	$0.360690\pi$
$54$	0	0
$55$	−3.69512e6	−2.99474
$56$	486912.	0.370504
$57$	0	0
$58$	−526760.	−0.354499
$59$	787635.	0.499279	0.249639	−	0.968339i	$-0.419688\pi$
$59$	787635.	0.499279	0.249639	+	0.968339i	$0.419688\pi$
$60$	0	0
$61$	3.10686e6	1.75254	0.876269	−	0.481822i	$-0.160025\pi$
$61$	3.10686e6	1.75254	0.876269	+	0.481822i	$0.160025\pi$
$62$	−261664.	−0.139435
$63$	0	0
$64$	262144.	0.125000
$65$	2.73812e6	1.23668
$66$	0	0
$67$	2.72600e6	1.10730	0.553649	−	0.832750i	$-0.313235\pi$
$67$	2.72600e6	1.10730	0.553649	+	0.832750i	$0.313235\pi$
$68$	−1.67750e6	−0.646967
$69$	0	0
$70$	−3.34752e6	−1.16649
$71$	1.80096e6	0.597172	0.298586	−	0.954383i	$-0.403485\pi$
$71$	1.80096e6	0.597172	0.298586	+	0.954383i	$0.403485\pi$
$72$	0	0
$73$	−1.43622e6	−0.432108	−0.216054	−	0.976381i	$-0.569319\pi$
$73$	−1.43622e6	−0.432108	−0.216054	+	0.976381i	$0.569319\pi$
$74$	−3.49355e6	−1.00220
$75$	0	0
$76$	−438976.	−0.114708
$77$	7.98650e6	1.99360
$78$	0	0
$79$	3.40211e6	0.776343	0.388171	−	0.921587i	$-0.373107\pi$
$79$	3.40211e6	0.776343	0.388171	+	0.921587i	$0.373107\pi$
$80$	−1.80224e6	−0.393548
$81$	0	0
$82$	230464.	0.0461588
$83$	9.45404e6	1.81486	0.907432	−	0.420199i	$-0.138040\pi$
$83$	9.45404e6	1.81486	0.907432	+	0.420199i	$0.138040\pi$
$84$	0	0
$85$	1.15328e7	2.03690
$86$	5.20218e6	0.881943
$87$	0	0
$88$	4.29978e6	0.672599
$89$	40980.0	0.00616179	0.00308090	−	0.999995i	$-0.499019\pi$
$89$	40980.0	0.00616179	0.00308090	+	0.999995i	$0.499019\pi$
$90$	0	0
$91$	−5.91807e6	−0.823257
$92$	4.10963e6	0.550232
$93$	0	0
$94$	−469888.	−0.0583508
$95$	3.01796e6	0.361144
$96$	0	0
$97$	4.28165e6	0.476332	0.238166	−	0.971224i	$-0.423454\pi$
$97$	4.28165e6	0.476332	0.238166	+	0.971224i	$0.423454\pi$
$98$	646864.	0.0694259
$99$	0	0
$100$	7.39040e6	0.739040
$101$	2.48364e6	0.239863	0.119932	−	0.992782i	$-0.461732\pi$
$101$	2.48364e6	0.239863	0.119932	+	0.992782i	$0.461732\pi$
$102$	0	0
$103$	−1.25032e7	−1.12743	−0.563716	−	0.825969i	$-0.690629\pi$
$103$	−1.25032e7	−1.12743	−0.563716	+	0.825969i	$0.690629\pi$
$104$	−3.18618e6	−0.277749
$105$	0	0
$106$	7.34962e6	0.599369
$107$	−2.22469e6	−0.175560	−0.0877802	−	0.996140i	$-0.527977\pi$
$107$	−2.22469e6	−0.175560	−0.0877802	+	0.996140i	$0.527977\pi$
$108$	0	0
$109$	1.41411e7	1.04590	0.522949	−	0.852364i	$-0.324832\pi$
$109$	1.41411e7	1.04590	0.522949	+	0.852364i	$0.324832\pi$
$110$	−2.95610e7	−2.11760
$111$	0	0
$112$	3.89530e6	0.261986
$113$	7.38354e6	0.481382	0.240691	−	0.970602i	$-0.422626\pi$
$113$	7.38354e6	0.481382	0.240691	+	0.970602i	$0.422626\pi$
$114$	0	0
$115$	−2.82537e7	−1.73234
$116$	−4.21408e6	−0.250669
$117$	0	0
$118$	6.30108e6	0.353043
$119$	−2.49267e7	−1.35597
$120$	0	0
$121$	5.10392e7	2.61912
$122$	2.48549e7	1.23923
$123$	0	0
$124$	−2.09331e6	−0.0985957
$125$	−1.64340e7	−0.752590
$126$	0	0
$127$	−2.24906e7	−0.974288	−0.487144	−	0.873322i	$-0.661961\pi$
$127$	−2.24906e7	−0.974288	−0.487144	+	0.873322i	$0.661961\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	2.19050e7	0.874462
$131$	3.90761e6	0.151866	0.0759332	−	0.997113i	$-0.475806\pi$
$131$	3.90761e6	0.151866	0.0759332	+	0.997113i	$0.475806\pi$
$132$	0	0
$133$	−6.52291e6	−0.240414
$134$	2.18080e7	0.782977
$135$	0	0
$136$	−1.34200e7	−0.457475
$137$	2.98680e7	0.992393	0.496197	−	0.868210i	$-0.334730\pi$
$137$	2.98680e7	0.992393	0.496197	+	0.868210i	$0.334730\pi$
$138$	0	0
$139$	9.35214e6	0.295365	0.147683	−	0.989035i	$-0.452819\pi$
$139$	9.35214e6	0.295365	0.147683	+	0.989035i	$0.452819\pi$
$140$	−2.67802e7	−0.824831
$141$	0	0
$142$	1.44077e7	0.422264
$143$	−5.22608e7	−1.49451
$144$	0	0
$145$	2.89718e7	0.789201
$146$	−1.14898e7	−0.305546
$147$	0	0
$148$	−2.79484e7	−0.708665
$149$	−2.48262e7	−0.614835	−0.307417	−	0.951575i	$-0.599465\pi$
$149$	−2.48262e7	−0.614835	−0.307417	+	0.951575i	$0.599465\pi$
$150$	0	0
$151$	−1.11303e7	−0.263080	−0.131540	−	0.991311i	$-0.541992\pi$
$151$	−1.11303e7	−0.263080	−0.131540	+	0.991311i	$0.541992\pi$
$152$	−3.51181e6	−0.0811107
$153$	0	0
$154$	6.38920e7	1.40969
$155$	1.43915e7	0.310417
$156$	0	0
$157$	4.52922e7	0.934059	0.467030	−	0.884242i	$-0.345324\pi$
$157$	4.52922e7	0.934059	0.467030	+	0.884242i	$0.345324\pi$
$158$	2.72169e7	0.548957
$159$	0	0
$160$	−1.44179e7	−0.278280
$161$	6.10666e7	1.15322
$162$	0	0
$163$	2.33565e7	0.422427	0.211214	−	0.977440i	$-0.432258\pi$
$163$	2.33565e7	0.422427	0.211214	+	0.977440i	$0.432258\pi$
$164$	1.84371e6	0.0326392
$165$	0	0
$166$	7.56323e7	1.28330
$167$	6.83692e7	1.13593	0.567967	−	0.823051i	$-0.307730\pi$
$167$	6.83692e7	1.13593	0.567967	+	0.823051i	$0.307730\pi$
$168$	0	0
$169$	−2.40228e7	−0.382842
$170$	9.22627e7	1.44031
$171$	0	0
$172$	4.16174e7	0.623628
$173$	−4.58676e7	−0.673511	−0.336755	−	0.941592i	$-0.609330\pi$
$173$	−4.58676e7	−0.673511	−0.336755	+	0.941592i	$0.609330\pi$
$174$	0	0
$175$	1.09817e8	1.54894
$176$	3.43982e7	0.475600
$177$	0	0
$178$	327840.	0.00435704
$179$	3.50479e7	0.456748	0.228374	−	0.973573i	$-0.426659\pi$
$179$	3.50479e7	0.456748	0.228374	+	0.973573i	$0.426659\pi$
$180$	0	0
$181$	1.30380e7	0.163431	0.0817155	−	0.996656i	$-0.473960\pi$
$181$	1.30380e7	0.163431	0.0817155	+	0.996656i	$0.473960\pi$
$182$	−4.73446e7	−0.582131
$183$	0	0
$184$	3.28771e7	0.389073
$185$	1.92145e8	2.23115
$186$	0	0
$187$	−2.20120e8	−2.46158
$188$	−3.75910e6	−0.0412603
$189$	0	0
$190$	2.41437e7	0.255368
$191$	7.52311e7	0.781233	0.390616	−	0.920554i	$-0.372262\pi$
$191$	7.52311e7	0.781233	0.390616	+	0.920554i	$0.372262\pi$
$192$	0	0
$193$	8.77425e7	0.878536	0.439268	−	0.898356i	$-0.355238\pi$
$193$	8.77425e7	0.878536	0.439268	+	0.898356i	$0.355238\pi$
$194$	3.42532e7	0.336818
$195$	0	0
$196$	5.17491e6	0.0490915
$197$	−1.69049e8	−1.57536	−0.787681	−	0.616083i	$-0.788719\pi$
$197$	−1.69049e8	−1.57536	−0.787681	+	0.616083i	$0.788719\pi$
$198$	0	0
$199$	1.39490e8	1.25475	0.627377	−	0.778716i	$-0.284128\pi$
$199$	1.39490e8	1.25475	0.627377	+	0.778716i	$0.284128\pi$
$200$	5.91232e7	0.522580
$201$	0	0
$202$	1.98691e7	0.169609
$203$	−6.26186e7	−0.525372
$204$	0	0
$205$	−1.26755e7	−0.102761
$206$	−1.00025e8	−0.797214
$207$	0	0
$208$	−2.54894e7	−0.196398
$209$	−5.76019e7	−0.436440
$210$	0	0
$211$	−2.25998e8	−1.65621	−0.828105	−	0.560573i	$-0.810581\pi$
$211$	−2.25998e8	−1.65621	−0.828105	+	0.560573i	$0.810581\pi$
$212$	5.87970e7	0.423818
$213$	0	0
$214$	−1.77975e7	−0.124140
$215$	−2.86120e8	−1.96342
$216$	0	0
$217$	−3.11053e7	−0.206645
$218$	1.13128e8	0.739561
$219$	0	0
$220$	−2.36488e8	−1.49737
$221$	1.63111e8	1.01651
$222$	0	0
$223$	1.59102e8	0.960746	0.480373	−	0.877064i	$-0.340501\pi$
$223$	1.59102e8	0.960746	0.480373	+	0.877064i	$0.340501\pi$
$224$	3.11624e7	0.185252
$225$	0	0
$226$	5.90683e7	0.340389
$227$	−2.73408e8	−1.55139	−0.775693	−	0.631110i	$-0.782600\pi$
$227$	−2.73408e8	−1.55139	−0.775693	+	0.631110i	$0.782600\pi$
$228$	0	0
$229$	9.36336e7	0.515238	0.257619	−	0.966247i	$-0.417062\pi$
$229$	9.36336e7	0.515238	0.257619	+	0.966247i	$0.417062\pi$
$230$	−2.26030e8	−1.22495
$231$	0	0
$232$	−3.37126e7	−0.177249
$233$	2.77939e7	0.143947	0.0719736	−	0.997407i	$-0.477070\pi$
$233$	2.77939e7	0.143947	0.0719736	+	0.997407i	$0.477070\pi$
$234$	0	0
$235$	2.58438e7	0.129903
$236$	5.04086e7	0.249639
$237$	0	0
$238$	−1.99413e8	−0.958815
$239$	−2.65907e8	−1.25990	−0.629952	−	0.776634i	$-0.716925\pi$
$239$	−2.65907e8	−1.25990	−0.629952	+	0.776634i	$0.716925\pi$
$240$	0	0
$241$	−1.15586e8	−0.531917	−0.265959	−	0.963984i	$-0.585688\pi$
$241$	−1.15586e8	−0.531917	−0.265959	+	0.963984i	$0.585688\pi$
$242$	4.08314e8	1.85200
$243$	0	0
$244$	1.98839e8	0.876269
$245$	−3.55775e7	−0.154559
$246$	0	0
$247$	4.26836e7	0.180228
$248$	−1.67465e7	−0.0697177
$249$	0	0
$250$	−1.31472e8	−0.532161
$251$	−9.91605e7	−0.395804	−0.197902	−	0.980222i	$-0.563413\pi$
$251$	−9.91605e7	−0.395804	−0.197902	+	0.980222i	$0.563413\pi$
$252$	0	0
$253$	5.39261e8	2.09352
$254$	−1.79925e8	−0.688926
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	3.72577e8	1.36915	0.684573	−	0.728944i	$-0.259989\pi$
$257$	3.72577e8	1.36915	0.684573	+	0.728944i	$0.259989\pi$
$258$	0	0
$259$	−4.15296e8	−1.48528
$260$	1.75240e8	0.618338
$261$	0	0
$262$	3.12609e7	0.107386
$263$	3.58686e8	1.21582	0.607911	−	0.794005i	$-0.292008\pi$
$263$	3.58686e8	1.21582	0.607911	+	0.794005i	$0.292008\pi$
$264$	0	0
$265$	−4.04229e8	−1.33434
$266$	−5.21833e7	−0.169999
$267$	0	0
$268$	1.74464e8	0.553649
$269$	−5.09148e8	−1.59482	−0.797408	−	0.603440i	$-0.793796\pi$
$269$	−5.09148e8	−1.59482	−0.797408	+	0.603440i	$0.793796\pi$
$270$	0	0
$271$	3.12525e8	0.953877	0.476938	−	0.878937i	$-0.341746\pi$
$271$	3.12525e8	0.953877	0.476938	+	0.878937i	$0.341746\pi$
$272$	−1.07360e8	−0.323484
$273$	0	0
$274$	2.38944e8	0.701728
$275$	9.69759e8	2.81190
$276$	0	0
$277$	3.42007e8	0.966844	0.483422	−	0.875388i	$-0.339394\pi$
$277$	3.42007e8	0.966844	0.483422	+	0.875388i	$0.339394\pi$
$278$	7.48171e7	0.208855
$279$	0	0
$280$	−2.14241e8	−0.583244
$281$	4.28706e8	1.15262	0.576311	−	0.817230i	$-0.304492\pi$
$281$	4.28706e8	1.15262	0.576311	+	0.817230i	$0.304492\pi$
$282$	0	0
$283$	4.52256e8	1.18613	0.593064	−	0.805155i	$-0.297918\pi$
$283$	4.52256e8	1.18613	0.593064	+	0.805155i	$0.297918\pi$
$284$	1.15261e8	0.298586
$285$	0	0
$286$	−4.18086e8	−1.05678
$287$	2.73964e7	0.0684080
$288$	0	0
$289$	2.76678e8	0.674267
$290$	2.31774e8	0.558049
$291$	0	0
$292$	−9.19183e7	−0.216054
$293$	−4.17395e8	−0.969416	−0.484708	−	0.874676i	$-0.661074\pi$
$293$	−4.17395e8	−0.969416	−0.484708	+	0.874676i	$0.661074\pi$
$294$	0	0
$295$	−3.46559e8	−0.785960
$296$	−2.23587e8	−0.501102
$297$	0	0
$298$	−1.98610e8	−0.434754
$299$	−3.99597e8	−0.864517
$300$	0	0
$301$	6.18409e8	1.30705
$302$	−8.90424e7	−0.186025
$303$	0	0
$304$	−2.80945e7	−0.0573539
$305$	−1.36702e9	−2.75883
$306$	0	0
$307$	3.15746e8	0.622807	0.311404	−	0.950278i	$-0.399201\pi$
$307$	3.15746e8	0.622807	0.311404	+	0.950278i	$0.399201\pi$
$308$	5.11136e8	0.996802
$309$	0	0
$310$	1.15132e8	0.219498
$311$	−5.25059e8	−0.989798	−0.494899	−	0.868950i	$-0.664795\pi$
$311$	−5.25059e8	−0.989798	−0.494899	+	0.868950i	$0.664795\pi$
$312$	0	0
$313$	9.13906e8	1.68460	0.842299	−	0.539011i	$-0.181202\pi$
$313$	9.13906e8	1.68460	0.842299	+	0.539011i	$0.181202\pi$
$314$	3.62337e8	0.660480
$315$	0	0
$316$	2.17735e8	0.388171
$317$	3.56880e8	0.629237	0.314619	−	0.949218i	$-0.398123\pi$
$317$	3.56880e8	0.629237	0.314619	+	0.949218i	$0.398123\pi$
$318$	0	0
$319$	−5.52966e8	−0.953743
$320$	−1.15343e8	−0.196774
$321$	0	0
$322$	4.88533e8	0.815451
$323$	1.79781e8	0.296849
$324$	0	0
$325$	−7.18601e8	−1.16117
$326$	1.86852e8	0.298701
$327$	0	0
$328$	1.47497e7	0.0230794
$329$	−5.58579e7	−0.0864767
$330$	0	0
$331$	−1.19720e9	−1.81455	−0.907274	−	0.420541i	$-0.861840\pi$
$331$	−1.19720e9	−1.81455	−0.907274	+	0.420541i	$0.861840\pi$
$332$	6.05058e8	0.907432
$333$	0	0
$334$	5.46954e8	0.803227
$335$	−1.19944e9	−1.74310
$336$	0	0
$337$	2.91487e8	0.414872	0.207436	−	0.978249i	$-0.433488\pi$
$337$	2.91487e8	0.414872	0.207436	+	0.978249i	$0.433488\pi$
$338$	−1.92182e8	−0.270710
$339$	0	0
$340$	7.38102e8	1.01845
$341$	−2.74682e8	−0.375137
$342$	0	0
$343$	−7.06293e8	−0.945052
$344$	3.32939e8	0.440971
$345$	0	0
$346$	−3.66941e8	−0.476244
$347$	−2.16207e8	−0.277791	−0.138895	−	0.990307i	$-0.544355\pi$
$347$	−2.16207e8	−0.277791	−0.138895	+	0.990307i	$0.544355\pi$
$348$	0	0
$349$	−1.03896e9	−1.30831	−0.654154	−	0.756362i	$-0.726975\pi$
$349$	−1.03896e9	−1.30831	−0.654154	+	0.756362i	$0.726975\pi$
$350$	8.78534e8	1.09527
$351$	0	0
$352$	2.75186e8	0.336300
$353$	−1.16693e8	−0.141200	−0.0705999	−	0.997505i	$-0.522491\pi$
$353$	−1.16693e8	−0.141200	−0.0705999	+	0.997505i	$0.522491\pi$
$354$	0	0
$355$	−7.92422e8	−0.940063
$356$	2.62272e6	0.00308090
$357$	0	0
$358$	2.80383e8	0.322970
$359$	1.28194e9	1.46230	0.731150	−	0.682217i	$-0.238984\pi$
$359$	1.28194e9	1.46230	0.731150	+	0.682217i	$0.238984\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	1.04304e8	0.115563
$363$	0	0
$364$	−3.78757e8	−0.411629
$365$	6.31938e8	0.680220
$366$	0	0
$367$	−1.91665e8	−0.202400	−0.101200	−	0.994866i	$-0.532268\pi$
$367$	−1.91665e8	−0.202400	−0.101200	+	0.994866i	$0.532268\pi$
$368$	2.63016e8	0.275116
$369$	0	0
$370$	1.53716e9	1.57766
$371$	8.73687e8	0.888274
$372$	0	0
$373$	1.00128e8	0.0999023	0.0499512	−	0.998752i	$-0.484093\pi$
$373$	1.00128e8	0.0999023	0.0499512	+	0.998752i	$0.484093\pi$
$374$	−1.76096e9	−1.74060
$375$	0	0
$376$	−3.00728e7	−0.0291754
$377$	4.09753e8	0.393847
$378$	0	0
$379$	−1.03285e9	−0.974543	−0.487272	−	0.873251i	$-0.662008\pi$
$379$	−1.03285e9	−0.974543	−0.487272	+	0.873251i	$0.662008\pi$
$380$	1.93149e8	0.180572
$381$	0	0
$382$	6.01848e8	0.552415
$383$	−1.41475e9	−1.28672	−0.643358	−	0.765566i	$-0.722459\pi$
$383$	−1.41475e9	−1.28672	−0.643358	+	0.765566i	$0.722459\pi$
$384$	0	0
$385$	−3.51406e9	−3.13831
$386$	7.01940e8	0.621219
$387$	0	0
$388$	2.74025e8	0.238166
$389$	−6.84402e8	−0.589505	−0.294752	−	0.955574i	$-0.595237\pi$
$389$	−6.84402e8	−0.589505	−0.294752	+	0.955574i	$0.595237\pi$
$390$	0	0
$391$	−1.68309e9	−1.42393
$392$	4.13993e7	0.0347130
$393$	0	0
$394$	−1.35239e9	−1.11395
$395$	−1.49693e9	−1.22211
$396$	0	0
$397$	−3.21394e8	−0.257793	−0.128897	−	0.991658i	$-0.541144\pi$
$397$	−3.21394e8	−0.257793	−0.128897	+	0.991658i	$0.541144\pi$
$398$	1.11592e9	0.887245
$399$	0	0
$400$	4.72986e8	0.369520
$401$	−1.49873e9	−1.16070	−0.580349	−	0.814368i	$-0.697084\pi$
$401$	−1.49873e9	−1.16070	−0.580349	+	0.814368i	$0.697084\pi$
$402$	0	0
$403$	2.03542e8	0.154912
$404$	1.58953e8	0.119932
$405$	0	0
$406$	−5.00949e8	−0.371494
$407$	−3.66736e9	−2.69633
$408$	0	0
$409$	−1.67043e9	−1.20725	−0.603623	−	0.797270i	$-0.706277\pi$
$409$	−1.67043e9	−1.20725	−0.603623	+	0.797270i	$0.706277\pi$
$410$	−1.01404e8	−0.0726628
$411$	0	0
$412$	−8.00203e8	−0.563716
$413$	7.49041e8	0.523215
$414$	0	0
$415$	−4.15978e9	−2.85694
$416$	−2.03915e8	−0.138875
$417$	0	0
$418$	−4.60815e8	−0.308610
$419$	8.15512e8	0.541603	0.270802	−	0.962635i	$-0.412711\pi$
$419$	8.15512e8	0.541603	0.270802	+	0.962635i	$0.412711\pi$
$420$	0	0
$421$	1.94494e8	0.127034	0.0635169	−	0.997981i	$-0.479768\pi$
$421$	1.94494e8	0.127034	0.0635169	+	0.997981i	$0.479768\pi$
$422$	−1.80798e9	−1.17112
$423$	0	0
$424$	4.70376e8	0.299685
$425$	−3.02672e9	−1.91254
$426$	0	0
$427$	2.95463e9	1.83656
$428$	−1.42380e8	−0.0877802
$429$	0	0
$430$	−2.28896e9	−1.38835
$431$	2.62496e9	1.57925	0.789627	−	0.613587i	$-0.210274\pi$
$431$	2.62496e9	1.57925	0.789627	+	0.613587i	$0.210274\pi$
$432$	0	0
$433$	1.90404e9	1.12712	0.563559	−	0.826076i	$-0.309432\pi$
$433$	1.90404e9	1.12712	0.563559	+	0.826076i	$0.309432\pi$
$434$	−2.48842e8	−0.146120
$435$	0	0
$436$	9.05028e8	0.522949
$437$	−4.40437e8	−0.252464
$438$	0	0
$439$	−1.51564e9	−0.855007	−0.427503	−	0.904014i	$-0.640607\pi$
$439$	−1.51564e9	−0.855007	−0.427503	+	0.904014i	$0.640607\pi$
$440$	−1.89190e9	−1.05880
$441$	0	0
$442$	1.30489e9	0.718779
$443$	2.83361e9	1.54856	0.774279	−	0.632844i	$-0.218113\pi$
$443$	2.83361e9	1.54856	0.774279	+	0.632844i	$0.218113\pi$
$444$	0	0
$445$	−1.80312e7	−0.00969984
$446$	1.27282e9	0.679350
$447$	0	0
$448$	2.49299e8	0.130993
$449$	−7.85104e8	−0.409322	−0.204661	−	0.978833i	$-0.565609\pi$
$449$	−7.85104e8	−0.409322	−0.204661	+	0.978833i	$0.565609\pi$
$450$	0	0
$451$	2.41930e8	0.124186
$452$	4.72546e8	0.240691
$453$	0	0
$454$	−2.18726e9	−1.09700
$455$	2.60395e9	1.29596
$456$	0	0
$457$	1.46139e8	0.0716242	0.0358121	−	0.999359i	$-0.488598\pi$
$457$	1.46139e8	0.0716242	0.0358121	+	0.999359i	$0.488598\pi$
$458$	7.49069e8	0.364328
$459$	0	0
$460$	−1.80824e9	−0.866170
$461$	−1.31176e9	−0.623593	−0.311797	−	0.950149i	$-0.600931\pi$
$461$	−1.31176e9	−0.623593	−0.311797	+	0.950149i	$0.600931\pi$
$462$	0	0
$463$	3.88164e9	1.81753	0.908764	−	0.417310i	$-0.137027\pi$
$463$	3.88164e9	1.81753	0.908764	+	0.417310i	$0.137027\pi$
$464$	−2.69701e8	−0.125334
$465$	0	0
$466$	2.22351e8	0.101786
$467$	−1.16917e9	−0.531211	−0.265606	−	0.964082i	$-0.585572\pi$
$467$	−1.16917e9	−0.531211	−0.265606	+	0.964082i	$0.585572\pi$
$468$	0	0
$469$	2.59243e9	1.16038
$470$	2.06751e8	0.0918554
$471$	0	0
$472$	4.03269e8	0.176522
$473$	5.46098e9	2.37278
$474$	0	0
$475$	−7.92043e8	−0.339095
$476$	−1.59531e9	−0.677984
$477$	0	0
$478$	−2.12726e9	−0.890886
$479$	1.99948e9	0.831271	0.415635	−	0.909531i	$-0.363559\pi$
$479$	1.99948e9	0.831271	0.415635	+	0.909531i	$0.363559\pi$
$480$	0	0
$481$	2.71755e9	1.11345
$482$	−9.24685e8	−0.376122
$483$	0	0
$484$	3.26651e9	1.30956
$485$	−1.88392e9	−0.749838
$486$	0	0
$487$	−1.34306e9	−0.526920	−0.263460	−	0.964670i	$-0.584864\pi$
$487$	−1.34306e9	−0.526920	−0.263460	+	0.964670i	$0.584864\pi$
$488$	1.59071e9	0.619616
$489$	0	0
$490$	−2.84620e8	−0.109290
$491$	−4.30613e9	−1.64173	−0.820865	−	0.571122i	$-0.806508\pi$
$491$	−4.30613e9	−1.64173	−0.820865	+	0.571122i	$0.806508\pi$
$492$	0	0
$493$	1.72586e9	0.648697
$494$	3.41468e8	0.127440
$495$	0	0
$496$	−1.33972e8	−0.0492979
$497$	1.71271e9	0.625802
$498$	0	0
$499$	2.28288e9	0.822492	0.411246	−	0.911524i	$-0.365094\pi$
$499$	2.28288e9	0.822492	0.411246	+	0.911524i	$0.365094\pi$
$500$	−1.05178e9	−0.376295
$501$	0	0
$502$	−7.93284e8	−0.279876
$503$	−1.75365e9	−0.614405	−0.307202	−	0.951644i	$-0.599393\pi$
$503$	−1.75365e9	−0.614405	−0.307202	+	0.951644i	$0.599393\pi$
$504$	0	0
$505$	−1.09280e9	−0.377591
$506$	4.31409e9	1.48034
$507$	0	0
$508$	−1.43940e9	−0.487144
$509$	−5.93337e9	−1.99429	−0.997147	−	0.0754798i	$-0.975951\pi$
$509$	−5.93337e9	−1.99429	−0.997147	+	0.0754798i	$0.975951\pi$
$510$	0	0
$511$	−1.36585e9	−0.452824
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	2.98061e9	0.968132
$515$	5.50140e9	1.77479
$516$	0	0
$517$	−4.93265e8	−0.156987
$518$	−3.32237e9	−1.05025
$519$	0	0
$520$	1.40192e9	0.437231
$521$	−3.43082e9	−1.06284	−0.531418	−	0.847110i	$-0.678341\pi$
$521$	−3.43082e9	−1.06284	−0.531418	+	0.847110i	$0.678341\pi$
$522$	0	0
$523$	5.51108e8	0.168454	0.0842269	−	0.996447i	$-0.473158\pi$
$523$	5.51108e8	0.168454	0.0842269	+	0.996447i	$0.473158\pi$
$524$	2.50087e8	0.0759332
$525$	0	0
$526$	2.86949e9	0.859716
$527$	8.57309e8	0.255153
$528$	0	0
$529$	7.18484e8	0.211019
$530$	−3.23383e9	−0.943522
$531$	0	0
$532$	−4.17466e8	−0.120207
$533$	−1.79272e8	−0.0512823
$534$	0	0
$535$	9.78864e8	0.276366
$536$	1.39571e9	0.391489
$537$	0	0
$538$	−4.07318e9	−1.12771
$539$	6.79045e8	0.186783
$540$	0	0
$541$	−1.74658e9	−0.474240	−0.237120	−	0.971480i	$-0.576203\pi$
$541$	−1.74658e9	−0.474240	−0.237120	+	0.971480i	$0.576203\pi$
$542$	2.50020e9	0.674493
$543$	0	0
$544$	−8.58882e8	−0.228738
$545$	−6.22206e9	−1.64644
$546$	0	0
$547$	−5.18146e9	−1.35362	−0.676810	−	0.736158i	$-0.736638\pi$
$547$	−5.18146e9	−1.35362	−0.676810	+	0.736158i	$0.736638\pi$
$548$	1.91155e9	0.496197
$549$	0	0
$550$	7.75807e9	1.98831
$551$	4.51631e8	0.115015
$552$	0	0
$553$	3.23541e9	0.813562
$554$	2.73606e9	0.683662
$555$	0	0
$556$	5.98537e8	0.147683
$557$	−1.08397e9	−0.265780	−0.132890	−	0.991131i	$-0.542426\pi$
$557$	−1.08397e9	−0.265780	−0.132890	+	0.991131i	$0.542426\pi$
$558$	0	0
$559$	−4.04664e9	−0.979836
$560$	−1.71393e9	−0.412416
$561$	0	0
$562$	3.42965e9	0.815027
$563$	−4.58547e9	−1.08294	−0.541471	−	0.840720i	$-0.682132\pi$
$563$	−4.58547e9	−1.08294	−0.541471	+	0.840720i	$0.682132\pi$
$564$	0	0
$565$	−3.24876e9	−0.757788
$566$	3.61805e9	0.838720
$567$	0	0
$568$	9.22090e8	0.211132
$569$	6.15694e9	1.40111	0.700555	−	0.713598i	$-0.252936\pi$
$569$	6.15694e9	1.40111	0.700555	+	0.713598i	$0.252936\pi$
$570$	0	0
$571$	3.85757e9	0.867136	0.433568	−	0.901121i	$-0.357254\pi$
$571$	3.85757e9	0.867136	0.433568	+	0.901121i	$0.357254\pi$
$572$	−3.34469e9	−0.747256
$573$	0	0
$574$	2.19171e8	0.0483718
$575$	7.41500e9	1.62657
$576$	0	0
$577$	−5.15448e9	−1.11704	−0.558521	−	0.829490i	$-0.688631\pi$
$577$	−5.15448e9	−1.11704	−0.558521	+	0.829490i	$0.688631\pi$
$578$	2.21342e9	0.476779
$579$	0	0
$580$	1.85420e9	0.394600
$581$	8.99079e9	1.90187
$582$	0	0
$583$	7.71527e9	1.61254
$584$	−7.35346e8	−0.152773
$585$	0	0
$586$	−3.33916e9	−0.685481
$587$	1.46585e9	0.299127	0.149564	−	0.988752i	$-0.452213\pi$
$587$	1.46585e9	0.299127	0.149564	+	0.988752i	$0.452213\pi$
$588$	0	0
$589$	2.24344e8	0.0452388
$590$	−2.77248e9	−0.555758
$591$	0	0
$592$	−1.78870e9	−0.354333
$593$	−4.20253e9	−0.827599	−0.413799	−	0.910368i	$-0.635798\pi$
$593$	−4.20253e9	−0.827599	−0.413799	+	0.910368i	$0.635798\pi$
$594$	0	0
$595$	1.09677e10	2.13456
$596$	−1.58888e9	−0.307417
$597$	0	0
$598$	−3.19678e9	−0.611306
$599$	1.57872e9	0.300132	0.150066	−	0.988676i	$-0.452051\pi$
$599$	1.57872e9	0.300132	0.150066	+	0.988676i	$0.452051\pi$
$600$	0	0
$601$	9.22428e9	1.73329	0.866647	−	0.498923i	$-0.166271\pi$
$601$	9.22428e9	1.73329	0.866647	+	0.498923i	$0.166271\pi$
$602$	4.94727e9	0.924225
$603$	0	0
$604$	−7.12339e8	−0.131540
$605$	−2.24573e10	−4.12300
$606$	0	0
$607$	3.36747e9	0.611143	0.305572	−	0.952169i	$-0.401152\pi$
$607$	3.36747e9	0.611143	0.305572	+	0.952169i	$0.401152\pi$
$608$	−2.24756e8	−0.0405554
$609$	0	0
$610$	−1.09362e10	−1.95079
$611$	3.65514e8	0.0648276
$612$	0	0
$613$	1.66253e9	0.291512	0.145756	−	0.989321i	$-0.453439\pi$
$613$	1.66253e9	0.291512	0.145756	+	0.989321i	$0.453439\pi$
$614$	2.52597e9	0.440391
$615$	0	0
$616$	4.08909e9	0.704845
$617$	4.02484e9	0.689843	0.344922	−	0.938632i	$-0.387906\pi$
$617$	4.02484e9	0.689843	0.344922	+	0.938632i	$0.387906\pi$
$618$	0	0
$619$	1.05076e10	1.78068	0.890338	−	0.455300i	$-0.150468\pi$
$619$	1.05076e10	1.78068	0.890338	+	0.455300i	$0.150468\pi$
$620$	9.21057e8	0.155209
$621$	0	0
$622$	−4.20047e9	−0.699893
$623$	3.89720e7	0.00645720
$624$	0	0
$625$	−1.79052e9	−0.293360
$626$	7.31124e9	1.19119
$627$	0	0
$628$	2.89870e9	0.467030
$629$	1.14462e10	1.83393
$630$	0	0
$631$	−1.43832e9	−0.227905	−0.113952	−	0.993486i	$-0.536351\pi$
$631$	−1.43832e9	−0.227905	−0.113952	+	0.993486i	$0.536351\pi$
$632$	1.74188e9	0.274479
$633$	0	0
$634$	2.85504e9	0.444938
$635$	9.89585e9	1.53372
$636$	0	0
$637$	−5.03179e8	−0.0771320
$638$	−4.42373e9	−0.674398
$639$	0	0
$640$	−9.22747e8	−0.139140
$641$	6.45356e9	0.967824	0.483912	−	0.875117i	$-0.339215\pi$
$641$	6.45356e9	0.967824	0.483912	+	0.875117i	$0.339215\pi$
$642$	0	0
$643$	−5.47199e9	−0.811721	−0.405860	−	0.913935i	$-0.633028\pi$
$643$	−5.47199e9	−0.811721	−0.405860	+	0.913935i	$0.633028\pi$
$644$	3.90826e9	0.576611
$645$	0	0
$646$	1.43825e9	0.209904
$647$	−9.50275e9	−1.37938	−0.689691	−	0.724104i	$-0.742254\pi$
$647$	−9.50275e9	−1.37938	−0.689691	+	0.724104i	$0.742254\pi$
$648$	0	0
$649$	6.61456e9	0.949827
$650$	−5.74881e9	−0.821072
$651$	0	0
$652$	1.49482e9	0.211214
$653$	−1.02097e10	−1.43489	−0.717445	−	0.696615i	$-0.754689\pi$
$653$	−1.02097e10	−1.43489	−0.717445	+	0.696615i	$0.754689\pi$
$654$	0	0
$655$	−1.71935e9	−0.239067
$656$	1.17998e8	0.0163196
$657$	0	0
$658$	−4.46863e8	−0.0611483
$659$	8.46676e9	1.15244	0.576220	−	0.817295i	$-0.304527\pi$
$659$	8.46676e9	1.15244	0.576220	+	0.817295i	$0.304527\pi$
$660$	0	0
$661$	−7.70863e9	−1.03818	−0.519089	−	0.854720i	$-0.673729\pi$
$661$	−7.70863e9	−1.03818	−0.519089	+	0.854720i	$0.673729\pi$
$662$	−9.57759e9	−1.28308
$663$	0	0
$664$	4.84047e9	0.641651
$665$	2.87008e9	0.378458
$666$	0	0
$667$	−4.22810e9	−0.551703
$668$	4.37563e9	0.567967
$669$	0	0
$670$	−9.59552e9	−1.23256
$671$	2.60914e10	3.33403
$672$	0	0
$673$	4.39307e9	0.555541	0.277770	−	0.960648i	$-0.410405\pi$
$673$	4.39307e9	0.555541	0.277770	+	0.960648i	$0.410405\pi$
$674$	2.33189e9	0.293359
$675$	0	0
$676$	−1.53746e9	−0.191421
$677$	7.42240e9	0.919357	0.459678	−	0.888085i	$-0.347965\pi$
$677$	7.42240e9	0.919357	0.459678	+	0.888085i	$0.347965\pi$
$678$	0	0
$679$	4.07185e9	0.499168
$680$	5.90481e9	0.720153
$681$	0	0
$682$	−2.19745e9	−0.265262
$683$	1.34040e10	1.60976	0.804882	−	0.593435i	$-0.202229\pi$
$683$	1.34040e10	1.60976	0.804882	+	0.593435i	$0.202229\pi$
$684$	0	0
$685$	−1.31419e10	−1.56222
$686$	−5.65035e9	−0.668253
$687$	0	0
$688$	2.66351e9	0.311814
$689$	−5.71709e9	−0.665898
$690$	0	0
$691$	−1.43025e10	−1.64907	−0.824535	−	0.565811i	$-0.808563\pi$
$691$	−1.43025e10	−1.64907	−0.824535	+	0.565811i	$0.808563\pi$
$692$	−2.93553e9	−0.336755
$693$	0	0
$694$	−1.72966e9	−0.196428
$695$	−4.11494e9	−0.464961
$696$	0	0
$697$	−7.55086e8	−0.0844660
$698$	−8.31168e9	−0.925113
$699$	0	0
$700$	7.02827e9	0.774471
$701$	−7.95614e9	−0.872347	−0.436174	−	0.899862i	$-0.643667\pi$
$701$	−7.95614e9	−0.872347	−0.436174	+	0.899862i	$0.643667\pi$
$702$	0	0
$703$	2.99528e9	0.325158
$704$	2.20149e9	0.237800
$705$	0	0
$706$	−9.33546e8	−0.0998434
$707$	2.36194e9	0.251363
$708$	0	0
$709$	1.07008e10	1.12760	0.563798	−	0.825912i	$-0.309339\pi$
$709$	1.07008e10	1.12760	0.563798	+	0.825912i	$0.309339\pi$
$710$	−6.33937e9	−0.664725
$711$	0	0
$712$	2.09818e7	0.00217852
$713$	−2.10028e9	−0.217002
$714$	0	0
$715$	2.29947e10	2.35265
$716$	2.24307e9	0.228374
$717$	0	0
$718$	1.02555e10	1.03400
$719$	−1.29294e9	−0.129726	−0.0648630	−	0.997894i	$-0.520661\pi$
$719$	−1.29294e9	−0.129726	−0.0648630	+	0.997894i	$0.520661\pi$
$720$	0	0
$721$	−1.18905e10	−1.18148
$722$	3.76367e8	0.0372161
$723$	0	0
$724$	8.34430e8	0.0817155
$725$	−7.60345e9	−0.741016
$726$	0	0
$727$	−4.13504e9	−0.399125	−0.199562	−	0.979885i	$-0.563952\pi$
$727$	−4.13504e9	−0.399125	−0.199562	+	0.979885i	$0.563952\pi$
$728$	−3.03005e9	−0.291065
$729$	0	0
$730$	5.05550e9	0.480988
$731$	−1.70443e10	−1.61387
$732$	0	0
$733$	7.76128e9	0.727896	0.363948	−	0.931419i	$-0.381429\pi$
$733$	7.76128e9	0.727896	0.363948	+	0.931419i	$0.381429\pi$
$734$	−1.53332e9	−0.143119
$735$	0	0
$736$	2.10413e9	0.194536
$737$	2.28930e10	2.10652
$738$	0	0
$739$	8.13711e9	0.741676	0.370838	−	0.928697i	$-0.379070\pi$
$739$	8.13711e9	0.741676	0.370838	+	0.928697i	$0.379070\pi$
$740$	1.22973e10	1.11558
$741$	0	0
$742$	6.98949e9	0.628105
$743$	1.46014e9	0.130597	0.0652987	−	0.997866i	$-0.479200\pi$
$743$	1.46014e9	0.130597	0.0652987	+	0.997866i	$0.479200\pi$
$744$	0	0
$745$	1.09235e10	0.967868
$746$	8.01026e8	0.0706416
$747$	0	0
$748$	−1.40877e10	−1.23079
$749$	−2.11568e9	−0.183977
$750$	0	0
$751$	1.24940e9	0.107637	0.0538187	−	0.998551i	$-0.482861\pi$
$751$	1.24940e9	0.107637	0.0538187	+	0.998551i	$0.482861\pi$
$752$	−2.40583e8	−0.0206301
$753$	0	0
$754$	3.27803e9	0.278492
$755$	4.89733e9	0.414138
$756$	0	0
$757$	−1.72633e9	−0.144640	−0.0723202	−	0.997381i	$-0.523040\pi$
$757$	−1.72633e9	−0.144640	−0.0723202	+	0.997381i	$0.523040\pi$
$758$	−8.26282e9	−0.689106
$759$	0	0
$760$	1.54520e9	0.127684
$761$	1.88538e10	1.55079	0.775393	−	0.631478i	$-0.217552\pi$
$761$	1.88538e10	1.55079	0.775393	+	0.631478i	$0.217552\pi$
$762$	0	0
$763$	1.34481e10	1.09604
$764$	4.81479e9	0.390616
$765$	0	0
$766$	−1.13180e10	−0.909845
$767$	−4.90145e9	−0.392230
$768$	0	0
$769$	1.02437e10	0.812299	0.406149	−	0.913807i	$-0.366871\pi$
$769$	1.02437e10	0.812299	0.406149	+	0.913807i	$0.366871\pi$
$770$	−2.81125e10	−2.21912
$771$	0	0
$772$	5.61552e9	0.439268
$773$	1.70435e10	1.32718	0.663590	−	0.748096i	$-0.269032\pi$
$773$	1.70435e10	1.32718	0.663590	+	0.748096i	$0.269032\pi$
$774$	0	0
$775$	−3.77696e9	−0.291465
$776$	2.19220e9	0.168409
$777$	0	0
$778$	−5.47521e9	−0.416843
$779$	−1.97594e8	−0.0149759
$780$	0	0
$781$	1.51244e10	1.13606
$782$	−1.34647e10	−1.00687
$783$	0	0
$784$	3.31194e8	0.0245458
$785$	−1.99286e10	−1.47039
$786$	0	0
$787$	−2.29012e10	−1.67474	−0.837370	−	0.546637i	$-0.815908\pi$
$787$	−2.29012e10	−1.67474	−0.837370	+	0.546637i	$0.815908\pi$
$788$	−1.08191e10	−0.787681
$789$	0	0
$790$	−1.19754e10	−0.864164
$791$	7.02174e9	0.504461
$792$	0	0
$793$	−1.93340e10	−1.37678
$794$	−2.57115e9	−0.182287
$795$	0	0
$796$	8.92739e9	0.627377
$797$	−4.91683e9	−0.344018	−0.172009	−	0.985095i	$-0.555026\pi$
$797$	−4.91683e9	−0.344018	−0.172009	+	0.985095i	$0.555026\pi$
$798$	0	0
$799$	1.53953e9	0.106776
$800$	3.78388e9	0.261290
$801$	0	0
$802$	−1.19899e10	−0.820737
$803$	−1.20614e10	−0.822041
$804$	0	0
$805$	−2.68693e10	−1.81539
$806$	1.62834e9	0.109540
$807$	0	0
$808$	1.27162e9	0.0848044
$809$	1.08294e10	0.719089	0.359545	−	0.933128i	$-0.382932\pi$
$809$	1.08294e10	0.719089	0.359545	+	0.933128i	$0.382932\pi$
$810$	0	0
$811$	1.60764e10	1.05832	0.529158	−	0.848523i	$-0.322508\pi$
$811$	1.60764e10	1.05832	0.529158	+	0.848523i	$0.322508\pi$
$812$	−4.00759e9	−0.262686
$813$	0	0
$814$	−2.93388e10	−1.90659
$815$	−1.02769e10	−0.664981
$816$	0	0
$817$	−4.46022e9	−0.286140
$818$	−1.33634e10	−0.853651
$819$	0	0
$820$	−8.11233e8	−0.0513804
$821$	−1.74122e10	−1.09813	−0.549064	−	0.835780i	$-0.685016\pi$
$821$	−1.74122e10	−1.09813	−0.549064	+	0.835780i	$0.685016\pi$
$822$	0	0
$823$	−1.26645e10	−0.791934	−0.395967	−	0.918265i	$-0.629591\pi$
$823$	−1.26645e10	−0.791934	−0.395967	+	0.918265i	$0.629591\pi$
$824$	−6.40163e9	−0.398607
$825$	0	0
$826$	5.99233e9	0.369969
$827$	−2.86299e10	−1.76015	−0.880077	−	0.474830i	$-0.842509\pi$
$827$	−2.86299e10	−1.76015	−0.880077	+	0.474830i	$0.842509\pi$
$828$	0	0
$829$	−2.75171e10	−1.67749	−0.838747	−	0.544521i	$-0.816712\pi$
$829$	−2.75171e10	−1.67749	−0.838747	+	0.544521i	$0.816712\pi$
$830$	−3.32782e10	−2.02016
$831$	0	0
$832$	−1.63132e9	−0.0981992
$833$	−2.11937e9	−0.127043
$834$	0	0
$835$	−3.00825e10	−1.78818
$836$	−3.68652e9	−0.218220
$837$	0	0
$838$	6.52410e9	0.382971
$839$	−9.77182e9	−0.571227	−0.285613	−	0.958345i	$-0.592197\pi$
$839$	−9.77182e9	−0.571227	−0.285613	+	0.958345i	$0.592197\pi$
$840$	0	0
$841$	−1.29143e10	−0.748661
$842$	1.55595e9	0.0898265
$843$	0	0
$844$	−1.44639e10	−0.828105
$845$	1.05700e10	0.602667
$846$	0	0
$847$	4.85383e10	2.74469
$848$	3.76301e9	0.211909
$849$	0	0
$850$	−2.42137e10	−1.35237
$851$	−2.80414e10	−1.55972
$852$	0	0
$853$	−3.51519e9	−0.193922	−0.0969611	−	0.995288i	$-0.530912\pi$
$853$	−3.51519e9	−0.193922	−0.0969611	+	0.995288i	$0.530912\pi$
$854$	2.36370e10	1.29864
$855$	0	0
$856$	−1.13904e9	−0.0620700
$857$	1.61455e10	0.876233	0.438117	−	0.898918i	$-0.355646\pi$
$857$	1.61455e10	0.876233	0.438117	+	0.898918i	$0.355646\pi$
$858$	0	0
$859$	1.97382e10	1.06250	0.531252	−	0.847214i	$-0.321722\pi$
$859$	1.97382e10	1.06250	0.531252	+	0.847214i	$0.321722\pi$
$860$	−1.83117e10	−0.981710
$861$	0	0
$862$	2.09997e10	1.11670
$863$	−8.05160e8	−0.0426427	−0.0213213	−	0.999773i	$-0.506787\pi$
$863$	−8.05160e8	−0.0426427	−0.0213213	+	0.999773i	$0.506787\pi$
$864$	0	0
$865$	2.01817e10	1.06024
$866$	1.52324e10	0.796993
$867$	0	0
$868$	−1.99074e9	−0.103323
$869$	2.85709e10	1.47691
$870$	0	0
$871$	−1.69639e10	−0.869886
$872$	7.24022e9	0.369781
$873$	0	0
$874$	−3.52350e9	−0.178519
$875$	−1.56287e10	−0.788671
$876$	0	0
$877$	6.23056e9	0.311909	0.155955	−	0.987764i	$-0.450155\pi$
$877$	6.23056e9	0.311909	0.155955	+	0.987764i	$0.450155\pi$
$878$	−1.21251e10	−0.604581
$879$	0	0
$880$	−1.51352e10	−0.748685
$881$	−9.81947e9	−0.483807	−0.241904	−	0.970300i	$-0.577772\pi$
$881$	−9.81947e9	−0.483807	−0.241904	+	0.970300i	$0.577772\pi$
$882$	0	0
$883$	−9.86220e9	−0.482071	−0.241036	−	0.970516i	$-0.577487\pi$
$883$	−9.86220e9	−0.482071	−0.241036	+	0.970516i	$0.577487\pi$
$884$	1.04391e10	0.508254
$885$	0	0
$886$	2.26689e10	1.09500
$887$	1.78336e10	0.858037	0.429019	−	0.903296i	$-0.358859\pi$
$887$	1.78336e10	0.858037	0.429019	+	0.903296i	$0.358859\pi$
$888$	0	0
$889$	−2.13885e10	−1.02100
$890$	−1.44250e8	−0.00685882
$891$	0	0
$892$	1.01825e10	0.480373
$893$	4.02870e8	0.0189315
$894$	0	0
$895$	−1.54211e10	−0.719009
$896$	1.99439e9	0.0926259
$897$	0	0
$898$	−6.28083e9	−0.289434
$899$	2.15366e9	0.0988594
$900$	0	0
$901$	−2.40801e10	−1.09679
$902$	1.93544e9	0.0878124
$903$	0	0
$904$	3.78037e9	0.170194
$905$	−5.73670e9	−0.257272
$906$	0	0
$907$	9.62665e8	0.0428400	0.0214200	−	0.999771i	$-0.493181\pi$
$907$	9.62665e8	0.0428400	0.0214200	+	0.999771i	$0.493181\pi$
$908$	−1.74981e10	−0.775693
$909$	0	0
$910$	2.08316e10	0.916385
$911$	−3.09938e8	−0.0135819	−0.00679096	−	0.999977i	$-0.502162\pi$
$911$	−3.09938e8	−0.0135819	−0.00679096	+	0.999977i	$0.502162\pi$
$912$	0	0
$913$	7.93950e10	3.45259
$914$	1.16911e9	0.0506460
$915$	0	0
$916$	5.99255e9	0.257619
$917$	3.71614e9	0.159147
$918$	0	0
$919$	−4.47873e9	−0.190349	−0.0951744	−	0.995461i	$-0.530341\pi$
$919$	−4.47873e9	−0.190349	−0.0951744	+	0.995461i	$0.530341\pi$
$920$	−1.44659e10	−0.612475
$921$	0	0
$922$	−1.04941e10	−0.440947
$923$	−1.12074e10	−0.469135
$924$	0	0
$925$	−5.04272e10	−2.09493
$926$	3.10531e10	1.28519
$927$	0	0
$928$	−2.15761e9	−0.0886247
$929$	2.62394e10	1.07374	0.536870	−	0.843665i	$-0.319606\pi$
$929$	2.62394e10	1.07374	0.536870	+	0.843665i	$0.319606\pi$
$930$	0	0
$931$	−5.54605e8	−0.0225247
$932$	1.77881e9	0.0719736
$933$	0	0
$934$	−9.35333e9	−0.375623
$935$	9.68528e10	3.87500
$936$	0	0
$937$	−5.22218e9	−0.207378	−0.103689	−	0.994610i	$-0.533065\pi$
$937$	−5.22218e9	−0.207378	−0.103689	+	0.994610i	$0.533065\pi$
$938$	2.07394e10	0.820515
$939$	0	0
$940$	1.65401e9	0.0649516
$941$	−3.43325e10	−1.34320	−0.671602	−	0.740913i	$-0.734393\pi$
$941$	−3.43325e10	−1.34320	−0.671602	+	0.740913i	$0.734393\pi$
$942$	0	0
$943$	1.84985e9	0.0718365
$944$	3.22615e9	0.124820
$945$	0	0
$946$	4.36879e10	1.67781
$947$	2.00808e10	0.768346	0.384173	−	0.923261i	$-0.374487\pi$
$947$	2.00808e10	0.768346	0.384173	+	0.923261i	$0.374487\pi$
$948$	0	0
$949$	8.93762e9	0.339461
$950$	−6.33634e9	−0.239776
$951$	0	0
$952$	−1.27625e10	−0.479407
$953$	−4.99401e10	−1.86906	−0.934532	−	0.355880i	$-0.884181\pi$
$953$	−4.99401e10	−1.86906	−0.934532	+	0.355880i	$0.884181\pi$
$954$	0	0
$955$	−3.31017e10	−1.22981
$956$	−1.70181e10	−0.629952
$957$	0	0
$958$	1.59958e10	0.587797
$959$	2.84044e10	1.03997
$960$	0	0
$961$	−2.64428e10	−0.961116
$962$	2.17404e10	0.787325
$963$	0	0
$964$	−7.39748e9	−0.265959
$965$	−3.86067e10	−1.38298
$966$	0	0
$967$	−3.94773e10	−1.40396	−0.701981	−	0.712196i	$-0.747701\pi$
$967$	−3.94773e10	−1.40396	−0.701981	+	0.712196i	$0.747701\pi$
$968$	2.61321e10	0.925999
$969$	0	0
$970$	−1.50714e10	−0.530215
$971$	1.83646e10	0.643746	0.321873	−	0.946783i	$-0.395688\pi$
$971$	1.83646e10	0.643746	0.321873	+	0.946783i	$0.395688\pi$
$972$	0	0
$973$	8.89389e9	0.309526
$974$	−1.07445e10	−0.372589
$975$	0	0
$976$	1.27257e10	0.438135
$977$	4.72867e10	1.62221	0.811107	−	0.584897i	$-0.198865\pi$
$977$	4.72867e10	1.62221	0.811107	+	0.584897i	$0.198865\pi$
$978$	0	0
$979$	3.44150e8	0.0117222
$980$	−2.27696e9	−0.0772795
$981$	0	0
$982$	−3.44490e10	−1.16088
$983$	1.03196e10	0.346517	0.173258	−	0.984876i	$-0.444570\pi$
$983$	1.03196e10	0.346517	0.173258	+	0.984876i	$0.444570\pi$
$984$	0	0
$985$	7.43815e10	2.47992
$986$	1.38069e10	0.458698
$987$	0	0
$988$	2.73175e9	0.0901138
$989$	4.17559e10	1.37256
$990$	0	0
$991$	−2.32803e10	−0.759856	−0.379928	−	0.925016i	$-0.624051\pi$
$991$	−2.32803e10	−0.759856	−0.379928	+	0.925016i	$0.624051\pi$
$992$	−1.07178e9	−0.0348589
$993$	0	0
$994$	1.37017e10	0.442509
$995$	−6.13758e10	−1.97522
$996$	0	0
$997$	−5.08802e9	−0.162598	−0.0812991	−	0.996690i	$-0.525907\pi$
$997$	−5.08802e9	−0.162598	−0.0812991	+	0.996690i	$0.525907\pi$
$998$	1.82631e10	0.581590
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.8.a.e.1.1		1
3.2	odd	2		38.8.a.a.1.1	✓	1
12.11	even	2		304.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.a.1.1	✓	1	3.2	odd	2
304.8.a.a.1.1		1	12.11	even	2
342.8.a.e.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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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