LMFDB - Embedded newform 342.10.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,10,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, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("342.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
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:	$176.142255968$
Analytic rank:	$1$
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)

$f(q)$

$=$

$q-16.0000 q^{2} +256.000 q^{4} +684.000 q^{5} +9149.00 q^{7} -4096.00 q^{8} +O(q^{10})$

\(q-16.0000 q^{2} +256.000 q^{4} +684.000 q^{5} +9149.00 q^{7} -4096.00 q^{8} -10944.0 q^{10} -5790.00 q^{11} -179881. q^{13} -146384. q^{14} +65536.0 q^{16} +594093. q^{17} +130321. q^{19} +175104. q^{20} +92640.0 q^{22} +1.74477e6 q^{23} -1.48527e6 q^{25} +2.87810e6 q^{26} +2.34214e6 q^{28} -4.31439e6 q^{29} +160232. q^{31} -1.04858e6 q^{32} -9.50549e6 q^{34} +6.25792e6 q^{35} -2.19431e7 q^{37} -2.08514e6 q^{38} -2.80166e6 q^{40} -294816. q^{41} -3.93931e7 q^{43} -1.48224e6 q^{44} -2.79163e7 q^{46} -4.65964e7 q^{47} +4.33506e7 q^{49} +2.37643e7 q^{50} -4.60495e7 q^{52} -2.21217e7 q^{53} -3.96036e6 q^{55} -3.74743e7 q^{56} +6.90302e7 q^{58} -3.30702e7 q^{59} +1.88536e8 q^{61} -2.56371e6 q^{62} +1.67772e7 q^{64} -1.23039e8 q^{65} -2.07691e7 q^{67} +1.52088e8 q^{68} -1.00127e8 q^{70} +2.32300e8 q^{71} -3.02218e6 q^{73} +3.51089e8 q^{74} +3.33622e7 q^{76} -5.29727e7 q^{77} -4.46379e8 q^{79} +4.48266e7 q^{80} +4.71706e6 q^{82} -7.94023e8 q^{83} +4.06360e8 q^{85} +6.30290e8 q^{86} +2.37158e7 q^{88} -9.09993e7 q^{89} -1.64573e9 q^{91} +4.46660e8 q^{92} +7.45542e8 q^{94} +8.91396e7 q^{95} -1.23974e8 q^{97} -6.93610e8 q^{98} +O(q^{100})\)

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 $

$2$	−16.0000	−0.707107
$2$	−16.0000	−0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	684.000	0.489431	0.244715	−	0.969595i	$-0.421306\pi$
$5$	684.000	0.489431	0.244715	+	0.969595i	$0.421306\pi$
$6$	0	0
$7$	9149.00	1.44023	0.720116	−	0.693854i	$-0.244089\pi$
$7$	9149.00	1.44023	0.720116	+	0.693854i	$0.244089\pi$
$8$	−4096.00	−0.353553
$9$	0	0
$10$	−10944.0	−0.346080
$11$	−5790.00	−0.119237	−0.0596186	−	0.998221i	$-0.518988\pi$
$11$	−5790.00	−0.119237	−0.0596186	+	0.998221i	$0.518988\pi$
$12$	0	0
$13$	−179881.	−1.74679	−0.873394	−	0.487014i	$-0.838086\pi$
$13$	−179881.	−1.74679	−0.873394	+	0.487014i	$0.838086\pi$
$14$	−146384.	−1.01840
$15$	0	0
$16$	65536.0	0.250000
$17$	594093.	1.72518	0.862590	−	0.505904i	$-0.168841\pi$
$17$	594093.	1.72518	0.862590	+	0.505904i	$0.168841\pi$
$18$	0	0
$19$	130321.	0.229416
$19$	130321.	0.229416
$20$	175104.	0.244715
$21$	0	0
$22$	92640.0	0.0843134
$23$	1.74477e6	1.30006	0.650028	−	0.759910i	$-0.274757\pi$
$23$	1.74477e6	1.30006	0.650028	+	0.759910i	$0.274757\pi$
$24$	0	0
$25$	−1.48527e6	−0.760458
$26$	2.87810e6	1.23517
$27$	0	0
$28$	2.34214e6	0.720116
$29$	−4.31439e6	−1.13273	−0.566367	−	0.824153i	$-0.691652\pi$
$29$	−4.31439e6	−1.13273	−0.566367	+	0.824153i	$0.691652\pi$
$30$	0	0
$31$	160232.	0.0311617	0.0155809	−	0.999879i	$-0.495040\pi$
$31$	160232.	0.0311617	0.0155809	+	0.999879i	$0.495040\pi$
$32$	−1.04858e6	−0.176777
$33$	0	0
$34$	−9.50549e6	−1.21989
$35$	6.25792e6	0.704894
$36$	0	0
$37$	−2.19431e7	−1.92482	−0.962410	−	0.271602i	$-0.912447\pi$
$37$	−2.19431e7	−1.92482	−0.962410	+	0.271602i	$0.912447\pi$
$38$	−2.08514e6	−0.162221
$39$	0	0
$40$	−2.80166e6	−0.173040
$41$	−294816.	−0.0162938	−0.00814692	−	0.999967i	$-0.502593\pi$
$41$	−294816.	−0.0162938	−0.00814692	+	0.999967i	$0.502593\pi$
$42$	0	0
$43$	−3.93931e7	−1.75717	−0.878583	−	0.477590i	$-0.841510\pi$
$43$	−3.93931e7	−1.75717	−0.878583	+	0.477590i	$0.841510\pi$
$44$	−1.48224e6	−0.0596186
$45$	0	0
$46$	−2.79163e7	−0.919279
$47$	−4.65964e7	−1.39287	−0.696437	−	0.717618i	$-0.745232\pi$
$47$	−4.65964e7	−1.39287	−0.696437	+	0.717618i	$0.745232\pi$
$48$	0	0
$49$	4.33506e7	1.07427
$50$	2.37643e7	0.537725
$51$	0	0
$52$	−4.60495e7	−0.873394
$53$	−2.21217e7	−0.385103	−0.192552	−	0.981287i	$-0.561676\pi$
$53$	−2.21217e7	−0.385103	−0.192552	+	0.981287i	$0.561676\pi$
$54$	0	0
$55$	−3.96036e6	−0.0583583
$56$	−3.74743e7	−0.509199
$57$	0	0
$58$	6.90302e7	0.800964
$59$	−3.30702e7	−0.355307	−0.177653	−	0.984093i	$-0.556851\pi$
$59$	−3.30702e7	−0.355307	−0.177653	+	0.984093i	$0.556851\pi$
$60$	0	0
$61$	1.88536e8	1.74345	0.871726	−	0.489994i	$-0.163001\pi$
$61$	1.88536e8	1.74345	0.871726	+	0.489994i	$0.163001\pi$
$62$	−2.56371e6	−0.0220347
$63$	0	0
$64$	1.67772e7	0.125000
$65$	−1.23039e8	−0.854931
$66$	0	0
$67$	−2.07691e7	−0.125916	−0.0629579	−	0.998016i	$-0.520053\pi$
$67$	−2.07691e7	−0.125916	−0.0629579	+	0.998016i	$0.520053\pi$
$68$	1.52088e8	0.862590
$69$	0	0
$70$	−1.00127e8	−0.498435
$71$	2.32300e8	1.08489	0.542446	−	0.840091i	$-0.317498\pi$
$71$	2.32300e8	1.08489	0.542446	+	0.840091i	$0.317498\pi$
$72$	0	0
$73$	−3.02218e6	−0.0124557	−0.00622785	−	0.999981i	$-0.501982\pi$
$73$	−3.02218e6	−0.0124557	−0.00622785	+	0.999981i	$0.501982\pi$
$74$	3.51089e8	1.36105
$75$	0	0
$76$	3.33622e7	0.114708
$77$	−5.29727e7	−0.171729
$78$	0	0
$79$	−4.46379e8	−1.28938	−0.644692	−	0.764442i	$-0.723014\pi$
$79$	−4.46379e8	−1.28938	−0.644692	+	0.764442i	$0.723014\pi$
$80$	4.48266e7	0.122358
$81$	0	0
$82$	4.71706e6	0.0115215
$83$	−7.94023e8	−1.83646	−0.918230	−	0.396047i	$-0.870382\pi$
$83$	−7.94023e8	−1.83646	−0.918230	+	0.396047i	$0.870382\pi$
$84$	0	0
$85$	4.06360e8	0.844356
$86$	6.30290e8	1.24250
$87$	0	0
$88$	2.37158e7	0.0421567
$89$	−9.09993e7	−0.153739	−0.0768693	−	0.997041i	$-0.524492\pi$
$89$	−9.09993e7	−0.153739	−0.0768693	+	0.997041i	$0.524492\pi$
$90$	0	0
$91$	−1.64573e9	−2.51578
$92$	4.46660e8	0.650028
$93$	0	0
$94$	7.45542e8	0.984910
$95$	8.91396e7	0.112283
$96$	0	0
$97$	−1.23974e8	−0.142187	−0.0710933	−	0.997470i	$-0.522649\pi$
$97$	−1.23974e8	−0.142187	−0.0710933	+	0.997470i	$0.522649\pi$
$98$	−6.93610e8	−0.759622
$99$	0	0
$100$	−3.80229e8	−0.380229
$101$	1.72437e9	1.64887	0.824433	−	0.565960i	$-0.191494\pi$
$101$	1.72437e9	1.64887	0.824433	+	0.565960i	$0.191494\pi$
$102$	0	0
$103$	2.23168e8	0.195373	0.0976863	−	0.995217i	$-0.468856\pi$
$103$	2.23168e8	0.195373	0.0976863	+	0.995217i	$0.468856\pi$
$104$	7.36793e8	0.617583
$105$	0	0
$106$	3.53947e8	0.272309
$107$	1.31384e8	0.0968983	0.0484491	−	0.998826i	$-0.484572\pi$
$107$	1.31384e8	0.0968983	0.0484491	+	0.998826i	$0.484572\pi$
$108$	0	0
$109$	4.16019e8	0.282289	0.141145	−	0.989989i	$-0.454922\pi$
$109$	4.16019e8	0.282289	0.141145	+	0.989989i	$0.454922\pi$
$110$	6.33658e7	0.0412655
$111$	0	0
$112$	5.99589e8	0.360058
$113$	1.39920e9	0.807284	0.403642	−	0.914917i	$-0.367744\pi$
$113$	1.39920e9	0.807284	0.403642	+	0.914917i	$0.367744\pi$
$114$	0	0
$115$	1.19342e9	0.636287
$116$	−1.10448e9	−0.566367
$117$	0	0
$118$	5.29124e8	0.251240
$119$	5.43536e9	2.48466
$120$	0	0
$121$	−2.32442e9	−0.985783
$122$	−3.01658e9	−1.23281
$123$	0	0
$124$	4.10194e7	0.0155809
$125$	−2.35186e9	−0.861622
$126$	0	0
$127$	−6.47098e8	−0.220726	−0.110363	−	0.993891i	$-0.535201\pi$
$127$	−6.47098e8	−0.220726	−0.110363	+	0.993891i	$0.535201\pi$
$128$	−2.68435e8	−0.0883883
$129$	0	0
$130$	1.96862e9	0.604528
$131$	−1.34166e9	−0.398036	−0.199018	−	0.979996i	$-0.563775\pi$
$131$	−1.34166e9	−0.398036	−0.199018	+	0.979996i	$0.563775\pi$
$132$	0	0
$133$	1.19231e9	0.330412
$134$	3.32305e8	0.0890359
$135$	0	0
$136$	−2.43340e9	−0.609943
$137$	−1.57846e9	−0.382818	−0.191409	−	0.981510i	$-0.561306\pi$
$137$	−1.57846e9	−0.382818	−0.191409	+	0.981510i	$0.561306\pi$
$138$	0	0
$139$	5.68725e9	1.29222	0.646109	−	0.763245i	$-0.276395\pi$
$139$	5.68725e9	1.29222	0.646109	+	0.763245i	$0.276395\pi$
$140$	1.60203e9	0.352447
$141$	0	0
$142$	−3.71680e9	−0.767135
$143$	1.04151e9	0.208282
$144$	0	0
$145$	−2.95104e9	−0.554395
$146$	4.83549e7	0.00880750
$147$	0	0
$148$	−5.61743e9	−0.962410
$149$	−6.27835e9	−1.04354	−0.521768	−	0.853087i	$-0.674727\pi$
$149$	−6.27835e9	−1.04354	−0.521768	+	0.853087i	$0.674727\pi$
$150$	0	0
$151$	−1.03030e9	−0.161275	−0.0806376	−	0.996743i	$-0.525696\pi$
$151$	−1.03030e9	−0.161275	−0.0806376	+	0.996743i	$0.525696\pi$
$152$	−5.33795e8	−0.0811107
$153$	0	0
$154$	8.47563e8	0.121431
$155$	1.09599e8	0.0152515
$156$	0	0
$157$	−1.42937e9	−0.187756	−0.0938782	−	0.995584i	$-0.529926\pi$
$157$	−1.42937e9	−0.187756	−0.0938782	+	0.995584i	$0.529926\pi$
$158$	7.14207e9	0.911732
$159$	0	0
$160$	−7.17226e8	−0.0865199
$161$	1.59629e10	1.87238
$162$	0	0
$163$	9.81973e9	1.08957	0.544785	−	0.838576i	$-0.316611\pi$
$163$	9.81973e9	1.08957	0.544785	+	0.838576i	$0.316611\pi$
$164$	−7.54729e7	−0.00814692
$165$	0	0
$166$	1.27044e10	1.29857
$167$	−2.44641e9	−0.243391	−0.121696	−	0.992567i	$-0.538833\pi$
$167$	−2.44641e9	−0.243391	−0.121696	+	0.992567i	$0.538833\pi$
$168$	0	0
$169$	2.17527e10	2.05127
$170$	−6.50175e9	−0.597050
$171$	0	0
$172$	−1.00846e10	−0.878583
$173$	1.20669e10	1.02420	0.512102	−	0.858925i	$-0.328867\pi$
$173$	1.20669e10	1.02420	0.512102	+	0.858925i	$0.328867\pi$
$174$	0	0
$175$	−1.35887e10	−1.09524
$176$	−3.79453e8	−0.0298093
$177$	0	0
$178$	1.45599e9	0.108710
$179$	−1.43460e10	−1.04446	−0.522230	−	0.852805i	$-0.674900\pi$
$179$	−1.43460e10	−1.04446	−0.522230	+	0.852805i	$0.674900\pi$
$180$	0	0
$181$	−1.85517e10	−1.28478	−0.642391	−	0.766377i	$-0.722058\pi$
$181$	−1.85517e10	−1.28478	−0.642391	+	0.766377i	$0.722058\pi$
$182$	2.63317e10	1.77893
$183$	0	0
$184$	−7.14657e9	−0.459639
$185$	−1.50091e10	−0.942066
$186$	0	0
$187$	−3.43980e9	−0.205705
$188$	−1.19287e10	−0.696437
$189$	0	0
$190$	−1.42623e9	−0.0793961
$191$	−1.15528e10	−0.628114	−0.314057	−	0.949404i	$-0.601688\pi$
$191$	−1.15528e10	−0.628114	−0.314057	+	0.949404i	$0.601688\pi$
$192$	0	0
$193$	1.13432e10	0.588475	0.294237	−	0.955732i	$-0.404934\pi$
$193$	1.13432e10	0.588475	0.294237	+	0.955732i	$0.404934\pi$
$194$	1.98359e9	0.100541
$195$	0	0
$196$	1.10978e10	0.537134
$197$	−8.08688e9	−0.382545	−0.191273	−	0.981537i	$-0.561261\pi$
$197$	−8.08688e9	−0.382545	−0.191273	+	0.981537i	$0.561261\pi$
$198$	0	0
$199$	−1.85057e10	−0.836503	−0.418252	−	0.908331i	$-0.637357\pi$
$199$	−1.85057e10	−0.836503	−0.418252	+	0.908331i	$0.637357\pi$
$200$	6.08366e9	0.268862
$201$	0	0
$202$	−2.75900e10	−1.16592
$203$	−3.94723e10	−1.63140
$204$	0	0
$205$	−2.01654e8	−0.00797471
$206$	−3.57068e9	−0.138149
$207$	0	0
$208$	−1.17887e10	−0.436697
$209$	−7.54559e8	−0.0273549
$210$	0	0
$211$	−2.80529e10	−0.974333	−0.487167	−	0.873309i	$-0.661970\pi$
$211$	−2.80529e10	−0.974333	−0.487167	+	0.873309i	$0.661970\pi$
$212$	−5.66316e9	−0.192552
$213$	0	0
$214$	−2.10215e9	−0.0685174
$215$	−2.69449e10	−0.860010
$216$	0	0
$217$	1.46596e9	0.0448801
$218$	−6.65631e9	−0.199609
$219$	0	0
$220$	−1.01385e9	−0.0291791
$221$	−1.06866e11	−3.01352
$222$	0	0
$223$	−3.73467e10	−1.01130	−0.505651	−	0.862738i	$-0.668748\pi$
$223$	−3.73467e10	−1.01130	−0.505651	+	0.862738i	$0.668748\pi$
$224$	−9.59342e9	−0.254599
$225$	0	0
$226$	−2.23872e10	−0.570836
$227$	4.26768e10	1.06678	0.533391	−	0.845869i	$-0.320918\pi$
$227$	4.26768e10	1.06678	0.533391	+	0.845869i	$0.320918\pi$
$228$	0	0
$229$	4.89097e10	1.17526	0.587632	−	0.809129i	$-0.300060\pi$
$229$	4.89097e10	1.17526	0.587632	+	0.809129i	$0.300060\pi$
$230$	−1.90947e10	−0.449923
$231$	0	0
$232$	1.76717e10	0.400482
$233$	5.16180e10	1.14736	0.573680	−	0.819079i	$-0.305515\pi$
$233$	5.16180e10	1.14736	0.573680	+	0.819079i	$0.305515\pi$
$234$	0	0
$235$	−3.18719e10	−0.681715
$236$	−8.46598e9	−0.177653
$237$	0	0
$238$	−8.69657e10	−1.75692
$239$	3.55464e10	0.704702	0.352351	−	0.935868i	$-0.385382\pi$
$239$	3.55464e10	0.704702	0.352351	+	0.935868i	$0.385382\pi$
$240$	0	0
$241$	−1.72210e10	−0.328838	−0.164419	−	0.986391i	$-0.552575\pi$
$241$	−1.72210e10	−0.328838	−0.164419	+	0.986391i	$0.552575\pi$
$242$	3.71908e10	0.697053
$243$	0	0
$244$	4.82652e10	0.871726
$245$	2.96518e10	0.525780
$246$	0	0
$247$	−2.34423e10	−0.400741
$248$	−6.56310e8	−0.0110173
$249$	0	0
$250$	3.76298e10	0.609259
$251$	−1.67656e10	−0.266616	−0.133308	−	0.991075i	$-0.542560\pi$
$251$	−1.67656e10	−0.266616	−0.133308	+	0.991075i	$0.542560\pi$
$252$	0	0
$253$	−1.01022e10	−0.155015
$254$	1.03536e10	0.156077
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	6.36805e10	0.910558	0.455279	−	0.890349i	$-0.349540\pi$
$257$	6.36805e10	0.910558	0.455279	+	0.890349i	$0.349540\pi$
$258$	0	0
$259$	−2.00757e11	−2.77219
$260$	−3.14979e10	−0.427466
$261$	0	0
$262$	2.14666e10	0.281454
$263$	1.82468e10	0.235172	0.117586	−	0.993063i	$-0.462484\pi$
$263$	1.82468e10	0.235172	0.117586	+	0.993063i	$0.462484\pi$
$264$	0	0
$265$	−1.51312e10	−0.188481
$266$	−1.90769e10	−0.233636
$267$	0	0
$268$	−5.31688e9	−0.0629579
$269$	−1.02940e11	−1.19867	−0.599335	−	0.800498i	$-0.704568\pi$
$269$	−1.02940e11	−1.19867	−0.599335	+	0.800498i	$0.704568\pi$
$270$	0	0
$271$	−1.04690e11	−1.17908	−0.589540	−	0.807739i	$-0.700691\pi$
$271$	−1.04690e11	−1.17908	−0.589540	+	0.807739i	$0.700691\pi$
$272$	3.89345e10	0.431295
$273$	0	0
$274$	2.52554e10	0.270693
$275$	8.59971e9	0.0906748
$276$	0	0
$277$	−1.25385e11	−1.27964	−0.639820	−	0.768525i	$-0.720991\pi$
$277$	−1.25385e11	−1.27964	−0.639820	+	0.768525i	$0.720991\pi$
$278$	−9.09961e10	−0.913736
$279$	0	0
$280$	−2.56324e10	−0.249218
$281$	−1.22584e11	−1.17289	−0.586443	−	0.809990i	$-0.699472\pi$
$281$	−1.22584e11	−1.17289	−0.586443	+	0.809990i	$0.699472\pi$
$282$	0	0
$283$	3.90941e10	0.362303	0.181152	−	0.983455i	$-0.442018\pi$
$283$	3.90941e10	0.362303	0.181152	+	0.983455i	$0.442018\pi$
$284$	5.94688e10	0.542446
$285$	0	0
$286$	−1.66642e10	−0.147278
$287$	−2.69727e9	−0.0234669
$288$	0	0
$289$	2.34359e11	1.97624
$290$	4.72167e10	0.392016
$291$	0	0
$292$	−7.73679e8	−0.00622785
$293$	−1.55860e11	−1.23546	−0.617732	−	0.786389i	$-0.711948\pi$
$293$	−1.55860e11	−1.23546	−0.617732	+	0.786389i	$0.711948\pi$
$294$	0	0
$295$	−2.26200e10	−0.173898
$296$	8.98789e10	0.680526
$297$	0	0
$298$	1.00454e11	0.737892
$299$	−3.13850e11	−2.27092
$300$	0	0
$301$	−3.60408e11	−2.53073
$302$	1.64848e10	0.114039
$303$	0	0
$304$	8.54072e9	0.0573539
$305$	1.28959e11	0.853298
$306$	0	0
$307$	−3.30267e10	−0.212199	−0.106099	−	0.994356i	$-0.533836\pi$
$307$	−3.30267e10	−0.212199	−0.106099	+	0.994356i	$0.533836\pi$
$308$	−1.35610e10	−0.0858646
$309$	0	0
$310$	−1.75358e9	−0.0107844
$311$	−1.31724e11	−0.798444	−0.399222	−	0.916854i	$-0.630720\pi$
$311$	−1.31724e11	−0.798444	−0.399222	+	0.916854i	$0.630720\pi$
$312$	0	0
$313$	−5.93622e10	−0.349591	−0.174796	−	0.984605i	$-0.555926\pi$
$313$	−5.93622e10	−0.349591	−0.174796	+	0.984605i	$0.555926\pi$
$314$	2.28698e10	0.132764
$315$	0	0
$316$	−1.14273e11	−0.644692
$317$	3.05552e11	1.69949	0.849744	−	0.527196i	$-0.176756\pi$
$317$	3.05552e11	1.69949	0.849744	+	0.527196i	$0.176756\pi$
$318$	0	0
$319$	2.49803e10	0.135064
$320$	1.14756e10	0.0611788
$321$	0	0
$322$	−2.55406e11	−1.32397
$323$	7.74228e10	0.395783
$324$	0	0
$325$	2.67172e11	1.32836
$326$	−1.57116e11	−0.770443
$327$	0	0
$328$	1.20757e9	0.00576075
$329$	−4.26310e11	−2.00606
$330$	0	0
$331$	−7.96190e10	−0.364578	−0.182289	−	0.983245i	$-0.558351\pi$
$331$	−7.96190e10	−0.364578	−0.182289	+	0.983245i	$0.558351\pi$
$332$	−2.03270e11	−0.918230
$333$	0	0
$334$	3.91425e10	0.172104
$335$	−1.42060e10	−0.0616271
$336$	0	0
$337$	1.04453e11	0.441152	0.220576	−	0.975370i	$-0.429206\pi$
$337$	1.04453e11	0.441152	0.220576	+	0.975370i	$0.429206\pi$
$338$	−3.48043e11	−1.45047
$339$	0	0
$340$	1.04028e11	0.422178
$341$	−9.27743e8	−0.00371564
$342$	0	0
$343$	2.74194e10	0.106963
$344$	1.61354e11	0.621252
$345$	0	0
$346$	−1.93070e11	−0.724222
$347$	2.75297e10	0.101934	0.0509670	−	0.998700i	$-0.483770\pi$
$347$	2.75297e10	0.101934	0.0509670	+	0.998700i	$0.483770\pi$
$348$	0	0
$349$	−3.70683e11	−1.33748	−0.668741	−	0.743496i	$-0.733166\pi$
$349$	−3.70683e11	−1.33748	−0.668741	+	0.743496i	$0.733166\pi$
$350$	2.17420e11	0.774448
$351$	0	0
$352$	6.07126e9	0.0210783
$353$	−3.04278e11	−1.04300	−0.521500	−	0.853251i	$-0.674627\pi$
$353$	−3.04278e11	−1.04300	−0.521500	+	0.853251i	$0.674627\pi$
$354$	0	0
$355$	1.58893e11	0.530979
$356$	−2.32958e10	−0.0768693
$357$	0	0
$358$	2.29536e11	0.738544
$359$	3.20122e11	1.01716	0.508581	−	0.861014i	$-0.330170\pi$
$359$	3.20122e11	1.01716	0.508581	+	0.861014i	$0.330170\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	2.96827e11	0.908478
$363$	0	0
$364$	−4.21307e11	−1.25789
$365$	−2.06717e9	−0.00609620
$366$	0	0
$367$	−4.06154e11	−1.16867	−0.584337	−	0.811511i	$-0.698645\pi$
$367$	−4.06154e11	−1.16867	−0.584337	+	0.811511i	$0.698645\pi$
$368$	1.14345e11	0.325014
$369$	0	0
$370$	2.40145e11	0.666141
$371$	−2.02391e11	−0.554638
$372$	0	0
$373$	−5.36019e11	−1.43381	−0.716903	−	0.697173i	$-0.754441\pi$
$373$	−5.36019e11	−1.43381	−0.716903	+	0.697173i	$0.754441\pi$
$374$	5.50368e10	0.145456
$375$	0	0
$376$	1.90859e11	0.492455
$377$	7.76076e11	1.97865
$378$	0	0
$379$	−1.75471e11	−0.436846	−0.218423	−	0.975854i	$-0.570091\pi$
$379$	−1.75471e11	−0.436846	−0.218423	+	0.975854i	$0.570091\pi$
$380$	2.28197e10	0.0561415
$381$	0	0
$382$	1.84845e11	0.444144
$383$	−9.02775e10	−0.214380	−0.107190	−	0.994239i	$-0.534185\pi$
$383$	−9.02775e10	−0.214380	−0.107190	+	0.994239i	$0.534185\pi$
$384$	0	0
$385$	−3.62333e10	−0.0840495
$386$	−1.81491e11	−0.416114
$387$	0	0
$388$	−3.17374e10	−0.0710933
$389$	−1.09377e11	−0.242188	−0.121094	−	0.992641i	$-0.538640\pi$
$389$	−1.09377e11	−0.242188	−0.121094	+	0.992641i	$0.538640\pi$
$390$	0	0
$391$	1.03655e12	2.24283
$392$	−1.77564e11	−0.379811
$393$	0	0
$394$	1.29390e11	0.270500
$395$	−3.05324e11	−0.631064
$396$	0	0
$397$	−9.38066e10	−0.189529	−0.0947646	−	0.995500i	$-0.530210\pi$
$397$	−9.38066e10	−0.189529	−0.0947646	+	0.995500i	$0.530210\pi$
$398$	2.96092e11	0.591497
$399$	0	0
$400$	−9.73386e10	−0.190114
$401$	5.27904e11	1.01954	0.509771	−	0.860310i	$-0.329730\pi$
$401$	5.27904e11	1.01954	0.509771	+	0.860310i	$0.329730\pi$
$402$	0	0
$403$	−2.88227e10	−0.0544329
$404$	4.41440e11	0.824433
$405$	0	0
$406$	6.31557e11	1.15357
$407$	1.27050e11	0.229510
$408$	0	0
$409$	−8.58268e11	−1.51659	−0.758295	−	0.651912i	$-0.773967\pi$
$409$	−8.58268e11	−1.51659	−0.758295	+	0.651912i	$0.773967\pi$
$410$	3.22647e9	0.00563897
$411$	0	0
$412$	5.71309e10	0.0976863
$413$	−3.02560e11	−0.511724
$414$	0	0
$415$	−5.43112e11	−0.898820
$416$	1.88619e11	0.308791
$417$	0	0
$418$	1.20729e10	0.0193428
$419$	−1.73865e11	−0.275581	−0.137790	−	0.990461i	$-0.544000\pi$
$419$	−1.73865e11	−0.275581	−0.137790	+	0.990461i	$0.544000\pi$
$420$	0	0
$421$	5.08316e11	0.788614	0.394307	−	0.918979i	$-0.370985\pi$
$421$	5.08316e11	0.788614	0.394307	+	0.918979i	$0.370985\pi$
$422$	4.48847e11	0.688958
$423$	0	0
$424$	9.06105e10	0.136155
$425$	−8.82388e11	−1.31193
$426$	0	0
$427$	1.72492e12	2.51097
$428$	3.36343e10	0.0484491
$429$	0	0
$430$	4.31119e11	0.608119
$431$	7.46649e11	1.04224	0.521121	−	0.853483i	$-0.325514\pi$
$431$	7.46649e11	1.04224	0.521121	+	0.853483i	$0.325514\pi$
$432$	0	0
$433$	−1.09877e12	−1.50214	−0.751070	−	0.660223i	$-0.770462\pi$
$433$	−1.09877e12	−1.50214	−0.751070	+	0.660223i	$0.770462\pi$
$434$	−2.34554e10	−0.0317350
$435$	0	0
$436$	1.06501e11	0.141145
$437$	2.27380e11	0.298253
$438$	0	0
$439$	6.74130e11	0.866270	0.433135	−	0.901329i	$-0.357407\pi$
$439$	6.74130e11	0.866270	0.433135	+	0.901329i	$0.357407\pi$
$440$	1.62216e10	0.0206328
$441$	0	0
$442$	1.70986e12	2.13088
$443$	4.75800e11	0.586959	0.293479	−	0.955965i	$-0.405187\pi$
$443$	4.75800e11	0.586959	0.293479	+	0.955965i	$0.405187\pi$
$444$	0	0
$445$	−6.22435e10	−0.0752444
$446$	5.97547e11	0.715098
$447$	0	0
$448$	1.53495e11	0.180029
$449$	−1.97114e11	−0.228880	−0.114440	−	0.993430i	$-0.536507\pi$
$449$	−1.97114e11	−0.228880	−0.114440	+	0.993430i	$0.536507\pi$
$450$	0	0
$451$	1.70698e9	0.00194283
$452$	3.58195e11	0.403642
$453$	0	0
$454$	−6.82829e11	−0.754329
$455$	−1.12568e12	−1.23130
$456$	0	0
$457$	2.87681e11	0.308523	0.154262	−	0.988030i	$-0.450700\pi$
$457$	2.87681e11	0.308523	0.154262	+	0.988030i	$0.450700\pi$
$458$	−7.82555e11	−0.831037
$459$	0	0
$460$	3.05516e11	0.318144
$461$	4.15817e11	0.428793	0.214397	−	0.976747i	$-0.431222\pi$
$461$	4.15817e11	0.428793	0.214397	+	0.976747i	$0.431222\pi$
$462$	0	0
$463$	1.06942e12	1.08151	0.540757	−	0.841179i	$-0.318138\pi$
$463$	1.06942e12	1.08151	0.540757	+	0.841179i	$0.318138\pi$
$464$	−2.82748e11	−0.283184
$465$	0	0
$466$	−8.25889e11	−0.811306
$467$	−1.42089e12	−1.38240	−0.691200	−	0.722663i	$-0.742918\pi$
$467$	−1.42089e12	−1.38240	−0.691200	+	0.722663i	$0.742918\pi$
$468$	0	0
$469$	−1.90016e11	−0.181348
$470$	5.09951e11	0.482045
$471$	0	0
$472$	1.35456e11	0.125620
$473$	2.28086e11	0.209519
$474$	0	0
$475$	−1.93562e11	−0.174461
$476$	1.39145e12	1.24233
$477$	0	0
$478$	−5.68743e11	−0.498299
$479$	−3.28883e11	−0.285451	−0.142725	−	0.989762i	$-0.545587\pi$
$479$	−3.28883e11	−0.285451	−0.142725	+	0.989762i	$0.545587\pi$
$480$	0	0
$481$	3.94714e12	3.36225
$482$	2.75536e11	0.232524
$483$	0	0
$484$	−5.95052e11	−0.492891
$485$	−8.47983e10	−0.0695904
$486$	0	0
$487$	−2.10189e12	−1.69328	−0.846641	−	0.532165i	$-0.821379\pi$
$487$	−2.10189e12	−1.69328	−0.846641	+	0.532165i	$0.821379\pi$
$488$	−7.72243e11	−0.616403
$489$	0	0
$490$	−4.74429e11	−0.371782
$491$	−8.74667e11	−0.679166	−0.339583	−	0.940576i	$-0.610286\pi$
$491$	−8.74667e11	−0.679166	−0.339583	+	0.940576i	$0.610286\pi$
$492$	0	0
$493$	−2.56315e12	−1.95417
$494$	3.75076e11	0.283366
$495$	0	0
$496$	1.05010e10	0.00779043
$497$	2.12531e12	1.56250
$498$	0	0
$499$	−1.00922e12	−0.728676	−0.364338	−	0.931267i	$-0.618705\pi$
$499$	−1.00922e12	−0.728676	−0.364338	+	0.931267i	$0.618705\pi$
$500$	−6.02077e11	−0.430811
$501$	0	0
$502$	2.68249e11	0.188526
$503$	−1.00939e12	−0.703077	−0.351539	−	0.936173i	$-0.614341\pi$
$503$	−1.00939e12	−0.703077	−0.351539	+	0.936173i	$0.614341\pi$
$504$	0	0
$505$	1.17947e12	0.807005
$506$	1.61635e11	0.109612
$507$	0	0
$508$	−1.65657e11	−0.110363
$509$	1.14121e12	0.753590	0.376795	−	0.926297i	$-0.377026\pi$
$509$	1.14121e12	0.753590	0.376795	+	0.926297i	$0.377026\pi$
$510$	0	0
$511$	−2.76500e10	−0.0179391
$512$	−6.87195e10	−0.0441942
$513$	0	0
$514$	−1.01889e12	−0.643862
$515$	1.52647e11	0.0956213
$516$	0	0
$517$	2.69793e11	0.166082
$518$	3.21212e12	1.96023
$519$	0	0
$520$	5.03966e11	0.302264
$521$	3.01416e11	0.179224	0.0896122	−	0.995977i	$-0.471437\pi$
$521$	3.01416e11	0.179224	0.0896122	+	0.995977i	$0.471437\pi$
$522$	0	0
$523$	−1.99206e12	−1.16425	−0.582123	−	0.813100i	$-0.697778\pi$
$523$	−1.99206e12	−1.16425	−0.582123	+	0.813100i	$0.697778\pi$
$524$	−3.43466e11	−0.199018
$525$	0	0
$526$	−2.91949e11	−0.166292
$527$	9.51927e10	0.0537596
$528$	0	0
$529$	1.24306e12	0.690147
$530$	2.42100e11	0.133276
$531$	0	0
$532$	3.05231e11	0.165206
$533$	5.30318e10	0.0284619
$534$	0	0
$535$	8.98668e10	0.0474250
$536$	8.50701e10	0.0445180
$537$	0	0
$538$	1.64704e12	0.847588
$539$	−2.51000e11	−0.128093
$540$	0	0
$541$	9.37595e11	0.470574	0.235287	−	0.971926i	$-0.424397\pi$
$541$	9.37595e11	0.470574	0.235287	+	0.971926i	$0.424397\pi$
$542$	1.67504e12	0.833735
$543$	0	0
$544$	−6.22952e11	−0.304972
$545$	2.84557e11	0.138161
$546$	0	0
$547$	3.43141e12	1.63881	0.819407	−	0.573212i	$-0.194303\pi$
$547$	3.43141e12	1.63881	0.819407	+	0.573212i	$0.194303\pi$
$548$	−4.04087e11	−0.191409
$549$	0	0
$550$	−1.37595e11	−0.0641168
$551$	−5.62255e11	−0.259867
$552$	0	0
$553$	−4.08393e12	−1.85701
$554$	2.00617e12	0.904842
$555$	0	0
$556$	1.45594e12	0.646109
$557$	−1.63929e11	−0.0721619	−0.0360809	−	0.999349i	$-0.511487\pi$
$557$	−1.63929e11	−0.0721619	−0.0360809	+	0.999349i	$0.511487\pi$
$558$	0	0
$559$	7.08608e12	3.06940
$560$	4.10119e11	0.176223
$561$	0	0
$562$	1.96135e12	0.829356
$563$	−1.96430e12	−0.823988	−0.411994	−	0.911187i	$-0.635168\pi$
$563$	−1.96430e12	−0.823988	−0.411994	+	0.911187i	$0.635168\pi$
$564$	0	0
$565$	9.57051e11	0.395109
$566$	−6.25506e11	−0.256187
$567$	0	0
$568$	−9.51501e11	−0.383567
$569$	1.16239e12	0.464886	0.232443	−	0.972610i	$-0.425328\pi$
$569$	1.16239e12	0.464886	0.232443	+	0.972610i	$0.425328\pi$
$570$	0	0
$571$	−2.08656e11	−0.0821425	−0.0410713	−	0.999156i	$-0.513077\pi$
$571$	−2.08656e11	−0.0821425	−0.0410713	+	0.999156i	$0.513077\pi$
$572$	2.66627e11	0.104141
$573$	0	0
$574$	4.31563e10	0.0165936
$575$	−2.59145e12	−0.988638
$576$	0	0
$577$	2.79792e11	0.105086	0.0525429	−	0.998619i	$-0.483267\pi$
$577$	2.79792e11	0.105086	0.0525429	+	0.998619i	$0.483267\pi$
$578$	−3.74974e12	−1.39742
$579$	0	0
$580$	−7.55466e11	−0.277197
$581$	−7.26452e12	−2.64493
$582$	0	0
$583$	1.28085e11	0.0459186
$584$	1.23789e10	0.00440375
$585$	0	0
$586$	2.49376e12	0.873605
$587$	−1.95046e12	−0.678056	−0.339028	−	0.940776i	$-0.610098\pi$
$587$	−1.95046e12	−0.678056	−0.339028	+	0.940776i	$0.610098\pi$
$588$	0	0
$589$	2.08816e10	0.00714899
$590$	3.61921e11	0.122964
$591$	0	0
$592$	−1.43806e12	−0.481205
$593$	2.07544e12	0.689231	0.344616	−	0.938744i	$-0.388009\pi$
$593$	2.07544e12	0.689231	0.344616	+	0.938744i	$0.388009\pi$
$594$	0	0
$595$	3.71778e12	1.21607
$596$	−1.60726e12	−0.521768
$597$	0	0
$598$	5.02161e12	1.60578
$599$	4.08326e12	1.29595	0.647973	−	0.761664i	$-0.275617\pi$
$599$	4.08326e12	1.29595	0.647973	+	0.761664i	$0.275617\pi$
$600$	0	0
$601$	−3.38666e12	−1.05886	−0.529428	−	0.848355i	$-0.677593\pi$
$601$	−3.38666e12	−1.05886	−0.529428	+	0.848355i	$0.677593\pi$
$602$	5.76653e12	1.78949
$603$	0	0
$604$	−2.63757e11	−0.0806376
$605$	−1.58991e12	−0.482472
$606$	0	0
$607$	5.51863e12	1.64999	0.824997	−	0.565138i	$-0.191177\pi$
$607$	5.51863e12	1.64999	0.824997	+	0.565138i	$0.191177\pi$
$608$	−1.36651e11	−0.0405554
$609$	0	0
$610$	−2.06334e12	−0.603373
$611$	8.38180e12	2.43305
$612$	0	0
$613$	−1.99422e12	−0.570428	−0.285214	−	0.958464i	$-0.592065\pi$
$613$	−1.99422e12	−0.570428	−0.285214	+	0.958464i	$0.592065\pi$
$614$	5.28428e11	0.150047
$615$	0	0
$616$	2.16976e11	0.0607154
$617$	6.55154e12	1.81995	0.909976	−	0.414661i	$-0.136100\pi$
$617$	6.55154e12	1.81995	0.909976	+	0.414661i	$0.136100\pi$
$618$	0	0
$619$	−5.30914e12	−1.45350	−0.726752	−	0.686900i	$-0.758971\pi$
$619$	−5.30914e12	−1.45350	−0.726752	+	0.686900i	$0.758971\pi$
$620$	2.80573e10	0.00762575
$621$	0	0
$622$	2.10759e12	0.564585
$623$	−8.32553e11	−0.221419
$624$	0	0
$625$	1.29224e12	0.338754
$626$	9.49796e11	0.247198
$627$	0	0
$628$	−3.65918e11	−0.0938782
$629$	−1.30362e13	−3.32066
$630$	0	0
$631$	−2.65589e12	−0.666926	−0.333463	−	0.942763i	$-0.608217\pi$
$631$	−2.65589e12	−0.666926	−0.333463	+	0.942763i	$0.608217\pi$
$632$	1.82837e12	0.455866
$633$	0	0
$634$	−4.88883e12	−1.20172
$635$	−4.42615e11	−0.108030
$636$	0	0
$637$	−7.79795e12	−1.87652
$638$	−3.99685e11	−0.0955047
$639$	0	0
$640$	−1.83610e11	−0.0432600
$641$	−4.14357e12	−0.969424	−0.484712	−	0.874674i	$-0.661076\pi$
$641$	−4.14357e12	−0.969424	−0.484712	+	0.874674i	$0.661076\pi$
$642$	0	0
$643$	−4.88535e12	−1.12706	−0.563529	−	0.826097i	$-0.690557\pi$
$643$	−4.88535e12	−1.12706	−0.563529	+	0.826097i	$0.690557\pi$
$644$	4.08650e12	0.936191
$645$	0	0
$646$	−1.23876e12	−0.279861
$647$	−6.46459e12	−1.45035	−0.725173	−	0.688567i	$-0.758240\pi$
$647$	−6.46459e12	−1.45035	−0.725173	+	0.688567i	$0.758240\pi$
$648$	0	0
$649$	1.91477e11	0.0423657
$650$	−4.27475e12	−0.939291
$651$	0	0
$652$	2.51385e12	0.544785
$653$	−9.17846e12	−1.97542	−0.987712	−	0.156282i	$-0.950049\pi$
$653$	−9.17846e12	−1.97542	−0.987712	+	0.156282i	$0.950049\pi$
$654$	0	0
$655$	−9.17698e11	−0.194811
$656$	−1.93211e10	−0.00407346
$657$	0	0
$658$	6.82096e12	1.41850
$659$	−3.84668e11	−0.0794514	−0.0397257	−	0.999211i	$-0.512648\pi$
$659$	−3.84668e11	−0.0794514	−0.0397257	+	0.999211i	$0.512648\pi$
$660$	0	0
$661$	−2.50094e12	−0.509562	−0.254781	−	0.966999i	$-0.582003\pi$
$661$	−2.50094e12	−0.509562	−0.254781	+	0.966999i	$0.582003\pi$
$662$	1.27390e12	0.257796
$663$	0	0
$664$	3.25232e12	0.649287
$665$	8.15538e11	0.161714
$666$	0	0
$667$	−7.52760e12	−1.47262
$668$	−6.26280e11	−0.121696
$669$	0	0
$670$	2.27297e11	0.0435769
$671$	−1.09162e12	−0.207884
$672$	0	0
$673$	8.42534e11	0.158314	0.0791571	−	0.996862i	$-0.474777\pi$
$673$	8.42534e11	0.158314	0.0791571	+	0.996862i	$0.474777\pi$
$674$	−1.67125e12	−0.311941
$675$	0	0
$676$	5.56868e12	1.02563
$677$	−5.86524e11	−0.107309	−0.0536546	−	0.998560i	$-0.517087\pi$
$677$	−5.86524e11	−0.107309	−0.0536546	+	0.998560i	$0.517087\pi$
$678$	0	0
$679$	−1.13424e12	−0.204782
$680$	−1.66445e12	−0.298525
$681$	0	0
$682$	1.48439e10	0.00262735
$683$	1.00974e13	1.77548	0.887739	−	0.460348i	$-0.152275\pi$
$683$	1.00974e13	1.77548	0.887739	+	0.460348i	$0.152275\pi$
$684$	0	0
$685$	−1.07967e12	−0.187363
$686$	−4.38711e11	−0.0756345
$687$	0	0
$688$	−2.58167e12	−0.439291
$689$	3.97927e12	0.672694
$690$	0	0
$691$	1.00969e13	1.68475	0.842377	−	0.538889i	$-0.181156\pi$
$691$	1.00969e13	1.68475	0.842377	+	0.538889i	$0.181156\pi$
$692$	3.08911e12	0.512102
$693$	0	0
$694$	−4.40475e11	−0.0720782
$695$	3.89008e12	0.632451
$696$	0	0
$697$	−1.75148e11	−0.0281098
$698$	5.93092e12	0.945742
$699$	0	0
$700$	−3.47871e12	−0.547618
$701$	−9.01232e12	−1.40963	−0.704816	−	0.709391i	$-0.748970\pi$
$701$	−9.01232e12	−1.40963	−0.704816	+	0.709391i	$0.748970\pi$
$702$	0	0
$703$	−2.85965e12	−0.441584
$704$	−9.71401e10	−0.0149046
$705$	0	0
$706$	4.86845e12	0.737512
$707$	1.57763e13	2.37475
$708$	0	0
$709$	1.20838e13	1.79596	0.897979	−	0.440039i	$-0.145035\pi$
$709$	1.20838e13	1.79596	0.897979	+	0.440039i	$0.145035\pi$
$710$	−2.54229e12	−0.375459
$711$	0	0
$712$	3.72733e11	0.0543548
$713$	2.79568e11	0.0405120
$714$	0	0
$715$	7.12394e11	0.101940
$716$	−3.67257e12	−0.522230
$717$	0	0
$718$	−5.12195e12	−0.719243
$719$	8.31774e12	1.16071	0.580357	−	0.814362i	$-0.302913\pi$
$719$	8.31774e12	1.16071	0.580357	+	0.814362i	$0.302913\pi$
$720$	0	0
$721$	2.04176e12	0.281382
$722$	−2.71737e11	−0.0372161
$723$	0	0
$724$	−4.74923e12	−0.642391
$725$	6.40803e12	0.861397
$726$	0	0
$727$	6.18564e11	0.0821258	0.0410629	−	0.999157i	$-0.486926\pi$
$727$	6.18564e11	0.0821258	0.0410629	+	0.999157i	$0.486926\pi$
$728$	6.74092e12	0.889463
$729$	0	0
$730$	3.30748e10	0.00431066
$731$	−2.34032e13	−3.03143
$732$	0	0
$733$	5.06624e12	0.648213	0.324107	−	0.946021i	$-0.394936\pi$
$733$	5.06624e12	0.648213	0.324107	+	0.946021i	$0.394936\pi$
$734$	6.49846e12	0.826377
$735$	0	0
$736$	−1.82952e12	−0.229820
$737$	1.20253e11	0.0150138
$738$	0	0
$739$	−3.41796e12	−0.421567	−0.210784	−	0.977533i	$-0.567602\pi$
$739$	−3.41796e12	−0.421567	−0.210784	+	0.977533i	$0.567602\pi$
$740$	−3.84232e12	−0.471033
$741$	0	0
$742$	3.23826e12	0.392188
$743$	−9.81930e12	−1.18204	−0.591018	−	0.806658i	$-0.701274\pi$
$743$	−9.81930e12	−1.18204	−0.591018	+	0.806658i	$0.701274\pi$
$744$	0	0
$745$	−4.29439e12	−0.510739
$746$	8.57630e12	1.01385
$747$	0	0
$748$	−8.80588e11	−0.102853
$749$	1.20203e12	0.139556
$750$	0	0
$751$	1.58426e13	1.81739	0.908693	−	0.417466i	$-0.137082\pi$
$751$	1.58426e13	1.81739	0.908693	+	0.417466i	$0.137082\pi$
$752$	−3.05374e12	−0.348218
$753$	0	0
$754$	−1.24172e13	−1.39911
$755$	−7.04726e11	−0.0789330
$756$	0	0
$757$	−3.26528e12	−0.361401	−0.180700	−	0.983538i	$-0.557836\pi$
$757$	−3.26528e12	−0.361401	−0.180700	+	0.983538i	$0.557836\pi$
$758$	2.80753e12	0.308897
$759$	0	0
$760$	−3.65116e11	−0.0396981
$761$	1.87390e12	0.202542	0.101271	−	0.994859i	$-0.467709\pi$
$761$	1.87390e12	0.202542	0.101271	+	0.994859i	$0.467709\pi$
$762$	0	0
$763$	3.80616e12	0.406562
$764$	−2.95753e12	−0.314057
$765$	0	0
$766$	1.44444e12	0.151590
$767$	5.94871e12	0.620645
$768$	0	0
$769$	4.14201e12	0.427113	0.213557	−	0.976931i	$-0.431495\pi$
$769$	4.14201e12	0.427113	0.213557	+	0.976931i	$0.431495\pi$
$770$	5.79733e11	0.0594320
$771$	0	0
$772$	2.90386e12	0.294237
$773$	−7.30455e12	−0.735844	−0.367922	−	0.929857i	$-0.619931\pi$
$773$	−7.30455e12	−0.735844	−0.367922	+	0.929857i	$0.619931\pi$
$774$	0	0
$775$	−2.37988e11	−0.0236972
$776$	5.07798e11	0.0502705
$777$	0	0
$778$	1.75003e12	0.171253
$779$	−3.84207e10	−0.00373807
$780$	0	0
$781$	−1.34502e12	−0.129359
$782$	−1.65849e13	−1.58592
$783$	0	0
$784$	2.84102e12	0.268567
$785$	−9.77686e11	−0.0918937
$786$	0	0
$787$	−6.77475e12	−0.629516	−0.314758	−	0.949172i	$-0.601923\pi$
$787$	−6.77475e12	−0.629516	−0.314758	+	0.949172i	$0.601923\pi$
$788$	−2.07024e12	−0.191273
$789$	0	0
$790$	4.88518e12	0.446230
$791$	1.28013e13	1.16268
$792$	0	0
$793$	−3.39140e13	−3.04544
$794$	1.50091e12	0.134017
$795$	0	0
$796$	−4.73747e12	−0.418252
$797$	8.80270e12	0.772776	0.386388	−	0.922336i	$-0.373723\pi$
$797$	8.80270e12	0.772776	0.386388	+	0.922336i	$0.373723\pi$
$798$	0	0
$799$	−2.76826e13	−2.40296
$800$	1.55742e12	0.134431
$801$	0	0
$802$	−8.44646e12	−0.720925
$803$	1.74984e10	0.00148518
$804$	0	0
$805$	1.09186e13	0.916401
$806$	4.61163e11	0.0384899
$807$	0	0
$808$	−7.06304e12	−0.582962
$809$	1.37181e13	1.12597	0.562984	−	0.826468i	$-0.309653\pi$
$809$	1.37181e13	1.12597	0.562984	+	0.826468i	$0.309653\pi$
$810$	0	0
$811$	8.04900e12	0.653353	0.326677	−	0.945136i	$-0.394071\pi$
$811$	8.04900e12	0.653353	0.326677	+	0.945136i	$0.394071\pi$
$812$	−1.01049e13	−0.815700
$813$	0	0
$814$	−2.03281e12	−0.162288
$815$	6.71670e12	0.533269
$816$	0	0
$817$	−5.13375e12	−0.403121
$818$	1.37323e13	1.07239
$819$	0	0
$820$	−5.16235e10	−0.00398735
$821$	−2.28223e13	−1.75314	−0.876568	−	0.481278i	$-0.840173\pi$
$821$	−2.28223e13	−1.75314	−0.876568	+	0.481278i	$0.840173\pi$
$822$	0	0
$823$	−8.82687e12	−0.670668	−0.335334	−	0.942099i	$-0.608849\pi$
$823$	−8.82687e12	−0.670668	−0.335334	+	0.942099i	$0.608849\pi$
$824$	−9.14094e11	−0.0690746
$825$	0	0
$826$	4.84095e12	0.361843
$827$	1.15347e13	0.857492	0.428746	−	0.903425i	$-0.358956\pi$
$827$	1.15347e13	0.857492	0.428746	+	0.903425i	$0.358956\pi$
$828$	0	0
$829$	−1.47019e13	−1.08113	−0.540566	−	0.841302i	$-0.681790\pi$
$829$	−1.47019e13	−1.08113	−0.540566	+	0.841302i	$0.681790\pi$
$830$	8.68979e12	0.635562
$831$	0	0
$832$	−3.01790e12	−0.218349
$833$	2.57543e13	1.85331
$834$	0	0
$835$	−1.67334e12	−0.119123
$836$	−1.93167e11	−0.0136774
$837$	0	0
$838$	2.78184e12	0.194865
$839$	3.67876e12	0.256314	0.128157	−	0.991754i	$-0.459094\pi$
$839$	3.67876e12	0.256314	0.128157	+	0.991754i	$0.459094\pi$
$840$	0	0
$841$	4.10679e12	0.283087
$842$	−8.13306e12	−0.557634
$843$	0	0
$844$	−7.18155e12	−0.487167
$845$	1.48788e13	1.00395
$846$	0	0
$847$	−2.12662e13	−1.41976
$848$	−1.44977e12	−0.0962758
$849$	0	0
$850$	1.41182e13	0.927672
$851$	−3.82856e13	−2.50237
$852$	0	0
$853$	7.86173e12	0.508449	0.254224	−	0.967145i	$-0.418180\pi$
$853$	7.86173e12	0.508449	0.254224	+	0.967145i	$0.418180\pi$
$854$	−2.75986e13	−1.77553
$855$	0	0
$856$	−5.38149e11	−0.0342587
$857$	−4.33710e12	−0.274654	−0.137327	−	0.990526i	$-0.543851\pi$
$857$	−4.33710e12	−0.274654	−0.137327	+	0.990526i	$0.543851\pi$
$858$	0	0
$859$	−1.21007e13	−0.758299	−0.379149	−	0.925335i	$-0.623783\pi$
$859$	−1.21007e13	−0.758299	−0.379149	+	0.925335i	$0.623783\pi$
$860$	−6.89790e12	−0.430005
$861$	0	0
$862$	−1.19464e13	−0.736976
$863$	−1.63159e13	−1.00129	−0.500647	−	0.865651i	$-0.666905\pi$
$863$	−1.63159e13	−1.00129	−0.500647	+	0.865651i	$0.666905\pi$
$864$	0	0
$865$	8.25373e12	0.501277
$866$	1.75803e13	1.06217
$867$	0	0
$868$	3.75286e11	0.0224401
$869$	2.58454e12	0.153742
$870$	0	0
$871$	3.73596e12	0.219948
$872$	−1.70401e12	−0.0998043
$873$	0	0
$874$	−3.63808e12	−0.210897
$875$	−2.15172e13	−1.24094
$876$	0	0
$877$	6.86374e12	0.391798	0.195899	−	0.980624i	$-0.437237\pi$
$877$	6.86374e12	0.391798	0.195899	+	0.980624i	$0.437237\pi$
$878$	−1.07861e13	−0.612545
$879$	0	0
$880$	−2.59546e11	−0.0145896
$881$	1.55190e13	0.867903	0.433951	−	0.900936i	$-0.357119\pi$
$881$	1.55190e13	0.867903	0.433951	+	0.900936i	$0.357119\pi$
$882$	0	0
$883$	−1.41378e13	−0.782633	−0.391317	−	0.920256i	$-0.627980\pi$
$883$	−1.41378e13	−0.782633	−0.391317	+	0.920256i	$0.627980\pi$
$884$	−2.73577e13	−1.50676
$885$	0	0
$886$	−7.61280e12	−0.415042
$887$	−7.90823e12	−0.428966	−0.214483	−	0.976728i	$-0.568807\pi$
$887$	−7.90823e12	−0.428966	−0.214483	+	0.976728i	$0.568807\pi$
$888$	0	0
$889$	−5.92030e12	−0.317897
$890$	9.95897e11	0.0532058
$891$	0	0
$892$	−9.56076e12	−0.505651
$893$	−6.07248e12	−0.319547
$894$	0	0
$895$	−9.81265e12	−0.511190
$896$	−2.45592e12	−0.127300
$897$	0	0
$898$	3.15382e12	0.161843
$899$	−6.91303e11	−0.0352980
$900$	0	0
$901$	−1.31423e13	−0.664372
$902$	−2.73118e10	−0.00137379
$903$	0	0
$904$	−5.73111e12	−0.285418
$905$	−1.26893e13	−0.628812
$906$	0	0
$907$	−4.29326e12	−0.210647	−0.105323	−	0.994438i	$-0.533588\pi$
$907$	−4.29326e12	−0.210647	−0.105323	+	0.994438i	$0.533588\pi$
$908$	1.09253e13	0.533391
$909$	0	0
$910$	1.80109e13	0.870660
$911$	−3.40286e12	−0.163686	−0.0818430	−	0.996645i	$-0.526081\pi$
$911$	−3.40286e12	−0.163686	−0.0818430	+	0.996645i	$0.526081\pi$
$912$	0	0
$913$	4.59739e12	0.218974
$914$	−4.60289e12	−0.218159
$915$	0	0
$916$	1.25209e13	0.587632
$917$	−1.22749e13	−0.573265
$918$	0	0
$919$	2.24723e13	1.03927	0.519634	−	0.854389i	$-0.326068\pi$
$919$	2.24723e13	1.03927	0.519634	+	0.854389i	$0.326068\pi$
$920$	−4.88825e12	−0.224962
$921$	0	0
$922$	−6.65307e12	−0.303202
$923$	−4.17864e13	−1.89508
$924$	0	0
$925$	3.25914e13	1.46374
$926$	−1.71107e13	−0.764746
$927$	0	0
$928$	4.52396e12	0.200241
$929$	1.47150e13	0.648170	0.324085	−	0.946028i	$-0.394944\pi$
$929$	1.47150e13	0.648170	0.324085	+	0.946028i	$0.394944\pi$
$930$	0	0
$931$	5.64949e12	0.246454
$932$	1.32142e13	0.573680
$933$	0	0
$934$	2.27342e13	0.977505
$935$	−2.35282e12	−0.100679
$936$	0	0
$937$	1.48535e13	0.629509	0.314754	−	0.949173i	$-0.398078\pi$
$937$	1.48535e13	0.629509	0.314754	+	0.949173i	$0.398078\pi$
$938$	3.04026e12	0.128232
$939$	0	0
$940$	−8.15921e12	−0.340857
$941$	2.17556e13	0.904521	0.452260	−	0.891886i	$-0.350618\pi$
$941$	2.17556e13	0.904521	0.452260	+	0.891886i	$0.350618\pi$
$942$	0	0
$943$	−5.14385e11	−0.0211829
$944$	−2.16729e12	−0.0888266
$945$	0	0
$946$	−3.64938e12	−0.148153
$947$	1.95060e13	0.788121	0.394061	−	0.919084i	$-0.371070\pi$
$947$	1.95060e13	0.788121	0.394061	+	0.919084i	$0.371070\pi$
$948$	0	0
$949$	5.43633e11	0.0217575
$950$	3.09699e12	0.123363
$951$	0	0
$952$	−2.22632e13	−0.878459
$953$	8.42406e12	0.330829	0.165414	−	0.986224i	$-0.447104\pi$
$953$	8.42406e12	0.330829	0.165414	+	0.986224i	$0.447104\pi$
$954$	0	0
$955$	−7.90214e12	−0.307418
$956$	9.09988e12	0.352351
$957$	0	0
$958$	5.26212e12	0.201844
$959$	−1.44414e13	−0.551346
$960$	0	0
$961$	−2.64139e13	−0.999029
$962$	−6.31543e13	−2.37747
$963$	0	0
$964$	−4.40858e12	−0.164419
$965$	7.75875e12	0.288017
$966$	0	0
$967$	2.23168e13	0.820754	0.410377	−	0.911916i	$-0.365397\pi$
$967$	2.23168e13	0.820754	0.410377	+	0.911916i	$0.365397\pi$
$968$	9.52084e12	0.348527
$969$	0	0
$970$	1.35677e12	0.0492079
$971$	3.10052e12	0.111930	0.0559652	−	0.998433i	$-0.482176\pi$
$971$	3.10052e12	0.111930	0.0559652	+	0.998433i	$0.482176\pi$
$972$	0	0
$973$	5.20327e13	1.86109
$974$	3.36302e13	1.19733
$975$	0	0
$976$	1.23559e13	0.435863
$977$	4.62584e13	1.62430	0.812149	−	0.583451i	$-0.198298\pi$
$977$	4.62584e13	1.62430	0.812149	+	0.583451i	$0.198298\pi$
$978$	0	0
$979$	5.26886e11	0.0183314
$980$	7.59086e12	0.262890
$981$	0	0
$982$	1.39947e13	0.480243
$983$	3.24323e13	1.10787	0.553933	−	0.832561i	$-0.313126\pi$
$983$	3.24323e13	1.10787	0.553933	+	0.832561i	$0.313126\pi$
$984$	0	0
$985$	−5.53143e12	−0.187229
$986$	4.10104e13	1.38181
$987$	0	0
$988$	−6.00122e12	−0.200370
$989$	−6.87319e13	−2.28441
$990$	0	0
$991$	−1.25266e11	−0.00412575	−0.00206288	−	0.999998i	$-0.500657\pi$
$991$	−1.25266e11	−0.00412575	−0.00206288	+	0.999998i	$0.500657\pi$
$992$	−1.68015e11	−0.00550867
$993$	0	0
$994$	−3.40050e13	−1.10485
$995$	−1.26579e13	−0.409410
$996$	0	0
$997$	1.07724e13	0.345291	0.172646	−	0.984984i	$-0.444768\pi$
$997$	1.07724e13	0.345291	0.172646	+	0.984984i	$0.444768\pi$
$998$	1.61476e13	0.515252
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.10.a.a.1.1		1
3.2	odd	2		38.10.a.a.1.1	✓	1
12.11	even	2		304.10.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.a.a.1.1	✓	1	3.2	odd	2
304.10.a.b.1.1		1	12.11	even	2
342.10.a.a.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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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