LMFDB - Embedded newform 384.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.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:	$119.955849786$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+27.0000 q^{3} -160.000 q^{5} +974.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q+27.0000 q^{3} -160.000 q^{5} +974.000 q^{7} +729.000 q^{9} -3956.00 q^{11} -574.000 q^{13} -4320.00 q^{15} -8474.00 q^{17} +53312.0 q^{19} +26298.0 q^{21} -98468.0 q^{23} -52525.0 q^{25} +19683.0 q^{27} -51060.0 q^{29} +205014. q^{31} -106812. q^{33} -155840. q^{35} -255674. q^{37} -15498.0 q^{39} -665394. q^{41} +396840. q^{43} -116640. q^{45} +549388. q^{47} +125133. q^{49} -228798. q^{51} +720060. q^{53} +632960. q^{55} +1.43942e6 q^{57} -1.04396e6 q^{59} +2.05573e6 q^{61} +710046. q^{63} +91840.0 q^{65} -2.09265e6 q^{67} -2.65864e6 q^{69} -2.72387e6 q^{71} -5.19001e6 q^{73} -1.41818e6 q^{75} -3.85314e6 q^{77} +3.64711e6 q^{79} +531441. q^{81} -2.10196e6 q^{83} +1.35584e6 q^{85} -1.37862e6 q^{87} +522514. q^{89} -559076. q^{91} +5.53538e6 q^{93} -8.52992e6 q^{95} -1.36199e6 q^{97} -2.88392e6 q^{99} +O(q^{100})$

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
$5$	−160.000	−0.572433	−0.286217	−	0.958165i	$-0.592398\pi$
$5$	−160.000	−0.572433	−0.286217	+	0.958165i	$0.592398\pi$
$6$	0	0
$7$	974.000	1.07329	0.536643	−	0.843809i	$-0.319692\pi$
$7$	974.000	1.07329	0.536643	+	0.843809i	$0.319692\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−3956.00	−0.896152	−0.448076	−	0.893995i	$-0.647891\pi$
$11$	−3956.00	−0.896152	−0.448076	+	0.893995i	$0.647891\pi$
$12$	0	0
$13$	−574.000	−0.0724620	−0.0362310	−	0.999343i	$-0.511535\pi$
$13$	−574.000	−0.0724620	−0.0362310	+	0.999343i	$0.511535\pi$
$14$	0	0
$15$	−4320.00	−0.330495
$16$	0	0
$17$	−8474.00	−0.418328	−0.209164	−	0.977881i	$-0.567074\pi$
$17$	−8474.00	−0.418328	−0.209164	+	0.977881i	$0.567074\pi$
$18$	0	0
$19$	53312.0	1.78315	0.891574	−	0.452875i	$-0.149602\pi$
$19$	53312.0	1.78315	0.891574	+	0.452875i	$0.149602\pi$
$20$	0	0
$21$	26298.0	0.619662
$22$	0	0
$23$	−98468.0	−1.68752	−0.843758	−	0.536724i	$-0.819661\pi$
$23$	−98468.0	−1.68752	−0.843758	+	0.536724i	$0.819661\pi$
$24$	0	0
$25$	−52525.0	−0.672320
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−51060.0	−0.388766	−0.194383	−	0.980926i	$-0.562270\pi$
$29$	−51060.0	−0.388766	−0.194383	+	0.980926i	$0.562270\pi$
$30$	0	0
$31$	205014.	1.23600	0.617999	−	0.786179i	$-0.287944\pi$
$31$	205014.	1.23600	0.617999	+	0.786179i	$0.287944\pi$
$32$	0	0
$33$	−106812.	−0.517394
$34$	0	0
$35$	−155840.	−0.614385
$36$	0	0
$37$	−255674.	−0.829814	−0.414907	−	0.909864i	$-0.636186\pi$
$37$	−255674.	−0.829814	−0.414907	+	0.909864i	$0.636186\pi$
$38$	0	0
$39$	−15498.0	−0.0418359
$40$	0	0
$41$	−665394.	−1.50777	−0.753885	−	0.657006i	$-0.771823\pi$
$41$	−665394.	−1.50777	−0.753885	+	0.657006i	$0.771823\pi$
$42$	0	0
$43$	396840.	0.761160	0.380580	−	0.924748i	$-0.375724\pi$
$43$	396840.	0.761160	0.380580	+	0.924748i	$0.375724\pi$
$44$	0	0
$45$	−116640.	−0.190811
$46$	0	0
$47$	549388.	0.771857	0.385928	−	0.922529i	$-0.373881\pi$
$47$	549388.	0.771857	0.385928	+	0.922529i	$0.373881\pi$
$48$	0	0
$49$	125133.	0.151945
$50$	0	0
$51$	−228798.	−0.241522
$52$	0	0
$53$	720060.	0.664359	0.332180	−	0.943216i	$-0.392216\pi$
$53$	720060.	0.664359	0.332180	+	0.943216i	$0.392216\pi$
$54$	0	0
$55$	632960.	0.512988
$56$	0	0
$57$	1.43942e6	1.02950
$58$	0	0
$59$	−1.04396e6	−0.661759	−0.330880	−	0.943673i	$-0.607345\pi$
$59$	−1.04396e6	−0.661759	−0.330880	+	0.943673i	$0.607345\pi$
$60$	0	0
$61$	2.05573e6	1.15961	0.579806	−	0.814755i	$-0.303128\pi$
$61$	2.05573e6	1.15961	0.579806	+	0.814755i	$0.303128\pi$
$62$	0	0
$63$	710046.	0.357762
$64$	0	0
$65$	91840.0	0.0414797
$66$	0	0
$67$	−2.09265e6	−0.850032	−0.425016	−	0.905186i	$-0.639732\pi$
$67$	−2.09265e6	−0.850032	−0.425016	+	0.905186i	$0.639732\pi$
$68$	0	0
$69$	−2.65864e6	−0.974287
$70$	0	0
$71$	−2.72387e6	−0.903196	−0.451598	−	0.892222i	$-0.649146\pi$
$71$	−2.72387e6	−0.903196	−0.451598	+	0.892222i	$0.649146\pi$
$72$	0	0
$73$	−5.19001e6	−1.56149	−0.780743	−	0.624852i	$-0.785159\pi$
$73$	−5.19001e6	−1.56149	−0.780743	+	0.624852i	$0.785159\pi$
$74$	0	0
$75$	−1.41818e6	−0.388164
$76$	0	0
$77$	−3.85314e6	−0.961829
$78$	0	0
$79$	3.64711e6	0.832250	0.416125	−	0.909307i	$-0.363388\pi$
$79$	3.64711e6	0.832250	0.416125	+	0.909307i	$0.363388\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−2.10196e6	−0.403506	−0.201753	−	0.979436i	$-0.564664\pi$
$83$	−2.10196e6	−0.403506	−0.201753	+	0.979436i	$0.564664\pi$
$84$	0	0
$85$	1.35584e6	0.239465
$86$	0	0
$87$	−1.37862e6	−0.224454
$88$	0	0
$89$	522514.	0.0785657	0.0392828	−	0.999228i	$-0.487493\pi$
$89$	522514.	0.0785657	0.0392828	+	0.999228i	$0.487493\pi$
$90$	0	0
$91$	−559076.	−0.0777725
$92$	0	0
$93$	5.53538e6	0.713603
$94$	0	0
$95$	−8.52992e6	−1.02073
$96$	0	0
$97$	−1.36199e6	−0.151521	−0.0757605	−	0.997126i	$-0.524138\pi$
$97$	−1.36199e6	−0.151521	−0.0757605	+	0.997126i	$0.524138\pi$
$98$	0	0
$99$	−2.88392e6	−0.298717
$100$	0	0
$101$	−1.02333e7	−0.988305	−0.494152	−	0.869375i	$-0.664522\pi$
$101$	−1.02333e7	−0.988305	−0.494152	+	0.869375i	$0.664522\pi$
$102$	0	0
$103$	−2.74567e6	−0.247581	−0.123791	−	0.992308i	$-0.539505\pi$
$103$	−2.74567e6	−0.247581	−0.123791	+	0.992308i	$0.539505\pi$
$104$	0	0
$105$	−4.20768e6	−0.354715
$106$	0	0
$107$	−5.90505e6	−0.465994	−0.232997	−	0.972477i	$-0.574853\pi$
$107$	−5.90505e6	−0.465994	−0.232997	+	0.972477i	$0.574853\pi$
$108$	0	0
$109$	2.48683e6	0.183930	0.0919650	−	0.995762i	$-0.470685\pi$
$109$	2.48683e6	0.183930	0.0919650	+	0.995762i	$0.470685\pi$
$110$	0	0
$111$	−6.90320e6	−0.479093
$112$	0	0
$113$	−1.78578e7	−1.16427	−0.582134	−	0.813093i	$-0.697782\pi$
$113$	−1.78578e7	−1.16427	−0.582134	+	0.813093i	$0.697782\pi$
$114$	0	0
$115$	1.57549e7	0.965990
$116$	0	0
$117$	−418446.	−0.0241540
$118$	0	0
$119$	−8.25368e6	−0.448986
$120$	0	0
$121$	−3.83724e6	−0.196911
$122$	0	0
$123$	−1.79656e7	−0.870512
$124$	0	0
$125$	2.09040e7	0.957292
$126$	0	0
$127$	−2.73553e6	−0.118503	−0.0592515	−	0.998243i	$-0.518871\pi$
$127$	−2.73553e6	−0.118503	−0.0592515	+	0.998243i	$0.518871\pi$
$128$	0	0
$129$	1.07147e7	0.439456
$130$	0	0
$131$	3.06060e7	1.18948	0.594740	−	0.803918i	$-0.297255\pi$
$131$	3.06060e7	1.18948	0.594740	+	0.803918i	$0.297255\pi$
$132$	0	0
$133$	5.19259e7	1.91383
$134$	0	0
$135$	−3.14928e6	−0.110165
$136$	0	0
$137$	1.24065e7	0.412218	0.206109	−	0.978529i	$-0.433920\pi$
$137$	1.24065e7	0.412218	0.206109	+	0.978529i	$0.433920\pi$
$138$	0	0
$139$	−4.36363e7	−1.37815	−0.689074	−	0.724691i	$-0.741983\pi$
$139$	−4.36363e7	−1.37815	−0.689074	+	0.724691i	$0.741983\pi$
$140$	0	0
$141$	1.48335e7	0.445632
$142$	0	0
$143$	2.27074e6	0.0649370
$144$	0	0
$145$	8.16960e6	0.222542
$146$	0	0
$147$	3.37859e6	0.0877253
$148$	0	0
$149$	3.96850e7	0.982821	0.491411	−	0.870928i	$-0.336481\pi$
$149$	3.96850e7	0.982821	0.491411	+	0.870928i	$0.336481\pi$
$150$	0	0
$151$	−5.42615e7	−1.28255	−0.641273	−	0.767313i	$-0.721593\pi$
$151$	−5.42615e7	−1.28255	−0.641273	+	0.767313i	$0.721593\pi$
$152$	0	0
$153$	−6.17755e6	−0.139443
$154$	0	0
$155$	−3.28022e7	−0.707526
$156$	0	0
$157$	4.84741e7	0.999679	0.499839	−	0.866118i	$-0.333392\pi$
$157$	4.84741e7	0.999679	0.499839	+	0.866118i	$0.333392\pi$
$158$	0	0
$159$	1.94416e7	0.383568
$160$	0	0
$161$	−9.59078e7	−1.81119
$162$	0	0
$163$	−8.38407e7	−1.51635	−0.758173	−	0.652054i	$-0.773908\pi$
$163$	−8.38407e7	−1.51635	−0.758173	+	0.652054i	$0.773908\pi$
$164$	0	0
$165$	1.70899e7	0.296174
$166$	0	0
$167$	−5.52460e7	−0.917895	−0.458947	−	0.888463i	$-0.651773\pi$
$167$	−5.52460e7	−0.917895	−0.458947	+	0.888463i	$0.651773\pi$
$168$	0	0
$169$	−6.24190e7	−0.994749
$170$	0	0
$171$	3.88644e7	0.594383
$172$	0	0
$173$	−1.45163e7	−0.213155	−0.106577	−	0.994304i	$-0.533989\pi$
$173$	−1.45163e7	−0.213155	−0.106577	+	0.994304i	$0.533989\pi$
$174$	0	0
$175$	−5.11594e7	−0.721592
$176$	0	0
$177$	−2.81868e7	−0.382067
$178$	0	0
$179$	−6.58271e7	−0.857865	−0.428932	−	0.903337i	$-0.641110\pi$
$179$	−6.58271e7	−0.857865	−0.428932	+	0.903337i	$0.641110\pi$
$180$	0	0
$181$	−1.00328e8	−1.25762	−0.628808	−	0.777561i	$-0.716457\pi$
$181$	−1.00328e8	−1.25762	−0.628808	+	0.777561i	$0.716457\pi$
$182$	0	0
$183$	5.55048e7	0.669502
$184$	0	0
$185$	4.09078e7	0.475013
$186$	0	0
$187$	3.35231e7	0.374886
$188$	0	0
$189$	1.91712e7	0.206554
$190$	0	0
$191$	−1.26036e8	−1.30881	−0.654407	−	0.756143i	$-0.727082\pi$
$191$	−1.26036e8	−1.30881	−0.654407	+	0.756143i	$0.727082\pi$
$192$	0	0
$193$	−1.07735e7	−0.107871	−0.0539357	−	0.998544i	$-0.517177\pi$
$193$	−1.07735e7	−0.107871	−0.0539357	+	0.998544i	$0.517177\pi$
$194$	0	0
$195$	2.47968e6	0.0239483
$196$	0	0
$197$	1.82754e8	1.70308	0.851541	−	0.524288i	$-0.175668\pi$
$197$	1.82754e8	1.70308	0.851541	+	0.524288i	$0.175668\pi$
$198$	0	0
$199$	−1.70902e8	−1.53731	−0.768653	−	0.639665i	$-0.779073\pi$
$199$	−1.70902e8	−1.53731	−0.768653	+	0.639665i	$0.779073\pi$
$200$	0	0
$201$	−5.65016e7	−0.490766
$202$	0	0
$203$	−4.97324e7	−0.417257
$204$	0	0
$205$	1.06463e8	0.863098
$206$	0	0
$207$	−7.17832e7	−0.562505
$208$	0	0
$209$	−2.10902e8	−1.59797
$210$	0	0
$211$	−7.80189e7	−0.571757	−0.285878	−	0.958266i	$-0.592285\pi$
$211$	−7.80189e7	−0.571757	−0.285878	+	0.958266i	$0.592285\pi$
$212$	0	0
$213$	−7.35444e7	−0.521460
$214$	0	0
$215$	−6.34944e7	−0.435713
$216$	0	0
$217$	1.99684e8	1.32658
$218$	0	0
$219$	−1.40130e8	−0.901525
$220$	0	0
$221$	4.86408e6	0.0303129
$222$	0	0
$223$	5.36223e7	0.323801	0.161901	−	0.986807i	$-0.448238\pi$
$223$	5.36223e7	0.323801	0.161901	+	0.986807i	$0.448238\pi$
$224$	0	0
$225$	−3.82907e7	−0.224107
$226$	0	0
$227$	−2.13091e8	−1.20914	−0.604568	−	0.796553i	$-0.706654\pi$
$227$	−2.13091e8	−1.20914	−0.604568	+	0.796553i	$0.706654\pi$
$228$	0	0
$229$	−1.16207e8	−0.639452	−0.319726	−	0.947510i	$-0.603591\pi$
$229$	−1.16207e8	−0.639452	−0.319726	+	0.947510i	$0.603591\pi$
$230$	0	0
$231$	−1.04035e8	−0.555312
$232$	0	0
$233$	−2.89375e8	−1.49870	−0.749351	−	0.662173i	$-0.769634\pi$
$233$	−2.89375e8	−1.49870	−0.749351	+	0.662173i	$0.769634\pi$
$234$	0	0
$235$	−8.79021e7	−0.441837
$236$	0	0
$237$	9.84720e7	0.480500
$238$	0	0
$239$	2.40117e8	1.13771	0.568854	−	0.822438i	$-0.307387\pi$
$239$	2.40117e8	1.13771	0.568854	+	0.822438i	$0.307387\pi$
$240$	0	0
$241$	−4.13712e8	−1.90387	−0.951937	−	0.306294i	$-0.900911\pi$
$241$	−4.13712e8	−1.90387	−0.951937	+	0.306294i	$0.900911\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−2.00213e7	−0.0869782
$246$	0	0
$247$	−3.06011e7	−0.129210
$248$	0	0
$249$	−5.67528e7	−0.232964
$250$	0	0
$251$	−3.10451e8	−1.23918	−0.619590	−	0.784925i	$-0.712701\pi$
$251$	−3.10451e8	−1.23918	−0.619590	+	0.784925i	$0.712701\pi$
$252$	0	0
$253$	3.89539e8	1.51227
$254$	0	0
$255$	3.66077e7	0.138255
$256$	0	0
$257$	2.45777e8	0.903181	0.451590	−	0.892225i	$-0.350857\pi$
$257$	2.45777e8	0.903181	0.451590	+	0.892225i	$0.350857\pi$
$258$	0	0
$259$	−2.49026e8	−0.890628
$260$	0	0
$261$	−3.72227e7	−0.129589
$262$	0	0
$263$	3.69594e8	1.25279	0.626397	−	0.779504i	$-0.284529\pi$
$263$	3.69594e8	1.25279	0.626397	+	0.779504i	$0.284529\pi$
$264$	0	0
$265$	−1.15210e8	−0.380302
$266$	0	0
$267$	1.41079e7	0.0453599
$268$	0	0
$269$	4.33145e8	1.35675	0.678375	−	0.734716i	$-0.262684\pi$
$269$	4.33145e8	1.35675	0.678375	+	0.734716i	$0.262684\pi$
$270$	0	0
$271$	−4.47319e8	−1.36529	−0.682644	−	0.730751i	$-0.739170\pi$
$271$	−4.47319e8	−1.36529	−0.682644	+	0.730751i	$0.739170\pi$
$272$	0	0
$273$	−1.50951e7	−0.0449020
$274$	0	0
$275$	2.07789e8	0.602501
$276$	0	0
$277$	3.49254e8	0.987329	0.493664	−	0.869653i	$-0.335657\pi$
$277$	3.49254e8	0.987329	0.493664	+	0.869653i	$0.335657\pi$
$278$	0	0
$279$	1.49455e8	0.411999
$280$	0	0
$281$	4.64823e8	1.24973	0.624864	−	0.780734i	$-0.285155\pi$
$281$	4.64823e8	1.24973	0.624864	+	0.780734i	$0.285155\pi$
$282$	0	0
$283$	3.59355e8	0.942478	0.471239	−	0.882005i	$-0.343807\pi$
$283$	3.59355e8	0.942478	0.471239	+	0.882005i	$0.343807\pi$
$284$	0	0
$285$	−2.30308e8	−0.589321
$286$	0	0
$287$	−6.48094e8	−1.61827
$288$	0	0
$289$	−3.38530e8	−0.825001
$290$	0	0
$291$	−3.67737e7	−0.0874807
$292$	0	0
$293$	−8.04828e8	−1.86925	−0.934623	−	0.355641i	$-0.884263\pi$
$293$	−8.04828e8	−1.86925	−0.934623	+	0.355641i	$0.884263\pi$
$294$	0	0
$295$	1.67033e8	0.378813
$296$	0	0
$297$	−7.78659e7	−0.172465
$298$	0	0
$299$	5.65206e7	0.122281
$300$	0	0
$301$	3.86522e8	0.816943
$302$	0	0
$303$	−2.76299e8	−0.570598
$304$	0	0
$305$	−3.28917e8	−0.663800
$306$	0	0
$307$	5.40976e8	1.06707	0.533536	−	0.845777i	$-0.320863\pi$
$307$	5.40976e8	1.06707	0.533536	+	0.845777i	$0.320863\pi$
$308$	0	0
$309$	−7.41331e7	−0.142941
$310$	0	0
$311$	2.47428e7	0.0466431	0.0233215	−	0.999728i	$-0.492576\pi$
$311$	2.47428e7	0.0466431	0.0233215	+	0.999728i	$0.492576\pi$
$312$	0	0
$313$	4.64343e7	0.0855920	0.0427960	−	0.999084i	$-0.486373\pi$
$313$	4.64343e7	0.0855920	0.0427960	+	0.999084i	$0.486373\pi$
$314$	0	0
$315$	−1.13607e8	−0.204795
$316$	0	0
$317$	3.10856e8	0.548089	0.274045	−	0.961717i	$-0.411638\pi$
$317$	3.10856e8	0.548089	0.274045	+	0.961717i	$0.411638\pi$
$318$	0	0
$319$	2.01993e8	0.348393
$320$	0	0
$321$	−1.59436e8	−0.269042
$322$	0	0
$323$	−4.51766e8	−0.745941
$324$	0	0
$325$	3.01494e7	0.0487176
$326$	0	0
$327$	6.71443e7	0.106192
$328$	0	0
$329$	5.35104e8	0.828424
$330$	0	0
$331$	9.20828e8	1.39566	0.697832	−	0.716262i	$-0.254148\pi$
$331$	9.20828e8	1.39566	0.697832	+	0.716262i	$0.254148\pi$
$332$	0	0
$333$	−1.86386e8	−0.276605
$334$	0	0
$335$	3.34824e8	0.486587
$336$	0	0
$337$	7.98845e8	1.13699	0.568497	−	0.822686i	$-0.307525\pi$
$337$	7.98845e8	1.13699	0.568497	+	0.822686i	$0.307525\pi$
$338$	0	0
$339$	−4.82160e8	−0.672190
$340$	0	0
$341$	−8.11035e8	−1.10764
$342$	0	0
$343$	−6.80251e8	−0.910207
$344$	0	0
$345$	4.25382e8	0.557715
$346$	0	0
$347$	2.45750e8	0.315748	0.157874	−	0.987459i	$-0.449536\pi$
$347$	2.45750e8	0.315748	0.157874	+	0.987459i	$0.449536\pi$
$348$	0	0
$349$	−4.91585e7	−0.0619027	−0.0309513	−	0.999521i	$-0.509854\pi$
$349$	−4.91585e7	−0.0619027	−0.0309513	+	0.999521i	$0.509854\pi$
$350$	0	0
$351$	−1.12980e7	−0.0139453
$352$	0	0
$353$	−1.10951e9	−1.34252	−0.671261	−	0.741221i	$-0.734247\pi$
$353$	−1.10951e9	−1.34252	−0.671261	+	0.741221i	$0.734247\pi$
$354$	0	0
$355$	4.35819e8	0.517019
$356$	0	0
$357$	−2.22849e8	−0.259222
$358$	0	0
$359$	−4.03877e8	−0.460700	−0.230350	−	0.973108i	$-0.573987\pi$
$359$	−4.03877e8	−0.460700	−0.230350	+	0.973108i	$0.573987\pi$
$360$	0	0
$361$	1.94830e9	2.17962
$362$	0	0
$363$	−1.03605e8	−0.113687
$364$	0	0
$365$	8.30402e8	0.893847
$366$	0	0
$367$	−3.56177e7	−0.0376127	−0.0188064	−	0.999823i	$-0.505987\pi$
$367$	−3.56177e7	−0.0376127	−0.0188064	+	0.999823i	$0.505987\pi$
$368$	0	0
$369$	−4.85072e8	−0.502590
$370$	0	0
$371$	7.01338e8	0.713048
$372$	0	0
$373$	−1.53631e9	−1.53284	−0.766422	−	0.642337i	$-0.777965\pi$
$373$	−1.53631e9	−1.53284	−0.766422	+	0.642337i	$0.777965\pi$
$374$	0	0
$375$	5.64408e8	0.552693
$376$	0	0
$377$	2.93084e7	0.0281707
$378$	0	0
$379$	−2.36572e8	−0.223217	−0.111608	−	0.993752i	$-0.535600\pi$
$379$	−2.36572e8	−0.223217	−0.111608	+	0.993752i	$0.535600\pi$
$380$	0	0
$381$	−7.38594e7	−0.0684177
$382$	0	0
$383$	8.94231e8	0.813306	0.406653	−	0.913583i	$-0.366696\pi$
$383$	8.94231e8	0.813306	0.406653	+	0.913583i	$0.366696\pi$
$384$	0	0
$385$	6.16503e8	0.550583
$386$	0	0
$387$	2.89296e8	0.253720
$388$	0	0
$389$	−2.22352e8	−0.191522	−0.0957608	−	0.995404i	$-0.530528\pi$
$389$	−2.22352e8	−0.191522	−0.0957608	+	0.995404i	$0.530528\pi$
$390$	0	0
$391$	8.34418e8	0.705935
$392$	0	0
$393$	8.26362e8	0.686747
$394$	0	0
$395$	−5.83538e8	−0.476408
$396$	0	0
$397$	8.54455e8	0.685366	0.342683	−	0.939451i	$-0.388664\pi$
$397$	8.54455e8	0.685366	0.342683	+	0.939451i	$0.388664\pi$
$398$	0	0
$399$	1.40200e9	1.10495
$400$	0	0
$401$	5.84757e8	0.452867	0.226433	−	0.974027i	$-0.427293\pi$
$401$	5.84757e8	0.452867	0.226433	+	0.974027i	$0.427293\pi$
$402$	0	0
$403$	−1.17678e8	−0.0895628
$404$	0	0
$405$	−8.50306e7	−0.0636037
$406$	0	0
$407$	1.01145e9	0.743639
$408$	0	0
$409$	2.33378e9	1.68666	0.843331	−	0.537395i	$-0.180592\pi$
$409$	2.33378e9	1.68666	0.843331	+	0.537395i	$0.180592\pi$
$410$	0	0
$411$	3.34975e8	0.237994
$412$	0	0
$413$	−1.01681e9	−0.710258
$414$	0	0
$415$	3.36313e8	0.230980
$416$	0	0
$417$	−1.17818e9	−0.795674
$418$	0	0
$419$	−2.83905e8	−0.188549	−0.0942744	−	0.995546i	$-0.530053\pi$
$419$	−2.83905e8	−0.188549	−0.0942744	+	0.995546i	$0.530053\pi$
$420$	0	0
$421$	1.76592e9	1.15341	0.576704	−	0.816953i	$-0.304339\pi$
$421$	1.76592e9	1.15341	0.576704	+	0.816953i	$0.304339\pi$
$422$	0	0
$423$	4.00504e8	0.257286
$424$	0	0
$425$	4.45097e8	0.281250
$426$	0	0
$427$	2.00228e9	1.24460
$428$	0	0
$429$	6.13101e7	0.0374914
$430$	0	0
$431$	1.86089e9	1.11956	0.559782	−	0.828640i	$-0.310885\pi$
$431$	1.86089e9	1.11956	0.559782	+	0.828640i	$0.310885\pi$
$432$	0	0
$433$	−1.51442e9	−0.896475	−0.448237	−	0.893915i	$-0.647948\pi$
$433$	−1.51442e9	−0.896475	−0.448237	+	0.893915i	$0.647948\pi$
$434$	0	0
$435$	2.20579e8	0.128485
$436$	0	0
$437$	−5.24953e9	−3.00909
$438$	0	0
$439$	−2.18093e9	−1.23032	−0.615158	−	0.788404i	$-0.710908\pi$
$439$	−2.18093e9	−1.23032	−0.615158	+	0.788404i	$0.710908\pi$
$440$	0	0
$441$	9.12220e7	0.0506482
$442$	0	0
$443$	2.66171e9	1.45461	0.727307	−	0.686312i	$-0.240772\pi$
$443$	2.66171e9	1.45461	0.727307	+	0.686312i	$0.240772\pi$
$444$	0	0
$445$	−8.36022e7	−0.0449736
$446$	0	0
$447$	1.07150e9	0.567432
$448$	0	0
$449$	−5.35177e8	−0.279020	−0.139510	−	0.990221i	$-0.544553\pi$
$449$	−5.35177e8	−0.279020	−0.139510	+	0.990221i	$0.544553\pi$
$450$	0	0
$451$	2.63230e9	1.35119
$452$	0	0
$453$	−1.46506e9	−0.740478
$454$	0	0
$455$	8.94522e7	0.0445196
$456$	0	0
$457$	3.07378e9	1.50649	0.753244	−	0.657741i	$-0.228488\pi$
$457$	3.07378e9	1.50649	0.753244	+	0.657741i	$0.228488\pi$
$458$	0	0
$459$	−1.66794e8	−0.0805073
$460$	0	0
$461$	−7.87759e8	−0.374490	−0.187245	−	0.982313i	$-0.559956\pi$
$461$	−7.87759e8	−0.374490	−0.187245	+	0.982313i	$0.559956\pi$
$462$	0	0
$463$	−7.31695e8	−0.342607	−0.171304	−	0.985218i	$-0.554798\pi$
$463$	−7.31695e8	−0.342607	−0.171304	+	0.985218i	$0.554798\pi$
$464$	0	0
$465$	−8.85660e8	−0.408490
$466$	0	0
$467$	3.07634e9	1.39774	0.698869	−	0.715250i	$-0.253687\pi$
$467$	3.07634e9	1.39774	0.698869	+	0.715250i	$0.253687\pi$
$468$	0	0
$469$	−2.03824e9	−0.912328
$470$	0	0
$471$	1.30880e9	0.577165
$472$	0	0
$473$	−1.56990e9	−0.682115
$474$	0	0
$475$	−2.80021e9	−1.19885
$476$	0	0
$477$	5.24924e8	0.221453
$478$	0	0
$479$	3.40125e9	1.41405	0.707024	−	0.707189i	$-0.250037\pi$
$479$	3.40125e9	1.41405	0.707024	+	0.707189i	$0.250037\pi$
$480$	0	0
$481$	1.46757e8	0.0601299
$482$	0	0
$483$	−2.58951e9	−1.04569
$484$	0	0
$485$	2.17918e8	0.0867357
$486$	0	0
$487$	2.20308e9	0.864329	0.432165	−	0.901795i	$-0.357750\pi$
$487$	2.20308e9	0.864329	0.432165	+	0.901795i	$0.357750\pi$
$488$	0	0
$489$	−2.26370e9	−0.875462
$490$	0	0
$491$	−3.67460e9	−1.40096	−0.700479	−	0.713673i	$-0.747030\pi$
$491$	−3.67460e9	−1.40096	−0.700479	+	0.713673i	$0.747030\pi$
$492$	0	0
$493$	4.32682e8	0.162632
$494$	0	0
$495$	4.61428e8	0.170996
$496$	0	0
$497$	−2.65305e9	−0.969388
$498$	0	0
$499$	2.82004e9	1.01602	0.508010	−	0.861351i	$-0.330381\pi$
$499$	2.82004e9	1.01602	0.508010	+	0.861351i	$0.330381\pi$
$500$	0	0
$501$	−1.49164e9	−0.529947
$502$	0	0
$503$	2.46294e9	0.862911	0.431456	−	0.902134i	$-0.358000\pi$
$503$	2.46294e9	0.862911	0.431456	+	0.902134i	$0.358000\pi$
$504$	0	0
$505$	1.63733e9	0.565739
$506$	0	0
$507$	−1.68531e9	−0.574319
$508$	0	0
$509$	8.44983e8	0.284011	0.142006	−	0.989866i	$-0.454645\pi$
$509$	8.44983e8	0.284011	0.142006	+	0.989866i	$0.454645\pi$
$510$	0	0
$511$	−5.05507e9	−1.67592
$512$	0	0
$513$	1.04934e9	0.343167
$514$	0	0
$515$	4.39307e8	0.141724
$516$	0	0
$517$	−2.17338e9	−0.691701
$518$	0	0
$519$	−3.91940e8	−0.123065
$520$	0	0
$521$	4.08230e9	1.26466	0.632328	−	0.774701i	$-0.282099\pi$
$521$	4.08230e9	1.26466	0.632328	+	0.774701i	$0.282099\pi$
$522$	0	0
$523$	4.38356e9	1.33990	0.669949	−	0.742407i	$-0.266316\pi$
$523$	4.38356e9	1.33990	0.669949	+	0.742407i	$0.266316\pi$
$524$	0	0
$525$	−1.38130e9	−0.416611
$526$	0	0
$527$	−1.73729e9	−0.517053
$528$	0	0
$529$	6.29112e9	1.84771
$530$	0	0
$531$	−7.61044e8	−0.220586
$532$	0	0
$533$	3.81936e8	0.109256
$534$	0	0
$535$	9.44808e8	0.266751
$536$	0	0
$537$	−1.77733e9	−0.495289
$538$	0	0
$539$	−4.95026e8	−0.136166
$540$	0	0
$541$	7.18661e9	1.95134	0.975671	−	0.219239i	$-0.0703573\pi$
$541$	7.18661e9	1.95134	0.975671	+	0.219239i	$0.0703573\pi$
$542$	0	0
$543$	−2.70886e9	−0.726085
$544$	0	0
$545$	−3.97892e8	−0.105288
$546$	0	0
$547$	2.29895e9	0.600585	0.300292	−	0.953847i	$-0.402916\pi$
$547$	2.29895e9	0.600585	0.300292	+	0.953847i	$0.402916\pi$
$548$	0	0
$549$	1.49863e9	0.386537
$550$	0	0
$551$	−2.72211e9	−0.693227
$552$	0	0
$553$	3.55229e9	0.893243
$554$	0	0
$555$	1.10451e9	0.274249
$556$	0	0
$557$	−2.30362e9	−0.564831	−0.282415	−	0.959292i	$-0.591136\pi$
$557$	−2.30362e9	−0.564831	−0.282415	+	0.959292i	$0.591136\pi$
$558$	0	0
$559$	−2.27786e8	−0.0551551
$560$	0	0
$561$	9.05125e8	0.216440
$562$	0	0
$563$	−6.41503e9	−1.51502	−0.757511	−	0.652822i	$-0.773585\pi$
$563$	−6.41503e9	−1.51502	−0.757511	+	0.652822i	$0.773585\pi$
$564$	0	0
$565$	2.85724e9	0.666466
$566$	0	0
$567$	5.17624e8	0.119254
$568$	0	0
$569$	3.92661e9	0.893563	0.446782	−	0.894643i	$-0.352570\pi$
$569$	3.92661e9	0.893563	0.446782	+	0.894643i	$0.352570\pi$
$570$	0	0
$571$	4.95437e9	1.11368	0.556842	−	0.830618i	$-0.312013\pi$
$571$	4.95437e9	1.11368	0.556842	+	0.830618i	$0.312013\pi$
$572$	0	0
$573$	−3.40297e9	−0.755644
$574$	0	0
$575$	5.17203e9	1.13455
$576$	0	0
$577$	−2.30307e9	−0.499105	−0.249553	−	0.968361i	$-0.580284\pi$
$577$	−2.30307e9	−0.499105	−0.249553	+	0.968361i	$0.580284\pi$
$578$	0	0
$579$	−2.90885e8	−0.0622796
$580$	0	0
$581$	−2.04731e9	−0.433078
$582$	0	0
$583$	−2.84856e9	−0.595367
$584$	0	0
$585$	6.69514e7	0.0138266
$586$	0	0
$587$	−7.91963e9	−1.61611	−0.808056	−	0.589106i	$-0.799480\pi$
$587$	−7.91963e9	−1.61611	−0.808056	+	0.589106i	$0.799480\pi$
$588$	0	0
$589$	1.09297e10	2.20397
$590$	0	0
$591$	4.93436e9	0.983275
$592$	0	0
$593$	−3.06920e9	−0.604412	−0.302206	−	0.953243i	$-0.597723\pi$
$593$	−3.06920e9	−0.604412	−0.302206	+	0.953243i	$0.597723\pi$
$594$	0	0
$595$	1.32059e9	0.257015
$596$	0	0
$597$	−4.61435e9	−0.887565
$598$	0	0
$599$	−4.83684e9	−0.919535	−0.459767	−	0.888039i	$-0.652067\pi$
$599$	−4.83684e9	−0.919535	−0.459767	+	0.888039i	$0.652067\pi$
$600$	0	0
$601$	−8.18969e9	−1.53889	−0.769444	−	0.638715i	$-0.779466\pi$
$601$	−8.18969e9	−1.53889	−0.769444	+	0.638715i	$0.779466\pi$
$602$	0	0
$603$	−1.52554e9	−0.283344
$604$	0	0
$605$	6.13958e8	0.112718
$606$	0	0
$607$	−4.60935e9	−0.836525	−0.418263	−	0.908326i	$-0.637361\pi$
$607$	−4.60935e9	−0.836525	−0.418263	+	0.908326i	$0.637361\pi$
$608$	0	0
$609$	−1.34278e9	−0.240903
$610$	0	0
$611$	−3.15349e8	−0.0559303
$612$	0	0
$613$	4.08038e9	0.715466	0.357733	−	0.933824i	$-0.383550\pi$
$613$	4.08038e9	0.715466	0.357733	+	0.933824i	$0.383550\pi$
$614$	0	0
$615$	2.87450e9	0.498310
$616$	0	0
$617$	−2.45151e9	−0.420180	−0.210090	−	0.977682i	$-0.567376\pi$
$617$	−2.45151e9	−0.420180	−0.210090	+	0.977682i	$0.567376\pi$
$618$	0	0
$619$	1.51355e9	0.256495	0.128248	−	0.991742i	$-0.459065\pi$
$619$	1.51355e9	0.256495	0.128248	+	0.991742i	$0.459065\pi$
$620$	0	0
$621$	−1.93815e9	−0.324762
$622$	0	0
$623$	5.08929e8	0.0843235
$624$	0	0
$625$	7.58876e8	0.124334
$626$	0	0
$627$	−5.69436e9	−0.922590
$628$	0	0
$629$	2.16658e9	0.347134
$630$	0	0
$631$	−1.14965e10	−1.82165	−0.910824	−	0.412794i	$-0.864553\pi$
$631$	−1.14965e10	−1.82165	−0.910824	+	0.412794i	$0.864553\pi$
$632$	0	0
$633$	−2.10651e9	−0.330104
$634$	0	0
$635$	4.37685e8	0.0678350
$636$	0	0
$637$	−7.18263e7	−0.0110102
$638$	0	0
$639$	−1.98570e9	−0.301065
$640$	0	0
$641$	−7.49495e9	−1.12400	−0.561999	−	0.827138i	$-0.689968\pi$
$641$	−7.49495e9	−1.12400	−0.561999	+	0.827138i	$0.689968\pi$
$642$	0	0
$643$	4.19179e9	0.621815	0.310907	−	0.950440i	$-0.399367\pi$
$643$	4.19179e9	0.621815	0.310907	+	0.950440i	$0.399367\pi$
$644$	0	0
$645$	−1.71435e9	−0.251559
$646$	0	0
$647$	8.05085e8	0.116863	0.0584314	−	0.998291i	$-0.481390\pi$
$647$	8.05085e8	0.116863	0.0584314	+	0.998291i	$0.481390\pi$
$648$	0	0
$649$	4.12989e9	0.593037
$650$	0	0
$651$	5.39146e9	0.765901
$652$	0	0
$653$	2.95775e9	0.415685	0.207843	−	0.978162i	$-0.433356\pi$
$653$	2.95775e9	0.415685	0.207843	+	0.978162i	$0.433356\pi$
$654$	0	0
$655$	−4.89696e9	−0.680898
$656$	0	0
$657$	−3.78352e9	−0.520495
$658$	0	0
$659$	−2.41602e9	−0.328853	−0.164426	−	0.986389i	$-0.552577\pi$
$659$	−2.41602e9	−0.328853	−0.164426	+	0.986389i	$0.552577\pi$
$660$	0	0
$661$	6.28842e9	0.846908	0.423454	−	0.905918i	$-0.360817\pi$
$661$	6.28842e9	0.846908	0.423454	+	0.905918i	$0.360817\pi$
$662$	0	0
$663$	1.31330e8	0.0175012
$664$	0	0
$665$	−8.30814e9	−1.09554
$666$	0	0
$667$	5.02778e9	0.656048
$668$	0	0
$669$	1.44780e9	0.186947
$670$	0	0
$671$	−8.13248e9	−1.03919
$672$	0	0
$673$	3.51073e9	0.443961	0.221981	−	0.975051i	$-0.428748\pi$
$673$	3.51073e9	0.443961	0.221981	+	0.975051i	$0.428748\pi$
$674$	0	0
$675$	−1.03385e9	−0.129388
$676$	0	0
$677$	1.36316e10	1.68845	0.844223	−	0.535992i	$-0.180062\pi$
$677$	1.36316e10	1.68845	0.844223	+	0.535992i	$0.180062\pi$
$678$	0	0
$679$	−1.32658e9	−0.162626
$680$	0	0
$681$	−5.75346e9	−0.698095
$682$	0	0
$683$	−1.63949e9	−0.196896	−0.0984480	−	0.995142i	$-0.531388\pi$
$683$	−1.63949e9	−0.196896	−0.0984480	+	0.995142i	$0.531388\pi$
$684$	0	0
$685$	−1.98504e9	−0.235967
$686$	0	0
$687$	−3.13759e9	−0.369188
$688$	0	0
$689$	−4.13314e8	−0.0481408
$690$	0	0
$691$	−8.86591e9	−1.02223	−0.511117	−	0.859511i	$-0.670768\pi$
$691$	−8.86591e9	−1.02223	−0.511117	+	0.859511i	$0.670768\pi$
$692$	0	0
$693$	−2.80894e9	−0.320610
$694$	0	0
$695$	6.98180e9	0.788898
$696$	0	0
$697$	5.63855e9	0.630743
$698$	0	0
$699$	−7.81312e9	−0.865276
$700$	0	0
$701$	−9.27328e9	−1.01676	−0.508382	−	0.861131i	$-0.669756\pi$
$701$	−9.27328e9	−1.01676	−0.508382	+	0.861131i	$0.669756\pi$
$702$	0	0
$703$	−1.36305e10	−1.47968
$704$	0	0
$705$	−2.37336e9	−0.255094
$706$	0	0
$707$	−9.96723e9	−1.06073
$708$	0	0
$709$	1.79094e10	1.88720	0.943602	−	0.331082i	$-0.107414\pi$
$709$	1.79094e10	1.88720	0.943602	+	0.331082i	$0.107414\pi$
$710$	0	0
$711$	2.65874e9	0.277417
$712$	0	0
$713$	−2.01873e10	−2.08576
$714$	0	0
$715$	−3.63319e8	−0.0371721
$716$	0	0
$717$	6.48317e9	0.656856
$718$	0	0
$719$	1.36386e10	1.36842	0.684208	−	0.729287i	$-0.260148\pi$
$719$	1.36386e10	1.36842	0.684208	+	0.729287i	$0.260148\pi$
$720$	0	0
$721$	−2.67428e9	−0.265726
$722$	0	0
$723$	−1.11702e10	−1.09920
$724$	0	0
$725$	2.68193e9	0.261375
$726$	0	0
$727$	1.03631e10	1.00027	0.500137	−	0.865947i	$-0.333283\pi$
$727$	1.03631e10	1.00027	0.500137	+	0.865947i	$0.333283\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−3.36282e9	−0.318415
$732$	0	0
$733$	−1.05901e10	−0.993196	−0.496598	−	0.867981i	$-0.665418\pi$
$733$	−1.05901e10	−0.993196	−0.496598	+	0.867981i	$0.665418\pi$
$734$	0	0
$735$	−5.40575e8	−0.0502169
$736$	0	0
$737$	8.27853e9	0.761758
$738$	0	0
$739$	−1.05206e10	−0.958929	−0.479465	−	0.877561i	$-0.659169\pi$
$739$	−1.05206e10	−0.958929	−0.479465	+	0.877561i	$0.659169\pi$
$740$	0	0
$741$	−8.26229e8	−0.0745997
$742$	0	0
$743$	−7.49764e9	−0.670600	−0.335300	−	0.942111i	$-0.608838\pi$
$743$	−7.49764e9	−0.670600	−0.335300	+	0.942111i	$0.608838\pi$
$744$	0	0
$745$	−6.34960e9	−0.562600
$746$	0	0
$747$	−1.53233e9	−0.134502
$748$	0	0
$749$	−5.75152e9	−0.500145
$750$	0	0
$751$	−9.53049e9	−0.821061	−0.410531	−	0.911847i	$-0.634657\pi$
$751$	−9.53049e9	−0.821061	−0.410531	+	0.911847i	$0.634657\pi$
$752$	0	0
$753$	−8.38217e9	−0.715441
$754$	0	0
$755$	8.68184e9	0.734172
$756$	0	0
$757$	−8.93979e8	−0.0749017	−0.0374509	−	0.999298i	$-0.511924\pi$
$757$	−8.93979e8	−0.0749017	−0.0374509	+	0.999298i	$0.511924\pi$
$758$	0	0
$759$	1.05176e10	0.873110
$760$	0	0
$761$	1.36522e10	1.12294	0.561471	−	0.827496i	$-0.310236\pi$
$761$	1.36522e10	1.12294	0.561471	+	0.827496i	$0.310236\pi$
$762$	0	0
$763$	2.42217e9	0.197410
$764$	0	0
$765$	9.88407e8	0.0798217
$766$	0	0
$767$	5.99231e8	0.0479524
$768$	0	0
$769$	1.45747e10	1.15573	0.577865	−	0.816132i	$-0.303886\pi$
$769$	1.45747e10	1.15573	0.577865	+	0.816132i	$0.303886\pi$
$770$	0	0
$771$	6.63597e9	0.521452
$772$	0	0
$773$	−4.53526e9	−0.353162	−0.176581	−	0.984286i	$-0.556504\pi$
$773$	−4.53526e9	−0.353162	−0.176581	+	0.984286i	$0.556504\pi$
$774$	0	0
$775$	−1.07684e10	−0.830986
$776$	0	0
$777$	−6.72371e9	−0.514204
$778$	0	0
$779$	−3.54735e10	−2.68858
$780$	0	0
$781$	1.07756e10	0.809401
$782$	0	0
$783$	−1.00501e9	−0.0748180
$784$	0	0
$785$	−7.75585e9	−0.572250
$786$	0	0
$787$	−8.62144e9	−0.630475	−0.315238	−	0.949013i	$-0.602084\pi$
$787$	−8.62144e9	−0.630475	−0.315238	+	0.949013i	$0.602084\pi$
$788$	0	0
$789$	9.97904e9	0.723301
$790$	0	0
$791$	−1.73935e10	−1.24959
$792$	0	0
$793$	−1.17999e9	−0.0840277
$794$	0	0
$795$	−3.11066e9	−0.219567
$796$	0	0
$797$	−1.05785e10	−0.740153	−0.370076	−	0.929001i	$-0.620669\pi$
$797$	−1.05785e10	−0.740153	−0.370076	+	0.929001i	$0.620669\pi$
$798$	0	0
$799$	−4.65551e9	−0.322889
$800$	0	0
$801$	3.80913e8	0.0261886
$802$	0	0
$803$	2.05317e10	1.39933
$804$	0	0
$805$	1.53453e10	1.03678
$806$	0	0
$807$	1.16949e10	0.783320
$808$	0	0
$809$	−2.53892e10	−1.68589	−0.842946	−	0.537998i	$-0.819181\pi$
$809$	−2.53892e10	−1.68589	−0.842946	+	0.537998i	$0.819181\pi$
$810$	0	0
$811$	−3.15391e9	−0.207623	−0.103812	−	0.994597i	$-0.533104\pi$
$811$	−3.15391e9	−0.207623	−0.103812	+	0.994597i	$0.533104\pi$
$812$	0	0
$813$	−1.20776e10	−0.788250
$814$	0	0
$815$	1.34145e10	0.868007
$816$	0	0
$817$	2.11563e10	1.35726
$818$	0	0
$819$	−4.07566e8	−0.0259242
$820$	0	0
$821$	−1.06670e10	−0.672731	−0.336366	−	0.941732i	$-0.609198\pi$
$821$	−1.06670e10	−0.672731	−0.336366	+	0.941732i	$0.609198\pi$
$822$	0	0
$823$	−2.08019e10	−1.30078	−0.650389	−	0.759601i	$-0.725394\pi$
$823$	−2.08019e10	−1.30078	−0.650389	+	0.759601i	$0.725394\pi$
$824$	0	0
$825$	5.61030e9	0.347854
$826$	0	0
$827$	−5.02808e9	−0.309124	−0.154562	−	0.987983i	$-0.549397\pi$
$827$	−5.02808e9	−0.309124	−0.154562	+	0.987983i	$0.549397\pi$
$828$	0	0
$829$	−7.43519e9	−0.453264	−0.226632	−	0.973980i	$-0.572771\pi$
$829$	−7.43519e9	−0.453264	−0.226632	+	0.973980i	$0.572771\pi$
$830$	0	0
$831$	9.42985e9	0.570035
$832$	0	0
$833$	−1.06038e9	−0.0635628
$834$	0	0
$835$	8.83935e9	0.525434
$836$	0	0
$837$	4.03529e9	0.237868
$838$	0	0
$839$	1.83882e10	1.07491	0.537455	−	0.843293i	$-0.319386\pi$
$839$	1.83882e10	1.07491	0.537455	+	0.843293i	$0.319386\pi$
$840$	0	0
$841$	−1.46428e10	−0.848861
$842$	0	0
$843$	1.25502e10	0.721530
$844$	0	0
$845$	9.98705e9	0.569428
$846$	0	0
$847$	−3.73747e9	−0.211342
$848$	0	0
$849$	9.70259e9	0.544140
$850$	0	0
$851$	2.51757e10	1.40032
$852$	0	0
$853$	1.66363e10	0.917772	0.458886	−	0.888495i	$-0.348249\pi$
$853$	1.66363e10	0.917772	0.458886	+	0.888495i	$0.348249\pi$
$854$	0	0
$855$	−6.21831e9	−0.340244
$856$	0	0
$857$	−2.03068e10	−1.10207	−0.551034	−	0.834483i	$-0.685767\pi$
$857$	−2.03068e10	−1.10207	−0.551034	+	0.834483i	$0.685767\pi$
$858$	0	0
$859$	5.78282e9	0.311289	0.155644	−	0.987813i	$-0.450255\pi$
$859$	5.78282e9	0.311289	0.155644	+	0.987813i	$0.450255\pi$
$860$	0	0
$861$	−1.74985e10	−0.934309
$862$	0	0
$863$	−1.50813e10	−0.798732	−0.399366	−	0.916792i	$-0.630770\pi$
$863$	−1.50813e10	−0.798732	−0.399366	+	0.916792i	$0.630770\pi$
$864$	0	0
$865$	2.32261e9	0.122017
$866$	0	0
$867$	−9.14031e9	−0.476315
$868$	0	0
$869$	−1.44280e10	−0.745823
$870$	0	0
$871$	1.20118e9	0.0615950
$872$	0	0
$873$	−9.92891e8	−0.0505070
$874$	0	0
$875$	2.03605e10	1.02745
$876$	0	0
$877$	−1.75861e10	−0.880382	−0.440191	−	0.897904i	$-0.645089\pi$
$877$	−1.75861e10	−0.880382	−0.440191	+	0.897904i	$0.645089\pi$
$878$	0	0
$879$	−2.17303e10	−1.07921
$880$	0	0
$881$	1.78503e10	0.879488	0.439744	−	0.898123i	$-0.355069\pi$
$881$	1.78503e10	0.879488	0.439744	+	0.898123i	$0.355069\pi$
$882$	0	0
$883$	7.69013e9	0.375899	0.187950	−	0.982179i	$-0.439816\pi$
$883$	7.69013e9	0.375899	0.187950	+	0.982179i	$0.439816\pi$
$884$	0	0
$885$	4.50989e9	0.218708
$886$	0	0
$887$	−3.82390e10	−1.83981	−0.919906	−	0.392139i	$-0.871735\pi$
$887$	−3.82390e10	−1.83981	−0.919906	+	0.392139i	$0.871735\pi$
$888$	0	0
$889$	−2.66441e9	−0.127188
$890$	0	0
$891$	−2.10238e9	−0.0995725
$892$	0	0
$893$	2.92890e10	1.37633
$894$	0	0
$895$	1.05323e10	0.491071
$896$	0	0
$897$	1.52606e9	0.0705988
$898$	0	0
$899$	−1.04680e10	−0.480513
$900$	0	0
$901$	−6.10179e9	−0.277920
$902$	0	0
$903$	1.04361e10	0.471662
$904$	0	0
$905$	1.60525e10	0.719901
$906$	0	0
$907$	4.12743e10	1.83677	0.918384	−	0.395690i	$-0.129494\pi$
$907$	4.12743e10	1.83677	0.918384	+	0.395690i	$0.129494\pi$
$908$	0	0
$909$	−7.46008e9	−0.329435
$910$	0	0
$911$	3.62507e10	1.58855	0.794277	−	0.607556i	$-0.207850\pi$
$911$	3.62507e10	1.58855	0.794277	+	0.607556i	$0.207850\pi$
$912$	0	0
$913$	8.31534e9	0.361603
$914$	0	0
$915$	−8.88077e9	−0.383245
$916$	0	0
$917$	2.98103e10	1.27665
$918$	0	0
$919$	4.06875e10	1.72924	0.864622	−	0.502423i	$-0.167558\pi$
$919$	4.06875e10	1.72924	0.864622	+	0.502423i	$0.167558\pi$
$920$	0	0
$921$	1.46063e10	0.616074
$922$	0	0
$923$	1.56350e9	0.0654473
$924$	0	0
$925$	1.34293e10	0.557900
$926$	0	0
$927$	−2.00159e9	−0.0825271
$928$	0	0
$929$	3.54623e10	1.45115	0.725574	−	0.688145i	$-0.241575\pi$
$929$	3.54623e10	1.45115	0.725574	+	0.688145i	$0.241575\pi$
$930$	0	0
$931$	6.67109e9	0.270940
$932$	0	0
$933$	6.68055e8	0.0269294
$934$	0	0
$935$	−5.36370e9	−0.214597
$936$	0	0
$937$	2.42277e10	0.962109	0.481055	−	0.876691i	$-0.340254\pi$
$937$	2.42277e10	0.962109	0.481055	+	0.876691i	$0.340254\pi$
$938$	0	0
$939$	1.25373e9	0.0494166
$940$	0	0
$941$	−4.07396e10	−1.59387	−0.796936	−	0.604064i	$-0.793547\pi$
$941$	−4.07396e10	−1.59387	−0.796936	+	0.604064i	$0.793547\pi$
$942$	0	0
$943$	6.55200e10	2.54439
$944$	0	0
$945$	−3.06740e9	−0.118238
$946$	0	0
$947$	1.04582e10	0.400157	0.200078	−	0.979780i	$-0.435880\pi$
$947$	1.04582e10	0.400157	0.200078	+	0.979780i	$0.435880\pi$
$948$	0	0
$949$	2.97907e9	0.113148
$950$	0	0
$951$	8.39310e9	0.316439
$952$	0	0
$953$	−1.91026e10	−0.714938	−0.357469	−	0.933925i	$-0.616360\pi$
$953$	−1.91026e10	−0.714938	−0.357469	+	0.933925i	$0.616360\pi$
$954$	0	0
$955$	2.01658e10	0.749209
$956$	0	0
$957$	5.45382e9	0.201145
$958$	0	0
$959$	1.20839e10	0.442428
$960$	0	0
$961$	1.45181e10	0.527690
$962$	0	0
$963$	−4.30478e9	−0.155331
$964$	0	0
$965$	1.72376e9	0.0617492
$966$	0	0
$967$	1.34833e10	0.479516	0.239758	−	0.970833i	$-0.422932\pi$
$967$	1.34833e10	0.479516	0.239758	+	0.970833i	$0.422932\pi$
$968$	0	0
$969$	−1.21977e10	−0.430669
$970$	0	0
$971$	−4.85424e10	−1.70159	−0.850793	−	0.525501i	$-0.823878\pi$
$971$	−4.85424e10	−1.70159	−0.850793	+	0.525501i	$0.823878\pi$
$972$	0	0
$973$	−4.25017e10	−1.47915
$974$	0	0
$975$	8.14032e8	0.0281271
$976$	0	0
$977$	3.14306e10	1.07825	0.539127	−	0.842224i	$-0.318754\pi$
$977$	3.14306e10	1.07825	0.539127	+	0.842224i	$0.318754\pi$
$978$	0	0
$979$	−2.06707e9	−0.0704068
$980$	0	0
$981$	1.81290e9	0.0613100
$982$	0	0
$983$	−2.14594e10	−0.720576	−0.360288	−	0.932841i	$-0.617322\pi$
$983$	−2.14594e10	−0.720576	−0.360288	+	0.932841i	$0.617322\pi$
$984$	0	0
$985$	−2.92407e10	−0.974901
$986$	0	0
$987$	1.44478e10	0.478291
$988$	0	0
$989$	−3.90760e10	−1.28447
$990$	0	0
$991$	−3.50008e10	−1.14241	−0.571203	−	0.820809i	$-0.693523\pi$
$991$	−3.50008e10	−1.14241	−0.571203	+	0.820809i	$0.693523\pi$
$992$	0	0
$993$	2.48624e10	0.805787
$994$	0	0
$995$	2.73443e10	0.880006
$996$	0	0
$997$	−3.12651e10	−0.999141	−0.499571	−	0.866273i	$-0.666509\pi$
$997$	−3.12651e10	−0.999141	−0.499571	+	0.866273i	$0.666509\pi$
$998$	0	0
$999$	−5.03243e9	−0.159698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.c.1.1	yes	1
4.3	odd	2		384.8.a.a.1.1	✓	1
8.3	odd	2		384.8.a.d.1.1	yes	1
8.5	even	2		384.8.a.b.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.a.1.1	✓	1	4.3	odd	2
384.8.a.b.1.1	yes	1	8.5	even	2
384.8.a.c.1.1	yes	1	1.1	even	1	trivial
384.8.a.d.1.1	yes	1	8.3	odd	2

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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