LMFDB - Embedded newform 378.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,8,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	378.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:	$118.081539633$
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	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +64.0000 q^{4} +297.000 q^{5} +343.000 q^{7} +512.000 q^{8} +O(q^{10})$

\(q+8.00000 q^{2} +64.0000 q^{4} +297.000 q^{5} +343.000 q^{7} +512.000 q^{8} +2376.00 q^{10} -378.000 q^{11} -4540.00 q^{13} +2744.00 q^{14} +4096.00 q^{16} -21603.0 q^{17} -43306.0 q^{19} +19008.0 q^{20} -3024.00 q^{22} -86094.0 q^{23} +10084.0 q^{25} -36320.0 q^{26} +21952.0 q^{28} -21570.0 q^{29} -298948. q^{31} +32768.0 q^{32} -172824. q^{34} +101871. q^{35} +452117. q^{37} -346448. q^{38} +152064. q^{40} -803109. q^{41} +201293. q^{43} -24192.0 q^{44} -688752. q^{46} +411081. q^{47} +117649. q^{49} +80672.0 q^{50} -290560. q^{52} -1.28383e6 q^{53} -112266. q^{55} +175616. q^{56} -172560. q^{58} +2.62802e6 q^{59} -3.25887e6 q^{61} -2.39158e6 q^{62} +262144. q^{64} -1.34838e6 q^{65} +4.15879e6 q^{67} -1.38259e6 q^{68} +814968. q^{70} -1.88928e6 q^{71} +2.20947e6 q^{73} +3.61694e6 q^{74} -2.77158e6 q^{76} -129654. q^{77} -2.47866e6 q^{79} +1.21651e6 q^{80} -6.42487e6 q^{82} -472785. q^{83} -6.41609e6 q^{85} +1.61034e6 q^{86} -193536. q^{88} +9.46151e6 q^{89} -1.55722e6 q^{91} -5.51002e6 q^{92} +3.28865e6 q^{94} -1.28619e7 q^{95} +1.37562e7 q^{97} +941192. q^{98} +O(q^{100})\)

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
$5$	297.000	1.06258	0.531290	−	0.847190i	$-0.321708\pi$
$5$	297.000	1.06258	0.531290	+	0.847190i	$0.321708\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	512.000	0.353553
$9$	0	0
$10$	2376.00	0.751357
$11$	−378.000	−0.0856283	−0.0428142	−	0.999083i	$-0.513632\pi$
$11$	−378.000	−0.0856283	−0.0428142	+	0.999083i	$0.513632\pi$
$12$	0	0
$13$	−4540.00	−0.573131	−0.286566	−	0.958061i	$-0.592514\pi$
$13$	−4540.00	−0.573131	−0.286566	+	0.958061i	$0.592514\pi$
$14$	2744.00	0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	−21603.0	−1.06646	−0.533228	−	0.845972i	$-0.679021\pi$
$17$	−21603.0	−1.06646	−0.533228	+	0.845972i	$0.679021\pi$
$18$	0	0
$19$	−43306.0	−1.44847	−0.724237	−	0.689552i	$-0.757808\pi$
$19$	−43306.0	−1.44847	−0.724237	+	0.689552i	$0.757808\pi$
$20$	19008.0	0.531290
$21$	0	0
$22$	−3024.00	−0.0605484
$23$	−86094.0	−1.47545	−0.737727	−	0.675100i	$-0.764101\pi$
$23$	−86094.0	−1.47545	−0.737727	+	0.675100i	$0.764101\pi$
$24$	0	0
$25$	10084.0	0.129075
$26$	−36320.0	−0.405265
$27$	0	0
$28$	21952.0	0.188982
$29$	−21570.0	−0.164232	−0.0821159	−	0.996623i	$-0.526168\pi$
$29$	−21570.0	−0.164232	−0.0821159	+	0.996623i	$0.526168\pi$
$30$	0	0
$31$	−298948.	−1.80231	−0.901155	−	0.433496i	$-0.857280\pi$
$31$	−298948.	−1.80231	−0.901155	+	0.433496i	$0.857280\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	−172824.	−0.754098
$35$	101871.	0.401617
$36$	0	0
$37$	452117.	1.46739	0.733694	−	0.679480i	$-0.237795\pi$
$37$	452117.	1.46739	0.733694	+	0.679480i	$0.237795\pi$
$38$	−346448.	−1.02423
$39$	0	0
$40$	152064.	0.375679
$41$	−803109.	−1.81983	−0.909915	−	0.414794i	$-0.863854\pi$
$41$	−803109.	−1.81983	−0.909915	+	0.414794i	$0.863854\pi$
$42$	0	0
$43$	201293.	0.386090	0.193045	−	0.981190i	$-0.438164\pi$
$43$	201293.	0.386090	0.193045	+	0.981190i	$0.438164\pi$
$44$	−24192.0	−0.0428142
$45$	0	0
$46$	−688752.	−1.04330
$47$	411081.	0.577544	0.288772	−	0.957398i	$-0.406753\pi$
$47$	411081.	0.577544	0.288772	+	0.957398i	$0.406753\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	80672.0	0.0912699
$51$	0	0
$52$	−290560.	−0.286566
$53$	−1.28383e6	−1.18452	−0.592258	−	0.805749i	$-0.701763\pi$
$53$	−1.28383e6	−1.18452	−0.592258	+	0.805749i	$0.701763\pi$
$54$	0	0
$55$	−112266.	−0.0909869
$56$	175616.	0.133631
$57$	0	0
$58$	−172560.	−0.116129
$59$	2.62802e6	1.66589	0.832946	−	0.553354i	$-0.186652\pi$
$59$	2.62802e6	1.66589	0.832946	+	0.553354i	$0.186652\pi$
$60$	0	0
$61$	−3.25887e6	−1.83829	−0.919143	−	0.393924i	$-0.871117\pi$
$61$	−3.25887e6	−1.83829	−0.919143	+	0.393924i	$0.871117\pi$
$62$	−2.39158e6	−1.27443
$63$	0	0
$64$	262144.	0.125000
$65$	−1.34838e6	−0.608998
$66$	0	0
$67$	4.15879e6	1.68929	0.844646	−	0.535325i	$-0.179811\pi$
$67$	4.15879e6	1.68929	0.844646	+	0.535325i	$0.179811\pi$
$68$	−1.38259e6	−0.533228
$69$	0	0
$70$	814968.	0.283986
$71$	−1.88928e6	−0.626458	−0.313229	−	0.949678i	$-0.601411\pi$
$71$	−1.88928e6	−0.626458	−0.313229	+	0.949678i	$0.601411\pi$
$72$	0	0
$73$	2.20947e6	0.664748	0.332374	−	0.943148i	$-0.392150\pi$
$73$	2.20947e6	0.664748	0.332374	+	0.943148i	$0.392150\pi$
$74$	3.61694e6	1.03760
$75$	0	0
$76$	−2.77158e6	−0.724237
$77$	−129654.	−0.0323645
$78$	0	0
$79$	−2.47866e6	−0.565615	−0.282808	−	0.959177i	$-0.591266\pi$
$79$	−2.47866e6	−0.565615	−0.282808	+	0.959177i	$0.591266\pi$
$80$	1.21651e6	0.265645
$81$	0	0
$82$	−6.42487e6	−1.28681
$83$	−472785.	−0.0907591	−0.0453796	−	0.998970i	$-0.514450\pi$
$83$	−472785.	−0.0907591	−0.0453796	+	0.998970i	$0.514450\pi$
$84$	0	0
$85$	−6.41609e6	−1.13319
$86$	1.61034e6	0.273007
$87$	0	0
$88$	−193536.	−0.0302742
$89$	9.46151e6	1.42264	0.711321	−	0.702867i	$-0.248097\pi$
$89$	9.46151e6	1.42264	0.711321	+	0.702867i	$0.248097\pi$
$90$	0	0
$91$	−1.55722e6	−0.216623
$92$	−5.51002e6	−0.737727
$93$	0	0
$94$	3.28865e6	0.408385
$95$	−1.28619e7	−1.53912
$96$	0	0
$97$	1.37562e7	1.53037	0.765185	−	0.643811i	$-0.222648\pi$
$97$	1.37562e7	1.53037	0.765185	+	0.643811i	$0.222648\pi$
$98$	941192.	0.101015
$99$	0	0
$100$	645376.	0.0645376
$101$	−1.44713e7	−1.39760	−0.698798	−	0.715319i	$-0.746281\pi$
$101$	−1.44713e7	−1.39760	−0.698798	+	0.715319i	$0.746281\pi$
$102$	0	0
$103$	6.40740e6	0.577766	0.288883	−	0.957364i	$-0.406716\pi$
$103$	6.40740e6	0.577766	0.288883	+	0.957364i	$0.406716\pi$
$104$	−2.32448e6	−0.202633
$105$	0	0
$106$	−1.02706e7	−0.837579
$107$	2.89911e6	0.228782	0.114391	−	0.993436i	$-0.463508\pi$
$107$	2.89911e6	0.228782	0.114391	+	0.993436i	$0.463508\pi$
$108$	0	0
$109$	7.85733e6	0.581142	0.290571	−	0.956853i	$-0.406155\pi$
$109$	7.85733e6	0.581142	0.290571	+	0.956853i	$0.406155\pi$
$110$	−898128.	−0.0643374
$111$	0	0
$112$	1.40493e6	0.0944911
$113$	1.42130e7	0.926641	0.463320	−	0.886191i	$-0.346658\pi$
$113$	1.42130e7	0.926641	0.463320	+	0.886191i	$0.346658\pi$
$114$	0	0
$115$	−2.55699e7	−1.56779
$116$	−1.38048e6	−0.0821159
$117$	0	0
$118$	2.10242e7	1.17796
$119$	−7.40983e6	−0.403082
$120$	0	0
$121$	−1.93443e7	−0.992668
$122$	−2.60710e7	−1.29986
$123$	0	0
$124$	−1.91327e7	−0.901155
$125$	−2.02082e7	−0.925427
$126$	0	0
$127$	1.44865e7	0.627554	0.313777	−	0.949497i	$-0.398406\pi$
$127$	1.44865e7	0.627554	0.313777	+	0.949497i	$0.398406\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	−1.07870e7	−0.430626
$131$	−1.60153e7	−0.622422	−0.311211	−	0.950341i	$-0.600735\pi$
$131$	−1.60153e7	−0.622422	−0.311211	+	0.950341i	$0.600735\pi$
$132$	0	0
$133$	−1.48540e7	−0.547471
$134$	3.32703e7	1.19451
$135$	0	0
$136$	−1.10607e7	−0.377049
$137$	5.55124e6	0.184445	0.0922227	−	0.995738i	$-0.470603\pi$
$137$	5.55124e6	0.184445	0.0922227	+	0.995738i	$0.470603\pi$
$138$	0	0
$139$	−4.53146e7	−1.43115	−0.715576	−	0.698535i	$-0.753836\pi$
$139$	−4.53146e7	−1.43115	−0.715576	+	0.698535i	$0.753836\pi$
$140$	6.51974e6	0.200809
$141$	0	0
$142$	−1.51142e7	−0.442973
$143$	1.71612e6	0.0490763
$144$	0	0
$145$	−6.40629e6	−0.174509
$146$	1.76757e7	0.470048
$147$	0	0
$148$	2.89355e7	0.733694
$149$	−1.22577e7	−0.303568	−0.151784	−	0.988414i	$-0.548502\pi$
$149$	−1.22577e7	−0.303568	−0.151784	+	0.988414i	$0.548502\pi$
$150$	0	0
$151$	−3.73113e7	−0.881904	−0.440952	−	0.897531i	$-0.645359\pi$
$151$	−3.73113e7	−0.881904	−0.440952	+	0.897531i	$0.645359\pi$
$152$	−2.21727e7	−0.512113
$153$	0	0
$154$	−1.03723e6	−0.0228851
$155$	−8.87876e7	−1.91510
$156$	0	0
$157$	−4.46267e7	−0.920334	−0.460167	−	0.887832i	$-0.652211\pi$
$157$	−4.46267e7	−0.920334	−0.460167	+	0.887832i	$0.652211\pi$
$158$	−1.98292e7	−0.399951
$159$	0	0
$160$	9.73210e6	0.187839
$161$	−2.95302e7	−0.557669
$162$	0	0
$163$	−4.51494e7	−0.816573	−0.408287	−	0.912854i	$-0.633874\pi$
$163$	−4.51494e7	−0.816573	−0.408287	+	0.912854i	$0.633874\pi$
$164$	−5.13990e7	−0.909915
$165$	0	0
$166$	−3.78228e6	−0.0641764
$167$	8.82797e7	1.46674	0.733370	−	0.679830i	$-0.237946\pi$
$167$	8.82797e7	1.46674	0.733370	+	0.679830i	$0.237946\pi$
$168$	0	0
$169$	−4.21369e7	−0.671521
$170$	−5.13287e7	−0.801289
$171$	0	0
$172$	1.28828e7	0.193045
$173$	1.05206e8	1.54483	0.772415	−	0.635118i	$-0.219048\pi$
$173$	1.05206e8	1.54483	0.772415	+	0.635118i	$0.219048\pi$
$174$	0	0
$175$	3.45881e6	0.0487858
$176$	−1.54829e6	−0.0214071
$177$	0	0
$178$	7.56921e7	1.00596
$179$	6.35782e7	0.828558	0.414279	−	0.910150i	$-0.364034\pi$
$179$	6.35782e7	0.828558	0.414279	+	0.910150i	$0.364034\pi$
$180$	0	0
$181$	−5.83582e7	−0.731521	−0.365761	−	0.930709i	$-0.619191\pi$
$181$	−5.83582e7	−0.731521	−0.365761	+	0.930709i	$0.619191\pi$
$182$	−1.24578e7	−0.153176
$183$	0	0
$184$	−4.40801e7	−0.521652
$185$	1.34279e8	1.55922
$186$	0	0
$187$	8.16593e6	0.0913188
$188$	2.63092e7	0.288772
$189$	0	0
$190$	−1.02895e8	−1.08832
$191$	4.62333e7	0.480107	0.240054	−	0.970760i	$-0.422835\pi$
$191$	4.62333e7	0.480107	0.240054	+	0.970760i	$0.422835\pi$
$192$	0	0
$193$	−1.31889e8	−1.32056	−0.660278	−	0.751021i	$-0.729561\pi$
$193$	−1.31889e8	−1.32056	−0.660278	+	0.751021i	$0.729561\pi$
$194$	1.10049e8	1.08213
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	8.68342e7	0.809206	0.404603	−	0.914492i	$-0.367410\pi$
$197$	8.68342e7	0.809206	0.404603	+	0.914492i	$0.367410\pi$
$198$	0	0
$199$	−8.01033e7	−0.720551	−0.360275	−	0.932846i	$-0.617317\pi$
$199$	−8.01033e7	−0.720551	−0.360275	+	0.932846i	$0.617317\pi$
$200$	5.16301e6	0.0456350
$201$	0	0
$202$	−1.15770e8	−0.988249
$203$	−7.39851e6	−0.0620738
$204$	0	0
$205$	−2.38523e8	−1.93371
$206$	5.12592e7	0.408542
$207$	0	0
$208$	−1.85958e7	−0.143283
$209$	1.63697e7	0.124030
$210$	0	0
$211$	−7.60511e7	−0.557336	−0.278668	−	0.960388i	$-0.589893\pi$
$211$	−7.60511e7	−0.557336	−0.278668	+	0.960388i	$0.589893\pi$
$212$	−8.21649e7	−0.592258
$213$	0	0
$214$	2.31929e7	0.161773
$215$	5.97840e7	0.410252
$216$	0	0
$217$	−1.02539e8	−0.681210
$218$	6.28587e7	0.410930
$219$	0	0
$220$	−7.18502e6	−0.0454934
$221$	9.80776e7	0.611219
$222$	0	0
$223$	1.88858e8	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$223$	1.88858e8	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$224$	1.12394e7	0.0668153
$225$	0	0
$226$	1.13704e8	0.655234
$227$	−1.32708e8	−0.753018	−0.376509	−	0.926413i	$-0.622876\pi$
$227$	−1.32708e8	−0.753018	−0.376509	+	0.926413i	$0.622876\pi$
$228$	0	0
$229$	−2.65989e8	−1.46366	−0.731830	−	0.681487i	$-0.761333\pi$
$229$	−2.65989e8	−1.46366	−0.731830	+	0.681487i	$0.761333\pi$
$230$	−2.04559e8	−1.10859
$231$	0	0
$232$	−1.10438e7	−0.0580647
$233$	−1.53411e8	−0.794531	−0.397266	−	0.917704i	$-0.630041\pi$
$233$	−1.53411e8	−0.794531	−0.397266	+	0.917704i	$0.630041\pi$
$234$	0	0
$235$	1.22091e8	0.613686
$236$	1.68193e8	0.832946
$237$	0	0
$238$	−5.92786e7	−0.285022
$239$	2.39550e8	1.13502	0.567509	−	0.823367i	$-0.307907\pi$
$239$	2.39550e8	1.13502	0.567509	+	0.823367i	$0.307907\pi$
$240$	0	0
$241$	3.17968e8	1.46327	0.731635	−	0.681697i	$-0.238758\pi$
$241$	3.17968e8	1.46327	0.731635	+	0.681697i	$0.238758\pi$
$242$	−1.54754e8	−0.701922
$243$	0	0
$244$	−2.08568e8	−0.919143
$245$	3.49418e7	0.151797
$246$	0	0
$247$	1.96609e8	0.830165
$248$	−1.53061e8	−0.637213
$249$	0	0
$250$	−1.61665e8	−0.654376
$251$	2.66731e8	1.06467	0.532335	−	0.846534i	$-0.321315\pi$
$251$	2.66731e8	1.06467	0.532335	+	0.846534i	$0.321315\pi$
$252$	0	0
$253$	3.25435e7	0.126341
$254$	1.15892e8	0.443748
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	3.94409e8	1.44937	0.724687	−	0.689078i	$-0.241984\pi$
$257$	3.94409e8	1.44937	0.724687	+	0.689078i	$0.241984\pi$
$258$	0	0
$259$	1.55076e8	0.554620
$260$	−8.62963e7	−0.304499
$261$	0	0
$262$	−1.28122e8	−0.440119
$263$	−3.41444e8	−1.15737	−0.578687	−	0.815549i	$-0.696435\pi$
$263$	−3.41444e8	−1.15737	−0.578687	+	0.815549i	$0.696435\pi$
$264$	0	0
$265$	−3.81296e8	−1.25864
$266$	−1.18832e8	−0.387121
$267$	0	0
$268$	2.66162e8	0.844646
$269$	−1.82551e8	−0.571810	−0.285905	−	0.958258i	$-0.592294\pi$
$269$	−1.82551e8	−0.571810	−0.285905	+	0.958258i	$0.592294\pi$
$270$	0	0
$271$	−3.60495e8	−1.10029	−0.550144	−	0.835070i	$-0.685427\pi$
$271$	−3.60495e8	−1.10029	−0.550144	+	0.835070i	$0.685427\pi$
$272$	−8.84859e7	−0.266614
$273$	0	0
$274$	4.44099e7	0.130423
$275$	−3.81175e6	−0.0110525
$276$	0	0
$277$	−1.85702e8	−0.524975	−0.262487	−	0.964935i	$-0.584543\pi$
$277$	−1.85702e8	−0.524975	−0.262487	+	0.964935i	$0.584543\pi$
$278$	−3.62516e8	−1.01198
$279$	0	0
$280$	5.21580e7	0.141993
$281$	1.61805e8	0.435030	0.217515	−	0.976057i	$-0.430205\pi$
$281$	1.61805e8	0.435030	0.217515	+	0.976057i	$0.430205\pi$
$282$	0	0
$283$	−2.98658e8	−0.783287	−0.391644	−	0.920117i	$-0.628093\pi$
$283$	−2.98658e8	−0.783287	−0.391644	+	0.920117i	$0.628093\pi$
$284$	−1.20914e8	−0.313229
$285$	0	0
$286$	1.37290e7	0.0347022
$287$	−2.75466e8	−0.687831
$288$	0	0
$289$	5.63509e7	0.137328
$290$	−5.12503e7	−0.123397
$291$	0	0
$292$	1.41406e8	0.332374
$293$	−2.15871e8	−0.501368	−0.250684	−	0.968069i	$-0.580656\pi$
$293$	−2.15871e8	−0.501368	−0.250684	+	0.968069i	$0.580656\pi$
$294$	0	0
$295$	7.80522e8	1.77014
$296$	2.31484e8	0.518800
$297$	0	0
$298$	−9.80613e7	−0.214655
$299$	3.90867e8	0.845628
$300$	0	0
$301$	6.90435e7	0.145928
$302$	−2.98490e8	−0.623600
$303$	0	0
$304$	−1.77381e8	−0.362118
$305$	−9.67886e8	−1.95333
$306$	0	0
$307$	−4.92521e8	−0.971495	−0.485747	−	0.874099i	$-0.661452\pi$
$307$	−4.92521e8	−0.971495	−0.485747	+	0.874099i	$0.661452\pi$
$308$	−8.29786e6	−0.0161822
$309$	0	0
$310$	−7.10300e8	−1.35418
$311$	−1.42817e8	−0.269227	−0.134613	−	0.990898i	$-0.542979\pi$
$311$	−1.42817e8	−0.269227	−0.134613	+	0.990898i	$0.542979\pi$
$312$	0	0
$313$	4.45232e8	0.820694	0.410347	−	0.911929i	$-0.365408\pi$
$313$	4.45232e8	0.820694	0.410347	+	0.911929i	$0.365408\pi$
$314$	−3.57013e8	−0.650775
$315$	0	0
$316$	−1.58634e8	−0.282808
$317$	4.97544e8	0.877252	0.438626	−	0.898670i	$-0.355465\pi$
$317$	4.97544e8	0.877252	0.438626	+	0.898670i	$0.355465\pi$
$318$	0	0
$319$	8.15346e6	0.0140629
$320$	7.78568e7	0.132822
$321$	0	0
$322$	−2.36242e8	−0.394331
$323$	9.35540e8	1.54473
$324$	0	0
$325$	−4.57814e7	−0.0739770
$326$	−3.61195e8	−0.577405
$327$	0	0
$328$	−4.11192e8	−0.643407
$329$	1.41001e8	0.218291
$330$	0	0
$331$	4.95526e8	0.751049	0.375524	−	0.926812i	$-0.377463\pi$
$331$	4.95526e8	0.751049	0.375524	+	0.926812i	$0.377463\pi$
$332$	−3.02582e7	−0.0453796
$333$	0	0
$334$	7.06238e8	1.03714
$335$	1.23516e9	1.79501
$336$	0	0
$337$	1.05451e8	0.150087	0.0750437	−	0.997180i	$-0.476090\pi$
$337$	1.05451e8	0.150087	0.0750437	+	0.997180i	$0.476090\pi$
$338$	−3.37095e8	−0.474837
$339$	0	0
$340$	−4.10630e8	−0.566597
$341$	1.13002e8	0.154329
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	1.03062e8	0.136504
$345$	0	0
$346$	8.41652e8	1.09236
$347$	5.72009e8	0.734936	0.367468	−	0.930036i	$-0.380225\pi$
$347$	5.72009e8	0.734936	0.367468	+	0.930036i	$0.380225\pi$
$348$	0	0
$349$	−4.98557e8	−0.627806	−0.313903	−	0.949455i	$-0.601637\pi$
$349$	−4.98557e8	−0.627806	−0.313903	+	0.949455i	$0.601637\pi$
$350$	2.76705e7	0.0344968
$351$	0	0
$352$	−1.23863e7	−0.0151371
$353$	−9.48941e8	−1.14823	−0.574114	−	0.818776i	$-0.694653\pi$
$353$	−9.48941e8	−1.14823	−0.574114	+	0.818776i	$0.694653\pi$
$354$	0	0
$355$	−5.61116e8	−0.665662
$356$	6.05537e8	0.711321
$357$	0	0
$358$	5.08626e8	0.585879
$359$	−1.21813e9	−1.38951	−0.694757	−	0.719244i	$-0.744488\pi$
$359$	−1.21813e9	−1.38951	−0.694757	+	0.719244i	$0.744488\pi$
$360$	0	0
$361$	9.81538e8	1.09807
$362$	−4.66866e8	−0.517264
$363$	0	0
$364$	−9.96621e7	−0.108312
$365$	6.56211e8	0.706348
$366$	0	0
$367$	−5.71901e8	−0.603934	−0.301967	−	0.953318i	$-0.597643\pi$
$367$	−5.71901e8	−0.603934	−0.301967	+	0.953318i	$0.597643\pi$
$368$	−3.52641e8	−0.368863
$369$	0	0
$370$	1.07423e9	1.10253
$371$	−4.40352e8	−0.447705
$372$	0	0
$373$	−9.28185e8	−0.926091	−0.463046	−	0.886334i	$-0.653243\pi$
$373$	−9.28185e8	−0.926091	−0.463046	+	0.886334i	$0.653243\pi$
$374$	6.53275e7	0.0645721
$375$	0	0
$376$	2.10473e8	0.204193
$377$	9.79278e7	0.0941264
$378$	0	0
$379$	1.61111e9	1.52016	0.760079	−	0.649830i	$-0.225160\pi$
$379$	1.61111e9	1.52016	0.760079	+	0.649830i	$0.225160\pi$
$380$	−8.23160e8	−0.769559
$381$	0	0
$382$	3.69867e8	0.339487
$383$	1.94419e8	0.176824	0.0884122	−	0.996084i	$-0.471821\pi$
$383$	1.94419e8	0.176824	0.0884122	+	0.996084i	$0.471821\pi$
$384$	0	0
$385$	−3.85072e7	−0.0343898
$386$	−1.05511e9	−0.933774
$387$	0	0
$388$	8.80395e8	0.765185
$389$	2.05416e9	1.76934	0.884669	−	0.466220i	$-0.154385\pi$
$389$	2.05416e9	1.76934	0.884669	+	0.466220i	$0.154385\pi$
$390$	0	0
$391$	1.85989e9	1.57351
$392$	6.02363e7	0.0505076
$393$	0	0
$394$	6.94674e8	0.572195
$395$	−7.36161e8	−0.601011
$396$	0	0
$397$	4.09863e8	0.328755	0.164378	−	0.986398i	$-0.447438\pi$
$397$	4.09863e8	0.328755	0.164378	+	0.986398i	$0.447438\pi$
$398$	−6.40826e8	−0.509506
$399$	0	0
$400$	4.13041e7	0.0322688
$401$	1.15674e9	0.895837	0.447919	−	0.894074i	$-0.352165\pi$
$401$	1.15674e9	0.895837	0.447919	+	0.894074i	$0.352165\pi$
$402$	0	0
$403$	1.35722e9	1.03296
$404$	−9.26161e8	−0.698798
$405$	0	0
$406$	−5.91881e7	−0.0438928
$407$	−1.70900e8	−0.125650
$408$	0	0
$409$	−2.10899e8	−0.152420	−0.0762101	−	0.997092i	$-0.524282\pi$
$409$	−2.10899e8	−0.152420	−0.0762101	+	0.997092i	$0.524282\pi$
$410$	−1.90819e9	−1.36734
$411$	0	0
$412$	4.10074e8	0.288883
$413$	9.01411e8	0.629648
$414$	0	0
$415$	−1.40417e8	−0.0964388
$416$	−1.48767e8	−0.101316
$417$	0	0
$418$	1.30957e8	0.0877027
$419$	1.06098e9	0.704627	0.352314	−	0.935882i	$-0.385395\pi$
$419$	1.06098e9	0.704627	0.352314	+	0.935882i	$0.385395\pi$
$420$	0	0
$421$	−1.92992e8	−0.126053	−0.0630264	−	0.998012i	$-0.520075\pi$
$421$	−1.92992e8	−0.126053	−0.0630264	+	0.998012i	$0.520075\pi$
$422$	−6.08409e8	−0.394096
$423$	0	0
$424$	−6.57319e8	−0.418789
$425$	−2.17845e8	−0.137653
$426$	0	0
$427$	−1.11779e9	−0.694807
$428$	1.85543e8	0.114391
$429$	0	0
$430$	4.78272e8	0.290092
$431$	−2.44939e9	−1.47363	−0.736813	−	0.676097i	$-0.763670\pi$
$431$	−2.44939e9	−1.47363	−0.736813	+	0.676097i	$0.763670\pi$
$432$	0	0
$433$	1.02204e9	0.605008	0.302504	−	0.953148i	$-0.402177\pi$
$433$	1.02204e9	0.605008	0.302504	+	0.953148i	$0.402177\pi$
$434$	−8.20313e8	−0.481688
$435$	0	0
$436$	5.02869e8	0.290571
$437$	3.72839e9	2.13715
$438$	0	0
$439$	1.14373e9	0.645207	0.322604	−	0.946534i	$-0.395442\pi$
$439$	1.14373e9	0.645207	0.322604	+	0.946534i	$0.395442\pi$
$440$	−5.74802e7	−0.0321687
$441$	0	0
$442$	7.84621e8	0.432197
$443$	1.56892e9	0.857410	0.428705	−	0.903444i	$-0.358970\pi$
$443$	1.56892e9	0.857410	0.428705	+	0.903444i	$0.358970\pi$
$444$	0	0
$445$	2.81007e9	1.51167
$446$	1.51086e9	0.806404
$447$	0	0
$448$	8.99154e7	0.0472456
$449$	−2.85442e9	−1.48818	−0.744090	−	0.668080i	$-0.767116\pi$
$449$	−2.85442e9	−1.48818	−0.744090	+	0.668080i	$0.767116\pi$
$450$	0	0
$451$	3.03575e8	0.155829
$452$	9.09632e8	0.463320
$453$	0	0
$454$	−1.06166e9	−0.532464
$455$	−4.62494e8	−0.230179
$456$	0	0
$457$	−1.21807e9	−0.596986	−0.298493	−	0.954412i	$-0.596484\pi$
$457$	−1.21807e9	−0.596986	−0.298493	+	0.954412i	$0.596484\pi$
$458$	−2.12792e9	−1.03496
$459$	0	0
$460$	−1.63647e9	−0.783893
$461$	−1.94684e9	−0.925504	−0.462752	−	0.886488i	$-0.653138\pi$
$461$	−1.94684e9	−0.925504	−0.462752	+	0.886488i	$0.653138\pi$
$462$	0	0
$463$	3.00086e9	1.40511	0.702557	−	0.711627i	$-0.252042\pi$
$463$	3.00086e9	1.40511	0.702557	+	0.711627i	$0.252042\pi$
$464$	−8.83507e7	−0.0410579
$465$	0	0
$466$	−1.22729e9	−0.561818
$467$	4.78778e8	0.217533	0.108767	−	0.994067i	$-0.465310\pi$
$467$	4.78778e8	0.217533	0.108767	+	0.994067i	$0.465310\pi$
$468$	0	0
$469$	1.42646e9	0.638493
$470$	9.76728e8	0.433942
$471$	0	0
$472$	1.34555e9	0.588982
$473$	−7.60888e7	−0.0330603
$474$	0	0
$475$	−4.36698e8	−0.186962
$476$	−4.74229e8	−0.201541
$477$	0	0
$478$	1.91640e9	0.802579
$479$	1.51839e9	0.631260	0.315630	−	0.948882i	$-0.397784\pi$
$479$	1.51839e9	0.631260	0.315630	+	0.948882i	$0.397784\pi$
$480$	0	0
$481$	−2.05261e9	−0.841006
$482$	2.54375e9	1.03469
$483$	0	0
$484$	−1.23803e9	−0.496334
$485$	4.08558e9	1.62614
$486$	0	0
$487$	2.42603e9	0.951796	0.475898	−	0.879500i	$-0.342123\pi$
$487$	2.42603e9	0.951796	0.475898	+	0.879500i	$0.342123\pi$
$488$	−1.66854e9	−0.649932
$489$	0	0
$490$	2.79534e8	0.107337
$491$	2.78337e9	1.06117	0.530586	−	0.847631i	$-0.321972\pi$
$491$	2.78337e9	1.06117	0.530586	+	0.847631i	$0.321972\pi$
$492$	0	0
$493$	4.65977e8	0.175146
$494$	1.57287e9	0.587016
$495$	0	0
$496$	−1.22449e9	−0.450578
$497$	−6.48023e8	−0.236779
$498$	0	0
$499$	3.56844e9	1.28566	0.642831	−	0.766008i	$-0.277760\pi$
$499$	3.56844e9	1.28566	0.642831	+	0.766008i	$0.277760\pi$
$500$	−1.29332e9	−0.462713
$501$	0	0
$502$	2.13384e9	0.752835
$503$	2.94013e9	1.03010	0.515050	−	0.857160i	$-0.327773\pi$
$503$	2.94013e9	1.03010	0.515050	+	0.857160i	$0.327773\pi$
$504$	0	0
$505$	−4.29796e9	−1.48506
$506$	2.60348e8	0.0893363
$507$	0	0
$508$	9.27137e8	0.313777
$509$	−4.14562e9	−1.39341	−0.696703	−	0.717360i	$-0.745350\pi$
$509$	−4.14562e9	−1.39341	−0.696703	+	0.717360i	$0.745350\pi$
$510$	0	0
$511$	7.57847e8	0.251251
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	3.15527e9	1.02486
$515$	1.90300e9	0.613922
$516$	0	0
$517$	−1.55389e8	−0.0494541
$518$	1.24061e9	0.392176
$519$	0	0
$520$	−6.90371e8	−0.215313
$521$	−6.22754e8	−0.192923	−0.0964617	−	0.995337i	$-0.530753\pi$
$521$	−6.22754e8	−0.192923	−0.0964617	+	0.995337i	$0.530753\pi$
$522$	0	0
$523$	−2.26690e8	−0.0692909	−0.0346454	−	0.999400i	$-0.511030\pi$
$523$	−2.26690e8	−0.0692909	−0.0346454	+	0.999400i	$0.511030\pi$
$524$	−1.02498e9	−0.311211
$525$	0	0
$526$	−2.73155e9	−0.818388
$527$	6.45817e9	1.92208
$528$	0	0
$529$	4.00735e9	1.17696
$530$	−3.05037e9	−0.889994
$531$	0	0
$532$	−9.50653e8	−0.273736
$533$	3.64611e9	1.04300
$534$	0	0
$535$	8.61036e8	0.243099
$536$	2.12930e9	0.597255
$537$	0	0
$538$	−1.46041e9	−0.404331
$539$	−4.44713e7	−0.0122326
$540$	0	0
$541$	−2.68144e9	−0.728078	−0.364039	−	0.931384i	$-0.618602\pi$
$541$	−2.68144e9	−0.728078	−0.364039	+	0.931384i	$0.618602\pi$
$542$	−2.88396e9	−0.778021
$543$	0	0
$544$	−7.07887e8	−0.188525
$545$	2.33363e9	0.617510
$546$	0	0
$547$	−3.04628e9	−0.795820	−0.397910	−	0.917425i	$-0.630264\pi$
$547$	−3.04628e9	−0.795820	−0.397910	+	0.917425i	$0.630264\pi$
$548$	3.55279e8	0.0922227
$549$	0	0
$550$	−3.04940e7	−0.00781529
$551$	9.34110e8	0.237885
$552$	0	0
$553$	−8.50179e8	−0.213783
$554$	−1.48562e9	−0.371213
$555$	0	0
$556$	−2.90013e9	−0.715576
$557$	−2.95597e9	−0.724781	−0.362390	−	0.932026i	$-0.618039\pi$
$557$	−2.95597e9	−0.724781	−0.362390	+	0.932026i	$0.618039\pi$
$558$	0	0
$559$	−9.13870e8	−0.221281
$560$	4.17264e8	0.100404
$561$	0	0
$562$	1.29444e9	0.307613
$563$	−2.08079e9	−0.491416	−0.245708	−	0.969344i	$-0.579020\pi$
$563$	−2.08079e9	−0.491416	−0.245708	+	0.969344i	$0.579020\pi$
$564$	0	0
$565$	4.22126e9	0.984629
$566$	−2.38926e9	−0.553868
$567$	0	0
$568$	−9.67311e8	−0.221486
$569$	8.57788e9	1.95203	0.976016	−	0.217700i	$-0.0698555\pi$
$569$	8.57788e9	1.95203	0.976016	+	0.217700i	$0.0698555\pi$
$570$	0	0
$571$	−8.53401e9	−1.91835	−0.959173	−	0.282822i	$-0.908730\pi$
$571$	−8.53401e9	−1.91835	−0.959173	+	0.282822i	$0.908730\pi$
$572$	1.09832e8	0.0245381
$573$	0	0
$574$	−2.20373e9	−0.486370
$575$	−8.68172e8	−0.190444
$576$	0	0
$577$	−3.43271e9	−0.743912	−0.371956	−	0.928250i	$-0.621313\pi$
$577$	−3.43271e9	−0.743912	−0.371956	+	0.928250i	$0.621313\pi$
$578$	4.50807e8	0.0971055
$579$	0	0
$580$	−4.10003e8	−0.0872547
$581$	−1.62165e8	−0.0343037
$582$	0	0
$583$	4.85286e8	0.101428
$584$	1.13125e9	0.235024
$585$	0	0
$586$	−1.72697e9	−0.354521
$587$	−1.77511e9	−0.362236	−0.181118	−	0.983461i	$-0.557972\pi$
$587$	−1.77511e9	−0.362236	−0.181118	+	0.983461i	$0.557972\pi$
$588$	0	0
$589$	1.29462e10	2.61060
$590$	6.24418e9	1.25168
$591$	0	0
$592$	1.85187e9	0.366847
$593$	−9.65664e9	−1.90167	−0.950834	−	0.309702i	$-0.899771\pi$
$593$	−9.65664e9	−1.90167	−0.950834	+	0.309702i	$0.899771\pi$
$594$	0	0
$595$	−2.20072e9	−0.428307
$596$	−7.84490e8	−0.151784
$597$	0	0
$598$	3.12693e9	0.597950
$599$	4.30993e9	0.819364	0.409682	−	0.912228i	$-0.365640\pi$
$599$	4.30993e9	0.819364	0.409682	+	0.912228i	$0.365640\pi$
$600$	0	0
$601$	8.50728e9	1.59856	0.799282	−	0.600956i	$-0.205213\pi$
$601$	8.50728e9	1.59856	0.799282	+	0.600956i	$0.205213\pi$
$602$	5.52348e8	0.103187
$603$	0	0
$604$	−2.38792e9	−0.440952
$605$	−5.74525e9	−1.05479
$606$	0	0
$607$	−1.06055e9	−0.192474	−0.0962369	−	0.995358i	$-0.530681\pi$
$607$	−1.06055e9	−0.192474	−0.0962369	+	0.995358i	$0.530681\pi$
$608$	−1.41905e9	−0.256056
$609$	0	0
$610$	−7.74308e9	−1.38121
$611$	−1.86631e9	−0.331008
$612$	0	0
$613$	−2.73570e9	−0.479685	−0.239843	−	0.970812i	$-0.577096\pi$
$613$	−2.73570e9	−0.479685	−0.239843	+	0.970812i	$0.577096\pi$
$614$	−3.94017e9	−0.686950
$615$	0	0
$616$	−6.63828e7	−0.0114426
$617$	−2.43808e9	−0.417879	−0.208939	−	0.977929i	$-0.567001\pi$
$617$	−2.43808e9	−0.417879	−0.208939	+	0.977929i	$0.567001\pi$
$618$	0	0
$619$	−1.17765e10	−1.99572	−0.997861	−	0.0653731i	$-0.979176\pi$
$619$	−1.17765e10	−1.99572	−0.997861	+	0.0653731i	$0.979176\pi$
$620$	−5.68240e9	−0.957549
$621$	0	0
$622$	−1.14253e9	−0.190372
$623$	3.24530e9	0.537708
$624$	0	0
$625$	−6.78964e9	−1.11241
$626$	3.56186e9	0.580319
$627$	0	0
$628$	−2.85611e9	−0.460167
$629$	−9.76708e9	−1.56490
$630$	0	0
$631$	−3.03709e9	−0.481233	−0.240617	−	0.970620i	$-0.577350\pi$
$631$	−3.03709e9	−0.481233	−0.240617	+	0.970620i	$0.577350\pi$
$632$	−1.26907e9	−0.199975
$633$	0	0
$634$	3.98036e9	0.620311
$635$	4.30250e9	0.666826
$636$	0	0
$637$	−5.34126e8	−0.0818759
$638$	6.52277e7	0.00994396
$639$	0	0
$640$	6.22854e8	0.0939196
$641$	1.14829e9	0.172207	0.0861033	−	0.996286i	$-0.472558\pi$
$641$	1.14829e9	0.172207	0.0861033	+	0.996286i	$0.472558\pi$
$642$	0	0
$643$	−1.18838e10	−1.76285	−0.881427	−	0.472320i	$-0.843417\pi$
$643$	−1.18838e10	−1.76285	−0.881427	+	0.472320i	$0.843417\pi$
$644$	−1.88994e9	−0.278834
$645$	0	0
$646$	7.48432e9	1.09229
$647$	−7.56322e9	−1.09785	−0.548923	−	0.835873i	$-0.684962\pi$
$647$	−7.56322e9	−1.09785	−0.548923	+	0.835873i	$0.684962\pi$
$648$	0	0
$649$	−9.93392e8	−0.142648
$650$	−3.66251e8	−0.0523097
$651$	0	0
$652$	−2.88956e9	−0.408287
$653$	1.59753e9	0.224519	0.112260	−	0.993679i	$-0.464191\pi$
$653$	1.59753e9	0.224519	0.112260	+	0.993679i	$0.464191\pi$
$654$	0	0
$655$	−4.75654e9	−0.661373
$656$	−3.28953e9	−0.454958
$657$	0	0
$658$	1.12801e9	0.154355
$659$	−5.29551e9	−0.720790	−0.360395	−	0.932800i	$-0.617358\pi$
$659$	−5.29551e9	−0.720790	−0.360395	+	0.932800i	$0.617358\pi$
$660$	0	0
$661$	−5.57482e9	−0.750803	−0.375401	−	0.926862i	$-0.622495\pi$
$661$	−5.57482e9	−0.750803	−0.375401	+	0.926862i	$0.622495\pi$
$662$	3.96421e9	0.531072
$663$	0	0
$664$	−2.42066e8	−0.0320882
$665$	−4.41163e9	−0.581732
$666$	0	0
$667$	1.85705e9	0.242316
$668$	5.64990e9	0.733370
$669$	0	0
$670$	9.88128e9	1.26926
$671$	1.23185e9	0.157409
$672$	0	0
$673$	−9.15597e9	−1.15785	−0.578924	−	0.815382i	$-0.696527\pi$
$673$	−9.15597e9	−1.15785	−0.578924	+	0.815382i	$0.696527\pi$
$674$	8.43605e8	0.106128
$675$	0	0
$676$	−2.69676e9	−0.335760
$677$	−7.86210e9	−0.973819	−0.486909	−	0.873453i	$-0.661876\pi$
$677$	−7.86210e9	−0.973819	−0.486909	+	0.873453i	$0.661876\pi$
$678$	0	0
$679$	4.71836e9	0.578425
$680$	−3.28504e9	−0.400645
$681$	0	0
$682$	9.04019e8	0.109127
$683$	4.07913e9	0.489886	0.244943	−	0.969538i	$-0.421231\pi$
$683$	4.07913e9	0.489886	0.244943	+	0.969538i	$0.421231\pi$
$684$	0	0
$685$	1.64872e9	0.195988
$686$	3.22829e8	0.0381802
$687$	0	0
$688$	8.24496e8	0.0965226
$689$	5.82857e9	0.678883
$690$	0	0
$691$	9.26382e9	1.06811	0.534056	−	0.845449i	$-0.320667\pi$
$691$	9.26382e9	1.06811	0.534056	+	0.845449i	$0.320667\pi$
$692$	6.73321e9	0.772415
$693$	0	0
$694$	4.57607e9	0.519678
$695$	−1.34584e10	−1.52071
$696$	0	0
$697$	1.73496e10	1.94077
$698$	−3.98846e9	−0.443926
$699$	0	0
$700$	2.21364e8	0.0243929
$701$	−9.78739e9	−1.07313	−0.536567	−	0.843858i	$-0.680279\pi$
$701$	−9.78739e9	−1.07313	−0.536567	+	0.843858i	$0.680279\pi$
$702$	0	0
$703$	−1.95794e10	−2.12547
$704$	−9.90904e7	−0.0107035
$705$	0	0
$706$	−7.59153e9	−0.811919
$707$	−4.96364e9	−0.528242
$708$	0	0
$709$	−7.94138e9	−0.836824	−0.418412	−	0.908257i	$-0.637413\pi$
$709$	−7.94138e9	−0.836824	−0.418412	+	0.908257i	$0.637413\pi$
$710$	−4.48893e9	−0.470694
$711$	0	0
$712$	4.84430e9	0.502980
$713$	2.57376e10	2.65923
$714$	0	0
$715$	5.09688e8	0.0521474
$716$	4.06901e9	0.414279
$717$	0	0
$718$	−9.74504e9	−0.982535
$719$	−1.37458e10	−1.37917	−0.689587	−	0.724203i	$-0.742208\pi$
$719$	−1.37458e10	−1.37917	−0.689587	+	0.724203i	$0.742208\pi$
$720$	0	0
$721$	2.19774e9	0.218375
$722$	7.85230e9	0.776456
$723$	0	0
$724$	−3.73493e9	−0.365761
$725$	−2.17512e8	−0.0211982
$726$	0	0
$727$	−1.23356e10	−1.19066	−0.595332	−	0.803480i	$-0.702979\pi$
$727$	−1.23356e10	−1.19066	−0.595332	+	0.803480i	$0.702979\pi$
$728$	−7.97297e8	−0.0765879
$729$	0	0
$730$	5.24969e9	0.499463
$731$	−4.34853e9	−0.411748
$732$	0	0
$733$	−3.24092e9	−0.303951	−0.151976	−	0.988384i	$-0.548564\pi$
$733$	−3.24092e9	−0.303951	−0.151976	+	0.988384i	$0.548564\pi$
$734$	−4.57521e9	−0.427046
$735$	0	0
$736$	−2.82113e9	−0.260826
$737$	−1.57202e9	−0.144651
$738$	0	0
$739$	2.73581e9	0.249362	0.124681	−	0.992197i	$-0.460209\pi$
$739$	2.73581e9	0.249362	0.124681	+	0.992197i	$0.460209\pi$
$740$	8.59384e9	0.779608
$741$	0	0
$742$	−3.52282e9	−0.316575
$743$	1.15155e10	1.02996	0.514980	−	0.857202i	$-0.327799\pi$
$743$	1.15155e10	1.02996	0.514980	+	0.857202i	$0.327799\pi$
$744$	0	0
$745$	−3.64052e9	−0.322565
$746$	−7.42548e9	−0.654845
$747$	0	0
$748$	5.22620e8	0.0456594
$749$	9.94395e8	0.0864714
$750$	0	0
$751$	−2.76930e9	−0.238578	−0.119289	−	0.992860i	$-0.538061\pi$
$751$	−2.76930e9	−0.238578	−0.119289	+	0.992860i	$0.538061\pi$
$752$	1.68379e9	0.144386
$753$	0	0
$754$	7.83422e8	0.0665574
$755$	−1.10815e10	−0.937093
$756$	0	0
$757$	−1.61093e10	−1.34971	−0.674855	−	0.737950i	$-0.735794\pi$
$757$	−1.61093e10	−1.34971	−0.674855	+	0.737950i	$0.735794\pi$
$758$	1.28889e10	1.07491
$759$	0	0
$760$	−6.58528e9	−0.544160
$761$	−9.60783e9	−0.790276	−0.395138	−	0.918622i	$-0.629303\pi$
$761$	−9.60783e9	−0.790276	−0.395138	+	0.918622i	$0.629303\pi$
$762$	0	0
$763$	2.69507e9	0.219651
$764$	2.95893e9	0.240054
$765$	0	0
$766$	1.55535e9	0.125034
$767$	−1.19312e10	−0.954775
$768$	0	0
$769$	−1.64537e10	−1.30473	−0.652366	−	0.757904i	$-0.726223\pi$
$769$	−1.64537e10	−1.30473	−0.652366	+	0.757904i	$0.726223\pi$
$770$	−3.08058e8	−0.0243173
$771$	0	0
$772$	−8.44087e9	−0.660278
$773$	−1.19951e10	−0.934063	−0.467032	−	0.884241i	$-0.654677\pi$
$773$	−1.19951e10	−0.934063	−0.467032	+	0.884241i	$0.654677\pi$
$774$	0	0
$775$	−3.01459e9	−0.232634
$776$	7.04316e9	0.541067
$777$	0	0
$778$	1.64333e10	1.25111
$779$	3.47794e10	2.63598
$780$	0	0
$781$	7.14148e8	0.0536426
$782$	1.48791e10	1.11264
$783$	0	0
$784$	4.81890e8	0.0357143
$785$	−1.32541e10	−0.977928
$786$	0	0
$787$	1.08309e10	0.792048	0.396024	−	0.918240i	$-0.370390\pi$
$787$	1.08309e10	0.792048	0.396024	+	0.918240i	$0.370390\pi$
$788$	5.55739e9	0.404603
$789$	0	0
$790$	−5.88928e9	−0.424979
$791$	4.87506e9	0.350237
$792$	0	0
$793$	1.47953e10	1.05358
$794$	3.27891e9	0.232465
$795$	0	0
$796$	−5.12661e9	−0.360275
$797$	1.41510e10	0.990112	0.495056	−	0.868861i	$-0.335148\pi$
$797$	1.41510e10	0.990112	0.495056	+	0.868861i	$0.335148\pi$
$798$	0	0
$799$	−8.88058e9	−0.615925
$800$	3.30433e8	0.0228175
$801$	0	0
$802$	9.25389e9	0.633453
$803$	−8.35178e8	−0.0569213
$804$	0	0
$805$	−8.77048e9	−0.592568
$806$	1.08578e10	0.730414
$807$	0	0
$808$	−7.40929e9	−0.494125
$809$	4.61529e9	0.306464	0.153232	−	0.988190i	$-0.451032\pi$
$809$	4.61529e9	0.306464	0.153232	+	0.988190i	$0.451032\pi$
$810$	0	0
$811$	−8.89149e9	−0.585331	−0.292666	−	0.956215i	$-0.594542\pi$
$811$	−8.89149e9	−0.585331	−0.292666	+	0.956215i	$0.594542\pi$
$812$	−4.73505e8	−0.0310369
$813$	0	0
$814$	−1.36720e9	−0.0888479
$815$	−1.34094e10	−0.867674
$816$	0	0
$817$	−8.71719e9	−0.559242
$818$	−1.68719e9	−0.107777
$819$	0	0
$820$	−1.52655e10	−0.966857
$821$	1.41358e10	0.891498	0.445749	−	0.895158i	$-0.352937\pi$
$821$	1.41358e10	0.891498	0.445749	+	0.895158i	$0.352937\pi$
$822$	0	0
$823$	2.73614e10	1.71095	0.855477	−	0.517842i	$-0.173264\pi$
$823$	2.73614e10	1.71095	0.855477	+	0.517842i	$0.173264\pi$
$824$	3.28059e9	0.204271
$825$	0	0
$826$	7.21129e9	0.445228
$827$	1.76474e10	1.08495	0.542477	−	0.840070i	$-0.317486\pi$
$827$	1.76474e10	1.08495	0.542477	+	0.840070i	$0.317486\pi$
$828$	0	0
$829$	−5.45703e9	−0.332672	−0.166336	−	0.986069i	$-0.553194\pi$
$829$	−5.45703e9	−0.332672	−0.166336	+	0.986069i	$0.553194\pi$
$830$	−1.12334e9	−0.0681925
$831$	0	0
$832$	−1.19013e9	−0.0716414
$833$	−2.54157e9	−0.152351
$834$	0	0
$835$	2.62191e10	1.55853
$836$	1.04766e9	0.0620152
$837$	0	0
$838$	8.48787e9	0.498247
$839$	5.82108e9	0.340280	0.170140	−	0.985420i	$-0.445578\pi$
$839$	5.82108e9	0.340280	0.170140	+	0.985420i	$0.445578\pi$
$840$	0	0
$841$	−1.67846e10	−0.973028
$842$	−1.54394e9	−0.0891328
$843$	0	0
$844$	−4.86727e9	−0.278668
$845$	−1.25147e10	−0.713544
$846$	0	0
$847$	−6.63509e9	−0.375193
$848$	−5.25855e9	−0.296129
$849$	0	0
$850$	−1.74276e9	−0.0973354
$851$	−3.89246e10	−2.16506
$852$	0	0
$853$	3.26104e10	1.79901	0.899507	−	0.436906i	$-0.143926\pi$
$853$	3.26104e10	1.79901	0.899507	+	0.436906i	$0.143926\pi$
$854$	−8.94235e9	−0.491303
$855$	0	0
$856$	1.48434e9	0.0808866
$857$	−1.04828e10	−0.568914	−0.284457	−	0.958689i	$-0.591813\pi$
$857$	−1.04828e10	−0.568914	−0.284457	+	0.958689i	$0.591813\pi$
$858$	0	0
$859$	−1.19907e10	−0.645459	−0.322729	−	0.946491i	$-0.604600\pi$
$859$	−1.19907e10	−0.645459	−0.322729	+	0.946491i	$0.604600\pi$
$860$	3.82618e9	0.205126
$861$	0	0
$862$	−1.95951e10	−1.04201
$863$	−5.20582e9	−0.275709	−0.137855	−	0.990452i	$-0.544021\pi$
$863$	−5.20582e9	−0.275709	−0.137855	+	0.990452i	$0.544021\pi$
$864$	0	0
$865$	3.12463e10	1.64151
$866$	8.17634e9	0.427805
$867$	0	0
$868$	−6.56251e9	−0.340605
$869$	9.36932e8	0.0484327
$870$	0	0
$871$	−1.88809e10	−0.968187
$872$	4.02296e9	0.205465
$873$	0	0
$874$	2.98271e10	1.51120
$875$	−6.93140e9	−0.349778
$876$	0	0
$877$	1.19857e10	0.600021	0.300011	−	0.953936i	$-0.403010\pi$
$877$	1.19857e10	0.600021	0.300011	+	0.953936i	$0.403010\pi$
$878$	9.14988e9	0.456231
$879$	0	0
$880$	−4.59842e8	−0.0227467
$881$	−8.62327e9	−0.424870	−0.212435	−	0.977175i	$-0.568139\pi$
$881$	−8.62327e9	−0.424870	−0.212435	+	0.977175i	$0.568139\pi$
$882$	0	0
$883$	−8.69375e9	−0.424957	−0.212478	−	0.977166i	$-0.568153\pi$
$883$	−8.69375e9	−0.424957	−0.212478	+	0.977166i	$0.568153\pi$
$884$	6.27697e9	0.305610
$885$	0	0
$886$	1.25514e10	0.606281
$887$	1.98166e10	0.953445	0.476723	−	0.879054i	$-0.341825\pi$
$887$	1.98166e10	0.953445	0.476723	+	0.879054i	$0.341825\pi$
$888$	0	0
$889$	4.96888e9	0.237193
$890$	2.24806e10	1.06891
$891$	0	0
$892$	1.20869e10	0.570214
$893$	−1.78023e10	−0.836557
$894$	0	0
$895$	1.88827e10	0.880408
$896$	7.19323e8	0.0334077
$897$	0	0
$898$	−2.28353e10	−1.05230
$899$	6.44831e9	0.295997
$900$	0	0
$901$	2.77345e10	1.26323
$902$	2.42860e9	0.110188
$903$	0	0
$904$	7.27706e9	0.327617
$905$	−1.73324e10	−0.777300
$906$	0	0
$907$	1.75483e10	0.780925	0.390463	−	0.920619i	$-0.372315\pi$
$907$	1.75483e10	0.780925	0.390463	+	0.920619i	$0.372315\pi$
$908$	−8.49329e9	−0.376509
$909$	0	0
$910$	−3.69995e9	−0.162761
$911$	3.45953e10	1.51601	0.758007	−	0.652246i	$-0.226173\pi$
$911$	3.45953e10	1.51601	0.758007	+	0.652246i	$0.226173\pi$
$912$	0	0
$913$	1.78713e8	0.00777155
$914$	−9.74454e9	−0.422133
$915$	0	0
$916$	−1.70233e10	−0.731830
$917$	−5.49324e9	−0.235253
$918$	0	0
$919$	−2.63708e10	−1.12078	−0.560388	−	0.828231i	$-0.689348\pi$
$919$	−2.63708e10	−1.12078	−0.560388	+	0.828231i	$0.689348\pi$
$920$	−1.30918e10	−0.554296
$921$	0	0
$922$	−1.55748e10	−0.654430
$923$	8.57733e9	0.359043
$924$	0	0
$925$	4.55915e9	0.189403
$926$	2.40068e10	0.993566
$927$	0	0
$928$	−7.06806e8	−0.0290323
$929$	6.45358e9	0.264086	0.132043	−	0.991244i	$-0.457846\pi$
$929$	6.45358e9	0.264086	0.132043	+	0.991244i	$0.457846\pi$
$930$	0	0
$931$	−5.09491e9	−0.206925
$932$	−9.81831e9	−0.397266
$933$	0	0
$934$	3.83023e9	0.153819
$935$	2.42528e9	0.0970335
$936$	0	0
$937$	−4.13326e10	−1.64136	−0.820680	−	0.571388i	$-0.806405\pi$
$937$	−4.13326e10	−1.64136	−0.820680	+	0.571388i	$0.806405\pi$
$938$	1.14117e10	0.451483
$939$	0	0
$940$	7.81383e9	0.306843
$941$	1.43128e10	0.559965	0.279982	−	0.960005i	$-0.409671\pi$
$941$	1.43128e10	0.559965	0.279982	+	0.960005i	$0.409671\pi$
$942$	0	0
$943$	6.91429e10	2.68508
$944$	1.07644e10	0.416473
$945$	0	0
$946$	−6.08710e8	−0.0233771
$947$	−1.68859e10	−0.646100	−0.323050	−	0.946382i	$-0.604708\pi$
$947$	−1.68859e10	−0.646100	−0.323050	+	0.946382i	$0.604708\pi$
$948$	0	0
$949$	−1.00310e10	−0.380988
$950$	−3.49358e9	−0.132202
$951$	0	0
$952$	−3.79383e9	−0.142511
$953$	−4.07066e10	−1.52349	−0.761746	−	0.647876i	$-0.775658\pi$
$953$	−4.07066e10	−1.52349	−0.761746	+	0.647876i	$0.775658\pi$
$954$	0	0
$955$	1.37313e10	0.510152
$956$	1.53312e10	0.567509
$957$	0	0
$958$	1.21471e10	0.446368
$959$	1.90407e9	0.0697138
$960$	0	0
$961$	6.18573e10	2.24832
$962$	−1.64209e10	−0.594681
$963$	0	0
$964$	2.03500e10	0.731635
$965$	−3.91709e10	−1.40320
$966$	0	0
$967$	3.76013e10	1.33724	0.668622	−	0.743603i	$-0.266885\pi$
$967$	3.76013e10	1.33724	0.668622	+	0.743603i	$0.266885\pi$
$968$	−9.90427e9	−0.350961
$969$	0	0
$970$	3.26847e10	1.14985
$971$	4.92140e10	1.72513	0.862565	−	0.505946i	$-0.168857\pi$
$971$	4.92140e10	1.72513	0.862565	+	0.505946i	$0.168857\pi$
$972$	0	0
$973$	−1.55429e10	−0.540925
$974$	1.94082e10	0.673022
$975$	0	0
$976$	−1.33483e10	−0.459572
$977$	−1.98045e10	−0.679412	−0.339706	−	0.940532i	$-0.610327\pi$
$977$	−1.98045e10	−0.679412	−0.339706	+	0.940532i	$0.610327\pi$
$978$	0	0
$979$	−3.57645e9	−0.121818
$980$	2.23627e9	0.0758985
$981$	0	0
$982$	2.22670e10	0.750362
$983$	−4.14936e10	−1.39330	−0.696649	−	0.717412i	$-0.745326\pi$
$983$	−4.14936e10	−1.39330	−0.696649	+	0.717412i	$0.745326\pi$
$984$	0	0
$985$	2.57898e10	0.859846
$986$	3.72781e9	0.123847
$987$	0	0
$988$	1.25830e10	0.415083
$989$	−1.73301e10	−0.569658
$990$	0	0
$991$	4.18413e10	1.36567	0.682837	−	0.730571i	$-0.260746\pi$
$991$	4.18413e10	1.36567	0.682837	+	0.730571i	$0.260746\pi$
$992$	−9.79593e9	−0.318607
$993$	0	0
$994$	−5.18418e9	−0.167428
$995$	−2.37907e10	−0.765642
$996$	0	0
$997$	3.75937e10	1.20138	0.600692	−	0.799480i	$-0.294892\pi$
$997$	3.75937e10	1.20138	0.600692	+	0.799480i	$0.294892\pi$
$998$	2.85475e10	0.909100
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.8.a.d.1.1	yes	1
3.2	odd	2		378.8.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.8.a.a.1.1	✓	1	3.2	odd	2
378.8.a.d.1.1	yes	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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