LMFDB - Embedded newform 392.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,10,Mod(1,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("392.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	392.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:	$201.894047776$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+60.0000 q^{3} +2074.00 q^{5} -16083.0 q^{9} +O(q^{10})$

$q+60.0000 q^{3} +2074.00 q^{5} -16083.0 q^{9} +93644.0 q^{11} +12242.0 q^{13} +124440. q^{15} +319598. q^{17} +553516. q^{19} -712936. q^{23} +2.34835e6 q^{25} -2.14596e6 q^{27} +2.07584e6 q^{29} +6.42045e6 q^{31} +5.61864e6 q^{33} -1.81978e7 q^{37} +734520. q^{39} -9.03383e6 q^{41} +1.95947e7 q^{43} -3.33561e7 q^{45} +1.84842e7 q^{47} +1.91759e7 q^{51} +1.02558e7 q^{53} +1.94218e8 q^{55} +3.32110e7 q^{57} -1.21667e8 q^{59} +4.59490e7 q^{61} +2.53899e7 q^{65} +5.05354e7 q^{67} -4.27762e7 q^{69} +2.67045e8 q^{71} +1.76213e8 q^{73} +1.40901e8 q^{75} -2.69686e8 q^{79} +1.87804e8 q^{81} +2.27033e8 q^{83} +6.62846e8 q^{85} +1.24550e8 q^{87} -7.21416e7 q^{89} +3.85227e8 q^{93} +1.14799e9 q^{95} -2.28777e8 q^{97} -1.50608e9 q^{99} +O(q^{100})$

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
$3$	60.0000	0.427667	0.213833	−	0.976870i	$-0.431405\pi$
$3$	60.0000	0.427667	0.213833	+	0.976870i	$0.431405\pi$
$4$	0	0
$5$	2074.00	1.48403	0.742017	−	0.670381i	$-0.233869\pi$
$5$	2074.00	1.48403	0.742017	+	0.670381i	$0.233869\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−16083.0	−0.817101
$10$	0	0
$11$	93644.0	1.92847	0.964235	−	0.265049i	$-0.0853881\pi$
$11$	93644.0	1.92847	0.964235	+	0.265049i	$0.0853881\pi$
$12$	0	0
$13$	12242.0	0.118880	0.0594398	−	0.998232i	$-0.481069\pi$
$13$	12242.0	0.118880	0.0594398	+	0.998232i	$0.481069\pi$
$14$	0	0
$15$	124440.	0.634672
$16$	0	0
$17$	319598.	0.928077	0.464038	−	0.885815i	$-0.346400\pi$
$17$	319598.	0.928077	0.464038	+	0.885815i	$0.346400\pi$
$18$	0	0
$19$	553516.	0.974404	0.487202	−	0.873289i	$-0.338018\pi$
$19$	553516.	0.974404	0.487202	+	0.873289i	$0.338018\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−712936.	−0.531221	−0.265611	−	0.964080i	$-0.585574\pi$
$23$	−712936.	−0.531221	−0.265611	+	0.964080i	$0.585574\pi$
$24$	0	0
$25$	2.34835e6	1.20236
$26$	0	0
$27$	−2.14596e6	−0.777114
$28$	0	0
$29$	2.07584e6	0.545007	0.272504	−	0.962155i	$-0.412148\pi$
$29$	2.07584e6	0.545007	0.272504	+	0.962155i	$0.412148\pi$
$30$	0	0
$31$	6.42045e6	1.24864	0.624321	−	0.781168i	$-0.285376\pi$
$31$	6.42045e6	1.24864	0.624321	+	0.781168i	$0.285376\pi$
$32$	0	0
$33$	5.61864e6	0.824743
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.81978e7	−1.59628	−0.798142	−	0.602470i	$-0.794183\pi$
$37$	−1.81978e7	−1.59628	−0.798142	+	0.602470i	$0.794183\pi$
$38$	0	0
$39$	734520.	0.0508409
$40$	0	0
$41$	−9.03383e6	−0.499281	−0.249640	−	0.968339i	$-0.580312\pi$
$41$	−9.03383e6	−0.499281	−0.249640	+	0.968339i	$0.580312\pi$
$42$	0	0
$43$	1.95947e7	0.874040	0.437020	−	0.899452i	$-0.356034\pi$
$43$	1.95947e7	0.874040	0.437020	+	0.899452i	$0.356034\pi$
$44$	0	0
$45$	−3.33561e7	−1.21261
$46$	0	0
$47$	1.84842e7	0.552535	0.276267	−	0.961081i	$-0.410902\pi$
$47$	1.84842e7	0.552535	0.276267	+	0.961081i	$0.410902\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.91759e7	0.396908
$52$	0	0
$53$	1.02558e7	0.178536	0.0892682	−	0.996008i	$-0.471547\pi$
$53$	1.02558e7	0.178536	0.0892682	+	0.996008i	$0.471547\pi$
$54$	0	0
$55$	1.94218e8	2.86191
$56$	0	0
$57$	3.32110e7	0.416720
$58$	0	0
$59$	−1.21667e8	−1.30719	−0.653593	−	0.756847i	$-0.726739\pi$
$59$	−1.21667e8	−1.30719	−0.653593	+	0.756847i	$0.726739\pi$
$60$	0	0
$61$	4.59490e7	0.424905	0.212452	−	0.977171i	$-0.431855\pi$
$61$	4.59490e7	0.424905	0.212452	+	0.977171i	$0.431855\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.53899e7	0.176421
$66$	0	0
$67$	5.05354e7	0.306379	0.153190	−	0.988197i	$-0.451045\pi$
$67$	5.05354e7	0.306379	0.153190	+	0.988197i	$0.451045\pi$
$68$	0	0
$69$	−4.27762e7	−0.227186
$70$	0	0
$71$	2.67045e8	1.24716	0.623579	−	0.781760i	$-0.285678\pi$
$71$	2.67045e8	1.24716	0.623579	+	0.781760i	$0.285678\pi$
$72$	0	0
$73$	1.76213e8	0.726250	0.363125	−	0.931740i	$-0.381710\pi$
$73$	1.76213e8	0.726250	0.363125	+	0.931740i	$0.381710\pi$
$74$	0	0
$75$	1.40901e8	0.514208
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.69686e8	−0.778997	−0.389499	−	0.921027i	$-0.627352\pi$
$79$	−2.69686e8	−0.778997	−0.389499	+	0.921027i	$0.627352\pi$
$80$	0	0
$81$	1.87804e8	0.484755
$82$	0	0
$83$	2.27033e8	0.525094	0.262547	−	0.964919i	$-0.415438\pi$
$83$	2.27033e8	0.525094	0.262547	+	0.964919i	$0.415438\pi$
$84$	0	0
$85$	6.62846e8	1.37730
$86$	0	0
$87$	1.24550e8	0.233082
$88$	0	0
$89$	−7.21416e7	−0.121880	−0.0609398	−	0.998141i	$-0.519410\pi$
$89$	−7.21416e7	−0.121880	−0.0609398	+	0.998141i	$0.519410\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.85227e8	0.534003
$94$	0	0
$95$	1.14799e9	1.44605
$96$	0	0
$97$	−2.28777e8	−0.262385	−0.131192	−	0.991357i	$-0.541881\pi$
$97$	−2.28777e8	−0.262385	−0.131192	+	0.991357i	$0.541881\pi$
$98$	0	0
$99$	−1.50608e9	−1.57575
$100$	0	0
$101$	8.03256e8	0.768082	0.384041	−	0.923316i	$-0.374532\pi$
$101$	8.03256e8	0.768082	0.384041	+	0.923316i	$0.374532\pi$
$102$	0	0
$103$	−7.81726e8	−0.684363	−0.342182	−	0.939634i	$-0.611166\pi$
$103$	−7.81726e8	−0.684363	−0.342182	+	0.939634i	$0.611166\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.00756e9	−0.743093	−0.371546	−	0.928414i	$-0.621172\pi$
$107$	−1.00756e9	−0.743093	−0.371546	+	0.928414i	$0.621172\pi$
$108$	0	0
$109$	−4.80692e8	−0.326173	−0.163086	−	0.986612i	$-0.552145\pi$
$109$	−4.80692e8	−0.326173	−0.163086	+	0.986612i	$0.552145\pi$
$110$	0	0
$111$	−1.09187e9	−0.682678
$112$	0	0
$113$	−2.89781e9	−1.67193	−0.835963	−	0.548786i	$-0.815090\pi$
$113$	−2.89781e9	−1.67193	−0.835963	+	0.548786i	$0.815090\pi$
$114$	0	0
$115$	−1.47863e9	−0.788350
$116$	0	0
$117$	−1.96888e8	−0.0971366
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.41125e9	2.71900
$122$	0	0
$123$	−5.42030e8	−0.213526
$124$	0	0
$125$	8.19699e8	0.300303
$126$	0	0
$127$	−4.24330e9	−1.44740	−0.723698	−	0.690117i	$-0.757559\pi$
$127$	−4.24330e9	−1.44740	−0.723698	+	0.690117i	$0.757559\pi$
$128$	0	0
$129$	1.17568e9	0.373798
$130$	0	0
$131$	−2.89728e9	−0.859546	−0.429773	−	0.902937i	$-0.641406\pi$
$131$	−2.89728e9	−0.859546	−0.429773	+	0.902937i	$0.641406\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.45072e9	−1.15326
$136$	0	0
$137$	−2.35617e9	−0.571432	−0.285716	−	0.958314i	$-0.592231\pi$
$137$	−2.35617e9	−0.571432	−0.285716	+	0.958314i	$0.592231\pi$
$138$	0	0
$139$	2.71527e9	0.616946	0.308473	−	0.951233i	$-0.400182\pi$
$139$	2.71527e9	0.616946	0.308473	+	0.951233i	$0.400182\pi$
$140$	0	0
$141$	1.10905e9	0.236301
$142$	0	0
$143$	1.14639e9	0.229256
$144$	0	0
$145$	4.30529e9	0.808809
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.67402e9	0.278242	0.139121	−	0.990275i	$-0.455572\pi$
$149$	1.67402e9	0.278242	0.139121	+	0.990275i	$0.455572\pi$
$150$	0	0
$151$	5.32709e9	0.833860	0.416930	−	0.908938i	$-0.363106\pi$
$151$	5.32709e9	0.833860	0.416930	+	0.908938i	$0.363106\pi$
$152$	0	0
$153$	−5.14009e9	−0.758333
$154$	0	0
$155$	1.33160e10	1.85303
$156$	0	0
$157$	1.15835e10	1.52156	0.760782	−	0.649008i	$-0.224816\pi$
$157$	1.15835e10	1.52156	0.760782	+	0.649008i	$0.224816\pi$
$158$	0	0
$159$	6.15346e8	0.0763541
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.48418e8	0.105234	0.0526169	−	0.998615i	$-0.483244\pi$
$163$	9.48418e8	0.105234	0.0526169	+	0.998615i	$0.483244\pi$
$164$	0	0
$165$	1.16531e10	1.22395
$166$	0	0
$167$	1.44718e10	1.43978	0.719891	−	0.694087i	$-0.244192\pi$
$167$	1.44718e10	1.43978	0.719891	+	0.694087i	$0.244192\pi$
$168$	0	0
$169$	−1.04546e10	−0.985868
$170$	0	0
$171$	−8.90220e9	−0.796186
$172$	0	0
$173$	1.39886e10	1.18732	0.593658	−	0.804717i	$-0.297683\pi$
$173$	1.39886e10	1.18732	0.593658	+	0.804717i	$0.297683\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.29999e9	−0.559040
$178$	0	0
$179$	−4.54924e9	−0.331207	−0.165604	−	0.986192i	$-0.552957\pi$
$179$	−4.54924e9	−0.331207	−0.165604	+	0.986192i	$0.552957\pi$
$180$	0	0
$181$	−1.56484e10	−1.08372	−0.541859	−	0.840469i	$-0.682279\pi$
$181$	−1.56484e10	−1.08372	−0.541859	+	0.840469i	$0.682279\pi$
$182$	0	0
$183$	2.75694e9	0.181718
$184$	0	0
$185$	−3.77421e10	−2.36894
$186$	0	0
$187$	2.99284e10	1.78977
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.02052e10	1.09853	0.549267	−	0.835647i	$-0.314907\pi$
$191$	2.02052e10	1.09853	0.549267	+	0.835647i	$0.314907\pi$
$192$	0	0
$193$	−7.10827e9	−0.368770	−0.184385	−	0.982854i	$-0.559029\pi$
$193$	−7.10827e9	−0.368770	−0.184385	+	0.982854i	$0.559029\pi$
$194$	0	0
$195$	1.52339e9	0.0754495
$196$	0	0
$197$	2.25924e10	1.06872	0.534359	−	0.845257i	$-0.320553\pi$
$197$	2.25924e10	1.06872	0.534359	+	0.845257i	$0.320553\pi$
$198$	0	0
$199$	−3.55506e10	−1.60697	−0.803485	−	0.595325i	$-0.797023\pi$
$199$	−3.55506e10	−1.60697	−0.803485	+	0.595325i	$0.797023\pi$
$200$	0	0
$201$	3.03213e9	0.131028
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.87362e10	−0.740949
$206$	0	0
$207$	1.14661e10	0.434061
$208$	0	0
$209$	5.18335e10	1.87911
$210$	0	0
$211$	−5.58480e9	−0.193971	−0.0969854	−	0.995286i	$-0.530920\pi$
$211$	−5.58480e9	−0.193971	−0.0969854	+	0.995286i	$0.530920\pi$
$212$	0	0
$213$	1.60227e10	0.533368
$214$	0	0
$215$	4.06395e10	1.29710
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.05728e10	0.310593
$220$	0	0
$221$	3.91252e9	0.110329
$222$	0	0
$223$	−4.74713e10	−1.28546	−0.642731	−	0.766092i	$-0.722199\pi$
$223$	−4.74713e10	−1.28546	−0.642731	+	0.766092i	$0.722199\pi$
$224$	0	0
$225$	−3.77685e10	−0.982446
$226$	0	0
$227$	3.37702e10	0.844146	0.422073	−	0.906562i	$-0.361303\pi$
$227$	3.37702e10	0.844146	0.422073	+	0.906562i	$0.361303\pi$
$228$	0	0
$229$	−7.28989e9	−0.175171	−0.0875854	−	0.996157i	$-0.527915\pi$
$229$	−7.28989e9	−0.175171	−0.0875854	+	0.996157i	$0.527915\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.79739e10	1.51092	0.755458	−	0.655197i	$-0.227414\pi$
$233$	6.79739e10	1.51092	0.755458	+	0.655197i	$0.227414\pi$
$234$	0	0
$235$	3.83362e10	0.819980
$236$	0	0
$237$	−1.61811e10	−0.333151
$238$	0	0
$239$	−3.11283e10	−0.617114	−0.308557	−	0.951206i	$-0.599846\pi$
$239$	−3.11283e10	−0.617114	−0.308557	+	0.951206i	$0.599846\pi$
$240$	0	0
$241$	−1.42372e10	−0.271861	−0.135931	−	0.990718i	$-0.543402\pi$
$241$	−1.42372e10	−0.271861	−0.135931	+	0.990718i	$0.543402\pi$
$242$	0	0
$243$	5.35072e10	0.984428
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.77614e9	0.115837
$248$	0	0
$249$	1.36220e10	0.224565
$250$	0	0
$251$	5.78389e10	0.919789	0.459894	−	0.887974i	$-0.347887\pi$
$251$	5.78389e10	0.919789	0.459894	+	0.887974i	$0.347887\pi$
$252$	0	0
$253$	−6.67622e10	−1.02444
$254$	0	0
$255$	3.97708e10	0.589024
$256$	0	0
$257$	1.87176e10	0.267641	0.133820	−	0.991006i	$-0.457276\pi$
$257$	1.87176e10	0.267641	0.133820	+	0.991006i	$0.457276\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.33857e10	−0.445326
$262$	0	0
$263$	−2.80437e10	−0.361439	−0.180719	−	0.983535i	$-0.557843\pi$
$263$	−2.80437e10	−0.361439	−0.180719	+	0.983535i	$0.557843\pi$
$264$	0	0
$265$	2.12705e10	0.264954
$266$	0	0
$267$	−4.32850e9	−0.0521238
$268$	0	0
$269$	4.46600e10	0.520036	0.260018	−	0.965604i	$-0.416271\pi$
$269$	4.46600e10	0.520036	0.260018	+	0.965604i	$0.416271\pi$
$270$	0	0
$271$	1.03375e11	1.16427	0.582137	−	0.813090i	$-0.302217\pi$
$271$	1.03375e11	1.16427	0.582137	+	0.813090i	$0.302217\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.19909e11	2.31871
$276$	0	0
$277$	1.81403e11	1.85133	0.925666	−	0.378341i	$-0.123505\pi$
$277$	1.81403e11	1.85133	0.925666	+	0.378341i	$0.123505\pi$
$278$	0	0
$279$	−1.03260e11	−1.02027
$280$	0	0
$281$	−1.25487e11	−1.20066	−0.600332	−	0.799751i	$-0.704965\pi$
$281$	−1.25487e11	−1.20066	−0.600332	+	0.799751i	$0.704965\pi$
$282$	0	0
$283$	−1.33561e11	−1.23777	−0.618886	−	0.785481i	$-0.712416\pi$
$283$	−1.33561e11	−1.23777	−0.618886	+	0.785481i	$0.712416\pi$
$284$	0	0
$285$	6.88795e10	0.618427
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.64450e10	−0.138673
$290$	0	0
$291$	−1.37266e10	−0.112213
$292$	0	0
$293$	3.50635e9	0.0277940	0.0138970	−	0.999903i	$-0.495576\pi$
$293$	3.50635e9	0.0277940	0.0138970	+	0.999903i	$0.495576\pi$
$294$	0	0
$295$	−2.52336e11	−1.93991
$296$	0	0
$297$	−2.00956e11	−1.49864
$298$	0	0
$299$	−8.72776e9	−0.0631513
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.81954e10	0.328483
$304$	0	0
$305$	9.52981e10	0.630573
$306$	0	0
$307$	2.94357e11	1.89126	0.945629	−	0.325246i	$-0.105447\pi$
$307$	2.94357e11	1.89126	0.945629	+	0.325246i	$0.105447\pi$
$308$	0	0
$309$	−4.69035e10	−0.292680
$310$	0	0
$311$	−2.40305e10	−0.145660	−0.0728301	−	0.997344i	$-0.523203\pi$
$311$	−2.40305e10	−0.145660	−0.0728301	+	0.997344i	$0.523203\pi$
$312$	0	0
$313$	2.55229e11	1.50308	0.751539	−	0.659689i	$-0.229312\pi$
$313$	2.55229e11	1.50308	0.751539	+	0.659689i	$0.229312\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.30255e11	1.28069	0.640343	−	0.768089i	$-0.278792\pi$
$317$	2.30255e11	1.28069	0.640343	+	0.768089i	$0.278792\pi$
$318$	0	0
$319$	1.94390e11	1.05103
$320$	0	0
$321$	−6.04535e10	−0.317796
$322$	0	0
$323$	1.76903e11	0.904322
$324$	0	0
$325$	2.87485e10	0.142936
$326$	0	0
$327$	−2.88415e10	−0.139493
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.21212e11	−0.555035	−0.277518	−	0.960721i	$-0.589512\pi$
$331$	−1.21212e11	−0.555035	−0.277518	+	0.960721i	$0.589512\pi$
$332$	0	0
$333$	2.92674e11	1.30432
$334$	0	0
$335$	1.04810e11	0.454677
$336$	0	0
$337$	2.52249e11	1.06536	0.532678	−	0.846318i	$-0.321186\pi$
$337$	2.52249e11	1.06536	0.532678	+	0.846318i	$0.321186\pi$
$338$	0	0
$339$	−1.73869e11	−0.715027
$340$	0	0
$341$	6.01236e11	2.40797
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8.87178e10	−0.337151
$346$	0	0
$347$	2.99996e11	1.11079	0.555397	−	0.831585i	$-0.312566\pi$
$347$	2.99996e11	1.11079	0.555397	+	0.831585i	$0.312566\pi$
$348$	0	0
$349$	1.25625e11	0.453275	0.226638	−	0.973979i	$-0.427227\pi$
$349$	1.25625e11	0.453275	0.226638	+	0.973979i	$0.427227\pi$
$350$	0	0
$351$	−2.62708e10	−0.0923830
$352$	0	0
$353$	4.31672e11	1.47968	0.739841	−	0.672782i	$-0.234901\pi$
$353$	4.31672e11	1.47968	0.739841	+	0.672782i	$0.234901\pi$
$354$	0	0
$355$	5.53851e11	1.85082
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.83615e11	0.583421	0.291711	−	0.956507i	$-0.405776\pi$
$359$	1.83615e11	0.583421	0.291711	+	0.956507i	$0.405776\pi$
$360$	0	0
$361$	−1.63077e10	−0.0505372
$362$	0	0
$363$	3.84675e11	1.16282
$364$	0	0
$365$	3.65467e11	1.07778
$366$	0	0
$367$	−3.77185e11	−1.08532	−0.542659	−	0.839953i	$-0.682582\pi$
$367$	−3.77185e11	−1.08532	−0.542659	+	0.839953i	$0.682582\pi$
$368$	0	0
$369$	1.45291e11	0.407963
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.69400e11	0.720623	0.360312	−	0.932832i	$-0.382670\pi$
$373$	2.69400e11	0.720623	0.360312	+	0.932832i	$0.382670\pi$
$374$	0	0
$375$	4.91819e10	0.128430
$376$	0	0
$377$	2.54124e10	0.0647903
$378$	0	0
$379$	−2.04102e11	−0.508124	−0.254062	−	0.967188i	$-0.581767\pi$
$379$	−2.04102e11	−0.508124	−0.254062	+	0.967188i	$0.581767\pi$
$380$	0	0
$381$	−2.54598e11	−0.619003
$382$	0	0
$383$	4.10631e10	0.0975117	0.0487559	−	0.998811i	$-0.484474\pi$
$383$	4.10631e10	0.0975117	0.0487559	+	0.998811i	$0.484474\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.15142e11	−0.714179
$388$	0	0
$389$	−2.86342e11	−0.634032	−0.317016	−	0.948420i	$-0.602681\pi$
$389$	−2.86342e11	−0.634032	−0.317016	+	0.948420i	$0.602681\pi$
$390$	0	0
$391$	−2.27853e11	−0.493014
$392$	0	0
$393$	−1.73837e11	−0.367599
$394$	0	0
$395$	−5.59328e11	−1.15606
$396$	0	0
$397$	−3.73016e11	−0.753651	−0.376826	−	0.926284i	$-0.622984\pi$
$397$	−3.73016e11	−0.753651	−0.376826	+	0.926284i	$0.622984\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.70676e11	0.909018	0.454509	−	0.890742i	$-0.349815\pi$
$401$	4.70676e11	0.909018	0.454509	+	0.890742i	$0.349815\pi$
$402$	0	0
$403$	7.85991e10	0.148438
$404$	0	0
$405$	3.89506e11	0.719393
$406$	0	0
$407$	−1.70411e12	−3.07838
$408$	0	0
$409$	8.60520e11	1.52057	0.760284	−	0.649590i	$-0.225060\pi$
$409$	8.60520e11	1.52057	0.760284	+	0.649590i	$0.225060\pi$
$410$	0	0
$411$	−1.41370e11	−0.244382
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.70866e11	0.779257
$416$	0	0
$417$	1.62916e11	0.263847
$418$	0	0
$419$	−8.46565e11	−1.34183	−0.670914	−	0.741535i	$-0.734098\pi$
$419$	−8.46565e11	−1.34183	−0.670914	+	0.741535i	$0.734098\pi$
$420$	0	0
$421$	−2.27835e11	−0.353468	−0.176734	−	0.984259i	$-0.556553\pi$
$421$	−2.27835e11	−0.353468	−0.176734	+	0.984259i	$0.556553\pi$
$422$	0	0
$423$	−2.97281e11	−0.451477
$424$	0	0
$425$	7.50528e11	1.11588
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.87834e10	0.0980451
$430$	0	0
$431$	−6.47351e11	−0.903633	−0.451817	−	0.892111i	$-0.649224\pi$
$431$	−6.47351e11	−0.903633	−0.451817	+	0.892111i	$0.649224\pi$
$432$	0	0
$433$	−5.69898e11	−0.779114	−0.389557	−	0.921002i	$-0.627372\pi$
$433$	−5.69898e11	−0.779114	−0.389557	+	0.921002i	$0.627372\pi$
$434$	0	0
$435$	2.58317e11	0.345901
$436$	0	0
$437$	−3.94621e11	−0.517624
$438$	0	0
$439$	−5.98042e11	−0.768496	−0.384248	−	0.923230i	$-0.625539\pi$
$439$	−5.98042e11	−0.768496	−0.384248	+	0.923230i	$0.625539\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.10867e11	0.383494	0.191747	−	0.981444i	$-0.438585\pi$
$443$	3.10867e11	0.383494	0.191747	+	0.981444i	$0.438585\pi$
$444$	0	0
$445$	−1.49622e11	−0.180873
$446$	0	0
$447$	1.00441e11	0.118995
$448$	0	0
$449$	7.47114e11	0.867517	0.433759	−	0.901029i	$-0.357187\pi$
$449$	7.47114e11	0.867517	0.433759	+	0.901029i	$0.357187\pi$
$450$	0	0
$451$	−8.45964e11	−0.962848
$452$	0	0
$453$	3.19625e11	0.356615
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.54275e12	−1.65452	−0.827260	−	0.561819i	$-0.810102\pi$
$457$	−1.54275e12	−1.65452	−0.827260	+	0.561819i	$0.810102\pi$
$458$	0	0
$459$	−6.85845e11	−0.721221
$460$	0	0
$461$	−1.62766e12	−1.67846	−0.839230	−	0.543777i	$-0.816994\pi$
$461$	−1.62766e12	−1.67846	−0.839230	+	0.543777i	$0.816994\pi$
$462$	0	0
$463$	−1.11591e12	−1.12854	−0.564268	−	0.825592i	$-0.690842\pi$
$463$	−1.11591e12	−1.12854	−0.564268	+	0.825592i	$0.690842\pi$
$464$	0	0
$465$	7.98961e11	0.792478
$466$	0	0
$467$	5.30194e11	0.515832	0.257916	−	0.966167i	$-0.416964\pi$
$467$	5.30194e11	0.515832	0.257916	+	0.966167i	$0.416964\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6.95008e11	0.650722
$472$	0	0
$473$	1.83493e12	1.68556
$474$	0	0
$475$	1.29985e12	1.17158
$476$	0	0
$477$	−1.64943e11	−0.145882
$478$	0	0
$479$	2.10019e12	1.82284	0.911422	−	0.411473i	$-0.134986\pi$
$479$	2.10019e12	1.82284	0.911422	+	0.411473i	$0.134986\pi$
$480$	0	0
$481$	−2.22777e11	−0.189766
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.74483e11	−0.389388
$486$	0	0
$487$	−1.05307e12	−0.848351	−0.424176	−	0.905580i	$-0.639436\pi$
$487$	−1.05307e12	−0.848351	−0.424176	+	0.905580i	$0.639436\pi$
$488$	0	0
$489$	5.69051e10	0.0450050
$490$	0	0
$491$	2.10556e12	1.63494	0.817470	−	0.575971i	$-0.195376\pi$
$491$	2.10556e12	1.63494	0.817470	+	0.575971i	$0.195376\pi$
$492$	0	0
$493$	6.63434e11	0.505809
$494$	0	0
$495$	−3.12360e12	−2.33847
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2.88807e11	0.208523	0.104262	−	0.994550i	$-0.466752\pi$
$499$	2.88807e11	0.208523	0.104262	+	0.994550i	$0.466752\pi$
$500$	0	0
$501$	8.68305e11	0.615747
$502$	0	0
$503$	−5.17681e11	−0.360584	−0.180292	−	0.983613i	$-0.557704\pi$
$503$	−5.17681e11	−0.360584	−0.180292	+	0.983613i	$0.557704\pi$
$504$	0	0
$505$	1.66595e12	1.13986
$506$	0	0
$507$	−6.27278e11	−0.421623
$508$	0	0
$509$	−6.01747e11	−0.397360	−0.198680	−	0.980064i	$-0.563665\pi$
$509$	−6.01747e11	−0.397360	−0.198680	+	0.980064i	$0.563665\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.18782e12	−0.757223
$514$	0	0
$515$	−1.62130e12	−1.01562
$516$	0	0
$517$	1.73093e12	1.06555
$518$	0	0
$519$	8.39316e11	0.507776
$520$	0	0
$521$	1.67285e11	0.0994691	0.0497345	−	0.998762i	$-0.484162\pi$
$521$	1.67285e11	0.0994691	0.0497345	+	0.998762i	$0.484162\pi$
$522$	0	0
$523$	1.57966e12	0.923222	0.461611	−	0.887083i	$-0.347272\pi$
$523$	1.57966e12	0.923222	0.461611	+	0.887083i	$0.347272\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.05196e12	1.15884
$528$	0	0
$529$	−1.29287e12	−0.717804
$530$	0	0
$531$	1.95676e12	1.06810
$532$	0	0
$533$	−1.10592e11	−0.0593543
$534$	0	0
$535$	−2.08968e12	−1.10277
$536$	0	0
$537$	−2.72954e11	−0.141646
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.08736e12	−1.54953	−0.774765	−	0.632250i	$-0.782132\pi$
$541$	−3.08736e12	−1.54953	−0.774765	+	0.632250i	$0.782132\pi$
$542$	0	0
$543$	−9.38904e11	−0.463470
$544$	0	0
$545$	−9.96955e11	−0.484051
$546$	0	0
$547$	2.62136e11	0.125194	0.0625969	−	0.998039i	$-0.480062\pi$
$547$	2.62136e11	0.125194	0.0625969	+	0.998039i	$0.480062\pi$
$548$	0	0
$549$	−7.38997e11	−0.347190
$550$	0	0
$551$	1.14901e12	0.531057
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.26453e12	−1.01312
$556$	0	0
$557$	3.64238e11	0.160338	0.0801691	−	0.996781i	$-0.474454\pi$
$557$	3.64238e11	0.160338	0.0801691	+	0.996781i	$0.474454\pi$
$558$	0	0
$559$	2.39879e11	0.103906
$560$	0	0
$561$	1.79571e12	0.765425
$562$	0	0
$563$	−3.04052e12	−1.27544	−0.637721	−	0.770267i	$-0.720123\pi$
$563$	−3.04052e12	−1.27544	−0.637721	+	0.770267i	$0.720123\pi$
$564$	0	0
$565$	−6.01006e12	−2.48119
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.35845e11	−0.294294	−0.147147	−	0.989115i	$-0.547009\pi$
$569$	−7.35845e11	−0.294294	−0.147147	+	0.989115i	$0.547009\pi$
$570$	0	0
$571$	1.44618e12	0.569324	0.284662	−	0.958628i	$-0.408119\pi$
$571$	1.44618e12	0.569324	0.284662	+	0.958628i	$0.408119\pi$
$572$	0	0
$573$	1.21231e12	0.469806
$574$	0	0
$575$	−1.67422e12	−0.638717
$576$	0	0
$577$	2.26945e12	0.852371	0.426186	−	0.904636i	$-0.359857\pi$
$577$	2.26945e12	0.852371	0.426186	+	0.904636i	$0.359857\pi$
$578$	0	0
$579$	−4.26496e11	−0.157711
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.60391e11	0.344302
$584$	0	0
$585$	−4.08346e11	−0.144154
$586$	0	0
$587$	−3.41977e12	−1.18885	−0.594423	−	0.804153i	$-0.702619\pi$
$587$	−3.41977e12	−1.18885	−0.594423	+	0.804153i	$0.702619\pi$
$588$	0	0
$589$	3.55382e12	1.21668
$590$	0	0
$591$	1.35554e12	0.457056
$592$	0	0
$593$	1.32482e12	0.439959	0.219979	−	0.975505i	$-0.429401\pi$
$593$	1.32482e12	0.439959	0.219979	+	0.975505i	$0.429401\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.13303e12	−0.687248
$598$	0	0
$599$	−4.19936e12	−1.33279	−0.666395	−	0.745599i	$-0.732164\pi$
$599$	−4.19936e12	−1.33279	−0.666395	+	0.745599i	$0.732164\pi$
$600$	0	0
$601$	−1.05682e12	−0.330418	−0.165209	−	0.986259i	$-0.552830\pi$
$601$	−1.05682e12	−0.330418	−0.165209	+	0.986259i	$0.552830\pi$
$602$	0	0
$603$	−8.12761e11	−0.250343
$604$	0	0
$605$	1.32969e13	4.03508
$606$	0	0
$607$	−5.97096e12	−1.78523	−0.892617	−	0.450816i	$-0.851133\pi$
$607$	−5.97096e12	−1.78523	−0.892617	+	0.450816i	$0.851133\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.26283e11	0.0656851
$612$	0	0
$613$	2.80650e12	0.802774	0.401387	−	0.915909i	$-0.368528\pi$
$613$	2.80650e12	0.802774	0.401387	+	0.915909i	$0.368528\pi$
$614$	0	0
$615$	−1.12417e12	−0.316879
$616$	0	0
$617$	1.48302e12	0.411968	0.205984	−	0.978555i	$-0.433961\pi$
$617$	1.48302e12	0.411968	0.205984	+	0.978555i	$0.433961\pi$
$618$	0	0
$619$	−1.53469e12	−0.420158	−0.210079	−	0.977684i	$-0.567372\pi$
$619$	−1.53469e12	−0.420158	−0.210079	+	0.977684i	$0.567372\pi$
$620$	0	0
$621$	1.52993e12	0.412819
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−2.88657e12	−0.756696
$626$	0	0
$627$	3.11001e12	0.803632
$628$	0	0
$629$	−5.81597e12	−1.48147
$630$	0	0
$631$	−4.43498e12	−1.11368	−0.556839	−	0.830620i	$-0.687986\pi$
$631$	−4.43498e12	−1.11368	−0.556839	+	0.830620i	$0.687986\pi$
$632$	0	0
$633$	−3.35088e11	−0.0829549
$634$	0	0
$635$	−8.80061e12	−2.14798
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.29488e12	−1.01905
$640$	0	0
$641$	−4.56257e12	−1.06745	−0.533725	−	0.845658i	$-0.679208\pi$
$641$	−4.56257e12	−1.06745	−0.533725	+	0.845658i	$0.679208\pi$
$642$	0	0
$643$	3.32818e12	0.767818	0.383909	−	0.923371i	$-0.374578\pi$
$643$	3.32818e12	0.767818	0.383909	+	0.923371i	$0.374578\pi$
$644$	0	0
$645$	2.43837e12	0.554729
$646$	0	0
$647$	−2.31374e12	−0.519093	−0.259547	−	0.965731i	$-0.583573\pi$
$647$	−2.31374e12	−0.519093	−0.259547	+	0.965731i	$0.583573\pi$
$648$	0	0
$649$	−1.13933e13	−2.52087
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.03697e12	1.51453	0.757263	−	0.653110i	$-0.226536\pi$
$653$	7.03697e12	1.51453	0.757263	+	0.653110i	$0.226536\pi$
$654$	0	0
$655$	−6.00895e12	−1.27559
$656$	0	0
$657$	−2.83404e12	−0.593419
$658$	0	0
$659$	2.20320e12	0.455060	0.227530	−	0.973771i	$-0.426935\pi$
$659$	2.20320e12	0.455060	0.227530	+	0.973771i	$0.426935\pi$
$660$	0	0
$661$	7.29570e12	1.48648	0.743242	−	0.669022i	$-0.233287\pi$
$661$	7.29570e12	1.48648	0.743242	+	0.669022i	$0.233287\pi$
$662$	0	0
$663$	2.34751e11	0.0471842
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.47994e12	−0.289519
$668$	0	0
$669$	−2.84828e12	−0.549749
$670$	0	0
$671$	4.30284e12	0.819416
$672$	0	0
$673$	−4.47079e12	−0.840073	−0.420036	−	0.907507i	$-0.637983\pi$
$673$	−4.47079e12	−0.840073	−0.420036	+	0.907507i	$0.637983\pi$
$674$	0	0
$675$	−5.03947e12	−0.934367
$676$	0	0
$677$	3.42095e12	0.625890	0.312945	−	0.949771i	$-0.398684\pi$
$677$	3.42095e12	0.625890	0.312945	+	0.949771i	$0.398684\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.02621e12	0.361013
$682$	0	0
$683$	−9.18730e12	−1.61546	−0.807728	−	0.589556i	$-0.799303\pi$
$683$	−9.18730e12	−1.61546	−0.807728	+	0.589556i	$0.799303\pi$
$684$	0	0
$685$	−4.88670e12	−0.848024
$686$	0	0
$687$	−4.37394e11	−0.0749147
$688$	0	0
$689$	1.25551e11	0.0212243
$690$	0	0
$691$	1.88811e12	0.315047	0.157524	−	0.987515i	$-0.449649\pi$
$691$	1.88811e12	0.315047	0.157524	+	0.987515i	$0.449649\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.63148e12	0.915568
$696$	0	0
$697$	−2.88720e12	−0.463371
$698$	0	0
$699$	4.07843e12	0.646169
$700$	0	0
$701$	1.61907e12	0.253242	0.126621	−	0.991951i	$-0.459587\pi$
$701$	1.61907e12	0.253242	0.126621	+	0.991951i	$0.459587\pi$
$702$	0	0
$703$	−1.00727e13	−1.55542
$704$	0	0
$705$	2.30017e12	0.350678
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.06375e13	−1.58099	−0.790497	−	0.612466i	$-0.790178\pi$
$709$	−1.06375e13	−1.58099	−0.790497	+	0.612466i	$0.790178\pi$
$710$	0	0
$711$	4.33735e12	0.636520
$712$	0	0
$713$	−4.57737e12	−0.663305
$714$	0	0
$715$	2.37761e12	0.340223
$716$	0	0
$717$	−1.86770e12	−0.263919
$718$	0	0
$719$	−1.32770e13	−1.85276	−0.926380	−	0.376589i	$-0.877097\pi$
$719$	−1.32770e13	−1.85276	−0.926380	+	0.376589i	$0.877097\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−8.54230e11	−0.116266
$724$	0	0
$725$	4.87480e12	0.655293
$726$	0	0
$727$	−2.60017e12	−0.345221	−0.172611	−	0.984990i	$-0.555220\pi$
$727$	−2.60017e12	−0.345221	−0.172611	+	0.984990i	$0.555220\pi$
$728$	0	0
$729$	−4.86117e11	−0.0637481
$730$	0	0
$731$	6.26244e12	0.811176
$732$	0	0
$733$	1.14818e13	1.46906	0.734532	−	0.678574i	$-0.237402\pi$
$733$	1.14818e13	1.46906	0.734532	+	0.678574i	$0.237402\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.73234e12	0.590843
$738$	0	0
$739$	−7.75984e12	−0.957090	−0.478545	−	0.878063i	$-0.658836\pi$
$739$	−7.75984e12	−0.957090	−0.478545	+	0.878063i	$0.658836\pi$
$740$	0	0
$741$	4.06569e11	0.0495395
$742$	0	0
$743$	−2.58115e12	−0.310717	−0.155358	−	0.987858i	$-0.549653\pi$
$743$	−2.58115e12	−0.310717	−0.155358	+	0.987858i	$0.549653\pi$
$744$	0	0
$745$	3.47191e12	0.412920
$746$	0	0
$747$	−3.65136e12	−0.429055
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8.39208e12	−0.962697	−0.481349	−	0.876529i	$-0.659853\pi$
$751$	−8.39208e12	−0.962697	−0.481349	+	0.876529i	$0.659853\pi$
$752$	0	0
$753$	3.47033e12	0.393363
$754$	0	0
$755$	1.10484e13	1.23748
$756$	0	0
$757$	8.15875e12	0.903009	0.451505	−	0.892269i	$-0.350887\pi$
$757$	8.15875e12	0.903009	0.451505	+	0.892269i	$0.350887\pi$
$758$	0	0
$759$	−4.00573e12	−0.438121
$760$	0	0
$761$	6.27433e12	0.678167	0.339083	−	0.940756i	$-0.389883\pi$
$761$	6.27433e12	0.678167	0.339083	+	0.940756i	$0.389883\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.06606e13	−1.12539
$766$	0	0
$767$	−1.48944e12	−0.155398
$768$	0	0
$769$	−6.12027e12	−0.631106	−0.315553	−	0.948908i	$-0.602190\pi$
$769$	−6.12027e12	−0.631106	−0.315553	+	0.948908i	$0.602190\pi$
$770$	0	0
$771$	1.12306e12	0.114461
$772$	0	0
$773$	6.62875e12	0.667765	0.333883	−	0.942615i	$-0.391641\pi$
$773$	6.62875e12	0.667765	0.333883	+	0.942615i	$0.391641\pi$
$774$	0	0
$775$	1.50775e13	1.50131
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.00037e12	−0.486501
$780$	0	0
$781$	2.50071e13	2.40511
$782$	0	0
$783$	−4.45467e12	−0.423533
$784$	0	0
$785$	2.40241e13	2.25805
$786$	0	0
$787$	−1.19503e13	−1.11043	−0.555216	−	0.831706i	$-0.687364\pi$
$787$	−1.19503e13	−1.11043	−0.555216	+	0.831706i	$0.687364\pi$
$788$	0	0
$789$	−1.68262e12	−0.154575
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.62507e11	0.0505125
$794$	0	0
$795$	1.27623e12	0.113312
$796$	0	0
$797$	1.18887e13	1.04369	0.521844	−	0.853041i	$-0.325244\pi$
$797$	1.18887e13	1.04369	0.521844	+	0.853041i	$0.325244\pi$
$798$	0	0
$799$	5.90751e12	0.512795
$800$	0	0
$801$	1.16025e12	0.0995879
$802$	0	0
$803$	1.65013e13	1.40055
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.67960e12	0.222402
$808$	0	0
$809$	−1.68063e13	−1.37944	−0.689722	−	0.724074i	$-0.742267\pi$
$809$	−1.68063e13	−1.37944	−0.689722	+	0.724074i	$0.742267\pi$
$810$	0	0
$811$	−1.98473e13	−1.61104	−0.805521	−	0.592567i	$-0.798114\pi$
$811$	−1.98473e13	−1.61104	−0.805521	+	0.592567i	$0.798114\pi$
$812$	0	0
$813$	6.20252e12	0.497922
$814$	0	0
$815$	1.96702e12	0.156171
$816$	0	0
$817$	1.08460e13	0.851668
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.83992e12	0.371787	0.185893	−	0.982570i	$-0.440482\pi$
$821$	4.83992e12	0.371787	0.185893	+	0.982570i	$0.440482\pi$
$822$	0	0
$823$	8.41664e12	0.639499	0.319749	−	0.947502i	$-0.396401\pi$
$823$	8.41664e12	0.639499	0.319749	+	0.947502i	$0.396401\pi$
$824$	0	0
$825$	1.31945e13	0.991634
$826$	0	0
$827$	−2.16658e13	−1.61064	−0.805321	−	0.592839i	$-0.798007\pi$
$827$	−2.16658e13	−1.61064	−0.805321	+	0.592839i	$0.798007\pi$
$828$	0	0
$829$	3.67734e12	0.270420	0.135210	−	0.990817i	$-0.456829\pi$
$829$	3.67734e12	0.270420	0.135210	+	0.990817i	$0.456829\pi$
$830$	0	0
$831$	1.08842e13	0.791754
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.00144e13	2.13669
$836$	0	0
$837$	−1.37780e13	−0.970337
$838$	0	0
$839$	1.46942e13	1.02381	0.511903	−	0.859043i	$-0.328941\pi$
$839$	1.46942e13	1.02381	0.511903	+	0.859043i	$0.328941\pi$
$840$	0	0
$841$	−1.01980e13	−0.702967
$842$	0	0
$843$	−7.52924e12	−0.513484
$844$	0	0
$845$	−2.16829e13	−1.46306
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.01366e12	−0.529354
$850$	0	0
$851$	1.29738e13	0.847979
$852$	0	0
$853$	1.99845e13	1.29248	0.646238	−	0.763136i	$-0.276341\pi$
$853$	1.99845e13	1.29248	0.646238	+	0.763136i	$0.276341\pi$
$854$	0	0
$855$	−1.84632e13	−1.18157
$856$	0	0
$857$	−2.11989e13	−1.34245	−0.671226	−	0.741252i	$-0.734232\pi$
$857$	−2.11989e13	−1.34245	−0.671226	+	0.741252i	$0.734232\pi$
$858$	0	0
$859$	−2.51809e13	−1.57798	−0.788992	−	0.614404i	$-0.789397\pi$
$859$	−2.51809e13	−1.57798	−0.788992	+	0.614404i	$0.789397\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.15905e13	1.32500	0.662498	−	0.749064i	$-0.269496\pi$
$863$	2.15905e13	1.32500	0.662498	+	0.749064i	$0.269496\pi$
$864$	0	0
$865$	2.90123e13	1.76202
$866$	0	0
$867$	−9.86700e11	−0.0593061
$868$	0	0
$869$	−2.52544e13	−1.50227
$870$	0	0
$871$	6.18655e11	0.0364222
$872$	0	0
$873$	3.67941e12	0.214395
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.63182e13	1.50230	0.751152	−	0.660129i	$-0.229498\pi$
$877$	2.63182e13	1.50230	0.751152	+	0.660129i	$0.229498\pi$
$878$	0	0
$879$	2.10381e11	0.0118866
$880$	0	0
$881$	−4.23513e12	−0.236851	−0.118425	−	0.992963i	$-0.537785\pi$
$881$	−4.23513e12	−0.236851	−0.118425	+	0.992963i	$0.537785\pi$
$882$	0	0
$883$	−2.56557e13	−1.42024	−0.710119	−	0.704081i	$-0.751359\pi$
$883$	−2.56557e13	−1.42024	−0.710119	+	0.704081i	$0.751359\pi$
$884$	0	0
$885$	−1.51402e13	−0.829634
$886$	0	0
$887$	3.14044e13	1.70347	0.851735	−	0.523973i	$-0.175551\pi$
$887$	3.14044e13	1.70347	0.851735	+	0.523973i	$0.175551\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.75867e13	0.934836
$892$	0	0
$893$	1.02313e13	0.538392
$894$	0	0
$895$	−9.43512e12	−0.491523
$896$	0	0
$897$	−5.23666e11	−0.0270077
$898$	0	0
$899$	1.33278e13	0.680519
$900$	0	0
$901$	3.27772e12	0.165695
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.24548e13	−1.60827
$906$	0	0
$907$	−3.87001e13	−1.89880	−0.949400	−	0.314070i	$-0.898307\pi$
$907$	−3.87001e13	−1.89880	−0.949400	+	0.314070i	$0.898307\pi$
$908$	0	0
$909$	−1.29188e13	−0.627601
$910$	0	0
$911$	1.73436e13	0.834269	0.417135	−	0.908845i	$-0.363034\pi$
$911$	1.73436e13	0.834269	0.417135	+	0.908845i	$0.363034\pi$
$912$	0	0
$913$	2.12602e13	1.01263
$914$	0	0
$915$	5.71789e12	0.269675
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.75232e12	−0.0810387	−0.0405194	−	0.999179i	$-0.512901\pi$
$919$	−1.75232e12	−0.0810387	−0.0405194	+	0.999179i	$0.512901\pi$
$920$	0	0
$921$	1.76614e13	0.808829
$922$	0	0
$923$	3.26916e12	0.148262
$924$	0	0
$925$	−4.27347e13	−1.91930
$926$	0	0
$927$	1.25725e13	0.559194
$928$	0	0
$929$	1.99977e13	0.880865	0.440432	−	0.897786i	$-0.354825\pi$
$929$	1.99977e13	0.880865	0.440432	+	0.897786i	$0.354825\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.44183e12	−0.0622940
$934$	0	0
$935$	6.20716e13	2.65608
$936$	0	0
$937$	−4.62137e12	−0.195859	−0.0979293	−	0.995193i	$-0.531222\pi$
$937$	−4.62137e12	−0.195859	−0.0979293	+	0.995193i	$0.531222\pi$
$938$	0	0
$939$	1.53138e13	0.642816
$940$	0	0
$941$	−1.64959e13	−0.685838	−0.342919	−	0.939365i	$-0.611416\pi$
$941$	−1.64959e13	−0.685838	−0.342919	+	0.939365i	$0.611416\pi$
$942$	0	0
$943$	6.44055e12	0.265228
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.99606e13	0.806490	0.403245	−	0.915092i	$-0.367882\pi$
$947$	1.99606e13	0.806490	0.403245	+	0.915092i	$0.367882\pi$
$948$	0	0
$949$	2.15720e12	0.0863363
$950$	0	0
$951$	1.38153e13	0.547707
$952$	0	0
$953$	−8.83087e12	−0.346805	−0.173402	−	0.984851i	$-0.555476\pi$
$953$	−8.83087e12	−0.346805	−0.173402	+	0.984851i	$0.555476\pi$
$954$	0	0
$955$	4.19056e13	1.63026
$956$	0	0
$957$	1.16634e13	0.449491
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.47825e13	0.559105
$962$	0	0
$963$	1.62046e13	0.607182
$964$	0	0
$965$	−1.47426e13	−0.547268
$966$	0	0
$967$	2.03562e13	0.748647	0.374323	−	0.927298i	$-0.377875\pi$
$967$	2.03562e13	0.748647	0.374323	+	0.927298i	$0.377875\pi$
$968$	0	0
$969$	1.06142e13	0.386748
$970$	0	0
$971$	−7.66888e12	−0.276851	−0.138425	−	0.990373i	$-0.544204\pi$
$971$	−7.66888e12	−0.276851	−0.138425	+	0.990373i	$0.544204\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.72491e12	0.0611288
$976$	0	0
$977$	6.23017e12	0.218763	0.109382	−	0.994000i	$-0.465113\pi$
$977$	6.23017e12	0.218763	0.109382	+	0.994000i	$0.465113\pi$
$978$	0	0
$979$	−6.75563e12	−0.235041
$980$	0	0
$981$	7.73097e12	0.266516
$982$	0	0
$983$	−5.22800e13	−1.78585	−0.892924	−	0.450207i	$-0.851350\pi$
$983$	−5.22800e13	−1.78585	−0.892924	+	0.450207i	$0.851350\pi$
$984$	0	0
$985$	4.68565e13	1.58601
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.39698e13	−0.464308
$990$	0	0
$991$	−3.73672e13	−1.23072	−0.615359	−	0.788247i	$-0.710989\pi$
$991$	−3.73672e13	−1.23072	−0.615359	+	0.788247i	$0.710989\pi$
$992$	0	0
$993$	−7.27273e12	−0.237370
$994$	0	0
$995$	−7.37319e13	−2.38480
$996$	0	0
$997$	−3.66230e13	−1.17389	−0.586943	−	0.809628i	$-0.699669\pi$
$997$	−3.66230e13	−1.17389	−0.586943	+	0.809628i	$0.699669\pi$
$998$	0	0
$999$	3.90517e13	1.24049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.10.a.b.1.1		1
7.6	odd	2		8.10.a.a.1.1	✓	1
21.20	even	2		72.10.a.e.1.1		1
28.27	even	2		16.10.a.c.1.1		1
35.13	even	4		200.10.c.b.49.1		2
35.27	even	4		200.10.c.b.49.2		2
35.34	odd	2		200.10.a.b.1.1		1
56.13	odd	2		64.10.a.f.1.1		1
56.27	even	2		64.10.a.d.1.1		1
84.83	odd	2		144.10.a.n.1.1		1
112.13	odd	4		256.10.b.i.129.1		2
112.27	even	4		256.10.b.c.129.1		2
112.69	odd	4		256.10.b.i.129.2		2
112.83	even	4		256.10.b.c.129.2		2
140.27	odd	4		400.10.c.g.49.1		2
140.83	odd	4		400.10.c.g.49.2		2
140.139	even	2		400.10.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.a.a.1.1	✓	1	7.6	odd	2
16.10.a.c.1.1		1	28.27	even	2
64.10.a.d.1.1		1	56.27	even	2
64.10.a.f.1.1		1	56.13	odd	2
72.10.a.e.1.1		1	21.20	even	2
144.10.a.n.1.1		1	84.83	odd	2
200.10.a.b.1.1		1	35.34	odd	2
200.10.c.b.49.1		2	35.13	even	4
200.10.c.b.49.2		2	35.27	even	4
256.10.b.c.129.1		2	112.27	even	4
256.10.b.c.129.2		2	112.83	even	4
256.10.b.i.129.1		2	112.13	odd	4
256.10.b.i.129.2		2	112.69	odd	4
392.10.a.b.1.1		1	1.1	even	1	trivial
400.10.a.d.1.1		1	140.139	even	2
400.10.c.g.49.1		2	140.27	odd	4
400.10.c.g.49.2		2	140.83	odd	4

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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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