LMFDB - Embedded newform 336.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,8,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("336.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	336.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:	$104.961368563$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} -122.000 q^{5} +343.000 q^{7} +729.000 q^{9} +1012.00 q^{11} +3126.00 q^{13} +3294.00 q^{15} -28294.0 q^{17} +22228.0 q^{19} -9261.00 q^{21} +108640. q^{23} -63241.0 q^{25} -19683.0 q^{27} -41354.0 q^{29} -46656.0 q^{31} -27324.0 q^{33} -41846.0 q^{35} -85714.0 q^{37} -84402.0 q^{39} -155694. q^{41} -926804. q^{43} -88938.0 q^{45} +529152. q^{47} +117649. q^{49} +763938. q^{51} -294066. q^{53} -123464. q^{55} -600156. q^{57} +667292. q^{59} +833430. q^{61} +250047. q^{63} -381372. q^{65} -1.15400e6 q^{67} -2.93328e6 q^{69} -3.84234e6 q^{71} +1.48369e6 q^{73} +1.70751e6 q^{75} +347116. q^{77} +3.76382e6 q^{79} +531441. q^{81} +5.39309e6 q^{83} +3.45187e6 q^{85} +1.11656e6 q^{87} +2.50269e6 q^{89} +1.07222e6 q^{91} +1.25971e6 q^{93} -2.71182e6 q^{95} -4.59755e6 q^{97} +737748. q^{99} +O(q^{100})$

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
$5$	−122.000	−0.436480	−0.218240	−	0.975895i	$-0.570032\pi$
$5$	−122.000	−0.436480	−0.218240	+	0.975895i	$0.570032\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	1012.00	0.229248	0.114624	−	0.993409i	$-0.463434\pi$
$11$	1012.00	0.229248	0.114624	+	0.993409i	$0.463434\pi$
$12$	0	0
$13$	3126.00	0.394627	0.197314	−	0.980340i	$-0.436778\pi$
$13$	3126.00	0.394627	0.197314	+	0.980340i	$0.436778\pi$
$14$	0	0
$15$	3294.00	0.252002
$16$	0	0
$17$	−28294.0	−1.39676	−0.698382	−	0.715725i	$-0.746096\pi$
$17$	−28294.0	−1.39676	−0.698382	+	0.715725i	$0.746096\pi$
$18$	0	0
$19$	22228.0	0.743469	0.371734	−	0.928339i	$-0.378763\pi$
$19$	22228.0	0.743469	0.371734	+	0.928339i	$0.378763\pi$
$20$	0	0
$21$	−9261.00	−0.218218
$22$	0	0
$23$	108640.	1.86184	0.930920	−	0.365223i	$-0.119007\pi$
$23$	108640.	1.86184	0.930920	+	0.365223i	$0.119007\pi$
$24$	0	0
$25$	−63241.0	−0.809485
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−41354.0	−0.314865	−0.157433	−	0.987530i	$-0.550322\pi$
$29$	−41354.0	−0.314865	−0.157433	+	0.987530i	$0.550322\pi$
$30$	0	0
$31$	−46656.0	−0.281282	−0.140641	−	0.990061i	$-0.544916\pi$
$31$	−46656.0	−0.281282	−0.140641	+	0.990061i	$0.544916\pi$
$32$	0	0
$33$	−27324.0	−0.132357
$34$	0	0
$35$	−41846.0	−0.164974
$36$	0	0
$37$	−85714.0	−0.278193	−0.139096	−	0.990279i	$-0.544420\pi$
$37$	−85714.0	−0.278193	−0.139096	+	0.990279i	$0.544420\pi$
$38$	0	0
$39$	−84402.0	−0.227838
$40$	0	0
$41$	−155694.	−0.352800	−0.176400	−	0.984319i	$-0.556445\pi$
$41$	−155694.	−0.352800	−0.176400	+	0.984319i	$0.556445\pi$
$42$	0	0
$43$	−926804.	−1.77766	−0.888829	−	0.458239i	$-0.848481\pi$
$43$	−926804.	−1.77766	−0.888829	+	0.458239i	$0.848481\pi$
$44$	0	0
$45$	−88938.0	−0.145493
$46$	0	0
$47$	529152.	0.743426	0.371713	−	0.928348i	$-0.378770\pi$
$47$	529152.	0.743426	0.371713	+	0.928348i	$0.378770\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	763938.	0.806422
$52$	0	0
$53$	−294066.	−0.271318	−0.135659	−	0.990756i	$-0.543315\pi$
$53$	−294066.	−0.271318	−0.135659	+	0.990756i	$0.543315\pi$
$54$	0	0
$55$	−123464.	−0.100062
$56$	0	0
$57$	−600156.	−0.429242
$58$	0	0
$59$	667292.	0.422994	0.211497	−	0.977379i	$-0.432166\pi$
$59$	667292.	0.422994	0.211497	+	0.977379i	$0.432166\pi$
$60$	0	0
$61$	833430.	0.470126	0.235063	−	0.971980i	$-0.424470\pi$
$61$	833430.	0.470126	0.235063	+	0.971980i	$0.424470\pi$
$62$	0	0
$63$	250047.	0.125988
$64$	0	0
$65$	−381372.	−0.172247
$66$	0	0
$67$	−1.15400e6	−0.468751	−0.234376	−	0.972146i	$-0.575305\pi$
$67$	−1.15400e6	−0.468751	−0.234376	+	0.972146i	$0.575305\pi$
$68$	0	0
$69$	−2.93328e6	−1.07493
$70$	0	0
$71$	−3.84234e6	−1.27406	−0.637032	−	0.770838i	$-0.719838\pi$
$71$	−3.84234e6	−1.27406	−0.637032	+	0.770838i	$0.719838\pi$
$72$	0	0
$73$	1.48369e6	0.446389	0.223194	−	0.974774i	$-0.428352\pi$
$73$	1.48369e6	0.446389	0.223194	+	0.974774i	$0.428352\pi$
$74$	0	0
$75$	1.70751e6	0.467356
$76$	0	0
$77$	347116.	0.0866477
$78$	0	0
$79$	3.76382e6	0.858884	0.429442	−	0.903094i	$-0.358710\pi$
$79$	3.76382e6	0.858884	0.429442	+	0.903094i	$0.358710\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.39309e6	1.03530	0.517648	−	0.855594i	$-0.326808\pi$
$83$	5.39309e6	1.03530	0.517648	+	0.855594i	$0.326808\pi$
$84$	0	0
$85$	3.45187e6	0.609660
$86$	0	0
$87$	1.11656e6	0.181787
$88$	0	0
$89$	2.50269e6	0.376307	0.188153	−	0.982140i	$-0.439750\pi$
$89$	2.50269e6	0.376307	0.188153	+	0.982140i	$0.439750\pi$
$90$	0	0
$91$	1.07222e6	0.149155
$92$	0	0
$93$	1.25971e6	0.162398
$94$	0	0
$95$	−2.71182e6	−0.324510
$96$	0	0
$97$	−4.59755e6	−0.511476	−0.255738	−	0.966746i	$-0.582318\pi$
$97$	−4.59755e6	−0.511476	−0.255738	+	0.966746i	$0.582318\pi$
$98$	0	0
$99$	737748.	0.0764161
$100$	0	0
$101$	2.05285e7	1.98258	0.991292	−	0.131679i	$-0.0420368\pi$
$101$	2.05285e7	1.98258	0.991292	+	0.131679i	$0.0420368\pi$
$102$	0	0
$103$	−1.42334e7	−1.28345	−0.641724	−	0.766936i	$-0.721781\pi$
$103$	−1.42334e7	−1.28345	−0.641724	+	0.766936i	$0.721781\pi$
$104$	0	0
$105$	1.12984e6	0.0952478
$106$	0	0
$107$	9.32024e6	0.735502	0.367751	−	0.929924i	$-0.380128\pi$
$107$	9.32024e6	0.735502	0.367751	+	0.929924i	$0.380128\pi$
$108$	0	0
$109$	4.30625e6	0.318498	0.159249	−	0.987238i	$-0.449093\pi$
$109$	4.30625e6	0.318498	0.159249	+	0.987238i	$0.449093\pi$
$110$	0	0
$111$	2.31428e6	0.160615
$112$	0	0
$113$	6.93570e6	0.452184	0.226092	−	0.974106i	$-0.427405\pi$
$113$	6.93570e6	0.452184	0.226092	+	0.974106i	$0.427405\pi$
$114$	0	0
$115$	−1.32541e7	−0.812657
$116$	0	0
$117$	2.27885e6	0.131542
$118$	0	0
$119$	−9.70484e6	−0.527927
$120$	0	0
$121$	−1.84630e7	−0.947445
$122$	0	0
$123$	4.20374e6	0.203689
$124$	0	0
$125$	1.72467e7	0.789805
$126$	0	0
$127$	3.14502e7	1.36242	0.681209	−	0.732089i	$-0.261455\pi$
$127$	3.14502e7	1.36242	0.681209	+	0.732089i	$0.261455\pi$
$128$	0	0
$129$	2.50237e7	1.02633
$130$	0	0
$131$	4.48802e6	0.174424	0.0872118	−	0.996190i	$-0.472204\pi$
$131$	4.48802e6	0.174424	0.0872118	+	0.996190i	$0.472204\pi$
$132$	0	0
$133$	7.62420e6	0.281005
$134$	0	0
$135$	2.40133e6	0.0840007
$136$	0	0
$137$	1.56256e7	0.519178	0.259589	−	0.965719i	$-0.416413\pi$
$137$	1.56256e7	0.519178	0.259589	+	0.965719i	$0.416413\pi$
$138$	0	0
$139$	5.42435e7	1.71315	0.856577	−	0.516020i	$-0.172587\pi$
$139$	5.42435e7	1.71315	0.856577	+	0.516020i	$0.172587\pi$
$140$	0	0
$141$	−1.42871e7	−0.429217
$142$	0	0
$143$	3.16351e6	0.0904677
$144$	0	0
$145$	5.04519e6	0.137432
$146$	0	0
$147$	−3.17652e6	−0.0824786
$148$	0	0
$149$	−3.24950e7	−0.804756	−0.402378	−	0.915474i	$-0.631816\pi$
$149$	−3.24950e7	−0.804756	−0.402378	+	0.915474i	$0.631816\pi$
$150$	0	0
$151$	7.63338e7	1.80425	0.902127	−	0.431470i	$-0.142005\pi$
$151$	7.63338e7	1.80425	0.902127	+	0.431470i	$0.142005\pi$
$152$	0	0
$153$	−2.06263e7	−0.465588
$154$	0	0
$155$	5.69203e6	0.122774
$156$	0	0
$157$	−2.92421e7	−0.603059	−0.301529	−	0.953457i	$-0.597497\pi$
$157$	−2.92421e7	−0.603059	−0.301529	+	0.953457i	$0.597497\pi$
$158$	0	0
$159$	7.93978e6	0.156646
$160$	0	0
$161$	3.72635e7	0.703709
$162$	0	0
$163$	9.50898e7	1.71980	0.859899	−	0.510465i	$-0.170527\pi$
$163$	9.50898e7	1.71980	0.859899	+	0.510465i	$0.170527\pi$
$164$	0	0
$165$	3.33353e6	0.0577711
$166$	0	0
$167$	2.29949e7	0.382054	0.191027	−	0.981585i	$-0.438818\pi$
$167$	2.29949e7	0.382054	0.191027	+	0.981585i	$0.438818\pi$
$168$	0	0
$169$	−5.29766e7	−0.844269
$170$	0	0
$171$	1.62042e7	0.247823
$172$	0	0
$173$	1.19486e8	1.75451	0.877255	−	0.480025i	$-0.159372\pi$
$173$	1.19486e8	1.75451	0.877255	+	0.480025i	$0.159372\pi$
$174$	0	0
$175$	−2.16917e7	−0.305956
$176$	0	0
$177$	−1.80169e7	−0.244216
$178$	0	0
$179$	−9.38255e7	−1.22274	−0.611372	−	0.791344i	$-0.709382\pi$
$179$	−9.38255e7	−1.22274	−0.611372	+	0.791344i	$0.709382\pi$
$180$	0	0
$181$	1.48834e8	1.86564	0.932819	−	0.360346i	$-0.117341\pi$
$181$	1.48834e8	1.86564	0.932819	+	0.360346i	$0.117341\pi$
$182$	0	0
$183$	−2.25026e7	−0.271428
$184$	0	0
$185$	1.04571e7	0.121426
$186$	0	0
$187$	−2.86335e7	−0.320206
$188$	0	0
$189$	−6.75127e6	−0.0727393
$190$	0	0
$191$	1.01406e8	1.05304	0.526522	−	0.850161i	$-0.323496\pi$
$191$	1.01406e8	1.05304	0.526522	+	0.850161i	$0.323496\pi$
$192$	0	0
$193$	7.05896e7	0.706789	0.353395	−	0.935474i	$-0.385027\pi$
$193$	7.05896e7	0.706789	0.353395	+	0.935474i	$0.385027\pi$
$194$	0	0
$195$	1.02970e7	0.0994469
$196$	0	0
$197$	5.02307e7	0.468099	0.234049	−	0.972225i	$-0.424802\pi$
$197$	5.02307e7	0.468099	0.234049	+	0.972225i	$0.424802\pi$
$198$	0	0
$199$	−2.08897e8	−1.87909	−0.939544	−	0.342429i	$-0.888750\pi$
$199$	−2.08897e8	−1.87909	−0.939544	+	0.342429i	$0.888750\pi$
$200$	0	0
$201$	3.11579e7	0.270634
$202$	0	0
$203$	−1.41844e7	−0.119008
$204$	0	0
$205$	1.89947e7	0.153990
$206$	0	0
$207$	7.91986e7	0.620613
$208$	0	0
$209$	2.24947e7	0.170439
$210$	0	0
$211$	8.44931e7	0.619202	0.309601	−	0.950866i	$-0.399804\pi$
$211$	8.44931e7	0.619202	0.309601	+	0.950866i	$0.399804\pi$
$212$	0	0
$213$	1.03743e8	0.735581
$214$	0	0
$215$	1.13070e8	0.775913
$216$	0	0
$217$	−1.60030e7	−0.106315
$218$	0	0
$219$	−4.00596e7	−0.257723
$220$	0	0
$221$	−8.84470e7	−0.551201
$222$	0	0
$223$	−1.50175e8	−0.906837	−0.453418	−	0.891298i	$-0.649796\pi$
$223$	−1.50175e8	−0.906837	−0.453418	+	0.891298i	$0.649796\pi$
$224$	0	0
$225$	−4.61027e7	−0.269828
$226$	0	0
$227$	3.02874e8	1.71859	0.859293	−	0.511484i	$-0.170904\pi$
$227$	3.02874e8	1.71859	0.859293	+	0.511484i	$0.170904\pi$
$228$	0	0
$229$	−5.31373e7	−0.292399	−0.146199	−	0.989255i	$-0.546704\pi$
$229$	−5.31373e7	−0.292399	−0.146199	+	0.989255i	$0.546704\pi$
$230$	0	0
$231$	−9.37213e6	−0.0500261
$232$	0	0
$233$	−1.36351e8	−0.706174	−0.353087	−	0.935590i	$-0.614868\pi$
$233$	−1.36351e8	−0.706174	−0.353087	+	0.935590i	$0.614868\pi$
$234$	0	0
$235$	−6.45565e7	−0.324491
$236$	0	0
$237$	−1.01623e8	−0.495877
$238$	0	0
$239$	−2.26802e8	−1.07462	−0.537309	−	0.843385i	$-0.680559\pi$
$239$	−2.26802e8	−1.07462	−0.537309	+	0.843385i	$0.680559\pi$
$240$	0	0
$241$	4.05053e8	1.86403	0.932014	−	0.362423i	$-0.118050\pi$
$241$	4.05053e8	1.86403	0.932014	+	0.362423i	$0.118050\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−1.43532e7	−0.0623544
$246$	0	0
$247$	6.94847e7	0.293393
$248$	0	0
$249$	−1.45613e8	−0.597728
$250$	0	0
$251$	3.08065e7	0.122966	0.0614828	−	0.998108i	$-0.480417\pi$
$251$	3.08065e7	0.122966	0.0614828	+	0.998108i	$0.480417\pi$
$252$	0	0
$253$	1.09944e8	0.426824
$254$	0	0
$255$	−9.32004e7	−0.351988
$256$	0	0
$257$	−2.75132e8	−1.01105	−0.505527	−	0.862811i	$-0.668702\pi$
$257$	−2.75132e8	−1.01105	−0.505527	+	0.862811i	$0.668702\pi$
$258$	0	0
$259$	−2.93999e7	−0.105147
$260$	0	0
$261$	−3.01471e7	−0.104955
$262$	0	0
$263$	4.83996e8	1.64058	0.820288	−	0.571950i	$-0.193813\pi$
$263$	4.83996e8	1.64058	0.820288	+	0.571950i	$0.193813\pi$
$264$	0	0
$265$	3.58761e7	0.118425
$266$	0	0
$267$	−6.75726e7	−0.217261
$268$	0	0
$269$	2.47375e8	0.774858	0.387429	−	0.921900i	$-0.373363\pi$
$269$	2.47375e8	0.774858	0.387429	+	0.921900i	$0.373363\pi$
$270$	0	0
$271$	2.28287e8	0.696768	0.348384	−	0.937352i	$-0.386731\pi$
$271$	2.28287e8	0.696768	0.348384	+	0.937352i	$0.386731\pi$
$272$	0	0
$273$	−2.89499e7	−0.0861148
$274$	0	0
$275$	−6.39999e7	−0.185573
$276$	0	0
$277$	−3.55116e8	−1.00390	−0.501950	−	0.864896i	$-0.667384\pi$
$277$	−3.55116e8	−1.00390	−0.501950	+	0.864896i	$0.667384\pi$
$278$	0	0
$279$	−3.40122e7	−0.0937606
$280$	0	0
$281$	2.28167e8	0.613453	0.306726	−	0.951798i	$-0.400766\pi$
$281$	2.28167e8	0.613453	0.306726	+	0.951798i	$0.400766\pi$
$282$	0	0
$283$	2.13791e8	0.560709	0.280354	−	0.959897i	$-0.409548\pi$
$283$	2.13791e8	0.560709	0.280354	+	0.959897i	$0.409548\pi$
$284$	0	0
$285$	7.32190e7	0.187356
$286$	0	0
$287$	−5.34030e7	−0.133346
$288$	0	0
$289$	3.90212e8	0.950950
$290$	0	0
$291$	1.24134e8	0.295301
$292$	0	0
$293$	3.78816e8	0.879816	0.439908	−	0.898043i	$-0.355011\pi$
$293$	3.78816e8	0.879816	0.439908	+	0.898043i	$0.355011\pi$
$294$	0	0
$295$	−8.14096e7	−0.184628
$296$	0	0
$297$	−1.99192e7	−0.0441189
$298$	0	0
$299$	3.39609e8	0.734733
$300$	0	0
$301$	−3.17894e8	−0.671892
$302$	0	0
$303$	−5.54269e8	−1.14465
$304$	0	0
$305$	−1.01678e8	−0.205201
$306$	0	0
$307$	−6.33814e8	−1.25019	−0.625097	−	0.780547i	$-0.714941\pi$
$307$	−6.33814e8	−1.25019	−0.625097	+	0.780547i	$0.714941\pi$
$308$	0	0
$309$	3.84302e8	0.740999
$310$	0	0
$311$	3.90195e8	0.735564	0.367782	−	0.929912i	$-0.380117\pi$
$311$	3.90195e8	0.735564	0.367782	+	0.929912i	$0.380117\pi$
$312$	0	0
$313$	−1.00444e9	−1.85148	−0.925741	−	0.378158i	$-0.876558\pi$
$313$	−1.00444e9	−1.85148	−0.925741	+	0.378158i	$0.876558\pi$
$314$	0	0
$315$	−3.05057e7	−0.0549914
$316$	0	0
$317$	2.42975e8	0.428404	0.214202	−	0.976789i	$-0.431285\pi$
$317$	2.42975e8	0.428404	0.214202	+	0.976789i	$0.431285\pi$
$318$	0	0
$319$	−4.18502e7	−0.0721823
$320$	0	0
$321$	−2.51647e8	−0.424642
$322$	0	0
$323$	−6.28919e8	−1.03845
$324$	0	0
$325$	−1.97691e8	−0.319445
$326$	0	0
$327$	−1.16269e8	−0.183885
$328$	0	0
$329$	1.81499e8	0.280989
$330$	0	0
$331$	2.83517e7	0.0429715	0.0214858	−	0.999769i	$-0.493160\pi$
$331$	2.83517e7	0.0429715	0.0214858	+	0.999769i	$0.493160\pi$
$332$	0	0
$333$	−6.24855e7	−0.0927309
$334$	0	0
$335$	1.40788e8	0.204601
$336$	0	0
$337$	−1.01675e9	−1.44714	−0.723572	−	0.690249i	$-0.757501\pi$
$337$	−1.01675e9	−1.44714	−0.723572	+	0.690249i	$0.757501\pi$
$338$	0	0
$339$	−1.87264e8	−0.261069
$340$	0	0
$341$	−4.72159e7	−0.0644834
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	3.57860e8	0.469188
$346$	0	0
$347$	9.70494e8	1.24692	0.623461	−	0.781854i	$-0.285726\pi$
$347$	9.70494e8	1.24692	0.623461	+	0.781854i	$0.285726\pi$
$348$	0	0
$349$	8.84145e8	1.11336	0.556678	−	0.830728i	$-0.312076\pi$
$349$	8.84145e8	1.11336	0.556678	+	0.830728i	$0.312076\pi$
$350$	0	0
$351$	−6.15291e7	−0.0759461
$352$	0	0
$353$	−1.09046e9	−1.31947	−0.659734	−	0.751499i	$-0.729331\pi$
$353$	−1.09046e9	−1.31947	−0.659734	+	0.751499i	$0.729331\pi$
$354$	0	0
$355$	4.68765e8	0.556104
$356$	0	0
$357$	2.62031e8	0.304799
$358$	0	0
$359$	−1.03373e9	−1.17917	−0.589585	−	0.807707i	$-0.700709\pi$
$359$	−1.03373e9	−1.17917	−0.589585	+	0.807707i	$0.700709\pi$
$360$	0	0
$361$	−3.99788e8	−0.447254
$362$	0	0
$363$	4.98502e8	0.547008
$364$	0	0
$365$	−1.81010e8	−0.194840
$366$	0	0
$367$	2.42183e8	0.255748	0.127874	−	0.991790i	$-0.459185\pi$
$367$	2.42183e8	0.255748	0.127874	+	0.991790i	$0.459185\pi$
$368$	0	0
$369$	−1.13501e8	−0.117600
$370$	0	0
$371$	−1.00865e8	−0.102549
$372$	0	0
$373$	−1.43769e9	−1.43444	−0.717222	−	0.696845i	$-0.754587\pi$
$373$	−1.43769e9	−1.43444	−0.717222	+	0.696845i	$0.754587\pi$
$374$	0	0
$375$	−4.65660e8	−0.455994
$376$	0	0
$377$	−1.29273e8	−0.124254
$378$	0	0
$379$	−1.20169e9	−1.13385	−0.566924	−	0.823770i	$-0.691867\pi$
$379$	−1.20169e9	−1.13385	−0.566924	+	0.823770i	$0.691867\pi$
$380$	0	0
$381$	−8.49155e8	−0.786592
$382$	0	0
$383$	−6.30859e8	−0.573768	−0.286884	−	0.957965i	$-0.592619\pi$
$383$	−6.30859e8	−0.573768	−0.286884	+	0.957965i	$0.592619\pi$
$384$	0	0
$385$	−4.23482e7	−0.0378200
$386$	0	0
$387$	−6.75640e8	−0.592553
$388$	0	0
$389$	−1.42186e9	−1.22471	−0.612354	−	0.790584i	$-0.709777\pi$
$389$	−1.42186e9	−1.22471	−0.612354	+	0.790584i	$0.709777\pi$
$390$	0	0
$391$	−3.07386e9	−2.60055
$392$	0	0
$393$	−1.21177e8	−0.100704
$394$	0	0
$395$	−4.59187e8	−0.374886
$396$	0	0
$397$	7.04969e7	0.0565462	0.0282731	−	0.999600i	$-0.490999\pi$
$397$	7.04969e7	0.0565462	0.0282731	+	0.999600i	$0.490999\pi$
$398$	0	0
$399$	−2.05854e8	−0.162238
$400$	0	0
$401$	2.02322e9	1.56689	0.783444	−	0.621462i	$-0.213461\pi$
$401$	2.02322e9	1.56689	0.783444	+	0.621462i	$0.213461\pi$
$402$	0	0
$403$	−1.45847e8	−0.111001
$404$	0	0
$405$	−6.48358e7	−0.0484978
$406$	0	0
$407$	−8.67426e7	−0.0637752
$408$	0	0
$409$	1.43726e9	1.03874	0.519368	−	0.854551i	$-0.326168\pi$
$409$	1.43726e9	1.03874	0.519368	+	0.854551i	$0.326168\pi$
$410$	0	0
$411$	−4.21892e8	−0.299747
$412$	0	0
$413$	2.28881e8	0.159877
$414$	0	0
$415$	−6.57957e8	−0.451886
$416$	0	0
$417$	−1.46458e9	−0.989090
$418$	0	0
$419$	−8.62098e7	−0.0572542	−0.0286271	−	0.999590i	$-0.509114\pi$
$419$	−8.62098e7	−0.0572542	−0.0286271	+	0.999590i	$0.509114\pi$
$420$	0	0
$421$	−1.18254e9	−0.772378	−0.386189	−	0.922420i	$-0.626209\pi$
$421$	−1.18254e9	−0.772378	−0.386189	+	0.922420i	$0.626209\pi$
$422$	0	0
$423$	3.85752e8	0.247809
$424$	0	0
$425$	1.78934e9	1.13066
$426$	0	0
$427$	2.85866e8	0.177691
$428$	0	0
$429$	−8.54148e7	−0.0522315
$430$	0	0
$431$	−6.04657e8	−0.363780	−0.181890	−	0.983319i	$-0.558222\pi$
$431$	−6.04657e8	−0.363780	−0.181890	+	0.983319i	$0.558222\pi$
$432$	0	0
$433$	7.16394e7	0.0424076	0.0212038	−	0.999775i	$-0.493250\pi$
$433$	7.16394e7	0.0424076	0.0212038	+	0.999775i	$0.493250\pi$
$434$	0	0
$435$	−1.36220e8	−0.0793467
$436$	0	0
$437$	2.41485e9	1.38422
$438$	0	0
$439$	−1.94917e8	−0.109958	−0.0549788	−	0.998488i	$-0.517509\pi$
$439$	−1.94917e8	−0.109958	−0.0549788	+	0.998488i	$0.517509\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	0	0
$443$	2.08018e9	1.13681	0.568405	−	0.822749i	$-0.307561\pi$
$443$	2.08018e9	1.13681	0.568405	+	0.822749i	$0.307561\pi$
$444$	0	0
$445$	−3.05328e8	−0.164251
$446$	0	0
$447$	8.77364e8	0.464626
$448$	0	0
$449$	1.78606e9	0.931180	0.465590	−	0.885000i	$-0.345842\pi$
$449$	1.78606e9	0.931180	0.465590	+	0.885000i	$0.345842\pi$
$450$	0	0
$451$	−1.57562e8	−0.0808788
$452$	0	0
$453$	−2.06101e9	−1.04169
$454$	0	0
$455$	−1.30811e8	−0.0651033
$456$	0	0
$457$	−2.24308e9	−1.09936	−0.549678	−	0.835377i	$-0.685249\pi$
$457$	−2.24308e9	−1.09936	−0.549678	+	0.835377i	$0.685249\pi$
$458$	0	0
$459$	5.56911e8	0.268807
$460$	0	0
$461$	−2.60254e9	−1.23721	−0.618607	−	0.785701i	$-0.712303\pi$
$461$	−2.60254e9	−1.23721	−0.618607	+	0.785701i	$0.712303\pi$
$462$	0	0
$463$	7.12557e8	0.333646	0.166823	−	0.985987i	$-0.446649\pi$
$463$	7.12557e8	0.333646	0.166823	+	0.985987i	$0.446649\pi$
$464$	0	0
$465$	−1.53685e8	−0.0708836
$466$	0	0
$467$	6.33469e8	0.287817	0.143908	−	0.989591i	$-0.454033\pi$
$467$	6.33469e8	0.287817	0.143908	+	0.989591i	$0.454033\pi$
$468$	0	0
$469$	−3.95821e8	−0.177171
$470$	0	0
$471$	7.89536e8	0.348176
$472$	0	0
$473$	−9.37926e8	−0.407525
$474$	0	0
$475$	−1.40572e9	−0.601827
$476$	0	0
$477$	−2.14374e8	−0.0904395
$478$	0	0
$479$	2.20552e8	0.0916931	0.0458466	−	0.998948i	$-0.485401\pi$
$479$	2.20552e8	0.0916931	0.0458466	+	0.998948i	$0.485401\pi$
$480$	0	0
$481$	−2.67942e8	−0.109782
$482$	0	0
$483$	−1.00612e9	−0.406287
$484$	0	0
$485$	5.60901e8	0.223249
$486$	0	0
$487$	−3.92932e9	−1.54158	−0.770791	−	0.637088i	$-0.780139\pi$
$487$	−3.92932e9	−1.54158	−0.770791	+	0.637088i	$0.780139\pi$
$488$	0	0
$489$	−2.56742e9	−0.992925
$490$	0	0
$491$	3.28904e9	1.25396	0.626980	−	0.779036i	$-0.284291\pi$
$491$	3.28904e9	1.25396	0.626980	+	0.779036i	$0.284291\pi$
$492$	0	0
$493$	1.17007e9	0.439792
$494$	0	0
$495$	−9.00053e7	−0.0333541
$496$	0	0
$497$	−1.31792e9	−0.481551
$498$	0	0
$499$	4.30630e9	1.55150	0.775752	−	0.631038i	$-0.217371\pi$
$499$	4.30630e9	1.55150	0.775752	+	0.631038i	$0.217371\pi$
$500$	0	0
$501$	−6.20863e8	−0.220579
$502$	0	0
$503$	2.56392e9	0.898291	0.449145	−	0.893459i	$-0.351729\pi$
$503$	2.56392e9	0.898291	0.449145	+	0.893459i	$0.351729\pi$
$504$	0	0
$505$	−2.50447e9	−0.865360
$506$	0	0
$507$	1.43037e9	0.487439
$508$	0	0
$509$	4.20386e9	1.41298	0.706490	−	0.707723i	$-0.250278\pi$
$509$	4.20386e9	1.41298	0.706490	+	0.707723i	$0.250278\pi$
$510$	0	0
$511$	5.08906e8	0.168719
$512$	0	0
$513$	−4.37514e8	−0.143081
$514$	0	0
$515$	1.73647e9	0.560200
$516$	0	0
$517$	5.35502e8	0.170429
$518$	0	0
$519$	−3.22612e9	−1.01297
$520$	0	0
$521$	4.50770e9	1.39644	0.698220	−	0.715883i	$-0.253976\pi$
$521$	4.50770e9	1.39644	0.698220	+	0.715883i	$0.253976\pi$
$522$	0	0
$523$	5.46352e9	1.67000	0.835000	−	0.550250i	$-0.185468\pi$
$523$	5.46352e9	1.67000	0.835000	+	0.550250i	$0.185468\pi$
$524$	0	0
$525$	5.85675e8	0.176644
$526$	0	0
$527$	1.32008e9	0.392884
$528$	0	0
$529$	8.39782e9	2.46645
$530$	0	0
$531$	4.86456e8	0.140998
$532$	0	0
$533$	−4.86699e8	−0.139224
$534$	0	0
$535$	−1.13707e9	−0.321032
$536$	0	0
$537$	2.53329e9	0.705951
$538$	0	0
$539$	1.19061e8	0.0327498
$540$	0	0
$541$	−6.58629e9	−1.78834	−0.894171	−	0.447725i	$-0.852234\pi$
$541$	−6.58629e9	−1.78834	−0.894171	+	0.447725i	$0.852234\pi$
$542$	0	0
$543$	−4.01852e9	−1.07713
$544$	0	0
$545$	−5.25362e8	−0.139018
$546$	0	0
$547$	1.71749e9	0.448681	0.224340	−	0.974511i	$-0.427977\pi$
$547$	1.71749e9	0.448681	0.224340	+	0.974511i	$0.427977\pi$
$548$	0	0
$549$	6.07570e8	0.156709
$550$	0	0
$551$	−9.19217e8	−0.234092
$552$	0	0
$553$	1.29099e9	0.324628
$554$	0	0
$555$	−2.82342e8	−0.0701051
$556$	0	0
$557$	4.94264e9	1.21190	0.605949	−	0.795504i	$-0.292794\pi$
$557$	4.94264e9	1.21190	0.605949	+	0.795504i	$0.292794\pi$
$558$	0	0
$559$	−2.89719e9	−0.701513
$560$	0	0
$561$	7.73105e8	0.184871
$562$	0	0
$563$	3.38308e9	0.798974	0.399487	−	0.916739i	$-0.369188\pi$
$563$	3.38308e9	0.798974	0.399487	+	0.916739i	$0.369188\pi$
$564$	0	0
$565$	−8.46155e8	−0.197370
$566$	0	0
$567$	1.82284e8	0.0419961
$568$	0	0
$569$	−2.24635e9	−0.511193	−0.255596	−	0.966784i	$-0.582272\pi$
$569$	−2.24635e9	−0.511193	−0.255596	+	0.966784i	$0.582272\pi$
$570$	0	0
$571$	6.81094e8	0.153102	0.0765509	−	0.997066i	$-0.475609\pi$
$571$	6.81094e8	0.153102	0.0765509	+	0.997066i	$0.475609\pi$
$572$	0	0
$573$	−2.73796e9	−0.607976
$574$	0	0
$575$	−6.87050e9	−1.50713
$576$	0	0
$577$	−7.48296e8	−0.162165	−0.0810827	−	0.996707i	$-0.525838\pi$
$577$	−7.48296e8	−0.162165	−0.0810827	+	0.996707i	$0.525838\pi$
$578$	0	0
$579$	−1.90592e9	−0.408065
$580$	0	0
$581$	1.84983e9	0.391305
$582$	0	0
$583$	−2.97595e8	−0.0621993
$584$	0	0
$585$	−2.78020e8	−0.0574157
$586$	0	0
$587$	6.17294e9	1.25968	0.629838	−	0.776727i	$-0.283121\pi$
$587$	6.17294e9	1.25968	0.629838	+	0.776727i	$0.283121\pi$
$588$	0	0
$589$	−1.03707e9	−0.209124
$590$	0	0
$591$	−1.35623e9	−0.270257
$592$	0	0
$593$	1.00425e10	1.97765	0.988826	−	0.149071i	$-0.0476284\pi$
$593$	1.00425e10	1.97765	0.988826	+	0.149071i	$0.0476284\pi$
$594$	0	0
$595$	1.18399e9	0.230430
$596$	0	0
$597$	5.64023e9	1.08489
$598$	0	0
$599$	1.32244e9	0.251410	0.125705	−	0.992068i	$-0.459881\pi$
$599$	1.32244e9	0.251410	0.125705	+	0.992068i	$0.459881\pi$
$600$	0	0
$601$	−4.48600e8	−0.0842944	−0.0421472	−	0.999111i	$-0.513420\pi$
$601$	−4.48600e8	−0.0842944	−0.0421472	+	0.999111i	$0.513420\pi$
$602$	0	0
$603$	−8.41263e8	−0.156250
$604$	0	0
$605$	2.25249e9	0.413541
$606$	0	0
$607$	−8.91500e8	−0.161793	−0.0808967	−	0.996722i	$-0.525778\pi$
$607$	−8.91500e8	−0.161793	−0.0808967	+	0.996722i	$0.525778\pi$
$608$	0	0
$609$	3.82979e8	0.0687092
$610$	0	0
$611$	1.65413e9	0.293376
$612$	0	0
$613$	3.51417e9	0.616185	0.308092	−	0.951356i	$-0.400309\pi$
$613$	3.51417e9	0.616185	0.308092	+	0.951356i	$0.400309\pi$
$614$	0	0
$615$	−5.12856e8	−0.0889063
$616$	0	0
$617$	3.78067e9	0.647993	0.323996	−	0.946058i	$-0.394973\pi$
$617$	3.78067e9	0.647993	0.323996	+	0.946058i	$0.394973\pi$
$618$	0	0
$619$	−2.68966e9	−0.455807	−0.227903	−	0.973684i	$-0.573187\pi$
$619$	−2.68966e9	−0.455807	−0.227903	+	0.973684i	$0.573187\pi$
$620$	0	0
$621$	−2.13836e9	−0.358311
$622$	0	0
$623$	8.58423e8	0.142231
$624$	0	0
$625$	2.83661e9	0.464750
$626$	0	0
$627$	−6.07358e8	−0.0984030
$628$	0	0
$629$	2.42519e9	0.388570
$630$	0	0
$631$	6.91824e9	1.09621	0.548104	−	0.836410i	$-0.315350\pi$
$631$	6.91824e9	1.09621	0.548104	+	0.836410i	$0.315350\pi$
$632$	0	0
$633$	−2.28131e9	−0.357497
$634$	0	0
$635$	−3.83692e9	−0.594668
$636$	0	0
$637$	3.67771e8	0.0563753
$638$	0	0
$639$	−2.80106e9	−0.424688
$640$	0	0
$641$	−5.49959e9	−0.824759	−0.412379	−	0.911012i	$-0.635302\pi$
$641$	−5.49959e9	−0.824759	−0.412379	+	0.911012i	$0.635302\pi$
$642$	0	0
$643$	1.35931e9	0.201641	0.100821	−	0.994905i	$-0.467853\pi$
$643$	1.35931e9	0.201641	0.100821	+	0.994905i	$0.467853\pi$
$644$	0	0
$645$	−3.05289e9	−0.447974
$646$	0	0
$647$	7.29185e9	1.05846	0.529228	−	0.848480i	$-0.322482\pi$
$647$	7.29185e9	1.05846	0.529228	+	0.848480i	$0.322482\pi$
$648$	0	0
$649$	6.75300e8	0.0969706
$650$	0	0
$651$	4.32081e8	0.0613807
$652$	0	0
$653$	−9.96884e8	−0.140103	−0.0700517	−	0.997543i	$-0.522316\pi$
$653$	−9.96884e8	−0.140103	−0.0700517	+	0.997543i	$0.522316\pi$
$654$	0	0
$655$	−5.47538e8	−0.0761325
$656$	0	0
$657$	1.08161e9	0.148796
$658$	0	0
$659$	1.35118e10	1.83914	0.919568	−	0.392932i	$-0.128539\pi$
$659$	1.35118e10	1.83914	0.919568	+	0.392932i	$0.128539\pi$
$660$	0	0
$661$	−7.79936e9	−1.05040	−0.525199	−	0.850979i	$-0.676009\pi$
$661$	−7.79936e9	−1.05040	−0.525199	+	0.850979i	$0.676009\pi$
$662$	0	0
$663$	2.38807e9	0.318236
$664$	0	0
$665$	−9.30153e8	−0.122653
$666$	0	0
$667$	−4.49270e9	−0.586228
$668$	0	0
$669$	4.05471e9	0.523563
$670$	0	0
$671$	8.43431e8	0.107776
$672$	0	0
$673$	−6.85231e9	−0.866531	−0.433265	−	0.901266i	$-0.642639\pi$
$673$	−6.85231e9	−0.866531	−0.433265	+	0.901266i	$0.642639\pi$
$674$	0	0
$675$	1.24477e9	0.155785
$676$	0	0
$677$	−7.65704e9	−0.948419	−0.474210	−	0.880412i	$-0.657266\pi$
$677$	−7.65704e9	−0.948419	−0.474210	+	0.880412i	$0.657266\pi$
$678$	0	0
$679$	−1.57696e9	−0.193320
$680$	0	0
$681$	−8.17759e9	−0.992226
$682$	0	0
$683$	5.52675e9	0.663739	0.331870	−	0.943325i	$-0.392321\pi$
$683$	5.52675e9	0.663739	0.331870	+	0.943325i	$0.392321\pi$
$684$	0	0
$685$	−1.90633e9	−0.226611
$686$	0	0
$687$	1.43471e9	0.168817
$688$	0	0
$689$	−9.19250e8	−0.107070
$690$	0	0
$691$	−1.51927e9	−0.175171	−0.0875854	−	0.996157i	$-0.527915\pi$
$691$	−1.51927e9	−0.175171	−0.0875854	+	0.996157i	$0.527915\pi$
$692$	0	0
$693$	2.53048e8	0.0288826
$694$	0	0
$695$	−6.61771e9	−0.747758
$696$	0	0
$697$	4.40521e9	0.492778
$698$	0	0
$699$	3.68147e9	0.407710
$700$	0	0
$701$	6.81967e9	0.747739	0.373869	−	0.927481i	$-0.378031\pi$
$701$	6.81967e9	0.747739	0.373869	+	0.927481i	$0.378031\pi$
$702$	0	0
$703$	−1.90525e9	−0.206828
$704$	0	0
$705$	1.74303e9	0.187345
$706$	0	0
$707$	7.04127e9	0.749347
$708$	0	0
$709$	1.21699e10	1.28241	0.641205	−	0.767370i	$-0.278435\pi$
$709$	1.21699e10	1.28241	0.641205	+	0.767370i	$0.278435\pi$
$710$	0	0
$711$	2.74383e9	0.286295
$712$	0	0
$713$	−5.06871e9	−0.523702
$714$	0	0
$715$	−3.85948e8	−0.0394874
$716$	0	0
$717$	6.12365e9	0.620431
$718$	0	0
$719$	−2.00025e9	−0.200693	−0.100347	−	0.994953i	$-0.531995\pi$
$719$	−2.00025e9	−0.200693	−0.100347	+	0.994953i	$0.531995\pi$
$720$	0	0
$721$	−4.88206e9	−0.485098
$722$	0	0
$723$	−1.09364e10	−1.07620
$724$	0	0
$725$	2.61527e9	0.254879
$726$	0	0
$727$	−1.21193e10	−1.16979	−0.584895	−	0.811109i	$-0.698864\pi$
$727$	−1.21193e10	−1.16979	−0.584895	+	0.811109i	$0.698864\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	2.62230e10	2.48297
$732$	0	0
$733$	1.30453e10	1.22346	0.611732	−	0.791065i	$-0.290473\pi$
$733$	1.30453e10	1.22346	0.611732	+	0.791065i	$0.290473\pi$
$734$	0	0
$735$	3.87536e8	0.0360003
$736$	0	0
$737$	−1.16784e9	−0.107460
$738$	0	0
$739$	−2.27587e9	−0.207440	−0.103720	−	0.994607i	$-0.533075\pi$
$739$	−2.27587e9	−0.207440	−0.103720	+	0.994607i	$0.533075\pi$
$740$	0	0
$741$	−1.87609e9	−0.169391
$742$	0	0
$743$	−2.13411e9	−0.190878	−0.0954390	−	0.995435i	$-0.530425\pi$
$743$	−2.13411e9	−0.190878	−0.0954390	+	0.995435i	$0.530425\pi$
$744$	0	0
$745$	3.96439e9	0.351260
$746$	0	0
$747$	3.93156e9	0.345099
$748$	0	0
$749$	3.19684e9	0.277994
$750$	0	0
$751$	−2.04177e10	−1.75901	−0.879503	−	0.475894i	$-0.842125\pi$
$751$	−2.04177e10	−1.75901	−0.879503	+	0.475894i	$0.842125\pi$
$752$	0	0
$753$	−8.31774e8	−0.0709942
$754$	0	0
$755$	−9.31273e9	−0.787522
$756$	0	0
$757$	8.04759e9	0.674265	0.337132	−	0.941457i	$-0.390543\pi$
$757$	8.04759e9	0.674265	0.337132	+	0.941457i	$0.390543\pi$
$758$	0	0
$759$	−2.96848e9	−0.246427
$760$	0	0
$761$	5.61734e9	0.462045	0.231023	−	0.972948i	$-0.425793\pi$
$761$	5.61734e9	0.462045	0.231023	+	0.972948i	$0.425793\pi$
$762$	0	0
$763$	1.47704e9	0.120381
$764$	0	0
$765$	2.51641e9	0.203220
$766$	0	0
$767$	2.08595e9	0.166925
$768$	0	0
$769$	−8.92653e9	−0.707849	−0.353924	−	0.935274i	$-0.615153\pi$
$769$	−8.92653e9	−0.707849	−0.353924	+	0.935274i	$0.615153\pi$
$770$	0	0
$771$	7.42855e9	0.583733
$772$	0	0
$773$	−1.79866e10	−1.40062	−0.700310	−	0.713839i	$-0.746955\pi$
$773$	−1.79866e10	−1.40062	−0.700310	+	0.713839i	$0.746955\pi$
$774$	0	0
$775$	2.95057e9	0.227693
$776$	0	0
$777$	7.93797e8	0.0607066
$778$	0	0
$779$	−3.46077e9	−0.262296
$780$	0	0
$781$	−3.88844e9	−0.292077
$782$	0	0
$783$	8.13971e8	0.0605958
$784$	0	0
$785$	3.56753e9	0.263223
$786$	0	0
$787$	1.31588e10	0.962289	0.481144	−	0.876641i	$-0.340221\pi$
$787$	1.31588e10	0.962289	0.481144	+	0.876641i	$0.340221\pi$
$788$	0	0
$789$	−1.30679e10	−0.947188
$790$	0	0
$791$	2.37894e9	0.170910
$792$	0	0
$793$	2.60530e9	0.185525
$794$	0	0
$795$	−9.68653e8	−0.0683728
$796$	0	0
$797$	2.50099e9	0.174988	0.0874939	−	0.996165i	$-0.472114\pi$
$797$	2.50099e9	0.174988	0.0874939	+	0.996165i	$0.472114\pi$
$798$	0	0
$799$	−1.49718e10	−1.03839
$800$	0	0
$801$	1.82446e9	0.125436
$802$	0	0
$803$	1.50149e9	0.102334
$804$	0	0
$805$	−4.54615e9	−0.307155
$806$	0	0
$807$	−6.67911e9	−0.447364
$808$	0	0
$809$	1.12119e10	0.744488	0.372244	−	0.928135i	$-0.378588\pi$
$809$	1.12119e10	0.744488	0.372244	+	0.928135i	$0.378588\pi$
$810$	0	0
$811$	2.38194e10	1.56804	0.784020	−	0.620736i	$-0.213166\pi$
$811$	2.38194e10	1.56804	0.784020	+	0.620736i	$0.213166\pi$
$812$	0	0
$813$	−6.16374e9	−0.402279
$814$	0	0
$815$	−1.16010e10	−0.750658
$816$	0	0
$817$	−2.06010e10	−1.32163
$818$	0	0
$819$	7.81647e8	0.0497184
$820$	0	0
$821$	−2.02921e10	−1.27975	−0.639877	−	0.768477i	$-0.721015\pi$
$821$	−2.02921e10	−1.27975	−0.639877	+	0.768477i	$0.721015\pi$
$822$	0	0
$823$	5.55929e9	0.347632	0.173816	−	0.984778i	$-0.444390\pi$
$823$	5.55929e9	0.347632	0.173816	+	0.984778i	$0.444390\pi$
$824$	0	0
$825$	1.72800e9	0.107141
$826$	0	0
$827$	−2.02195e10	−1.24308	−0.621542	−	0.783381i	$-0.713493\pi$
$827$	−2.02195e10	−1.24308	−0.621542	+	0.783381i	$0.713493\pi$
$828$	0	0
$829$	3.74799e9	0.228485	0.114243	−	0.993453i	$-0.463556\pi$
$829$	3.74799e9	0.228485	0.114243	+	0.993453i	$0.463556\pi$
$830$	0	0
$831$	9.58813e9	0.579602
$832$	0	0
$833$	−3.32876e9	−0.199538
$834$	0	0
$835$	−2.80538e9	−0.166759
$836$	0	0
$837$	9.18330e8	0.0541327
$838$	0	0
$839$	−2.03224e10	−1.18798	−0.593988	−	0.804474i	$-0.702447\pi$
$839$	−2.03224e10	−1.18798	−0.593988	+	0.804474i	$0.702447\pi$
$840$	0	0
$841$	−1.55397e10	−0.900860
$842$	0	0
$843$	−6.16051e9	−0.354177
$844$	0	0
$845$	6.46315e9	0.368507
$846$	0	0
$847$	−6.33282e9	−0.358101
$848$	0	0
$849$	−5.77236e9	−0.323725
$850$	0	0
$851$	−9.31197e9	−0.517950
$852$	0	0
$853$	3.86410e8	0.0213170	0.0106585	−	0.999943i	$-0.496607\pi$
$853$	3.86410e8	0.0213170	0.0106585	+	0.999943i	$0.496607\pi$
$854$	0	0
$855$	−1.97691e9	−0.108170
$856$	0	0
$857$	−4.64244e9	−0.251950	−0.125975	−	0.992033i	$-0.540206\pi$
$857$	−4.64244e9	−0.251950	−0.125975	+	0.992033i	$0.540206\pi$
$858$	0	0
$859$	2.73307e10	1.47121	0.735606	−	0.677410i	$-0.236898\pi$
$859$	2.73307e10	1.47121	0.735606	+	0.677410i	$0.236898\pi$
$860$	0	0
$861$	1.44188e9	0.0769872
$862$	0	0
$863$	−1.84334e10	−0.976264	−0.488132	−	0.872770i	$-0.662322\pi$
$863$	−1.84334e10	−0.976264	−0.488132	+	0.872770i	$0.662322\pi$
$864$	0	0
$865$	−1.45773e10	−0.765809
$866$	0	0
$867$	−1.05357e10	−0.549032
$868$	0	0
$869$	3.80899e9	0.196898
$870$	0	0
$871$	−3.60739e9	−0.184982
$872$	0	0
$873$	−3.35161e9	−0.170492
$874$	0	0
$875$	5.91560e9	0.298518
$876$	0	0
$877$	−1.30287e10	−0.652235	−0.326117	−	0.945329i	$-0.605740\pi$
$877$	−1.30287e10	−0.652235	−0.326117	+	0.945329i	$0.605740\pi$
$878$	0	0
$879$	−1.02280e10	−0.507962
$880$	0	0
$881$	3.46348e10	1.70646	0.853232	−	0.521531i	$-0.174639\pi$
$881$	3.46348e10	1.70646	0.853232	+	0.521531i	$0.174639\pi$
$882$	0	0
$883$	−3.40346e10	−1.66363	−0.831817	−	0.555049i	$-0.812699\pi$
$883$	−3.40346e10	−1.66363	−0.831817	+	0.555049i	$0.812699\pi$
$884$	0	0
$885$	2.19806e9	0.106595
$886$	0	0
$887$	4.78148e9	0.230054	0.115027	−	0.993362i	$-0.463305\pi$
$887$	4.78148e9	0.230054	0.115027	+	0.993362i	$0.463305\pi$
$888$	0	0
$889$	1.07874e10	0.514945
$890$	0	0
$891$	5.37818e8	0.0254720
$892$	0	0
$893$	1.17620e10	0.552714
$894$	0	0
$895$	1.14467e10	0.533704
$896$	0	0
$897$	−9.16943e9	−0.424198
$898$	0	0
$899$	1.92941e9	0.0885658
$900$	0	0
$901$	8.32030e9	0.378968
$902$	0	0
$903$	8.58313e9	0.387917
$904$	0	0
$905$	−1.81578e10	−0.814314
$906$	0	0
$907$	2.78639e10	1.23999	0.619993	−	0.784608i	$-0.287136\pi$
$907$	2.78639e10	1.23999	0.619993	+	0.784608i	$0.287136\pi$
$908$	0	0
$909$	1.49653e10	0.660862
$910$	0	0
$911$	−3.21050e10	−1.40689	−0.703443	−	0.710752i	$-0.748355\pi$
$911$	−3.21050e10	−1.40689	−0.703443	+	0.710752i	$0.748355\pi$
$912$	0	0
$913$	5.45781e9	0.237340
$914$	0	0
$915$	2.74532e9	0.118473
$916$	0	0
$917$	1.53939e9	0.0659259
$918$	0	0
$919$	−2.82454e10	−1.20045	−0.600224	−	0.799832i	$-0.704922\pi$
$919$	−2.82454e10	−1.20045	−0.600224	+	0.799832i	$0.704922\pi$
$920$	0	0
$921$	1.71130e10	0.721800
$922$	0	0
$923$	−1.20111e10	−0.502780
$924$	0	0
$925$	5.42064e9	0.225193
$926$	0	0
$927$	−1.03761e10	−0.427816
$928$	0	0
$929$	1.02974e10	0.421379	0.210690	−	0.977553i	$-0.432429\pi$
$929$	1.02974e10	0.421379	0.210690	+	0.977553i	$0.432429\pi$
$930$	0	0
$931$	2.61510e9	0.106210
$932$	0	0
$933$	−1.05353e10	−0.424678
$934$	0	0
$935$	3.49329e9	0.139764
$936$	0	0
$937$	−3.96648e10	−1.57513	−0.787565	−	0.616231i	$-0.788659\pi$
$937$	−3.96648e10	−1.57513	−0.787565	+	0.616231i	$0.788659\pi$
$938$	0	0
$939$	2.71199e10	1.06895
$940$	0	0
$941$	2.86294e9	0.112008	0.0560040	−	0.998431i	$-0.482164\pi$
$941$	2.86294e9	0.112008	0.0560040	+	0.998431i	$0.482164\pi$
$942$	0	0
$943$	−1.69146e10	−0.656857
$944$	0	0
$945$	8.23655e8	0.0317493
$946$	0	0
$947$	−6.84247e9	−0.261811	−0.130906	−	0.991395i	$-0.541788\pi$
$947$	−6.84247e9	−0.261811	−0.130906	+	0.991395i	$0.541788\pi$
$948$	0	0
$949$	4.63801e9	0.176157
$950$	0	0
$951$	−6.56031e9	−0.247339
$952$	0	0
$953$	−2.50283e10	−0.936711	−0.468356	−	0.883540i	$-0.655153\pi$
$953$	−2.50283e10	−0.936711	−0.468356	+	0.883540i	$0.655153\pi$
$954$	0	0
$955$	−1.23715e10	−0.459633
$956$	0	0
$957$	1.12996e9	0.0416745
$958$	0	0
$959$	5.35960e9	0.196231
$960$	0	0
$961$	−2.53358e10	−0.920881
$962$	0	0
$963$	6.79446e9	0.245167
$964$	0	0
$965$	−8.61193e9	−0.308500
$966$	0	0
$967$	5.13292e10	1.82546	0.912730	−	0.408564i	$-0.133970\pi$
$967$	5.13292e10	1.82546	0.912730	+	0.408564i	$0.133970\pi$
$968$	0	0
$969$	1.69808e10	0.599550
$970$	0	0
$971$	−3.62532e10	−1.27081	−0.635403	−	0.772181i	$-0.719166\pi$
$971$	−3.62532e10	−1.27081	−0.635403	+	0.772181i	$0.719166\pi$
$972$	0	0
$973$	1.86055e10	0.647511
$974$	0	0
$975$	5.33767e9	0.184432
$976$	0	0
$977$	−3.57974e10	−1.22806	−0.614031	−	0.789282i	$-0.710453\pi$
$977$	−3.57974e10	−1.22806	−0.614031	+	0.789282i	$0.710453\pi$
$978$	0	0
$979$	2.53272e9	0.0862677
$980$	0	0
$981$	3.13925e9	0.106166
$982$	0	0
$983$	−4.89020e10	−1.64206	−0.821030	−	0.570885i	$-0.806600\pi$
$983$	−4.89020e10	−1.64206	−0.821030	+	0.570885i	$0.806600\pi$
$984$	0	0
$985$	−6.12815e9	−0.204316
$986$	0	0
$987$	−4.90048e9	−0.162229
$988$	0	0
$989$	−1.00688e11	−3.30972
$990$	0	0
$991$	6.89882e9	0.225173	0.112587	−	0.993642i	$-0.464086\pi$
$991$	6.89882e9	0.225173	0.112587	+	0.993642i	$0.464086\pi$
$992$	0	0
$993$	−7.65495e8	−0.0248096
$994$	0	0
$995$	2.54855e10	0.820185
$996$	0	0
$997$	−5.53031e9	−0.176732	−0.0883662	−	0.996088i	$-0.528165\pi$
$997$	−5.53031e9	−0.176732	−0.0883662	+	0.996088i	$0.528165\pi$
$998$	0	0
$999$	1.68711e9	0.0535382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.8.a.d.1.1		1
4.3	odd	2		42.8.a.c.1.1	✓	1
12.11	even	2		126.8.a.g.1.1		1
28.3	even	6		294.8.e.n.79.1		2
28.11	odd	6		294.8.e.j.79.1		2
28.19	even	6		294.8.e.n.67.1		2
28.23	odd	6		294.8.e.j.67.1		2
28.27	even	2		294.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.8.a.c.1.1	✓	1	4.3	odd	2
126.8.a.g.1.1		1	12.11	even	2
294.8.a.c.1.1		1	28.27	even	2
294.8.e.j.67.1		2	28.23	odd	6
294.8.e.j.79.1		2	28.11	odd	6
294.8.e.n.67.1		2	28.19	even	6
294.8.e.n.79.1		2	28.3	even	6
336.8.a.d.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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