LMFDB - Embedded newform 16.34.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,34,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 34, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 34);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 34 $
Character orbit:	$[\chi]$	$=$	16.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:	$110.372526210$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.33006e8 q^{3} +5.38799e11 q^{5} +3.33473e13 q^{7} +1.21314e16 q^{9} +O(q^{10})$

$q+1.33006e8 q^{3} +5.38799e11 q^{5} +3.33473e13 q^{7} +1.21314e16 q^{9} +8.58823e16 q^{11} +1.14405e18 q^{13} +7.16633e19 q^{15} -1.39114e20 q^{17} -8.06950e19 q^{19} +4.43538e21 q^{21} +1.41204e22 q^{23} +1.73889e23 q^{25} +8.74160e23 q^{27} -1.63269e24 q^{29} +1.89408e24 q^{31} +1.14228e25 q^{33} +1.79675e25 q^{35} -9.64442e25 q^{37} +1.52166e26 q^{39} +6.41768e26 q^{41} +8.17976e26 q^{43} +6.53640e27 q^{45} +6.22925e27 q^{47} -6.61895e27 q^{49} -1.85029e28 q^{51} -2.13221e28 q^{53} +4.62733e28 q^{55} -1.07329e28 q^{57} -2.98988e29 q^{59} -4.55882e29 q^{61} +4.04550e29 q^{63} +6.16415e29 q^{65} -1.17233e30 q^{67} +1.87809e30 q^{69} -2.59152e30 q^{71} -2.82517e30 q^{73} +2.31282e31 q^{75} +2.86394e30 q^{77} -9.20688e29 q^{79} +4.88289e31 q^{81} +1.61992e31 q^{83} -7.49543e31 q^{85} -2.17156e32 q^{87} -2.03492e32 q^{89} +3.81511e31 q^{91} +2.51923e32 q^{93} -4.34784e31 q^{95} +2.26807e32 q^{97} +1.04187e33 q^{99} +O(q^{100})$

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
$3$	1.33006e8	1.78389	0.891947	−	0.452140i	$-0.149339\pi$
$3$	1.33006e8	1.78389	0.891947	+	0.452140i	$0.149339\pi$
$4$	0	0
$5$	5.38799e11	1.57914	0.789572	−	0.613658i	$-0.210303\pi$
$5$	5.38799e11	1.57914	0.789572	+	0.613658i	$0.210303\pi$
$6$	0	0
$7$	3.33473e13	0.379265	0.189633	−	0.981855i	$-0.439270\pi$
$7$	3.33473e13	0.379265	0.189633	+	0.981855i	$0.439270\pi$
$8$	0	0
$9$	1.21314e16	2.18228
$10$	0	0
$11$	8.58823e16	0.563539	0.281770	−	0.959482i	$-0.409079\pi$
$11$	8.58823e16	0.563539	0.281770	+	0.959482i	$0.409079\pi$
$12$	0	0
$13$	1.14405e18	0.476849	0.238425	−	0.971161i	$-0.423369\pi$
$13$	1.14405e18	0.476849	0.238425	+	0.971161i	$0.423369\pi$
$14$	0	0
$15$	7.16633e19	2.81703
$16$	0	0
$17$	−1.39114e20	−0.693366	−0.346683	−	0.937982i	$-0.612692\pi$
$17$	−1.39114e20	−0.693366	−0.346683	+	0.937982i	$0.612692\pi$
$18$	0	0
$19$	−8.06950e19	−0.0641818	−0.0320909	−	0.999485i	$-0.510217\pi$
$19$	−8.06950e19	−0.0641818	−0.0320909	+	0.999485i	$0.510217\pi$
$20$	0	0
$21$	4.43538e21	0.676569
$22$	0	0
$23$	1.41204e22	0.480106	0.240053	−	0.970760i	$-0.422835\pi$
$23$	1.41204e22	0.480106	0.240053	+	0.970760i	$0.422835\pi$
$24$	0	0
$25$	1.73889e23	1.49370
$26$	0	0
$27$	8.74160e23	2.10906
$28$	0	0
$29$	−1.63269e24	−1.21153	−0.605767	−	0.795642i	$-0.707134\pi$
$29$	−1.63269e24	−1.21153	−0.605767	+	0.795642i	$0.707134\pi$
$30$	0	0
$31$	1.89408e24	0.467660	0.233830	−	0.972278i	$-0.424874\pi$
$31$	1.89408e24	0.467660	0.233830	+	0.972278i	$0.424874\pi$
$32$	0	0
$33$	1.14228e25	1.00529
$34$	0	0
$35$	1.79675e25	0.598915
$36$	0	0
$37$	−9.64442e25	−1.28513	−0.642566	−	0.766230i	$-0.722130\pi$
$37$	−9.64442e25	−1.28513	−0.642566	+	0.766230i	$0.722130\pi$
$38$	0	0
$39$	1.52166e26	0.850649
$40$	0	0
$41$	6.41768e26	1.57197	0.785986	−	0.618245i	$-0.212156\pi$
$41$	6.41768e26	1.57197	0.785986	+	0.618245i	$0.212156\pi$
$42$	0	0
$43$	8.17976e26	0.913084	0.456542	−	0.889702i	$-0.349088\pi$
$43$	8.17976e26	0.913084	0.456542	+	0.889702i	$0.349088\pi$
$44$	0	0
$45$	6.53640e27	3.44613
$46$	0	0
$47$	6.22925e27	1.60258	0.801292	−	0.598273i	$-0.204146\pi$
$47$	6.22925e27	1.60258	0.801292	+	0.598273i	$0.204146\pi$
$48$	0	0
$49$	−6.61895e27	−0.856158
$50$	0	0
$51$	−1.85029e28	−1.23689
$52$	0	0
$53$	−2.13221e28	−0.755577	−0.377788	−	0.925892i	$-0.623315\pi$
$53$	−2.13221e28	−0.755577	−0.377788	+	0.925892i	$0.623315\pi$
$54$	0	0
$55$	4.62733e28	0.889910
$56$	0	0
$57$	−1.07329e28	−0.114494
$58$	0	0
$59$	−2.98988e29	−1.80549	−0.902746	−	0.430175i	$-0.858452\pi$
$59$	−2.98988e29	−1.80549	−0.902746	+	0.430175i	$0.858452\pi$
$60$	0	0
$61$	−4.55882e29	−1.58822	−0.794109	−	0.607775i	$-0.792062\pi$
$61$	−4.55882e29	−1.58822	−0.794109	+	0.607775i	$0.792062\pi$
$62$	0	0
$63$	4.04550e29	0.827663
$64$	0	0
$65$	6.16415e29	0.753014
$66$	0	0
$67$	−1.17233e30	−0.868593	−0.434297	−	0.900770i	$-0.643003\pi$
$67$	−1.17233e30	−0.868593	−0.434297	+	0.900770i	$0.643003\pi$
$68$	0	0
$69$	1.87809e30	0.856458
$70$	0	0
$71$	−2.59152e30	−0.737552	−0.368776	−	0.929518i	$-0.620223\pi$
$71$	−2.59152e30	−0.737552	−0.368776	+	0.929518i	$0.620223\pi$
$72$	0	0
$73$	−2.82517e30	−0.508415	−0.254207	−	0.967150i	$-0.581815\pi$
$73$	−2.82517e30	−0.508415	−0.254207	+	0.967150i	$0.581815\pi$
$74$	0	0
$75$	2.31282e31	2.66460
$76$	0	0
$77$	2.86394e30	0.213731
$78$	0	0
$79$	−9.20688e29	−0.0450055	−0.0225028	−	0.999747i	$-0.507163\pi$
$79$	−9.20688e29	−0.0450055	−0.0225028	+	0.999747i	$0.507163\pi$
$80$	0	0
$81$	4.88289e31	1.58006
$82$	0	0
$83$	1.61992e31	0.350514	0.175257	−	0.984523i	$-0.443924\pi$
$83$	1.61992e31	0.350514	0.175257	+	0.984523i	$0.443924\pi$
$84$	0	0
$85$	−7.49543e31	−1.09493
$86$	0	0
$87$	−2.17156e32	−2.16125
$88$	0	0
$89$	−2.03492e32	−1.39191	−0.695955	−	0.718085i	$-0.745019\pi$
$89$	−2.03492e32	−1.39191	−0.695955	+	0.718085i	$0.745019\pi$
$90$	0	0
$91$	3.81511e31	0.180852
$92$	0	0
$93$	2.51923e32	0.834256
$94$	0	0
$95$	−4.34784e31	−0.101352
$96$	0	0
$97$	2.26807e32	0.374906	0.187453	−	0.982274i	$-0.439977\pi$
$97$	2.26807e32	0.374906	0.187453	+	0.982274i	$0.439977\pi$
$98$	0	0
$99$	1.04187e33	1.22980
$100$	0	0
$101$	7.19854e32	0.610860	0.305430	−	0.952215i	$-0.401200\pi$
$101$	7.19854e32	0.610860	0.305430	+	0.952215i	$0.401200\pi$
$102$	0	0
$103$	2.42535e33	1.48922	0.744611	−	0.667499i	$-0.232635\pi$
$103$	2.42535e33	1.48922	0.744611	+	0.667499i	$0.232635\pi$
$104$	0	0
$105$	2.38978e33	1.06840
$106$	0	0
$107$	4.14367e33	1.35692	0.678458	−	0.734639i	$-0.262648\pi$
$107$	4.14367e33	1.35692	0.678458	+	0.734639i	$0.262648\pi$
$108$	0	0
$109$	9.64390e32	0.232657	0.116328	−	0.993211i	$-0.462888\pi$
$109$	9.64390e32	0.232657	0.116328	+	0.993211i	$0.462888\pi$
$110$	0	0
$111$	−1.28276e34	−2.29254
$112$	0	0
$113$	−1.19165e33	−0.158619	−0.0793095	−	0.996850i	$-0.525272\pi$
$113$	−1.19165e33	−0.158619	−0.0793095	+	0.996850i	$0.525272\pi$
$114$	0	0
$115$	7.60804e33	0.758156
$116$	0	0
$117$	1.38790e34	1.04062
$118$	0	0
$119$	−4.63907e33	−0.262970
$120$	0	0
$121$	−1.58494e34	−0.682424
$122$	0	0
$123$	8.53587e34	2.80423
$124$	0	0
$125$	3.09669e34	0.779618
$126$	0	0
$127$	2.18147e34	0.422656	0.211328	−	0.977415i	$-0.432221\pi$
$127$	2.18147e34	0.422656	0.211328	+	0.977415i	$0.432221\pi$
$128$	0	0
$129$	1.08795e35	1.62885
$130$	0	0
$131$	−1.54600e35	−1.79569	−0.897847	−	0.440309i	$-0.854869\pi$
$131$	−1.54600e35	−1.79569	−0.897847	+	0.440309i	$0.854869\pi$
$132$	0	0
$133$	−2.69096e33	−0.0243420
$134$	0	0
$135$	4.70997e35	3.33051
$136$	0	0
$137$	1.66110e35	0.921522	0.460761	−	0.887524i	$-0.347577\pi$
$137$	1.66110e35	0.921522	0.460761	+	0.887524i	$0.347577\pi$
$138$	0	0
$139$	1.53290e35	0.669531	0.334766	−	0.942301i	$-0.391343\pi$
$139$	1.53290e35	0.669531	0.334766	+	0.942301i	$0.391343\pi$
$140$	0	0
$141$	8.28524e35	2.85884
$142$	0	0
$143$	9.82539e34	0.268723
$144$	0	0
$145$	−8.79690e35	−1.91319
$146$	0	0
$147$	−8.80357e35	−1.52729
$148$	0	0
$149$	3.23613e35	0.449213	0.224606	−	0.974450i	$-0.427890\pi$
$149$	3.23613e35	0.449213	0.224606	+	0.974450i	$0.427890\pi$
$150$	0	0
$151$	7.40848e35	0.825295	0.412648	−	0.910891i	$-0.364604\pi$
$151$	7.40848e35	0.825295	0.412648	+	0.910891i	$0.364604\pi$
$152$	0	0
$153$	−1.68765e36	−1.51312
$154$	0	0
$155$	1.02053e36	0.738502
$156$	0	0
$157$	5.54604e35	0.324817	0.162408	−	0.986724i	$-0.448074\pi$
$157$	5.54604e35	0.324817	0.162408	+	0.986724i	$0.448074\pi$
$158$	0	0
$159$	−2.83596e36	−1.34787
$160$	0	0
$161$	4.70876e35	0.182087
$162$	0	0
$163$	2.15178e36	0.678737	0.339369	−	0.940653i	$-0.389787\pi$
$163$	2.15178e36	0.678737	0.339369	+	0.940653i	$0.389787\pi$
$164$	0	0
$165$	6.15460e36	1.58750
$166$	0	0
$167$	−9.54101e35	−0.201731	−0.100866	−	0.994900i	$-0.532161\pi$
$167$	−9.54101e35	−0.201731	−0.100866	+	0.994900i	$0.532161\pi$
$168$	0	0
$169$	−4.44727e36	−0.772615
$170$	0	0
$171$	−9.78945e35	−0.140063
$172$	0	0
$173$	8.72939e36	1.03092	0.515458	−	0.856915i	$-0.327622\pi$
$173$	8.72939e36	1.03092	0.515458	+	0.856915i	$0.327622\pi$
$174$	0	0
$175$	5.79874e36	0.566507
$176$	0	0
$177$	−3.97671e37	−3.22081
$178$	0	0
$179$	−1.44217e37	−0.970377	−0.485189	−	0.874409i	$-0.661249\pi$
$179$	−1.44217e37	−0.970377	−0.485189	+	0.874409i	$0.661249\pi$
$180$	0	0
$181$	−6.19356e36	−0.346932	−0.173466	−	0.984840i	$-0.555497\pi$
$181$	−6.19356e36	−0.346932	−0.173466	+	0.984840i	$0.555497\pi$
$182$	0	0
$183$	−6.06348e37	−2.83321
$184$	0	0
$185$	−5.19641e37	−2.02941
$186$	0	0
$187$	−1.19474e37	−0.390739
$188$	0	0
$189$	2.91509e37	0.799893
$190$	0	0
$191$	2.37825e37	0.548538	0.274269	−	0.961653i	$-0.411564\pi$
$191$	2.37825e37	0.548538	0.274269	+	0.961653i	$0.411564\pi$
$192$	0	0
$193$	7.58741e37	1.47366	0.736832	−	0.676076i	$-0.236321\pi$
$193$	7.58741e37	1.47366	0.736832	+	0.676076i	$0.236321\pi$
$194$	0	0
$195$	8.19867e37	1.34330
$196$	0	0
$197$	−1.46746e37	−0.203177	−0.101589	−	0.994827i	$-0.532393\pi$
$197$	−1.46746e37	−0.203177	−0.101589	+	0.994827i	$0.532393\pi$
$198$	0	0
$199$	−5.46042e37	−0.639956	−0.319978	−	0.947425i	$-0.603676\pi$
$199$	−5.46042e37	−0.639956	−0.319978	+	0.947425i	$0.603676\pi$
$200$	0	0
$201$	−1.55927e38	−1.54948
$202$	0	0
$203$	−5.44457e37	−0.459493
$204$	0	0
$205$	3.45784e38	2.48237
$206$	0	0
$207$	1.71300e38	1.04772
$208$	0	0
$209$	−6.93027e36	−0.0361690
$210$	0	0
$211$	−2.20449e37	−0.0983213	−0.0491607	−	0.998791i	$-0.515655\pi$
$211$	−2.20449e37	−0.0983213	−0.0491607	+	0.998791i	$0.515655\pi$
$212$	0	0
$213$	−3.44687e38	−1.31571
$214$	0	0
$215$	4.40725e38	1.44189
$216$	0	0
$217$	6.31624e37	0.177367
$218$	0	0
$219$	−3.75764e38	−0.906958
$220$	0	0
$221$	−1.59154e38	−0.330631
$222$	0	0
$223$	3.00366e38	0.537799	0.268900	−	0.963168i	$-0.413340\pi$
$223$	3.00366e38	0.537799	0.268900	+	0.963168i	$0.413340\pi$
$224$	0	0
$225$	2.10952e39	3.25966
$226$	0	0
$227$	−7.55693e38	−1.00906	−0.504532	−	0.863393i	$-0.668335\pi$
$227$	−7.55693e38	−1.00906	−0.504532	+	0.863393i	$0.668335\pi$
$228$	0	0
$229$	−4.47790e38	−0.517355	−0.258678	−	0.965964i	$-0.583287\pi$
$229$	−4.47790e38	−0.517355	−0.258678	+	0.965964i	$0.583287\pi$
$230$	0	0
$231$	3.80920e38	0.381273
$232$	0	0
$233$	−8.40517e38	−0.729750	−0.364875	−	0.931057i	$-0.618888\pi$
$233$	−8.40517e38	−0.729750	−0.364875	+	0.931057i	$0.618888\pi$
$234$	0	0
$235$	3.35631e39	2.53071
$236$	0	0
$237$	−1.22457e38	−0.0802851
$238$	0	0
$239$	−1.59672e39	−0.911306	−0.455653	−	0.890158i	$-0.650594\pi$
$239$	−1.59672e39	−0.911306	−0.455653	+	0.890158i	$0.650594\pi$
$240$	0	0
$241$	3.46992e39	1.72599	0.862996	−	0.505212i	$-0.168586\pi$
$241$	3.46992e39	1.72599	0.862996	+	0.505212i	$0.168586\pi$
$242$	0	0
$243$	1.63500e39	0.709603
$244$	0	0
$245$	−3.56628e39	−1.35200
$246$	0	0
$247$	−9.23194e37	−0.0306051
$248$	0	0
$249$	2.15459e39	0.625280
$250$	0	0
$251$	1.48419e39	0.377461	0.188731	−	0.982029i	$-0.439563\pi$
$251$	1.48419e39	0.377461	0.188731	+	0.982029i	$0.439563\pi$
$252$	0	0
$253$	1.21269e39	0.270558
$254$	0	0
$255$	−9.96934e39	−1.95323
$256$	0	0
$257$	−1.36871e39	−0.235730	−0.117865	−	0.993030i	$-0.537605\pi$
$257$	−1.36871e39	−0.235730	−0.117865	+	0.993030i	$0.537605\pi$
$258$	0	0
$259$	−3.21616e39	−0.487406
$260$	0	0
$261$	−1.98068e40	−2.64391
$262$	0	0
$263$	−6.59217e39	−0.775816	−0.387908	−	0.921698i	$-0.626802\pi$
$263$	−6.59217e39	−0.775816	−0.387908	+	0.921698i	$0.626802\pi$
$264$	0	0
$265$	−1.14883e40	−1.19316
$266$	0	0
$267$	−2.70655e40	−2.48302
$268$	0	0
$269$	5.00218e39	0.405739	0.202870	−	0.979206i	$-0.434973\pi$
$269$	5.00218e39	0.405739	0.202870	+	0.979206i	$0.434973\pi$
$270$	0	0
$271$	2.49179e40	1.78862	0.894308	−	0.447451i	$-0.147668\pi$
$271$	2.49179e40	1.78862	0.894308	+	0.447451i	$0.147668\pi$
$272$	0	0
$273$	5.07431e39	0.322622
$274$	0	0
$275$	1.49340e40	0.841757
$276$	0	0
$277$	3.71250e40	1.85673	0.928367	−	0.371665i	$-0.121213\pi$
$277$	3.71250e40	1.85673	0.928367	+	0.371665i	$0.121213\pi$
$278$	0	0
$279$	2.29779e40	1.02056
$280$	0	0
$281$	−2.42056e40	−0.955570	−0.477785	−	0.878477i	$-0.658560\pi$
$281$	−2.42056e40	−0.955570	−0.477785	+	0.878477i	$0.658560\pi$
$282$	0	0
$283$	−2.77995e40	−0.976249	−0.488125	−	0.872774i	$-0.662319\pi$
$283$	−2.77995e40	−0.976249	−0.488125	+	0.872774i	$0.662319\pi$
$284$	0	0
$285$	−5.78287e39	−0.180802
$286$	0	0
$287$	2.14012e40	0.596194
$288$	0	0
$289$	−2.09019e40	−0.519243
$290$	0	0
$291$	3.01666e40	0.668792
$292$	0	0
$293$	1.14215e40	0.226155	0.113078	−	0.993586i	$-0.463929\pi$
$293$	1.14215e40	0.226155	0.113078	+	0.993586i	$0.463929\pi$
$294$	0	0
$295$	−1.61094e41	−2.85113
$296$	0	0
$297$	7.50749e40	1.18854
$298$	0	0
$299$	1.61545e40	0.228938
$300$	0	0
$301$	2.72773e40	0.346301
$302$	0	0
$303$	9.57446e40	1.08971
$304$	0	0
$305$	−2.45629e41	−2.50803
$306$	0	0
$307$	−1.67395e40	−0.153447	−0.0767236	−	0.997052i	$-0.524446\pi$
$307$	−1.67395e40	−0.153447	−0.0767236	+	0.997052i	$0.524446\pi$
$308$	0	0
$309$	3.22584e41	2.65661
$310$	0	0
$311$	2.59200e41	1.91906	0.959528	−	0.281612i	$-0.0908690\pi$
$311$	2.59200e41	1.91906	0.959528	+	0.281612i	$0.0908690\pi$
$312$	0	0
$313$	−1.65212e41	−1.10042	−0.550212	−	0.835025i	$-0.685453\pi$
$313$	−1.65212e41	−1.10042	−0.550212	+	0.835025i	$0.685453\pi$
$314$	0	0
$315$	2.17971e41	1.30700
$316$	0	0
$317$	5.16289e40	0.278879	0.139439	−	0.990231i	$-0.455470\pi$
$317$	5.16289e40	0.278879	0.139439	+	0.990231i	$0.455470\pi$
$318$	0	0
$319$	−1.40219e41	−0.682748
$320$	0	0
$321$	5.51131e41	2.42060
$322$	0	0
$323$	1.12258e40	0.0445015
$324$	0	0
$325$	1.98939e41	0.712268
$326$	0	0
$327$	1.28269e41	0.415035
$328$	0	0
$329$	2.07729e41	0.607805
$330$	0	0
$331$	2.93097e40	0.0775977	0.0387989	−	0.999247i	$-0.487647\pi$
$331$	2.93097e40	0.0775977	0.0387989	+	0.999247i	$0.487647\pi$
$332$	0	0
$333$	−1.17001e42	−2.80452
$334$	0	0
$335$	−6.31652e41	−1.37163
$336$	0	0
$337$	−5.29809e41	−1.04286	−0.521429	−	0.853294i	$-0.674601\pi$
$337$	−5.29809e41	−1.04286	−0.521429	+	0.853294i	$0.674601\pi$
$338$	0	0
$339$	−1.58496e41	−0.282960
$340$	0	0
$341$	1.62668e41	0.263545
$342$	0	0
$343$	−4.78532e41	−0.703976
$344$	0	0
$345$	1.01191e42	1.35247
$346$	0	0
$347$	−2.52378e40	−0.0306630	−0.0153315	−	0.999882i	$-0.504880\pi$
$347$	−2.52378e40	−0.0306630	−0.0153315	+	0.999882i	$0.504880\pi$
$348$	0	0
$349$	−1.27613e42	−1.41018	−0.705092	−	0.709116i	$-0.749094\pi$
$349$	−1.27613e42	−1.41018	−0.705092	+	0.709116i	$0.749094\pi$
$350$	0	0
$351$	1.00009e42	1.00570
$352$	0	0
$353$	−6.97275e41	−0.638442	−0.319221	−	0.947680i	$-0.603421\pi$
$353$	−6.97275e41	−0.638442	−0.319221	+	0.947680i	$0.603421\pi$
$354$	0	0
$355$	−1.39631e42	−1.16470
$356$	0	0
$357$	−6.17022e41	−0.469110
$358$	0	0
$359$	−8.87202e41	−0.615126	−0.307563	−	0.951528i	$-0.599513\pi$
$359$	−8.87202e41	−0.615126	−0.307563	+	0.951528i	$0.599513\pi$
$360$	0	0
$361$	−1.57426e42	−0.995881
$362$	0	0
$363$	−2.10806e42	−1.21737
$364$	0	0
$365$	−1.52220e42	−0.802860
$366$	0	0
$367$	3.14846e42	1.51743	0.758714	−	0.651424i	$-0.225828\pi$
$367$	3.14846e42	1.51743	0.758714	+	0.651424i	$0.225828\pi$
$368$	0	0
$369$	7.78556e42	3.43048
$370$	0	0
$371$	−7.11035e41	−0.286564
$372$	0	0
$373$	−9.37335e41	−0.345700	−0.172850	−	0.984948i	$-0.555298\pi$
$373$	−9.37335e41	−0.345700	−0.172850	+	0.984948i	$0.555298\pi$
$374$	0	0
$375$	4.11877e42	1.39076
$376$	0	0
$377$	−1.86788e42	−0.577720
$378$	0	0
$379$	−3.51502e42	−0.996278	−0.498139	−	0.867097i	$-0.665983\pi$
$379$	−3.51502e42	−0.996278	−0.498139	+	0.867097i	$0.665983\pi$
$380$	0	0
$381$	2.90147e42	0.753974
$382$	0	0
$383$	−4.14787e42	−0.988657	−0.494329	−	0.869275i	$-0.664586\pi$
$383$	−4.14787e42	−0.988657	−0.494329	+	0.869275i	$0.664586\pi$
$384$	0	0
$385$	1.54309e42	0.337512
$386$	0	0
$387$	9.92321e42	1.99260
$388$	0	0
$389$	−1.63044e42	−0.300703	−0.150351	−	0.988633i	$-0.548041\pi$
$389$	−1.63044e42	−0.300703	−0.150351	+	0.988633i	$0.548041\pi$
$390$	0	0
$391$	−1.96434e42	−0.332889
$392$	0	0
$393$	−2.05626e43	−3.20333
$394$	0	0
$395$	−4.96066e41	−0.0710702
$396$	0	0
$397$	7.05205e42	0.929548	0.464774	−	0.885429i	$-0.346136\pi$
$397$	7.05205e42	0.929548	0.464774	+	0.885429i	$0.346136\pi$
$398$	0	0
$399$	−3.57913e41	−0.0434235
$400$	0	0
$401$	9.06730e42	1.01297	0.506484	−	0.862249i	$-0.330945\pi$
$401$	9.06730e42	1.01297	0.506484	+	0.862249i	$0.330945\pi$
$402$	0	0
$403$	2.16693e42	0.223003
$404$	0	0
$405$	2.63090e43	2.49515
$406$	0	0
$407$	−8.28285e42	−0.724222
$408$	0	0
$409$	1.80287e42	0.145388	0.0726942	−	0.997354i	$-0.476840\pi$
$409$	1.80287e42	0.145388	0.0726942	+	0.997354i	$0.476840\pi$
$410$	0	0
$411$	2.20935e43	1.64390
$412$	0	0
$413$	−9.97044e42	−0.684760
$414$	0	0
$415$	8.72813e42	0.553512
$416$	0	0
$417$	2.03885e43	1.19437
$418$	0	0
$419$	−2.18879e43	−1.18488	−0.592438	−	0.805616i	$-0.701834\pi$
$419$	−2.18879e43	−1.18488	−0.592438	+	0.805616i	$0.701834\pi$
$420$	0	0
$421$	1.25344e43	0.627263	0.313632	−	0.949545i	$-0.398454\pi$
$421$	1.25344e43	0.627263	0.313632	+	0.949545i	$0.398454\pi$
$422$	0	0
$423$	7.55696e43	3.49729
$424$	0	0
$425$	−2.41904e43	−1.03568
$426$	0	0
$427$	−1.52024e43	−0.602356
$428$	0	0
$429$	1.30683e43	0.479374
$430$	0	0
$431$	−1.38369e43	−0.470072	−0.235036	−	0.971987i	$-0.575521\pi$
$431$	−1.38369e43	−0.470072	−0.235036	+	0.971987i	$0.575521\pi$
$432$	0	0
$433$	−4.86806e43	−1.53216	−0.766082	−	0.642743i	$-0.777796\pi$
$433$	−4.86806e43	−1.53216	−0.766082	+	0.642743i	$0.777796\pi$
$434$	0	0
$435$	−1.17004e44	−3.41293
$436$	0	0
$437$	−1.13944e42	−0.0308141
$438$	0	0
$439$	−2.44480e43	−0.613167	−0.306584	−	0.951844i	$-0.599186\pi$
$439$	−2.44480e43	−0.613167	−0.306584	+	0.951844i	$0.599186\pi$
$440$	0	0
$441$	−8.02973e43	−1.86837
$442$	0	0
$443$	−3.59045e43	−0.775331	−0.387666	−	0.921800i	$-0.626718\pi$
$443$	−3.59045e43	−0.775331	−0.387666	+	0.921800i	$0.626718\pi$
$444$	0	0
$445$	−1.09641e44	−2.19803
$446$	0	0
$447$	4.30423e43	0.801348
$448$	0	0
$449$	8.52869e43	1.47509	0.737544	−	0.675299i	$-0.235986\pi$
$449$	8.52869e43	1.47509	0.737544	+	0.675299i	$0.235986\pi$
$450$	0	0
$451$	5.51165e43	0.885867
$452$	0	0
$453$	9.85369e43	1.47224
$454$	0	0
$455$	2.05558e43	0.285592
$456$	0	0
$457$	6.07893e43	0.785615	0.392807	−	0.919621i	$-0.371504\pi$
$457$	6.07893e43	0.785615	0.392807	+	0.919621i	$0.371504\pi$
$458$	0	0
$459$	−1.21608e44	−1.46235
$460$	0	0
$461$	−1.43000e44	−1.60055	−0.800277	−	0.599631i	$-0.795314\pi$
$461$	−1.43000e44	−1.60055	−0.800277	+	0.599631i	$0.795314\pi$
$462$	0	0
$463$	−1.40454e43	−0.146369	−0.0731844	−	0.997318i	$-0.523316\pi$
$463$	−1.40454e43	−0.146369	−0.0731844	+	0.997318i	$0.523316\pi$
$464$	0	0
$465$	1.35736e44	1.31741
$466$	0	0
$467$	5.13667e43	0.464465	0.232233	−	0.972660i	$-0.425397\pi$
$467$	5.13667e43	0.464465	0.232233	+	0.972660i	$0.425397\pi$
$468$	0	0
$469$	−3.90941e43	−0.329427
$470$	0	0
$471$	7.37654e43	0.579439
$472$	0	0
$473$	7.02496e43	0.514559
$474$	0	0
$475$	−1.40320e43	−0.0958682
$476$	0	0
$477$	−2.58668e44	−1.64888
$478$	0	0
$479$	1.67218e44	0.994823	0.497412	−	0.867515i	$-0.334284\pi$
$479$	1.67218e44	0.994823	0.497412	+	0.867515i	$0.334284\pi$
$480$	0	0
$481$	−1.10337e44	−0.612814
$482$	0	0
$483$	6.26292e43	0.324825
$484$	0	0
$485$	1.22203e44	0.592030
$486$	0	0
$487$	−1.67480e44	−0.758117	−0.379058	−	0.925373i	$-0.623752\pi$
$487$	−1.67480e44	−0.758117	−0.379058	+	0.925373i	$0.623752\pi$
$488$	0	0
$489$	2.86198e44	1.21080
$490$	0	0
$491$	−2.77223e44	−1.09644	−0.548220	−	0.836334i	$-0.684694\pi$
$491$	−2.77223e44	−1.09644	−0.548220	+	0.836334i	$0.684694\pi$
$492$	0	0
$493$	2.27129e44	0.840037
$494$	0	0
$495$	5.61361e44	1.94203
$496$	0	0
$497$	−8.64204e43	−0.279728
$498$	0	0
$499$	−4.56174e44	−1.38188	−0.690942	−	0.722910i	$-0.742804\pi$
$499$	−4.56174e44	−1.38188	−0.690942	+	0.722910i	$0.742804\pi$
$500$	0	0
$501$	−1.26901e44	−0.359867
$502$	0	0
$503$	5.38096e44	1.42885	0.714427	−	0.699710i	$-0.246688\pi$
$503$	5.38096e44	1.42885	0.714427	+	0.699710i	$0.246688\pi$
$504$	0	0
$505$	3.87857e44	0.964636
$506$	0	0
$507$	−5.91512e44	−1.37826
$508$	0	0
$509$	6.21555e43	0.135718	0.0678588	−	0.997695i	$-0.478383\pi$
$509$	6.21555e43	0.135718	0.0678588	+	0.997695i	$0.478383\pi$
$510$	0	0
$511$	−9.42120e43	−0.192824
$512$	0	0
$513$	−7.05404e43	−0.135363
$514$	0	0
$515$	1.30677e45	2.35170
$516$	0	0
$517$	5.34982e44	0.903119
$518$	0	0
$519$	1.16106e45	1.83904
$520$	0	0
$521$	−7.97253e44	−1.18515	−0.592576	−	0.805515i	$-0.701889\pi$
$521$	−7.97253e44	−1.18515	−0.592576	+	0.805515i	$0.701889\pi$
$522$	0	0
$523$	−1.35653e44	−0.189301	−0.0946504	−	0.995511i	$-0.530173\pi$
$523$	−1.35653e44	−0.189301	−0.0946504	+	0.995511i	$0.530173\pi$
$524$	0	0
$525$	7.71264e44	1.01059
$526$	0	0
$527$	−2.63492e44	−0.324260
$528$	0	0
$529$	−6.65620e44	−0.769499
$530$	0	0
$531$	−3.62715e45	−3.94009
$532$	0	0
$533$	7.34218e44	0.749593
$534$	0	0
$535$	2.23261e45	2.14277
$536$	0	0
$537$	−1.91816e45	−1.73105
$538$	0	0
$539$	−5.68450e44	−0.482479
$540$	0	0
$541$	1.73198e45	1.38289	0.691445	−	0.722429i	$-0.256974\pi$
$541$	1.73198e45	1.38289	0.691445	+	0.722429i	$0.256974\pi$
$542$	0	0
$543$	−8.23778e44	−0.618889
$544$	0	0
$545$	5.19613e44	0.367399
$546$	0	0
$547$	1.10462e45	0.735227	0.367614	−	0.929979i	$-0.380175\pi$
$547$	1.10462e45	0.735227	0.367614	+	0.929979i	$0.380175\pi$
$548$	0	0
$549$	−5.53049e45	−3.46593
$550$	0	0
$551$	1.31750e44	0.0777586
$552$	0	0
$553$	−3.07025e43	−0.0170690
$554$	0	0
$555$	−6.91151e45	−3.62025
$556$	0	0
$557$	1.13068e45	0.558120	0.279060	−	0.960274i	$-0.409977\pi$
$557$	1.13068e45	0.558120	0.279060	+	0.960274i	$0.409977\pi$
$558$	0	0
$559$	9.35808e44	0.435404
$560$	0	0
$561$	−1.58907e45	−0.697037
$562$	0	0
$563$	−2.85343e45	−1.18026	−0.590131	−	0.807307i	$-0.700924\pi$
$563$	−2.85343e45	−1.18026	−0.590131	+	0.807307i	$0.700924\pi$
$564$	0	0
$565$	−6.42061e44	−0.250482
$566$	0	0
$567$	1.62831e45	0.599263
$568$	0	0
$569$	4.66376e44	0.161951	0.0809756	−	0.996716i	$-0.474196\pi$
$569$	4.66376e44	0.161951	0.0809756	+	0.996716i	$0.474196\pi$
$570$	0	0
$571$	5.04939e45	1.65479	0.827396	−	0.561619i	$-0.189821\pi$
$571$	5.04939e45	1.65479	0.827396	+	0.561619i	$0.189821\pi$
$572$	0	0
$573$	3.16320e45	0.978533
$574$	0	0
$575$	2.45538e45	0.717132
$576$	0	0
$577$	2.74486e44	0.0757040	0.0378520	−	0.999283i	$-0.487948\pi$
$577$	2.74486e44	0.0757040	0.0378520	+	0.999283i	$0.487948\pi$
$578$	0	0
$579$	1.00917e46	2.62886
$580$	0	0
$581$	5.40200e44	0.132938
$582$	0	0
$583$	−1.83119e45	−0.425797
$584$	0	0
$585$	7.47799e45	1.64329
$586$	0	0
$587$	−3.25926e45	−0.677002	−0.338501	−	0.940966i	$-0.609920\pi$
$587$	−3.25926e45	−0.677002	−0.338501	+	0.940966i	$0.609920\pi$
$588$	0	0
$589$	−1.52843e44	−0.0300153
$590$	0	0
$591$	−1.95181e45	−0.362446
$592$	0	0
$593$	1.56542e45	0.274933	0.137467	−	0.990506i	$-0.456104\pi$
$593$	1.56542e45	0.274933	0.137467	+	0.990506i	$0.456104\pi$
$594$	0	0
$595$	−2.49953e45	−0.415267
$596$	0	0
$597$	−7.26266e45	−1.14161
$598$	0	0
$599$	−2.66724e45	−0.396753	−0.198376	−	0.980126i	$-0.563567\pi$
$599$	−2.66724e45	−0.396753	−0.198376	+	0.980126i	$0.563567\pi$
$600$	0	0
$601$	1.01736e46	1.43234	0.716170	−	0.697926i	$-0.245893\pi$
$601$	1.01736e46	1.43234	0.716170	+	0.697926i	$0.245893\pi$
$602$	0	0
$603$	−1.42221e46	−1.89551
$604$	0	0
$605$	−8.53964e45	−1.07765
$606$	0	0
$607$	1.91910e45	0.229343	0.114671	−	0.993403i	$-0.463419\pi$
$607$	1.91910e45	0.229343	0.114671	+	0.993403i	$0.463419\pi$
$608$	0	0
$609$	−7.24158e45	−0.819687
$610$	0	0
$611$	7.12659e45	0.764191
$612$	0	0
$613$	−9.21265e45	−0.936023	−0.468012	−	0.883722i	$-0.655029\pi$
$613$	−9.21265e45	−0.936023	−0.468012	+	0.883722i	$0.655029\pi$
$614$	0	0
$615$	4.59912e46	4.42828
$616$	0	0
$617$	1.85344e46	1.69150	0.845752	−	0.533577i	$-0.179152\pi$
$617$	1.85344e46	1.69150	0.845752	+	0.533577i	$0.179152\pi$
$618$	0	0
$619$	−1.79796e46	−1.55555	−0.777777	−	0.628540i	$-0.783653\pi$
$619$	−1.79796e46	−1.55555	−0.777777	+	0.628540i	$0.783653\pi$
$620$	0	0
$621$	1.23435e46	1.01257
$622$	0	0
$623$	−6.78590e45	−0.527904
$624$	0	0
$625$	−3.55844e45	−0.262567
$626$	0	0
$627$	−9.21764e44	−0.0645217
$628$	0	0
$629$	1.34167e46	0.891067
$630$	0	0
$631$	1.17228e46	0.738831	0.369416	−	0.929264i	$-0.379558\pi$
$631$	1.17228e46	0.738831	0.369416	+	0.929264i	$0.379558\pi$
$632$	0	0
$633$	−2.93209e45	−0.175395
$634$	0	0
$635$	1.17537e46	0.667435
$636$	0	0
$637$	−7.57244e45	−0.408258
$638$	0	0
$639$	−3.14389e46	−1.60954
$640$	0	0
$641$	−1.41216e46	−0.686638	−0.343319	−	0.939219i	$-0.611551\pi$
$641$	−1.41216e46	−0.686638	−0.343319	+	0.939219i	$0.611551\pi$
$642$	0	0
$643$	4.07405e45	0.188168	0.0940838	−	0.995564i	$-0.470008\pi$
$643$	4.07405e45	0.188168	0.0940838	+	0.995564i	$0.470008\pi$
$644$	0	0
$645$	5.86188e46	2.57218
$646$	0	0
$647$	3.65004e45	0.152187	0.0760933	−	0.997101i	$-0.475755\pi$
$647$	3.65004e45	0.152187	0.0760933	+	0.997101i	$0.475755\pi$
$648$	0	0
$649$	−2.56778e46	−1.01747
$650$	0	0
$651$	8.40096e45	0.316404
$652$	0	0
$653$	−5.21698e46	−1.86789	−0.933946	−	0.357414i	$-0.883659\pi$
$653$	−5.21698e46	−1.86789	−0.933946	+	0.357414i	$0.883659\pi$
$654$	0	0
$655$	−8.32982e46	−2.83566
$656$	0	0
$657$	−3.42734e46	−1.10950
$658$	0	0
$659$	5.05202e45	0.155545	0.0777724	−	0.996971i	$-0.475219\pi$
$659$	5.05202e45	0.155545	0.0777724	+	0.996971i	$0.475219\pi$
$660$	0	0
$661$	−5.49372e46	−1.60895	−0.804475	−	0.593987i	$-0.797553\pi$
$661$	−5.49372e46	−1.60895	−0.804475	+	0.593987i	$0.797553\pi$
$662$	0	0
$663$	−2.11683e46	−0.589811
$664$	0	0
$665$	−1.44989e45	−0.0384395
$666$	0	0
$667$	−2.30541e46	−0.581665
$668$	0	0
$669$	3.99503e46	0.959377
$670$	0	0
$671$	−3.91522e46	−0.895023
$672$	0	0
$673$	3.75226e45	0.0816666	0.0408333	−	0.999166i	$-0.486999\pi$
$673$	3.75226e45	0.0816666	0.0408333	+	0.999166i	$0.486999\pi$
$674$	0	0
$675$	1.52007e47	3.15030
$676$	0	0
$677$	4.64616e46	0.917025	0.458513	−	0.888688i	$-0.348382\pi$
$677$	4.64616e46	0.917025	0.458513	+	0.888688i	$0.348382\pi$
$678$	0	0
$679$	7.56339e45	0.142189
$680$	0	0
$681$	−1.00511e47	−1.80006
$682$	0	0
$683$	−8.92876e46	−1.52352	−0.761762	−	0.647857i	$-0.775666\pi$
$683$	−8.92876e46	−1.52352	−0.761762	+	0.647857i	$0.775666\pi$
$684$	0	0
$685$	8.94997e46	1.45522
$686$	0	0
$687$	−5.95585e46	−0.922907
$688$	0	0
$689$	−2.43937e46	−0.360296
$690$	0	0
$691$	1.40554e47	1.97904	0.989522	−	0.144382i	$-0.0461193\pi$
$691$	1.40554e47	1.97904	0.989522	+	0.144382i	$0.0461193\pi$
$692$	0	0
$693$	3.47437e46	0.466420
$694$	0	0
$695$	8.25928e46	1.05729
$696$	0	0
$697$	−8.92787e46	−1.08995
$698$	0	0
$699$	−1.11793e47	−1.30180
$700$	0	0
$701$	−5.92277e46	−0.657926	−0.328963	−	0.944343i	$-0.606699\pi$
$701$	−5.92277e46	−0.657926	−0.328963	+	0.944343i	$0.606699\pi$
$702$	0	0
$703$	7.78257e45	0.0824821
$704$	0	0
$705$	4.46408e47	4.51452
$706$	0	0
$707$	2.40052e46	0.231678
$708$	0	0
$709$	6.19740e46	0.570882	0.285441	−	0.958396i	$-0.407860\pi$
$709$	6.19740e46	0.570882	0.285441	+	0.958396i	$0.407860\pi$
$710$	0	0
$711$	−1.11693e46	−0.0982146
$712$	0	0
$713$	2.67451e46	0.224526
$714$	0	0
$715$	5.29391e46	0.424353
$716$	0	0
$717$	−2.12373e47	−1.62567
$718$	0	0
$719$	2.33935e46	0.171028	0.0855142	−	0.996337i	$-0.472747\pi$
$719$	2.33935e46	0.171028	0.0855142	+	0.996337i	$0.472747\pi$
$720$	0	0
$721$	8.08787e46	0.564810
$722$	0	0
$723$	4.61519e47	3.07899
$724$	0	0
$725$	−2.83907e47	−1.80967
$726$	0	0
$727$	−3.68313e46	−0.224336	−0.112168	−	0.993689i	$-0.535780\pi$
$727$	−3.68313e46	−0.224336	−0.112168	+	0.993689i	$0.535780\pi$
$728$	0	0
$729$	−5.39781e46	−0.314205
$730$	0	0
$731$	−1.13792e47	−0.633102
$732$	0	0
$733$	1.06749e47	0.567740	0.283870	−	0.958863i	$-0.408382\pi$
$733$	1.06749e47	0.567740	0.283870	+	0.958863i	$0.408382\pi$
$734$	0	0
$735$	−4.74336e47	−2.41182
$736$	0	0
$737$	−1.00683e47	−0.489486
$738$	0	0
$739$	−1.86383e47	−0.866509	−0.433254	−	0.901272i	$-0.642635\pi$
$739$	−1.86383e47	−0.866509	−0.433254	+	0.901272i	$0.642635\pi$
$740$	0	0
$741$	−1.22790e46	−0.0545962
$742$	0	0
$743$	1.41560e47	0.602040	0.301020	−	0.953618i	$-0.402673\pi$
$743$	1.41560e47	0.602040	0.301020	+	0.953618i	$0.402673\pi$
$744$	0	0
$745$	1.74362e47	0.709372
$746$	0	0
$747$	1.96520e47	0.764920
$748$	0	0
$749$	1.38180e47	0.514631
$750$	0	0
$751$	−2.61886e47	−0.933368	−0.466684	−	0.884424i	$-0.654551\pi$
$751$	−2.61886e47	−0.933368	−0.466684	+	0.884424i	$0.654551\pi$
$752$	0	0
$753$	1.97405e47	0.673351
$754$	0	0
$755$	3.99168e47	1.30326
$756$	0	0
$757$	2.21676e47	0.692846	0.346423	−	0.938078i	$-0.387396\pi$
$757$	2.21676e47	0.692846	0.346423	+	0.938078i	$0.387396\pi$
$758$	0	0
$759$	1.61294e47	0.482648
$760$	0	0
$761$	1.15570e47	0.331131	0.165565	−	0.986199i	$-0.447055\pi$
$761$	1.15570e47	0.331131	0.165565	+	0.986199i	$0.447055\pi$
$762$	0	0
$763$	3.21598e46	0.0882386
$764$	0	0
$765$	−9.09302e47	−2.38943
$766$	0	0
$767$	−3.42058e47	−0.860947
$768$	0	0
$769$	−3.82649e47	−0.922605	−0.461302	−	0.887243i	$-0.652618\pi$
$769$	−3.82649e47	−0.922605	−0.461302	+	0.887243i	$0.652618\pi$
$770$	0	0
$771$	−1.82046e47	−0.420518
$772$	0	0
$773$	7.89533e47	1.74746	0.873732	−	0.486407i	$-0.161693\pi$
$773$	7.89533e47	1.74746	0.873732	+	0.486407i	$0.161693\pi$
$774$	0	0
$775$	3.29360e47	0.698542
$776$	0	0
$777$	−4.27767e47	−0.869481
$778$	0	0
$779$	−5.17875e46	−0.100892
$780$	0	0
$781$	−2.22566e47	−0.415639
$782$	0	0
$783$	−1.42723e48	−2.55520
$784$	0	0
$785$	2.98820e47	0.512933
$786$	0	0
$787$	1.15500e48	1.90107	0.950534	−	0.310620i	$-0.100537\pi$
$787$	1.15500e48	1.90107	0.950534	+	0.310620i	$0.100537\pi$
$788$	0	0
$789$	−8.76795e47	−1.38397
$790$	0	0
$791$	−3.97384e46	−0.0601587
$792$	0	0
$793$	−5.21554e47	−0.757341
$794$	0	0
$795$	−1.52801e48	−2.12848
$796$	0	0
$797$	4.08413e47	0.545805	0.272902	−	0.962042i	$-0.412016\pi$
$797$	4.08413e47	0.545805	0.272902	+	0.962042i	$0.412016\pi$
$798$	0	0
$799$	−8.66573e47	−1.11118
$800$	0	0
$801$	−2.46864e48	−3.03754
$802$	0	0
$803$	−2.42632e47	−0.286512
$804$	0	0
$805$	2.53708e47	0.287542
$806$	0	0
$807$	6.65318e47	0.723796
$808$	0	0
$809$	7.61932e47	0.795730	0.397865	−	0.917444i	$-0.369751\pi$
$809$	7.61932e47	0.795730	0.397865	+	0.917444i	$0.369751\pi$
$810$	0	0
$811$	1.03654e48	1.03930	0.519651	−	0.854378i	$-0.326062\pi$
$811$	1.03654e48	1.03930	0.519651	+	0.854378i	$0.326062\pi$
$812$	0	0
$813$	3.31421e48	3.19070
$814$	0	0
$815$	1.15938e48	1.07182
$816$	0	0
$817$	−6.60065e46	−0.0586034
$818$	0	0
$819$	4.62827e47	0.394670
$820$	0	0
$821$	8.66860e47	0.710047	0.355023	−	0.934857i	$-0.384473\pi$
$821$	8.66860e47	0.710047	0.355023	+	0.934857i	$0.384473\pi$
$822$	0	0
$823$	−6.30492e47	−0.496115	−0.248058	−	0.968745i	$-0.579792\pi$
$823$	−6.30492e47	−0.496115	−0.248058	+	0.968745i	$0.579792\pi$
$824$	0	0
$825$	1.98630e48	1.50161
$826$	0	0
$827$	−2.94601e47	−0.213990	−0.106995	−	0.994260i	$-0.534123\pi$
$827$	−2.94601e47	−0.213990	−0.106995	+	0.994260i	$0.534123\pi$
$828$	0	0
$829$	−1.63460e48	−1.14094	−0.570470	−	0.821318i	$-0.693239\pi$
$829$	−1.63460e48	−1.14094	−0.570470	+	0.821318i	$0.693239\pi$
$830$	0	0
$831$	4.93782e48	3.31222
$832$	0	0
$833$	9.20787e47	0.593631
$834$	0	0
$835$	−5.14069e47	−0.318562
$836$	0	0
$837$	1.65573e48	0.986323
$838$	0	0
$839$	2.42256e48	1.38740	0.693702	−	0.720262i	$-0.255979\pi$
$839$	2.42256e48	1.38740	0.693702	+	0.720262i	$0.255979\pi$
$840$	0	0
$841$	8.49591e47	0.467817
$842$	0	0
$843$	−3.21949e48	−1.70464
$844$	0	0
$845$	−2.39619e48	−1.22007
$846$	0	0
$847$	−5.28535e47	−0.258820
$848$	0	0
$849$	−3.69749e48	−1.74153
$850$	0	0
$851$	−1.36183e48	−0.616999
$852$	0	0
$853$	−2.74926e48	−1.19827	−0.599137	−	0.800647i	$-0.704489\pi$
$853$	−2.74926e48	−1.19827	−0.599137	+	0.800647i	$0.704489\pi$
$854$	0	0
$855$	−5.27455e47	−0.221179
$856$	0	0
$857$	−7.23564e47	−0.291940	−0.145970	−	0.989289i	$-0.546630\pi$
$857$	−7.23564e47	−0.291940	−0.145970	+	0.989289i	$0.546630\pi$
$858$	0	0
$859$	8.35479e47	0.324376	0.162188	−	0.986760i	$-0.448145\pi$
$859$	8.35479e47	0.324376	0.162188	+	0.986760i	$0.448145\pi$
$860$	0	0
$861$	2.84648e48	1.06355
$862$	0	0
$863$	−6.91469e46	−0.0248653	−0.0124327	−	0.999923i	$-0.503958\pi$
$863$	−6.91469e46	−0.0248653	−0.0124327	+	0.999923i	$0.503958\pi$
$864$	0	0
$865$	4.70339e48	1.62796
$866$	0	0
$867$	−2.78007e48	−0.926275
$868$	0	0
$869$	−7.90708e46	−0.0253624
$870$	0	0
$871$	−1.34121e48	−0.414188
$872$	0	0
$873$	2.75149e48	0.818149
$874$	0	0
$875$	1.03266e48	0.295682
$876$	0	0
$877$	4.76181e48	1.31304	0.656521	−	0.754308i	$-0.272027\pi$
$877$	4.76181e48	1.31304	0.656521	+	0.754308i	$0.272027\pi$
$878$	0	0
$879$	1.51912e48	0.403437
$880$	0	0
$881$	1.74378e48	0.446055	0.223028	−	0.974812i	$-0.428406\pi$
$881$	1.74378e48	0.446055	0.223028	+	0.974812i	$0.428406\pi$
$882$	0	0
$883$	7.49080e47	0.184577	0.0922883	−	0.995732i	$-0.470582\pi$
$883$	7.49080e47	0.184577	0.0922883	+	0.995732i	$0.470582\pi$
$884$	0	0
$885$	−2.14265e49	−5.08612
$886$	0	0
$887$	−1.36136e47	−0.0311338	−0.0155669	−	0.999879i	$-0.504955\pi$
$887$	−1.36136e47	−0.0311338	−0.0155669	+	0.999879i	$0.504955\pi$
$888$	0	0
$889$	7.27461e47	0.160299
$890$	0	0
$891$	4.19354e48	0.890427
$892$	0	0
$893$	−5.02669e47	−0.102857
$894$	0	0
$895$	−7.77039e48	−1.53237
$896$	0	0
$897$	2.14863e48	0.408401
$898$	0	0
$899$	−3.09244e48	−0.566586
$900$	0	0
$901$	2.96620e48	0.523891
$902$	0	0
$903$	3.62803e48	0.617765
$904$	0	0
$905$	−3.33708e48	−0.547855
$906$	0	0
$907$	6.26172e48	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$907$	6.26172e48	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$908$	0	0
$909$	8.73285e48	1.33307
$910$	0	0
$911$	1.10745e49	1.63031	0.815155	−	0.579243i	$-0.196652\pi$
$911$	1.10745e49	1.63031	0.815155	+	0.579243i	$0.196652\pi$
$912$	0	0
$913$	1.39123e48	0.197528
$914$	0	0
$915$	−3.26700e49	−4.47405
$916$	0	0
$917$	−5.15548e48	−0.681044
$918$	0	0
$919$	−6.88858e48	−0.877857	−0.438929	−	0.898522i	$-0.644642\pi$
$919$	−6.88858e48	−0.877857	−0.438929	+	0.898522i	$0.644642\pi$
$920$	0	0
$921$	−2.22644e48	−0.273733
$922$	0	0
$923$	−2.96484e48	−0.351701
$924$	0	0
$925$	−1.67706e49	−1.91960
$926$	0	0
$927$	2.94229e49	3.24990
$928$	0	0
$929$	−1.62943e49	−1.73691	−0.868456	−	0.495767i	$-0.834887\pi$
$929$	−1.62943e49	−1.73691	−0.868456	+	0.495767i	$0.834887\pi$
$930$	0	0
$931$	5.34116e47	0.0549498
$932$	0	0
$933$	3.44750e49	3.42339
$934$	0	0
$935$	−6.43725e48	−0.617033
$936$	0	0
$937$	−1.49919e49	−1.38724	−0.693621	−	0.720340i	$-0.743986\pi$
$937$	−1.49919e49	−1.38724	−0.693621	+	0.720340i	$0.743986\pi$
$938$	0	0
$939$	−2.19741e49	−1.96304
$940$	0	0
$941$	8.73939e48	0.753794	0.376897	−	0.926255i	$-0.376991\pi$
$941$	8.73939e48	0.753794	0.376897	+	0.926255i	$0.376991\pi$
$942$	0	0
$943$	9.06201e48	0.754712
$944$	0	0
$945$	1.57065e49	1.26315
$946$	0	0
$947$	−7.24316e48	−0.562540	−0.281270	−	0.959629i	$-0.590756\pi$
$947$	−7.24316e48	−0.562540	−0.281270	+	0.959629i	$0.590756\pi$
$948$	0	0
$949$	−3.23215e48	−0.242437
$950$	0	0
$951$	6.86694e48	0.497490
$952$	0	0
$953$	8.48504e48	0.593774	0.296887	−	0.954913i	$-0.404052\pi$
$953$	8.48504e48	0.593774	0.296887	+	0.954913i	$0.404052\pi$
$954$	0	0
$955$	1.28140e49	0.866220
$956$	0	0
$957$	−1.86499e49	−1.21795
$958$	0	0
$959$	5.53931e48	0.349501
$960$	0	0
$961$	−1.28159e49	−0.781294
$962$	0	0
$963$	5.02686e49	2.96117
$964$	0	0
$965$	4.08809e49	2.32713
$966$	0	0
$967$	−3.55845e49	−1.95760	−0.978800	−	0.204820i	$-0.934339\pi$
$967$	−3.55845e49	−1.95760	−0.978800	+	0.204820i	$0.934339\pi$
$968$	0	0
$969$	1.49309e48	0.0793860
$970$	0	0
$971$	1.83055e49	0.940728	0.470364	−	0.882473i	$-0.344123\pi$
$971$	1.83055e49	0.940728	0.470364	+	0.882473i	$0.344123\pi$
$972$	0	0
$973$	5.11183e48	0.253930
$974$	0	0
$975$	2.64599e49	1.27061
$976$	0	0
$977$	9.99310e48	0.463916	0.231958	−	0.972726i	$-0.425487\pi$
$977$	9.99310e48	0.463916	0.231958	+	0.972726i	$0.425487\pi$
$978$	0	0
$979$	−1.74763e49	−0.784396
$980$	0	0
$981$	1.16994e49	0.507722
$982$	0	0
$983$	2.70444e49	1.13487	0.567434	−	0.823419i	$-0.307936\pi$
$983$	2.70444e49	1.13487	0.567434	+	0.823419i	$0.307936\pi$
$984$	0	0
$985$	−7.90668e48	−0.320846
$986$	0	0
$987$	2.76291e49	1.08426
$988$	0	0
$989$	1.15501e49	0.438377
$990$	0	0
$991$	−2.95819e49	−1.08595	−0.542977	−	0.839748i	$-0.682703\pi$
$991$	−2.95819e49	−1.08595	−0.542977	+	0.839748i	$0.682703\pi$
$992$	0	0
$993$	3.89835e48	0.138426
$994$	0	0
$995$	−2.94207e49	−1.01058
$996$	0	0
$997$	4.00799e49	1.33185	0.665927	−	0.746017i	$-0.268036\pi$
$997$	4.00799e49	1.33185	0.665927	+	0.746017i	$0.268036\pi$
$998$	0	0
$999$	−8.43077e49	−2.71042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.34.a.a.1.1		1
4.3	odd	2		2.34.a.a.1.1	✓	1
12.11	even	2		18.34.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.34.a.a.1.1	✓	1	4.3	odd	2
16.34.a.a.1.1		1	1.1	even	1	trivial
18.34.a.c.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 34 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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