LMFDB - Embedded newform 3234.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.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:	$25.8236200137$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.14336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 14 $ x^4 - 8*x^2 + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.60804$ of defining polynomial
Character	$\chi$	$=$	3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.27411 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.27411 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.27411 q^{10} -1.00000 q^{11} +1.00000 q^{12} -4.63029 q^{13} -2.27411 q^{15} +1.00000 q^{16} +0.554318 q^{17} -1.00000 q^{18} +4.07597 q^{19} -2.27411 q^{20} +1.00000 q^{22} -6.93587 q^{23} -1.00000 q^{24} +0.171573 q^{25} +4.63029 q^{26} +1.00000 q^{27} -2.82843 q^{29} +2.27411 q^{30} +7.68832 q^{31} -1.00000 q^{32} -1.00000 q^{33} -0.554318 q^{34} +1.00000 q^{36} +10.8729 q^{37} -4.07597 q^{38} -4.63029 q^{39} +2.27411 q^{40} -0.554318 q^{41} -8.59272 q^{43} -1.00000 q^{44} -2.27411 q^{45} +6.93587 q^{46} +10.9044 q^{47} +1.00000 q^{48} -0.171573 q^{50} +0.554318 q^{51} -4.63029 q^{52} -0.440778 q^{53} -1.00000 q^{54} +2.27411 q^{55} +4.07597 q^{57} +2.82843 q^{58} +5.17157 q^{59} -2.27411 q^{60} +2.19814 q^{61} -7.68832 q^{62} +1.00000 q^{64} +10.5298 q^{65} +1.00000 q^{66} -11.1409 q^{67} +0.554318 q^{68} -6.93587 q^{69} +3.60373 q^{71} -1.00000 q^{72} +14.3631 q^{73} -10.8729 q^{74} +0.171573 q^{75} +4.07597 q^{76} +4.63029 q^{78} +15.7014 q^{79} -2.27411 q^{80} +1.00000 q^{81} +0.554318 q^{82} -6.51675 q^{83} -1.26058 q^{85} +8.59272 q^{86} -2.82843 q^{87} +1.00000 q^{88} -4.23401 q^{89} +2.27411 q^{90} -6.93587 q^{92} +7.68832 q^{93} -10.9044 q^{94} -9.26921 q^{95} -1.00000 q^{96} +3.84637 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{11} + 4 q^{12} + 4 q^{16} - 4 q^{18} + 4 q^{22} - 8 q^{23} - 4 q^{24} + 12 q^{25} + 4 q^{27} + 16 q^{31} - 4 q^{32} - 4 q^{33} + 4 q^{36} + 8 q^{37} + 8 q^{43} - 4 q^{44} + 8 q^{46} + 16 q^{47} + 4 q^{48} - 12 q^{50} + 8 q^{53} - 4 q^{54} + 32 q^{59} + 16 q^{61} - 16 q^{62} + 4 q^{64} - 16 q^{65} + 4 q^{66} + 16 q^{67} - 8 q^{69} - 4 q^{72} - 8 q^{74} + 12 q^{75} + 16 q^{79} + 4 q^{81} + 32 q^{85} - 8 q^{86} + 4 q^{88} + 16 q^{89} - 8 q^{92} + 16 q^{93} - 16 q^{94} - 16 q^{95} - 4 q^{96} - 16 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^11 + 4 * q^12 + 4 * q^16 - 4 * q^18 + 4 * q^22 - 8 * q^23 - 4 * q^24 + 12 * q^25 + 4 * q^27 + 16 * q^31 - 4 * q^32 - 4 * q^33 + 4 * q^36 + 8 * q^37 + 8 * q^43 - 4 * q^44 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 12 * q^50 + 8 * q^53 - 4 * q^54 + 32 * q^59 + 16 * q^61 - 16 * q^62 + 4 * q^64 - 16 * q^65 + 4 * q^66 + 16 * q^67 - 8 * q^69 - 4 * q^72 - 8 * q^74 + 12 * q^75 + 16 * q^79 + 4 * q^81 + 32 * q^85 - 8 * q^86 + 4 * q^88 + 16 * q^89 - 8 * q^92 + 16 * q^93 - 16 * q^94 - 16 * q^95 - 4 * q^96 - 16 * q^97 - 4 * q^99

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.27411	−1.01701	−0.508506	−	0.861058i	$-0.669802\pi$
$5$	−2.27411	−1.01701	−0.508506	+	0.861058i	$0.669802\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.27411	0.719136
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−4.63029	−1.28421	−0.642106	−	0.766616i	$-0.721939\pi$
$13$	−4.63029	−1.28421	−0.642106	+	0.766616i	$0.721939\pi$
$14$	0	0
$15$	−2.27411	−0.587172
$16$	1.00000	0.250000
$17$	0.554318	0.134442	0.0672209	−	0.997738i	$-0.478587\pi$
$17$	0.554318	0.134442	0.0672209	+	0.997738i	$0.478587\pi$
$18$	−1.00000	−0.235702
$19$	4.07597	0.935092	0.467546	−	0.883969i	$-0.345138\pi$
$19$	4.07597	0.935092	0.467546	+	0.883969i	$0.345138\pi$
$20$	−2.27411	−0.508506
$21$	0	0
$22$	1.00000	0.213201
$23$	−6.93587	−1.44623	−0.723114	−	0.690729i	$-0.757290\pi$
$23$	−6.93587	−1.44623	−0.723114	+	0.690729i	$0.757290\pi$
$24$	−1.00000	−0.204124
$25$	0.171573	0.0343146
$26$	4.63029	0.908075
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	2.27411	0.415194
$31$	7.68832	1.38086	0.690432	−	0.723398i	$-0.257421\pi$
$31$	7.68832	1.38086	0.690432	+	0.723398i	$0.257421\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−0.554318	−0.0950647
$35$	0	0
$36$	1.00000	0.166667
$37$	10.8729	1.78750	0.893749	−	0.448567i	$-0.148065\pi$
$37$	10.8729	1.78750	0.893749	+	0.448567i	$0.148065\pi$
$38$	−4.07597	−0.661210
$39$	−4.63029	−0.741440
$40$	2.27411	0.359568
$41$	−0.554318	−0.0865699	−0.0432850	−	0.999063i	$-0.513782\pi$
$41$	−0.554318	−0.0865699	−0.0432850	+	0.999063i	$0.513782\pi$
$42$	0	0
$43$	−8.59272	−1.31038	−0.655189	−	0.755465i	$-0.727411\pi$
$43$	−8.59272	−1.31038	−0.655189	+	0.755465i	$0.727411\pi$
$44$	−1.00000	−0.150756
$45$	−2.27411	−0.339004
$46$	6.93587	1.02264
$47$	10.9044	1.59057	0.795285	−	0.606236i	$-0.207321\pi$
$47$	10.9044	1.59057	0.795285	+	0.606236i	$0.207321\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−0.171573	−0.0242641
$51$	0.554318	0.0776200
$52$	−4.63029	−0.642106
$53$	−0.440778	−0.0605455	−0.0302728	−	0.999542i	$-0.509638\pi$
$53$	−0.440778	−0.0605455	−0.0302728	+	0.999542i	$0.509638\pi$
$54$	−1.00000	−0.136083
$55$	2.27411	0.306641
$56$	0	0
$57$	4.07597	0.539876
$58$	2.82843	0.371391
$59$	5.17157	0.673281	0.336641	−	0.941633i	$-0.390709\pi$
$59$	5.17157	0.673281	0.336641	+	0.941633i	$0.390709\pi$
$60$	−2.27411	−0.293586
$61$	2.19814	0.281443	0.140721	−	0.990049i	$-0.455058\pi$
$61$	2.19814	0.281443	0.140721	+	0.990049i	$0.455058\pi$
$62$	−7.68832	−0.976418
$63$	0	0
$64$	1.00000	0.125000
$65$	10.5298	1.30606
$66$	1.00000	0.123091
$67$	−11.1409	−1.36108	−0.680541	−	0.732710i	$-0.738255\pi$
$67$	−11.1409	−1.36108	−0.680541	+	0.732710i	$0.738255\pi$
$68$	0.554318	0.0672209
$69$	−6.93587	−0.834980
$70$	0	0
$71$	3.60373	0.427683	0.213842	−	0.976868i	$-0.431402\pi$
$71$	3.60373	0.427683	0.213842	+	0.976868i	$0.431402\pi$
$72$	−1.00000	−0.117851
$73$	14.3631	1.68108	0.840538	−	0.541753i	$-0.182239\pi$
$73$	14.3631	1.68108	0.840538	+	0.541753i	$0.182239\pi$
$74$	−10.8729	−1.26395
$75$	0.171573	0.0198115
$76$	4.07597	0.467546
$77$	0	0
$78$	4.63029	0.524277
$79$	15.7014	1.76654	0.883270	−	0.468864i	$-0.155337\pi$
$79$	15.7014	1.76654	0.883270	+	0.468864i	$0.155337\pi$
$80$	−2.27411	−0.254253
$81$	1.00000	0.111111
$82$	0.554318	0.0612142
$83$	−6.51675	−0.715306	−0.357653	−	0.933855i	$-0.616423\pi$
$83$	−6.51675	−0.715306	−0.357653	+	0.933855i	$0.616423\pi$
$84$	0	0
$85$	−1.26058	−0.136729
$86$	8.59272	0.926577
$87$	−2.82843	−0.303239
$88$	1.00000	0.106600
$89$	−4.23401	−0.448805	−0.224402	−	0.974497i	$-0.572043\pi$
$89$	−4.23401	−0.448805	−0.224402	+	0.974497i	$0.572043\pi$
$90$	2.27411	0.239712
$91$	0	0
$92$	−6.93587	−0.723114
$93$	7.68832	0.797242
$94$	−10.9044	−1.12470
$95$	−9.26921	−0.951000
$96$	−1.00000	−0.102062
$97$	3.84637	0.390539	0.195270	−	0.980750i	$-0.437442\pi$
$97$	3.84637	0.390539	0.195270	+	0.980750i	$0.437442\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0.171573	0.0171573
$101$	10.7377	1.06844	0.534222	−	0.845344i	$-0.320605\pi$
$101$	10.7377	1.06844	0.534222	+	0.845344i	$0.320605\pi$
$102$	−0.554318	−0.0548856
$103$	−9.49712	−0.935779	−0.467890	−	0.883787i	$-0.654985\pi$
$103$	−9.49712	−0.935779	−0.467890	+	0.883787i	$0.654985\pi$
$104$	4.63029	0.454037
$105$	0	0
$106$	0.440778	0.0428122
$107$	−7.64823	−0.739382	−0.369691	−	0.929155i	$-0.620536\pi$
$107$	−7.64823	−0.739382	−0.369691	+	0.929155i	$0.620536\pi$
$108$	1.00000	0.0962250
$109$	−1.22470	−0.117305	−0.0586526	−	0.998278i	$-0.518680\pi$
$109$	−1.22470	−0.117305	−0.0586526	+	0.998278i	$0.518680\pi$
$110$	−2.27411	−0.216828
$111$	10.8729	1.03201
$112$	0	0
$113$	3.37665	0.317648	0.158824	−	0.987307i	$-0.449230\pi$
$113$	3.37665	0.317648	0.158824	+	0.987307i	$0.449230\pi$
$114$	−4.07597	−0.381750
$115$	15.7729	1.47083
$116$	−2.82843	−0.262613
$117$	−4.63029	−0.428070
$118$	−5.17157	−0.476082
$119$	0	0
$120$	2.27411	0.207597
$121$	1.00000	0.0909091
$122$	−2.19814	−0.199010
$123$	−0.554318	−0.0499812
$124$	7.68832	0.690432
$125$	10.9804	0.982114
$126$	0	0
$127$	18.3656	1.62969	0.814844	−	0.579681i	$-0.196823\pi$
$127$	18.3656	1.62969	0.814844	+	0.579681i	$0.196823\pi$
$128$	−1.00000	−0.0883883
$129$	−8.59272	−0.756547
$130$	−10.5298	−0.923523
$131$	10.5167	0.918853	0.459426	−	0.888216i	$-0.348055\pi$
$131$	10.5167	0.918853	0.459426	+	0.888216i	$0.348055\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	11.1409	0.962431
$135$	−2.27411	−0.195724
$136$	−0.554318	−0.0475324
$137$	−1.60373	−0.137015	−0.0685077	−	0.997651i	$-0.521824\pi$
$137$	−1.60373	−0.137015	−0.0685077	+	0.997651i	$0.521824\pi$
$138$	6.93587	0.590420
$139$	−11.4526	−0.971398	−0.485699	−	0.874126i	$-0.661435\pi$
$139$	−11.4526	−0.971398	−0.485699	+	0.874126i	$0.661435\pi$
$140$	0	0
$141$	10.9044	0.918316
$142$	−3.60373	−0.302418
$143$	4.63029	0.387204
$144$	1.00000	0.0833333
$145$	6.43215	0.534161
$146$	−14.3631	−1.18870
$147$	0	0
$148$	10.8729	0.893749
$149$	−6.99257	−0.572854	−0.286427	−	0.958102i	$-0.592468\pi$
$149$	−6.99257	−0.572854	−0.286427	+	0.958102i	$0.592468\pi$
$150$	−0.171573	−0.0140089
$151$	19.5286	1.58921	0.794607	−	0.607124i	$-0.207677\pi$
$151$	19.5286	1.58921	0.794607	+	0.607124i	$0.207677\pi$
$152$	−4.07597	−0.330605
$153$	0.554318	0.0448139
$154$	0	0
$155$	−17.4841	−1.40436
$156$	−4.63029	−0.370720
$157$	9.81490	0.783314	0.391657	−	0.920111i	$-0.371902\pi$
$157$	9.81490	0.783314	0.391657	+	0.920111i	$0.371902\pi$
$158$	−15.7014	−1.24913
$159$	−0.440778	−0.0349560
$160$	2.27411	0.179784
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	13.6569	1.06969	0.534844	−	0.844951i	$-0.320370\pi$
$163$	13.6569	1.06969	0.534844	+	0.844951i	$0.320370\pi$
$164$	−0.554318	−0.0432850
$165$	2.27411	0.177039
$166$	6.51675	0.505798
$167$	−6.03588	−0.467070	−0.233535	−	0.972348i	$-0.575029\pi$
$167$	−6.03588	−0.467070	−0.233535	+	0.972348i	$0.575029\pi$
$168$	0	0
$169$	8.43958	0.649199
$170$	1.26058	0.0966820
$171$	4.07597	0.311697
$172$	−8.59272	−0.655189
$173$	3.68222	0.279954	0.139977	−	0.990155i	$-0.455297\pi$
$173$	3.68222	0.279954	0.139977	+	0.990155i	$0.455297\pi$
$174$	2.82843	0.214423
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	5.17157	0.388719
$178$	4.23401	0.317353
$179$	−10.6049	−0.792649	−0.396324	−	0.918111i	$-0.629714\pi$
$179$	−10.6049	−0.792649	−0.396324	+	0.918111i	$0.629714\pi$
$180$	−2.27411	−0.169502
$181$	20.7692	1.54376	0.771881	−	0.635767i	$-0.219316\pi$
$181$	20.7692	1.54376	0.771881	+	0.635767i	$0.219316\pi$
$182$	0	0
$183$	2.19814	0.162491
$184$	6.93587	0.511319
$185$	−24.7262	−1.81791
$186$	−7.68832	−0.563735
$187$	−0.554318	−0.0405357
$188$	10.9044	0.795285
$189$	0	0
$190$	9.26921	0.672459
$191$	16.6817	1.20705	0.603524	−	0.797345i	$-0.293763\pi$
$191$	16.6817	1.20705	0.603524	+	0.797345i	$0.293763\pi$
$192$	1.00000	0.0721688
$193$	5.43958	0.391550	0.195775	−	0.980649i	$-0.437278\pi$
$193$	5.43958	0.391550	0.195775	+	0.980649i	$0.437278\pi$
$194$	−3.84637	−0.276153
$195$	10.5298	0.754054
$196$	0	0
$197$	−10.3963	−0.740704	−0.370352	−	0.928892i	$-0.620763\pi$
$197$	−10.3963	−0.740704	−0.370352	+	0.928892i	$0.620763\pi$
$198$	1.00000	0.0710669
$199$	6.03147	0.427559	0.213780	−	0.976882i	$-0.431423\pi$
$199$	6.03147	0.427559	0.213780	+	0.976882i	$0.431423\pi$
$200$	−0.171573	−0.0121320
$201$	−11.1409	−0.785821
$202$	−10.7377	−0.755504
$203$	0	0
$204$	0.554318	0.0388100
$205$	1.26058	0.0880427
$206$	9.49712	0.661696
$207$	−6.93587	−0.482076
$208$	−4.63029	−0.321053
$209$	−4.07597	−0.281941
$210$	0	0
$211$	−2.16057	−0.148740	−0.0743699	−	0.997231i	$-0.523695\pi$
$211$	−2.16057	−0.148740	−0.0743699	+	0.997231i	$0.523695\pi$
$212$	−0.440778	−0.0302728
$213$	3.60373	0.246923
$214$	7.64823	0.522822
$215$	19.5408	1.33267
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	1.22470	0.0829473
$219$	14.3631	0.970569
$220$	2.27411	0.153320
$221$	−2.56665	−0.172652
$222$	−10.8729	−0.729743
$223$	−1.17598	−0.0787496	−0.0393748	−	0.999225i	$-0.512537\pi$
$223$	−1.17598	−0.0787496	−0.0393748	+	0.999225i	$0.512537\pi$
$224$	0	0
$225$	0.171573	0.0114382
$226$	−3.37665	−0.224611
$227$	−22.0453	−1.46320	−0.731600	−	0.681734i	$-0.761226\pi$
$227$	−22.0453	−1.46320	−0.731600	+	0.681734i	$0.761226\pi$
$228$	4.07597	0.269938
$229$	−0.554318	−0.0366304	−0.0183152	−	0.999832i	$-0.505830\pi$
$229$	−0.554318	−0.0366304	−0.0183152	+	0.999832i	$0.505830\pi$
$230$	−15.7729	−1.04004
$231$	0	0
$232$	2.82843	0.185695
$233$	−0.343146	−0.0224802	−0.0112401	−	0.999937i	$-0.503578\pi$
$233$	−0.343146	−0.0224802	−0.0112401	+	0.999937i	$0.503578\pi$
$234$	4.63029	0.302692
$235$	−24.7978	−1.61763
$236$	5.17157	0.336641
$237$	15.7014	1.01991
$238$	0	0
$239$	−5.56785	−0.360154	−0.180077	−	0.983653i	$-0.557635\pi$
$239$	−5.56785	−0.360154	−0.180077	+	0.983653i	$0.557635\pi$
$240$	−2.27411	−0.146793
$241$	−1.94077	−0.125016	−0.0625080	−	0.998044i	$-0.519910\pi$
$241$	−1.94077	−0.125016	−0.0625080	+	0.998044i	$0.519910\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	2.19814	0.140721
$245$	0	0
$246$	0.554318	0.0353420
$247$	−18.8729	−1.20086
$248$	−7.68832	−0.488209
$249$	−6.51675	−0.412982
$250$	−10.9804	−0.694460
$251$	23.4657	1.48114	0.740569	−	0.671980i	$-0.234556\pi$
$251$	23.4657	1.48114	0.740569	+	0.671980i	$0.234556\pi$
$252$	0	0
$253$	6.93587	0.436054
$254$	−18.3656	−1.15236
$255$	−1.26058	−0.0789405
$256$	1.00000	0.0625000
$257$	22.6083	1.41027	0.705133	−	0.709075i	$-0.250887\pi$
$257$	22.6083	1.41027	0.705133	+	0.709075i	$0.250887\pi$
$258$	8.59272	0.534959
$259$	0	0
$260$	10.5298	0.653030
$261$	−2.82843	−0.175075
$262$	−10.5167	−0.649727
$263$	−31.2299	−1.92572	−0.962861	−	0.269999i	$-0.912977\pi$
$263$	−31.2299	−1.92572	−0.962861	+	0.269999i	$0.912977\pi$
$264$	1.00000	0.0615457
$265$	1.00238	0.0615756
$266$	0	0
$267$	−4.23401	−0.259118
$268$	−11.1409	−0.680541
$269$	28.4792	1.73641	0.868203	−	0.496209i	$-0.165275\pi$
$269$	28.4792	1.73641	0.868203	+	0.496209i	$0.165275\pi$
$270$	2.27411	0.138398
$271$	9.94687	0.604229	0.302115	−	0.953272i	$-0.402307\pi$
$271$	9.94687	0.604229	0.302115	+	0.953272i	$0.402307\pi$
$272$	0.554318	0.0336105
$273$	0	0
$274$	1.60373	0.0968846
$275$	−0.171573	−0.0103462
$276$	−6.93587	−0.417490
$277$	26.5262	1.59381	0.796903	−	0.604108i	$-0.206470\pi$
$277$	26.5262	1.59381	0.796903	+	0.604108i	$0.206470\pi$
$278$	11.4526	0.686882
$279$	7.68832	0.460288
$280$	0	0
$281$	−5.11844	−0.305341	−0.152670	−	0.988277i	$-0.548787\pi$
$281$	−5.11844	−0.305341	−0.152670	+	0.988277i	$0.548787\pi$
$282$	−10.9044	−0.649348
$283$	−2.48205	−0.147543	−0.0737714	−	0.997275i	$-0.523504\pi$
$283$	−2.48205	−0.147543	−0.0737714	+	0.997275i	$0.523504\pi$
$284$	3.60373	0.213842
$285$	−9.26921	−0.549060
$286$	−4.63029	−0.273795
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−16.6927	−0.981925
$290$	−6.43215	−0.377709
$291$	3.84637	0.225478
$292$	14.3631	0.840538
$293$	2.02537	0.118323	0.0591617	−	0.998248i	$-0.481157\pi$
$293$	2.02537	0.118323	0.0591617	+	0.998248i	$0.481157\pi$
$294$	0	0
$295$	−11.7607	−0.684736
$296$	−10.8729	−0.631976
$297$	−1.00000	−0.0580259
$298$	6.99257	0.405069
$299$	32.1151	1.85726
$300$	0.171573	0.00990576
$301$	0	0
$302$	−19.5286	−1.12374
$303$	10.7377	0.616866
$304$	4.07597	0.233773
$305$	−4.99880	−0.286231
$306$	−0.554318	−0.0316882
$307$	5.88477	0.335862	0.167931	−	0.985799i	$-0.446291\pi$
$307$	5.88477	0.335862	0.167931	+	0.985799i	$0.446291\pi$
$308$	0	0
$309$	−9.49712	−0.540272
$310$	17.4841	0.993029
$311$	17.4624	0.990203	0.495102	−	0.868835i	$-0.335131\pi$
$311$	17.4624	0.990203	0.495102	+	0.868835i	$0.335131\pi$
$312$	4.63029	0.262139
$313$	−8.87987	−0.501920	−0.250960	−	0.967997i	$-0.580746\pi$
$313$	−8.87987	−0.501920	−0.250960	+	0.967997i	$0.580746\pi$
$314$	−9.81490	−0.553887
$315$	0	0
$316$	15.7014	0.883270
$317$	−3.96937	−0.222942	−0.111471	−	0.993768i	$-0.535556\pi$
$317$	−3.96937	−0.222942	−0.111471	+	0.993768i	$0.535556\pi$
$318$	0.440778	0.0247176
$319$	2.82843	0.158362
$320$	−2.27411	−0.127127
$321$	−7.64823	−0.426882
$322$	0	0
$323$	2.25938	0.125715
$324$	1.00000	0.0555556
$325$	−0.794432	−0.0440672
$326$	−13.6569	−0.756383
$327$	−1.22470	−0.0677262
$328$	0.554318	0.0306071
$329$	0	0
$330$	−2.27411	−0.125186
$331$	−13.2692	−0.729341	−0.364671	−	0.931137i	$-0.618818\pi$
$331$	−13.2692	−0.729341	−0.364671	+	0.931137i	$0.618818\pi$
$332$	−6.51675	−0.357653
$333$	10.8729	0.595833
$334$	6.03588	0.330269
$335$	25.3357	1.38424
$336$	0	0
$337$	−3.88036	−0.211377	−0.105688	−	0.994399i	$-0.533705\pi$
$337$	−3.88036	−0.211377	−0.105688	+	0.994399i	$0.533705\pi$
$338$	−8.43958	−0.459053
$339$	3.37665	0.183394
$340$	−1.26058	−0.0683645
$341$	−7.68832	−0.416346
$342$	−4.07597	−0.220403
$343$	0	0
$344$	8.59272	0.463289
$345$	15.7729	0.849185
$346$	−3.68222	−0.197958
$347$	19.1768	1.02947	0.514733	−	0.857351i	$-0.327891\pi$
$347$	19.1768	1.02947	0.514733	+	0.857351i	$0.327891\pi$
$348$	−2.82843	−0.151620
$349$	−30.7602	−1.64656	−0.823279	−	0.567638i	$-0.807858\pi$
$349$	−30.7602	−1.64656	−0.823279	+	0.567638i	$0.807858\pi$
$350$	0	0
$351$	−4.63029	−0.247147
$352$	1.00000	0.0533002
$353$	−1.19071	−0.0633749	−0.0316875	−	0.999498i	$-0.510088\pi$
$353$	−1.19071	−0.0633749	−0.0316875	+	0.999498i	$0.510088\pi$
$354$	−5.17157	−0.274866
$355$	−8.19526	−0.434959
$356$	−4.23401	−0.224402
$357$	0	0
$358$	10.6049	0.560487
$359$	−24.6915	−1.30317	−0.651585	−	0.758576i	$-0.725895\pi$
$359$	−24.6915	−1.30317	−0.651585	+	0.758576i	$0.725895\pi$
$360$	2.27411	0.119856
$361$	−2.38645	−0.125603
$362$	−20.7692	−1.09160
$363$	1.00000	0.0524864
$364$	0	0
$365$	−32.6633	−1.70967
$366$	−2.19814	−0.114898
$367$	−7.21691	−0.376720	−0.188360	−	0.982100i	$-0.560317\pi$
$367$	−7.21691	−0.376720	−0.188360	+	0.982100i	$0.560317\pi$
$368$	−6.93587	−0.361557
$369$	−0.554318	−0.0288566
$370$	24.7262	1.28546
$371$	0	0
$372$	7.68832	0.398621
$373$	1.70998	0.0885396	0.0442698	−	0.999020i	$-0.485904\pi$
$373$	1.70998	0.0885396	0.0442698	+	0.999020i	$0.485904\pi$
$374$	0.554318	0.0286631
$375$	10.9804	0.567024
$376$	−10.9044	−0.562351
$377$	13.0964	0.674501
$378$	0	0
$379$	−37.2745	−1.91466	−0.957330	−	0.288997i	$-0.906678\pi$
$379$	−37.2745	−1.91466	−0.957330	+	0.288997i	$0.906678\pi$
$380$	−9.26921	−0.475500
$381$	18.3656	0.940900
$382$	−16.6817	−0.853511
$383$	11.4019	0.582609	0.291304	−	0.956630i	$-0.405911\pi$
$383$	11.4019	0.582609	0.291304	+	0.956630i	$0.405911\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−5.43958	−0.276867
$387$	−8.59272	−0.436793
$388$	3.84637	0.195270
$389$	38.6250	1.95837	0.979183	−	0.202978i	$-0.0650620\pi$
$389$	38.6250	1.95837	0.979183	+	0.202978i	$0.0650620\pi$
$390$	−10.5298	−0.533196
$391$	−3.84468	−0.194434
$392$	0	0
$393$	10.5167	0.530500
$394$	10.3963	0.523757
$395$	−35.7066	−1.79659
$396$	−1.00000	−0.0502519
$397$	15.7929	0.792622	0.396311	−	0.918116i	$-0.370290\pi$
$397$	15.7929	0.792622	0.396311	+	0.918116i	$0.370290\pi$
$398$	−6.03147	−0.302330
$399$	0	0
$400$	0.171573	0.00857864
$401$	−21.7100	−1.08414	−0.542072	−	0.840332i	$-0.682360\pi$
$401$	−21.7100	−1.08414	−0.542072	+	0.840332i	$0.682360\pi$
$402$	11.1409	0.555660
$403$	−35.5992	−1.77332
$404$	10.7377	0.534222
$405$	−2.27411	−0.113001
$406$	0	0
$407$	−10.8729	−0.538951
$408$	−0.554318	−0.0274428
$409$	37.7520	1.86671	0.933357	−	0.358949i	$-0.116865\pi$
$409$	37.7520	1.86671	0.933357	+	0.358949i	$0.116865\pi$
$410$	−1.26058	−0.0622556
$411$	−1.60373	−0.0791059
$412$	−9.49712	−0.467890
$413$	0	0
$414$	6.93587	0.340879
$415$	14.8198	0.727475
$416$	4.63029	0.227019
$417$	−11.4526	−0.560837
$418$	4.07597	0.199362
$419$	15.5408	0.759217	0.379609	−	0.925147i	$-0.376059\pi$
$419$	15.5408	0.759217	0.379609	+	0.925147i	$0.376059\pi$
$420$	0	0
$421$	29.6176	1.44347	0.721737	−	0.692168i	$-0.243344\pi$
$421$	29.6176	1.44347	0.721737	+	0.692168i	$0.243344\pi$
$422$	2.16057	0.105175
$423$	10.9044	0.530190
$424$	0.440778	0.0214061
$425$	0.0951059	0.00461331
$426$	−3.60373	−0.174601
$427$	0	0
$428$	−7.64823	−0.369691
$429$	4.63029	0.223552
$430$	−19.5408	−0.942340
$431$	24.7978	1.19447	0.597234	−	0.802067i	$-0.296266\pi$
$431$	24.7978	1.19447	0.597234	+	0.802067i	$0.296266\pi$
$432$	1.00000	0.0481125
$433$	−11.2859	−0.542368	−0.271184	−	0.962528i	$-0.587415\pi$
$433$	−11.2859	−0.542368	−0.271184	+	0.962528i	$0.587415\pi$
$434$	0	0
$435$	6.43215	0.308398
$436$	−1.22470	−0.0586526
$437$	−28.2704	−1.35236
$438$	−14.3631	−0.686296
$439$	−1.17157	−0.0559161	−0.0279581	−	0.999609i	$-0.508900\pi$
$439$	−1.17157	−0.0559161	−0.0279581	+	0.999609i	$0.508900\pi$
$440$	−2.27411	−0.108414
$441$	0	0
$442$	2.56665	0.122083
$443$	16.6915	0.793039	0.396519	−	0.918026i	$-0.370218\pi$
$443$	16.6915	0.793039	0.396519	+	0.918026i	$0.370218\pi$
$444$	10.8729	0.516006
$445$	9.62861	0.456440
$446$	1.17598	0.0556844
$447$	−6.99257	−0.330737
$448$	0	0
$449$	−27.5547	−1.30038	−0.650192	−	0.759770i	$-0.725312\pi$
$449$	−27.5547	−1.30038	−0.650192	+	0.759770i	$0.725312\pi$
$450$	−0.171573	−0.00808802
$451$	0.554318	0.0261018
$452$	3.37665	0.158824
$453$	19.5286	0.917533
$454$	22.0453	1.03464
$455$	0	0
$456$	−4.07597	−0.190875
$457$	−18.1953	−0.851139	−0.425569	−	0.904926i	$-0.639926\pi$
$457$	−18.1953	−0.851139	−0.425569	+	0.904926i	$0.639926\pi$
$458$	0.554318	0.0259016
$459$	0.554318	0.0258733
$460$	15.7729	0.735416
$461$	−19.6193	−0.913761	−0.456881	−	0.889528i	$-0.651033\pi$
$461$	−19.6193	−0.913761	−0.456881	+	0.889528i	$0.651033\pi$
$462$	0	0
$463$	35.6176	1.65529	0.827645	−	0.561252i	$-0.189680\pi$
$463$	35.6176	1.65529	0.827645	+	0.561252i	$0.189680\pi$
$464$	−2.82843	−0.131306
$465$	−17.4841	−0.810805
$466$	0.343146	0.0158959
$467$	35.0744	1.62305	0.811526	−	0.584317i	$-0.198638\pi$
$467$	35.0744	1.62305	0.811526	+	0.584317i	$0.198638\pi$
$468$	−4.63029	−0.214035
$469$	0	0
$470$	24.7978	1.14384
$471$	9.81490	0.452247
$472$	−5.17157	−0.238041
$473$	8.59272	0.395094
$474$	−15.7014	−0.721187
$475$	0.699326	0.0320873
$476$	0	0
$477$	−0.440778	−0.0201818
$478$	5.56785	0.254667
$479$	18.7482	0.856629	0.428314	−	0.903630i	$-0.359108\pi$
$479$	18.7482	0.856629	0.428314	+	0.903630i	$0.359108\pi$
$480$	2.27411	0.103798
$481$	−50.3448	−2.29553
$482$	1.94077	0.0883997
$483$	0	0
$484$	1.00000	0.0454545
$485$	−8.74706	−0.397183
$486$	−1.00000	−0.0453609
$487$	29.9301	1.35626	0.678131	−	0.734941i	$-0.262790\pi$
$487$	29.9301	1.35626	0.678131	+	0.734941i	$0.262790\pi$
$488$	−2.19814	−0.0995050
$489$	13.6569	0.617584
$490$	0	0
$491$	−8.68629	−0.392007	−0.196003	−	0.980603i	$-0.562796\pi$
$491$	−8.68629	−0.392007	−0.196003	+	0.980603i	$0.562796\pi$
$492$	−0.554318	−0.0249906
$493$	−1.56785	−0.0706123
$494$	18.8729	0.849133
$495$	2.27411	0.102214
$496$	7.68832	0.345216
$497$	0	0
$498$	6.51675	0.292023
$499$	23.8076	1.06578	0.532888	−	0.846186i	$-0.321107\pi$
$499$	23.8076	1.06578	0.532888	+	0.846186i	$0.321107\pi$
$500$	10.9804	0.491057
$501$	−6.03588	−0.269663
$502$	−23.4657	−1.04732
$503$	−32.2531	−1.43810	−0.719048	−	0.694960i	$-0.755422\pi$
$503$	−32.2531	−1.43810	−0.719048	+	0.694960i	$0.755422\pi$
$504$	0	0
$505$	−24.4188	−1.08662
$506$	−6.93587	−0.308337
$507$	8.43958	0.374815
$508$	18.3656	0.814844
$509$	35.1066	1.55607	0.778036	−	0.628219i	$-0.216216\pi$
$509$	35.1066	1.55607	0.778036	+	0.628219i	$0.216216\pi$
$510$	1.26058	0.0558194
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.07597	0.179959
$514$	−22.6083	−0.997209
$515$	21.5975	0.951699
$516$	−8.59272	−0.378273
$517$	−10.9044	−0.479575
$518$	0	0
$519$	3.68222	0.161632
$520$	−10.5298	−0.461762
$521$	31.0763	1.36148	0.680739	−	0.732526i	$-0.261659\pi$
$521$	31.0763	1.36148	0.680739	+	0.732526i	$0.261659\pi$
$522$	2.82843	0.123797
$523$	−18.3822	−0.803800	−0.401900	−	0.915684i	$-0.631650\pi$
$523$	−18.3822	−0.803800	−0.401900	+	0.915684i	$0.631650\pi$
$524$	10.5167	0.459426
$525$	0	0
$526$	31.2299	1.36169
$527$	4.26177	0.185646
$528$	−1.00000	−0.0435194
$529$	25.1063	1.09158
$530$	−1.00238	−0.0435405
$531$	5.17157	0.224427
$532$	0	0
$533$	2.56665	0.111174
$534$	4.23401	0.183224
$535$	17.3929	0.751961
$536$	11.1409	0.481215
$537$	−10.6049	−0.457636
$538$	−28.4792	−1.22782
$539$	0	0
$540$	−2.27411	−0.0978621
$541$	−20.2311	−0.869805	−0.434902	−	0.900478i	$-0.643217\pi$
$541$	−20.2311	−0.869805	−0.434902	+	0.900478i	$0.643217\pi$
$542$	−9.94687	−0.427255
$543$	20.7692	0.891292
$544$	−0.554318	−0.0237662
$545$	2.78511	0.119301
$546$	0	0
$547$	31.9249	1.36501	0.682504	−	0.730882i	$-0.260891\pi$
$547$	31.9249	1.36501	0.682504	+	0.730882i	$0.260891\pi$
$548$	−1.60373	−0.0685077
$549$	2.19814	0.0938142
$550$	0.171573	0.00731589
$551$	−11.5286	−0.491134
$552$	6.93587	0.295210
$553$	0	0
$554$	−26.5262	−1.12699
$555$	−24.7262	−1.04957
$556$	−11.4526	−0.485699
$557$	39.4386	1.67107	0.835533	−	0.549440i	$-0.185159\pi$
$557$	39.4386	1.67107	0.835533	+	0.549440i	$0.185159\pi$
$558$	−7.68832	−0.325473
$559$	39.7868	1.68280
$560$	0	0
$561$	−0.554318	−0.0234033
$562$	5.11844	0.215909
$563$	−44.6098	−1.88008	−0.940040	−	0.341065i	$-0.889213\pi$
$563$	−44.6098	−1.88008	−0.940040	+	0.341065i	$0.889213\pi$
$564$	10.9044	0.459158
$565$	−7.67886	−0.323052
$566$	2.48205	0.104329
$567$	0	0
$568$	−3.60373	−0.151209
$569$	35.1682	1.47433	0.737164	−	0.675714i	$-0.236165\pi$
$569$	35.1682	1.47433	0.737164	+	0.675714i	$0.236165\pi$
$570$	9.26921	0.388244
$571$	−9.96412	−0.416986	−0.208493	−	0.978024i	$-0.566856\pi$
$571$	−9.96412	−0.416986	−0.208493	+	0.978024i	$0.566856\pi$
$572$	4.63029	0.193602
$573$	16.6817	0.696889
$574$	0	0
$575$	−1.19001	−0.0496267
$576$	1.00000	0.0416667
$577$	2.12657	0.0885305	0.0442652	−	0.999020i	$-0.485905\pi$
$577$	2.12657	0.0885305	0.0442652	+	0.999020i	$0.485905\pi$
$578$	16.6927	0.694326
$579$	5.43958	0.226061
$580$	6.43215	0.267081
$581$	0	0
$582$	−3.84637	−0.159437
$583$	0.440778	0.0182552
$584$	−14.3631	−0.594350
$585$	10.5298	0.435353
$586$	−2.02537	−0.0836672
$587$	−25.8260	−1.06596	−0.532978	−	0.846129i	$-0.678927\pi$
$587$	−25.8260	−1.06596	−0.532978	+	0.846129i	$0.678927\pi$
$588$	0	0
$589$	31.3374	1.29123
$590$	11.7607	0.484181
$591$	−10.3963	−0.427646
$592$	10.8729	0.446875
$593$	−18.8975	−0.776026	−0.388013	−	0.921654i	$-0.626838\pi$
$593$	−18.8975	−0.776026	−0.388013	+	0.921654i	$0.626838\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−6.99257	−0.286427
$597$	6.03147	0.246852
$598$	−32.1151	−1.31328
$599$	3.49510	0.142806	0.0714030	−	0.997448i	$-0.477252\pi$
$599$	3.49510	0.142806	0.0714030	+	0.997448i	$0.477252\pi$
$600$	−0.171573	−0.00700443
$601$	−23.6409	−0.964334	−0.482167	−	0.876079i	$-0.660150\pi$
$601$	−23.6409	−0.964334	−0.482167	+	0.876079i	$0.660150\pi$
$602$	0	0
$603$	−11.1409	−0.453694
$604$	19.5286	0.794607
$605$	−2.27411	−0.0924557
$606$	−10.7377	−0.436190
$607$	−8.19526	−0.332636	−0.166318	−	0.986072i	$-0.553188\pi$
$607$	−8.19526	−0.332636	−0.166318	+	0.986072i	$0.553188\pi$
$608$	−4.07597	−0.165302
$609$	0	0
$610$	4.99880	0.202396
$611$	−50.4905	−2.04263
$612$	0.554318	0.0224070
$613$	26.3692	1.06504	0.532521	−	0.846417i	$-0.321245\pi$
$613$	26.3692	1.06504	0.532521	+	0.846417i	$0.321245\pi$
$614$	−5.88477	−0.237490
$615$	1.26058	0.0508315
$616$	0	0
$617$	−2.35534	−0.0948226	−0.0474113	−	0.998875i	$-0.515097\pi$
$617$	−2.35534	−0.0948226	−0.0474113	+	0.998875i	$0.515097\pi$
$618$	9.49712	0.382030
$619$	10.2051	0.410177	0.205088	−	0.978743i	$-0.434252\pi$
$619$	10.2051	0.410177	0.205088	+	0.978743i	$0.434252\pi$
$620$	−17.4841	−0.702178
$621$	−6.93587	−0.278327
$622$	−17.4624	−0.700179
$623$	0	0
$624$	−4.63029	−0.185360
$625$	−25.8284	−1.03314
$626$	8.87987	0.354911
$627$	−4.07597	−0.162779
$628$	9.81490	0.391657
$629$	6.02706	0.240315
$630$	0	0
$631$	−9.86550	−0.392739	−0.196370	−	0.980530i	$-0.562915\pi$
$631$	−9.86550	−0.392739	−0.196370	+	0.980530i	$0.562915\pi$
$632$	−15.7014	−0.624566
$633$	−2.16057	−0.0858749
$634$	3.96937	0.157644
$635$	−41.7655	−1.65741
$636$	−0.440778	−0.0174780
$637$	0	0
$638$	−2.82843	−0.111979
$639$	3.60373	0.142561
$640$	2.27411	0.0898921
$641$	−13.3063	−0.525566	−0.262783	−	0.964855i	$-0.584640\pi$
$641$	−13.3063	−0.525566	−0.262783	+	0.964855i	$0.584640\pi$
$642$	7.64823	0.301851
$643$	12.2531	0.483217	0.241609	−	0.970374i	$-0.422325\pi$
$643$	12.2531	0.483217	0.241609	+	0.970374i	$0.422325\pi$
$644$	0	0
$645$	19.5408	0.769418
$646$	−2.25938	−0.0888943
$647$	−22.3873	−0.880136	−0.440068	−	0.897965i	$-0.645046\pi$
$647$	−22.3873	−0.880136	−0.440068	+	0.897965i	$0.645046\pi$
$648$	−1.00000	−0.0392837
$649$	−5.17157	−0.203002
$650$	0.794432	0.0311602
$651$	0	0
$652$	13.6569	0.534844
$653$	14.1953	0.555504	0.277752	−	0.960653i	$-0.410411\pi$
$653$	14.1953	0.555504	0.277752	+	0.960653i	$0.410411\pi$
$654$	1.22470	0.0478896
$655$	−23.9162	−0.934485
$656$	−0.554318	−0.0216425
$657$	14.3631	0.560359
$658$	0	0
$659$	−21.7152	−0.845905	−0.422953	−	0.906152i	$-0.639006\pi$
$659$	−21.7152	−0.845905	−0.422953	+	0.906152i	$0.639006\pi$
$660$	2.27411	0.0885196
$661$	−46.5029	−1.80875	−0.904376	−	0.426737i	$-0.859663\pi$
$661$	−46.5029	−1.80875	−0.904376	+	0.426737i	$0.859663\pi$
$662$	13.2692	0.515722
$663$	−2.56665	−0.0996805
$664$	6.51675	0.252899
$665$	0	0
$666$	−10.8729	−0.421317
$667$	19.6176	0.759596
$668$	−6.03588	−0.233535
$669$	−1.17598	−0.0454661
$670$	−25.3357	−0.978804
$671$	−2.19814	−0.0848582
$672$	0	0
$673$	−29.7152	−1.14544	−0.572719	−	0.819752i	$-0.694111\pi$
$673$	−29.7152	−1.14544	−0.572719	+	0.819752i	$0.694111\pi$
$674$	3.88036	0.149466
$675$	0.171573	0.00660384
$676$	8.43958	0.324599
$677$	−43.5147	−1.67241	−0.836203	−	0.548420i	$-0.815229\pi$
$677$	−43.5147	−1.67241	−0.836203	+	0.548420i	$0.815229\pi$
$678$	−3.37665	−0.129679
$679$	0	0
$680$	1.26058	0.0483410
$681$	−22.0453	−0.844779
$682$	7.68832	0.294401
$683$	−29.3582	−1.12336	−0.561680	−	0.827354i	$-0.689845\pi$
$683$	−29.3582	−1.12336	−0.561680	+	0.827354i	$0.689845\pi$
$684$	4.07597	0.155849
$685$	3.64705	0.139346
$686$	0	0
$687$	−0.554318	−0.0211485
$688$	−8.59272	−0.327594
$689$	2.04093	0.0777533
$690$	−15.7729	−0.600465
$691$	−6.84063	−0.260230	−0.130115	−	0.991499i	$-0.541535\pi$
$691$	−6.84063	−0.260230	−0.130115	+	0.991499i	$0.541535\pi$
$692$	3.68222	0.139977
$693$	0	0
$694$	−19.1768	−0.727942
$695$	26.0445	0.987924
$696$	2.82843	0.107211
$697$	−0.307268	−0.0116386
$698$	30.7602	1.16429
$699$	−0.343146	−0.0129790
$700$	0	0
$701$	−24.1929	−0.913752	−0.456876	−	0.889530i	$-0.651032\pi$
$701$	−24.1929	−0.913752	−0.456876	+	0.889530i	$0.651032\pi$
$702$	4.63029	0.174759
$703$	44.3178	1.67148
$704$	−1.00000	−0.0376889
$705$	−24.7978	−0.933939
$706$	1.19071	0.0448128
$707$	0	0
$708$	5.17157	0.194360
$709$	−32.8509	−1.23374	−0.616871	−	0.787064i	$-0.711600\pi$
$709$	−32.8509	−1.23374	−0.616871	+	0.787064i	$0.711600\pi$
$710$	8.19526	0.307563
$711$	15.7014	0.588847
$712$	4.23401	0.158676
$713$	−53.3252	−1.99704
$714$	0	0
$715$	−10.5298	−0.393792
$716$	−10.6049	−0.396324
$717$	−5.56785	−0.207935
$718$	24.6915	0.921480
$719$	−6.43132	−0.239848	−0.119924	−	0.992783i	$-0.538265\pi$
$719$	−6.43132	−0.239848	−0.119924	+	0.992783i	$0.538265\pi$
$720$	−2.27411	−0.0847510
$721$	0	0
$722$	2.38645	0.0888146
$723$	−1.94077	−0.0721781
$724$	20.7692	0.771881
$725$	−0.485281	−0.0180229
$726$	−1.00000	−0.0371135
$727$	2.71776	0.100796	0.0503981	−	0.998729i	$-0.483951\pi$
$727$	2.71776	0.100796	0.0503981	+	0.998729i	$0.483951\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	32.6633	1.20892
$731$	−4.76310	−0.176170
$732$	2.19814	0.0812455
$733$	−47.7799	−1.76479	−0.882395	−	0.470510i	$-0.844070\pi$
$733$	−47.7799	−1.76479	−0.882395	+	0.470510i	$0.844070\pi$
$734$	7.21691	0.266381
$735$	0	0
$736$	6.93587	0.255659
$737$	11.1409	0.410382
$738$	0.554318	0.0204047
$739$	−35.5425	−1.30745	−0.653725	−	0.756732i	$-0.726795\pi$
$739$	−35.5425	−1.30745	−0.653725	+	0.756732i	$0.726795\pi$
$740$	−24.7262	−0.908954
$741$	−18.8729	−0.693314
$742$	0	0
$743$	20.4992	0.752041	0.376020	−	0.926611i	$-0.377292\pi$
$743$	20.4992	0.752041	0.376020	+	0.926611i	$0.377292\pi$
$744$	−7.68832	−0.281868
$745$	15.9019	0.582599
$746$	−1.70998	−0.0626069
$747$	−6.51675	−0.238435
$748$	−0.554318	−0.0202679
$749$	0	0
$750$	−10.9804	−0.400946
$751$	−45.2682	−1.65186	−0.825930	−	0.563772i	$-0.809350\pi$
$751$	−45.2682	−1.65186	−0.825930	+	0.563772i	$0.809350\pi$
$752$	10.9044	0.397643
$753$	23.4657	0.855136
$754$	−13.0964	−0.476944
$755$	−44.4101	−1.61625
$756$	0	0
$757$	−40.6188	−1.47632	−0.738158	−	0.674628i	$-0.764304\pi$
$757$	−40.6188	−1.47632	−0.738158	+	0.674628i	$0.764304\pi$
$758$	37.2745	1.35387
$759$	6.93587	0.251756
$760$	9.26921	0.336229
$761$	29.1164	1.05547	0.527734	−	0.849409i	$-0.323042\pi$
$761$	29.1164	1.05547	0.527734	+	0.849409i	$0.323042\pi$
$762$	−18.3656	−0.665317
$763$	0	0
$764$	16.6817	0.603524
$765$	−1.26058	−0.0455763
$766$	−11.4019	−0.411967
$767$	−23.9459	−0.864636
$768$	1.00000	0.0360844
$769$	−22.9987	−0.829353	−0.414677	−	0.909969i	$-0.636105\pi$
$769$	−22.9987	−0.829353	−0.414677	+	0.909969i	$0.636105\pi$
$770$	0	0
$771$	22.6083	0.814217
$772$	5.43958	0.195775
$773$	−51.0755	−1.83706	−0.918529	−	0.395355i	$-0.870622\pi$
$773$	−51.0755	−1.83706	−0.918529	+	0.395355i	$0.870622\pi$
$774$	8.59272	0.308859
$775$	1.31911	0.0473837
$776$	−3.84637	−0.138076
$777$	0	0
$778$	−38.6250	−1.38477
$779$	−2.25938	−0.0809508
$780$	10.5298	0.377027
$781$	−3.60373	−0.128951
$782$	3.84468	0.137485
$783$	−2.82843	−0.101080
$784$	0	0
$785$	−22.3201	−0.796640
$786$	−10.5167	−0.375120
$787$	29.5024	1.05165	0.525823	−	0.850594i	$-0.323757\pi$
$787$	29.5024	1.05165	0.525823	+	0.850594i	$0.323757\pi$
$788$	−10.3963	−0.370352
$789$	−31.2299	−1.11182
$790$	35.7066	1.27038
$791$	0	0
$792$	1.00000	0.0355335
$793$	−10.1780	−0.361432
$794$	−15.7929	−0.560469
$795$	1.00238	0.0355507
$796$	6.03147	0.213780
$797$	−16.3990	−0.580882	−0.290441	−	0.956893i	$-0.593802\pi$
$797$	−16.3990	−0.580882	−0.290441	+	0.956893i	$0.593802\pi$
$798$	0	0
$799$	6.04450	0.213839
$800$	−0.171573	−0.00606602
$801$	−4.23401	−0.149602
$802$	21.7100	0.766606
$803$	−14.3631	−0.506863
$804$	−11.1409	−0.392911
$805$	0	0
$806$	35.5992	1.25393
$807$	28.4792	1.00251
$808$	−10.7377	−0.377752
$809$	17.0730	0.600253	0.300126	−	0.953899i	$-0.402971\pi$
$809$	17.0730	0.600253	0.300126	+	0.953899i	$0.402971\pi$
$810$	2.27411	0.0799041
$811$	−26.9166	−0.945170	−0.472585	−	0.881285i	$-0.656679\pi$
$811$	−26.9166	−0.945170	−0.472585	+	0.881285i	$0.656679\pi$
$812$	0	0
$813$	9.94687	0.348852
$814$	10.8729	0.381096
$815$	−31.0572	−1.08789
$816$	0.554318	0.0194050
$817$	−35.0237	−1.22532
$818$	−37.7520	−1.31997
$819$	0	0
$820$	1.26058	0.0440213
$821$	−21.2917	−0.743086	−0.371543	−	0.928416i	$-0.621171\pi$
$821$	−21.2917	−0.743086	−0.371543	+	0.928416i	$0.621171\pi$
$822$	1.60373	0.0559363
$823$	4.18425	0.145854	0.0729269	−	0.997337i	$-0.476766\pi$
$823$	4.18425	0.145854	0.0729269	+	0.997337i	$0.476766\pi$
$824$	9.49712	0.330848
$825$	−0.171573	−0.00597340
$826$	0	0
$827$	−8.23213	−0.286259	−0.143130	−	0.989704i	$-0.545717\pi$
$827$	−8.23213	−0.286259	−0.143130	+	0.989704i	$0.545717\pi$
$828$	−6.93587	−0.241038
$829$	−42.4954	−1.47593	−0.737964	−	0.674840i	$-0.764213\pi$
$829$	−42.4954	−1.47593	−0.737964	+	0.674840i	$0.764213\pi$
$830$	−14.8198	−0.514403
$831$	26.5262	0.920184
$832$	−4.63029	−0.160526
$833$	0	0
$834$	11.4526	0.396572
$835$	13.7262	0.475016
$836$	−4.07597	−0.140970
$837$	7.68832	0.265747
$838$	−15.5408	−0.536848
$839$	−14.4330	−0.498282	−0.249141	−	0.968467i	$-0.580148\pi$
$839$	−14.4330	−0.498282	−0.249141	+	0.968467i	$0.580148\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	−29.6176	−1.02069
$843$	−5.11844	−0.176289
$844$	−2.16057	−0.0743699
$845$	−19.1925	−0.660243
$846$	−10.9044	−0.374901
$847$	0	0
$848$	−0.440778	−0.0151364
$849$	−2.48205	−0.0851839
$850$	−0.0951059	−0.00326211
$851$	−75.4132	−2.58513
$852$	3.60373	0.123462
$853$	52.1866	1.78684	0.893418	−	0.449226i	$-0.148300\pi$
$853$	52.1866	1.78684	0.893418	+	0.449226i	$0.148300\pi$
$854$	0	0
$855$	−9.26921	−0.317000
$856$	7.64823	0.261411
$857$	25.6474	0.876098	0.438049	−	0.898951i	$-0.355670\pi$
$857$	25.6474	0.876098	0.438049	+	0.898951i	$0.355670\pi$
$858$	−4.63029	−0.158075
$859$	2.32928	0.0794738	0.0397369	−	0.999210i	$-0.487348\pi$
$859$	2.32928	0.0794738	0.0397369	+	0.999210i	$0.487348\pi$
$860$	19.5408	0.666335
$861$	0	0
$862$	−24.7978	−0.844616
$863$	−4.46685	−0.152053	−0.0760266	−	0.997106i	$-0.524223\pi$
$863$	−4.46685	−0.152053	−0.0760266	+	0.997106i	$0.524223\pi$
$864$	−1.00000	−0.0340207
$865$	−8.37378	−0.284717
$866$	11.2859	0.383512
$867$	−16.6927	−0.566915
$868$	0	0
$869$	−15.7014	−0.532632
$870$	−6.43215	−0.218070
$871$	51.5858	1.74792
$872$	1.22470	0.0414736
$873$	3.84637	0.130180
$874$	28.2704	0.956261
$875$	0	0
$876$	14.3631	0.485285
$877$	−45.8497	−1.54824	−0.774118	−	0.633042i	$-0.781806\pi$
$877$	−45.8497	−1.54824	−0.774118	+	0.633042i	$0.781806\pi$
$878$	1.17157	0.0395387
$879$	2.02537	0.0683140
$880$	2.27411	0.0766602
$881$	39.6418	1.33557	0.667783	−	0.744356i	$-0.267244\pi$
$881$	39.6418	1.33557	0.667783	+	0.744356i	$0.267244\pi$
$882$	0	0
$883$	32.2843	1.08645	0.543226	−	0.839586i	$-0.317203\pi$
$883$	32.2843	1.08645	0.543226	+	0.839586i	$0.317203\pi$
$884$	−2.56665	−0.0863259
$885$	−11.7607	−0.395332
$886$	−16.6915	−0.560763
$887$	23.4027	0.785786	0.392893	−	0.919584i	$-0.371474\pi$
$887$	23.4027	0.785786	0.392893	+	0.919584i	$0.371474\pi$
$888$	−10.8729	−0.364872
$889$	0	0
$890$	−9.62861	−0.322752
$891$	−1.00000	−0.0335013
$892$	−1.17598	−0.0393748
$893$	44.4460	1.48733
$894$	6.99257	0.233867
$895$	24.1167	0.806134
$896$	0	0
$897$	32.1151	1.07229
$898$	27.5547	0.919511
$899$	−21.7459	−0.725265
$900$	0.171573	0.00571910
$901$	−0.244331	−0.00813985
$902$	−0.554318	−0.0184568
$903$	0	0
$904$	−3.37665	−0.112306
$905$	−47.2314	−1.57003
$906$	−19.5286	−0.648794
$907$	11.4692	0.380829	0.190415	−	0.981704i	$-0.439017\pi$
$907$	11.4692	0.380829	0.190415	+	0.981704i	$0.439017\pi$
$908$	−22.0453	−0.731600
$909$	10.7377	0.356148
$910$	0	0
$911$	32.1213	1.06423	0.532113	−	0.846673i	$-0.321398\pi$
$911$	32.1213	1.06423	0.532113	+	0.846673i	$0.321398\pi$
$912$	4.07597	0.134969
$913$	6.51675	0.215673
$914$	18.1953	0.601846
$915$	−4.99880	−0.165255
$916$	−0.554318	−0.0183152
$917$	0	0
$918$	−0.554318	−0.0182952
$919$	35.2965	1.16432	0.582161	−	0.813073i	$-0.302207\pi$
$919$	35.2965	1.16432	0.582161	+	0.813073i	$0.302207\pi$
$920$	−15.7729	−0.520018
$921$	5.88477	0.193910
$922$	19.6193	0.646127
$923$	−16.6863	−0.549236
$924$	0	0
$925$	1.86550	0.0613373
$926$	−35.6176	−1.17047
$927$	−9.49712	−0.311926
$928$	2.82843	0.0928477
$929$	−28.6614	−0.940350	−0.470175	−	0.882573i	$-0.655809\pi$
$929$	−28.6614	−0.940350	−0.470175	+	0.882573i	$0.655809\pi$
$930$	17.4841	0.573326
$931$	0	0
$932$	−0.343146	−0.0112401
$933$	17.4624	0.571694
$934$	−35.0744	−1.14767
$935$	1.26058	0.0412254
$936$	4.63029	0.151346
$937$	12.1753	0.397750	0.198875	−	0.980025i	$-0.436271\pi$
$937$	12.1753	0.397750	0.198875	+	0.980025i	$0.436271\pi$
$938$	0	0
$939$	−8.87987	−0.289783
$940$	−24.7978	−0.808815
$941$	−32.9935	−1.07556	−0.537779	−	0.843086i	$-0.680737\pi$
$941$	−32.9935	−1.07556	−0.537779	+	0.843086i	$0.680737\pi$
$942$	−9.81490	−0.319787
$943$	3.84468	0.125200
$944$	5.17157	0.168320
$945$	0	0
$946$	−8.59272	−0.279373
$947$	54.6748	1.77669	0.888346	−	0.459175i	$-0.151855\pi$
$947$	54.6748	1.77669	0.888346	+	0.459175i	$0.151855\pi$
$948$	15.7014	0.509956
$949$	−66.5054	−2.15886
$950$	−0.699326	−0.0226891
$951$	−3.96937	−0.128716
$952$	0	0
$953$	12.9122	0.418267	0.209133	−	0.977887i	$-0.432936\pi$
$953$	12.9122	0.418267	0.209133	+	0.977887i	$0.432936\pi$
$954$	0.440778	0.0142707
$955$	−37.9361	−1.22758
$956$	−5.56785	−0.180077
$957$	2.82843	0.0914301
$958$	−18.7482	−0.605728
$959$	0	0
$960$	−2.27411	−0.0733966
$961$	28.1103	0.906784
$962$	50.3448	1.62318
$963$	−7.64823	−0.246461
$964$	−1.94077	−0.0625080
$965$	−12.3702	−0.398211
$966$	0	0
$967$	−51.6003	−1.65936	−0.829678	−	0.558243i	$-0.811476\pi$
$967$	−51.6003	−1.65936	−0.829678	+	0.558243i	$0.811476\pi$
$968$	−1.00000	−0.0321412
$969$	2.25938	0.0725819
$970$	8.74706	0.280851
$971$	45.9485	1.47456	0.737279	−	0.675588i	$-0.236110\pi$
$971$	45.9485	1.47456	0.737279	+	0.675588i	$0.236110\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−29.9301	−0.959023
$975$	−0.794432	−0.0254422
$976$	2.19814	0.0703607
$977$	62.2328	1.99100	0.995502	−	0.0947399i	$-0.0302020\pi$
$977$	62.2328	1.99100	0.995502	+	0.0947399i	$0.0302020\pi$
$978$	−13.6569	−0.436698
$979$	4.23401	0.135320
$980$	0	0
$981$	−1.22470	−0.0391017
$982$	8.68629	0.277191
$983$	26.4763	0.844463	0.422232	−	0.906488i	$-0.361247\pi$
$983$	26.4763	0.844463	0.422232	+	0.906488i	$0.361247\pi$
$984$	0.554318	0.0176710
$985$	23.6423	0.753305
$986$	1.56785	0.0499304
$987$	0	0
$988$	−18.8729	−0.600428
$989$	59.5980	1.89511
$990$	−2.27411	−0.0722759
$991$	46.5858	1.47985	0.739923	−	0.672692i	$-0.234862\pi$
$991$	46.5858	1.47985	0.739923	+	0.672692i	$0.234862\pi$
$992$	−7.68832	−0.244104
$993$	−13.2692	−0.421086
$994$	0	0
$995$	−13.7162	−0.434833
$996$	−6.51675	−0.206491
$997$	−54.1457	−1.71481	−0.857406	−	0.514641i	$-0.827925\pi$
$997$	−54.1457	−1.71481	−0.857406	+	0.514641i	$0.827925\pi$
$998$	−23.8076	−0.753617
$999$	10.8729	0.344004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bk.1.2	yes	4
3.2	odd	2		9702.2.a.ec.1.3		4
7.6	odd	2		3234.2.a.bj.1.3	✓	4
21.20	even	2		9702.2.a.eb.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bj.1.3	✓	4	7.6	odd	2
3234.2.a.bk.1.2	yes	4	1.1	even	1	trivial
9702.2.a.eb.1.2		4	21.20	even	2
9702.2.a.ec.1.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.27411 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.27411 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.27411 q^{10} -1.00000 q^{11} +1.00000 q^{12} -4.63029 q^{13} -2.27411 q^{15} +1.00000 q^{16} +0.554318 q^{17} -1.00000 q^{18} +4.07597 q^{19} -2.27411 q^{20} +1.00000 q^{22} -6.93587 q^{23} -1.00000 q^{24} +0.171573 q^{25} +4.63029 q^{26} +1.00000 q^{27} -2.82843 q^{29} +2.27411 q^{30} +7.68832 q^{31} -1.00000 q^{32} -1.00000 q^{33} -0.554318 q^{34} +1.00000 q^{36} +10.8729 q^{37} -4.07597 q^{38} -4.63029 q^{39} +2.27411 q^{40} -0.554318 q^{41} -8.59272 q^{43} -1.00000 q^{44} -2.27411 q^{45} +6.93587 q^{46} +10.9044 q^{47} +1.00000 q^{48} -0.171573 q^{50} +0.554318 q^{51} -4.63029 q^{52} -0.440778 q^{53} -1.00000 q^{54} +2.27411 q^{55} +4.07597 q^{57} +2.82843 q^{58} +5.17157 q^{59} -2.27411 q^{60} +2.19814 q^{61} -7.68832 q^{62} +1.00000 q^{64} +10.5298 q^{65} +1.00000 q^{66} -11.1409 q^{67} +0.554318 q^{68} -6.93587 q^{69} +3.60373 q^{71} -1.00000 q^{72} +14.3631 q^{73} -10.8729 q^{74} +0.171573 q^{75} +4.07597 q^{76} +4.63029 q^{78} +15.7014 q^{79} -2.27411 q^{80} +1.00000 q^{81} +0.554318 q^{82} -6.51675 q^{83} -1.26058 q^{85} +8.59272 q^{86} -2.82843 q^{87} +1.00000 q^{88} -4.23401 q^{89} +2.27411 q^{90} -6.93587 q^{92} +7.68832 q^{93} -10.9044 q^{94} -9.26921 q^{95} -1.00000 q^{96} +3.84637 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{11} + 4 q^{12} + 4 q^{16} - 4 q^{18} + 4 q^{22} - 8 q^{23} - 4 q^{24} + 12 q^{25} + 4 q^{27} + 16 q^{31} - 4 q^{32} - 4 q^{33} + 4 q^{36} + 8 q^{37} + 8 q^{43} - 4 q^{44} + 8 q^{46} + 16 q^{47} + 4 q^{48} - 12 q^{50} + 8 q^{53} - 4 q^{54} + 32 q^{59} + 16 q^{61} - 16 q^{62} + 4 q^{64} - 16 q^{65} + 4 q^{66} + 16 q^{67} - 8 q^{69} - 4 q^{72} - 8 q^{74} + 12 q^{75} + 16 q^{79} + 4 q^{81} + 32 q^{85} - 8 q^{86} + 4 q^{88} + 16 q^{89} - 8 q^{92} + 16 q^{93} - 16 q^{94} - 16 q^{95} - 4 q^{96} - 16 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^11 + 4 * q^12 + 4 * q^16 - 4 * q^18 + 4 * q^22 - 8 * q^23 - 4 * q^24 + 12 * q^25 + 4 * q^27 + 16 * q^31 - 4 * q^32 - 4 * q^33 + 4 * q^36 + 8 * q^37 + 8 * q^43 - 4 * q^44 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 12 * q^50 + 8 * q^53 - 4 * q^54 + 32 * q^59 + 16 * q^61 - 16 * q^62 + 4 * q^64 - 16 * q^65 + 4 * q^66 + 16 * q^67 - 8 * q^69 - 4 * q^72 - 8 * q^74 + 12 * q^75 + 16 * q^79 + 4 * q^81 + 32 * q^85 - 8 * q^86 + 4 * q^88 + 16 * q^89 - 8 * q^92 + 16 * q^93 - 16 * q^94 - 16 * q^95 - 4 * q^96 - 16 * q^97 - 4 * q^99

Introduction

L-functions

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