LMFDB - Embedded newform 18.16.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,16,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	18.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:	$25.6848309180$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-128.000 q^{2} +16384.0 q^{4} +114810. q^{5} -3.03453e6 q^{7} -2.09715e6 q^{8} +O(q^{10})$

\(q-128.000 q^{2} +16384.0 q^{4} +114810. q^{5} -3.03453e6 q^{7} -2.09715e6 q^{8} -1.46957e7 q^{10} +1.03452e8 q^{11} -1.04366e8 q^{13} +3.88420e8 q^{14} +2.68435e8 q^{16} -9.97690e8 q^{17} +4.93402e9 q^{19} +1.88105e9 q^{20} -1.32418e10 q^{22} -8.32492e9 q^{23} -1.73362e10 q^{25} +1.33588e10 q^{26} -4.97177e10 q^{28} -1.04128e11 q^{29} -2.96697e11 q^{31} -3.43597e10 q^{32} +1.27704e11 q^{34} -3.48394e11 q^{35} -1.78337e11 q^{37} -6.31554e11 q^{38} -2.40774e11 q^{40} +1.79088e12 q^{41} -2.86346e12 q^{43} +1.69495e12 q^{44} +1.06559e12 q^{46} -4.33291e12 q^{47} +4.46080e12 q^{49} +2.21904e12 q^{50} -1.70993e12 q^{52} -9.73232e12 q^{53} +1.18773e13 q^{55} +6.36387e12 q^{56} +1.33284e13 q^{58} +1.35148e13 q^{59} +5.35266e12 q^{61} +3.79772e13 q^{62} +4.39805e12 q^{64} -1.19822e13 q^{65} -5.32339e13 q^{67} -1.63461e13 q^{68} +4.45945e13 q^{70} +2.02297e13 q^{71} +2.62642e13 q^{73} +2.28272e13 q^{74} +8.08389e13 q^{76} -3.13927e14 q^{77} -3.39031e14 q^{79} +3.08191e13 q^{80} -2.29233e14 q^{82} -1.31685e14 q^{83} -1.14545e14 q^{85} +3.66523e14 q^{86} -2.16954e14 q^{88} +3.93521e13 q^{89} +3.16701e14 q^{91} -1.36395e14 q^{92} +5.54612e14 q^{94} +5.66474e14 q^{95} +1.12875e15 q^{97} -5.70982e14 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−128.000	−0.707107
$2$	−128.000	−0.707107
$3$	0	0
$3$	0	0
$4$	16384.0	0.500000
$5$	114810.	0.657211	0.328605	−	0.944467i	$-0.393421\pi$
$5$	114810.	0.657211	0.328605	+	0.944467i	$0.393421\pi$
$6$	0	0
$7$	−3.03453e6	−1.39269	−0.696347	−	0.717705i	$-0.745193\pi$
$7$	−3.03453e6	−1.39269	−0.696347	+	0.717705i	$0.745193\pi$
$8$	−2.09715e6	−0.353553
$9$	0	0
$10$	−1.46957e7	−0.464718
$11$	1.03452e8	1.60064	0.800318	−	0.599576i	$-0.204664\pi$
$11$	1.03452e8	1.60064	0.800318	+	0.599576i	$0.204664\pi$
$12$	0	0
$13$	−1.04366e8	−0.461300	−0.230650	−	0.973037i	$-0.574085\pi$
$13$	−1.04366e8	−0.461300	−0.230650	+	0.973037i	$0.574085\pi$
$14$	3.88420e8	0.984784
$15$	0	0
$16$	2.68435e8	0.250000
$17$	−9.97690e8	−0.589697	−0.294848	−	0.955544i	$-0.595269\pi$
$17$	−9.97690e8	−0.589697	−0.294848	+	0.955544i	$0.595269\pi$
$18$	0	0
$19$	4.93402e9	1.26633	0.633167	−	0.774015i	$-0.281754\pi$
$19$	4.93402e9	1.26633	0.633167	+	0.774015i	$0.281754\pi$
$20$	1.88105e9	0.328605
$21$	0	0
$22$	−1.32418e10	−1.13182
$23$	−8.32492e9	−0.509825	−0.254913	−	0.966964i	$-0.582047\pi$
$23$	−8.32492e9	−0.509825	−0.254913	+	0.966964i	$0.582047\pi$
$24$	0	0
$25$	−1.73362e10	−0.568074
$26$	1.33588e10	0.326188
$27$	0	0
$28$	−4.97177e10	−0.696347
$29$	−1.04128e11	−1.12094	−0.560472	−	0.828174i	$-0.689380\pi$
$29$	−1.04128e11	−1.12094	−0.560472	+	0.828174i	$0.689380\pi$
$30$	0	0
$31$	−2.96697e11	−1.93687	−0.968434	−	0.249270i	$-0.919809\pi$
$31$	−2.96697e11	−1.93687	−0.968434	+	0.249270i	$0.919809\pi$
$32$	−3.43597e10	−0.176777
$33$	0	0
$34$	1.27704e11	0.416978
$35$	−3.48394e11	−0.915294
$36$	0	0
$37$	−1.78337e11	−0.308837	−0.154419	−	0.988006i	$-0.549350\pi$
$37$	−1.78337e11	−0.308837	−0.154419	+	0.988006i	$0.549350\pi$
$38$	−6.31554e11	−0.895434
$39$	0	0
$40$	−2.40774e11	−0.232359
$41$	1.79088e12	1.43611	0.718056	−	0.695986i	$-0.245032\pi$
$41$	1.79088e12	1.43611	0.718056	+	0.695986i	$0.245032\pi$
$42$	0	0
$43$	−2.86346e12	−1.60649	−0.803244	−	0.595650i	$-0.796895\pi$
$43$	−2.86346e12	−1.60649	−0.803244	+	0.595650i	$0.796895\pi$
$44$	1.69495e12	0.800318
$45$	0	0
$46$	1.06559e12	0.360501
$47$	−4.33291e12	−1.24751	−0.623757	−	0.781618i	$-0.714395\pi$
$47$	−4.33291e12	−1.24751	−0.623757	+	0.781618i	$0.714395\pi$
$48$	0	0
$49$	4.46080e12	0.939598
$50$	2.21904e12	0.401689
$51$	0	0
$52$	−1.70993e12	−0.230650
$53$	−9.73232e12	−1.13801	−0.569007	−	0.822333i	$-0.692672\pi$
$53$	−9.73232e12	−1.13801	−0.569007	+	0.822333i	$0.692672\pi$
$54$	0	0
$55$	1.18773e13	1.05196
$56$	6.36387e12	0.492392
$57$	0	0
$58$	1.33284e13	0.792626
$59$	1.35148e13	0.707002	0.353501	−	0.935434i	$-0.384991\pi$
$59$	1.35148e13	0.707002	0.353501	+	0.935434i	$0.384991\pi$
$60$	0	0
$61$	5.35266e12	0.218070	0.109035	−	0.994038i	$-0.465224\pi$
$61$	5.35266e12	0.218070	0.109035	+	0.994038i	$0.465224\pi$
$62$	3.79772e13	1.36957
$63$	0	0
$64$	4.39805e12	0.125000
$65$	−1.19822e13	−0.303171
$66$	0	0
$67$	−5.32339e13	−1.07307	−0.536534	−	0.843879i	$-0.680267\pi$
$67$	−5.32339e13	−1.07307	−0.536534	+	0.843879i	$0.680267\pi$
$68$	−1.63461e13	−0.294848
$69$	0	0
$70$	4.45945e13	0.647210
$71$	2.02297e13	0.263968	0.131984	−	0.991252i	$-0.457865\pi$
$71$	2.02297e13	0.263968	0.131984	+	0.991252i	$0.457865\pi$
$72$	0	0
$73$	2.62642e13	0.278254	0.139127	−	0.990275i	$-0.455570\pi$
$73$	2.62642e13	0.278254	0.139127	+	0.990275i	$0.455570\pi$
$74$	2.28272e13	0.218381
$75$	0	0
$76$	8.08389e13	0.633167
$77$	−3.13927e14	−2.22920
$78$	0	0
$79$	−3.39031e14	−1.98626	−0.993131	−	0.117005i	$-0.962671\pi$
$79$	−3.39031e14	−1.98626	−0.993131	+	0.117005i	$0.962671\pi$
$80$	3.08191e13	0.164303
$81$	0	0
$82$	−2.29233e14	−1.01548
$83$	−1.31685e14	−0.532660	−0.266330	−	0.963882i	$-0.585811\pi$
$83$	−1.31685e14	−0.532660	−0.266330	+	0.963882i	$0.585811\pi$
$84$	0	0
$85$	−1.14545e14	−0.387555
$86$	3.66523e14	1.13596
$87$	0	0
$88$	−2.16954e14	−0.565910
$89$	3.93521e13	0.0943069	0.0471534	−	0.998888i	$-0.484985\pi$
$89$	3.93521e13	0.0943069	0.0471534	+	0.998888i	$0.484985\pi$
$90$	0	0
$91$	3.16701e14	0.642450
$92$	−1.36395e14	−0.254913
$93$	0	0
$94$	5.54612e14	0.882126
$95$	5.66474e14	0.832249
$96$	0	0
$97$	1.12875e15	1.41844	0.709219	−	0.704989i	$-0.249048\pi$
$97$	1.12875e15	1.41844	0.709219	+	0.704989i	$0.249048\pi$
$98$	−5.70982e14	−0.664396
$99$	0	0
$100$	−2.84037e14	−0.284037
$101$	−3.79528e13	−0.0352236	−0.0176118	−	0.999845i	$-0.505606\pi$
$101$	−3.79528e13	−0.0352236	−0.0176118	+	0.999845i	$0.505606\pi$
$102$	0	0
$103$	−2.07297e14	−0.166079	−0.0830393	−	0.996546i	$-0.526463\pi$
$103$	−2.07297e14	−0.166079	−0.0830393	+	0.996546i	$0.526463\pi$
$104$	2.18871e14	0.163094
$105$	0	0
$106$	1.24574e15	0.804697
$107$	1.99692e15	1.20221	0.601107	−	0.799169i	$-0.294727\pi$
$107$	1.99692e15	1.20221	0.601107	+	0.799169i	$0.294727\pi$
$108$	0	0
$109$	1.35603e13	0.00710510	0.00355255	−	0.999994i	$-0.498869\pi$
$109$	1.35603e13	0.00710510	0.00355255	+	0.999994i	$0.498869\pi$
$110$	−1.52029e15	−0.743845
$111$	0	0
$112$	−8.14575e14	−0.348174
$113$	7.08794e14	0.283421	0.141710	−	0.989908i	$-0.454740\pi$
$113$	7.08794e14	0.283421	0.141710	+	0.989908i	$0.454740\pi$
$114$	0	0
$115$	−9.55784e14	−0.335063
$116$	−1.70604e15	−0.560472
$117$	0	0
$118$	−1.72990e15	−0.499926
$119$	3.02752e15	0.821267
$120$	0	0
$121$	6.52501e15	1.56203
$122$	−6.85141e14	−0.154199
$123$	0	0
$124$	−4.86108e15	−0.968434
$125$	−5.49410e15	−1.03056
$126$	0	0
$127$	−1.23021e15	−0.204858	−0.102429	−	0.994740i	$-0.532661\pi$
$127$	−1.23021e15	−0.204858	−0.102429	+	0.994740i	$0.532661\pi$
$128$	−5.62950e14	−0.0883883
$129$	0	0
$130$	1.53373e15	0.214374
$131$	−7.94836e15	−1.04892	−0.524460	−	0.851435i	$-0.675733\pi$
$131$	−7.94836e15	−1.04892	−0.524460	+	0.851435i	$0.675733\pi$
$132$	0	0
$133$	−1.49724e16	−1.76362
$134$	6.81394e15	0.758774
$135$	0	0
$136$	2.09231e15	0.208489
$137$	1.78131e16	1.68010	0.840050	−	0.542508i	$-0.182525\pi$
$137$	1.78131e16	1.68010	0.840050	+	0.542508i	$0.182525\pi$
$138$	0	0
$139$	3.89941e15	0.329904	0.164952	−	0.986302i	$-0.447253\pi$
$139$	3.89941e15	0.329904	0.164952	+	0.986302i	$0.447253\pi$
$140$	−5.70809e15	−0.457647
$141$	0	0
$142$	−2.58940e15	−0.186653
$143$	−1.07968e16	−0.738373
$144$	0	0
$145$	−1.19550e16	−0.736696
$146$	−3.36181e15	−0.196756
$147$	0	0
$148$	−2.92188e15	−0.154419
$149$	3.48726e15	0.175222	0.0876108	−	0.996155i	$-0.472077\pi$
$149$	3.48726e15	0.175222	0.0876108	+	0.996155i	$0.472077\pi$
$150$	0	0
$151$	6.85712e15	0.311756	0.155878	−	0.987776i	$-0.450179\pi$
$151$	6.85712e15	0.311756	0.155878	+	0.987776i	$0.450179\pi$
$152$	−1.03474e16	−0.447717
$153$	0	0
$154$	4.01827e16	1.57628
$155$	−3.40637e16	−1.27293
$156$	0	0
$157$	3.69836e16	1.25534	0.627670	−	0.778480i	$-0.284009\pi$
$157$	3.69836e16	1.25534	0.627670	+	0.778480i	$0.284009\pi$
$158$	4.33960e16	1.40450
$159$	0	0
$160$	−3.94484e15	−0.116180
$161$	2.52622e16	0.710031
$162$	0	0
$163$	7.42535e15	0.190244	0.0951218	−	0.995466i	$-0.469676\pi$
$163$	7.42535e15	0.190244	0.0951218	+	0.995466i	$0.469676\pi$
$164$	2.93418e16	0.718056
$165$	0	0
$166$	1.68557e16	0.376647
$167$	1.47365e16	0.314789	0.157395	−	0.987536i	$-0.449691\pi$
$167$	1.47365e16	0.314789	0.157395	+	0.987536i	$0.449691\pi$
$168$	0	0
$169$	−4.02937e16	−0.787203
$170$	1.46617e16	0.274043
$171$	0	0
$172$	−4.69149e16	−0.803244
$173$	−3.40039e16	−0.557421	−0.278710	−	0.960375i	$-0.589907\pi$
$173$	−3.40039e16	−0.557421	−0.278710	+	0.960375i	$0.589907\pi$
$174$	0	0
$175$	5.26073e16	0.791153
$176$	2.77701e16	0.400159
$177$	0	0
$178$	−5.03707e15	−0.0666850
$179$	−3.81276e16	−0.483996	−0.241998	−	0.970277i	$-0.577803\pi$
$179$	−3.81276e16	−0.483996	−0.241998	+	0.970277i	$0.577803\pi$
$180$	0	0
$181$	−5.14124e16	−0.600452	−0.300226	−	0.953868i	$-0.597062\pi$
$181$	−5.14124e16	−0.600452	−0.300226	+	0.953868i	$0.597062\pi$
$182$	−4.05377e16	−0.454280
$183$	0	0
$184$	1.74586e16	0.180250
$185$	−2.04749e16	−0.202971
$186$	0	0
$187$	−1.03213e17	−0.943889
$188$	−7.09904e16	−0.623757
$189$	0	0
$190$	−7.25087e16	−0.588489
$191$	−6.75568e16	−0.527131	−0.263566	−	0.964641i	$-0.584899\pi$
$191$	−6.75568e16	−0.527131	−0.263566	+	0.964641i	$0.584899\pi$
$192$	0	0
$193$	−2.07235e16	−0.149549	−0.0747746	−	0.997200i	$-0.523824\pi$
$193$	−2.07235e16	−0.149549	−0.0747746	+	0.997200i	$0.523824\pi$
$194$	−1.44480e17	−1.00299
$195$	0	0
$196$	7.30857e16	0.469799
$197$	−1.71244e16	−0.105954	−0.0529772	−	0.998596i	$-0.516871\pi$
$197$	−1.71244e16	−0.105954	−0.0529772	+	0.998596i	$0.516871\pi$
$198$	0	0
$199$	−1.05743e17	−0.606533	−0.303267	−	0.952906i	$-0.598077\pi$
$199$	−1.05743e17	−0.606533	−0.303267	+	0.952906i	$0.598077\pi$
$200$	3.63567e16	0.200844
$201$	0	0
$202$	4.85796e15	0.0249069
$203$	3.15980e17	1.56113
$204$	0	0
$205$	2.05611e17	0.943828
$206$	2.65340e16	0.117435
$207$	0	0
$208$	−2.80155e16	−0.115325
$209$	5.10432e17	2.02694
$210$	0	0
$211$	−3.13093e17	−1.15759	−0.578795	−	0.815473i	$-0.696477\pi$
$211$	−3.13093e17	−1.15759	−0.578795	+	0.815473i	$0.696477\pi$
$212$	−1.59454e17	−0.569007
$213$	0	0
$214$	−2.55605e17	−0.850093
$215$	−3.28754e17	−1.05580
$216$	0	0
$217$	9.00334e17	2.69747
$218$	−1.73572e15	−0.00502406
$219$	0	0
$220$	1.94598e17	0.525978
$221$	1.04125e17	0.272027
$222$	0	0
$223$	3.35017e17	0.818052	0.409026	−	0.912523i	$-0.365869\pi$
$223$	3.35017e17	0.818052	0.409026	+	0.912523i	$0.365869\pi$
$224$	1.04266e17	0.246196
$225$	0	0
$226$	−9.07256e16	−0.200409
$227$	−4.67560e17	−0.999180	−0.499590	−	0.866262i	$-0.666516\pi$
$227$	−4.67560e17	−0.999180	−0.499590	+	0.866262i	$0.666516\pi$
$228$	0	0
$229$	3.61186e17	0.722711	0.361355	−	0.932428i	$-0.382314\pi$
$229$	3.61186e17	0.722711	0.361355	+	0.932428i	$0.382314\pi$
$230$	1.22340e17	0.236925
$231$	0	0
$232$	2.18373e17	0.396313
$233$	−3.61787e17	−0.635747	−0.317873	−	0.948133i	$-0.602969\pi$
$233$	−3.61787e17	−0.635747	−0.317873	+	0.948133i	$0.602969\pi$
$234$	0	0
$235$	−4.97461e17	−0.819880
$236$	2.21427e17	0.353501
$237$	0	0
$238$	−3.87522e17	−0.580724
$239$	1.06842e18	1.55152	0.775762	−	0.631026i	$-0.217366\pi$
$239$	1.06842e18	1.55152	0.775762	+	0.631026i	$0.217366\pi$
$240$	0	0
$241$	−1.00588e18	−1.37220	−0.686102	−	0.727505i	$-0.740680\pi$
$241$	−1.00588e18	−1.37220	−0.686102	+	0.727505i	$0.740680\pi$
$242$	−8.35201e17	−1.10453
$243$	0	0
$244$	8.76980e16	0.109035
$245$	5.12144e17	0.617514
$246$	0	0
$247$	−5.14943e17	−0.584160
$248$	6.22218e17	0.684786
$249$	0	0
$250$	7.03244e17	0.728713
$251$	−1.74766e18	−1.75754	−0.878768	−	0.477249i	$-0.841634\pi$
$251$	−1.74766e18	−1.75754	−0.878768	+	0.477249i	$0.841634\pi$
$252$	0	0
$253$	−8.61227e17	−0.816044
$254$	1.57467e17	0.144856
$255$	0	0
$256$	7.20576e16	0.0625000
$257$	2.77264e17	0.233558	0.116779	−	0.993158i	$-0.462743\pi$
$257$	2.77264e17	0.233558	0.116779	+	0.993158i	$0.462743\pi$
$258$	0	0
$259$	5.41170e17	0.430116
$260$	−1.96317e17	−0.151586
$261$	0	0
$262$	1.01739e18	0.741699
$263$	−1.73303e18	−1.22783	−0.613914	−	0.789373i	$-0.710406\pi$
$263$	−1.73303e18	−1.22783	−0.613914	+	0.789373i	$0.710406\pi$
$264$	0	0
$265$	−1.11737e18	−0.747914
$266$	1.91647e18	1.24707
$267$	0	0
$268$	−8.72184e17	−0.536534
$269$	5.11271e16	0.0305850	0.0152925	−	0.999883i	$-0.495132\pi$
$269$	5.11271e16	0.0305850	0.0152925	+	0.999883i	$0.495132\pi$
$270$	0	0
$271$	2.08455e17	0.117962	0.0589812	−	0.998259i	$-0.481215\pi$
$271$	2.08455e17	0.117962	0.0589812	+	0.998259i	$0.481215\pi$
$272$	−2.67815e17	−0.147424
$273$	0	0
$274$	−2.28008e18	−1.18801
$275$	−1.79346e18	−0.909279
$276$	0	0
$277$	3.29723e18	1.58326	0.791628	−	0.611003i	$-0.209234\pi$
$277$	3.29723e18	1.58326	0.791628	+	0.611003i	$0.209234\pi$
$278$	−4.99124e17	−0.233277
$279$	0	0
$280$	7.30636e17	0.323605
$281$	3.98328e18	1.71768	0.858841	−	0.512242i	$-0.171185\pi$
$281$	3.98328e18	1.71768	0.858841	+	0.512242i	$0.171185\pi$
$282$	0	0
$283$	−2.19051e18	−0.895668	−0.447834	−	0.894117i	$-0.647804\pi$
$283$	−2.19051e18	−0.895668	−0.447834	+	0.894117i	$0.647804\pi$
$284$	3.31443e17	0.131984
$285$	0	0
$286$	1.38199e18	0.522108
$287$	−5.43448e18	−2.00006
$288$	0	0
$289$	−1.86704e18	−0.652258
$290$	1.53024e18	0.520923
$291$	0	0
$292$	4.30312e17	0.139127
$293$	5.27183e18	1.66132	0.830662	−	0.556778i	$-0.187962\pi$
$293$	5.27183e18	1.66132	0.830662	+	0.556778i	$0.187962\pi$
$294$	0	0
$295$	1.55164e18	0.464649
$296$	3.74001e17	0.109190
$297$	0	0
$298$	−4.46369e17	−0.123900
$299$	8.68837e17	0.235182
$300$	0	0
$301$	8.68925e18	2.23735
$302$	−8.77712e17	−0.220445
$303$	0	0
$304$	1.32446e18	0.316584
$305$	6.14539e17	0.143318
$306$	0	0
$307$	4.77067e18	1.05935	0.529677	−	0.848199i	$-0.322313\pi$
$307$	4.77067e18	1.05935	0.529677	+	0.848199i	$0.322313\pi$
$308$	−5.14338e18	−1.11460
$309$	0	0
$310$	4.36016e18	0.900098
$311$	6.46666e18	1.30310	0.651549	−	0.758607i	$-0.274120\pi$
$311$	6.46666e18	1.30310	0.651549	+	0.758607i	$0.274120\pi$
$312$	0	0
$313$	−6.00129e18	−1.15256	−0.576278	−	0.817254i	$-0.695496\pi$
$313$	−6.00129e18	−1.15256	−0.576278	+	0.817254i	$0.695496\pi$
$314$	−4.73389e18	−0.887659
$315$	0	0
$316$	−5.55469e18	−0.993131
$317$	9.53943e17	0.166563	0.0832814	−	0.996526i	$-0.473460\pi$
$317$	9.53943e17	0.166563	0.0832814	+	0.996526i	$0.473460\pi$
$318$	0	0
$319$	−1.07722e19	−1.79422
$320$	5.04940e17	0.0821513
$321$	0	0
$322$	−3.23356e18	−0.502068
$323$	−4.92262e18	−0.746753
$324$	0	0
$325$	1.80931e18	0.262052
$326$	−9.50445e17	−0.134523
$327$	0	0
$328$	−3.75575e18	−0.507742
$329$	1.31483e19	1.73741
$330$	0	0
$331$	−5.29071e18	−0.668042	−0.334021	−	0.942566i	$-0.608406\pi$
$331$	−5.29071e18	−0.668042	−0.334021	+	0.942566i	$0.608406\pi$
$332$	−2.15752e18	−0.266330
$333$	0	0
$334$	−1.88627e18	−0.222590
$335$	−6.11179e18	−0.705232
$336$	0	0
$337$	6.22409e18	0.686834	0.343417	−	0.939183i	$-0.388416\pi$
$337$	6.22409e18	0.686834	0.343417	+	0.939183i	$0.388416\pi$
$338$	5.15759e18	0.556636
$339$	0	0
$340$	−1.87670e18	−0.193777
$341$	−3.06938e19	−3.10022
$342$	0	0
$343$	8.70190e17	0.0841217
$344$	6.00511e18	0.567979
$345$	0	0
$346$	4.35250e18	0.394156
$347$	3.22035e18	0.285386	0.142693	−	0.989767i	$-0.454424\pi$
$347$	3.22035e18	0.285386	0.142693	+	0.989767i	$0.454424\pi$
$348$	0	0
$349$	−6.17407e18	−0.524060	−0.262030	−	0.965060i	$-0.584392\pi$
$349$	−6.17407e18	−0.524060	−0.262030	+	0.965060i	$0.584392\pi$
$350$	−6.73374e18	−0.559430
$351$	0	0
$352$	−3.55457e18	−0.282955
$353$	6.14267e18	0.478681	0.239341	−	0.970936i	$-0.423069\pi$
$353$	6.14267e18	0.478681	0.239341	+	0.970936i	$0.423069\pi$
$354$	0	0
$355$	2.32257e18	0.173483
$356$	6.44746e17	0.0471534
$357$	0	0
$358$	4.88033e18	0.342237
$359$	5.81103e18	0.399066	0.199533	−	0.979891i	$-0.436057\pi$
$359$	5.81103e18	0.399066	0.199533	+	0.979891i	$0.436057\pi$
$360$	0	0
$361$	9.16338e18	0.603603
$362$	6.58079e18	0.424584
$363$	0	0
$364$	5.18883e18	0.321225
$365$	3.01539e18	0.182872
$366$	0	0
$367$	−2.77147e17	−0.0161330	−0.00806650	−	0.999967i	$-0.502568\pi$
$367$	−2.77147e17	−0.0161330	−0.00806650	+	0.999967i	$0.502568\pi$
$368$	−2.23470e18	−0.127456
$369$	0	0
$370$	2.62079e18	0.143522
$371$	2.95330e19	1.58490
$372$	0	0
$373$	2.30498e19	1.18809	0.594047	−	0.804430i	$-0.297529\pi$
$373$	2.30498e19	1.18809	0.594047	+	0.804430i	$0.297529\pi$
$374$	1.32112e19	0.667431
$375$	0	0
$376$	9.08677e18	0.441063
$377$	1.08674e19	0.517091
$378$	0	0
$379$	−1.34397e19	−0.614604	−0.307302	−	0.951612i	$-0.599426\pi$
$379$	−1.34397e19	−0.614604	−0.307302	+	0.951612i	$0.599426\pi$
$380$	9.28112e18	0.416124
$381$	0	0
$382$	8.64727e18	0.372738
$383$	−2.64151e19	−1.11651	−0.558254	−	0.829670i	$-0.688529\pi$
$383$	−2.64151e19	−1.11651	−0.558254	+	0.829670i	$0.688529\pi$
$384$	0	0
$385$	−3.60420e19	−1.46505
$386$	2.65261e18	0.105747
$387$	0	0
$388$	1.84935e19	0.709219
$389$	−3.95275e19	−1.48688	−0.743442	−	0.668800i	$-0.766808\pi$
$389$	−3.95275e19	−1.48688	−0.743442	+	0.668800i	$0.766808\pi$
$390$	0	0
$391$	8.30569e18	0.300642
$392$	−9.35497e18	−0.332198
$393$	0	0
$394$	2.19192e18	0.0749210
$395$	−3.89242e19	−1.30539
$396$	0	0
$397$	2.82890e18	0.0913457	0.0456729	−	0.998956i	$-0.485457\pi$
$397$	2.82890e18	0.0913457	0.0456729	+	0.998956i	$0.485457\pi$
$398$	1.35351e19	0.428884
$399$	0	0
$400$	−4.65366e18	−0.142018
$401$	4.63681e19	1.38879	0.694395	−	0.719594i	$-0.255672\pi$
$401$	4.63681e19	1.38879	0.694395	+	0.719594i	$0.255672\pi$
$402$	0	0
$403$	3.09650e19	0.893477
$404$	−6.21819e17	−0.0176118
$405$	0	0
$406$	−4.04454e19	−1.10389
$407$	−1.84493e19	−0.494336
$408$	0	0
$409$	−1.17092e19	−0.302415	−0.151207	−	0.988502i	$-0.548316\pi$
$409$	−1.17092e19	−0.302415	−0.151207	+	0.988502i	$0.548316\pi$
$410$	−2.63182e19	−0.667387
$411$	0	0
$412$	−3.39635e18	−0.0830393
$413$	−4.10112e19	−0.984638
$414$	0	0
$415$	−1.51187e19	−0.350070
$416$	3.58598e18	0.0815470
$417$	0	0
$418$	−6.53353e19	−1.43326
$419$	4.20102e19	0.905210	0.452605	−	0.891711i	$-0.350495\pi$
$419$	4.20102e19	0.905210	0.452605	+	0.891711i	$0.350495\pi$
$420$	0	0
$421$	−1.59718e19	−0.332077	−0.166039	−	0.986119i	$-0.553098\pi$
$421$	−1.59718e19	−0.332077	−0.166039	+	0.986119i	$0.553098\pi$
$422$	4.00759e19	0.818540
$423$	0	0
$424$	2.04101e19	0.402348
$425$	1.72962e19	0.334991
$426$	0	0
$427$	−1.62428e19	−0.303705
$428$	3.27175e19	0.601107
$429$	0	0
$430$	4.20805e19	0.746564
$431$	9.76365e19	1.70229	0.851143	−	0.524934i	$-0.175910\pi$
$431$	9.76365e19	1.70229	0.851143	+	0.524934i	$0.175910\pi$
$432$	0	0
$433$	−6.91455e19	−1.16441	−0.582204	−	0.813043i	$-0.697809\pi$
$433$	−6.91455e19	−1.16441	−0.582204	+	0.813043i	$0.697809\pi$
$434$	−1.15243e20	−1.90740
$435$	0	0
$436$	2.22172e17	0.00355255
$437$	−4.10753e19	−0.645609
$438$	0	0
$439$	5.48990e19	0.833836	0.416918	−	0.908944i	$-0.363110\pi$
$439$	5.48990e19	0.833836	0.416918	+	0.908944i	$0.363110\pi$
$440$	−2.49085e19	−0.371922
$441$	0	0
$442$	−1.33280e19	−0.192352
$443$	−3.80504e19	−0.539923	−0.269961	−	0.962871i	$-0.587011\pi$
$443$	−3.80504e19	−0.539923	−0.269961	+	0.962871i	$0.587011\pi$
$444$	0	0
$445$	4.51802e18	0.0619795
$446$	−4.28822e19	−0.578450
$447$	0	0
$448$	−1.33460e19	−0.174087
$449$	−1.11994e18	−0.0143663	−0.00718317	−	0.999974i	$-0.502286\pi$
$449$	−1.11994e18	−0.0143663	−0.00718317	+	0.999974i	$0.502286\pi$
$450$	0	0
$451$	1.85270e20	2.29869
$452$	1.16129e19	0.141710
$453$	0	0
$454$	5.98477e19	0.706527
$455$	3.63604e19	0.422225
$456$	0	0
$457$	−9.70734e19	−1.09076	−0.545379	−	0.838189i	$-0.683614\pi$
$457$	−9.70734e19	−1.09076	−0.545379	+	0.838189i	$0.683614\pi$
$458$	−4.62318e19	−0.511034
$459$	0	0
$460$	−1.56596e19	−0.167531
$461$	9.78165e19	1.02957	0.514784	−	0.857320i	$-0.327872\pi$
$461$	9.78165e19	1.02957	0.514784	+	0.857320i	$0.327872\pi$
$462$	0	0
$463$	−2.17159e19	−0.221269	−0.110634	−	0.993861i	$-0.535288\pi$
$463$	−2.17159e19	−0.221269	−0.110634	+	0.993861i	$0.535288\pi$
$464$	−2.79517e19	−0.280236
$465$	0	0
$466$	4.63088e19	0.449541
$467$	1.98443e20	1.89565	0.947826	−	0.318788i	$-0.103276\pi$
$467$	1.98443e20	1.89565	0.947826	+	0.318788i	$0.103276\pi$
$468$	0	0
$469$	1.61540e20	1.49446
$470$	6.36750e19	0.579743
$471$	0	0
$472$	−2.83427e19	−0.249963
$473$	−2.96230e20	−2.57140
$474$	0	0
$475$	−8.55373e19	−0.719372
$476$	4.96028e19	0.410634
$477$	0	0
$478$	−1.36758e20	−1.09709
$479$	3.78661e19	0.299043	0.149522	−	0.988758i	$-0.452227\pi$
$479$	3.78661e19	0.299043	0.149522	+	0.988758i	$0.452227\pi$
$480$	0	0
$481$	1.86123e19	0.142467
$482$	1.28753e20	0.970295
$483$	0	0
$484$	1.06906e20	0.781017
$485$	1.29592e20	0.932212
$486$	0	0
$487$	3.89265e19	0.271505	0.135752	−	0.990743i	$-0.456655\pi$
$487$	3.89265e19	0.271505	0.135752	+	0.990743i	$0.456655\pi$
$488$	−1.12253e19	−0.0770994
$489$	0	0
$490$	−6.55545e19	−0.436648
$491$	−1.40516e20	−0.921753	−0.460876	−	0.887464i	$-0.652465\pi$
$491$	−1.40516e20	−0.921753	−0.460876	+	0.887464i	$0.652465\pi$
$492$	0	0
$493$	1.03888e20	0.661016
$494$	6.59127e19	0.413063
$495$	0	0
$496$	−7.96439e19	−0.484217
$497$	−6.13875e19	−0.367627
$498$	0	0
$499$	−1.53703e20	−0.893157	−0.446578	−	0.894745i	$-0.647358\pi$
$499$	−1.53703e20	−0.893157	−0.446578	+	0.894745i	$0.647358\pi$
$500$	−9.00153e19	−0.515278
$501$	0	0
$502$	2.23701e20	1.24277
$503$	−5.62265e18	−0.0307738	−0.0153869	−	0.999882i	$-0.504898\pi$
$503$	−5.62265e18	−0.0307738	−0.0153869	+	0.999882i	$0.504898\pi$
$504$	0	0
$505$	−4.35737e18	−0.0231493
$506$	1.10237e20	0.577031
$507$	0	0
$508$	−2.01558e19	−0.102429
$509$	−3.29307e20	−1.64899	−0.824495	−	0.565869i	$-0.808541\pi$
$509$	−3.29307e20	−1.64899	−0.824495	+	0.565869i	$0.808541\pi$
$510$	0	0
$511$	−7.96993e19	−0.387523
$512$	−9.22337e18	−0.0441942
$513$	0	0
$514$	−3.54898e19	−0.165151
$515$	−2.37998e19	−0.109149
$516$	0	0
$517$	−4.48247e20	−1.99682
$518$	−6.92698e19	−0.304138
$519$	0	0
$520$	2.51286e19	0.107187
$521$	−1.71775e20	−0.722231	−0.361116	−	0.932521i	$-0.617604\pi$
$521$	−1.71775e20	−0.722231	−0.361116	+	0.932521i	$0.617604\pi$
$522$	0	0
$523$	−1.59414e20	−0.651275	−0.325637	−	0.945495i	$-0.605579\pi$
$523$	−1.59414e20	−0.651275	−0.325637	+	0.945495i	$0.605579\pi$
$524$	−1.30226e20	−0.524460
$525$	0	0
$526$	2.21828e20	0.868205
$527$	2.96011e20	1.14216
$528$	0	0
$529$	−1.97331e20	−0.740078
$530$	1.43023e20	0.528855
$531$	0	0
$532$	−2.45308e20	−0.881809
$533$	−1.86907e20	−0.662478
$534$	0	0
$535$	2.29266e20	0.790108
$536$	1.11640e20	0.379387
$537$	0	0
$538$	−6.54427e18	−0.0216269
$539$	4.61477e20	1.50395
$540$	0	0
$541$	4.74121e19	0.150283	0.0751415	−	0.997173i	$-0.476059\pi$
$541$	4.74121e19	0.150283	0.0751415	+	0.997173i	$0.476059\pi$
$542$	−2.66823e19	−0.0834120
$543$	0	0
$544$	3.42804e19	0.104245
$545$	1.55686e18	0.00466955
$546$	0	0
$547$	−2.80986e20	−0.819934	−0.409967	−	0.912100i	$-0.634460\pi$
$547$	−2.80986e20	−0.819934	−0.409967	+	0.912100i	$0.634460\pi$
$548$	2.91850e20	0.840050
$549$	0	0
$550$	2.29563e20	0.642958
$551$	−5.13770e20	−1.41949
$552$	0	0
$553$	1.02880e21	2.76626
$554$	−4.22046e20	−1.11953
$555$	0	0
$556$	6.38879e19	0.164952
$557$	3.49642e20	0.890654	0.445327	−	0.895368i	$-0.353087\pi$
$557$	3.49642e20	0.890654	0.445327	+	0.895368i	$0.353087\pi$
$558$	0	0
$559$	2.98847e20	0.741073
$560$	−9.35213e19	−0.228823
$561$	0	0
$562$	−5.09859e20	−1.21459
$563$	−2.68215e20	−0.630479	−0.315240	−	0.949012i	$-0.602085\pi$
$563$	−2.68215e20	−0.630479	−0.315240	+	0.949012i	$0.602085\pi$
$564$	0	0
$565$	8.13766e19	0.186267
$566$	2.80385e20	0.633333
$567$	0	0
$568$	−4.24247e19	−0.0933267
$569$	7.95691e20	1.72744	0.863719	−	0.503974i	$-0.168129\pi$
$569$	7.95691e20	1.72744	0.863719	+	0.503974i	$0.168129\pi$
$570$	0	0
$571$	8.84470e20	1.87030	0.935152	−	0.354246i	$-0.115262\pi$
$571$	8.84470e20	1.87030	0.935152	+	0.354246i	$0.115262\pi$
$572$	−1.76895e20	−0.369186
$573$	0	0
$574$	6.95614e20	1.41426
$575$	1.44323e20	0.289618
$576$	0	0
$577$	3.15722e20	0.617286	0.308643	−	0.951178i	$-0.400125\pi$
$577$	3.15722e20	0.617286	0.308643	+	0.951178i	$0.400125\pi$
$578$	2.38981e20	0.461216
$579$	0	0
$580$	−1.95870e20	−0.368348
$581$	3.99601e20	0.741832
$582$	0	0
$583$	−1.00682e21	−1.82154
$584$	−5.50799e19	−0.0983778
$585$	0	0
$586$	−6.74794e20	−1.17473
$587$	−7.33397e20	−1.26053	−0.630265	−	0.776380i	$-0.717054\pi$
$587$	−7.33397e20	−1.26053	−0.630265	+	0.776380i	$0.717054\pi$
$588$	0	0
$589$	−1.46391e21	−2.45272
$590$	−1.98610e20	−0.328557
$591$	0	0
$592$	−4.78721e19	−0.0772093
$593$	3.25571e20	0.518485	0.259242	−	0.965812i	$-0.416527\pi$
$593$	3.25571e20	0.518485	0.259242	+	0.965812i	$0.416527\pi$
$594$	0	0
$595$	3.47589e20	0.539746
$596$	5.71353e19	0.0876108
$597$	0	0
$598$	−1.11211e20	−0.166299
$599$	−6.61173e20	−0.976368	−0.488184	−	0.872741i	$-0.662341\pi$
$599$	−6.61173e20	−0.976368	−0.488184	+	0.872741i	$0.662341\pi$
$600$	0	0
$601$	−2.49467e20	−0.359297	−0.179649	−	0.983731i	$-0.557496\pi$
$601$	−2.49467e20	−0.359297	−0.179649	+	0.983731i	$0.557496\pi$
$602$	−1.11222e21	−1.58204
$603$	0	0
$604$	1.12347e20	0.155878
$605$	7.49136e20	1.02659
$606$	0	0
$607$	1.08917e21	1.45606	0.728031	−	0.685544i	$-0.240436\pi$
$607$	1.08917e21	1.45606	0.728031	+	0.685544i	$0.240436\pi$
$608$	−1.69531e20	−0.223858
$609$	0	0
$610$	−7.86610e19	−0.101341
$611$	4.52208e20	0.575478
$612$	0	0
$613$	1.99450e20	0.247674	0.123837	−	0.992303i	$-0.460480\pi$
$613$	1.99450e20	0.247674	0.123837	+	0.992303i	$0.460480\pi$
$614$	−6.10645e20	−0.749076
$615$	0	0
$616$	6.58353e20	0.788140
$617$	−4.21127e20	−0.498052	−0.249026	−	0.968497i	$-0.580110\pi$
$617$	−4.21127e20	−0.498052	−0.249026	+	0.968497i	$0.580110\pi$
$618$	0	0
$619$	−1.27272e21	−1.46910	−0.734552	−	0.678552i	$-0.762608\pi$
$619$	−1.27272e21	−1.46910	−0.734552	+	0.678552i	$0.762608\pi$
$620$	−5.58100e20	−0.636465
$621$	0	0
$622$	−8.27732e20	−0.921429
$623$	−1.19415e20	−0.131341
$624$	0	0
$625$	−1.01717e20	−0.109218
$626$	7.68165e20	0.814980
$627$	0	0
$628$	6.05938e20	0.627670
$629$	1.77925e20	0.182120
$630$	0	0
$631$	−3.42884e20	−0.342711	−0.171355	−	0.985209i	$-0.554815\pi$
$631$	−3.42884e20	−0.342711	−0.171355	+	0.985209i	$0.554815\pi$
$632$	7.11000e20	0.702250
$633$	0	0
$634$	−1.22105e20	−0.117778
$635$	−1.41241e20	−0.134635
$636$	0	0
$637$	−4.65555e20	−0.433436
$638$	1.37885e21	1.26871
$639$	0	0
$640$	−6.46323e19	−0.0580898
$641$	−1.00075e21	−0.888980	−0.444490	−	0.895784i	$-0.646615\pi$
$641$	−1.00075e21	−0.888980	−0.444490	+	0.895784i	$0.646615\pi$
$642$	0	0
$643$	−1.35729e20	−0.117785	−0.0588927	−	0.998264i	$-0.518757\pi$
$643$	−1.35729e20	−0.117785	−0.0588927	+	0.998264i	$0.518757\pi$
$644$	4.13896e20	0.355015
$645$	0	0
$646$	6.30095e20	0.528034
$647$	−2.15462e21	−1.78479	−0.892397	−	0.451251i	$-0.850978\pi$
$647$	−2.15462e21	−1.78479	−0.892397	+	0.451251i	$0.850978\pi$
$648$	0	0
$649$	1.39813e21	1.13165
$650$	−2.31592e20	−0.185299
$651$	0	0
$652$	1.21657e20	0.0951218
$653$	2.06101e21	1.59306	0.796529	−	0.604601i	$-0.206667\pi$
$653$	2.06101e21	1.59306	0.796529	+	0.604601i	$0.206667\pi$
$654$	0	0
$655$	−9.12551e20	−0.689362
$656$	4.80736e20	0.359028
$657$	0	0
$658$	−1.68299e21	−1.22853
$659$	1.61309e21	1.16418	0.582089	−	0.813125i	$-0.302236\pi$
$659$	1.61309e21	1.16418	0.582089	+	0.813125i	$0.302236\pi$
$660$	0	0
$661$	1.60888e21	1.13504	0.567521	−	0.823359i	$-0.307903\pi$
$661$	1.60888e21	1.13504	0.567521	+	0.823359i	$0.307903\pi$
$662$	6.77211e20	0.472377
$663$	0	0
$664$	2.76163e20	0.188324
$665$	−1.71898e21	−1.15907
$666$	0	0
$667$	8.66859e20	0.571485
$668$	2.41442e20	0.157395
$669$	0	0
$670$	7.82309e20	0.498674
$671$	5.53742e20	0.349051
$672$	0	0
$673$	1.67294e21	1.03126	0.515628	−	0.856812i	$-0.327559\pi$
$673$	1.67294e21	1.03126	0.515628	+	0.856812i	$0.327559\pi$
$674$	−7.96684e20	−0.485665
$675$	0	0
$676$	−6.60171e20	−0.393601
$677$	−7.27000e20	−0.428666	−0.214333	−	0.976761i	$-0.568758\pi$
$677$	−7.27000e20	−0.428666	−0.214333	+	0.976761i	$0.568758\pi$
$678$	0	0
$679$	−3.42523e21	−1.97545
$680$	2.40218e20	0.137021
$681$	0	0
$682$	3.92880e21	2.19219
$683$	−2.85909e21	−1.57787	−0.788936	−	0.614475i	$-0.789368\pi$
$683$	−2.85909e21	−1.57787	−0.788936	+	0.614475i	$0.789368\pi$
$684$	0	0
$685$	2.04512e21	1.10418
$686$	−1.11384e20	−0.0594830
$687$	0	0
$688$	−7.68654e20	−0.401622
$689$	1.01572e21	0.524965
$690$	0	0
$691$	1.71274e21	0.866178	0.433089	−	0.901351i	$-0.357424\pi$
$691$	1.71274e21	0.866178	0.433089	+	0.901351i	$0.357424\pi$
$692$	−5.57120e20	−0.278710
$693$	0	0
$694$	−4.12205e20	−0.201798
$695$	4.47691e20	0.216817
$696$	0	0
$697$	−1.78675e21	−0.846870
$698$	7.90281e20	0.370566
$699$	0	0
$700$	8.61918e20	0.395577
$701$	8.95519e20	0.406621	0.203311	−	0.979114i	$-0.434830\pi$
$701$	8.95519e20	0.406621	0.203311	+	0.979114i	$0.434830\pi$
$702$	0	0
$703$	−8.79920e20	−0.391091
$704$	4.54985e20	0.200079
$705$	0	0
$706$	−7.86261e20	−0.338479
$707$	1.15169e20	0.0490557
$708$	0	0
$709$	−3.99642e21	−1.66657	−0.833286	−	0.552842i	$-0.813543\pi$
$709$	−3.99642e21	−1.66657	−0.833286	+	0.552842i	$0.813543\pi$
$710$	−2.97289e20	−0.122671
$711$	0	0
$712$	−8.25274e19	−0.0333425
$713$	2.46998e21	0.987464
$714$	0	0
$715$	−1.23958e21	−0.485267
$716$	−6.24683e20	−0.241998
$717$	0	0
$718$	−7.43812e20	−0.282183
$719$	2.66669e20	0.100117	0.0500583	−	0.998746i	$-0.484059\pi$
$719$	2.66669e20	0.100117	0.0500583	+	0.998746i	$0.484059\pi$
$720$	0	0
$721$	6.29049e20	0.231297
$722$	−1.17291e21	−0.426812
$723$	0	0
$724$	−8.42341e20	−0.300226
$725$	1.80519e21	0.636779
$726$	0	0
$727$	−1.69447e21	−0.585499	−0.292750	−	0.956189i	$-0.594570\pi$
$727$	−1.69447e21	−0.585499	−0.292750	+	0.956189i	$0.594570\pi$
$728$	−6.64170e20	−0.227140
$729$	0	0
$730$	−3.85970e20	−0.129310
$731$	2.85684e21	0.947341
$732$	0	0
$733$	3.16239e21	1.02739	0.513695	−	0.857973i	$-0.328276\pi$
$733$	3.16239e21	1.02739	0.513695	+	0.857973i	$0.328276\pi$
$734$	3.54749e19	0.0114078
$735$	0	0
$736$	2.86042e20	0.0901252
$737$	−5.50714e21	−1.71759
$738$	0	0
$739$	2.01381e21	0.615440	0.307720	−	0.951477i	$-0.400434\pi$
$739$	2.01381e21	0.615440	0.307720	+	0.951477i	$0.400434\pi$
$740$	−3.35461e20	−0.101486
$741$	0	0
$742$	−3.78022e21	−1.12070
$743$	−2.11411e21	−0.620458	−0.310229	−	0.950662i	$-0.600406\pi$
$743$	−2.11411e21	−0.620458	−0.310229	+	0.950662i	$0.600406\pi$
$744$	0	0
$745$	4.00372e20	0.115157
$746$	−2.95038e21	−0.840110
$747$	0	0
$748$	−1.69104e21	−0.471945
$749$	−6.05970e21	−1.67432
$750$	0	0
$751$	4.95087e21	1.34086	0.670428	−	0.741974i	$-0.266110\pi$
$751$	4.95087e21	1.34086	0.670428	+	0.741974i	$0.266110\pi$
$752$	−1.16311e21	−0.311879
$753$	0	0
$754$	−1.39103e21	−0.365638
$755$	7.87266e20	0.204889
$756$	0	0
$757$	−4.85879e21	−1.23968	−0.619839	−	0.784729i	$-0.712802\pi$
$757$	−4.85879e21	−1.23968	−0.619839	+	0.784729i	$0.712802\pi$
$758$	1.72028e21	0.434590
$759$	0	0
$760$	−1.18798e21	−0.294244
$761$	3.69092e21	0.905211	0.452605	−	0.891711i	$-0.350495\pi$
$761$	3.69092e21	0.905211	0.452605	+	0.891711i	$0.350495\pi$
$762$	0	0
$763$	−4.11491e19	−0.00989523
$764$	−1.10685e21	−0.263566
$765$	0	0
$766$	3.38114e21	0.789491
$767$	−1.41049e21	−0.326140
$768$	0	0
$769$	−2.33815e21	−0.530181	−0.265091	−	0.964224i	$-0.585402\pi$
$769$	−2.33815e21	−0.530181	−0.265091	+	0.964224i	$0.585402\pi$
$770$	4.61337e21	1.03595
$771$	0	0
$772$	−3.39534e20	−0.0747746
$773$	5.20682e21	1.13560	0.567801	−	0.823166i	$-0.307794\pi$
$773$	5.20682e21	1.13560	0.567801	+	0.823166i	$0.307794\pi$
$774$	0	0
$775$	5.14361e21	1.10028
$776$	−2.36716e21	−0.501493
$777$	0	0
$778$	5.05952e21	1.05139
$779$	8.83624e21	1.81860
$780$	0	0
$781$	2.09279e21	0.422516
$782$	−1.06313e21	−0.212586
$783$	0	0
$784$	1.19744e21	0.234899
$785$	4.24608e21	0.825023
$786$	0	0
$787$	−1.92196e21	−0.366381	−0.183191	−	0.983077i	$-0.558643\pi$
$787$	−1.92196e21	−0.366381	−0.183191	+	0.983077i	$0.558643\pi$
$788$	−2.80566e20	−0.0529772
$789$	0	0
$790$	4.98230e21	0.923052
$791$	−2.15085e21	−0.394718
$792$	0	0
$793$	−5.58635e20	−0.100596
$794$	−3.62099e20	−0.0645912
$795$	0	0
$796$	−1.73250e21	−0.303267
$797$	6.23880e20	0.108184	0.0540921	−	0.998536i	$-0.482774\pi$
$797$	6.23880e20	0.108184	0.0540921	+	0.998536i	$0.482774\pi$
$798$	0	0
$799$	4.32290e21	0.735655
$800$	5.95669e20	0.100422
$801$	0	0
$802$	−5.93511e21	−0.982022
$803$	2.71707e21	0.445384
$804$	0	0
$805$	2.90035e21	0.466640
$806$	−3.96352e21	−0.631783
$807$	0	0
$808$	7.95929e19	0.0124534
$809$	−5.72858e21	−0.888042	−0.444021	−	0.896016i	$-0.646448\pi$
$809$	−5.72858e21	−0.888042	−0.444021	+	0.896016i	$0.646448\pi$
$810$	0	0
$811$	4.80014e21	0.730462	0.365231	−	0.930917i	$-0.380990\pi$
$811$	4.80014e21	0.730462	0.365231	+	0.930917i	$0.380990\pi$
$812$	5.17702e21	0.780566
$813$	0	0
$814$	2.36151e21	0.349548
$815$	8.52505e20	0.125030
$816$	0	0
$817$	−1.41284e22	−2.03435
$818$	1.49878e21	0.213840
$819$	0	0
$820$	3.36873e21	0.471914
$821$	−8.89775e21	−1.23511	−0.617556	−	0.786527i	$-0.711877\pi$
$821$	−8.89775e21	−1.23511	−0.617556	+	0.786527i	$0.711877\pi$
$822$	0	0
$823$	−1.39946e21	−0.190749	−0.0953744	−	0.995441i	$-0.530405\pi$
$823$	−1.39946e21	−0.190749	−0.0953744	+	0.995441i	$0.530405\pi$
$824$	4.34733e20	0.0587177
$825$	0	0
$826$	5.24943e21	0.696244
$827$	−6.61343e21	−0.869232	−0.434616	−	0.900616i	$-0.643116\pi$
$827$	−6.61343e21	−0.869232	−0.434616	+	0.900616i	$0.643116\pi$
$828$	0	0
$829$	4.14735e21	0.535318	0.267659	−	0.963514i	$-0.413750\pi$
$829$	4.14735e21	0.535318	0.267659	+	0.963514i	$0.413750\pi$
$830$	1.93520e21	0.247537
$831$	0	0
$832$	−4.59006e20	−0.0576625
$833$	−4.45049e21	−0.554078
$834$	0	0
$835$	1.69190e21	0.206883
$836$	8.36292e21	1.01347
$837$	0	0
$838$	−5.37730e21	−0.640080
$839$	−9.57811e21	−1.12997	−0.564983	−	0.825103i	$-0.691117\pi$
$839$	−9.57811e21	−1.12997	−0.564983	+	0.825103i	$0.691117\pi$
$840$	0	0
$841$	2.21350e21	0.256513
$842$	2.04440e21	0.234814
$843$	0	0
$844$	−5.12972e21	−0.578795
$845$	−4.62612e21	−0.517358
$846$	0	0
$847$	−1.98003e22	−2.17544
$848$	−2.61250e21	−0.284503
$849$	0	0
$850$	−2.21391e21	−0.236875
$851$	1.48465e21	0.157453
$852$	0	0
$853$	−8.15361e21	−0.849634	−0.424817	−	0.905279i	$-0.639662\pi$
$853$	−8.15361e21	−0.849634	−0.424817	+	0.905279i	$0.639662\pi$
$854$	2.07908e21	0.214752
$855$	0	0
$856$	−4.18784e21	−0.425047
$857$	1.53552e22	1.54490	0.772449	−	0.635076i	$-0.219031\pi$
$857$	1.53552e22	1.54490	0.772449	+	0.635076i	$0.219031\pi$
$858$	0	0
$859$	−4.87617e21	−0.482092	−0.241046	−	0.970514i	$-0.577490\pi$
$859$	−4.87617e21	−0.482092	−0.241046	+	0.970514i	$0.577490\pi$
$860$	−5.38630e21	−0.527901
$861$	0	0
$862$	−1.24975e22	−1.20370
$863$	4.76667e21	0.455128	0.227564	−	0.973763i	$-0.426924\pi$
$863$	4.76667e21	0.455128	0.227564	+	0.973763i	$0.426924\pi$
$864$	0	0
$865$	−3.90399e21	−0.366343
$866$	8.85063e21	0.823360
$867$	0	0
$868$	1.47511e22	1.34873
$869$	−3.50734e22	−3.17928
$870$	0	0
$871$	5.55580e21	0.495006
$872$	−2.84380e19	−0.00251203
$873$	0	0
$874$	5.25764e21	0.456515
$875$	1.66720e22	1.43525
$876$	0	0
$877$	1.57944e22	1.33661	0.668306	−	0.743887i	$-0.267020\pi$
$877$	1.57944e22	1.33661	0.668306	+	0.743887i	$0.267020\pi$
$878$	−7.02707e21	−0.589611
$879$	0	0
$880$	3.18829e21	0.262989
$881$	1.12789e22	0.922461	0.461230	−	0.887280i	$-0.347408\pi$
$881$	1.12789e22	0.922461	0.461230	+	0.887280i	$0.347408\pi$
$882$	0	0
$883$	4.48307e21	0.360471	0.180236	−	0.983623i	$-0.442314\pi$
$883$	4.48307e21	0.360471	0.180236	+	0.983623i	$0.442314\pi$
$884$	1.70598e21	0.136013
$885$	0	0
$886$	4.87045e21	0.381783
$887$	8.56246e21	0.665536	0.332768	−	0.943009i	$-0.392017\pi$
$887$	8.56246e21	0.665536	0.332768	+	0.943009i	$0.392017\pi$
$888$	0	0
$889$	3.73312e21	0.285304
$890$	−5.78307e20	−0.0438261
$891$	0	0
$892$	5.48893e21	0.409026
$893$	−2.13786e22	−1.57977
$894$	0	0
$895$	−4.37743e21	−0.318087
$896$	1.70829e21	0.123098
$897$	0	0
$898$	1.43352e20	0.0101585
$899$	3.08945e22	2.17112
$900$	0	0
$901$	9.70983e21	0.671082
$902$	−2.37145e22	−1.62542
$903$	0	0
$904$	−1.48645e21	−0.100204
$905$	−5.90266e21	−0.394624
$906$	0	0
$907$	7.34638e21	0.483079	0.241540	−	0.970391i	$-0.422348\pi$
$907$	7.34638e21	0.483079	0.241540	+	0.970391i	$0.422348\pi$
$908$	−7.66051e21	−0.499590
$909$	0	0
$910$	−4.65414e21	−0.298558
$911$	1.18999e22	0.757106	0.378553	−	0.925580i	$-0.376422\pi$
$911$	1.18999e22	0.757106	0.378553	+	0.925580i	$0.376422\pi$
$912$	0	0
$913$	−1.36230e22	−0.852594
$914$	1.24254e22	0.771283
$915$	0	0
$916$	5.91767e21	0.361355
$917$	2.41195e22	1.46083
$918$	0	0
$919$	1.92629e22	1.14777	0.573886	−	0.818935i	$-0.305435\pi$
$919$	1.92629e22	1.14777	0.573886	+	0.818935i	$0.305435\pi$
$920$	2.00442e21	0.118463
$921$	0	0
$922$	−1.25205e22	−0.728015
$923$	−2.11129e21	−0.121768
$924$	0	0
$925$	3.09170e21	0.175442
$926$	2.77964e21	0.156461
$927$	0	0
$928$	3.57782e21	0.198157
$929$	−8.78109e21	−0.482426	−0.241213	−	0.970472i	$-0.577545\pi$
$929$	−8.78109e21	−0.482426	−0.241213	+	0.970472i	$0.577545\pi$
$930$	0	0
$931$	2.20096e22	1.18985
$932$	−5.92753e21	−0.317873
$933$	0	0
$934$	−2.54007e22	−1.34043
$935$	−1.18499e22	−0.620334
$936$	0	0
$937$	−2.35424e22	−1.21284	−0.606421	−	0.795144i	$-0.707395\pi$
$937$	−2.35424e22	−1.21284	−0.606421	+	0.795144i	$0.707395\pi$
$938$	−2.06771e22	−1.05674
$939$	0	0
$940$	−8.15040e21	−0.409940
$941$	2.31660e22	1.15592	0.577961	−	0.816064i	$-0.303848\pi$
$941$	2.31660e22	1.15592	0.577961	+	0.816064i	$0.303848\pi$
$942$	0	0
$943$	−1.49090e22	−0.732166
$944$	3.62786e21	0.176751
$945$	0	0
$946$	3.79174e22	1.81826
$947$	−2.02578e22	−0.963755	−0.481877	−	0.876239i	$-0.660045\pi$
$947$	−2.02578e22	−0.963755	−0.481877	+	0.876239i	$0.660045\pi$
$948$	0	0
$949$	−2.74108e21	−0.128359
$950$	1.09488e22	0.508673
$951$	0	0
$952$	−6.34916e21	−0.290362
$953$	1.66351e22	0.754794	0.377397	−	0.926052i	$-0.376819\pi$
$953$	1.66351e22	0.754794	0.377397	+	0.926052i	$0.376819\pi$
$954$	0	0
$955$	−7.75620e21	−0.346436
$956$	1.75050e22	0.775762
$957$	0	0
$958$	−4.84686e21	−0.211456
$959$	−5.40544e22	−2.33987
$960$	0	0
$961$	6.45637e22	2.75146
$962$	−2.38238e21	−0.100739
$963$	0	0
$964$	−1.64804e22	−0.686102
$965$	−2.37927e21	−0.0982853
$966$	0	0
$967$	−2.10794e22	−0.857355	−0.428677	−	0.903458i	$-0.641020\pi$
$967$	−2.10794e22	−0.857355	−0.428677	+	0.903458i	$0.641020\pi$
$968$	−1.36839e22	−0.552263
$969$	0	0
$970$	−1.65878e22	−0.659174
$971$	3.09261e22	1.21950	0.609749	−	0.792594i	$-0.291270\pi$
$971$	3.09261e22	1.21950	0.609749	+	0.792594i	$0.291270\pi$
$972$	0	0
$973$	−1.18329e22	−0.459456
$974$	−4.98259e21	−0.191983
$975$	0	0
$976$	1.43684e21	0.0545175
$977$	−7.85725e21	−0.295843	−0.147922	−	0.988999i	$-0.547258\pi$
$977$	−7.85725e21	−0.295843	−0.147922	+	0.988999i	$0.547258\pi$
$978$	0	0
$979$	4.07105e21	0.150951
$980$	8.39097e21	0.308757
$981$	0	0
$982$	1.79860e22	0.651778
$983$	−1.27981e22	−0.460251	−0.230125	−	0.973161i	$-0.573914\pi$
$983$	−1.27981e22	−0.460251	−0.230125	+	0.973161i	$0.573914\pi$
$984$	0	0
$985$	−1.96605e21	−0.0696343
$986$	−1.32976e22	−0.467409
$987$	0	0
$988$	−8.43682e21	−0.292080
$989$	2.38381e22	0.819028
$990$	0	0
$991$	−5.39385e22	−1.82535	−0.912675	−	0.408686i	$-0.865987\pi$
$991$	−5.39385e22	−1.82535	−0.912675	+	0.408686i	$0.865987\pi$
$992$	1.01944e22	0.342393
$993$	0	0
$994$	7.85760e21	0.259951
$995$	−1.21404e22	−0.398620
$996$	0	0
$997$	−1.99678e22	−0.645828	−0.322914	−	0.946428i	$-0.604663\pi$
$997$	−1.99678e22	−0.645828	−0.322914	+	0.946428i	$0.604663\pi$
$998$	1.96740e22	0.631557
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.16.a.b.1.1		1
3.2	odd	2		6.16.a.b.1.1	✓	1
4.3	odd	2		144.16.a.j.1.1		1
12.11	even	2		48.16.a.d.1.1		1
15.2	even	4		150.16.c.a.49.2		2
15.8	even	4		150.16.c.a.49.1		2
15.14	odd	2		150.16.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.16.a.b.1.1	✓	1	3.2	odd	2
18.16.a.b.1.1		1	1.1	even	1	trivial
48.16.a.d.1.1		1	12.11	even	2
144.16.a.j.1.1		1	4.3	odd	2
150.16.a.f.1.1		1	15.14	odd	2
150.16.c.a.49.1		2	15.8	even	4
150.16.c.a.49.2		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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