LMFDB - Embedded newform 9.18.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,18,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	9.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:	$16.4899878610$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+528.000 q^{2} +147712. q^{4} +1.02585e6 q^{5} +3.22599e6 q^{7} +8.78592e6 q^{8} +O(q^{10})$

\(q+528.000 q^{2} +147712. q^{4} +1.02585e6 q^{5} +3.22599e6 q^{7} +8.78592e6 q^{8} +5.41649e8 q^{10} +7.53618e8 q^{11} +2.54106e9 q^{13} +1.70332e9 q^{14} -1.47219e10 q^{16} +5.42974e9 q^{17} +1.48750e9 q^{19} +1.51530e11 q^{20} +3.97910e11 q^{22} +3.17092e11 q^{23} +2.89429e11 q^{25} +1.34168e12 q^{26} +4.76518e11 q^{28} -2.43341e12 q^{29} -8.84972e12 q^{31} -8.92477e12 q^{32} +2.86690e12 q^{34} +3.30938e12 q^{35} +1.26917e13 q^{37} +7.85400e11 q^{38} +9.01304e12 q^{40} -4.88642e13 q^{41} -9.10200e13 q^{43} +1.11318e14 q^{44} +1.67424e14 q^{46} +4.93050e13 q^{47} -2.22223e14 q^{49} +1.52818e14 q^{50} +3.75346e14 q^{52} -2.29405e13 q^{53} +7.73099e14 q^{55} +2.83433e13 q^{56} -1.28484e15 q^{58} -3.26951e13 q^{59} -1.30829e15 q^{61} -4.67265e15 q^{62} -2.78265e15 q^{64} +2.60675e15 q^{65} +5.19614e15 q^{67} +8.02038e14 q^{68} +1.74735e15 q^{70} +3.70949e15 q^{71} +3.40237e15 q^{73} +6.70119e15 q^{74} +2.19722e14 q^{76} +2.43117e15 q^{77} +2.36653e15 q^{79} -1.51025e16 q^{80} -2.58003e16 q^{82} +2.97668e16 q^{83} +5.57010e15 q^{85} -4.80585e16 q^{86} +6.62123e15 q^{88} -2.91672e16 q^{89} +8.19745e15 q^{91} +4.68383e16 q^{92} +2.60330e16 q^{94} +1.52595e15 q^{95} -6.37699e16 q^{97} -1.17334e17 q^{98} +O(q^{100})\)

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 $

$2$	528.000	1.45841	0.729204	−	0.684297i	$-0.239891\pi$
$2$	528.000	1.45841	0.729204	+	0.684297i	$0.239891\pi$
$3$	0	0
$3$	0	0
$4$	147712.	1.12695
$5$	1.02585e6	1.17446	0.587231	−	0.809420i	$-0.300218\pi$
$5$	1.02585e6	1.17446	0.587231	+	0.809420i	$0.300218\pi$
$6$	0	0
$7$	3.22599e6	0.211510	0.105755	−	0.994392i	$-0.466274\pi$
$7$	3.22599e6	0.211510	0.105755	+	0.994392i	$0.466274\pi$
$8$	8.78592e6	0.185149
$9$	0	0
$10$	5.41649e8	1.71284
$11$	7.53618e8	1.06002	0.530009	−	0.847992i	$-0.322188\pi$
$11$	7.53618e8	1.06002	0.530009	+	0.847992i	$0.322188\pi$
$12$	0	0
$13$	2.54106e9	0.863967	0.431984	−	0.901881i	$-0.357814\pi$
$13$	2.54106e9	0.863967	0.431984	+	0.901881i	$0.357814\pi$
$14$	1.70332e9	0.308467
$15$	0	0
$16$	−1.47219e10	−0.856930
$17$	5.42974e9	0.188783	0.0943916	−	0.995535i	$-0.469909\pi$
$17$	5.42974e9	0.188783	0.0943916	+	0.995535i	$0.469909\pi$
$18$	0	0
$19$	1.48750e9	0.0200933	0.0100467	−	0.999950i	$-0.496802\pi$
$19$	1.48750e9	0.0200933	0.0100467	+	0.999950i	$0.496802\pi$
$20$	1.51530e11	1.32356
$21$	0	0
$22$	3.97910e11	1.54594
$23$	3.17092e11	0.844303	0.422152	−	0.906525i	$-0.361275\pi$
$23$	3.17092e11	0.844303	0.422152	+	0.906525i	$0.361275\pi$
$24$	0	0
$25$	2.89429e11	0.379360
$26$	1.34168e12	1.26002
$27$	0	0
$28$	4.76518e11	0.238361
$29$	−2.43341e12	−0.903301	−0.451650	−	0.892195i	$-0.649165\pi$
$29$	−2.43341e12	−0.903301	−0.451650	+	0.892195i	$0.649165\pi$
$30$	0	0
$31$	−8.84972e12	−1.86361	−0.931805	−	0.362958i	$-0.881767\pi$
$31$	−8.84972e12	−1.86361	−0.931805	+	0.362958i	$0.881767\pi$
$32$	−8.92477e12	−1.43490
$33$	0	0
$34$	2.86690e12	0.275323
$35$	3.30938e12	0.248410
$36$	0	0
$37$	1.26917e13	0.594023	0.297012	−	0.954874i	$-0.404010\pi$
$37$	1.26917e13	0.594023	0.297012	+	0.954874i	$0.404010\pi$
$38$	7.85400e11	0.0293042
$39$	0	0
$40$	9.01304e12	0.217451
$41$	−4.88642e13	−0.955713	−0.477857	−	0.878438i	$-0.658586\pi$
$41$	−4.88642e13	−0.955713	−0.477857	+	0.878438i	$0.658586\pi$
$42$	0	0
$43$	−9.10200e13	−1.18756	−0.593779	−	0.804628i	$-0.702365\pi$
$43$	−9.10200e13	−1.18756	−0.593779	+	0.804628i	$0.702365\pi$
$44$	1.11318e14	1.19459
$45$	0	0
$46$	1.67424e14	1.23134
$47$	4.93050e13	0.302036	0.151018	−	0.988531i	$-0.451745\pi$
$47$	4.93050e13	0.302036	0.151018	+	0.988531i	$0.451745\pi$
$48$	0	0
$49$	−2.22223e14	−0.955264
$50$	1.52818e14	0.553262
$51$	0	0
$52$	3.75346e14	0.973650
$53$	−2.29405e13	−0.0506124	−0.0253062	−	0.999680i	$-0.508056\pi$
$53$	−2.29405e13	−0.0506124	−0.0253062	+	0.999680i	$0.508056\pi$
$54$	0	0
$55$	7.73099e14	1.24495
$56$	2.83433e13	0.0391609
$57$	0	0
$58$	−1.28484e15	−1.31738
$59$	−3.26951e13	−0.0289895	−0.0144947	−	0.999895i	$-0.504614\pi$
$59$	−3.26951e13	−0.0289895	−0.0144947	+	0.999895i	$0.504614\pi$
$60$	0	0
$61$	−1.30829e15	−0.873774	−0.436887	−	0.899516i	$-0.643919\pi$
$61$	−1.30829e15	−0.873774	−0.436887	+	0.899516i	$0.643919\pi$
$62$	−4.67265e15	−2.71790
$63$	0	0
$64$	−2.78265e15	−1.23574
$65$	2.60675e15	1.01470
$66$	0	0
$67$	5.19614e15	1.56331	0.781655	−	0.623711i	$-0.214376\pi$
$67$	5.19614e15	1.56331	0.781655	+	0.623711i	$0.214376\pi$
$68$	8.02038e14	0.212750
$69$	0	0
$70$	1.74735e15	0.362283
$71$	3.70949e15	0.681739	0.340870	−	0.940111i	$-0.389279\pi$
$71$	3.70949e15	0.681739	0.340870	+	0.940111i	$0.389279\pi$
$72$	0	0
$73$	3.40237e15	0.493785	0.246892	−	0.969043i	$-0.420591\pi$
$73$	3.40237e15	0.493785	0.246892	+	0.969043i	$0.420591\pi$
$74$	6.70119e15	0.866328
$75$	0	0
$76$	2.19722e14	0.0226442
$77$	2.43117e15	0.224204
$78$	0	0
$79$	2.36653e15	0.175502	0.0877511	−	0.996142i	$-0.472032\pi$
$79$	2.36653e15	0.175502	0.0877511	+	0.996142i	$0.472032\pi$
$80$	−1.51025e16	−1.00643
$81$	0	0
$82$	−2.58003e16	−1.39382
$83$	2.97668e16	1.45067	0.725333	−	0.688398i	$-0.241686\pi$
$83$	2.97668e16	1.45067	0.725333	+	0.688398i	$0.241686\pi$
$84$	0	0
$85$	5.57010e15	0.221719
$86$	−4.80585e16	−1.73194
$87$	0	0
$88$	6.62123e15	0.196262
$89$	−2.91672e16	−0.785379	−0.392690	−	0.919671i	$-0.628455\pi$
$89$	−2.91672e16	−0.785379	−0.392690	+	0.919671i	$0.628455\pi$
$90$	0	0
$91$	8.19745e15	0.182737
$92$	4.68383e16	0.951490
$93$	0	0
$94$	2.60330e16	0.440492
$95$	1.52595e15	0.0235988
$96$	0	0
$97$	−6.37699e16	−0.826144	−0.413072	−	0.910698i	$-0.635544\pi$
$97$	−6.37699e16	−0.826144	−0.413072	+	0.910698i	$0.635544\pi$
$98$	−1.17334e17	−1.39316
$99$	0	0
$100$	4.27521e16	0.427521
$101$	1.60611e17	1.47586	0.737929	−	0.674878i	$-0.235804\pi$
$101$	1.60611e17	1.47586	0.737929	+	0.674878i	$0.235804\pi$
$102$	0	0
$103$	−9.07136e16	−0.705596	−0.352798	−	0.935700i	$-0.614770\pi$
$103$	−9.07136e16	−0.705596	−0.352798	+	0.935700i	$0.614770\pi$
$104$	2.23256e16	0.159963
$105$	0	0
$106$	−1.21126e16	−0.0738135
$107$	−1.95549e17	−1.10026	−0.550129	−	0.835080i	$-0.685421\pi$
$107$	−1.95549e17	−1.10026	−0.550129	+	0.835080i	$0.685421\pi$
$108$	0	0
$109$	2.13756e17	1.02753	0.513763	−	0.857932i	$-0.328251\pi$
$109$	2.13756e17	1.02753	0.513763	+	0.857932i	$0.328251\pi$
$110$	4.08196e17	1.81565
$111$	0	0
$112$	−4.74929e16	−0.181249
$113$	2.81383e17	0.995706	0.497853	−	0.867262i	$-0.334122\pi$
$113$	2.81383e17	0.995706	0.497853	+	0.867262i	$0.334122\pi$
$114$	0	0
$115$	3.25289e17	0.991602
$116$	−3.59444e17	−1.01798
$117$	0	0
$118$	−1.72630e16	−0.0422785
$119$	1.75163e16	0.0399295
$120$	0	0
$121$	6.24934e16	0.123640
$122$	−6.90775e17	−1.27432
$123$	0	0
$124$	−1.30721e18	−2.10020
$125$	−4.85751e17	−0.728918
$126$	0	0
$127$	−8.70305e17	−1.14114	−0.570571	−	0.821248i	$-0.693278\pi$
$127$	−8.70305e17	−1.14114	−0.570571	+	0.821248i	$0.693278\pi$
$128$	−2.99449e17	−0.367315
$129$	0	0
$130$	1.37636e18	1.47984
$131$	−1.31783e17	−0.132756	−0.0663781	−	0.997795i	$-0.521144\pi$
$131$	−1.31783e17	−0.132756	−0.0663781	+	0.997795i	$0.521144\pi$
$132$	0	0
$133$	4.79866e15	0.00424993
$134$	2.74356e18	2.27994
$135$	0	0
$136$	4.77053e16	0.0349531
$137$	−1.87128e18	−1.28829	−0.644147	−	0.764902i	$-0.722787\pi$
$137$	−1.87128e18	−1.28829	−0.644147	+	0.764902i	$0.722787\pi$
$138$	0	0
$139$	1.58706e18	0.965980	0.482990	−	0.875626i	$-0.339551\pi$
$139$	1.58706e18	0.965980	0.482990	+	0.875626i	$0.339551\pi$
$140$	4.88836e17	0.279946
$141$	0	0
$142$	1.95861e18	0.994254
$143$	1.91499e18	0.915821
$144$	0	0
$145$	−2.49631e18	−1.06089
$146$	1.79645e18	0.720139
$147$	0	0
$148$	1.87471e18	0.669436
$149$	4.77240e17	0.160936	0.0804680	−	0.996757i	$-0.474359\pi$
$149$	4.77240e17	0.160936	0.0804680	+	0.996757i	$0.474359\pi$
$150$	0	0
$151$	−3.92964e18	−1.18317	−0.591587	−	0.806241i	$-0.701499\pi$
$151$	−3.92964e18	−1.18317	−0.591587	+	0.806241i	$0.701499\pi$
$152$	1.30691e16	0.00372026
$153$	0	0
$154$	1.28366e18	0.326981
$155$	−9.07849e18	−2.18874
$156$	0	0
$157$	−2.29453e18	−0.496075	−0.248038	−	0.968750i	$-0.579786\pi$
$157$	−2.29453e18	−0.496075	−0.248038	+	0.968750i	$0.579786\pi$
$158$	1.24953e18	0.255954
$159$	0	0
$160$	−9.15548e18	−1.68524
$161$	1.02294e18	0.178578
$162$	0	0
$163$	8.01044e18	1.25910	0.629552	−	0.776958i	$-0.283238\pi$
$163$	8.01044e18	1.25910	0.629552	+	0.776958i	$0.283238\pi$
$164$	−7.21782e18	−1.07704
$165$	0	0
$166$	1.57168e19	2.11566
$167$	8.61477e18	1.10193	0.550965	−	0.834528i	$-0.314260\pi$
$167$	8.61477e18	1.10193	0.550965	+	0.834528i	$0.314260\pi$
$168$	0	0
$169$	−2.19341e18	−0.253561
$170$	2.94101e18	0.323356
$171$	0	0
$172$	−1.34447e19	−1.33832
$173$	−2.31430e18	−0.219294	−0.109647	−	0.993971i	$-0.534972\pi$
$173$	−2.31430e18	−0.219294	−0.109647	+	0.993971i	$0.534972\pi$
$174$	0	0
$175$	9.33695e17	0.0802383
$176$	−1.10947e19	−0.908362
$177$	0	0
$178$	−1.54003e19	−1.14540
$179$	7.90307e18	0.560461	0.280230	−	0.959933i	$-0.409589\pi$
$179$	7.90307e18	0.560461	0.280230	+	0.959933i	$0.409589\pi$
$180$	0	0
$181$	1.40729e19	0.908060	0.454030	−	0.890986i	$-0.349986\pi$
$181$	1.40729e19	0.908060	0.454030	+	0.890986i	$0.349986\pi$
$182$	4.32826e18	0.266505
$183$	0	0
$184$	2.78594e18	0.156322
$185$	1.30197e19	0.697658
$186$	0	0
$187$	4.09195e18	0.200114
$188$	7.28294e18	0.340381
$189$	0	0
$190$	8.05703e17	0.0344167
$191$	2.82501e19	1.15408	0.577040	−	0.816716i	$-0.304208\pi$
$191$	2.82501e19	1.15408	0.577040	+	0.816716i	$0.304208\pi$
$192$	0	0
$193$	4.91755e19	1.83870	0.919351	−	0.393437i	$-0.128714\pi$
$193$	4.91755e19	1.83870	0.919351	+	0.393437i	$0.128714\pi$
$194$	−3.36705e19	−1.20486
$195$	0	0
$196$	−3.28251e19	−1.07654
$197$	−1.29458e19	−0.406598	−0.203299	−	0.979117i	$-0.565166\pi$
$197$	−1.29458e19	−0.406598	−0.203299	+	0.979117i	$0.565166\pi$
$198$	0	0
$199$	−5.51755e19	−1.59036	−0.795179	−	0.606375i	$-0.792623\pi$
$199$	−5.51755e19	−1.59036	−0.795179	+	0.606375i	$0.792623\pi$
$200$	2.54290e18	0.0702383
$201$	0	0
$202$	8.48028e19	2.15240
$203$	−7.85016e18	−0.191057
$204$	0	0
$205$	−5.01273e19	−1.12245
$206$	−4.78968e19	−1.02905
$207$	0	0
$208$	−3.74094e19	−0.740359
$209$	1.12101e18	0.0212993
$210$	0	0
$211$	1.73510e19	0.304035	0.152017	−	0.988378i	$-0.451423\pi$
$211$	1.73510e19	0.304035	0.152017	+	0.988378i	$0.451423\pi$
$212$	−3.38858e18	−0.0570378
$213$	0	0
$214$	−1.03250e20	−1.60462
$215$	−9.33728e19	−1.39474
$216$	0	0
$217$	−2.85491e19	−0.394171
$218$	1.12863e20	1.49855
$219$	0	0
$220$	1.14196e20	1.40300
$221$	1.37973e19	0.163102
$222$	0	0
$223$	9.48415e19	1.03850	0.519251	−	0.854622i	$-0.326211\pi$
$223$	9.48415e19	1.03850	0.519251	+	0.854622i	$0.326211\pi$
$224$	−2.87912e19	−0.303496
$225$	0	0
$226$	1.48570e20	1.45214
$227$	1.83782e20	1.73015	0.865073	−	0.501647i	$-0.167272\pi$
$227$	1.83782e20	1.73015	0.865073	+	0.501647i	$0.167272\pi$
$228$	0	0
$229$	−9.14908e19	−0.799422	−0.399711	−	0.916641i	$-0.630889\pi$
$229$	−9.14908e19	−0.799422	−0.399711	+	0.916641i	$0.630889\pi$
$230$	1.71752e20	1.44616
$231$	0	0
$232$	−2.13798e19	−0.167246
$233$	−9.87497e19	−0.744750	−0.372375	−	0.928082i	$-0.621456\pi$
$233$	−9.87497e19	−0.744750	−0.372375	+	0.928082i	$0.621456\pi$
$234$	0	0
$235$	5.05795e19	0.354730
$236$	−4.82946e18	−0.0326698
$237$	0	0
$238$	9.24861e18	0.0582334
$239$	−1.89337e20	−1.15041	−0.575206	−	0.818009i	$-0.695078\pi$
$239$	−1.89337e20	−1.15041	−0.575206	+	0.818009i	$0.695078\pi$
$240$	0	0
$241$	−1.38762e20	−0.785463	−0.392731	−	0.919653i	$-0.628470\pi$
$241$	−1.38762e20	−0.785463	−0.392731	+	0.919653i	$0.628470\pi$
$242$	3.29965e19	0.180317
$243$	0	0
$244$	−1.93250e20	−0.984702
$245$	−2.27968e20	−1.12192
$246$	0	0
$247$	3.77983e18	0.0173600
$248$	−7.77529e19	−0.345046
$249$	0	0
$250$	−2.56476e20	−1.06306
$251$	3.34508e20	1.34023	0.670115	−	0.742257i	$-0.266245\pi$
$251$	3.34508e20	1.34023	0.670115	+	0.742257i	$0.266245\pi$
$252$	0	0
$253$	2.38966e20	0.894977
$254$	−4.59521e20	−1.66425
$255$	0	0
$256$	2.06618e20	0.700048
$257$	−4.91382e19	−0.161060	−0.0805300	−	0.996752i	$-0.525661\pi$
$257$	−4.91382e19	−0.161060	−0.0805300	+	0.996752i	$0.525661\pi$
$258$	0	0
$259$	4.09432e19	0.125642
$260$	3.85048e20	1.14352
$261$	0	0
$262$	−6.95816e19	−0.193613
$263$	−1.78845e19	−0.0481784	−0.0240892	−	0.999710i	$-0.507669\pi$
$263$	−1.78845e19	−0.0481784	−0.0240892	+	0.999710i	$0.507669\pi$
$264$	0	0
$265$	−2.35335e19	−0.0594423
$266$	2.53369e18	0.00619812
$267$	0	0
$268$	7.67533e20	1.76178
$269$	−1.15237e20	−0.256270	−0.128135	−	0.991757i	$-0.540899\pi$
$269$	−1.15237e20	−0.256270	−0.128135	+	0.991757i	$0.540899\pi$
$270$	0	0
$271$	−1.47067e20	−0.307098	−0.153549	−	0.988141i	$-0.549070\pi$
$271$	−1.47067e20	−0.307098	−0.153549	+	0.988141i	$0.549070\pi$
$272$	−7.99363e19	−0.161774
$273$	0	0
$274$	−9.88038e20	−1.87886
$275$	2.18119e20	0.402129
$276$	0	0
$277$	−1.78744e20	−0.309852	−0.154926	−	0.987926i	$-0.549514\pi$
$277$	−1.78744e20	−0.309852	−0.154926	+	0.987926i	$0.549514\pi$
$278$	8.37969e20	1.40879
$279$	0	0
$280$	2.90760e19	0.0459929
$281$	−2.80546e20	−0.430527	−0.215263	−	0.976556i	$-0.569061\pi$
$281$	−2.80546e20	−0.430527	−0.215263	+	0.976556i	$0.569061\pi$
$282$	0	0
$283$	3.39877e20	0.491062	0.245531	−	0.969389i	$-0.421038\pi$
$283$	3.39877e20	0.491062	0.245531	+	0.969389i	$0.421038\pi$
$284$	5.47936e20	0.768288
$285$	0	0
$286$	1.01112e21	1.33564
$287$	−1.57635e20	−0.202142
$288$	0	0
$289$	−7.97758e20	−0.964361
$290$	−1.31805e21	−1.54721
$291$	0	0
$292$	5.02571e20	0.556472
$293$	7.64887e20	0.822664	0.411332	−	0.911486i	$-0.365064\pi$
$293$	7.64887e20	0.822664	0.411332	+	0.911486i	$0.365064\pi$
$294$	0	0
$295$	−3.35403e19	−0.0340470
$296$	1.11508e20	0.109983
$297$	0	0
$298$	2.51983e20	0.234710
$299$	8.05751e20	0.729450
$300$	0	0
$301$	−2.93630e20	−0.251180
$302$	−2.07485e21	−1.72555
$303$	0	0
$304$	−2.18989e19	−0.0172185
$305$	−1.34211e21	−1.02621
$306$	0	0
$307$	1.38472e21	1.00158	0.500789	−	0.865569i	$-0.333043\pi$
$307$	1.38472e21	1.00158	0.500789	+	0.865569i	$0.333043\pi$
$308$	3.59112e20	0.252667
$309$	0	0
$310$	−4.79344e21	−3.19207
$311$	2.34733e21	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$311$	2.34733e21	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$312$	0	0
$313$	1.58424e21	0.972065	0.486033	−	0.873941i	$-0.338444\pi$
$313$	1.58424e21	0.972065	0.486033	+	0.873941i	$0.338444\pi$
$314$	−1.21151e21	−0.723480
$315$	0	0
$316$	3.49565e20	0.197783
$317$	−8.98378e20	−0.494829	−0.247415	−	0.968910i	$-0.579581\pi$
$317$	−8.98378e20	−0.494829	−0.247415	+	0.968910i	$0.579581\pi$
$318$	0	0
$319$	−1.83386e21	−0.957516
$320$	−2.85458e21	−1.45133
$321$	0	0
$322$	5.40110e20	0.260440
$323$	8.07674e18	0.00379328
$324$	0	0
$325$	7.35457e20	0.327755
$326$	4.22951e21	1.83629
$327$	0	0
$328$	−4.29317e20	−0.176950
$329$	1.59058e20	0.0638836
$330$	0	0
$331$	4.00469e21	1.52767	0.763837	−	0.645409i	$-0.223313\pi$
$331$	4.00469e21	1.52767	0.763837	+	0.645409i	$0.223313\pi$
$332$	4.39691e21	1.63483
$333$	0	0
$334$	4.54860e21	1.60706
$335$	5.33046e21	1.83605
$336$	0	0
$337$	−1.06753e21	−0.349564	−0.174782	−	0.984607i	$-0.555922\pi$
$337$	−1.06753e21	−0.349564	−0.174782	+	0.984607i	$0.555922\pi$
$338$	−1.15812e21	−0.369795
$339$	0	0
$340$	8.22771e20	0.249867
$341$	−6.66931e21	−1.97546
$342$	0	0
$343$	−1.46736e21	−0.413557
$344$	−7.99694e20	−0.219876
$345$	0	0
$346$	−1.22195e21	−0.319821
$347$	1.61841e21	0.413321	0.206660	−	0.978413i	$-0.433740\pi$
$347$	1.61841e21	0.413321	0.206660	+	0.978413i	$0.433740\pi$
$348$	0	0
$349$	−5.60078e21	−1.36217	−0.681086	−	0.732203i	$-0.738492\pi$
$349$	−5.60078e21	−1.36217	−0.681086	+	0.732203i	$0.738492\pi$
$350$	4.92991e20	0.117020
$351$	0	0
$352$	−6.72587e21	−1.52102
$353$	−5.10774e21	−1.12757	−0.563785	−	0.825921i	$-0.690656\pi$
$353$	−5.10774e21	−1.12757	−0.563785	+	0.825921i	$0.690656\pi$
$354$	0	0
$355$	3.80538e21	0.800676
$356$	−4.30834e21	−0.885086
$357$	0	0
$358$	4.17282e21	0.817380
$359$	−5.84965e21	−1.11899	−0.559496	−	0.828833i	$-0.689005\pi$
$359$	−5.84965e21	−1.11899	−0.559496	+	0.828833i	$0.689005\pi$
$360$	0	0
$361$	−5.47817e21	−0.999596
$362$	7.43047e21	1.32432
$363$	0	0
$364$	1.21086e21	0.205936
$365$	3.49032e21	0.579931
$366$	0	0
$367$	4.29535e20	0.0681297	0.0340649	−	0.999420i	$-0.489155\pi$
$367$	4.29535e20	0.0681297	0.0340649	+	0.999420i	$0.489155\pi$
$368$	−4.66821e21	−0.723509
$369$	0	0
$370$	6.87442e21	1.01747
$371$	−7.40057e19	−0.0107050
$372$	0	0
$373$	−6.54734e21	−0.904774	−0.452387	−	0.891822i	$-0.649427\pi$
$373$	−6.54734e21	−0.904774	−0.452387	+	0.891822i	$0.649427\pi$
$374$	2.16055e21	0.291847
$375$	0	0
$376$	4.33190e20	0.0559218
$377$	−6.18345e21	−0.780422
$378$	0	0
$379$	1.51850e21	0.183223	0.0916117	−	0.995795i	$-0.470798\pi$
$379$	1.51850e21	0.183223	0.0916117	+	0.995795i	$0.470798\pi$
$380$	2.25401e20	0.0265948
$381$	0	0
$382$	1.49160e22	1.68312
$383$	8.04597e21	0.887951	0.443975	−	0.896039i	$-0.353568\pi$
$383$	8.04597e21	0.887951	0.443975	+	0.896039i	$0.353568\pi$
$384$	0	0
$385$	2.49401e21	0.263319
$386$	2.59647e22	2.68158
$387$	0	0
$388$	−9.41958e21	−0.931026
$389$	−6.84040e21	−0.661470	−0.330735	−	0.943724i	$-0.607297\pi$
$389$	−6.84040e21	−0.661470	−0.330735	+	0.943724i	$0.607297\pi$
$390$	0	0
$391$	1.72173e21	0.159390
$392$	−1.95244e21	−0.176867
$393$	0	0
$394$	−6.83536e21	−0.592985
$395$	2.42771e21	0.206121
$396$	0	0
$397$	6.23458e21	0.507093	0.253547	−	0.967323i	$-0.418403\pi$
$397$	6.23458e21	0.507093	0.253547	+	0.967323i	$0.418403\pi$
$398$	−2.91327e22	−2.31939
$399$	0	0
$400$	−4.26095e21	−0.325085
$401$	2.32048e21	0.173320	0.0866602	−	0.996238i	$-0.472381\pi$
$401$	2.32048e21	0.173320	0.0866602	+	0.996238i	$0.472381\pi$
$402$	0	0
$403$	−2.24877e22	−1.61010
$404$	2.37242e22	1.66322
$405$	0	0
$406$	−4.14489e21	−0.278639
$407$	9.56466e21	0.629676
$408$	0	0
$409$	2.38590e22	1.50662	0.753312	−	0.657663i	$-0.228455\pi$
$409$	2.38590e22	1.50662	0.753312	+	0.657663i	$0.228455\pi$
$410$	−2.64672e22	−1.63699
$411$	0	0
$412$	−1.33995e22	−0.795173
$413$	−1.05474e20	−0.00613155
$414$	0	0
$415$	3.05362e22	1.70375
$416$	−2.26784e22	−1.23971
$417$	0	0
$418$	5.91892e20	0.0310630
$419$	2.61839e22	1.34653	0.673263	−	0.739403i	$-0.264892\pi$
$419$	2.61839e22	1.34653	0.673263	+	0.739403i	$0.264892\pi$
$420$	0	0
$421$	1.11903e22	0.552642	0.276321	−	0.961065i	$-0.410885\pi$
$421$	1.11903e22	0.552642	0.276321	+	0.961065i	$0.410885\pi$
$422$	9.16132e21	0.443407
$423$	0	0
$424$	−2.01553e20	−0.00937086
$425$	1.57152e21	0.0716168
$426$	0	0
$427$	−4.22052e21	−0.184812
$428$	−2.88850e22	−1.23994
$429$	0	0
$430$	−4.93009e22	−2.03410
$431$	−1.74296e22	−0.705068	−0.352534	−	0.935799i	$-0.614680\pi$
$431$	−1.74296e22	−0.705068	−0.352534	+	0.935799i	$0.614680\pi$
$432$	0	0
$433$	−4.51413e22	−1.75560	−0.877802	−	0.479024i	$-0.840991\pi$
$433$	−4.51413e22	−1.75560	−0.877802	+	0.479024i	$0.840991\pi$
$434$	−1.50739e22	−0.574863
$435$	0	0
$436$	3.15743e22	1.15797
$437$	4.71674e20	0.0169648
$438$	0	0
$439$	3.85266e22	1.33294	0.666472	−	0.745530i	$-0.267804\pi$
$439$	3.85266e22	1.33294	0.666472	+	0.745530i	$0.267804\pi$
$440$	6.79239e21	0.230502
$441$	0	0
$442$	7.28499e21	0.237870
$443$	3.84966e21	0.123308	0.0616539	−	0.998098i	$-0.480363\pi$
$443$	3.84966e21	0.123308	0.0616539	+	0.998098i	$0.480363\pi$
$444$	0	0
$445$	−2.99212e22	−0.922398
$446$	5.00763e22	1.51456
$447$	0	0
$448$	−8.97679e21	−0.261371
$449$	2.77614e21	0.0793136	0.0396568	−	0.999213i	$-0.487374\pi$
$449$	2.77614e21	0.0793136	0.0396568	+	0.999213i	$0.487374\pi$
$450$	0	0
$451$	−3.68249e22	−1.01307
$452$	4.15636e22	1.12211
$453$	0	0
$454$	9.70368e22	2.52326
$455$	8.40936e21	0.214618
$456$	0	0
$457$	−1.34139e22	−0.329814	−0.164907	−	0.986309i	$-0.552732\pi$
$457$	−1.34139e22	−0.329814	−0.164907	+	0.986309i	$0.552732\pi$
$458$	−4.83071e22	−1.16588
$459$	0	0
$460$	4.80490e22	1.11749
$461$	4.32728e22	0.988000	0.494000	−	0.869462i	$-0.335534\pi$
$461$	4.32728e22	0.988000	0.494000	+	0.869462i	$0.335534\pi$
$462$	0	0
$463$	3.85721e22	0.848859	0.424429	−	0.905461i	$-0.360475\pi$
$463$	3.85721e22	0.848859	0.424429	+	0.905461i	$0.360475\pi$
$464$	3.58245e22	0.774065
$465$	0	0
$466$	−5.21398e22	−1.08615
$467$	−4.96724e22	−1.01607	−0.508033	−	0.861338i	$-0.669627\pi$
$467$	−4.96724e22	−1.01607	−0.508033	+	0.861338i	$0.669627\pi$
$468$	0	0
$469$	1.67627e22	0.330655
$470$	2.67060e22	0.517341
$471$	0	0
$472$	−2.87256e20	−0.00536739
$473$	−6.85943e22	−1.25883
$474$	0	0
$475$	4.30525e20	0.00762260
$476$	2.58737e21	0.0449986
$477$	0	0
$478$	−9.99699e22	−1.67777
$479$	4.26513e22	0.703202	0.351601	−	0.936150i	$-0.385637\pi$
$479$	4.26513e22	0.703202	0.351601	+	0.936150i	$0.385637\pi$
$480$	0	0
$481$	3.22503e22	0.513217
$482$	−7.32663e22	−1.14553
$483$	0	0
$484$	9.23103e21	0.139336
$485$	−6.54183e22	−0.970275
$486$	0	0
$487$	−9.28126e22	−1.32926	−0.664631	−	0.747172i	$-0.731411\pi$
$487$	−9.28126e22	−1.32926	−0.664631	+	0.747172i	$0.731411\pi$
$488$	−1.14945e22	−0.161779
$489$	0	0
$490$	−1.20367e23	−1.63622
$491$	−1.29877e21	−0.0173516	−0.00867579	−	0.999962i	$-0.502762\pi$
$491$	−1.29877e21	−0.0173516	−0.00867579	+	0.999962i	$0.502762\pi$
$492$	0	0
$493$	−1.32128e22	−0.170528
$494$	1.99575e21	0.0253179
$495$	0	0
$496$	1.30285e23	1.59698
$497$	1.19668e22	0.144194
$498$	0	0
$499$	−7.66788e22	−0.892937	−0.446468	−	0.894799i	$-0.647318\pi$
$499$	−7.66788e22	−0.892937	−0.446468	+	0.894799i	$0.647318\pi$
$500$	−7.17512e22	−0.821456
$501$	0	0
$502$	1.76620e23	1.95460
$503$	−6.67712e22	−0.726543	−0.363272	−	0.931683i	$-0.618340\pi$
$503$	−6.67712e22	−0.726543	−0.363272	+	0.931683i	$0.618340\pi$
$504$	0	0
$505$	1.64763e23	1.73334
$506$	1.26174e23	1.30524
$507$	0	0
$508$	−1.28555e23	−1.28601
$509$	1.15659e23	1.13783	0.568916	−	0.822395i	$-0.307363\pi$
$509$	1.15659e23	1.13783	0.568916	+	0.822395i	$0.307363\pi$
$510$	0	0
$511$	1.09760e22	0.104440
$512$	1.48344e23	1.38827
$513$	0	0
$514$	−2.59450e22	−0.234891
$515$	−9.30585e22	−0.828695
$516$	0	0
$517$	3.71571e22	0.320164
$518$	2.16180e22	0.183237
$519$	0	0
$520$	2.29027e22	0.187870
$521$	1.79855e23	1.45145	0.725725	−	0.687985i	$-0.241504\pi$
$521$	1.79855e23	1.45145	0.725725	+	0.687985i	$0.241504\pi$
$522$	0	0
$523$	9.37211e22	0.732104	0.366052	−	0.930594i	$-0.380709\pi$
$523$	9.37211e22	0.732104	0.366052	+	0.930594i	$0.380709\pi$
$524$	−1.94660e22	−0.149610
$525$	0	0
$526$	−9.44299e21	−0.0702637
$527$	−4.80517e22	−0.351818
$528$	0	0
$529$	−4.05028e22	−0.287152
$530$	−1.24257e22	−0.0866912
$531$	0	0
$532$	7.08820e20	0.00478947
$533$	−1.24167e23	−0.825705
$534$	0	0
$535$	−2.00604e23	−1.29221
$536$	4.56529e22	0.289446
$537$	0	0
$538$	−6.08452e22	−0.373746
$539$	−1.67472e23	−1.01260
$540$	0	0
$541$	−1.52396e23	−0.892889	−0.446444	−	0.894811i	$-0.647310\pi$
$541$	−1.52396e23	−0.892889	−0.446444	+	0.894811i	$0.647310\pi$
$542$	−7.76514e22	−0.447874
$543$	0	0
$544$	−4.84592e22	−0.270886
$545$	2.19281e23	1.20679
$546$	0	0
$547$	2.04979e23	1.09349	0.546747	−	0.837298i	$-0.315866\pi$
$547$	2.04979e23	1.09349	0.546747	+	0.837298i	$0.315866\pi$
$548$	−2.76411e23	−1.45185
$549$	0	0
$550$	1.15167e23	0.586468
$551$	−3.61970e21	−0.0181503
$552$	0	0
$553$	7.63442e21	0.0371204
$554$	−9.43770e22	−0.451890
$555$	0	0
$556$	2.34428e23	1.08861
$557$	4.07359e23	1.86298	0.931490	−	0.363767i	$-0.118509\pi$
$557$	4.07359e23	1.86298	0.931490	+	0.363767i	$0.118509\pi$
$558$	0	0
$559$	−2.31288e23	−1.02601
$560$	−4.87206e22	−0.212870
$561$	0	0
$562$	−1.48128e23	−0.627884
$563$	−2.21297e23	−0.923962	−0.461981	−	0.886890i	$-0.652861\pi$
$563$	−2.21297e23	−0.923962	−0.461981	+	0.886890i	$0.652861\pi$
$564$	0	0
$565$	2.88657e23	1.16942
$566$	1.79455e23	0.716169
$567$	0	0
$568$	3.25913e22	0.126224
$569$	−4.88686e23	−1.86456	−0.932278	−	0.361743i	$-0.882182\pi$
$569$	−4.88686e23	−1.86456	−0.932278	+	0.361743i	$0.882182\pi$
$570$	0	0
$571$	3.42218e20	0.00126735	0.000633673	−	1.00000i	$-0.499798\pi$
$571$	3.42218e20	0.00126735	0.000633673		1.00000i	$0.499798\pi$
$572$	2.82867e23	1.03209
$573$	0	0
$574$	−8.32315e22	−0.294806
$575$	9.17755e22	0.320295
$576$	0	0
$577$	−1.95911e23	−0.663840	−0.331920	−	0.943308i	$-0.607696\pi$
$577$	−1.95911e23	−0.663840	−0.331920	+	0.943308i	$0.607696\pi$
$578$	−4.21216e23	−1.40643
$579$	0	0
$580$	−3.68736e23	−1.19558
$581$	9.60273e22	0.306830
$582$	0	0
$583$	−1.72883e22	−0.0536501
$584$	2.98930e22	0.0914239
$585$	0	0
$586$	4.03860e23	1.19978
$587$	4.80676e23	1.40744	0.703719	−	0.710479i	$-0.251522\pi$
$587$	4.80676e23	1.40744	0.703719	+	0.710479i	$0.251522\pi$
$588$	0	0
$589$	−1.31640e22	−0.0374461
$590$	−1.77093e22	−0.0496545
$591$	0	0
$592$	−1.86846e23	−0.509036
$593$	6.40804e23	1.72092	0.860459	−	0.509519i	$-0.170177\pi$
$593$	6.40804e23	1.72092	0.860459	+	0.509519i	$0.170177\pi$
$594$	0	0
$595$	1.79691e22	0.0468956
$596$	7.04940e22	0.181367
$597$	0	0
$598$	4.25436e23	1.06384
$599$	−3.60950e23	−0.889854	−0.444927	−	0.895567i	$-0.646770\pi$
$599$	−3.60950e23	−0.889854	−0.444927	+	0.895567i	$0.646770\pi$
$600$	0	0
$601$	6.53955e23	1.56717	0.783583	−	0.621287i	$-0.213390\pi$
$601$	6.53955e23	1.56717	0.783583	+	0.621287i	$0.213390\pi$
$602$	−1.55036e23	−0.366323
$603$	0	0
$604$	−5.80455e23	−1.33338
$605$	6.41089e22	0.145210
$606$	0	0
$607$	4.08715e23	0.900154	0.450077	−	0.892990i	$-0.351397\pi$
$607$	4.08715e23	0.900154	0.450077	+	0.892990i	$0.351397\pi$
$608$	−1.32756e22	−0.0288319
$609$	0	0
$610$	−7.08631e23	−1.49664
$611$	1.25287e23	0.260949
$612$	0	0
$613$	1.80349e23	0.365343	0.182671	−	0.983174i	$-0.441526\pi$
$613$	1.80349e23	0.365343	0.182671	+	0.983174i	$0.441526\pi$
$614$	7.31131e23	1.46071
$615$	0	0
$616$	2.13600e22	0.0415113
$617$	−4.42851e23	−0.848856	−0.424428	−	0.905462i	$-0.639525\pi$
$617$	−4.42851e23	−0.848856	−0.424428	+	0.905462i	$0.639525\pi$
$618$	0	0
$619$	1.71098e22	0.0319062	0.0159531	−	0.999873i	$-0.494922\pi$
$619$	1.71098e22	0.0319062	0.0159531	+	0.999873i	$0.494922\pi$
$620$	−1.34100e24	−2.46661
$621$	0	0
$622$	1.23939e24	2.21815
$623$	−9.40931e22	−0.166115
$624$	0	0
$625$	−7.19124e23	−1.23545
$626$	8.36481e23	1.41767
$627$	0	0
$628$	−3.38930e23	−0.559053
$629$	6.89124e22	0.112142
$630$	0	0
$631$	−2.82293e23	−0.447147	−0.223574	−	0.974687i	$-0.571772\pi$
$631$	−2.82293e23	−0.447147	−0.223574	+	0.974687i	$0.571772\pi$
$632$	2.07922e22	0.0324941
$633$	0	0
$634$	−4.74344e23	−0.721663
$635$	−8.92802e23	−1.34023
$636$	0	0
$637$	−5.64684e23	−0.825316
$638$	−9.68279e23	−1.39645
$639$	0	0
$640$	−3.07190e23	−0.431397
$641$	3.77613e23	0.523304	0.261652	−	0.965162i	$-0.415733\pi$
$641$	3.77613e23	0.523304	0.261652	+	0.965162i	$0.415733\pi$
$642$	0	0
$643$	−4.65536e23	−0.628289	−0.314145	−	0.949375i	$-0.601718\pi$
$643$	−4.65536e23	−0.628289	−0.314145	+	0.949375i	$0.601718\pi$
$644$	1.51100e23	0.201249
$645$	0	0
$646$	4.26452e21	0.00553215
$647$	−1.00412e24	−1.28558	−0.642788	−	0.766045i	$-0.722222\pi$
$647$	−1.00412e24	−1.28558	−0.642788	+	0.766045i	$0.722222\pi$
$648$	0	0
$649$	−2.46396e22	−0.0307294
$650$	3.88321e23	0.478000
$651$	0	0
$652$	1.18324e24	1.41895
$653$	−1.26098e23	−0.149261	−0.0746304	−	0.997211i	$-0.523778\pi$
$653$	−1.26098e23	−0.149261	−0.0746304	+	0.997211i	$0.523778\pi$
$654$	0	0
$655$	−1.35190e23	−0.155917
$656$	7.19375e23	0.818979
$657$	0	0
$658$	8.39824e22	0.0931683
$659$	1.06105e24	1.16201	0.581005	−	0.813900i	$-0.302660\pi$
$659$	1.06105e24	1.16201	0.581005	+	0.813900i	$0.302660\pi$
$660$	0	0
$661$	1.57511e24	1.68112	0.840560	−	0.541719i	$-0.182226\pi$
$661$	1.57511e24	1.68112	0.840560	+	0.541719i	$0.182226\pi$
$662$	2.11448e24	2.22797
$663$	0	0
$664$	2.61528e23	0.268590
$665$	4.92271e21	0.00499137
$666$	0	0
$667$	−7.71615e23	−0.762660
$668$	1.27251e24	1.24182
$669$	0	0
$670$	2.81449e24	2.67771
$671$	−9.85948e23	−0.926217
$672$	0	0
$673$	2.91330e23	0.266844	0.133422	−	0.991059i	$-0.457403\pi$
$673$	2.91330e23	0.266844	0.133422	+	0.991059i	$0.457403\pi$
$674$	−5.63657e23	−0.509807
$675$	0	0
$676$	−3.23993e23	−0.285751
$677$	−1.33861e24	−1.16587	−0.582937	−	0.812517i	$-0.698097\pi$
$677$	−1.33861e24	−1.16587	−0.582937	+	0.812517i	$0.698097\pi$
$678$	0	0
$679$	−2.05721e23	−0.174737
$680$	4.89385e22	0.0410511
$681$	0	0
$682$	−3.52140e24	−2.88103
$683$	1.05295e24	0.850812	0.425406	−	0.905003i	$-0.360131\pi$
$683$	1.05295e24	0.850812	0.425406	+	0.905003i	$0.360131\pi$
$684$	0	0
$685$	−1.91966e24	−1.51305
$686$	−7.74764e23	−0.603135
$687$	0	0
$688$	1.33999e24	1.01765
$689$	−5.82932e22	−0.0437275
$690$	0	0
$691$	−5.71311e22	−0.0418128	−0.0209064	−	0.999781i	$-0.506655\pi$
$691$	−5.71311e22	−0.0418128	−0.0209064	+	0.999781i	$0.506655\pi$
$692$	−3.41850e23	−0.247134
$693$	0	0
$694$	8.54519e23	0.602790
$695$	1.62809e24	1.13451
$696$	0	0
$697$	−2.65320e23	−0.180423
$698$	−2.95721e24	−1.98660
$699$	0	0
$700$	1.37918e23	0.0904248
$701$	3.18403e23	0.206241	0.103120	−	0.994669i	$-0.467117\pi$
$701$	3.18403e23	0.206241	0.103120	+	0.994669i	$0.467117\pi$
$702$	0	0
$703$	1.88788e22	0.0119359
$704$	−2.09705e24	−1.30991
$705$	0	0
$706$	−2.69689e24	−1.64446
$707$	5.18131e23	0.312158
$708$	0	0
$709$	2.12997e24	1.25279	0.626397	−	0.779504i	$-0.284529\pi$
$709$	2.12997e24	1.25279	0.626397	+	0.779504i	$0.284529\pi$
$710$	2.00924e24	1.16771
$711$	0	0
$712$	−2.56261e23	−0.145413
$713$	−2.80617e24	−1.57345
$714$	0	0
$715$	1.96450e24	1.07560
$716$	1.16738e24	0.631613
$717$	0	0
$718$	−3.08861e24	−1.63195
$719$	2.34234e24	1.22308	0.611540	−	0.791213i	$-0.290550\pi$
$719$	2.34234e24	1.22308	0.611540	+	0.791213i	$0.290550\pi$
$720$	0	0
$721$	−2.92641e23	−0.149240
$722$	−2.89248e24	−1.45782
$723$	0	0
$724$	2.07873e24	1.02334
$725$	−7.04299e23	−0.342676
$726$	0	0
$727$	3.58185e24	1.70241	0.851207	−	0.524830i	$-0.175871\pi$
$727$	3.58185e24	1.70241	0.851207	+	0.524830i	$0.175871\pi$
$728$	7.20222e22	0.0338337
$729$	0	0
$730$	1.84289e24	0.845776
$731$	−4.94215e23	−0.224191
$732$	0	0
$733$	−3.29013e24	−1.45824	−0.729119	−	0.684387i	$-0.760070\pi$
$733$	−3.29013e24	−1.45824	−0.729119	+	0.684387i	$0.760070\pi$
$734$	2.26794e23	0.0993609
$735$	0	0
$736$	−2.82997e24	−1.21149
$737$	3.91591e24	1.65714
$738$	0	0
$739$	1.74130e24	0.720103	0.360052	−	0.932932i	$-0.382759\pi$
$739$	1.74130e24	0.720103	0.360052	+	0.932932i	$0.382759\pi$
$740$	1.92317e24	0.786227
$741$	0	0
$742$	−3.90750e22	−0.0156123
$743$	4.61544e24	1.82309	0.911545	−	0.411199i	$-0.134890\pi$
$743$	4.61544e24	1.82309	0.911545	+	0.411199i	$0.134890\pi$
$744$	0	0
$745$	4.89576e23	0.189013
$746$	−3.45700e24	−1.31953
$747$	0	0
$748$	6.04431e23	0.225519
$749$	−6.30841e23	−0.232715
$750$	0	0
$751$	−8.19607e22	−0.0295574	−0.0147787	−	0.999891i	$-0.504704\pi$
$751$	−8.19607e22	−0.0295574	−0.0147787	+	0.999891i	$0.504704\pi$
$752$	−7.25865e23	−0.258824
$753$	0	0
$754$	−3.26486e24	−1.13817
$755$	−4.03122e24	−1.38959
$756$	0	0
$757$	−2.51608e24	−0.848025	−0.424013	−	0.905656i	$-0.639379\pi$
$757$	−2.51608e24	−0.848025	−0.424013	+	0.905656i	$0.639379\pi$
$758$	8.01767e23	0.267215
$759$	0	0
$760$	1.34069e22	0.00436931
$761$	5.27745e23	0.170081	0.0850403	−	0.996378i	$-0.472898\pi$
$761$	5.27745e23	0.170081	0.0850403	+	0.996378i	$0.472898\pi$
$762$	0	0
$763$	6.89574e23	0.217331
$764$	4.17287e24	1.30059
$765$	0	0
$766$	4.24827e24	1.29499
$767$	−8.30803e22	−0.0250460
$768$	0	0
$769$	−7.51719e23	−0.221657	−0.110828	−	0.993840i	$-0.535350\pi$
$769$	−7.51719e23	−0.221657	−0.110828	+	0.993840i	$0.535350\pi$
$770$	1.31684e24	0.384027
$771$	0	0
$772$	7.26381e24	2.07213
$773$	−2.81914e24	−0.795409	−0.397704	−	0.917514i	$-0.630193\pi$
$773$	−2.81914e24	−0.795409	−0.397704	+	0.917514i	$0.630193\pi$
$774$	0	0
$775$	−2.56136e24	−0.706980
$776$	−5.60277e23	−0.152960
$777$	0	0
$778$	−3.61173e24	−0.964693
$779$	−7.26854e22	−0.0192034
$780$	0	0
$781$	2.79554e24	0.722656
$782$	9.09072e23	0.232456
$783$	0	0
$784$	3.27156e24	0.818594
$785$	−2.35385e24	−0.582621
$786$	0	0
$787$	−3.20732e24	−0.776887	−0.388443	−	0.921473i	$-0.626987\pi$
$787$	−3.20732e24	−0.776887	−0.388443	+	0.921473i	$0.626987\pi$
$788$	−1.91224e24	−0.458216
$789$	0	0
$790$	1.28183e24	0.300608
$791$	9.07739e23	0.210601
$792$	0	0
$793$	−3.32444e24	−0.754912
$794$	3.29186e24	0.739549
$795$	0	0
$796$	−8.15008e24	−1.79226
$797$	−2.36328e24	−0.514186	−0.257093	−	0.966387i	$-0.582765\pi$
$797$	−2.36328e24	−0.514186	−0.257093	+	0.966387i	$0.582765\pi$
$798$	0	0
$799$	2.67713e23	0.0570194
$800$	−2.58309e24	−0.544345
$801$	0	0
$802$	1.22521e24	0.252772
$803$	2.56409e24	0.523421
$804$	0	0
$805$	1.04938e24	0.209733
$806$	−1.18735e25	−2.34818
$807$	0	0
$808$	1.41112e24	0.273254
$809$	−9.60721e24	−1.84092	−0.920460	−	0.390836i	$-0.872186\pi$
$809$	−9.60721e24	−1.84092	−0.920460	+	0.390836i	$0.872186\pi$
$810$	0	0
$811$	−6.64616e24	−1.24708	−0.623539	−	0.781792i	$-0.714306\pi$
$811$	−6.64616e24	−1.24708	−0.623539	+	0.781792i	$0.714306\pi$
$812$	−1.15956e24	−0.215312
$813$	0	0
$814$	5.05014e24	0.918324
$815$	8.21751e24	1.47877
$816$	0	0
$817$	−1.35392e23	−0.0238620
$818$	1.25976e25	2.19727
$819$	0	0
$820$	−7.40440e24	−1.26495
$821$	5.09458e24	0.861373	0.430687	−	0.902502i	$-0.358271\pi$
$821$	5.09458e24	0.861373	0.430687	+	0.902502i	$0.358271\pi$
$822$	0	0
$823$	5.78198e24	0.957586	0.478793	−	0.877928i	$-0.341074\pi$
$823$	5.78198e24	0.957586	0.478793	+	0.877928i	$0.341074\pi$
$824$	−7.97002e23	−0.130641
$825$	0	0
$826$	−5.56903e22	−0.00894230
$827$	−8.38907e24	−1.33327	−0.666633	−	0.745386i	$-0.732265\pi$
$827$	−8.38907e24	−1.33327	−0.666633	+	0.745386i	$0.732265\pi$
$828$	0	0
$829$	5.02769e24	0.782808	0.391404	−	0.920219i	$-0.371989\pi$
$829$	5.02769e24	0.782808	0.391404	+	0.920219i	$0.371989\pi$
$830$	1.61231e25	2.48477
$831$	0	0
$832$	−7.07088e24	−1.06764
$833$	−1.20662e24	−0.180338
$834$	0	0
$835$	8.83747e24	1.29417
$836$	1.65586e23	0.0240033
$837$	0	0
$838$	1.38251e25	1.96378
$839$	−9.38328e23	−0.131940	−0.0659702	−	0.997822i	$-0.521014\pi$
$839$	−9.38328e23	−0.131940	−0.0659702	+	0.997822i	$0.521014\pi$
$840$	0	0
$841$	−1.33566e24	−0.184048
$842$	5.90847e24	0.805977
$843$	0	0
$844$	2.56295e24	0.342633
$845$	−2.25011e24	−0.297798
$846$	0	0
$847$	2.01603e23	0.0261510
$848$	3.37728e23	0.0433713
$849$	0	0
$850$	8.29764e23	0.104447
$851$	4.02442e24	0.501536
$852$	0	0
$853$	2.33188e24	0.284865	0.142433	−	0.989805i	$-0.454508\pi$
$853$	2.33188e24	0.284865	0.142433	+	0.989805i	$0.454508\pi$
$854$	−2.22843e24	−0.269531
$855$	0	0
$856$	−1.71808e24	−0.203712
$857$	1.09008e25	1.27974	0.639872	−	0.768482i	$-0.278988\pi$
$857$	1.09008e25	1.27974	0.639872	+	0.768482i	$0.278988\pi$
$858$	0	0
$859$	−1.47838e25	−1.70155	−0.850773	−	0.525533i	$-0.823866\pi$
$859$	−1.47838e25	−1.70155	−0.850773	+	0.525533i	$0.823866\pi$
$860$	−1.37923e25	−1.57181
$861$	0	0
$862$	−9.20284e24	−1.02828
$863$	−5.86087e24	−0.648441	−0.324220	−	0.945982i	$-0.605102\pi$
$863$	−5.86087e24	−0.648441	−0.324220	+	0.945982i	$0.605102\pi$
$864$	0	0
$865$	−2.37412e24	−0.257553
$866$	−2.38346e25	−2.56039
$867$	0	0
$868$	−4.21705e24	−0.444213
$869$	1.78346e24	0.186036
$870$	0	0
$871$	1.32037e25	1.35065
$872$	1.87804e24	0.190246
$873$	0	0
$874$	2.49044e23	0.0247417
$875$	−1.56703e24	−0.154173
$876$	0	0
$877$	−1.08364e25	−1.04565	−0.522827	−	0.852439i	$-0.675122\pi$
$877$	−1.08364e25	−1.04565	−0.522827	+	0.852439i	$0.675122\pi$
$878$	2.03420e25	1.94398
$879$	0	0
$880$	−1.13815e25	−1.06684
$881$	−5.00054e24	−0.464217	−0.232109	−	0.972690i	$-0.574563\pi$
$881$	−5.00054e24	−0.464217	−0.232109	+	0.972690i	$0.574563\pi$
$882$	0	0
$883$	−1.04108e24	−0.0948025	−0.0474012	−	0.998876i	$-0.515094\pi$
$883$	−1.04108e24	−0.0948025	−0.0474012	+	0.998876i	$0.515094\pi$
$884$	2.03803e24	0.183809
$885$	0	0
$886$	2.03262e24	0.179833
$887$	−1.47651e25	−1.29386	−0.646929	−	0.762550i	$-0.723947\pi$
$887$	−1.47651e25	−1.29386	−0.646929	+	0.762550i	$0.723947\pi$
$888$	0	0
$889$	−2.80760e24	−0.241363
$890$	−1.57984e25	−1.34523
$891$	0	0
$892$	1.40092e25	1.17034
$893$	7.33412e22	0.00606891
$894$	0	0
$895$	8.10737e24	0.658240
$896$	−9.66021e23	−0.0776906
$897$	0	0
$898$	1.46580e24	0.115672
$899$	2.15350e25	1.68340
$900$	0	0
$901$	−1.24561e23	−0.00955478
$902$	−1.94436e25	−1.47747
$903$	0	0
$904$	2.47221e24	0.184354
$905$	1.44366e25	1.06648
$906$	0	0
$907$	−2.99747e23	−0.0217317	−0.0108658	−	0.999941i	$-0.503459\pi$
$907$	−2.99747e23	−0.0217317	−0.0108658	+	0.999941i	$0.503459\pi$
$908$	2.71468e25	1.94979
$909$	0	0
$910$	4.44014e24	0.313000
$911$	1.71431e24	0.119724	0.0598621	−	0.998207i	$-0.480934\pi$
$911$	1.71431e24	0.119724	0.0598621	+	0.998207i	$0.480934\pi$
$912$	0	0
$913$	2.24328e25	1.53773
$914$	−7.08256e24	−0.481003
$915$	0	0
$916$	−1.35143e25	−0.900911
$917$	−4.25132e23	−0.0280792
$918$	0	0
$919$	−2.42539e25	−1.57253	−0.786265	−	0.617889i	$-0.787988\pi$
$919$	−2.42539e25	−1.57253	−0.786265	+	0.617889i	$0.787988\pi$
$920$	2.85796e24	0.183594
$921$	0	0
$922$	2.28480e25	1.44091
$923$	9.42605e24	0.589000
$924$	0	0
$925$	3.67333e24	0.225349
$926$	2.03661e25	1.23798
$927$	0	0
$928$	2.17176e25	1.29615
$929$	−1.40138e25	−0.828745	−0.414373	−	0.910107i	$-0.635999\pi$
$929$	−1.40138e25	−0.828745	−0.414373	+	0.910107i	$0.635999\pi$
$930$	0	0
$931$	−3.30557e23	−0.0191944
$932$	−1.45865e25	−0.839298
$933$	0	0
$934$	−2.62270e25	−1.48184
$935$	4.19773e24	0.235026
$936$	0	0
$937$	−2.43217e25	−1.33724	−0.668618	−	0.743606i	$-0.733114\pi$
$937$	−2.43217e25	−1.33724	−0.668618	+	0.743606i	$0.733114\pi$
$938$	8.85072e24	0.482230
$939$	0	0
$940$	7.47120e24	0.399764
$941$	−5.98394e23	−0.0317304	−0.0158652	−	0.999874i	$-0.505050\pi$
$941$	−5.98394e23	−0.0317304	−0.0158652	+	0.999874i	$0.505050\pi$
$942$	0	0
$943$	−1.54944e25	−0.806912
$944$	4.81335e23	0.0248419
$945$	0	0
$946$	−3.62178e25	−1.83589
$947$	−3.62477e25	−1.82098	−0.910491	−	0.413530i	$-0.864296\pi$
$947$	−3.62477e25	−1.82098	−0.910491	+	0.413530i	$0.864296\pi$
$948$	0	0
$949$	8.64565e24	0.426614
$950$	2.27317e23	0.0111169
$951$	0	0
$952$	1.53897e23	0.00739291
$953$	1.88157e25	0.895839	0.447919	−	0.894074i	$-0.352165\pi$
$953$	1.88157e25	0.895839	0.447919	+	0.894074i	$0.352165\pi$
$954$	0	0
$955$	2.89803e25	1.35542
$956$	−2.79673e25	−1.29646
$957$	0	0
$958$	2.25199e25	1.02556
$959$	−6.03674e24	−0.272486
$960$	0	0
$961$	5.57675e25	2.47305
$962$	1.70282e25	0.748479
$963$	0	0
$964$	−2.04968e25	−0.885180
$965$	5.04467e25	2.15949
$966$	0	0
$967$	−1.81212e24	−0.0762186	−0.0381093	−	0.999274i	$-0.512134\pi$
$967$	−1.81212e24	−0.0762186	−0.0381093	+	0.999274i	$0.512134\pi$
$968$	5.49062e23	0.0228919
$969$	0	0
$970$	−3.45409e25	−1.41506
$971$	−2.03685e25	−0.827170	−0.413585	−	0.910465i	$-0.635724\pi$
$971$	−2.03685e25	−0.827170	−0.413585	+	0.910465i	$0.635724\pi$
$972$	0	0
$973$	5.11985e24	0.204314
$974$	−4.90050e25	−1.93861
$975$	0	0
$976$	1.92605e25	0.748763
$977$	2.52590e25	0.973449	0.486724	−	0.873556i	$-0.338192\pi$
$977$	2.52590e25	0.973449	0.486724	+	0.873556i	$0.338192\pi$
$978$	0	0
$979$	−2.19809e25	−0.832517
$980$	−3.36736e25	−1.26435
$981$	0	0
$982$	−6.85750e23	−0.0253057
$983$	1.51840e24	0.0555495	0.0277747	−	0.999614i	$-0.491158\pi$
$983$	1.51840e24	0.0555495	0.0277747	+	0.999614i	$0.491158\pi$
$984$	0	0
$985$	−1.32804e25	−0.477533
$986$	−6.97635e24	−0.248699
$987$	0	0
$988$	5.58327e23	0.0195639
$989$	−2.88617e25	−1.00266
$990$	0	0
$991$	−6.69650e24	−0.228677	−0.114338	−	0.993442i	$-0.536475\pi$
$991$	−6.69650e24	−0.228677	−0.114338	+	0.993442i	$0.536475\pi$
$992$	7.89818e25	2.67410
$993$	0	0
$994$	6.31846e24	0.210294
$995$	−5.66018e25	−1.86781
$996$	0	0
$997$	−3.45326e25	−1.12026	−0.560132	−	0.828404i	$-0.689249\pi$
$997$	−3.45326e25	−1.12026	−0.560132	+	0.828404i	$0.689249\pi$
$998$	−4.04864e25	−1.30227
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.18.a.b.1.1		1
3.2	odd	2		1.18.a.a.1.1	✓	1
12.11	even	2		16.18.a.b.1.1		1
15.2	even	4		25.18.b.a.24.1		2
15.8	even	4		25.18.b.a.24.2		2
15.14	odd	2		25.18.a.a.1.1		1
21.20	even	2		49.18.a.a.1.1		1
24.5	odd	2		64.18.a.d.1.1		1
24.11	even	2		64.18.a.b.1.1		1
33.32	even	2		121.18.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.18.a.a.1.1	✓	1	3.2	odd	2
9.18.a.b.1.1		1	1.1	even	1	trivial
16.18.a.b.1.1		1	12.11	even	2
25.18.a.a.1.1		1	15.14	odd	2
25.18.b.a.24.1		2	15.2	even	4
25.18.b.a.24.2		2	15.8	even	4
49.18.a.a.1.1		1	21.20	even	2
64.18.a.b.1.1		1	24.11	even	2
64.18.a.d.1.1		1	24.5	odd	2
121.18.a.b.1.1		1	33.32	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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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