LMFDB - Embedded newform 2151.2.a.j.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-1.25296$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25296 q^{2} -0.430099 q^{4} +1.67805 q^{5} -0.437804 q^{7} -3.04481 q^{8} +O(q^{10})$	\(q+1.25296 q^{2} -0.430099 q^{4} +1.67805 q^{5} -0.437804 q^{7} -3.04481 q^{8} +2.10252 q^{10} -3.01794 q^{11} +3.79768 q^{13} -0.548550 q^{14} -2.95482 q^{16} -4.87279 q^{17} -7.63286 q^{19} -0.721726 q^{20} -3.78135 q^{22} -0.261247 q^{23} -2.18416 q^{25} +4.75833 q^{26} +0.188299 q^{28} -0.689712 q^{29} +8.79905 q^{31} +2.38736 q^{32} -6.10540 q^{34} -0.734656 q^{35} -6.63032 q^{37} -9.56364 q^{38} -5.10934 q^{40} +0.687470 q^{41} -7.99902 q^{43} +1.29801 q^{44} -0.327332 q^{46} -0.785372 q^{47} -6.80833 q^{49} -2.73665 q^{50} -1.63338 q^{52} -12.0733 q^{53} -5.06424 q^{55} +1.33303 q^{56} -0.864179 q^{58} +11.3147 q^{59} -1.44871 q^{61} +11.0248 q^{62} +8.90089 q^{64} +6.37269 q^{65} +6.28276 q^{67} +2.09578 q^{68} -0.920493 q^{70} -11.4266 q^{71} +9.10142 q^{73} -8.30751 q^{74} +3.28288 q^{76} +1.32127 q^{77} +5.17440 q^{79} -4.95832 q^{80} +0.861370 q^{82} +1.59126 q^{83} -8.17678 q^{85} -10.0224 q^{86} +9.18904 q^{88} -16.4673 q^{89} -1.66264 q^{91} +0.112362 q^{92} -0.984037 q^{94} -12.8083 q^{95} +2.60081 q^{97} -8.53054 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) $ 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8	$ 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8} + 4 q^{10} - 12 q^{11} - 4 q^{13} - 20 q^{14} + 22 q^{16} - 24 q^{17} - 4 q^{19} - 40 q^{20} - 6 q^{22} - 12 q^{23} + 22 q^{25} - 30 q^{26} - 12 q^{28} - 24 q^{29} - 4 q^{31} - 28 q^{32} + 8 q^{34} - 20 q^{35} - 10 q^{37} - 26 q^{38} + 6 q^{40} - 66 q^{41} + 8 q^{43} - 36 q^{44} - 12 q^{46} - 28 q^{47} + 18 q^{49} - 28 q^{50} - 18 q^{52} - 28 q^{53} - 4 q^{55} - 60 q^{56} - 54 q^{59} - 4 q^{61} - 20 q^{62} + 22 q^{64} - 42 q^{65} + 12 q^{67} - 12 q^{68} + 20 q^{70} - 36 q^{71} + 14 q^{73} - 50 q^{76} - 8 q^{77} - 12 q^{79} - 88 q^{80} - 8 q^{82} - 20 q^{83} + 4 q^{85} - 18 q^{86} - 10 q^{88} - 130 q^{89} - 6 q^{91} + 46 q^{92} - 26 q^{94} - 2 q^{97} - 12 q^{98}+O(q^{100}) $ 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8 + 4 * q^10 - 12 * q^11 - 4 * q^13 - 20 * q^14 + 22 * q^16 - 24 * q^17 - 4 * q^19 - 40 * q^20 - 6 * q^22 - 12 * q^23 + 22 * q^25 - 30 * q^26 - 12 * q^28 - 24 * q^29 - 4 * q^31 - 28 * q^32 + 8 * q^34 - 20 * q^35 - 10 * q^37 - 26 * q^38 + 6 * q^40 - 66 * q^41 + 8 * q^43 - 36 * q^44 - 12 * q^46 - 28 * q^47 + 18 * q^49 - 28 * q^50 - 18 * q^52 - 28 * q^53 - 4 * q^55 - 60 * q^56 - 54 * q^59 - 4 * q^61 - 20 * q^62 + 22 * q^64 - 42 * q^65 + 12 * q^67 - 12 * q^68 + 20 * q^70 - 36 * q^71 + 14 * q^73 - 50 * q^76 - 8 * q^77 - 12 * q^79 - 88 * q^80 - 8 * q^82 - 20 * q^83 + 4 * q^85 - 18 * q^86 - 10 * q^88 - 130 * q^89 - 6 * q^91 + 46 * q^92 - 26 * q^94 - 2 * q^97 - 12 * q^98

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 $

$2$	1.25296	0.885974	0.442987	−	0.896528i	$-0.353919\pi$
$2$	1.25296	0.885974	0.442987	+	0.896528i	$0.353919\pi$
$3$	0	0
$3$	0	0
$4$	−0.430099	−0.215049
$5$	1.67805	0.750446	0.375223	−	0.926935i	$-0.377566\pi$
$5$	1.67805	0.750446	0.375223	+	0.926935i	$0.377566\pi$
$6$	0	0
$7$	−0.437804	−0.165474	−0.0827372	−	0.996571i	$-0.526366\pi$
$7$	−0.437804	−0.165474	−0.0827372	+	0.996571i	$0.526366\pi$
$8$	−3.04481	−1.07650
$9$	0	0
$10$	2.10252	0.664876
$11$	−3.01794	−0.909942	−0.454971	−	0.890506i	$-0.650350\pi$
$11$	−3.01794	−0.909942	−0.454971	+	0.890506i	$0.650350\pi$
$12$	0	0
$13$	3.79768	1.05329	0.526643	−	0.850086i	$-0.323450\pi$
$13$	3.79768	1.05329	0.526643	+	0.850086i	$0.323450\pi$
$14$	−0.548550	−0.146606
$15$	0	0
$16$	−2.95482	−0.738704
$17$	−4.87279	−1.18183	−0.590913	−	0.806735i	$-0.701232\pi$
$17$	−4.87279	−1.18183	−0.590913	+	0.806735i	$0.701232\pi$
$18$	0	0
$19$	−7.63286	−1.75110	−0.875549	−	0.483129i	$-0.839500\pi$
$19$	−7.63286	−1.75110	−0.875549	+	0.483129i	$0.839500\pi$
$20$	−0.721726	−0.161383
$21$	0	0
$22$	−3.78135	−0.806186
$23$	−0.261247	−0.0544738	−0.0272369	−	0.999629i	$-0.508671\pi$
$23$	−0.261247	−0.0544738	−0.0272369	+	0.999629i	$0.508671\pi$
$24$	0	0
$25$	−2.18416	−0.436831
$26$	4.75833	0.933185
$27$	0	0
$28$	0.188299	0.0355852
$29$	−0.689712	−0.128076	−0.0640381	−	0.997947i	$-0.520398\pi$
$29$	−0.689712	−0.128076	−0.0640381	+	0.997947i	$0.520398\pi$
$30$	0	0
$31$	8.79905	1.58036	0.790178	−	0.612877i	$-0.209988\pi$
$31$	8.79905	1.58036	0.790178	+	0.612877i	$0.209988\pi$
$32$	2.38736	0.422030
$33$	0	0
$34$	−6.10540	−1.04707
$35$	−0.734656	−0.124180
$36$	0	0
$37$	−6.63032	−1.09002	−0.545009	−	0.838430i	$-0.683474\pi$
$37$	−6.63032	−1.09002	−0.545009	+	0.838430i	$0.683474\pi$
$38$	−9.56364	−1.55143
$39$	0	0
$40$	−5.10934	−0.807857
$41$	0.687470	0.107365	0.0536824	−	0.998558i	$-0.482904\pi$
$41$	0.687470	0.107365	0.0536824	+	0.998558i	$0.482904\pi$
$42$	0	0
$43$	−7.99902	−1.21984	−0.609920	−	0.792463i	$-0.708798\pi$
$43$	−7.99902	−1.21984	−0.609920	+	0.792463i	$0.708798\pi$
$44$	1.29801	0.195683
$45$	0	0
$46$	−0.327332	−0.0482624
$47$	−0.785372	−0.114558	−0.0572791	−	0.998358i	$-0.518243\pi$
$47$	−0.785372	−0.114558	−0.0572791	+	0.998358i	$0.518243\pi$
$48$	0	0
$49$	−6.80833	−0.972618
$50$	−2.73665	−0.387021
$51$	0	0
$52$	−1.63338	−0.226509
$53$	−12.0733	−1.65840	−0.829199	−	0.558954i	$-0.811203\pi$
$53$	−12.0733	−1.65840	−0.829199	+	0.558954i	$0.811203\pi$
$54$	0	0
$55$	−5.06424	−0.682862
$56$	1.33303	0.178134
$57$	0	0
$58$	−0.864179	−0.113472
$59$	11.3147	1.47305	0.736525	−	0.676411i	$-0.236466\pi$
$59$	11.3147	1.47305	0.736525	+	0.676411i	$0.236466\pi$
$60$	0	0
$61$	−1.44871	−0.185488	−0.0927439	−	0.995690i	$-0.529564\pi$
$61$	−1.44871	−0.185488	−0.0927439	+	0.995690i	$0.529564\pi$
$62$	11.0248	1.40016
$63$	0	0
$64$	8.90089	1.11261
$65$	6.37269	0.790434
$66$	0	0
$67$	6.28276	0.767561	0.383780	−	0.923424i	$-0.374622\pi$
$67$	6.28276	0.767561	0.383780	+	0.923424i	$0.374622\pi$
$68$	2.09578	0.254151
$69$	0	0
$70$	−0.920493	−0.110020
$71$	−11.4266	−1.35609	−0.678044	−	0.735022i	$-0.737172\pi$
$71$	−11.4266	−1.35609	−0.678044	+	0.735022i	$0.737172\pi$
$72$	0	0
$73$	9.10142	1.06524	0.532620	−	0.846354i	$-0.321207\pi$
$73$	9.10142	1.06524	0.532620	+	0.846354i	$0.321207\pi$
$74$	−8.30751	−0.965728
$75$	0	0
$76$	3.28288	0.376573
$77$	1.32127	0.150572
$78$	0	0
$79$	5.17440	0.582165	0.291083	−	0.956698i	$-0.405985\pi$
$79$	5.17440	0.582165	0.291083	+	0.956698i	$0.405985\pi$
$80$	−4.95832	−0.554357
$81$	0	0
$82$	0.861370	0.0951224
$83$	1.59126	0.174663	0.0873317	−	0.996179i	$-0.472166\pi$
$83$	1.59126	0.174663	0.0873317	+	0.996179i	$0.472166\pi$
$84$	0	0
$85$	−8.17678	−0.886896
$86$	−10.0224	−1.08075
$87$	0	0
$88$	9.18904	0.979555
$89$	−16.4673	−1.74553	−0.872766	−	0.488138i	$-0.837676\pi$
$89$	−16.4673	−1.74553	−0.872766	+	0.488138i	$0.837676\pi$
$90$	0	0
$91$	−1.66264	−0.174292
$92$	0.112362	0.0117146
$93$	0	0
$94$	−0.984037	−0.101496
$95$	−12.8083	−1.31410
$96$	0	0
$97$	2.60081	0.264073	0.132036	−	0.991245i	$-0.457848\pi$
$97$	2.60081	0.264073	0.132036	+	0.991245i	$0.457848\pi$
$98$	−8.53054	−0.861715
$99$	0	0
$100$	0.939403	0.0939403
$101$	−6.29708	−0.626583	−0.313292	−	0.949657i	$-0.601432\pi$
$101$	−6.29708	−0.626583	−0.313292	+	0.949657i	$0.601432\pi$
$102$	0	0
$103$	2.23873	0.220589	0.110294	−	0.993899i	$-0.464821\pi$
$103$	2.23873	0.220589	0.110294	+	0.993899i	$0.464821\pi$
$104$	−11.5632	−1.13387
$105$	0	0
$106$	−15.1273	−1.46930
$107$	11.8071	1.14143	0.570716	−	0.821148i	$-0.306666\pi$
$107$	11.8071	1.14143	0.570716	+	0.821148i	$0.306666\pi$
$108$	0	0
$109$	−8.93712	−0.856020	−0.428010	−	0.903774i	$-0.640785\pi$
$109$	−8.93712	−0.856020	−0.428010	+	0.903774i	$0.640785\pi$
$110$	−6.34528	−0.604999
$111$	0	0
$112$	1.29363	0.122237
$113$	−18.4672	−1.73725	−0.868625	−	0.495470i	$-0.834996\pi$
$113$	−18.4672	−1.73725	−0.868625	+	0.495470i	$0.834996\pi$
$114$	0	0
$115$	−0.438385	−0.0408796
$116$	0.296644	0.0275427
$117$	0	0
$118$	14.1768	1.30508
$119$	2.13333	0.195562
$120$	0	0
$121$	−1.89205	−0.172005
$122$	−1.81517	−0.164337
$123$	0	0
$124$	−3.78446	−0.339855
$125$	−12.0554	−1.07826
$126$	0	0
$127$	18.2247	1.61718	0.808592	−	0.588369i	$-0.200230\pi$
$127$	18.2247	1.61718	0.808592	+	0.588369i	$0.200230\pi$
$128$	6.37772	0.563716
$129$	0	0
$130$	7.98470	0.700305
$131$	0.446703	0.0390286	0.0195143	−	0.999810i	$-0.493788\pi$
$131$	0.446703	0.0390286	0.0195143	+	0.999810i	$0.493788\pi$
$132$	0	0
$133$	3.34170	0.289762
$134$	7.87202	0.680039
$135$	0	0
$136$	14.8367	1.27224
$137$	−8.14824	−0.696152	−0.348076	−	0.937466i	$-0.613165\pi$
$137$	−8.14824	−0.696152	−0.348076	+	0.937466i	$0.613165\pi$
$138$	0	0
$139$	−7.92325	−0.672041	−0.336021	−	0.941855i	$-0.609081\pi$
$139$	−7.92325	−0.672041	−0.336021	+	0.941855i	$0.609081\pi$
$140$	0.315975	0.0267048
$141$	0	0
$142$	−14.3170	−1.20146
$143$	−11.4612	−0.958430
$144$	0	0
$145$	−1.15737	−0.0961143
$146$	11.4037	0.943776
$147$	0	0
$148$	2.85169	0.234408
$149$	5.04368	0.413194	0.206597	−	0.978426i	$-0.433761\pi$
$149$	5.04368	0.413194	0.206597	+	0.978426i	$0.433761\pi$
$150$	0	0
$151$	−14.2381	−1.15868	−0.579340	−	0.815086i	$-0.696689\pi$
$151$	−14.2381	−1.15868	−0.579340	+	0.815086i	$0.696689\pi$
$152$	23.2406	1.88506
$153$	0	0
$154$	1.65549	0.133403
$155$	14.7652	1.18597
$156$	0	0
$157$	18.0010	1.43664	0.718318	−	0.695715i	$-0.244912\pi$
$157$	18.0010	1.43664	0.718318	+	0.695715i	$0.244912\pi$
$158$	6.48330	0.515783
$159$	0	0
$160$	4.00610	0.316710
$161$	0.114375	0.00901402
$162$	0	0
$163$	10.7570	0.842556	0.421278	−	0.906932i	$-0.361582\pi$
$163$	10.7570	0.842556	0.421278	+	0.906932i	$0.361582\pi$
$164$	−0.295680	−0.0230887
$165$	0	0
$166$	1.99378	0.154747
$167$	−3.56318	−0.275727	−0.137863	−	0.990451i	$-0.544024\pi$
$167$	−3.56318	−0.275727	−0.137863	+	0.990451i	$0.544024\pi$
$168$	0	0
$169$	1.42236	0.109412
$170$	−10.2451	−0.785767
$171$	0	0
$172$	3.44037	0.262326
$173$	10.6387	0.808845	0.404423	−	0.914572i	$-0.367472\pi$
$173$	10.6387	0.808845	0.404423	+	0.914572i	$0.367472\pi$
$174$	0	0
$175$	0.956233	0.0722844
$176$	8.91745	0.672178
$177$	0	0
$178$	−20.6328	−1.54650
$179$	−2.79518	−0.208922	−0.104461	−	0.994529i	$-0.533312\pi$
$179$	−2.79518	−0.208922	−0.104461	+	0.994529i	$0.533312\pi$
$180$	0	0
$181$	2.45006	0.182112	0.0910559	−	0.995846i	$-0.470976\pi$
$181$	2.45006	0.182112	0.0910559	+	0.995846i	$0.470976\pi$
$182$	−2.08322	−0.154418
$183$	0	0
$184$	0.795448	0.0586412
$185$	−11.1260	−0.817999
$186$	0	0
$187$	14.7058	1.07539
$188$	0.337788	0.0246357
$189$	0	0
$190$	−16.0483	−1.16426
$191$	17.0298	1.23223	0.616116	−	0.787656i	$-0.288705\pi$
$191$	17.0298	1.23223	0.616116	+	0.787656i	$0.288705\pi$
$192$	0	0
$193$	−5.17425	−0.372451	−0.186225	−	0.982507i	$-0.559625\pi$
$193$	−5.17425	−0.372451	−0.186225	+	0.982507i	$0.559625\pi$
$194$	3.25871	0.233961
$195$	0	0
$196$	2.92825	0.209161
$197$	−20.2790	−1.44482	−0.722411	−	0.691464i	$-0.756966\pi$
$197$	−20.2790	−1.44482	−0.722411	+	0.691464i	$0.756966\pi$
$198$	0	0
$199$	18.8206	1.33415	0.667077	−	0.744988i	$-0.267545\pi$
$199$	18.8206	1.33415	0.667077	+	0.744988i	$0.267545\pi$
$200$	6.65034	0.470250
$201$	0	0
$202$	−7.88998	−0.555137
$203$	0.301959	0.0211933
$204$	0	0
$205$	1.15361	0.0805714
$206$	2.80503	0.195436
$207$	0	0
$208$	−11.2214	−0.778067
$209$	23.0355	1.59340
$210$	0	0
$211$	27.8687	1.91856	0.959280	−	0.282458i	$-0.0911497\pi$
$211$	27.8687	1.91856	0.959280	+	0.282458i	$0.0911497\pi$
$212$	5.19272	0.356638
$213$	0	0
$214$	14.7937	1.01128
$215$	−13.4227	−0.915423
$216$	0	0
$217$	−3.85226	−0.261509
$218$	−11.1978	−0.758412
$219$	0	0
$220$	2.17813	0.146849
$221$	−18.5053	−1.24480
$222$	0	0
$223$	−12.8425	−0.859997	−0.429999	−	0.902830i	$-0.641486\pi$
$223$	−12.8425	−0.859997	−0.429999	+	0.902830i	$0.641486\pi$
$224$	−1.04520	−0.0698351
$225$	0	0
$226$	−23.1386	−1.53916
$227$	−3.74994	−0.248892	−0.124446	−	0.992226i	$-0.539715\pi$
$227$	−3.74994	−0.248892	−0.124446	+	0.992226i	$0.539715\pi$
$228$	0	0
$229$	−4.60000	−0.303977	−0.151988	−	0.988382i	$-0.548568\pi$
$229$	−4.60000	−0.303977	−0.151988	+	0.988382i	$0.548568\pi$
$230$	−0.549278	−0.0362183
$231$	0	0
$232$	2.10004	0.137874
$233$	−28.5275	−1.86890	−0.934449	−	0.356097i	$-0.884107\pi$
$233$	−28.5275	−1.86890	−0.934449	+	0.356097i	$0.884107\pi$
$234$	0	0
$235$	−1.31789	−0.0859698
$236$	−4.86644	−0.316778
$237$	0	0
$238$	2.67297	0.173263
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	13.0672	0.841732	0.420866	−	0.907123i	$-0.361726\pi$
$241$	13.0672	0.841732	0.420866	+	0.907123i	$0.361726\pi$
$242$	−2.37066	−0.152392
$243$	0	0
$244$	0.623087	0.0398890
$245$	−11.4247	−0.729897
$246$	0	0
$247$	−28.9871	−1.84441
$248$	−26.7914	−1.70126
$249$	0	0
$250$	−15.1048	−0.955314
$251$	13.0772	0.825426	0.412713	−	0.910861i	$-0.364581\pi$
$251$	13.0772	0.825426	0.412713	+	0.910861i	$0.364581\pi$
$252$	0	0
$253$	0.788428	0.0495680
$254$	22.8348	1.43278
$255$	0	0
$256$	−9.81078	−0.613174
$257$	22.4358	1.39951	0.699753	−	0.714385i	$-0.253294\pi$
$257$	22.4358	1.39951	0.699753	+	0.714385i	$0.253294\pi$
$258$	0	0
$259$	2.90278	0.180370
$260$	−2.74088	−0.169982
$261$	0	0
$262$	0.559699	0.0345783
$263$	6.21793	0.383414	0.191707	−	0.981452i	$-0.438598\pi$
$263$	6.21793	0.383414	0.191707	+	0.981452i	$0.438598\pi$
$264$	0	0
$265$	−20.2596	−1.24454
$266$	4.18700	0.256722
$267$	0	0
$268$	−2.70221	−0.165064
$269$	18.5021	1.12809	0.564045	−	0.825744i	$-0.309244\pi$
$269$	18.5021	1.12809	0.564045	+	0.825744i	$0.309244\pi$
$270$	0	0
$271$	24.5430	1.49088	0.745442	−	0.666571i	$-0.232239\pi$
$271$	24.5430	1.49088	0.745442	+	0.666571i	$0.232239\pi$
$272$	14.3982	0.873020
$273$	0	0
$274$	−10.2094	−0.616772
$275$	6.59165	0.397491
$276$	0	0
$277$	1.77904	0.106892	0.0534462	−	0.998571i	$-0.482979\pi$
$277$	1.77904	0.106892	0.0534462	+	0.998571i	$0.482979\pi$
$278$	−9.92749	−0.595411
$279$	0	0
$280$	2.23689	0.133680
$281$	−12.6890	−0.756963	−0.378482	−	0.925609i	$-0.623554\pi$
$281$	−12.6890	−0.756963	−0.378482	+	0.925609i	$0.623554\pi$
$282$	0	0
$283$	24.2666	1.44250	0.721250	−	0.692675i	$-0.243568\pi$
$283$	24.2666	1.44250	0.721250	+	0.692675i	$0.243568\pi$
$284$	4.91456	0.291626
$285$	0	0
$286$	−14.3603	−0.849144
$287$	−0.300977	−0.0177661
$288$	0	0
$289$	6.74410	0.396712
$290$	−1.45013	−0.0851548
$291$	0	0
$292$	−3.91451	−0.229079
$293$	15.3635	0.897545	0.448772	−	0.893646i	$-0.351861\pi$
$293$	15.3635	0.897545	0.448772	+	0.893646i	$0.351861\pi$
$294$	0	0
$295$	18.9866	1.10544
$296$	20.1881	1.17341
$297$	0	0
$298$	6.31951	0.366080
$299$	−0.992133	−0.0573765
$300$	0	0
$301$	3.50201	0.201852
$302$	−17.8397	−1.02656
$303$	0	0
$304$	22.5537	1.29354
$305$	−2.43100	−0.139198
$306$	0	0
$307$	−5.88479	−0.335863	−0.167931	−	0.985799i	$-0.553709\pi$
$307$	−5.88479	−0.335863	−0.167931	+	0.985799i	$0.553709\pi$
$308$	−0.568275	−0.0323805
$309$	0	0
$310$	18.5002	1.05074
$311$	−19.4787	−1.10453	−0.552267	−	0.833668i	$-0.686237\pi$
$311$	−19.4787	−1.10453	−0.552267	+	0.833668i	$0.686237\pi$
$312$	0	0
$313$	16.4195	0.928087	0.464044	−	0.885812i	$-0.346398\pi$
$313$	16.4195	0.928087	0.464044	+	0.885812i	$0.346398\pi$
$314$	22.5545	1.27282
$315$	0	0
$316$	−2.22550	−0.125194
$317$	−32.9447	−1.85036	−0.925178	−	0.379533i	$-0.876085\pi$
$317$	−32.9447	−1.85036	−0.925178	+	0.379533i	$0.876085\pi$
$318$	0	0
$319$	2.08151	0.116542
$320$	14.9361	0.834955
$321$	0	0
$322$	0.143307	0.00798619
$323$	37.1933	2.06949
$324$	0	0
$325$	−8.29472	−0.460108
$326$	13.4781	0.746483
$327$	0	0
$328$	−2.09321	−0.115578
$329$	0.343839	0.0189565
$330$	0	0
$331$	29.5028	1.62162	0.810810	−	0.585310i	$-0.199027\pi$
$331$	29.5028	1.62162	0.810810	+	0.585310i	$0.199027\pi$
$332$	−0.684399	−0.0375613
$333$	0	0
$334$	−4.46451	−0.244287
$335$	10.5428	0.576013
$336$	0	0
$337$	−15.6639	−0.853268	−0.426634	−	0.904424i	$-0.640301\pi$
$337$	−15.6639	−0.853268	−0.426634	+	0.904424i	$0.640301\pi$
$338$	1.78216	0.0969367
$339$	0	0
$340$	3.51682	0.190726
$341$	−26.5550	−1.43803
$342$	0	0
$343$	6.04534	0.326418
$344$	24.3555	1.31316
$345$	0	0
$346$	13.3298	0.716616
$347$	−23.0843	−1.23923	−0.619616	−	0.784905i	$-0.712712\pi$
$347$	−23.0843	−1.23923	−0.619616	+	0.784905i	$0.712712\pi$
$348$	0	0
$349$	−8.93132	−0.478083	−0.239041	−	0.971009i	$-0.576833\pi$
$349$	−8.93132	−0.478083	−0.239041	+	0.971009i	$0.576833\pi$
$350$	1.19812	0.0640421
$351$	0	0
$352$	−7.20490	−0.384023
$353$	28.7892	1.53230	0.766148	−	0.642664i	$-0.222171\pi$
$353$	28.7892	1.53230	0.766148	+	0.642664i	$0.222171\pi$
$354$	0	0
$355$	−19.1744	−1.01767
$356$	7.08258	0.375376
$357$	0	0
$358$	−3.50225	−0.185099
$359$	−32.7177	−1.72677	−0.863387	−	0.504543i	$-0.831661\pi$
$359$	−32.7177	−1.72677	−0.863387	+	0.504543i	$0.831661\pi$
$360$	0	0
$361$	39.2605	2.06634
$362$	3.06982	0.161346
$363$	0	0
$364$	0.715099	0.0374814
$365$	15.2726	0.799405
$366$	0	0
$367$	−36.0851	−1.88363	−0.941814	−	0.336136i	$-0.890880\pi$
$367$	−36.0851	−1.88363	−0.941814	+	0.336136i	$0.890880\pi$
$368$	0.771938	0.0402400
$369$	0	0
$370$	−13.9404	−0.724727
$371$	5.28575	0.274422
$372$	0	0
$373$	−34.8068	−1.80223	−0.901114	−	0.433582i	$-0.857249\pi$
$373$	−34.8068	−1.80223	−0.901114	+	0.433582i	$0.857249\pi$
$374$	18.4257	0.952771
$375$	0	0
$376$	2.39131	0.123322
$377$	−2.61930	−0.134901
$378$	0	0
$379$	14.3593	0.737588	0.368794	−	0.929511i	$-0.379771\pi$
$379$	14.3593	0.737588	0.368794	+	0.929511i	$0.379771\pi$
$380$	5.50884	0.282597
$381$	0	0
$382$	21.3376	1.09173
$383$	8.25786	0.421957	0.210978	−	0.977491i	$-0.432335\pi$
$383$	8.25786	0.421957	0.210978	+	0.977491i	$0.432335\pi$
$384$	0	0
$385$	2.21715	0.112996
$386$	−6.48311	−0.329982
$387$	0	0
$388$	−1.11861	−0.0567886
$389$	−28.9801	−1.46935	−0.734676	−	0.678418i	$-0.762666\pi$
$389$	−28.9801	−1.46935	−0.734676	+	0.678418i	$0.762666\pi$
$390$	0	0
$391$	1.27300	0.0643786
$392$	20.7301	1.04703
$393$	0	0
$394$	−25.4088	−1.28007
$395$	8.68288	0.436883
$396$	0	0
$397$	3.66863	0.184123	0.0920615	−	0.995753i	$-0.470654\pi$
$397$	3.66863	0.184123	0.0920615	+	0.995753i	$0.470654\pi$
$398$	23.5814	1.18203
$399$	0	0
$400$	6.45378	0.322689
$401$	5.67684	0.283488	0.141744	−	0.989903i	$-0.454729\pi$
$401$	5.67684	0.283488	0.141744	+	0.989903i	$0.454729\pi$
$402$	0	0
$403$	33.4160	1.66457
$404$	2.70837	0.134746
$405$	0	0
$406$	0.378341	0.0187768
$407$	20.0099	0.991854
$408$	0	0
$409$	16.3901	0.810440	0.405220	−	0.914219i	$-0.367195\pi$
$409$	16.3901	0.810440	0.405220	+	0.914219i	$0.367195\pi$
$410$	1.44542	0.0713842
$411$	0	0
$412$	−0.962876	−0.0474375
$413$	−4.95363	−0.243752
$414$	0	0
$415$	2.67021	0.131075
$416$	9.06643	0.444518
$417$	0	0
$418$	28.8625	1.41171
$419$	−33.3282	−1.62819	−0.814094	−	0.580733i	$-0.802766\pi$
$419$	−33.3282	−1.62819	−0.814094	+	0.580733i	$0.802766\pi$
$420$	0	0
$421$	26.0005	1.26719	0.633594	−	0.773666i	$-0.281579\pi$
$421$	26.0005	1.26719	0.633594	+	0.773666i	$0.281579\pi$
$422$	34.9183	1.69979
$423$	0	0
$424$	36.7610	1.78527
$425$	10.6429	0.516258
$426$	0	0
$427$	0.634249	0.0306935
$428$	−5.07820	−0.245464
$429$	0	0
$430$	−16.8181	−0.811042
$431$	14.5215	0.699478	0.349739	−	0.936847i	$-0.386270\pi$
$431$	14.5215	0.699478	0.349739	+	0.936847i	$0.386270\pi$
$432$	0	0
$433$	8.16667	0.392465	0.196232	−	0.980557i	$-0.437129\pi$
$433$	8.16667	0.392465	0.196232	+	0.980557i	$0.437129\pi$
$434$	−4.82672	−0.231690
$435$	0	0
$436$	3.84384	0.184087
$437$	1.99406	0.0953890
$438$	0	0
$439$	−13.2750	−0.633581	−0.316791	−	0.948496i	$-0.602605\pi$
$439$	−13.2750	−0.633581	−0.316791	+	0.948496i	$0.602605\pi$
$440$	15.4197	0.735103
$441$	0	0
$442$	−23.1863	−1.10286
$443$	7.07100	0.335953	0.167977	−	0.985791i	$-0.446277\pi$
$443$	7.07100	0.335953	0.167977	+	0.985791i	$0.446277\pi$
$444$	0	0
$445$	−27.6330	−1.30993
$446$	−16.0911	−0.761936
$447$	0	0
$448$	−3.89685	−0.184109
$449$	13.1410	0.620163	0.310081	−	0.950710i	$-0.399644\pi$
$449$	13.1410	0.620163	0.310081	+	0.950710i	$0.399644\pi$
$450$	0	0
$451$	−2.07474	−0.0976957
$452$	7.94273	0.373595
$453$	0	0
$454$	−4.69852	−0.220512
$455$	−2.78999	−0.130797
$456$	0	0
$457$	30.8808	1.44454	0.722272	−	0.691609i	$-0.243098\pi$
$457$	30.8808	1.44454	0.722272	+	0.691609i	$0.243098\pi$
$458$	−5.76361	−0.269316
$459$	0	0
$460$	0.188549	0.00879115
$461$	4.10612	0.191241	0.0956205	−	0.995418i	$-0.469516\pi$
$461$	4.10612	0.191241	0.0956205	+	0.995418i	$0.469516\pi$
$462$	0	0
$463$	−33.2340	−1.54452	−0.772259	−	0.635308i	$-0.780873\pi$
$463$	−33.2340	−1.54452	−0.772259	+	0.635308i	$0.780873\pi$
$464$	2.03797	0.0946105
$465$	0	0
$466$	−35.7437	−1.65580
$467$	−24.3933	−1.12879	−0.564393	−	0.825506i	$-0.690890\pi$
$467$	−24.3933	−1.12879	−0.564393	+	0.825506i	$0.690890\pi$
$468$	0	0
$469$	−2.75062	−0.127012
$470$	−1.65126	−0.0761670
$471$	0	0
$472$	−34.4511	−1.58574
$473$	24.1406	1.10998
$474$	0	0
$475$	16.6714	0.764934
$476$	−0.917542	−0.0420555
$477$	0	0
$478$	−1.25296	−0.0573089
$479$	−24.1619	−1.10399	−0.551994	−	0.833848i	$-0.686133\pi$
$479$	−24.1619	−1.10399	−0.551994	+	0.833848i	$0.686133\pi$
$480$	0	0
$481$	−25.1798	−1.14810
$482$	16.3726	0.745753
$483$	0	0
$484$	0.813770	0.0369896
$485$	4.36429	0.198172
$486$	0	0
$487$	−10.8428	−0.491334	−0.245667	−	0.969354i	$-0.579007\pi$
$487$	−10.8428	−0.491334	−0.245667	+	0.969354i	$0.579007\pi$
$488$	4.41103	0.199678
$489$	0	0
$490$	−14.3147	−0.646670
$491$	8.33251	0.376041	0.188020	−	0.982165i	$-0.439793\pi$
$491$	8.33251	0.376041	0.188020	+	0.982165i	$0.439793\pi$
$492$	0	0
$493$	3.36082	0.151364
$494$	−36.3196	−1.63410
$495$	0	0
$496$	−25.9996	−1.16742
$497$	5.00261	0.224398
$498$	0	0
$499$	−5.29529	−0.237050	−0.118525	−	0.992951i	$-0.537817\pi$
$499$	−5.29529	−0.237050	−0.118525	+	0.992951i	$0.537817\pi$
$500$	5.18500	0.231880
$501$	0	0
$502$	16.3852	0.731306
$503$	13.6978	0.610756	0.305378	−	0.952231i	$-0.401217\pi$
$503$	13.6978	0.610756	0.305378	+	0.952231i	$0.401217\pi$
$504$	0	0
$505$	−10.5668	−0.470217
$506$	0.987866	0.0439160
$507$	0	0
$508$	−7.83844	−0.347775
$509$	19.4548	0.862319	0.431159	−	0.902276i	$-0.358105\pi$
$509$	19.4548	0.862319	0.431159	+	0.902276i	$0.358105\pi$
$510$	0	0
$511$	−3.98464	−0.176270
$512$	−25.0479	−1.10697
$513$	0	0
$514$	28.1111	1.23993
$515$	3.75670	0.165540
$516$	0	0
$517$	2.37020	0.104241
$518$	3.63706	0.159803
$519$	0	0
$520$	−19.4036	−0.850905
$521$	10.3167	0.451983	0.225992	−	0.974129i	$-0.427438\pi$
$521$	10.3167	0.451983	0.225992	+	0.974129i	$0.427438\pi$
$522$	0	0
$523$	1.78034	0.0778489	0.0389245	−	0.999242i	$-0.487607\pi$
$523$	1.78034	0.0778489	0.0389245	+	0.999242i	$0.487607\pi$
$524$	−0.192126	−0.00839308
$525$	0	0
$526$	7.79079	0.339695
$527$	−42.8760	−1.86771
$528$	0	0
$529$	−22.9317	−0.997033
$530$	−25.3844	−1.10263
$531$	0	0
$532$	−1.43726	−0.0623132
$533$	2.61079	0.113086
$534$	0	0
$535$	19.8128	0.856582
$536$	−19.1298	−0.826281
$537$	0	0
$538$	23.1823	0.999459
$539$	20.5471	0.885027
$540$	0	0
$541$	7.99971	0.343934	0.171967	−	0.985103i	$-0.444988\pi$
$541$	7.99971	0.343934	0.171967	+	0.985103i	$0.444988\pi$
$542$	30.7514	1.32088
$543$	0	0
$544$	−11.6331	−0.498765
$545$	−14.9969	−0.642397
$546$	0	0
$547$	−43.8106	−1.87321	−0.936603	−	0.350392i	$-0.886048\pi$
$547$	−43.8106	−1.87321	−0.936603	+	0.350392i	$0.886048\pi$
$548$	3.50455	0.149707
$549$	0	0
$550$	8.25905	0.352167
$551$	5.26447	0.224274
$552$	0	0
$553$	−2.26537	−0.0963334
$554$	2.22906	0.0947039
$555$	0	0
$556$	3.40778	0.144522
$557$	43.3503	1.83681	0.918405	−	0.395641i	$-0.129478\pi$
$557$	43.3503	1.83681	0.918405	+	0.395641i	$0.129478\pi$
$558$	0	0
$559$	−30.3777	−1.28484
$560$	2.17078	0.0917320
$561$	0	0
$562$	−15.8988	−0.670650
$563$	−0.411863	−0.0173579	−0.00867897	−	0.999962i	$-0.502763\pi$
$563$	−0.411863	−0.0173579	−0.00867897	+	0.999962i	$0.502763\pi$
$564$	0	0
$565$	−30.9889	−1.30371
$566$	30.4050	1.27802
$567$	0	0
$568$	34.7918	1.45983
$569$	17.9961	0.754436	0.377218	−	0.926124i	$-0.376881\pi$
$569$	17.9961	0.754436	0.377218	+	0.926124i	$0.376881\pi$
$570$	0	0
$571$	20.8344	0.871891	0.435946	−	0.899973i	$-0.356414\pi$
$571$	20.8344	0.871891	0.435946	+	0.899973i	$0.356414\pi$
$572$	4.92943	0.206110
$573$	0	0
$574$	−0.377111	−0.0157403
$575$	0.570605	0.0237959
$576$	0	0
$577$	28.4066	1.18258	0.591291	−	0.806458i	$-0.298618\pi$
$577$	28.4066	1.18258	0.591291	+	0.806458i	$0.298618\pi$
$578$	8.45007	0.351477
$579$	0	0
$580$	0.497783	0.0206693
$581$	−0.696660	−0.0289023
$582$	0	0
$583$	36.4365	1.50905
$584$	−27.7121	−1.14673
$585$	0	0
$586$	19.2498	0.795202
$587$	8.95210	0.369493	0.184746	−	0.982786i	$-0.440854\pi$
$587$	8.95210	0.369493	0.184746	+	0.982786i	$0.440854\pi$
$588$	0	0
$589$	−67.1619	−2.76736
$590$	23.7894	0.979395
$591$	0	0
$592$	19.5914	0.805201
$593$	−18.0335	−0.740547	−0.370273	−	0.928923i	$-0.620736\pi$
$593$	−18.0335	−0.740547	−0.370273	+	0.928923i	$0.620736\pi$
$594$	0	0
$595$	3.57983	0.146759
$596$	−2.16928	−0.0888572
$597$	0	0
$598$	−1.24310	−0.0508341
$599$	−14.5505	−0.594517	−0.297259	−	0.954797i	$-0.596072\pi$
$599$	−14.5505	−0.594517	−0.297259	+	0.954797i	$0.596072\pi$
$600$	0	0
$601$	21.6209	0.881936	0.440968	−	0.897523i	$-0.354635\pi$
$601$	21.6209	0.881936	0.440968	+	0.897523i	$0.354635\pi$
$602$	4.38786	0.178836
$603$	0	0
$604$	6.12379	0.249173
$605$	−3.17496	−0.129080
$606$	0	0
$607$	−24.8442	−1.00840	−0.504198	−	0.863588i	$-0.668212\pi$
$607$	−24.8442	−1.00840	−0.504198	+	0.863588i	$0.668212\pi$
$608$	−18.2224	−0.739015
$609$	0	0
$610$	−3.04593	−0.123326
$611$	−2.98259	−0.120663
$612$	0	0
$613$	−33.5479	−1.35499	−0.677494	−	0.735528i	$-0.736934\pi$
$613$	−33.5479	−1.35499	−0.677494	+	0.735528i	$0.736934\pi$
$614$	−7.37339	−0.297566
$615$	0	0
$616$	−4.02300	−0.162091
$617$	2.01405	0.0810826	0.0405413	−	0.999178i	$-0.487092\pi$
$617$	2.01405	0.0810826	0.0405413	+	0.999178i	$0.487092\pi$
$618$	0	0
$619$	−16.5501	−0.665205	−0.332603	−	0.943067i	$-0.607927\pi$
$619$	−16.5501	−0.665205	−0.332603	+	0.943067i	$0.607927\pi$
$620$	−6.35051	−0.255043
$621$	0	0
$622$	−24.4059	−0.978588
$623$	7.20946	0.288841
$624$	0	0
$625$	−9.30868	−0.372347
$626$	20.5730	0.822262
$627$	0	0
$628$	−7.74221	−0.308948
$629$	32.3082	1.28821
$630$	0	0
$631$	−12.6654	−0.504204	−0.252102	−	0.967701i	$-0.581122\pi$
$631$	−12.6654	−0.504204	−0.252102	+	0.967701i	$0.581122\pi$
$632$	−15.7551	−0.626702
$633$	0	0
$634$	−41.2782	−1.63937
$635$	30.5820	1.21361
$636$	0	0
$637$	−25.8558	−1.02445
$638$	2.60804	0.103253
$639$	0	0
$640$	10.7021	0.423038
$641$	−16.4831	−0.651043	−0.325522	−	0.945535i	$-0.605540\pi$
$641$	−16.4831	−0.651043	−0.325522	+	0.945535i	$0.605540\pi$
$642$	0	0
$643$	4.60496	0.181602	0.0908009	−	0.995869i	$-0.471057\pi$
$643$	4.60496	0.181602	0.0908009	+	0.995869i	$0.471057\pi$
$644$	−0.0491926	−0.00193846
$645$	0	0
$646$	46.6017	1.83352
$647$	34.5212	1.35717	0.678584	−	0.734523i	$-0.262594\pi$
$647$	34.5212	1.35717	0.678584	+	0.734523i	$0.262594\pi$
$648$	0	0
$649$	−34.1471	−1.34039
$650$	−10.3929	−0.407644
$651$	0	0
$652$	−4.62659	−0.181191
$653$	−21.9138	−0.857554	−0.428777	−	0.903410i	$-0.641055\pi$
$653$	−21.9138	−0.857554	−0.428777	+	0.903410i	$0.641055\pi$
$654$	0	0
$655$	0.749588	0.0292888
$656$	−2.03135	−0.0793108
$657$	0	0
$658$	0.430816	0.0167949
$659$	−2.42455	−0.0944471	−0.0472235	−	0.998884i	$-0.515037\pi$
$659$	−2.42455	−0.0944471	−0.0472235	+	0.998884i	$0.515037\pi$
$660$	0	0
$661$	22.1526	0.861638	0.430819	−	0.902438i	$-0.358225\pi$
$661$	22.1526	0.861638	0.430819	+	0.902438i	$0.358225\pi$
$662$	36.9657	1.43671
$663$	0	0
$664$	−4.84508	−0.188026
$665$	5.60753	0.217451
$666$	0	0
$667$	0.180185	0.00697680
$668$	1.53252	0.0592949
$669$	0	0
$670$	13.2096	0.510332
$671$	4.37210	0.168783
$672$	0	0
$673$	23.3640	0.900615	0.450307	−	0.892874i	$-0.351314\pi$
$673$	23.3640	0.900615	0.450307	+	0.892874i	$0.351314\pi$
$674$	−19.6262	−0.755974
$675$	0	0
$676$	−0.611757	−0.0235291
$677$	11.8456	0.455264	0.227632	−	0.973747i	$-0.426902\pi$
$677$	11.8456	0.455264	0.227632	+	0.973747i	$0.426902\pi$
$678$	0	0
$679$	−1.13865	−0.0436973
$680$	24.8967	0.954746
$681$	0	0
$682$	−33.2723	−1.27406
$683$	5.41333	0.207135	0.103568	−	0.994622i	$-0.466974\pi$
$683$	5.41333	0.207135	0.103568	+	0.994622i	$0.466974\pi$
$684$	0	0
$685$	−13.6731	−0.522424
$686$	7.57456	0.289198
$687$	0	0
$688$	23.6357	0.901101
$689$	−45.8506	−1.74677
$690$	0	0
$691$	−31.7264	−1.20693	−0.603464	−	0.797390i	$-0.706213\pi$
$691$	−31.7264	−1.20693	−0.603464	+	0.797390i	$0.706213\pi$
$692$	−4.57569	−0.173942
$693$	0	0
$694$	−28.9237	−1.09793
$695$	−13.2956	−0.504331
$696$	0	0
$697$	−3.34990	−0.126886
$698$	−11.1906	−0.423569
$699$	0	0
$700$	−0.411275	−0.0155447
$701$	−12.1452	−0.458717	−0.229359	−	0.973342i	$-0.573663\pi$
$701$	−12.1452	−0.458717	−0.229359	+	0.973342i	$0.573663\pi$
$702$	0	0
$703$	50.6083	1.90873
$704$	−26.8623	−1.01241
$705$	0	0
$706$	36.0717	1.35758
$707$	2.75689	0.103684
$708$	0	0
$709$	−38.9674	−1.46345	−0.731726	−	0.681599i	$-0.761285\pi$
$709$	−38.9674	−1.46345	−0.731726	+	0.681599i	$0.761285\pi$
$710$	−24.0247	−0.901629
$711$	0	0
$712$	50.1399	1.87907
$713$	−2.29873	−0.0860881
$714$	0	0
$715$	−19.2324	−0.719250
$716$	1.20221	0.0449285
$717$	0	0
$718$	−40.9938	−1.52988
$719$	−16.5847	−0.618506	−0.309253	−	0.950980i	$-0.600079\pi$
$719$	−16.5847	−0.618506	−0.309253	+	0.950980i	$0.600079\pi$
$720$	0	0
$721$	−0.980126	−0.0365018
$722$	49.1918	1.83073
$723$	0	0
$724$	−1.05377	−0.0391630
$725$	1.50644	0.0559477
$726$	0	0
$727$	11.7059	0.434146	0.217073	−	0.976155i	$-0.430349\pi$
$727$	11.7059	0.434146	0.217073	+	0.976155i	$0.430349\pi$
$728$	5.06242	0.187626
$729$	0	0
$730$	19.1359	0.708253
$731$	38.9776	1.44164
$732$	0	0
$733$	−18.9548	−0.700112	−0.350056	−	0.936729i	$-0.613837\pi$
$733$	−18.9548	−0.700112	−0.350056	+	0.936729i	$0.613837\pi$
$734$	−45.2131	−1.66885
$735$	0	0
$736$	−0.623691	−0.0229896
$737$	−18.9610	−0.698436
$738$	0	0
$739$	−0.435521	−0.0160209	−0.00801045	−	0.999968i	$-0.502550\pi$
$739$	−0.435521	−0.0160209	−0.00801045	+	0.999968i	$0.502550\pi$
$740$	4.78528	0.175910
$741$	0	0
$742$	6.62282	0.243131
$743$	11.5604	0.424110	0.212055	−	0.977258i	$-0.431984\pi$
$743$	11.5604	0.424110	0.212055	+	0.977258i	$0.431984\pi$
$744$	0	0
$745$	8.46354	0.310080
$746$	−43.6114	−1.59673
$747$	0	0
$748$	−6.32494	−0.231263
$749$	−5.16918	−0.188878
$750$	0	0
$751$	−18.9573	−0.691762	−0.345881	−	0.938278i	$-0.612420\pi$
$751$	−18.9573	−0.691762	−0.345881	+	0.938278i	$0.612420\pi$
$752$	2.32063	0.0846247
$753$	0	0
$754$	−3.28187	−0.119519
$755$	−23.8922	−0.869526
$756$	0	0
$757$	−36.1482	−1.31383	−0.656914	−	0.753965i	$-0.728139\pi$
$757$	−36.1482	−1.31383	−0.656914	+	0.753965i	$0.728139\pi$
$758$	17.9916	0.653484
$759$	0	0
$760$	38.9988	1.41464
$761$	−33.4183	−1.21141	−0.605707	−	0.795688i	$-0.707110\pi$
$761$	−33.4183	−1.21141	−0.605707	+	0.795688i	$0.707110\pi$
$762$	0	0
$763$	3.91271	0.141650
$764$	−7.32449	−0.264991
$765$	0	0
$766$	10.3467	0.373843
$767$	42.9696	1.55154
$768$	0	0
$769$	46.7975	1.68756	0.843781	−	0.536688i	$-0.180325\pi$
$769$	46.7975	1.68756	0.843781	+	0.536688i	$0.180325\pi$
$770$	2.77799	0.100112
$771$	0	0
$772$	2.22544	0.0800953
$773$	−49.1281	−1.76702	−0.883508	−	0.468416i	$-0.844825\pi$
$773$	−49.1281	−1.76702	−0.883508	+	0.468416i	$0.844825\pi$
$774$	0	0
$775$	−19.2185	−0.690349
$776$	−7.91898	−0.284275
$777$	0	0
$778$	−36.3109	−1.30181
$779$	−5.24736	−0.188006
$780$	0	0
$781$	34.4847	1.23396
$782$	1.59502	0.0570378
$783$	0	0
$784$	20.1174	0.718477
$785$	30.2065	1.07812
$786$	0	0
$787$	−25.2734	−0.900901	−0.450450	−	0.892801i	$-0.648737\pi$
$787$	−25.2734	−0.900901	−0.450450	+	0.892801i	$0.648737\pi$
$788$	8.72199	0.310708
$789$	0	0
$790$	10.8793	0.387067
$791$	8.08503	0.287470
$792$	0	0
$793$	−5.50172	−0.195372
$794$	4.59663	0.163128
$795$	0	0
$796$	−8.09471	−0.286909
$797$	16.3302	0.578447	0.289223	−	0.957262i	$-0.406603\pi$
$797$	16.3302	0.578447	0.289223	+	0.957262i	$0.406603\pi$
$798$	0	0
$799$	3.82695	0.135388
$800$	−5.21437	−0.184356
$801$	0	0
$802$	7.11284	0.251163
$803$	−27.4675	−0.969308
$804$	0	0
$805$	0.191927	0.00676454
$806$	41.8688	1.47476
$807$	0	0
$808$	19.1734	0.674519
$809$	6.61012	0.232400	0.116200	−	0.993226i	$-0.462929\pi$
$809$	6.61012	0.232400	0.116200	+	0.993226i	$0.462929\pi$
$810$	0	0
$811$	8.25580	0.289900	0.144950	−	0.989439i	$-0.453698\pi$
$811$	8.25580	0.289900	0.144950	+	0.989439i	$0.453698\pi$
$812$	−0.129872	−0.00455762
$813$	0	0
$814$	25.0715	0.878757
$815$	18.0508	0.632292
$816$	0	0
$817$	61.0554	2.13606
$818$	20.5361	0.718029
$819$	0	0
$820$	−0.496165	−0.0173268
$821$	−32.1939	−1.12358	−0.561788	−	0.827282i	$-0.689886\pi$
$821$	−32.1939	−1.12358	−0.561788	+	0.827282i	$0.689886\pi$
$822$	0	0
$823$	−34.2382	−1.19347	−0.596735	−	0.802439i	$-0.703536\pi$
$823$	−34.2382	−1.19347	−0.596735	+	0.802439i	$0.703536\pi$
$824$	−6.81651	−0.237464
$825$	0	0
$826$	−6.20668	−0.215958
$827$	−35.3643	−1.22974	−0.614869	−	0.788629i	$-0.710791\pi$
$827$	−35.3643	−1.22974	−0.614869	+	0.788629i	$0.710791\pi$
$828$	0	0
$829$	6.39310	0.222041	0.111021	−	0.993818i	$-0.464588\pi$
$829$	6.39310	0.222041	0.111021	+	0.993818i	$0.464588\pi$
$830$	3.34566	0.116129
$831$	0	0
$832$	33.8027	1.17190
$833$	33.1756	1.14947
$834$	0	0
$835$	−5.97918	−0.206918
$836$	−9.90754	−0.342659
$837$	0	0
$838$	−41.7588	−1.44253
$839$	−47.5595	−1.64194	−0.820968	−	0.570974i	$-0.806566\pi$
$839$	−47.5595	−1.64194	−0.820968	+	0.570974i	$0.806566\pi$
$840$	0	0
$841$	−28.5243	−0.983596
$842$	32.5775	1.12270
$843$	0	0
$844$	−11.9863	−0.412585
$845$	2.38679	0.0821081
$846$	0	0
$847$	0.828349	0.0284624
$848$	35.6744	1.22507
$849$	0	0
$850$	13.3351	0.457392
$851$	1.73215	0.0593775
$852$	0	0
$853$	27.9031	0.955384	0.477692	−	0.878527i	$-0.341474\pi$
$853$	27.9031	0.955384	0.477692	+	0.878527i	$0.341474\pi$
$854$	0.794687	0.0271936
$855$	0	0
$856$	−35.9502	−1.22875
$857$	−41.0114	−1.40092	−0.700462	−	0.713690i	$-0.747023\pi$
$857$	−41.0114	−1.40092	−0.700462	+	0.713690i	$0.747023\pi$
$858$	0	0
$859$	17.8846	0.610215	0.305107	−	0.952318i	$-0.401308\pi$
$859$	17.8846	0.610215	0.305107	+	0.952318i	$0.401308\pi$
$860$	5.77311	0.196861
$861$	0	0
$862$	18.1949	0.619719
$863$	−14.1941	−0.483174	−0.241587	−	0.970379i	$-0.577668\pi$
$863$	−14.1941	−0.483174	−0.241587	+	0.970379i	$0.577668\pi$
$864$	0	0
$865$	17.8522	0.606995
$866$	10.2325	0.347714
$867$	0	0
$868$	1.65685	0.0562373
$869$	−15.6160	−0.529737
$870$	0	0
$871$	23.8599	0.808461
$872$	27.2118	0.921508
$873$	0	0
$874$	2.49848	0.0845122
$875$	5.27789	0.178425
$876$	0	0
$877$	16.4208	0.554490	0.277245	−	0.960799i	$-0.410579\pi$
$877$	16.4208	0.554490	0.277245	+	0.960799i	$0.410579\pi$
$878$	−16.6330	−0.561337
$879$	0	0
$880$	14.9639	0.504433
$881$	−13.1795	−0.444029	−0.222014	−	0.975043i	$-0.571263\pi$
$881$	−13.1795	−0.444029	−0.222014	+	0.975043i	$0.571263\pi$
$882$	0	0
$883$	−57.2316	−1.92600	−0.962999	−	0.269506i	$-0.913140\pi$
$883$	−57.2316	−1.92600	−0.962999	+	0.269506i	$0.913140\pi$
$884$	7.95911	0.267694
$885$	0	0
$886$	8.85966	0.297646
$887$	38.0535	1.27771	0.638857	−	0.769326i	$-0.279408\pi$
$887$	38.0535	1.27771	0.638857	+	0.769326i	$0.279408\pi$
$888$	0	0
$889$	−7.97887	−0.267603
$890$	−34.6229	−1.16056
$891$	0	0
$892$	5.52354	0.184942
$893$	5.99463	0.200603
$894$	0	0
$895$	−4.69045	−0.156785
$896$	−2.79219	−0.0932806
$897$	0	0
$898$	16.4651	0.549448
$899$	−6.06881	−0.202406
$900$	0	0
$901$	58.8308	1.95994
$902$	−2.59956	−0.0865559
$903$	0	0
$904$	56.2292	1.87015
$905$	4.11132	0.136665
$906$	0	0
$907$	−23.2951	−0.773500	−0.386750	−	0.922185i	$-0.626402\pi$
$907$	−23.2951	−0.773500	−0.386750	+	0.922185i	$0.626402\pi$
$908$	1.61285	0.0535242
$909$	0	0
$910$	−3.49574	−0.115883
$911$	29.0411	0.962175	0.481088	−	0.876673i	$-0.340242\pi$
$911$	29.0411	0.962175	0.481088	+	0.876673i	$0.340242\pi$
$912$	0	0
$913$	−4.80232	−0.158934
$914$	38.6924	1.27983
$915$	0	0
$916$	1.97846	0.0653701
$917$	−0.195568	−0.00645823
$918$	0	0
$919$	−37.6246	−1.24112	−0.620561	−	0.784158i	$-0.713095\pi$
$919$	−37.6246	−1.24112	−0.620561	+	0.784158i	$0.713095\pi$
$920$	1.33480	0.0440070
$921$	0	0
$922$	5.14479	0.169435
$923$	−43.3945	−1.42835
$924$	0	0
$925$	14.4817	0.476154
$926$	−41.6408	−1.36840
$927$	0	0
$928$	−1.64659	−0.0540520
$929$	−53.0432	−1.74029	−0.870145	−	0.492796i	$-0.835975\pi$
$929$	−53.0432	−1.74029	−0.870145	+	0.492796i	$0.835975\pi$
$930$	0	0
$931$	51.9670	1.70315
$932$	12.2696	0.401906
$933$	0	0
$934$	−30.5637	−1.00008
$935$	24.6770	0.807024
$936$	0	0
$937$	−12.0063	−0.392228	−0.196114	−	0.980581i	$-0.562832\pi$
$937$	−12.0063	−0.392228	−0.196114	+	0.980581i	$0.562832\pi$
$938$	−3.44640	−0.112529
$939$	0	0
$940$	0.566824	0.0184878
$941$	30.1289	0.982173	0.491086	−	0.871111i	$-0.336600\pi$
$941$	30.1289	0.982173	0.491086	+	0.871111i	$0.336600\pi$
$942$	0	0
$943$	−0.179600	−0.00584857
$944$	−33.4329	−1.08815
$945$	0	0
$946$	30.2471	0.983417
$947$	−16.5223	−0.536902	−0.268451	−	0.963293i	$-0.586512\pi$
$947$	−16.5223	−0.536902	−0.268451	+	0.963293i	$0.586512\pi$
$948$	0	0
$949$	34.5643	1.12200
$950$	20.8885	0.677712
$951$	0	0
$952$	−6.49558	−0.210523
$953$	−51.7178	−1.67531	−0.837653	−	0.546204i	$-0.816073\pi$
$953$	−51.7178	−1.67531	−0.837653	+	0.546204i	$0.816073\pi$
$954$	0	0
$955$	28.5768	0.924723
$956$	0.430099	0.0139104
$957$	0	0
$958$	−30.2739	−0.978104
$959$	3.56734	0.115195
$960$	0	0
$961$	46.4233	1.49753
$962$	−31.5492	−1.01719
$963$	0	0
$964$	−5.62019	−0.181014
$965$	−8.68263	−0.279504
$966$	0	0
$967$	−31.8965	−1.02572	−0.512862	−	0.858471i	$-0.671415\pi$
$967$	−31.8965	−1.02572	−0.512862	+	0.858471i	$0.671415\pi$
$968$	5.76094	0.185164
$969$	0	0
$970$	5.46826	0.175575
$971$	−52.5319	−1.68583	−0.842915	−	0.538047i	$-0.819162\pi$
$971$	−52.5319	−1.68583	−0.842915	+	0.538047i	$0.819162\pi$
$972$	0	0
$973$	3.46883	0.111206
$974$	−13.5856	−0.435310
$975$	0	0
$976$	4.28066	0.137021
$977$	−52.8123	−1.68961	−0.844807	−	0.535071i	$-0.820285\pi$
$977$	−52.8123	−1.68961	−0.844807	+	0.535071i	$0.820285\pi$
$978$	0	0
$979$	49.6974	1.58833
$980$	4.91375	0.156964
$981$	0	0
$982$	10.4403	0.333163
$983$	9.10617	0.290442	0.145221	−	0.989399i	$-0.453611\pi$
$983$	9.10617	0.290442	0.145221	+	0.989399i	$0.453611\pi$
$984$	0	0
$985$	−34.0292	−1.08426
$986$	4.21097	0.134104
$987$	0	0
$988$	12.4673	0.396639
$989$	2.08972	0.0664493
$990$	0	0
$991$	15.1055	0.479843	0.239921	−	0.970792i	$-0.422878\pi$
$991$	15.1055	0.479843	0.239921	+	0.970792i	$0.422878\pi$
$992$	21.0065	0.666957
$993$	0	0
$994$	6.26806	0.198811
$995$	31.5818	1.00121
$996$	0	0
$997$	7.86287	0.249019	0.124510	−	0.992218i	$-0.460264\pi$
$997$	7.86287	0.249019	0.124510	+	0.992218i	$0.460264\pi$
$998$	−6.63477	−0.210020
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.j.1.15	✓	20
3.2	odd	2		2151.2.a.k.1.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.15	✓	20	1.1	even	1	trivial
2151.2.a.k.1.6	yes	20	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+1.25296 q^{2} -0.430099 q^{4} +1.67805 q^{5} -0.437804 q^{7} -3.04481 q^{8} +O(q^{10})\)	\(q+1.25296 q^{2} -0.430099 q^{4} +1.67805 q^{5} -0.437804 q^{7} -3.04481 q^{8} +2.10252 q^{10} -3.01794 q^{11} +3.79768 q^{13} -0.548550 q^{14} -2.95482 q^{16} -4.87279 q^{17} -7.63286 q^{19} -0.721726 q^{20} -3.78135 q^{22} -0.261247 q^{23} -2.18416 q^{25} +4.75833 q^{26} +0.188299 q^{28} -0.689712 q^{29} +8.79905 q^{31} +2.38736 q^{32} -6.10540 q^{34} -0.734656 q^{35} -6.63032 q^{37} -9.56364 q^{38} -5.10934 q^{40} +0.687470 q^{41} -7.99902 q^{43} +1.29801 q^{44} -0.327332 q^{46} -0.785372 q^{47} -6.80833 q^{49} -2.73665 q^{50} -1.63338 q^{52} -12.0733 q^{53} -5.06424 q^{55} +1.33303 q^{56} -0.864179 q^{58} +11.3147 q^{59} -1.44871 q^{61} +11.0248 q^{62} +8.90089 q^{64} +6.37269 q^{65} +6.28276 q^{67} +2.09578 q^{68} -0.920493 q^{70} -11.4266 q^{71} +9.10142 q^{73} -8.30751 q^{74} +3.28288 q^{76} +1.32127 q^{77} +5.17440 q^{79} -4.95832 q^{80} +0.861370 q^{82} +1.59126 q^{83} -8.17678 q^{85} -10.0224 q^{86} +9.18904 q^{88} -16.4673 q^{89} -1.66264 q^{91} +0.112362 q^{92} -0.984037 q^{94} -12.8083 q^{95} +2.60081 q^{97} -8.53054 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8	\( 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8} + 4 q^{10} - 12 q^{11} - 4 q^{13} - 20 q^{14} + 22 q^{16} - 24 q^{17} - 4 q^{19} - 40 q^{20} - 6 q^{22} - 12 q^{23} + 22 q^{25} - 30 q^{26} - 12 q^{28} - 24 q^{29} - 4 q^{31} - 28 q^{32} + 8 q^{34} - 20 q^{35} - 10 q^{37} - 26 q^{38} + 6 q^{40} - 66 q^{41} + 8 q^{43} - 36 q^{44} - 12 q^{46} - 28 q^{47} + 18 q^{49} - 28 q^{50} - 18 q^{52} - 28 q^{53} - 4 q^{55} - 60 q^{56} - 54 q^{59} - 4 q^{61} - 20 q^{62} + 22 q^{64} - 42 q^{65} + 12 q^{67} - 12 q^{68} + 20 q^{70} - 36 q^{71} + 14 q^{73} - 50 q^{76} - 8 q^{77} - 12 q^{79} - 88 q^{80} - 8 q^{82} - 20 q^{83} + 4 q^{85} - 18 q^{86} - 10 q^{88} - 130 q^{89} - 6 q^{91} + 46 q^{92} - 26 q^{94} - 2 q^{97} - 12 q^{98}+O(q^{100}) \) 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8 + 4 * q^10 - 12 * q^11 - 4 * q^13 - 20 * q^14 + 22 * q^16 - 24 * q^17 - 4 * q^19 - 40 * q^20 - 6 * q^22 - 12 * q^23 + 22 * q^25 - 30 * q^26 - 12 * q^28 - 24 * q^29 - 4 * q^31 - 28 * q^32 + 8 * q^34 - 20 * q^35 - 10 * q^37 - 26 * q^38 + 6 * q^40 - 66 * q^41 + 8 * q^43 - 36 * q^44 - 12 * q^46 - 28 * q^47 + 18 * q^49 - 28 * q^50 - 18 * q^52 - 28 * q^53 - 4 * q^55 - 60 * q^56 - 54 * q^59 - 4 * q^61 - 20 * q^62 + 22 * q^64 - 42 * q^65 + 12 * q^67 - 12 * q^68 + 20 * q^70 - 36 * q^71 + 14 * q^73 - 50 * q^76 - 8 * q^77 - 12 * q^79 - 88 * q^80 - 8 * q^82 - 20 * q^83 + 4 * q^85 - 18 * q^86 - 10 * q^88 - 130 * q^89 - 6 * q^91 + 46 * q^92 - 26 * q^94 - 2 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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