LMFDB - Embedded newform 8281.2.a.cp.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1 $ x^12 - 16x^10 + 88x^8 - 197x^6 + 172x^4 - 36*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.180824$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.180824 q^{2} -1.82601 q^{3} -1.96730 q^{4} +2.68664 q^{5} -0.330186 q^{6} -0.717383 q^{8} +0.334323 q^{9} +O(q^{10})$	\(q+0.180824 q^{2} -1.82601 q^{3} -1.96730 q^{4} +2.68664 q^{5} -0.330186 q^{6} -0.717383 q^{8} +0.334323 q^{9} +0.485809 q^{10} +2.69424 q^{11} +3.59232 q^{12} -4.90584 q^{15} +3.80489 q^{16} +4.76493 q^{17} +0.0604535 q^{18} +0.188424 q^{19} -5.28544 q^{20} +0.487183 q^{22} +4.39929 q^{23} +1.30995 q^{24} +2.21804 q^{25} +4.86756 q^{27} +7.08560 q^{29} -0.887093 q^{30} -3.69931 q^{31} +2.12278 q^{32} -4.91972 q^{33} +0.861613 q^{34} -0.657715 q^{36} +7.95413 q^{37} +0.0340716 q^{38} -1.92735 q^{40} +5.42958 q^{41} -8.01065 q^{43} -5.30039 q^{44} +0.898206 q^{45} +0.795496 q^{46} -1.84889 q^{47} -6.94777 q^{48} +0.401075 q^{50} -8.70083 q^{51} -7.07244 q^{53} +0.880171 q^{54} +7.23846 q^{55} -0.344066 q^{57} +1.28125 q^{58} +7.58888 q^{59} +9.65128 q^{60} -0.411564 q^{61} -0.668922 q^{62} -7.22592 q^{64} -0.889602 q^{66} +11.4010 q^{67} -9.37407 q^{68} -8.03315 q^{69} -3.34488 q^{71} -0.239838 q^{72} -14.2158 q^{73} +1.43830 q^{74} -4.05018 q^{75} -0.370688 q^{76} +9.11059 q^{79} +10.2224 q^{80} -9.89120 q^{81} +0.981797 q^{82} +16.5866 q^{83} +12.8017 q^{85} -1.44852 q^{86} -12.9384 q^{87} -1.93280 q^{88} -5.89165 q^{89} +0.162417 q^{90} -8.65473 q^{92} +6.75498 q^{93} -0.334323 q^{94} +0.506229 q^{95} -3.87622 q^{96} -0.451094 q^{97} +0.900747 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) $ 12 * q + 6 * q^3 + 8 * q^4 + 2 * q^9	$ 12 q + 6 q^{3} + 8 q^{4} + 2 q^{9} + 24 q^{10} - 2 q^{12} + 16 q^{16} + 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} + 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} + 38 q^{38} + 2 q^{40} + 22 q^{43} - 38 q^{48} + 8 q^{51} + 16 q^{53} + 30 q^{55} - 10 q^{61} + 82 q^{62} - 2 q^{64} + 68 q^{66} + 22 q^{68} + 14 q^{69} + 66 q^{74} + 2 q^{75} + 70 q^{79} - 28 q^{81} + 10 q^{82} - 20 q^{87} - 28 q^{88} - 66 q^{92} - 2 q^{94} + 4 q^{95}+O(q^{100}) $ 12 * q + 6 * q^3 + 8 * q^4 + 2 * q^9 + 24 * q^10 - 2 * q^12 + 16 * q^16 + 34 * q^17 + 30 * q^22 + 6 * q^23 - 10 * q^25 + 12 * q^27 + 2 * q^29 + 22 * q^30 - 26 * q^36 + 38 * q^38 + 2 * q^40 + 22 * q^43 - 38 * q^48 + 8 * q^51 + 16 * q^53 + 30 * q^55 - 10 * q^61 + 82 * q^62 - 2 * q^64 + 68 * q^66 + 22 * q^68 + 14 * q^69 + 66 * q^74 + 2 * q^75 + 70 * q^79 - 28 * q^81 + 10 * q^82 - 20 * q^87 - 28 * q^88 - 66 * q^92 - 2 * q^94 + 4 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.180824	0.127862	0.0639308	−	0.997954i	$-0.479636\pi$
$2$	0.180824	0.127862	0.0639308	+	0.997954i	$0.479636\pi$
$3$	−1.82601	−1.05425	−0.527125	−	0.849788i	$-0.676730\pi$
$3$	−1.82601	−1.05425	−0.527125	+	0.849788i	$0.676730\pi$
$4$	−1.96730	−0.983651
$5$	2.68664	1.20150	0.600751	−	0.799436i	$-0.294868\pi$
$5$	2.68664	1.20150	0.600751	+	0.799436i	$0.294868\pi$
$6$	−0.330186	−0.134798
$7$	0	0
$7$	0	0
$8$	−0.717383	−0.253633
$9$	0.334323	0.111441
$10$	0.485809	0.153626
$11$	2.69424	0.812345	0.406172	−	0.913796i	$-0.366863\pi$
$11$	2.69424	0.812345	0.406172	+	0.913796i	$0.366863\pi$
$12$	3.59232	1.03701
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−4.90584	−1.26668
$16$	3.80489	0.951221
$17$	4.76493	1.15567	0.577833	−	0.816155i	$-0.303898\pi$
$17$	4.76493	1.15567	0.577833	+	0.816155i	$0.303898\pi$
$18$	0.0604535	0.0142490
$19$	0.188424	0.0432275	0.0216138	−	0.999766i	$-0.493120\pi$
$19$	0.188424	0.0432275	0.0216138	+	0.999766i	$0.493120\pi$
$20$	−5.28544	−1.18186
$21$	0	0
$22$	0.487183	0.103868
$23$	4.39929	0.917315	0.458657	−	0.888613i	$-0.348331\pi$
$23$	4.39929	0.917315	0.458657	+	0.888613i	$0.348331\pi$
$24$	1.30995	0.267392
$25$	2.21804	0.443609
$26$	0	0
$27$	4.86756	0.936762
$28$	0	0
$29$	7.08560	1.31576	0.657882	−	0.753121i	$-0.271453\pi$
$29$	7.08560	1.31576	0.657882	+	0.753121i	$0.271453\pi$
$30$	−0.887093	−0.161960
$31$	−3.69931	−0.664415	−0.332207	−	0.943206i	$-0.607793\pi$
$31$	−3.69931	−0.664415	−0.332207	+	0.943206i	$0.607793\pi$
$32$	2.12278	0.375258
$33$	−4.91972	−0.856414
$34$	0.861613	0.147765
$35$	0	0
$36$	−0.657715	−0.109619
$37$	7.95413	1.30765	0.653826	−	0.756645i	$-0.273163\pi$
$37$	7.95413	1.30765	0.653826	+	0.756645i	$0.273163\pi$
$38$	0.0340716	0.00552715
$39$	0	0
$40$	−1.92735	−0.304741
$41$	5.42958	0.847958	0.423979	−	0.905672i	$-0.360633\pi$
$41$	5.42958	0.847958	0.423979	+	0.905672i	$0.360633\pi$
$42$	0	0
$43$	−8.01065	−1.22161	−0.610807	−	0.791780i	$-0.709155\pi$
$43$	−8.01065	−1.22161	−0.610807	+	0.791780i	$0.709155\pi$
$44$	−5.30039	−0.799064
$45$	0.898206	0.133897
$46$	0.795496	0.117289
$47$	−1.84889	−0.269688	−0.134844	−	0.990867i	$-0.543053\pi$
$47$	−1.84889	−0.269688	−0.134844	+	0.990867i	$0.543053\pi$
$48$	−6.94777	−1.00282
$49$	0	0
$50$	0.401075	0.0567206
$51$	−8.70083	−1.21836
$52$	0	0
$53$	−7.07244	−0.971474	−0.485737	−	0.874105i	$-0.661449\pi$
$53$	−7.07244	−0.971474	−0.485737	+	0.874105i	$0.661449\pi$
$54$	0.880171	0.119776
$55$	7.23846	0.976034
$56$	0	0
$57$	−0.344066	−0.0455726
$58$	1.28125	0.168236
$59$	7.58888	0.987988	0.493994	−	0.869465i	$-0.335536\pi$
$59$	7.58888	0.987988	0.493994	+	0.869465i	$0.335536\pi$
$60$	9.65128	1.24597
$61$	−0.411564	−0.0526954	−0.0263477	−	0.999653i	$-0.508388\pi$
$61$	−0.411564	−0.0526954	−0.0263477	+	0.999653i	$0.508388\pi$
$62$	−0.668922	−0.0849532
$63$	0	0
$64$	−7.22592	−0.903240
$65$	0	0
$66$	−0.889602	−0.109502
$67$	11.4010	1.39286	0.696429	−	0.717626i	$-0.254771\pi$
$67$	11.4010	1.39286	0.696429	+	0.717626i	$0.254771\pi$
$68$	−9.37407	−1.13677
$69$	−8.03315	−0.967078
$70$	0	0
$71$	−3.34488	−0.396965	−0.198482	−	0.980104i	$-0.563601\pi$
$71$	−3.34488	−0.396965	−0.198482	+	0.980104i	$0.563601\pi$
$72$	−0.239838	−0.0282651
$73$	−14.2158	−1.66383	−0.831917	−	0.554900i	$-0.812756\pi$
$73$	−14.2158	−1.66383	−0.831917	+	0.554900i	$0.812756\pi$
$74$	1.43830	0.167199
$75$	−4.05018	−0.467674
$76$	−0.370688	−0.0425208
$77$	0	0
$78$	0	0
$79$	9.11059	1.02502	0.512511	−	0.858681i	$-0.328715\pi$
$79$	9.11059	1.02502	0.512511	+	0.858681i	$0.328715\pi$
$80$	10.2224	1.14290
$81$	−9.89120	−1.09902
$82$	0.981797	0.108421
$83$	16.5866	1.82061	0.910307	−	0.413934i	$-0.135845\pi$
$83$	16.5866	1.82061	0.910307	+	0.413934i	$0.135845\pi$
$84$	0	0
$85$	12.8017	1.38854
$86$	−1.44852	−0.156198
$87$	−12.9384	−1.38714
$88$	−1.93280	−0.206037
$89$	−5.89165	−0.624513	−0.312257	−	0.949998i	$-0.601085\pi$
$89$	−5.89165	−0.624513	−0.312257	+	0.949998i	$0.601085\pi$
$90$	0.162417	0.0171203
$91$	0	0
$92$	−8.65473	−0.902318
$93$	6.75498	0.700459
$94$	−0.334323	−0.0344828
$95$	0.506229	0.0519380
$96$	−3.87622	−0.395615
$97$	−0.451094	−0.0458016	−0.0229008	−	0.999738i	$-0.507290\pi$
$97$	−0.451094	−0.0458016	−0.0229008	+	0.999738i	$0.507290\pi$
$98$	0	0
$99$	0.900747	0.0905285
$100$	−4.36356	−0.436356
$101$	−7.65680	−0.761880	−0.380940	−	0.924600i	$-0.624400\pi$
$101$	−7.65680	−0.761880	−0.380940	+	0.924600i	$0.624400\pi$
$102$	−1.57332	−0.155782
$103$	5.15740	0.508173	0.254087	−	0.967181i	$-0.418225\pi$
$103$	5.15740	0.508173	0.254087	+	0.967181i	$0.418225\pi$
$104$	0	0
$105$	0	0
$106$	−1.27887	−0.124214
$107$	8.03289	0.776569	0.388284	−	0.921540i	$-0.373068\pi$
$107$	8.03289	0.776569	0.388284	+	0.921540i	$0.373068\pi$
$108$	−9.57597	−0.921448
$109$	1.33356	0.127732	0.0638660	−	0.997958i	$-0.479657\pi$
$109$	1.33356	0.127732	0.0638660	+	0.997958i	$0.479657\pi$
$110$	1.30889	0.124797
$111$	−14.5243	−1.37859
$112$	0	0
$113$	−19.9383	−1.87564	−0.937821	−	0.347119i	$-0.887160\pi$
$113$	−19.9383	−1.87564	−0.937821	+	0.347119i	$0.887160\pi$
$114$	−0.0622152	−0.00582699
$115$	11.8193	1.10216
$116$	−13.9395	−1.29425
$117$	0	0
$118$	1.37225	0.126326
$119$	0	0
$120$	3.51937	0.321273
$121$	−3.74106	−0.340096
$122$	−0.0744205	−0.00673772
$123$	−9.91448	−0.893959
$124$	7.27765	0.653553
$125$	−7.47412	−0.668505
$126$	0	0
$127$	−7.96722	−0.706976	−0.353488	−	0.935439i	$-0.615005\pi$
$127$	−7.96722	−0.706976	−0.353488	+	0.935439i	$0.615005\pi$
$128$	−5.55218	−0.490748
$129$	14.6276	1.28788
$130$	0	0
$131$	10.0179	0.875271	0.437636	−	0.899152i	$-0.355816\pi$
$131$	10.0179	0.875271	0.437636	+	0.899152i	$0.355816\pi$
$132$	9.67858	0.842412
$133$	0	0
$134$	2.06158	0.178093
$135$	13.0774	1.12552
$136$	−3.41828	−0.293115
$137$	−5.06696	−0.432899	−0.216450	−	0.976294i	$-0.569448\pi$
$137$	−5.06696	−0.432899	−0.216450	+	0.976294i	$0.569448\pi$
$138$	−1.45259	−0.123652
$139$	7.72578	0.655292	0.327646	−	0.944800i	$-0.393745\pi$
$139$	7.72578	0.655292	0.327646	+	0.944800i	$0.393745\pi$
$140$	0	0
$141$	3.37610	0.284319
$142$	−0.604834	−0.0507566
$143$	0	0
$144$	1.27206	0.106005
$145$	19.0365	1.58089
$146$	−2.57055	−0.212741
$147$	0	0
$148$	−15.6482	−1.28627
$149$	14.3185	1.17301	0.586507	−	0.809944i	$-0.300502\pi$
$149$	14.3185	1.17301	0.586507	+	0.809944i	$0.300502\pi$
$150$	−0.732368	−0.0597976
$151$	−6.47249	−0.526724	−0.263362	−	0.964697i	$-0.584831\pi$
$151$	−6.47249	−0.526724	−0.263362	+	0.964697i	$0.584831\pi$
$152$	−0.135172	−0.0109639
$153$	1.59303	0.128789
$154$	0	0
$155$	−9.93871	−0.798296
$156$	0	0
$157$	15.9187	1.27045	0.635227	−	0.772326i	$-0.280907\pi$
$157$	15.9187	1.27045	0.635227	+	0.772326i	$0.280907\pi$
$158$	1.64741	0.131061
$159$	12.9144	1.02418
$160$	5.70315	0.450873
$161$	0	0
$162$	−1.78856	−0.140523
$163$	4.78162	0.374525	0.187263	−	0.982310i	$-0.440038\pi$
$163$	4.78162	0.374525	0.187263	+	0.982310i	$0.440038\pi$
$164$	−10.6816	−0.834095
$165$	−13.2175	−1.02898
$166$	2.99925	0.232787
$167$	2.71042	0.209739	0.104869	−	0.994486i	$-0.466558\pi$
$167$	2.71042	0.209739	0.104869	+	0.994486i	$0.466558\pi$
$168$	0	0
$169$	0	0
$170$	2.31485	0.177541
$171$	0.0629946	0.00481732
$172$	15.7594	1.20164
$173$	−0.899816	−0.0684118	−0.0342059	−	0.999415i	$-0.510890\pi$
$173$	−0.899816	−0.0684118	−0.0342059	+	0.999415i	$0.510890\pi$
$174$	−2.33957	−0.177362
$175$	0	0
$176$	10.2513	0.772720
$177$	−13.8574	−1.04159
$178$	−1.06535	−0.0798513
$179$	−11.0558	−0.826351	−0.413175	−	0.910651i	$-0.635580\pi$
$179$	−11.0558	−0.826351	−0.413175	+	0.910651i	$0.635580\pi$
$180$	−1.76704	−0.131708
$181$	3.52898	0.262307	0.131153	−	0.991362i	$-0.458132\pi$
$181$	3.52898	0.262307	0.131153	+	0.991362i	$0.458132\pi$
$182$	0	0
$183$	0.751521	0.0555540
$184$	−3.15597	−0.232661
$185$	21.3699	1.57115
$186$	1.22146	0.0895619
$187$	12.8379	0.938799
$188$	3.63732	0.265279
$189$	0	0
$190$	0.0915382	0.00664088
$191$	−20.4004	−1.47612	−0.738059	−	0.674736i	$-0.764258\pi$
$191$	−20.4004	−1.47612	−0.738059	+	0.674736i	$0.764258\pi$
$192$	13.1946	0.952240
$193$	−17.2646	−1.24273	−0.621365	−	0.783521i	$-0.713422\pi$
$193$	−17.2646	−1.24273	−0.621365	+	0.783521i	$0.713422\pi$
$194$	−0.0815684	−0.00585627
$195$	0	0
$196$	0	0
$197$	−4.95672	−0.353152	−0.176576	−	0.984287i	$-0.556502\pi$
$197$	−4.95672	−0.353152	−0.176576	+	0.984287i	$0.556502\pi$
$198$	0.162877	0.0115751
$199$	7.18195	0.509115	0.254557	−	0.967058i	$-0.418070\pi$
$199$	7.18195	0.509115	0.254557	+	0.967058i	$0.418070\pi$
$200$	−1.59119	−0.112514
$201$	−20.8184	−1.46842
$202$	−1.38453	−0.0974153
$203$	0	0
$204$	17.1172	1.19844
$205$	14.5873	1.01882
$206$	0.932580	0.0649759
$207$	1.47078	0.102226
$208$	0	0
$209$	0.507661	0.0351157
$210$	0	0
$211$	−17.5927	−1.21113	−0.605566	−	0.795795i	$-0.707053\pi$
$211$	−17.5927	−1.21113	−0.605566	+	0.795795i	$0.707053\pi$
$212$	13.9136	0.955592
$213$	6.10780	0.418499
$214$	1.45254	0.0992934
$215$	−21.5218	−1.46777
$216$	−3.49190	−0.237594
$217$	0	0
$218$	0.241140	0.0163320
$219$	25.9582	1.75410
$220$	−14.2403	−0.960078
$221$	0	0
$222$	−2.62635	−0.176269
$223$	14.1054	0.944569	0.472284	−	0.881446i	$-0.343430\pi$
$223$	14.1054	0.944569	0.472284	+	0.881446i	$0.343430\pi$
$224$	0	0
$225$	0.741543	0.0494362
$226$	−3.60533	−0.239823
$227$	2.86877	0.190407	0.0952035	−	0.995458i	$-0.469650\pi$
$227$	2.86877	0.190407	0.0952035	+	0.995458i	$0.469650\pi$
$228$	0.676881	0.0448275
$229$	8.77411	0.579810	0.289905	−	0.957055i	$-0.406376\pi$
$229$	8.77411	0.579810	0.289905	+	0.957055i	$0.406376\pi$
$230$	2.13721	0.140924
$231$	0	0
$232$	−5.08309	−0.333721
$233$	5.10743	0.334599	0.167299	−	0.985906i	$-0.446495\pi$
$233$	5.10743	0.334599	0.167299	+	0.985906i	$0.446495\pi$
$234$	0	0
$235$	−4.96730	−0.324031
$236$	−14.9296	−0.971836
$237$	−16.6361	−1.08063
$238$	0	0
$239$	2.49797	0.161580	0.0807901	−	0.996731i	$-0.474256\pi$
$239$	2.49797	0.161580	0.0807901	+	0.996731i	$0.474256\pi$
$240$	−18.6662	−1.20490
$241$	−7.98512	−0.514367	−0.257183	−	0.966363i	$-0.582794\pi$
$241$	−7.98512	−0.514367	−0.257183	+	0.966363i	$0.582794\pi$
$242$	−0.676472	−0.0434853
$243$	3.45877	0.221880
$244$	0.809671	0.0518339
$245$	0	0
$246$	−1.79277	−0.114303
$247$	0	0
$248$	2.65382	0.168518
$249$	−30.2873	−1.91938
$250$	−1.35150	−0.0854762
$251$	25.2570	1.59421	0.797105	−	0.603841i	$-0.206364\pi$
$251$	25.2570	1.59421	0.797105	+	0.603841i	$0.206364\pi$
$252$	0	0
$253$	11.8527	0.745176
$254$	−1.44066	−0.0903952
$255$	−23.3760	−1.46386
$256$	13.4479	0.840493
$257$	−3.37363	−0.210442	−0.105221	−	0.994449i	$-0.533555\pi$
$257$	−3.37363	−0.210442	−0.105221	+	0.994449i	$0.533555\pi$
$258$	2.64501	0.164671
$259$	0	0
$260$	0	0
$261$	2.36888	0.146630
$262$	1.81148	0.111914
$263$	−0.158935	−0.00980037	−0.00490019	−	0.999988i	$-0.501560\pi$
$263$	−0.158935	−0.00980037	−0.00490019	+	0.999988i	$0.501560\pi$
$264$	3.52932	0.217215
$265$	−19.0011	−1.16723
$266$	0	0
$267$	10.7582	0.658392
$268$	−22.4293	−1.37009
$269$	−23.3266	−1.42225	−0.711124	−	0.703066i	$-0.751814\pi$
$269$	−23.3266	−1.42225	−0.711124	+	0.703066i	$0.751814\pi$
$270$	2.36470	0.143911
$271$	11.8210	0.718074	0.359037	−	0.933323i	$-0.383105\pi$
$271$	11.8210	0.718074	0.359037	+	0.933323i	$0.383105\pi$
$272$	18.1300	1.09929
$273$	0	0
$274$	−0.916226	−0.0553513
$275$	5.97595	0.360363
$276$	15.8036	0.951268
$277$	−27.3653	−1.64422	−0.822111	−	0.569327i	$-0.807204\pi$
$277$	−27.3653	−1.64422	−0.822111	+	0.569327i	$0.807204\pi$
$278$	1.39701	0.0837868
$279$	−1.23676	−0.0740431
$280$	0	0
$281$	28.5383	1.70245	0.851225	−	0.524801i	$-0.175860\pi$
$281$	28.5383	1.70245	0.851225	+	0.524801i	$0.175860\pi$
$282$	0.610478	0.0363534
$283$	17.9721	1.06833	0.534165	−	0.845380i	$-0.320626\pi$
$283$	17.9721	1.06833	0.534165	+	0.845380i	$0.320626\pi$
$284$	6.58040	0.390475
$285$	−0.924381	−0.0547556
$286$	0	0
$287$	0	0
$288$	0.709694	0.0418191
$289$	5.70459	0.335564
$290$	3.44225	0.202136
$291$	0.823703	0.0482863
$292$	27.9668	1.63663
$293$	14.8891	0.869828	0.434914	−	0.900472i	$-0.356779\pi$
$293$	14.8891	0.869828	0.434914	+	0.900472i	$0.356779\pi$
$294$	0	0
$295$	20.3886	1.18707
$296$	−5.70616	−0.331664
$297$	13.1144	0.760974
$298$	2.58912	0.149984
$299$	0	0
$300$	7.96793	0.460028
$301$	0	0
$302$	−1.17038	−0.0673478
$303$	13.9814	0.803211
$304$	0.716934	0.0411190
$305$	−1.10572	−0.0633136
$306$	0.288057	0.0164671
$307$	−23.5161	−1.34214	−0.671068	−	0.741396i	$-0.734164\pi$
$307$	−23.5161	−1.34214	−0.671068	+	0.741396i	$0.734164\pi$
$308$	0	0
$309$	−9.41747	−0.535741
$310$	−1.79715	−0.102072
$311$	1.63090	0.0924799	0.0462399	−	0.998930i	$-0.485276\pi$
$311$	1.63090	0.0924799	0.0462399	+	0.998930i	$0.485276\pi$
$312$	0	0
$313$	−0.696734	−0.0393817	−0.0196909	−	0.999806i	$-0.506268\pi$
$313$	−0.696734	−0.0393817	−0.0196909	+	0.999806i	$0.506268\pi$
$314$	2.87849	0.162442
$315$	0	0
$316$	−17.9233	−1.00826
$317$	−21.4288	−1.20356	−0.601780	−	0.798662i	$-0.705542\pi$
$317$	−21.4288	−1.20356	−0.601780	+	0.798662i	$0.705542\pi$
$318$	2.33522	0.130953
$319$	19.0903	1.06885
$320$	−19.4135	−1.08525
$321$	−14.6682	−0.818697
$322$	0	0
$323$	0.897830	0.0499566
$324$	19.4590	1.08105
$325$	0	0
$326$	0.864630	0.0478874
$327$	−2.43510	−0.134661
$328$	−3.89509	−0.215070
$329$	0	0
$330$	−2.39004	−0.131568
$331$	−1.52046	−0.0835722	−0.0417861	−	0.999127i	$-0.513305\pi$
$331$	−1.52046	−0.0835722	−0.0417861	+	0.999127i	$0.513305\pi$
$332$	−32.6308	−1.79085
$333$	2.65925	0.145726
$334$	0.490108	0.0268175
$335$	30.6305	1.67352
$336$	0	0
$337$	32.2304	1.75570	0.877850	−	0.478936i	$-0.158977\pi$
$337$	32.2304	1.75570	0.877850	+	0.478936i	$0.158977\pi$
$338$	0	0
$339$	36.4077	1.97739
$340$	−25.1848	−1.36584
$341$	−9.96683	−0.539734
$342$	0.0113909	0.000615951 0
$343$	0	0
$344$	5.74670	0.309841
$345$	−21.5822	−1.16195
$346$	−0.162708	−0.00874724
$347$	8.18431	0.439357	0.219678	−	0.975572i	$-0.429499\pi$
$347$	8.18431	0.439357	0.219678	+	0.975572i	$0.429499\pi$
$348$	25.4538	1.36446
$349$	21.8493	1.16956	0.584782	−	0.811190i	$-0.301180\pi$
$349$	21.8493	1.16956	0.584782	+	0.811190i	$0.301180\pi$
$350$	0	0
$351$	0	0
$352$	5.71928	0.304839
$353$	0.567179	0.0301879	0.0150940	−	0.999886i	$-0.495195\pi$
$353$	0.567179	0.0301879	0.0150940	+	0.999886i	$0.495195\pi$
$354$	−2.50575	−0.133179
$355$	−8.98650	−0.476954
$356$	11.5907	0.614303
$357$	0	0
$358$	−1.99915	−0.105659
$359$	−32.4043	−1.71024	−0.855118	−	0.518434i	$-0.826515\pi$
$359$	−32.4043	−1.71024	−0.855118	+	0.518434i	$0.826515\pi$
$360$	−0.644358	−0.0339606
$361$	−18.9645	−0.998131
$362$	0.638123	0.0335390
$363$	6.83122	0.358546
$364$	0	0
$365$	−38.1928	−1.99910
$366$	0.135893	0.00710323
$367$	−7.86888	−0.410752	−0.205376	−	0.978683i	$-0.565842\pi$
$367$	−7.86888	−0.410752	−0.205376	+	0.978683i	$0.565842\pi$
$368$	16.7388	0.872569
$369$	1.81523	0.0944973
$370$	3.86419	0.200889
$371$	0	0
$372$	−13.2891	−0.689007
$373$	−2.09163	−0.108300	−0.0541502	−	0.998533i	$-0.517245\pi$
$373$	−2.09163	−0.108300	−0.0541502	+	0.998533i	$0.517245\pi$
$374$	2.32139	0.120036
$375$	13.6478	0.704771
$376$	1.32636	0.0684018
$377$	0	0
$378$	0	0
$379$	14.3163	0.735381	0.367691	−	0.929948i	$-0.380149\pi$
$379$	14.3163	0.735381	0.367691	+	0.929948i	$0.380149\pi$
$380$	−0.995906	−0.0510889
$381$	14.5482	0.745329
$382$	−3.68887	−0.188739
$383$	25.1873	1.28701	0.643507	−	0.765441i	$-0.277479\pi$
$383$	25.1873	1.28701	0.643507	+	0.765441i	$0.277479\pi$
$384$	10.1383	0.517370
$385$	0	0
$386$	−3.12184	−0.158898
$387$	−2.67815	−0.136138
$388$	0.887438	0.0450528
$389$	28.1023	1.42484	0.712422	−	0.701752i	$-0.247598\pi$
$389$	28.1023	1.42484	0.712422	+	0.701752i	$0.247598\pi$
$390$	0	0
$391$	20.9623	1.06011
$392$	0	0
$393$	−18.2929	−0.922754
$394$	−0.896292	−0.0451546
$395$	24.4769	1.23157
$396$	−1.77204	−0.0890485
$397$	21.7765	1.09293	0.546465	−	0.837482i	$-0.315973\pi$
$397$	21.7765	1.09293	0.546465	+	0.837482i	$0.315973\pi$
$398$	1.29867	0.0650963
$399$	0	0
$400$	8.43941	0.421970
$401$	−20.5290	−1.02517	−0.512584	−	0.858637i	$-0.671312\pi$
$401$	−20.5290	−1.02517	−0.512584	+	0.858637i	$0.671312\pi$
$402$	−3.76447	−0.187754
$403$	0	0
$404$	15.0632	0.749424
$405$	−26.5741	−1.32048
$406$	0	0
$407$	21.4304	1.06226
$408$	6.24182	0.309016
$409$	6.26862	0.309963	0.154982	−	0.987917i	$-0.450468\pi$
$409$	6.26862	0.309963	0.154982	+	0.987917i	$0.450468\pi$
$410$	2.63774	0.130269
$411$	9.25233	0.456384
$412$	−10.1462	−0.499865
$413$	0	0
$414$	0.265953	0.0130709
$415$	44.5622	2.18747
$416$	0	0
$417$	−14.1074	−0.690841
$418$	0.0917972	0.00448995
$419$	34.1635	1.66899	0.834497	−	0.551013i	$-0.185758\pi$
$419$	34.1635	1.66899	0.834497	+	0.551013i	$0.185758\pi$
$420$	0	0
$421$	11.5233	0.561613	0.280806	−	0.959764i	$-0.409398\pi$
$421$	11.5233	0.561613	0.280806	+	0.959764i	$0.409398\pi$
$422$	−3.18118	−0.154858
$423$	−0.618126	−0.0300543
$424$	5.07364	0.246398
$425$	10.5688	0.512664
$426$	1.10444	0.0535101
$427$	0	0
$428$	−15.8031	−0.763873
$429$	0	0
$430$	−3.89164	−0.187672
$431$	8.77001	0.422436	0.211218	−	0.977439i	$-0.432257\pi$
$431$	8.77001	0.422436	0.211218	+	0.977439i	$0.432257\pi$
$432$	18.5205	0.891069
$433$	22.1069	1.06239	0.531196	−	0.847249i	$-0.321743\pi$
$433$	22.1069	1.06239	0.531196	+	0.847249i	$0.321743\pi$
$434$	0	0
$435$	−34.7609	−1.66666
$436$	−2.62352	−0.125644
$437$	0.828933	0.0396533
$438$	4.69387	0.224282
$439$	−10.3709	−0.494978	−0.247489	−	0.968891i	$-0.579605\pi$
$439$	−10.3709	−0.494978	−0.247489	+	0.968891i	$0.579605\pi$
$440$	−5.19275	−0.247555
$441$	0	0
$442$	0	0
$443$	35.8137	1.70156	0.850780	−	0.525522i	$-0.176130\pi$
$443$	35.8137	1.70156	0.850780	+	0.525522i	$0.176130\pi$
$444$	28.5738	1.35605
$445$	−15.8287	−0.750354
$446$	2.55059	0.120774
$447$	−26.1457	−1.23665
$448$	0	0
$449$	−22.7502	−1.07365	−0.536825	−	0.843693i	$-0.680377\pi$
$449$	−22.7502	−1.07365	−0.536825	+	0.843693i	$0.680377\pi$
$450$	0.134089	0.00632100
$451$	14.6286	0.688834
$452$	39.2248	1.84498
$453$	11.8188	0.555298
$454$	0.518742	0.0243458
$455$	0	0
$456$	0.246827	0.0115587
$457$	31.3172	1.46496	0.732478	−	0.680791i	$-0.238364\pi$
$457$	31.3172	1.46496	0.732478	+	0.680791i	$0.238364\pi$
$458$	1.58657	0.0741354
$459$	23.1936	1.08258
$460$	−23.2522	−1.08414
$461$	8.40753	0.391578	0.195789	−	0.980646i	$-0.437273\pi$
$461$	8.40753	0.391578	0.195789	+	0.980646i	$0.437273\pi$
$462$	0	0
$463$	10.0392	0.466563	0.233281	−	0.972409i	$-0.425054\pi$
$463$	10.0392	0.466563	0.233281	+	0.972409i	$0.425054\pi$
$464$	26.9599	1.25158
$465$	18.1482	0.841603
$466$	0.923545	0.0427824
$467$	−26.3513	−1.21939	−0.609696	−	0.792635i	$-0.708709\pi$
$467$	−26.3513	−1.21939	−0.609696	+	0.792635i	$0.708709\pi$
$468$	0	0
$469$	0	0
$470$	−0.898206	−0.0414312
$471$	−29.0678	−1.33937
$472$	−5.44413	−0.250586
$473$	−21.5826	−0.992371
$474$	−3.00819	−0.138171
$475$	0.417934	0.0191761
$476$	0	0
$477$	−2.36448	−0.108262
$478$	0.451692	0.0206599
$479$	−8.58414	−0.392220	−0.196110	−	0.980582i	$-0.562831\pi$
$479$	−8.58414	−0.392220	−0.196110	+	0.980582i	$0.562831\pi$
$480$	−10.4140	−0.475333
$481$	0	0
$482$	−1.44390	−0.0657678
$483$	0	0
$484$	7.35979	0.334536
$485$	−1.21193	−0.0550308
$486$	0.625428	0.0283700
$487$	21.2562	0.963212	0.481606	−	0.876388i	$-0.340054\pi$
$487$	21.2562	0.963212	0.481606	+	0.876388i	$0.340054\pi$
$488$	0.295249	0.0133653
$489$	−8.73130	−0.394843
$490$	0	0
$491$	−22.4535	−1.01331	−0.506657	−	0.862148i	$-0.669119\pi$
$491$	−22.4535	−1.01331	−0.506657	+	0.862148i	$0.669119\pi$
$492$	19.5048	0.879344
$493$	33.7624	1.52058
$494$	0	0
$495$	2.41999	0.108770
$496$	−14.0754	−0.632006
$497$	0	0
$498$	−5.47667	−0.245415
$499$	38.8780	1.74042	0.870210	−	0.492681i	$-0.163983\pi$
$499$	38.8780	1.74042	0.870210	+	0.492681i	$0.163983\pi$
$500$	14.7039	0.657576
$501$	−4.94926	−0.221117
$502$	4.56707	0.203838
$503$	5.45701	0.243316	0.121658	−	0.992572i	$-0.461179\pi$
$503$	5.45701	0.243316	0.121658	+	0.992572i	$0.461179\pi$
$504$	0	0
$505$	−20.5711	−0.915401
$506$	2.14326	0.0952794
$507$	0	0
$508$	15.6739	0.695418
$509$	−10.8925	−0.482800	−0.241400	−	0.970426i	$-0.577607\pi$
$509$	−10.8925	−0.482800	−0.241400	+	0.970426i	$0.577607\pi$
$510$	−4.22694	−0.187172
$511$	0	0
$512$	13.5360	0.598214
$513$	0.917168	0.0404939
$514$	−0.610033	−0.0269074
$515$	13.8561	0.610572
$516$	−28.7768	−1.26683
$517$	−4.98136	−0.219080
$518$	0	0
$519$	1.64308	0.0721230
$520$	0	0
$521$	27.8961	1.22215	0.611074	−	0.791573i	$-0.290738\pi$
$521$	27.8961	1.22215	0.611074	+	0.791573i	$0.290738\pi$
$522$	0.428350	0.0187484
$523$	16.7236	0.731272	0.365636	−	0.930758i	$-0.380852\pi$
$523$	16.7236	0.731272	0.365636	+	0.930758i	$0.380852\pi$
$524$	−19.7083	−0.860962
$525$	0	0
$526$	−0.0287393	−0.00125309
$527$	−17.6269	−0.767842
$528$	−18.7190	−0.814639
$529$	−3.64627	−0.158534
$530$	−3.43585	−0.149244
$531$	2.53714	0.110102
$532$	0	0
$533$	0	0
$534$	1.94534	0.0841832
$535$	21.5815	0.933049
$536$	−8.17890	−0.353275
$537$	20.1881	0.871179
$538$	−4.21800	−0.181851
$539$	0	0
$540$	−25.7272	−1.10712
$541$	11.1605	0.479828	0.239914	−	0.970794i	$-0.422881\pi$
$541$	11.1605	0.479828	0.239914	+	0.970794i	$0.422881\pi$
$542$	2.13751	0.0918141
$543$	−6.44396	−0.276537
$544$	10.1149	0.433673
$545$	3.58280	0.153470
$546$	0	0
$547$	36.6556	1.56728	0.783640	−	0.621215i	$-0.213361\pi$
$547$	36.6556	1.56728	0.783640	+	0.621215i	$0.213361\pi$
$548$	9.96824	0.425822
$549$	−0.137595	−0.00587242
$550$	1.08059	0.0460767
$551$	1.33510	0.0568772
$552$	5.76285	0.245283
$553$	0	0
$554$	−4.94830	−0.210233
$555$	−39.0217	−1.65638
$556$	−15.1990	−0.644579
$557$	−33.0776	−1.40154	−0.700772	−	0.713386i	$-0.747161\pi$
$557$	−33.0776	−1.40154	−0.700772	+	0.713386i	$0.747161\pi$
$558$	−0.223636	−0.00946727
$559$	0	0
$560$	0	0
$561$	−23.4421	−0.989728
$562$	5.16039	0.217678
$563$	17.7967	0.750043	0.375021	−	0.927016i	$-0.377635\pi$
$563$	17.7967	0.750043	0.375021	+	0.927016i	$0.377635\pi$
$564$	−6.64180	−0.279670
$565$	−53.5672	−2.25359
$566$	3.24978	0.136598
$567$	0	0
$568$	2.39956	0.100683
$569$	−8.22094	−0.344640	−0.172320	−	0.985041i	$-0.555126\pi$
$569$	−8.22094	−0.344640	−0.172320	+	0.985041i	$0.555126\pi$
$570$	−0.167150	−0.00700114
$571$	−25.7553	−1.07782	−0.538912	−	0.842362i	$-0.681165\pi$
$571$	−25.7553	−1.07782	−0.538912	+	0.842362i	$0.681165\pi$
$572$	0	0
$573$	37.2513	1.55620
$574$	0	0
$575$	9.75781	0.406929
$576$	−2.41579	−0.100658
$577$	−0.769393	−0.0320302	−0.0160151	−	0.999872i	$-0.505098\pi$
$577$	−0.769393	−0.0320302	−0.0160151	+	0.999872i	$0.505098\pi$
$578$	1.03152	0.0429058
$579$	31.5253	1.31015
$580$	−37.4505	−1.55505
$581$	0	0
$582$	0.148945	0.00617397
$583$	−19.0549	−0.789172
$584$	10.1982	0.422003
$585$	0	0
$586$	2.69229	0.111218
$587$	−12.0929	−0.499127	−0.249563	−	0.968358i	$-0.580287\pi$
$587$	−12.0929	−0.499127	−0.249563	+	0.968358i	$0.580287\pi$
$588$	0	0
$589$	−0.697040	−0.0287210
$590$	3.68674	0.151781
$591$	9.05103	0.372310
$592$	30.2646	1.24387
$593$	−15.9481	−0.654911	−0.327456	−	0.944867i	$-0.606191\pi$
$593$	−15.9481	−0.654911	−0.327456	+	0.944867i	$0.606191\pi$
$594$	2.37139	0.0972994
$595$	0	0
$596$	−28.1688	−1.15384
$597$	−13.1143	−0.536734
$598$	0	0
$599$	−7.11022	−0.290516	−0.145258	−	0.989394i	$-0.546401\pi$
$599$	−7.11022	−0.290516	−0.145258	+	0.989394i	$0.546401\pi$
$600$	2.90553	0.118618
$601$	20.7905	0.848064	0.424032	−	0.905647i	$-0.360614\pi$
$601$	20.7905	0.848064	0.424032	+	0.905647i	$0.360614\pi$
$602$	0	0
$603$	3.81163	0.155221
$604$	12.7333	0.518112
$605$	−10.0509	−0.408626
$606$	2.52817	0.102700
$607$	−7.71405	−0.313104	−0.156552	−	0.987670i	$-0.550038\pi$
$607$	−7.71405	−0.313104	−0.156552	+	0.987670i	$0.550038\pi$
$608$	0.399983	0.0162215
$609$	0	0
$610$	−0.199941	−0.00809539
$611$	0	0
$612$	−3.13397	−0.126683
$613$	20.4378	0.825476	0.412738	−	0.910850i	$-0.364572\pi$
$613$	20.4378	0.825476	0.412738	+	0.910850i	$0.364572\pi$
$614$	−4.25227	−0.171608
$615$	−26.6367	−1.07409
$616$	0	0
$617$	4.59812	0.185113	0.0925567	−	0.995707i	$-0.470496\pi$
$617$	4.59812	0.185113	0.0925567	+	0.995707i	$0.470496\pi$
$618$	−1.70290	−0.0685008
$619$	10.0528	0.404057	0.202028	−	0.979380i	$-0.435247\pi$
$619$	10.0528	0.404057	0.202028	+	0.979380i	$0.435247\pi$
$620$	19.5525	0.785245
$621$	21.4138	0.859306
$622$	0.294906	0.0118246
$623$	0	0
$624$	0	0
$625$	−31.1705	−1.24682
$626$	−0.125986	−0.00503541
$627$	−0.926996	−0.0370207
$628$	−31.3170	−1.24968
$629$	37.9009	1.51121
$630$	0	0
$631$	−7.27372	−0.289562	−0.144781	−	0.989464i	$-0.546248\pi$
$631$	−7.27372	−0.289562	−0.144781	+	0.989464i	$0.546248\pi$
$632$	−6.53578	−0.259979
$633$	32.1245	1.27684
$634$	−3.87483	−0.153889
$635$	−21.4051	−0.849434
$636$	−25.4065	−1.00743
$637$	0	0
$638$	3.45199	0.136665
$639$	−1.11827	−0.0442381
$640$	−14.9167	−0.589635
$641$	3.85033	0.152079	0.0760394	−	0.997105i	$-0.475773\pi$
$641$	3.85033	0.152079	0.0760394	+	0.997105i	$0.475773\pi$
$642$	−2.65235	−0.104680
$643$	−2.87709	−0.113461	−0.0567307	−	0.998390i	$-0.518068\pi$
$643$	−2.87709	−0.113461	−0.0567307	+	0.998390i	$0.518068\pi$
$644$	0	0
$645$	39.2990	1.54740
$646$	0.162349	0.00638754
$647$	37.1001	1.45856	0.729278	−	0.684218i	$-0.239856\pi$
$647$	37.1001	1.45856	0.729278	+	0.684218i	$0.239856\pi$
$648$	7.09577	0.278748
$649$	20.4463	0.802587
$650$	0	0
$651$	0	0
$652$	−9.40689	−0.368402
$653$	20.0950	0.786377	0.393189	−	0.919458i	$-0.371372\pi$
$653$	20.0950	0.786377	0.393189	+	0.919458i	$0.371372\pi$
$654$	−0.440324	−0.0172180
$655$	26.9146	1.05164
$656$	20.6589	0.806596
$657$	−4.75267	−0.185419
$658$	0	0
$659$	9.91058	0.386061	0.193031	−	0.981193i	$-0.438168\pi$
$659$	9.91058	0.386061	0.193031	+	0.981193i	$0.438168\pi$
$660$	26.0029	1.01216
$661$	47.2266	1.83690	0.918450	−	0.395537i	$-0.129442\pi$
$661$	47.2266	1.83690	0.918450	+	0.395537i	$0.129442\pi$
$662$	−0.274936	−0.0106857
$663$	0	0
$664$	−11.8989	−0.461768
$665$	0	0
$666$	0.480855	0.0186328
$667$	31.1716	1.20697
$668$	−5.33222	−0.206310
$669$	−25.7567	−0.995811
$670$	5.53872	0.213979
$671$	−1.10885	−0.0428068
$672$	0	0
$673$	6.91689	0.266627	0.133313	−	0.991074i	$-0.457438\pi$
$673$	6.91689	0.266627	0.133313	+	0.991074i	$0.457438\pi$
$674$	5.82801	0.224487
$675$	10.7965	0.415556
$676$	0	0
$677$	−12.3291	−0.473844	−0.236922	−	0.971529i	$-0.576139\pi$
$677$	−12.3291	−0.473844	−0.236922	+	0.971529i	$0.576139\pi$
$678$	6.58337	0.252833
$679$	0	0
$680$	−9.18369	−0.352179
$681$	−5.23841	−0.200736
$682$	−1.80224	−0.0690113
$683$	−24.5364	−0.938859	−0.469430	−	0.882970i	$-0.655540\pi$
$683$	−24.5364	−0.938859	−0.469430	+	0.882970i	$0.655540\pi$
$684$	−0.123930	−0.00473856
$685$	−13.6131	−0.520130
$686$	0	0
$687$	−16.0216	−0.611264
$688$	−30.4796	−1.16202
$689$	0	0
$690$	−3.90258	−0.148569
$691$	9.10716	0.346453	0.173226	−	0.984882i	$-0.444581\pi$
$691$	9.10716	0.346453	0.173226	+	0.984882i	$0.444581\pi$
$692$	1.77021	0.0672933
$693$	0	0
$694$	1.47992	0.0561769
$695$	20.7564	0.787336
$696$	9.28179	0.351825
$697$	25.8716	0.979956
$698$	3.95087	0.149542
$699$	−9.32623	−0.352750
$700$	0	0
$701$	−0.286950	−0.0108380	−0.00541898	−	0.999985i	$-0.501725\pi$
$701$	−0.286950	−0.0108380	−0.00541898	+	0.999985i	$0.501725\pi$
$702$	0	0
$703$	1.49875	0.0565265
$704$	−19.4684	−0.733742
$705$	9.07036	0.341609
$706$	0.102559	0.00385988
$707$	0	0
$708$	27.2617	1.02456
$709$	−18.5848	−0.697967	−0.348984	−	0.937129i	$-0.613473\pi$
$709$	−18.5848	−0.697967	−0.348984	+	0.937129i	$0.613473\pi$
$710$	−1.62497	−0.0609841
$711$	3.04588	0.114229
$712$	4.22656	0.158397
$713$	−16.2743	−0.609478
$714$	0	0
$715$	0	0
$716$	21.7501	0.812841
$717$	−4.56132	−0.170346
$718$	−5.85947	−0.218674
$719$	41.6949	1.55496	0.777479	−	0.628909i	$-0.216498\pi$
$719$	41.6949	1.55496	0.777479	+	0.628909i	$0.216498\pi$
$720$	3.41757	0.127365
$721$	0	0
$722$	−3.42923	−0.127623
$723$	14.5809	0.542271
$724$	−6.94257	−0.258019
$725$	15.7162	0.583684
$726$	1.23525	0.0458443
$727$	−32.7039	−1.21292	−0.606461	−	0.795113i	$-0.707411\pi$
$727$	−32.7039	−1.21292	−0.606461	+	0.795113i	$0.707411\pi$
$728$	0	0
$729$	23.3578	0.865105
$730$	−6.90616	−0.255608
$731$	−38.1702	−1.41178
$732$	−1.47847	−0.0546458
$733$	−9.93531	−0.366969	−0.183484	−	0.983023i	$-0.558738\pi$
$733$	−9.93531	−0.366969	−0.183484	+	0.983023i	$0.558738\pi$
$734$	−1.42288	−0.0525195
$735$	0	0
$736$	9.33871	0.344230
$737$	30.7171	1.13148
$738$	0.328237	0.0120826
$739$	−10.4022	−0.382649	−0.191325	−	0.981527i	$-0.561278\pi$
$739$	−10.4022	−0.382649	−0.191325	+	0.981527i	$0.561278\pi$
$740$	−42.0411	−1.54546
$741$	0	0
$742$	0	0
$743$	−1.70863	−0.0626837	−0.0313419	−	0.999509i	$-0.509978\pi$
$743$	−1.70863	−0.0626837	−0.0313419	+	0.999509i	$0.509978\pi$
$744$	−4.84590	−0.177659
$745$	38.4686	1.40938
$746$	−0.378216	−0.0138475
$747$	5.54528	0.202891
$748$	−25.2560	−0.923451
$749$	0	0
$750$	2.46785	0.0901132
$751$	29.9812	1.09403	0.547015	−	0.837123i	$-0.315764\pi$
$751$	29.9812	1.09403	0.547015	+	0.837123i	$0.315764\pi$
$752$	−7.03481	−0.256533
$753$	−46.1197	−1.68069
$754$	0	0
$755$	−17.3893	−0.632860
$756$	0	0
$757$	8.40458	0.305470	0.152735	−	0.988267i	$-0.451192\pi$
$757$	8.40458	0.305470	0.152735	+	0.988267i	$0.451192\pi$
$758$	2.58874	0.0940271
$759$	−21.6433	−0.785601
$760$	−0.363160	−0.0131732
$761$	−51.0590	−1.85089	−0.925444	−	0.378885i	$-0.876308\pi$
$761$	−51.0590	−1.85089	−0.925444	+	0.378885i	$0.876308\pi$
$762$	2.63067	0.0952990
$763$	0	0
$764$	40.1337	1.45199
$765$	4.27989	0.154740
$766$	4.55447	0.164560
$767$	0	0
$768$	−24.5560	−0.886088
$769$	−0.704439	−0.0254027	−0.0127014	−	0.999919i	$-0.504043\pi$
$769$	−0.704439	−0.0254027	−0.0127014	+	0.999919i	$0.504043\pi$
$770$	0	0
$771$	6.16030	0.221858
$772$	33.9646	1.22241
$773$	−1.26521	−0.0455066	−0.0227533	−	0.999741i	$-0.507243\pi$
$773$	−1.26521	−0.0455066	−0.0227533	+	0.999741i	$0.507243\pi$
$774$	−0.484272	−0.0174068
$775$	−8.20522	−0.294740
$776$	0.323607	0.0116168
$777$	0	0
$778$	5.08156	0.182183
$779$	1.02307	0.0366551
$780$	0	0
$781$	−9.01193	−0.322472
$782$	3.79048	0.135547
$783$	34.4896	1.23256
$784$	0	0
$785$	42.7679	1.52645
$786$	−3.30779	−0.117985
$787$	−43.7969	−1.56119	−0.780595	−	0.625037i	$-0.785084\pi$
$787$	−43.7969	−1.56119	−0.780595	+	0.625037i	$0.785084\pi$
$788$	9.75137	0.347378
$789$	0.290218	0.0103320
$790$	4.42600	0.157470
$791$	0	0
$792$	−0.646180	−0.0229610
$793$	0	0
$794$	3.93770	0.139744
$795$	34.6963	1.23055
$796$	−14.1291	−0.500792
$797$	13.6837	0.484700	0.242350	−	0.970189i	$-0.422082\pi$
$797$	13.6837	0.484700	0.242350	+	0.970189i	$0.422082\pi$
$798$	0	0
$799$	−8.80983	−0.311669
$800$	4.70842	0.166468
$801$	−1.96971	−0.0695964
$802$	−3.71212	−0.131080
$803$	−38.3008	−1.35161
$804$	40.9561	1.44441
$805$	0	0
$806$	0	0
$807$	42.5947	1.49940
$808$	5.49285	0.193238
$809$	9.11375	0.320422	0.160211	−	0.987083i	$-0.448782\pi$
$809$	9.11375	0.320422	0.160211	+	0.987083i	$0.448782\pi$
$810$	−4.80523	−0.168839
$811$	−2.31899	−0.0814309	−0.0407154	−	0.999171i	$-0.512964\pi$
$811$	−2.31899	−0.0814309	−0.0407154	+	0.999171i	$0.512964\pi$
$812$	0	0
$813$	−21.5853	−0.757028
$814$	3.87512	0.135823
$815$	12.8465	0.449993
$816$	−33.1057	−1.15893
$817$	−1.50940	−0.0528073
$818$	1.13352	0.0396325
$819$	0	0
$820$	−28.6977	−1.00217
$821$	−10.1447	−0.354053	−0.177026	−	0.984206i	$-0.556648\pi$
$821$	−10.1447	−0.354053	−0.177026	+	0.984206i	$0.556648\pi$
$822$	1.67304	0.0583540
$823$	−26.8178	−0.934811	−0.467405	−	0.884043i	$-0.654811\pi$
$823$	−26.8178	−0.934811	−0.467405	+	0.884043i	$0.654811\pi$
$824$	−3.69983	−0.128890
$825$	−10.9122	−0.379913
$826$	0	0
$827$	−33.6015	−1.16844	−0.584219	−	0.811596i	$-0.698599\pi$
$827$	−33.6015	−1.16844	−0.584219	+	0.811596i	$0.698599\pi$
$828$	−2.89348	−0.100555
$829$	11.8666	0.412142	0.206071	−	0.978537i	$-0.433932\pi$
$829$	11.8666	0.412142	0.206071	+	0.978537i	$0.433932\pi$
$830$	8.05791	0.279694
$831$	49.9694	1.73342
$832$	0	0
$833$	0	0
$834$	−2.55095	−0.0883322
$835$	7.28193	0.252001
$836$	−0.998723	−0.0345416
$837$	−18.0066	−0.622399
$838$	6.17756	0.213400
$839$	−34.6177	−1.19514	−0.597568	−	0.801818i	$-0.703866\pi$
$839$	−34.6177	−1.19514	−0.597568	+	0.801818i	$0.703866\pi$
$840$	0	0
$841$	21.2058	0.731234
$842$	2.08369	0.0718088
$843$	−52.1112	−1.79481
$844$	34.6102	1.19133
$845$	0	0
$846$	−0.111772	−0.00384280
$847$	0	0
$848$	−26.9098	−0.924087
$849$	−32.8173	−1.12629
$850$	1.91110	0.0655500
$851$	34.9925	1.19953
$852$	−12.0159	−0.411658
$853$	29.1897	0.999436	0.499718	−	0.866188i	$-0.333437\pi$
$853$	29.1897	0.999436	0.499718	+	0.866188i	$0.333437\pi$
$854$	0	0
$855$	0.169244	0.00578802
$856$	−5.76265	−0.196963
$857$	−25.3198	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$857$	−25.3198	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$858$	0	0
$859$	26.0849	0.890004	0.445002	−	0.895530i	$-0.353203\pi$
$859$	26.0849	0.890004	0.445002	+	0.895530i	$0.353203\pi$
$860$	42.3398	1.44378
$861$	0	0
$862$	1.58583	0.0540134
$863$	−36.0337	−1.22660	−0.613302	−	0.789849i	$-0.710159\pi$
$863$	−36.0337	−1.22660	−0.613302	+	0.789849i	$0.710159\pi$
$864$	10.3328	0.351527
$865$	−2.41748	−0.0821969
$866$	3.99746	0.135839
$867$	−10.4166	−0.353768
$868$	0	0
$869$	24.5461	0.832671
$870$	−6.28559	−0.213101
$871$	0	0
$872$	−0.956674	−0.0323971
$873$	−0.150811	−0.00510418
$874$	0.149891	0.00507013
$875$	0	0
$876$	−51.0677	−1.72542
$877$	9.12168	0.308017	0.154009	−	0.988070i	$-0.450782\pi$
$877$	9.12168	0.308017	0.154009	+	0.988070i	$0.450782\pi$
$878$	−1.87531	−0.0632887
$879$	−27.1876	−0.917015
$880$	27.5415	0.928425
$881$	13.0331	0.439095	0.219548	−	0.975602i	$-0.429542\pi$
$881$	13.0331	0.439095	0.219548	+	0.975602i	$0.429542\pi$
$882$	0	0
$883$	2.13222	0.0717548	0.0358774	−	0.999356i	$-0.488577\pi$
$883$	2.13222	0.0717548	0.0358774	+	0.999356i	$0.488577\pi$
$884$	0	0
$885$	−37.2299	−1.25147
$886$	6.47597	0.217564
$887$	−47.1715	−1.58386	−0.791932	−	0.610610i	$-0.790924\pi$
$887$	−47.1715	−1.58386	−0.791932	+	0.610610i	$0.790924\pi$
$888$	10.4195	0.349656
$889$	0	0
$890$	−2.86221	−0.0959416
$891$	−26.6493	−0.892785
$892$	−27.7496	−0.929126
$893$	−0.348376	−0.0116580
$894$	−4.72777	−0.158120
$895$	−29.7030	−0.992863
$896$	0	0
$897$	0	0
$898$	−4.11378	−0.137279
$899$	−26.2118	−0.874213
$900$	−1.45884	−0.0486280
$901$	−33.6997	−1.12270
$902$	2.64520	0.0880755
$903$	0	0
$904$	14.3034	0.475725
$905$	9.48110	0.315162
$906$	2.13713	0.0710013
$907$	16.0431	0.532702	0.266351	−	0.963876i	$-0.414182\pi$
$907$	16.0431	0.532702	0.266351	+	0.963876i	$0.414182\pi$
$908$	−5.64374	−0.187294
$909$	−2.55984	−0.0849047
$910$	0	0
$911$	−24.4319	−0.809466	−0.404733	−	0.914435i	$-0.632636\pi$
$911$	−24.4319	−0.809466	−0.404733	+	0.914435i	$0.632636\pi$
$912$	−1.30913	−0.0433496
$913$	44.6883	1.47897
$914$	5.66289	0.187312
$915$	2.01907	0.0667483
$916$	−17.2613	−0.570331
$917$	0	0
$918$	4.19395	0.138421
$919$	15.9160	0.525020	0.262510	−	0.964929i	$-0.415450\pi$
$919$	15.9160	0.525020	0.262510	+	0.964929i	$0.415450\pi$
$920$	−8.47897	−0.279543
$921$	42.9407	1.41494
$922$	1.52028	0.0500678
$923$	0	0
$924$	0	0
$925$	17.6426	0.580086
$926$	1.81533	0.0596555
$927$	1.72424	0.0566314
$928$	15.0412	0.493751
$929$	−47.3235	−1.55263	−0.776317	−	0.630342i	$-0.782914\pi$
$929$	−47.3235	−1.55263	−0.776317	+	0.630342i	$0.782914\pi$
$930$	3.28163	0.107609
$931$	0	0
$932$	−10.0479	−0.329129
$933$	−2.97805	−0.0974968
$934$	−4.76494	−0.155914
$935$	34.4908	1.12797
$936$	0	0
$937$	−29.7044	−0.970401	−0.485200	−	0.874403i	$-0.661253\pi$
$937$	−29.7044	−0.970401	−0.485200	+	0.874403i	$0.661253\pi$
$938$	0	0
$939$	1.27224	0.0415181
$940$	9.77219	0.318734
$941$	40.4224	1.31773	0.658866	−	0.752260i	$-0.271036\pi$
$941$	40.4224	1.31773	0.658866	+	0.752260i	$0.271036\pi$
$942$	−5.25615	−0.171255
$943$	23.8863	0.777844
$944$	28.8748	0.939795
$945$	0	0
$946$	−3.90265	−0.126886
$947$	45.9228	1.49229	0.746146	−	0.665783i	$-0.231902\pi$
$947$	45.9228	1.49229	0.746146	+	0.665783i	$0.231902\pi$
$948$	32.7282	1.06296
$949$	0	0
$950$	0.0755724	0.00245189
$951$	39.1292	1.26885
$952$	0	0
$953$	12.3893	0.401329	0.200664	−	0.979660i	$-0.435690\pi$
$953$	12.3893	0.401329	0.200664	+	0.979660i	$0.435690\pi$
$954$	−0.427554	−0.0138426
$955$	−54.8085	−1.77356
$956$	−4.91426	−0.158939
$957$	−34.8592	−1.12684
$958$	−1.55222	−0.0501499
$959$	0	0
$960$	35.4492	1.14412
$961$	−17.3151	−0.558553
$962$	0	0
$963$	2.68558	0.0865416
$964$	15.7091	0.505958
$965$	−46.3837	−1.49314
$966$	0	0
$967$	9.44932	0.303870	0.151935	−	0.988391i	$-0.451450\pi$
$967$	9.44932	0.303870	0.151935	+	0.988391i	$0.451450\pi$
$968$	2.68377	0.0862596
$969$	−1.63945	−0.0526667
$970$	−0.219145	−0.00703633
$971$	−20.9763	−0.673163	−0.336581	−	0.941654i	$-0.609271\pi$
$971$	−20.9763	−0.673163	−0.336581	+	0.941654i	$0.609271\pi$
$972$	−6.80445	−0.218253
$973$	0	0
$974$	3.84363	0.123158
$975$	0	0
$976$	−1.56595	−0.0501249
$977$	13.1843	0.421802	0.210901	−	0.977507i	$-0.432360\pi$
$977$	13.1843	0.421802	0.210901	+	0.977507i	$0.432360\pi$
$978$	−1.57883	−0.0504853
$979$	−15.8735	−0.507320
$980$	0	0
$981$	0.445840	0.0142346
$982$	−4.06013	−0.129564
$983$	23.3588	0.745032	0.372516	−	0.928026i	$-0.378495\pi$
$983$	23.3588	0.745032	0.372516	+	0.928026i	$0.378495\pi$
$984$	7.11248	0.226738
$985$	−13.3169	−0.424313
$986$	6.10505	0.194424
$987$	0	0
$988$	0	0
$989$	−35.2412	−1.12060
$990$	0.437591	0.0139076
$991$	39.4929	1.25453	0.627267	−	0.778804i	$-0.284173\pi$
$991$	39.4929	1.25453	0.627267	+	0.778804i	$0.284173\pi$
$992$	−7.85281	−0.249327
$993$	2.77639	0.0881060
$994$	0	0
$995$	19.2953	0.611703
$996$	59.5843	1.88800
$997$	−51.4791	−1.63036	−0.815180	−	0.579208i	$-0.803362\pi$
$997$	−51.4791	−1.63036	−0.815180	+	0.579208i	$0.803362\pi$
$998$	7.03007	0.222533
$999$	38.7172	1.22496

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cp.1.7		12
7.2	even	3		1183.2.e.j.508.6		24
7.4	even	3		1183.2.e.j.170.6		24
7.6	odd	2		8281.2.a.co.1.7		12
13.2	odd	12		637.2.q.g.589.4		12
13.7	odd	12		637.2.q.g.491.4		12
13.12	even	2	inner	8281.2.a.cp.1.6		12
91.2	odd	12		91.2.k.b.4.3	✓	12
91.20	even	12		637.2.q.i.491.4		12
91.25	even	6		1183.2.e.j.170.7		24
91.33	even	12		637.2.u.g.361.3		12
91.41	even	12		637.2.q.i.589.4		12
91.46	odd	12		91.2.k.b.23.4	yes	12
91.51	even	6		1183.2.e.j.508.7		24
91.54	even	12		637.2.k.i.459.3		12
91.59	even	12		637.2.k.i.569.4		12
91.67	odd	12		91.2.u.b.30.3	yes	12
91.72	odd	12		91.2.u.b.88.3	yes	12
91.80	even	12		637.2.u.g.30.3		12
91.90	odd	2		8281.2.a.co.1.6		12
273.2	even	12		819.2.bm.f.550.4		12
273.137	even	12		819.2.bm.f.478.3		12
273.158	even	12		819.2.do.e.667.4		12
273.254	even	12		819.2.do.e.361.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.k.b.4.3	✓	12	91.2	odd	12
91.2.k.b.23.4	yes	12	91.46	odd	12
91.2.u.b.30.3	yes	12	91.67	odd	12
91.2.u.b.88.3	yes	12	91.72	odd	12
637.2.k.i.459.3		12	91.54	even	12
637.2.k.i.569.4		12	91.59	even	12
637.2.q.g.491.4		12	13.7	odd	12
637.2.q.g.589.4		12	13.2	odd	12
637.2.q.i.491.4		12	91.20	even	12
637.2.q.i.589.4		12	91.41	even	12
637.2.u.g.30.3		12	91.80	even	12
637.2.u.g.361.3		12	91.33	even	12
819.2.bm.f.478.3		12	273.137	even	12
819.2.bm.f.550.4		12	273.2	even	12
819.2.do.e.361.4		12	273.254	even	12
819.2.do.e.667.4		12	273.158	even	12
1183.2.e.j.170.6		24	7.4	even	3
1183.2.e.j.170.7		24	91.25	even	6
1183.2.e.j.508.6		24	7.2	even	3
1183.2.e.j.508.7		24	91.51	even	6
8281.2.a.co.1.6		12	91.90	odd	2
8281.2.a.co.1.7		12	7.6	odd	2
8281.2.a.cp.1.6		12	13.12	even	2	inner
8281.2.a.cp.1.7		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+0.180824 q^{2} -1.82601 q^{3} -1.96730 q^{4} +2.68664 q^{5} -0.330186 q^{6} -0.717383 q^{8} +0.334323 q^{9} +O(q^{10})\)	\(q+0.180824 q^{2} -1.82601 q^{3} -1.96730 q^{4} +2.68664 q^{5} -0.330186 q^{6} -0.717383 q^{8} +0.334323 q^{9} +0.485809 q^{10} +2.69424 q^{11} +3.59232 q^{12} -4.90584 q^{15} +3.80489 q^{16} +4.76493 q^{17} +0.0604535 q^{18} +0.188424 q^{19} -5.28544 q^{20} +0.487183 q^{22} +4.39929 q^{23} +1.30995 q^{24} +2.21804 q^{25} +4.86756 q^{27} +7.08560 q^{29} -0.887093 q^{30} -3.69931 q^{31} +2.12278 q^{32} -4.91972 q^{33} +0.861613 q^{34} -0.657715 q^{36} +7.95413 q^{37} +0.0340716 q^{38} -1.92735 q^{40} +5.42958 q^{41} -8.01065 q^{43} -5.30039 q^{44} +0.898206 q^{45} +0.795496 q^{46} -1.84889 q^{47} -6.94777 q^{48} +0.401075 q^{50} -8.70083 q^{51} -7.07244 q^{53} +0.880171 q^{54} +7.23846 q^{55} -0.344066 q^{57} +1.28125 q^{58} +7.58888 q^{59} +9.65128 q^{60} -0.411564 q^{61} -0.668922 q^{62} -7.22592 q^{64} -0.889602 q^{66} +11.4010 q^{67} -9.37407 q^{68} -8.03315 q^{69} -3.34488 q^{71} -0.239838 q^{72} -14.2158 q^{73} +1.43830 q^{74} -4.05018 q^{75} -0.370688 q^{76} +9.11059 q^{79} +10.2224 q^{80} -9.89120 q^{81} +0.981797 q^{82} +16.5866 q^{83} +12.8017 q^{85} -1.44852 q^{86} -12.9384 q^{87} -1.93280 q^{88} -5.89165 q^{89} +0.162417 q^{90} -8.65473 q^{92} +6.75498 q^{93} -0.334323 q^{94} +0.506229 q^{95} -3.87622 q^{96} -0.451094 q^{97} +0.900747 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) \) 12 * q + 6 * q^3 + 8 * q^4 + 2 * q^9	\( 12 q + 6 q^{3} + 8 q^{4} + 2 q^{9} + 24 q^{10} - 2 q^{12} + 16 q^{16} + 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} + 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} + 38 q^{38} + 2 q^{40} + 22 q^{43} - 38 q^{48} + 8 q^{51} + 16 q^{53} + 30 q^{55} - 10 q^{61} + 82 q^{62} - 2 q^{64} + 68 q^{66} + 22 q^{68} + 14 q^{69} + 66 q^{74} + 2 q^{75} + 70 q^{79} - 28 q^{81} + 10 q^{82} - 20 q^{87} - 28 q^{88} - 66 q^{92} - 2 q^{94} + 4 q^{95}+O(q^{100}) \) 12 * q + 6 * q^3 + 8 * q^4 + 2 * q^9 + 24 * q^10 - 2 * q^12 + 16 * q^16 + 34 * q^17 + 30 * q^22 + 6 * q^23 - 10 * q^25 + 12 * q^27 + 2 * q^29 + 22 * q^30 - 26 * q^36 + 38 * q^38 + 2 * q^40 + 22 * q^43 - 38 * q^48 + 8 * q^51 + 16 * q^53 + 30 * q^55 - 10 * q^61 + 82 * q^62 - 2 * q^64 + 68 * q^66 + 22 * q^68 + 14 * q^69 + 66 * q^74 + 2 * q^75 + 70 * q^79 - 28 * q^81 + 10 * q^82 - 20 * q^87 - 28 * q^88 - 66 * q^92 - 2 * q^94 + 4 * q^95

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