LMFDB - Embedded newform 8281.2.a.cp.1.11

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			1.11
Root			$2.30327$ 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)

$f(q)$	$=$	$q+2.30327 q^{2} -1.47336 q^{3} +3.30504 q^{4} +0.847292 q^{5} -3.39354 q^{6} +3.00585 q^{8} -0.829208 q^{9} +O(q^{10})$	\(q+2.30327 q^{2} -1.47336 q^{3} +3.30504 q^{4} +0.847292 q^{5} -3.39354 q^{6} +3.00585 q^{8} -0.829208 q^{9} +1.95154 q^{10} -1.50340 q^{11} -4.86951 q^{12} -1.24837 q^{15} +0.313194 q^{16} +2.07140 q^{17} -1.90989 q^{18} -0.0474272 q^{19} +2.80033 q^{20} -3.46274 q^{22} -7.81870 q^{23} -4.42870 q^{24} -4.28210 q^{25} +5.64180 q^{27} +1.35971 q^{29} -2.87532 q^{30} +7.86105 q^{31} -5.29033 q^{32} +2.21505 q^{33} +4.77099 q^{34} -2.74056 q^{36} +6.70219 q^{37} -0.109237 q^{38} +2.54683 q^{40} +10.0184 q^{41} +9.26566 q^{43} -4.96880 q^{44} -0.702581 q^{45} -18.0086 q^{46} +0.360014 q^{47} -0.461448 q^{48} -9.86281 q^{50} -3.05192 q^{51} +2.71181 q^{53} +12.9946 q^{54} -1.27382 q^{55} +0.0698773 q^{57} +3.13177 q^{58} +1.64120 q^{59} -4.12590 q^{60} +4.52194 q^{61} +18.1061 q^{62} -12.8114 q^{64} +5.10186 q^{66} -2.04266 q^{67} +6.84606 q^{68} +11.5198 q^{69} +14.2139 q^{71} -2.49247 q^{72} +6.76150 q^{73} +15.4369 q^{74} +6.30907 q^{75} -0.156749 q^{76} +11.6590 q^{79} +0.265367 q^{80} -5.82479 q^{81} +23.0751 q^{82} +11.5362 q^{83} +1.75508 q^{85} +21.3413 q^{86} -2.00334 q^{87} -4.51900 q^{88} -17.5112 q^{89} -1.61823 q^{90} -25.8411 q^{92} -11.5822 q^{93} +0.829208 q^{94} -0.0401846 q^{95} +7.79456 q^{96} +0.426229 q^{97} +1.24663 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.30327	1.62866	0.814328	−	0.580405i	$-0.197106\pi$
$2$	2.30327	1.62866	0.814328	+	0.580405i	$0.197106\pi$
$3$	−1.47336	−0.850645	−0.425323	−	0.905042i	$-0.639839\pi$
$3$	−1.47336	−0.850645	−0.425323	+	0.905042i	$0.639839\pi$
$4$	3.30504	1.65252
$5$	0.847292	0.378920	0.189460	−	0.981888i	$-0.439326\pi$
$5$	0.847292	0.378920	0.189460	+	0.981888i	$0.439326\pi$
$6$	−3.39354	−1.38541
$7$	0	0
$7$	0	0
$8$	3.00585	1.06273
$9$	−0.829208	−0.276403
$10$	1.95154	0.617131
$11$	−1.50340	−0.453293	−0.226646	−	0.973977i	$-0.572776\pi$
$11$	−1.50340	−0.453293	−0.226646	+	0.973977i	$0.572776\pi$
$12$	−4.86951	−1.40571
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.24837	−0.322327
$16$	0.313194	0.0782985
$17$	2.07140	0.502389	0.251194	−	0.967937i	$-0.419177\pi$
$17$	2.07140	0.502389	0.251194	+	0.967937i	$0.419177\pi$
$18$	−1.90989	−0.450164
$19$	−0.0474272	−0.0108805	−0.00544027	−	0.999985i	$-0.501732\pi$
$19$	−0.0474272	−0.0108805	−0.00544027	+	0.999985i	$0.501732\pi$
$20$	2.80033	0.626173
$21$	0	0
$22$	−3.46274	−0.738258
$23$	−7.81870	−1.63031	−0.815156	−	0.579241i	$-0.803349\pi$
$23$	−7.81870	−1.63031	−0.815156	+	0.579241i	$0.803349\pi$
$24$	−4.42870	−0.904004
$25$	−4.28210	−0.856419
$26$	0	0
$27$	5.64180	1.08577
$28$	0	0
$29$	1.35971	0.252491	0.126246	−	0.991999i	$-0.459707\pi$
$29$	1.35971	0.252491	0.126246	+	0.991999i	$0.459707\pi$
$30$	−2.87532	−0.524959
$31$	7.86105	1.41189	0.705943	−	0.708269i	$-0.250523\pi$
$31$	7.86105	1.41189	0.705943	+	0.708269i	$0.250523\pi$
$32$	−5.29033	−0.935206
$33$	2.21505	0.385591
$34$	4.77099	0.818218
$35$	0	0
$36$	−2.74056	−0.456760
$37$	6.70219	1.10183	0.550917	−	0.834560i	$-0.314278\pi$
$37$	6.70219	1.10183	0.550917	+	0.834560i	$0.314278\pi$
$38$	−0.109237	−0.0177206
$39$	0	0
$40$	2.54683	0.402689
$41$	10.0184	1.56462	0.782309	−	0.622891i	$-0.214042\pi$
$41$	10.0184	1.56462	0.782309	+	0.622891i	$0.214042\pi$
$42$	0	0
$43$	9.26566	1.41300	0.706500	−	0.707713i	$-0.250273\pi$
$43$	9.26566	1.41300	0.706500	+	0.707713i	$0.250273\pi$
$44$	−4.96880	−0.749075
$45$	−0.702581	−0.104735
$46$	−18.0086	−2.65522
$47$	0.360014	0.0525134	0.0262567	−	0.999655i	$-0.491641\pi$
$47$	0.360014	0.0525134	0.0262567	+	0.999655i	$0.491641\pi$
$48$	−0.461448	−0.0666042
$49$	0	0
$50$	−9.86281	−1.39481
$51$	−3.05192	−0.427355
$52$	0	0
$53$	2.71181	0.372496	0.186248	−	0.982503i	$-0.440367\pi$
$53$	2.71181	0.372496	0.186248	+	0.982503i	$0.440367\pi$
$54$	12.9946	1.76834
$55$	−1.27382	−0.171762
$56$	0	0
$57$	0.0698773	0.00925548
$58$	3.13177	0.411222
$59$	1.64120	0.213666	0.106833	−	0.994277i	$-0.465929\pi$
$59$	1.64120	0.213666	0.106833	+	0.994277i	$0.465929\pi$
$60$	−4.12590	−0.532651
$61$	4.52194	0.578975	0.289488	−	0.957182i	$-0.406515\pi$
$61$	4.52194	0.578975	0.289488	+	0.957182i	$0.406515\pi$
$62$	18.1061	2.29948
$63$	0	0
$64$	−12.8114	−1.60143
$65$	0	0
$66$	5.10186	0.627995
$67$	−2.04266	−0.249551	−0.124775	−	0.992185i	$-0.539821\pi$
$67$	−2.04266	−0.249551	−0.124775	+	0.992185i	$0.539821\pi$
$68$	6.84606	0.830206
$69$	11.5198	1.38682
$70$	0	0
$71$	14.2139	1.68688	0.843442	−	0.537220i	$-0.180526\pi$
$71$	14.2139	1.68688	0.843442	+	0.537220i	$0.180526\pi$
$72$	−2.49247	−0.293741
$73$	6.76150	0.791373	0.395687	−	0.918386i	$-0.370507\pi$
$73$	6.76150	0.791373	0.395687	+	0.918386i	$0.370507\pi$
$74$	15.4369	1.79451
$75$	6.30907	0.728509
$76$	−0.156749	−0.0179803
$77$	0	0
$78$	0	0
$79$	11.6590	1.31175	0.655873	−	0.754871i	$-0.272301\pi$
$79$	11.6590	1.31175	0.655873	+	0.754871i	$0.272301\pi$
$80$	0.265367	0.0296689
$81$	−5.82479	−0.647199
$82$	23.0751	2.54822
$83$	11.5362	1.26627	0.633133	−	0.774043i	$-0.281768\pi$
$83$	11.5362	1.26627	0.633133	+	0.774043i	$0.281768\pi$
$84$	0	0
$85$	1.75508	0.190365
$86$	21.3413	2.30129
$87$	−2.00334	−0.214781
$88$	−4.51900	−0.481727
$89$	−17.5112	−1.85619	−0.928093	−	0.372350i	$-0.878552\pi$
$89$	−17.5112	−1.85619	−0.928093	+	0.372350i	$0.878552\pi$
$90$	−1.61823	−0.170576
$91$	0	0
$92$	−25.8411	−2.69412
$93$	−11.5822	−1.20101
$94$	0.829208	0.0855262
$95$	−0.0401846	−0.00412286
$96$	7.79456	0.795529
$97$	0.426229	0.0432770	0.0216385	−	0.999766i	$-0.493112\pi$
$97$	0.426229	0.0432770	0.0216385	+	0.999766i	$0.493112\pi$
$98$	0	0
$99$	1.24663	0.125291
$100$	−14.1525	−1.41525
$101$	9.66997	0.962198	0.481099	−	0.876666i	$-0.340238\pi$
$101$	9.66997	0.962198	0.481099	+	0.876666i	$0.340238\pi$
$102$	−7.02939	−0.696013
$103$	−9.97823	−0.983185	−0.491592	−	0.870825i	$-0.663585\pi$
$103$	−9.97823	−0.983185	−0.491592	+	0.870825i	$0.663585\pi$
$104$	0	0
$105$	0	0
$106$	6.24603	0.606668
$107$	9.86223	0.953417	0.476709	−	0.879061i	$-0.341830\pi$
$107$	9.86223	0.953417	0.476709	+	0.879061i	$0.341830\pi$
$108$	18.6464	1.79425
$109$	11.6055	1.11161	0.555803	−	0.831314i	$-0.312411\pi$
$109$	11.6055	1.11161	0.555803	+	0.831314i	$0.312411\pi$
$110$	−2.93395	−0.279741
$111$	−9.87475	−0.937269
$112$	0	0
$113$	−3.47758	−0.327143	−0.163572	−	0.986531i	$-0.552301\pi$
$113$	−3.47758	−0.327143	−0.163572	+	0.986531i	$0.552301\pi$
$114$	0.160946	0.0150740
$115$	−6.62472	−0.617758
$116$	4.49388	0.417247
$117$	0	0
$118$	3.78011	0.347987
$119$	0	0
$120$	−3.75240	−0.342546
$121$	−8.73978	−0.794526
$122$	10.4152	0.942951
$123$	−14.7608	−1.33093
$124$	25.9811	2.33317
$125$	−7.86464	−0.703435
$126$	0	0
$127$	−15.6998	−1.39313	−0.696567	−	0.717491i	$-0.745290\pi$
$127$	−15.6998	−1.39313	−0.696567	+	0.717491i	$0.745290\pi$
$128$	−18.9275	−1.67297
$129$	−13.6517	−1.20196
$130$	0	0
$131$	−2.54517	−0.222373	−0.111186	−	0.993800i	$-0.535465\pi$
$131$	−2.54517	−0.222373	−0.111186	+	0.993800i	$0.535465\pi$
$132$	7.32083	0.637197
$133$	0	0
$134$	−4.70479	−0.406432
$135$	4.78025	0.411419
$136$	6.22632	0.533902
$137$	−1.86472	−0.159314	−0.0796571	−	0.996822i	$-0.525383\pi$
$137$	−1.86472	−0.159314	−0.0796571	+	0.996822i	$0.525383\pi$
$138$	26.5331	2.25865
$139$	15.6092	1.32396	0.661979	−	0.749522i	$-0.269717\pi$
$139$	15.6092	1.32396	0.661979	+	0.749522i	$0.269717\pi$
$140$	0	0
$141$	−0.530430	−0.0446703
$142$	32.7385	2.74735
$143$	0	0
$144$	−0.259703	−0.0216419
$145$	1.15207	0.0956741
$146$	15.5735	1.28887
$147$	0	0
$148$	22.1510	1.82080
$149$	−6.36363	−0.521329	−0.260664	−	0.965429i	$-0.583942\pi$
$149$	−6.36363	−0.521329	−0.260664	+	0.965429i	$0.583942\pi$
$150$	14.5315	1.18649
$151$	0.664094	0.0540432	0.0270216	−	0.999635i	$-0.491398\pi$
$151$	0.664094	0.0540432	0.0270216	+	0.999635i	$0.491398\pi$
$152$	−0.142559	−0.0115630
$153$	−1.71762	−0.138861
$154$	0	0
$155$	6.66060	0.534992
$156$	0	0
$157$	−16.5760	−1.32291	−0.661453	−	0.749986i	$-0.730060\pi$
$157$	−16.5760	−1.32291	−0.661453	+	0.749986i	$0.730060\pi$
$158$	26.8539	2.13638
$159$	−3.99548	−0.316862
$160$	−4.48245	−0.354369
$161$	0	0
$162$	−13.4160	−1.05406
$163$	9.05127	0.708950	0.354475	−	0.935065i	$-0.384660\pi$
$163$	9.05127	0.708950	0.354475	+	0.935065i	$0.384660\pi$
$164$	33.1113	2.58556
$165$	1.87680	0.146108
$166$	26.5710	2.06231
$167$	−2.65761	−0.205652	−0.102826	−	0.994699i	$-0.532788\pi$
$167$	−2.65761	−0.205652	−0.102826	+	0.994699i	$0.532788\pi$
$168$	0	0
$169$	0	0
$170$	4.04242	0.310039
$171$	0.0393270	0.00300741
$172$	30.6234	2.33501
$173$	19.5870	1.48918	0.744588	−	0.667525i	$-0.232646\pi$
$173$	19.5870	1.48918	0.744588	+	0.667525i	$0.232646\pi$
$174$	−4.61423	−0.349804
$175$	0	0
$176$	−0.470856	−0.0354921
$177$	−2.41807	−0.181754
$178$	−40.3330	−3.02309
$179$	−2.89332	−0.216257	−0.108129	−	0.994137i	$-0.534486\pi$
$179$	−2.89332	−0.216257	−0.108129	+	0.994137i	$0.534486\pi$
$180$	−2.32205	−0.173076
$181$	1.36804	0.101686	0.0508429	−	0.998707i	$-0.483809\pi$
$181$	1.36804	0.101686	0.0508429	+	0.998707i	$0.483809\pi$
$182$	0	0
$183$	−6.66245	−0.492503
$184$	−23.5018	−1.73258
$185$	5.67871	0.417507
$186$	−26.6768	−1.95604
$187$	−3.11415	−0.227729
$188$	1.18986	0.0867794
$189$	0	0
$190$	−0.0925559	−0.00671471
$191$	−1.51325	−0.109495	−0.0547475	−	0.998500i	$-0.517435\pi$
$191$	−1.51325	−0.109495	−0.0547475	+	0.998500i	$0.517435\pi$
$192$	18.8758	1.36225
$193$	6.95394	0.500556	0.250278	−	0.968174i	$-0.419478\pi$
$193$	6.95394	0.500556	0.250278	+	0.968174i	$0.419478\pi$
$194$	0.981719	0.0704834
$195$	0	0
$196$	0	0
$197$	15.4772	1.10271	0.551353	−	0.834272i	$-0.314112\pi$
$197$	15.4772	1.10271	0.551353	+	0.834272i	$0.314112\pi$
$198$	2.87133	0.204056
$199$	6.61529	0.468945	0.234473	−	0.972123i	$-0.424664\pi$
$199$	6.61529	0.468945	0.234473	+	0.972123i	$0.424664\pi$
$200$	−12.8713	−0.910140
$201$	3.00958	0.212279
$202$	22.2725	1.56709
$203$	0	0
$204$	−10.0867	−0.706211
$205$	8.48854	0.592865
$206$	−22.9825	−1.60127
$207$	6.48333	0.450622
$208$	0	0
$209$	0.0713021	0.00493207
$210$	0	0
$211$	−8.09428	−0.557234	−0.278617	−	0.960402i	$-0.589876\pi$
$211$	−8.09428	−0.557234	−0.278617	+	0.960402i	$0.589876\pi$
$212$	8.96264	0.615557
$213$	−20.9423	−1.43494
$214$	22.7153	1.55279
$215$	7.85072	0.535415
$216$	16.9584	1.15387
$217$	0	0
$218$	26.7306	1.81042
$219$	−9.96212	−0.673178
$220$	−4.21002	−0.283840
$221$	0	0
$222$	−22.7442	−1.52649
$223$	−16.0581	−1.07533	−0.537664	−	0.843159i	$-0.680693\pi$
$223$	−16.0581	−1.07533	−0.537664	+	0.843159i	$0.680693\pi$
$224$	0	0
$225$	3.55075	0.236716
$226$	−8.00979	−0.532804
$227$	1.29581	0.0860057	0.0430029	−	0.999075i	$-0.486308\pi$
$227$	1.29581	0.0860057	0.0430029	+	0.999075i	$0.486308\pi$
$228$	0.230947	0.0152948
$229$	−20.8175	−1.37566	−0.687831	−	0.725871i	$-0.741437\pi$
$229$	−20.8175	−1.37566	−0.687831	+	0.725871i	$0.741437\pi$
$230$	−15.2585	−1.00612
$231$	0	0
$232$	4.08707	0.268330
$233$	−13.3043	−0.871591	−0.435796	−	0.900046i	$-0.643533\pi$
$233$	−13.3043	−0.871591	−0.435796	+	0.900046i	$0.643533\pi$
$234$	0	0
$235$	0.305037	0.0198984
$236$	5.42421	0.353086
$237$	−17.1780	−1.11583
$238$	0	0
$239$	13.3652	0.864525	0.432263	−	0.901748i	$-0.357715\pi$
$239$	13.3652	0.864525	0.432263	+	0.901748i	$0.357715\pi$
$240$	−0.390981	−0.0252377
$241$	−0.834153	−0.0537325	−0.0268663	−	0.999639i	$-0.508553\pi$
$241$	−0.834153	−0.0537325	−0.0268663	+	0.999639i	$0.508553\pi$
$242$	−20.1300	−1.29401
$243$	−8.34339	−0.535229
$244$	14.9452	0.956767
$245$	0	0
$246$	−33.9980	−2.16763
$247$	0	0
$248$	23.6291	1.50045
$249$	−16.9970	−1.07714
$250$	−18.1144	−1.14565
$251$	27.2721	1.72140	0.860699	−	0.509114i	$-0.170027\pi$
$251$	27.2721	1.72140	0.860699	+	0.509114i	$0.170027\pi$
$252$	0	0
$253$	11.7547	0.739009
$254$	−36.1609	−2.26894
$255$	−2.58587	−0.161933
$256$	−17.9721	−1.12326
$257$	6.55188	0.408695	0.204348	−	0.978898i	$-0.434493\pi$
$257$	6.55188	0.408695	0.204348	+	0.978898i	$0.434493\pi$
$258$	−31.4434	−1.95758
$259$	0	0
$260$	0	0
$261$	−1.12748	−0.0697893
$262$	−5.86221	−0.362168
$263$	−22.5891	−1.39290	−0.696450	−	0.717605i	$-0.745238\pi$
$263$	−22.5891	−1.39290	−0.696450	+	0.717605i	$0.745238\pi$
$264$	6.65811	0.409779
$265$	2.29770	0.141146
$266$	0	0
$267$	25.8003	1.57896
$268$	−6.75107	−0.412387
$269$	16.0013	0.975617	0.487808	−	0.872951i	$-0.337797\pi$
$269$	16.0013	0.975617	0.487808	+	0.872951i	$0.337797\pi$
$270$	11.0102	0.670059
$271$	8.75935	0.532093	0.266046	−	0.963960i	$-0.414283\pi$
$271$	8.75935	0.532093	0.266046	+	0.963960i	$0.414283\pi$
$272$	0.648750	0.0393363
$273$	0	0
$274$	−4.29496	−0.259468
$275$	6.43771	0.388209
$276$	38.0733	2.29174
$277$	−19.9183	−1.19677	−0.598387	−	0.801208i	$-0.704191\pi$
$277$	−19.9183	−1.19677	−0.598387	+	0.801208i	$0.704191\pi$
$278$	35.9522	2.15627
$279$	−6.51844	−0.390249
$280$	0	0
$281$	14.0234	0.836566	0.418283	−	0.908317i	$-0.362632\pi$
$281$	14.0234	0.836566	0.418283	+	0.908317i	$0.362632\pi$
$282$	−1.22172	−0.0727525
$283$	1.01259	0.0601922	0.0300961	−	0.999547i	$-0.490419\pi$
$283$	1.01259	0.0601922	0.0300961	+	0.999547i	$0.490419\pi$
$284$	46.9776	2.78761
$285$	0.0592065	0.00350709
$286$	0	0
$287$	0	0
$288$	4.38678	0.258493
$289$	−12.7093	−0.747606
$290$	2.65352	0.155820
$291$	−0.627989	−0.0368134
$292$	22.3470	1.30776
$293$	0.199235	0.0116394	0.00581972	−	0.999983i	$-0.498148\pi$
$293$	0.199235	0.0116394	0.00581972	+	0.999983i	$0.498148\pi$
$294$	0	0
$295$	1.39057	0.0809622
$296$	20.1458	1.17095
$297$	−8.48190	−0.492170
$298$	−14.6571	−0.849065
$299$	0	0
$300$	20.8517	1.20387
$301$	0	0
$302$	1.52958	0.0880177
$303$	−14.2474	−0.818489
$304$	−0.0148539	−0.000851930 0
$305$	3.83140	0.219386
$306$	−3.95614	−0.226157
$307$	27.2004	1.55241	0.776204	−	0.630482i	$-0.217143\pi$
$307$	27.2004	1.55241	0.776204	+	0.630482i	$0.217143\pi$
$308$	0	0
$309$	14.7015	0.836341
$310$	15.3411	0.871318
$311$	27.1009	1.53675	0.768376	−	0.639999i	$-0.221065\pi$
$311$	27.1009	1.53675	0.768376	+	0.639999i	$0.221065\pi$
$312$	0	0
$313$	−22.0785	−1.24795	−0.623975	−	0.781445i	$-0.714483\pi$
$313$	−22.0785	−1.24795	−0.623975	+	0.781445i	$0.714483\pi$
$314$	−38.1789	−2.15456
$315$	0	0
$316$	38.5336	2.16768
$317$	−7.06823	−0.396991	−0.198496	−	0.980102i	$-0.563606\pi$
$317$	−7.06823	−0.396991	−0.198496	+	0.980102i	$0.563606\pi$
$318$	−9.20265	−0.516059
$319$	−2.04419	−0.114453
$320$	−10.8550	−0.606813
$321$	−14.5306	−0.811020
$322$	0	0
$323$	−0.0982407	−0.00546626
$324$	−19.2512	−1.06951
$325$	0	0
$326$	20.8475	1.15464
$327$	−17.0991	−0.945582
$328$	30.1139	1.66276
$329$	0	0
$330$	4.32276	0.237960
$331$	−6.58858	−0.362141	−0.181071	−	0.983470i	$-0.557956\pi$
$331$	−6.58858	−0.362141	−0.181071	+	0.983470i	$0.557956\pi$
$332$	38.1277	2.09253
$333$	−5.55751	−0.304550
$334$	−6.12118	−0.334936
$335$	−1.73073	−0.0945599
$336$	0	0
$337$	4.22290	0.230036	0.115018	−	0.993363i	$-0.463307\pi$
$337$	4.22290	0.230036	0.115018	+	0.993363i	$0.463307\pi$
$338$	0	0
$339$	5.12373	0.278283
$340$	5.80061	0.314582
$341$	−11.8183	−0.639998
$342$	0.0905805	0.00489803
$343$	0	0
$344$	27.8512	1.50163
$345$	9.76060	0.525493
$346$	45.1142	2.42535
$347$	−9.09478	−0.488233	−0.244117	−	0.969746i	$-0.578498\pi$
$347$	−9.09478	−0.488233	−0.244117	+	0.969746i	$0.578498\pi$
$348$	−6.62111	−0.354929
$349$	9.22053	0.493564	0.246782	−	0.969071i	$-0.420627\pi$
$349$	9.22053	0.493564	0.246782	+	0.969071i	$0.420627\pi$
$350$	0	0
$351$	0	0
$352$	7.95349	0.423922
$353$	2.15449	0.114672	0.0573359	−	0.998355i	$-0.481739\pi$
$353$	2.15449	0.114672	0.0573359	+	0.998355i	$0.481739\pi$
$354$	−5.56947	−0.296014
$355$	12.0433	0.639195
$356$	−57.8752	−3.06738
$357$	0	0
$358$	−6.66410	−0.352208
$359$	8.55756	0.451651	0.225825	−	0.974168i	$-0.427492\pi$
$359$	8.55756	0.451651	0.225825	+	0.974168i	$0.427492\pi$
$360$	−2.11185	−0.111304
$361$	−18.9978	−0.999882
$362$	3.15096	0.165611
$363$	12.8769	0.675860
$364$	0	0
$365$	5.72896	0.299867
$366$	−15.3454	−0.802117
$367$	2.29823	0.119967	0.0599833	−	0.998199i	$-0.480895\pi$
$367$	2.29823	0.119967	0.0599833	+	0.998199i	$0.480895\pi$
$368$	−2.44877	−0.127651
$369$	−8.30736	−0.432464
$370$	13.0796	0.679975
$371$	0	0
$372$	−38.2795	−1.98470
$373$	11.7684	0.609343	0.304672	−	0.952457i	$-0.401453\pi$
$373$	11.7684	0.609343	0.304672	+	0.952457i	$0.401453\pi$
$374$	−7.17271	−0.370892
$375$	11.5875	0.598374
$376$	1.08215	0.0558074
$377$	0	0
$378$	0	0
$379$	7.99093	0.410466	0.205233	−	0.978713i	$-0.434205\pi$
$379$	7.99093	0.410466	0.205233	+	0.978713i	$0.434205\pi$
$380$	−0.132812	−0.00681310
$381$	23.1315	1.18506
$382$	−3.48542	−0.178329
$383$	28.2446	1.44323	0.721616	−	0.692294i	$-0.243400\pi$
$383$	28.2446	1.44323	0.721616	+	0.692294i	$0.243400\pi$
$384$	27.8870	1.42310
$385$	0	0
$386$	16.0168	0.815233
$387$	−7.68316	−0.390557
$388$	1.40870	0.0715161
$389$	7.68086	0.389435	0.194717	−	0.980859i	$-0.437621\pi$
$389$	7.68086	0.389435	0.194717	+	0.980859i	$0.437621\pi$
$390$	0	0
$391$	−16.1957	−0.819050
$392$	0	0
$393$	3.74996	0.189160
$394$	35.6481	1.79593
$395$	9.87862	0.497047
$396$	4.12017	0.207046
$397$	7.45281	0.374046	0.187023	−	0.982356i	$-0.440116\pi$
$397$	7.45281	0.374046	0.187023	+	0.982356i	$0.440116\pi$
$398$	15.2368	0.763750
$399$	0	0
$400$	−1.34113	−0.0670563
$401$	18.1982	0.908777	0.454389	−	0.890804i	$-0.349858\pi$
$401$	18.1982	0.908777	0.454389	+	0.890804i	$0.349858\pi$
$402$	6.93186	0.345730
$403$	0	0
$404$	31.9596	1.59005
$405$	−4.93530	−0.245237
$406$	0	0
$407$	−10.0761	−0.499453
$408$	−9.17361	−0.454161
$409$	29.2825	1.44793	0.723964	−	0.689838i	$-0.242318\pi$
$409$	29.2825	1.44793	0.723964	+	0.689838i	$0.242318\pi$
$410$	19.5514	0.965573
$411$	2.74741	0.135520
$412$	−32.9784	−1.62473
$413$	0	0
$414$	14.9328	0.733909
$415$	9.77456	0.479814
$416$	0	0
$417$	−22.9980	−1.12622
$418$	0.164228	0.00803264
$419$	−20.7393	−1.01318	−0.506591	−	0.862187i	$-0.669095\pi$
$419$	−20.7393	−1.01318	−0.506591	+	0.862187i	$0.669095\pi$
$420$	0	0
$421$	−24.8696	−1.21207	−0.606036	−	0.795437i	$-0.707241\pi$
$421$	−24.8696	−1.21207	−0.606036	+	0.795437i	$0.707241\pi$
$422$	−18.6433	−0.907541
$423$	−0.298526	−0.0145148
$424$	8.15130	0.395862
$425$	−8.86994	−0.430255
$426$	−48.2356	−2.33702
$427$	0	0
$428$	32.5950	1.57554
$429$	0	0
$430$	18.0823	0.872006
$431$	−21.1688	−1.01966	−0.509832	−	0.860274i	$-0.670292\pi$
$431$	−21.1688	−1.01966	−0.509832	+	0.860274i	$0.670292\pi$
$432$	1.76698	0.0850138
$433$	−23.4296	−1.12595	−0.562977	−	0.826472i	$-0.690344\pi$
$433$	−23.4296	−1.12595	−0.562977	+	0.826472i	$0.690344\pi$
$434$	0	0
$435$	−1.69741	−0.0813848
$436$	38.3566	1.83695
$437$	0.370819	0.0177387
$438$	−22.9454	−1.09637
$439$	−12.0384	−0.574561	−0.287280	−	0.957847i	$-0.592751\pi$
$439$	−12.0384	−0.574561	−0.287280	+	0.957847i	$0.592751\pi$
$440$	−3.82891	−0.182536
$441$	0	0
$442$	0	0
$443$	15.7331	0.747503	0.373752	−	0.927529i	$-0.378071\pi$
$443$	15.7331	0.747503	0.373752	+	0.927529i	$0.378071\pi$
$444$	−32.6364	−1.54885
$445$	−14.8371	−0.703346
$446$	−36.9860	−1.75134
$447$	9.37592	0.443466
$448$	0	0
$449$	−26.0012	−1.22707	−0.613536	−	0.789667i	$-0.710253\pi$
$449$	−26.0012	−1.22707	−0.613536	+	0.789667i	$0.710253\pi$
$450$	8.17832	0.385530
$451$	−15.0617	−0.709230
$452$	−11.4935	−0.540610
$453$	−0.978449	−0.0459716
$454$	2.98459	0.140074
$455$	0	0
$456$	0.210041	0.00983605
$457$	30.7958	1.44057	0.720284	−	0.693679i	$-0.244011\pi$
$457$	30.7958	1.44057	0.720284	+	0.693679i	$0.244011\pi$
$458$	−47.9483	−2.24048
$459$	11.6864	0.545476
$460$	−21.8949	−1.02086
$461$	−34.0958	−1.58800	−0.794000	−	0.607918i	$-0.792005\pi$
$461$	−34.0958	−1.58800	−0.794000	+	0.607918i	$0.792005\pi$
$462$	0	0
$463$	−1.69184	−0.0786263	−0.0393131	−	0.999227i	$-0.512517\pi$
$463$	−1.69184	−0.0786263	−0.0393131	+	0.999227i	$0.512517\pi$
$464$	0.425852	0.0197697
$465$	−9.81347	−0.455089
$466$	−30.6433	−1.41952
$467$	28.3524	1.31199	0.655996	−	0.754764i	$-0.272249\pi$
$467$	28.3524	1.31199	0.655996	+	0.754764i	$0.272249\pi$
$468$	0	0
$469$	0	0
$470$	0.702581	0.0324076
$471$	24.4224	1.12532
$472$	4.93318	0.227068
$473$	−13.9300	−0.640503
$474$	−39.5655	−1.81730
$475$	0.203088	0.00931830
$476$	0	0
$477$	−2.24866	−0.102959
$478$	30.7837	1.40801
$479$	−6.28246	−0.287053	−0.143526	−	0.989646i	$-0.545844\pi$
$479$	−6.28246	−0.287053	−0.143526	+	0.989646i	$0.545844\pi$
$480$	6.60426	0.301442
$481$	0	0
$482$	−1.92128	−0.0875117
$483$	0	0
$484$	−28.8853	−1.31297
$485$	0.361140	0.0163985
$486$	−19.2171	−0.871703
$487$	13.0176	0.589883	0.294942	−	0.955515i	$-0.404700\pi$
$487$	13.0176	0.589883	0.294942	+	0.955515i	$0.404700\pi$
$488$	13.5923	0.615293
$489$	−13.3358	−0.603065
$490$	0	0
$491$	−12.3523	−0.557453	−0.278726	−	0.960371i	$-0.589912\pi$
$491$	−12.3523	−0.557453	−0.278726	+	0.960371i	$0.589912\pi$
$492$	−48.7849	−2.19939
$493$	2.81650	0.126849
$494$	0	0
$495$	1.05626	0.0474754
$496$	2.46203	0.110549
$497$	0	0
$498$	−39.1487	−1.75430
$499$	9.15340	0.409763	0.204881	−	0.978787i	$-0.434319\pi$
$499$	9.15340	0.409763	0.204881	+	0.978787i	$0.434319\pi$
$500$	−25.9929	−1.16244
$501$	3.91562	0.174937
$502$	62.8149	2.80357
$503$	22.5037	1.00339	0.501696	−	0.865044i	$-0.332710\pi$
$503$	22.5037	1.00339	0.501696	+	0.865044i	$0.332710\pi$
$504$	0	0
$505$	8.19329	0.364596
$506$	27.0741	1.20359
$507$	0	0
$508$	−51.8885	−2.30218
$509$	−38.6606	−1.71360	−0.856800	−	0.515649i	$-0.827551\pi$
$509$	−38.6606	−1.71360	−0.856800	+	0.515649i	$0.827551\pi$
$510$	−5.95594	−0.263734
$511$	0	0
$512$	−3.53972	−0.156435
$513$	−0.267575	−0.0118137
$514$	15.0907	0.665623
$515$	−8.45447	−0.372549
$516$	−45.1193	−1.98626
$517$	−0.541245	−0.0238039
$518$	0	0
$519$	−28.8588	−1.26676
$520$	0	0
$521$	40.2351	1.76273	0.881366	−	0.472434i	$-0.156625\pi$
$521$	40.2351	1.76273	0.881366	+	0.472434i	$0.156625\pi$
$522$	−2.59689	−0.113663
$523$	−0.732146	−0.0320145	−0.0160073	−	0.999872i	$-0.505095\pi$
$523$	−0.732146	−0.0320145	−0.0160073	+	0.999872i	$0.505095\pi$
$524$	−8.41189	−0.367475
$525$	0	0
$526$	−52.0286	−2.26856
$527$	16.2834	0.709316
$528$	0.693741	0.0301912
$529$	38.1321	1.65792
$530$	5.29221	0.229879
$531$	−1.36089	−0.0590577
$532$	0	0
$533$	0	0
$534$	59.4251	2.57157
$535$	8.35618	0.361269
$536$	−6.13993	−0.265204
$537$	4.26291	0.183958
$538$	36.8553	1.58894
$539$	0	0
$540$	15.7989	0.679877
$541$	23.6537	1.01695	0.508476	−	0.861076i	$-0.330209\pi$
$541$	23.6537	1.01695	0.508476	+	0.861076i	$0.330209\pi$
$542$	20.1751	0.866595
$543$	−2.01562	−0.0864985
$544$	−10.9584	−0.469837
$545$	9.83325	0.421210
$546$	0	0
$547$	−12.9472	−0.553582	−0.276791	−	0.960930i	$-0.589271\pi$
$547$	−12.9472	−0.553582	−0.276791	+	0.960930i	$0.589271\pi$
$548$	−6.16298	−0.263270
$549$	−3.74963	−0.160030
$550$	14.8278	0.632258
$551$	−0.0644871	−0.00274724
$552$	34.6267	1.47381
$553$	0	0
$554$	−45.8771	−1.94913
$555$	−8.36679	−0.355150
$556$	51.5891	2.18787
$557$	−6.40680	−0.271465	−0.135732	−	0.990746i	$-0.543339\pi$
$557$	−6.40680	−0.271465	−0.135732	+	0.990746i	$0.543339\pi$
$558$	−15.0137	−0.635581
$559$	0	0
$560$	0	0
$561$	4.58827	0.193717
$562$	32.2996	1.36248
$563$	7.32084	0.308537	0.154268	−	0.988029i	$-0.450698\pi$
$563$	7.32084	0.308537	0.154268	+	0.988029i	$0.450698\pi$
$564$	−1.75309	−0.0738185
$565$	−2.94652	−0.123961
$566$	2.33226	0.0980324
$567$	0	0
$568$	42.7249	1.79270
$569$	−4.31743	−0.180996	−0.0904981	−	0.995897i	$-0.528846\pi$
$569$	−4.31743	−0.180996	−0.0904981	+	0.995897i	$0.528846\pi$
$570$	0.136368	0.00571184
$571$	34.1695	1.42995	0.714974	−	0.699152i	$-0.246439\pi$
$571$	34.1695	1.42995	0.714974	+	0.699152i	$0.246439\pi$
$572$	0	0
$573$	2.22956	0.0931413
$574$	0	0
$575$	33.4804	1.39623
$576$	10.6233	0.442639
$577$	6.35656	0.264627	0.132314	−	0.991208i	$-0.457759\pi$
$577$	6.35656	0.264627	0.132314	+	0.991208i	$0.457759\pi$
$578$	−29.2729	−1.21759
$579$	−10.2457	−0.425795
$580$	3.80763	0.158103
$581$	0	0
$582$	−1.44643	−0.0599563
$583$	−4.07695	−0.168850
$584$	20.3240	0.841014
$585$	0	0
$586$	0.458892	0.0189566
$587$	−31.4120	−1.29651	−0.648256	−	0.761422i	$-0.724501\pi$
$587$	−31.4120	−1.29651	−0.648256	+	0.761422i	$0.724501\pi$
$588$	0	0
$589$	−0.372827	−0.0153621
$590$	3.20286	0.131860
$591$	−22.8035	−0.938011
$592$	2.09909	0.0862719
$593$	−0.473013	−0.0194243	−0.00971215	−	0.999953i	$-0.503092\pi$
$593$	−0.473013	−0.0194243	−0.00971215	+	0.999953i	$0.503092\pi$
$594$	−19.5361	−0.801575
$595$	0	0
$596$	−21.0320	−0.861505
$597$	−9.74670	−0.398906
$598$	0	0
$599$	−9.62695	−0.393347	−0.196673	−	0.980469i	$-0.563014\pi$
$599$	−9.62695	−0.393347	−0.196673	+	0.980469i	$0.563014\pi$
$600$	18.9641	0.774207
$601$	−41.0799	−1.67568	−0.837842	−	0.545914i	$-0.816183\pi$
$601$	−41.0799	−1.67568	−0.837842	+	0.545914i	$0.816183\pi$
$602$	0	0
$603$	1.69379	0.0689765
$604$	2.19485	0.0893073
$605$	−7.40514	−0.301062
$606$	−32.8155	−1.33304
$607$	19.0858	0.774668	0.387334	−	0.921939i	$-0.373396\pi$
$607$	19.0858	0.774668	0.387334	+	0.921939i	$0.373396\pi$
$608$	0.250905	0.0101755
$609$	0	0
$610$	8.82474	0.357303
$611$	0	0
$612$	−5.67680	−0.229471
$613$	−38.0048	−1.53500	−0.767500	−	0.641049i	$-0.778499\pi$
$613$	−38.0048	−1.53500	−0.767500	+	0.641049i	$0.778499\pi$
$614$	62.6498	2.52834
$615$	−12.5067	−0.504318
$616$	0	0
$617$	8.31519	0.334757	0.167378	−	0.985893i	$-0.446470\pi$
$617$	8.31519	0.334757	0.167378	+	0.985893i	$0.446470\pi$
$618$	33.8616	1.36211
$619$	−44.4728	−1.78751	−0.893756	−	0.448553i	$-0.851940\pi$
$619$	−44.4728	−1.78751	−0.893756	+	0.448553i	$0.851940\pi$
$620$	22.0135	0.884085
$621$	−44.1116	−1.77014
$622$	62.4206	2.50284
$623$	0	0
$624$	0	0
$625$	14.7468	0.589874
$626$	−50.8526	−2.03248
$627$	−0.105054	−0.00419544
$628$	−54.7842	−2.18613
$629$	13.8829	0.553549
$630$	0	0
$631$	11.7524	0.467858	0.233929	−	0.972254i	$-0.424842\pi$
$631$	11.7524	0.467858	0.233929	+	0.972254i	$0.424842\pi$
$632$	35.0453	1.39403
$633$	11.9258	0.474008
$634$	−16.2800	−0.646562
$635$	−13.3023	−0.527887
$636$	−13.2052	−0.523620
$637$	0	0
$638$	−4.70831	−0.186404
$639$	−11.7863	−0.466259
$640$	−16.0371	−0.633921
$641$	10.4868	0.414205	0.207102	−	0.978319i	$-0.433597\pi$
$641$	10.4868	0.414205	0.207102	+	0.978319i	$0.433597\pi$
$642$	−33.4679	−1.32087
$643$	31.2822	1.23365	0.616825	−	0.787101i	$-0.288419\pi$
$643$	31.2822	1.23365	0.616825	+	0.787101i	$0.288419\pi$
$644$	0	0
$645$	−11.5669	−0.455448
$646$	−0.226275	−0.00890265
$647$	−26.8675	−1.05627	−0.528135	−	0.849160i	$-0.677109\pi$
$647$	−26.8675	−1.05627	−0.528135	+	0.849160i	$0.677109\pi$
$648$	−17.5084	−0.687796
$649$	−2.46738	−0.0968530
$650$	0	0
$651$	0	0
$652$	29.9148	1.17155
$653$	−4.14161	−0.162074	−0.0810369	−	0.996711i	$-0.525823\pi$
$653$	−4.14161	−0.162074	−0.0810369	+	0.996711i	$0.525823\pi$
$654$	−39.3838	−1.54003
$655$	−2.15650	−0.0842615
$656$	3.13771	0.122507
$657$	−5.60668	−0.218738
$658$	0	0
$659$	21.4551	0.835773	0.417887	−	0.908499i	$-0.362771\pi$
$659$	21.4551	0.835773	0.417887	+	0.908499i	$0.362771\pi$
$660$	6.20288	0.241447
$661$	42.3872	1.64867	0.824335	−	0.566102i	$-0.191549\pi$
$661$	42.3872	1.64867	0.824335	+	0.566102i	$0.191549\pi$
$662$	−15.1753	−0.589803
$663$	0	0
$664$	34.6762	1.34570
$665$	0	0
$666$	−12.8004	−0.496006
$667$	−10.6312	−0.411640
$668$	−8.78349	−0.339844
$669$	23.6593	0.914722
$670$	−3.98633	−0.154005
$671$	−6.79830	−0.262445
$672$	0	0
$673$	−29.5856	−1.14044	−0.570220	−	0.821492i	$-0.693142\pi$
$673$	−29.5856	−1.14044	−0.570220	+	0.821492i	$0.693142\pi$
$674$	9.72645	0.374649
$675$	−24.1588	−0.929871
$676$	0	0
$677$	32.1659	1.23624	0.618118	−	0.786085i	$-0.287895\pi$
$677$	32.1659	1.23624	0.618118	+	0.786085i	$0.287895\pi$
$678$	11.8013	0.453227
$679$	0	0
$680$	5.27551	0.202306
$681$	−1.90919	−0.0731604
$682$	−27.2207	−1.04234
$683$	−8.60236	−0.329160	−0.164580	−	0.986364i	$-0.552627\pi$
$683$	−8.60236	−0.329160	−0.164580	+	0.986364i	$0.552627\pi$
$684$	0.129977	0.00496980
$685$	−1.57997	−0.0603674
$686$	0	0
$687$	30.6717	1.17020
$688$	2.90195	0.110636
$689$	0	0
$690$	22.4813	0.855847
$691$	20.4420	0.777651	0.388826	−	0.921311i	$-0.372881\pi$
$691$	20.4420	0.777651	0.388826	+	0.921311i	$0.372881\pi$
$692$	64.7359	2.46089
$693$	0	0
$694$	−20.9477	−0.795164
$695$	13.2256	0.501675
$696$	−6.02174	−0.228253
$697$	20.7522	0.786046
$698$	21.2373	0.803845
$699$	19.6020	0.741415
$700$	0	0
$701$	25.1373	0.949422	0.474711	−	0.880142i	$-0.342553\pi$
$701$	25.1373	0.949422	0.474711	+	0.880142i	$0.342553\pi$
$702$	0	0
$703$	−0.317866	−0.0119885
$704$	19.2607	0.725915
$705$	−0.449429	−0.0169265
$706$	4.96236	0.186761
$707$	0	0
$708$	−7.99182	−0.300351
$709$	−29.4929	−1.10763	−0.553814	−	0.832640i	$-0.686828\pi$
$709$	−29.4929	−1.10763	−0.553814	+	0.832640i	$0.686828\pi$
$710$	27.7390	1.04103
$711$	−9.66777	−0.362570
$712$	−52.6360	−1.97262
$713$	−61.4632	−2.30182
$714$	0	0
$715$	0	0
$716$	−9.56254	−0.357369
$717$	−19.6918	−0.735404
$718$	19.7104	0.735584
$719$	−8.33153	−0.310713	−0.155357	−	0.987858i	$-0.549653\pi$
$719$	−8.33153	−0.310713	−0.155357	+	0.987858i	$0.549653\pi$
$720$	−0.220044	−0.00820055
$721$	0	0
$722$	−43.7569	−1.62846
$723$	1.22901	0.0457073
$724$	4.52143	0.168038
$725$	−5.82240	−0.216239
$726$	29.6588	1.10074
$727$	−9.66141	−0.358322	−0.179161	−	0.983820i	$-0.557338\pi$
$727$	−9.66141	−0.358322	−0.179161	+	0.983820i	$0.557338\pi$
$728$	0	0
$729$	29.7672	1.10249
$730$	13.1953	0.488381
$731$	19.1929	0.709875
$732$	−22.0197	−0.813870
$733$	−14.0179	−0.517762	−0.258881	−	0.965909i	$-0.583354\pi$
$733$	−14.0179	−0.517762	−0.258881	+	0.965909i	$0.583354\pi$
$734$	5.29344	0.195384
$735$	0	0
$736$	41.3635	1.52468
$737$	3.07094	0.113120
$738$	−19.1341	−0.704335
$739$	−38.8147	−1.42782	−0.713910	−	0.700237i	$-0.753077\pi$
$739$	−38.8147	−1.42782	−0.713910	+	0.700237i	$0.753077\pi$
$740$	18.7683	0.689938
$741$	0	0
$742$	0	0
$743$	−34.3942	−1.26180	−0.630901	−	0.775863i	$-0.717315\pi$
$743$	−34.3942	−1.26180	−0.630901	+	0.775863i	$0.717315\pi$
$744$	−34.8142	−1.27635
$745$	−5.39185	−0.197542
$746$	27.1057	0.992410
$747$	−9.56594	−0.349999
$748$	−10.2924	−0.376327
$749$	0	0
$750$	26.6890	0.974545
$751$	48.1470	1.75691	0.878454	−	0.477827i	$-0.158575\pi$
$751$	48.1470	1.75691	0.878454	+	0.477827i	$0.158575\pi$
$752$	0.112754	0.00411172
$753$	−40.1816	−1.46430
$754$	0	0
$755$	0.562681	0.0204781
$756$	0	0
$757$	−6.90638	−0.251016	−0.125508	−	0.992093i	$-0.540056\pi$
$757$	−6.90638	−0.251016	−0.125508	+	0.992093i	$0.540056\pi$
$758$	18.4052	0.668508
$759$	−17.3188	−0.628634
$760$	−0.120789	−0.00438147
$761$	−31.9730	−1.15902	−0.579511	−	0.814965i	$-0.696756\pi$
$761$	−31.9730	−1.15902	−0.579511	+	0.814965i	$0.696756\pi$
$762$	53.2781	1.93006
$763$	0	0
$764$	−5.00135	−0.180942
$765$	−1.45533	−0.0526174
$766$	65.0548	2.35053
$767$	0	0
$768$	26.4795	0.955495
$769$	−14.3950	−0.519099	−0.259549	−	0.965730i	$-0.583574\pi$
$769$	−14.3950	−0.519099	−0.259549	+	0.965730i	$0.583574\pi$
$770$	0	0
$771$	−9.65328	−0.347654
$772$	22.9830	0.827177
$773$	37.2771	1.34076	0.670382	−	0.742016i	$-0.266130\pi$
$773$	37.2771	1.34076	0.670382	+	0.742016i	$0.266130\pi$
$774$	−17.6964	−0.636083
$775$	−33.6618	−1.20917
$776$	1.28118	0.0459917
$777$	0	0
$778$	17.6911	0.634255
$779$	−0.475146	−0.0170239
$780$	0	0
$781$	−21.3693	−0.764652
$782$	−37.3029	−1.33395
$783$	7.67121	0.274147
$784$	0	0
$785$	−14.0447	−0.501276
$786$	8.63715	0.308077
$787$	−14.3486	−0.511472	−0.255736	−	0.966747i	$-0.582318\pi$
$787$	−14.3486	−0.511472	−0.255736	+	0.966747i	$0.582318\pi$
$788$	51.1527	1.82224
$789$	33.2818	1.18486
$790$	22.7531	0.809518
$791$	0	0
$792$	3.74719	0.133150
$793$	0	0
$794$	17.1658	0.609192
$795$	−3.38534	−0.120066
$796$	21.8638	0.774940
$797$	11.0844	0.392629	0.196314	−	0.980541i	$-0.437103\pi$
$797$	11.0844	0.392629	0.196314	+	0.980541i	$0.437103\pi$
$798$	0	0
$799$	0.745733	0.0263821
$800$	22.6537	0.800929
$801$	14.5204	0.513054
$802$	41.9154	1.48008
$803$	−10.1652	−0.358724
$804$	9.94676	0.350795
$805$	0	0
$806$	0	0
$807$	−23.5757	−0.829904
$808$	29.0665	1.02255
$809$	−42.5536	−1.49610	−0.748052	−	0.663640i	$-0.769011\pi$
$809$	−42.5536	−1.49610	−0.748052	+	0.663640i	$0.769011\pi$
$810$	−11.3673	−0.399406
$811$	−16.3622	−0.574554	−0.287277	−	0.957848i	$-0.592750\pi$
$811$	−16.3622	−0.574554	−0.287277	+	0.957848i	$0.592750\pi$
$812$	0	0
$813$	−12.9057	−0.452622
$814$	−23.2079	−0.813437
$815$	7.66906	0.268636
$816$	−0.955843	−0.0334612
$817$	−0.439444	−0.0153742
$818$	67.4455	2.35818
$819$	0	0
$820$	28.0549	0.979721
$821$	3.10550	0.108383	0.0541913	−	0.998531i	$-0.482742\pi$
$821$	3.10550	0.108383	0.0541913	+	0.998531i	$0.482742\pi$
$822$	6.32802	0.220715
$823$	49.0164	1.70860	0.854301	−	0.519778i	$-0.173985\pi$
$823$	49.0164	1.70860	0.854301	+	0.519778i	$0.173985\pi$
$824$	−29.9930	−1.04486
$825$	−9.48507	−0.330228
$826$	0	0
$827$	−13.0887	−0.455140	−0.227570	−	0.973762i	$-0.573078\pi$
$827$	−13.0887	−0.455140	−0.227570	+	0.973762i	$0.573078\pi$
$828$	21.4276	0.744662
$829$	49.2565	1.71075	0.855374	−	0.518010i	$-0.173327\pi$
$829$	49.2565	1.71075	0.855374	+	0.518010i	$0.173327\pi$
$830$	22.5134	0.781452
$831$	29.3468	1.01803
$832$	0	0
$833$	0	0
$834$	−52.9706	−1.83422
$835$	−2.25177	−0.0779257
$836$	0.235656	0.00815034
$837$	44.3505	1.53298
$838$	−47.7682	−1.65012
$839$	−17.2636	−0.596007	−0.298004	−	0.954565i	$-0.596321\pi$
$839$	−17.2636	−0.596007	−0.298004	+	0.954565i	$0.596321\pi$
$840$	0	0
$841$	−27.1512	−0.936248
$842$	−57.2814	−1.97405
$843$	−20.6615	−0.711621
$844$	−26.7519	−0.920839
$845$	0	0
$846$	−0.687585	−0.0236397
$847$	0	0
$848$	0.849323	0.0291659
$849$	−1.49191	−0.0512022
$850$	−20.4298	−0.700738
$851$	−52.4024	−1.79633
$852$	−69.2149	−2.37126
$853$	−52.4163	−1.79470	−0.897350	−	0.441319i	$-0.854511\pi$
$853$	−52.4163	−1.79470	−0.897350	+	0.441319i	$0.854511\pi$
$854$	0	0
$855$	0.0333214	0.00113957
$856$	29.6443	1.01322
$857$	10.1271	0.345935	0.172967	−	0.984928i	$-0.444664\pi$
$857$	10.1271	0.345935	0.172967	+	0.984928i	$0.444664\pi$
$858$	0	0
$859$	−0.510237	−0.0174090	−0.00870452	−	0.999962i	$-0.502771\pi$
$859$	−0.510237	−0.0174090	−0.00870452	+	0.999962i	$0.502771\pi$
$860$	25.9469	0.884782
$861$	0	0
$862$	−48.7573	−1.66068
$863$	−20.4991	−0.697797	−0.348898	−	0.937161i	$-0.613444\pi$
$863$	−20.4991	−0.697797	−0.348898	+	0.937161i	$0.613444\pi$
$864$	−29.8470	−1.01541
$865$	16.5959	0.564279
$866$	−53.9646	−1.83379
$867$	18.7254	0.635947
$868$	0	0
$869$	−17.5282	−0.594605
$870$	−3.90960	−0.132548
$871$	0	0
$872$	34.8844	1.18133
$873$	−0.353433	−0.0119619
$874$	0.854095	0.0288902
$875$	0	0
$876$	−32.9252	−1.11244
$877$	11.2906	0.381256	0.190628	−	0.981662i	$-0.438948\pi$
$877$	11.2906	0.381256	0.190628	+	0.981662i	$0.438948\pi$
$878$	−27.7276	−0.935762
$879$	−0.293545	−0.00990103
$880$	−0.398953	−0.0134487
$881$	22.5268	0.758947	0.379474	−	0.925203i	$-0.376105\pi$
$881$	22.5268	0.758947	0.379474	+	0.925203i	$0.376105\pi$
$882$	0	0
$883$	28.0268	0.943178	0.471589	−	0.881819i	$-0.343681\pi$
$883$	28.0268	0.943178	0.471589	+	0.881819i	$0.343681\pi$
$884$	0	0
$885$	−2.04881	−0.0688701
$886$	36.2376	1.21743
$887$	−20.6235	−0.692470	−0.346235	−	0.938148i	$-0.612540\pi$
$887$	−20.6235	−0.692470	−0.346235	+	0.938148i	$0.612540\pi$
$888$	−29.6820	−0.996062
$889$	0	0
$890$	−34.1738	−1.14551
$891$	8.75700	0.293371
$892$	−53.0725	−1.77700
$893$	−0.0170744	−0.000571374 0
$894$	21.5952	0.722253
$895$	−2.45149	−0.0819443
$896$	0	0
$897$	0	0
$898$	−59.8876	−1.99848
$899$	10.6887	0.356489
$900$	11.7353	0.391178
$901$	5.61725	0.187138
$902$	−34.6912	−1.15509
$903$	0	0
$904$	−10.4531	−0.347664
$905$	1.15913	0.0385308
$906$	−2.25363	−0.0748718
$907$	41.4631	1.37676	0.688379	−	0.725351i	$-0.258322\pi$
$907$	41.4631	1.37676	0.688379	+	0.725351i	$0.258322\pi$
$908$	4.28269	0.142126
$909$	−8.01841	−0.265954
$910$	0	0
$911$	40.8187	1.35239	0.676193	−	0.736725i	$-0.263629\pi$
$911$	40.8187	1.35239	0.676193	+	0.736725i	$0.263629\pi$
$912$	0.0218852	0.000724690 0
$913$	−17.3436	−0.573990
$914$	70.9310	2.34619
$915$	−5.64504	−0.186619
$916$	−68.8027	−2.27331
$917$	0	0
$918$	26.9170	0.888393
$919$	−48.7678	−1.60870	−0.804350	−	0.594155i	$-0.797486\pi$
$919$	−48.7678	−1.60870	−0.804350	+	0.594155i	$0.797486\pi$
$920$	−19.9129	−0.656509
$921$	−40.0760	−1.32055
$922$	−78.5317	−2.58630
$923$	0	0
$924$	0	0
$925$	−28.6994	−0.943631
$926$	−3.89675	−0.128055
$927$	8.27403	0.271755
$928$	−7.19330	−0.236132
$929$	29.3829	0.964023	0.482012	−	0.876165i	$-0.339906\pi$
$929$	29.3829	0.964023	0.482012	+	0.876165i	$0.339906\pi$
$930$	−22.6030	−0.741183
$931$	0	0
$932$	−43.9711	−1.44032
$933$	−39.9294	−1.30723
$934$	65.3031	2.13678
$935$	−2.63859	−0.0862912
$936$	0	0
$937$	−21.0196	−0.686681	−0.343340	−	0.939211i	$-0.611558\pi$
$937$	−21.0196	−0.686681	−0.343340	+	0.939211i	$0.611558\pi$
$938$	0	0
$939$	32.5296	1.06156
$940$	1.00816	0.0328825
$941$	24.1033	0.785744	0.392872	−	0.919593i	$-0.371482\pi$
$941$	24.1033	0.785744	0.392872	+	0.919593i	$0.371482\pi$
$942$	56.2513	1.83277
$943$	−78.3312	−2.55081
$944$	0.514013	0.0167297
$945$	0	0
$946$	−32.0845	−1.04316
$947$	3.34046	0.108550	0.0542751	−	0.998526i	$-0.482715\pi$
$947$	3.34046	0.108550	0.0542751	+	0.998526i	$0.482715\pi$
$948$	−56.7739	−1.84393
$949$	0	0
$950$	0.467765	0.0151763
$951$	10.4141	0.337699
$952$	0	0
$953$	−4.97124	−0.161034	−0.0805171	−	0.996753i	$-0.525657\pi$
$953$	−4.97124	−0.161034	−0.0805171	+	0.996753i	$0.525657\pi$
$954$	−5.17925	−0.167685
$955$	−1.28216	−0.0414898
$956$	44.1726	1.42864
$957$	3.01183	0.0973585
$958$	−14.4702	−0.467510
$959$	0	0
$960$	15.9933	0.516183
$961$	30.7961	0.993423
$962$	0	0
$963$	−8.17783	−0.263527
$964$	−2.75691	−0.0887939
$965$	5.89202	0.189671
$966$	0	0
$967$	−47.4943	−1.52731	−0.763657	−	0.645623i	$-0.776598\pi$
$967$	−47.4943	−1.52731	−0.763657	+	0.645623i	$0.776598\pi$
$968$	−26.2705	−0.844364
$969$	0.144744	0.00464985
$970$	0.831803	0.0267076
$971$	34.4715	1.10624	0.553121	−	0.833101i	$-0.313437\pi$
$971$	34.4715	1.10624	0.553121	+	0.833101i	$0.313437\pi$
$972$	−27.5752	−0.884476
$973$	0	0
$974$	29.9830	0.960717
$975$	0	0
$976$	1.41624	0.0453329
$977$	13.3481	0.427044	0.213522	−	0.976938i	$-0.431506\pi$
$977$	13.3481	0.427044	0.213522	+	0.976938i	$0.431506\pi$
$978$	−30.7159	−0.982185
$979$	26.3264	0.841395
$980$	0	0
$981$	−9.62337	−0.307251
$982$	−28.4507	−0.907898
$983$	−12.5344	−0.399785	−0.199893	−	0.979818i	$-0.564059\pi$
$983$	−12.5344	−0.399785	−0.199893	+	0.979818i	$0.564059\pi$
$984$	−44.3686	−1.41442
$985$	13.1137	0.417838
$986$	6.48715	0.206593
$987$	0	0
$988$	0	0
$989$	−72.4455	−2.30363
$990$	2.43285	0.0773211
$991$	−10.4119	−0.330745	−0.165373	−	0.986231i	$-0.552883\pi$
$991$	−10.4119	−0.330745	−0.165373	+	0.986231i	$0.552883\pi$
$992$	−41.5875	−1.32040
$993$	9.70736	0.308054
$994$	0	0
$995$	5.60508	0.177693
$996$	−56.1759	−1.78000
$997$	5.75270	0.182190	0.0910949	−	0.995842i	$-0.470963\pi$
$997$	5.75270	0.182190	0.0910949	+	0.995842i	$0.470963\pi$
$998$	21.0827	0.667362
$999$	37.8125	1.19633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cp.1.11		12
7.2	even	3		1183.2.e.j.508.2		24
7.4	even	3		1183.2.e.j.170.2		24
7.6	odd	2		8281.2.a.co.1.11		12
13.6	odd	12		637.2.q.g.491.1		12
13.11	odd	12		637.2.q.g.589.1		12
13.12	even	2	inner	8281.2.a.cp.1.2		12
91.6	even	12		637.2.q.i.491.1		12
91.11	odd	12		91.2.u.b.30.6	yes	12
91.19	even	12		637.2.u.g.361.6		12
91.24	even	12		637.2.u.g.30.6		12
91.25	even	6		1183.2.e.j.170.11		24
91.32	odd	12		91.2.k.b.23.1	yes	12
91.37	odd	12		91.2.k.b.4.6	✓	12
91.45	even	12		637.2.k.i.569.1		12
91.51	even	6		1183.2.e.j.508.11		24
91.58	odd	12		91.2.u.b.88.6	yes	12
91.76	even	12		637.2.q.i.589.1		12
91.89	even	12		637.2.k.i.459.6		12
91.90	odd	2		8281.2.a.co.1.2		12
273.11	even	12		819.2.do.e.667.1		12
273.32	even	12		819.2.bm.f.478.6		12
273.128	even	12		819.2.bm.f.550.1		12
273.149	even	12		819.2.do.e.361.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.k.b.4.6	✓	12	91.37	odd	12
91.2.k.b.23.1	yes	12	91.32	odd	12
91.2.u.b.30.6	yes	12	91.11	odd	12
91.2.u.b.88.6	yes	12	91.58	odd	12
637.2.k.i.459.6		12	91.89	even	12
637.2.k.i.569.1		12	91.45	even	12
637.2.q.g.491.1		12	13.6	odd	12
637.2.q.g.589.1		12	13.11	odd	12
637.2.q.i.491.1		12	91.6	even	12
637.2.q.i.589.1		12	91.76	even	12
637.2.u.g.30.6		12	91.24	even	12
637.2.u.g.361.6		12	91.19	even	12
819.2.bm.f.478.6		12	273.32	even	12
819.2.bm.f.550.1		12	273.128	even	12
819.2.do.e.361.1		12	273.149	even	12
819.2.do.e.667.1		12	273.11	even	12
1183.2.e.j.170.2		24	7.4	even	3
1183.2.e.j.170.11		24	91.25	even	6
1183.2.e.j.508.2		24	7.2	even	3
1183.2.e.j.508.11		24	91.51	even	6
8281.2.a.co.1.2		12	91.90	odd	2
8281.2.a.co.1.11		12	7.6	odd	2
8281.2.a.cp.1.2		12	13.12	even	2	inner
8281.2.a.cp.1.11		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)

\(f(q)\)	\(=\)	\(q+2.30327 q^{2} -1.47336 q^{3} +3.30504 q^{4} +0.847292 q^{5} -3.39354 q^{6} +3.00585 q^{8} -0.829208 q^{9} +O(q^{10})\)	\(q+2.30327 q^{2} -1.47336 q^{3} +3.30504 q^{4} +0.847292 q^{5} -3.39354 q^{6} +3.00585 q^{8} -0.829208 q^{9} +1.95154 q^{10} -1.50340 q^{11} -4.86951 q^{12} -1.24837 q^{15} +0.313194 q^{16} +2.07140 q^{17} -1.90989 q^{18} -0.0474272 q^{19} +2.80033 q^{20} -3.46274 q^{22} -7.81870 q^{23} -4.42870 q^{24} -4.28210 q^{25} +5.64180 q^{27} +1.35971 q^{29} -2.87532 q^{30} +7.86105 q^{31} -5.29033 q^{32} +2.21505 q^{33} +4.77099 q^{34} -2.74056 q^{36} +6.70219 q^{37} -0.109237 q^{38} +2.54683 q^{40} +10.0184 q^{41} +9.26566 q^{43} -4.96880 q^{44} -0.702581 q^{45} -18.0086 q^{46} +0.360014 q^{47} -0.461448 q^{48} -9.86281 q^{50} -3.05192 q^{51} +2.71181 q^{53} +12.9946 q^{54} -1.27382 q^{55} +0.0698773 q^{57} +3.13177 q^{58} +1.64120 q^{59} -4.12590 q^{60} +4.52194 q^{61} +18.1061 q^{62} -12.8114 q^{64} +5.10186 q^{66} -2.04266 q^{67} +6.84606 q^{68} +11.5198 q^{69} +14.2139 q^{71} -2.49247 q^{72} +6.76150 q^{73} +15.4369 q^{74} +6.30907 q^{75} -0.156749 q^{76} +11.6590 q^{79} +0.265367 q^{80} -5.82479 q^{81} +23.0751 q^{82} +11.5362 q^{83} +1.75508 q^{85} +21.3413 q^{86} -2.00334 q^{87} -4.51900 q^{88} -17.5112 q^{89} -1.61823 q^{90} -25.8411 q^{92} -11.5822 q^{93} +0.829208 q^{94} -0.0401846 q^{95} +7.79456 q^{96} +0.426229 q^{97} +1.24663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more