LMFDB - Embedded newform 8281.2.a.co.1.1

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:	$1$
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			1.1
Root			$-2.58860$ 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)

$f(q)$	$=$	$q-2.58860 q^{2} -0.518466 q^{3} +4.70085 q^{4} +1.61205 q^{5} +1.34210 q^{6} -6.99143 q^{8} -2.73119 q^{9} +O(q^{10})$	\(q-2.58860 q^{2} -0.518466 q^{3} +4.70085 q^{4} +1.61205 q^{5} +1.34210 q^{6} -6.99143 q^{8} -2.73119 q^{9} -4.17296 q^{10} -2.70496 q^{11} -2.43723 q^{12} -0.835795 q^{15} +8.69632 q^{16} -3.12661 q^{17} +7.06997 q^{18} +3.68150 q^{19} +7.57803 q^{20} +7.00205 q^{22} +1.98604 q^{23} +3.62482 q^{24} -2.40128 q^{25} +2.97143 q^{27} -5.37271 q^{29} +2.16354 q^{30} +10.4780 q^{31} -8.52843 q^{32} +1.40243 q^{33} +8.09354 q^{34} -12.8389 q^{36} -5.95346 q^{37} -9.52994 q^{38} -11.2706 q^{40} -7.70150 q^{41} -3.35600 q^{43} -12.7156 q^{44} -4.40283 q^{45} -5.14106 q^{46} +1.05508 q^{47} -4.50874 q^{48} +6.21596 q^{50} +1.62104 q^{51} +7.26568 q^{53} -7.69184 q^{54} -4.36054 q^{55} -1.90873 q^{57} +13.9078 q^{58} +11.4241 q^{59} -3.92895 q^{60} +2.92507 q^{61} -27.1235 q^{62} +4.68406 q^{64} -3.63033 q^{66} +13.5818 q^{67} -14.6977 q^{68} -1.02969 q^{69} +1.35111 q^{71} +19.0949 q^{72} +9.10335 q^{73} +15.4111 q^{74} +1.24498 q^{75} +17.3062 q^{76} -6.20578 q^{79} +14.0189 q^{80} +6.65300 q^{81} +19.9361 q^{82} -2.69672 q^{83} -5.04026 q^{85} +8.68734 q^{86} +2.78557 q^{87} +18.9115 q^{88} +1.75988 q^{89} +11.3972 q^{90} +9.33607 q^{92} -5.43251 q^{93} -2.73119 q^{94} +5.93478 q^{95} +4.42170 q^{96} +15.4820 q^{97} +7.38776 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.58860	−1.83042	−0.915209	−	0.402981i	$-0.867974\pi$
$2$	−2.58860	−1.83042	−0.915209	+	0.402981i	$0.867974\pi$
$3$	−0.518466	−0.299336	−0.149668	−	0.988736i	$-0.547821\pi$
$3$	−0.518466	−0.299336	−0.149668	+	0.988736i	$0.547821\pi$
$4$	4.70085	2.35043
$5$	1.61205	0.720932	0.360466	−	0.932772i	$-0.382618\pi$
$5$	1.61205	0.720932	0.360466	+	0.932772i	$0.382618\pi$
$6$	1.34210	0.547910
$7$	0	0
$7$	0	0
$8$	−6.99143	−2.47184
$9$	−2.73119	−0.910398
$10$	−4.17296	−1.31961
$11$	−2.70496	−0.815575	−0.407788	−	0.913077i	$-0.633700\pi$
$11$	−2.70496	−0.815575	−0.407788	+	0.913077i	$0.633700\pi$
$12$	−2.43723	−0.703568
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.835795	−0.215801
$16$	8.69632	2.17408
$17$	−3.12661	−0.758314	−0.379157	−	0.925332i	$-0.623786\pi$
$17$	−3.12661	−0.758314	−0.379157	+	0.925332i	$0.623786\pi$
$18$	7.06997	1.66641
$19$	3.68150	0.844595	0.422297	−	0.906457i	$-0.361224\pi$
$19$	3.68150	0.844595	0.422297	+	0.906457i	$0.361224\pi$
$20$	7.57803	1.69450
$21$	0	0
$22$	7.00205	1.49284
$23$	1.98604	0.414117	0.207059	−	0.978329i	$-0.433611\pi$
$23$	1.98604	0.414117	0.207059	+	0.978329i	$0.433611\pi$
$24$	3.62482	0.739913
$25$	−2.40128	−0.480257
$26$	0	0
$27$	2.97143	0.571852
$28$	0	0
$29$	−5.37271	−0.997687	−0.498844	−	0.866692i	$-0.666242\pi$
$29$	−5.37271	−0.997687	−0.498844	+	0.866692i	$0.666242\pi$
$30$	2.16354	0.395006
$31$	10.4780	1.88191	0.940956	−	0.338529i	$-0.109929\pi$
$31$	10.4780	1.88191	0.940956	+	0.338529i	$0.109929\pi$
$32$	−8.52843	−1.50763
$33$	1.40243	0.244131
$34$	8.09354	1.38803
$35$	0	0
$36$	−12.8389	−2.13982
$37$	−5.95346	−0.978743	−0.489371	−	0.872075i	$-0.662774\pi$
$37$	−5.95346	−0.978743	−0.489371	+	0.872075i	$0.662774\pi$
$38$	−9.52994	−1.54596
$39$	0	0
$40$	−11.2706	−1.78203
$41$	−7.70150	−1.20277	−0.601386	−	0.798958i	$-0.705385\pi$
$41$	−7.70150	−1.20277	−0.601386	+	0.798958i	$0.705385\pi$
$42$	0	0
$43$	−3.35600	−0.511785	−0.255892	−	0.966705i	$-0.582369\pi$
$43$	−3.35600	−0.511785	−0.255892	+	0.966705i	$0.582369\pi$
$44$	−12.7156	−1.91695
$45$	−4.40283	−0.656335
$46$	−5.14106	−0.758008
$47$	1.05508	0.153900	0.0769500	−	0.997035i	$-0.475482\pi$
$47$	1.05508	0.153900	0.0769500	+	0.997035i	$0.475482\pi$
$48$	−4.50874	−0.650781
$49$	0	0
$50$	6.21596	0.879070
$51$	1.62104	0.226991
$52$	0	0
$53$	7.26568	0.998017	0.499009	−	0.866597i	$-0.333698\pi$
$53$	7.26568	0.998017	0.499009	+	0.866597i	$0.333698\pi$
$54$	−7.69184	−1.04673
$55$	−4.36054	−0.587975
$56$	0	0
$57$	−1.90873	−0.252818
$58$	13.9078	1.82618
$59$	11.4241	1.48729	0.743643	−	0.668577i	$-0.233096\pi$
$59$	11.4241	1.48729	0.743643	+	0.668577i	$0.233096\pi$
$60$	−3.92895	−0.507225
$61$	2.92507	0.374517	0.187259	−	0.982311i	$-0.440040\pi$
$61$	2.92507	0.374517	0.187259	+	0.982311i	$0.440040\pi$
$62$	−27.1235	−3.44468
$63$	0	0
$64$	4.68406	0.585507
$65$	0	0
$66$	−3.63033	−0.446862
$67$	13.5818	1.65928	0.829642	−	0.558296i	$-0.188545\pi$
$67$	13.5818	1.65928	0.829642	+	0.558296i	$0.188545\pi$
$68$	−14.6977	−1.78236
$69$	−1.02969	−0.123960
$70$	0	0
$71$	1.35111	0.160347	0.0801736	−	0.996781i	$-0.474453\pi$
$71$	1.35111	0.160347	0.0801736	+	0.996781i	$0.474453\pi$
$72$	19.0949	2.25036
$73$	9.10335	1.06547	0.532733	−	0.846283i	$-0.321165\pi$
$73$	9.10335	1.06547	0.532733	+	0.846283i	$0.321165\pi$
$74$	15.4111	1.79151
$75$	1.24498	0.143758
$76$	17.3062	1.98516
$77$	0	0
$78$	0	0
$79$	−6.20578	−0.698205	−0.349102	−	0.937085i	$-0.613513\pi$
$79$	−6.20578	−0.698205	−0.349102	+	0.937085i	$0.613513\pi$
$80$	14.0189	1.56736
$81$	6.65300	0.739222
$82$	19.9361	2.20158
$83$	−2.69672	−0.296003	−0.148002	−	0.988987i	$-0.547284\pi$
$83$	−2.69672	−0.296003	−0.148002	+	0.988987i	$0.547284\pi$
$84$	0	0
$85$	−5.04026	−0.546693
$86$	8.68734	0.936780
$87$	2.78557	0.298644
$88$	18.9115	2.01597
$89$	1.75988	0.186546	0.0932732	−	0.995641i	$-0.470267\pi$
$89$	1.75988	0.186546	0.0932732	+	0.995641i	$0.470267\pi$
$90$	11.3972	1.20137
$91$	0	0
$92$	9.33607	0.973353
$93$	−5.43251	−0.563325
$94$	−2.73119	−0.281701
$95$	5.93478	0.608896
$96$	4.42170	0.451288
$97$	15.4820	1.57196	0.785981	−	0.618250i	$-0.212158\pi$
$97$	15.4820	1.57196	0.785981	+	0.618250i	$0.212158\pi$
$98$	0	0
$99$	7.38776	0.742498
$100$	−11.2881	−1.12881
$101$	1.27930	0.127295	0.0636477	−	0.997972i	$-0.479727\pi$
$101$	1.27930	0.127295	0.0636477	+	0.997972i	$0.479727\pi$
$102$	−4.19622	−0.415488
$103$	11.4673	1.12991	0.564956	−	0.825121i	$-0.308893\pi$
$103$	11.4673	1.12991	0.564956	+	0.825121i	$0.308893\pi$
$104$	0	0
$105$	0	0
$106$	−18.8079	−1.82679
$107$	−5.13525	−0.496444	−0.248222	−	0.968703i	$-0.579846\pi$
$107$	−5.13525	−0.496444	−0.248222	+	0.968703i	$0.579846\pi$
$108$	13.9682	1.34410
$109$	−1.72783	−0.165496	−0.0827481	−	0.996570i	$-0.526370\pi$
$109$	−1.72783	−0.165496	−0.0827481	+	0.996570i	$0.526370\pi$
$110$	11.2877	1.07624
$111$	3.08667	0.292973
$112$	0	0
$113$	−8.59113	−0.808185	−0.404093	−	0.914718i	$-0.632413\pi$
$113$	−8.59113	−0.808185	−0.404093	+	0.914718i	$0.632413\pi$
$114$	4.94095	0.462762
$115$	3.20160	0.298551
$116$	−25.2563	−2.34499
$117$	0	0
$118$	−29.5723	−2.72235
$119$	0	0
$120$	5.84340	0.533427
$121$	−3.68321	−0.334837
$122$	−7.57184	−0.685522
$123$	3.99297	0.360034
$124$	49.2557	4.42330
$125$	−11.9313	−1.06716
$126$	0	0
$127$	−3.12412	−0.277221	−0.138610	−	0.990347i	$-0.544264\pi$
$127$	−3.12412	−0.277221	−0.138610	+	0.990347i	$0.544264\pi$
$128$	4.93170	0.435904
$129$	1.73997	0.153196
$130$	0	0
$131$	−10.2092	−0.891982	−0.445991	−	0.895038i	$-0.647149\pi$
$131$	−10.2092	−0.891982	−0.445991	+	0.895038i	$0.647149\pi$
$132$	6.59261	0.573813
$133$	0	0
$134$	−35.1579	−3.03718
$135$	4.79010	0.412266
$136$	21.8595	1.87443
$137$	9.99261	0.853726	0.426863	−	0.904316i	$-0.359619\pi$
$137$	9.99261	0.853726	0.426863	+	0.904316i	$0.359619\pi$
$138$	2.66546	0.226899
$139$	1.66420	0.141156	0.0705778	−	0.997506i	$-0.477516\pi$
$139$	1.66420	0.141156	0.0705778	+	0.997506i	$0.477516\pi$
$140$	0	0
$141$	−0.547025	−0.0460679
$142$	−3.49748	−0.293502
$143$	0	0
$144$	−23.7513	−1.97928
$145$	−8.66110	−0.719265
$146$	−23.5649	−1.95025
$147$	0	0
$148$	−27.9863	−2.30046
$149$	−19.7980	−1.62192	−0.810959	−	0.585103i	$-0.801054\pi$
$149$	−19.7980	−1.62192	−0.810959	+	0.585103i	$0.801054\pi$
$150$	−3.22276	−0.263138
$151$	−7.53493	−0.613184	−0.306592	−	0.951841i	$-0.599189\pi$
$151$	−7.53493	−0.613184	−0.306592	+	0.951841i	$0.599189\pi$
$152$	−25.7390	−2.08771
$153$	8.53937	0.690367
$154$	0	0
$155$	16.8912	1.35673
$156$	0	0
$157$	−14.0045	−1.11768	−0.558839	−	0.829276i	$-0.688753\pi$
$157$	−14.0045	−1.11768	−0.558839	+	0.829276i	$0.688753\pi$
$158$	16.0643	1.27801
$159$	−3.76700	−0.298743
$160$	−13.7483	−1.08690
$161$	0	0
$162$	−17.2219	−1.35308
$163$	7.16995	0.561594	0.280797	−	0.959767i	$-0.409401\pi$
$163$	7.16995	0.561594	0.280797	+	0.959767i	$0.409401\pi$
$164$	−36.2036	−2.82703
$165$	2.26079	0.176002
$166$	6.98072	0.541809
$167$	−17.9805	−1.39138	−0.695688	−	0.718344i	$-0.744901\pi$
$167$	−17.9805	−1.39138	−0.695688	+	0.718344i	$0.744901\pi$
$168$	0	0
$169$	0	0
$170$	13.0472	1.00068
$171$	−10.0549	−0.768917
$172$	−15.7761	−1.20291
$173$	12.8116	0.974047	0.487023	−	0.873389i	$-0.338083\pi$
$173$	12.8116	0.974047	0.487023	+	0.873389i	$0.338083\pi$
$174$	−7.21072	−0.546643
$175$	0	0
$176$	−23.5232	−1.77312
$177$	−5.92298	−0.445199
$178$	−4.55561	−0.341458
$179$	1.84022	0.137545	0.0687723	−	0.997632i	$-0.478092\pi$
$179$	1.84022	0.137545	0.0687723	+	0.997632i	$0.478092\pi$
$180$	−20.6971	−1.54267
$181$	−3.29928	−0.245234	−0.122617	−	0.992454i	$-0.539129\pi$
$181$	−3.29928	−0.245234	−0.122617	+	0.992454i	$0.539129\pi$
$182$	0	0
$183$	−1.51655	−0.112107
$184$	−13.8852	−1.02363
$185$	−9.59730	−0.705607
$186$	14.0626	1.03112
$187$	8.45734	0.618462
$188$	4.95980	0.361731
$189$	0	0
$190$	−15.3628	−1.11453
$191$	4.89614	0.354272	0.177136	−	0.984186i	$-0.443317\pi$
$191$	4.89614	0.354272	0.177136	+	0.984186i	$0.443317\pi$
$192$	−2.42852	−0.175264
$193$	−3.01910	−0.217320	−0.108660	−	0.994079i	$-0.534656\pi$
$193$	−3.01910	−0.217320	−0.108660	+	0.994079i	$0.534656\pi$
$194$	−40.0768	−2.87735
$195$	0	0
$196$	0	0
$197$	−4.64991	−0.331292	−0.165646	−	0.986185i	$-0.552971\pi$
$197$	−4.64991	−0.331292	−0.165646	+	0.986185i	$0.552971\pi$
$198$	−19.1240	−1.35908
$199$	−0.410721	−0.0291152	−0.0145576	−	0.999894i	$-0.504634\pi$
$199$	−0.410721	−0.0291152	−0.0145576	+	0.999894i	$0.504634\pi$
$200$	16.7884	1.18712
$201$	−7.04171	−0.496684
$202$	−3.31160	−0.233004
$203$	0	0
$204$	7.62027	0.533525
$205$	−12.4152	−0.867118
$206$	−29.6844	−2.06821
$207$	−5.42425	−0.377012
$208$	0	0
$209$	−9.95831	−0.688831
$210$	0	0
$211$	−7.51600	−0.517423	−0.258711	−	0.965955i	$-0.583298\pi$
$211$	−7.51600	−0.517423	−0.258711	+	0.965955i	$0.583298\pi$
$212$	34.1549	2.34577
$213$	−0.700504	−0.0479977
$214$	13.2931	0.908699
$215$	−5.41005	−0.368962
$216$	−20.7745	−1.41353
$217$	0	0
$218$	4.47267	0.302927
$219$	−4.71978	−0.318933
$220$	−20.4982	−1.38199
$221$	0	0
$222$	−7.99014	−0.536263
$223$	−22.5794	−1.51203	−0.756016	−	0.654553i	$-0.772857\pi$
$223$	−22.5794	−1.51203	−0.756016	+	0.654553i	$0.772857\pi$
$224$	0	0
$225$	6.55837	0.437225
$226$	22.2390	1.47932
$227$	13.6717	0.907424	0.453712	−	0.891148i	$-0.350099\pi$
$227$	13.6717	0.907424	0.453712	+	0.891148i	$0.350099\pi$
$228$	−8.97268	−0.594230
$229$	7.93086	0.524086	0.262043	−	0.965056i	$-0.415604\pi$
$229$	7.93086	0.524086	0.262043	+	0.965056i	$0.415604\pi$
$230$	−8.28766	−0.546472
$231$	0	0
$232$	37.5629	2.46613
$233$	−6.57171	−0.430527	−0.215263	−	0.976556i	$-0.569061\pi$
$233$	−6.57171	−0.430527	−0.215263	+	0.976556i	$0.569061\pi$
$234$	0	0
$235$	1.70085	0.110951
$236$	53.7028	3.49576
$237$	3.21749	0.208998
$238$	0	0
$239$	9.39284	0.607572	0.303786	−	0.952740i	$-0.401749\pi$
$239$	9.39284	0.607572	0.303786	+	0.952740i	$0.401749\pi$
$240$	−7.26833	−0.469169
$241$	−10.0858	−0.649686	−0.324843	−	0.945768i	$-0.605311\pi$
$241$	−10.0858	−0.649686	−0.324843	+	0.945768i	$0.605311\pi$
$242$	9.53435	0.612891
$243$	−12.3636	−0.793128
$244$	13.7503	0.880275
$245$	0	0
$246$	−10.3362	−0.659012
$247$	0	0
$248$	−73.2565	−4.65179
$249$	1.39816	0.0886045
$250$	30.8853	1.95336
$251$	10.3485	0.653194	0.326597	−	0.945164i	$-0.394098\pi$
$251$	10.3485	0.653194	0.326597	+	0.945164i	$0.394098\pi$
$252$	0	0
$253$	−5.37215	−0.337744
$254$	8.08709	0.507429
$255$	2.61320	0.163645
$256$	−22.1343	−1.38339
$257$	−7.98658	−0.498189	−0.249095	−	0.968479i	$-0.580133\pi$
$257$	−7.98658	−0.498189	−0.249095	+	0.968479i	$0.580133\pi$
$258$	−4.50409	−0.280412
$259$	0	0
$260$	0	0
$261$	14.6739	0.908292
$262$	26.4275	1.63270
$263$	5.05934	0.311972	0.155986	−	0.987759i	$-0.450144\pi$
$263$	5.05934	0.311972	0.155986	+	0.987759i	$0.450144\pi$
$264$	−9.80498	−0.603455
$265$	11.7127	0.719503
$266$	0	0
$267$	−0.912435	−0.0558401
$268$	63.8461	3.90002
$269$	−13.8902	−0.846902	−0.423451	−	0.905919i	$-0.639181\pi$
$269$	−13.8902	−0.846902	−0.423451	+	0.905919i	$0.639181\pi$
$270$	−12.3997	−0.754619
$271$	−8.32721	−0.505842	−0.252921	−	0.967487i	$-0.581391\pi$
$271$	−8.32721	−0.505842	−0.252921	+	0.967487i	$0.581391\pi$
$272$	−27.1900	−1.64863
$273$	0	0
$274$	−25.8669	−1.56267
$275$	6.49537	0.391685
$276$	−4.84043	−0.291360
$277$	−23.2116	−1.39465	−0.697325	−	0.716755i	$-0.745626\pi$
$277$	−23.2116	−1.39465	−0.697325	+	0.716755i	$0.745626\pi$
$278$	−4.30795	−0.258374
$279$	−28.6176	−1.71329
$280$	0	0
$281$	−27.1595	−1.62020	−0.810100	−	0.586292i	$-0.800587\pi$
$281$	−27.1595	−1.62020	−0.810100	+	0.586292i	$0.800587\pi$
$282$	1.41603	0.0843234
$283$	16.1513	0.960092	0.480046	−	0.877243i	$-0.340620\pi$
$283$	16.1513	0.960092	0.480046	+	0.877243i	$0.340620\pi$
$284$	6.35136	0.376884
$285$	−3.07698	−0.182265
$286$	0	0
$287$	0	0
$288$	23.2928	1.37254
$289$	−7.22433	−0.424960
$290$	22.4201	1.31656
$291$	−8.02691	−0.470546
$292$	42.7935	2.50430
$293$	−14.6452	−0.855582	−0.427791	−	0.903878i	$-0.640708\pi$
$293$	−14.6452	−0.855582	−0.427791	+	0.903878i	$0.640708\pi$
$294$	0	0
$295$	18.4162	1.07223
$296$	41.6232	2.41930
$297$	−8.03758	−0.466388
$298$	51.2492	2.96879
$299$	0	0
$300$	5.85248	0.337893
$301$	0	0
$302$	19.5049	1.12238
$303$	−0.663274	−0.0381041
$304$	32.0155	1.83622
$305$	4.71537	0.270001
$306$	−22.1050	−1.26366
$307$	−8.97844	−0.512427	−0.256213	−	0.966620i	$-0.582475\pi$
$307$	−8.97844	−0.512427	−0.256213	+	0.966620i	$0.582475\pi$
$308$	0	0
$309$	−5.94543	−0.338224
$310$	−43.7245	−2.48338
$311$	−12.1816	−0.690755	−0.345378	−	0.938464i	$-0.612249\pi$
$311$	−12.1816	−0.690755	−0.345378	+	0.938464i	$0.612249\pi$
$312$	0	0
$313$	−13.1240	−0.741810	−0.370905	−	0.928671i	$-0.620953\pi$
$313$	−13.1240	−0.741810	−0.370905	+	0.928671i	$0.620953\pi$
$314$	36.2520	2.04582
$315$	0	0
$316$	−29.1725	−1.64108
$317$	16.7155	0.938836	0.469418	−	0.882976i	$-0.344464\pi$
$317$	16.7155	0.938836	0.469418	+	0.882976i	$0.344464\pi$
$318$	9.75127	0.546824
$319$	14.5330	0.813689
$320$	7.55095	0.422111
$321$	2.66245	0.148604
$322$	0	0
$323$	−11.5106	−0.640468
$324$	31.2748	1.73749
$325$	0	0
$326$	−18.5601	−1.02795
$327$	0.895821	0.0495390
$328$	53.8445	2.97307
$329$	0	0
$330$	−5.85228	−0.322157
$331$	3.96665	0.218027	0.109013	−	0.994040i	$-0.465231\pi$
$331$	3.96665	0.218027	0.109013	+	0.994040i	$0.465231\pi$
$332$	−12.6769	−0.695733
$333$	16.2600	0.891045
$334$	46.5445	2.54680
$335$	21.8946	1.19623
$336$	0	0
$337$	13.7032	0.746461	0.373230	−	0.927739i	$-0.378250\pi$
$337$	13.7032	0.746461	0.373230	+	0.927739i	$0.378250\pi$
$338$	0	0
$339$	4.45421	0.241919
$340$	−23.6935	−1.28496
$341$	−28.3426	−1.53484
$342$	26.0281	1.40744
$343$	0	0
$344$	23.4632	1.26505
$345$	−1.65992	−0.0893671
$346$	−33.1641	−1.78291
$347$	−26.3979	−1.41711	−0.708556	−	0.705655i	$-0.750653\pi$
$347$	−26.3979	−1.41711	−0.708556	+	0.705655i	$0.750653\pi$
$348$	13.0945	0.701941
$349$	4.89024	0.261769	0.130884	−	0.991398i	$-0.458218\pi$
$349$	4.89024	0.261769	0.130884	+	0.991398i	$0.458218\pi$
$350$	0	0
$351$	0	0
$352$	23.0690	1.22958
$353$	13.5577	0.721605	0.360802	−	0.932642i	$-0.382503\pi$
$353$	13.5577	0.721605	0.360802	+	0.932642i	$0.382503\pi$
$354$	15.3322	0.814899
$355$	2.17806	0.115599
$356$	8.27291	0.438464
$357$	0	0
$358$	−4.76360	−0.251764
$359$	−8.58568	−0.453135	−0.226567	−	0.973996i	$-0.572750\pi$
$359$	−8.58568	−0.453135	−0.226567	+	0.973996i	$0.572750\pi$
$360$	30.7821	1.62236
$361$	−5.44653	−0.286659
$362$	8.54053	0.448880
$363$	1.90962	0.100229
$364$	0	0
$365$	14.6751	0.768130
$366$	3.92574	0.205202
$367$	1.66322	0.0868196	0.0434098	−	0.999057i	$-0.486178\pi$
$367$	1.66322	0.0868196	0.0434098	+	0.999057i	$0.486178\pi$
$368$	17.2712	0.900324
$369$	21.0343	1.09500
$370$	24.8436	1.29156
$371$	0	0
$372$	−25.5374	−1.32405
$373$	13.9635	0.723002	0.361501	−	0.932372i	$-0.382264\pi$
$373$	13.9635	0.723002	0.361501	+	0.932372i	$0.382264\pi$
$374$	−21.8927	−1.13204
$375$	6.18595	0.319441
$376$	−7.37655	−0.380417
$377$	0	0
$378$	0	0
$379$	31.5758	1.62194	0.810969	−	0.585089i	$-0.198941\pi$
$379$	31.5758	1.62194	0.810969	+	0.585089i	$0.198941\pi$
$380$	27.8985	1.43116
$381$	1.61975	0.0829822
$382$	−12.6741	−0.648466
$383$	−31.9082	−1.63043	−0.815217	−	0.579156i	$-0.803382\pi$
$383$	−31.9082	−1.63043	−0.815217	+	0.579156i	$0.803382\pi$
$384$	−2.55692	−0.130482
$385$	0	0
$386$	7.81525	0.397786
$387$	9.16588	0.465928
$388$	72.7788	3.69478
$389$	−25.4150	−1.28859	−0.644296	−	0.764776i	$-0.722850\pi$
$389$	−25.4150	−1.28859	−0.644296	+	0.764776i	$0.722850\pi$
$390$	0	0
$391$	−6.20956	−0.314031
$392$	0	0
$393$	5.29312	0.267003
$394$	12.0368	0.606403
$395$	−10.0041	−0.503358
$396$	34.7288	1.74519
$397$	4.15897	0.208733	0.104366	−	0.994539i	$-0.466719\pi$
$397$	4.15897	0.208733	0.104366	+	0.994539i	$0.466719\pi$
$398$	1.06319	0.0532930
$399$	0	0
$400$	−20.8823	−1.04412
$401$	19.6013	0.978844	0.489422	−	0.872047i	$-0.337208\pi$
$401$	19.6013	0.978844	0.489422	+	0.872047i	$0.337208\pi$
$402$	18.2282	0.909139
$403$	0	0
$404$	6.01381	0.299198
$405$	10.7250	0.532929
$406$	0	0
$407$	16.1039	0.798238
$408$	−11.3334	−0.561086
$409$	−17.6337	−0.871930	−0.435965	−	0.899964i	$-0.643593\pi$
$409$	−17.6337	−0.871930	−0.435965	+	0.899964i	$0.643593\pi$
$410$	32.1381	1.58719
$411$	−5.18083	−0.255551
$412$	53.9063	2.65577
$413$	0	0
$414$	14.0412	0.690088
$415$	−4.34725	−0.213398
$416$	0	0
$417$	−0.862831	−0.0422530
$418$	25.7781	1.26085
$419$	−29.8911	−1.46027	−0.730137	−	0.683301i	$-0.760544\pi$
$419$	−29.8911	−1.46027	−0.730137	+	0.683301i	$0.760544\pi$
$420$	0	0
$421$	12.8528	0.626407	0.313203	−	0.949686i	$-0.398598\pi$
$421$	12.8528	0.626407	0.313203	+	0.949686i	$0.398598\pi$
$422$	19.4559	0.947100
$423$	−2.88164	−0.140110
$424$	−50.7975	−2.46694
$425$	7.50787	0.364185
$426$	1.81332	0.0878559
$427$	0	0
$428$	−24.1401	−1.16685
$429$	0	0
$430$	14.0045	0.675355
$431$	8.97060	0.432098	0.216049	−	0.976382i	$-0.430683\pi$
$431$	8.97060	0.432098	0.216049	+	0.976382i	$0.430683\pi$
$432$	25.8405	1.24325
$433$	3.45062	0.165826	0.0829132	−	0.996557i	$-0.473578\pi$
$433$	3.45062	0.165826	0.0829132	+	0.996557i	$0.473578\pi$
$434$	0	0
$435$	4.49048	0.215302
$436$	−8.12228	−0.388987
$437$	7.31161	0.349762
$438$	12.2176	0.583780
$439$	38.5144	1.83819	0.919096	−	0.394034i	$-0.128921\pi$
$439$	38.5144	1.83819	0.919096	+	0.394034i	$0.128921\pi$
$440$	30.4864	1.45338
$441$	0	0
$442$	0	0
$443$	−15.0399	−0.714569	−0.357284	−	0.933996i	$-0.616297\pi$
$443$	−15.0399	−0.714569	−0.357284	+	0.933996i	$0.616297\pi$
$444$	14.5100	0.688612
$445$	2.83701	0.134487
$446$	58.4492	2.76765
$447$	10.2646	0.485499
$448$	0	0
$449$	−38.9235	−1.83691	−0.918456	−	0.395522i	$-0.870564\pi$
$449$	−38.9235	−1.83691	−0.918456	+	0.395522i	$0.870564\pi$
$450$	−16.9770	−0.800303
$451$	20.8322	0.980952
$452$	−40.3856	−1.89958
$453$	3.90660	0.183548
$454$	−35.3906	−1.66097
$455$	0	0
$456$	13.3448	0.624927
$457$	−13.9396	−0.652069	−0.326034	−	0.945358i	$-0.605713\pi$
$457$	−13.9396	−0.652069	−0.326034	+	0.945358i	$0.605713\pi$
$458$	−20.5298	−0.959295
$459$	−9.29049	−0.433643
$460$	15.0502	0.701721
$461$	−37.4635	−1.74485	−0.872424	−	0.488749i	$-0.837453\pi$
$461$	−37.4635	−1.74485	−0.872424	+	0.488749i	$0.837453\pi$
$462$	0	0
$463$	6.75275	0.313827	0.156913	−	0.987612i	$-0.449846\pi$
$463$	6.75275	0.313827	0.156913	+	0.987612i	$0.449846\pi$
$464$	−46.7228	−2.16905
$465$	−8.75749	−0.406119
$466$	17.0115	0.788044
$467$	−5.05032	−0.233701	−0.116851	−	0.993150i	$-0.537280\pi$
$467$	−5.05032	−0.233701	−0.116851	+	0.993150i	$0.537280\pi$
$468$	0	0
$469$	0	0
$470$	−4.40283	−0.203087
$471$	7.26084	0.334562
$472$	−79.8705	−3.67634
$473$	9.07783	0.417399
$474$	−8.32878	−0.382554
$475$	−8.84033	−0.405622
$476$	0	0
$477$	−19.8440	−0.908593
$478$	−24.3143	−1.11211
$479$	9.45319	0.431927	0.215964	−	0.976401i	$-0.430711\pi$
$479$	9.45319	0.431927	0.215964	+	0.976401i	$0.430711\pi$
$480$	7.12801	0.325348
$481$	0	0
$482$	26.1082	1.18920
$483$	0	0
$484$	−17.3142	−0.787010
$485$	24.9579	1.13328
$486$	32.0045	1.45175
$487$	−39.9996	−1.81255	−0.906277	−	0.422684i	$-0.861088\pi$
$487$	−39.9996	−1.81255	−0.906277	+	0.422684i	$0.861088\pi$
$488$	−20.4504	−0.925748
$489$	−3.71737	−0.168105
$490$	0	0
$491$	−6.76097	−0.305118	−0.152559	−	0.988294i	$-0.548751\pi$
$491$	−6.76097	−0.305118	−0.152559	+	0.988294i	$0.548751\pi$
$492$	18.7704	0.846233
$493$	16.7984	0.756560
$494$	0	0
$495$	11.9095	0.535291
$496$	91.1203	4.09142
$497$	0	0
$498$	−3.61927	−0.162183
$499$	−11.3575	−0.508433	−0.254217	−	0.967147i	$-0.581818\pi$
$499$	−11.3575	−0.508433	−0.254217	+	0.967147i	$0.581818\pi$
$500$	−56.0871	−2.50829
$501$	9.32230	0.416490
$502$	−26.7882	−1.19562
$503$	13.9285	0.621040	0.310520	−	0.950567i	$-0.399497\pi$
$503$	13.9285	0.621040	0.310520	+	0.950567i	$0.399497\pi$
$504$	0	0
$505$	2.06230	0.0917713
$506$	13.9063	0.618212
$507$	0	0
$508$	−14.6860	−0.651587
$509$	−19.8149	−0.878281	−0.439141	−	0.898418i	$-0.644717\pi$
$509$	−19.8149	−0.878281	−0.439141	+	0.898418i	$0.644717\pi$
$510$	−6.76454	−0.299539
$511$	0	0
$512$	47.4335	2.09628
$513$	10.9393	0.482983
$514$	20.6741	0.911894
$515$	18.4860	0.814590
$516$	8.17934	0.360076
$517$	−2.85396	−0.125517
$518$	0	0
$519$	−6.64237	−0.291568
$520$	0	0
$521$	31.0951	1.36230	0.681151	−	0.732143i	$-0.261480\pi$
$521$	31.0951	1.36230	0.681151	+	0.732143i	$0.261480\pi$
$522$	−37.9849	−1.66255
$523$	−22.7202	−0.993485	−0.496742	−	0.867898i	$-0.665471\pi$
$523$	−22.7202	−0.993485	−0.496742	+	0.867898i	$0.665471\pi$
$524$	−47.9919	−2.09654
$525$	0	0
$526$	−13.0966	−0.571040
$527$	−32.7607	−1.42708
$528$	12.1960	0.530761
$529$	−19.0557	−0.828507
$530$	−30.3194	−1.31699
$531$	−31.2013	−1.35402
$532$	0	0
$533$	0	0
$534$	2.36193	0.102211
$535$	−8.27830	−0.357902
$536$	−94.9564	−4.10149
$537$	−0.954091	−0.0411721
$538$	35.9563	1.55018
$539$	0	0
$540$	22.5176	0.969002
$541$	−2.09872	−0.0902310	−0.0451155	−	0.998982i	$-0.514366\pi$
$541$	−2.09872	−0.0902310	−0.0451155	+	0.998982i	$0.514366\pi$
$542$	21.5558	0.925902
$543$	1.71057	0.0734074
$544$	26.6650	1.14325
$545$	−2.78536	−0.119312
$546$	0	0
$547$	25.3770	1.08504	0.542521	−	0.840042i	$-0.317470\pi$
$547$	25.3770	1.08504	0.542521	+	0.840042i	$0.317470\pi$
$548$	46.9738	2.00662
$549$	−7.98894	−0.340959
$550$	−16.8139	−0.716948
$551$	−19.7797	−0.842642
$552$	7.19902	0.306411
$553$	0	0
$554$	60.0855	2.55279
$555$	4.97587	0.211214
$556$	7.82316	0.331776
$557$	44.2503	1.87495	0.937473	−	0.348058i	$-0.113159\pi$
$557$	44.2503	1.87495	0.937473	+	0.348058i	$0.113159\pi$
$558$	74.0794	3.13603
$559$	0	0
$560$	0	0
$561$	−4.38484	−0.185128
$562$	70.3051	2.96564
$563$	38.8907	1.63905	0.819523	−	0.573046i	$-0.194238\pi$
$563$	38.8907	1.63905	0.819523	+	0.573046i	$0.194238\pi$
$564$	−2.57149	−0.108279
$565$	−13.8494	−0.582647
$566$	−41.8092	−1.75737
$567$	0	0
$568$	−9.44618	−0.396353
$569$	46.1579	1.93504	0.967520	−	0.252796i	$-0.0813500\pi$
$569$	46.1579	1.93504	0.967520	+	0.252796i	$0.0813500\pi$
$570$	7.96508	0.333620
$571$	21.1368	0.884548	0.442274	−	0.896880i	$-0.354172\pi$
$571$	21.1368	0.884548	0.442274	+	0.896880i	$0.354172\pi$
$572$	0	0
$573$	−2.53848	−0.106047
$574$	0	0
$575$	−4.76904	−0.198883
$576$	−12.7931	−0.533045
$577$	25.3304	1.05452	0.527259	−	0.849705i	$-0.323220\pi$
$577$	25.3304	1.05452	0.527259	+	0.849705i	$0.323220\pi$
$578$	18.7009	0.777855
$579$	1.56530	0.0650517
$580$	−40.7146	−1.69058
$581$	0	0
$582$	20.7785	0.861295
$583$	−19.6533	−0.813958
$584$	−63.6455	−2.63367
$585$	0	0
$586$	37.9106	1.56607
$587$	3.56287	0.147056	0.0735278	−	0.997293i	$-0.476574\pi$
$587$	3.56287	0.147056	0.0735278	+	0.997293i	$0.476574\pi$
$588$	0	0
$589$	38.5749	1.58945
$590$	−47.6722	−1.96263
$591$	2.41082	0.0991679
$592$	−51.7732	−2.12786
$593$	25.3536	1.04115	0.520573	−	0.853817i	$-0.325718\pi$
$593$	25.3536	1.04115	0.520573	+	0.853817i	$0.325718\pi$
$594$	20.8061	0.853685
$595$	0	0
$596$	−93.0677	−3.81220
$597$	0.212945	0.00871524
$598$	0	0
$599$	10.9216	0.446243	0.223122	−	0.974791i	$-0.428375\pi$
$599$	10.9216	0.446243	0.223122	+	0.974791i	$0.428375\pi$
$600$	−8.70421	−0.355348
$601$	−24.2564	−0.989439	−0.494720	−	0.869053i	$-0.664729\pi$
$601$	−24.2564	−0.989439	−0.494720	+	0.869053i	$0.664729\pi$
$602$	0	0
$603$	−37.0946	−1.51061
$604$	−35.4206	−1.44124
$605$	−5.93753	−0.241395
$606$	1.71695	0.0697464
$607$	9.85447	0.399981	0.199990	−	0.979798i	$-0.435909\pi$
$607$	9.85447	0.399981	0.199990	+	0.979798i	$0.435909\pi$
$608$	−31.3974	−1.27333
$609$	0	0
$610$	−12.2062	−0.494215
$611$	0	0
$612$	40.1423	1.62266
$613$	−3.67688	−0.148508	−0.0742540	−	0.997239i	$-0.523658\pi$
$613$	−3.67688	−0.148508	−0.0742540	+	0.997239i	$0.523658\pi$
$614$	23.2416	0.937955
$615$	6.43688	0.259560
$616$	0	0
$617$	18.7468	0.754718	0.377359	−	0.926067i	$-0.376832\pi$
$617$	18.7468	0.754718	0.377359	+	0.926067i	$0.376832\pi$
$618$	15.3903	0.619090
$619$	−15.8945	−0.638854	−0.319427	−	0.947611i	$-0.603490\pi$
$619$	−15.8945	−0.638854	−0.319427	+	0.947611i	$0.603490\pi$
$620$	79.4029	3.18890
$621$	5.90137	0.236814
$622$	31.5333	1.26437
$623$	0	0
$624$	0	0
$625$	−7.22743	−0.289097
$626$	33.9727	1.35782
$627$	5.16304	0.206192
$628$	−65.8330	−2.62702
$629$	18.6141	0.742194
$630$	0	0
$631$	19.7451	0.786040	0.393020	−	0.919530i	$-0.371430\pi$
$631$	19.7451	0.786040	0.393020	+	0.919530i	$0.371430\pi$
$632$	43.3873	1.72585
$633$	3.89679	0.154883
$634$	−43.2698	−1.71846
$635$	−5.03624	−0.199857
$636$	−17.7081	−0.702173
$637$	0	0
$638$	−37.6200	−1.48939
$639$	−3.69014	−0.145980
$640$	7.95016	0.314258
$641$	−29.7786	−1.17618	−0.588092	−	0.808794i	$-0.700121\pi$
$641$	−29.7786	−1.17618	−0.588092	+	0.808794i	$0.700121\pi$
$642$	−6.89203	−0.272007
$643$	−11.5725	−0.456373	−0.228187	−	0.973617i	$-0.573280\pi$
$643$	−11.5725	−0.456373	−0.228187	+	0.973617i	$0.573280\pi$
$644$	0	0
$645$	2.80493	0.110444
$646$	29.7964	1.17232
$647$	−25.5065	−1.00276	−0.501382	−	0.865226i	$-0.667175\pi$
$647$	−25.5065	−1.00276	−0.501382	+	0.865226i	$0.667175\pi$
$648$	−46.5140	−1.82724
$649$	−30.9016	−1.21299
$650$	0	0
$651$	0	0
$652$	33.7049	1.31999
$653$	−44.8293	−1.75430	−0.877152	−	0.480212i	$-0.840560\pi$
$653$	−44.8293	−1.75430	−0.877152	+	0.480212i	$0.840560\pi$
$654$	−2.31892	−0.0906771
$655$	−16.4578	−0.643058
$656$	−66.9747	−2.61492
$657$	−24.8630	−0.969999
$658$	0	0
$659$	41.1734	1.60389	0.801944	−	0.597399i	$-0.203799\pi$
$659$	41.1734	1.60389	0.801944	+	0.597399i	$0.203799\pi$
$660$	10.6276	0.413680
$661$	−21.8938	−0.851569	−0.425785	−	0.904825i	$-0.640002\pi$
$661$	−21.8938	−0.851569	−0.425785	+	0.904825i	$0.640002\pi$
$662$	−10.2681	−0.399080
$663$	0	0
$664$	18.8539	0.731673
$665$	0	0
$666$	−42.0908	−1.63098
$667$	−10.6704	−0.413160
$668$	−84.5239	−3.27033
$669$	11.7067	0.452606
$670$	−56.6764	−2.18960
$671$	−7.91219	−0.305447
$672$	0	0
$673$	35.6688	1.37493	0.687466	−	0.726217i	$-0.258723\pi$
$673$	35.6688	1.37493	0.687466	+	0.726217i	$0.258723\pi$
$674$	−35.4721	−1.36633
$675$	−7.13524	−0.274635
$676$	0	0
$677$	−2.55532	−0.0982089	−0.0491044	−	0.998794i	$-0.515637\pi$
$677$	−2.55532	−0.0982089	−0.0491044	+	0.998794i	$0.515637\pi$
$678$	−11.5302	−0.442813
$679$	0	0
$680$	35.2386	1.35134
$681$	−7.08832	−0.271625
$682$	73.3678	2.80940
$683$	−35.7399	−1.36755	−0.683775	−	0.729693i	$-0.739663\pi$
$683$	−35.7399	−1.36755	−0.683775	+	0.729693i	$0.739663\pi$
$684$	−47.2666	−1.80728
$685$	16.1086	0.615479
$686$	0	0
$687$	−4.11188	−0.156878
$688$	−29.1848	−1.11266
$689$	0	0
$690$	4.29687	0.163579
$691$	−26.0292	−0.990197	−0.495099	−	0.868837i	$-0.664868\pi$
$691$	−26.0292	−0.990197	−0.495099	+	0.868837i	$0.664868\pi$
$692$	60.2254	2.28943
$693$	0	0
$694$	68.3335	2.59391
$695$	2.68278	0.101764
$696$	−19.4751	−0.738202
$697$	24.0796	0.912079
$698$	−12.6589	−0.479146
$699$	3.40721	0.128872
$700$	0	0
$701$	1.12731	0.0425779	0.0212890	−	0.999773i	$-0.493223\pi$
$701$	1.12731	0.0425779	0.0212890	+	0.999773i	$0.493223\pi$
$702$	0	0
$703$	−21.9177	−0.826641
$704$	−12.6702	−0.477525
$705$	−0.881834	−0.0332118
$706$	−35.0955	−1.32084
$707$	0	0
$708$	−27.8431	−1.04641
$709$	−6.05031	−0.227224	−0.113612	−	0.993525i	$-0.536242\pi$
$709$	−6.05031	−0.227224	−0.113612	+	0.993525i	$0.536242\pi$
$710$	−5.63813	−0.211595
$711$	16.9492	0.635644
$712$	−12.3040	−0.461114
$713$	20.8098	0.779332
$714$	0	0
$715$	0	0
$716$	8.65061	0.323288
$717$	−4.86986	−0.181868
$718$	22.2249	0.829425
$719$	−47.1177	−1.75719	−0.878597	−	0.477563i	$-0.841520\pi$
$719$	−47.1177	−1.75719	−0.878597	+	0.477563i	$0.841520\pi$
$720$	−38.2884	−1.42692
$721$	0	0
$722$	14.0989	0.524706
$723$	5.22916	0.194475
$724$	−15.5095	−0.576404
$725$	12.9014	0.479146
$726$	−4.94324	−0.183461
$727$	17.9215	0.664671	0.332335	−	0.943161i	$-0.392163\pi$
$727$	17.9215	0.664671	0.332335	+	0.943161i	$0.392163\pi$
$728$	0	0
$729$	−13.5489	−0.501810
$730$	−37.9880	−1.40600
$731$	10.4929	0.388093
$732$	−7.12908	−0.263498
$733$	45.2685	1.67203	0.836016	−	0.548705i	$-0.184879\pi$
$733$	45.2685	1.67203	0.836016	+	0.548705i	$0.184879\pi$
$734$	−4.30542	−0.158916
$735$	0	0
$736$	−16.9378	−0.624335
$737$	−36.7382	−1.35327
$738$	−54.4494	−2.00431
$739$	−19.2613	−0.708539	−0.354270	−	0.935143i	$-0.615270\pi$
$739$	−19.2613	−0.708539	−0.354270	+	0.935143i	$0.615270\pi$
$740$	−45.1155	−1.65848
$741$	0	0
$742$	0	0
$743$	−34.8853	−1.27982	−0.639908	−	0.768452i	$-0.721028\pi$
$743$	−34.8853	−1.27982	−0.639908	+	0.768452i	$0.721028\pi$
$744$	37.9810	1.39245
$745$	−31.9155	−1.16929
$746$	−36.1459	−1.32339
$747$	7.36525	0.269480
$748$	39.7567	1.45365
$749$	0	0
$750$	−16.0130	−0.584711
$751$	−24.9668	−0.911051	−0.455526	−	0.890223i	$-0.650549\pi$
$751$	−24.9668	−0.911051	−0.455526	+	0.890223i	$0.650549\pi$
$752$	9.17535	0.334591
$753$	−5.36536	−0.195525
$754$	0	0
$755$	−12.1467	−0.442064
$756$	0	0
$757$	−10.6049	−0.385440	−0.192720	−	0.981254i	$-0.561731\pi$
$757$	−10.6049	−0.385440	−0.192720	+	0.981254i	$0.561731\pi$
$758$	−81.7370	−2.96882
$759$	2.78527	0.101099
$760$	−41.4926	−1.50510
$761$	−32.6388	−1.18316	−0.591578	−	0.806248i	$-0.701495\pi$
$761$	−32.6388	−1.18316	−0.591578	+	0.806248i	$0.701495\pi$
$762$	−4.19288	−0.151892
$763$	0	0
$764$	23.0160	0.832691
$765$	13.7659	0.497708
$766$	82.5976	2.98437
$767$	0	0
$768$	11.4759	0.414100
$769$	−52.1752	−1.88149	−0.940744	−	0.339119i	$-0.889871\pi$
$769$	−52.1752	−1.88149	−0.940744	+	0.339119i	$0.889871\pi$
$770$	0	0
$771$	4.14077	0.149126
$772$	−14.1924	−0.510794
$773$	−35.7057	−1.28425	−0.642123	−	0.766602i	$-0.721946\pi$
$773$	−35.7057	−1.28425	−0.642123	+	0.766602i	$0.721946\pi$
$774$	−23.7268	−0.852842
$775$	−25.1607	−0.903800
$776$	−108.242	−3.88565
$777$	0	0
$778$	65.7893	2.35866
$779$	−28.3531	−1.01586
$780$	0	0
$781$	−3.65469	−0.130775
$782$	16.0741	0.574808
$783$	−15.9646	−0.570529
$784$	0	0
$785$	−22.5760	−0.805770
$786$	−13.7018	−0.488726
$787$	−6.10621	−0.217663	−0.108831	−	0.994060i	$-0.534711\pi$
$787$	−6.10621	−0.217663	−0.108831	+	0.994060i	$0.534711\pi$
$788$	−21.8586	−0.778679
$789$	−2.62310	−0.0933847
$790$	25.8965	0.921356
$791$	0	0
$792$	−51.6510	−1.83534
$793$	0	0
$794$	−10.7659	−0.382068
$795$	−6.07261	−0.215373
$796$	−1.93074	−0.0684332
$797$	46.2299	1.63755	0.818773	−	0.574117i	$-0.194654\pi$
$797$	46.2299	1.63755	0.818773	+	0.574117i	$0.194654\pi$
$798$	0	0
$799$	−3.29884	−0.116704
$800$	20.4792	0.724048
$801$	−4.80656	−0.169831
$802$	−50.7400	−1.79169
$803$	−24.6242	−0.868968
$804$	−33.1020	−1.16742
$805$	0	0
$806$	0	0
$807$	7.20161	0.253509
$808$	−8.94415	−0.314654
$809$	39.2879	1.38129	0.690644	−	0.723195i	$-0.257327\pi$
$809$	39.2879	1.38129	0.690644	+	0.723195i	$0.257327\pi$
$810$	−27.7627	−0.975482
$811$	−6.90664	−0.242525	−0.121262	−	0.992620i	$-0.538694\pi$
$811$	−6.90664	−0.242525	−0.121262	+	0.992620i	$0.538694\pi$
$812$	0	0
$813$	4.31738	0.151417
$814$	−41.6864	−1.46111
$815$	11.5583	0.404871
$816$	14.0971	0.493496
$817$	−12.3551	−0.432251
$818$	45.6466	1.59600
$819$	0	0
$820$	−58.3622	−2.03810
$821$	−1.91049	−0.0666765	−0.0333382	−	0.999444i	$-0.510614\pi$
$821$	−1.91049	−0.0666765	−0.0333382	+	0.999444i	$0.510614\pi$
$822$	13.4111	0.467765
$823$	1.57969	0.0550645	0.0275322	−	0.999621i	$-0.491235\pi$
$823$	1.57969	0.0550645	0.0275322	+	0.999621i	$0.491235\pi$
$824$	−80.1732	−2.79296
$825$	−3.36763	−0.117246
$826$	0	0
$827$	−32.5050	−1.13031	−0.565155	−	0.824985i	$-0.691184\pi$
$827$	−32.5050	−1.13031	−0.565155	+	0.824985i	$0.691184\pi$
$828$	−25.4986	−0.886138
$829$	35.0538	1.21747	0.608735	−	0.793373i	$-0.291677\pi$
$829$	35.0538	1.21747	0.608735	+	0.793373i	$0.291677\pi$
$830$	11.2533	0.390608
$831$	12.0344	0.417469
$832$	0	0
$833$	0	0
$834$	2.23353	0.0773407
$835$	−28.9856	−1.00309
$836$	−46.8126	−1.61905
$837$	31.1347	1.07617
$838$	77.3760	2.67291
$839$	5.35487	0.184871	0.0924354	−	0.995719i	$-0.470535\pi$
$839$	5.35487	0.184871	0.0924354	+	0.995719i	$0.470535\pi$
$840$	0	0
$841$	−0.133978	−0.00461993
$842$	−33.2708	−1.14659
$843$	14.0813	0.484985
$844$	−35.3316	−1.21616
$845$	0	0
$846$	7.45942	0.256460
$847$	0	0
$848$	63.1846	2.16977
$849$	−8.37387	−0.287391
$850$	−19.4349	−0.666611
$851$	−11.8238	−0.405314
$852$	−3.29297	−0.112815
$853$	49.6270	1.69920	0.849598	−	0.527431i	$-0.176845\pi$
$853$	49.6270	1.69920	0.849598	+	0.527431i	$0.176845\pi$
$854$	0	0
$855$	−16.2090	−0.554337
$856$	35.9028	1.22713
$857$	−5.88392	−0.200991	−0.100496	−	0.994938i	$-0.532043\pi$
$857$	−5.88392	−0.200991	−0.100496	+	0.994938i	$0.532043\pi$
$858$	0	0
$859$	−43.3862	−1.48032	−0.740159	−	0.672432i	$-0.765250\pi$
$859$	−43.3862	−1.48032	−0.740159	+	0.672432i	$0.765250\pi$
$860$	−25.4318	−0.867219
$861$	0	0
$862$	−23.2213	−0.790920
$863$	31.1272	1.05958	0.529792	−	0.848128i	$-0.322270\pi$
$863$	31.1272	1.05958	0.529792	+	0.848128i	$0.322270\pi$
$864$	−25.3416	−0.862139
$865$	20.6530	0.702222
$866$	−8.93228	−0.303531
$867$	3.74557	0.127206
$868$	0	0
$869$	16.7864	0.569439
$870$	−11.6241	−0.394093
$871$	0	0
$872$	12.0800	0.409081
$873$	−42.2844	−1.43111
$874$	−18.9268	−0.640209
$875$	0	0
$876$	−22.1870	−0.749629
$877$	−29.9106	−1.01001	−0.505004	−	0.863117i	$-0.668509\pi$
$877$	−29.9106	−1.01001	−0.505004	+	0.863117i	$0.668509\pi$
$878$	−99.6984	−3.36466
$879$	7.59304	0.256107
$880$	−37.9206	−1.27830
$881$	−14.5695	−0.490860	−0.245430	−	0.969414i	$-0.578929\pi$
$881$	−14.5695	−0.490860	−0.245430	+	0.969414i	$0.578929\pi$
$882$	0	0
$883$	−48.9296	−1.64661	−0.823307	−	0.567597i	$-0.807873\pi$
$883$	−48.9296	−1.64661	−0.823307	+	0.567597i	$0.807873\pi$
$884$	0	0
$885$	−9.54817	−0.320958
$886$	38.9324	1.30796
$887$	−54.5902	−1.83296	−0.916480	−	0.400080i	$-0.868982\pi$
$887$	−54.5902	−1.83296	−0.916480	+	0.400080i	$0.868982\pi$
$888$	−21.5802	−0.724184
$889$	0	0
$890$	−7.34389	−0.246168
$891$	−17.9961	−0.602891
$892$	−106.143	−3.55392
$893$	3.88430	0.129983
$894$	−26.5710	−0.888666
$895$	2.96653	0.0991603
$896$	0	0
$897$	0	0
$898$	100.757	3.36232
$899$	−56.2955	−1.87756
$900$	30.8299	1.02766
$901$	−22.7169	−0.756810
$902$	−53.9263	−1.79555
$903$	0	0
$904$	60.0643	1.99771
$905$	−5.31862	−0.176797
$906$	−10.1126	−0.335970
$907$	22.7255	0.754589	0.377295	−	0.926093i	$-0.376854\pi$
$907$	22.7255	0.754589	0.377295	+	0.926093i	$0.376854\pi$
$908$	64.2688	2.13283
$909$	−3.49402	−0.115889
$910$	0	0
$911$	−42.2359	−1.39934	−0.699669	−	0.714467i	$-0.746669\pi$
$911$	−42.2359	−1.39934	−0.699669	+	0.714467i	$0.746669\pi$
$912$	−16.5990	−0.549646
$913$	7.29450	0.241413
$914$	36.0842	1.19356
$915$	−2.44476	−0.0808213
$916$	37.2818	1.23182
$917$	0	0
$918$	24.0494	0.793747
$919$	30.6940	1.01250	0.506251	−	0.862386i	$-0.331031\pi$
$919$	30.6940	1.01250	0.506251	+	0.862386i	$0.331031\pi$
$920$	−22.3838	−0.737971
$921$	4.65502	0.153388
$922$	96.9780	3.19380
$923$	0	0
$924$	0	0
$925$	14.2959	0.470048
$926$	−17.4802	−0.574434
$927$	−31.3195	−1.02867
$928$	45.8208	1.50414
$929$	−37.4250	−1.22787	−0.613936	−	0.789355i	$-0.710415\pi$
$929$	−37.4250	−1.22787	−0.613936	+	0.789355i	$0.710415\pi$
$930$	22.6696	0.743367
$931$	0	0
$932$	−30.8926	−1.01192
$933$	6.31574	0.206768
$934$	13.0733	0.427770
$935$	13.6337	0.445869
$936$	0	0
$937$	−44.3386	−1.44848	−0.724239	−	0.689549i	$-0.757809\pi$
$937$	−44.3386	−1.44848	−0.724239	+	0.689549i	$0.757809\pi$
$938$	0	0
$939$	6.80433	0.222051
$940$	7.99546	0.260783
$941$	−27.5052	−0.896646	−0.448323	−	0.893872i	$-0.647978\pi$
$941$	−27.5052	−0.896646	−0.448323	+	0.893872i	$0.647978\pi$
$942$	−18.7954	−0.612388
$943$	−15.2955	−0.498089
$944$	99.3472	3.23348
$945$	0	0
$946$	−23.4989	−0.764014
$947$	−5.08330	−0.165185	−0.0825925	−	0.996583i	$-0.526320\pi$
$947$	−5.08330	−0.165185	−0.0825925	+	0.996583i	$0.526320\pi$
$948$	15.1249	0.491235
$949$	0	0
$950$	22.8841	0.742458
$951$	−8.66642	−0.281028
$952$	0	0
$953$	9.81437	0.317919	0.158959	−	0.987285i	$-0.449186\pi$
$953$	9.81437	0.317919	0.158959	+	0.987285i	$0.449186\pi$
$954$	51.3681	1.66310
$955$	7.89284	0.255406
$956$	44.1543	1.42805
$957$	−7.53484	−0.243567
$958$	−24.4705	−0.790607
$959$	0	0
$960$	−3.91491	−0.126353
$961$	78.7893	2.54159
$962$	0	0
$963$	14.0254	0.451961
$964$	−47.4121	−1.52704
$965$	−4.86695	−0.156673
$966$	0	0
$967$	2.69619	0.0867036	0.0433518	−	0.999060i	$-0.486196\pi$
$967$	2.69619	0.0867036	0.0433518	+	0.999060i	$0.486196\pi$
$968$	25.7509	0.827665
$969$	5.96786	0.191715
$970$	−64.6060	−2.07437
$971$	−24.9240	−0.799850	−0.399925	−	0.916548i	$-0.630964\pi$
$971$	−24.9240	−0.799850	−0.399925	+	0.916548i	$0.630964\pi$
$972$	−58.1196	−1.86419
$973$	0	0
$974$	103.543	3.31773
$975$	0	0
$976$	25.4373	0.814230
$977$	28.3129	0.905811	0.452906	−	0.891558i	$-0.350387\pi$
$977$	28.3129	0.905811	0.452906	+	0.891558i	$0.350387\pi$
$978$	9.62280	0.307703
$979$	−4.76039	−0.152143
$980$	0	0
$981$	4.71904	0.150667
$982$	17.5015	0.558494
$983$	37.8517	1.20728	0.603641	−	0.797256i	$-0.293716\pi$
$983$	37.8517	1.20728	0.603641	+	0.797256i	$0.293716\pi$
$984$	−27.9166	−0.889947
$985$	−7.49591	−0.238839
$986$	−43.4842	−1.38482
$987$	0	0
$988$	0	0
$989$	−6.66514	−0.211939
$990$	−30.8289	−0.979805
$991$	58.4158	1.85564	0.927820	−	0.373028i	$-0.121681\pi$
$991$	58.4158	1.85564	0.927820	+	0.373028i	$0.121681\pi$
$992$	−89.3612	−2.83722
$993$	−2.05657	−0.0652633
$994$	0	0
$995$	−0.662104	−0.0209901
$996$	6.57252	0.208258
$997$	−28.0588	−0.888632	−0.444316	−	0.895870i	$-0.646553\pi$
$997$	−28.0588	−0.888632	−0.444316	+	0.895870i	$0.646553\pi$
$998$	29.4001	0.930645
$999$	−17.6903	−0.559696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.co.1.1		12
7.3	odd	6		1183.2.e.j.170.12		24
7.5	odd	6		1183.2.e.j.508.12		24
7.6	odd	2		8281.2.a.cp.1.1		12
13.6	odd	12		637.2.q.i.491.6		12
13.11	odd	12		637.2.q.i.589.6		12
13.12	even	2	inner	8281.2.a.co.1.12		12
91.6	even	12		637.2.q.g.491.6		12
91.11	odd	12		637.2.u.g.30.1		12
91.12	odd	6		1183.2.e.j.508.1		24
91.19	even	12		91.2.u.b.88.1	yes	12
91.24	even	12		91.2.u.b.30.1	yes	12
91.32	odd	12		637.2.k.i.569.6		12
91.37	odd	12		637.2.k.i.459.1		12
91.38	odd	6		1183.2.e.j.170.1		24
91.45	even	12		91.2.k.b.23.6	yes	12
91.58	odd	12		637.2.u.g.361.1		12
91.76	even	12		637.2.q.g.589.6		12
91.89	even	12		91.2.k.b.4.1	✓	12
91.90	odd	2		8281.2.a.cp.1.12		12
273.89	odd	12		819.2.bm.f.550.6		12
273.110	odd	12		819.2.do.e.361.6		12
273.206	odd	12		819.2.do.e.667.6		12
273.227	odd	12		819.2.bm.f.478.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.k.b.4.1	✓	12	91.89	even	12
91.2.k.b.23.6	yes	12	91.45	even	12
91.2.u.b.30.1	yes	12	91.24	even	12
91.2.u.b.88.1	yes	12	91.19	even	12
637.2.k.i.459.1		12	91.37	odd	12
637.2.k.i.569.6		12	91.32	odd	12
637.2.q.g.491.6		12	91.6	even	12
637.2.q.g.589.6		12	91.76	even	12
637.2.q.i.491.6		12	13.6	odd	12
637.2.q.i.589.6		12	13.11	odd	12
637.2.u.g.30.1		12	91.11	odd	12
637.2.u.g.361.1		12	91.58	odd	12
819.2.bm.f.478.1		12	273.227	odd	12
819.2.bm.f.550.6		12	273.89	odd	12
819.2.do.e.361.6		12	273.110	odd	12
819.2.do.e.667.6		12	273.206	odd	12
1183.2.e.j.170.1		24	91.38	odd	6
1183.2.e.j.170.12		24	7.3	odd	6
1183.2.e.j.508.1		24	91.12	odd	6
1183.2.e.j.508.12		24	7.5	odd	6
8281.2.a.co.1.1		12	1.1	even	1	trivial
8281.2.a.co.1.12		12	13.12	even	2	inner
8281.2.a.cp.1.1		12	7.6	odd	2
8281.2.a.cp.1.12		12	91.90	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.58860 q^{2} -0.518466 q^{3} +4.70085 q^{4} +1.61205 q^{5} +1.34210 q^{6} -6.99143 q^{8} -2.73119 q^{9} +O(q^{10})\)	\(q-2.58860 q^{2} -0.518466 q^{3} +4.70085 q^{4} +1.61205 q^{5} +1.34210 q^{6} -6.99143 q^{8} -2.73119 q^{9} -4.17296 q^{10} -2.70496 q^{11} -2.43723 q^{12} -0.835795 q^{15} +8.69632 q^{16} -3.12661 q^{17} +7.06997 q^{18} +3.68150 q^{19} +7.57803 q^{20} +7.00205 q^{22} +1.98604 q^{23} +3.62482 q^{24} -2.40128 q^{25} +2.97143 q^{27} -5.37271 q^{29} +2.16354 q^{30} +10.4780 q^{31} -8.52843 q^{32} +1.40243 q^{33} +8.09354 q^{34} -12.8389 q^{36} -5.95346 q^{37} -9.52994 q^{38} -11.2706 q^{40} -7.70150 q^{41} -3.35600 q^{43} -12.7156 q^{44} -4.40283 q^{45} -5.14106 q^{46} +1.05508 q^{47} -4.50874 q^{48} +6.21596 q^{50} +1.62104 q^{51} +7.26568 q^{53} -7.69184 q^{54} -4.36054 q^{55} -1.90873 q^{57} +13.9078 q^{58} +11.4241 q^{59} -3.92895 q^{60} +2.92507 q^{61} -27.1235 q^{62} +4.68406 q^{64} -3.63033 q^{66} +13.5818 q^{67} -14.6977 q^{68} -1.02969 q^{69} +1.35111 q^{71} +19.0949 q^{72} +9.10335 q^{73} +15.4111 q^{74} +1.24498 q^{75} +17.3062 q^{76} -6.20578 q^{79} +14.0189 q^{80} +6.65300 q^{81} +19.9361 q^{82} -2.69672 q^{83} -5.04026 q^{85} +8.68734 q^{86} +2.78557 q^{87} +18.9115 q^{88} +1.75988 q^{89} +11.3972 q^{90} +9.33607 q^{92} -5.43251 q^{93} -2.73119 q^{94} +5.93478 q^{95} +4.42170 q^{96} +15.4820 q^{97} +7.38776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) \) 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