LMFDB - Embedded newform 8281.2.a.co.1.5

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			1.5
Root			$-0.499987$ 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)

$f(q)$	$=$	$q-0.499987 q^{2} -0.849601 q^{3} -1.75001 q^{4} -1.04248 q^{5} +0.424789 q^{6} +1.87496 q^{8} -2.27818 q^{9} +O(q^{10})$	\(q-0.499987 q^{2} -0.849601 q^{3} -1.75001 q^{4} -1.04248 q^{5} +0.424789 q^{6} +1.87496 q^{8} -2.27818 q^{9} +0.521224 q^{10} -3.96730 q^{11} +1.48681 q^{12} +0.885688 q^{15} +2.56257 q^{16} +0.142035 q^{17} +1.13906 q^{18} +5.50977 q^{19} +1.82435 q^{20} +1.98360 q^{22} +4.39098 q^{23} -1.59297 q^{24} -3.91325 q^{25} +4.48435 q^{27} -8.39759 q^{29} -0.442832 q^{30} -2.84652 q^{31} -5.03117 q^{32} +3.37063 q^{33} -0.0710158 q^{34} +3.98684 q^{36} -0.843187 q^{37} -2.75481 q^{38} -1.95460 q^{40} -12.0974 q^{41} +4.82323 q^{43} +6.94284 q^{44} +2.37494 q^{45} -2.19543 q^{46} +4.55648 q^{47} -2.17717 q^{48} +1.95657 q^{50} -0.120673 q^{51} -0.279600 q^{53} -2.24211 q^{54} +4.13582 q^{55} -4.68111 q^{57} +4.19868 q^{58} +10.7815 q^{59} -1.54997 q^{60} +5.86354 q^{61} +1.42322 q^{62} -2.60963 q^{64} -1.68527 q^{66} +5.14447 q^{67} -0.248564 q^{68} -3.73058 q^{69} +3.69880 q^{71} -4.27148 q^{72} +6.61281 q^{73} +0.421582 q^{74} +3.32470 q^{75} -9.64216 q^{76} +11.9227 q^{79} -2.67142 q^{80} +3.02462 q^{81} +6.04853 q^{82} -2.87321 q^{83} -0.148068 q^{85} -2.41155 q^{86} +7.13461 q^{87} -7.43852 q^{88} -1.74765 q^{89} -1.18744 q^{90} -7.68427 q^{92} +2.41841 q^{93} -2.27818 q^{94} -5.74379 q^{95} +4.27449 q^{96} +2.70291 q^{97} +9.03822 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.499987	−0.353544	−0.176772	−	0.984252i	$-0.556566\pi$
$2$	−0.499987	−0.353544	−0.176772	+	0.984252i	$0.556566\pi$
$3$	−0.849601	−0.490518	−0.245259	−	0.969458i	$-0.578873\pi$
$3$	−0.849601	−0.490518	−0.245259	+	0.969458i	$0.578873\pi$
$4$	−1.75001	−0.875007
$5$	−1.04248	−0.466209	−0.233105	−	0.972452i	$-0.574888\pi$
$5$	−1.04248	−0.466209	−0.233105	+	0.972452i	$0.574888\pi$
$6$	0.424789	0.173420
$7$	0	0
$7$	0	0
$8$	1.87496	0.662897
$9$	−2.27818	−0.759392
$10$	0.521224	0.164825
$11$	−3.96730	−1.19619	−0.598094	−	0.801426i	$-0.704075\pi$
$11$	−3.96730	−1.19619	−0.598094	+	0.801426i	$0.704075\pi$
$12$	1.48681	0.429206
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.885688	0.228684
$16$	2.56257	0.640643
$17$	0.142035	0.0344486	0.0172243	−	0.999852i	$-0.494517\pi$
$17$	0.142035	0.0344486	0.0172243	+	0.999852i	$0.494517\pi$
$18$	1.13906	0.268479
$19$	5.50977	1.26403	0.632014	−	0.774957i	$-0.282229\pi$
$19$	5.50977	1.26403	0.632014	+	0.774957i	$0.282229\pi$
$20$	1.82435	0.407936
$21$	0	0
$22$	1.98360	0.422905
$23$	4.39098	0.915582	0.457791	−	0.889060i	$-0.348641\pi$
$23$	4.39098	0.915582	0.457791	+	0.889060i	$0.348641\pi$
$24$	−1.59297	−0.325163
$25$	−3.91325	−0.782649
$26$	0	0
$27$	4.48435	0.863013
$28$	0	0
$29$	−8.39759	−1.55939	−0.779697	−	0.626157i	$-0.784627\pi$
$29$	−8.39759	−1.55939	−0.779697	+	0.626157i	$0.784627\pi$
$30$	−0.442832	−0.0808498
$31$	−2.84652	−0.511251	−0.255625	−	0.966776i	$-0.582281\pi$
$31$	−2.84652	−0.511251	−0.255625	+	0.966776i	$0.582281\pi$
$32$	−5.03117	−0.889393
$33$	3.37063	0.586751
$34$	−0.0710158	−0.0121791
$35$	0	0
$36$	3.98684	0.664473
$37$	−0.843187	−0.138619	−0.0693095	−	0.997595i	$-0.522080\pi$
$37$	−0.843187	−0.138619	−0.0693095	+	0.997595i	$0.522080\pi$
$38$	−2.75481	−0.446889
$39$	0	0
$40$	−1.95460	−0.309049
$41$	−12.0974	−1.88929	−0.944647	−	0.328089i	$-0.893595\pi$
$41$	−12.0974	−1.88929	−0.944647	+	0.328089i	$0.893595\pi$
$42$	0	0
$43$	4.82323	0.735536	0.367768	−	0.929918i	$-0.380122\pi$
$43$	4.82323	0.735536	0.367768	+	0.929918i	$0.380122\pi$
$44$	6.94284	1.04667
$45$	2.37494	0.354036
$46$	−2.19543	−0.323699
$47$	4.55648	0.664630	0.332315	−	0.943168i	$-0.392170\pi$
$47$	4.55648	0.664630	0.332315	+	0.943168i	$0.392170\pi$
$48$	−2.17717	−0.314247
$49$	0	0
$50$	1.95657	0.276701
$51$	−0.120673	−0.0168977
$52$	0	0
$53$	−0.279600	−0.0384060	−0.0192030	−	0.999816i	$-0.506113\pi$
$53$	−0.279600	−0.0384060	−0.0192030	+	0.999816i	$0.506113\pi$
$54$	−2.24211	−0.305113
$55$	4.13582	0.557673
$56$	0	0
$57$	−4.68111	−0.620028
$58$	4.19868	0.551314
$59$	10.7815	1.40363	0.701815	−	0.712359i	$-0.252373\pi$
$59$	10.7815	1.40363	0.701815	+	0.712359i	$0.252373\pi$
$60$	−1.54997	−0.200100
$61$	5.86354	0.750749	0.375374	−	0.926873i	$-0.377514\pi$
$61$	5.86354	0.750749	0.375374	+	0.926873i	$0.377514\pi$
$62$	1.42322	0.180750
$63$	0	0
$64$	−2.60963	−0.326204
$65$	0	0
$66$	−1.68527	−0.207442
$67$	5.14447	0.628497	0.314248	−	0.949341i	$-0.398247\pi$
$67$	5.14447	0.628497	0.314248	+	0.949341i	$0.398247\pi$
$68$	−0.248564	−0.0301428
$69$	−3.73058	−0.449109
$70$	0	0
$71$	3.69880	0.438967	0.219484	−	0.975616i	$-0.429563\pi$
$71$	3.69880	0.438967	0.219484	+	0.975616i	$0.429563\pi$
$72$	−4.27148	−0.503399
$73$	6.61281	0.773970	0.386985	−	0.922086i	$-0.373516\pi$
$73$	6.61281	0.773970	0.386985	+	0.922086i	$0.373516\pi$
$74$	0.421582	0.0490079
$75$	3.32470	0.383903
$76$	−9.64216	−1.10603
$77$	0	0
$78$	0	0
$79$	11.9227	1.34141	0.670705	−	0.741725i	$-0.265992\pi$
$79$	11.9227	1.34141	0.670705	+	0.741725i	$0.265992\pi$
$80$	−2.67142	−0.298674
$81$	3.02462	0.336069
$82$	6.04853	0.667948
$83$	−2.87321	−0.315376	−0.157688	−	0.987489i	$-0.550404\pi$
$83$	−2.87321	−0.315376	−0.157688	+	0.987489i	$0.550404\pi$
$84$	0	0
$85$	−0.148068	−0.0160603
$86$	−2.41155	−0.260044
$87$	7.13461	0.764910
$88$	−7.43852	−0.792949
$89$	−1.74765	−0.185250	−0.0926252	−	0.995701i	$-0.529526\pi$
$89$	−1.74765	−0.185250	−0.0926252	+	0.995701i	$0.529526\pi$
$90$	−1.18744	−0.125167
$91$	0	0
$92$	−7.68427	−0.801141
$93$	2.41841	0.250777
$94$	−2.27818	−0.234976
$95$	−5.74379	−0.589301
$96$	4.27449	0.436263
$97$	2.70291	0.274438	0.137219	−	0.990541i	$-0.456184\pi$
$97$	2.70291	0.274438	0.137219	+	0.990541i	$0.456184\pi$
$98$	0	0
$99$	9.03822	0.908376
$100$	6.84823	0.684823
$101$	−11.4722	−1.14153	−0.570765	−	0.821114i	$-0.693353\pi$
$101$	−11.4722	−1.14153	−0.570765	+	0.821114i	$0.693353\pi$
$102$	0.0603351	0.00597407
$103$	−4.16950	−0.410834	−0.205417	−	0.978675i	$-0.565855\pi$
$103$	−4.16950	−0.410834	−0.205417	+	0.978675i	$0.565855\pi$
$104$	0	0
$105$	0	0
$106$	0.139796	0.0135782
$107$	8.48742	0.820510	0.410255	−	0.911971i	$-0.365440\pi$
$107$	8.48742	0.820510	0.410255	+	0.911971i	$0.365440\pi$
$108$	−7.84767	−0.755142
$109$	6.43036	0.615917	0.307958	−	0.951400i	$-0.400354\pi$
$109$	6.43036	0.615917	0.307958	+	0.951400i	$0.400354\pi$
$110$	−2.06785	−0.197162
$111$	0.716373	0.0679951
$112$	0	0
$113$	10.9633	1.03134	0.515670	−	0.856788i	$-0.327543\pi$
$113$	10.9633	1.03134	0.515670	+	0.856788i	$0.327543\pi$
$114$	2.34049	0.219207
$115$	−4.57749	−0.426853
$116$	14.6959	1.36448
$117$	0	0
$118$	−5.39060	−0.496245
$119$	0	0
$120$	1.66063	0.151594
$121$	4.73951	0.430864
$122$	−2.93169	−0.265423
$123$	10.2780	0.926732
$124$	4.98145	0.447348
$125$	9.29184	0.831087
$126$	0	0
$127$	−2.00787	−0.178170	−0.0890849	−	0.996024i	$-0.528394\pi$
$127$	−2.00787	−0.178170	−0.0890849	+	0.996024i	$0.528394\pi$
$128$	11.3671	1.00472
$129$	−4.09782	−0.360793
$130$	0	0
$131$	12.4502	1.08778	0.543890	−	0.839156i	$-0.316951\pi$
$131$	12.4502	1.08778	0.543890	+	0.839156i	$0.316951\pi$
$132$	−5.89864	−0.513411
$133$	0	0
$134$	−2.57217	−0.222201
$135$	−4.67482	−0.402344
$136$	0.266310	0.0228359
$137$	−5.24518	−0.448126	−0.224063	−	0.974575i	$-0.571932\pi$
$137$	−5.24518	−0.448126	−0.224063	+	0.974575i	$0.571932\pi$
$138$	1.86524	0.158780
$139$	20.7385	1.75902	0.879510	−	0.475881i	$-0.157871\pi$
$139$	20.7385	1.75902	0.879510	+	0.475881i	$0.157871\pi$
$140$	0	0
$141$	−3.87119	−0.326013
$142$	−1.84935	−0.155194
$143$	0	0
$144$	−5.83800	−0.486500
$145$	8.75428	0.727004
$146$	−3.30631	−0.273633
$147$	0	0
$148$	1.47559	0.121293
$149$	0.0113760	0.000931956 0	0.000465978	−	1.00000i	$-0.499852\pi$
$149$	0.0113760	0.000931956 0	0.000465978		1.00000i	$0.499852\pi$
$150$	−1.66231	−0.135727
$151$	18.9054	1.53850	0.769251	−	0.638947i	$-0.220630\pi$
$151$	18.9054	1.53850	0.769251	+	0.638947i	$0.220630\pi$
$152$	10.3306	0.837920
$153$	−0.323582	−0.0261600
$154$	0	0
$155$	2.96743	0.238350
$156$	0	0
$157$	−19.7937	−1.57971	−0.789856	−	0.613292i	$-0.789845\pi$
$157$	−19.7937	−1.57971	−0.789856	+	0.613292i	$0.789845\pi$
$158$	−5.96119	−0.474247
$159$	0.237549	0.0188388
$160$	5.24486	0.414643
$161$	0	0
$162$	−1.51227	−0.118815
$163$	−8.93255	−0.699651	−0.349825	−	0.936815i	$-0.613759\pi$
$163$	−8.93255	−0.699651	−0.349825	+	0.936815i	$0.613759\pi$
$164$	21.1706	1.65314
$165$	−3.51380	−0.273549
$166$	1.43657	0.111499
$167$	6.13469	0.474716	0.237358	−	0.971422i	$-0.423719\pi$
$167$	6.13469	0.474716	0.237358	+	0.971422i	$0.423719\pi$
$168$	0	0
$169$	0	0
$170$	0.0740322	0.00567801
$171$	−12.5522	−0.959893
$172$	−8.44072	−0.643599
$173$	−24.2628	−1.84466	−0.922332	−	0.386399i	$-0.873719\pi$
$173$	−24.2628	−1.84466	−0.922332	+	0.386399i	$0.873719\pi$
$174$	−3.56721	−0.270429
$175$	0	0
$176$	−10.1665	−0.766330
$177$	−9.15997	−0.688506
$178$	0.873801	0.0654942
$179$	−4.13675	−0.309195	−0.154598	−	0.987978i	$-0.549408\pi$
$179$	−4.13675	−0.309195	−0.154598	+	0.987978i	$0.549408\pi$
$180$	−4.15618	−0.309784
$181$	−7.86568	−0.584651	−0.292326	−	0.956319i	$-0.594429\pi$
$181$	−7.86568	−0.584651	−0.292326	+	0.956319i	$0.594429\pi$
$182$	0	0
$183$	−4.98167	−0.368256
$184$	8.23289	0.606937
$185$	0.879001	0.0646254
$186$	−1.20917	−0.0886608
$187$	−0.563498	−0.0412070
$188$	−7.97389	−0.581556
$189$	0	0
$190$	2.87182	0.208344
$191$	−6.47866	−0.468780	−0.234390	−	0.972143i	$-0.575309\pi$
$191$	−6.47866	−0.468780	−0.234390	+	0.972143i	$0.575309\pi$
$192$	2.21715	0.160009
$193$	4.82928	0.347619	0.173810	−	0.984779i	$-0.444392\pi$
$193$	4.82928	0.347619	0.173810	+	0.984779i	$0.444392\pi$
$194$	−1.35142	−0.0970261
$195$	0	0
$196$	0	0
$197$	−25.8362	−1.84075	−0.920377	−	0.391032i	$-0.872118\pi$
$197$	−25.8362	−1.84075	−0.920377	+	0.391032i	$0.872118\pi$
$198$	−4.51899	−0.321151
$199$	−17.1146	−1.21322	−0.606612	−	0.794998i	$-0.707472\pi$
$199$	−17.1146	−1.21322	−0.606612	+	0.794998i	$0.707472\pi$
$200$	−7.33717	−0.518816
$201$	−4.37075	−0.308289
$202$	5.73596	0.403581
$203$	0	0
$204$	0.211180	0.0147856
$205$	12.6112	0.880806
$206$	2.08470	0.145248
$207$	−10.0034	−0.695286
$208$	0	0
$209$	−21.8589	−1.51201
$210$	0	0
$211$	18.2911	1.25921	0.629607	−	0.776914i	$-0.283216\pi$
$211$	18.2911	1.25921	0.629607	+	0.776914i	$0.283216\pi$
$212$	0.489304	0.0336055
$213$	−3.14251	−0.215321
$214$	−4.24360	−0.290086
$215$	−5.02810	−0.342913
$216$	8.40796	0.572089
$217$	0	0
$218$	−3.21509	−0.217754
$219$	−5.61825	−0.379646
$220$	−7.23773	−0.487968
$221$	0	0
$222$	−0.358177	−0.0240392
$223$	11.5087	0.770679	0.385340	−	0.922775i	$-0.374084\pi$
$223$	11.5087	0.770679	0.385340	+	0.922775i	$0.374084\pi$
$224$	0	0
$225$	8.91507	0.594338
$226$	−5.48150	−0.364624
$227$	−17.9045	−1.18836	−0.594181	−	0.804332i	$-0.702524\pi$
$227$	−17.9045	−1.18836	−0.594181	+	0.804332i	$0.702524\pi$
$228$	8.19200	0.542528
$229$	3.86350	0.255307	0.127654	−	0.991819i	$-0.459255\pi$
$229$	3.86350	0.255307	0.127654	+	0.991819i	$0.459255\pi$
$230$	2.28868	0.150911
$231$	0	0
$232$	−15.7451	−1.03372
$233$	25.0642	1.64201	0.821004	−	0.570922i	$-0.193414\pi$
$233$	25.0642	1.64201	0.821004	+	0.570922i	$0.193414\pi$
$234$	0	0
$235$	−4.75001	−0.309857
$236$	−18.8678	−1.22819
$237$	−10.1295	−0.657985
$238$	0	0
$239$	7.80462	0.504839	0.252419	−	0.967618i	$-0.418774\pi$
$239$	7.80462	0.504839	0.252419	+	0.967618i	$0.418774\pi$
$240$	2.26964	0.146505
$241$	21.7653	1.40202	0.701012	−	0.713150i	$-0.252732\pi$
$241$	21.7653	1.40202	0.701012	+	0.713150i	$0.252732\pi$
$242$	−2.36969	−0.152329
$243$	−16.0228	−1.02786
$244$	−10.2613	−0.656910
$245$	0	0
$246$	−5.13884	−0.327640
$247$	0	0
$248$	−5.33711	−0.338907
$249$	2.44109	0.154698
$250$	−4.64579	−0.293826
$251$	−7.67980	−0.484745	−0.242372	−	0.970183i	$-0.577926\pi$
$251$	−7.67980	−0.484745	−0.242372	+	0.970183i	$0.577926\pi$
$252$	0	0
$253$	−17.4203	−1.09521
$254$	1.00391	0.0629909
$255$	0.125799	0.00787784
$256$	−0.464141	−0.0290088
$257$	−13.6237	−0.849826	−0.424913	−	0.905234i	$-0.639695\pi$
$257$	−13.6237	−0.849826	−0.424913	+	0.905234i	$0.639695\pi$
$258$	2.04886	0.127556
$259$	0	0
$260$	0	0
$261$	19.1312	1.18419
$262$	−6.22494	−0.384578
$263$	−11.7232	−0.722880	−0.361440	−	0.932395i	$-0.617715\pi$
$263$	−11.7232	−0.722880	−0.361440	+	0.932395i	$0.617715\pi$
$264$	6.31978	0.388956
$265$	0.291476	0.0179052
$266$	0	0
$267$	1.48481	0.0908686
$268$	−9.00289	−0.549939
$269$	−9.19876	−0.560858	−0.280429	−	0.959875i	$-0.590477\pi$
$269$	−9.19876	−0.560858	−0.280429	+	0.959875i	$0.590477\pi$
$270$	2.33735	0.142246
$271$	−2.56369	−0.155733	−0.0778665	−	0.996964i	$-0.524811\pi$
$271$	−2.56369	−0.155733	−0.0778665	+	0.996964i	$0.524811\pi$
$272$	0.363976	0.0220693
$273$	0	0
$274$	2.62252	0.158432
$275$	15.5250	0.936195
$276$	6.52857	0.392974
$277$	−0.933882	−0.0561115	−0.0280558	−	0.999606i	$-0.508932\pi$
$277$	−0.933882	−0.0561115	−0.0280558	+	0.999606i	$0.508932\pi$
$278$	−10.3690	−0.621891
$279$	6.48488	0.388240
$280$	0	0
$281$	−6.45288	−0.384947	−0.192473	−	0.981302i	$-0.561651\pi$
$281$	−6.45288	−0.384947	−0.192473	+	0.981302i	$0.561651\pi$
$282$	1.93554	0.115260
$283$	−22.1746	−1.31814	−0.659071	−	0.752081i	$-0.729050\pi$
$283$	−22.1746	−1.31814	−0.659071	+	0.752081i	$0.729050\pi$
$284$	−6.47296	−0.384099
$285$	4.87994	0.289062
$286$	0	0
$287$	0	0
$288$	11.4619	0.675398
$289$	−16.9798	−0.998813
$290$	−4.37702	−0.257028
$291$	−2.29639	−0.134617
$292$	−11.5725	−0.677229
$293$	24.2026	1.41393	0.706964	−	0.707249i	$-0.250064\pi$
$293$	24.2026	1.41393	0.706964	+	0.707249i	$0.250064\pi$
$294$	0	0
$295$	−11.2394	−0.654385
$296$	−1.58094	−0.0918902
$297$	−17.7908	−1.03233
$298$	−0.00568784	−0.000329487 0
$299$	0	0
$300$	−5.81827	−0.335918
$301$	0	0
$302$	−9.45246	−0.543928
$303$	9.74683	0.559940
$304$	14.1192	0.809791
$305$	−6.11259	−0.350006
$306$	0.161787	0.00924872
$307$	24.2924	1.38644	0.693220	−	0.720726i	$-0.256191\pi$
$307$	24.2924	1.38644	0.693220	+	0.720726i	$0.256191\pi$
$308$	0	0
$309$	3.54242	0.201521
$310$	−1.48367	−0.0842671
$311$	−3.98711	−0.226088	−0.113044	−	0.993590i	$-0.536060\pi$
$311$	−3.98711	−0.226088	−0.113044	+	0.993590i	$0.536060\pi$
$312$	0	0
$313$	−28.4754	−1.60953	−0.804763	−	0.593597i	$-0.797707\pi$
$313$	−28.4754	−1.60953	−0.804763	+	0.593597i	$0.797707\pi$
$314$	9.89661	0.558498
$315$	0	0
$316$	−20.8649	−1.17374
$317$	16.8161	0.944487	0.472244	−	0.881468i	$-0.343444\pi$
$317$	16.8161	0.944487	0.472244	+	0.881468i	$0.343444\pi$
$318$	−0.118771	−0.00666036
$319$	33.3158	1.86533
$320$	2.72048	0.152079
$321$	−7.21093	−0.402475
$322$	0	0
$323$	0.782582	0.0435440
$324$	−5.29313	−0.294063
$325$	0	0
$326$	4.46615	0.247357
$327$	−5.46324	−0.302118
$328$	−22.6821	−1.25241
$329$	0	0
$330$	1.75685	0.0967115
$331$	6.20917	0.341287	0.170644	−	0.985333i	$-0.445415\pi$
$331$	6.20917	0.341287	0.170644	+	0.985333i	$0.445415\pi$
$332$	5.02816	0.275956
$333$	1.92093	0.105266
$334$	−3.06726	−0.167833
$335$	−5.36298	−0.293011
$336$	0	0
$337$	−7.69650	−0.419255	−0.209628	−	0.977781i	$-0.567225\pi$
$337$	−7.69650	−0.419255	−0.209628	+	0.977781i	$0.567225\pi$
$338$	0	0
$339$	−9.31442	−0.505890
$340$	0.259122	0.0140528
$341$	11.2930	0.611552
$342$	6.27594	0.339364
$343$	0	0
$344$	9.04335	0.487585
$345$	3.88904	0.209379
$346$	12.1311	0.652170
$347$	30.4094	1.63246	0.816231	−	0.577725i	$-0.196059\pi$
$347$	30.4094	1.63246	0.816231	+	0.577725i	$0.196059\pi$
$348$	−12.4857	−0.669302
$349$	−16.1581	−0.864924	−0.432462	−	0.901652i	$-0.642355\pi$
$349$	−16.1581	−0.864924	−0.432462	+	0.901652i	$0.642355\pi$
$350$	0	0
$351$	0	0
$352$	19.9602	1.06388
$353$	−11.8424	−0.630306	−0.315153	−	0.949041i	$-0.602056\pi$
$353$	−11.8424	−0.630306	−0.315153	+	0.949041i	$0.602056\pi$
$354$	4.57986	0.243417
$355$	−3.85591	−0.204651
$356$	3.05841	0.162095
$357$	0	0
$358$	2.06832	0.109314
$359$	−31.3653	−1.65540	−0.827698	−	0.561174i	$-0.810350\pi$
$359$	−31.3653	−1.65540	−0.827698	+	0.561174i	$0.810350\pi$
$360$	4.45292	0.234689
$361$	11.3575	0.597765
$362$	3.93273	0.206700
$363$	−4.02669	−0.211346
$364$	0	0
$365$	−6.89369	−0.360832
$366$	2.49077	0.130195
$367$	−24.0774	−1.25683	−0.628415	−	0.777878i	$-0.716296\pi$
$367$	−24.0774	−1.25683	−0.628415	+	0.777878i	$0.716296\pi$
$368$	11.2522	0.586562
$369$	27.5600	1.43472
$370$	−0.439489	−0.0228479
$371$	0	0
$372$	−4.23225	−0.219432
$373$	−18.3922	−0.952314	−0.476157	−	0.879360i	$-0.657971\pi$
$373$	−18.3922	−0.952314	−0.476157	+	0.879360i	$0.657971\pi$
$374$	0.281741	0.0145685
$375$	−7.89436	−0.407663
$376$	8.54320	0.440582
$377$	0	0
$378$	0	0
$379$	−8.13740	−0.417990	−0.208995	−	0.977917i	$-0.567019\pi$
$379$	−8.13740	−0.417990	−0.208995	+	0.977917i	$0.567019\pi$
$380$	10.0517	0.515642
$381$	1.70589	0.0873955
$382$	3.23925	0.165734
$383$	22.3711	1.14311	0.571555	−	0.820564i	$-0.306340\pi$
$383$	22.3711	1.14311	0.571555	+	0.820564i	$0.306340\pi$
$384$	−9.65752	−0.492833
$385$	0	0
$386$	−2.41458	−0.122899
$387$	−10.9882	−0.558560
$388$	−4.73012	−0.240136
$389$	−21.3946	−1.08475	−0.542374	−	0.840137i	$-0.682475\pi$
$389$	−21.3946	−1.08475	−0.542374	+	0.840137i	$0.682475\pi$
$390$	0	0
$391$	0.623674	0.0315406
$392$	0	0
$393$	−10.5777	−0.533576
$394$	12.9178	0.650788
$395$	−12.4291	−0.625377
$396$	−15.8170	−0.794835
$397$	1.19673	0.0600622	0.0300311	−	0.999549i	$-0.490439\pi$
$397$	1.19673	0.0600622	0.0300311	+	0.999549i	$0.490439\pi$
$398$	8.55708	0.428928
$399$	0	0
$400$	−10.0280	−0.501399
$401$	36.2749	1.81148	0.905741	−	0.423831i	$-0.139315\pi$
$401$	36.2749	1.81148	0.905741	+	0.423831i	$0.139315\pi$
$402$	2.18532	0.108994
$403$	0	0
$404$	20.0766	0.998846
$405$	−3.15309	−0.156679
$406$	0	0
$407$	3.34518	0.165814
$408$	−0.226257	−0.0112014
$409$	14.6723	0.725500	0.362750	−	0.931887i	$-0.381838\pi$
$409$	14.6723	0.725500	0.362750	+	0.931887i	$0.381838\pi$
$410$	−6.30544	−0.311404
$411$	4.45631	0.219814
$412$	7.29669	0.359482
$413$	0	0
$414$	5.00158	0.245814
$415$	2.99525	0.147031
$416$	0	0
$417$	−17.6195	−0.862830
$418$	10.9292	0.534563
$419$	5.93348	0.289870	0.144935	−	0.989441i	$-0.453703\pi$
$419$	5.93348	0.289870	0.144935	+	0.989441i	$0.453703\pi$
$420$	0	0
$421$	−2.63174	−0.128263	−0.0641317	−	0.997941i	$-0.520428\pi$
$421$	−2.63174	−0.128263	−0.0641317	+	0.997941i	$0.520428\pi$
$422$	−9.14532	−0.445187
$423$	−10.3805	−0.504715
$424$	−0.524238	−0.0254593
$425$	−0.555819	−0.0269612
$426$	1.57121	0.0761255
$427$	0	0
$428$	−14.8531	−0.717952
$429$	0	0
$430$	2.51398	0.121235
$431$	−18.8377	−0.907378	−0.453689	−	0.891160i	$-0.649892\pi$
$431$	−18.8377	−0.907378	−0.453689	+	0.891160i	$0.649892\pi$
$432$	11.4915	0.552884
$433$	−19.1355	−0.919591	−0.459796	−	0.888025i	$-0.652077\pi$
$433$	−19.1355	−0.919591	−0.459796	+	0.888025i	$0.652077\pi$
$434$	0	0
$435$	−7.43765	−0.356608
$436$	−11.2532	−0.538931
$437$	24.1933	1.15732
$438$	2.80905	0.134222
$439$	1.26511	0.0603803	0.0301901	−	0.999544i	$-0.490389\pi$
$439$	1.26511	0.0603803	0.0301901	+	0.999544i	$0.490389\pi$
$440$	7.75448	0.369680
$441$	0	0
$442$	0	0
$443$	−20.9392	−0.994853	−0.497426	−	0.867506i	$-0.665722\pi$
$443$	−20.9392	−0.994853	−0.497426	+	0.867506i	$0.665722\pi$
$444$	−1.25366	−0.0594961
$445$	1.82188	0.0863654
$446$	−5.75419	−0.272469
$447$	−0.00966505	−0.000457141 0
$448$	0	0
$449$	17.8632	0.843018	0.421509	−	0.906824i	$-0.361501\pi$
$449$	17.8632	0.843018	0.421509	+	0.906824i	$0.361501\pi$
$450$	−4.45741	−0.210125
$451$	47.9940	2.25995
$452$	−19.1859	−0.902429
$453$	−16.0621	−0.754662
$454$	8.95199	0.420138
$455$	0	0
$456$	−8.77687	−0.411015
$457$	6.56597	0.307143	0.153571	−	0.988138i	$-0.450922\pi$
$457$	6.56597	0.307143	0.153571	+	0.988138i	$0.450922\pi$
$458$	−1.93170	−0.0902624
$459$	0.636936	0.0297296
$460$	8.01066	0.373499
$461$	−5.11364	−0.238166	−0.119083	−	0.992884i	$-0.537995\pi$
$461$	−5.11364	−0.238166	−0.119083	+	0.992884i	$0.537995\pi$
$462$	0	0
$463$	33.3239	1.54869	0.774347	−	0.632761i	$-0.218079\pi$
$463$	33.3239	1.54869	0.774347	+	0.632761i	$0.218079\pi$
$464$	−21.5194	−0.999015
$465$	−2.52113	−0.116915
$466$	−12.5317	−0.580522
$467$	12.9494	0.599229	0.299614	−	0.954060i	$-0.403142\pi$
$467$	12.9494	0.599229	0.299614	+	0.954060i	$0.403142\pi$
$468$	0	0
$469$	0	0
$470$	2.37494	0.109548
$471$	16.8168	0.774877
$472$	20.2148	0.930463
$473$	−19.1352	−0.879838
$474$	5.06464	0.232627
$475$	−21.5611	−0.989290
$476$	0	0
$477$	0.636979	0.0291652
$478$	−3.90221	−0.178483
$479$	−27.0119	−1.23421	−0.617104	−	0.786882i	$-0.711694\pi$
$479$	−27.0119	−1.23421	−0.617104	+	0.786882i	$0.711694\pi$
$480$	−4.45605	−0.203390
$481$	0	0
$482$	−10.8823	−0.495677
$483$	0	0
$484$	−8.29420	−0.377009
$485$	−2.81771	−0.127946
$486$	8.01117	0.363394
$487$	32.0838	1.45386	0.726928	−	0.686713i	$-0.240947\pi$
$487$	32.0838	1.45386	0.726928	+	0.686713i	$0.240947\pi$
$488$	10.9939	0.497669
$489$	7.58910	0.343191
$490$	0	0
$491$	−28.6040	−1.29088	−0.645440	−	0.763811i	$-0.723326\pi$
$491$	−28.6040	−1.29088	−0.645440	+	0.763811i	$0.723326\pi$
$492$	−17.9866	−0.810897
$493$	−1.19276	−0.0537190
$494$	0	0
$495$	−9.42212	−0.423493
$496$	−7.29442	−0.327529
$497$	0	0
$498$	−1.22051	−0.0546924
$499$	−1.79816	−0.0804969	−0.0402484	−	0.999190i	$-0.512815\pi$
$499$	−1.79816	−0.0804969	−0.0402484	+	0.999190i	$0.512815\pi$
$500$	−16.2608	−0.727207
$501$	−5.21204	−0.232857
$502$	3.83980	0.171379
$503$	−29.0772	−1.29649	−0.648245	−	0.761432i	$-0.724497\pi$
$503$	−29.0772	−1.29649	−0.648245	+	0.761432i	$0.724497\pi$
$504$	0	0
$505$	11.9595	0.532191
$506$	8.70994	0.387204
$507$	0	0
$508$	3.51380	0.155900
$509$	23.1913	1.02794	0.513969	−	0.857809i	$-0.328175\pi$
$509$	23.1913	1.02794	0.513969	+	0.857809i	$0.328175\pi$
$510$	−0.0628979	−0.00278516
$511$	0	0
$512$	−22.5022	−0.994464
$513$	24.7077	1.09087
$514$	6.81169	0.300451
$515$	4.34660	0.191534
$516$	7.17125	0.315696
$517$	−18.0769	−0.795022
$518$	0	0
$519$	20.6137	0.904840
$520$	0	0
$521$	−33.2510	−1.45675	−0.728376	−	0.685178i	$-0.759725\pi$
$521$	−33.2510	−1.45675	−0.728376	+	0.685178i	$0.759725\pi$
$522$	−9.56535	−0.418664
$523$	−38.7121	−1.69276	−0.846380	−	0.532579i	$-0.821223\pi$
$523$	−38.7121	−1.69276	−0.846380	+	0.532579i	$0.821223\pi$
$524$	−21.7881	−0.951816
$525$	0	0
$526$	5.86142	0.255570
$527$	−0.404307	−0.0176119
$528$	8.63748	0.375898
$529$	−3.71931	−0.161709
$530$	−0.145734	−0.00633029
$531$	−24.5622	−1.06591
$532$	0	0
$533$	0	0
$534$	−0.742383	−0.0321260
$535$	−8.84793	−0.382529
$536$	9.64566	0.416629
$537$	3.51459	0.151666
$538$	4.59926	0.198288
$539$	0	0
$540$	8.18100	0.352054
$541$	−22.6675	−0.974551	−0.487275	−	0.873248i	$-0.662009\pi$
$541$	−22.6675	−0.974551	−0.487275	+	0.873248i	$0.662009\pi$
$542$	1.28181	0.0550585
$543$	6.68269	0.286782
$544$	−0.714603	−0.0306384
$545$	−6.70349	−0.287146
$546$	0	0
$547$	−9.21134	−0.393848	−0.196924	−	0.980419i	$-0.563095\pi$
$547$	−9.21134	−0.393848	−0.196924	+	0.980419i	$0.563095\pi$
$548$	9.17913	0.392113
$549$	−13.3582	−0.570113
$550$	−7.76231	−0.330986
$551$	−46.2688	−1.97112
$552$	−6.99468	−0.297713
$553$	0	0
$554$	0.466928	0.0198379
$555$	−0.746801	−0.0316999
$556$	−36.2927	−1.53915
$557$	11.3281	0.479986	0.239993	−	0.970775i	$-0.422855\pi$
$557$	11.3281	0.479986	0.239993	+	0.970775i	$0.422855\pi$
$558$	−3.24236	−0.137260
$559$	0	0
$560$	0	0
$561$	0.478748	0.0202128
$562$	3.22636	0.136096
$563$	−32.6386	−1.37555	−0.687777	−	0.725922i	$-0.741414\pi$
$563$	−32.6386	−1.37555	−0.687777	+	0.725922i	$0.741414\pi$
$564$	6.77463	0.285264
$565$	−11.4290	−0.480820
$566$	11.0870	0.466021
$567$	0	0
$568$	6.93510	0.290990
$569$	35.0091	1.46766	0.733829	−	0.679335i	$-0.237731\pi$
$569$	35.0091	1.46766	0.733829	+	0.679335i	$0.237731\pi$
$570$	−2.43990	−0.102196
$571$	26.2546	1.09872	0.549360	−	0.835586i	$-0.314872\pi$
$571$	26.2546	1.09872	0.549360	+	0.835586i	$0.314872\pi$
$572$	0	0
$573$	5.50428	0.229945
$574$	0	0
$575$	−17.1830	−0.716580
$576$	5.94520	0.247717
$577$	−24.5727	−1.02297	−0.511487	−	0.859291i	$-0.670905\pi$
$577$	−24.5727	−1.02297	−0.511487	+	0.859291i	$0.670905\pi$
$578$	8.48969	0.353124
$579$	−4.10296	−0.170513
$580$	−15.3201	−0.636133
$581$	0	0
$582$	1.14817	0.0475930
$583$	1.10926	0.0459408
$584$	12.3987	0.513063
$585$	0	0
$586$	−12.1010	−0.499886
$587$	−20.5279	−0.847279	−0.423639	−	0.905831i	$-0.639248\pi$
$587$	−20.5279	−0.847279	−0.423639	+	0.905831i	$0.639248\pi$
$588$	0	0
$589$	−15.6837	−0.646235
$590$	5.61957	0.231354
$591$	21.9505	0.902923
$592$	−2.16073	−0.0888054
$593$	−38.2835	−1.57211	−0.786057	−	0.618154i	$-0.787881\pi$
$593$	−38.2835	−1.57211	−0.786057	+	0.618154i	$0.787881\pi$
$594$	8.89515	0.364972
$595$	0	0
$596$	−0.0199081	−0.000815468 0
$597$	14.5406	0.595107
$598$	0	0
$599$	14.0713	0.574939	0.287470	−	0.957790i	$-0.407186\pi$
$599$	14.0713	0.574939	0.287470	+	0.957790i	$0.407186\pi$
$600$	6.23367	0.254488
$601$	20.2342	0.825369	0.412685	−	0.910874i	$-0.364591\pi$
$601$	20.2342	0.825369	0.412685	+	0.910874i	$0.364591\pi$
$602$	0	0
$603$	−11.7200	−0.477276
$604$	−33.0847	−1.34620
$605$	−4.94082	−0.200873
$606$	−4.87328	−0.197964
$607$	−6.55127	−0.265908	−0.132954	−	0.991122i	$-0.542446\pi$
$607$	−6.55127	−0.265908	−0.132954	+	0.991122i	$0.542446\pi$
$608$	−27.7205	−1.12422
$609$	0	0
$610$	3.05621	0.123742
$611$	0	0
$612$	0.566272	0.0228902
$613$	−33.3244	−1.34596	−0.672980	−	0.739660i	$-0.734986\pi$
$613$	−33.3244	−1.34596	−0.672980	+	0.739660i	$0.734986\pi$
$614$	−12.1459	−0.490168
$615$	−10.7145	−0.432051
$616$	0	0
$617$	−6.76038	−0.272162	−0.136081	−	0.990698i	$-0.543451\pi$
$617$	−6.76038	−0.272162	−0.136081	+	0.990698i	$0.543451\pi$
$618$	−1.77116	−0.0712466
$619$	17.6186	0.708152	0.354076	−	0.935217i	$-0.384795\pi$
$619$	17.6186	0.708152	0.354076	+	0.935217i	$0.384795\pi$
$620$	−5.19304	−0.208558
$621$	19.6907	0.790159
$622$	1.99350	0.0799321
$623$	0	0
$624$	0	0
$625$	9.87972	0.395189
$626$	14.2373	0.569038
$627$	18.5714	0.741669
$628$	34.6393	1.38226
$629$	−0.119762	−0.00477524
$630$	0	0
$631$	15.7519	0.627074	0.313537	−	0.949576i	$-0.398486\pi$
$631$	15.7519	0.627074	0.313537	+	0.949576i	$0.398486\pi$
$632$	22.3546	0.889216
$633$	−15.5402	−0.617667
$634$	−8.40783	−0.333918
$635$	2.09316	0.0830644
$636$	−0.415713	−0.0164841
$637$	0	0
$638$	−16.6575	−0.659475
$639$	−8.42653	−0.333348
$640$	−11.8499	−0.468410
$641$	−20.9405	−0.827099	−0.413550	−	0.910482i	$-0.635711\pi$
$641$	−20.9405	−0.827099	−0.413550	+	0.910482i	$0.635711\pi$
$642$	3.60537	0.142293
$643$	−18.9315	−0.746586	−0.373293	−	0.927713i	$-0.621771\pi$
$643$	−18.9315	−0.746586	−0.373293	+	0.927713i	$0.621771\pi$
$644$	0	0
$645$	4.27188	0.168205
$646$	−0.391280	−0.0153947
$647$	37.6768	1.48123	0.740614	−	0.671930i	$-0.234535\pi$
$647$	37.6768	1.48123	0.740614	+	0.671930i	$0.234535\pi$
$648$	5.67104	0.222779
$649$	−42.7735	−1.67901
$650$	0	0
$651$	0	0
$652$	15.6321	0.612199
$653$	29.0326	1.13613	0.568066	−	0.822983i	$-0.307692\pi$
$653$	29.0326	1.13613	0.568066	+	0.822983i	$0.307692\pi$
$654$	2.73155	0.106812
$655$	−12.9790	−0.507133
$656$	−31.0004	−1.21036
$657$	−15.0651	−0.587747
$658$	0	0
$659$	−1.41830	−0.0552493	−0.0276247	−	0.999618i	$-0.508794\pi$
$659$	−1.41830	−0.0552493	−0.0276247	+	0.999618i	$0.508794\pi$
$660$	6.14919	0.239357
$661$	4.59298	0.178646	0.0893231	−	0.996003i	$-0.471530\pi$
$661$	4.59298	0.178646	0.0893231	+	0.996003i	$0.471530\pi$
$662$	−3.10450	−0.120660
$663$	0	0
$664$	−5.38715	−0.209062
$665$	0	0
$666$	−0.960439	−0.0372162
$667$	−36.8736	−1.42775
$668$	−10.7358	−0.415380
$669$	−9.77780	−0.378032
$670$	2.68142	0.103592
$671$	−23.2624	−0.898036
$672$	0	0
$673$	−4.20223	−0.161984	−0.0809920	−	0.996715i	$-0.525809\pi$
$673$	−4.20223	−0.161984	−0.0809920	+	0.996715i	$0.525809\pi$
$674$	3.84815	0.148225
$675$	−17.5484	−0.675436
$676$	0	0
$677$	−8.08708	−0.310812	−0.155406	−	0.987851i	$-0.549669\pi$
$677$	−8.08708	−0.310812	−0.155406	+	0.987851i	$0.549669\pi$
$678$	4.65709	0.178854
$679$	0	0
$680$	−0.277622	−0.0106463
$681$	15.2117	0.582912
$682$	−5.64636	−0.216210
$683$	−24.6865	−0.944604	−0.472302	−	0.881437i	$-0.656577\pi$
$683$	−24.6865	−0.944604	−0.472302	+	0.881437i	$0.656577\pi$
$684$	21.9666	0.839912
$685$	5.46797	0.208920
$686$	0	0
$687$	−3.28244	−0.125233
$688$	12.3599	0.471216
$689$	0	0
$690$	−1.94447	−0.0740246
$691$	−11.2567	−0.428225	−0.214113	−	0.976809i	$-0.568686\pi$
$691$	−11.2567	−0.428225	−0.214113	+	0.976809i	$0.568686\pi$
$692$	42.4602	1.61409
$693$	0	0
$694$	−15.2043	−0.577147
$695$	−21.6194	−0.820071
$696$	13.3771	0.507057
$697$	−1.71826	−0.0650836
$698$	8.07884	0.305789
$699$	−21.2946	−0.805434
$700$	0	0
$701$	−22.2305	−0.839635	−0.419818	−	0.907608i	$-0.637906\pi$
$701$	−22.2305	−0.839635	−0.419818	+	0.907608i	$0.637906\pi$
$702$	0	0
$703$	−4.64576	−0.175218
$704$	10.3532	0.390201
$705$	4.03562	0.151990
$706$	5.92103	0.222841
$707$	0	0
$708$	16.0301	0.602447
$709$	−23.7741	−0.892854	−0.446427	−	0.894820i	$-0.647304\pi$
$709$	−23.7741	−0.892854	−0.446427	+	0.894820i	$0.647304\pi$
$710$	1.92790	0.0723530
$711$	−27.1620	−1.01866
$712$	−3.27677	−0.122802
$713$	−12.4990	−0.468092
$714$	0	0
$715$	0	0
$716$	7.23937	0.270548
$717$	−6.63082	−0.247632
$718$	15.6822	0.585255
$719$	−20.7808	−0.774992	−0.387496	−	0.921871i	$-0.626660\pi$
$719$	−20.7808	−0.774992	−0.387496	+	0.921871i	$0.626660\pi$
$720$	6.08597	0.226811
$721$	0	0
$722$	−5.67861	−0.211336
$723$	−18.4918	−0.687717
$724$	13.7650	0.511574
$725$	32.8618	1.22046
$726$	2.01329	0.0747203
$727$	−26.7719	−0.992915	−0.496457	−	0.868061i	$-0.665366\pi$
$727$	−26.7719	−0.992915	−0.496457	+	0.868061i	$0.665366\pi$
$728$	0	0
$729$	4.53910	0.168115
$730$	3.44675	0.127570
$731$	0.685069	0.0253382
$732$	8.71799	0.322226
$733$	5.25647	0.194152	0.0970761	−	0.995277i	$-0.469051\pi$
$733$	5.25647	0.194152	0.0970761	+	0.995277i	$0.469051\pi$
$734$	12.0384	0.444345
$735$	0	0
$736$	−22.0917	−0.814312
$737$	−20.4097	−0.751800
$738$	−13.7796	−0.507235
$739$	7.15001	0.263017	0.131509	−	0.991315i	$-0.458018\pi$
$739$	7.15001	0.263017	0.131509	+	0.991315i	$0.458018\pi$
$740$	−1.53826	−0.0565477
$741$	0	0
$742$	0	0
$743$	−0.713641	−0.0261810	−0.0130905	−	0.999914i	$-0.504167\pi$
$743$	−0.713641	−0.0261810	−0.0130905	+	0.999914i	$0.504167\pi$
$744$	4.53441	0.166240
$745$	−0.0118592	−0.000434486 0
$746$	9.19587	0.336685
$747$	6.54569	0.239494
$748$	0.986128	0.0360564
$749$	0	0
$750$	3.94707	0.144127
$751$	−25.7013	−0.937854	−0.468927	−	0.883237i	$-0.655359\pi$
$751$	−25.7013	−0.937854	−0.468927	+	0.883237i	$0.655359\pi$
$752$	11.6763	0.425791
$753$	6.52477	0.237776
$754$	0	0
$755$	−19.7084	−0.717263
$756$	0	0
$757$	16.3885	0.595650	0.297825	−	0.954621i	$-0.403739\pi$
$757$	16.3885	0.595650	0.297825	+	0.954621i	$0.403739\pi$
$758$	4.06859	0.147778
$759$	14.8004	0.537219
$760$	−10.7694	−0.390646
$761$	8.31998	0.301599	0.150800	−	0.988564i	$-0.451815\pi$
$761$	8.31998	0.301599	0.150800	+	0.988564i	$0.451815\pi$
$762$	−0.852923	−0.0308981
$763$	0	0
$764$	11.3378	0.410185
$765$	0.337326	0.0121960
$766$	−11.1853	−0.404140
$767$	0	0
$768$	0.394335	0.0142293
$769$	−25.5588	−0.921675	−0.460838	−	0.887485i	$-0.652451\pi$
$769$	−25.5588	−0.921675	−0.460838	+	0.887485i	$0.652451\pi$
$770$	0	0
$771$	11.5748	0.416855
$772$	−8.45131	−0.304169
$773$	−8.40077	−0.302155	−0.151077	−	0.988522i	$-0.548274\pi$
$773$	−8.40077	−0.302155	−0.151077	+	0.988522i	$0.548274\pi$
$774$	5.49394	0.197476
$775$	11.1391	0.400130
$776$	5.06783	0.181925
$777$	0	0
$778$	10.6970	0.383506
$779$	−66.6537	−2.38812
$780$	0	0
$781$	−14.6743	−0.525087
$782$	−0.311829	−0.0111510
$783$	−37.6577	−1.34578
$784$	0	0
$785$	20.6345	0.736476
$786$	5.28872	0.188642
$787$	−35.0644	−1.24991	−0.624956	−	0.780660i	$-0.714883\pi$
$787$	−35.0644	−1.24991	−0.624956	+	0.780660i	$0.714883\pi$
$788$	45.2137	1.61067
$789$	9.96001	0.354586
$790$	6.21440	0.221098
$791$	0	0
$792$	16.9463	0.602160
$793$	0	0
$794$	−0.598349	−0.0212346
$795$	−0.247639	−0.00878284
$796$	29.9508	1.06158
$797$	24.5752	0.870500	0.435250	−	0.900310i	$-0.356660\pi$
$797$	24.5752	0.870500	0.435250	+	0.900310i	$0.356660\pi$
$798$	0	0
$799$	0.647181	0.0228956
$800$	19.6882	0.696083
$801$	3.98145	0.140678
$802$	−18.1370	−0.640439
$803$	−26.2350	−0.925814
$804$	7.64887	0.269755
$805$	0	0
$806$	0	0
$807$	7.81528	0.275111
$808$	−21.5099	−0.756717
$809$	−31.8012	−1.11807	−0.559035	−	0.829144i	$-0.688828\pi$
$809$	−31.8012	−1.11807	−0.559035	+	0.829144i	$0.688828\pi$
$810$	1.57651	0.0553927
$811$	−10.3633	−0.363905	−0.181953	−	0.983307i	$-0.558242\pi$
$811$	−10.3633	−0.363905	−0.181953	+	0.983307i	$0.558242\pi$
$812$	0	0
$813$	2.17811	0.0763898
$814$	−1.67254	−0.0586226
$815$	9.31196	0.326184
$816$	−0.309235	−0.0108254
$817$	26.5749	0.929737
$818$	−7.33597	−0.256496
$819$	0	0
$820$	−22.0698	−0.770711
$821$	−4.05592	−0.141552	−0.0707762	−	0.997492i	$-0.522548\pi$
$821$	−4.05592	−0.141552	−0.0707762	+	0.997492i	$0.522548\pi$
$822$	−2.22810	−0.0777138
$823$	8.71697	0.303854	0.151927	−	0.988392i	$-0.451452\pi$
$823$	8.71697	0.303854	0.151927	+	0.988392i	$0.451452\pi$
$824$	−7.81764	−0.272340
$825$	−13.1901	−0.459220
$826$	0	0
$827$	23.5788	0.819915	0.409957	−	0.912105i	$-0.365544\pi$
$827$	23.5788	0.819915	0.409957	+	0.912105i	$0.365544\pi$
$828$	17.5061	0.608380
$829$	29.0406	1.00862	0.504311	−	0.863522i	$-0.331747\pi$
$829$	29.0406	1.00862	0.504311	+	0.863522i	$0.331747\pi$
$830$	−1.49759	−0.0519820
$831$	0.793427	0.0275237
$832$	0	0
$833$	0	0
$834$	8.80951	0.305048
$835$	−6.39526	−0.221317
$836$	38.2534	1.32302
$837$	−12.7648	−0.441216
$838$	−2.96666	−0.102482
$839$	11.5574	0.399007	0.199503	−	0.979897i	$-0.436067\pi$
$839$	11.5574	0.399007	0.199503	+	0.979897i	$0.436067\pi$
$840$	0	0
$841$	41.5196	1.43171
$842$	1.31584	0.0453467
$843$	5.48238	0.188823
$844$	−32.0097	−1.10182
$845$	0	0
$846$	5.19009	0.178439
$847$	0	0
$848$	−0.716496	−0.0246046
$849$	18.8395	0.646572
$850$	0.277902	0.00953197
$851$	−3.70241	−0.126917
$852$	5.49943	0.188408
$853$	−1.51845	−0.0519906	−0.0259953	−	0.999662i	$-0.508275\pi$
$853$	−1.51845	−0.0519906	−0.0259953	+	0.999662i	$0.508275\pi$
$854$	0	0
$855$	13.0854	0.447511
$856$	15.9136	0.543914
$857$	6.00119	0.204997	0.102498	−	0.994733i	$-0.467316\pi$
$857$	6.00119	0.204997	0.102498	+	0.994733i	$0.467316\pi$
$858$	0	0
$859$	−1.67604	−0.0571858	−0.0285929	−	0.999591i	$-0.509103\pi$
$859$	−1.67604	−0.0571858	−0.0285929	+	0.999591i	$0.509103\pi$
$860$	8.79924	0.300051
$861$	0	0
$862$	9.41858	0.320798
$863$	−42.0504	−1.43141	−0.715706	−	0.698401i	$-0.753895\pi$
$863$	−42.0504	−1.43141	−0.715706	+	0.698401i	$0.753895\pi$
$864$	−22.5615	−0.767558
$865$	25.2933	0.859999
$866$	9.56747	0.325116
$867$	14.4261	0.489936
$868$	0	0
$869$	−47.3010	−1.60458
$870$	3.71873	0.126077
$871$	0	0
$872$	12.0566	0.408290
$873$	−6.15770	−0.208407
$874$	−12.0963	−0.409164
$875$	0	0
$876$	9.83201	0.332193
$877$	30.8630	1.04217	0.521084	−	0.853505i	$-0.325528\pi$
$877$	30.8630	1.04217	0.521084	+	0.853505i	$0.325528\pi$
$878$	−0.632537	−0.0213471
$879$	−20.5625	−0.693557
$880$	10.5983	0.357270
$881$	55.7338	1.87772	0.938860	−	0.344298i	$-0.111883\pi$
$881$	55.7338	1.87772	0.938860	+	0.344298i	$0.111883\pi$
$882$	0	0
$883$	2.92007	0.0982681	0.0491341	−	0.998792i	$-0.484354\pi$
$883$	2.92007	0.0982681	0.0491341	+	0.998792i	$0.484354\pi$
$884$	0	0
$885$	9.54904	0.320988
$886$	10.4693	0.351724
$887$	−1.87909	−0.0630937	−0.0315468	−	0.999502i	$-0.510043\pi$
$887$	−1.87909	−0.0630937	−0.0315468	+	0.999502i	$0.510043\pi$
$888$	1.34317	0.0450737
$889$	0	0
$890$	−0.910916	−0.0305340
$891$	−11.9996	−0.402002
$892$	−20.1404	−0.674349
$893$	25.1051	0.840111
$894$	0.00483239	0.000161619 0
$895$	4.31246	0.144150
$896$	0	0
$897$	0	0
$898$	−8.93138	−0.298044
$899$	23.9039	0.797241
$900$	−15.6015	−0.520050
$901$	−0.0397131	−0.00132304
$902$	−23.9964	−0.798991
$903$	0	0
$904$	20.5557	0.683672
$905$	8.19978	0.272570
$906$	8.03082	0.266806
$907$	−35.6164	−1.18262	−0.591311	−	0.806444i	$-0.701389\pi$
$907$	−35.6164	−1.18262	−0.591311	+	0.806444i	$0.701389\pi$
$908$	31.3331	1.03982
$909$	26.1358	0.866869
$910$	0	0
$911$	44.2844	1.46721	0.733604	−	0.679577i	$-0.237837\pi$
$911$	44.2844	1.46721	0.733604	+	0.679577i	$0.237837\pi$
$912$	−11.9957	−0.397217
$913$	11.3989	0.377249
$914$	−3.28290	−0.108589
$915$	5.19327	0.171684
$916$	−6.76118	−0.223396
$917$	0	0
$918$	−0.318459	−0.0105107
$919$	26.1128	0.861382	0.430691	−	0.902499i	$-0.358270\pi$
$919$	26.1128	0.861382	0.430691	+	0.902499i	$0.358270\pi$
$920$	−8.58259	−0.282959
$921$	−20.6389	−0.680074
$922$	2.55675	0.0842021
$923$	0	0
$924$	0	0
$925$	3.29960	0.108490
$926$	−16.6615	−0.547531
$927$	9.49887	0.311984
$928$	42.2497	1.38691
$929$	10.0231	0.328849	0.164424	−	0.986390i	$-0.447423\pi$
$929$	10.0231	0.328849	0.164424	+	0.986390i	$0.447423\pi$
$930$	1.26053	0.0413345
$931$	0	0
$932$	−43.8626	−1.43677
$933$	3.38745	0.110900
$934$	−6.47455	−0.211854
$935$	0.587432	0.0192111
$936$	0	0
$937$	0.916838	0.0299518	0.0149759	−	0.999888i	$-0.495233\pi$
$937$	0.916838	0.0299518	0.0149759	+	0.999888i	$0.495233\pi$
$938$	0	0
$939$	24.1928	0.789501
$940$	8.31259	0.271127
$941$	17.0896	0.557107	0.278553	−	0.960421i	$-0.410145\pi$
$941$	17.0896	0.557107	0.278553	+	0.960421i	$0.410145\pi$
$942$	−8.40817	−0.273953
$943$	−53.1193	−1.72980
$944$	27.6284	0.899227
$945$	0	0
$946$	9.56735	0.311062
$947$	21.7555	0.706957	0.353479	−	0.935443i	$-0.384999\pi$
$947$	21.7555	0.706957	0.353479	+	0.935443i	$0.384999\pi$
$948$	17.7268	0.575741
$949$	0	0
$950$	10.7802	0.349757
$951$	−14.2870	−0.463288
$952$	0	0
$953$	−17.8081	−0.576862	−0.288431	−	0.957501i	$-0.593134\pi$
$953$	−17.8081	−0.576862	−0.288431	+	0.957501i	$0.593134\pi$
$954$	−0.318481	−0.0103112
$955$	6.75385	0.218549
$956$	−13.6582	−0.441737
$957$	−28.3052	−0.914976
$958$	13.5056	0.436347
$959$	0	0
$960$	−2.31132	−0.0745975
$961$	−22.8973	−0.738623
$962$	0	0
$963$	−19.3359	−0.623089
$964$	−38.0895	−1.22678
$965$	−5.03441	−0.162063
$966$	0	0
$967$	35.3813	1.13779	0.568893	−	0.822412i	$-0.307372\pi$
$967$	35.3813	1.13779	0.568893	+	0.822412i	$0.307372\pi$
$968$	8.88637	0.285619
$969$	−0.664883	−0.0213591
$970$	1.40882	0.0452344
$971$	−49.7067	−1.59516	−0.797582	−	0.603211i	$-0.793888\pi$
$971$	−49.7067	−1.59516	−0.797582	+	0.603211i	$0.793888\pi$
$972$	28.0401	0.899385
$973$	0	0
$974$	−16.0415	−0.514002
$975$	0	0
$976$	15.0257	0.480962
$977$	61.2801	1.96052	0.980262	−	0.197702i	$-0.0633477\pi$
$977$	61.2801	1.96052	0.980262	+	0.197702i	$0.0633477\pi$
$978$	−3.79445	−0.121333
$979$	6.93346	0.221594
$980$	0	0
$981$	−14.6495	−0.467723
$982$	14.3016	0.456383
$983$	−6.98962	−0.222934	−0.111467	−	0.993768i	$-0.535555\pi$
$983$	−6.98962	−0.222934	−0.111467	+	0.993768i	$0.535555\pi$
$984$	19.2707	0.614328
$985$	26.9336	0.858176
$986$	0.596362	0.0189920
$987$	0	0
$988$	0	0
$989$	21.1787	0.673443
$990$	4.71094	0.149723
$991$	−32.7341	−1.03983	−0.519917	−	0.854217i	$-0.674037\pi$
$991$	−32.7341	−1.03983	−0.519917	+	0.854217i	$0.674037\pi$
$992$	14.3213	0.454703
$993$	−5.27532	−0.167407
$994$	0	0
$995$	17.8416	0.565616
$996$	−4.27193	−0.135361
$997$	−57.7897	−1.83022	−0.915109	−	0.403207i	$-0.867896\pi$
$997$	−57.7897	−1.83022	−0.915109	+	0.403207i	$0.867896\pi$
$998$	0.899058	0.0284592
$999$	−3.78114	−0.119630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.co.1.5		12
7.3	odd	6		1183.2.e.j.170.8		24
7.5	odd	6		1183.2.e.j.508.8		24
7.6	odd	2		8281.2.a.cp.1.5		12
13.2	odd	12		637.2.q.i.589.3		12
13.7	odd	12		637.2.q.i.491.3		12
13.12	even	2	inner	8281.2.a.co.1.8		12
91.2	odd	12		637.2.k.i.459.4		12
91.12	odd	6		1183.2.e.j.508.5		24
91.20	even	12		637.2.q.g.491.3		12
91.33	even	12		91.2.u.b.88.4	yes	12
91.38	odd	6		1183.2.e.j.170.5		24
91.41	even	12		637.2.q.g.589.3		12
91.46	odd	12		637.2.k.i.569.3		12
91.54	even	12		91.2.k.b.4.4	✓	12
91.59	even	12		91.2.k.b.23.3	yes	12
91.67	odd	12		637.2.u.g.30.4		12
91.72	odd	12		637.2.u.g.361.4		12
91.80	even	12		91.2.u.b.30.4	yes	12
91.90	odd	2		8281.2.a.cp.1.8		12
273.59	odd	12		819.2.bm.f.478.4		12
273.80	odd	12		819.2.do.e.667.3		12
273.215	odd	12		819.2.do.e.361.3		12
273.236	odd	12		819.2.bm.f.550.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.k.b.4.4	✓	12	91.54	even	12
91.2.k.b.23.3	yes	12	91.59	even	12
91.2.u.b.30.4	yes	12	91.80	even	12
91.2.u.b.88.4	yes	12	91.33	even	12
637.2.k.i.459.4		12	91.2	odd	12
637.2.k.i.569.3		12	91.46	odd	12
637.2.q.g.491.3		12	91.20	even	12
637.2.q.g.589.3		12	91.41	even	12
637.2.q.i.491.3		12	13.7	odd	12
637.2.q.i.589.3		12	13.2	odd	12
637.2.u.g.30.4		12	91.67	odd	12
637.2.u.g.361.4		12	91.72	odd	12
819.2.bm.f.478.4		12	273.59	odd	12
819.2.bm.f.550.3		12	273.236	odd	12
819.2.do.e.361.3		12	273.215	odd	12
819.2.do.e.667.3		12	273.80	odd	12
1183.2.e.j.170.5		24	91.38	odd	6
1183.2.e.j.170.8		24	7.3	odd	6
1183.2.e.j.508.5		24	91.12	odd	6
1183.2.e.j.508.8		24	7.5	odd	6
8281.2.a.co.1.5		12	1.1	even	1	trivial
8281.2.a.co.1.8		12	13.12	even	2	inner
8281.2.a.cp.1.5		12	7.6	odd	2
8281.2.a.cp.1.8		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)

\(f(q)\)	\(=\)	\(q-0.499987 q^{2} -0.849601 q^{3} -1.75001 q^{4} -1.04248 q^{5} +0.424789 q^{6} +1.87496 q^{8} -2.27818 q^{9} +O(q^{10})\)	\(q-0.499987 q^{2} -0.849601 q^{3} -1.75001 q^{4} -1.04248 q^{5} +0.424789 q^{6} +1.87496 q^{8} -2.27818 q^{9} +0.521224 q^{10} -3.96730 q^{11} +1.48681 q^{12} +0.885688 q^{15} +2.56257 q^{16} +0.142035 q^{17} +1.13906 q^{18} +5.50977 q^{19} +1.82435 q^{20} +1.98360 q^{22} +4.39098 q^{23} -1.59297 q^{24} -3.91325 q^{25} +4.48435 q^{27} -8.39759 q^{29} -0.442832 q^{30} -2.84652 q^{31} -5.03117 q^{32} +3.37063 q^{33} -0.0710158 q^{34} +3.98684 q^{36} -0.843187 q^{37} -2.75481 q^{38} -1.95460 q^{40} -12.0974 q^{41} +4.82323 q^{43} +6.94284 q^{44} +2.37494 q^{45} -2.19543 q^{46} +4.55648 q^{47} -2.17717 q^{48} +1.95657 q^{50} -0.120673 q^{51} -0.279600 q^{53} -2.24211 q^{54} +4.13582 q^{55} -4.68111 q^{57} +4.19868 q^{58} +10.7815 q^{59} -1.54997 q^{60} +5.86354 q^{61} +1.42322 q^{62} -2.60963 q^{64} -1.68527 q^{66} +5.14447 q^{67} -0.248564 q^{68} -3.73058 q^{69} +3.69880 q^{71} -4.27148 q^{72} +6.61281 q^{73} +0.421582 q^{74} +3.32470 q^{75} -9.64216 q^{76} +11.9227 q^{79} -2.67142 q^{80} +3.02462 q^{81} +6.04853 q^{82} -2.87321 q^{83} -0.148068 q^{85} -2.41155 q^{86} +7.13461 q^{87} -7.43852 q^{88} -1.74765 q^{89} -1.18744 q^{90} -7.68427 q^{92} +2.41841 q^{93} -2.27818 q^{94} -5.74379 q^{95} +4.27449 q^{96} +2.70291 q^{97} +9.03822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) \) 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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