LMFDB - Embedded newform 8281.2.a.ck.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 3 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
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			1.8
Root			$2.28481$ 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)

$f(q)$	$=$	$q+2.28481 q^{2} +3.15042 q^{3} +3.22037 q^{4} +2.12499 q^{5} +7.19813 q^{6} +2.78832 q^{8} +6.92516 q^{9} +O(q^{10})$	\(q+2.28481 q^{2} +3.15042 q^{3} +3.22037 q^{4} +2.12499 q^{5} +7.19813 q^{6} +2.78832 q^{8} +6.92516 q^{9} +4.85521 q^{10} +0.308465 q^{11} +10.1455 q^{12} +6.69462 q^{15} -0.0699499 q^{16} -1.77474 q^{17} +15.8227 q^{18} +1.78131 q^{19} +6.84326 q^{20} +0.704786 q^{22} +1.15042 q^{23} +8.78439 q^{24} -0.484414 q^{25} +12.3659 q^{27} +2.01052 q^{29} +15.2960 q^{30} -4.60485 q^{31} -5.73646 q^{32} +0.971796 q^{33} -4.05494 q^{34} +22.3016 q^{36} -5.54142 q^{37} +4.06995 q^{38} +5.92516 q^{40} -6.72984 q^{41} +1.52611 q^{43} +0.993373 q^{44} +14.7159 q^{45} +2.62850 q^{46} +9.51816 q^{47} -0.220372 q^{48} -1.10680 q^{50} -5.59116 q^{51} +7.44074 q^{53} +28.2538 q^{54} +0.655486 q^{55} +5.61186 q^{57} +4.59367 q^{58} -8.12106 q^{59} +21.5592 q^{60} -3.44074 q^{61} -10.5212 q^{62} -12.9669 q^{64} +2.22037 q^{66} +12.6149 q^{67} -5.71531 q^{68} +3.62431 q^{69} -1.35070 q^{71} +19.3096 q^{72} +11.8812 q^{73} -12.6611 q^{74} -1.52611 q^{75} +5.73646 q^{76} -7.92516 q^{79} -0.148643 q^{80} +18.1823 q^{81} -15.3764 q^{82} -11.2290 q^{83} -3.77130 q^{85} +3.48687 q^{86} +6.33399 q^{87} +0.860100 q^{88} -1.65917 q^{89} +33.6231 q^{90} +3.70479 q^{92} -14.5072 q^{93} +21.7472 q^{94} +3.78526 q^{95} -18.0723 q^{96} -7.66641 q^{97} +2.13617 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9	$ 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9} - 6 q^{10} + 18 q^{12} - 2 q^{16} + 8 q^{17} - 18 q^{22} - 12 q^{23} + 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} + 34 q^{38} + 4 q^{40} - 8 q^{43} + 18 q^{48} + 16 q^{51} + 20 q^{53} + 12 q^{55} + 12 q^{61} - 22 q^{62} - 44 q^{64} - 2 q^{66} + 2 q^{68} + 28 q^{69} - 42 q^{74} + 8 q^{75} - 20 q^{79} + 24 q^{81} - 16 q^{82} + 68 q^{87} + 4 q^{88} + 108 q^{90} + 6 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100}) $ 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9 - 6 * q^10 + 18 * q^12 - 2 * q^16 + 8 * q^17 - 18 * q^22 - 12 * q^23 + 16 * q^27 - 8 * q^29 + 38 * q^30 + 28 * q^36 + 34 * q^38 + 4 * q^40 - 8 * q^43 + 18 * q^48 + 16 * q^51 + 20 * q^53 + 12 * q^55 + 12 * q^61 - 22 * q^62 - 44 * q^64 - 2 * q^66 + 2 * q^68 + 28 * q^69 - 42 * q^74 + 8 * q^75 - 20 * q^79 + 24 * q^81 - 16 * q^82 + 68 * q^87 + 4 * q^88 + 108 * q^90 + 6 * q^92 + 26 * q^94 - 16 * 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 $

$2$	2.28481	1.61561	0.807803	−	0.589452i	$-0.200656\pi$
$2$	2.28481	1.61561	0.807803	+	0.589452i	$0.200656\pi$
$3$	3.15042	1.81890	0.909448	−	0.415817i	$-0.136504\pi$
$3$	3.15042	1.81890	0.909448	+	0.415817i	$0.136504\pi$
$4$	3.22037	1.61019
$5$	2.12499	0.950325	0.475162	−	0.879898i	$-0.342389\pi$
$5$	2.12499	0.950325	0.475162	+	0.879898i	$0.342389\pi$
$6$	7.19813	2.93862
$7$	0	0
$7$	0	0
$8$	2.78832	0.985820
$9$	6.92516	2.30839
$10$	4.85521	1.53535
$11$	0.308465	0.0930058	0.0465029	−	0.998918i	$-0.485192\pi$
$11$	0.308465	0.0930058	0.0465029	+	0.998918i	$0.485192\pi$
$12$	10.1455	2.92876
$13$	0	0
$13$	0	0
$14$	0	0
$15$	6.69462	1.72854
$16$	−0.0699499	−0.0174875
$17$	−1.77474	−0.430437	−0.215218	−	0.976566i	$-0.569046\pi$
$17$	−1.77474	−0.430437	−0.215218	+	0.976566i	$0.569046\pi$
$18$	15.8227	3.72944
$19$	1.78131	0.408659	0.204330	−	0.978902i	$-0.434499\pi$
$19$	1.78131	0.408659	0.204330	+	0.978902i	$0.434499\pi$
$20$	6.84326	1.53020
$21$	0	0
$22$	0.704786	0.150261
$23$	1.15042	0.239880	0.119940	−	0.992781i	$-0.461730\pi$
$23$	1.15042	0.239880	0.119940	+	0.992781i	$0.461730\pi$
$24$	8.78439	1.79311
$25$	−0.484414	−0.0968828
$26$	0	0
$27$	12.3659	2.37982
$28$	0	0
$29$	2.01052	0.373345	0.186672	−	0.982422i	$-0.440230\pi$
$29$	2.01052	0.373345	0.186672	+	0.982422i	$0.440230\pi$
$30$	15.2960	2.79265
$31$	−4.60485	−0.827055	−0.413527	−	0.910492i	$-0.635703\pi$
$31$	−4.60485	−0.827055	−0.413527	+	0.910492i	$0.635703\pi$
$32$	−5.73646	−1.01407
$33$	0.971796	0.169168
$34$	−4.05494	−0.695416
$35$	0	0
$36$	22.3016	3.71693
$37$	−5.54142	−0.911004	−0.455502	−	0.890235i	$-0.650540\pi$
$37$	−5.54142	−0.911004	−0.455502	+	0.890235i	$0.650540\pi$
$38$	4.06995	0.660233
$39$	0	0
$40$	5.92516	0.936850
$41$	−6.72984	−1.05102	−0.525512	−	0.850786i	$-0.676126\pi$
$41$	−6.72984	−1.05102	−0.525512	+	0.850786i	$0.676126\pi$
$42$	0	0
$43$	1.52611	0.232729	0.116365	−	0.993207i	$-0.462876\pi$
$43$	1.52611	0.232729	0.116365	+	0.993207i	$0.462876\pi$
$44$	0.993373	0.149757
$45$	14.7159	2.19372
$46$	2.62850	0.387551
$47$	9.51816	1.38837	0.694183	−	0.719798i	$-0.255766\pi$
$47$	9.51816	1.38837	0.694183	+	0.719798i	$0.255766\pi$
$48$	−0.220372	−0.0318079
$49$	0	0
$50$	−1.10680	−0.156524
$51$	−5.59116	−0.782920
$52$	0	0
$53$	7.44074	1.02206	0.511032	−	0.859561i	$-0.329263\pi$
$53$	7.44074	1.02206	0.511032	+	0.859561i	$0.329263\pi$
$54$	28.2538	3.84485
$55$	0.655486	0.0883857
$56$	0	0
$57$	5.61186	0.743309
$58$	4.59367	0.603178
$59$	−8.12106	−1.05727	−0.528636	−	0.848849i	$-0.677296\pi$
$59$	−8.12106	−1.05727	−0.528636	+	0.848849i	$0.677296\pi$
$60$	21.5592	2.78328
$61$	−3.44074	−0.440542	−0.220271	−	0.975439i	$-0.570694\pi$
$61$	−3.44074	−0.440542	−0.220271	+	0.975439i	$0.570694\pi$
$62$	−10.5212	−1.33620
$63$	0	0
$64$	−12.9669	−1.62086
$65$	0	0
$66$	2.22037	0.273309
$67$	12.6149	1.54116	0.770580	−	0.637343i	$-0.219966\pi$
$67$	12.6149	1.54116	0.770580	+	0.637343i	$0.219966\pi$
$68$	−5.71531	−0.693083
$69$	3.62431	0.436316
$70$	0	0
$71$	−1.35070	−0.160299	−0.0801494	−	0.996783i	$-0.525540\pi$
$71$	−1.35070	−0.160299	−0.0801494	+	0.996783i	$0.525540\pi$
$72$	19.3096	2.27565
$73$	11.8812	1.39059	0.695293	−	0.718726i	$-0.255275\pi$
$73$	11.8812	1.39059	0.695293	+	0.718726i	$0.255275\pi$
$74$	−12.6611	−1.47182
$75$	−1.52611	−0.176220
$76$	5.73646	0.658018
$77$	0	0
$78$	0	0
$79$	−7.92516	−0.891650	−0.445825	−	0.895120i	$-0.647090\pi$
$79$	−7.92516	−0.891650	−0.445825	+	0.895120i	$0.647090\pi$
$80$	−0.148643	−0.0166188
$81$	18.1823	2.02026
$82$	−15.3764	−1.69804
$83$	−11.2290	−1.23255	−0.616273	−	0.787533i	$-0.711358\pi$
$83$	−11.2290	−1.23255	−0.616273	+	0.787533i	$0.711358\pi$
$84$	0	0
$85$	−3.77130	−0.409055
$86$	3.48687	0.375999
$87$	6.33399	0.679075
$88$	0.860100	0.0916870
$89$	−1.65917	−0.175871	−0.0879357	−	0.996126i	$-0.528027\pi$
$89$	−1.65917	−0.175871	−0.0879357	+	0.996126i	$0.528027\pi$
$90$	33.6231	3.54418
$91$	0	0
$92$	3.70479	0.386251
$93$	−14.5072	−1.50433
$94$	21.7472	2.24305
$95$	3.78526	0.388359
$96$	−18.0723	−1.84449
$97$	−7.66641	−0.778406	−0.389203	−	0.921152i	$-0.627250\pi$
$97$	−7.66641	−0.778406	−0.389203	+	0.921152i	$0.627250\pi$
$98$	0	0
$99$	2.13617	0.214693
$100$	−1.55999	−0.155999
$101$	9.11727	0.907203	0.453601	−	0.891205i	$-0.350139\pi$
$101$	9.11727	0.907203	0.453601	+	0.891205i	$0.350139\pi$
$102$	−12.7748	−1.26489
$103$	6.04169	0.595306	0.297653	−	0.954674i	$-0.403796\pi$
$103$	6.04169	0.595306	0.297653	+	0.954674i	$0.403796\pi$
$104$	0	0
$105$	0	0
$106$	17.0007	1.65125
$107$	12.0861	1.16841	0.584204	−	0.811607i	$-0.301407\pi$
$107$	12.0861	1.16841	0.584204	+	0.811607i	$0.301407\pi$
$108$	39.8228	3.83195
$109$	1.36188	0.130445	0.0652223	−	0.997871i	$-0.479224\pi$
$109$	1.36188	0.130445	0.0652223	+	0.997871i	$0.479224\pi$
$110$	1.49766	0.142797
$111$	−17.4578	−1.65702
$112$	0	0
$113$	−9.42009	−0.886168	−0.443084	−	0.896480i	$-0.646116\pi$
$113$	−9.42009	−0.886168	−0.443084	+	0.896480i	$0.646116\pi$
$114$	12.8221	1.20090
$115$	2.44464	0.227963
$116$	6.47463	0.601154
$117$	0	0
$118$	−18.5551	−1.70814
$119$	0	0
$120$	18.6667	1.70403
$121$	−10.9048	−0.991350
$122$	−7.86146	−0.711743
$123$	−21.2018	−1.91170
$124$	−14.8293	−1.33171
$125$	−11.6543	−1.04239
$126$	0	0
$127$	−13.3998	−1.18904	−0.594519	−	0.804081i	$-0.702658\pi$
$127$	−13.3998	−1.18904	−0.594519	+	0.804081i	$0.702658\pi$
$128$	−18.1539	−1.60459
$129$	4.80788	0.423311
$130$	0	0
$131$	13.3971	1.17051	0.585254	−	0.810850i	$-0.300995\pi$
$131$	13.3971	1.17051	0.585254	+	0.810850i	$0.300995\pi$
$132$	3.12954	0.272392
$133$	0	0
$134$	28.8228	2.48991
$135$	26.2774	2.26160
$136$	−4.94853	−0.424333
$137$	0.501044	0.0428071	0.0214036	−	0.999771i	$-0.493187\pi$
$137$	0.501044	0.0428071	0.0214036	+	0.999771i	$0.493187\pi$
$138$	8.28088	0.704915
$139$	1.41936	0.120388	0.0601941	−	0.998187i	$-0.480828\pi$
$139$	1.41936	0.120388	0.0601941	+	0.998187i	$0.480828\pi$
$140$	0	0
$141$	29.9862	2.52529
$142$	−3.08610	−0.258980
$143$	0	0
$144$	−0.484414	−0.0403678
$145$	4.27234	0.354799
$146$	27.1463	2.24664
$147$	0	0
$148$	−17.8454	−1.46689
$149$	−21.0909	−1.72783	−0.863916	−	0.503637i	$-0.831995\pi$
$149$	−21.0909	−1.72783	−0.863916	+	0.503637i	$0.831995\pi$
$150$	−3.48687	−0.284702
$151$	−17.5042	−1.42447	−0.712236	−	0.701940i	$-0.752318\pi$
$151$	−17.5042	−1.42447	−0.712236	+	0.701940i	$0.752318\pi$
$152$	4.96685	0.402865
$153$	−12.2903	−0.993614
$154$	0	0
$155$	−9.78526	−0.785971
$156$	0	0
$157$	0.0755789	0.00603185	0.00301593	−	0.999995i	$-0.499040\pi$
$157$	0.0755789	0.00603185	0.00301593	+	0.999995i	$0.499040\pi$
$158$	−18.1075	−1.44056
$159$	23.4415	1.85903
$160$	−12.1899	−0.963699
$161$	0	0
$162$	41.5432	3.26394
$163$	−10.0817	−0.789660	−0.394830	−	0.918754i	$-0.629197\pi$
$163$	−10.0817	−0.789660	−0.394830	+	0.918754i	$0.629197\pi$
$164$	−21.6726	−1.69234
$165$	2.06506	0.160764
$166$	−25.6562	−1.99131
$167$	−5.84989	−0.452678	−0.226339	−	0.974049i	$-0.572676\pi$
$167$	−5.84989	−0.452678	−0.226339	+	0.974049i	$0.572676\pi$
$168$	0	0
$169$	0	0
$170$	−8.61671	−0.660871
$171$	12.3358	0.943344
$172$	4.91464	0.374737
$173$	16.9902	1.29174	0.645871	−	0.763447i	$-0.276494\pi$
$173$	16.9902	1.29174	0.645871	+	0.763447i	$0.276494\pi$
$174$	14.4720	1.09712
$175$	0	0
$176$	−0.0215771	−0.00162644
$177$	−25.5848	−1.92307
$178$	−3.79089	−0.284139
$179$	−15.3016	−1.14369	−0.571847	−	0.820360i	$-0.693773\pi$
$179$	−15.3016	−1.14369	−0.571847	+	0.820360i	$0.693773\pi$
$180$	47.3907	3.53229
$181$	5.84958	0.434796	0.217398	−	0.976083i	$-0.430243\pi$
$181$	5.84958	0.434796	0.217398	+	0.976083i	$0.430243\pi$
$182$	0	0
$183$	−10.8398	−0.801301
$184$	3.20775	0.236478
$185$	−11.7755	−0.865750
$186$	−33.1463	−2.43040
$187$	−0.547444	−0.0400331
$188$	30.6520	2.23553
$189$	0	0
$190$	8.64861	0.627436
$191$	−26.8179	−1.94048	−0.970238	−	0.242155i	$-0.922146\pi$
$191$	−26.8179	−1.94048	−0.970238	+	0.242155i	$0.922146\pi$
$192$	−40.8511	−2.94817
$193$	−0.213984	−0.0154029	−0.00770145	−	0.999970i	$-0.502451\pi$
$193$	−0.213984	−0.0154029	−0.00770145	+	0.999970i	$0.502451\pi$
$194$	−17.5163	−1.25760
$195$	0	0
$196$	0	0
$197$	11.2290	0.800035	0.400017	−	0.916508i	$-0.369004\pi$
$197$	11.2290	0.800035	0.400017	+	0.916508i	$0.369004\pi$
$198$	4.88075	0.346860
$199$	20.4201	1.44754	0.723771	−	0.690040i	$-0.242407\pi$
$199$	20.4201	1.44754	0.723771	+	0.690040i	$0.242407\pi$
$200$	−1.35070	−0.0955090
$201$	39.7424	2.80321
$202$	20.8313	1.46568
$203$	0	0
$204$	−18.0056	−1.26065
$205$	−14.3008	−0.998814
$206$	13.8041	0.961780
$207$	7.96685	0.553734
$208$	0	0
$209$	0.549471	0.0380077
$210$	0	0
$211$	8.41738	0.579476	0.289738	−	0.957106i	$-0.406432\pi$
$211$	8.41738	0.579476	0.289738	+	0.957106i	$0.406432\pi$
$212$	23.9620	1.64571
$213$	−4.25528	−0.291567
$214$	27.6145	1.88769
$215$	3.24297	0.221168
$216$	34.4801	2.34607
$217$	0	0
$218$	3.11164	0.210747
$219$	37.4307	2.52933
$220$	2.11091	0.142317
$221$	0	0
$222$	−39.8879	−2.67710
$223$	13.6091	0.911333	0.455666	−	0.890151i	$-0.349401\pi$
$223$	13.6091	0.911333	0.455666	+	0.890151i	$0.349401\pi$
$224$	0	0
$225$	−3.35464	−0.223643
$226$	−21.5232	−1.43170
$227$	−3.60276	−0.239123	−0.119562	−	0.992827i	$-0.538149\pi$
$227$	−3.60276	−0.239123	−0.119562	+	0.992827i	$0.538149\pi$
$228$	18.0723	1.19687
$229$	−18.3842	−1.21486	−0.607430	−	0.794373i	$-0.707799\pi$
$229$	−18.3842	−1.21486	−0.607430	+	0.794373i	$0.707799\pi$
$230$	5.58554	0.368299
$231$	0	0
$232$	5.60598	0.368051
$233$	20.2697	1.32791	0.663955	−	0.747772i	$-0.268877\pi$
$233$	20.2697	1.32791	0.663955	+	0.747772i	$0.268877\pi$
$234$	0	0
$235$	20.2260	1.31940
$236$	−26.1528	−1.70240
$237$	−24.9676	−1.62182
$238$	0	0
$239$	−20.8097	−1.34607	−0.673033	−	0.739612i	$-0.735009\pi$
$239$	−20.8097	−1.34607	−0.673033	+	0.739612i	$0.735009\pi$
$240$	−0.468288	−0.0302278
$241$	12.7147	0.819027	0.409514	−	0.912304i	$-0.365698\pi$
$241$	12.7147	0.819027	0.409514	+	0.912304i	$0.365698\pi$
$242$	−24.9155	−1.60163
$243$	20.1843	1.29482
$244$	−11.0805	−0.709355
$245$	0	0
$246$	−48.4422	−3.08856
$247$	0	0
$248$	−12.8398	−0.815328
$249$	−35.3762	−2.24187
$250$	−26.6280	−1.68410
$251$	13.7436	0.867486	0.433743	−	0.901037i	$-0.357193\pi$
$251$	13.7436	0.867486	0.433743	+	0.901037i	$0.357193\pi$
$252$	0	0
$253$	0.354865	0.0223102
$254$	−30.6160	−1.92102
$255$	−11.8812	−0.744028
$256$	−15.5446	−0.971536
$257$	−7.33473	−0.457528	−0.228764	−	0.973482i	$-0.573468\pi$
$257$	−7.33473	−0.457528	−0.228764	+	0.973482i	$0.573468\pi$
$258$	10.9851	0.683904
$259$	0	0
$260$	0	0
$261$	13.9232	0.861823
$262$	30.6098	1.89108
$263$	−6.67885	−0.411835	−0.205918	−	0.978569i	$-0.566018\pi$
$263$	−6.67885	−0.411835	−0.205918	+	0.978569i	$0.566018\pi$
$264$	2.70968	0.166769
$265$	15.8115	0.971293
$266$	0	0
$267$	−5.22708	−0.319892
$268$	40.6248	2.48156
$269$	16.2253	0.989272	0.494636	−	0.869100i	$-0.335301\pi$
$269$	16.2253	0.989272	0.494636	+	0.869100i	$0.335301\pi$
$270$	60.0390	3.65386
$271$	18.7381	1.13826	0.569129	−	0.822249i	$-0.307281\pi$
$271$	18.7381	1.13826	0.569129	+	0.822249i	$0.307281\pi$
$272$	0.124143	0.00752725
$273$	0	0
$274$	1.14479	0.0691595
$275$	−0.149425	−0.00901066
$276$	11.6716	0.702550
$277$	30.0326	1.80449	0.902243	−	0.431227i	$-0.141919\pi$
$277$	30.0326	1.80449	0.902243	+	0.431227i	$0.141919\pi$
$278$	3.24297	0.194500
$279$	−31.8893	−1.90916
$280$	0	0
$281$	−2.23065	−0.133070	−0.0665348	−	0.997784i	$-0.521194\pi$
$281$	−2.23065	−0.133070	−0.0665348	+	0.997784i	$0.521194\pi$
$282$	68.5129	4.07988
$283$	13.7755	0.818867	0.409433	−	0.912340i	$-0.365726\pi$
$283$	13.7755	0.818867	0.409433	+	0.912340i	$0.365726\pi$
$284$	−4.34976	−0.258111
$285$	11.9252	0.706385
$286$	0	0
$287$	0	0
$288$	−39.7259	−2.34087
$289$	−13.8503	−0.814724
$290$	9.76150	0.573215
$291$	−24.1524	−1.41584
$292$	38.2618	2.23910
$293$	1.01231	0.0591400	0.0295700	−	0.999563i	$-0.490586\pi$
$293$	1.01231	0.0591400	0.0295700	+	0.999563i	$0.490586\pi$
$294$	0	0
$295$	−17.2572	−1.00475
$296$	−15.4513	−0.898087
$297$	3.81445	0.221337
$298$	−48.1887	−2.79150
$299$	0	0
$300$	−4.91464	−0.283747
$301$	0	0
$302$	−39.9939	−2.30139
$303$	28.7233	1.65011
$304$	−0.124602	−0.00714642
$305$	−7.31155	−0.418658
$306$	−28.0811	−1.60529
$307$	−24.0527	−1.37276	−0.686379	−	0.727244i	$-0.740801\pi$
$307$	−24.0527	−1.37276	−0.686379	+	0.727244i	$0.740801\pi$
$308$	0	0
$309$	19.0339	1.08280
$310$	−22.3575	−1.26982
$311$	−8.99095	−0.509830	−0.254915	−	0.966963i	$-0.582047\pi$
$311$	−8.99095	−0.509830	−0.254915	+	0.966963i	$0.582047\pi$
$312$	0	0
$313$	−15.2361	−0.861197	−0.430598	−	0.902544i	$-0.641697\pi$
$313$	−15.2361	−0.861197	−0.430598	+	0.902544i	$0.641697\pi$
$314$	0.172684	0.00974510
$315$	0	0
$316$	−25.5220	−1.43572
$317$	−6.83550	−0.383920	−0.191960	−	0.981403i	$-0.561484\pi$
$317$	−6.83550	−0.383920	−0.191960	+	0.981403i	$0.561484\pi$
$318$	53.5594	3.00346
$319$	0.620176	0.0347232
$320$	−27.5544	−1.54034
$321$	38.0763	2.12521
$322$	0	0
$323$	−3.16135	−0.175902
$324$	58.5539	3.25299
$325$	0	0
$326$	−23.0348	−1.27578
$327$	4.29050	0.237265
$328$	−18.7649	−1.03612
$329$	0	0
$330$	4.71827	0.259732
$331$	−13.8145	−0.759316	−0.379658	−	0.925127i	$-0.623958\pi$
$331$	−13.8145	−0.759316	−0.379658	+	0.925127i	$0.623958\pi$
$332$	−36.1616	−1.98463
$333$	−38.3752	−2.10295
$334$	−13.3659	−0.731350
$335$	26.8066	1.46460
$336$	0	0
$337$	27.0432	1.47314	0.736568	−	0.676364i	$-0.236445\pi$
$337$	27.0432	1.47314	0.736568	+	0.676364i	$0.236445\pi$
$338$	0	0
$339$	−29.6773	−1.61185
$340$	−12.1450	−0.658654
$341$	−1.42044	−0.0769209
$342$	28.1850	1.52407
$343$	0	0
$344$	4.25528	0.229429
$345$	7.70163	0.414642
$346$	38.8195	2.08695
$347$	−19.3114	−1.03669	−0.518344	−	0.855172i	$-0.673451\pi$
$347$	−19.3114	−1.03669	−0.518344	+	0.855172i	$0.673451\pi$
$348$	20.3978	1.09344
$349$	14.1573	0.757821	0.378911	−	0.925433i	$-0.376299\pi$
$349$	14.1573	0.757821	0.378911	+	0.925433i	$0.376299\pi$
$350$	0	0
$351$	0	0
$352$	−1.76950	−0.0943147
$353$	16.9647	0.902940	0.451470	−	0.892286i	$-0.350900\pi$
$353$	16.9647	0.902940	0.451470	+	0.892286i	$0.350900\pi$
$354$	−58.4564	−3.10692
$355$	−2.87023	−0.152336
$356$	−5.34313	−0.283186
$357$	0	0
$358$	−34.9613	−1.84776
$359$	−22.7633	−1.20140	−0.600700	−	0.799475i	$-0.705111\pi$
$359$	−22.7633	−1.20140	−0.600700	+	0.799475i	$0.705111\pi$
$360$	41.0326	2.16261
$361$	−15.8270	−0.832997
$362$	13.3652	0.702459
$363$	−34.3549	−1.80316
$364$	0	0
$365$	25.2474	1.32151
$366$	−24.7669	−1.29459
$367$	16.5834	0.865644	0.432822	−	0.901479i	$-0.357518\pi$
$367$	16.5834	0.865644	0.432822	+	0.901479i	$0.357518\pi$
$368$	−0.0804719	−0.00419489
$369$	−46.6052	−2.42617
$370$	−26.9048	−1.39871
$371$	0	0
$372$	−46.7186	−2.42225
$373$	27.6461	1.43146	0.715730	−	0.698377i	$-0.246094\pi$
$373$	27.6461	1.43146	0.715730	+	0.698377i	$0.246094\pi$
$374$	−1.25081	−0.0646777
$375$	−36.7161	−1.89601
$376$	26.5397	1.36868
$377$	0	0
$378$	0	0
$379$	−9.24228	−0.474744	−0.237372	−	0.971419i	$-0.576286\pi$
$379$	−9.24228	−0.474744	−0.237372	+	0.971419i	$0.576286\pi$
$380$	12.1899	0.625330
$381$	−42.2150	−2.16274
$382$	−61.2739	−3.13505
$383$	7.64825	0.390808	0.195404	−	0.980723i	$-0.437398\pi$
$383$	7.64825	0.390808	0.195404	+	0.980723i	$0.437398\pi$
$384$	−57.1925	−2.91859
$385$	0	0
$386$	−0.488913	−0.0248850
$387$	10.5685	0.537229
$388$	−24.6887	−1.25338
$389$	6.53737	0.331458	0.165729	−	0.986171i	$-0.447002\pi$
$389$	6.53737	0.331458	0.165729	+	0.986171i	$0.447002\pi$
$390$	0	0
$391$	−2.04169	−0.103253
$392$	0	0
$393$	42.2064	2.12903
$394$	25.6562	1.29254
$395$	−16.8409	−0.847357
$396$	6.87926	0.345696
$397$	28.9307	1.45199	0.725996	−	0.687699i	$-0.241379\pi$
$397$	28.9307	1.45199	0.725996	+	0.687699i	$0.241379\pi$
$398$	46.6561	2.33866
$399$	0	0
$400$	0.0338847	0.00169423
$401$	26.7432	1.33549	0.667747	−	0.744388i	$-0.267259\pi$
$401$	26.7432	1.33549	0.667747	+	0.744388i	$0.267259\pi$
$402$	90.8040	4.52889
$403$	0	0
$404$	29.3610	1.46076
$405$	38.6373	1.91990
$406$	0	0
$407$	−1.70934	−0.0847287
$408$	−15.5900	−0.771818
$409$	−34.7725	−1.71939	−0.859694	−	0.510810i	$-0.829346\pi$
$409$	−34.7725	−1.71939	−0.859694	+	0.510810i	$0.829346\pi$
$410$	−32.6748	−1.61369
$411$	1.57850	0.0778617
$412$	19.4565	0.958553
$413$	0	0
$414$	18.2028	0.894617
$415$	−23.8616	−1.17132
$416$	0	0
$417$	4.47157	0.218974
$418$	1.25544	0.0614055
$419$	−4.19246	−0.204815	−0.102407	−	0.994743i	$-0.532655\pi$
$419$	−4.19246	−0.204815	−0.102407	+	0.994743i	$0.532655\pi$
$420$	0	0
$421$	−20.9526	−1.02117	−0.510584	−	0.859828i	$-0.670571\pi$
$421$	−20.9526	−1.02117	−0.510584	+	0.859828i	$0.670571\pi$
$422$	19.2321	0.936206
$423$	65.9147	3.20488
$424$	20.7472	1.00757
$425$	0.859706	0.0417019
$426$	−9.72252	−0.471058
$427$	0	0
$428$	38.9217	1.88135
$429$	0	0
$430$	7.40957	0.357321
$431$	16.8943	0.813768	0.406884	−	0.913480i	$-0.366615\pi$
$431$	16.8943	0.813768	0.406884	+	0.913480i	$0.366615\pi$
$432$	−0.864993	−0.0416170
$433$	−3.42241	−0.164471	−0.0822353	−	0.996613i	$-0.526206\pi$
$433$	−3.42241	−0.164471	−0.0822353	+	0.996613i	$0.526206\pi$
$434$	0	0
$435$	13.4597	0.645342
$436$	4.38576	0.210040
$437$	2.04925	0.0980290
$438$	85.5222	4.08641
$439$	18.0651	0.862198	0.431099	−	0.902305i	$-0.358126\pi$
$439$	18.0651	0.862198	0.431099	+	0.902305i	$0.358126\pi$
$440$	1.82771	0.0871324
$441$	0	0
$442$	0	0
$443$	6.44346	0.306138	0.153069	−	0.988216i	$-0.451084\pi$
$443$	6.44346	0.306138	0.153069	+	0.988216i	$0.451084\pi$
$444$	−56.2207	−2.66811
$445$	−3.52571	−0.167135
$446$	31.0943	1.47236
$447$	−66.4451	−3.14275
$448$	0	0
$449$	1.75306	0.0827322	0.0413661	−	0.999144i	$-0.486829\pi$
$449$	1.75306	0.0827322	0.0413661	+	0.999144i	$0.486829\pi$
$450$	−7.66473	−0.361319
$451$	−2.07592	−0.0977513
$452$	−30.3362	−1.42689
$453$	−55.1457	−2.59097
$454$	−8.23163	−0.386330
$455$	0	0
$456$	15.6477	0.732770
$457$	32.7100	1.53011	0.765054	−	0.643966i	$-0.222712\pi$
$457$	32.7100	1.53011	0.765054	+	0.643966i	$0.222712\pi$
$458$	−42.0044	−1.96274
$459$	−21.9462	−1.02436
$460$	7.87264	0.367064
$461$	7.66641	0.357060	0.178530	−	0.983934i	$-0.442866\pi$
$461$	7.66641	0.357060	0.178530	+	0.983934i	$0.442866\pi$
$462$	0	0
$463$	−14.4720	−0.672570	−0.336285	−	0.941760i	$-0.609171\pi$
$463$	−14.4720	−0.672570	−0.336285	+	0.941760i	$0.609171\pi$
$464$	−0.140636	−0.00652885
$465$	−30.8277	−1.42960
$466$	46.3124	2.14538
$467$	−3.37603	−0.156224	−0.0781120	−	0.996945i	$-0.524889\pi$
$467$	−3.37603	−0.156224	−0.0781120	+	0.996945i	$0.524889\pi$
$468$	0	0
$469$	0	0
$470$	46.2126	2.13163
$471$	0.238105	0.0109713
$472$	−22.6441	−1.04228
$473$	0.470751	0.0216452
$474$	−57.0463	−2.62022
$475$	−0.862889	−0.0395921
$476$	0	0
$477$	51.5283	2.35932
$478$	−47.5463	−2.17472
$479$	0.145225	0.00663549	0.00331775	−	0.999994i	$-0.498944\pi$
$479$	0.145225	0.00663549	0.00331775	+	0.999994i	$0.498944\pi$
$480$	−38.4034	−1.75287
$481$	0	0
$482$	29.0508	1.32323
$483$	0	0
$484$	−35.1177	−1.59626
$485$	−16.2911	−0.739739
$486$	46.1174	2.09193
$487$	−16.4215	−0.744127	−0.372064	−	0.928207i	$-0.621350\pi$
$487$	−16.4215	−0.744127	−0.372064	+	0.928207i	$0.621350\pi$
$488$	−9.59390	−0.434295
$489$	−31.7616	−1.43631
$490$	0	0
$491$	18.2077	0.821701	0.410850	−	0.911703i	$-0.365232\pi$
$491$	18.2077	0.821701	0.410850	+	0.911703i	$0.365232\pi$
$492$	−68.2778	−3.07820
$493$	−3.56814	−0.160701
$494$	0	0
$495$	4.53934	0.204028
$496$	0.322108	0.0144631
$497$	0	0
$498$	−80.8279	−3.62199
$499$	37.5720	1.68195	0.840976	−	0.541072i	$-0.181982\pi$
$499$	37.5720	1.68195	0.840976	+	0.541072i	$0.181982\pi$
$500$	−37.5313	−1.67845
$501$	−18.4296	−0.823374
$502$	31.4015	1.40152
$503$	−4.20535	−0.187507	−0.0937537	−	0.995595i	$-0.529887\pi$
$503$	−4.20535	−0.187507	−0.0937537	+	0.995595i	$0.529887\pi$
$504$	0	0
$505$	19.3741	0.862137
$506$	0.810801	0.0360445
$507$	0	0
$508$	−43.1523	−1.91457
$509$	−8.43746	−0.373984	−0.186992	−	0.982361i	$-0.559874\pi$
$509$	−8.43746	−0.373984	−0.186992	+	0.982361i	$0.559874\pi$
$510$	−27.1463	−1.20206
$511$	0	0
$512$	0.791350	0.0349731
$513$	22.0274	0.972535
$514$	−16.7585	−0.739185
$515$	12.8385	0.565734
$516$	15.4832	0.681609
$517$	2.93602	0.129126
$518$	0	0
$519$	53.5263	2.34955
$520$	0	0
$521$	25.8280	1.13155	0.565773	−	0.824561i	$-0.308578\pi$
$521$	25.8280	1.13155	0.565773	+	0.824561i	$0.308578\pi$
$522$	31.8119	1.39237
$523$	−0.756404	−0.0330752	−0.0165376	−	0.999863i	$-0.505264\pi$
$523$	−0.756404	−0.0330752	−0.0165376	+	0.999863i	$0.505264\pi$
$524$	43.1436	1.88473
$525$	0	0
$526$	−15.2599	−0.665364
$527$	8.17238	0.355995
$528$	−0.0679770	−0.00295832
$529$	−21.6765	−0.942458
$530$	36.1264	1.56923
$531$	−56.2396	−2.44059
$532$	0	0
$533$	0	0
$534$	−11.9429	−0.516819
$535$	25.6829	1.11037
$536$	35.1745	1.51931
$537$	−48.2064	−2.08026
$538$	37.0717	1.59827
$539$	0	0
$540$	84.6231	3.64160
$541$	22.4126	0.963593	0.481797	−	0.876283i	$-0.339984\pi$
$541$	22.4126	0.963593	0.481797	+	0.876283i	$0.339984\pi$
$542$	42.8130	1.83898
$543$	18.4286	0.790849
$544$	10.1807	0.436494
$545$	2.89398	0.123965
$546$	0	0
$547$	−11.8059	−0.504784	−0.252392	−	0.967625i	$-0.581217\pi$
$547$	−11.8059	−0.504784	−0.252392	+	0.967625i	$0.581217\pi$
$548$	1.61355	0.0689274
$549$	−23.8277	−1.01694
$550$	−0.341408	−0.0145577
$551$	3.58135	0.152571
$552$	10.1058	0.430129
$553$	0	0
$554$	68.6190	2.91534
$555$	−37.0977	−1.57471
$556$	4.57086	0.193848
$557$	8.76733	0.371484	0.185742	−	0.982599i	$-0.440531\pi$
$557$	8.76733	0.371484	0.185742	+	0.982599i	$0.440531\pi$
$558$	−72.8611	−3.08445
$559$	0	0
$560$	0	0
$561$	−1.72468	−0.0728161
$562$	−5.09662	−0.214988
$563$	36.7758	1.54992	0.774958	−	0.632013i	$-0.217771\pi$
$563$	36.7758	1.54992	0.774958	+	0.632013i	$0.217771\pi$
$564$	96.5668	4.06619
$565$	−20.0176	−0.842147
$566$	31.4744	1.32297
$567$	0	0
$568$	−3.76619	−0.158026
$569$	−35.7837	−1.50013	−0.750065	−	0.661364i	$-0.769978\pi$
$569$	−35.7837	−1.50013	−0.750065	+	0.661364i	$0.769978\pi$
$570$	27.2468	1.14124
$571$	14.9384	0.625152	0.312576	−	0.949893i	$-0.398808\pi$
$571$	14.9384	0.625152	0.312576	+	0.949893i	$0.398808\pi$
$572$	0	0
$573$	−84.4877	−3.52952
$574$	0	0
$575$	−0.557280	−0.0232402
$576$	−89.7975	−3.74156
$577$	16.8462	0.701316	0.350658	−	0.936504i	$-0.385958\pi$
$577$	16.8462	0.701316	0.350658	+	0.936504i	$0.385958\pi$
$578$	−31.6454	−1.31627
$579$	−0.674139	−0.0280163
$580$	13.7585	0.571292
$581$	0	0
$582$	−55.1838	−2.28744
$583$	2.29521	0.0950579
$584$	33.1285	1.37087
$585$	0	0
$586$	2.31295	0.0955470
$587$	−36.8833	−1.52234	−0.761168	−	0.648555i	$-0.775374\pi$
$587$	−36.8833	−1.52234	−0.761168	+	0.648555i	$0.775374\pi$
$588$	0	0
$589$	−8.20264	−0.337984
$590$	−39.4294	−1.62328
$591$	35.3762	1.45518
$592$	0.387622	0.0159312
$593$	−16.1535	−0.663345	−0.331673	−	0.943395i	$-0.607613\pi$
$593$	−16.1535	−0.663345	−0.331673	+	0.943395i	$0.607613\pi$
$594$	8.71531	0.357593
$595$	0	0
$596$	−67.9204	−2.78213
$597$	64.3319	2.63293
$598$	0	0
$599$	−2.48476	−0.101524	−0.0507622	−	0.998711i	$-0.516165\pi$
$599$	−2.48476	−0.101524	−0.0507622	+	0.998711i	$0.516165\pi$
$600$	−4.25528	−0.173721
$601$	9.55999	0.389960	0.194980	−	0.980807i	$-0.437536\pi$
$601$	9.55999	0.389960	0.194980	+	0.980807i	$0.437536\pi$
$602$	0	0
$603$	87.3605	3.55759
$604$	−56.3701	−2.29367
$605$	−23.1727	−0.942104
$606$	65.6273	2.66593
$607$	−19.4859	−0.790908	−0.395454	−	0.918486i	$-0.629413\pi$
$607$	−19.4859	−0.790908	−0.395454	+	0.918486i	$0.629413\pi$
$608$	−10.2184	−0.414411
$609$	0	0
$610$	−16.7055	−0.676387
$611$	0	0
$612$	−39.5794	−1.59990
$613$	14.7682	0.596481	0.298241	−	0.954491i	$-0.403600\pi$
$613$	14.7682	0.596481	0.298241	+	0.954491i	$0.403600\pi$
$614$	−54.9558	−2.21784
$615$	−45.0537	−1.81674
$616$	0	0
$617$	−30.9478	−1.24591	−0.622955	−	0.782257i	$-0.714068\pi$
$617$	−30.9478	−1.24591	−0.622955	+	0.782257i	$0.714068\pi$
$618$	43.4889	1.74938
$619$	−13.0825	−0.525828	−0.262914	−	0.964819i	$-0.584684\pi$
$619$	−13.0825	−0.525828	−0.262914	+	0.964819i	$0.584684\pi$
$620$	−31.5122	−1.26556
$621$	14.2260	0.570870
$622$	−20.5426	−0.823685
$623$	0	0
$624$	0	0
$625$	−22.3433	−0.893731
$626$	−34.8117	−1.39136
$627$	1.73106	0.0691321
$628$	0.243392	0.00971240
$629$	9.83456	0.392130
$630$	0	0
$631$	−35.3591	−1.40762	−0.703812	−	0.710387i	$-0.748520\pi$
$631$	−35.3591	−1.40762	−0.703812	+	0.710387i	$0.748520\pi$
$632$	−22.0979	−0.879007
$633$	26.5183	1.05401
$634$	−15.6178	−0.620264
$635$	−28.4744	−1.12997
$636$	75.4903	2.99338
$637$	0	0
$638$	1.41699	0.0560990
$639$	−9.35382	−0.370031
$640$	−38.5769	−1.52489
$641$	−21.2376	−0.838835	−0.419417	−	0.907794i	$-0.637766\pi$
$641$	−21.2376	−0.838835	−0.419417	+	0.907794i	$0.637766\pi$
$642$	86.9973	3.43351
$643$	25.4808	1.00486	0.502432	−	0.864617i	$-0.332439\pi$
$643$	25.4808	1.00486	0.502432	+	0.864617i	$0.332439\pi$
$644$	0	0
$645$	10.2167	0.402283
$646$	−7.22308	−0.284188
$647$	22.7856	0.895794	0.447897	−	0.894085i	$-0.352173\pi$
$647$	22.7856	0.895794	0.447897	+	0.894085i	$0.352173\pi$
$648$	50.6982	1.99161
$649$	−2.50506	−0.0983324
$650$	0	0
$651$	0	0
$652$	−32.4669	−1.27150
$653$	16.2786	0.637029	0.318515	−	0.947918i	$-0.396816\pi$
$653$	16.2786	0.637029	0.318515	+	0.947918i	$0.396816\pi$
$654$	9.80299	0.383327
$655$	28.4687	1.11236
$656$	0.470751	0.0183798
$657$	82.2790	3.21001
$658$	0	0
$659$	20.5596	0.800888	0.400444	−	0.916321i	$-0.368856\pi$
$659$	20.5596	0.800888	0.400444	+	0.916321i	$0.368856\pi$
$660$	6.65025	0.258861
$661$	−27.7726	−1.08023	−0.540115	−	0.841591i	$-0.681619\pi$
$661$	−27.7726	−1.08023	−0.540115	+	0.841591i	$0.681619\pi$
$662$	−31.5636	−1.22676
$663$	0	0
$664$	−31.3101	−1.21507
$665$	0	0
$666$	−87.6802	−3.39754
$667$	2.31295	0.0895577
$668$	−18.8388	−0.728895
$669$	42.8744	1.65762
$670$	61.2482	2.36622
$671$	−1.06135	−0.0409730
$672$	0	0
$673$	−5.20337	−0.200575	−0.100288	−	0.994958i	$-0.531976\pi$
$673$	−5.20337	−0.200575	−0.100288	+	0.994958i	$0.531976\pi$
$674$	61.7886	2.38001
$675$	−5.99021	−0.230563
$676$	0	0
$677$	44.8478	1.72364	0.861821	−	0.507212i	$-0.169324\pi$
$677$	44.8478	1.72364	0.861821	+	0.507212i	$0.169324\pi$
$678$	−67.8070	−2.60411
$679$	0	0
$680$	−10.5156	−0.403254
$681$	−11.3502	−0.434941
$682$	−3.24543	−0.124274
$683$	18.9738	0.726013	0.363006	−	0.931787i	$-0.381750\pi$
$683$	18.9738	0.726013	0.363006	+	0.931787i	$0.381750\pi$
$684$	39.7259	1.51896
$685$	1.06471	0.0406807
$686$	0	0
$687$	−57.9179	−2.20970
$688$	−0.106751	−0.00406985
$689$	0	0
$690$	17.5968	0.669898
$691$	37.4222	1.42361	0.711803	−	0.702379i	$-0.247879\pi$
$691$	37.4222	1.42361	0.711803	+	0.702379i	$0.247879\pi$
$692$	54.7148	2.07994
$693$	0	0
$694$	−44.1229	−1.67488
$695$	3.01612	0.114408
$696$	17.6612	0.669446
$697$	11.9437	0.452399
$698$	32.3467	1.22434
$699$	63.8580	2.41533
$700$	0	0
$701$	−42.5513	−1.60714	−0.803570	−	0.595210i	$-0.797069\pi$
$701$	−42.5513	−1.60714	−0.803570	+	0.595210i	$0.797069\pi$
$702$	0	0
$703$	−9.87096	−0.372290
$704$	−3.99982	−0.150749
$705$	63.7204	2.39985
$706$	38.7612	1.45880
$707$	0	0
$708$	−82.3924	−3.09650
$709$	−50.3321	−1.89026	−0.945131	−	0.326690i	$-0.894067\pi$
$709$	−50.3321	−1.89026	−0.945131	+	0.326690i	$0.894067\pi$
$710$	−6.55794	−0.246115
$711$	−54.8830	−2.05827
$712$	−4.62629	−0.173378
$713$	−5.29752	−0.198394
$714$	0	0
$715$	0	0
$716$	−49.2768	−1.84156
$717$	−65.5593	−2.44836
$718$	−52.0098	−1.94099
$719$	28.9232	1.07865	0.539326	−	0.842097i	$-0.318679\pi$
$719$	28.9232	1.07865	0.539326	+	0.842097i	$0.318679\pi$
$720$	−1.02938	−0.0383625
$721$	0	0
$722$	−36.1616	−1.34580
$723$	40.0567	1.48973
$724$	18.8378	0.700102
$725$	−0.973925	−0.0361707
$726$	−78.4945	−2.91320
$727$	−19.8593	−0.736539	−0.368269	−	0.929719i	$-0.620050\pi$
$727$	−19.8593	−0.736539	−0.368269	+	0.929719i	$0.620050\pi$
$728$	0	0
$729$	9.04209	0.334892
$730$	57.6856	2.13504
$731$	−2.70844	−0.100175
$732$	−34.9082	−1.29024
$733$	20.3501	0.751649	0.375824	−	0.926691i	$-0.377360\pi$
$733$	20.3501	0.751649	0.375824	+	0.926691i	$0.377360\pi$
$734$	37.8899	1.39854
$735$	0	0
$736$	−6.59935	−0.243255
$737$	3.89127	0.143337
$738$	−106.484	−3.91974
$739$	19.0011	0.698967	0.349483	−	0.936943i	$-0.386357\pi$
$739$	19.0011	0.698967	0.349483	+	0.936943i	$0.386357\pi$
$740$	−37.9214	−1.39402
$741$	0	0
$742$	0	0
$743$	8.15098	0.299030	0.149515	−	0.988759i	$-0.452229\pi$
$743$	8.15098	0.299030	0.149515	+	0.988759i	$0.452229\pi$
$744$	−40.4508	−1.48300
$745$	−44.8179	−1.64200
$746$	63.1662	2.31268
$747$	−77.7628	−2.84519
$748$	−1.76297	−0.0644607
$749$	0	0
$750$	−83.8893	−3.06320
$751$	36.7427	1.34076	0.670379	−	0.742018i	$-0.266131\pi$
$751$	36.7427	1.34076	0.670379	+	0.742018i	$0.266131\pi$
$752$	−0.665794	−0.0242790
$753$	43.2980	1.57787
$754$	0	0
$755$	−37.1963	−1.35371
$756$	0	0
$757$	38.3971	1.39557	0.697783	−	0.716310i	$-0.254170\pi$
$757$	38.3971	1.39557	0.697783	+	0.716310i	$0.254170\pi$
$758$	−21.1169	−0.767000
$759$	1.11797	0.0405799
$760$	10.5545	0.382852
$761$	12.4618	0.451739	0.225870	−	0.974158i	$-0.427478\pi$
$761$	12.4618	0.451739	0.225870	+	0.974158i	$0.427478\pi$
$762$	−96.4533	−3.49414
$763$	0	0
$764$	−86.3636	−3.12453
$765$	−26.1168	−0.944256
$766$	17.4748	0.631391
$767$	0	0
$768$	−48.9720	−1.76712
$769$	4.81390	0.173594	0.0867969	−	0.996226i	$-0.472337\pi$
$769$	4.81390	0.173594	0.0867969	+	0.996226i	$0.472337\pi$
$770$	0	0
$771$	−23.1075	−0.832196
$772$	−0.689108	−0.0248015
$773$	28.2744	1.01696	0.508480	−	0.861074i	$-0.330208\pi$
$773$	28.2744	1.01696	0.508480	+	0.861074i	$0.330208\pi$
$774$	24.1471	0.867951
$775$	2.23065	0.0801274
$776$	−21.3764	−0.767369
$777$	0	0
$778$	14.9367	0.535505
$779$	−11.9879	−0.429511
$780$	0	0
$781$	−0.416645	−0.0149087
$782$	−4.66489	−0.166816
$783$	24.8619	0.888492
$784$	0	0
$785$	0.160604	0.00573222
$786$	96.4338	3.43968
$787$	−49.6582	−1.77012	−0.885062	−	0.465473i	$-0.845884\pi$
$787$	−49.6582	−1.77012	−0.885062	+	0.465473i	$0.845884\pi$
$788$	36.1616	1.28820
$789$	−21.0412	−0.749086
$790$	−38.4783	−1.36900
$791$	0	0
$792$	5.95633	0.211649
$793$	0	0
$794$	66.1014	2.34585
$795$	49.8129	1.76668
$796$	65.7603	2.33081
$797$	5.37263	0.190308	0.0951542	−	0.995463i	$-0.469666\pi$
$797$	5.37263	0.190308	0.0951542	+	0.995463i	$0.469666\pi$
$798$	0	0
$799$	−16.8922	−0.597604
$800$	2.77882	0.0982462
$801$	−11.4900	−0.405979
$802$	61.1033	2.15763
$803$	3.66493	0.129333
$804$	127.985	4.51369
$805$	0	0
$806$	0	0
$807$	51.1164	1.79938
$808$	25.4219	0.894339
$809$	41.2369	1.44981	0.724905	−	0.688848i	$-0.241883\pi$
$809$	41.2369	1.44981	0.724905	+	0.688848i	$0.241883\pi$
$810$	88.2790	3.10181
$811$	19.4366	0.682512	0.341256	−	0.939970i	$-0.389148\pi$
$811$	19.4366	0.682512	0.341256	+	0.939970i	$0.389148\pi$
$812$	0	0
$813$	59.0328	2.07037
$814$	−3.90551	−0.136888
$815$	−21.4235	−0.750434
$816$	0.391101	0.0136913
$817$	2.71846	0.0951070
$818$	−79.4486	−2.77785
$819$	0	0
$820$	−46.0540	−1.60828
$821$	−20.1906	−0.704657	−0.352329	−	0.935876i	$-0.614610\pi$
$821$	−20.1906	−0.704657	−0.352329	+	0.935876i	$0.614610\pi$
$822$	3.60658	0.125794
$823$	−42.8097	−1.49225	−0.746127	−	0.665804i	$-0.768089\pi$
$823$	−42.8097	−1.49225	−0.746127	+	0.665804i	$0.768089\pi$
$824$	16.8462	0.586865
$825$	−0.470751	−0.0163895
$826$	0	0
$827$	33.5376	1.16622	0.583109	−	0.812394i	$-0.301836\pi$
$827$	33.5376	1.16622	0.583109	+	0.812394i	$0.301836\pi$
$828$	25.6562	0.891615
$829$	−39.7898	−1.38196	−0.690978	−	0.722876i	$-0.742820\pi$
$829$	−39.7898	−1.38196	−0.690978	+	0.722876i	$0.742820\pi$
$830$	−54.5192	−1.89239
$831$	94.6155	3.28218
$832$	0	0
$833$	0	0
$834$	10.2167	0.353776
$835$	−12.4310	−0.430191
$836$	1.76950	0.0611994
$837$	−56.9431	−1.96824
$838$	−9.57898	−0.330901
$839$	−36.7098	−1.26736	−0.633682	−	0.773594i	$-0.718457\pi$
$839$	−36.7098	−1.26736	−0.633682	+	0.773594i	$0.718457\pi$
$840$	0	0
$841$	−24.9578	−0.860614
$842$	−47.8728	−1.64981
$843$	−7.02749	−0.242040
$844$	27.1071	0.933065
$845$	0	0
$846$	150.603	5.17783
$847$	0	0
$848$	−0.520479	−0.0178733
$849$	43.3985	1.48943
$850$	1.96427	0.0673739
$851$	−6.37497	−0.218531
$852$	−13.7036	−0.469477
$853$	11.7156	0.401136	0.200568	−	0.979680i	$-0.435721\pi$
$853$	11.7156	0.401136	0.200568	+	0.979680i	$0.435721\pi$
$854$	0	0
$855$	26.2135	0.896483
$856$	33.6999	1.15184
$857$	−27.6905	−0.945889	−0.472945	−	0.881092i	$-0.656809\pi$
$857$	−27.6905	−0.945889	−0.472945	+	0.881092i	$0.656809\pi$
$858$	0	0
$859$	−38.5638	−1.31578	−0.657890	−	0.753114i	$-0.728551\pi$
$859$	−38.5638	−1.31578	−0.657890	+	0.753114i	$0.728551\pi$
$860$	10.4436	0.356122
$861$	0	0
$862$	38.6002	1.31473
$863$	−17.8532	−0.607730	−0.303865	−	0.952715i	$-0.598277\pi$
$863$	−17.8532	−0.607730	−0.303865	+	0.952715i	$0.598277\pi$
$864$	−70.9366	−2.41331
$865$	36.1040	1.22757
$866$	−7.81957	−0.265720
$867$	−43.6343	−1.48190
$868$	0	0
$869$	−2.44464	−0.0829286
$870$	30.7528	1.04262
$871$	0	0
$872$	3.79736	0.128595
$873$	−53.0911	−1.79686
$874$	4.68216	0.158376
$875$	0	0
$876$	120.541	4.07270
$877$	8.92034	0.301219	0.150609	−	0.988593i	$-0.451876\pi$
$877$	8.92034	0.301219	0.150609	+	0.988593i	$0.451876\pi$
$878$	41.2753	1.39297
$879$	3.18922	0.107570
$880$	−0.0458512	−0.00154564
$881$	−54.6144	−1.84001	−0.920003	−	0.391911i	$-0.871814\pi$
$881$	−54.6144	−1.84001	−0.920003	+	0.391911i	$0.871814\pi$
$882$	0	0
$883$	7.51632	0.252944	0.126472	−	0.991970i	$-0.459635\pi$
$883$	7.51632	0.252944	0.126472	+	0.991970i	$0.459635\pi$
$884$	0	0
$885$	−54.3674	−1.82754
$886$	14.7221	0.494598
$887$	45.0783	1.51358	0.756790	−	0.653658i	$-0.226766\pi$
$887$	45.0783	1.51358	0.756790	+	0.653658i	$0.226766\pi$
$888$	−48.6780	−1.63353
$889$	0	0
$890$	−8.05560	−0.270024
$891$	5.60862	0.187896
$892$	43.8264	1.46742
$893$	16.9547	0.567369
$894$	−151.815	−5.07744
$895$	−32.5157	−1.08688
$896$	0	0
$897$	0	0
$898$	4.00542	0.133663
$899$	−9.25815	−0.308776
$900$	−10.8032	−0.360107
$901$	−13.2054	−0.439934
$902$	−4.74309	−0.157928
$903$	0	0
$904$	−26.2662	−0.873602
$905$	12.4303	0.413197
$906$	−125.998	−4.18599
$907$	6.36590	0.211376	0.105688	−	0.994399i	$-0.466295\pi$
$907$	6.36590	0.211376	0.105688	+	0.994399i	$0.466295\pi$
$908$	−11.6022	−0.385033
$909$	63.1385	2.09417
$910$	0	0
$911$	20.9161	0.692982	0.346491	−	0.938053i	$-0.387373\pi$
$911$	20.9161	0.692982	0.346491	+	0.938053i	$0.387373\pi$
$912$	−0.392549	−0.0129986
$913$	−3.46376	−0.114634
$914$	74.7362	2.47205
$915$	−23.0345	−0.761496
$916$	−59.2038	−1.95615
$917$	0	0
$918$	−50.1430	−1.65496
$919$	4.88652	0.161191	0.0805957	−	0.996747i	$-0.474318\pi$
$919$	4.88652	0.161191	0.0805957	+	0.996747i	$0.474318\pi$
$920$	6.81643	0.224731
$921$	−75.7760	−2.49690
$922$	17.5163	0.576869
$923$	0	0
$924$	0	0
$925$	2.68434	0.0882606
$926$	−33.0658	−1.08661
$927$	41.8397	1.37420
$928$	−11.5333	−0.378599
$929$	−51.5350	−1.69081	−0.845404	−	0.534128i	$-0.820640\pi$
$929$	−51.5350	−1.69081	−0.845404	+	0.534128i	$0.820640\pi$
$930$	−70.4355	−2.30967
$931$	0	0
$932$	65.2759	2.13818
$933$	−28.3253	−0.927328
$934$	−7.71359	−0.252397
$935$	−1.16331	−0.0380444
$936$	0	0
$937$	20.3565	0.665016	0.332508	−	0.943100i	$-0.392105\pi$
$937$	20.3565	0.665016	0.332508	+	0.943100i	$0.392105\pi$
$938$	0	0
$939$	−48.0002	−1.56643
$940$	65.1352	2.12448
$941$	−34.2635	−1.11696	−0.558479	−	0.829519i	$-0.688615\pi$
$941$	−34.2635	−1.11696	−0.558479	+	0.829519i	$0.688615\pi$
$942$	0.544026	0.0177253
$943$	−7.74215	−0.252119
$944$	0.568067	0.0184890
$945$	0	0
$946$	1.07558	0.0349701
$947$	−55.3155	−1.79751	−0.898756	−	0.438448i	$-0.855528\pi$
$947$	−55.3155	−1.79751	−0.898756	+	0.438448i	$0.855528\pi$
$948$	−80.4049	−2.61143
$949$	0	0
$950$	−1.97154	−0.0639652
$951$	−21.5347	−0.698311
$952$	0	0
$953$	14.8378	0.480644	0.240322	−	0.970693i	$-0.422747\pi$
$953$	14.8378	0.480644	0.240322	+	0.970693i	$0.422747\pi$
$954$	117.733	3.81173
$955$	−56.9878	−1.84408
$956$	−67.0149	−2.16742
$957$	1.95382	0.0631579
$958$	0.331812	0.0107203
$959$	0	0
$960$	−86.8081	−2.80172
$961$	−9.79539	−0.315980
$962$	0	0
$963$	83.6981	2.69714
$964$	40.9461	1.31879
$965$	−0.454714	−0.0146378
$966$	0	0
$967$	−3.09473	−0.0995199	−0.0497600	−	0.998761i	$-0.515846\pi$
$967$	−3.09473	−0.0995199	−0.0497600	+	0.998761i	$0.515846\pi$
$968$	−30.4062	−0.977293
$969$	−9.95957	−0.319948
$970$	−37.2220	−1.19513
$971$	54.9013	1.76187	0.880933	−	0.473241i	$-0.156916\pi$
$971$	54.9013	1.76187	0.880933	+	0.473241i	$0.156916\pi$
$972$	65.0010	2.08491
$973$	0	0
$974$	−37.5200	−1.20222
$975$	0	0
$976$	0.240680	0.00770397
$977$	43.2085	1.38236	0.691181	−	0.722682i	$-0.257091\pi$
$977$	43.2085	1.38236	0.691181	+	0.722682i	$0.257091\pi$
$978$	−72.5694	−2.32051
$979$	−0.511795	−0.0163570
$980$	0	0
$981$	9.43124	0.301116
$982$	41.6011	1.32755
$983$	−24.8836	−0.793663	−0.396831	−	0.917892i	$-0.629890\pi$
$983$	−24.8836	−0.793663	−0.396831	+	0.917892i	$0.629890\pi$
$984$	−59.1175	−1.88460
$985$	23.8616	0.760293
$986$	−8.15254	−0.259630
$987$	0	0
$988$	0	0
$989$	1.75567	0.0558270
$990$	10.3716	0.329629
$991$	4.78328	0.151946	0.0759730	−	0.997110i	$-0.475794\pi$
$991$	4.78328	0.151946	0.0759730	+	0.997110i	$0.475794\pi$
$992$	26.4155	0.838694
$993$	−43.5216	−1.38112
$994$	0	0
$995$	43.3925	1.37564
$996$	−113.924	−3.60983
$997$	3.44074	0.108969	0.0544847	−	0.998515i	$-0.482648\pi$
$997$	3.44074	0.108969	0.0544847	+	0.998515i	$0.482648\pi$
$998$	85.8449	2.71737
$999$	−68.5247	−2.16802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.ck.1.8		8
7.2	even	3		1183.2.e.i.508.1		16
7.4	even	3		1183.2.e.i.170.1		16
7.6	odd	2		8281.2.a.cj.1.8		8
13.5	odd	4		637.2.c.f.246.1		8
13.8	odd	4		637.2.c.f.246.8		8
13.12	even	2	inner	8281.2.a.ck.1.1		8
91.5	even	12		637.2.r.f.116.1		16
91.18	odd	12		91.2.r.a.51.8	yes	16
91.25	even	6		1183.2.e.i.170.8		16
91.31	even	12		637.2.r.f.324.8		16
91.34	even	4		637.2.c.e.246.8		8
91.44	odd	12		91.2.r.a.25.1	✓	16
91.47	even	12		637.2.r.f.116.8		16
91.51	even	6		1183.2.e.i.508.8		16
91.60	odd	12		91.2.r.a.51.1	yes	16
91.73	even	12		637.2.r.f.324.1		16
91.83	even	4		637.2.c.e.246.1		8
91.86	odd	12		91.2.r.a.25.8	yes	16
91.90	odd	2		8281.2.a.cj.1.1		8
273.44	even	12		819.2.dl.e.298.8		16
273.86	even	12		819.2.dl.e.298.1		16
273.200	even	12		819.2.dl.e.415.1		16
273.242	even	12		819.2.dl.e.415.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.r.a.25.1	✓	16	91.44	odd	12
91.2.r.a.25.8	yes	16	91.86	odd	12
91.2.r.a.51.1	yes	16	91.60	odd	12
91.2.r.a.51.8	yes	16	91.18	odd	12
637.2.c.e.246.1		8	91.83	even	4
637.2.c.e.246.8		8	91.34	even	4
637.2.c.f.246.1		8	13.5	odd	4
637.2.c.f.246.8		8	13.8	odd	4
637.2.r.f.116.1		16	91.5	even	12
637.2.r.f.116.8		16	91.47	even	12
637.2.r.f.324.1		16	91.73	even	12
637.2.r.f.324.8		16	91.31	even	12
819.2.dl.e.298.1		16	273.86	even	12
819.2.dl.e.298.8		16	273.44	even	12
819.2.dl.e.415.1		16	273.200	even	12
819.2.dl.e.415.8		16	273.242	even	12
1183.2.e.i.170.1		16	7.4	even	3
1183.2.e.i.170.8		16	91.25	even	6
1183.2.e.i.508.1		16	7.2	even	3
1183.2.e.i.508.8		16	91.51	even	6
8281.2.a.cj.1.1		8	91.90	odd	2
8281.2.a.cj.1.8		8	7.6	odd	2
8281.2.a.ck.1.1		8	13.12	even	2	inner
8281.2.a.ck.1.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.28481 q^{2} +3.15042 q^{3} +3.22037 q^{4} +2.12499 q^{5} +7.19813 q^{6} +2.78832 q^{8} +6.92516 q^{9} +O(q^{10})\)	\(q+2.28481 q^{2} +3.15042 q^{3} +3.22037 q^{4} +2.12499 q^{5} +7.19813 q^{6} +2.78832 q^{8} +6.92516 q^{9} +4.85521 q^{10} +0.308465 q^{11} +10.1455 q^{12} +6.69462 q^{15} -0.0699499 q^{16} -1.77474 q^{17} +15.8227 q^{18} +1.78131 q^{19} +6.84326 q^{20} +0.704786 q^{22} +1.15042 q^{23} +8.78439 q^{24} -0.484414 q^{25} +12.3659 q^{27} +2.01052 q^{29} +15.2960 q^{30} -4.60485 q^{31} -5.73646 q^{32} +0.971796 q^{33} -4.05494 q^{34} +22.3016 q^{36} -5.54142 q^{37} +4.06995 q^{38} +5.92516 q^{40} -6.72984 q^{41} +1.52611 q^{43} +0.993373 q^{44} +14.7159 q^{45} +2.62850 q^{46} +9.51816 q^{47} -0.220372 q^{48} -1.10680 q^{50} -5.59116 q^{51} +7.44074 q^{53} +28.2538 q^{54} +0.655486 q^{55} +5.61186 q^{57} +4.59367 q^{58} -8.12106 q^{59} +21.5592 q^{60} -3.44074 q^{61} -10.5212 q^{62} -12.9669 q^{64} +2.22037 q^{66} +12.6149 q^{67} -5.71531 q^{68} +3.62431 q^{69} -1.35070 q^{71} +19.3096 q^{72} +11.8812 q^{73} -12.6611 q^{74} -1.52611 q^{75} +5.73646 q^{76} -7.92516 q^{79} -0.148643 q^{80} +18.1823 q^{81} -15.3764 q^{82} -11.2290 q^{83} -3.77130 q^{85} +3.48687 q^{86} +6.33399 q^{87} +0.860100 q^{88} -1.65917 q^{89} +33.6231 q^{90} +3.70479 q^{92} -14.5072 q^{93} +21.7472 q^{94} +3.78526 q^{95} -18.0723 q^{96} -7.66641 q^{97} +2.13617 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9	\( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9} - 6 q^{10} + 18 q^{12} - 2 q^{16} + 8 q^{17} - 18 q^{22} - 12 q^{23} + 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} + 34 q^{38} + 4 q^{40} - 8 q^{43} + 18 q^{48} + 16 q^{51} + 20 q^{53} + 12 q^{55} + 12 q^{61} - 22 q^{62} - 44 q^{64} - 2 q^{66} + 2 q^{68} + 28 q^{69} - 42 q^{74} + 8 q^{75} - 20 q^{79} + 24 q^{81} - 16 q^{82} + 68 q^{87} + 4 q^{88} + 108 q^{90} + 6 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100}) \) 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9 - 6 * q^10 + 18 * q^12 - 2 * q^16 + 8 * q^17 - 18 * q^22 - 12 * q^23 + 16 * q^27 - 8 * q^29 + 38 * q^30 + 28 * q^36 + 34 * q^38 + 4 * q^40 - 8 * q^43 + 18 * q^48 + 16 * q^51 + 20 * q^53 + 12 * q^55 + 12 * q^61 - 22 * q^62 - 44 * q^64 - 2 * q^66 + 2 * q^68 + 28 * q^69 - 42 * q^74 + 8 * q^75 - 20 * q^79 + 24 * q^81 - 16 * q^82 + 68 * q^87 + 4 * q^88 + 108 * q^90 + 6 * q^92 + 26 * q^94 - 16 * q^95

Introduction

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