LMFDB - Embedded newform 8281.2.a.bn.1.4

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:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{23})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} + 121 $ x^4 - 24*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.68406$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} -1.00000 q^{4} +2.68406 q^{5} -1.41421 q^{6} +3.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.41421 q^{3} -1.00000 q^{4} +2.68406 q^{5} -1.41421 q^{6} +3.00000 q^{8} -1.00000 q^{9} -2.68406 q^{10} -5.79583 q^{11} -1.41421 q^{12} +3.79583 q^{15} -1.00000 q^{16} +5.51249 q^{17} +1.00000 q^{18} +2.82843 q^{19} -2.68406 q^{20} +5.79583 q^{22} +1.79583 q^{23} +4.24264 q^{24} +2.20417 q^{25} -5.65685 q^{27} -8.79583 q^{29} -3.79583 q^{30} +1.41421 q^{31} -5.00000 q^{32} -8.19654 q^{33} -5.51249 q^{34} +1.00000 q^{36} +6.79583 q^{37} -2.82843 q^{38} +8.05217 q^{40} -9.75513 q^{41} -1.79583 q^{43} +5.79583 q^{44} -2.68406 q^{45} -1.79583 q^{46} -2.82843 q^{47} -1.41421 q^{48} -2.20417 q^{50} +7.79583 q^{51} +6.59166 q^{53} +5.65685 q^{54} -15.5563 q^{55} +4.00000 q^{57} +8.79583 q^{58} -1.12548 q^{59} -3.79583 q^{60} +1.55858 q^{61} -1.41421 q^{62} +7.00000 q^{64} +8.19654 q^{66} -5.79583 q^{67} -5.51249 q^{68} +2.53969 q^{69} -6.00000 q^{71} -3.00000 q^{72} +5.80122 q^{73} -6.79583 q^{74} +3.11716 q^{75} -2.82843 q^{76} -11.7958 q^{79} -2.68406 q^{80} -5.00000 q^{81} +9.75513 q^{82} +9.89949 q^{83} +14.7958 q^{85} +1.79583 q^{86} -12.4392 q^{87} -17.3875 q^{88} -12.1504 q^{89} +2.68406 q^{90} -1.79583 q^{92} +2.00000 q^{93} +2.82843 q^{94} +7.59166 q^{95} -7.07107 q^{96} +4.24264 q^{97} +5.79583 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{4} + 12 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^4 + 12 * q^8 - 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{4} + 12 q^{8} - 4 q^{9} - 4 q^{11} - 4 q^{15} - 4 q^{16} + 4 q^{18} + 4 q^{22} - 12 q^{23} + 28 q^{25} - 16 q^{29} + 4 q^{30} - 20 q^{32} + 4 q^{36} + 8 q^{37} + 12 q^{43} + 4 q^{44} + 12 q^{46} - 28 q^{50} + 12 q^{51} - 12 q^{53} + 16 q^{57} + 16 q^{58} + 4 q^{60} + 28 q^{64} - 4 q^{67} - 24 q^{71} - 12 q^{72} - 8 q^{74} - 28 q^{79} - 20 q^{81} + 40 q^{85} - 12 q^{86} - 12 q^{88} + 12 q^{92} + 8 q^{93} - 8 q^{95} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^4 + 12 * q^8 - 4 * q^9 - 4 * q^11 - 4 * q^15 - 4 * q^16 + 4 * q^18 + 4 * q^22 - 12 * q^23 + 28 * q^25 - 16 * q^29 + 4 * q^30 - 20 * q^32 + 4 * q^36 + 8 * q^37 + 12 * q^43 + 4 * q^44 + 12 * q^46 - 28 * q^50 + 12 * q^51 - 12 * q^53 + 16 * q^57 + 16 * q^58 + 4 * q^60 + 28 * q^64 - 4 * q^67 - 24 * q^71 - 12 * q^72 - 8 * q^74 - 28 * q^79 - 20 * q^81 + 40 * q^85 - 12 * q^86 - 12 * q^88 + 12 * q^92 + 8 * q^93 - 8 * q^95 + 4 * q^99

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	−1.00000	−0.500000
$5$	2.68406	1.20035	0.600174	−	0.799870i	$-0.295098\pi$
$5$	2.68406	1.20035	0.600174	+	0.799870i	$0.295098\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	3.00000	1.06066
$9$	−1.00000	−0.333333
$10$	−2.68406	−0.848774
$11$	−5.79583	−1.74751	−0.873754	−	0.486367i	$-0.838322\pi$
$11$	−5.79583	−1.74751	−0.873754	+	0.486367i	$0.838322\pi$
$12$	−1.41421	−0.408248
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.79583	0.980079
$16$	−1.00000	−0.250000
$17$	5.51249	1.33697	0.668487	−	0.743724i	$-0.266942\pi$
$17$	5.51249	1.33697	0.668487	+	0.743724i	$0.266942\pi$
$18$	1.00000	0.235702
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−2.68406	−0.600174
$21$	0	0
$22$	5.79583	1.23568
$23$	1.79583	0.374457	0.187228	−	0.982316i	$-0.440050\pi$
$23$	1.79583	0.374457	0.187228	+	0.982316i	$0.440050\pi$
$24$	4.24264	0.866025
$25$	2.20417	0.440834
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−8.79583	−1.63334	−0.816672	−	0.577101i	$-0.804184\pi$
$29$	−8.79583	−1.63334	−0.816672	+	0.577101i	$0.804184\pi$
$30$	−3.79583	−0.693021
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	−5.00000	−0.883883
$33$	−8.19654	−1.42684
$34$	−5.51249	−0.945383
$35$	0	0
$36$	1.00000	0.166667
$37$	6.79583	1.11723	0.558614	−	0.829428i	$-0.311333\pi$
$37$	6.79583	1.11723	0.558614	+	0.829428i	$0.311333\pi$
$38$	−2.82843	−0.458831
$39$	0	0
$40$	8.05217	1.27316
$41$	−9.75513	−1.52349	−0.761747	−	0.647874i	$-0.775658\pi$
$41$	−9.75513	−1.52349	−0.761747	+	0.647874i	$0.775658\pi$
$42$	0	0
$43$	−1.79583	−0.273862	−0.136931	−	0.990581i	$-0.543724\pi$
$43$	−1.79583	−0.273862	−0.136931	+	0.990581i	$0.543724\pi$
$44$	5.79583	0.873754
$45$	−2.68406	−0.400116
$46$	−1.79583	−0.264781
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	−1.41421	−0.204124
$49$	0	0
$50$	−2.20417	−0.311716
$51$	7.79583	1.09163
$52$	0	0
$53$	6.59166	0.905435	0.452717	−	0.891654i	$-0.350455\pi$
$53$	6.59166	0.905435	0.452717	+	0.891654i	$0.350455\pi$
$54$	5.65685	0.769800
$55$	−15.5563	−2.09762
$56$	0	0
$57$	4.00000	0.529813
$58$	8.79583	1.15495
$59$	−1.12548	−0.146524	−0.0732622	−	0.997313i	$-0.523341\pi$
$59$	−1.12548	−0.146524	−0.0732622	+	0.997313i	$0.523341\pi$
$60$	−3.79583	−0.490040
$61$	1.55858	0.199556	0.0997780	−	0.995010i	$-0.468187\pi$
$61$	1.55858	0.199556	0.0997780	+	0.995010i	$0.468187\pi$
$62$	−1.41421	−0.179605
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	8.19654	1.00892
$67$	−5.79583	−0.708074	−0.354037	−	0.935232i	$-0.615191\pi$
$67$	−5.79583	−0.708074	−0.354037	+	0.935232i	$0.615191\pi$
$68$	−5.51249	−0.668487
$69$	2.53969	0.305743
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−3.00000	−0.353553
$73$	5.80122	0.678982	0.339491	−	0.940609i	$-0.389745\pi$
$73$	5.80122	0.678982	0.339491	+	0.940609i	$0.389745\pi$
$74$	−6.79583	−0.789999
$75$	3.11716	0.359939
$76$	−2.82843	−0.324443
$77$	0	0
$78$	0	0
$79$	−11.7958	−1.32713	−0.663567	−	0.748117i	$-0.730958\pi$
$79$	−11.7958	−1.32713	−0.663567	+	0.748117i	$0.730958\pi$
$80$	−2.68406	−0.300087
$81$	−5.00000	−0.555556
$82$	9.75513	1.07727
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	14.7958	1.60483
$86$	1.79583	0.193649
$87$	−12.4392	−1.33362
$88$	−17.3875	−1.85351
$89$	−12.1504	−1.28794	−0.643972	−	0.765049i	$-0.722715\pi$
$89$	−12.1504	−1.28794	−0.643972	+	0.765049i	$0.722715\pi$
$90$	2.68406	0.282925
$91$	0	0
$92$	−1.79583	−0.187228
$93$	2.00000	0.207390
$94$	2.82843	0.291730
$95$	7.59166	0.778888
$96$	−7.07107	−0.721688
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	0	0
$99$	5.79583	0.582503
$100$	−2.20417	−0.220417
$101$	−2.97280	−0.295804	−0.147902	−	0.989002i	$-0.547252\pi$
$101$	−2.97280	−0.295804	−0.147902	+	0.989002i	$0.547252\pi$
$102$	−7.79583	−0.771902
$103$	−8.19654	−0.807629	−0.403815	−	0.914841i	$-0.632316\pi$
$103$	−8.19654	−0.807629	−0.403815	+	0.914841i	$0.632316\pi$
$104$	0	0
$105$	0	0
$106$	−6.59166	−0.640239
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	5.65685	0.544331
$109$	17.5917	1.68498	0.842488	−	0.538715i	$-0.181090\pi$
$109$	17.5917	1.68498	0.842488	+	0.538715i	$0.181090\pi$
$110$	15.5563	1.48324
$111$	9.61076	0.912213
$112$	0	0
$113$	−16.5917	−1.56081	−0.780406	−	0.625273i	$-0.784988\pi$
$113$	−16.5917	−1.56081	−0.780406	+	0.625273i	$0.784988\pi$
$114$	−4.00000	−0.374634
$115$	4.82012	0.449478
$116$	8.79583	0.816672
$117$	0	0
$118$	1.12548	0.103608
$119$	0	0
$120$	11.3875	1.03953
$121$	22.5917	2.05379
$122$	−1.55858	−0.141107
$123$	−13.7958	−1.24393
$124$	−1.41421	−0.127000
$125$	−7.50417	−0.671194
$126$	0	0
$127$	7.59166	0.673651	0.336826	−	0.941567i	$-0.390647\pi$
$127$	7.59166	0.673651	0.336826	+	0.941567i	$0.390647\pi$
$128$	3.00000	0.265165
$129$	−2.53969	−0.223607
$130$	0	0
$131$	−8.19654	−0.716135	−0.358068	−	0.933696i	$-0.616564\pi$
$131$	−8.19654	−0.716135	−0.358068	+	0.933696i	$0.616564\pi$
$132$	8.19654	0.713418
$133$	0	0
$134$	5.79583	0.500684
$135$	−15.1833	−1.30677
$136$	16.5375	1.41808
$137$	9.20417	0.786365	0.393183	−	0.919460i	$-0.371374\pi$
$137$	9.20417	0.786365	0.393183	+	0.919460i	$0.371374\pi$
$138$	−2.53969	−0.216193
$139$	−15.2676	−1.29498	−0.647491	−	0.762073i	$-0.724182\pi$
$139$	−15.2676	−1.29498	−0.647491	+	0.762073i	$0.724182\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	6.00000	0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	−23.6085	−1.96058
$146$	−5.80122	−0.480113
$147$	0	0
$148$	−6.79583	−0.558614
$149$	−4.59166	−0.376164	−0.188082	−	0.982153i	$-0.560227\pi$
$149$	−4.59166	−0.376164	−0.188082	+	0.982153i	$0.560227\pi$
$150$	−3.11716	−0.254515
$151$	17.5917	1.43159	0.715795	−	0.698311i	$-0.246065\pi$
$151$	17.5917	1.43159	0.715795	+	0.698311i	$0.246065\pi$
$152$	8.48528	0.688247
$153$	−5.51249	−0.445658
$154$	0	0
$155$	3.79583	0.304889
$156$	0	0
$157$	−6.92670	−0.552811	−0.276405	−	0.961041i	$-0.589143\pi$
$157$	−6.92670	−0.552811	−0.276405	+	0.961041i	$0.589143\pi$
$158$	11.7958	0.938426
$159$	9.32202	0.739284
$160$	−13.4203	−1.06097
$161$	0	0
$162$	5.00000	0.392837
$163$	13.3875	1.04859	0.524295	−	0.851537i	$-0.324329\pi$
$163$	13.3875	1.04859	0.524295	+	0.851537i	$0.324329\pi$
$164$	9.75513	0.761747
$165$	−22.0000	−1.71270
$166$	−9.89949	−0.768350
$167$	−15.2676	−1.18144	−0.590722	−	0.806875i	$-0.701157\pi$
$167$	−15.2676	−1.18144	−0.590722	+	0.806875i	$0.701157\pi$
$168$	0	0
$169$	0	0
$170$	−14.7958	−1.13479
$171$	−2.82843	−0.216295
$172$	1.79583	0.136931
$173$	9.32202	0.708740	0.354370	−	0.935105i	$-0.384695\pi$
$173$	9.32202	0.708740	0.354370	+	0.935105i	$0.384695\pi$
$174$	12.4392	0.943012
$175$	0	0
$176$	5.79583	0.436877
$177$	−1.59166	−0.119637
$178$	12.1504	0.910714
$179$	−0.408337	−0.0305205	−0.0152603	−	0.999884i	$-0.504858\pi$
$179$	−0.408337	−0.0305205	−0.0152603	+	0.999884i	$0.504858\pi$
$180$	2.68406	0.200058
$181$	−16.5375	−1.22922	−0.614610	−	0.788831i	$-0.710686\pi$
$181$	−16.5375	−1.22922	−0.614610	+	0.788831i	$0.710686\pi$
$182$	0	0
$183$	2.20417	0.162937
$184$	5.38749	0.397171
$185$	18.2404	1.34106
$186$	−2.00000	−0.146647
$187$	−31.9494	−2.33637
$188$	2.82843	0.206284
$189$	0	0
$190$	−7.59166	−0.550757
$191$	−25.7958	−1.86652	−0.933260	−	0.359200i	$-0.883049\pi$
$191$	−25.7958	−1.86652	−0.933260	+	0.359200i	$0.883049\pi$
$192$	9.89949	0.714435
$193$	−3.40834	−0.245337	−0.122669	−	0.992448i	$-0.539145\pi$
$193$	−3.40834	−0.245337	−0.122669	+	0.992448i	$0.539145\pi$
$194$	−4.24264	−0.304604
$195$	0	0
$196$	0	0
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	−5.79583	−0.411892
$199$	−22.0499	−1.56308	−0.781539	−	0.623856i	$-0.785565\pi$
$199$	−22.0499	−1.56308	−0.781539	+	0.623856i	$0.785565\pi$
$200$	6.61251	0.467575
$201$	−8.19654	−0.578140
$202$	2.97280	0.209165
$203$	0	0
$204$	−7.79583	−0.545817
$205$	−26.1833	−1.82872
$206$	8.19654	0.571080
$207$	−1.79583	−0.124819
$208$	0	0
$209$	−16.3931	−1.13393
$210$	0	0
$211$	−1.79583	−0.123630	−0.0618151	−	0.998088i	$-0.519689\pi$
$211$	−1.79583	−0.123630	−0.0618151	+	0.998088i	$0.519689\pi$
$212$	−6.59166	−0.452717
$213$	−8.48528	−0.581402
$214$	6.00000	0.410152
$215$	−4.82012	−0.328729
$216$	−16.9706	−1.15470
$217$	0	0
$218$	−17.5917	−1.19146
$219$	8.20417	0.554386
$220$	15.5563	1.04881
$221$	0	0
$222$	−9.61076	−0.645032
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	0	0
$225$	−2.20417	−0.146945
$226$	16.5917	1.10366
$227$	−20.9245	−1.38881	−0.694403	−	0.719587i	$-0.744331\pi$
$227$	−20.9245	−1.38881	−0.694403	+	0.719587i	$0.744331\pi$
$228$	−4.00000	−0.264906
$229$	−12.7279	−0.841085	−0.420542	−	0.907273i	$-0.638160\pi$
$229$	−12.7279	−0.841085	−0.420542	+	0.907273i	$0.638160\pi$
$230$	−4.82012	−0.317829
$231$	0	0
$232$	−26.3875	−1.73242
$233$	−21.1833	−1.38777	−0.693883	−	0.720088i	$-0.744101\pi$
$233$	−21.1833	−1.38777	−0.693883	+	0.720088i	$0.744101\pi$
$234$	0	0
$235$	−7.59166	−0.495225
$236$	1.12548	0.0732622
$237$	−16.6818	−1.08360
$238$	0	0
$239$	19.7958	1.28049	0.640243	−	0.768172i	$-0.278834\pi$
$239$	19.7958	1.28049	0.640243	+	0.768172i	$0.278834\pi$
$240$	−3.79583	−0.245020
$241$	−4.38701	−0.282592	−0.141296	−	0.989967i	$-0.545127\pi$
$241$	−4.38701	−0.282592	−0.141296	+	0.989967i	$0.545127\pi$
$242$	−22.5917	−1.45225
$243$	9.89949	0.635053
$244$	−1.55858	−0.0997780
$245$	0	0
$246$	13.7958	0.879590
$247$	0	0
$248$	4.24264	0.269408
$249$	14.0000	0.887214
$250$	7.50417	0.474606
$251$	−3.11716	−0.196754	−0.0983769	−	0.995149i	$-0.531365\pi$
$251$	−3.11716	−0.196754	−0.0983769	+	0.995149i	$0.531365\pi$
$252$	0	0
$253$	−10.4083	−0.654367
$254$	−7.59166	−0.476343
$255$	20.9245	1.31034
$256$	−17.0000	−1.06250
$257$	15.4120	0.961373	0.480686	−	0.876893i	$-0.340388\pi$
$257$	15.4120	0.961373	0.480686	+	0.876893i	$0.340388\pi$
$258$	2.53969	0.158114
$259$	0	0
$260$	0	0
$261$	8.79583	0.544448
$262$	8.19654	0.506384
$263$	−25.3875	−1.56546	−0.782730	−	0.622361i	$-0.786173\pi$
$263$	−25.3875	−1.56546	−0.782730	+	0.622361i	$0.786173\pi$
$264$	−24.5896	−1.51339
$265$	17.6924	1.08684
$266$	0	0
$267$	−17.1833	−1.05160
$268$	5.79583	0.354037
$269$	−1.12548	−0.0686215	−0.0343107	−	0.999411i	$-0.510924\pi$
$269$	−1.12548	−0.0686215	−0.0343107	+	0.999411i	$0.510924\pi$
$270$	15.1833	0.924028
$271$	23.1754	1.40781	0.703903	−	0.710296i	$-0.251439\pi$
$271$	23.1754	1.40781	0.703903	+	0.710296i	$0.251439\pi$
$272$	−5.51249	−0.334244
$273$	0	0
$274$	−9.20417	−0.556044
$275$	−12.7750	−0.770361
$276$	−2.53969	−0.152871
$277$	−6.18333	−0.371520	−0.185760	−	0.982595i	$-0.559475\pi$
$277$	−6.18333	−0.371520	−0.185760	+	0.982595i	$0.559475\pi$
$278$	15.2676	0.915690
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	−15.2042	−0.907005	−0.453502	−	0.891255i	$-0.649826\pi$
$281$	−15.2042	−0.907005	−0.453502	+	0.891255i	$0.649826\pi$
$282$	4.00000	0.238197
$283$	−29.4097	−1.74823	−0.874114	−	0.485721i	$-0.838557\pi$
$283$	−29.4097	−1.74823	−0.874114	+	0.485721i	$0.838557\pi$
$284$	6.00000	0.356034
$285$	10.7362	0.635960
$286$	0	0
$287$	0	0
$288$	5.00000	0.294628
$289$	13.3875	0.787500
$290$	23.6085	1.38634
$291$	6.00000	0.351726
$292$	−5.80122	−0.339491
$293$	9.75513	0.569901	0.284950	−	0.958542i	$-0.408023\pi$
$293$	9.75513	0.569901	0.284950	+	0.958542i	$0.408023\pi$
$294$	0	0
$295$	−3.02084	−0.175880
$296$	20.3875	1.18500
$297$	32.7862	1.90245
$298$	4.59166	0.265988
$299$	0	0
$300$	−3.11716	−0.179970
$301$	0	0
$302$	−17.5917	−1.01229
$303$	−4.20417	−0.241523
$304$	−2.82843	−0.162221
$305$	4.18333	0.239537
$306$	5.51249	0.315128
$307$	−34.2004	−1.95192	−0.975960	−	0.217951i	$-0.930063\pi$
$307$	−34.2004	−1.95192	−0.975960	+	0.217951i	$0.930063\pi$
$308$	0	0
$309$	−11.5917	−0.659427
$310$	−3.79583	−0.215589
$311$	11.8617	0.672616	0.336308	−	0.941752i	$-0.390822\pi$
$311$	11.8617	0.672616	0.336308	+	0.941752i	$0.390822\pi$
$312$	0	0
$313$	1.70295	0.0962565	0.0481283	−	0.998841i	$-0.484674\pi$
$313$	1.70295	0.0962565	0.0481283	+	0.998841i	$0.484674\pi$
$314$	6.92670	0.390896
$315$	0	0
$316$	11.7958	0.663567
$317$	12.5917	0.707218	0.353609	−	0.935393i	$-0.384954\pi$
$317$	12.5917	0.707218	0.353609	+	0.935393i	$0.384954\pi$
$318$	−9.32202	−0.522753
$319$	50.9792	2.85428
$320$	18.7884	1.05030
$321$	−8.48528	−0.473602
$322$	0	0
$323$	15.5917	0.867543
$324$	5.00000	0.277778
$325$	0	0
$326$	−13.3875	−0.741465
$327$	24.8784	1.37578
$328$	−29.2654	−1.61591
$329$	0	0
$330$	22.0000	1.21106
$331$	−0.612505	−0.0336663	−0.0168332	−	0.999858i	$-0.505358\pi$
$331$	−0.612505	−0.0336663	−0.0168332	+	0.999858i	$0.505358\pi$
$332$	−9.89949	−0.543305
$333$	−6.79583	−0.372409
$334$	15.2676	0.835407
$335$	−15.5563	−0.849934
$336$	0	0
$337$	−29.9792	−1.63307	−0.816534	−	0.577297i	$-0.804108\pi$
$337$	−29.9792	−1.63307	−0.816534	+	0.577297i	$0.804108\pi$
$338$	0	0
$339$	−23.4642	−1.27440
$340$	−14.7958	−0.802417
$341$	−8.19654	−0.443868
$342$	2.82843	0.152944
$343$	0	0
$344$	−5.38749	−0.290474
$345$	6.81667	0.366997
$346$	−9.32202	−0.501155
$347$	14.9792	0.804123	0.402062	−	0.915613i	$-0.368294\pi$
$347$	14.9792	0.804123	0.402062	+	0.915613i	$0.368294\pi$
$348$	12.4392	0.666810
$349$	−13.0167	−0.696766	−0.348383	−	0.937352i	$-0.613269\pi$
$349$	−13.0167	−0.696766	−0.348383	+	0.937352i	$0.613269\pi$
$350$	0	0
$351$	0	0
$352$	28.9792	1.54459
$353$	15.1232	0.804929	0.402464	−	0.915436i	$-0.368154\pi$
$353$	15.1232	0.804929	0.402464	+	0.915436i	$0.368154\pi$
$354$	1.59166	0.0845959
$355$	−16.1043	−0.854730
$356$	12.1504	0.643972
$357$	0	0
$358$	0.408337	0.0215813
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	−8.05217	−0.424387
$361$	−11.0000	−0.578947
$362$	16.5375	0.869189
$363$	31.9494	1.67691
$364$	0	0
$365$	15.5708	0.815014
$366$	−2.20417	−0.115214
$367$	21.2132	1.10732	0.553660	−	0.832743i	$-0.313231\pi$
$367$	21.2132	1.10732	0.553660	+	0.832743i	$0.313231\pi$
$368$	−1.79583	−0.0936142
$369$	9.75513	0.507832
$370$	−18.2404	−0.948274
$371$	0	0
$372$	−2.00000	−0.103695
$373$	−12.5917	−0.651972	−0.325986	−	0.945375i	$-0.605696\pi$
$373$	−12.5917	−0.651972	−0.325986	+	0.945375i	$0.605696\pi$
$374$	31.9494	1.65207
$375$	−10.6125	−0.548027
$376$	−8.48528	−0.437595
$377$	0	0
$378$	0	0
$379$	3.38749	0.174004	0.0870020	−	0.996208i	$-0.472271\pi$
$379$	3.38749	0.174004	0.0870020	+	0.996208i	$0.472271\pi$
$380$	−7.59166	−0.389444
$381$	10.7362	0.550034
$382$	25.7958	1.31983
$383$	14.9789	0.765385	0.382692	−	0.923876i	$-0.374997\pi$
$383$	14.9789	0.765385	0.382692	+	0.923876i	$0.374997\pi$
$384$	4.24264	0.216506
$385$	0	0
$386$	3.40834	0.173480
$387$	1.79583	0.0912872
$388$	−4.24264	−0.215387
$389$	28.3875	1.43930	0.719652	−	0.694335i	$-0.244302\pi$
$389$	28.3875	1.43930	0.719652	+	0.694335i	$0.244302\pi$
$390$	0	0
$391$	9.89949	0.500639
$392$	0	0
$393$	−11.5917	−0.584722
$394$	−8.00000	−0.403034
$395$	−31.6607	−1.59302
$396$	−5.79583	−0.291251
$397$	−7.07107	−0.354887	−0.177443	−	0.984131i	$-0.556783\pi$
$397$	−7.07107	−0.354887	−0.177443	+	0.984131i	$0.556783\pi$
$398$	22.0499	1.10526
$399$	0	0
$400$	−2.20417	−0.110208
$401$	−24.3875	−1.21785	−0.608927	−	0.793227i	$-0.708400\pi$
$401$	−24.3875	−1.21785	−0.608927	+	0.793227i	$0.708400\pi$
$402$	8.19654	0.408806
$403$	0	0
$404$	2.97280	0.147902
$405$	−13.4203	−0.666860
$406$	0	0
$407$	−39.3875	−1.95237
$408$	23.3875	1.15785
$409$	28.6879	1.41853	0.709263	−	0.704944i	$-0.249028\pi$
$409$	28.6879	1.41853	0.709263	+	0.704944i	$0.249028\pi$
$410$	26.1833	1.29310
$411$	13.0167	0.642064
$412$	8.19654	0.403815
$413$	0	0
$414$	1.79583	0.0882603
$415$	26.5708	1.30431
$416$	0	0
$417$	−21.5917	−1.05735
$418$	16.3931	0.801812
$419$	28.5435	1.39444	0.697221	−	0.716856i	$-0.254419\pi$
$419$	28.5435	1.39444	0.697221	+	0.716856i	$0.254419\pi$
$420$	0	0
$421$	6.59166	0.321258	0.160629	−	0.987015i	$-0.448648\pi$
$421$	6.59166	0.321258	0.160629	+	0.987015i	$0.448648\pi$
$422$	1.79583	0.0874197
$423$	2.82843	0.137523
$424$	19.7750	0.960358
$425$	12.1504	0.589383
$426$	8.48528	0.411113
$427$	0	0
$428$	6.00000	0.290021
$429$	0	0
$430$	4.82012	0.232447
$431$	5.18333	0.249672	0.124836	−	0.992177i	$-0.460160\pi$
$431$	5.18333	0.249672	0.124836	+	0.992177i	$0.460160\pi$
$432$	5.65685	0.272166
$433$	13.4203	0.644938	0.322469	−	0.946580i	$-0.395487\pi$
$433$	13.4203	0.644938	0.322469	+	0.946580i	$0.395487\pi$
$434$	0	0
$435$	−33.3875	−1.60081
$436$	−17.5917	−0.842488
$437$	5.07938	0.242980
$438$	−8.20417	−0.392010
$439$	34.2004	1.63230	0.816148	−	0.577843i	$-0.196106\pi$
$439$	34.2004	1.63230	0.816148	+	0.577843i	$0.196106\pi$
$440$	−46.6690	−2.22486
$441$	0	0
$442$	0	0
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	−9.61076	−0.456106
$445$	−32.6125	−1.54598
$446$	−5.65685	−0.267860
$447$	−6.49359	−0.307136
$448$	0	0
$449$	−4.40834	−0.208042	−0.104021	−	0.994575i	$-0.533171\pi$
$449$	−4.40834	−0.208042	−0.104021	+	0.994575i	$0.533171\pi$
$450$	2.20417	0.103905
$451$	56.5391	2.66232
$452$	16.5917	0.780406
$453$	24.8784	1.16889
$454$	20.9245	0.982034
$455$	0	0
$456$	12.0000	0.561951
$457$	14.5917	0.682569	0.341285	−	0.939960i	$-0.389138\pi$
$457$	14.5917	0.682569	0.341285	+	0.939960i	$0.389138\pi$
$458$	12.7279	0.594737
$459$	−31.1833	−1.45551
$460$	−4.82012	−0.224739
$461$	−30.9683	−1.44234	−0.721169	−	0.692759i	$-0.756395\pi$
$461$	−30.9683	−1.44234	−0.721169	+	0.692759i	$0.756395\pi$
$462$	0	0
$463$	−11.3875	−0.529222	−0.264611	−	0.964355i	$-0.585244\pi$
$463$	−11.3875	−0.529222	−0.264611	+	0.964355i	$0.585244\pi$
$464$	8.79583	0.408336
$465$	5.36812	0.248940
$466$	21.1833	0.981299
$467$	−4.82012	−0.223048	−0.111524	−	0.993762i	$-0.535573\pi$
$467$	−4.82012	−0.223048	−0.111524	+	0.993762i	$0.535573\pi$
$468$	0	0
$469$	0	0
$470$	7.59166	0.350177
$471$	−9.79583	−0.451368
$472$	−3.37643	−0.155413
$473$	10.4083	0.478576
$474$	16.6818	0.766222
$475$	6.23433	0.286051
$476$	0	0
$477$	−6.59166	−0.301812
$478$	−19.7958	−0.905440
$479$	7.35981	0.336278	0.168139	−	0.985763i	$-0.446224\pi$
$479$	7.35981	0.336278	0.168139	+	0.985763i	$0.446224\pi$
$480$	−18.9792	−0.866276
$481$	0	0
$482$	4.38701	0.199823
$483$	0	0
$484$	−22.5917	−1.02689
$485$	11.3875	0.517079
$486$	−9.89949	−0.449050
$487$	−19.5917	−0.887783	−0.443891	−	0.896081i	$-0.646402\pi$
$487$	−19.5917	−0.887783	−0.443891	+	0.896081i	$0.646402\pi$
$488$	4.67575	0.211661
$489$	18.9328	0.856170
$490$	0	0
$491$	−9.59166	−0.432866	−0.216433	−	0.976298i	$-0.569442\pi$
$491$	−9.59166	−0.432866	−0.216433	+	0.976298i	$0.569442\pi$
$492$	13.7958	0.621964
$493$	−48.4869	−2.18374
$494$	0	0
$495$	15.5563	0.699206
$496$	−1.41421	−0.0635001
$497$	0	0
$498$	−14.0000	−0.627355
$499$	−16.2042	−0.725398	−0.362699	−	0.931906i	$-0.618145\pi$
$499$	−16.2042	−0.725398	−0.362699	+	0.931906i	$0.618145\pi$
$500$	7.50417	0.335597
$501$	−21.5917	−0.964644
$502$	3.11716	0.139126
$503$	−28.5435	−1.27269	−0.636347	−	0.771403i	$-0.719555\pi$
$503$	−28.5435	−1.27269	−0.636347	+	0.771403i	$0.719555\pi$
$504$	0	0
$505$	−7.97916	−0.355068
$506$	10.4083	0.462707
$507$	0	0
$508$	−7.59166	−0.336826
$509$	−26.4370	−1.17180	−0.585899	−	0.810384i	$-0.699258\pi$
$509$	−26.4370	−1.17180	−0.585899	+	0.810384i	$0.699258\pi$
$510$	−20.9245	−0.926551
$511$	0	0
$512$	11.0000	0.486136
$513$	−16.0000	−0.706417
$514$	−15.4120	−0.679793
$515$	−22.0000	−0.969436
$516$	2.53969	0.111804
$517$	16.3931	0.720967
$518$	0	0
$519$	13.1833	0.578684
$520$	0	0
$521$	25.0227	1.09627	0.548133	−	0.836391i	$-0.315339\pi$
$521$	25.0227	1.09627	0.548133	+	0.836391i	$0.315339\pi$
$522$	−8.79583	−0.384983
$523$	28.5730	1.24941	0.624705	−	0.780861i	$-0.285219\pi$
$523$	28.5730	1.24941	0.624705	+	0.780861i	$0.285219\pi$
$524$	8.19654	0.358068
$525$	0	0
$526$	25.3875	1.10695
$527$	7.79583	0.339592
$528$	8.19654	0.356709
$529$	−19.7750	−0.859782
$530$	−17.6924	−0.768509
$531$	1.12548	0.0488415
$532$	0	0
$533$	0	0
$534$	17.1833	0.743595
$535$	−16.1043	−0.696252
$536$	−17.3875	−0.751025
$537$	−0.577476	−0.0249199
$538$	1.12548	0.0485227
$539$	0	0
$540$	15.1833	0.653386
$541$	12.5917	0.541358	0.270679	−	0.962670i	$-0.412752\pi$
$541$	12.5917	0.541358	0.270679	+	0.962670i	$0.412752\pi$
$542$	−23.1754	−0.995469
$543$	−23.3875	−1.00365
$544$	−27.5624	−1.18173
$545$	47.2170	2.02256
$546$	0	0
$547$	−36.9792	−1.58111	−0.790557	−	0.612388i	$-0.790209\pi$
$547$	−36.9792	−1.58111	−0.790557	+	0.612388i	$0.790209\pi$
$548$	−9.20417	−0.393183
$549$	−1.55858	−0.0665187
$550$	12.7750	0.544727
$551$	−24.8784	−1.05985
$552$	7.61907	0.324289
$553$	0	0
$554$	6.18333	0.262704
$555$	25.7958	1.09497
$556$	15.2676	0.647491
$557$	−20.5917	−0.872497	−0.436248	−	0.899826i	$-0.643693\pi$
$557$	−20.5917	−0.872497	−0.436248	+	0.899826i	$0.643693\pi$
$558$	1.41421	0.0598684
$559$	0	0
$560$	0	0
$561$	−45.1833	−1.90764
$562$	15.2042	0.641349
$563$	−1.12548	−0.0474331	−0.0237166	−	0.999719i	$-0.507550\pi$
$563$	−1.12548	−0.0474331	−0.0237166	+	0.999719i	$0.507550\pi$
$564$	4.00000	0.168430
$565$	−44.5330	−1.87352
$566$	29.4097	1.23618
$567$	0	0
$568$	−18.0000	−0.755263
$569$	6.40834	0.268651	0.134326	−	0.990937i	$-0.457113\pi$
$569$	6.40834	0.268651	0.134326	+	0.990937i	$0.457113\pi$
$570$	−10.7362	−0.449691
$571$	15.1833	0.635402	0.317701	−	0.948191i	$-0.397089\pi$
$571$	15.1833	0.635402	0.317701	+	0.948191i	$0.397089\pi$
$572$	0	0
$573$	−36.4808	−1.52401
$574$	0	0
$575$	3.95832	0.165073
$576$	−7.00000	−0.291667
$577$	9.17765	0.382071	0.191035	−	0.981583i	$-0.438816\pi$
$577$	9.17765	0.382071	0.191035	+	0.981583i	$0.438816\pi$
$578$	−13.3875	−0.556846
$579$	−4.82012	−0.200317
$580$	23.6085	0.980291
$581$	0	0
$582$	−6.00000	−0.248708
$583$	−38.2042	−1.58225
$584$	17.4037	0.720169
$585$	0	0
$586$	−9.75513	−0.402981
$587$	−3.11716	−0.128659	−0.0643296	−	0.997929i	$-0.520491\pi$
$587$	−3.11716	−0.128659	−0.0643296	+	0.997929i	$0.520491\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	3.02084	0.124366
$591$	11.3137	0.465384
$592$	−6.79583	−0.279307
$593$	1.55858	0.0640033	0.0320017	−	0.999488i	$-0.489812\pi$
$593$	1.55858	0.0640033	0.0320017	+	0.999488i	$0.489812\pi$
$594$	−32.7862	−1.34523
$595$	0	0
$596$	4.59166	0.188082
$597$	−31.1833	−1.27625
$598$	0	0
$599$	−14.4083	−0.588709	−0.294354	−	0.955696i	$-0.595105\pi$
$599$	−14.4083	−0.588709	−0.294354	+	0.955696i	$0.595105\pi$
$600$	9.35149	0.381773
$601$	−30.9683	−1.26322	−0.631612	−	0.775284i	$-0.717607\pi$
$601$	−30.9683	−1.26322	−0.631612	+	0.775284i	$0.717607\pi$
$602$	0	0
$603$	5.79583	0.236025
$604$	−17.5917	−0.715795
$605$	60.6373	2.46526
$606$	4.20417	0.170783
$607$	−35.9033	−1.45727	−0.728636	−	0.684901i	$-0.759845\pi$
$607$	−35.9033	−1.45727	−0.728636	+	0.684901i	$0.759845\pi$
$608$	−14.1421	−0.573539
$609$	0	0
$610$	−4.18333	−0.169378
$611$	0	0
$612$	5.51249	0.222829
$613$	5.97916	0.241496	0.120748	−	0.992683i	$-0.461471\pi$
$613$	5.97916	0.241496	0.120748	+	0.992683i	$0.461471\pi$
$614$	34.2004	1.38022
$615$	−37.0288	−1.49315
$616$	0	0
$617$	24.3875	0.981804	0.490902	−	0.871215i	$-0.336667\pi$
$617$	24.3875	0.981804	0.490902	+	0.871215i	$0.336667\pi$
$618$	11.5917	0.466285
$619$	−33.9116	−1.36302	−0.681512	−	0.731807i	$-0.738677\pi$
$619$	−33.9116	−1.36302	−0.681512	+	0.731807i	$0.738677\pi$
$620$	−3.79583	−0.152444
$621$	−10.1588	−0.407657
$622$	−11.8617	−0.475611
$623$	0	0
$624$	0	0
$625$	−31.1625	−1.24650
$626$	−1.70295	−0.0680636
$627$	−23.1833	−0.925853
$628$	6.92670	0.276405
$629$	37.4619	1.49370
$630$	0	0
$631$	20.4083	0.812443	0.406222	−	0.913775i	$-0.366846\pi$
$631$	20.4083	0.812443	0.406222	+	0.913775i	$0.366846\pi$
$632$	−35.3875	−1.40764
$633$	−2.53969	−0.100944
$634$	−12.5917	−0.500079
$635$	20.3765	0.808615
$636$	−9.32202	−0.369642
$637$	0	0
$638$	−50.9792	−2.01828
$639$	6.00000	0.237356
$640$	8.05217	0.318290
$641$	4.79583	0.189424	0.0947120	−	0.995505i	$-0.469807\pi$
$641$	4.79583	0.189424	0.0947120	+	0.995505i	$0.469807\pi$
$642$	8.48528	0.334887
$643$	−5.65685	−0.223085	−0.111542	−	0.993760i	$-0.535579\pi$
$643$	−5.65685	−0.223085	−0.111542	+	0.993760i	$0.535579\pi$
$644$	0	0
$645$	−6.81667	−0.268406
$646$	−15.5917	−0.613446
$647$	−1.96221	−0.0771426	−0.0385713	−	0.999256i	$-0.512281\pi$
$647$	−1.96221	−0.0771426	−0.0385713	+	0.999256i	$0.512281\pi$
$648$	−15.0000	−0.589256
$649$	6.52307	0.256053
$650$	0	0
$651$	0	0
$652$	−13.3875	−0.524295
$653$	25.1833	0.985500	0.492750	−	0.870171i	$-0.335992\pi$
$653$	25.1833	0.985500	0.492750	+	0.870171i	$0.335992\pi$
$654$	−24.8784	−0.972821
$655$	−22.0000	−0.859611
$656$	9.75513	0.380874
$657$	−5.80122	−0.226327
$658$	0	0
$659$	−37.5917	−1.46436	−0.732182	−	0.681109i	$-0.761498\pi$
$659$	−37.5917	−1.46436	−0.732182	+	0.681109i	$0.761498\pi$
$660$	22.0000	0.856349
$661$	−27.8512	−1.08328	−0.541642	−	0.840609i	$-0.682197\pi$
$661$	−27.8512	−1.08328	−0.541642	+	0.840609i	$0.682197\pi$
$662$	0.612505	0.0238057
$663$	0	0
$664$	29.6985	1.15252
$665$	0	0
$666$	6.79583	0.263333
$667$	−15.7958	−0.611617
$668$	15.2676	0.590722
$669$	8.00000	0.309298
$670$	15.5563	0.600994
$671$	−9.03328	−0.348726
$672$	0	0
$673$	23.9792	0.924329	0.462164	−	0.886794i	$-0.347073\pi$
$673$	23.9792	0.924329	0.462164	+	0.886794i	$0.347073\pi$
$674$	29.9792	1.15475
$675$	−12.4687	−0.479919
$676$	0	0
$677$	34.7779	1.33662	0.668311	−	0.743882i	$-0.267018\pi$
$677$	34.7779	1.33662	0.668311	+	0.743882i	$0.267018\pi$
$678$	23.4642	0.901135
$679$	0	0
$680$	44.3875	1.70218
$681$	−29.5917	−1.13395
$682$	8.19654	0.313862
$683$	−50.7750	−1.94285	−0.971425	−	0.237345i	$-0.923723\pi$
$683$	−50.7750	−1.94285	−0.971425	+	0.237345i	$0.923723\pi$
$684$	2.82843	0.108148
$685$	24.7045	0.943911
$686$	0	0
$687$	−18.0000	−0.686743
$688$	1.79583	0.0684654
$689$	0	0
$690$	−6.81667	−0.259506
$691$	−11.0250	−0.419410	−0.209705	−	0.977765i	$-0.567250\pi$
$691$	−11.0250	−0.419410	−0.209705	+	0.977765i	$0.567250\pi$
$692$	−9.32202	−0.354370
$693$	0	0
$694$	−14.9792	−0.568601
$695$	−40.9792	−1.55443
$696$	−37.3176	−1.41452
$697$	−53.7750	−2.03687
$698$	13.0167	0.492688
$699$	−29.9577	−1.13311
$700$	0	0
$701$	−10.4083	−0.393117	−0.196559	−	0.980492i	$-0.562977\pi$
$701$	−10.4083	−0.393117	−0.196559	+	0.980492i	$0.562977\pi$
$702$	0	0
$703$	19.2215	0.724953
$704$	−40.5708	−1.52907
$705$	−10.7362	−0.404350
$706$	−15.1232	−0.569171
$707$	0	0
$708$	1.59166	0.0598184
$709$	18.7958	0.705892	0.352946	−	0.935644i	$-0.385180\pi$
$709$	18.7958	0.705892	0.352946	+	0.935644i	$0.385180\pi$
$710$	16.1043	0.604385
$711$	11.7958	0.442378
$712$	−36.4513	−1.36607
$713$	2.53969	0.0951121
$714$	0	0
$715$	0	0
$716$	0.408337	0.0152603
$717$	27.9955	1.04551
$718$	−4.00000	−0.149279
$719$	29.1210	1.08603	0.543015	−	0.839723i	$-0.317283\pi$
$719$	29.1210	1.08603	0.543015	+	0.839723i	$0.317283\pi$
$720$	2.68406	0.100029
$721$	0	0
$722$	11.0000	0.409378
$723$	−6.20417	−0.230736
$724$	16.5375	0.614610
$725$	−19.3875	−0.720033
$726$	−31.9494	−1.18575
$727$	−35.3259	−1.31016	−0.655082	−	0.755558i	$-0.727366\pi$
$727$	−35.3259	−1.31016	−0.655082	+	0.755558i	$0.727366\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	−15.5708	−0.576302
$731$	−9.89949	−0.366146
$732$	−2.20417	−0.0814684
$733$	−0.692369	−0.0255732	−0.0127866	−	0.999918i	$-0.504070\pi$
$733$	−0.692369	−0.0255732	−0.0127866	+	0.999918i	$0.504070\pi$
$734$	−21.2132	−0.782994
$735$	0	0
$736$	−8.97916	−0.330976
$737$	33.5917	1.23736
$738$	−9.75513	−0.359091
$739$	−15.1833	−0.558528	−0.279264	−	0.960214i	$-0.590090\pi$
$739$	−15.1833	−0.558528	−0.279264	+	0.960214i	$0.590090\pi$
$740$	−18.2404	−0.670531
$741$	0	0
$742$	0	0
$743$	−9.18333	−0.336904	−0.168452	−	0.985710i	$-0.553877\pi$
$743$	−9.18333	−0.336904	−0.168452	+	0.985710i	$0.553877\pi$
$744$	6.00000	0.219971
$745$	−12.3243	−0.451527
$746$	12.5917	0.461014
$747$	−9.89949	−0.362204
$748$	31.9494	1.16819
$749$	0	0
$750$	10.6125	0.387514
$751$	1.79583	0.0655308	0.0327654	−	0.999463i	$-0.489569\pi$
$751$	1.79583	0.0655308	0.0327654	+	0.999463i	$0.489569\pi$
$752$	2.82843	0.103142
$753$	−4.40834	−0.160649
$754$	0	0
$755$	47.2170	1.71840
$756$	0	0
$757$	−25.1833	−0.915304	−0.457652	−	0.889132i	$-0.651309\pi$
$757$	−25.1833	−0.915304	−0.457652	+	0.889132i	$0.651309\pi$
$758$	−3.38749	−0.123039
$759$	−14.7196	−0.534288
$760$	22.7750	0.826136
$761$	−9.32202	−0.337923	−0.168961	−	0.985623i	$-0.554041\pi$
$761$	−9.32202	−0.337923	−0.168961	+	0.985623i	$0.554041\pi$
$762$	−10.7362	−0.388933
$763$	0	0
$764$	25.7958	0.933260
$765$	−14.7958	−0.534944
$766$	−14.9789	−0.541209
$767$	0	0
$768$	−24.0416	−0.867528
$769$	−4.24264	−0.152994	−0.0764968	−	0.997070i	$-0.524373\pi$
$769$	−4.24264	−0.152994	−0.0764968	+	0.997070i	$0.524373\pi$
$770$	0	0
$771$	21.7958	0.784958
$772$	3.40834	0.122669
$773$	−7.64854	−0.275099	−0.137549	−	0.990495i	$-0.543923\pi$
$773$	−7.64854	−0.275099	−0.137549	+	0.990495i	$0.543923\pi$
$774$	−1.79583	−0.0645498
$775$	3.11716	0.111972
$776$	12.7279	0.456906
$777$	0	0
$778$	−28.3875	−1.01774
$779$	−27.5917	−0.988574
$780$	0	0
$781$	34.7750	1.24435
$782$	−9.89949	−0.354005
$783$	49.7567	1.77816
$784$	0	0
$785$	−18.5917	−0.663565
$786$	11.5917	0.413461
$787$	−25.4264	−0.906352	−0.453176	−	0.891421i	$-0.649709\pi$
$787$	−25.4264	−0.906352	−0.453176	+	0.891421i	$0.649709\pi$
$788$	−8.00000	−0.284988
$789$	−35.9033	−1.27819
$790$	31.6607	1.12644
$791$	0	0
$792$	17.3875	0.617838
$793$	0	0
$794$	7.07107	0.250943
$795$	25.0208	0.887398
$796$	22.0499	0.781539
$797$	−26.2926	−0.931331	−0.465666	−	0.884961i	$-0.654185\pi$
$797$	−26.2926	−0.931331	−0.465666	+	0.884961i	$0.654185\pi$
$798$	0	0
$799$	−15.5917	−0.551593
$800$	−11.0208	−0.389646
$801$	12.1504	0.429315
$802$	24.3875	0.861152
$803$	−33.6229	−1.18653
$804$	8.19654	0.289070
$805$	0	0
$806$	0	0
$807$	−1.59166	−0.0560292
$808$	−8.91839	−0.313748
$809$	39.3667	1.38406	0.692029	−	0.721870i	$-0.256717\pi$
$809$	39.3667	1.38406	0.692029	+	0.721870i	$0.256717\pi$
$810$	13.4203	0.471541
$811$	38.4725	1.35095	0.675476	−	0.737382i	$-0.263938\pi$
$811$	38.4725	1.35095	0.675476	+	0.737382i	$0.263938\pi$
$812$	0	0
$813$	32.7750	1.14947
$814$	39.3875	1.38053
$815$	35.9328	1.25867
$816$	−7.79583	−0.272909
$817$	−5.07938	−0.177705
$818$	−28.6879	−1.00305
$819$	0	0
$820$	26.1833	0.914361
$821$	40.3667	1.40881	0.704403	−	0.709800i	$-0.251215\pi$
$821$	40.3667	1.40881	0.704403	+	0.709800i	$0.251215\pi$
$822$	−13.0167	−0.454008
$823$	36.7750	1.28190	0.640948	−	0.767584i	$-0.278542\pi$
$823$	36.7750	1.28190	0.640948	+	0.767584i	$0.278542\pi$
$824$	−24.5896	−0.856620
$825$	−18.0666	−0.628997
$826$	0	0
$827$	−26.9792	−0.938157	−0.469079	−	0.883156i	$-0.655414\pi$
$827$	−26.9792	−0.938157	−0.469079	+	0.883156i	$0.655414\pi$
$828$	1.79583	0.0624095
$829$	20.2026	0.701666	0.350833	−	0.936438i	$-0.385898\pi$
$829$	20.2026	0.701666	0.350833	+	0.936438i	$0.385898\pi$
$830$	−26.5708	−0.922287
$831$	−8.74454	−0.303345
$832$	0	0
$833$	0	0
$834$	21.5917	0.747658
$835$	−40.9792	−1.41814
$836$	16.3931	0.566967
$837$	−8.00000	−0.276520
$838$	−28.5435	−0.986020
$839$	−5.65685	−0.195296	−0.0976481	−	0.995221i	$-0.531132\pi$
$839$	−5.65685	−0.195296	−0.0976481	+	0.995221i	$0.531132\pi$
$840$	0	0
$841$	48.3667	1.66782
$842$	−6.59166	−0.227164
$843$	−21.5019	−0.740566
$844$	1.79583	0.0618151
$845$	0	0
$846$	−2.82843	−0.0972433
$847$	0	0
$848$	−6.59166	−0.226359
$849$	−41.5917	−1.42742
$850$	−12.1504	−0.416757
$851$	12.2042	0.418354
$852$	8.48528	0.290701
$853$	13.1610	0.450625	0.225313	−	0.974287i	$-0.427660\pi$
$853$	13.1610	0.450625	0.225313	+	0.974287i	$0.427660\pi$
$854$	0	0
$855$	−7.59166	−0.259629
$856$	−18.0000	−0.615227
$857$	−37.7507	−1.28954	−0.644769	−	0.764377i	$-0.723046\pi$
$857$	−37.7507	−1.28954	−0.644769	+	0.764377i	$0.723046\pi$
$858$	0	0
$859$	23.7529	0.810438	0.405219	−	0.914220i	$-0.367195\pi$
$859$	23.7529	0.810438	0.405219	+	0.914220i	$0.367195\pi$
$860$	4.82012	0.164365
$861$	0	0
$862$	−5.18333	−0.176545
$863$	−17.7958	−0.605777	−0.302889	−	0.953026i	$-0.597951\pi$
$863$	−17.7958	−0.605777	−0.302889	+	0.953026i	$0.597951\pi$
$864$	28.2843	0.962250
$865$	25.0208	0.850734
$866$	−13.4203	−0.456040
$867$	18.9328	0.642991
$868$	0	0
$869$	68.3667	2.31918
$870$	33.3875	1.13194
$871$	0	0
$872$	52.7750	1.78719
$873$	−4.24264	−0.143592
$874$	−5.07938	−0.171813
$875$	0	0
$876$	−8.20417	−0.277193
$877$	4.79583	0.161944	0.0809719	−	0.996716i	$-0.474198\pi$
$877$	4.79583	0.161944	0.0809719	+	0.996716i	$0.474198\pi$
$878$	−34.2004	−1.15421
$879$	13.7958	0.465322
$880$	15.5563	0.524404
$881$	−23.3198	−0.785664	−0.392832	−	0.919610i	$-0.628505\pi$
$881$	−23.3198	−0.785664	−0.392832	+	0.919610i	$0.628505\pi$
$882$	0	0
$883$	−16.6125	−0.559055	−0.279528	−	0.960138i	$-0.590178\pi$
$883$	−16.6125	−0.559055	−0.279528	+	0.960138i	$0.590178\pi$
$884$	0	0
$885$	−4.27212	−0.143606
$886$	−10.0000	−0.335957
$887$	−28.5730	−0.959388	−0.479694	−	0.877436i	$-0.659252\pi$
$887$	−28.5730	−0.959388	−0.479694	+	0.877436i	$0.659252\pi$
$888$	28.8323	0.967548
$889$	0	0
$890$	32.6125	1.09317
$891$	28.9792	0.970838
$892$	−5.65685	−0.189405
$893$	−8.00000	−0.267710
$894$	6.49359	0.217178
$895$	−1.09600	−0.0366352
$896$	0	0
$897$	0	0
$898$	4.40834	0.147108
$899$	−12.4392	−0.414870
$900$	2.20417	0.0734723
$901$	36.3364	1.21054
$902$	−56.5391	−1.88255
$903$	0	0
$904$	−49.7750	−1.65549
$905$	−44.3875	−1.47549
$906$	−24.8784	−0.826528
$907$	49.3875	1.63988	0.819942	−	0.572446i	$-0.194005\pi$
$907$	49.3875	1.63988	0.819942	+	0.572446i	$0.194005\pi$
$908$	20.9245	0.694403
$909$	2.97280	0.0986014
$910$	0	0
$911$	−43.1833	−1.43073	−0.715364	−	0.698752i	$-0.753739\pi$
$911$	−43.1833	−1.43073	−0.715364	+	0.698752i	$0.753739\pi$
$912$	−4.00000	−0.132453
$913$	−57.3758	−1.89886
$914$	−14.5917	−0.482649
$915$	5.91612	0.195581
$916$	12.7279	0.420542
$917$	0	0
$918$	31.1833	1.02920
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	14.4603	0.476744
$921$	−48.3667	−1.59374
$922$	30.9683	1.01989
$923$	0	0
$924$	0	0
$925$	14.9792	0.492512
$926$	11.3875	0.374216
$927$	8.19654	0.269210
$928$	43.9792	1.44369
$929$	28.1399	0.923240	0.461620	−	0.887078i	$-0.347268\pi$
$929$	28.1399	0.923240	0.461620	+	0.887078i	$0.347268\pi$
$930$	−5.36812	−0.176027
$931$	0	0
$932$	21.1833	0.693883
$933$	16.7750	0.549188
$934$	4.82012	0.157719
$935$	−85.7541	−2.80446
$936$	0	0
$937$	6.63796	0.216853	0.108426	−	0.994104i	$-0.465419\pi$
$937$	6.63796	0.216853	0.108426	+	0.994104i	$0.465419\pi$
$938$	0	0
$939$	2.40834	0.0785931
$940$	7.59166	0.247613
$941$	23.4642	0.764910	0.382455	−	0.923974i	$-0.375079\pi$
$941$	23.4642	0.764910	0.382455	+	0.923974i	$0.375079\pi$
$942$	9.79583	0.319165
$943$	−17.5186	−0.570483
$944$	1.12548	0.0366311
$945$	0	0
$946$	−10.4083	−0.338404
$947$	−28.2042	−0.916512	−0.458256	−	0.888820i	$-0.651526\pi$
$947$	−28.2042	−0.916512	−0.458256	+	0.888820i	$0.651526\pi$
$948$	16.6818	0.541800
$949$	0	0
$950$	−6.23433	−0.202268
$951$	17.8073	0.577441
$952$	0	0
$953$	−4.81667	−0.156027	−0.0780137	−	0.996952i	$-0.524858\pi$
$953$	−4.81667	−0.156027	−0.0780137	+	0.996952i	$0.524858\pi$
$954$	6.59166	0.213413
$955$	−69.2375	−2.24047
$956$	−19.7958	−0.640243
$957$	72.0954	2.33051
$958$	−7.35981	−0.237785
$959$	0	0
$960$	26.5708	0.857570
$961$	−29.0000	−0.935484
$962$	0	0
$963$	6.00000	0.193347
$964$	4.38701	0.141296
$965$	−9.14817	−0.294490
$966$	0	0
$967$	−11.3875	−0.366197	−0.183099	−	0.983095i	$-0.558613\pi$
$967$	−11.3875	−0.366197	−0.183099	+	0.983095i	$0.558613\pi$
$968$	67.7750	2.17837
$969$	22.0499	0.708346
$970$	−11.3875	−0.365630
$971$	−22.6274	−0.726148	−0.363074	−	0.931760i	$-0.618273\pi$
$971$	−22.6274	−0.726148	−0.363074	+	0.931760i	$0.618273\pi$
$972$	−9.89949	−0.317526
$973$	0	0
$974$	19.5917	0.627757
$975$	0	0
$976$	−1.55858	−0.0498890
$977$	−0.387495	−0.0123970	−0.00619852	−	0.999981i	$-0.501973\pi$
$977$	−0.387495	−0.0123970	−0.00619852	+	0.999981i	$0.501973\pi$
$978$	−18.9328	−0.605403
$979$	70.4219	2.25069
$980$	0	0
$981$	−17.5917	−0.561659
$982$	9.59166	0.306082
$983$	−41.5602	−1.32556	−0.662782	−	0.748812i	$-0.730624\pi$
$983$	−41.5602	−1.32556	−0.662782	+	0.748812i	$0.730624\pi$
$984$	−41.3875	−1.31939
$985$	21.4725	0.684170
$986$	48.4869	1.54414
$987$	0	0
$988$	0	0
$989$	−3.22501	−0.102549
$990$	−15.5563	−0.494413
$991$	18.2042	0.578274	0.289137	−	0.957288i	$-0.406632\pi$
$991$	18.2042	0.578274	0.289137	+	0.957288i	$0.406632\pi$
$992$	−7.07107	−0.224507
$993$	−0.866213	−0.0274885
$994$	0	0
$995$	−59.1833	−1.87624
$996$	−14.0000	−0.443607
$997$	−5.51249	−0.174582	−0.0872911	−	0.996183i	$-0.527821\pi$
$997$	−5.51249	−0.174582	−0.0872911	+	0.996183i	$0.527821\pi$
$998$	16.2042	0.512934
$999$	−38.4430	−1.21628

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bn.1.4		4
7.6	odd	2	inner	8281.2.a.bn.1.1		4
13.4	even	6		637.2.f.g.393.1	yes	8
13.10	even	6		637.2.f.g.295.1	✓	8
13.12	even	2		8281.2.a.bv.1.3		4
91.4	even	6		637.2.h.k.471.2		8
91.10	odd	6		637.2.g.h.373.1		8
91.17	odd	6		637.2.h.k.471.3		8
91.23	even	6		637.2.h.k.165.2		8
91.30	even	6		637.2.g.h.263.4		8
91.62	odd	6		637.2.f.g.295.4	yes	8
91.69	odd	6		637.2.f.g.393.4	yes	8
91.75	odd	6		637.2.h.k.165.3		8
91.82	odd	6		637.2.g.h.263.1		8
91.88	even	6		637.2.g.h.373.4		8
91.90	odd	2		8281.2.a.bv.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.f.g.295.1	✓	8	13.10	even	6
637.2.f.g.295.4	yes	8	91.62	odd	6
637.2.f.g.393.1	yes	8	13.4	even	6
637.2.f.g.393.4	yes	8	91.69	odd	6
637.2.g.h.263.1		8	91.82	odd	6
637.2.g.h.263.4		8	91.30	even	6
637.2.g.h.373.1		8	91.10	odd	6
637.2.g.h.373.4		8	91.88	even	6
637.2.h.k.165.2		8	91.23	even	6
637.2.h.k.165.3		8	91.75	odd	6
637.2.h.k.471.2		8	91.4	even	6
637.2.h.k.471.3		8	91.17	odd	6
8281.2.a.bn.1.1		4	7.6	odd	2	inner
8281.2.a.bn.1.4		4	1.1	even	1	trivial
8281.2.a.bv.1.2		4	91.90	odd	2
8281.2.a.bv.1.3		4	13.12	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} -1.00000 q^{4} +2.68406 q^{5} -1.41421 q^{6} +3.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} -1.00000 q^{4} +2.68406 q^{5} -1.41421 q^{6} +3.00000 q^{8} -1.00000 q^{9} -2.68406 q^{10} -5.79583 q^{11} -1.41421 q^{12} +3.79583 q^{15} -1.00000 q^{16} +5.51249 q^{17} +1.00000 q^{18} +2.82843 q^{19} -2.68406 q^{20} +5.79583 q^{22} +1.79583 q^{23} +4.24264 q^{24} +2.20417 q^{25} -5.65685 q^{27} -8.79583 q^{29} -3.79583 q^{30} +1.41421 q^{31} -5.00000 q^{32} -8.19654 q^{33} -5.51249 q^{34} +1.00000 q^{36} +6.79583 q^{37} -2.82843 q^{38} +8.05217 q^{40} -9.75513 q^{41} -1.79583 q^{43} +5.79583 q^{44} -2.68406 q^{45} -1.79583 q^{46} -2.82843 q^{47} -1.41421 q^{48} -2.20417 q^{50} +7.79583 q^{51} +6.59166 q^{53} +5.65685 q^{54} -15.5563 q^{55} +4.00000 q^{57} +8.79583 q^{58} -1.12548 q^{59} -3.79583 q^{60} +1.55858 q^{61} -1.41421 q^{62} +7.00000 q^{64} +8.19654 q^{66} -5.79583 q^{67} -5.51249 q^{68} +2.53969 q^{69} -6.00000 q^{71} -3.00000 q^{72} +5.80122 q^{73} -6.79583 q^{74} +3.11716 q^{75} -2.82843 q^{76} -11.7958 q^{79} -2.68406 q^{80} -5.00000 q^{81} +9.75513 q^{82} +9.89949 q^{83} +14.7958 q^{85} +1.79583 q^{86} -12.4392 q^{87} -17.3875 q^{88} -12.1504 q^{89} +2.68406 q^{90} -1.79583 q^{92} +2.00000 q^{93} +2.82843 q^{94} +7.59166 q^{95} -7.07107 q^{96} +4.24264 q^{97} +5.79583 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{4} + 12 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^4 + 12 * q^8 - 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{4} + 12 q^{8} - 4 q^{9} - 4 q^{11} - 4 q^{15} - 4 q^{16} + 4 q^{18} + 4 q^{22} - 12 q^{23} + 28 q^{25} - 16 q^{29} + 4 q^{30} - 20 q^{32} + 4 q^{36} + 8 q^{37} + 12 q^{43} + 4 q^{44} + 12 q^{46} - 28 q^{50} + 12 q^{51} - 12 q^{53} + 16 q^{57} + 16 q^{58} + 4 q^{60} + 28 q^{64} - 4 q^{67} - 24 q^{71} - 12 q^{72} - 8 q^{74} - 28 q^{79} - 20 q^{81} + 40 q^{85} - 12 q^{86} - 12 q^{88} + 12 q^{92} + 8 q^{93} - 8 q^{95} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^4 + 12 * q^8 - 4 * q^9 - 4 * q^11 - 4 * q^15 - 4 * q^16 + 4 * q^18 + 4 * q^22 - 12 * q^23 + 28 * q^25 - 16 * q^29 + 4 * q^30 - 20 * q^32 + 4 * q^36 + 8 * q^37 + 12 * q^43 + 4 * q^44 + 12 * q^46 - 28 * q^50 + 12 * q^51 - 12 * q^53 + 16 * q^57 + 16 * q^58 + 4 * q^60 + 28 * q^64 - 4 * q^67 - 24 * q^71 - 12 * q^72 - 8 * q^74 - 28 * q^79 - 20 * q^81 + 40 * q^85 - 12 * q^86 - 12 * q^88 + 12 * q^92 + 8 * q^93 - 8 * q^95 + 4 * q^99

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