LMFDB - Embedded newform 8281.2.a.z.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+2.61803 q^{2} -2.23607 q^{3} +4.85410 q^{4} -2.23607 q^{5} -5.85410 q^{6} +7.47214 q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+2.61803 q^{2} -2.23607 q^{3} +4.85410 q^{4} -2.23607 q^{5} -5.85410 q^{6} +7.47214 q^{8} +2.00000 q^{9} -5.85410 q^{10} +3.00000 q^{11} -10.8541 q^{12} +5.00000 q^{15} +9.85410 q^{16} +1.47214 q^{17} +5.23607 q^{18} -3.00000 q^{19} -10.8541 q^{20} +7.85410 q^{22} -8.23607 q^{23} -16.7082 q^{24} +2.23607 q^{27} +4.47214 q^{29} +13.0902 q^{30} -5.00000 q^{31} +10.8541 q^{32} -6.70820 q^{33} +3.85410 q^{34} +9.70820 q^{36} -4.70820 q^{37} -7.85410 q^{38} -16.7082 q^{40} +4.47214 q^{41} -8.00000 q^{43} +14.5623 q^{44} -4.47214 q^{45} -21.5623 q^{46} +7.47214 q^{47} -22.0344 q^{48} -3.29180 q^{51} -7.47214 q^{53} +5.85410 q^{54} -6.70820 q^{55} +6.70820 q^{57} +11.7082 q^{58} +1.47214 q^{59} +24.2705 q^{60} +3.00000 q^{61} -13.0902 q^{62} +8.70820 q^{64} -17.5623 q^{66} +3.00000 q^{67} +7.14590 q^{68} +18.4164 q^{69} +8.94427 q^{71} +14.9443 q^{72} +2.70820 q^{73} -12.3262 q^{74} -14.5623 q^{76} -2.70820 q^{79} -22.0344 q^{80} -11.0000 q^{81} +11.7082 q^{82} -3.29180 q^{85} -20.9443 q^{86} -10.0000 q^{87} +22.4164 q^{88} -2.23607 q^{89} -11.7082 q^{90} -39.9787 q^{92} +11.1803 q^{93} +19.5623 q^{94} +6.70820 q^{95} -24.2705 q^{96} -9.41641 q^{97} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9	$ 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9} - 5 q^{10} + 6 q^{11} - 15 q^{12} + 10 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} - 6 q^{19} - 15 q^{20} + 9 q^{22} - 12 q^{23} - 20 q^{24} + 15 q^{30} - 10 q^{31} + 15 q^{32} + q^{34} + 6 q^{36} + 4 q^{37} - 9 q^{38} - 20 q^{40} - 16 q^{43} + 9 q^{44} - 23 q^{46} + 6 q^{47} - 15 q^{48} - 20 q^{51} - 6 q^{53} + 5 q^{54} + 10 q^{58} - 6 q^{59} + 15 q^{60} + 6 q^{61} - 15 q^{62} + 4 q^{64} - 15 q^{66} + 6 q^{67} + 21 q^{68} + 10 q^{69} + 12 q^{72} - 8 q^{73} - 9 q^{74} - 9 q^{76} + 8 q^{79} - 15 q^{80} - 22 q^{81} + 10 q^{82} - 20 q^{85} - 24 q^{86} - 20 q^{87} + 18 q^{88} - 10 q^{90} - 33 q^{92} + 19 q^{94} - 15 q^{96} + 8 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9 - 5 * q^10 + 6 * q^11 - 15 * q^12 + 10 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 - 6 * q^19 - 15 * q^20 + 9 * q^22 - 12 * q^23 - 20 * q^24 + 15 * q^30 - 10 * q^31 + 15 * q^32 + q^34 + 6 * q^36 + 4 * q^37 - 9 * q^38 - 20 * q^40 - 16 * q^43 + 9 * q^44 - 23 * q^46 + 6 * q^47 - 15 * q^48 - 20 * q^51 - 6 * q^53 + 5 * q^54 + 10 * q^58 - 6 * q^59 + 15 * q^60 + 6 * q^61 - 15 * q^62 + 4 * q^64 - 15 * q^66 + 6 * q^67 + 21 * q^68 + 10 * q^69 + 12 * q^72 - 8 * q^73 - 9 * q^74 - 9 * q^76 + 8 * q^79 - 15 * q^80 - 22 * q^81 + 10 * q^82 - 20 * q^85 - 24 * q^86 - 20 * q^87 + 18 * q^88 - 10 * q^90 - 33 * q^92 + 19 * q^94 - 15 * q^96 + 8 * q^97 + 12 * 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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	4.85410	2.42705
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	−5.85410	−2.38993
$7$	0	0
$7$	0	0
$8$	7.47214	2.64180
$9$	2.00000	0.666667
$10$	−5.85410	−1.85123
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−10.8541	−3.13331
$13$	0	0
$13$	0	0
$14$	0	0
$15$	5.00000	1.29099
$16$	9.85410	2.46353
$17$	1.47214	0.357045	0.178523	−	0.983936i	$-0.442868\pi$
$17$	1.47214	0.357045	0.178523	+	0.983936i	$0.442868\pi$
$18$	5.23607	1.23415
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−10.8541	−2.42705
$21$	0	0
$22$	7.85410	1.67450
$23$	−8.23607	−1.71734	−0.858669	−	0.512530i	$-0.828708\pi$
$23$	−8.23607	−1.71734	−0.858669	+	0.512530i	$0.828708\pi$
$24$	−16.7082	−3.41055
$25$	0	0
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	13.0902	2.38993
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	10.8541	1.91875
$33$	−6.70820	−1.16775
$34$	3.85410	0.660973
$35$	0	0
$36$	9.70820	1.61803
$37$	−4.70820	−0.774024	−0.387012	−	0.922075i	$-0.626493\pi$
$37$	−4.70820	−0.774024	−0.387012	+	0.922075i	$0.626493\pi$
$38$	−7.85410	−1.27410
$39$	0	0
$40$	−16.7082	−2.64180
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	14.5623	2.19535
$45$	−4.47214	−0.666667
$46$	−21.5623	−3.17919
$47$	7.47214	1.08992	0.544962	−	0.838461i	$-0.316544\pi$
$47$	7.47214	1.08992	0.544962	+	0.838461i	$0.316544\pi$
$48$	−22.0344	−3.18040
$49$	0	0
$50$	0	0
$51$	−3.29180	−0.460944
$52$	0	0
$53$	−7.47214	−1.02638	−0.513188	−	0.858276i	$-0.671536\pi$
$53$	−7.47214	−1.02638	−0.513188	+	0.858276i	$0.671536\pi$
$54$	5.85410	0.796642
$55$	−6.70820	−0.904534
$56$	0	0
$57$	6.70820	0.888523
$58$	11.7082	1.53736
$59$	1.47214	0.191656	0.0958279	−	0.995398i	$-0.469450\pi$
$59$	1.47214	0.191656	0.0958279	+	0.995398i	$0.469450\pi$
$60$	24.2705	3.13331
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	−13.0902	−1.66245
$63$	0	0
$64$	8.70820	1.08853
$65$	0	0
$66$	−17.5623	−2.16177
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	7.14590	0.866567
$69$	18.4164	2.21707
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	14.9443	1.76120
$73$	2.70820	0.316971	0.158486	−	0.987361i	$-0.449339\pi$
$73$	2.70820	0.316971	0.158486	+	0.987361i	$0.449339\pi$
$74$	−12.3262	−1.43290
$75$	0	0
$76$	−14.5623	−1.67041
$77$	0	0
$78$	0	0
$79$	−2.70820	−0.304697	−0.152348	−	0.988327i	$-0.548684\pi$
$79$	−2.70820	−0.304697	−0.152348	+	0.988327i	$0.548684\pi$
$80$	−22.0344	−2.46353
$81$	−11.0000	−1.22222
$82$	11.7082	1.29295
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−3.29180	−0.357045
$86$	−20.9443	−2.25848
$87$	−10.0000	−1.07211
$88$	22.4164	2.38960
$89$	−2.23607	−0.237023	−0.118511	−	0.992953i	$-0.537812\pi$
$89$	−2.23607	−0.237023	−0.118511	+	0.992953i	$0.537812\pi$
$90$	−11.7082	−1.23415
$91$	0	0
$92$	−39.9787	−4.16807
$93$	11.1803	1.15935
$94$	19.5623	2.01770
$95$	6.70820	0.688247
$96$	−24.2705	−2.47710
$97$	−9.41641	−0.956091	−0.478046	−	0.878335i	$-0.658655\pi$
$97$	−9.41641	−0.956091	−0.478046	+	0.878335i	$0.658655\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	−8.61803	−0.853313
$103$	−2.70820	−0.266847	−0.133424	−	0.991059i	$-0.542597\pi$
$103$	−2.70820	−0.266847	−0.133424	+	0.991059i	$0.542597\pi$
$104$	0	0
$105$	0	0
$106$	−19.5623	−1.90006
$107$	−9.76393	−0.943915	−0.471957	−	0.881621i	$-0.656452\pi$
$107$	−9.76393	−0.943915	−0.471957	+	0.881621i	$0.656452\pi$
$108$	10.8541	1.04444
$109$	2.70820	0.259399	0.129699	−	0.991553i	$-0.458599\pi$
$109$	2.70820	0.259399	0.129699	+	0.991553i	$0.458599\pi$
$110$	−17.5623	−1.67450
$111$	10.5279	0.999261
$112$	0	0
$113$	2.94427	0.276974	0.138487	−	0.990364i	$-0.455776\pi$
$113$	2.94427	0.276974	0.138487	+	0.990364i	$0.455776\pi$
$114$	17.5623	1.64486
$115$	18.4164	1.71734
$116$	21.7082	2.01556
$117$	0	0
$118$	3.85410	0.354799
$119$	0	0
$120$	37.3607	3.41055
$121$	−2.00000	−0.181818
$122$	7.85410	0.711077
$123$	−10.0000	−0.901670
$124$	−24.2705	−2.17956
$125$	11.1803	1.00000
$126$	0	0
$127$	−11.4164	−1.01304	−0.506521	−	0.862228i	$-0.669069\pi$
$127$	−11.4164	−1.01304	−0.506521	+	0.862228i	$0.669069\pi$
$128$	1.09017	0.0963583
$129$	17.8885	1.57500
$130$	0	0
$131$	−8.23607	−0.719589	−0.359794	−	0.933032i	$-0.617153\pi$
$131$	−8.23607	−0.719589	−0.359794	+	0.933032i	$0.617153\pi$
$132$	−32.5623	−2.83418
$133$	0	0
$134$	7.85410	0.678491
$135$	−5.00000	−0.430331
$136$	11.0000	0.943242
$137$	8.23607	0.703655	0.351827	−	0.936065i	$-0.385560\pi$
$137$	8.23607	0.703655	0.351827	+	0.936065i	$0.385560\pi$
$138$	48.2148	4.10431
$139$	−23.4164	−1.98615	−0.993077	−	0.117466i	$-0.962523\pi$
$139$	−23.4164	−1.98615	−0.993077	+	0.117466i	$0.962523\pi$
$140$	0	0
$141$	−16.7082	−1.40708
$142$	23.4164	1.96506
$143$	0	0
$144$	19.7082	1.64235
$145$	−10.0000	−0.830455
$146$	7.09017	0.586787
$147$	0	0
$148$	−22.8541	−1.87860
$149$	−0.708204	−0.0580183	−0.0290092	−	0.999579i	$-0.509235\pi$
$149$	−0.708204	−0.0580183	−0.0290092	+	0.999579i	$0.509235\pi$
$150$	0	0
$151$	−20.4164	−1.66146	−0.830732	−	0.556673i	$-0.812078\pi$
$151$	−20.4164	−1.66146	−0.830732	+	0.556673i	$0.812078\pi$
$152$	−22.4164	−1.81821
$153$	2.94427	0.238030
$154$	0	0
$155$	11.1803	0.898027
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−7.09017	−0.564064
$159$	16.7082	1.32505
$160$	−24.2705	−1.91875
$161$	0	0
$162$	−28.7984	−2.26261
$163$	16.4164	1.28583	0.642916	−	0.765937i	$-0.277724\pi$
$163$	16.4164	1.28583	0.642916	+	0.765937i	$0.277724\pi$
$164$	21.7082	1.69513
$165$	15.0000	1.16775
$166$	0	0
$167$	−22.4721	−1.73895	−0.869473	−	0.493980i	$-0.835541\pi$
$167$	−22.4721	−1.73895	−0.869473	+	0.493980i	$0.835541\pi$
$168$	0	0
$169$	0	0
$170$	−8.61803	−0.660973
$171$	−6.00000	−0.458831
$172$	−38.8328	−2.96097
$173$	−16.4164	−1.24812	−0.624058	−	0.781378i	$-0.714517\pi$
$173$	−16.4164	−1.24812	−0.624058	+	0.781378i	$0.714517\pi$
$174$	−26.1803	−1.98473
$175$	0	0
$176$	29.5623	2.22834
$177$	−3.29180	−0.247427
$178$	−5.85410	−0.438783
$179$	−20.1246	−1.50418	−0.752092	−	0.659058i	$-0.770955\pi$
$179$	−20.1246	−1.50418	−0.752092	+	0.659058i	$0.770955\pi$
$180$	−21.7082	−1.61803
$181$	−25.4164	−1.88919	−0.944593	−	0.328243i	$-0.893544\pi$
$181$	−25.4164	−1.88919	−0.944593	+	0.328243i	$0.893544\pi$
$182$	0	0
$183$	−6.70820	−0.495885
$184$	−61.5410	−4.53686
$185$	10.5279	0.774024
$186$	29.2705	2.14622
$187$	4.41641	0.322960
$188$	36.2705	2.64530
$189$	0	0
$190$	17.5623	1.27410
$191$	11.1803	0.808981	0.404491	−	0.914542i	$-0.367449\pi$
$191$	11.1803	0.808981	0.404491	+	0.914542i	$0.367449\pi$
$192$	−19.4721	−1.40528
$193$	−0.708204	−0.0509776	−0.0254888	−	0.999675i	$-0.508114\pi$
$193$	−0.708204	−0.0509776	−0.0254888	+	0.999675i	$0.508114\pi$
$194$	−24.6525	−1.76994
$195$	0	0
$196$	0	0
$197$	9.05573	0.645194	0.322597	−	0.946536i	$-0.395444\pi$
$197$	9.05573	0.645194	0.322597	+	0.946536i	$0.395444\pi$
$198$	15.7082	1.11633
$199$	−20.7082	−1.46797	−0.733983	−	0.679168i	$-0.762341\pi$
$199$	−20.7082	−1.46797	−0.733983	+	0.679168i	$0.762341\pi$
$200$	0	0
$201$	−6.70820	−0.473160
$202$	−23.5623	−1.65784
$203$	0	0
$204$	−15.9787	−1.11873
$205$	−10.0000	−0.698430
$206$	−7.09017	−0.493996
$207$	−16.4721	−1.14489
$208$	0	0
$209$	−9.00000	−0.622543
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−36.2705	−2.49107
$213$	−20.0000	−1.37038
$214$	−25.5623	−1.74740
$215$	17.8885	1.21999
$216$	16.7082	1.13685
$217$	0	0
$218$	7.09017	0.480207
$219$	−6.05573	−0.409208
$220$	−32.5623	−2.19535
$221$	0	0
$222$	27.5623	1.84986
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	7.70820	0.512742
$227$	−5.94427	−0.394535	−0.197268	−	0.980350i	$-0.563207\pi$
$227$	−5.94427	−0.394535	−0.197268	+	0.980350i	$0.563207\pi$
$228$	32.5623	2.15649
$229$	−24.1246	−1.59420	−0.797100	−	0.603848i	$-0.793633\pi$
$229$	−24.1246	−1.59420	−0.797100	+	0.603848i	$0.793633\pi$
$230$	48.2148	3.17919
$231$	0	0
$232$	33.4164	2.19389
$233$	11.9443	0.782495	0.391248	−	0.920285i	$-0.372044\pi$
$233$	11.9443	0.782495	0.391248	+	0.920285i	$0.372044\pi$
$234$	0	0
$235$	−16.7082	−1.08992
$236$	7.14590	0.465158
$237$	6.05573	0.393362
$238$	0	0
$239$	−19.4164	−1.25594	−0.627972	−	0.778236i	$-0.716115\pi$
$239$	−19.4164	−1.25594	−0.627972	+	0.778236i	$0.716115\pi$
$240$	49.2705	3.18040
$241$	−4.70820	−0.303282	−0.151641	−	0.988436i	$-0.548456\pi$
$241$	−4.70820	−0.303282	−0.151641	+	0.988436i	$0.548456\pi$
$242$	−5.23607	−0.336587
$243$	17.8885	1.14755
$244$	14.5623	0.932256
$245$	0	0
$246$	−26.1803	−1.66920
$247$	0	0
$248$	−37.3607	−2.37241
$249$	0	0
$250$	29.2705	1.85123
$251$	1.52786	0.0964379	0.0482190	−	0.998837i	$-0.484645\pi$
$251$	1.52786	0.0964379	0.0482190	+	0.998837i	$0.484645\pi$
$252$	0	0
$253$	−24.7082	−1.55339
$254$	−29.8885	−1.87537
$255$	7.36068	0.460944
$256$	−14.5623	−0.910144
$257$	−0.0557281	−0.00347622	−0.00173811	−	0.999998i	$-0.500553\pi$
$257$	−0.0557281	−0.00347622	−0.00173811	+	0.999998i	$0.500553\pi$
$258$	46.8328	2.91568
$259$	0	0
$260$	0	0
$261$	8.94427	0.553637
$262$	−21.5623	−1.33212
$263$	26.1246	1.61091	0.805456	−	0.592655i	$-0.201920\pi$
$263$	26.1246	1.61091	0.805456	+	0.592655i	$0.201920\pi$
$264$	−50.1246	−3.08496
$265$	16.7082	1.02638
$266$	0	0
$267$	5.00000	0.305995
$268$	14.5623	0.889534
$269$	13.4721	0.821411	0.410705	−	0.911768i	$-0.365283\pi$
$269$	13.4721	0.821411	0.410705	+	0.911768i	$0.365283\pi$
$270$	−13.0902	−0.796642
$271$	20.4164	1.24021	0.620104	−	0.784519i	$-0.287090\pi$
$271$	20.4164	1.24021	0.620104	+	0.784519i	$0.287090\pi$
$272$	14.5066	0.879590
$273$	0	0
$274$	21.5623	1.30263
$275$	0	0
$276$	89.3951	5.38095
$277$	−0.416408	−0.0250195	−0.0125098	−	0.999922i	$-0.503982\pi$
$277$	−0.416408	−0.0250195	−0.0125098	+	0.999922i	$0.503982\pi$
$278$	−61.3050	−3.67683
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−26.9443	−1.60736	−0.803680	−	0.595061i	$-0.797128\pi$
$281$	−26.9443	−1.60736	−0.803680	+	0.595061i	$0.797128\pi$
$282$	−43.7426	−2.60484
$283$	26.1246	1.55295	0.776473	−	0.630150i	$-0.217007\pi$
$283$	26.1246	1.55295	0.776473	+	0.630150i	$0.217007\pi$
$284$	43.4164	2.57629
$285$	−15.0000	−0.888523
$286$	0	0
$287$	0	0
$288$	21.7082	1.27917
$289$	−14.8328	−0.872519
$290$	−26.1803	−1.53736
$291$	21.0557	1.23431
$292$	13.1459	0.769305
$293$	14.9443	0.873054	0.436527	−	0.899691i	$-0.356208\pi$
$293$	14.9443	0.873054	0.436527	+	0.899691i	$0.356208\pi$
$294$	0	0
$295$	−3.29180	−0.191656
$296$	−35.1803	−2.04482
$297$	6.70820	0.389249
$298$	−1.85410	−0.107405
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−53.4508	−3.07575
$303$	20.1246	1.15613
$304$	−29.5623	−1.69551
$305$	−6.70820	−0.384111
$306$	7.70820	0.440649
$307$	−19.4164	−1.10815	−0.554076	−	0.832466i	$-0.686928\pi$
$307$	−19.4164	−1.10815	−0.554076	+	0.832466i	$0.686928\pi$
$308$	0	0
$309$	6.05573	0.344498
$310$	29.2705	1.66245
$311$	−27.7639	−1.57435	−0.787174	−	0.616731i	$-0.788457\pi$
$311$	−27.7639	−1.57435	−0.787174	+	0.616731i	$0.788457\pi$
$312$	0	0
$313$	−5.58359	−0.315603	−0.157802	−	0.987471i	$-0.550441\pi$
$313$	−5.58359	−0.315603	−0.157802	+	0.987471i	$0.550441\pi$
$314$	18.3262	1.03421
$315$	0	0
$316$	−13.1459	−0.739515
$317$	8.23607	0.462584	0.231292	−	0.972884i	$-0.425705\pi$
$317$	8.23607	0.462584	0.231292	+	0.972884i	$0.425705\pi$
$318$	43.7426	2.45297
$319$	13.4164	0.751175
$320$	−19.4721	−1.08853
$321$	21.8328	1.21859
$322$	0	0
$323$	−4.41641	−0.245736
$324$	−53.3951	−2.96640
$325$	0	0
$326$	42.9787	2.38037
$327$	−6.05573	−0.334883
$328$	33.4164	1.84511
$329$	0	0
$330$	39.2705	2.16177
$331$	1.58359	0.0870421	0.0435210	−	0.999053i	$-0.486142\pi$
$331$	1.58359	0.0870421	0.0435210	+	0.999053i	$0.486142\pi$
$332$	0	0
$333$	−9.41641	−0.516016
$334$	−58.8328	−3.21919
$335$	−6.70820	−0.366508
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−6.58359	−0.357572
$340$	−15.9787	−0.866567
$341$	−15.0000	−0.812296
$342$	−15.7082	−0.849402
$343$	0	0
$344$	−59.7771	−3.22296
$345$	−41.1803	−2.21707
$346$	−42.9787	−2.31055
$347$	23.0689	1.23840	0.619201	−	0.785232i	$-0.287456\pi$
$347$	23.0689	1.23840	0.619201	+	0.785232i	$0.287456\pi$
$348$	−48.5410	−2.60207
$349$	29.4164	1.57462	0.787312	−	0.616555i	$-0.211472\pi$
$349$	29.4164	1.57462	0.787312	+	0.616555i	$0.211472\pi$
$350$	0	0
$351$	0	0
$352$	32.5623	1.73558
$353$	−17.2918	−0.920349	−0.460175	−	0.887828i	$-0.652213\pi$
$353$	−17.2918	−0.920349	−0.460175	+	0.887828i	$0.652213\pi$
$354$	−8.61803	−0.458043
$355$	−20.0000	−1.06149
$356$	−10.8541	−0.575266
$357$	0	0
$358$	−52.6869	−2.78459
$359$	−11.9443	−0.630395	−0.315197	−	0.949026i	$-0.602071\pi$
$359$	−11.9443	−0.630395	−0.315197	+	0.949026i	$0.602071\pi$
$360$	−33.4164	−1.76120
$361$	−10.0000	−0.526316
$362$	−66.5410	−3.49732
$363$	4.47214	0.234726
$364$	0	0
$365$	−6.05573	−0.316971
$366$	−17.5623	−0.917996
$367$	−12.7082	−0.663363	−0.331681	−	0.943391i	$-0.607616\pi$
$367$	−12.7082	−0.663363	−0.331681	+	0.943391i	$0.607616\pi$
$368$	−81.1591	−4.23071
$369$	8.94427	0.465620
$370$	27.5623	1.43290
$371$	0	0
$372$	54.2705	2.81379
$373$	−1.58359	−0.0819953	−0.0409976	−	0.999159i	$-0.513054\pi$
$373$	−1.58359	−0.0819953	−0.0409976	+	0.999159i	$0.513054\pi$
$374$	11.5623	0.597873
$375$	−25.0000	−1.29099
$376$	55.8328	2.87936
$377$	0	0
$378$	0	0
$379$	15.4164	0.791888	0.395944	−	0.918275i	$-0.370417\pi$
$379$	15.4164	0.791888	0.395944	+	0.918275i	$0.370417\pi$
$380$	32.5623	1.67041
$381$	25.5279	1.30783
$382$	29.2705	1.49761
$383$	−15.0000	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$383$	−15.0000	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$384$	−2.43769	−0.124398
$385$	0	0
$386$	−1.85410	−0.0943713
$387$	−16.0000	−0.813326
$388$	−45.7082	−2.32048
$389$	1.47214	0.0746403	0.0373201	−	0.999303i	$-0.488118\pi$
$389$	1.47214	0.0746403	0.0373201	+	0.999303i	$0.488118\pi$
$390$	0	0
$391$	−12.1246	−0.613168
$392$	0	0
$393$	18.4164	0.928985
$394$	23.7082	1.19440
$395$	6.05573	0.304697
$396$	29.1246	1.46357
$397$	26.1246	1.31116	0.655578	−	0.755127i	$-0.272425\pi$
$397$	26.1246	1.31116	0.655578	+	0.755127i	$0.272425\pi$
$398$	−54.2148	−2.71754
$399$	0	0
$400$	0	0
$401$	−14.2361	−0.710915	−0.355458	−	0.934692i	$-0.615675\pi$
$401$	−14.2361	−0.710915	−0.355458	+	0.934692i	$0.615675\pi$
$402$	−17.5623	−0.875928
$403$	0	0
$404$	−43.6869	−2.17351
$405$	24.5967	1.22222
$406$	0	0
$407$	−14.1246	−0.700131
$408$	−24.5967	−1.21772
$409$	−8.70820	−0.430593	−0.215296	−	0.976549i	$-0.569072\pi$
$409$	−8.70820	−0.430593	−0.215296	+	0.976549i	$0.569072\pi$
$410$	−26.1803	−1.29295
$411$	−18.4164	−0.908414
$412$	−13.1459	−0.647652
$413$	0	0
$414$	−43.1246	−2.11946
$415$	0	0
$416$	0	0
$417$	52.3607	2.56411
$418$	−23.5623	−1.15247
$419$	32.9443	1.60943	0.804717	−	0.593659i	$-0.202317\pi$
$419$	32.9443	1.60943	0.804717	+	0.593659i	$0.202317\pi$
$420$	0	0
$421$	−13.4164	−0.653876	−0.326938	−	0.945046i	$-0.606017\pi$
$421$	−13.4164	−0.653876	−0.326938	+	0.945046i	$0.606017\pi$
$422$	10.4721	0.509776
$423$	14.9443	0.726615
$424$	−55.8328	−2.71148
$425$	0	0
$426$	−52.3607	−2.53688
$427$	0	0
$428$	−47.3951	−2.29093
$429$	0	0
$430$	46.8328	2.25848
$431$	31.3607	1.51059	0.755295	−	0.655385i	$-0.227493\pi$
$431$	31.3607	1.51059	0.755295	+	0.655385i	$0.227493\pi$
$432$	22.0344	1.06013
$433$	29.4164	1.41366	0.706831	−	0.707382i	$-0.250124\pi$
$433$	29.4164	1.41366	0.706831	+	0.707382i	$0.250124\pi$
$434$	0	0
$435$	22.3607	1.07211
$436$	13.1459	0.629574
$437$	24.7082	1.18195
$438$	−15.8541	−0.757538
$439$	24.1246	1.15140	0.575702	−	0.817659i	$-0.304729\pi$
$439$	24.1246	1.15140	0.575702	+	0.817659i	$0.304729\pi$
$440$	−50.1246	−2.38960
$441$	0	0
$442$	0	0
$443$	2.23607	0.106239	0.0531194	−	0.998588i	$-0.483084\pi$
$443$	2.23607	0.106239	0.0531194	+	0.998588i	$0.483084\pi$
$444$	51.1033	2.42526
$445$	5.00000	0.237023
$446$	10.4721	0.495870
$447$	1.58359	0.0749013
$448$	0	0
$449$	34.3607	1.62158	0.810790	−	0.585337i	$-0.199038\pi$
$449$	34.3607	1.62158	0.810790	+	0.585337i	$0.199038\pi$
$450$	0	0
$451$	13.4164	0.631754
$452$	14.2918	0.672230
$453$	45.6525	2.14494
$454$	−15.5623	−0.730375
$455$	0	0
$456$	50.1246	2.34730
$457$	6.12461	0.286497	0.143249	−	0.989687i	$-0.454245\pi$
$457$	6.12461	0.286497	0.143249	+	0.989687i	$0.454245\pi$
$458$	−63.1591	−2.95123
$459$	3.29180	0.153648
$460$	89.3951	4.16807
$461$	−34.3607	−1.60034	−0.800168	−	0.599776i	$-0.795256\pi$
$461$	−34.3607	−1.60034	−0.800168	+	0.599776i	$0.795256\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	44.0689	2.04585
$465$	−25.0000	−1.15935
$466$	31.2705	1.44858
$467$	9.65248	0.446663	0.223332	−	0.974743i	$-0.428307\pi$
$467$	9.65248	0.446663	0.223332	+	0.974743i	$0.428307\pi$
$468$	0	0
$469$	0	0
$470$	−43.7426	−2.01770
$471$	−15.6525	−0.721228
$472$	11.0000	0.506316
$473$	−24.0000	−1.10352
$474$	15.8541	0.728203
$475$	0	0
$476$	0	0
$477$	−14.9443	−0.684251
$478$	−50.8328	−2.32504
$479$	−23.8328	−1.08895	−0.544475	−	0.838777i	$-0.683271\pi$
$479$	−23.8328	−1.08895	−0.544475	+	0.838777i	$0.683271\pi$
$480$	54.2705	2.47710
$481$	0	0
$482$	−12.3262	−0.561445
$483$	0	0
$484$	−9.70820	−0.441282
$485$	21.0557	0.956091
$486$	46.8328	2.12438
$487$	21.8328	0.989339	0.494670	−	0.869081i	$-0.335289\pi$
$487$	21.8328	0.989339	0.494670	+	0.869081i	$0.335289\pi$
$488$	22.4164	1.01474
$489$	−36.7082	−1.66000
$490$	0	0
$491$	−25.5279	−1.15206	−0.576028	−	0.817430i	$-0.695398\pi$
$491$	−25.5279	−1.15206	−0.576028	+	0.817430i	$0.695398\pi$
$492$	−48.5410	−2.18840
$493$	6.58359	0.296510
$494$	0	0
$495$	−13.4164	−0.603023
$496$	−49.2705	−2.21231
$497$	0	0
$498$	0	0
$499$	−26.4164	−1.18256	−0.591280	−	0.806466i	$-0.701377\pi$
$499$	−26.4164	−1.18256	−0.591280	+	0.806466i	$0.701377\pi$
$500$	54.2705	2.42705
$501$	50.2492	2.24497
$502$	4.00000	0.178529
$503$	20.9443	0.933859	0.466929	−	0.884295i	$-0.345360\pi$
$503$	20.9443	0.933859	0.466929	+	0.884295i	$0.345360\pi$
$504$	0	0
$505$	20.1246	0.895533
$506$	−64.6869	−2.87568
$507$	0	0
$508$	−55.4164	−2.45871
$509$	20.2361	0.896948	0.448474	−	0.893796i	$-0.351968\pi$
$509$	20.2361	0.896948	0.448474	+	0.893796i	$0.351968\pi$
$510$	19.2705	0.853313
$511$	0	0
$512$	−40.3050	−1.78124
$513$	−6.70820	−0.296174
$514$	−0.145898	−0.00643529
$515$	6.05573	0.266847
$516$	86.8328	3.82260
$517$	22.4164	0.985872
$518$	0	0
$519$	36.7082	1.61131
$520$	0	0
$521$	−17.9443	−0.786153	−0.393076	−	0.919506i	$-0.628589\pi$
$521$	−17.9443	−0.786153	−0.393076	+	0.919506i	$0.628589\pi$
$522$	23.4164	1.02491
$523$	32.7082	1.43023	0.715115	−	0.699007i	$-0.246374\pi$
$523$	32.7082	1.43023	0.715115	+	0.699007i	$0.246374\pi$
$524$	−39.9787	−1.74648
$525$	0	0
$526$	68.3951	2.98217
$527$	−7.36068	−0.320636
$528$	−66.1033	−2.87678
$529$	44.8328	1.94925
$530$	43.7426	1.90006
$531$	2.94427	0.127771
$532$	0	0
$533$	0	0
$534$	13.0902	0.566467
$535$	21.8328	0.943915
$536$	22.4164	0.968241
$537$	45.0000	1.94189
$538$	35.2705	1.52062
$539$	0	0
$540$	−24.2705	−1.04444
$541$	−1.29180	−0.0555387	−0.0277693	−	0.999614i	$-0.508840\pi$
$541$	−1.29180	−0.0555387	−0.0277693	+	0.999614i	$0.508840\pi$
$542$	53.4508	2.29591
$543$	56.8328	2.43893
$544$	15.9787	0.685082
$545$	−6.05573	−0.259399
$546$	0	0
$547$	−4.58359	−0.195980	−0.0979901	−	0.995187i	$-0.531241\pi$
$547$	−4.58359	−0.195980	−0.0979901	+	0.995187i	$0.531241\pi$
$548$	39.9787	1.70781
$549$	6.00000	0.256074
$550$	0	0
$551$	−13.4164	−0.571558
$552$	137.610	5.85707
$553$	0	0
$554$	−1.09017	−0.0463169
$555$	−23.5410	−0.999261
$556$	−113.666	−4.82050
$557$	18.7082	0.792692	0.396346	−	0.918101i	$-0.370278\pi$
$557$	18.7082	0.792692	0.396346	+	0.918101i	$0.370278\pi$
$558$	−26.1803	−1.10830
$559$	0	0
$560$	0	0
$561$	−9.87539	−0.416939
$562$	−70.5410	−2.97559
$563$	12.5967	0.530890	0.265445	−	0.964126i	$-0.414481\pi$
$563$	12.5967	0.530890	0.265445	+	0.964126i	$0.414481\pi$
$564$	−81.1033	−3.41507
$565$	−6.58359	−0.276974
$566$	68.3951	2.87486
$567$	0	0
$568$	66.8328	2.80424
$569$	25.4721	1.06785	0.533924	−	0.845533i	$-0.320717\pi$
$569$	25.4721	1.06785	0.533924	+	0.845533i	$0.320717\pi$
$570$	−39.2705	−1.64486
$571$	−36.1246	−1.51177	−0.755884	−	0.654706i	$-0.772793\pi$
$571$	−36.1246	−1.51177	−0.755884	+	0.654706i	$0.772793\pi$
$572$	0	0
$573$	−25.0000	−1.04439
$574$	0	0
$575$	0	0
$576$	17.4164	0.725684
$577$	19.2918	0.803128	0.401564	−	0.915831i	$-0.368467\pi$
$577$	19.2918	0.803128	0.401564	+	0.915831i	$0.368467\pi$
$578$	−38.8328	−1.61523
$579$	1.58359	0.0658118
$580$	−48.5410	−2.01556
$581$	0	0
$582$	55.1246	2.28499
$583$	−22.4164	−0.928393
$584$	20.2361	0.837374
$585$	0	0
$586$	39.1246	1.61622
$587$	6.11146	0.252247	0.126123	−	0.992015i	$-0.459746\pi$
$587$	6.11146	0.252247	0.126123	+	0.992015i	$0.459746\pi$
$588$	0	0
$589$	15.0000	0.618064
$590$	−8.61803	−0.354799
$591$	−20.2492	−0.832942
$592$	−46.3951	−1.90683
$593$	−27.7639	−1.14013	−0.570064	−	0.821600i	$-0.693082\pi$
$593$	−27.7639	−1.14013	−0.570064	+	0.821600i	$0.693082\pi$
$594$	17.5623	0.720590
$595$	0	0
$596$	−3.43769	−0.140813
$597$	46.3050	1.89514
$598$	0	0
$599$	−17.0689	−0.697416	−0.348708	−	0.937231i	$-0.613379\pi$
$599$	−17.0689	−0.697416	−0.348708	+	0.937231i	$0.613379\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	−99.1033	−4.03246
$605$	4.47214	0.181818
$606$	52.6869	2.14026
$607$	24.1246	0.979188	0.489594	−	0.871951i	$-0.337145\pi$
$607$	24.1246	0.979188	0.489594	+	0.871951i	$0.337145\pi$
$608$	−32.5623	−1.32058
$609$	0	0
$610$	−17.5623	−0.711077
$611$	0	0
$612$	14.2918	0.577712
$613$	18.1246	0.732046	0.366023	−	0.930606i	$-0.380719\pi$
$613$	18.1246	0.732046	0.366023	+	0.930606i	$0.380719\pi$
$614$	−50.8328	−2.05145
$615$	22.3607	0.901670
$616$	0	0
$617$	4.47214	0.180041	0.0900207	−	0.995940i	$-0.471307\pi$
$617$	4.47214	0.180041	0.0900207	+	0.995940i	$0.471307\pi$
$618$	15.8541	0.637746
$619$	−17.0000	−0.683288	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000	−0.683288	−0.341644	+	0.939829i	$0.610984\pi$
$620$	54.2705	2.17956
$621$	−18.4164	−0.739025
$622$	−72.6869	−2.91448
$623$	0	0
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−14.6180	−0.584254
$627$	20.1246	0.803700
$628$	33.9787	1.35590
$629$	−6.93112	−0.276362
$630$	0	0
$631$	−22.8328	−0.908960	−0.454480	−	0.890757i	$-0.650175\pi$
$631$	−22.8328	−0.908960	−0.454480	+	0.890757i	$0.650175\pi$
$632$	−20.2361	−0.804948
$633$	−8.94427	−0.355503
$634$	21.5623	0.856349
$635$	25.5279	1.01304
$636$	81.1033	3.21596
$637$	0	0
$638$	35.1246	1.39060
$639$	17.8885	0.707660
$640$	−2.43769	−0.0963583
$641$	11.9443	0.471770	0.235885	−	0.971781i	$-0.424201\pi$
$641$	11.9443	0.471770	0.235885	+	0.971781i	$0.424201\pi$
$642$	57.1591	2.25589
$643$	−34.8328	−1.37367	−0.686836	−	0.726812i	$-0.741001\pi$
$643$	−34.8328	−1.37367	−0.686836	+	0.726812i	$0.741001\pi$
$644$	0	0
$645$	−40.0000	−1.57500
$646$	−11.5623	−0.454913
$647$	20.2361	0.795562	0.397781	−	0.917480i	$-0.369780\pi$
$647$	20.2361	0.795562	0.397781	+	0.917480i	$0.369780\pi$
$648$	−82.1935	−3.22887
$649$	4.41641	0.173359
$650$	0	0
$651$	0	0
$652$	79.6869	3.12078
$653$	4.52786	0.177189	0.0885945	−	0.996068i	$-0.471762\pi$
$653$	4.52786	0.177189	0.0885945	+	0.996068i	$0.471762\pi$
$654$	−15.8541	−0.619944
$655$	18.4164	0.719589
$656$	44.0689	1.72060
$657$	5.41641	0.211314
$658$	0	0
$659$	−8.94427	−0.348419	−0.174210	−	0.984709i	$-0.555737\pi$
$659$	−8.94427	−0.348419	−0.174210	+	0.984709i	$0.555737\pi$
$660$	72.8115	2.83418
$661$	6.70820	0.260919	0.130459	−	0.991454i	$-0.458355\pi$
$661$	6.70820	0.260919	0.130459	+	0.991454i	$0.458355\pi$
$662$	4.14590	0.161135
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−24.6525	−0.955264
$667$	−36.8328	−1.42617
$668$	−109.082	−4.22051
$669$	−8.94427	−0.345806
$670$	−17.5623	−0.678491
$671$	9.00000	0.347441
$672$	0	0
$673$	−9.41641	−0.362976	−0.181488	−	0.983393i	$-0.558091\pi$
$673$	−9.41641	−0.362976	−0.181488	+	0.983393i	$0.558091\pi$
$674$	47.1246	1.81517
$675$	0	0
$676$	0	0
$677$	−2.88854	−0.111016	−0.0555079	−	0.998458i	$-0.517678\pi$
$677$	−2.88854	−0.111016	−0.0555079	+	0.998458i	$0.517678\pi$
$678$	−17.2361	−0.661947
$679$	0	0
$680$	−24.5967	−0.943242
$681$	13.2918	0.509343
$682$	−39.2705	−1.50375
$683$	13.4721	0.515497	0.257748	−	0.966212i	$-0.417019\pi$
$683$	13.4721	0.515497	0.257748	+	0.966212i	$0.417019\pi$
$684$	−29.1246	−1.11361
$685$	−18.4164	−0.703655
$686$	0	0
$687$	53.9443	2.05810
$688$	−78.8328	−3.00547
$689$	0	0
$690$	−107.812	−4.10431
$691$	51.8328	1.97181	0.985907	−	0.167297i	$-0.0535037\pi$
$691$	51.8328	1.97181	0.985907	+	0.167297i	$0.0535037\pi$
$692$	−79.6869	−3.02924
$693$	0	0
$694$	60.3951	2.29257
$695$	52.3607	1.98615
$696$	−74.7214	−2.83231
$697$	6.58359	0.249371
$698$	77.0132	2.91499
$699$	−26.7082	−1.01020
$700$	0	0
$701$	−22.3607	−0.844551	−0.422276	−	0.906467i	$-0.638769\pi$
$701$	−22.3607	−0.844551	−0.422276	+	0.906467i	$0.638769\pi$
$702$	0	0
$703$	14.1246	0.532720
$704$	26.1246	0.984608
$705$	37.3607	1.40708
$706$	−45.2705	−1.70378
$707$	0	0
$708$	−15.9787	−0.600517
$709$	50.1246	1.88247	0.941235	−	0.337753i	$-0.109667\pi$
$709$	50.1246	1.88247	0.941235	+	0.337753i	$0.109667\pi$
$710$	−52.3607	−1.96506
$711$	−5.41641	−0.203131
$712$	−16.7082	−0.626166
$713$	41.1803	1.54222
$714$	0	0
$715$	0	0
$716$	−97.6869	−3.65073
$717$	43.4164	1.62142
$718$	−31.2705	−1.16701
$719$	24.7082	0.921461	0.460730	−	0.887540i	$-0.347588\pi$
$719$	24.7082	0.921461	0.460730	+	0.887540i	$0.347588\pi$
$720$	−44.0689	−1.64235
$721$	0	0
$722$	−26.1803	−0.974331
$723$	10.5279	0.391535
$724$	−123.374	−4.58515
$725$	0	0
$726$	11.7082	0.434532
$727$	−38.8328	−1.44023	−0.720115	−	0.693855i	$-0.755911\pi$
$727$	−38.8328	−1.44023	−0.720115	+	0.693855i	$0.755911\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	−15.8541	−0.586787
$731$	−11.7771	−0.435591
$732$	−32.5623	−1.20354
$733$	−28.7082	−1.06036	−0.530181	−	0.847885i	$-0.677876\pi$
$733$	−28.7082	−1.06036	−0.530181	+	0.847885i	$0.677876\pi$
$734$	−33.2705	−1.22804
$735$	0	0
$736$	−89.3951	−3.29515
$737$	9.00000	0.331519
$738$	23.4164	0.861970
$739$	17.8328	0.655991	0.327995	−	0.944679i	$-0.393627\pi$
$739$	17.8328	0.655991	0.327995	+	0.944679i	$0.393627\pi$
$740$	51.1033	1.87860
$741$	0	0
$742$	0	0
$743$	−32.9443	−1.20861	−0.604304	−	0.796754i	$-0.706549\pi$
$743$	−32.9443	−1.20861	−0.604304	+	0.796754i	$0.706549\pi$
$744$	83.5410	3.06276
$745$	1.58359	0.0580183
$746$	−4.14590	−0.151792
$747$	0	0
$748$	21.4377	0.783840
$749$	0	0
$750$	−65.4508	−2.38993
$751$	−10.1246	−0.369452	−0.184726	−	0.982790i	$-0.559140\pi$
$751$	−10.1246	−0.369452	−0.184726	+	0.982790i	$0.559140\pi$
$752$	73.6312	2.68505
$753$	−3.41641	−0.124501
$754$	0	0
$755$	45.6525	1.66146
$756$	0	0
$757$	−52.8328	−1.92024	−0.960121	−	0.279586i	$-0.909803\pi$
$757$	−52.8328	−1.92024	−0.960121	+	0.279586i	$0.909803\pi$
$758$	40.3607	1.46597
$759$	55.2492	2.00542
$760$	50.1246	1.81821
$761$	33.5410	1.21586	0.607931	−	0.793990i	$-0.292000\pi$
$761$	33.5410	1.21586	0.607931	+	0.793990i	$0.292000\pi$
$762$	66.8328	2.42110
$763$	0	0
$764$	54.2705	1.96344
$765$	−6.58359	−0.238030
$766$	−39.2705	−1.41890
$767$	0	0
$768$	32.5623	1.17499
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	0.124612	0.00448778
$772$	−3.43769	−0.123725
$773$	47.0689	1.69295	0.846475	−	0.532428i	$-0.178720\pi$
$773$	47.0689	1.69295	0.846475	+	0.532428i	$0.178720\pi$
$774$	−41.8885	−1.50565
$775$	0	0
$776$	−70.3607	−2.52580
$777$	0	0
$778$	3.85410	0.138176
$779$	−13.4164	−0.480693
$780$	0	0
$781$	26.8328	0.960154
$782$	−31.7426	−1.13511
$783$	10.0000	0.357371
$784$	0	0
$785$	−15.6525	−0.558661
$786$	48.2148	1.71976
$787$	−20.4164	−0.727766	−0.363883	−	0.931445i	$-0.618549\pi$
$787$	−20.4164	−0.727766	−0.363883	+	0.931445i	$0.618549\pi$
$788$	43.9574	1.56592
$789$	−58.4164	−2.07968
$790$	15.8541	0.564064
$791$	0	0
$792$	44.8328	1.59306
$793$	0	0
$794$	68.3951	2.42725
$795$	−37.3607	−1.32505
$796$	−100.520	−3.56283
$797$	9.05573	0.320770	0.160385	−	0.987055i	$-0.448726\pi$
$797$	9.05573	0.320770	0.160385	+	0.987055i	$0.448726\pi$
$798$	0	0
$799$	11.0000	0.389152
$800$	0	0
$801$	−4.47214	−0.158015
$802$	−37.2705	−1.31607
$803$	8.12461	0.286711
$804$	−32.5623	−1.14838
$805$	0	0
$806$	0	0
$807$	−30.1246	−1.06044
$808$	−67.2492	−2.36582
$809$	22.4164	0.788119	0.394059	−	0.919085i	$-0.371070\pi$
$809$	22.4164	0.788119	0.394059	+	0.919085i	$0.371070\pi$
$810$	64.3951	2.26261
$811$	14.8328	0.520851	0.260425	−	0.965494i	$-0.416137\pi$
$811$	14.8328	0.520851	0.260425	+	0.965494i	$0.416137\pi$
$812$	0	0
$813$	−45.6525	−1.60110
$814$	−36.9787	−1.29610
$815$	−36.7082	−1.28583
$816$	−32.4377	−1.13555
$817$	24.0000	0.839654
$818$	−22.7984	−0.797126
$819$	0	0
$820$	−48.5410	−1.69513
$821$	−38.2361	−1.33445	−0.667224	−	0.744857i	$-0.732518\pi$
$821$	−38.2361	−1.33445	−0.667224	+	0.744857i	$0.732518\pi$
$822$	−48.2148	−1.68168
$823$	34.1246	1.18951	0.594755	−	0.803907i	$-0.297249\pi$
$823$	34.1246	1.18951	0.594755	+	0.803907i	$0.297249\pi$
$824$	−20.2361	−0.704957
$825$	0	0
$826$	0	0
$827$	−26.8328	−0.933068	−0.466534	−	0.884503i	$-0.654498\pi$
$827$	−26.8328	−0.933068	−0.466534	+	0.884503i	$0.654498\pi$
$828$	−79.9574	−2.77871
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	0.931116	0.0323001
$832$	0	0
$833$	0	0
$834$	137.082	4.74676
$835$	50.2492	1.73895
$836$	−43.6869	−1.51094
$837$	−11.1803	−0.386449
$838$	86.2492	2.97943
$839$	5.88854	0.203295	0.101648	−	0.994820i	$-0.467589\pi$
$839$	5.88854	0.203295	0.101648	+	0.994820i	$0.467589\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−35.1246	−1.21047
$843$	60.2492	2.07509
$844$	19.4164	0.668340
$845$	0	0
$846$	39.1246	1.34513
$847$	0	0
$848$	−73.6312	−2.52851
$849$	−58.4164	−2.00485
$850$	0	0
$851$	38.7771	1.32926
$852$	−97.0820	−3.32598
$853$	52.2492	1.78898	0.894490	−	0.447089i	$-0.147539\pi$
$853$	52.2492	1.78898	0.894490	+	0.447089i	$0.147539\pi$
$854$	0	0
$855$	13.4164	0.458831
$856$	−72.9574	−2.49363
$857$	−1.36068	−0.0464799	−0.0232400	−	0.999730i	$-0.507398\pi$
$857$	−1.36068	−0.0464799	−0.0232400	+	0.999730i	$0.507398\pi$
$858$	0	0
$859$	20.7082	0.706555	0.353277	−	0.935519i	$-0.385067\pi$
$859$	20.7082	0.706555	0.353277	+	0.935519i	$0.385067\pi$
$860$	86.8328	2.96097
$861$	0	0
$862$	82.1033	2.79645
$863$	−23.9443	−0.815072	−0.407536	−	0.913189i	$-0.633612\pi$
$863$	−23.9443	−0.815072	−0.407536	+	0.913189i	$0.633612\pi$
$864$	24.2705	0.825700
$865$	36.7082	1.24812
$866$	77.0132	2.61701
$867$	33.1672	1.12642
$868$	0	0
$869$	−8.12461	−0.275609
$870$	58.5410	1.98473
$871$	0	0
$872$	20.2361	0.685280
$873$	−18.8328	−0.637394
$874$	64.6869	2.18807
$875$	0	0
$876$	−29.3951	−0.993169
$877$	−32.1246	−1.08477	−0.542386	−	0.840130i	$-0.682479\pi$
$877$	−32.1246	−1.08477	−0.542386	+	0.840130i	$0.682479\pi$
$878$	63.1591	2.13151
$879$	−33.4164	−1.12711
$880$	−66.1033	−2.22834
$881$	43.3050	1.45898	0.729490	−	0.683991i	$-0.239757\pi$
$881$	43.3050	1.45898	0.729490	+	0.683991i	$0.239757\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	7.36068	0.247427
$886$	5.85410	0.196672
$887$	−20.2361	−0.679461	−0.339730	−	0.940523i	$-0.610336\pi$
$887$	−20.2361	−0.679461	−0.339730	+	0.940523i	$0.610336\pi$
$888$	78.6656	2.63985
$889$	0	0
$890$	13.0902	0.438783
$891$	−33.0000	−1.10554
$892$	19.4164	0.650109
$893$	−22.4164	−0.750136
$894$	4.14590	0.138660
$895$	45.0000	1.50418
$896$	0	0
$897$	0	0
$898$	89.9574	3.00192
$899$	−22.3607	−0.745770
$900$	0	0
$901$	−11.0000	−0.366463
$902$	35.1246	1.16952
$903$	0	0
$904$	22.0000	0.731709
$905$	56.8328	1.88919
$906$	119.520	3.97078
$907$	18.7082	0.621196	0.310598	−	0.950541i	$-0.399471\pi$
$907$	18.7082	0.621196	0.310598	+	0.950541i	$0.399471\pi$
$908$	−28.8541	−0.957557
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−34.2492	−1.13473	−0.567364	−	0.823467i	$-0.692037\pi$
$911$	−34.2492	−1.13473	−0.567364	+	0.823467i	$0.692037\pi$
$912$	66.1033	2.18890
$913$	0	0
$914$	16.0344	0.530372
$915$	15.0000	0.495885
$916$	−117.103	−3.86920
$917$	0	0
$918$	8.61803	0.284438
$919$	20.1246	0.663850	0.331925	−	0.943306i	$-0.392302\pi$
$919$	20.1246	0.663850	0.331925	+	0.943306i	$0.392302\pi$
$920$	137.610	4.53686
$921$	43.4164	1.43062
$922$	−89.9574	−2.96259
$923$	0	0
$924$	0	0
$925$	0	0
$926$	62.8328	2.06481
$927$	−5.41641	−0.177898
$928$	48.5410	1.59344
$929$	24.8197	0.814307	0.407153	−	0.913360i	$-0.366521\pi$
$929$	24.8197	0.814307	0.407153	+	0.913360i	$0.366521\pi$
$930$	−65.4508	−2.14622
$931$	0	0
$932$	57.9787	1.89916
$933$	62.0820	2.03247
$934$	25.2705	0.826876
$935$	−9.87539	−0.322960
$936$	0	0
$937$	25.4164	0.830318	0.415159	−	0.909749i	$-0.363726\pi$
$937$	25.4164	0.830318	0.415159	+	0.909749i	$0.363726\pi$
$938$	0	0
$939$	12.4853	0.407442
$940$	−81.1033	−2.64530
$941$	−50.2361	−1.63765	−0.818825	−	0.574044i	$-0.805374\pi$
$941$	−50.2361	−1.63765	−0.818825	+	0.574044i	$0.805374\pi$
$942$	−40.9787	−1.33516
$943$	−36.8328	−1.19944
$944$	14.5066	0.472149
$945$	0	0
$946$	−62.8328	−2.04287
$947$	22.5279	0.732057	0.366029	−	0.930604i	$-0.380717\pi$
$947$	22.5279	0.732057	0.366029	+	0.930604i	$0.380717\pi$
$948$	29.3951	0.954709
$949$	0	0
$950$	0	0
$951$	−18.4164	−0.597193
$952$	0	0
$953$	−41.7771	−1.35329	−0.676646	−	0.736308i	$-0.736567\pi$
$953$	−41.7771	−1.35329	−0.676646	+	0.736308i	$0.736567\pi$
$954$	−39.1246	−1.26671
$955$	−25.0000	−0.808981
$956$	−94.2492	−3.04824
$957$	−30.0000	−0.969762
$958$	−62.3951	−2.01589
$959$	0	0
$960$	43.5410	1.40528
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−19.5279	−0.629277
$964$	−22.8541	−0.736081
$965$	1.58359	0.0509776
$966$	0	0
$967$	−16.5836	−0.533292	−0.266646	−	0.963794i	$-0.585916\pi$
$967$	−16.5836	−0.533292	−0.266646	+	0.963794i	$0.585916\pi$
$968$	−14.9443	−0.480327
$969$	9.87539	0.317243
$970$	55.1246	1.76994
$971$	−11.2918	−0.362371	−0.181185	−	0.983449i	$-0.557993\pi$
$971$	−11.2918	−0.362371	−0.181185	+	0.983449i	$0.557993\pi$
$972$	86.8328	2.78516
$973$	0	0
$974$	57.1591	1.83149
$975$	0	0
$976$	29.5623	0.946266
$977$	2.34752	0.0751040	0.0375520	−	0.999295i	$-0.488044\pi$
$977$	2.34752	0.0751040	0.0375520	+	0.999295i	$0.488044\pi$
$978$	−96.1033	−3.07305
$979$	−6.70820	−0.214395
$980$	0	0
$981$	5.41641	0.172933
$982$	−66.8328	−2.13272
$983$	−7.47214	−0.238324	−0.119162	−	0.992875i	$-0.538021\pi$
$983$	−7.47214	−0.238324	−0.119162	+	0.992875i	$0.538021\pi$
$984$	−74.7214	−2.38203
$985$	−20.2492	−0.645194
$986$	17.2361	0.548908
$987$	0	0
$988$	0	0
$989$	65.8885	2.09513
$990$	−35.1246	−1.11633
$991$	30.7082	0.975478	0.487739	−	0.872989i	$-0.337822\pi$
$991$	30.7082	0.975478	0.487739	+	0.872989i	$0.337822\pi$
$992$	−54.2705	−1.72309
$993$	−3.54102	−0.112371
$994$	0	0
$995$	46.3050	1.46797
$996$	0	0
$997$	26.4164	0.836616	0.418308	−	0.908305i	$-0.362623\pi$
$997$	26.4164	0.836616	0.418308	+	0.908305i	$0.362623\pi$
$998$	−69.1591	−2.18919
$999$	−10.5279	−0.333087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.z.1.2		2
7.2	even	3		1183.2.e.d.508.1		4
7.4	even	3		1183.2.e.d.170.1		4
7.6	odd	2		8281.2.a.ba.1.2		2
13.12	even	2		637.2.a.f.1.1		2
39.38	odd	2		5733.2.a.v.1.2		2
91.12	odd	6		637.2.e.h.508.2		4
91.25	even	6		91.2.e.b.79.2	yes	4
91.38	odd	6		637.2.e.h.79.2		4
91.51	even	6		91.2.e.b.53.2	✓	4
91.90	odd	2		637.2.a.e.1.1		2
273.116	odd	6		819.2.j.c.352.1		4
273.233	odd	6		819.2.j.c.235.1		4
273.272	even	2		5733.2.a.w.1.2		2
364.51	odd	6		1456.2.r.j.417.1		4
364.207	odd	6		1456.2.r.j.625.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.b.53.2	✓	4	91.51	even	6
91.2.e.b.79.2	yes	4	91.25	even	6
637.2.a.e.1.1		2	91.90	odd	2
637.2.a.f.1.1		2	13.12	even	2
637.2.e.h.79.2		4	91.38	odd	6
637.2.e.h.508.2		4	91.12	odd	6
819.2.j.c.235.1		4	273.233	odd	6
819.2.j.c.352.1		4	273.116	odd	6
1183.2.e.d.170.1		4	7.4	even	3
1183.2.e.d.508.1		4	7.2	even	3
1456.2.r.j.417.1		4	364.51	odd	6
1456.2.r.j.625.1		4	364.207	odd	6
5733.2.a.v.1.2		2	39.38	odd	2
5733.2.a.w.1.2		2	273.272	even	2
8281.2.a.z.1.2		2	1.1	even	1	trivial
8281.2.a.ba.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} -2.23607 q^{3} +4.85410 q^{4} -2.23607 q^{5} -5.85410 q^{6} +7.47214 q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+2.61803 q^{2} -2.23607 q^{3} +4.85410 q^{4} -2.23607 q^{5} -5.85410 q^{6} +7.47214 q^{8} +2.00000 q^{9} -5.85410 q^{10} +3.00000 q^{11} -10.8541 q^{12} +5.00000 q^{15} +9.85410 q^{16} +1.47214 q^{17} +5.23607 q^{18} -3.00000 q^{19} -10.8541 q^{20} +7.85410 q^{22} -8.23607 q^{23} -16.7082 q^{24} +2.23607 q^{27} +4.47214 q^{29} +13.0902 q^{30} -5.00000 q^{31} +10.8541 q^{32} -6.70820 q^{33} +3.85410 q^{34} +9.70820 q^{36} -4.70820 q^{37} -7.85410 q^{38} -16.7082 q^{40} +4.47214 q^{41} -8.00000 q^{43} +14.5623 q^{44} -4.47214 q^{45} -21.5623 q^{46} +7.47214 q^{47} -22.0344 q^{48} -3.29180 q^{51} -7.47214 q^{53} +5.85410 q^{54} -6.70820 q^{55} +6.70820 q^{57} +11.7082 q^{58} +1.47214 q^{59} +24.2705 q^{60} +3.00000 q^{61} -13.0902 q^{62} +8.70820 q^{64} -17.5623 q^{66} +3.00000 q^{67} +7.14590 q^{68} +18.4164 q^{69} +8.94427 q^{71} +14.9443 q^{72} +2.70820 q^{73} -12.3262 q^{74} -14.5623 q^{76} -2.70820 q^{79} -22.0344 q^{80} -11.0000 q^{81} +11.7082 q^{82} -3.29180 q^{85} -20.9443 q^{86} -10.0000 q^{87} +22.4164 q^{88} -2.23607 q^{89} -11.7082 q^{90} -39.9787 q^{92} +11.1803 q^{93} +19.5623 q^{94} +6.70820 q^{95} -24.2705 q^{96} -9.41641 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9	\( 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9} - 5 q^{10} + 6 q^{11} - 15 q^{12} + 10 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} - 6 q^{19} - 15 q^{20} + 9 q^{22} - 12 q^{23} - 20 q^{24} + 15 q^{30} - 10 q^{31} + 15 q^{32} + q^{34} + 6 q^{36} + 4 q^{37} - 9 q^{38} - 20 q^{40} - 16 q^{43} + 9 q^{44} - 23 q^{46} + 6 q^{47} - 15 q^{48} - 20 q^{51} - 6 q^{53} + 5 q^{54} + 10 q^{58} - 6 q^{59} + 15 q^{60} + 6 q^{61} - 15 q^{62} + 4 q^{64} - 15 q^{66} + 6 q^{67} + 21 q^{68} + 10 q^{69} + 12 q^{72} - 8 q^{73} - 9 q^{74} - 9 q^{76} + 8 q^{79} - 15 q^{80} - 22 q^{81} + 10 q^{82} - 20 q^{85} - 24 q^{86} - 20 q^{87} + 18 q^{88} - 10 q^{90} - 33 q^{92} + 19 q^{94} - 15 q^{96} + 8 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9 - 5 * q^10 + 6 * q^11 - 15 * q^12 + 10 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 - 6 * q^19 - 15 * q^20 + 9 * q^22 - 12 * q^23 - 20 * q^24 + 15 * q^30 - 10 * q^31 + 15 * q^32 + q^34 + 6 * q^36 + 4 * q^37 - 9 * q^38 - 20 * q^40 - 16 * q^43 + 9 * q^44 - 23 * q^46 + 6 * q^47 - 15 * q^48 - 20 * q^51 - 6 * q^53 + 5 * q^54 + 10 * q^58 - 6 * q^59 + 15 * q^60 + 6 * q^61 - 15 * q^62 + 4 * q^64 - 15 * q^66 + 6 * q^67 + 21 * q^68 + 10 * q^69 + 12 * q^72 - 8 * q^73 - 9 * q^74 - 9 * q^76 + 8 * q^79 - 15 * q^80 - 22 * q^81 + 10 * q^82 - 20 * q^85 - 24 * q^86 - 20 * q^87 + 18 * q^88 - 10 * q^90 - 33 * q^92 + 19 * q^94 - 15 * q^96 + 8 * q^97 + 12 * q^99

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