LMFDB - Embedded newform 8673.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8673,2,Mod(1,8673)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8673, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8673.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8673 = 3 \cdot 7^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8673.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:	$69.2542536731$
Analytic rank:	$0$
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 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8673.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -2.23607 q^{5} -1.61803 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -2.23607 q^{5} -1.61803 q^{6} +2.23607 q^{8} +1.00000 q^{9} +3.61803 q^{10} -2.23607 q^{11} +0.618034 q^{12} +1.76393 q^{13} -2.23607 q^{15} -4.85410 q^{16} +4.85410 q^{17} -1.61803 q^{18} +8.09017 q^{19} -1.38197 q^{20} +3.61803 q^{22} -2.38197 q^{23} +2.23607 q^{24} -2.85410 q^{26} +1.00000 q^{27} +8.61803 q^{29} +3.61803 q^{30} -9.56231 q^{31} +3.38197 q^{32} -2.23607 q^{33} -7.85410 q^{34} +0.618034 q^{36} -6.85410 q^{37} -13.0902 q^{38} +1.76393 q^{39} -5.00000 q^{40} +3.09017 q^{41} +4.70820 q^{43} -1.38197 q^{44} -2.23607 q^{45} +3.85410 q^{46} +4.14590 q^{47} -4.85410 q^{48} +4.85410 q^{51} +1.09017 q^{52} +1.76393 q^{53} -1.61803 q^{54} +5.00000 q^{55} +8.09017 q^{57} -13.9443 q^{58} +1.00000 q^{59} -1.38197 q^{60} +9.85410 q^{61} +15.4721 q^{62} +4.23607 q^{64} -3.94427 q^{65} +3.61803 q^{66} -2.70820 q^{67} +3.00000 q^{68} -2.38197 q^{69} +9.94427 q^{71} +2.23607 q^{72} +5.85410 q^{73} +11.0902 q^{74} +5.00000 q^{76} -2.85410 q^{78} -3.00000 q^{79} +10.8541 q^{80} +1.00000 q^{81} -5.00000 q^{82} -0.618034 q^{83} -10.8541 q^{85} -7.61803 q^{86} +8.61803 q^{87} -5.00000 q^{88} -10.7984 q^{89} +3.61803 q^{90} -1.47214 q^{92} -9.56231 q^{93} -6.70820 q^{94} -18.0902 q^{95} +3.38197 q^{96} -3.00000 q^{97} -2.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + 2 * q^9	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + 2 q^{9} + 5 q^{10} - q^{12} + 8 q^{13} - 3 q^{16} + 3 q^{17} - q^{18} + 5 q^{19} - 5 q^{20} + 5 q^{22} - 7 q^{23} + q^{26} + 2 q^{27} + 15 q^{29} + 5 q^{30} + q^{31} + 9 q^{32} - 9 q^{34} - q^{36} - 7 q^{37} - 15 q^{38} + 8 q^{39} - 10 q^{40} - 5 q^{41} - 4 q^{43} - 5 q^{44} + q^{46} + 15 q^{47} - 3 q^{48} + 3 q^{51} - 9 q^{52} + 8 q^{53} - q^{54} + 10 q^{55} + 5 q^{57} - 10 q^{58} + 2 q^{59} - 5 q^{60} + 13 q^{61} + 22 q^{62} + 4 q^{64} + 10 q^{65} + 5 q^{66} + 8 q^{67} + 6 q^{68} - 7 q^{69} + 2 q^{71} + 5 q^{73} + 11 q^{74} + 10 q^{76} + q^{78} - 6 q^{79} + 15 q^{80} + 2 q^{81} - 10 q^{82} + q^{83} - 15 q^{85} - 13 q^{86} + 15 q^{87} - 10 q^{88} + 3 q^{89} + 5 q^{90} + 6 q^{92} + q^{93} - 25 q^{95} + 9 q^{96} - 6 q^{97}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + 2 * q^9 + 5 * q^10 - q^12 + 8 * q^13 - 3 * q^16 + 3 * q^17 - q^18 + 5 * q^19 - 5 * q^20 + 5 * q^22 - 7 * q^23 + q^26 + 2 * q^27 + 15 * q^29 + 5 * q^30 + q^31 + 9 * q^32 - 9 * q^34 - q^36 - 7 * q^37 - 15 * q^38 + 8 * q^39 - 10 * q^40 - 5 * q^41 - 4 * q^43 - 5 * q^44 + q^46 + 15 * q^47 - 3 * q^48 + 3 * q^51 - 9 * q^52 + 8 * q^53 - q^54 + 10 * q^55 + 5 * q^57 - 10 * q^58 + 2 * q^59 - 5 * q^60 + 13 * q^61 + 22 * q^62 + 4 * q^64 + 10 * q^65 + 5 * q^66 + 8 * q^67 + 6 * q^68 - 7 * q^69 + 2 * q^71 + 5 * q^73 + 11 * q^74 + 10 * q^76 + q^78 - 6 * q^79 + 15 * q^80 + 2 * q^81 - 10 * q^82 + q^83 - 15 * q^85 - 13 * q^86 + 15 * q^87 - 10 * q^88 + 3 * q^89 + 5 * q^90 + 6 * q^92 + q^93 - 25 * q^95 + 9 * q^96 - 6 * q^97

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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.618034	0.309017
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	−1.61803	−0.660560
$7$	0	0
$7$	0	0
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	3.61803	1.14412
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0.618034	0.178411
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	0	0
$15$	−2.23607	−0.577350
$16$	−4.85410	−1.21353
$17$	4.85410	1.17729	0.588646	−	0.808391i	$-0.299661\pi$
$17$	4.85410	1.17729	0.588646	+	0.808391i	$0.299661\pi$
$18$	−1.61803	−0.381374
$19$	8.09017	1.85601	0.928006	−	0.372565i	$-0.121522\pi$
$19$	8.09017	1.85601	0.928006	+	0.372565i	$0.121522\pi$
$20$	−1.38197	−0.309017
$21$	0	0
$22$	3.61803	0.771367
$23$	−2.38197	−0.496674	−0.248337	−	0.968674i	$-0.579884\pi$
$23$	−2.38197	−0.496674	−0.248337	+	0.968674i	$0.579884\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−2.85410	−0.559735
$27$	1.00000	0.192450
$28$	0	0
$29$	8.61803	1.60033	0.800164	−	0.599781i	$-0.204746\pi$
$29$	8.61803	1.60033	0.800164	+	0.599781i	$0.204746\pi$
$30$	3.61803	0.660560
$31$	−9.56231	−1.71744	−0.858720	−	0.512444i	$-0.828740\pi$
$31$	−9.56231	−1.71744	−0.858720	+	0.512444i	$0.828740\pi$
$32$	3.38197	0.597853
$33$	−2.23607	−0.389249
$34$	−7.85410	−1.34697
$35$	0	0
$36$	0.618034	0.103006
$37$	−6.85410	−1.12681	−0.563404	−	0.826182i	$-0.690508\pi$
$37$	−6.85410	−1.12681	−0.563404	+	0.826182i	$0.690508\pi$
$38$	−13.0902	−2.12351
$39$	1.76393	0.282455
$40$	−5.00000	−0.790569
$41$	3.09017	0.482603	0.241302	−	0.970450i	$-0.422426\pi$
$41$	3.09017	0.482603	0.241302	+	0.970450i	$0.422426\pi$
$42$	0	0
$43$	4.70820	0.717994	0.358997	−	0.933339i	$-0.383119\pi$
$43$	4.70820	0.717994	0.358997	+	0.933339i	$0.383119\pi$
$44$	−1.38197	−0.208339
$45$	−2.23607	−0.333333
$46$	3.85410	0.568256
$47$	4.14590	0.604741	0.302371	−	0.953190i	$-0.402222\pi$
$47$	4.14590	0.604741	0.302371	+	0.953190i	$0.402222\pi$
$48$	−4.85410	−0.700629
$49$	0	0
$50$	0	0
$51$	4.85410	0.679710
$52$	1.09017	0.151179
$53$	1.76393	0.242295	0.121147	−	0.992635i	$-0.461343\pi$
$53$	1.76393	0.242295	0.121147	+	0.992635i	$0.461343\pi$
$54$	−1.61803	−0.220187
$55$	5.00000	0.674200
$56$	0	0
$57$	8.09017	1.07157
$58$	−13.9443	−1.83097
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	−1.38197	−0.178411
$61$	9.85410	1.26169	0.630844	−	0.775909i	$-0.282709\pi$
$61$	9.85410	1.26169	0.630844	+	0.775909i	$0.282709\pi$
$62$	15.4721	1.96496
$63$	0	0
$64$	4.23607	0.529508
$65$	−3.94427	−0.489227
$66$	3.61803	0.445349
$67$	−2.70820	−0.330860	−0.165430	−	0.986222i	$-0.552901\pi$
$67$	−2.70820	−0.330860	−0.165430	+	0.986222i	$0.552901\pi$
$68$	3.00000	0.363803
$69$	−2.38197	−0.286755
$70$	0	0
$71$	9.94427	1.18017	0.590084	−	0.807342i	$-0.299095\pi$
$71$	9.94427	1.18017	0.590084	+	0.807342i	$0.299095\pi$
$72$	2.23607	0.263523
$73$	5.85410	0.685171	0.342585	−	0.939487i	$-0.388697\pi$
$73$	5.85410	0.685171	0.342585	+	0.939487i	$0.388697\pi$
$74$	11.0902	1.28921
$75$	0	0
$76$	5.00000	0.573539
$77$	0	0
$78$	−2.85410	−0.323163
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	10.8541	1.21353
$81$	1.00000	0.111111
$82$	−5.00000	−0.552158
$83$	−0.618034	−0.0678380	−0.0339190	−	0.999425i	$-0.510799\pi$
$83$	−0.618034	−0.0678380	−0.0339190	+	0.999425i	$0.510799\pi$
$84$	0	0
$85$	−10.8541	−1.17729
$86$	−7.61803	−0.821474
$87$	8.61803	0.923950
$88$	−5.00000	−0.533002
$89$	−10.7984	−1.14463	−0.572313	−	0.820035i	$-0.693954\pi$
$89$	−10.7984	−1.14463	−0.572313	+	0.820035i	$0.693954\pi$
$90$	3.61803	0.381374
$91$	0	0
$92$	−1.47214	−0.153481
$93$	−9.56231	−0.991565
$94$	−6.70820	−0.691898
$95$	−18.0902	−1.85601
$96$	3.38197	0.345170
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	0	0
$99$	−2.23607	−0.224733
$100$	0	0
$101$	9.70820	0.966002	0.483001	−	0.875620i	$-0.339547\pi$
$101$	9.70820	0.966002	0.483001	+	0.875620i	$0.339547\pi$
$102$	−7.85410	−0.777672
$103$	−1.23607	−0.121793	−0.0608967	−	0.998144i	$-0.519396\pi$
$103$	−1.23607	−0.121793	−0.0608967	+	0.998144i	$0.519396\pi$
$104$	3.94427	0.386768
$105$	0	0
$106$	−2.85410	−0.277215
$107$	12.0902	1.16880	0.584400	−	0.811465i	$-0.301330\pi$
$107$	12.0902	1.16880	0.584400	+	0.811465i	$0.301330\pi$
$108$	0.618034	0.0594703
$109$	−10.8541	−1.03963	−0.519817	−	0.854278i	$-0.674000\pi$
$109$	−10.8541	−1.03963	−0.519817	+	0.854278i	$0.674000\pi$
$110$	−8.09017	−0.771367
$111$	−6.85410	−0.650563
$112$	0	0
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	−13.0902	−1.22601
$115$	5.32624	0.496674
$116$	5.32624	0.494529
$117$	1.76393	0.163076
$118$	−1.61803	−0.148952
$119$	0	0
$120$	−5.00000	−0.456435
$121$	−6.00000	−0.545455
$122$	−15.9443	−1.44353
$123$	3.09017	0.278631
$124$	−5.90983	−0.530718
$125$	11.1803	1.00000
$126$	0	0
$127$	−1.94427	−0.172526	−0.0862631	−	0.996272i	$-0.527493\pi$
$127$	−1.94427	−0.172526	−0.0862631	+	0.996272i	$0.527493\pi$
$128$	−13.6180	−1.20368
$129$	4.70820	0.414534
$130$	6.38197	0.559735
$131$	−10.6525	−0.930711	−0.465356	−	0.885124i	$-0.654074\pi$
$131$	−10.6525	−0.930711	−0.465356	+	0.885124i	$0.654074\pi$
$132$	−1.38197	−0.120285
$133$	0	0
$134$	4.38197	0.378544
$135$	−2.23607	−0.192450
$136$	10.8541	0.930732
$137$	7.76393	0.663317	0.331659	−	0.943399i	$-0.392392\pi$
$137$	7.76393	0.663317	0.331659	+	0.943399i	$0.392392\pi$
$138$	3.85410	0.328083
$139$	−16.2361	−1.37713	−0.688563	−	0.725177i	$-0.741758\pi$
$139$	−16.2361	−1.37713	−0.688563	+	0.725177i	$0.741758\pi$
$140$	0	0
$141$	4.14590	0.349148
$142$	−16.0902	−1.35026
$143$	−3.94427	−0.329837
$144$	−4.85410	−0.404508
$145$	−19.2705	−1.60033
$146$	−9.47214	−0.783920
$147$	0	0
$148$	−4.23607	−0.348203
$149$	7.90983	0.647999	0.323999	−	0.946057i	$-0.394972\pi$
$149$	7.90983	0.647999	0.323999	+	0.946057i	$0.394972\pi$
$150$	0	0
$151$	−17.5623	−1.42920	−0.714600	−	0.699533i	$-0.753391\pi$
$151$	−17.5623	−1.42920	−0.714600	+	0.699533i	$0.753391\pi$
$152$	18.0902	1.46731
$153$	4.85410	0.392431
$154$	0	0
$155$	21.3820	1.71744
$156$	1.09017	0.0872835
$157$	−9.00000	−0.718278	−0.359139	−	0.933284i	$-0.616930\pi$
$157$	−9.00000	−0.718278	−0.359139	+	0.933284i	$0.616930\pi$
$158$	4.85410	0.386172
$159$	1.76393	0.139889
$160$	−7.56231	−0.597853
$161$	0	0
$162$	−1.61803	−0.127125
$163$	1.56231	0.122369	0.0611846	−	0.998126i	$-0.480512\pi$
$163$	1.56231	0.122369	0.0611846	+	0.998126i	$0.480512\pi$
$164$	1.90983	0.149133
$165$	5.00000	0.389249
$166$	1.00000	0.0776151
$167$	22.0344	1.70508	0.852538	−	0.522665i	$-0.175062\pi$
$167$	22.0344	1.70508	0.852538	+	0.522665i	$0.175062\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	17.5623	1.34697
$171$	8.09017	0.618671
$172$	2.90983	0.221872
$173$	−14.6180	−1.11139	−0.555694	−	0.831387i	$-0.687547\pi$
$173$	−14.6180	−1.11139	−0.555694	+	0.831387i	$0.687547\pi$
$174$	−13.9443	−1.05711
$175$	0	0
$176$	10.8541	0.818159
$177$	1.00000	0.0751646
$178$	17.4721	1.30959
$179$	−9.47214	−0.707981	−0.353990	−	0.935249i	$-0.615175\pi$
$179$	−9.47214	−0.707981	−0.353990	+	0.935249i	$0.615175\pi$
$180$	−1.38197	−0.103006
$181$	11.2705	0.837730	0.418865	−	0.908048i	$-0.362428\pi$
$181$	11.2705	0.837730	0.418865	+	0.908048i	$0.362428\pi$
$182$	0	0
$183$	9.85410	0.728436
$184$	−5.32624	−0.392655
$185$	15.3262	1.12681
$186$	15.4721	1.13447
$187$	−10.8541	−0.793731
$188$	2.56231	0.186875
$189$	0	0
$190$	29.2705	2.12351
$191$	−10.4164	−0.753705	−0.376852	−	0.926273i	$-0.622994\pi$
$191$	−10.4164	−0.753705	−0.376852	+	0.926273i	$0.622994\pi$
$192$	4.23607	0.305712
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	4.85410	0.348504
$195$	−3.94427	−0.282455
$196$	0	0
$197$	10.6525	0.758957	0.379479	−	0.925200i	$-0.376103\pi$
$197$	10.6525	0.758957	0.379479	+	0.925200i	$0.376103\pi$
$198$	3.61803	0.257122
$199$	−3.56231	−0.252525	−0.126263	−	0.991997i	$-0.540298\pi$
$199$	−3.56231	−0.252525	−0.126263	+	0.991997i	$0.540298\pi$
$200$	0	0
$201$	−2.70820	−0.191022
$202$	−15.7082	−1.10523
$203$	0	0
$204$	3.00000	0.210042
$205$	−6.90983	−0.482603
$206$	2.00000	0.139347
$207$	−2.38197	−0.165558
$208$	−8.56231	−0.593689
$209$	−18.0902	−1.25132
$210$	0	0
$211$	−8.85410	−0.609542	−0.304771	−	0.952426i	$-0.598580\pi$
$211$	−8.85410	−0.609542	−0.304771	+	0.952426i	$0.598580\pi$
$212$	1.09017	0.0748732
$213$	9.94427	0.681370
$214$	−19.5623	−1.33725
$215$	−10.5279	−0.717994
$216$	2.23607	0.152145
$217$	0	0
$218$	17.5623	1.18947
$219$	5.85410	0.395584
$220$	3.09017	0.208339
$221$	8.56231	0.575963
$222$	11.0902	0.744323
$223$	18.4721	1.23699	0.618493	−	0.785790i	$-0.287744\pi$
$223$	18.4721	1.23699	0.618493	+	0.785790i	$0.287744\pi$
$224$	0	0
$225$	0	0
$226$	14.5623	0.968670
$227$	28.8541	1.91511	0.957557	−	0.288244i	$-0.0930714\pi$
$227$	28.8541	1.91511	0.957557	+	0.288244i	$0.0930714\pi$
$228$	5.00000	0.331133
$229$	15.8541	1.04767	0.523834	−	0.851820i	$-0.324501\pi$
$229$	15.8541	1.04767	0.523834	+	0.851820i	$0.324501\pi$
$230$	−8.61803	−0.568256
$231$	0	0
$232$	19.2705	1.26517
$233$	−21.7082	−1.42215	−0.711076	−	0.703115i	$-0.751792\pi$
$233$	−21.7082	−1.42215	−0.711076	+	0.703115i	$0.751792\pi$
$234$	−2.85410	−0.186578
$235$	−9.27051	−0.604741
$236$	0.618034	0.0402306
$237$	−3.00000	−0.194871
$238$	0	0
$239$	7.47214	0.483332	0.241666	−	0.970359i	$-0.422306\pi$
$239$	7.47214	0.483332	0.241666	+	0.970359i	$0.422306\pi$
$240$	10.8541	0.700629
$241$	−3.41641	−0.220070	−0.110035	−	0.993928i	$-0.535096\pi$
$241$	−3.41641	−0.220070	−0.110035	+	0.993928i	$0.535096\pi$
$242$	9.70820	0.624067
$243$	1.00000	0.0641500
$244$	6.09017	0.389883
$245$	0	0
$246$	−5.00000	−0.318788
$247$	14.2705	0.908011
$248$	−21.3820	−1.35776
$249$	−0.618034	−0.0391663
$250$	−18.0902	−1.14412
$251$	7.18034	0.453219	0.226610	−	0.973986i	$-0.427236\pi$
$251$	7.18034	0.453219	0.226610	+	0.973986i	$0.427236\pi$
$252$	0	0
$253$	5.32624	0.334858
$254$	3.14590	0.197391
$255$	−10.8541	−0.679710
$256$	13.5623	0.847644
$257$	−7.41641	−0.462623	−0.231311	−	0.972880i	$-0.574302\pi$
$257$	−7.41641	−0.462623	−0.231311	+	0.972880i	$0.574302\pi$
$258$	−7.61803	−0.474278
$259$	0	0
$260$	−2.43769	−0.151179
$261$	8.61803	0.533443
$262$	17.2361	1.06485
$263$	−0.618034	−0.0381096	−0.0190548	−	0.999818i	$-0.506066\pi$
$263$	−0.618034	−0.0381096	−0.0190548	+	0.999818i	$0.506066\pi$
$264$	−5.00000	−0.307729
$265$	−3.94427	−0.242295
$266$	0	0
$267$	−10.7984	−0.660850
$268$	−1.67376	−0.102241
$269$	2.52786	0.154127	0.0770633	−	0.997026i	$-0.475446\pi$
$269$	2.52786	0.154127	0.0770633	+	0.997026i	$0.475446\pi$
$270$	3.61803	0.220187
$271$	12.2361	0.743288	0.371644	−	0.928375i	$-0.378794\pi$
$271$	12.2361	0.743288	0.371644	+	0.928375i	$0.378794\pi$
$272$	−23.5623	−1.42867
$273$	0	0
$274$	−12.5623	−0.758917
$275$	0	0
$276$	−1.47214	−0.0886122
$277$	3.47214	0.208620	0.104310	−	0.994545i	$-0.466737\pi$
$277$	3.47214	0.208620	0.104310	+	0.994545i	$0.466737\pi$
$278$	26.2705	1.57560
$279$	−9.56231	−0.572480
$280$	0	0
$281$	−3.70820	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$281$	−3.70820	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$282$	−6.70820	−0.399468
$283$	−10.2705	−0.610518	−0.305259	−	0.952269i	$-0.598743\pi$
$283$	−10.2705	−0.610518	−0.305259	+	0.952269i	$0.598743\pi$
$284$	6.14590	0.364692
$285$	−18.0902	−1.07157
$286$	6.38197	0.377374
$287$	0	0
$288$	3.38197	0.199284
$289$	6.56231	0.386018
$290$	31.1803	1.83097
$291$	−3.00000	−0.175863
$292$	3.61803	0.211729
$293$	−21.6180	−1.26294	−0.631470	−	0.775401i	$-0.717548\pi$
$293$	−21.6180	−1.26294	−0.631470	+	0.775401i	$0.717548\pi$
$294$	0	0
$295$	−2.23607	−0.130189
$296$	−15.3262	−0.890819
$297$	−2.23607	−0.129750
$298$	−12.7984	−0.741390
$299$	−4.20163	−0.242986
$300$	0	0
$301$	0	0
$302$	28.4164	1.63518
$303$	9.70820	0.557722
$304$	−39.2705	−2.25232
$305$	−22.0344	−1.26169
$306$	−7.85410	−0.448989
$307$	−9.88854	−0.564369	−0.282185	−	0.959360i	$-0.591059\pi$
$307$	−9.88854	−0.564369	−0.282185	+	0.959360i	$0.591059\pi$
$308$	0	0
$309$	−1.23607	−0.0703175
$310$	−34.5967	−1.96496
$311$	28.4508	1.61330	0.806650	−	0.591030i	$-0.201278\pi$
$311$	28.4508	1.61330	0.806650	+	0.591030i	$0.201278\pi$
$312$	3.94427	0.223300
$313$	−3.79837	−0.214697	−0.107348	−	0.994221i	$-0.534236\pi$
$313$	−3.79837	−0.214697	−0.107348	+	0.994221i	$0.534236\pi$
$314$	14.5623	0.821798
$315$	0	0
$316$	−1.85410	−0.104301
$317$	−6.81966	−0.383030	−0.191515	−	0.981490i	$-0.561340\pi$
$317$	−6.81966	−0.383030	−0.191515	+	0.981490i	$0.561340\pi$
$318$	−2.85410	−0.160050
$319$	−19.2705	−1.07894
$320$	−9.47214	−0.529508
$321$	12.0902	0.674807
$322$	0	0
$323$	39.2705	2.18507
$324$	0.618034	0.0343352
$325$	0	0
$326$	−2.52786	−0.140005
$327$	−10.8541	−0.600233
$328$	6.90983	0.381532
$329$	0	0
$330$	−8.09017	−0.445349
$331$	29.1246	1.60083	0.800417	−	0.599444i	$-0.204612\pi$
$331$	29.1246	1.60083	0.800417	+	0.599444i	$0.204612\pi$
$332$	−0.381966	−0.0209631
$333$	−6.85410	−0.375602
$334$	−35.6525	−1.95082
$335$	6.05573	0.330860
$336$	0	0
$337$	−26.0902	−1.42122	−0.710611	−	0.703585i	$-0.751581\pi$
$337$	−26.0902	−1.42122	−0.710611	+	0.703585i	$0.751581\pi$
$338$	16.0000	0.870285
$339$	−9.00000	−0.488813
$340$	−6.70820	−0.363803
$341$	21.3820	1.15790
$342$	−13.0902	−0.707835
$343$	0	0
$344$	10.5279	0.567624
$345$	5.32624	0.286755
$346$	23.6525	1.27156
$347$	1.96556	0.105517	0.0527583	−	0.998607i	$-0.483199\pi$
$347$	1.96556	0.105517	0.0527583	+	0.998607i	$0.483199\pi$
$348$	5.32624	0.285516
$349$	27.0000	1.44528	0.722638	−	0.691226i	$-0.242929\pi$
$349$	27.0000	1.44528	0.722638	+	0.691226i	$0.242929\pi$
$350$	0	0
$351$	1.76393	0.0941517
$352$	−7.56231	−0.403072
$353$	−13.0344	−0.693753	−0.346877	−	0.937911i	$-0.612758\pi$
$353$	−13.0344	−0.693753	−0.346877	+	0.937911i	$0.612758\pi$
$354$	−1.61803	−0.0859975
$355$	−22.2361	−1.18017
$356$	−6.67376	−0.353709
$357$	0	0
$358$	15.3262	0.810017
$359$	13.8885	0.733009	0.366505	−	0.930416i	$-0.380554\pi$
$359$	13.8885	0.733009	0.366505	+	0.930416i	$0.380554\pi$
$360$	−5.00000	−0.263523
$361$	46.4508	2.44478
$362$	−18.2361	−0.958466
$363$	−6.00000	−0.314918
$364$	0	0
$365$	−13.0902	−0.685171
$366$	−15.9443	−0.833420
$367$	23.8328	1.24406	0.622031	−	0.782992i	$-0.286308\pi$
$367$	23.8328	1.24406	0.622031	+	0.782992i	$0.286308\pi$
$368$	11.5623	0.602727
$369$	3.09017	0.160868
$370$	−24.7984	−1.28921
$371$	0	0
$372$	−5.90983	−0.306410
$373$	18.6738	0.966891	0.483445	−	0.875375i	$-0.339385\pi$
$373$	18.6738	0.966891	0.483445	+	0.875375i	$0.339385\pi$
$374$	17.5623	0.908125
$375$	11.1803	0.577350
$376$	9.27051	0.478090
$377$	15.2016	0.782924
$378$	0	0
$379$	−4.41641	−0.226856	−0.113428	−	0.993546i	$-0.536183\pi$
$379$	−4.41641	−0.226856	−0.113428	+	0.993546i	$0.536183\pi$
$380$	−11.1803	−0.573539
$381$	−1.94427	−0.0996081
$382$	16.8541	0.862331
$383$	13.7639	0.703304	0.351652	−	0.936131i	$-0.385620\pi$
$383$	13.7639	0.703304	0.351652	+	0.936131i	$0.385620\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	−12.9443	−0.658846
$387$	4.70820	0.239331
$388$	−1.85410	−0.0941278
$389$	23.5279	1.19291	0.596455	−	0.802646i	$-0.296575\pi$
$389$	23.5279	1.19291	0.596455	+	0.802646i	$0.296575\pi$
$390$	6.38197	0.323163
$391$	−11.5623	−0.584731
$392$	0	0
$393$	−10.6525	−0.537346
$394$	−17.2361	−0.868341
$395$	6.70820	0.337526
$396$	−1.38197	−0.0694464
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	5.76393	0.288920
$399$	0	0
$400$	0	0
$401$	−5.90983	−0.295123	−0.147561	−	0.989053i	$-0.547142\pi$
$401$	−5.90983	−0.295123	−0.147561	+	0.989053i	$0.547142\pi$
$402$	4.38197	0.218553
$403$	−16.8673	−0.840218
$404$	6.00000	0.298511
$405$	−2.23607	−0.111111
$406$	0	0
$407$	15.3262	0.759693
$408$	10.8541	0.537358
$409$	37.4164	1.85012	0.925061	−	0.379818i	$-0.124013\pi$
$409$	37.4164	1.85012	0.925061	+	0.379818i	$0.124013\pi$
$410$	11.1803	0.552158
$411$	7.76393	0.382967
$412$	−0.763932	−0.0376362
$413$	0	0
$414$	3.85410	0.189419
$415$	1.38197	0.0678380
$416$	5.96556	0.292486
$417$	−16.2361	−0.795084
$418$	29.2705	1.43167
$419$	−31.3050	−1.52935	−0.764673	−	0.644418i	$-0.777100\pi$
$419$	−31.3050	−1.52935	−0.764673	+	0.644418i	$0.777100\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	14.3262	0.697390
$423$	4.14590	0.201580
$424$	3.94427	0.191551
$425$	0	0
$426$	−16.0902	−0.779571
$427$	0	0
$428$	7.47214	0.361179
$429$	−3.94427	−0.190431
$430$	17.0344	0.821474
$431$	14.6180	0.704126	0.352063	−	0.935976i	$-0.385480\pi$
$431$	14.6180	0.704126	0.352063	+	0.935976i	$0.385480\pi$
$432$	−4.85410	−0.233543
$433$	−19.3262	−0.928760	−0.464380	−	0.885636i	$-0.653723\pi$
$433$	−19.3262	−0.928760	−0.464380	+	0.885636i	$0.653723\pi$
$434$	0	0
$435$	−19.2705	−0.923950
$436$	−6.70820	−0.321265
$437$	−19.2705	−0.921833
$438$	−9.47214	−0.452596
$439$	32.3820	1.54551	0.772753	−	0.634706i	$-0.218879\pi$
$439$	32.3820	1.54551	0.772753	+	0.634706i	$0.218879\pi$
$440$	11.1803	0.533002
$441$	0	0
$442$	−13.8541	−0.658972
$443$	5.61803	0.266921	0.133460	−	0.991054i	$-0.457391\pi$
$443$	5.61803	0.266921	0.133460	+	0.991054i	$0.457391\pi$
$444$	−4.23607	−0.201035
$445$	24.1459	1.14463
$446$	−29.8885	−1.41526
$447$	7.90983	0.374122
$448$	0	0
$449$	−28.8885	−1.36333	−0.681667	−	0.731662i	$-0.738745\pi$
$449$	−28.8885	−1.36333	−0.681667	+	0.731662i	$0.738745\pi$
$450$	0	0
$451$	−6.90983	−0.325371
$452$	−5.56231	−0.261629
$453$	−17.5623	−0.825149
$454$	−46.6869	−2.19113
$455$	0	0
$456$	18.0902	0.847150
$457$	−26.6869	−1.24836	−0.624181	−	0.781280i	$-0.714567\pi$
$457$	−26.6869	−1.24836	−0.624181	+	0.781280i	$0.714567\pi$
$458$	−25.6525	−1.19866
$459$	4.85410	0.226570
$460$	3.29180	0.153481
$461$	18.7426	0.872932	0.436466	−	0.899721i	$-0.356230\pi$
$461$	18.7426	0.872932	0.436466	+	0.899721i	$0.356230\pi$
$462$	0	0
$463$	10.8541	0.504433	0.252216	−	0.967671i	$-0.418841\pi$
$463$	10.8541	0.504433	0.252216	+	0.967671i	$0.418841\pi$
$464$	−41.8328	−1.94204
$465$	21.3820	0.991565
$466$	35.1246	1.62712
$467$	38.8328	1.79697	0.898484	−	0.439006i	$-0.144669\pi$
$467$	38.8328	1.79697	0.898484	+	0.439006i	$0.144669\pi$
$468$	1.09017	0.0503931
$469$	0	0
$470$	15.0000	0.691898
$471$	−9.00000	−0.414698
$472$	2.23607	0.102923
$473$	−10.5279	−0.484072
$474$	4.85410	0.222956
$475$	0	0
$476$	0	0
$477$	1.76393	0.0807649
$478$	−12.0902	−0.552992
$479$	26.0902	1.19209	0.596045	−	0.802951i	$-0.296738\pi$
$479$	26.0902	1.19209	0.596045	+	0.802951i	$0.296738\pi$
$480$	−7.56231	−0.345170
$481$	−12.0902	−0.551264
$482$	5.52786	0.251787
$483$	0	0
$484$	−3.70820	−0.168555
$485$	6.70820	0.304604
$486$	−1.61803	−0.0733955
$487$	36.7426	1.66497	0.832484	−	0.554049i	$-0.186918\pi$
$487$	36.7426	1.66497	0.832484	+	0.554049i	$0.186918\pi$
$488$	22.0344	0.997452
$489$	1.56231	0.0706499
$490$	0	0
$491$	26.5066	1.19623	0.598113	−	0.801412i	$-0.295918\pi$
$491$	26.5066	1.19623	0.598113	+	0.801412i	$0.295918\pi$
$492$	1.90983	0.0861018
$493$	41.8328	1.88406
$494$	−23.0902	−1.03888
$495$	5.00000	0.224733
$496$	46.4164	2.08416
$497$	0	0
$498$	1.00000	0.0448111
$499$	12.4164	0.555835	0.277917	−	0.960605i	$-0.410356\pi$
$499$	12.4164	0.555835	0.277917	+	0.960605i	$0.410356\pi$
$500$	6.90983	0.309017
$501$	22.0344	0.984426
$502$	−11.6180	−0.518538
$503$	20.7984	0.927354	0.463677	−	0.886004i	$-0.346530\pi$
$503$	20.7984	0.927354	0.463677	+	0.886004i	$0.346530\pi$
$504$	0	0
$505$	−21.7082	−0.966002
$506$	−8.61803	−0.383118
$507$	−9.88854	−0.439166
$508$	−1.20163	−0.0533135
$509$	29.9230	1.32631	0.663157	−	0.748481i	$-0.269216\pi$
$509$	29.9230	1.32631	0.663157	+	0.748481i	$0.269216\pi$
$510$	17.5623	0.777672
$511$	0	0
$512$	5.29180	0.233867
$513$	8.09017	0.357190
$514$	12.0000	0.529297
$515$	2.76393	0.121793
$516$	2.90983	0.128098
$517$	−9.27051	−0.407717
$518$	0	0
$519$	−14.6180	−0.641660
$520$	−8.81966	−0.386768
$521$	14.5066	0.635545	0.317772	−	0.948167i	$-0.397065\pi$
$521$	14.5066	0.635545	0.317772	+	0.948167i	$0.397065\pi$
$522$	−13.9443	−0.610324
$523$	−31.9443	−1.39683	−0.698413	−	0.715695i	$-0.746110\pi$
$523$	−31.9443	−1.39683	−0.698413	+	0.715695i	$0.746110\pi$
$524$	−6.58359	−0.287606
$525$	0	0
$526$	1.00000	0.0436021
$527$	−46.4164	−2.02193
$528$	10.8541	0.472364
$529$	−17.3262	−0.753315
$530$	6.38197	0.277215
$531$	1.00000	0.0433963
$532$	0	0
$533$	5.45085	0.236103
$534$	17.4721	0.756093
$535$	−27.0344	−1.16880
$536$	−6.05573	−0.261568
$537$	−9.47214	−0.408753
$538$	−4.09017	−0.176340
$539$	0	0
$540$	−1.38197	−0.0594703
$541$	14.1246	0.607264	0.303632	−	0.952789i	$-0.401801\pi$
$541$	14.1246	0.607264	0.303632	+	0.952789i	$0.401801\pi$
$542$	−19.7984	−0.850413
$543$	11.2705	0.483664
$544$	16.4164	0.703848
$545$	24.2705	1.03963
$546$	0	0
$547$	32.5623	1.39226	0.696132	−	0.717914i	$-0.254903\pi$
$547$	32.5623	1.39226	0.696132	+	0.717914i	$0.254903\pi$
$548$	4.79837	0.204976
$549$	9.85410	0.420563
$550$	0	0
$551$	69.7214	2.97023
$552$	−5.32624	−0.226700
$553$	0	0
$554$	−5.61803	−0.238687
$555$	15.3262	0.650563
$556$	−10.0344	−0.425555
$557$	1.41641	0.0600151	0.0300076	−	0.999550i	$-0.490447\pi$
$557$	1.41641	0.0600151	0.0300076	+	0.999550i	$0.490447\pi$
$558$	15.4721	0.654988
$559$	8.30495	0.351262
$560$	0	0
$561$	−10.8541	−0.458261
$562$	6.00000	0.253095
$563$	−28.5967	−1.20521	−0.602605	−	0.798040i	$-0.705870\pi$
$563$	−28.5967	−1.20521	−0.602605	+	0.798040i	$0.705870\pi$
$564$	2.56231	0.107893
$565$	20.1246	0.846649
$566$	16.6180	0.698508
$567$	0	0
$568$	22.2361	0.933005
$569$	26.5066	1.11121	0.555607	−	0.831445i	$-0.312486\pi$
$569$	26.5066	1.11121	0.555607	+	0.831445i	$0.312486\pi$
$570$	29.2705	1.22601
$571$	28.4164	1.18919	0.594595	−	0.804025i	$-0.297312\pi$
$571$	28.4164	1.18919	0.594595	+	0.804025i	$0.297312\pi$
$572$	−2.43769	−0.101925
$573$	−10.4164	−0.435152
$574$	0	0
$575$	0	0
$576$	4.23607	0.176503
$577$	−21.4721	−0.893897	−0.446948	−	0.894560i	$-0.647489\pi$
$577$	−21.4721	−0.893897	−0.446948	+	0.894560i	$0.647489\pi$
$578$	−10.6180	−0.441652
$579$	8.00000	0.332469
$580$	−11.9098	−0.494529
$581$	0	0
$582$	4.85410	0.201209
$583$	−3.94427	−0.163355
$584$	13.0902	0.541675
$585$	−3.94427	−0.163076
$586$	34.9787	1.44496
$587$	−29.3607	−1.21184	−0.605922	−	0.795524i	$-0.707196\pi$
$587$	−29.3607	−1.21184	−0.605922	+	0.795524i	$0.707196\pi$
$588$	0	0
$589$	−77.3607	−3.18759
$590$	3.61803	0.148952
$591$	10.6525	0.438184
$592$	33.2705	1.36741
$593$	−4.90983	−0.201623	−0.100811	−	0.994906i	$-0.532144\pi$
$593$	−4.90983	−0.201623	−0.100811	+	0.994906i	$0.532144\pi$
$594$	3.61803	0.148450
$595$	0	0
$596$	4.88854	0.200243
$597$	−3.56231	−0.145795
$598$	6.79837	0.278006
$599$	−28.6525	−1.17071	−0.585354	−	0.810778i	$-0.699045\pi$
$599$	−28.6525	−1.17071	−0.585354	+	0.810778i	$0.699045\pi$
$600$	0	0
$601$	−25.6180	−1.04498	−0.522491	−	0.852645i	$-0.674997\pi$
$601$	−25.6180	−1.04498	−0.522491	+	0.852645i	$0.674997\pi$
$602$	0	0
$603$	−2.70820	−0.110287
$604$	−10.8541	−0.441647
$605$	13.4164	0.545455
$606$	−15.7082	−0.638102
$607$	−17.6525	−0.716492	−0.358246	−	0.933627i	$-0.616625\pi$
$607$	−17.6525	−0.716492	−0.358246	+	0.933627i	$0.616625\pi$
$608$	27.3607	1.10962
$609$	0	0
$610$	35.6525	1.44353
$611$	7.31308	0.295856
$612$	3.00000	0.121268
$613$	−7.34752	−0.296764	−0.148382	−	0.988930i	$-0.547406\pi$
$613$	−7.34752	−0.296764	−0.148382	+	0.988930i	$0.547406\pi$
$614$	16.0000	0.645707
$615$	−6.90983	−0.278631
$616$	0	0
$617$	28.8541	1.16162	0.580811	−	0.814038i	$-0.302735\pi$
$617$	28.8541	1.16162	0.580811	+	0.814038i	$0.302735\pi$
$618$	2.00000	0.0804518
$619$	28.1246	1.13042	0.565212	−	0.824946i	$-0.308794\pi$
$619$	28.1246	1.13042	0.565212	+	0.824946i	$0.308794\pi$
$620$	13.2148	0.530718
$621$	−2.38197	−0.0955850
$622$	−46.0344	−1.84581
$623$	0	0
$624$	−8.56231	−0.342767
$625$	−25.0000	−1.00000
$626$	6.14590	0.245639
$627$	−18.0902	−0.722452
$628$	−5.56231	−0.221960
$629$	−33.2705	−1.32658
$630$	0	0
$631$	−13.5836	−0.540754	−0.270377	−	0.962754i	$-0.587148\pi$
$631$	−13.5836	−0.540754	−0.270377	+	0.962754i	$0.587148\pi$
$632$	−6.70820	−0.266838
$633$	−8.85410	−0.351919
$634$	11.0344	0.438234
$635$	4.34752	0.172526
$636$	1.09017	0.0432281
$637$	0	0
$638$	31.1803	1.23444
$639$	9.94427	0.393389
$640$	30.4508	1.20368
$641$	−10.0557	−0.397177	−0.198589	−	0.980083i	$-0.563636\pi$
$641$	−10.0557	−0.397177	−0.198589	+	0.980083i	$0.563636\pi$
$642$	−19.5623	−0.772063
$643$	13.1459	0.518424	0.259212	−	0.965821i	$-0.416537\pi$
$643$	13.1459	0.518424	0.259212	+	0.965821i	$0.416537\pi$
$644$	0	0
$645$	−10.5279	−0.414534
$646$	−63.5410	−2.49999
$647$	−27.0557	−1.06367	−0.531835	−	0.846848i	$-0.678497\pi$
$647$	−27.0557	−1.06367	−0.531835	+	0.846848i	$0.678497\pi$
$648$	2.23607	0.0878410
$649$	−2.23607	−0.0877733
$650$	0	0
$651$	0	0
$652$	0.965558	0.0378142
$653$	28.9098	1.13133	0.565665	−	0.824635i	$-0.308620\pi$
$653$	28.9098	1.13133	0.565665	+	0.824635i	$0.308620\pi$
$654$	17.5623	0.686741
$655$	23.8197	0.930711
$656$	−15.0000	−0.585652
$657$	5.85410	0.228390
$658$	0	0
$659$	−4.85410	−0.189089	−0.0945445	−	0.995521i	$-0.530139\pi$
$659$	−4.85410	−0.189089	−0.0945445	+	0.995521i	$0.530139\pi$
$660$	3.09017	0.120285
$661$	11.7426	0.456736	0.228368	−	0.973575i	$-0.426661\pi$
$661$	11.7426	0.456736	0.228368	+	0.973575i	$0.426661\pi$
$662$	−47.1246	−1.83155
$663$	8.56231	0.332532
$664$	−1.38197	−0.0536307
$665$	0	0
$666$	11.0902	0.429735
$667$	−20.5279	−0.794842
$668$	13.6180	0.526898
$669$	18.4721	0.714174
$670$	−9.79837	−0.378544
$671$	−22.0344	−0.850630
$672$	0	0
$673$	0.472136	0.0181995	0.00909975	−	0.999959i	$-0.497103\pi$
$673$	0.472136	0.0181995	0.00909975	+	0.999959i	$0.497103\pi$
$674$	42.2148	1.62605
$675$	0	0
$676$	−6.11146	−0.235056
$677$	50.1803	1.92859	0.964294	−	0.264836i	$-0.0853177\pi$
$677$	50.1803	1.92859	0.964294	+	0.264836i	$0.0853177\pi$
$678$	14.5623	0.559262
$679$	0	0
$680$	−24.2705	−0.930732
$681$	28.8541	1.10569
$682$	−34.5967	−1.32478
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	5.00000	0.191180
$685$	−17.3607	−0.663317
$686$	0	0
$687$	15.8541	0.604872
$688$	−22.8541	−0.871304
$689$	3.11146	0.118537
$690$	−8.61803	−0.328083
$691$	5.12461	0.194949	0.0974747	−	0.995238i	$-0.468923\pi$
$691$	5.12461	0.194949	0.0974747	+	0.995238i	$0.468923\pi$
$692$	−9.03444	−0.343438
$693$	0	0
$694$	−3.18034	−0.120724
$695$	36.3050	1.37713
$696$	19.2705	0.730447
$697$	15.0000	0.568166
$698$	−43.6869	−1.65357
$699$	−21.7082	−0.821080
$700$	0	0
$701$	42.7984	1.61647	0.808236	−	0.588859i	$-0.200422\pi$
$701$	42.7984	1.61647	0.808236	+	0.588859i	$0.200422\pi$
$702$	−2.85410	−0.107721
$703$	−55.4508	−2.09137
$704$	−9.47214	−0.356995
$705$	−9.27051	−0.349148
$706$	21.0902	0.793739
$707$	0	0
$708$	0.618034	0.0232271
$709$	−14.2918	−0.536740	−0.268370	−	0.963316i	$-0.586485\pi$
$709$	−14.2918	−0.536740	−0.268370	+	0.963316i	$0.586485\pi$
$710$	35.9787	1.35026
$711$	−3.00000	−0.112509
$712$	−24.1459	−0.904906
$713$	22.7771	0.853009
$714$	0	0
$715$	8.81966	0.329837
$716$	−5.85410	−0.218778
$717$	7.47214	0.279052
$718$	−22.4721	−0.838653
$719$	−12.3820	−0.461769	−0.230885	−	0.972981i	$-0.574162\pi$
$719$	−12.3820	−0.461769	−0.230885	+	0.972981i	$0.574162\pi$
$720$	10.8541	0.404508
$721$	0	0
$722$	−75.1591	−2.79713
$723$	−3.41641	−0.127058
$724$	6.96556	0.258873
$725$	0	0
$726$	9.70820	0.360305
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	21.1803	0.783920
$731$	22.8541	0.845289
$732$	6.09017	0.225099
$733$	−29.4721	−1.08858	−0.544289	−	0.838898i	$-0.683201\pi$
$733$	−29.4721	−1.08858	−0.544289	+	0.838898i	$0.683201\pi$
$734$	−38.5623	−1.42336
$735$	0	0
$736$	−8.05573	−0.296938
$737$	6.05573	0.223066
$738$	−5.00000	−0.184053
$739$	−39.1033	−1.43844	−0.719220	−	0.694783i	$-0.755500\pi$
$739$	−39.1033	−1.43844	−0.719220	+	0.694783i	$0.755500\pi$
$740$	9.47214	0.348203
$741$	14.2705	0.524240
$742$	0	0
$743$	−22.3820	−0.821115	−0.410557	−	0.911835i	$-0.634666\pi$
$743$	−22.3820	−0.821115	−0.410557	+	0.911835i	$0.634666\pi$
$744$	−21.3820	−0.783901
$745$	−17.6869	−0.647999
$746$	−30.2148	−1.10624
$747$	−0.618034	−0.0226127
$748$	−6.70820	−0.245276
$749$	0	0
$750$	−18.0902	−0.660560
$751$	−45.1246	−1.64662	−0.823310	−	0.567592i	$-0.807875\pi$
$751$	−45.1246	−1.64662	−0.823310	+	0.567592i	$0.807875\pi$
$752$	−20.1246	−0.733869
$753$	7.18034	0.261666
$754$	−24.5967	−0.895761
$755$	39.2705	1.42920
$756$	0	0
$757$	−43.6180	−1.58532	−0.792662	−	0.609661i	$-0.791306\pi$
$757$	−43.6180	−1.58532	−0.792662	+	0.609661i	$0.791306\pi$
$758$	7.14590	0.259551
$759$	5.32624	0.193330
$760$	−40.4508	−1.46731
$761$	−17.2918	−0.626827	−0.313414	−	0.949617i	$-0.601473\pi$
$761$	−17.2918	−0.626827	−0.313414	+	0.949617i	$0.601473\pi$
$762$	3.14590	0.113964
$763$	0	0
$764$	−6.43769	−0.232908
$765$	−10.8541	−0.392431
$766$	−22.2705	−0.804666
$767$	1.76393	0.0636919
$768$	13.5623	0.489388
$769$	35.9787	1.29743	0.648713	−	0.761033i	$-0.275308\pi$
$769$	35.9787	1.29743	0.648713	+	0.761033i	$0.275308\pi$
$770$	0	0
$771$	−7.41641	−0.267095
$772$	4.94427	0.177948
$773$	2.65248	0.0954029	0.0477015	−	0.998862i	$-0.484810\pi$
$773$	2.65248	0.0954029	0.0477015	+	0.998862i	$0.484810\pi$
$774$	−7.61803	−0.273825
$775$	0	0
$776$	−6.70820	−0.240810
$777$	0	0
$778$	−38.0689	−1.36484
$779$	25.0000	0.895718
$780$	−2.43769	−0.0872835
$781$	−22.2361	−0.795669
$782$	18.7082	0.669004
$783$	8.61803	0.307983
$784$	0	0
$785$	20.1246	0.718278
$786$	17.2361	0.614790
$787$	17.7082	0.631229	0.315615	−	0.948887i	$-0.397789\pi$
$787$	17.7082	0.631229	0.315615	+	0.948887i	$0.397789\pi$
$788$	6.58359	0.234531
$789$	−0.618034	−0.0220026
$790$	−10.8541	−0.386172
$791$	0	0
$792$	−5.00000	−0.177667
$793$	17.3820	0.617252
$794$	4.85410	0.172266
$795$	−3.94427	−0.139889
$796$	−2.20163	−0.0780346
$797$	−3.23607	−0.114627	−0.0573137	−	0.998356i	$-0.518254\pi$
$797$	−3.23607	−0.114627	−0.0573137	+	0.998356i	$0.518254\pi$
$798$	0	0
$799$	20.1246	0.711958
$800$	0	0
$801$	−10.7984	−0.381542
$802$	9.56231	0.337657
$803$	−13.0902	−0.461942
$804$	−1.67376	−0.0590290
$805$	0	0
$806$	27.2918	0.961313
$807$	2.52786	0.0889850
$808$	21.7082	0.763692
$809$	23.3262	0.820107	0.410053	−	0.912062i	$-0.365510\pi$
$809$	23.3262	0.820107	0.410053	+	0.912062i	$0.365510\pi$
$810$	3.61803	0.127125
$811$	16.6525	0.584748	0.292374	−	0.956304i	$-0.405555\pi$
$811$	16.6525	0.584748	0.292374	+	0.956304i	$0.405555\pi$
$812$	0	0
$813$	12.2361	0.429138
$814$	−24.7984	−0.869183
$815$	−3.49342	−0.122369
$816$	−23.5623	−0.824846
$817$	38.0902	1.33261
$818$	−60.5410	−2.11677
$819$	0	0
$820$	−4.27051	−0.149133
$821$	23.9230	0.834918	0.417459	−	0.908696i	$-0.362921\pi$
$821$	23.9230	0.834918	0.417459	+	0.908696i	$0.362921\pi$
$822$	−12.5623	−0.438161
$823$	24.7082	0.861274	0.430637	−	0.902525i	$-0.358289\pi$
$823$	24.7082	0.861274	0.430637	+	0.902525i	$0.358289\pi$
$824$	−2.76393	−0.0962861
$825$	0	0
$826$	0	0
$827$	−25.1803	−0.875606	−0.437803	−	0.899071i	$-0.644243\pi$
$827$	−25.1803	−0.875606	−0.437803	+	0.899071i	$0.644243\pi$
$828$	−1.47214	−0.0511603
$829$	−14.6869	−0.510098	−0.255049	−	0.966928i	$-0.582092\pi$
$829$	−14.6869	−0.510098	−0.255049	+	0.966928i	$0.582092\pi$
$830$	−2.23607	−0.0776151
$831$	3.47214	0.120447
$832$	7.47214	0.259050
$833$	0	0
$834$	26.2705	0.909673
$835$	−49.2705	−1.70508
$836$	−11.1803	−0.386680
$837$	−9.56231	−0.330522
$838$	50.6525	1.74976
$839$	47.1803	1.62885	0.814423	−	0.580271i	$-0.197054\pi$
$839$	47.1803	1.62885	0.814423	+	0.580271i	$0.197054\pi$
$840$	0	0
$841$	45.2705	1.56105
$842$	−1.61803	−0.0557611
$843$	−3.70820	−0.127717
$844$	−5.47214	−0.188359
$845$	22.1115	0.760657
$846$	−6.70820	−0.230633
$847$	0	0
$848$	−8.56231	−0.294031
$849$	−10.2705	−0.352483
$850$	0	0
$851$	16.3262	0.559656
$852$	6.14590	0.210555
$853$	33.0344	1.13108	0.565539	−	0.824722i	$-0.308668\pi$
$853$	33.0344	1.13108	0.565539	+	0.824722i	$0.308668\pi$
$854$	0	0
$855$	−18.0902	−0.618671
$856$	27.0344	0.924018
$857$	−42.7771	−1.46124	−0.730619	−	0.682786i	$-0.760768\pi$
$857$	−42.7771	−1.46124	−0.730619	+	0.682786i	$0.760768\pi$
$858$	6.38197	0.217877
$859$	−24.4721	−0.834979	−0.417489	−	0.908682i	$-0.637090\pi$
$859$	−24.4721	−0.834979	−0.417489	+	0.908682i	$0.637090\pi$
$860$	−6.50658	−0.221872
$861$	0	0
$862$	−23.6525	−0.805607
$863$	43.7984	1.49091	0.745457	−	0.666554i	$-0.232231\pi$
$863$	43.7984	1.49091	0.745457	+	0.666554i	$0.232231\pi$
$864$	3.38197	0.115057
$865$	32.6869	1.11139
$866$	31.2705	1.06262
$867$	6.56231	0.222868
$868$	0	0
$869$	6.70820	0.227560
$870$	31.1803	1.05711
$871$	−4.77709	−0.161865
$872$	−24.2705	−0.821903
$873$	−3.00000	−0.101535
$874$	31.1803	1.05469
$875$	0	0
$876$	3.61803	0.122242
$877$	−51.8885	−1.75215	−0.876076	−	0.482173i	$-0.839848\pi$
$877$	−51.8885	−1.75215	−0.876076	+	0.482173i	$0.839848\pi$
$878$	−52.3951	−1.76825
$879$	−21.6180	−0.729158
$880$	−24.2705	−0.818159
$881$	2.29180	0.0772126	0.0386063	−	0.999254i	$-0.487708\pi$
$881$	2.29180	0.0772126	0.0386063	+	0.999254i	$0.487708\pi$
$882$	0	0
$883$	3.41641	0.114971	0.0574856	−	0.998346i	$-0.481692\pi$
$883$	3.41641	0.114971	0.0574856	+	0.998346i	$0.481692\pi$
$884$	5.29180	0.177982
$885$	−2.23607	−0.0751646
$886$	−9.09017	−0.305390
$887$	31.4721	1.05673	0.528365	−	0.849017i	$-0.322805\pi$
$887$	31.4721	1.05673	0.528365	+	0.849017i	$0.322805\pi$
$888$	−15.3262	−0.514315
$889$	0	0
$890$	−39.0689	−1.30959
$891$	−2.23607	−0.0749111
$892$	11.4164	0.382250
$893$	33.5410	1.12241
$894$	−12.7984	−0.428042
$895$	21.1803	0.707981
$896$	0	0
$897$	−4.20163	−0.140288
$898$	46.7426	1.55982
$899$	−82.4083	−2.74847
$900$	0	0
$901$	8.56231	0.285252
$902$	11.1803	0.372265
$903$	0	0
$904$	−20.1246	−0.669335
$905$	−25.2016	−0.837730
$906$	28.4164	0.944072
$907$	−2.05573	−0.0682593	−0.0341297	−	0.999417i	$-0.510866\pi$
$907$	−2.05573	−0.0682593	−0.0341297	+	0.999417i	$0.510866\pi$
$908$	17.8328	0.591803
$909$	9.70820	0.322001
$910$	0	0
$911$	42.1033	1.39495	0.697473	−	0.716611i	$-0.254308\pi$
$911$	42.1033	1.39495	0.697473	+	0.716611i	$0.254308\pi$
$912$	−39.2705	−1.30038
$913$	1.38197	0.0457364
$914$	43.1803	1.42828
$915$	−22.0344	−0.728436
$916$	9.79837	0.323747
$917$	0	0
$918$	−7.85410	−0.259224
$919$	35.5967	1.17423	0.587114	−	0.809504i	$-0.300264\pi$
$919$	35.5967	1.17423	0.587114	+	0.809504i	$0.300264\pi$
$920$	11.9098	0.392655
$921$	−9.88854	−0.325839
$922$	−30.3262	−0.998741
$923$	17.5410	0.577370
$924$	0	0
$925$	0	0
$926$	−17.5623	−0.577133
$927$	−1.23607	−0.0405978
$928$	29.1459	0.956761
$929$	35.2918	1.15789	0.578943	−	0.815368i	$-0.303465\pi$
$929$	35.2918	1.15789	0.578943	+	0.815368i	$0.303465\pi$
$930$	−34.5967	−1.13447
$931$	0	0
$932$	−13.4164	−0.439469
$933$	28.4508	0.931439
$934$	−62.8328	−2.05595
$935$	24.2705	0.793731
$936$	3.94427	0.128923
$937$	37.7214	1.23230	0.616152	−	0.787628i	$-0.288691\pi$
$937$	37.7214	1.23230	0.616152	+	0.787628i	$0.288691\pi$
$938$	0	0
$939$	−3.79837	−0.123955
$940$	−5.72949	−0.186875
$941$	22.3050	0.727121	0.363560	−	0.931571i	$-0.381561\pi$
$941$	22.3050	0.727121	0.363560	+	0.931571i	$0.381561\pi$
$942$	14.5623	0.474466
$943$	−7.36068	−0.239697
$944$	−4.85410	−0.157988
$945$	0	0
$946$	17.0344	0.553837
$947$	−4.03444	−0.131102	−0.0655509	−	0.997849i	$-0.520880\pi$
$947$	−4.03444	−0.131102	−0.0655509	+	0.997849i	$0.520880\pi$
$948$	−1.85410	−0.0602184
$949$	10.3262	0.335204
$950$	0	0
$951$	−6.81966	−0.221143
$952$	0	0
$953$	−12.9443	−0.419306	−0.209653	−	0.977776i	$-0.567233\pi$
$953$	−12.9443	−0.419306	−0.209653	+	0.977776i	$0.567233\pi$
$954$	−2.85410	−0.0924050
$955$	23.2918	0.753705
$956$	4.61803	0.149358
$957$	−19.2705	−0.622927
$958$	−42.2148	−1.36390
$959$	0	0
$960$	−9.47214	−0.305712
$961$	60.4377	1.94960
$962$	19.5623	0.630714
$963$	12.0902	0.389600
$964$	−2.11146	−0.0680054
$965$	−17.8885	−0.575853
$966$	0	0
$967$	−18.2705	−0.587540	−0.293770	−	0.955876i	$-0.594910\pi$
$967$	−18.2705	−0.587540	−0.293770	+	0.955876i	$0.594910\pi$
$968$	−13.4164	−0.431220
$969$	39.2705	1.26155
$970$	−10.8541	−0.348504
$971$	−22.2016	−0.712484	−0.356242	−	0.934394i	$-0.615942\pi$
$971$	−22.2016	−0.712484	−0.356242	+	0.934394i	$0.615942\pi$
$972$	0.618034	0.0198234
$973$	0	0
$974$	−59.4508	−1.90493
$975$	0	0
$976$	−47.8328	−1.53109
$977$	14.8885	0.476327	0.238163	−	0.971225i	$-0.423455\pi$
$977$	14.8885	0.476327	0.238163	+	0.971225i	$0.423455\pi$
$978$	−2.52786	−0.0808322
$979$	24.1459	0.771706
$980$	0	0
$981$	−10.8541	−0.346545
$982$	−42.8885	−1.36863
$983$	58.5967	1.86895	0.934473	−	0.356034i	$-0.115871\pi$
$983$	58.5967	1.86895	0.934473	+	0.356034i	$0.115871\pi$
$984$	6.90983	0.220277
$985$	−23.8197	−0.758957
$986$	−67.6869	−2.15559
$987$	0	0
$988$	8.81966	0.280591
$989$	−11.2148	−0.356609
$990$	−8.09017	−0.257122
$991$	−18.7426	−0.595380	−0.297690	−	0.954663i	$-0.596216\pi$
$991$	−18.7426	−0.595380	−0.297690	+	0.954663i	$0.596216\pi$
$992$	−32.3394	−1.02678
$993$	29.1246	0.924242
$994$	0	0
$995$	7.96556	0.252525
$996$	−0.381966	−0.0121031
$997$	−1.78522	−0.0565384	−0.0282692	−	0.999600i	$-0.509000\pi$
$997$	−1.78522	−0.0565384	−0.0282692	+	0.999600i	$0.509000\pi$
$998$	−20.0902	−0.635943
$999$	−6.85410	−0.216854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8673.2.a.k.1.1		2
7.6	odd	2		177.2.a.b.1.1	✓	2
21.20	even	2		531.2.a.b.1.2		2
28.27	even	2		2832.2.a.o.1.2		2
35.34	odd	2		4425.2.a.t.1.2		2
84.83	odd	2		8496.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.b.1.1	✓	2	7.6	odd	2
531.2.a.b.1.2		2	21.20	even	2
2832.2.a.o.1.2		2	28.27	even	2
4425.2.a.t.1.2		2	35.34	odd	2
8496.2.a.bb.1.1		2	84.83	odd	2
8673.2.a.k.1.1		2	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:	\( N \)	\(=\)	\( 8673 = 3 \cdot 7^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8673.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -2.23607 q^{5} -1.61803 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -2.23607 q^{5} -1.61803 q^{6} +2.23607 q^{8} +1.00000 q^{9} +3.61803 q^{10} -2.23607 q^{11} +0.618034 q^{12} +1.76393 q^{13} -2.23607 q^{15} -4.85410 q^{16} +4.85410 q^{17} -1.61803 q^{18} +8.09017 q^{19} -1.38197 q^{20} +3.61803 q^{22} -2.38197 q^{23} +2.23607 q^{24} -2.85410 q^{26} +1.00000 q^{27} +8.61803 q^{29} +3.61803 q^{30} -9.56231 q^{31} +3.38197 q^{32} -2.23607 q^{33} -7.85410 q^{34} +0.618034 q^{36} -6.85410 q^{37} -13.0902 q^{38} +1.76393 q^{39} -5.00000 q^{40} +3.09017 q^{41} +4.70820 q^{43} -1.38197 q^{44} -2.23607 q^{45} +3.85410 q^{46} +4.14590 q^{47} -4.85410 q^{48} +4.85410 q^{51} +1.09017 q^{52} +1.76393 q^{53} -1.61803 q^{54} +5.00000 q^{55} +8.09017 q^{57} -13.9443 q^{58} +1.00000 q^{59} -1.38197 q^{60} +9.85410 q^{61} +15.4721 q^{62} +4.23607 q^{64} -3.94427 q^{65} +3.61803 q^{66} -2.70820 q^{67} +3.00000 q^{68} -2.38197 q^{69} +9.94427 q^{71} +2.23607 q^{72} +5.85410 q^{73} +11.0902 q^{74} +5.00000 q^{76} -2.85410 q^{78} -3.00000 q^{79} +10.8541 q^{80} +1.00000 q^{81} -5.00000 q^{82} -0.618034 q^{83} -10.8541 q^{85} -7.61803 q^{86} +8.61803 q^{87} -5.00000 q^{88} -10.7984 q^{89} +3.61803 q^{90} -1.47214 q^{92} -9.56231 q^{93} -6.70820 q^{94} -18.0902 q^{95} +3.38197 q^{96} -3.00000 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + 2 q^{9} + 5 q^{10} - q^{12} + 8 q^{13} - 3 q^{16} + 3 q^{17} - q^{18} + 5 q^{19} - 5 q^{20} + 5 q^{22} - 7 q^{23} + q^{26} + 2 q^{27} + 15 q^{29} + 5 q^{30} + q^{31} + 9 q^{32} - 9 q^{34} - q^{36} - 7 q^{37} - 15 q^{38} + 8 q^{39} - 10 q^{40} - 5 q^{41} - 4 q^{43} - 5 q^{44} + q^{46} + 15 q^{47} - 3 q^{48} + 3 q^{51} - 9 q^{52} + 8 q^{53} - q^{54} + 10 q^{55} + 5 q^{57} - 10 q^{58} + 2 q^{59} - 5 q^{60} + 13 q^{61} + 22 q^{62} + 4 q^{64} + 10 q^{65} + 5 q^{66} + 8 q^{67} + 6 q^{68} - 7 q^{69} + 2 q^{71} + 5 q^{73} + 11 q^{74} + 10 q^{76} + q^{78} - 6 q^{79} + 15 q^{80} + 2 q^{81} - 10 q^{82} + q^{83} - 15 q^{85} - 13 q^{86} + 15 q^{87} - 10 q^{88} + 3 q^{89} + 5 q^{90} + 6 q^{92} + q^{93} - 25 q^{95} + 9 q^{96} - 6 q^{97}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + 2 * q^9 + 5 * q^10 - q^12 + 8 * q^13 - 3 * q^16 + 3 * q^17 - q^18 + 5 * q^19 - 5 * q^20 + 5 * q^22 - 7 * q^23 + q^26 + 2 * q^27 + 15 * q^29 + 5 * q^30 + q^31 + 9 * q^32 - 9 * q^34 - q^36 - 7 * q^37 - 15 * q^38 + 8 * q^39 - 10 * q^40 - 5 * q^41 - 4 * q^43 - 5 * q^44 + q^46 + 15 * q^47 - 3 * q^48 + 3 * q^51 - 9 * q^52 + 8 * q^53 - q^54 + 10 * q^55 + 5 * q^57 - 10 * q^58 + 2 * q^59 - 5 * q^60 + 13 * q^61 + 22 * q^62 + 4 * q^64 + 10 * q^65 + 5 * q^66 + 8 * q^67 + 6 * q^68 - 7 * q^69 + 2 * q^71 + 5 * q^73 + 11 * q^74 + 10 * q^76 + q^78 - 6 * q^79 + 15 * q^80 + 2 * q^81 - 10 * q^82 + q^83 - 15 * q^85 - 13 * q^86 + 15 * q^87 - 10 * q^88 + 3 * q^89 + 5 * q^90 + 6 * q^92 + q^93 - 25 * q^95 + 9 * q^96 - 6 * q^97

Introduction

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