LMFDB - Embedded newform 8034.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.53113$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.53113 q^{17} -1.00000 q^{18} -1.00000 q^{21} -3.00000 q^{22} +1.53113 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +2.53113 q^{34} +1.00000 q^{36} -7.53113 q^{37} -1.00000 q^{39} +3.53113 q^{41} +1.00000 q^{42} +3.53113 q^{43} +3.00000 q^{44} -1.53113 q^{46} -0.468871 q^{47} -1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} +2.53113 q^{51} +1.00000 q^{52} -2.53113 q^{53} +1.00000 q^{54} -1.00000 q^{56} -9.06226 q^{59} -2.46887 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -12.5311 q^{67} -2.53113 q^{68} -1.53113 q^{69} +1.06226 q^{71} -1.00000 q^{72} +7.59339 q^{73} +7.53113 q^{74} +5.00000 q^{75} +3.00000 q^{77} +1.00000 q^{78} -9.06226 q^{79} +1.00000 q^{81} -3.53113 q^{82} +9.06226 q^{83} -1.00000 q^{84} -3.53113 q^{86} -3.00000 q^{88} -13.0623 q^{89} +1.00000 q^{91} +1.53113 q^{92} -4.00000 q^{93} +0.468871 q^{94} +1.00000 q^{96} -18.1245 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{17} - 2 q^{18} - 2 q^{21} - 6 q^{22} - 5 q^{23} + 2 q^{24} - 10 q^{25} - 2 q^{26} - 2 q^{27} + 2 q^{28} + 8 q^{31} - 2 q^{32} - 6 q^{33} - 3 q^{34} + 2 q^{36} - 7 q^{37} - 2 q^{39} - q^{41} + 2 q^{42} - q^{43} + 6 q^{44} + 5 q^{46} - 9 q^{47} - 2 q^{48} - 12 q^{49} + 10 q^{50} - 3 q^{51} + 2 q^{52} + 3 q^{53} + 2 q^{54} - 2 q^{56} - 2 q^{59} - 13 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} + 6 q^{66} - 17 q^{67} + 3 q^{68} + 5 q^{69} - 14 q^{71} - 2 q^{72} - 9 q^{73} + 7 q^{74} + 10 q^{75} + 6 q^{77} + 2 q^{78} - 2 q^{79} + 2 q^{81} + q^{82} + 2 q^{83} - 2 q^{84} + q^{86} - 6 q^{88} - 10 q^{89} + 2 q^{91} - 5 q^{92} - 8 q^{93} + 9 q^{94} + 2 q^{96} - 4 q^{97} + 12 q^{98} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 3 * q^17 - 2 * q^18 - 2 * q^21 - 6 * q^22 - 5 * q^23 + 2 * q^24 - 10 * q^25 - 2 * q^26 - 2 * q^27 + 2 * q^28 + 8 * q^31 - 2 * q^32 - 6 * q^33 - 3 * q^34 + 2 * q^36 - 7 * q^37 - 2 * q^39 - q^41 + 2 * q^42 - q^43 + 6 * q^44 + 5 * q^46 - 9 * q^47 - 2 * q^48 - 12 * q^49 + 10 * q^50 - 3 * q^51 + 2 * q^52 + 3 * q^53 + 2 * q^54 - 2 * q^56 - 2 * q^59 - 13 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 + 6 * q^66 - 17 * q^67 + 3 * q^68 + 5 * q^69 - 14 * q^71 - 2 * q^72 - 9 * q^73 + 7 * q^74 + 10 * q^75 + 6 * q^77 + 2 * q^78 - 2 * q^79 + 2 * q^81 + q^82 + 2 * q^83 - 2 * q^84 + q^86 - 6 * q^88 - 10 * q^89 + 2 * q^91 - 5 * q^92 - 8 * q^93 + 9 * q^94 + 2 * q^96 - 4 * q^97 + 12 * q^98 + 6 * 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
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.53113	−0.613889	−0.306944	−	0.951727i	$-0.599307\pi$
$17$	−2.53113	−0.613889	−0.306944	+	0.951727i	$0.599307\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−3.00000	−0.639602
$23$	1.53113	0.319262	0.159631	−	0.987177i	$-0.448969\pi$
$23$	1.53113	0.319262	0.159631	+	0.987177i	$0.448969\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	2.53113	0.434085
$35$	0	0
$36$	1.00000	0.166667
$37$	−7.53113	−1.23811	−0.619055	−	0.785348i	$-0.712484\pi$
$37$	−7.53113	−1.23811	−0.619055	+	0.785348i	$0.712484\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.53113	0.551470	0.275735	−	0.961234i	$-0.411079\pi$
$41$	3.53113	0.551470	0.275735	+	0.961234i	$0.411079\pi$
$42$	1.00000	0.154303
$43$	3.53113	0.538492	0.269246	−	0.963071i	$-0.413226\pi$
$43$	3.53113	0.538492	0.269246	+	0.963071i	$0.413226\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−1.53113	−0.225753
$47$	−0.468871	−0.0683919	−0.0341959	−	0.999415i	$-0.510887\pi$
$47$	−0.468871	−0.0683919	−0.0341959	+	0.999415i	$0.510887\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	5.00000	0.707107
$51$	2.53113	0.354429
$52$	1.00000	0.138675
$53$	−2.53113	−0.347677	−0.173839	−	0.984774i	$-0.555617\pi$
$53$	−2.53113	−0.347677	−0.173839	+	0.984774i	$0.555617\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	0	0
$59$	−9.06226	−1.17981	−0.589903	−	0.807474i	$-0.700834\pi$
$59$	−9.06226	−1.17981	−0.589903	+	0.807474i	$0.700834\pi$
$60$	0	0
$61$	−2.46887	−0.316107	−0.158053	−	0.987431i	$-0.550522\pi$
$61$	−2.46887	−0.316107	−0.158053	+	0.987431i	$0.550522\pi$
$62$	−4.00000	−0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	−12.5311	−1.53092	−0.765461	−	0.643483i	$-0.777489\pi$
$67$	−12.5311	−1.53092	−0.765461	+	0.643483i	$0.777489\pi$
$68$	−2.53113	−0.306944
$69$	−1.53113	−0.184326
$70$	0	0
$71$	1.06226	0.126067	0.0630334	−	0.998011i	$-0.479923\pi$
$71$	1.06226	0.126067	0.0630334	+	0.998011i	$0.479923\pi$
$72$	−1.00000	−0.117851
$73$	7.59339	0.888739	0.444369	−	0.895844i	$-0.353428\pi$
$73$	7.59339	0.888739	0.444369	+	0.895844i	$0.353428\pi$
$74$	7.53113	0.875476
$75$	5.00000	0.577350
$76$	0	0
$77$	3.00000	0.341882
$78$	1.00000	0.113228
$79$	−9.06226	−1.01958	−0.509792	−	0.860298i	$-0.670278\pi$
$79$	−9.06226	−1.01958	−0.509792	+	0.860298i	$0.670278\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.53113	−0.389948
$83$	9.06226	0.994712	0.497356	−	0.867547i	$-0.334304\pi$
$83$	9.06226	0.994712	0.497356	+	0.867547i	$0.334304\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−3.53113	−0.380771
$87$	0	0
$88$	−3.00000	−0.319801
$89$	−13.0623	−1.38460	−0.692298	−	0.721611i	$-0.743402\pi$
$89$	−13.0623	−1.38460	−0.692298	+	0.721611i	$0.743402\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	1.53113	0.159631
$93$	−4.00000	−0.414781
$94$	0.468871	0.0483604
$95$	0	0
$96$	1.00000	0.102062
$97$	−18.1245	−1.84027	−0.920133	−	0.391606i	$-0.871920\pi$
$97$	−18.1245	−1.84027	−0.920133	+	0.391606i	$0.871920\pi$
$98$	6.00000	0.606092
$99$	3.00000	0.301511
$100$	−5.00000	−0.500000
$101$	4.06226	0.404210	0.202105	−	0.979364i	$-0.435222\pi$
$101$	4.06226	0.404210	0.202105	+	0.979364i	$0.435222\pi$
$102$	−2.53113	−0.250619
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	2.53113	0.245845
$107$	8.53113	0.824735	0.412368	−	0.911018i	$-0.364702\pi$
$107$	8.53113	0.824735	0.412368	+	0.911018i	$0.364702\pi$
$108$	−1.00000	−0.0962250
$109$	−10.4689	−1.00274	−0.501368	−	0.865234i	$-0.667170\pi$
$109$	−10.4689	−1.00274	−0.501368	+	0.865234i	$0.667170\pi$
$110$	0	0
$111$	7.53113	0.714823
$112$	1.00000	0.0944911
$113$	0.468871	0.0441077	0.0220538	−	0.999757i	$-0.492979\pi$
$113$	0.468871	0.0441077	0.0220538	+	0.999757i	$0.492979\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	9.06226	0.834248
$119$	−2.53113	−0.232028
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.46887	0.223521
$123$	−3.53113	−0.318391
$124$	4.00000	0.359211
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	15.0000	1.33103	0.665517	−	0.746382i	$-0.268211\pi$
$127$	15.0000	1.33103	0.665517	+	0.746382i	$0.268211\pi$
$128$	−1.00000	−0.0883883
$129$	−3.53113	−0.310899
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	12.5311	1.08252
$135$	0	0
$136$	2.53113	0.217043
$137$	−4.59339	−0.392440	−0.196220	−	0.980560i	$-0.562867\pi$
$137$	−4.59339	−0.392440	−0.196220	+	0.980560i	$0.562867\pi$
$138$	1.53113	0.130338
$139$	−7.46887	−0.633501	−0.316751	−	0.948509i	$-0.602592\pi$
$139$	−7.46887	−0.633501	−0.316751	+	0.948509i	$0.602592\pi$
$140$	0	0
$141$	0.468871	0.0394861
$142$	−1.06226	−0.0891427
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	−7.59339	−0.628433
$147$	6.00000	0.494872
$148$	−7.53113	−0.619055
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	−5.00000	−0.408248
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−2.53113	−0.204630
$154$	−3.00000	−0.241747
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	12.5311	1.00009	0.500046	−	0.865999i	$-0.333316\pi$
$157$	12.5311	1.00009	0.500046	+	0.865999i	$0.333316\pi$
$158$	9.06226	0.720955
$159$	2.53113	0.200732
$160$	0	0
$161$	1.53113	0.120670
$162$	−1.00000	−0.0785674
$163$	16.5934	1.29969	0.649847	−	0.760065i	$-0.274833\pi$
$163$	16.5934	1.29969	0.649847	+	0.760065i	$0.274833\pi$
$164$	3.53113	0.275735
$165$	0	0
$166$	−9.06226	−0.703368
$167$	−2.53113	−0.195865	−0.0979323	−	0.995193i	$-0.531223\pi$
$167$	−2.53113	−0.195865	−0.0979323	+	0.995193i	$0.531223\pi$
$168$	1.00000	0.0771517
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	3.53113	0.269246
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	3.00000	0.226134
$177$	9.06226	0.681161
$178$	13.0623	0.979058
$179$	−16.0623	−1.20055	−0.600275	−	0.799794i	$-0.704942\pi$
$179$	−16.0623	−1.20055	−0.600275	+	0.799794i	$0.704942\pi$
$180$	0	0
$181$	−6.53113	−0.485455	−0.242727	−	0.970095i	$-0.578042\pi$
$181$	−6.53113	−0.485455	−0.242727	+	0.970095i	$0.578042\pi$
$182$	−1.00000	−0.0741249
$183$	2.46887	0.182504
$184$	−1.53113	−0.112876
$185$	0	0
$186$	4.00000	0.293294
$187$	−7.59339	−0.555283
$188$	−0.468871	−0.0341959
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−0.531129	−0.0384311	−0.0192156	−	0.999815i	$-0.506117\pi$
$191$	−0.531129	−0.0384311	−0.0192156	+	0.999815i	$0.506117\pi$
$192$	−1.00000	−0.0721688
$193$	4.59339	0.330639	0.165320	−	0.986240i	$-0.447134\pi$
$193$	4.59339	0.330639	0.165320	+	0.986240i	$0.447134\pi$
$194$	18.1245	1.30126
$195$	0	0
$196$	−6.00000	−0.428571
$197$	5.06226	0.360671	0.180335	−	0.983605i	$-0.442282\pi$
$197$	5.06226	0.360671	0.180335	+	0.983605i	$0.442282\pi$
$198$	−3.00000	−0.213201
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	5.00000	0.353553
$201$	12.5311	0.883878
$202$	−4.06226	−0.285819
$203$	0	0
$204$	2.53113	0.177214
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	1.53113	0.106421
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−7.53113	−0.518464	−0.259232	−	0.965815i	$-0.583469\pi$
$211$	−7.53113	−0.518464	−0.259232	+	0.965815i	$0.583469\pi$
$212$	−2.53113	−0.173839
$213$	−1.06226	−0.0727847
$214$	−8.53113	−0.583176
$215$	0	0
$216$	1.00000	0.0680414
$217$	4.00000	0.271538
$218$	10.4689	0.709041
$219$	−7.59339	−0.513114
$220$	0	0
$221$	−2.53113	−0.170262
$222$	−7.53113	−0.505456
$223$	17.1245	1.14674	0.573371	−	0.819296i	$-0.305635\pi$
$223$	17.1245	1.14674	0.573371	+	0.819296i	$0.305635\pi$
$224$	−1.00000	−0.0668153
$225$	−5.00000	−0.333333
$226$	−0.468871	−0.0311888
$227$	1.59339	0.105757	0.0528784	−	0.998601i	$-0.483160\pi$
$227$	1.59339	0.105757	0.0528784	+	0.998601i	$0.483160\pi$
$228$	0	0
$229$	−23.1868	−1.53223	−0.766113	−	0.642706i	$-0.777812\pi$
$229$	−23.1868	−1.53223	−0.766113	+	0.642706i	$0.777812\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	17.5311	1.14850	0.574251	−	0.818679i	$-0.305293\pi$
$233$	17.5311	1.14850	0.574251	+	0.818679i	$0.305293\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−9.06226	−0.589903
$237$	9.06226	0.588657
$238$	2.53113	0.164069
$239$	3.46887	0.224383	0.112191	−	0.993687i	$-0.464213\pi$
$239$	3.46887	0.224383	0.112191	+	0.993687i	$0.464213\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	−2.46887	−0.158053
$245$	0	0
$246$	3.53113	0.225137
$247$	0	0
$248$	−4.00000	−0.254000
$249$	−9.06226	−0.574297
$250$	0	0
$251$	14.1245	0.891532	0.445766	−	0.895150i	$-0.352931\pi$
$251$	14.1245	0.891532	0.445766	+	0.895150i	$0.352931\pi$
$252$	1.00000	0.0629941
$253$	4.59339	0.288784
$254$	−15.0000	−0.941184
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.0623	0.690045	0.345022	−	0.938594i	$-0.387871\pi$
$257$	11.0623	0.690045	0.345022	+	0.938594i	$0.387871\pi$
$258$	3.53113	0.219838
$259$	−7.53113	−0.467962
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	22.5311	1.38933	0.694664	−	0.719334i	$-0.255553\pi$
$263$	22.5311	1.38933	0.694664	+	0.719334i	$0.255553\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	0	0
$267$	13.0623	0.799397
$268$	−12.5311	−0.765461
$269$	−27.0623	−1.65001	−0.825007	−	0.565122i	$-0.808829\pi$
$269$	−27.0623	−1.65001	−0.825007	+	0.565122i	$0.808829\pi$
$270$	0	0
$271$	−18.0000	−1.09342	−0.546711	−	0.837321i	$-0.684120\pi$
$271$	−18.0000	−1.09342	−0.546711	+	0.837321i	$0.684120\pi$
$272$	−2.53113	−0.153472
$273$	−1.00000	−0.0605228
$274$	4.59339	0.277497
$275$	−15.0000	−0.904534
$276$	−1.53113	−0.0921631
$277$	−10.9377	−0.657185	−0.328593	−	0.944472i	$-0.606574\pi$
$277$	−10.9377	−0.657185	−0.328593	+	0.944472i	$0.606574\pi$
$278$	7.46887	0.447953
$279$	4.00000	0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−0.468871	−0.0279209
$283$	1.40661	0.0836145	0.0418072	−	0.999126i	$-0.486688\pi$
$283$	1.40661	0.0836145	0.0418072	+	0.999126i	$0.486688\pi$
$284$	1.06226	0.0630334
$285$	0	0
$286$	−3.00000	−0.177394
$287$	3.53113	0.208436
$288$	−1.00000	−0.0589256
$289$	−10.5934	−0.623140
$290$	0	0
$291$	18.1245	1.06248
$292$	7.59339	0.444369
$293$	0.937742	0.0547835	0.0273917	−	0.999625i	$-0.491280\pi$
$293$	0.937742	0.0547835	0.0273917	+	0.999625i	$0.491280\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	7.53113	0.437738
$297$	−3.00000	−0.174078
$298$	−9.00000	−0.521356
$299$	1.53113	0.0885475
$300$	5.00000	0.288675
$301$	3.53113	0.203531
$302$	12.0000	0.690522
$303$	−4.06226	−0.233371
$304$	0	0
$305$	0	0
$306$	2.53113	0.144695
$307$	−24.5311	−1.40007	−0.700033	−	0.714111i	$-0.746831\pi$
$307$	−24.5311	−1.40007	−0.700033	+	0.714111i	$0.746831\pi$
$308$	3.00000	0.170941
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	10.5934	0.600696	0.300348	−	0.953830i	$-0.402897\pi$
$311$	10.5934	0.600696	0.300348	+	0.953830i	$0.402897\pi$
$312$	1.00000	0.0566139
$313$	−16.5934	−0.937914	−0.468957	−	0.883221i	$-0.655370\pi$
$313$	−16.5934	−0.937914	−0.468957	+	0.883221i	$0.655370\pi$
$314$	−12.5311	−0.707173
$315$	0	0
$316$	−9.06226	−0.509792
$317$	−31.5934	−1.77446	−0.887231	−	0.461326i	$-0.847374\pi$
$317$	−31.5934	−1.77446	−0.887231	+	0.461326i	$0.847374\pi$
$318$	−2.53113	−0.141939
$319$	0	0
$320$	0	0
$321$	−8.53113	−0.476161
$322$	−1.53113	−0.0853265
$323$	0	0
$324$	1.00000	0.0555556
$325$	−5.00000	−0.277350
$326$	−16.5934	−0.919023
$327$	10.4689	0.578930
$328$	−3.53113	−0.194974
$329$	−0.468871	−0.0258497
$330$	0	0
$331$	−25.7179	−1.41358	−0.706792	−	0.707422i	$-0.749858\pi$
$331$	−25.7179	−1.41358	−0.706792	+	0.707422i	$0.749858\pi$
$332$	9.06226	0.497356
$333$	−7.53113	−0.412703
$334$	2.53113	0.138497
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−1.93774	−0.105556	−0.0527778	−	0.998606i	$-0.516807\pi$
$337$	−1.93774	−0.105556	−0.0527778	+	0.998606i	$0.516807\pi$
$338$	−1.00000	−0.0543928
$339$	−0.468871	−0.0254656
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−3.53113	−0.190386
$345$	0	0
$346$	−21.0000	−1.12897
$347$	16.4689	0.884095	0.442048	−	0.896992i	$-0.354252\pi$
$347$	16.4689	0.884095	0.442048	+	0.896992i	$0.354252\pi$
$348$	0	0
$349$	34.2490	1.83331	0.916654	−	0.399681i	$-0.130879\pi$
$349$	34.2490	1.83331	0.916654	+	0.399681i	$0.130879\pi$
$350$	5.00000	0.267261
$351$	−1.00000	−0.0533761
$352$	−3.00000	−0.159901
$353$	−21.7179	−1.15593	−0.577964	−	0.816063i	$-0.696152\pi$
$353$	−21.7179	−1.15593	−0.577964	+	0.816063i	$0.696152\pi$
$354$	−9.06226	−0.481654
$355$	0	0
$356$	−13.0623	−0.692298
$357$	2.53113	0.133962
$358$	16.0623	0.848917
$359$	7.59339	0.400764	0.200382	−	0.979718i	$-0.435782\pi$
$359$	7.59339	0.400764	0.200382	+	0.979718i	$0.435782\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	6.53113	0.343269
$363$	2.00000	0.104973
$364$	1.00000	0.0524142
$365$	0	0
$366$	−2.46887	−0.129050
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	1.53113	0.0798156
$369$	3.53113	0.183823
$370$	0	0
$371$	−2.53113	−0.131410
$372$	−4.00000	−0.207390
$373$	31.6556	1.63907	0.819534	−	0.573031i	$-0.194233\pi$
$373$	31.6556	1.63907	0.819534	+	0.573031i	$0.194233\pi$
$374$	7.59339	0.392645
$375$	0	0
$376$	0.468871	0.0241802
$377$	0	0
$378$	1.00000	0.0514344
$379$	28.6556	1.47194	0.735971	−	0.677013i	$-0.236726\pi$
$379$	28.6556	1.47194	0.735971	+	0.677013i	$0.236726\pi$
$380$	0	0
$381$	−15.0000	−0.768473
$382$	0.531129	0.0271749
$383$	−37.7802	−1.93048	−0.965238	−	0.261373i	$-0.915825\pi$
$383$	−37.7802	−1.93048	−0.965238	+	0.261373i	$0.915825\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−4.59339	−0.233797
$387$	3.53113	0.179497
$388$	−18.1245	−0.920133
$389$	−29.1245	−1.47667	−0.738336	−	0.674433i	$-0.764388\pi$
$389$	−29.1245	−1.47667	−0.738336	+	0.674433i	$0.764388\pi$
$390$	0	0
$391$	−3.87548	−0.195992
$392$	6.00000	0.303046
$393$	12.0000	0.605320
$394$	−5.06226	−0.255033
$395$	0	0
$396$	3.00000	0.150756
$397$	36.1245	1.81304	0.906519	−	0.422166i	$-0.138730\pi$
$397$	36.1245	1.81304	0.906519	+	0.422166i	$0.138730\pi$
$398$	5.00000	0.250627
$399$	0	0
$400$	−5.00000	−0.250000
$401$	29.5311	1.47471	0.737357	−	0.675503i	$-0.236074\pi$
$401$	29.5311	1.47471	0.737357	+	0.675503i	$0.236074\pi$
$402$	−12.5311	−0.624996
$403$	4.00000	0.199254
$404$	4.06226	0.202105
$405$	0	0
$406$	0	0
$407$	−22.5934	−1.11991
$408$	−2.53113	−0.125310
$409$	14.1245	0.698412	0.349206	−	0.937046i	$-0.386451\pi$
$409$	14.1245	0.698412	0.349206	+	0.937046i	$0.386451\pi$
$410$	0	0
$411$	4.59339	0.226575
$412$	1.00000	0.0492665
$413$	−9.06226	−0.445925
$414$	−1.53113	−0.0752509
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	7.46887	0.365752
$418$	0	0
$419$	−27.2490	−1.33120	−0.665601	−	0.746308i	$-0.731825\pi$
$419$	−27.2490	−1.33120	−0.665601	+	0.746308i	$0.731825\pi$
$420$	0	0
$421$	−37.5934	−1.83219	−0.916095	−	0.400962i	$-0.868676\pi$
$421$	−37.5934	−1.83219	−0.916095	+	0.400962i	$0.868676\pi$
$422$	7.53113	0.366610
$423$	−0.468871	−0.0227973
$424$	2.53113	0.122922
$425$	12.6556	0.613889
$426$	1.06226	0.0514666
$427$	−2.46887	−0.119477
$428$	8.53113	0.412368
$429$	−3.00000	−0.144841
$430$	0	0
$431$	15.0623	0.725523	0.362762	−	0.931882i	$-0.381834\pi$
$431$	15.0623	0.725523	0.362762	+	0.931882i	$0.381834\pi$
$432$	−1.00000	−0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−10.4689	−0.501368
$437$	0	0
$438$	7.59339	0.362826
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	2.53113	0.120394
$443$	36.2490	1.72224	0.861122	−	0.508399i	$-0.169762\pi$
$443$	36.2490	1.72224	0.861122	+	0.508399i	$0.169762\pi$
$444$	7.53113	0.357412
$445$	0	0
$446$	−17.1245	−0.810869
$447$	−9.00000	−0.425685
$448$	1.00000	0.0472456
$449$	28.6556	1.35234	0.676172	−	0.736744i	$-0.263638\pi$
$449$	28.6556	1.35234	0.676172	+	0.736744i	$0.263638\pi$
$450$	5.00000	0.235702
$451$	10.5934	0.498823
$452$	0.468871	0.0220538
$453$	12.0000	0.563809
$454$	−1.59339	−0.0747813
$455$	0	0
$456$	0	0
$457$	−41.7802	−1.95439	−0.977197	−	0.212336i	$-0.931893\pi$
$457$	−41.7802	−1.95439	−0.977197	+	0.212336i	$0.931893\pi$
$458$	23.1868	1.08345
$459$	2.53113	0.118143
$460$	0	0
$461$	14.4689	0.673883	0.336941	−	0.941526i	$-0.390608\pi$
$461$	14.4689	0.673883	0.336941	+	0.941526i	$0.390608\pi$
$462$	3.00000	0.139573
$463$	−37.0623	−1.72243	−0.861215	−	0.508242i	$-0.830296\pi$
$463$	−37.0623	−1.72243	−0.861215	+	0.508242i	$0.830296\pi$
$464$	0	0
$465$	0	0
$466$	−17.5311	−0.812114
$467$	−15.9377	−0.737511	−0.368755	−	0.929526i	$-0.620216\pi$
$467$	−15.9377	−0.737511	−0.368755	+	0.929526i	$0.620216\pi$
$468$	1.00000	0.0462250
$469$	−12.5311	−0.578634
$470$	0	0
$471$	−12.5311	−0.577404
$472$	9.06226	0.417124
$473$	10.5934	0.487084
$474$	−9.06226	−0.416243
$475$	0	0
$476$	−2.53113	−0.116014
$477$	−2.53113	−0.115892
$478$	−3.46887	−0.158662
$479$	−23.6556	−1.08085	−0.540427	−	0.841391i	$-0.681737\pi$
$479$	−23.6556	−1.08085	−0.540427	+	0.841391i	$0.681737\pi$
$480$	0	0
$481$	−7.53113	−0.343390
$482$	−5.00000	−0.227744
$483$	−1.53113	−0.0696688
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	−33.1868	−1.50384	−0.751918	−	0.659257i	$-0.770871\pi$
$487$	−33.1868	−1.50384	−0.751918	+	0.659257i	$0.770871\pi$
$488$	2.46887	0.111761
$489$	−16.5934	−0.750379
$490$	0	0
$491$	−30.6556	−1.38347	−0.691735	−	0.722151i	$-0.743153\pi$
$491$	−30.6556	−1.38347	−0.691735	+	0.722151i	$0.743153\pi$
$492$	−3.53113	−0.159196
$493$	0	0
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	1.06226	0.0476488
$498$	9.06226	0.406089
$499$	9.18677	0.411256	0.205628	−	0.978630i	$-0.434076\pi$
$499$	9.18677	0.411256	0.205628	+	0.978630i	$0.434076\pi$
$500$	0	0
$501$	2.53113	0.113082
$502$	−14.1245	−0.630408
$503$	−20.7179	−0.923766	−0.461883	−	0.886941i	$-0.652826\pi$
$503$	−20.7179	−0.923766	−0.461883	+	0.886941i	$0.652826\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	−4.59339	−0.204201
$507$	−1.00000	−0.0444116
$508$	15.0000	0.665517
$509$	18.0623	0.800595	0.400298	−	0.916385i	$-0.368907\pi$
$509$	18.0623	0.800595	0.400298	+	0.916385i	$0.368907\pi$
$510$	0	0
$511$	7.59339	0.335912
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−11.0623	−0.487935
$515$	0	0
$516$	−3.53113	−0.155449
$517$	−1.40661	−0.0618628
$518$	7.53113	0.330899
$519$	−21.0000	−0.921798
$520$	0	0
$521$	−1.40661	−0.0616249	−0.0308124	−	0.999525i	$-0.509809\pi$
$521$	−1.40661	−0.0616249	−0.0308124	+	0.999525i	$0.509809\pi$
$522$	0	0
$523$	−38.7802	−1.69574	−0.847869	−	0.530206i	$-0.822115\pi$
$523$	−38.7802	−1.69574	−0.847869	+	0.530206i	$0.822115\pi$
$524$	−12.0000	−0.524222
$525$	5.00000	0.218218
$526$	−22.5311	−0.982404
$527$	−10.1245	−0.441031
$528$	−3.00000	−0.130558
$529$	−20.6556	−0.898071
$530$	0	0
$531$	−9.06226	−0.393268
$532$	0	0
$533$	3.53113	0.152950
$534$	−13.0623	−0.565259
$535$	0	0
$536$	12.5311	0.541262
$537$	16.0623	0.693138
$538$	27.0623	1.16674
$539$	−18.0000	−0.775315
$540$	0	0
$541$	34.5311	1.48461	0.742305	−	0.670063i	$-0.233733\pi$
$541$	34.5311	1.48461	0.742305	+	0.670063i	$0.233733\pi$
$542$	18.0000	0.773166
$543$	6.53113	0.280278
$544$	2.53113	0.108521
$545$	0	0
$546$	1.00000	0.0427960
$547$	−21.1868	−0.905881	−0.452941	−	0.891541i	$-0.649625\pi$
$547$	−21.1868	−0.905881	−0.452941	+	0.891541i	$0.649625\pi$
$548$	−4.59339	−0.196220
$549$	−2.46887	−0.105369
$550$	15.0000	0.639602
$551$	0	0
$552$	1.53113	0.0651692
$553$	−9.06226	−0.385366
$554$	10.9377	0.464700
$555$	0	0
$556$	−7.46887	−0.316751
$557$	43.3113	1.83516	0.917579	−	0.397553i	$-0.130141\pi$
$557$	43.3113	1.83516	0.917579	+	0.397553i	$0.130141\pi$
$558$	−4.00000	−0.169334
$559$	3.53113	0.149351
$560$	0	0
$561$	7.59339	0.320593
$562$	−6.00000	−0.253095
$563$	−26.9377	−1.13529	−0.567645	−	0.823273i	$-0.692145\pi$
$563$	−26.9377	−1.13529	−0.567645	+	0.823273i	$0.692145\pi$
$564$	0.468871	0.0197430
$565$	0	0
$566$	−1.40661	−0.0591244
$567$	1.00000	0.0419961
$568$	−1.06226	−0.0445713
$569$	−31.6556	−1.32707	−0.663537	−	0.748144i	$-0.730945\pi$
$569$	−31.6556	−1.32707	−0.663537	+	0.748144i	$0.730945\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	3.00000	0.125436
$573$	0.531129	0.0221882
$574$	−3.53113	−0.147386
$575$	−7.65564	−0.319262
$576$	1.00000	0.0416667
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	10.5934	0.440627
$579$	−4.59339	−0.190895
$580$	0	0
$581$	9.06226	0.375966
$582$	−18.1245	−0.751285
$583$	−7.59339	−0.314486
$584$	−7.59339	−0.314217
$585$	0	0
$586$	−0.937742	−0.0387378
$587$	9.06226	0.374039	0.187020	−	0.982356i	$-0.440117\pi$
$587$	9.06226	0.374039	0.187020	+	0.982356i	$0.440117\pi$
$588$	6.00000	0.247436
$589$	0	0
$590$	0	0
$591$	−5.06226	−0.208233
$592$	−7.53113	−0.309527
$593$	4.40661	0.180958	0.0904790	−	0.995898i	$-0.471160\pi$
$593$	4.40661	0.180958	0.0904790	+	0.995898i	$0.471160\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	9.00000	0.368654
$597$	5.00000	0.204636
$598$	−1.53113	−0.0626125
$599$	19.5934	0.800564	0.400282	−	0.916392i	$-0.368912\pi$
$599$	19.5934	0.800564	0.400282	+	0.916392i	$0.368912\pi$
$600$	−5.00000	−0.204124
$601$	9.18677	0.374736	0.187368	−	0.982290i	$-0.440004\pi$
$601$	9.18677	0.374736	0.187368	+	0.982290i	$0.440004\pi$
$602$	−3.53113	−0.143918
$603$	−12.5311	−0.510307
$604$	−12.0000	−0.488273
$605$	0	0
$606$	4.06226	0.165018
$607$	−42.0000	−1.70473	−0.852364	−	0.522949i	$-0.824832\pi$
$607$	−42.0000	−1.70473	−0.852364	+	0.522949i	$0.824832\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.468871	−0.0189685
$612$	−2.53113	−0.102315
$613$	9.59339	0.387473	0.193737	−	0.981054i	$-0.437939\pi$
$613$	9.59339	0.387473	0.193737	+	0.981054i	$0.437939\pi$
$614$	24.5311	0.989996
$615$	0	0
$616$	−3.00000	−0.120873
$617$	−9.06226	−0.364833	−0.182416	−	0.983221i	$-0.558392\pi$
$617$	−9.06226	−0.364833	−0.182416	+	0.983221i	$0.558392\pi$
$618$	1.00000	0.0402259
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−1.53113	−0.0614421
$622$	−10.5934	−0.424756
$623$	−13.0623	−0.523328
$624$	−1.00000	−0.0400320
$625$	25.0000	1.00000
$626$	16.5934	0.663205
$627$	0	0
$628$	12.5311	0.500046
$629$	19.0623	0.760062
$630$	0	0
$631$	25.2490	1.00515	0.502574	−	0.864534i	$-0.332386\pi$
$631$	25.2490	1.00515	0.502574	+	0.864534i	$0.332386\pi$
$632$	9.06226	0.360477
$633$	7.53113	0.299335
$634$	31.5934	1.25473
$635$	0	0
$636$	2.53113	0.100366
$637$	−6.00000	−0.237729
$638$	0	0
$639$	1.06226	0.0420223
$640$	0	0
$641$	−9.34436	−0.369080	−0.184540	−	0.982825i	$-0.559080\pi$
$641$	−9.34436	−0.369080	−0.184540	+	0.982825i	$0.559080\pi$
$642$	8.53113	0.336697
$643$	−9.40661	−0.370961	−0.185480	−	0.982648i	$-0.559384\pi$
$643$	−9.40661	−0.370961	−0.185480	+	0.982648i	$0.559384\pi$
$644$	1.53113	0.0603349
$645$	0	0
$646$	0	0
$647$	−1.40661	−0.0552997	−0.0276498	−	0.999618i	$-0.508802\pi$
$647$	−1.40661	−0.0552997	−0.0276498	+	0.999618i	$0.508802\pi$
$648$	−1.00000	−0.0392837
$649$	−27.1868	−1.06717
$650$	5.00000	0.196116
$651$	−4.00000	−0.156772
$652$	16.5934	0.649847
$653$	−7.40661	−0.289843	−0.144922	−	0.989443i	$-0.546293\pi$
$653$	−7.40661	−0.289843	−0.144922	+	0.989443i	$0.546293\pi$
$654$	−10.4689	−0.409365
$655$	0	0
$656$	3.53113	0.137867
$657$	7.59339	0.296246
$658$	0.468871	0.0182785
$659$	22.6556	0.882539	0.441269	−	0.897375i	$-0.354528\pi$
$659$	22.6556	0.882539	0.441269	+	0.897375i	$0.354528\pi$
$660$	0	0
$661$	−24.1245	−0.938335	−0.469167	−	0.883109i	$-0.655446\pi$
$661$	−24.1245	−0.938335	−0.469167	+	0.883109i	$0.655446\pi$
$662$	25.7179	0.999555
$663$	2.53113	0.0983009
$664$	−9.06226	−0.351684
$665$	0	0
$666$	7.53113	0.291825
$667$	0	0
$668$	−2.53113	−0.0979323
$669$	−17.1245	−0.662072
$670$	0	0
$671$	−7.40661	−0.285929
$672$	1.00000	0.0385758
$673$	−24.0623	−0.927532	−0.463766	−	0.885958i	$-0.653502\pi$
$673$	−24.0623	−0.927532	−0.463766	+	0.885958i	$0.653502\pi$
$674$	1.93774	0.0746390
$675$	5.00000	0.192450
$676$	1.00000	0.0384615
$677$	23.0623	0.886355	0.443177	−	0.896434i	$-0.353851\pi$
$677$	23.0623	0.886355	0.443177	+	0.896434i	$0.353851\pi$
$678$	0.468871	0.0180069
$679$	−18.1245	−0.695555
$680$	0	0
$681$	−1.59339	−0.0610587
$682$	−12.0000	−0.459504
$683$	−28.5311	−1.09171	−0.545857	−	0.837879i	$-0.683796\pi$
$683$	−28.5311	−1.09171	−0.545857	+	0.837879i	$0.683796\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	23.1868	0.884631
$688$	3.53113	0.134623
$689$	−2.53113	−0.0964283
$690$	0	0
$691$	26.1245	0.993823	0.496912	−	0.867801i	$-0.334467\pi$
$691$	26.1245	0.993823	0.496912	+	0.867801i	$0.334467\pi$
$692$	21.0000	0.798300
$693$	3.00000	0.113961
$694$	−16.4689	−0.625150
$695$	0	0
$696$	0	0
$697$	−8.93774	−0.338541
$698$	−34.2490	−1.29634
$699$	−17.5311	−0.663088
$700$	−5.00000	−0.188982
$701$	−0.937742	−0.0354180	−0.0177090	−	0.999843i	$-0.505637\pi$
$701$	−0.937742	−0.0354180	−0.0177090	+	0.999843i	$0.505637\pi$
$702$	1.00000	0.0377426
$703$	0	0
$704$	3.00000	0.113067
$705$	0	0
$706$	21.7179	0.817364
$707$	4.06226	0.152777
$708$	9.06226	0.340581
$709$	0.813227	0.0305414	0.0152707	−	0.999883i	$-0.495139\pi$
$709$	0.813227	0.0305414	0.0152707	+	0.999883i	$0.495139\pi$
$710$	0	0
$711$	−9.06226	−0.339861
$712$	13.0623	0.489529
$713$	6.12452	0.229365
$714$	−2.53113	−0.0947251
$715$	0	0
$716$	−16.0623	−0.600275
$717$	−3.46887	−0.129547
$718$	−7.59339	−0.283383
$719$	−22.5311	−0.840269	−0.420135	−	0.907462i	$-0.638017\pi$
$719$	−22.5311	−0.840269	−0.420135	+	0.907462i	$0.638017\pi$
$720$	0	0
$721$	1.00000	0.0372419
$722$	19.0000	0.707107
$723$	−5.00000	−0.185952
$724$	−6.53113	−0.242727
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	−2.40661	−0.0892563	−0.0446282	−	0.999004i	$-0.514210\pi$
$727$	−2.40661	−0.0892563	−0.0446282	+	0.999004i	$0.514210\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.93774	−0.330574
$732$	2.46887	0.0912521
$733$	−37.5311	−1.38624	−0.693122	−	0.720820i	$-0.743765\pi$
$733$	−37.5311	−1.38624	−0.693122	+	0.720820i	$0.743765\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−1.53113	−0.0564382
$737$	−37.5934	−1.38477
$738$	−3.53113	−0.129983
$739$	−30.7179	−1.12998	−0.564988	−	0.825099i	$-0.691119\pi$
$739$	−30.7179	−1.12998	−0.564988	+	0.825099i	$0.691119\pi$
$740$	0	0
$741$	0	0
$742$	2.53113	0.0929207
$743$	17.0623	0.625954	0.312977	−	0.949761i	$-0.398674\pi$
$743$	17.0623	0.625954	0.312977	+	0.949761i	$0.398674\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	−31.6556	−1.15900
$747$	9.06226	0.331571
$748$	−7.59339	−0.277642
$749$	8.53113	0.311721
$750$	0	0
$751$	5.18677	0.189268	0.0946340	−	0.995512i	$-0.469832\pi$
$751$	5.18677	0.189268	0.0946340	+	0.995512i	$0.469832\pi$
$752$	−0.468871	−0.0170980
$753$	−14.1245	−0.514726
$754$	0	0
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	13.4066	0.487272	0.243636	−	0.969867i	$-0.421660\pi$
$757$	13.4066	0.487272	0.243636	+	0.969867i	$0.421660\pi$
$758$	−28.6556	−1.04082
$759$	−4.59339	−0.166729
$760$	0	0
$761$	−34.2490	−1.24153	−0.620763	−	0.783998i	$-0.713177\pi$
$761$	−34.2490	−1.24153	−0.620763	+	0.783998i	$0.713177\pi$
$762$	15.0000	0.543393
$763$	−10.4689	−0.378999
$764$	−0.531129	−0.0192156
$765$	0	0
$766$	37.7802	1.36505
$767$	−9.06226	−0.327219
$768$	−1.00000	−0.0360844
$769$	38.1868	1.37705	0.688525	−	0.725212i	$-0.258258\pi$
$769$	38.1868	1.37705	0.688525	+	0.725212i	$0.258258\pi$
$770$	0	0
$771$	−11.0623	−0.398397
$772$	4.59339	0.165320
$773$	−20.0623	−0.721589	−0.360795	−	0.932645i	$-0.617494\pi$
$773$	−20.0623	−0.721589	−0.360795	+	0.932645i	$0.617494\pi$
$774$	−3.53113	−0.126924
$775$	−20.0000	−0.718421
$776$	18.1245	0.650632
$777$	7.53113	0.270178
$778$	29.1245	1.04416
$779$	0	0
$780$	0	0
$781$	3.18677	0.114032
$782$	3.87548	0.138587
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−12.0000	−0.428026
$787$	10.5934	0.377613	0.188807	−	0.982014i	$-0.439538\pi$
$787$	10.5934	0.377613	0.188807	+	0.982014i	$0.439538\pi$
$788$	5.06226	0.180335
$789$	−22.5311	−0.802129
$790$	0	0
$791$	0.468871	0.0166711
$792$	−3.00000	−0.106600
$793$	−2.46887	−0.0876722
$794$	−36.1245	−1.28201
$795$	0	0
$796$	−5.00000	−0.177220
$797$	−24.9377	−0.883340	−0.441670	−	0.897178i	$-0.645614\pi$
$797$	−24.9377	−0.883340	−0.441670	+	0.897178i	$0.645614\pi$
$798$	0	0
$799$	1.18677	0.0419850
$800$	5.00000	0.176777
$801$	−13.0623	−0.461532
$802$	−29.5311	−1.04278
$803$	22.7802	0.803894
$804$	12.5311	0.441939
$805$	0	0
$806$	−4.00000	−0.140894
$807$	27.0623	0.952637
$808$	−4.06226	−0.142910
$809$	43.3113	1.52274	0.761372	−	0.648315i	$-0.224526\pi$
$809$	43.3113	1.52274	0.761372	+	0.648315i	$0.224526\pi$
$810$	0	0
$811$	−33.1868	−1.16535	−0.582673	−	0.812707i	$-0.697993\pi$
$811$	−33.1868	−1.16535	−0.582673	+	0.812707i	$0.697993\pi$
$812$	0	0
$813$	18.0000	0.631288
$814$	22.5934	0.791898
$815$	0	0
$816$	2.53113	0.0886072
$817$	0	0
$818$	−14.1245	−0.493852
$819$	1.00000	0.0349428
$820$	0	0
$821$	−52.3113	−1.82568	−0.912838	−	0.408321i	$-0.866114\pi$
$821$	−52.3113	−1.82568	−0.912838	+	0.408321i	$0.866114\pi$
$822$	−4.59339	−0.160213
$823$	−41.6556	−1.45202	−0.726012	−	0.687682i	$-0.758628\pi$
$823$	−41.6556	−1.45202	−0.726012	+	0.687682i	$0.758628\pi$
$824$	−1.00000	−0.0348367
$825$	15.0000	0.522233
$826$	9.06226	0.315316
$827$	0.468871	0.0163042	0.00815212	−	0.999967i	$-0.497405\pi$
$827$	0.468871	0.0163042	0.00815212	+	0.999967i	$0.497405\pi$
$828$	1.53113	0.0532104
$829$	−33.3113	−1.15695	−0.578474	−	0.815701i	$-0.696352\pi$
$829$	−33.3113	−1.15695	−0.578474	+	0.815701i	$0.696352\pi$
$830$	0	0
$831$	10.9377	0.379426
$832$	1.00000	0.0346688
$833$	15.1868	0.526191
$834$	−7.46887	−0.258626
$835$	0	0
$836$	0	0
$837$	−4.00000	−0.138260
$838$	27.2490	0.941302
$839$	35.0623	1.21048	0.605242	−	0.796042i	$-0.293076\pi$
$839$	35.0623	1.21048	0.605242	+	0.796042i	$0.293076\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	37.5934	1.29555
$843$	−6.00000	−0.206651
$844$	−7.53113	−0.259232
$845$	0	0
$846$	0.468871	0.0161201
$847$	−2.00000	−0.0687208
$848$	−2.53113	−0.0869193
$849$	−1.40661	−0.0482748
$850$	−12.6556	−0.434085
$851$	−11.5311	−0.395282
$852$	−1.06226	−0.0363924
$853$	32.7802	1.12237	0.561186	−	0.827690i	$-0.310345\pi$
$853$	32.7802	1.12237	0.561186	+	0.827690i	$0.310345\pi$
$854$	2.46887	0.0844830
$855$	0	0
$856$	−8.53113	−0.291588
$857$	−1.59339	−0.0544291	−0.0272145	−	0.999630i	$-0.508664\pi$
$857$	−1.59339	−0.0544291	−0.0272145	+	0.999630i	$0.508664\pi$
$858$	3.00000	0.102418
$859$	−17.7802	−0.606651	−0.303326	−	0.952887i	$-0.598097\pi$
$859$	−17.7802	−0.606651	−0.303326	+	0.952887i	$0.598097\pi$
$860$	0	0
$861$	−3.53113	−0.120341
$862$	−15.0623	−0.513023
$863$	−9.65564	−0.328682	−0.164341	−	0.986404i	$-0.552550\pi$
$863$	−9.65564	−0.328682	−0.164341	+	0.986404i	$0.552550\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−16.0000	−0.543702
$867$	10.5934	0.359770
$868$	4.00000	0.135769
$869$	−27.1868	−0.922248
$870$	0	0
$871$	−12.5311	−0.424601
$872$	10.4689	0.354521
$873$	−18.1245	−0.613422
$874$	0	0
$875$	0	0
$876$	−7.59339	−0.256557
$877$	−13.4066	−0.452709	−0.226355	−	0.974045i	$-0.572681\pi$
$877$	−13.4066	−0.452709	−0.226355	+	0.974045i	$0.572681\pi$
$878$	−1.00000	−0.0337484
$879$	−0.937742	−0.0316293
$880$	0	0
$881$	−28.9377	−0.974937	−0.487469	−	0.873140i	$-0.662080\pi$
$881$	−28.9377	−0.974937	−0.487469	+	0.873140i	$0.662080\pi$
$882$	6.00000	0.202031
$883$	−15.5934	−0.524759	−0.262380	−	0.964965i	$-0.584507\pi$
$883$	−15.5934	−0.524759	−0.262380	+	0.964965i	$0.584507\pi$
$884$	−2.53113	−0.0851311
$885$	0	0
$886$	−36.2490	−1.21781
$887$	48.8424	1.63997	0.819984	−	0.572387i	$-0.193982\pi$
$887$	48.8424	1.63997	0.819984	+	0.572387i	$0.193982\pi$
$888$	−7.53113	−0.252728
$889$	15.0000	0.503084
$890$	0	0
$891$	3.00000	0.100504
$892$	17.1245	0.573371
$893$	0	0
$894$	9.00000	0.301005
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−1.53113	−0.0511229
$898$	−28.6556	−0.956251
$899$	0	0
$900$	−5.00000	−0.166667
$901$	6.40661	0.213435
$902$	−10.5934	−0.352721
$903$	−3.53113	−0.117509
$904$	−0.468871	−0.0155944
$905$	0	0
$906$	−12.0000	−0.398673
$907$	19.5934	0.650588	0.325294	−	0.945613i	$-0.394537\pi$
$907$	19.5934	0.650588	0.325294	+	0.945613i	$0.394537\pi$
$908$	1.59339	0.0528784
$909$	4.06226	0.134737
$910$	0	0
$911$	−11.0623	−0.366509	−0.183254	−	0.983066i	$-0.558663\pi$
$911$	−11.0623	−0.366509	−0.183254	+	0.983066i	$0.558663\pi$
$912$	0	0
$913$	27.1868	0.899751
$914$	41.7802	1.38196
$915$	0	0
$916$	−23.1868	−0.766113
$917$	−12.0000	−0.396275
$918$	−2.53113	−0.0835397
$919$	−56.1868	−1.85343	−0.926715	−	0.375764i	$-0.877380\pi$
$919$	−56.1868	−1.85343	−0.926715	+	0.375764i	$0.877380\pi$
$920$	0	0
$921$	24.5311	0.808328
$922$	−14.4689	−0.476507
$923$	1.06226	0.0349646
$924$	−3.00000	−0.0986928
$925$	37.6556	1.23811
$926$	37.0623	1.21794
$927$	1.00000	0.0328443
$928$	0	0
$929$	−23.5311	−0.772031	−0.386016	−	0.922492i	$-0.626149\pi$
$929$	−23.5311	−0.772031	−0.386016	+	0.922492i	$0.626149\pi$
$930$	0	0
$931$	0	0
$932$	17.5311	0.574251
$933$	−10.5934	−0.346812
$934$	15.9377	0.521499
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	40.2490	1.31488	0.657439	−	0.753508i	$-0.271640\pi$
$937$	40.2490	1.31488	0.657439	+	0.753508i	$0.271640\pi$
$938$	12.5311	0.409156
$939$	16.5934	0.541505
$940$	0	0
$941$	−16.6556	−0.542958	−0.271479	−	0.962444i	$-0.587513\pi$
$941$	−16.6556	−0.542958	−0.271479	+	0.962444i	$0.587513\pi$
$942$	12.5311	0.408286
$943$	5.40661	0.176064
$944$	−9.06226	−0.294951
$945$	0	0
$946$	−10.5934	−0.344421
$947$	−24.7179	−0.803224	−0.401612	−	0.915810i	$-0.631550\pi$
$947$	−24.7179	−0.803224	−0.401612	+	0.915810i	$0.631550\pi$
$948$	9.06226	0.294328
$949$	7.59339	0.246492
$950$	0	0
$951$	31.5934	1.02449
$952$	2.53113	0.0820344
$953$	−20.6556	−0.669102	−0.334551	−	0.942378i	$-0.608585\pi$
$953$	−20.6556	−0.669102	−0.334551	+	0.942378i	$0.608585\pi$
$954$	2.53113	0.0819483
$955$	0	0
$956$	3.46887	0.112191
$957$	0	0
$958$	23.6556	0.764279
$959$	−4.59339	−0.148328
$960$	0	0
$961$	−15.0000	−0.483871
$962$	7.53113	0.242813
$963$	8.53113	0.274912
$964$	5.00000	0.161039
$965$	0	0
$966$	1.53113	0.0492633
$967$	−24.2490	−0.779796	−0.389898	−	0.920858i	$-0.627490\pi$
$967$	−24.2490	−0.779796	−0.389898	+	0.920858i	$0.627490\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	0	0
$971$	−18.1245	−0.581643	−0.290822	−	0.956777i	$-0.593929\pi$
$971$	−18.1245	−0.581643	−0.290822	+	0.956777i	$0.593929\pi$
$972$	−1.00000	−0.0320750
$973$	−7.46887	−0.239441
$974$	33.1868	1.06337
$975$	5.00000	0.160128
$976$	−2.46887	−0.0790266
$977$	21.5311	0.688842	0.344421	−	0.938815i	$-0.388075\pi$
$977$	21.5311	0.688842	0.344421	+	0.938815i	$0.388075\pi$
$978$	16.5934	0.530598
$979$	−39.1868	−1.25241
$980$	0	0
$981$	−10.4689	−0.334245
$982$	30.6556	0.978261
$983$	−4.40661	−0.140549	−0.0702746	−	0.997528i	$-0.522388\pi$
$983$	−4.40661	−0.140549	−0.0702746	+	0.997528i	$0.522388\pi$
$984$	3.53113	0.112568
$985$	0	0
$986$	0	0
$987$	0.468871	0.0149243
$988$	0	0
$989$	5.40661	0.171920
$990$	0	0
$991$	33.3113	1.05817	0.529084	−	0.848569i	$-0.322536\pi$
$991$	33.3113	1.05817	0.529084	+	0.848569i	$0.322536\pi$
$992$	−4.00000	−0.127000
$993$	25.7179	0.816133
$994$	−1.06226	−0.0336928
$995$	0	0
$996$	−9.06226	−0.287149
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−9.18677	−0.290802
$999$	7.53113	0.238274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.m.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.m.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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.53113 q^{17} -1.00000 q^{18} -1.00000 q^{21} -3.00000 q^{22} +1.53113 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +2.53113 q^{34} +1.00000 q^{36} -7.53113 q^{37} -1.00000 q^{39} +3.53113 q^{41} +1.00000 q^{42} +3.53113 q^{43} +3.00000 q^{44} -1.53113 q^{46} -0.468871 q^{47} -1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} +2.53113 q^{51} +1.00000 q^{52} -2.53113 q^{53} +1.00000 q^{54} -1.00000 q^{56} -9.06226 q^{59} -2.46887 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -12.5311 q^{67} -2.53113 q^{68} -1.53113 q^{69} +1.06226 q^{71} -1.00000 q^{72} +7.59339 q^{73} +7.53113 q^{74} +5.00000 q^{75} +3.00000 q^{77} +1.00000 q^{78} -9.06226 q^{79} +1.00000 q^{81} -3.53113 q^{82} +9.06226 q^{83} -1.00000 q^{84} -3.53113 q^{86} -3.00000 q^{88} -13.0623 q^{89} +1.00000 q^{91} +1.53113 q^{92} -4.00000 q^{93} +0.468871 q^{94} +1.00000 q^{96} -18.1245 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{17} - 2 q^{18} - 2 q^{21} - 6 q^{22} - 5 q^{23} + 2 q^{24} - 10 q^{25} - 2 q^{26} - 2 q^{27} + 2 q^{28} + 8 q^{31} - 2 q^{32} - 6 q^{33} - 3 q^{34} + 2 q^{36} - 7 q^{37} - 2 q^{39} - q^{41} + 2 q^{42} - q^{43} + 6 q^{44} + 5 q^{46} - 9 q^{47} - 2 q^{48} - 12 q^{49} + 10 q^{50} - 3 q^{51} + 2 q^{52} + 3 q^{53} + 2 q^{54} - 2 q^{56} - 2 q^{59} - 13 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} + 6 q^{66} - 17 q^{67} + 3 q^{68} + 5 q^{69} - 14 q^{71} - 2 q^{72} - 9 q^{73} + 7 q^{74} + 10 q^{75} + 6 q^{77} + 2 q^{78} - 2 q^{79} + 2 q^{81} + q^{82} + 2 q^{83} - 2 q^{84} + q^{86} - 6 q^{88} - 10 q^{89} + 2 q^{91} - 5 q^{92} - 8 q^{93} + 9 q^{94} + 2 q^{96} - 4 q^{97} + 12 q^{98} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 3 * q^17 - 2 * q^18 - 2 * q^21 - 6 * q^22 - 5 * q^23 + 2 * q^24 - 10 * q^25 - 2 * q^26 - 2 * q^27 + 2 * q^28 + 8 * q^31 - 2 * q^32 - 6 * q^33 - 3 * q^34 + 2 * q^36 - 7 * q^37 - 2 * q^39 - q^41 + 2 * q^42 - q^43 + 6 * q^44 + 5 * q^46 - 9 * q^47 - 2 * q^48 - 12 * q^49 + 10 * q^50 - 3 * q^51 + 2 * q^52 + 3 * q^53 + 2 * q^54 - 2 * q^56 - 2 * q^59 - 13 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 + 6 * q^66 - 17 * q^67 + 3 * q^68 + 5 * q^69 - 14 * q^71 - 2 * q^72 - 9 * q^73 + 7 * q^74 + 10 * q^75 + 6 * q^77 + 2 * q^78 - 2 * q^79 + 2 * q^81 + q^82 + 2 * q^83 - 2 * q^84 + q^86 - 6 * q^88 - 10 * q^89 + 2 * q^91 - 5 * q^92 - 8 * q^93 + 9 * q^94 + 2 * q^96 - 4 * q^97 + 12 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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