LMFDB - Embedded newform 5290.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5290,2,Mod(1,5290)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5290, 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("5290.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5290 = 2 \cdot 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5290.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:	$42.2408626693$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	5290.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +0.732051 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +0.732051 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.46410 q^{11} +1.73205 q^{12} +0.732051 q^{14} -1.73205 q^{15} +1.00000 q^{16} -3.73205 q^{17} -4.46410 q^{19} -1.00000 q^{20} +1.26795 q^{21} -3.46410 q^{22} +1.73205 q^{24} +1.00000 q^{25} -5.19615 q^{27} +0.732051 q^{28} -0.732051 q^{29} -1.73205 q^{30} +0.196152 q^{31} +1.00000 q^{32} -6.00000 q^{33} -3.73205 q^{34} -0.732051 q^{35} -4.53590 q^{37} -4.46410 q^{38} -1.00000 q^{40} -8.39230 q^{41} +1.26795 q^{42} -9.19615 q^{43} -3.46410 q^{44} +10.1962 q^{47} +1.73205 q^{48} -6.46410 q^{49} +1.00000 q^{50} -6.46410 q^{51} +8.19615 q^{53} -5.19615 q^{54} +3.46410 q^{55} +0.732051 q^{56} -7.73205 q^{57} -0.732051 q^{58} -5.92820 q^{59} -1.73205 q^{60} +4.00000 q^{61} +0.196152 q^{62} +1.00000 q^{64} -6.00000 q^{66} +1.73205 q^{67} -3.73205 q^{68} -0.732051 q^{70} +5.26795 q^{71} -1.19615 q^{73} -4.53590 q^{74} +1.73205 q^{75} -4.46410 q^{76} -2.53590 q^{77} +2.19615 q^{79} -1.00000 q^{80} -9.00000 q^{81} -8.39230 q^{82} -16.6603 q^{83} +1.26795 q^{84} +3.73205 q^{85} -9.19615 q^{86} -1.26795 q^{87} -3.46410 q^{88} -6.92820 q^{89} +0.339746 q^{93} +10.1962 q^{94} +4.46410 q^{95} +1.73205 q^{96} +6.39230 q^{97} -6.46410 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{19} - 2 q^{20} + 6 q^{21} + 2 q^{25} - 2 q^{28} + 2 q^{29} - 10 q^{31} + 2 q^{32} - 12 q^{33} - 4 q^{34} + 2 q^{35} - 16 q^{37} - 2 q^{38} - 2 q^{40} + 4 q^{41} + 6 q^{42} - 8 q^{43} + 10 q^{47} - 6 q^{49} + 2 q^{50} - 6 q^{51} + 6 q^{53} - 2 q^{56} - 12 q^{57} + 2 q^{58} + 2 q^{59} + 8 q^{61} - 10 q^{62} + 2 q^{64} - 12 q^{66} - 4 q^{68} + 2 q^{70} + 14 q^{71} + 8 q^{73} - 16 q^{74} - 2 q^{76} - 12 q^{77} - 6 q^{79} - 2 q^{80} - 18 q^{81} + 4 q^{82} - 16 q^{83} + 6 q^{84} + 4 q^{85} - 8 q^{86} - 6 q^{87} + 18 q^{93} + 10 q^{94} + 2 q^{95} - 8 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^19 - 2 * q^20 + 6 * q^21 + 2 * q^25 - 2 * q^28 + 2 * q^29 - 10 * q^31 + 2 * q^32 - 12 * q^33 - 4 * q^34 + 2 * q^35 - 16 * q^37 - 2 * q^38 - 2 * q^40 + 4 * q^41 + 6 * q^42 - 8 * q^43 + 10 * q^47 - 6 * q^49 + 2 * q^50 - 6 * q^51 + 6 * q^53 - 2 * q^56 - 12 * q^57 + 2 * q^58 + 2 * q^59 + 8 * q^61 - 10 * q^62 + 2 * q^64 - 12 * q^66 - 4 * q^68 + 2 * q^70 + 14 * q^71 + 8 * q^73 - 16 * q^74 - 2 * q^76 - 12 * q^77 - 6 * q^79 - 2 * q^80 - 18 * q^81 + 4 * q^82 - 16 * q^83 + 6 * q^84 + 4 * q^85 - 8 * q^86 - 6 * q^87 + 18 * q^93 + 10 * q^94 + 2 * q^95 - 8 * q^97 - 6 * q^98

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
$3$	1.73205	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.73205	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.73205	0.707107
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	1.73205	0.500000
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0.732051	0.195649
$15$	−1.73205	−0.447214
$16$	1.00000	0.250000
$17$	−3.73205	−0.905155	−0.452578	−	0.891725i	$-0.649495\pi$
$17$	−3.73205	−0.905155	−0.452578	+	0.891725i	$0.649495\pi$
$18$	0	0
$19$	−4.46410	−1.02414	−0.512068	−	0.858945i	$-0.671120\pi$
$19$	−4.46410	−1.02414	−0.512068	+	0.858945i	$0.671120\pi$
$20$	−1.00000	−0.223607
$21$	1.26795	0.276689
$22$	−3.46410	−0.738549
$23$	0	0
$23$	0	0
$24$	1.73205	0.353553
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.19615	−1.00000
$28$	0.732051	0.138345
$29$	−0.732051	−0.135938	−0.0679692	−	0.997687i	$-0.521652\pi$
$29$	−0.732051	−0.135938	−0.0679692	+	0.997687i	$0.521652\pi$
$30$	−1.73205	−0.316228
$31$	0.196152	0.0352300	0.0176150	−	0.999845i	$-0.494393\pi$
$31$	0.196152	0.0352300	0.0176150	+	0.999845i	$0.494393\pi$
$32$	1.00000	0.176777
$33$	−6.00000	−1.04447
$34$	−3.73205	−0.640041
$35$	−0.732051	−0.123739
$36$	0	0
$37$	−4.53590	−0.745697	−0.372849	−	0.927892i	$-0.621619\pi$
$37$	−4.53590	−0.745697	−0.372849	+	0.927892i	$0.621619\pi$
$38$	−4.46410	−0.724173
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−8.39230	−1.31066	−0.655329	−	0.755344i	$-0.727470\pi$
$41$	−8.39230	−1.31066	−0.655329	+	0.755344i	$0.727470\pi$
$42$	1.26795	0.195649
$43$	−9.19615	−1.40240	−0.701200	−	0.712965i	$-0.747352\pi$
$43$	−9.19615	−1.40240	−0.701200	+	0.712965i	$0.747352\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	0	0
$47$	10.1962	1.48726	0.743631	−	0.668590i	$-0.233102\pi$
$47$	10.1962	1.48726	0.743631	+	0.668590i	$0.233102\pi$
$48$	1.73205	0.250000
$49$	−6.46410	−0.923443
$50$	1.00000	0.141421
$51$	−6.46410	−0.905155
$52$	0	0
$53$	8.19615	1.12583	0.562914	−	0.826515i	$-0.309680\pi$
$53$	8.19615	1.12583	0.562914	+	0.826515i	$0.309680\pi$
$54$	−5.19615	−0.707107
$55$	3.46410	0.467099
$56$	0.732051	0.0978244
$57$	−7.73205	−1.02414
$58$	−0.732051	−0.0961230
$59$	−5.92820	−0.771786	−0.385893	−	0.922543i	$-0.626107\pi$
$59$	−5.92820	−0.771786	−0.385893	+	0.922543i	$0.626107\pi$
$60$	−1.73205	−0.223607
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0.196152	0.0249114
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	1.73205	0.211604	0.105802	−	0.994387i	$-0.466259\pi$
$67$	1.73205	0.211604	0.105802	+	0.994387i	$0.466259\pi$
$68$	−3.73205	−0.452578
$69$	0	0
$70$	−0.732051	−0.0874968
$71$	5.26795	0.625191	0.312595	−	0.949886i	$-0.398802\pi$
$71$	5.26795	0.625191	0.312595	+	0.949886i	$0.398802\pi$
$72$	0	0
$73$	−1.19615	−0.139999	−0.0699995	−	0.997547i	$-0.522300\pi$
$73$	−1.19615	−0.139999	−0.0699995	+	0.997547i	$0.522300\pi$
$74$	−4.53590	−0.527287
$75$	1.73205	0.200000
$76$	−4.46410	−0.512068
$77$	−2.53590	−0.288992
$78$	0	0
$79$	2.19615	0.247086	0.123543	−	0.992339i	$-0.460574\pi$
$79$	2.19615	0.247086	0.123543	+	0.992339i	$0.460574\pi$
$80$	−1.00000	−0.111803
$81$	−9.00000	−1.00000
$82$	−8.39230	−0.926775
$83$	−16.6603	−1.82870	−0.914350	−	0.404924i	$-0.867298\pi$
$83$	−16.6603	−1.82870	−0.914350	+	0.404924i	$0.867298\pi$
$84$	1.26795	0.138345
$85$	3.73205	0.404798
$86$	−9.19615	−0.991647
$87$	−1.26795	−0.135938
$88$	−3.46410	−0.369274
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.339746	0.0352300
$94$	10.1962	1.05165
$95$	4.46410	0.458007
$96$	1.73205	0.176777
$97$	6.39230	0.649040	0.324520	−	0.945879i	$-0.394797\pi$
$97$	6.39230	0.649040	0.324520	+	0.945879i	$0.394797\pi$
$98$	−6.46410	−0.652973
$99$	0	0
$100$	1.00000	0.100000
$101$	17.8564	1.77678	0.888389	−	0.459091i	$-0.151825\pi$
$101$	17.8564	1.77678	0.888389	+	0.459091i	$0.151825\pi$
$102$	−6.46410	−0.640041
$103$	−3.07180	−0.302673	−0.151337	−	0.988482i	$-0.548358\pi$
$103$	−3.07180	−0.302673	−0.151337	+	0.988482i	$0.548358\pi$
$104$	0	0
$105$	−1.26795	−0.123739
$106$	8.19615	0.796081
$107$	10.9282	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$107$	10.9282	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$108$	−5.19615	−0.500000
$109$	−5.26795	−0.504578	−0.252289	−	0.967652i	$-0.581183\pi$
$109$	−5.26795	−0.504578	−0.252289	+	0.967652i	$0.581183\pi$
$110$	3.46410	0.330289
$111$	−7.85641	−0.745697
$112$	0.732051	0.0691723
$113$	11.1962	1.05325	0.526623	−	0.850099i	$-0.323458\pi$
$113$	11.1962	1.05325	0.526623	+	0.850099i	$0.323458\pi$
$114$	−7.73205	−0.724173
$115$	0	0
$116$	−0.732051	−0.0679692
$117$	0	0
$118$	−5.92820	−0.545735
$119$	−2.73205	−0.250447
$120$	−1.73205	−0.158114
$121$	1.00000	0.0909091
$122$	4.00000	0.362143
$123$	−14.5359	−1.31066
$124$	0.196152	0.0176150
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.6603	−1.38962	−0.694811	−	0.719192i	$-0.744512\pi$
$127$	−15.6603	−1.38962	−0.694811	+	0.719192i	$0.744512\pi$
$128$	1.00000	0.0883883
$129$	−15.9282	−1.40240
$130$	0	0
$131$	−11.5359	−1.00790	−0.503948	−	0.863734i	$-0.668120\pi$
$131$	−11.5359	−1.00790	−0.503948	+	0.863734i	$0.668120\pi$
$132$	−6.00000	−0.522233
$133$	−3.26795	−0.283367
$134$	1.73205	0.149626
$135$	5.19615	0.447214
$136$	−3.73205	−0.320021
$137$	−21.5885	−1.84443	−0.922213	−	0.386682i	$-0.873621\pi$
$137$	−21.5885	−1.84443	−0.922213	+	0.386682i	$0.873621\pi$
$138$	0	0
$139$	8.46410	0.717916	0.358958	−	0.933354i	$-0.383132\pi$
$139$	8.46410	0.717916	0.358958	+	0.933354i	$0.383132\pi$
$140$	−0.732051	−0.0618696
$141$	17.6603	1.48726
$142$	5.26795	0.442076
$143$	0	0
$144$	0	0
$145$	0.732051	0.0607935
$146$	−1.19615	−0.0989943
$147$	−11.1962	−0.923443
$148$	−4.53590	−0.372849
$149$	−9.80385	−0.803162	−0.401581	−	0.915823i	$-0.631539\pi$
$149$	−9.80385	−0.803162	−0.401581	+	0.915823i	$0.631539\pi$
$150$	1.73205	0.141421
$151$	6.39230	0.520198	0.260099	−	0.965582i	$-0.416245\pi$
$151$	6.39230	0.520198	0.260099	+	0.965582i	$0.416245\pi$
$152$	−4.46410	−0.362086
$153$	0	0
$154$	−2.53590	−0.204349
$155$	−0.196152	−0.0157553
$156$	0	0
$157$	15.3205	1.22271	0.611355	−	0.791357i	$-0.290625\pi$
$157$	15.3205	1.22271	0.611355	+	0.791357i	$0.290625\pi$
$158$	2.19615	0.174717
$159$	14.1962	1.12583
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−9.00000	−0.707107
$163$	15.3205	1.19999	0.599997	−	0.800002i	$-0.295168\pi$
$163$	15.3205	1.19999	0.599997	+	0.800002i	$0.295168\pi$
$164$	−8.39230	−0.655329
$165$	6.00000	0.467099
$166$	−16.6603	−1.29309
$167$	16.1962	1.25330	0.626648	−	0.779302i	$-0.284426\pi$
$167$	16.1962	1.25330	0.626648	+	0.779302i	$0.284426\pi$
$168$	1.26795	0.0978244
$169$	−13.0000	−1.00000
$170$	3.73205	0.286235
$171$	0	0
$172$	−9.19615	−0.701200
$173$	−4.33975	−0.329945	−0.164972	−	0.986298i	$-0.552754\pi$
$173$	−4.33975	−0.329945	−0.164972	+	0.986298i	$0.552754\pi$
$174$	−1.26795	−0.0961230
$175$	0.732051	0.0553378
$176$	−3.46410	−0.261116
$177$	−10.2679	−0.771786
$178$	−6.92820	−0.519291
$179$	−3.92820	−0.293608	−0.146804	−	0.989166i	$-0.546899\pi$
$179$	−3.92820	−0.293608	−0.146804	+	0.989166i	$0.546899\pi$
$180$	0	0
$181$	−26.3923	−1.96172	−0.980862	−	0.194703i	$-0.937626\pi$
$181$	−26.3923	−1.96172	−0.980862	+	0.194703i	$0.937626\pi$
$182$	0	0
$183$	6.92820	0.512148
$184$	0	0
$185$	4.53590	0.333486
$186$	0.339746	0.0249114
$187$	12.9282	0.945404
$188$	10.1962	0.743631
$189$	−3.80385	−0.276689
$190$	4.46410	0.323860
$191$	−9.80385	−0.709382	−0.354691	−	0.934984i	$-0.615414\pi$
$191$	−9.80385	−0.709382	−0.354691	+	0.934984i	$0.615414\pi$
$192$	1.73205	0.125000
$193$	13.7321	0.988455	0.494227	−	0.869333i	$-0.335451\pi$
$193$	13.7321	0.988455	0.494227	+	0.869333i	$0.335451\pi$
$194$	6.39230	0.458941
$195$	0	0
$196$	−6.46410	−0.461722
$197$	−0.535898	−0.0381812	−0.0190906	−	0.999818i	$-0.506077\pi$
$197$	−0.535898	−0.0381812	−0.0190906	+	0.999818i	$0.506077\pi$
$198$	0	0
$199$	15.8038	1.12031	0.560153	−	0.828389i	$-0.310743\pi$
$199$	15.8038	1.12031	0.560153	+	0.828389i	$0.310743\pi$
$200$	1.00000	0.0707107
$201$	3.00000	0.211604
$202$	17.8564	1.25637
$203$	−0.535898	−0.0376127
$204$	−6.46410	−0.452578
$205$	8.39230	0.586144
$206$	−3.07180	−0.214022
$207$	0	0
$208$	0	0
$209$	15.4641	1.06967
$210$	−1.26795	−0.0874968
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	8.19615	0.562914
$213$	9.12436	0.625191
$214$	10.9282	0.747037
$215$	9.19615	0.627172
$216$	−5.19615	−0.353553
$217$	0.143594	0.00974776
$218$	−5.26795	−0.356791
$219$	−2.07180	−0.139999
$220$	3.46410	0.233550
$221$	0	0
$222$	−7.85641	−0.527287
$223$	−0.928203	−0.0621571	−0.0310785	−	0.999517i	$-0.509894\pi$
$223$	−0.928203	−0.0621571	−0.0310785	+	0.999517i	$0.509894\pi$
$224$	0.732051	0.0489122
$225$	0	0
$226$	11.1962	0.744757
$227$	21.0526	1.39731	0.698654	−	0.715460i	$-0.253783\pi$
$227$	21.0526	1.39731	0.698654	+	0.715460i	$0.253783\pi$
$228$	−7.73205	−0.512068
$229$	25.5167	1.68619	0.843094	−	0.537766i	$-0.180732\pi$
$229$	25.5167	1.68619	0.843094	+	0.537766i	$0.180732\pi$
$230$	0	0
$231$	−4.39230	−0.288992
$232$	−0.732051	−0.0480615
$233$	−10.1244	−0.663269	−0.331634	−	0.943408i	$-0.607600\pi$
$233$	−10.1244	−0.663269	−0.331634	+	0.943408i	$0.607600\pi$
$234$	0	0
$235$	−10.1962	−0.665124
$236$	−5.92820	−0.385893
$237$	3.80385	0.247086
$238$	−2.73205	−0.177093
$239$	−30.2487	−1.95663	−0.978313	−	0.207131i	$-0.933587\pi$
$239$	−30.2487	−1.95663	−0.978313	+	0.207131i	$0.933587\pi$
$240$	−1.73205	−0.111803
$241$	−13.9282	−0.897194	−0.448597	−	0.893734i	$-0.648076\pi$
$241$	−13.9282	−0.897194	−0.448597	+	0.893734i	$0.648076\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	4.00000	0.256074
$245$	6.46410	0.412976
$246$	−14.5359	−0.926775
$247$	0	0
$248$	0.196152	0.0124557
$249$	−28.8564	−1.82870
$250$	−1.00000	−0.0632456
$251$	−1.00000	−0.0631194	−0.0315597	−	0.999502i	$-0.510047\pi$
$251$	−1.00000	−0.0631194	−0.0315597	+	0.999502i	$0.510047\pi$
$252$	0	0
$253$	0	0
$254$	−15.6603	−0.982612
$255$	6.46410	0.404798
$256$	1.00000	0.0625000
$257$	−13.1962	−0.823153	−0.411577	−	0.911375i	$-0.635022\pi$
$257$	−13.1962	−0.823153	−0.411577	+	0.911375i	$0.635022\pi$
$258$	−15.9282	−0.991647
$259$	−3.32051	−0.206326
$260$	0	0
$261$	0	0
$262$	−11.5359	−0.712690
$263$	−7.07180	−0.436066	−0.218033	−	0.975941i	$-0.569964\pi$
$263$	−7.07180	−0.436066	−0.218033	+	0.975941i	$0.569964\pi$
$264$	−6.00000	−0.369274
$265$	−8.19615	−0.503486
$266$	−3.26795	−0.200371
$267$	−12.0000	−0.734388
$268$	1.73205	0.105802
$269$	−27.7128	−1.68968	−0.844840	−	0.535019i	$-0.820304\pi$
$269$	−27.7128	−1.68968	−0.844840	+	0.535019i	$0.820304\pi$
$270$	5.19615	0.316228
$271$	31.4641	1.91131	0.955654	−	0.294492i	$-0.0951503\pi$
$271$	31.4641	1.91131	0.955654	+	0.294492i	$0.0951503\pi$
$272$	−3.73205	−0.226289
$273$	0	0
$274$	−21.5885	−1.30421
$275$	−3.46410	−0.208893
$276$	0	0
$277$	10.5885	0.636199	0.318099	−	0.948057i	$-0.396955\pi$
$277$	10.5885	0.636199	0.318099	+	0.948057i	$0.396955\pi$
$278$	8.46410	0.507643
$279$	0	0
$280$	−0.732051	−0.0437484
$281$	6.46410	0.385616	0.192808	−	0.981237i	$-0.438241\pi$
$281$	6.46410	0.385616	0.192808	+	0.981237i	$0.438241\pi$
$282$	17.6603	1.05165
$283$	12.8038	0.761110	0.380555	−	0.924758i	$-0.375733\pi$
$283$	12.8038	0.761110	0.380555	+	0.924758i	$0.375733\pi$
$284$	5.26795	0.312595
$285$	7.73205	0.458007
$286$	0	0
$287$	−6.14359	−0.362645
$288$	0	0
$289$	−3.07180	−0.180694
$290$	0.732051	0.0429875
$291$	11.0718	0.649040
$292$	−1.19615	−0.0699995
$293$	26.7846	1.56477	0.782387	−	0.622793i	$-0.214002\pi$
$293$	26.7846	1.56477	0.782387	+	0.622793i	$0.214002\pi$
$294$	−11.1962	−0.652973
$295$	5.92820	0.345153
$296$	−4.53590	−0.263644
$297$	18.0000	1.04447
$298$	−9.80385	−0.567922
$299$	0	0
$300$	1.73205	0.100000
$301$	−6.73205	−0.388029
$302$	6.39230	0.367836
$303$	30.9282	1.77678
$304$	−4.46410	−0.256034
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−28.2679	−1.61334	−0.806669	−	0.591004i	$-0.798732\pi$
$307$	−28.2679	−1.61334	−0.806669	+	0.591004i	$0.798732\pi$
$308$	−2.53590	−0.144496
$309$	−5.32051	−0.302673
$310$	−0.196152	−0.0111407
$311$	−32.7321	−1.85606	−0.928032	−	0.372500i	$-0.878501\pi$
$311$	−32.7321	−1.85606	−0.928032	+	0.372500i	$0.878501\pi$
$312$	0	0
$313$	13.5885	0.768065	0.384033	−	0.923320i	$-0.374535\pi$
$313$	13.5885	0.768065	0.384033	+	0.923320i	$0.374535\pi$
$314$	15.3205	0.864586
$315$	0	0
$316$	2.19615	0.123543
$317$	−0.339746	−0.0190820	−0.00954102	−	0.999954i	$-0.503037\pi$
$317$	−0.339746	−0.0190820	−0.00954102	+	0.999954i	$0.503037\pi$
$318$	14.1962	0.796081
$319$	2.53590	0.141983
$320$	−1.00000	−0.0559017
$321$	18.9282	1.05647
$322$	0	0
$323$	16.6603	0.927001
$324$	−9.00000	−0.500000
$325$	0	0
$326$	15.3205	0.848524
$327$	−9.12436	−0.504578
$328$	−8.39230	−0.463388
$329$	7.46410	0.411509
$330$	6.00000	0.330289
$331$	−4.32051	−0.237477	−0.118738	−	0.992926i	$-0.537885\pi$
$331$	−4.32051	−0.237477	−0.118738	+	0.992926i	$0.537885\pi$
$332$	−16.6603	−0.914350
$333$	0	0
$334$	16.1962	0.886214
$335$	−1.73205	−0.0946320
$336$	1.26795	0.0691723
$337$	13.0526	0.711018	0.355509	−	0.934673i	$-0.384308\pi$
$337$	13.0526	0.711018	0.355509	+	0.934673i	$0.384308\pi$
$338$	−13.0000	−0.707107
$339$	19.3923	1.05325
$340$	3.73205	0.202399
$341$	−0.679492	−0.0367966
$342$	0	0
$343$	−9.85641	−0.532196
$344$	−9.19615	−0.495823
$345$	0	0
$346$	−4.33975	−0.233306
$347$	−35.0526	−1.88172	−0.940860	−	0.338796i	$-0.889980\pi$
$347$	−35.0526	−1.88172	−0.940860	+	0.338796i	$0.889980\pi$
$348$	−1.26795	−0.0679692
$349$	31.3205	1.67655	0.838274	−	0.545249i	$-0.183565\pi$
$349$	31.3205	1.67655	0.838274	+	0.545249i	$0.183565\pi$
$350$	0.732051	0.0391298
$351$	0	0
$352$	−3.46410	−0.184637
$353$	−19.9808	−1.06347	−0.531734	−	0.846911i	$-0.678460\pi$
$353$	−19.9808	−1.06347	−0.531734	+	0.846911i	$0.678460\pi$
$354$	−10.2679	−0.545735
$355$	−5.26795	−0.279594
$356$	−6.92820	−0.367194
$357$	−4.73205	−0.250447
$358$	−3.92820	−0.207612
$359$	−21.2679	−1.12248	−0.561240	−	0.827653i	$-0.689675\pi$
$359$	−21.2679	−1.12248	−0.561240	+	0.827653i	$0.689675\pi$
$360$	0	0
$361$	0.928203	0.0488528
$362$	−26.3923	−1.38715
$363$	1.73205	0.0909091
$364$	0	0
$365$	1.19615	0.0626095
$366$	6.92820	0.362143
$367$	−18.0526	−0.942336	−0.471168	−	0.882044i	$-0.656167\pi$
$367$	−18.0526	−0.942336	−0.471168	+	0.882044i	$0.656167\pi$
$368$	0	0
$369$	0	0
$370$	4.53590	0.235810
$371$	6.00000	0.311504
$372$	0.339746	0.0176150
$373$	7.26795	0.376320	0.188160	−	0.982138i	$-0.439748\pi$
$373$	7.26795	0.376320	0.188160	+	0.982138i	$0.439748\pi$
$374$	12.9282	0.668501
$375$	−1.73205	−0.0894427
$376$	10.1962	0.525826
$377$	0	0
$378$	−3.80385	−0.195649
$379$	16.3205	0.838328	0.419164	−	0.907910i	$-0.362323\pi$
$379$	16.3205	0.838328	0.419164	+	0.907910i	$0.362323\pi$
$380$	4.46410	0.229004
$381$	−27.1244	−1.38962
$382$	−9.80385	−0.501608
$383$	−1.60770	−0.0821494	−0.0410747	−	0.999156i	$-0.513078\pi$
$383$	−1.60770	−0.0821494	−0.0410747	+	0.999156i	$0.513078\pi$
$384$	1.73205	0.0883883
$385$	2.53590	0.129241
$386$	13.7321	0.698943
$387$	0	0
$388$	6.39230	0.324520
$389$	11.0718	0.561362	0.280681	−	0.959801i	$-0.409440\pi$
$389$	11.0718	0.561362	0.280681	+	0.959801i	$0.409440\pi$
$390$	0	0
$391$	0	0
$392$	−6.46410	−0.326486
$393$	−19.9808	−1.00790
$394$	−0.535898	−0.0269982
$395$	−2.19615	−0.110500
$396$	0	0
$397$	27.1244	1.36133	0.680666	−	0.732594i	$-0.261690\pi$
$397$	27.1244	1.36133	0.680666	+	0.732594i	$0.261690\pi$
$398$	15.8038	0.792175
$399$	−5.66025	−0.283367
$400$	1.00000	0.0500000
$401$	20.8564	1.04152	0.520760	−	0.853703i	$-0.325649\pi$
$401$	20.8564	1.04152	0.520760	+	0.853703i	$0.325649\pi$
$402$	3.00000	0.149626
$403$	0	0
$404$	17.8564	0.888389
$405$	9.00000	0.447214
$406$	−0.535898	−0.0265962
$407$	15.7128	0.778855
$408$	−6.46410	−0.320021
$409$	−8.32051	−0.411423	−0.205711	−	0.978613i	$-0.565951\pi$
$409$	−8.32051	−0.411423	−0.205711	+	0.978613i	$0.565951\pi$
$410$	8.39230	0.414466
$411$	−37.3923	−1.84443
$412$	−3.07180	−0.151337
$413$	−4.33975	−0.213545
$414$	0	0
$415$	16.6603	0.817820
$416$	0	0
$417$	14.6603	0.717916
$418$	15.4641	0.756374
$419$	−8.32051	−0.406483	−0.203242	−	0.979129i	$-0.565148\pi$
$419$	−8.32051	−0.406483	−0.203242	+	0.979129i	$0.565148\pi$
$420$	−1.26795	−0.0618696
$421$	−31.6603	−1.54303	−0.771513	−	0.636213i	$-0.780500\pi$
$421$	−31.6603	−1.54303	−0.771513	+	0.636213i	$0.780500\pi$
$422$	11.0000	0.535472
$423$	0	0
$424$	8.19615	0.398040
$425$	−3.73205	−0.181031
$426$	9.12436	0.442076
$427$	2.92820	0.141706
$428$	10.9282	0.528235
$429$	0	0
$430$	9.19615	0.443478
$431$	26.9282	1.29709	0.648543	−	0.761178i	$-0.275379\pi$
$431$	26.9282	1.29709	0.648543	+	0.761178i	$0.275379\pi$
$432$	−5.19615	−0.250000
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0.143594	0.00689271
$435$	1.26795	0.0607935
$436$	−5.26795	−0.252289
$437$	0	0
$438$	−2.07180	−0.0989943
$439$	−17.6603	−0.842878	−0.421439	−	0.906857i	$-0.638475\pi$
$439$	−17.6603	−0.842878	−0.421439	+	0.906857i	$0.638475\pi$
$440$	3.46410	0.165145
$441$	0	0
$442$	0	0
$443$	−22.5167	−1.06980	−0.534899	−	0.844916i	$-0.679651\pi$
$443$	−22.5167	−1.06980	−0.534899	+	0.844916i	$0.679651\pi$
$444$	−7.85641	−0.372849
$445$	6.92820	0.328428
$446$	−0.928203	−0.0439517
$447$	−16.9808	−0.803162
$448$	0.732051	0.0345861
$449$	−19.0000	−0.896665	−0.448333	−	0.893867i	$-0.647982\pi$
$449$	−19.0000	−0.896665	−0.448333	+	0.893867i	$0.647982\pi$
$450$	0	0
$451$	29.0718	1.36894
$452$	11.1962	0.526623
$453$	11.0718	0.520198
$454$	21.0526	0.988046
$455$	0	0
$456$	−7.73205	−0.362086
$457$	37.7128	1.76413	0.882065	−	0.471127i	$-0.156153\pi$
$457$	37.7128	1.76413	0.882065	+	0.471127i	$0.156153\pi$
$458$	25.5167	1.19232
$459$	19.3923	0.905155
$460$	0	0
$461$	22.3923	1.04291	0.521457	−	0.853278i	$-0.325389\pi$
$461$	22.3923	1.04291	0.521457	+	0.853278i	$0.325389\pi$
$462$	−4.39230	−0.204349
$463$	−16.5359	−0.768488	−0.384244	−	0.923232i	$-0.625538\pi$
$463$	−16.5359	−0.768488	−0.384244	+	0.923232i	$0.625538\pi$
$464$	−0.732051	−0.0339846
$465$	−0.339746	−0.0157553
$466$	−10.1244	−0.469002
$467$	−15.3205	−0.708949	−0.354474	−	0.935066i	$-0.615340\pi$
$467$	−15.3205	−0.708949	−0.354474	+	0.935066i	$0.615340\pi$
$468$	0	0
$469$	1.26795	0.0585485
$470$	−10.1962	−0.470313
$471$	26.5359	1.22271
$472$	−5.92820	−0.272868
$473$	31.8564	1.46476
$474$	3.80385	0.174717
$475$	−4.46410	−0.204827
$476$	−2.73205	−0.125223
$477$	0	0
$478$	−30.2487	−1.38354
$479$	−11.2679	−0.514846	−0.257423	−	0.966299i	$-0.582873\pi$
$479$	−11.2679	−0.514846	−0.257423	+	0.966299i	$0.582873\pi$
$480$	−1.73205	−0.0790569
$481$	0	0
$482$	−13.9282	−0.634412
$483$	0	0
$484$	1.00000	0.0454545
$485$	−6.39230	−0.290260
$486$	0	0
$487$	13.8564	0.627894	0.313947	−	0.949441i	$-0.398349\pi$
$487$	13.8564	0.627894	0.313947	+	0.949441i	$0.398349\pi$
$488$	4.00000	0.181071
$489$	26.5359	1.19999
$490$	6.46410	0.292018
$491$	15.4641	0.697885	0.348943	−	0.937144i	$-0.386541\pi$
$491$	15.4641	0.697885	0.348943	+	0.937144i	$0.386541\pi$
$492$	−14.5359	−0.655329
$493$	2.73205	0.123045
$494$	0	0
$495$	0	0
$496$	0.196152	0.00880750
$497$	3.85641	0.172983
$498$	−28.8564	−1.29309
$499$	15.0000	0.671492	0.335746	−	0.941953i	$-0.391012\pi$
$499$	15.0000	0.671492	0.335746	+	0.941953i	$0.391012\pi$
$500$	−1.00000	−0.0447214
$501$	28.0526	1.25330
$502$	−1.00000	−0.0446322
$503$	−1.85641	−0.0827731	−0.0413865	−	0.999143i	$-0.513178\pi$
$503$	−1.85641	−0.0827731	−0.0413865	+	0.999143i	$0.513178\pi$
$504$	0	0
$505$	−17.8564	−0.794600
$506$	0	0
$507$	−22.5167	−1.00000
$508$	−15.6603	−0.694811
$509$	−34.9808	−1.55050	−0.775248	−	0.631658i	$-0.782375\pi$
$509$	−34.9808	−1.55050	−0.775248	+	0.631658i	$0.782375\pi$
$510$	6.46410	0.286235
$511$	−0.875644	−0.0387362
$512$	1.00000	0.0441942
$513$	23.1962	1.02414
$514$	−13.1962	−0.582057
$515$	3.07180	0.135360
$516$	−15.9282	−0.701200
$517$	−35.3205	−1.55339
$518$	−3.32051	−0.145895
$519$	−7.51666	−0.329945
$520$	0	0
$521$	29.1769	1.27826	0.639132	−	0.769097i	$-0.279294\pi$
$521$	29.1769	1.27826	0.639132	+	0.769097i	$0.279294\pi$
$522$	0	0
$523$	36.3923	1.59132	0.795662	−	0.605741i	$-0.207123\pi$
$523$	36.3923	1.59132	0.795662	+	0.605741i	$0.207123\pi$
$524$	−11.5359	−0.503948
$525$	1.26795	0.0553378
$526$	−7.07180	−0.308345
$527$	−0.732051	−0.0318886
$528$	−6.00000	−0.261116
$529$	0	0
$530$	−8.19615	−0.356018
$531$	0	0
$532$	−3.26795	−0.141684
$533$	0	0
$534$	−12.0000	−0.519291
$535$	−10.9282	−0.472467
$536$	1.73205	0.0748132
$537$	−6.80385	−0.293608
$538$	−27.7128	−1.19478
$539$	22.3923	0.964505
$540$	5.19615	0.223607
$541$	5.51666	0.237180	0.118590	−	0.992943i	$-0.462163\pi$
$541$	5.51666	0.237180	0.118590	+	0.992943i	$0.462163\pi$
$542$	31.4641	1.35150
$543$	−45.7128	−1.96172
$544$	−3.73205	−0.160010
$545$	5.26795	0.225654
$546$	0	0
$547$	−5.60770	−0.239768	−0.119884	−	0.992788i	$-0.538252\pi$
$547$	−5.60770	−0.239768	−0.119884	+	0.992788i	$0.538252\pi$
$548$	−21.5885	−0.922213
$549$	0	0
$550$	−3.46410	−0.147710
$551$	3.26795	0.139219
$552$	0	0
$553$	1.60770	0.0683662
$554$	10.5885	0.449860
$555$	7.85641	0.333486
$556$	8.46410	0.358958
$557$	12.3397	0.522852	0.261426	−	0.965224i	$-0.415807\pi$
$557$	12.3397	0.522852	0.261426	+	0.965224i	$0.415807\pi$
$558$	0	0
$559$	0	0
$560$	−0.732051	−0.0309348
$561$	22.3923	0.945404
$562$	6.46410	0.272672
$563$	−3.05256	−0.128650	−0.0643250	−	0.997929i	$-0.520489\pi$
$563$	−3.05256	−0.128650	−0.0643250	+	0.997929i	$0.520489\pi$
$564$	17.6603	0.743631
$565$	−11.1962	−0.471026
$566$	12.8038	0.538186
$567$	−6.58846	−0.276689
$568$	5.26795	0.221038
$569$	−36.7128	−1.53908	−0.769541	−	0.638598i	$-0.779515\pi$
$569$	−36.7128	−1.53908	−0.769541	+	0.638598i	$0.779515\pi$
$570$	7.73205	0.323860
$571$	31.0000	1.29731	0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000	1.29731	0.648655	+	0.761083i	$0.275332\pi$
$572$	0	0
$573$	−16.9808	−0.709382
$574$	−6.14359	−0.256429
$575$	0	0
$576$	0	0
$577$	−17.3397	−0.721863	−0.360932	−	0.932592i	$-0.617541\pi$
$577$	−17.3397	−0.721863	−0.360932	+	0.932592i	$0.617541\pi$
$578$	−3.07180	−0.127770
$579$	23.7846	0.988455
$580$	0.732051	0.0303968
$581$	−12.1962	−0.505982
$582$	11.0718	0.458941
$583$	−28.3923	−1.17589
$584$	−1.19615	−0.0494971
$585$	0	0
$586$	26.7846	1.10646
$587$	−15.0526	−0.621286	−0.310643	−	0.950527i	$-0.600544\pi$
$587$	−15.0526	−0.621286	−0.310643	+	0.950527i	$0.600544\pi$
$588$	−11.1962	−0.461722
$589$	−0.875644	−0.0360803
$590$	5.92820	0.244060
$591$	−0.928203	−0.0381812
$592$	−4.53590	−0.186424
$593$	−19.5885	−0.804402	−0.402201	−	0.915551i	$-0.631755\pi$
$593$	−19.5885	−0.804402	−0.402201	+	0.915551i	$0.631755\pi$
$594$	18.0000	0.738549
$595$	2.73205	0.112003
$596$	−9.80385	−0.401581
$597$	27.3731	1.12031
$598$	0	0
$599$	−5.12436	−0.209375	−0.104688	−	0.994505i	$-0.533384\pi$
$599$	−5.12436	−0.209375	−0.104688	+	0.994505i	$0.533384\pi$
$600$	1.73205	0.0707107
$601$	−26.2487	−1.07071	−0.535354	−	0.844628i	$-0.679822\pi$
$601$	−26.2487	−1.07071	−0.535354	+	0.844628i	$0.679822\pi$
$602$	−6.73205	−0.274378
$603$	0	0
$604$	6.39230	0.260099
$605$	−1.00000	−0.0406558
$606$	30.9282	1.25637
$607$	−6.87564	−0.279074	−0.139537	−	0.990217i	$-0.544561\pi$
$607$	−6.87564	−0.279074	−0.139537	+	0.990217i	$0.544561\pi$
$608$	−4.46410	−0.181043
$609$	−0.928203	−0.0376127
$610$	−4.00000	−0.161955
$611$	0	0
$612$	0	0
$613$	−26.7321	−1.07970	−0.539849	−	0.841762i	$-0.681519\pi$
$613$	−26.7321	−1.07970	−0.539849	+	0.841762i	$0.681519\pi$
$614$	−28.2679	−1.14080
$615$	14.5359	0.586144
$616$	−2.53590	−0.102174
$617$	−10.9474	−0.440727	−0.220364	−	0.975418i	$-0.570724\pi$
$617$	−10.9474	−0.440727	−0.220364	+	0.975418i	$0.570724\pi$
$618$	−5.32051	−0.214022
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	−0.196152	−0.00787767
$621$	0	0
$622$	−32.7321	−1.31244
$623$	−5.07180	−0.203197
$624$	0	0
$625$	1.00000	0.0400000
$626$	13.5885	0.543104
$627$	26.7846	1.06967
$628$	15.3205	0.611355
$629$	16.9282	0.674972
$630$	0	0
$631$	24.5885	0.978851	0.489426	−	0.872045i	$-0.337207\pi$
$631$	24.5885	0.978851	0.489426	+	0.872045i	$0.337207\pi$
$632$	2.19615	0.0873583
$633$	19.0526	0.757271
$634$	−0.339746	−0.0134930
$635$	15.6603	0.621458
$636$	14.1962	0.562914
$637$	0	0
$638$	2.53590	0.100397
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−32.0718	−1.26676	−0.633380	−	0.773841i	$-0.718333\pi$
$641$	−32.0718	−1.26676	−0.633380	+	0.773841i	$0.718333\pi$
$642$	18.9282	0.747037
$643$	4.67949	0.184541	0.0922706	−	0.995734i	$-0.470588\pi$
$643$	4.67949	0.184541	0.0922706	+	0.995734i	$0.470588\pi$
$644$	0	0
$645$	15.9282	0.627172
$646$	16.6603	0.655489
$647$	4.78461	0.188102	0.0940512	−	0.995567i	$-0.470018\pi$
$647$	4.78461	0.188102	0.0940512	+	0.995567i	$0.470018\pi$
$648$	−9.00000	−0.353553
$649$	20.5359	0.806105
$650$	0	0
$651$	0.248711	0.00974776
$652$	15.3205	0.599997
$653$	42.1962	1.65126	0.825632	−	0.564210i	$-0.190819\pi$
$653$	42.1962	1.65126	0.825632	+	0.564210i	$0.190819\pi$
$654$	−9.12436	−0.356791
$655$	11.5359	0.450745
$656$	−8.39230	−0.327664
$657$	0	0
$658$	7.46410	0.290981
$659$	−40.7128	−1.58595	−0.792973	−	0.609257i	$-0.791468\pi$
$659$	−40.7128	−1.58595	−0.792973	+	0.609257i	$0.791468\pi$
$660$	6.00000	0.233550
$661$	−25.5167	−0.992483	−0.496242	−	0.868185i	$-0.665287\pi$
$661$	−25.5167	−0.992483	−0.496242	+	0.868185i	$0.665287\pi$
$662$	−4.32051	−0.167921
$663$	0	0
$664$	−16.6603	−0.646543
$665$	3.26795	0.126726
$666$	0	0
$667$	0	0
$668$	16.1962	0.626648
$669$	−1.60770	−0.0621571
$670$	−1.73205	−0.0669150
$671$	−13.8564	−0.534921
$672$	1.26795	0.0489122
$673$	24.5167	0.945048	0.472524	−	0.881318i	$-0.343343\pi$
$673$	24.5167	0.945048	0.472524	+	0.881318i	$0.343343\pi$
$674$	13.0526	0.502766
$675$	−5.19615	−0.200000
$676$	−13.0000	−0.500000
$677$	26.1962	1.00680	0.503400	−	0.864054i	$-0.332082\pi$
$677$	26.1962	1.00680	0.503400	+	0.864054i	$0.332082\pi$
$678$	19.3923	0.744757
$679$	4.67949	0.179582
$680$	3.73205	0.143118
$681$	36.4641	1.39731
$682$	−0.679492	−0.0260191
$683$	23.0526	0.882082	0.441041	−	0.897487i	$-0.354609\pi$
$683$	23.0526	0.882082	0.441041	+	0.897487i	$0.354609\pi$
$684$	0	0
$685$	21.5885	0.824853
$686$	−9.85641	−0.376319
$687$	44.1962	1.68619
$688$	−9.19615	−0.350600
$689$	0	0
$690$	0	0
$691$	37.3205	1.41974	0.709870	−	0.704333i	$-0.248754\pi$
$691$	37.3205	1.41974	0.709870	+	0.704333i	$0.248754\pi$
$692$	−4.33975	−0.164972
$693$	0	0
$694$	−35.0526	−1.33058
$695$	−8.46410	−0.321062
$696$	−1.26795	−0.0480615
$697$	31.3205	1.18635
$698$	31.3205	1.18550
$699$	−17.5359	−0.663269
$700$	0.732051	0.0276689
$701$	39.6603	1.49795	0.748974	−	0.662600i	$-0.230547\pi$
$701$	39.6603	1.49795	0.748974	+	0.662600i	$0.230547\pi$
$702$	0	0
$703$	20.2487	0.763695
$704$	−3.46410	−0.130558
$705$	−17.6603	−0.665124
$706$	−19.9808	−0.751986
$707$	13.0718	0.491616
$708$	−10.2679	−0.385893
$709$	−27.5167	−1.03341	−0.516705	−	0.856164i	$-0.672842\pi$
$709$	−27.5167	−1.03341	−0.516705	+	0.856164i	$0.672842\pi$
$710$	−5.26795	−0.197703
$711$	0	0
$712$	−6.92820	−0.259645
$713$	0	0
$714$	−4.73205	−0.177093
$715$	0	0
$716$	−3.92820	−0.146804
$717$	−52.3923	−1.95663
$718$	−21.2679	−0.793713
$719$	−50.9808	−1.90126	−0.950631	−	0.310324i	$-0.899562\pi$
$719$	−50.9808	−1.90126	−0.950631	+	0.310324i	$0.899562\pi$
$720$	0	0
$721$	−2.24871	−0.0837464
$722$	0.928203	0.0345441
$723$	−24.1244	−0.897194
$724$	−26.3923	−0.980862
$725$	−0.732051	−0.0271877
$726$	1.73205	0.0642824
$727$	4.67949	0.173553	0.0867764	−	0.996228i	$-0.472343\pi$
$727$	4.67949	0.173553	0.0867764	+	0.996228i	$0.472343\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	1.19615	0.0442716
$731$	34.3205	1.26939
$732$	6.92820	0.256074
$733$	3.41154	0.126008	0.0630041	−	0.998013i	$-0.479932\pi$
$733$	3.41154	0.126008	0.0630041	+	0.998013i	$0.479932\pi$
$734$	−18.0526	−0.666332
$735$	11.1962	0.412976
$736$	0	0
$737$	−6.00000	−0.221013
$738$	0	0
$739$	−16.9282	−0.622714	−0.311357	−	0.950293i	$-0.600783\pi$
$739$	−16.9282	−0.622714	−0.311357	+	0.950293i	$0.600783\pi$
$740$	4.53590	0.166743
$741$	0	0
$742$	6.00000	0.220267
$743$	−20.5885	−0.755317	−0.377659	−	0.925945i	$-0.623271\pi$
$743$	−20.5885	−0.755317	−0.377659	+	0.925945i	$0.623271\pi$
$744$	0.339746	0.0124557
$745$	9.80385	0.359185
$746$	7.26795	0.266099
$747$	0	0
$748$	12.9282	0.472702
$749$	8.00000	0.292314
$750$	−1.73205	−0.0632456
$751$	−25.1769	−0.918719	−0.459359	−	0.888251i	$-0.651921\pi$
$751$	−25.1769	−0.918719	−0.459359	+	0.888251i	$0.651921\pi$
$752$	10.1962	0.371815
$753$	−1.73205	−0.0631194
$754$	0	0
$755$	−6.39230	−0.232640
$756$	−3.80385	−0.138345
$757$	18.0526	0.656131	0.328066	−	0.944655i	$-0.393603\pi$
$757$	18.0526	0.656131	0.328066	+	0.944655i	$0.393603\pi$
$758$	16.3205	0.592788
$759$	0	0
$760$	4.46410	0.161930
$761$	52.1769	1.89141	0.945706	−	0.325024i	$-0.105372\pi$
$761$	52.1769	1.89141	0.945706	+	0.325024i	$0.105372\pi$
$762$	−27.1244	−0.982612
$763$	−3.85641	−0.139611
$764$	−9.80385	−0.354691
$765$	0	0
$766$	−1.60770	−0.0580884
$767$	0	0
$768$	1.73205	0.0625000
$769$	46.6410	1.68192	0.840959	−	0.541099i	$-0.181992\pi$
$769$	46.6410	1.68192	0.840959	+	0.541099i	$0.181992\pi$
$770$	2.53590	0.0913874
$771$	−22.8564	−0.823153
$772$	13.7321	0.494227
$773$	−14.1436	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$773$	−14.1436	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$774$	0	0
$775$	0.196152	0.00704600
$776$	6.39230	0.229470
$777$	−5.75129	−0.206326
$778$	11.0718	0.396943
$779$	37.4641	1.34229
$780$	0	0
$781$	−18.2487	−0.652990
$782$	0	0
$783$	3.80385	0.135938
$784$	−6.46410	−0.230861
$785$	−15.3205	−0.546812
$786$	−19.9808	−0.712690
$787$	−8.80385	−0.313823	−0.156912	−	0.987613i	$-0.550154\pi$
$787$	−8.80385	−0.313823	−0.156912	+	0.987613i	$0.550154\pi$
$788$	−0.535898	−0.0190906
$789$	−12.2487	−0.436066
$790$	−2.19615	−0.0781356
$791$	8.19615	0.291422
$792$	0	0
$793$	0	0
$794$	27.1244	0.962607
$795$	−14.1962	−0.503486
$796$	15.8038	0.560153
$797$	−29.9090	−1.05943	−0.529715	−	0.848176i	$-0.677701\pi$
$797$	−29.9090	−1.05943	−0.529715	+	0.848176i	$0.677701\pi$
$798$	−5.66025	−0.200371
$799$	−38.0526	−1.34620
$800$	1.00000	0.0353553
$801$	0	0
$802$	20.8564	0.736465
$803$	4.14359	0.146224
$804$	3.00000	0.105802
$805$	0	0
$806$	0	0
$807$	−48.0000	−1.68968
$808$	17.8564	0.628186
$809$	−15.1436	−0.532420	−0.266210	−	0.963915i	$-0.585772\pi$
$809$	−15.1436	−0.532420	−0.266210	+	0.963915i	$0.585772\pi$
$810$	9.00000	0.316228
$811$	40.7128	1.42962	0.714810	−	0.699319i	$-0.246513\pi$
$811$	40.7128	1.42962	0.714810	+	0.699319i	$0.246513\pi$
$812$	−0.535898	−0.0188063
$813$	54.4974	1.91131
$814$	15.7128	0.550734
$815$	−15.3205	−0.536654
$816$	−6.46410	−0.226289
$817$	41.0526	1.43625
$818$	−8.32051	−0.290920
$819$	0	0
$820$	8.39230	0.293072
$821$	−17.4641	−0.609501	−0.304751	−	0.952432i	$-0.598573\pi$
$821$	−17.4641	−0.609501	−0.304751	+	0.952432i	$0.598573\pi$
$822$	−37.3923	−1.30421
$823$	13.6603	0.476167	0.238083	−	0.971245i	$-0.423481\pi$
$823$	13.6603	0.476167	0.238083	+	0.971245i	$0.423481\pi$
$824$	−3.07180	−0.107011
$825$	−6.00000	−0.208893
$826$	−4.33975	−0.150999
$827$	−35.9808	−1.25117	−0.625587	−	0.780155i	$-0.715140\pi$
$827$	−35.9808	−1.25117	−0.625587	+	0.780155i	$0.715140\pi$
$828$	0	0
$829$	−11.0718	−0.384539	−0.192270	−	0.981342i	$-0.561585\pi$
$829$	−11.0718	−0.384539	−0.192270	+	0.981342i	$0.561585\pi$
$830$	16.6603	0.578286
$831$	18.3397	0.636199
$832$	0	0
$833$	24.1244	0.835859
$834$	14.6603	0.507643
$835$	−16.1962	−0.560491
$836$	15.4641	0.534837
$837$	−1.01924	−0.0352300
$838$	−8.32051	−0.287427
$839$	11.4641	0.395785	0.197892	−	0.980224i	$-0.436590\pi$
$839$	11.4641	0.395785	0.197892	+	0.980224i	$0.436590\pi$
$840$	−1.26795	−0.0437484
$841$	−28.4641	−0.981521
$842$	−31.6603	−1.09108
$843$	11.1962	0.385616
$844$	11.0000	0.378636
$845$	13.0000	0.447214
$846$	0	0
$847$	0.732051	0.0251536
$848$	8.19615	0.281457
$849$	22.1769	0.761110
$850$	−3.73205	−0.128008
$851$	0	0
$852$	9.12436	0.312595
$853$	29.3205	1.00392	0.501958	−	0.864892i	$-0.332613\pi$
$853$	29.3205	1.00392	0.501958	+	0.864892i	$0.332613\pi$
$854$	2.92820	0.100201
$855$	0	0
$856$	10.9282	0.373518
$857$	5.19615	0.177497	0.0887486	−	0.996054i	$-0.471713\pi$
$857$	5.19615	0.177497	0.0887486	+	0.996054i	$0.471713\pi$
$858$	0	0
$859$	−39.3923	−1.34405	−0.672024	−	0.740529i	$-0.734575\pi$
$859$	−39.3923	−1.34405	−0.672024	+	0.740529i	$0.734575\pi$
$860$	9.19615	0.313586
$861$	−10.6410	−0.362645
$862$	26.9282	0.917178
$863$	8.87564	0.302130	0.151065	−	0.988524i	$-0.451730\pi$
$863$	8.87564	0.302130	0.151065	+	0.988524i	$0.451730\pi$
$864$	−5.19615	−0.176777
$865$	4.33975	0.147556
$866$	4.00000	0.135926
$867$	−5.32051	−0.180694
$868$	0.143594	0.00487388
$869$	−7.60770	−0.258073
$870$	1.26795	0.0429875
$871$	0	0
$872$	−5.26795	−0.178395
$873$	0	0
$874$	0	0
$875$	−0.732051	−0.0247478
$876$	−2.07180	−0.0699995
$877$	−3.60770	−0.121823	−0.0609116	−	0.998143i	$-0.519401\pi$
$877$	−3.60770	−0.121823	−0.0609116	+	0.998143i	$0.519401\pi$
$878$	−17.6603	−0.596005
$879$	46.3923	1.56477
$880$	3.46410	0.116775
$881$	−52.3923	−1.76514	−0.882571	−	0.470180i	$-0.844189\pi$
$881$	−52.3923	−1.76514	−0.882571	+	0.470180i	$0.844189\pi$
$882$	0	0
$883$	−11.3397	−0.381613	−0.190806	−	0.981628i	$-0.561110\pi$
$883$	−11.3397	−0.381613	−0.190806	+	0.981628i	$0.561110\pi$
$884$	0	0
$885$	10.2679	0.345153
$886$	−22.5167	−0.756462
$887$	28.7321	0.964728	0.482364	−	0.875971i	$-0.339778\pi$
$887$	28.7321	0.964728	0.482364	+	0.875971i	$0.339778\pi$
$888$	−7.85641	−0.263644
$889$	−11.4641	−0.384494
$890$	6.92820	0.232234
$891$	31.1769	1.04447
$892$	−0.928203	−0.0310785
$893$	−45.5167	−1.52316
$894$	−16.9808	−0.567922
$895$	3.92820	0.131305
$896$	0.732051	0.0244561
$897$	0	0
$898$	−19.0000	−0.634038
$899$	−0.143594	−0.00478911
$900$	0	0
$901$	−30.5885	−1.01905
$902$	29.0718	0.967985
$903$	−11.6603	−0.388029
$904$	11.1962	0.372378
$905$	26.3923	0.877310
$906$	11.0718	0.367836
$907$	59.3013	1.96907	0.984533	−	0.175198i	$-0.0560566\pi$
$907$	59.3013	1.96907	0.984533	+	0.175198i	$0.0560566\pi$
$908$	21.0526	0.698654
$909$	0	0
$910$	0	0
$911$	20.9282	0.693382	0.346691	−	0.937979i	$-0.387305\pi$
$911$	20.9282	0.693382	0.346691	+	0.937979i	$0.387305\pi$
$912$	−7.73205	−0.256034
$913$	57.7128	1.91002
$914$	37.7128	1.24743
$915$	−6.92820	−0.229039
$916$	25.5167	0.843094
$917$	−8.44486	−0.278874
$918$	19.3923	0.640041
$919$	36.1962	1.19400	0.597000	−	0.802241i	$-0.296359\pi$
$919$	36.1962	1.19400	0.597000	+	0.802241i	$0.296359\pi$
$920$	0	0
$921$	−48.9615	−1.61334
$922$	22.3923	0.737451
$923$	0	0
$924$	−4.39230	−0.144496
$925$	−4.53590	−0.149139
$926$	−16.5359	−0.543403
$927$	0	0
$928$	−0.732051	−0.0240307
$929$	22.7846	0.747539	0.373769	−	0.927522i	$-0.378065\pi$
$929$	22.7846	0.747539	0.373769	+	0.927522i	$0.378065\pi$
$930$	−0.339746	−0.0111407
$931$	28.8564	0.945731
$932$	−10.1244	−0.331634
$933$	−56.6936	−1.85606
$934$	−15.3205	−0.501302
$935$	−12.9282	−0.422797
$936$	0	0
$937$	−40.5167	−1.32362	−0.661811	−	0.749671i	$-0.730212\pi$
$937$	−40.5167	−1.32362	−0.661811	+	0.749671i	$0.730212\pi$
$938$	1.26795	0.0414000
$939$	23.5359	0.768065
$940$	−10.1962	−0.332562
$941$	−23.8038	−0.775983	−0.387992	−	0.921663i	$-0.626831\pi$
$941$	−23.8038	−0.775983	−0.387992	+	0.921663i	$0.626831\pi$
$942$	26.5359	0.864586
$943$	0	0
$944$	−5.92820	−0.192947
$945$	3.80385	0.123739
$946$	31.8564	1.03574
$947$	−40.2679	−1.30853	−0.654266	−	0.756264i	$-0.727022\pi$
$947$	−40.2679	−1.30853	−0.654266	+	0.756264i	$0.727022\pi$
$948$	3.80385	0.123543
$949$	0	0
$950$	−4.46410	−0.144835
$951$	−0.588457	−0.0190820
$952$	−2.73205	−0.0885463
$953$	31.1769	1.00992	0.504960	−	0.863143i	$-0.331507\pi$
$953$	31.1769	1.00992	0.504960	+	0.863143i	$0.331507\pi$
$954$	0	0
$955$	9.80385	0.317245
$956$	−30.2487	−0.978313
$957$	4.39230	0.141983
$958$	−11.2679	−0.364051
$959$	−15.8038	−0.510333
$960$	−1.73205	−0.0559017
$961$	−30.9615	−0.998759
$962$	0	0
$963$	0	0
$964$	−13.9282	−0.448597
$965$	−13.7321	−0.442050
$966$	0	0
$967$	−46.7846	−1.50449	−0.752246	−	0.658883i	$-0.771029\pi$
$967$	−46.7846	−1.50449	−0.752246	+	0.658883i	$0.771029\pi$
$968$	1.00000	0.0321412
$969$	28.8564	0.927001
$970$	−6.39230	−0.205245
$971$	21.9282	0.703710	0.351855	−	0.936055i	$-0.385551\pi$
$971$	21.9282	0.703710	0.351855	+	0.936055i	$0.385551\pi$
$972$	0	0
$973$	6.19615	0.198640
$974$	13.8564	0.443988
$975$	0	0
$976$	4.00000	0.128037
$977$	−58.7846	−1.88069	−0.940343	−	0.340228i	$-0.889496\pi$
$977$	−58.7846	−1.88069	−0.940343	+	0.340228i	$0.889496\pi$
$978$	26.5359	0.848524
$979$	24.0000	0.767043
$980$	6.46410	0.206488
$981$	0	0
$982$	15.4641	0.493479
$983$	−7.85641	−0.250580	−0.125290	−	0.992120i	$-0.539986\pi$
$983$	−7.85641	−0.250580	−0.125290	+	0.992120i	$0.539986\pi$
$984$	−14.5359	−0.463388
$985$	0.535898	0.0170751
$986$	2.73205	0.0870062
$987$	12.9282	0.411509
$988$	0	0
$989$	0	0
$990$	0	0
$991$	13.3205	0.423140	0.211570	−	0.977363i	$-0.432142\pi$
$991$	13.3205	0.423140	0.211570	+	0.977363i	$0.432142\pi$
$992$	0.196152	0.00622785
$993$	−7.48334	−0.237477
$994$	3.85641	0.122318
$995$	−15.8038	−0.501016
$996$	−28.8564	−0.914350
$997$	−40.9282	−1.29621	−0.648105	−	0.761551i	$-0.724438\pi$
$997$	−40.9282	−1.29621	−0.648105	+	0.761551i	$0.724438\pi$
$998$	15.0000	0.474817
$999$	23.5692	0.745697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5290.2.a.m.1.2	✓	2
23.22	odd	2		5290.2.a.n.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5290.2.a.m.1.2	✓	2	1.1	even	1	trivial
5290.2.a.n.1.2	yes	2	23.22	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:	\( N \)	\(=\)	\( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5290.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +0.732051 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +0.732051 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.46410 q^{11} +1.73205 q^{12} +0.732051 q^{14} -1.73205 q^{15} +1.00000 q^{16} -3.73205 q^{17} -4.46410 q^{19} -1.00000 q^{20} +1.26795 q^{21} -3.46410 q^{22} +1.73205 q^{24} +1.00000 q^{25} -5.19615 q^{27} +0.732051 q^{28} -0.732051 q^{29} -1.73205 q^{30} +0.196152 q^{31} +1.00000 q^{32} -6.00000 q^{33} -3.73205 q^{34} -0.732051 q^{35} -4.53590 q^{37} -4.46410 q^{38} -1.00000 q^{40} -8.39230 q^{41} +1.26795 q^{42} -9.19615 q^{43} -3.46410 q^{44} +10.1962 q^{47} +1.73205 q^{48} -6.46410 q^{49} +1.00000 q^{50} -6.46410 q^{51} +8.19615 q^{53} -5.19615 q^{54} +3.46410 q^{55} +0.732051 q^{56} -7.73205 q^{57} -0.732051 q^{58} -5.92820 q^{59} -1.73205 q^{60} +4.00000 q^{61} +0.196152 q^{62} +1.00000 q^{64} -6.00000 q^{66} +1.73205 q^{67} -3.73205 q^{68} -0.732051 q^{70} +5.26795 q^{71} -1.19615 q^{73} -4.53590 q^{74} +1.73205 q^{75} -4.46410 q^{76} -2.53590 q^{77} +2.19615 q^{79} -1.00000 q^{80} -9.00000 q^{81} -8.39230 q^{82} -16.6603 q^{83} +1.26795 q^{84} +3.73205 q^{85} -9.19615 q^{86} -1.26795 q^{87} -3.46410 q^{88} -6.92820 q^{89} +0.339746 q^{93} +10.1962 q^{94} +4.46410 q^{95} +1.73205 q^{96} +6.39230 q^{97} -6.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{19} - 2 q^{20} + 6 q^{21} + 2 q^{25} - 2 q^{28} + 2 q^{29} - 10 q^{31} + 2 q^{32} - 12 q^{33} - 4 q^{34} + 2 q^{35} - 16 q^{37} - 2 q^{38} - 2 q^{40} + 4 q^{41} + 6 q^{42} - 8 q^{43} + 10 q^{47} - 6 q^{49} + 2 q^{50} - 6 q^{51} + 6 q^{53} - 2 q^{56} - 12 q^{57} + 2 q^{58} + 2 q^{59} + 8 q^{61} - 10 q^{62} + 2 q^{64} - 12 q^{66} - 4 q^{68} + 2 q^{70} + 14 q^{71} + 8 q^{73} - 16 q^{74} - 2 q^{76} - 12 q^{77} - 6 q^{79} - 2 q^{80} - 18 q^{81} + 4 q^{82} - 16 q^{83} + 6 q^{84} + 4 q^{85} - 8 q^{86} - 6 q^{87} + 18 q^{93} + 10 q^{94} + 2 q^{95} - 8 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^19 - 2 * q^20 + 6 * q^21 + 2 * q^25 - 2 * q^28 + 2 * q^29 - 10 * q^31 + 2 * q^32 - 12 * q^33 - 4 * q^34 + 2 * q^35 - 16 * q^37 - 2 * q^38 - 2 * q^40 + 4 * q^41 + 6 * q^42 - 8 * q^43 + 10 * q^47 - 6 * q^49 + 2 * q^50 - 6 * q^51 + 6 * q^53 - 2 * q^56 - 12 * q^57 + 2 * q^58 + 2 * q^59 + 8 * q^61 - 10 * q^62 + 2 * q^64 - 12 * q^66 - 4 * q^68 + 2 * q^70 + 14 * q^71 + 8 * q^73 - 16 * q^74 - 2 * q^76 - 12 * q^77 - 6 * q^79 - 2 * q^80 - 18 * q^81 + 4 * q^82 - 16 * q^83 + 6 * q^84 + 4 * q^85 - 8 * q^86 - 6 * q^87 + 18 * q^93 + 10 * q^94 + 2 * q^95 - 8 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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