LMFDB - Embedded newform 2001.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,2,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, 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("2001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2001.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:	$15.9780654445$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.63931$ of defining polynomial
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63931 q^{2} +1.00000 q^{3} +4.96597 q^{4} -1.13791 q^{5} -2.63931 q^{6} +1.99715 q^{7} -7.82811 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.63931 q^{2} +1.00000 q^{3} +4.96597 q^{4} -1.13791 q^{5} -2.63931 q^{6} +1.99715 q^{7} -7.82811 q^{8} +1.00000 q^{9} +3.00330 q^{10} +1.82707 q^{11} +4.96597 q^{12} +7.15732 q^{13} -5.27109 q^{14} -1.13791 q^{15} +10.7289 q^{16} +7.66347 q^{17} -2.63931 q^{18} +7.80709 q^{19} -5.65083 q^{20} +1.99715 q^{21} -4.82220 q^{22} +1.00000 q^{23} -7.82811 q^{24} -3.70516 q^{25} -18.8904 q^{26} +1.00000 q^{27} +9.91777 q^{28} +1.00000 q^{29} +3.00330 q^{30} -8.74793 q^{31} -12.6607 q^{32} +1.82707 q^{33} -20.2263 q^{34} -2.27258 q^{35} +4.96597 q^{36} +3.85152 q^{37} -20.6053 q^{38} +7.15732 q^{39} +8.90770 q^{40} -1.74746 q^{41} -5.27109 q^{42} +7.55495 q^{43} +9.07316 q^{44} -1.13791 q^{45} -2.63931 q^{46} -12.7120 q^{47} +10.7289 q^{48} -3.01140 q^{49} +9.77907 q^{50} +7.66347 q^{51} +35.5430 q^{52} +0.821375 q^{53} -2.63931 q^{54} -2.07904 q^{55} -15.6339 q^{56} +7.80709 q^{57} -2.63931 q^{58} +1.89394 q^{59} -5.65083 q^{60} -6.35225 q^{61} +23.0885 q^{62} +1.99715 q^{63} +11.9577 q^{64} -8.14439 q^{65} -4.82220 q^{66} -3.73113 q^{67} +38.0565 q^{68} +1.00000 q^{69} +5.99804 q^{70} +7.52382 q^{71} -7.82811 q^{72} -11.6746 q^{73} -10.1654 q^{74} -3.70516 q^{75} +38.7697 q^{76} +3.64893 q^{77} -18.8904 q^{78} +10.2423 q^{79} -12.2085 q^{80} +1.00000 q^{81} +4.61210 q^{82} +3.44576 q^{83} +9.91777 q^{84} -8.72034 q^{85} -19.9399 q^{86} +1.00000 q^{87} -14.3025 q^{88} -3.74628 q^{89} +3.00330 q^{90} +14.2942 q^{91} +4.96597 q^{92} -8.74793 q^{93} +33.5509 q^{94} -8.88377 q^{95} -12.6607 q^{96} -12.1906 q^{97} +7.94803 q^{98} +1.82707 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) $ 20 * q + 2 * q^2 + 20 * q^3 + 30 * q^4 - q^5 + 2 * q^6 + 9 * q^7 + 6 * q^8 + 20 * q^9	\( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9} + 7 q^{10} + 30 q^{12} + 21 q^{13} - q^{14} - q^{15} + 58 q^{16} - 4 q^{17} + 2 q^{18} + 7 q^{19} - 20 q^{20} + 9 q^{21} + 7 q^{22} + 20 q^{23} + 6 q^{24} + 47 q^{25} + 8 q^{26} + 20 q^{27} + 11 q^{28} + 20 q^{29} + 7 q^{30} + 28 q^{31} + 14 q^{32} + 16 q^{34} + 9 q^{35} + 30 q^{36} + 14 q^{37} - 20 q^{38} + 21 q^{39} + 34 q^{40} + 7 q^{41} - q^{42} + 3 q^{43} - q^{44} - q^{45} + 2 q^{46} + 3 q^{47} + 58 q^{48} + 35 q^{49} - 24 q^{50} - 4 q^{51} + 73 q^{52} - 19 q^{53} + 2 q^{54} + 29 q^{55} - 30 q^{56} + 7 q^{57} + 2 q^{58} + 20 q^{59} - 20 q^{60} + 15 q^{61} + 12 q^{62} + 9 q^{63} + 82 q^{64} - 28 q^{65} + 7 q^{66} + 20 q^{67} - 23 q^{68} + 20 q^{69} - 24 q^{70} + 63 q^{71} + 6 q^{72} + 19 q^{73} + 16 q^{74} + 47 q^{75} - 44 q^{76} - 7 q^{77} + 8 q^{78} + 32 q^{79} - 56 q^{80} + 20 q^{81} - 20 q^{82} - 21 q^{83} + 11 q^{84} + 4 q^{85} - 6 q^{86} + 20 q^{87} + 55 q^{88} - 13 q^{89} + 7 q^{90} + 70 q^{91} + 30 q^{92} + 28 q^{93} - 12 q^{94} + 9 q^{95} + 14 q^{96} - 9 q^{97} + 31 q^{98}+O(q^{100}) \) 20 * q + 2 * q^2 + 20 * q^3 + 30 * q^4 - q^5 + 2 * q^6 + 9 * q^7 + 6 * q^8 + 20 * q^9 + 7 * q^10 + 30 * q^12 + 21 * q^13 - q^14 - q^15 + 58 * q^16 - 4 * q^17 + 2 * q^18 + 7 * q^19 - 20 * q^20 + 9 * q^21 + 7 * q^22 + 20 * q^23 + 6 * q^24 + 47 * q^25 + 8 * q^26 + 20 * q^27 + 11 * q^28 + 20 * q^29 + 7 * q^30 + 28 * q^31 + 14 * q^32 + 16 * q^34 + 9 * q^35 + 30 * q^36 + 14 * q^37 - 20 * q^38 + 21 * q^39 + 34 * q^40 + 7 * q^41 - q^42 + 3 * q^43 - q^44 - q^45 + 2 * q^46 + 3 * q^47 + 58 * q^48 + 35 * q^49 - 24 * q^50 - 4 * q^51 + 73 * q^52 - 19 * q^53 + 2 * q^54 + 29 * q^55 - 30 * q^56 + 7 * q^57 + 2 * q^58 + 20 * q^59 - 20 * q^60 + 15 * q^61 + 12 * q^62 + 9 * q^63 + 82 * q^64 - 28 * q^65 + 7 * q^66 + 20 * q^67 - 23 * q^68 + 20 * q^69 - 24 * q^70 + 63 * q^71 + 6 * q^72 + 19 * q^73 + 16 * q^74 + 47 * q^75 - 44 * q^76 - 7 * q^77 + 8 * q^78 + 32 * q^79 - 56 * q^80 + 20 * q^81 - 20 * q^82 - 21 * q^83 + 11 * q^84 + 4 * q^85 - 6 * q^86 + 20 * q^87 + 55 * q^88 - 13 * q^89 + 7 * q^90 + 70 * q^91 + 30 * q^92 + 28 * q^93 - 12 * q^94 + 9 * q^95 + 14 * q^96 - 9 * q^97 + 31 * 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 $

$2$	−2.63931	−1.86628	−0.933138	−	0.359519i	$-0.882941\pi$
$2$	−2.63931	−1.86628	−0.933138	+	0.359519i	$0.882941\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.96597	2.48298
$5$	−1.13791	−0.508889	−0.254445	−	0.967087i	$-0.581893\pi$
$5$	−1.13791	−0.508889	−0.254445	+	0.967087i	$0.581893\pi$
$6$	−2.63931	−1.07749
$7$	1.99715	0.754851	0.377425	−	0.926040i	$-0.376809\pi$
$7$	1.99715	0.754851	0.377425	+	0.926040i	$0.376809\pi$
$8$	−7.82811	−2.76766
$9$	1.00000	0.333333
$10$	3.00330	0.949728
$11$	1.82707	0.550882	0.275441	−	0.961318i	$-0.411176\pi$
$11$	1.82707	0.550882	0.275441	+	0.961318i	$0.411176\pi$
$12$	4.96597	1.43355
$13$	7.15732	1.98508	0.992542	−	0.121906i	$-0.0389008\pi$
$13$	7.15732	1.98508	0.992542	+	0.121906i	$0.0389008\pi$
$14$	−5.27109	−1.40876
$15$	−1.13791	−0.293807
$16$	10.7289	2.68222
$17$	7.66347	1.85866	0.929332	−	0.369246i	$-0.120384\pi$
$17$	7.66347	1.85866	0.929332	+	0.369246i	$0.120384\pi$
$18$	−2.63931	−0.622092
$19$	7.80709	1.79107	0.895534	−	0.444992i	$-0.146794\pi$
$19$	7.80709	1.79107	0.895534	+	0.444992i	$0.146794\pi$
$20$	−5.65083	−1.26356
$21$	1.99715	0.435813
$22$	−4.82220	−1.02810
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−7.82811	−1.59791
$25$	−3.70516	−0.741032
$26$	−18.8904	−3.70471
$27$	1.00000	0.192450
$28$	9.91777	1.87428
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	3.00330	0.548326
$31$	−8.74793	−1.57118	−0.785588	−	0.618750i	$-0.787639\pi$
$31$	−8.74793	−1.57118	−0.785588	+	0.618750i	$0.787639\pi$
$32$	−12.6607	−2.23811
$33$	1.82707	0.318052
$34$	−20.2263	−3.46878
$35$	−2.27258	−0.384135
$36$	4.96597	0.827661
$37$	3.85152	0.633186	0.316593	−	0.948561i	$-0.397461\pi$
$37$	3.85152	0.633186	0.316593	+	0.948561i	$0.397461\pi$
$38$	−20.6053	−3.34263
$39$	7.15732	1.14609
$40$	8.90770	1.40843
$41$	−1.74746	−0.272908	−0.136454	−	0.990646i	$-0.543571\pi$
$41$	−1.74746	−0.272908	−0.136454	+	0.990646i	$0.543571\pi$
$42$	−5.27109	−0.813347
$43$	7.55495	1.15212	0.576060	−	0.817408i	$-0.304590\pi$
$43$	7.55495	1.15212	0.576060	+	0.817408i	$0.304590\pi$
$44$	9.07316	1.36783
$45$	−1.13791	−0.169630
$46$	−2.63931	−0.389145
$47$	−12.7120	−1.85423	−0.927117	−	0.374772i	$-0.877721\pi$
$47$	−12.7120	−1.85423	−0.927117	+	0.374772i	$0.877721\pi$
$48$	10.7289	1.54858
$49$	−3.01140	−0.430200
$50$	9.77907	1.38297
$51$	7.66347	1.07310
$52$	35.5430	4.92893
$53$	0.821375	0.112825	0.0564123	−	0.998408i	$-0.482034\pi$
$53$	0.821375	0.112825	0.0564123	+	0.998408i	$0.482034\pi$
$54$	−2.63931	−0.359165
$55$	−2.07904	−0.280338
$56$	−15.6339	−2.08917
$57$	7.80709	1.03407
$58$	−2.63931	−0.346559
$59$	1.89394	0.246570	0.123285	−	0.992371i	$-0.460657\pi$
$59$	1.89394	0.246570	0.123285	+	0.992371i	$0.460657\pi$
$60$	−5.65083	−0.729519
$61$	−6.35225	−0.813322	−0.406661	−	0.913579i	$-0.633307\pi$
$61$	−6.35225	−0.813322	−0.406661	+	0.913579i	$0.633307\pi$
$62$	23.0885	2.93225
$63$	1.99715	0.251617
$64$	11.9577	1.49471
$65$	−8.14439	−1.01019
$66$	−4.82220	−0.593572
$67$	−3.73113	−0.455830	−0.227915	−	0.973681i	$-0.573191\pi$
$67$	−3.73113	−0.455830	−0.227915	+	0.973681i	$0.573191\pi$
$68$	38.0565	4.61503
$69$	1.00000	0.120386
$70$	5.99804	0.716903
$71$	7.52382	0.892914	0.446457	−	0.894805i	$-0.352686\pi$
$71$	7.52382	0.892914	0.446457	+	0.894805i	$0.352686\pi$
$72$	−7.82811	−0.922552
$73$	−11.6746	−1.36641	−0.683204	−	0.730227i	$-0.739414\pi$
$73$	−11.6746	−1.36641	−0.683204	+	0.730227i	$0.739414\pi$
$74$	−10.1654	−1.18170
$75$	−3.70516	−0.427835
$76$	38.7697	4.44719
$77$	3.64893	0.415834
$78$	−18.8904	−2.13892
$79$	10.2423	1.15235	0.576173	−	0.817328i	$-0.304546\pi$
$79$	10.2423	1.15235	0.576173	+	0.817328i	$0.304546\pi$
$80$	−12.2085	−1.36495
$81$	1.00000	0.111111
$82$	4.61210	0.509321
$83$	3.44576	0.378221	0.189111	−	0.981956i	$-0.439440\pi$
$83$	3.44576	0.378221	0.189111	+	0.981956i	$0.439440\pi$
$84$	9.91777	1.08212
$85$	−8.72034	−0.945854
$86$	−19.9399	−2.15017
$87$	1.00000	0.107211
$88$	−14.3025	−1.52465
$89$	−3.74628	−0.397105	−0.198553	−	0.980090i	$-0.563624\pi$
$89$	−3.74628	−0.397105	−0.198553	+	0.980090i	$0.563624\pi$
$90$	3.00330	0.316576
$91$	14.2942	1.49844
$92$	4.96597	0.517738
$93$	−8.74793	−0.907118
$94$	33.5509	3.46051
$95$	−8.88377	−0.911456
$96$	−12.6607	−1.29217
$97$	−12.1906	−1.23777	−0.618883	−	0.785484i	$-0.712414\pi$
$97$	−12.1906	−1.23777	−0.618883	+	0.785484i	$0.712414\pi$
$98$	7.94803	0.802872
$99$	1.82707	0.183627
$100$	−18.3997	−1.83997
$101$	12.0017	1.19422	0.597108	−	0.802161i	$-0.296316\pi$
$101$	12.0017	1.19422	0.597108	+	0.802161i	$0.296316\pi$
$102$	−20.2263	−2.00270
$103$	−0.310984	−0.0306422	−0.0153211	−	0.999883i	$-0.504877\pi$
$103$	−0.310984	−0.0306422	−0.0153211	+	0.999883i	$0.504877\pi$
$104$	−56.0283	−5.49403
$105$	−2.27258	−0.221781
$106$	−2.16787	−0.210562
$107$	−3.70187	−0.357873	−0.178936	−	0.983861i	$-0.557266\pi$
$107$	−3.70187	−0.357873	−0.178936	+	0.983861i	$0.557266\pi$
$108$	4.96597	0.477850
$109$	8.23947	0.789198	0.394599	−	0.918853i	$-0.370883\pi$
$109$	8.23947	0.789198	0.394599	+	0.918853i	$0.370883\pi$
$110$	5.48724	0.523188
$111$	3.85152	0.365570
$112$	21.4272	2.02468
$113$	−17.3427	−1.63146	−0.815732	−	0.578430i	$-0.803666\pi$
$113$	−17.3427	−1.63146	−0.815732	+	0.578430i	$0.803666\pi$
$114$	−20.6053	−1.92987
$115$	−1.13791	−0.106111
$116$	4.96597	0.461078
$117$	7.15732	0.661694
$118$	−4.99869	−0.460167
$119$	15.3051	1.40301
$120$	8.90770	0.813158
$121$	−7.66182	−0.696529
$122$	16.7656	1.51788
$123$	−1.74746	−0.157563
$124$	−43.4419	−3.90120
$125$	9.90570	0.885992
$126$	−5.27109	−0.469586
$127$	−4.30110	−0.381661	−0.190831	−	0.981623i	$-0.561118\pi$
$127$	−4.30110	−0.381661	−0.190831	+	0.981623i	$0.561118\pi$
$128$	−6.23869	−0.551427
$129$	7.55495	0.665176
$130$	21.4956	1.88529
$131$	−5.69493	−0.497569	−0.248784	−	0.968559i	$-0.580031\pi$
$131$	−5.69493	−0.497569	−0.248784	+	0.968559i	$0.580031\pi$
$132$	9.07316	0.789718
$133$	15.5919	1.35199
$134$	9.84761	0.850704
$135$	−1.13791	−0.0979358
$136$	−59.9905	−5.14414
$137$	−0.133263	−0.0113855	−0.00569273	−	0.999984i	$-0.501812\pi$
$137$	−0.133263	−0.0113855	−0.00569273	+	0.999984i	$0.501812\pi$
$138$	−2.63931	−0.224673
$139$	−3.70378	−0.314150	−0.157075	−	0.987587i	$-0.550207\pi$
$139$	−3.70378	−0.314150	−0.157075	+	0.987587i	$0.550207\pi$
$140$	−11.2855	−0.953802
$141$	−12.7120	−1.07054
$142$	−19.8577	−1.66642
$143$	13.0769	1.09355
$144$	10.7289	0.894074
$145$	−1.13791	−0.0944984
$146$	30.8129	2.55009
$147$	−3.01140	−0.248376
$148$	19.1265	1.57219
$149$	−3.47160	−0.284405	−0.142202	−	0.989838i	$-0.545418\pi$
$149$	−3.47160	−0.284405	−0.142202	+	0.989838i	$0.545418\pi$
$150$	9.77907	0.798458
$151$	1.43748	0.116981	0.0584903	−	0.998288i	$-0.481371\pi$
$151$	1.43748	0.116981	0.0584903	+	0.998288i	$0.481371\pi$
$152$	−61.1147	−4.95706
$153$	7.66347	0.619555
$154$	−9.63065	−0.776060
$155$	9.95437	0.799554
$156$	35.5430	2.84572
$157$	−7.24011	−0.577824	−0.288912	−	0.957356i	$-0.593293\pi$
$157$	−7.24011	−0.577824	−0.288912	+	0.957356i	$0.593293\pi$
$158$	−27.0325	−2.15059
$159$	0.821375	0.0651393
$160$	14.4067	1.13895
$161$	1.99715	0.157397
$162$	−2.63931	−0.207364
$163$	−22.7977	−1.78566	−0.892829	−	0.450396i	$-0.851283\pi$
$163$	−22.7977	−1.78566	−0.892829	+	0.450396i	$0.851283\pi$
$164$	−8.67784	−0.677626
$165$	−2.07904	−0.161853
$166$	−9.09443	−0.705865
$167$	−18.4991	−1.43150	−0.715752	−	0.698355i	$-0.753916\pi$
$167$	−18.4991	−1.43150	−0.715752	+	0.698355i	$0.753916\pi$
$168$	−15.6339	−1.20618
$169$	38.2272	2.94056
$170$	23.0157	1.76522
$171$	7.80709	0.597023
$172$	37.5176	2.86069
$173$	14.4105	1.09561	0.547806	−	0.836605i	$-0.315463\pi$
$173$	14.4105	1.09561	0.547806	+	0.836605i	$0.315463\pi$
$174$	−2.63931	−0.200086
$175$	−7.39975	−0.559368
$176$	19.6024	1.47759
$177$	1.89394	0.142357
$178$	9.88761	0.741108
$179$	7.85751	0.587298	0.293649	−	0.955913i	$-0.405130\pi$
$179$	7.85751	0.587298	0.293649	+	0.955913i	$0.405130\pi$
$180$	−5.65083	−0.421188
$181$	10.1822	0.756839	0.378420	−	0.925634i	$-0.376468\pi$
$181$	10.1822	0.756839	0.378420	+	0.925634i	$0.376468\pi$
$182$	−37.7269	−2.79650
$183$	−6.35225	−0.469572
$184$	−7.82811	−0.577096
$185$	−4.38269	−0.322222
$186$	23.0885	1.69293
$187$	14.0017	1.02390
$188$	−63.1273	−4.60403
$189$	1.99715	0.145271
$190$	23.4470	1.70103
$191$	6.02951	0.436280	0.218140	−	0.975917i	$-0.430001\pi$
$191$	6.02951	0.436280	0.218140	+	0.975917i	$0.430001\pi$
$192$	11.9577	0.862971
$193$	−10.8433	−0.780517	−0.390259	−	0.920705i	$-0.627614\pi$
$193$	−10.8433	−0.780517	−0.390259	+	0.920705i	$0.627614\pi$
$194$	32.1747	2.31001
$195$	−8.14439	−0.583232
$196$	−14.9545	−1.06818
$197$	−3.72493	−0.265390	−0.132695	−	0.991157i	$-0.542363\pi$
$197$	−3.72493	−0.265390	−0.132695	+	0.991157i	$0.542363\pi$
$198$	−4.82220	−0.342699
$199$	23.8729	1.69231	0.846153	−	0.532940i	$-0.178913\pi$
$199$	23.8729	1.69231	0.846153	+	0.532940i	$0.178913\pi$
$200$	29.0044	2.05092
$201$	−3.73113	−0.263173
$202$	−31.6763	−2.22874
$203$	1.99715	0.140172
$204$	38.0565	2.66449
$205$	1.98846	0.138880
$206$	0.820784	0.0571867
$207$	1.00000	0.0695048
$208$	76.7901	5.32444
$209$	14.2641	0.986668
$210$	5.99804	0.413904
$211$	−14.8626	−1.02318	−0.511591	−	0.859229i	$-0.670944\pi$
$211$	−14.8626	−1.02318	−0.511591	+	0.859229i	$0.670944\pi$
$212$	4.07892	0.280142
$213$	7.52382	0.515524
$214$	9.77038	0.667889
$215$	−8.59686	−0.586301
$216$	−7.82811	−0.532636
$217$	−17.4709	−1.18600
$218$	−21.7465	−1.47286
$219$	−11.6746	−0.788896
$220$	−10.3245	−0.696075
$221$	54.8499	3.68960
$222$	−10.1654	−0.682255
$223$	1.42650	0.0955252	0.0477626	−	0.998859i	$-0.484791\pi$
$223$	1.42650	0.0955252	0.0477626	+	0.998859i	$0.484791\pi$
$224$	−25.2852	−1.68944
$225$	−3.70516	−0.247011
$226$	45.7728	3.04476
$227$	21.5865	1.43275	0.716374	−	0.697717i	$-0.245801\pi$
$227$	21.5865	1.43275	0.716374	+	0.697717i	$0.245801\pi$
$228$	38.7697	2.56759
$229$	21.3452	1.41053	0.705264	−	0.708944i	$-0.250828\pi$
$229$	21.3452	1.41053	0.705264	+	0.708944i	$0.250828\pi$
$230$	3.00330	0.198032
$231$	3.64893	0.240082
$232$	−7.82811	−0.513941
$233$	−14.8297	−0.971526	−0.485763	−	0.874091i	$-0.661458\pi$
$233$	−14.8297	−0.971526	−0.485763	+	0.874091i	$0.661458\pi$
$234$	−18.8904	−1.23490
$235$	14.4651	0.943600
$236$	9.40523	0.612228
$237$	10.2423	0.665307
$238$	−40.3949	−2.61841
$239$	3.91325	0.253127	0.126563	−	0.991959i	$-0.459605\pi$
$239$	3.91325	0.253127	0.126563	+	0.991959i	$0.459605\pi$
$240$	−12.2085	−0.788057
$241$	−8.98398	−0.578709	−0.289355	−	0.957222i	$-0.593441\pi$
$241$	−8.98398	−0.578709	−0.289355	+	0.957222i	$0.593441\pi$
$242$	20.2219	1.29991
$243$	1.00000	0.0641500
$244$	−31.5451	−2.01947
$245$	3.42671	0.218924
$246$	4.61210	0.294057
$247$	55.8778	3.55542
$248$	68.4798	4.34847
$249$	3.44576	0.218366
$250$	−26.1442	−1.65351
$251$	−21.2258	−1.33976	−0.669880	−	0.742469i	$-0.733655\pi$
$251$	−21.2258	−1.33976	−0.669880	+	0.742469i	$0.733655\pi$
$252$	9.91777	0.624761
$253$	1.82707	0.114867
$254$	11.3519	0.712285
$255$	−8.72034	−0.546089
$256$	−7.44951	−0.465594
$257$	−12.4232	−0.774939	−0.387469	−	0.921883i	$-0.626651\pi$
$257$	−12.4232	−0.774939	−0.387469	+	0.921883i	$0.626651\pi$
$258$	−19.9399	−1.24140
$259$	7.69206	0.477961
$260$	−40.4448	−2.50828
$261$	1.00000	0.0618984
$262$	15.0307	0.928600
$263$	−18.1329	−1.11812	−0.559062	−	0.829126i	$-0.688839\pi$
$263$	−18.1329	−1.11812	−0.559062	+	0.829126i	$0.688839\pi$
$264$	−14.3025	−0.880258
$265$	−0.934652	−0.0574152
$266$	−41.1519	−2.52318
$267$	−3.74628	−0.229269
$268$	−18.5287	−1.13182
$269$	−11.4338	−0.697134	−0.348567	−	0.937284i	$-0.613332\pi$
$269$	−11.4338	−0.697134	−0.348567	+	0.937284i	$0.613332\pi$
$270$	3.00330	0.182775
$271$	−20.2276	−1.22874	−0.614369	−	0.789019i	$-0.710589\pi$
$271$	−20.2276	−1.22874	−0.614369	+	0.789019i	$0.710589\pi$
$272$	82.2205	4.98535
$273$	14.2942	0.865126
$274$	0.351724	0.0212484
$275$	−6.76958	−0.408221
$276$	4.96597	0.298916
$277$	27.5724	1.65667	0.828333	−	0.560235i	$-0.189289\pi$
$277$	27.5724	1.65667	0.828333	+	0.560235i	$0.189289\pi$
$278$	9.77543	0.586291
$279$	−8.74793	−0.523725
$280$	17.7900	1.06315
$281$	16.0187	0.955596	0.477798	−	0.878470i	$-0.341435\pi$
$281$	16.0187	0.955596	0.477798	+	0.878470i	$0.341435\pi$
$282$	33.5509	1.99793
$283$	14.7763	0.878361	0.439181	−	0.898399i	$-0.355269\pi$
$283$	14.7763	0.878361	0.439181	+	0.898399i	$0.355269\pi$
$284$	37.3631	2.21709
$285$	−8.88377	−0.526229
$286$	−34.5141	−2.04086
$287$	−3.48994	−0.206005
$288$	−12.6607	−0.746037
$289$	41.7287	2.45463
$290$	3.00330	0.176360
$291$	−12.1906	−0.714624
$292$	−57.9757	−3.39277
$293$	−11.7733	−0.687804	−0.343902	−	0.939005i	$-0.611749\pi$
$293$	−11.7733	−0.687804	−0.343902	+	0.939005i	$0.611749\pi$
$294$	7.94803	0.463539
$295$	−2.15513	−0.125477
$296$	−30.1501	−1.75244
$297$	1.82707	0.106017
$298$	9.16264	0.530778
$299$	7.15732	0.413918
$300$	−18.3997	−1.06231
$301$	15.0883	0.869678
$302$	−3.79396	−0.218318
$303$	12.0017	0.689481
$304$	83.7614	4.80405
$305$	7.22829	0.413891
$306$	−20.2263	−1.15626
$307$	10.9992	0.627759	0.313880	−	0.949463i	$-0.398371\pi$
$307$	10.9992	0.627759	0.313880	+	0.949463i	$0.398371\pi$
$308$	18.1204	1.03251
$309$	−0.310984	−0.0176913
$310$	−26.2727	−1.49219
$311$	22.8722	1.29696	0.648482	−	0.761230i	$-0.275404\pi$
$311$	22.8722	1.29696	0.648482	+	0.761230i	$0.275404\pi$
$312$	−56.0283	−3.17198
$313$	11.0644	0.625399	0.312699	−	0.949852i	$-0.398767\pi$
$313$	11.0644	0.625399	0.312699	+	0.949852i	$0.398767\pi$
$314$	19.1089	1.07838
$315$	−2.27258	−0.128045
$316$	50.8628	2.86125
$317$	21.5018	1.20766	0.603831	−	0.797112i	$-0.293640\pi$
$317$	21.5018	1.20766	0.603831	+	0.797112i	$0.293640\pi$
$318$	−2.16787	−0.121568
$319$	1.82707	0.102296
$320$	−13.6068	−0.760642
$321$	−3.70187	−0.206618
$322$	−5.27109	−0.293747
$323$	59.8294	3.32899
$324$	4.96597	0.275887
$325$	−26.5190	−1.47101
$326$	60.1704	3.33253
$327$	8.23947	0.455644
$328$	13.6793	0.755315
$329$	−25.3877	−1.39967
$330$	5.48724	0.302063
$331$	−16.4775	−0.905686	−0.452843	−	0.891590i	$-0.649590\pi$
$331$	−16.4775	−0.905686	−0.452843	+	0.891590i	$0.649590\pi$
$332$	17.1115	0.939117
$333$	3.85152	0.211062
$334$	48.8249	2.67158
$335$	4.24569	0.231967
$336$	21.4272	1.16895
$337$	−30.2999	−1.65054	−0.825271	−	0.564736i	$-0.808978\pi$
$337$	−30.2999	−1.65054	−0.825271	+	0.564736i	$0.808978\pi$
$338$	−100.894	−5.48789
$339$	−17.3427	−0.941927
$340$	−43.3049	−2.34854
$341$	−15.9831	−0.865532
$342$	−20.6053	−1.11421
$343$	−19.9942	−1.07959
$344$	−59.1410	−3.18867
$345$	−1.13791	−0.0612631
$346$	−38.0339	−2.04472
$347$	32.4355	1.74123	0.870613	−	0.491968i	$-0.163722\pi$
$347$	32.4355	1.74123	0.870613	+	0.491968i	$0.163722\pi$
$348$	4.96597	0.266204
$349$	−12.4906	−0.668604	−0.334302	−	0.942466i	$-0.608501\pi$
$349$	−12.4906	−0.668604	−0.334302	+	0.942466i	$0.608501\pi$
$350$	19.5302	1.04394
$351$	7.15732	0.382029
$352$	−23.1319	−1.23294
$353$	−14.8565	−0.790734	−0.395367	−	0.918523i	$-0.629383\pi$
$353$	−14.8565	−0.790734	−0.395367	+	0.918523i	$0.629383\pi$
$354$	−4.99869	−0.265677
$355$	−8.56144	−0.454394
$356$	−18.6039	−0.986006
$357$	15.3051	0.810030
$358$	−20.7384	−1.09606
$359$	9.70922	0.512433	0.256217	−	0.966619i	$-0.417524\pi$
$359$	9.70922	0.512433	0.256217	+	0.966619i	$0.417524\pi$
$360$	8.90770	0.469477
$361$	41.9506	2.20793
$362$	−26.8741	−1.41247
$363$	−7.66182	−0.402141
$364$	70.9846	3.72061
$365$	13.2847	0.695351
$366$	16.7656	0.876350
$367$	−22.1799	−1.15778	−0.578890	−	0.815406i	$-0.696514\pi$
$367$	−22.1799	−1.15778	−0.578890	+	0.815406i	$0.696514\pi$
$368$	10.7289	0.559282
$369$	−1.74746	−0.0909693
$370$	11.5673	0.601355
$371$	1.64041	0.0851657
$372$	−43.4419	−2.25236
$373$	2.92841	0.151627	0.0758137	−	0.997122i	$-0.475845\pi$
$373$	2.92841	0.151627	0.0758137	+	0.997122i	$0.475845\pi$
$374$	−36.9548	−1.91089
$375$	9.90570	0.511528
$376$	99.5109	5.13188
$377$	7.15732	0.368621
$378$	−5.27109	−0.271116
$379$	−0.0109094	−0.000560380 0	−0.000280190	−	1.00000i	$-0.500089\pi$
$379$	−0.0109094	−0.000560380 0	−0.000280190		1.00000i	$0.500089\pi$
$380$	−44.1165	−2.26313
$381$	−4.30110	−0.220352
$382$	−15.9138	−0.814219
$383$	8.65848	0.442428	0.221214	−	0.975225i	$-0.428998\pi$
$383$	8.65848	0.442428	0.221214	+	0.975225i	$0.428998\pi$
$384$	−6.23869	−0.318367
$385$	−4.15215	−0.211613
$386$	28.6188	1.45666
$387$	7.55495	0.384040
$388$	−60.5380	−3.07335
$389$	−21.5514	−1.09270	−0.546350	−	0.837557i	$-0.683983\pi$
$389$	−21.5514	−1.09270	−0.546350	+	0.837557i	$0.683983\pi$
$390$	21.4956	1.08847
$391$	7.66347	0.387558
$392$	23.5736	1.19065
$393$	−5.69493	−0.287271
$394$	9.83125	0.495291
$395$	−11.6548	−0.586416
$396$	9.07316	0.455944
$397$	−17.6986	−0.888268	−0.444134	−	0.895960i	$-0.646489\pi$
$397$	−17.6986	−0.888268	−0.444134	+	0.895960i	$0.646489\pi$
$398$	−63.0081	−3.15831
$399$	15.5919	0.780571
$400$	−39.7522	−1.98761
$401$	−20.7434	−1.03587	−0.517937	−	0.855419i	$-0.673300\pi$
$401$	−20.7434	−1.03587	−0.517937	+	0.855419i	$0.673300\pi$
$402$	9.84761	0.491154
$403$	−62.6118	−3.11891
$404$	59.6001	2.96522
$405$	−1.13791	−0.0565433
$406$	−5.27109	−0.261600
$407$	7.03700	0.348811
$408$	−59.9905	−2.96997
$409$	1.47889	0.0731262	0.0365631	−	0.999331i	$-0.488359\pi$
$409$	1.47889	0.0731262	0.0365631	+	0.999331i	$0.488359\pi$
$410$	−5.24816	−0.259188
$411$	−0.133263	−0.00657340
$412$	−1.54434	−0.0760840
$413$	3.78247	0.186123
$414$	−2.63931	−0.129715
$415$	−3.92097	−0.192473
$416$	−90.6165	−4.44284
$417$	−3.70378	−0.181375
$418$	−37.6474	−1.84139
$419$	−34.4168	−1.68137	−0.840685	−	0.541525i	$-0.817847\pi$
$419$	−34.4168	−1.68137	−0.840685	+	0.541525i	$0.817847\pi$
$420$	−11.2855	−0.550678
$421$	−12.6704	−0.617519	−0.308760	−	0.951140i	$-0.599914\pi$
$421$	−12.6704	−0.617519	−0.308760	+	0.951140i	$0.599914\pi$
$422$	39.2270	1.90954
$423$	−12.7120	−0.618078
$424$	−6.42982	−0.312260
$425$	−28.3944	−1.37733
$426$	−19.8577	−0.962109
$427$	−12.6864	−0.613937
$428$	−18.3833	−0.888592
$429$	13.0769	0.631359
$430$	22.6898	1.09420
$431$	−15.8151	−0.761786	−0.380893	−	0.924619i	$-0.624383\pi$
$431$	−15.8151	−0.761786	−0.380893	+	0.924619i	$0.624383\pi$
$432$	10.7289	0.516194
$433$	−0.469588	−0.0225670	−0.0112835	−	0.999936i	$-0.503592\pi$
$433$	−0.469588	−0.0225670	−0.0112835	+	0.999936i	$0.503592\pi$
$434$	46.1112	2.21341
$435$	−1.13791	−0.0545587
$436$	40.9169	1.95957
$437$	7.80709	0.373464
$438$	30.8129	1.47230
$439$	−2.69041	−0.128406	−0.0642032	−	0.997937i	$-0.520451\pi$
$439$	−2.69041	−0.128406	−0.0642032	+	0.997937i	$0.520451\pi$
$440$	16.2750	0.775879
$441$	−3.01140	−0.143400
$442$	−144.766	−6.88581
$443$	−27.1858	−1.29164	−0.645818	−	0.763491i	$-0.723483\pi$
$443$	−27.1858	−1.29164	−0.645818	+	0.763491i	$0.723483\pi$
$444$	19.1265	0.907705
$445$	4.26294	0.202083
$446$	−3.76497	−0.178276
$447$	−3.47160	−0.164201
$448$	23.8812	1.12828
$449$	16.1482	0.762079	0.381039	−	0.924559i	$-0.375566\pi$
$449$	16.1482	0.762079	0.381039	+	0.924559i	$0.375566\pi$
$450$	9.77907	0.460990
$451$	−3.19274	−0.150340
$452$	−86.1233	−4.05090
$453$	1.43748	0.0675387
$454$	−56.9735	−2.67390
$455$	−16.2656	−0.762541
$456$	−61.1147	−2.86196
$457$	−34.2345	−1.60142	−0.800710	−	0.599052i	$-0.795544\pi$
$457$	−34.2345	−1.60142	−0.800710	+	0.599052i	$0.795544\pi$
$458$	−56.3366	−2.63244
$459$	7.66347	0.357700
$460$	−5.65083	−0.263471
$461$	11.4102	0.531425	0.265712	−	0.964052i	$-0.414393\pi$
$461$	11.4102	0.531425	0.265712	+	0.964052i	$0.414393\pi$
$462$	−9.63065	−0.448059
$463$	−35.8584	−1.66648	−0.833240	−	0.552912i	$-0.813517\pi$
$463$	−35.8584	−1.66648	−0.833240	+	0.552912i	$0.813517\pi$
$464$	10.7289	0.498076
$465$	9.95437	0.461623
$466$	39.1402	1.81313
$467$	27.2125	1.25924	0.629622	−	0.776902i	$-0.283210\pi$
$467$	27.2125	1.25924	0.629622	+	0.776902i	$0.283210\pi$
$468$	35.5430	1.64298
$469$	−7.45161	−0.344083
$470$	−38.1779	−1.76102
$471$	−7.24011	−0.333607
$472$	−14.8259	−0.682419
$473$	13.8034	0.634682
$474$	−27.0325	−1.24165
$475$	−28.9265	−1.32724
$476$	76.0045	3.48366
$477$	0.821375	0.0376082
$478$	−10.3283	−0.472404
$479$	24.6021	1.12410	0.562050	−	0.827103i	$-0.310013\pi$
$479$	24.6021	1.12410	0.562050	+	0.827103i	$0.310013\pi$
$480$	14.4067	0.657574
$481$	27.5666	1.25693
$482$	23.7115	1.08003
$483$	1.99715	0.0908733
$484$	−38.0483	−1.72947
$485$	13.8718	0.629885
$486$	−2.63931	−0.119722
$487$	16.9386	0.767563	0.383782	−	0.923424i	$-0.374622\pi$
$487$	16.9386	0.767563	0.383782	+	0.923424i	$0.374622\pi$
$488$	49.7261	2.25100
$489$	−22.7977	−1.03095
$490$	−9.04415	−0.408573
$491$	4.83267	0.218095	0.109048	−	0.994037i	$-0.465220\pi$
$491$	4.83267	0.218095	0.109048	+	0.994037i	$0.465220\pi$
$492$	−8.67784	−0.391227
$493$	7.66347	0.345145
$494$	−147.479	−6.63539
$495$	−2.07904	−0.0934460
$496$	−93.8556	−4.21424
$497$	15.0262	0.674016
$498$	−9.09443	−0.407531
$499$	−2.59207	−0.116037	−0.0580186	−	0.998316i	$-0.518478\pi$
$499$	−2.59207	−0.116037	−0.0580186	+	0.998316i	$0.518478\pi$
$500$	49.1914	2.19990
$501$	−18.4991	−0.826479
$502$	56.0215	2.50036
$503$	13.6316	0.607802	0.303901	−	0.952704i	$-0.401711\pi$
$503$	13.6316	0.607802	0.303901	+	0.952704i	$0.401711\pi$
$504$	−15.6339	−0.696389
$505$	−13.6569	−0.607724
$506$	−4.82220	−0.214373
$507$	38.2272	1.69773
$508$	−21.3591	−0.947658
$509$	−12.1764	−0.539708	−0.269854	−	0.962901i	$-0.586975\pi$
$509$	−12.1764	−0.539708	−0.269854	+	0.962901i	$0.586975\pi$
$510$	23.0157	1.01915
$511$	−23.3159	−1.03143
$512$	32.1389	1.42035
$513$	7.80709	0.344691
$514$	32.7887	1.44625
$515$	0.353872	0.0155935
$516$	37.5176	1.65162
$517$	−23.2257	−1.02146
$518$	−20.3017	−0.892007
$519$	14.4105	0.632552
$520$	63.7552	2.79585
$521$	33.5095	1.46808	0.734040	−	0.679106i	$-0.237633\pi$
$521$	33.5095	1.46808	0.734040	+	0.679106i	$0.237633\pi$
$522$	−2.63931	−0.115520
$523$	14.7417	0.644611	0.322305	−	0.946636i	$-0.395542\pi$
$523$	14.7417	0.644611	0.322305	+	0.946636i	$0.395542\pi$
$524$	−28.2809	−1.23545
$525$	−7.39975	−0.322951
$526$	47.8584	2.08673
$527$	−67.0395	−2.92029
$528$	19.6024	0.853086
$529$	1.00000	0.0434783
$530$	2.46684	0.107153
$531$	1.89394	0.0821898
$532$	77.4289	3.35697
$533$	−12.5072	−0.541745
$534$	9.88761	0.427879
$535$	4.21239	0.182118
$536$	29.2077	1.26158
$537$	7.85751	0.339077
$538$	30.1775	1.30104
$539$	−5.50204	−0.236990
$540$	−5.65083	−0.243173
$541$	7.55303	0.324730	0.162365	−	0.986731i	$-0.448088\pi$
$541$	7.55303	0.324730	0.162365	+	0.986731i	$0.448088\pi$
$542$	53.3869	2.29316
$543$	10.1822	0.436961
$544$	−97.0246	−4.15990
$545$	−9.37579	−0.401615
$546$	−37.7269	−1.61456
$547$	−10.9149	−0.466688	−0.233344	−	0.972394i	$-0.574967\pi$
$547$	−10.9149	−0.466688	−0.233344	+	0.972394i	$0.574967\pi$
$548$	−0.661781	−0.0282699
$549$	−6.35225	−0.271107
$550$	17.8670	0.761853
$551$	7.80709	0.332593
$552$	−7.82811	−0.333187
$553$	20.4553	0.869849
$554$	−72.7722	−3.09180
$555$	−4.38269	−0.186035
$556$	−18.3928	−0.780030
$557$	12.8768	0.545606	0.272803	−	0.962070i	$-0.412049\pi$
$557$	12.8768	0.545606	0.272803	+	0.962070i	$0.412049\pi$
$558$	23.0885	0.977415
$559$	54.0732	2.28705
$560$	−24.3822	−1.03034
$561$	14.0017	0.591152
$562$	−42.2784	−1.78340
$563$	27.0936	1.14186	0.570930	−	0.820999i	$-0.306583\pi$
$563$	27.0936	1.14186	0.570930	+	0.820999i	$0.306583\pi$
$564$	−63.1273	−2.65814
$565$	19.7345	0.830235
$566$	−38.9993	−1.63926
$567$	1.99715	0.0838723
$568$	−58.8973	−2.47128
$569$	−11.0896	−0.464899	−0.232450	−	0.972608i	$-0.574674\pi$
$569$	−11.0896	−0.464899	−0.232450	+	0.972608i	$0.574674\pi$
$570$	23.4470	0.982089
$571$	28.4667	1.19129	0.595647	−	0.803246i	$-0.296896\pi$
$571$	28.4667	1.19129	0.595647	+	0.803246i	$0.296896\pi$
$572$	64.9395	2.71526
$573$	6.02951	0.251886
$574$	9.21104	0.384462
$575$	−3.70516	−0.154516
$576$	11.9577	0.498236
$577$	−1.56409	−0.0651138	−0.0325569	−	0.999470i	$-0.510365\pi$
$577$	−1.56409	−0.0651138	−0.0325569	+	0.999470i	$0.510365\pi$
$578$	−110.135	−4.58102
$579$	−10.8433	−0.450632
$580$	−5.65083	−0.234638
$581$	6.88169	0.285501
$582$	32.1747	1.33369
$583$	1.50071	0.0621530
$584$	91.3901	3.78175
$585$	−8.14439	−0.336729
$586$	31.0735	1.28363
$587$	26.8963	1.11013	0.555065	−	0.831807i	$-0.312693\pi$
$587$	26.8963	1.11013	0.555065	+	0.831807i	$0.312693\pi$
$588$	−14.9545	−0.616714
$589$	−68.2959	−2.81408
$590$	5.68806	0.234174
$591$	−3.72493	−0.153223
$592$	41.3226	1.69835
$593$	−10.6925	−0.439087	−0.219543	−	0.975603i	$-0.570457\pi$
$593$	−10.6925	−0.439087	−0.219543	+	0.975603i	$0.570457\pi$
$594$	−4.82220	−0.197857
$595$	−17.4158	−0.713979
$596$	−17.2399	−0.706172
$597$	23.8729	0.977053
$598$	−18.8904	−0.772486
$599$	13.6418	0.557390	0.278695	−	0.960380i	$-0.410098\pi$
$599$	13.6418	0.557390	0.278695	+	0.960380i	$0.410098\pi$
$600$	29.0044	1.18410
$601$	28.6114	1.16708	0.583542	−	0.812083i	$-0.301666\pi$
$601$	28.6114	1.16708	0.583542	+	0.812083i	$0.301666\pi$
$602$	−39.8229	−1.62306
$603$	−3.73113	−0.151943
$604$	7.13848	0.290461
$605$	8.71847	0.354456
$606$	−31.6763	−1.28676
$607$	19.7862	0.803096	0.401548	−	0.915838i	$-0.368472\pi$
$607$	19.7862	0.803096	0.401548	+	0.915838i	$0.368472\pi$
$608$	−98.8430	−4.00861
$609$	1.99715	0.0809285
$610$	−19.0777	−0.772434
$611$	−90.9838	−3.68081
$612$	38.0565	1.53834
$613$	−8.54960	−0.345315	−0.172657	−	0.984982i	$-0.555235\pi$
$613$	−8.54960	−0.345315	−0.172657	+	0.984982i	$0.555235\pi$
$614$	−29.0304	−1.17157
$615$	1.98846	0.0801824
$616$	−28.5642	−1.15088
$617$	−32.6012	−1.31247	−0.656237	−	0.754555i	$-0.727853\pi$
$617$	−32.6012	−1.31247	−0.656237	+	0.754555i	$0.727853\pi$
$618$	0.820784	0.0330168
$619$	21.9824	0.883549	0.441774	−	0.897126i	$-0.354349\pi$
$619$	21.9824	0.883549	0.441774	+	0.897126i	$0.354349\pi$
$620$	49.4331	1.98528
$621$	1.00000	0.0401286
$622$	−60.3669	−2.42049
$623$	−7.48188	−0.299755
$624$	76.7901	3.07406
$625$	7.25399	0.290159
$626$	−29.2025	−1.16717
$627$	14.2641	0.569653
$628$	−35.9542	−1.43473
$629$	29.5160	1.17688
$630$	5.99804	0.238968
$631$	−18.3394	−0.730080	−0.365040	−	0.930992i	$-0.618945\pi$
$631$	−18.3394	−0.730080	−0.365040	+	0.930992i	$0.618945\pi$
$632$	−80.1776	−3.18929
$633$	−14.8626	−0.590734
$634$	−56.7500	−2.25383
$635$	4.89427	0.194223
$636$	4.07892	0.161740
$637$	−21.5536	−0.853984
$638$	−4.82220	−0.190913
$639$	7.52382	0.297638
$640$	7.09907	0.280615
$641$	45.0095	1.77777	0.888885	−	0.458131i	$-0.151481\pi$
$641$	45.0095	1.77777	0.888885	+	0.458131i	$0.151481\pi$
$642$	9.77038	0.385606
$643$	2.90860	0.114704	0.0573520	−	0.998354i	$-0.481734\pi$
$643$	2.90860	0.114704	0.0573520	+	0.998354i	$0.481734\pi$
$644$	9.91777	0.390815
$645$	−8.59686	−0.338501
$646$	−157.908	−6.21282
$647$	−27.1195	−1.06618	−0.533089	−	0.846059i	$-0.678969\pi$
$647$	−27.1195	−1.06618	−0.533089	+	0.846059i	$0.678969\pi$
$648$	−7.82811	−0.307517
$649$	3.46035	0.135831
$650$	69.9919	2.74531
$651$	−17.4709	−0.684739
$652$	−113.213	−4.43376
$653$	3.77083	0.147564	0.0737820	−	0.997274i	$-0.476493\pi$
$653$	3.77083	0.147564	0.0737820	+	0.997274i	$0.476493\pi$
$654$	−21.7465	−0.850357
$655$	6.48033	0.253207
$656$	−18.7483	−0.732000
$657$	−11.6746	−0.455469
$658$	67.0061	2.61217
$659$	−45.4659	−1.77110	−0.885550	−	0.464544i	$-0.846218\pi$
$659$	−45.4659	−1.77110	−0.885550	+	0.464544i	$0.846218\pi$
$660$	−10.3245	−0.401879
$661$	−14.0990	−0.548388	−0.274194	−	0.961674i	$-0.588411\pi$
$661$	−14.0990	−0.548388	−0.274194	+	0.961674i	$0.588411\pi$
$662$	43.4893	1.69026
$663$	54.8499	2.13019
$664$	−26.9738	−1.04679
$665$	−17.7422	−0.688013
$666$	−10.1654	−0.393900
$667$	1.00000	0.0387202
$668$	−91.8659	−3.55440
$669$	1.42650	0.0551515
$670$	−11.2057	−0.432914
$671$	−11.6060	−0.448045
$672$	−25.2852	−0.975399
$673$	42.0594	1.62127	0.810636	−	0.585551i	$-0.199122\pi$
$673$	42.0594	1.62127	0.810636	+	0.585551i	$0.199122\pi$
$674$	79.9710	3.08037
$675$	−3.70516	−0.142612
$676$	189.835	7.30135
$677$	9.52489	0.366071	0.183036	−	0.983106i	$-0.441408\pi$
$677$	9.52489	0.366071	0.183036	+	0.983106i	$0.441408\pi$
$678$	45.7728	1.75789
$679$	−24.3464	−0.934328
$680$	68.2638	2.61780
$681$	21.5865	0.827197
$682$	42.1843	1.61532
$683$	43.9788	1.68280	0.841401	−	0.540411i	$-0.181731\pi$
$683$	43.9788	1.68280	0.841401	+	0.540411i	$0.181731\pi$
$684$	38.7697	1.48240
$685$	0.151642	0.00579394
$686$	52.7710	2.01481
$687$	21.3452	0.814369
$688$	81.0563	3.09024
$689$	5.87885	0.223966
$690$	3.00330	0.114334
$691$	−26.2731	−0.999476	−0.499738	−	0.866177i	$-0.666570\pi$
$691$	−26.2731	−0.999476	−0.499738	+	0.866177i	$0.666570\pi$
$692$	71.5622	2.72039
$693$	3.64893	0.138611
$694$	−85.6073	−3.24961
$695$	4.21457	0.159868
$696$	−7.82811	−0.296724
$697$	−13.3916	−0.507244
$698$	32.9665	1.24780
$699$	−14.8297	−0.560911
$700$	−36.7469	−1.38890
$701$	28.9579	1.09372	0.546862	−	0.837222i	$-0.315822\pi$
$701$	28.9579	1.09372	0.546862	+	0.837222i	$0.315822\pi$
$702$	−18.8904	−0.712972
$703$	30.0692	1.13408
$704$	21.8475	0.823409
$705$	14.4651	0.544788
$706$	39.2110	1.47573
$707$	23.9692	0.901455
$708$	9.40523	0.353470
$709$	−29.0553	−1.09119	−0.545597	−	0.838048i	$-0.683697\pi$
$709$	−29.0553	−1.09119	−0.545597	+	0.838048i	$0.683697\pi$
$710$	22.5963	0.848025
$711$	10.2423	0.384115
$712$	29.3263	1.09905
$713$	−8.74793	−0.327613
$714$	−40.3949	−1.51174
$715$	−14.8804	−0.556494
$716$	39.0202	1.45825
$717$	3.91325	0.146143
$718$	−25.6257	−0.956341
$719$	−34.6009	−1.29040	−0.645199	−	0.764015i	$-0.723225\pi$
$719$	−34.6009	−1.29040	−0.645199	+	0.764015i	$0.723225\pi$
$720$	−12.2085	−0.454985
$721$	−0.621081	−0.0231303
$722$	−110.721	−4.12060
$723$	−8.98398	−0.334118
$724$	50.5646	1.87922
$725$	−3.70516	−0.137606
$726$	20.2219	0.750506
$727$	−38.2126	−1.41723	−0.708613	−	0.705597i	$-0.750679\pi$
$727$	−38.2126	−1.41723	−0.708613	+	0.705597i	$0.750679\pi$
$728$	−111.897	−4.14717
$729$	1.00000	0.0370370
$730$	−35.0624	−1.29772
$731$	57.8971	2.14140
$732$	−31.5451	−1.16594
$733$	45.5939	1.68405	0.842025	−	0.539439i	$-0.181364\pi$
$733$	45.5939	1.68405	0.842025	+	0.539439i	$0.181364\pi$
$734$	58.5396	2.16074
$735$	3.42671	0.126396
$736$	−12.6607	−0.466679
$737$	−6.81703	−0.251108
$738$	4.61210	0.169774
$739$	31.8440	1.17140	0.585700	−	0.810528i	$-0.300820\pi$
$739$	31.8440	1.17140	0.585700	+	0.810528i	$0.300820\pi$
$740$	−21.7643	−0.800071
$741$	55.8778	2.05272
$742$	−4.32955	−0.158943
$743$	1.55885	0.0571885	0.0285943	−	0.999591i	$-0.490897\pi$
$743$	1.55885	0.0571885	0.0285943	+	0.999591i	$0.490897\pi$
$744$	68.4798	2.51059
$745$	3.95038	0.144731
$746$	−7.72899	−0.282978
$747$	3.44576	0.126074
$748$	69.5319	2.54234
$749$	−7.39317	−0.270141
$750$	−26.1442	−0.954652
$751$	1.74830	0.0637962	0.0318981	−	0.999491i	$-0.489845\pi$
$751$	1.74830	0.0637962	0.0318981	+	0.999491i	$0.489845\pi$
$752$	−136.386	−4.97347
$753$	−21.2258	−0.773511
$754$	−18.8904	−0.687948
$755$	−1.63573	−0.0595301
$756$	9.91777	0.360706
$757$	−26.0434	−0.946565	−0.473282	−	0.880911i	$-0.656931\pi$
$757$	−26.0434	−0.946565	−0.473282	+	0.880911i	$0.656931\pi$
$758$	0.0287934	0.00104582
$759$	1.82707	0.0663184
$760$	69.5432	2.52260
$761$	−48.3248	−1.75177	−0.875887	−	0.482517i	$-0.839723\pi$
$761$	−48.3248	−1.75177	−0.875887	+	0.482517i	$0.839723\pi$
$762$	11.3519	0.411238
$763$	16.4554	0.595727
$764$	29.9423	1.08328
$765$	−8.72034	−0.315285
$766$	−22.8524	−0.825692
$767$	13.5555	0.489461
$768$	−7.44951	−0.268811
$769$	−16.5182	−0.595661	−0.297830	−	0.954619i	$-0.596263\pi$
$769$	−16.5182	−0.595661	−0.297830	+	0.954619i	$0.596263\pi$
$770$	10.9588	0.394929
$771$	−12.4232	−0.447411
$772$	−53.8474	−1.93801
$773$	−55.1574	−1.98387	−0.991937	−	0.126728i	$-0.959552\pi$
$773$	−55.1574	−1.98387	−0.991937	+	0.126728i	$0.959552\pi$
$774$	−19.9399	−0.716724
$775$	32.4125	1.16429
$776$	95.4292	3.42571
$777$	7.69206	0.275951
$778$	56.8809	2.03928
$779$	−13.6426	−0.488797
$780$	−40.4448	−1.44816
$781$	13.7465	0.491890
$782$	−20.2263	−0.723290
$783$	1.00000	0.0357371
$784$	−32.3090	−1.15389
$785$	8.23861	0.294048
$786$	15.0307	0.536128
$787$	−22.2422	−0.792850	−0.396425	−	0.918067i	$-0.629749\pi$
$787$	−22.2422	−0.792850	−0.396425	+	0.918067i	$0.629749\pi$
$788$	−18.4979	−0.658959
$789$	−18.1329	−0.645549
$790$	30.7606	1.09441
$791$	−34.6359	−1.23151
$792$	−14.3025	−0.508217
$793$	−45.4651	−1.61451
$794$	46.7121	1.65775
$795$	−0.934652	−0.0331487
$796$	118.552	4.20197
$797$	−10.8748	−0.385204	−0.192602	−	0.981277i	$-0.561693\pi$
$797$	−10.8748	−0.385204	−0.192602	+	0.981277i	$0.561693\pi$
$798$	−41.1519	−1.45676
$799$	−97.4179	−3.44640
$800$	46.9098	1.65851
$801$	−3.74628	−0.132368
$802$	54.7482	1.93323
$803$	−21.3303	−0.752730
$804$	−18.5287	−0.653455
$805$	−2.27258	−0.0800978
$806$	165.252	5.82075
$807$	−11.4338	−0.402491
$808$	−93.9508	−3.30518
$809$	13.7637	0.483905	0.241952	−	0.970288i	$-0.422212\pi$
$809$	13.7637	0.483905	0.241952	+	0.970288i	$0.422212\pi$
$810$	3.00330	0.105525
$811$	12.2997	0.431901	0.215951	−	0.976404i	$-0.430715\pi$
$811$	12.2997	0.431901	0.215951	+	0.976404i	$0.430715\pi$
$812$	9.91777	0.348045
$813$	−20.2276	−0.709412
$814$	−18.5728	−0.650977
$815$	25.9418	0.908702
$816$	82.2205	2.87829
$817$	58.9822	2.06352
$818$	−3.90324	−0.136474
$819$	14.2942	0.499480
$820$	9.87462	0.344837
$821$	29.6079	1.03332	0.516661	−	0.856190i	$-0.327175\pi$
$821$	29.6079	1.03332	0.516661	+	0.856190i	$0.327175\pi$
$822$	0.351724	0.0122678
$823$	−5.38068	−0.187559	−0.0937794	−	0.995593i	$-0.529895\pi$
$823$	−5.38068	−0.187559	−0.0937794	+	0.995593i	$0.529895\pi$
$824$	2.43442	0.0848069
$825$	−6.76958	−0.235687
$826$	−9.98312	−0.347357
$827$	15.9369	0.554180	0.277090	−	0.960844i	$-0.410630\pi$
$827$	15.9369	0.554180	0.277090	+	0.960844i	$0.410630\pi$
$828$	4.96597	0.172579
$829$	−16.5493	−0.574782	−0.287391	−	0.957813i	$-0.592788\pi$
$829$	−16.5493	−0.574782	−0.287391	+	0.957813i	$0.592788\pi$
$830$	10.3487	0.359207
$831$	27.5724	0.956477
$832$	85.5849	2.96712
$833$	−23.0778	−0.799598
$834$	9.77543	0.338495
$835$	21.0503	0.728477
$836$	70.8350	2.44988
$837$	−8.74793	−0.302373
$838$	90.8366	3.13790
$839$	−9.07828	−0.313417	−0.156708	−	0.987645i	$-0.550088\pi$
$839$	−9.07828	−0.313417	−0.156708	+	0.987645i	$0.550088\pi$
$840$	17.7900	0.613813
$841$	1.00000	0.0344828
$842$	33.4412	1.15246
$843$	16.0187	0.551714
$844$	−73.8070	−2.54054
$845$	−43.4992	−1.49642
$846$	33.5509	1.15350
$847$	−15.3018	−0.525775
$848$	8.81245	0.302621
$849$	14.7763	0.507122
$850$	74.9416	2.57047
$851$	3.85152	0.132028
$852$	37.3631	1.28004
$853$	−12.8921	−0.441417	−0.220708	−	0.975340i	$-0.570837\pi$
$853$	−12.8921	−0.441417	−0.220708	+	0.975340i	$0.570837\pi$
$854$	33.4833	1.14577
$855$	−8.88377	−0.303819
$856$	28.9786	0.990469
$857$	−55.4025	−1.89251	−0.946256	−	0.323418i	$-0.895168\pi$
$857$	−55.4025	−1.89251	−0.946256	+	0.323418i	$0.895168\pi$
$858$	−34.5141	−1.17829
$859$	7.14952	0.243939	0.121969	−	0.992534i	$-0.461079\pi$
$859$	7.14952	0.243939	0.121969	+	0.992534i	$0.461079\pi$
$860$	−42.6917	−1.45578
$861$	−3.48994	−0.118937
$862$	41.7409	1.42170
$863$	24.2776	0.826418	0.413209	−	0.910636i	$-0.364408\pi$
$863$	24.2776	0.826418	0.413209	+	0.910636i	$0.364408\pi$
$864$	−12.6607	−0.430725
$865$	−16.3979	−0.557546
$866$	1.23939	0.0421162
$867$	41.7287	1.41718
$868$	−86.7600	−2.94483
$869$	18.7133	0.634806
$870$	3.00330	0.101821
$871$	−26.7049	−0.904860
$872$	−64.4995	−2.18423
$873$	−12.1906	−0.412588
$874$	−20.6053	−0.696986
$875$	19.7831	0.668792
$876$	−57.9757	−1.95882
$877$	−13.5313	−0.456919	−0.228460	−	0.973553i	$-0.573369\pi$
$877$	−13.5313	−0.456919	−0.228460	+	0.973553i	$0.573369\pi$
$878$	7.10084	0.239642
$879$	−11.7733	−0.397104
$880$	−22.3058	−0.751929
$881$	−3.41548	−0.115071	−0.0575353	−	0.998343i	$-0.518324\pi$
$881$	−3.41548	−0.115071	−0.0575353	+	0.998343i	$0.518324\pi$
$882$	7.94803	0.267624
$883$	−40.0276	−1.34704	−0.673519	−	0.739170i	$-0.735218\pi$
$883$	−40.0276	−1.34704	−0.673519	+	0.739170i	$0.735218\pi$
$884$	272.383	9.16122
$885$	−2.15513	−0.0724440
$886$	71.7518	2.41055
$887$	30.4519	1.02247	0.511237	−	0.859440i	$-0.329188\pi$
$887$	30.4519	1.02247	0.511237	+	0.859440i	$0.329188\pi$
$888$	−30.1501	−1.01177
$889$	−8.58993	−0.288097
$890$	−11.2512	−0.377142
$891$	1.82707	0.0612091
$892$	7.08393	0.237187
$893$	−99.2436	−3.32106
$894$	9.16264	0.306445
$895$	−8.94115	−0.298870
$896$	−12.4596	−0.416245
$897$	7.15732	0.238976
$898$	−42.6200	−1.42225
$899$	−8.74793	−0.291760
$900$	−18.3997	−0.613323
$901$	6.29458	0.209703
$902$	8.42663	0.280576
$903$	15.0883	0.502109
$904$	135.761	4.51533
$905$	−11.5865	−0.385148
$906$	−3.79396	−0.126046
$907$	−11.4541	−0.380328	−0.190164	−	0.981752i	$-0.560902\pi$
$907$	−11.4541	−0.380328	−0.190164	+	0.981752i	$0.560902\pi$
$908$	107.198	3.55749
$909$	12.0017	0.398072
$910$	42.9299	1.42311
$911$	31.0278	1.02800	0.513998	−	0.857791i	$-0.328164\pi$
$911$	31.0278	1.02800	0.513998	+	0.857791i	$0.328164\pi$
$912$	83.7614	2.77362
$913$	6.29564	0.208355
$914$	90.3554	2.98869
$915$	7.22829	0.238960
$916$	105.999	3.50232
$917$	−11.3736	−0.375590
$918$	−20.2263	−0.667567
$919$	−15.7969	−0.521092	−0.260546	−	0.965461i	$-0.583903\pi$
$919$	−15.7969	−0.521092	−0.260546	+	0.965461i	$0.583903\pi$
$920$	8.90770	0.293678
$921$	10.9992	0.362437
$922$	−30.1150	−0.991784
$923$	53.8504	1.77251
$924$	18.1204	0.596119
$925$	−14.2705	−0.469211
$926$	94.6414	3.11011
$927$	−0.310984	−0.0102141
$928$	−12.6607	−0.415607
$929$	−40.0724	−1.31473	−0.657366	−	0.753572i	$-0.728329\pi$
$929$	−40.0724	−1.31473	−0.657366	+	0.753572i	$0.728329\pi$
$930$	−26.2727	−0.861516
$931$	−23.5103	−0.770519
$932$	−73.6438	−2.41228
$933$	22.8722	0.748802
$934$	−71.8223	−2.35010
$935$	−15.9327	−0.521054
$936$	−56.0283	−1.83134
$937$	50.5610	1.65176	0.825878	−	0.563849i	$-0.190680\pi$
$937$	50.5610	1.65176	0.825878	+	0.563849i	$0.190680\pi$
$938$	19.6671	0.642154
$939$	11.0644	0.361074
$940$	71.8333	2.34294
$941$	13.7738	0.449012	0.224506	−	0.974473i	$-0.427923\pi$
$941$	13.7738	0.449012	0.224506	+	0.974473i	$0.427923\pi$
$942$	19.1089	0.622602
$943$	−1.74746	−0.0569052
$944$	20.3198	0.661355
$945$	−2.27258	−0.0739269
$946$	−36.4315	−1.18449
$947$	−9.78085	−0.317835	−0.158917	−	0.987292i	$-0.550800\pi$
$947$	−9.78085	−0.317835	−0.158917	+	0.987292i	$0.550800\pi$
$948$	50.8628	1.65195
$949$	−83.5588	−2.71243
$950$	76.3460	2.47699
$951$	21.5018	0.697244
$952$	−119.810	−3.88306
$953$	−11.2357	−0.363961	−0.181981	−	0.983302i	$-0.558251\pi$
$953$	−11.2357	−0.363961	−0.181981	+	0.983302i	$0.558251\pi$
$954$	−2.16787	−0.0701872
$955$	−6.86105	−0.222018
$956$	19.4330	0.628510
$957$	1.82707	0.0590608
$958$	−64.9327	−2.09788
$959$	−0.266147	−0.00859432
$960$	−13.6068	−0.439157
$961$	45.5263	1.46859
$962$	−72.7568	−2.34577
$963$	−3.70187	−0.119291
$964$	−44.6142	−1.43693
$965$	12.3387	0.397197
$966$	−5.27109	−0.169595
$967$	17.9352	0.576758	0.288379	−	0.957516i	$-0.406884\pi$
$967$	17.9352	0.576758	0.288379	+	0.957516i	$0.406884\pi$
$968$	59.9776	1.92775
$969$	59.8294	1.92200
$970$	−36.6120	−1.17554
$971$	4.44805	0.142745	0.0713723	−	0.997450i	$-0.477262\pi$
$971$	4.44805	0.142745	0.0713723	+	0.997450i	$0.477262\pi$
$972$	4.96597	0.159283
$973$	−7.39699	−0.237137
$974$	−44.7064	−1.43248
$975$	−26.5190	−0.849288
$976$	−68.1526	−2.18151
$977$	−5.96596	−0.190868	−0.0954340	−	0.995436i	$-0.530424\pi$
$977$	−5.96596	−0.190868	−0.0954340	+	0.995436i	$0.530424\pi$
$978$	60.1704	1.92404
$979$	−6.84472	−0.218758
$980$	17.0169	0.543586
$981$	8.23947	0.263066
$982$	−12.7549	−0.407026
$983$	46.8322	1.49372	0.746858	−	0.664984i	$-0.231562\pi$
$983$	46.8322	1.49372	0.746858	+	0.664984i	$0.231562\pi$
$984$	13.6793	0.436081
$985$	4.23864	0.135054
$986$	−20.2263	−0.644136
$987$	−25.3877	−0.808100
$988$	277.487	8.82805
$989$	7.55495	0.240233
$990$	5.48724	0.174396
$991$	24.5843	0.780945	0.390472	−	0.920615i	$-0.372312\pi$
$991$	24.5843	0.780945	0.390472	+	0.920615i	$0.372312\pi$
$992$	110.755	3.51647
$993$	−16.4775	−0.522898
$994$	−39.6588	−1.25790
$995$	−27.1653	−0.861196
$996$	17.1115	0.542199
$997$	29.1589	0.923471	0.461736	−	0.887018i	$-0.347227\pi$
$997$	29.1589	0.923471	0.461736	+	0.887018i	$0.347227\pi$
$998$	6.84129	0.216557
$999$	3.85152	0.121857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.2.a.o.1.2	✓	20
3.2	odd	2		6003.2.a.s.1.19		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.2	✓	20	1.1	even	1	trivial
6003.2.a.s.1.19		20	3.2	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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.63931 q^{2} +1.00000 q^{3} +4.96597 q^{4} -1.13791 q^{5} -2.63931 q^{6} +1.99715 q^{7} -7.82811 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.63931 q^{2} +1.00000 q^{3} +4.96597 q^{4} -1.13791 q^{5} -2.63931 q^{6} +1.99715 q^{7} -7.82811 q^{8} +1.00000 q^{9} +3.00330 q^{10} +1.82707 q^{11} +4.96597 q^{12} +7.15732 q^{13} -5.27109 q^{14} -1.13791 q^{15} +10.7289 q^{16} +7.66347 q^{17} -2.63931 q^{18} +7.80709 q^{19} -5.65083 q^{20} +1.99715 q^{21} -4.82220 q^{22} +1.00000 q^{23} -7.82811 q^{24} -3.70516 q^{25} -18.8904 q^{26} +1.00000 q^{27} +9.91777 q^{28} +1.00000 q^{29} +3.00330 q^{30} -8.74793 q^{31} -12.6607 q^{32} +1.82707 q^{33} -20.2263 q^{34} -2.27258 q^{35} +4.96597 q^{36} +3.85152 q^{37} -20.6053 q^{38} +7.15732 q^{39} +8.90770 q^{40} -1.74746 q^{41} -5.27109 q^{42} +7.55495 q^{43} +9.07316 q^{44} -1.13791 q^{45} -2.63931 q^{46} -12.7120 q^{47} +10.7289 q^{48} -3.01140 q^{49} +9.77907 q^{50} +7.66347 q^{51} +35.5430 q^{52} +0.821375 q^{53} -2.63931 q^{54} -2.07904 q^{55} -15.6339 q^{56} +7.80709 q^{57} -2.63931 q^{58} +1.89394 q^{59} -5.65083 q^{60} -6.35225 q^{61} +23.0885 q^{62} +1.99715 q^{63} +11.9577 q^{64} -8.14439 q^{65} -4.82220 q^{66} -3.73113 q^{67} +38.0565 q^{68} +1.00000 q^{69} +5.99804 q^{70} +7.52382 q^{71} -7.82811 q^{72} -11.6746 q^{73} -10.1654 q^{74} -3.70516 q^{75} +38.7697 q^{76} +3.64893 q^{77} -18.8904 q^{78} +10.2423 q^{79} -12.2085 q^{80} +1.00000 q^{81} +4.61210 q^{82} +3.44576 q^{83} +9.91777 q^{84} -8.72034 q^{85} -19.9399 q^{86} +1.00000 q^{87} -14.3025 q^{88} -3.74628 q^{89} +3.00330 q^{90} +14.2942 q^{91} +4.96597 q^{92} -8.74793 q^{93} +33.5509 q^{94} -8.88377 q^{95} -12.6607 q^{96} -12.1906 q^{97} +7.94803 q^{98} +1.82707 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) \) 20 * q + 2 * q^2 + 20 * q^3 + 30 * q^4 - q^5 + 2 * q^6 + 9 * q^7 + 6 * q^8 + 20 * q^9	\( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9} + 7 q^{10} + 30 q^{12} + 21 q^{13} - q^{14} - q^{15} + 58 q^{16} - 4 q^{17} + 2 q^{18} + 7 q^{19} - 20 q^{20} + 9 q^{21} + 7 q^{22} + 20 q^{23} + 6 q^{24} + 47 q^{25} + 8 q^{26} + 20 q^{27} + 11 q^{28} + 20 q^{29} + 7 q^{30} + 28 q^{31} + 14 q^{32} + 16 q^{34} + 9 q^{35} + 30 q^{36} + 14 q^{37} - 20 q^{38} + 21 q^{39} + 34 q^{40} + 7 q^{41} - q^{42} + 3 q^{43} - q^{44} - q^{45} + 2 q^{46} + 3 q^{47} + 58 q^{48} + 35 q^{49} - 24 q^{50} - 4 q^{51} + 73 q^{52} - 19 q^{53} + 2 q^{54} + 29 q^{55} - 30 q^{56} + 7 q^{57} + 2 q^{58} + 20 q^{59} - 20 q^{60} + 15 q^{61} + 12 q^{62} + 9 q^{63} + 82 q^{64} - 28 q^{65} + 7 q^{66} + 20 q^{67} - 23 q^{68} + 20 q^{69} - 24 q^{70} + 63 q^{71} + 6 q^{72} + 19 q^{73} + 16 q^{74} + 47 q^{75} - 44 q^{76} - 7 q^{77} + 8 q^{78} + 32 q^{79} - 56 q^{80} + 20 q^{81} - 20 q^{82} - 21 q^{83} + 11 q^{84} + 4 q^{85} - 6 q^{86} + 20 q^{87} + 55 q^{88} - 13 q^{89} + 7 q^{90} + 70 q^{91} + 30 q^{92} + 28 q^{93} - 12 q^{94} + 9 q^{95} + 14 q^{96} - 9 q^{97} + 31 q^{98}+O(q^{100}) \) 20 * q + 2 * q^2 + 20 * q^3 + 30 * q^4 - q^5 + 2 * q^6 + 9 * q^7 + 6 * q^8 + 20 * q^9 + 7 * q^10 + 30 * q^12 + 21 * q^13 - q^14 - q^15 + 58 * q^16 - 4 * q^17 + 2 * q^18 + 7 * q^19 - 20 * q^20 + 9 * q^21 + 7 * q^22 + 20 * q^23 + 6 * q^24 + 47 * q^25 + 8 * q^26 + 20 * q^27 + 11 * q^28 + 20 * q^29 + 7 * q^30 + 28 * q^31 + 14 * q^32 + 16 * q^34 + 9 * q^35 + 30 * q^36 + 14 * q^37 - 20 * q^38 + 21 * q^39 + 34 * q^40 + 7 * q^41 - q^42 + 3 * q^43 - q^44 - q^45 + 2 * q^46 + 3 * q^47 + 58 * q^48 + 35 * q^49 - 24 * q^50 - 4 * q^51 + 73 * q^52 - 19 * q^53 + 2 * q^54 + 29 * q^55 - 30 * q^56 + 7 * q^57 + 2 * q^58 + 20 * q^59 - 20 * q^60 + 15 * q^61 + 12 * q^62 + 9 * q^63 + 82 * q^64 - 28 * q^65 + 7 * q^66 + 20 * q^67 - 23 * q^68 + 20 * q^69 - 24 * q^70 + 63 * q^71 + 6 * q^72 + 19 * q^73 + 16 * q^74 + 47 * q^75 - 44 * q^76 - 7 * q^77 + 8 * q^78 + 32 * q^79 - 56 * q^80 + 20 * q^81 - 20 * q^82 - 21 * q^83 + 11 * q^84 + 4 * q^85 - 6 * q^86 + 20 * q^87 + 55 * q^88 - 13 * q^89 + 7 * q^90 + 70 * q^91 + 30 * q^92 + 28 * q^93 - 12 * q^94 + 9 * q^95 + 14 * q^96 - 9 * q^97 + 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more