LMFDB - Embedded newform 4010.2.a.m.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, 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("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
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} - 4 x^{19} - 35 x^{18} + 154 x^{17} + 460 x^{16} - 2392 x^{15} - 2591 x^{14} + 19157 x^{13} + \cdots + 104 $ x^20 - 4x^19 - 35x^18 + 154x^17 + 460x^16 - 2392x^15 - 2591x^14 + 19157x^13 + 2776x^12 - 83577x^11 + 34362x^10 + 190617x^9 - 150697x^8 - 189098x^7 + 211360x^6 + 37365x^5 - 68876x^4 - 8984x^3 + 7920x^2 + 1972*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$0.622776$ of defining polynomial
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.622776 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.622776 q^{6} +3.30687 q^{7} -1.00000 q^{8} -2.61215 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.622776 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.622776 q^{6} +3.30687 q^{7} -1.00000 q^{8} -2.61215 q^{9} -1.00000 q^{10} +1.88207 q^{11} +0.622776 q^{12} +1.73738 q^{13} -3.30687 q^{14} +0.622776 q^{15} +1.00000 q^{16} +5.95406 q^{17} +2.61215 q^{18} +2.09140 q^{19} +1.00000 q^{20} +2.05944 q^{21} -1.88207 q^{22} +5.93930 q^{23} -0.622776 q^{24} +1.00000 q^{25} -1.73738 q^{26} -3.49511 q^{27} +3.30687 q^{28} +4.40127 q^{29} -0.622776 q^{30} +2.17180 q^{31} -1.00000 q^{32} +1.17211 q^{33} -5.95406 q^{34} +3.30687 q^{35} -2.61215 q^{36} -2.94159 q^{37} -2.09140 q^{38} +1.08200 q^{39} -1.00000 q^{40} -10.4206 q^{41} -2.05944 q^{42} -6.51163 q^{43} +1.88207 q^{44} -2.61215 q^{45} -5.93930 q^{46} -2.61949 q^{47} +0.622776 q^{48} +3.93538 q^{49} -1.00000 q^{50} +3.70805 q^{51} +1.73738 q^{52} +11.8216 q^{53} +3.49511 q^{54} +1.88207 q^{55} -3.30687 q^{56} +1.30247 q^{57} -4.40127 q^{58} +3.12873 q^{59} +0.622776 q^{60} -8.50657 q^{61} -2.17180 q^{62} -8.63804 q^{63} +1.00000 q^{64} +1.73738 q^{65} -1.17211 q^{66} +5.94457 q^{67} +5.95406 q^{68} +3.69886 q^{69} -3.30687 q^{70} -10.5321 q^{71} +2.61215 q^{72} +3.03116 q^{73} +2.94159 q^{74} +0.622776 q^{75} +2.09140 q^{76} +6.22375 q^{77} -1.08200 q^{78} +6.13016 q^{79} +1.00000 q^{80} +5.65977 q^{81} +10.4206 q^{82} -9.86206 q^{83} +2.05944 q^{84} +5.95406 q^{85} +6.51163 q^{86} +2.74101 q^{87} -1.88207 q^{88} -11.4350 q^{89} +2.61215 q^{90} +5.74530 q^{91} +5.93930 q^{92} +1.35255 q^{93} +2.61949 q^{94} +2.09140 q^{95} -0.622776 q^{96} +12.2492 q^{97} -3.93538 q^{98} -4.91624 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) $ 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 - 9 * q^13 - 11 * q^14 + 4 * q^15 + 20 * q^16 - 11 * q^17 - 26 * q^18 + 17 * q^19 + 20 * q^20 - 2 * q^21 - 10 * q^22 - 3 * q^23 - 4 * q^24 + 20 * q^25 + 9 * q^26 - 2 * q^27 + 11 * q^28 + 6 * q^29 - 4 * q^30 + 28 * q^31 - 20 * q^32 + 2 * q^33 + 11 * q^34 + 11 * q^35 + 26 * q^36 + 33 * q^37 - 17 * q^38 + 36 * q^39 - 20 * q^40 + 32 * q^41 + 2 * q^42 + 30 * q^43 + 10 * q^44 + 26 * q^45 + 3 * q^46 + 13 * q^47 + 4 * q^48 + 43 * q^49 - 20 * q^50 + 43 * q^51 - 9 * q^52 + 2 * q^54 + 10 * q^55 - 11 * q^56 + 19 * q^57 - 6 * q^58 + 52 * q^59 + 4 * q^60 + 25 * q^61 - 28 * q^62 + 16 * q^63 + 20 * q^64 - 9 * q^65 - 2 * q^66 + 40 * q^67 - 11 * q^68 + 39 * q^69 - 11 * q^70 + 25 * q^71 - 26 * q^72 - 5 * q^73 - 33 * q^74 + 4 * q^75 + 17 * q^76 - 9 * q^77 - 36 * q^78 + 40 * q^79 + 20 * q^80 + 48 * q^81 - 32 * q^82 + 10 * q^83 - 2 * q^84 - 11 * q^85 - 30 * q^86 + 10 * q^87 - 10 * q^88 + 17 * q^89 - 26 * q^90 + 88 * q^91 - 3 * q^92 - 4 * q^93 - 13 * q^94 + 17 * q^95 - 4 * q^96 - 2 * q^97 - 43 * q^98 + 50 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.622776	0.359560	0.179780	−	0.983707i	$-0.442461\pi$
$3$	0.622776	0.359560	0.179780	+	0.983707i	$0.442461\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.622776	−0.254247
$7$	3.30687	1.24988	0.624939	−	0.780673i	$-0.285124\pi$
$7$	3.30687	1.24988	0.624939	+	0.780673i	$0.285124\pi$
$8$	−1.00000	−0.353553
$9$	−2.61215	−0.870717
$10$	−1.00000	−0.316228
$11$	1.88207	0.567465	0.283732	−	0.958904i	$-0.408427\pi$
$11$	1.88207	0.567465	0.283732	+	0.958904i	$0.408427\pi$
$12$	0.622776	0.179780
$13$	1.73738	0.481864	0.240932	−	0.970542i	$-0.422547\pi$
$13$	1.73738	0.481864	0.240932	+	0.970542i	$0.422547\pi$
$14$	−3.30687	−0.883798
$15$	0.622776	0.160800
$16$	1.00000	0.250000
$17$	5.95406	1.44407	0.722036	−	0.691856i	$-0.243207\pi$
$17$	5.95406	1.44407	0.722036	+	0.691856i	$0.243207\pi$
$18$	2.61215	0.615690
$19$	2.09140	0.479800	0.239900	−	0.970798i	$-0.422885\pi$
$19$	2.09140	0.479800	0.239900	+	0.970798i	$0.422885\pi$
$20$	1.00000	0.223607
$21$	2.05944	0.449407
$22$	−1.88207	−0.401258
$23$	5.93930	1.23843	0.619215	−	0.785221i	$-0.287451\pi$
$23$	5.93930	1.23843	0.619215	+	0.785221i	$0.287451\pi$
$24$	−0.622776	−0.127124
$25$	1.00000	0.200000
$26$	−1.73738	−0.340729
$27$	−3.49511	−0.672635
$28$	3.30687	0.624939
$29$	4.40127	0.817296	0.408648	−	0.912692i	$-0.366000\pi$
$29$	4.40127	0.817296	0.408648	+	0.912692i	$0.366000\pi$
$30$	−0.622776	−0.113703
$31$	2.17180	0.390068	0.195034	−	0.980797i	$-0.437518\pi$
$31$	2.17180	0.390068	0.195034	+	0.980797i	$0.437518\pi$
$32$	−1.00000	−0.176777
$33$	1.17211	0.204038
$34$	−5.95406	−1.02111
$35$	3.30687	0.558963
$36$	−2.61215	−0.435358
$37$	−2.94159	−0.483595	−0.241797	−	0.970327i	$-0.577737\pi$
$37$	−2.94159	−0.483595	−0.241797	+	0.970327i	$0.577737\pi$
$38$	−2.09140	−0.339270
$39$	1.08200	0.173259
$40$	−1.00000	−0.158114
$41$	−10.4206	−1.62742	−0.813710	−	0.581271i	$-0.802556\pi$
$41$	−10.4206	−1.62742	−0.813710	+	0.581271i	$0.802556\pi$
$42$	−2.05944	−0.317778
$43$	−6.51163	−0.993014	−0.496507	−	0.868033i	$-0.665384\pi$
$43$	−6.51163	−0.993014	−0.496507	+	0.868033i	$0.665384\pi$
$44$	1.88207	0.283732
$45$	−2.61215	−0.389396
$46$	−5.93930	−0.875702
$47$	−2.61949	−0.382091	−0.191046	−	0.981581i	$-0.561188\pi$
$47$	−2.61949	−0.382091	−0.191046	+	0.981581i	$0.561188\pi$
$48$	0.622776	0.0898900
$49$	3.93538	0.562197
$50$	−1.00000	−0.141421
$51$	3.70805	0.519230
$52$	1.73738	0.240932
$53$	11.8216	1.62383	0.811914	−	0.583777i	$-0.198426\pi$
$53$	11.8216	1.62383	0.811914	+	0.583777i	$0.198426\pi$
$54$	3.49511	0.475625
$55$	1.88207	0.253778
$56$	−3.30687	−0.441899
$57$	1.30247	0.172517
$58$	−4.40127	−0.577915
$59$	3.12873	0.407325	0.203663	−	0.979041i	$-0.434715\pi$
$59$	3.12873	0.407325	0.203663	+	0.979041i	$0.434715\pi$
$60$	0.622776	0.0804001
$61$	−8.50657	−1.08915	−0.544577	−	0.838711i	$-0.683310\pi$
$61$	−8.50657	−1.08915	−0.544577	+	0.838711i	$0.683310\pi$
$62$	−2.17180	−0.275819
$63$	−8.63804	−1.08829
$64$	1.00000	0.125000
$65$	1.73738	0.215496
$66$	−1.17211	−0.144276
$67$	5.94457	0.726245	0.363122	−	0.931741i	$-0.381711\pi$
$67$	5.94457	0.726245	0.363122	+	0.931741i	$0.381711\pi$
$68$	5.95406	0.722036
$69$	3.69886	0.445290
$70$	−3.30687	−0.395246
$71$	−10.5321	−1.24993	−0.624966	−	0.780652i	$-0.714887\pi$
$71$	−10.5321	−1.24993	−0.624966	+	0.780652i	$0.714887\pi$
$72$	2.61215	0.307845
$73$	3.03116	0.354771	0.177385	−	0.984141i	$-0.443236\pi$
$73$	3.03116	0.354771	0.177385	+	0.984141i	$0.443236\pi$
$74$	2.94159	0.341953
$75$	0.622776	0.0719120
$76$	2.09140	0.239900
$77$	6.22375	0.709262
$78$	−1.08200	−0.122513
$79$	6.13016	0.689697	0.344849	−	0.938658i	$-0.387930\pi$
$79$	6.13016	0.689697	0.344849	+	0.938658i	$0.387930\pi$
$80$	1.00000	0.111803
$81$	5.65977	0.628864
$82$	10.4206	1.15076
$83$	−9.86206	−1.08250	−0.541251	−	0.840861i	$-0.682049\pi$
$83$	−9.86206	−1.08250	−0.541251	+	0.840861i	$0.682049\pi$
$84$	2.05944	0.224703
$85$	5.95406	0.645808
$86$	6.51163	0.702167
$87$	2.74101	0.293867
$88$	−1.88207	−0.200629
$89$	−11.4350	−1.21210	−0.606052	−	0.795425i	$-0.707248\pi$
$89$	−11.4350	−1.21210	−0.606052	+	0.795425i	$0.707248\pi$
$90$	2.61215	0.275345
$91$	5.74530	0.602271
$92$	5.93930	0.619215
$93$	1.35255	0.140253
$94$	2.61949	0.270179
$95$	2.09140	0.214573
$96$	−0.622776	−0.0635618
$97$	12.2492	1.24372	0.621860	−	0.783129i	$-0.286377\pi$
$97$	12.2492	1.24372	0.621860	+	0.783129i	$0.286377\pi$
$98$	−3.93538	−0.397534
$99$	−4.91624	−0.494101
$100$	1.00000	0.100000
$101$	9.28587	0.923979	0.461990	−	0.886885i	$-0.347136\pi$
$101$	9.28587	0.923979	0.461990	+	0.886885i	$0.347136\pi$
$102$	−3.70805	−0.367151
$103$	5.60026	0.551810	0.275905	−	0.961185i	$-0.411022\pi$
$103$	5.60026	0.551810	0.275905	+	0.961185i	$0.411022\pi$
$104$	−1.73738	−0.170365
$105$	2.05944	0.200981
$106$	−11.8216	−1.14822
$107$	−0.371470	−0.0359113	−0.0179557	−	0.999839i	$-0.505716\pi$
$107$	−0.371470	−0.0359113	−0.0179557	+	0.999839i	$0.505716\pi$
$108$	−3.49511	−0.336318
$109$	10.2489	0.981665	0.490833	−	0.871254i	$-0.336693\pi$
$109$	10.2489	0.981665	0.490833	+	0.871254i	$0.336693\pi$
$110$	−1.88207	−0.179448
$111$	−1.83195	−0.173881
$112$	3.30687	0.312470
$113$	0.217315	0.0204433	0.0102216	−	0.999948i	$-0.496746\pi$
$113$	0.217315	0.0204433	0.0102216	+	0.999948i	$0.496746\pi$
$114$	−1.30247	−0.121988
$115$	5.93930	0.553843
$116$	4.40127	0.408648
$117$	−4.53831	−0.419567
$118$	−3.12873	−0.288023
$119$	19.6893	1.80491
$120$	−0.622776	−0.0568514
$121$	−7.45782	−0.677984
$122$	8.50657	0.770148
$123$	−6.48969	−0.585155
$124$	2.17180	0.195034
$125$	1.00000	0.0894427
$126$	8.63804	0.769537
$127$	2.30038	0.204125	0.102063	−	0.994778i	$-0.467456\pi$
$127$	2.30038	0.204125	0.102063	+	0.994778i	$0.467456\pi$
$128$	−1.00000	−0.0883883
$129$	−4.05529	−0.357048
$130$	−1.73738	−0.152379
$131$	−16.0399	−1.40142	−0.700708	−	0.713448i	$-0.747132\pi$
$131$	−16.0399	−1.40142	−0.700708	+	0.713448i	$0.747132\pi$
$132$	1.17211	0.102019
$133$	6.91598	0.599692
$134$	−5.94457	−0.513533
$135$	−3.49511	−0.300812
$136$	−5.95406	−0.510556
$137$	3.13771	0.268072	0.134036	−	0.990976i	$-0.457206\pi$
$137$	3.13771	0.268072	0.134036	+	0.990976i	$0.457206\pi$
$138$	−3.69886	−0.314868
$139$	−11.4704	−0.972910	−0.486455	−	0.873706i	$-0.661710\pi$
$139$	−11.4704	−0.972910	−0.486455	+	0.873706i	$0.661710\pi$
$140$	3.30687	0.279481
$141$	−1.63135	−0.137385
$142$	10.5321	0.883835
$143$	3.26988	0.273441
$144$	−2.61215	−0.217679
$145$	4.40127	0.365506
$146$	−3.03116	−0.250861
$147$	2.45086	0.202144
$148$	−2.94159	−0.241797
$149$	6.06177	0.496600	0.248300	−	0.968683i	$-0.420128\pi$
$149$	6.06177	0.496600	0.248300	+	0.968683i	$0.420128\pi$
$150$	−0.622776	−0.0508495
$151$	15.4833	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$151$	15.4833	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$152$	−2.09140	−0.169635
$153$	−15.5529	−1.25738
$154$	−6.22375	−0.501524
$155$	2.17180	0.174443
$156$	1.08200	0.0866295
$157$	−5.89782	−0.470697	−0.235349	−	0.971911i	$-0.575623\pi$
$157$	−5.89782	−0.470697	−0.235349	+	0.971911i	$0.575623\pi$
$158$	−6.13016	−0.487690
$159$	7.36224	0.583864
$160$	−1.00000	−0.0790569
$161$	19.6405	1.54789
$162$	−5.65977	−0.444674
$163$	−3.57835	−0.280278	−0.140139	−	0.990132i	$-0.544755\pi$
$163$	−3.57835	−0.280278	−0.140139	+	0.990132i	$0.544755\pi$
$164$	−10.4206	−0.813710
$165$	1.17211	0.0912484
$166$	9.86206	0.765444
$167$	−21.4986	−1.66361	−0.831804	−	0.555069i	$-0.812692\pi$
$167$	−21.4986	−1.66361	−0.831804	+	0.555069i	$0.812692\pi$
$168$	−2.05944	−0.158889
$169$	−9.98149	−0.767807
$170$	−5.95406	−0.456655
$171$	−5.46305	−0.417770
$172$	−6.51163	−0.496507
$173$	−14.6980	−1.11746	−0.558732	−	0.829348i	$-0.688712\pi$
$173$	−14.6980	−1.11746	−0.558732	+	0.829348i	$0.688712\pi$
$174$	−2.74101	−0.207795
$175$	3.30687	0.249976
$176$	1.88207	0.141866
$177$	1.94850	0.146458
$178$	11.4350	0.857087
$179$	8.79442	0.657325	0.328663	−	0.944447i	$-0.393402\pi$
$179$	8.79442	0.657325	0.328663	+	0.944447i	$0.393402\pi$
$180$	−2.61215	−0.194698
$181$	−9.10260	−0.676591	−0.338295	−	0.941040i	$-0.609850\pi$
$181$	−9.10260	−0.676591	−0.338295	+	0.941040i	$0.609850\pi$
$182$	−5.74530	−0.425870
$183$	−5.29769	−0.391616
$184$	−5.93930	−0.437851
$185$	−2.94159	−0.216270
$186$	−1.35255	−0.0991736
$187$	11.2059	0.819459
$188$	−2.61949	−0.191046
$189$	−11.5579	−0.840712
$190$	−2.09140	−0.151726
$191$	−15.5253	−1.12337	−0.561686	−	0.827350i	$-0.689847\pi$
$191$	−15.5253	−1.12337	−0.561686	+	0.827350i	$0.689847\pi$
$192$	0.622776	0.0449450
$193$	22.4801	1.61815	0.809077	−	0.587703i	$-0.199968\pi$
$193$	22.4801	1.61815	0.809077	+	0.587703i	$0.199968\pi$
$194$	−12.2492	−0.879442
$195$	1.08200	0.0774838
$196$	3.93538	0.281099
$197$	7.12866	0.507896	0.253948	−	0.967218i	$-0.418271\pi$
$197$	7.12866	0.507896	0.253948	+	0.967218i	$0.418271\pi$
$198$	4.91624	0.349382
$199$	28.0040	1.98515	0.992576	−	0.121623i	$-0.0388098\pi$
$199$	28.0040	1.98515	0.992576	+	0.121623i	$0.0388098\pi$
$200$	−1.00000	−0.0707107
$201$	3.70214	0.261129
$202$	−9.28587	−0.653352
$203$	14.5544	1.02152
$204$	3.70805	0.259615
$205$	−10.4206	−0.727804
$206$	−5.60026	−0.390189
$207$	−15.5143	−1.07832
$208$	1.73738	0.120466
$209$	3.93615	0.272269
$210$	−2.05944	−0.142115
$211$	12.6929	0.873812	0.436906	−	0.899507i	$-0.356074\pi$
$211$	12.6929	0.873812	0.436906	+	0.899507i	$0.356074\pi$
$212$	11.8216	0.811914
$213$	−6.55915	−0.449426
$214$	0.371470	0.0253932
$215$	−6.51163	−0.444089
$216$	3.49511	0.237812
$217$	7.18187	0.487537
$218$	−10.2489	−0.694142
$219$	1.88774	0.127561
$220$	1.88207	0.126889
$221$	10.3445	0.695846
$222$	1.83195	0.122953
$223$	13.7262	0.919176	0.459588	−	0.888132i	$-0.347997\pi$
$223$	13.7262	0.919176	0.459588	+	0.888132i	$0.347997\pi$
$224$	−3.30687	−0.220949
$225$	−2.61215	−0.174143
$226$	−0.217315	−0.0144556
$227$	−27.3065	−1.81239	−0.906197	−	0.422856i	$-0.861028\pi$
$227$	−27.3065	−1.81239	−0.906197	+	0.422856i	$0.861028\pi$
$228$	1.30247	0.0862584
$229$	11.8029	0.779955	0.389978	−	0.920824i	$-0.372483\pi$
$229$	11.8029	0.779955	0.389978	+	0.920824i	$0.372483\pi$
$230$	−5.93930	−0.391626
$231$	3.87600	0.255022
$232$	−4.40127	−0.288958
$233$	−11.5539	−0.756921	−0.378461	−	0.925617i	$-0.623546\pi$
$233$	−11.5539	−0.756921	−0.378461	+	0.925617i	$0.623546\pi$
$234$	4.53831	0.296679
$235$	−2.61949	−0.170876
$236$	3.12873	0.203663
$237$	3.81772	0.247988
$238$	−19.6893	−1.27627
$239$	15.1107	0.977430	0.488715	−	0.872443i	$-0.337466\pi$
$239$	15.1107	0.977430	0.488715	+	0.872443i	$0.337466\pi$
$240$	0.622776	0.0402000
$241$	23.6346	1.52244	0.761219	−	0.648495i	$-0.224601\pi$
$241$	23.6346	1.52244	0.761219	+	0.648495i	$0.224601\pi$
$242$	7.45782	0.479407
$243$	14.0101	0.898749
$244$	−8.50657	−0.544577
$245$	3.93538	0.251422
$246$	6.48969	0.413767
$247$	3.63356	0.231198
$248$	−2.17180	−0.137910
$249$	−6.14186	−0.389224
$250$	−1.00000	−0.0632456
$251$	11.1732	0.705245	0.352623	−	0.935766i	$-0.385290\pi$
$251$	11.1732	0.705245	0.352623	+	0.935766i	$0.385290\pi$
$252$	−8.63804	−0.544145
$253$	11.1782	0.702765
$254$	−2.30038	−0.144338
$255$	3.70805	0.232207
$256$	1.00000	0.0625000
$257$	−29.6432	−1.84909	−0.924547	−	0.381068i	$-0.875556\pi$
$257$	−29.6432	−1.84909	−0.924547	+	0.381068i	$0.875556\pi$
$258$	4.05529	0.252471
$259$	−9.72745	−0.604435
$260$	1.73738	0.107748
$261$	−11.4968	−0.711633
$262$	16.0399	0.990950
$263$	−11.1634	−0.688365	−0.344183	−	0.938903i	$-0.611844\pi$
$263$	−11.1634	−0.688365	−0.344183	+	0.938903i	$0.611844\pi$
$264$	−1.17211	−0.0721382
$265$	11.8216	0.726198
$266$	−6.91598	−0.424046
$267$	−7.12143	−0.435824
$268$	5.94457	0.363122
$269$	14.0248	0.855105	0.427552	−	0.903991i	$-0.359376\pi$
$269$	14.0248	0.855105	0.427552	+	0.903991i	$0.359376\pi$
$270$	3.49511	0.212706
$271$	1.93695	0.117661	0.0588306	−	0.998268i	$-0.481263\pi$
$271$	1.93695	0.117661	0.0588306	+	0.998268i	$0.481263\pi$
$272$	5.95406	0.361018
$273$	3.57804	0.216553
$274$	−3.13771	−0.189556
$275$	1.88207	0.113493
$276$	3.69886	0.222645
$277$	8.27155	0.496989	0.248495	−	0.968633i	$-0.420064\pi$
$277$	8.27155	0.496989	0.248495	+	0.968633i	$0.420064\pi$
$278$	11.4704	0.687951
$279$	−5.67308	−0.339638
$280$	−3.30687	−0.197623
$281$	3.42586	0.204370	0.102185	−	0.994765i	$-0.467417\pi$
$281$	3.42586	0.204370	0.102185	+	0.994765i	$0.467417\pi$
$282$	1.63135	0.0971457
$283$	−7.83197	−0.465562	−0.232781	−	0.972529i	$-0.574783\pi$
$283$	−7.83197	−0.465562	−0.232781	+	0.972529i	$0.574783\pi$
$284$	−10.5321	−0.624966
$285$	1.30247	0.0771519
$286$	−3.26988	−0.193352
$287$	−34.4595	−2.03408
$288$	2.61215	0.153922
$289$	18.4508	1.08534
$290$	−4.40127	−0.258452
$291$	7.62852	0.447192
$292$	3.03116	0.177385
$293$	0.497689	0.0290753	0.0145376	−	0.999894i	$-0.495372\pi$
$293$	0.497689	0.0290753	0.0145376	+	0.999894i	$0.495372\pi$
$294$	−2.45086	−0.142937
$295$	3.12873	0.182161
$296$	2.94159	0.170976
$297$	−6.57804	−0.381697
$298$	−6.06177	−0.351149
$299$	10.3189	0.596755
$300$	0.622776	0.0359560
$301$	−21.5331	−1.24115
$302$	−15.4833	−0.890964
$303$	5.78302	0.332226
$304$	2.09140	0.119950
$305$	−8.50657	−0.487085
$306$	15.5529	0.889100
$307$	11.1679	0.637385	0.318692	−	0.947858i	$-0.396756\pi$
$307$	11.1679	0.637385	0.318692	+	0.947858i	$0.396756\pi$
$308$	6.22375	0.354631
$309$	3.48771	0.198409
$310$	−2.17180	−0.123350
$311$	−13.8100	−0.783094	−0.391547	−	0.920158i	$-0.628060\pi$
$311$	−13.8100	−0.783094	−0.391547	+	0.920158i	$0.628060\pi$
$312$	−1.08200	−0.0612563
$313$	−13.6084	−0.769191	−0.384595	−	0.923085i	$-0.625659\pi$
$313$	−13.6084	−0.769191	−0.384595	+	0.923085i	$0.625659\pi$
$314$	5.89782	0.332833
$315$	−8.63804	−0.486698
$316$	6.13016	0.344849
$317$	−16.8087	−0.944070	−0.472035	−	0.881580i	$-0.656480\pi$
$317$	−16.8087	−0.944070	−0.472035	+	0.881580i	$0.656480\pi$
$318$	−7.36224	−0.412854
$319$	8.28349	0.463786
$320$	1.00000	0.0559017
$321$	−0.231343	−0.0129123
$322$	−19.6405	−1.09452
$323$	12.4523	0.692865
$324$	5.65977	0.314432
$325$	1.73738	0.0963728
$326$	3.57835	0.198187
$327$	6.38276	0.352968
$328$	10.4206	0.575380
$329$	−8.66230	−0.477568
$330$	−1.17211	−0.0645224
$331$	−8.69922	−0.478152	−0.239076	−	0.971001i	$-0.576845\pi$
$331$	−8.69922	−0.478152	−0.239076	+	0.971001i	$0.576845\pi$
$332$	−9.86206	−0.541251
$333$	7.68387	0.421074
$334$	21.4986	1.17635
$335$	5.94457	0.324787
$336$	2.05944	0.112352
$337$	8.90404	0.485034	0.242517	−	0.970147i	$-0.422027\pi$
$337$	8.90404	0.485034	0.242517	+	0.970147i	$0.422027\pi$
$338$	9.98149	0.542922
$339$	0.135339	0.00735059
$340$	5.95406	0.322904
$341$	4.08748	0.221350
$342$	5.46305	0.295408
$343$	−10.1343	−0.547200
$344$	6.51163	0.351083
$345$	3.69886	0.199140
$346$	14.6980	0.790167
$347$	10.2349	0.549439	0.274720	−	0.961524i	$-0.411415\pi$
$347$	10.2349	0.549439	0.274720	+	0.961524i	$0.411415\pi$
$348$	2.74101	0.146933
$349$	33.4660	1.79140	0.895698	−	0.444663i	$-0.146677\pi$
$349$	33.4660	1.79140	0.895698	+	0.444663i	$0.146677\pi$
$350$	−3.30687	−0.176760
$351$	−6.07236	−0.324118
$352$	−1.88207	−0.100315
$353$	9.58252	0.510026	0.255013	−	0.966938i	$-0.417920\pi$
$353$	9.58252	0.510026	0.255013	+	0.966938i	$0.417920\pi$
$354$	−1.94850	−0.103561
$355$	−10.5321	−0.558987
$356$	−11.4350	−0.606052
$357$	12.2620	0.648975
$358$	−8.79442	−0.464799
$359$	30.4741	1.60836	0.804181	−	0.594384i	$-0.202604\pi$
$359$	30.4741	1.60836	0.804181	+	0.594384i	$0.202604\pi$
$360$	2.61215	0.137672
$361$	−14.6261	−0.769792
$362$	9.10260	0.478422
$363$	−4.64456	−0.243776
$364$	5.74530	0.301136
$365$	3.03116	0.158658
$366$	5.29769	0.276915
$367$	−7.90669	−0.412726	−0.206363	−	0.978476i	$-0.566163\pi$
$367$	−7.90669	−0.412726	−0.206363	+	0.978476i	$0.566163\pi$
$368$	5.93930	0.309608
$369$	27.2201	1.41702
$370$	2.94159	0.152926
$371$	39.0926	2.02959
$372$	1.35255	0.0701264
$373$	21.5711	1.11691	0.558454	−	0.829535i	$-0.311395\pi$
$373$	21.5711	1.11691	0.558454	+	0.829535i	$0.311395\pi$
$374$	−11.2059	−0.579445
$375$	0.622776	0.0321600
$376$	2.61949	0.135090
$377$	7.64670	0.393825
$378$	11.5579	0.594473
$379$	3.72506	0.191343	0.0956717	−	0.995413i	$-0.469500\pi$
$379$	3.72506	0.191343	0.0956717	+	0.995413i	$0.469500\pi$
$380$	2.09140	0.107286
$381$	1.43262	0.0733953
$382$	15.5253	0.794345
$383$	−25.2621	−1.29083	−0.645417	−	0.763831i	$-0.723316\pi$
$383$	−25.2621	−1.29083	−0.645417	+	0.763831i	$0.723316\pi$
$384$	−0.622776	−0.0317809
$385$	6.22375	0.317192
$386$	−22.4801	−1.14421
$387$	17.0093	0.864633
$388$	12.2492	0.621860
$389$	19.6944	0.998548	0.499274	−	0.866444i	$-0.333600\pi$
$389$	19.6944	0.998548	0.499274	+	0.866444i	$0.333600\pi$
$390$	−1.08200	−0.0547893
$391$	35.3629	1.78838
$392$	−3.93538	−0.198767
$393$	−9.98929	−0.503893
$394$	−7.12866	−0.359137
$395$	6.13016	0.308442
$396$	−4.91624	−0.247050
$397$	−21.5556	−1.08185	−0.540923	−	0.841072i	$-0.681925\pi$
$397$	−21.5556	−1.08185	−0.540923	+	0.841072i	$0.681925\pi$
$398$	−28.0040	−1.40372
$399$	4.30711	0.215625
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−3.70214	−0.184646
$403$	3.77326	0.187959
$404$	9.28587	0.461990
$405$	5.65977	0.281236
$406$	−14.5544	−0.722324
$407$	−5.53627	−0.274423
$408$	−3.70805	−0.183576
$409$	11.0596	0.546862	0.273431	−	0.961892i	$-0.411841\pi$
$409$	11.0596	0.546862	0.273431	+	0.961892i	$0.411841\pi$
$410$	10.4206	0.514635
$411$	1.95409	0.0963881
$412$	5.60026	0.275905
$413$	10.3463	0.509107
$414$	15.5143	0.762488
$415$	−9.86206	−0.484109
$416$	−1.73738	−0.0851823
$417$	−7.14352	−0.349820
$418$	−3.93615	−0.192524
$419$	2.24274	0.109565	0.0547825	−	0.998498i	$-0.482553\pi$
$419$	2.24274	0.109565	0.0547825	+	0.998498i	$0.482553\pi$
$420$	2.05944	0.100490
$421$	−8.53014	−0.415734	−0.207867	−	0.978157i	$-0.566652\pi$
$421$	−8.53014	−0.415734	−0.207867	+	0.978157i	$0.566652\pi$
$422$	−12.6929	−0.617879
$423$	6.84249	0.332693
$424$	−11.8216	−0.574110
$425$	5.95406	0.288814
$426$	6.55915	0.317792
$427$	−28.1301	−1.36131
$428$	−0.371470	−0.0179557
$429$	2.03640	0.0983184
$430$	6.51163	0.314018
$431$	5.56609	0.268109	0.134055	−	0.990974i	$-0.457200\pi$
$431$	5.56609	0.268109	0.134055	+	0.990974i	$0.457200\pi$
$432$	−3.49511	−0.168159
$433$	−13.5511	−0.651226	−0.325613	−	0.945503i	$-0.605571\pi$
$433$	−13.5511	−0.651226	−0.325613	+	0.945503i	$0.605571\pi$
$434$	−7.18187	−0.344741
$435$	2.74101	0.131421
$436$	10.2489	0.490833
$437$	12.4214	0.594198
$438$	−1.88774	−0.0901995
$439$	−21.6716	−1.03433	−0.517164	−	0.855886i	$-0.673012\pi$
$439$	−21.6716	−1.03433	−0.517164	+	0.855886i	$0.673012\pi$
$440$	−1.88207	−0.0897240
$441$	−10.2798	−0.489515
$442$	−10.3445	−0.492037
$443$	26.0198	1.23624	0.618120	−	0.786084i	$-0.287895\pi$
$443$	26.0198	1.23624	0.618120	+	0.786084i	$0.287895\pi$
$444$	−1.83195	−0.0869406
$445$	−11.4350	−0.542070
$446$	−13.7262	−0.649956
$447$	3.77513	0.178558
$448$	3.30687	0.156235
$449$	−14.1385	−0.667238	−0.333619	−	0.942708i	$-0.608270\pi$
$449$	−14.1385	−0.667238	−0.333619	+	0.942708i	$0.608270\pi$
$450$	2.61215	0.123138
$451$	−19.6122	−0.923503
$452$	0.217315	0.0102216
$453$	9.64264	0.453051
$454$	27.3065	1.28156
$455$	5.74530	0.269344
$456$	−1.30247	−0.0609939
$457$	−14.4250	−0.674775	−0.337388	−	0.941366i	$-0.609543\pi$
$457$	−14.4250	−0.674775	−0.337388	+	0.941366i	$0.609543\pi$
$458$	−11.8029	−0.551512
$459$	−20.8101	−0.971333
$460$	5.93930	0.276921
$461$	32.2623	1.50260	0.751302	−	0.659959i	$-0.229426\pi$
$461$	32.2623	1.50260	0.751302	+	0.659959i	$0.229426\pi$
$462$	−3.87600	−0.180328
$463$	28.6447	1.33123	0.665617	−	0.746293i	$-0.268168\pi$
$463$	28.6447	1.33123	0.665617	+	0.746293i	$0.268168\pi$
$464$	4.40127	0.204324
$465$	1.35255	0.0627229
$466$	11.5539	0.535224
$467$	−8.91661	−0.412611	−0.206306	−	0.978488i	$-0.566144\pi$
$467$	−8.91661	−0.412611	−0.206306	+	0.978488i	$0.566144\pi$
$468$	−4.53831	−0.209783
$469$	19.6579	0.907718
$470$	2.61949	0.120828
$471$	−3.67302	−0.169244
$472$	−3.12873	−0.144011
$473$	−12.2553	−0.563500
$474$	−3.81772	−0.175354
$475$	2.09140	0.0959599
$476$	19.6893	0.902457
$477$	−30.8799	−1.41389
$478$	−15.1107	−0.691148
$479$	−21.9610	−1.00342	−0.501712	−	0.865035i	$-0.667296\pi$
$479$	−21.9610	−1.00342	−0.501712	+	0.865035i	$0.667296\pi$
$480$	−0.622776	−0.0284257
$481$	−5.11067	−0.233027
$482$	−23.6346	−1.07653
$483$	12.2316	0.556559
$484$	−7.45782	−0.338992
$485$	12.2492	0.556208
$486$	−14.0101	−0.635512
$487$	8.87070	0.401970	0.200985	−	0.979594i	$-0.435586\pi$
$487$	8.87070	0.401970	0.200985	+	0.979594i	$0.435586\pi$
$488$	8.50657	0.385074
$489$	−2.22851	−0.100777
$490$	−3.93538	−0.177782
$491$	−16.3036	−0.735771	−0.367886	−	0.929871i	$-0.619918\pi$
$491$	−16.3036	−0.735771	−0.367886	+	0.929871i	$0.619918\pi$
$492$	−6.48969	−0.292578
$493$	26.2054	1.18023
$494$	−3.63356	−0.163482
$495$	−4.91624	−0.220969
$496$	2.17180	0.0975169
$497$	−34.8283	−1.56226
$498$	6.14186	0.275223
$499$	−18.0426	−0.807698	−0.403849	−	0.914826i	$-0.632328\pi$
$499$	−18.0426	−0.807698	−0.403849	+	0.914826i	$0.632328\pi$
$500$	1.00000	0.0447214
$501$	−13.3888	−0.598167
$502$	−11.1732	−0.498684
$503$	−3.40138	−0.151660	−0.0758301	−	0.997121i	$-0.524161\pi$
$503$	−3.40138	−0.151660	−0.0758301	+	0.997121i	$0.524161\pi$
$504$	8.63804	0.384769
$505$	9.28587	0.413216
$506$	−11.1782	−0.496930
$507$	−6.21624	−0.276073
$508$	2.30038	0.102063
$509$	−23.6153	−1.04673	−0.523365	−	0.852109i	$-0.675323\pi$
$509$	−23.6153	−1.04673	−0.523365	+	0.852109i	$0.675323\pi$
$510$	−3.70805	−0.164195
$511$	10.0237	0.443421
$512$	−1.00000	−0.0441942
$513$	−7.30968	−0.322730
$514$	29.6432	1.30751
$515$	5.60026	0.246777
$516$	−4.05529	−0.178524
$517$	−4.93005	−0.216823
$518$	9.72745	0.427400
$519$	−9.15354	−0.401796
$520$	−1.73738	−0.0761894
$521$	−7.58569	−0.332335	−0.166168	−	0.986098i	$-0.553139\pi$
$521$	−7.58569	−0.332335	−0.166168	+	0.986098i	$0.553139\pi$
$522$	11.4968	0.503200
$523$	21.5492	0.942280	0.471140	−	0.882058i	$-0.343843\pi$
$523$	21.5492	0.942280	0.471140	+	0.882058i	$0.343843\pi$
$524$	−16.0399	−0.700708
$525$	2.05944	0.0898813
$526$	11.1634	0.486748
$527$	12.9310	0.563285
$528$	1.17211	0.0510094
$529$	12.2753	0.533709
$530$	−11.8216	−0.513499
$531$	−8.17270	−0.354665
$532$	6.91598	0.299846
$533$	−18.1045	−0.784195
$534$	7.12143	0.308174
$535$	−0.371470	−0.0160600
$536$	−5.94457	−0.256766
$537$	5.47695	0.236348
$538$	−14.0248	−0.604650
$539$	7.40665	0.319027
$540$	−3.49511	−0.150406
$541$	8.83118	0.379682	0.189841	−	0.981815i	$-0.439203\pi$
$541$	8.83118	0.379682	0.189841	+	0.981815i	$0.439203\pi$
$542$	−1.93695	−0.0831991
$543$	−5.66888	−0.243275
$544$	−5.95406	−0.255278
$545$	10.2489	0.439014
$546$	−3.57804	−0.153126
$547$	−35.2301	−1.50633	−0.753166	−	0.657831i	$-0.771474\pi$
$547$	−35.2301	−1.50633	−0.753166	+	0.657831i	$0.771474\pi$
$548$	3.13771	0.134036
$549$	22.2204	0.948345
$550$	−1.88207	−0.0802516
$551$	9.20481	0.392138
$552$	−3.69886	−0.157434
$553$	20.2717	0.862038
$554$	−8.27155	−0.351424
$555$	−1.83195	−0.0777621
$556$	−11.4704	−0.486455
$557$	14.9217	0.632251	0.316126	−	0.948717i	$-0.397618\pi$
$557$	14.9217	0.632251	0.316126	+	0.948717i	$0.397618\pi$
$558$	5.67308	0.240160
$559$	−11.3132	−0.478497
$560$	3.30687	0.139741
$561$	6.97879	0.294645
$562$	−3.42586	−0.144511
$563$	7.61634	0.320990	0.160495	−	0.987037i	$-0.448691\pi$
$563$	7.61634	0.320990	0.160495	+	0.987037i	$0.448691\pi$
$564$	−1.63135	−0.0686924
$565$	0.217315	0.00914252
$566$	7.83197	0.329202
$567$	18.7161	0.786004
$568$	10.5321	0.441918
$569$	29.8839	1.25280	0.626400	−	0.779502i	$-0.284528\pi$
$569$	29.8839	1.25280	0.626400	+	0.779502i	$0.284528\pi$
$570$	−1.30247	−0.0545546
$571$	33.6078	1.40644	0.703222	−	0.710971i	$-0.251744\pi$
$571$	33.6078	1.40644	0.703222	+	0.710971i	$0.251744\pi$
$572$	3.26988	0.136720
$573$	−9.66880	−0.403920
$574$	34.4595	1.43831
$575$	5.93930	0.247686
$576$	−2.61215	−0.108840
$577$	−36.7310	−1.52913	−0.764565	−	0.644547i	$-0.777046\pi$
$577$	−36.7310	−1.52913	−0.764565	+	0.644547i	$0.777046\pi$
$578$	−18.4508	−0.767452
$579$	14.0001	0.581823
$580$	4.40127	0.182753
$581$	−32.6125	−1.35300
$582$	−7.62852	−0.316212
$583$	22.2491	0.921465
$584$	−3.03116	−0.125430
$585$	−4.53831	−0.187636
$586$	−0.497689	−0.0205593
$587$	2.47508	0.102157	0.0510787	−	0.998695i	$-0.483734\pi$
$587$	2.47508	0.102157	0.0510787	+	0.998695i	$0.483734\pi$
$588$	2.45086	0.101072
$589$	4.54211	0.187154
$590$	−3.12873	−0.128808
$591$	4.43956	0.182619
$592$	−2.94159	−0.120899
$593$	−20.5166	−0.842514	−0.421257	−	0.906941i	$-0.638411\pi$
$593$	−20.5166	−0.842514	−0.421257	+	0.906941i	$0.638411\pi$
$594$	6.57804	0.269900
$595$	19.6893	0.807182
$596$	6.06177	0.248300
$597$	17.4402	0.713782
$598$	−10.3189	−0.421969
$599$	32.1319	1.31287	0.656437	−	0.754381i	$-0.272063\pi$
$599$	32.1319	1.31287	0.656437	+	0.754381i	$0.272063\pi$
$600$	−0.622776	−0.0254247
$601$	−26.4267	−1.07797	−0.538984	−	0.842316i	$-0.681192\pi$
$601$	−26.4267	−1.07797	−0.538984	+	0.842316i	$0.681192\pi$
$602$	21.5331	0.877623
$603$	−15.5281	−0.632353
$604$	15.4833	0.630007
$605$	−7.45782	−0.303204
$606$	−5.78302	−0.234919
$607$	22.4640	0.911787	0.455893	−	0.890034i	$-0.349320\pi$
$607$	22.4640	0.911787	0.455893	+	0.890034i	$0.349320\pi$
$608$	−2.09140	−0.0848174
$609$	9.06415	0.367298
$610$	8.50657	0.344421
$611$	−4.55105	−0.184116
$612$	−15.5529	−0.628688
$613$	21.9844	0.887943	0.443972	−	0.896041i	$-0.353569\pi$
$613$	21.9844	0.887943	0.443972	+	0.896041i	$0.353569\pi$
$614$	−11.1679	−0.450699
$615$	−6.48969	−0.261689
$616$	−6.22375	−0.250762
$617$	−35.3569	−1.42341	−0.711707	−	0.702476i	$-0.752078\pi$
$617$	−35.3569	−1.42341	−0.711707	+	0.702476i	$0.752078\pi$
$618$	−3.48771	−0.140296
$619$	13.6132	0.547162	0.273581	−	0.961849i	$-0.411792\pi$
$619$	13.6132	0.547162	0.273581	+	0.961849i	$0.411792\pi$
$620$	2.17180	0.0872217
$621$	−20.7585	−0.833011
$622$	13.8100	0.553731
$623$	−37.8139	−1.51498
$624$	1.08200	0.0433148
$625$	1.00000	0.0400000
$626$	13.6084	0.543900
$627$	2.45134	0.0978972
$628$	−5.89782	−0.235349
$629$	−17.5144	−0.698345
$630$	8.63804	0.344148
$631$	−1.21018	−0.0481763	−0.0240882	−	0.999710i	$-0.507668\pi$
$631$	−1.21018	−0.0481763	−0.0240882	+	0.999710i	$0.507668\pi$
$632$	−6.13016	−0.243845
$633$	7.90481	0.314188
$634$	16.8087	0.667558
$635$	2.30038	0.0912876
$636$	7.36224	0.291932
$637$	6.83727	0.270903
$638$	−8.28349	−0.327946
$639$	27.5115	1.08834
$640$	−1.00000	−0.0395285
$641$	1.79749	0.0709966	0.0354983	−	0.999370i	$-0.488698\pi$
$641$	1.79749	0.0709966	0.0354983	+	0.999370i	$0.488698\pi$
$642$	0.231343	0.00913037
$643$	−25.8411	−1.01907	−0.509536	−	0.860449i	$-0.670183\pi$
$643$	−25.8411	−1.01907	−0.509536	+	0.860449i	$0.670183\pi$
$644$	19.6405	0.773944
$645$	−4.05529	−0.159677
$646$	−12.4523	−0.489929
$647$	−11.5353	−0.453499	−0.226749	−	0.973953i	$-0.572810\pi$
$647$	−11.5353	−0.453499	−0.226749	+	0.973953i	$0.572810\pi$
$648$	−5.65977	−0.222337
$649$	5.88847	0.231143
$650$	−1.73738	−0.0681458
$651$	4.47270	0.175299
$652$	−3.57835	−0.140139
$653$	−15.3938	−0.602405	−0.301203	−	0.953560i	$-0.597388\pi$
$653$	−15.3938	−0.602405	−0.301203	+	0.953560i	$0.597388\pi$
$654$	−6.38276	−0.249586
$655$	−16.0399	−0.626732
$656$	−10.4206	−0.406855
$657$	−7.91785	−0.308905
$658$	8.66230	0.337691
$659$	−28.5855	−1.11353	−0.556767	−	0.830669i	$-0.687958\pi$
$659$	−28.5855	−1.11353	−0.556767	+	0.830669i	$0.687958\pi$
$660$	1.17211	0.0456242
$661$	−13.8258	−0.537762	−0.268881	−	0.963173i	$-0.586654\pi$
$661$	−13.8258	−0.537762	−0.268881	+	0.963173i	$0.586654\pi$
$662$	8.69922	0.338105
$663$	6.44230	0.250198
$664$	9.86206	0.382722
$665$	6.91598	0.268190
$666$	−7.68387	−0.297744
$667$	26.1405	1.01216
$668$	−21.4986	−0.831804
$669$	8.54837	0.330499
$670$	−5.94457	−0.229659
$671$	−16.0099	−0.618056
$672$	−2.05944	−0.0794446
$673$	−42.7962	−1.64967	−0.824836	−	0.565372i	$-0.808733\pi$
$673$	−42.7962	−1.64967	−0.824836	+	0.565372i	$0.808733\pi$
$674$	−8.90404	−0.342971
$675$	−3.49511	−0.134527
$676$	−9.98149	−0.383904
$677$	−11.1563	−0.428773	−0.214387	−	0.976749i	$-0.568775\pi$
$677$	−11.1563	−0.428773	−0.214387	+	0.976749i	$0.568775\pi$
$678$	−0.135339	−0.00519765
$679$	40.5065	1.55450
$680$	−5.95406	−0.228328
$681$	−17.0058	−0.651665
$682$	−4.08748	−0.156518
$683$	−8.29565	−0.317424	−0.158712	−	0.987325i	$-0.550734\pi$
$683$	−8.29565	−0.317424	−0.158712	+	0.987325i	$0.550734\pi$
$684$	−5.46305	−0.208885
$685$	3.13771	0.119886
$686$	10.1343	0.386929
$687$	7.35055	0.280441
$688$	−6.51163	−0.248253
$689$	20.5387	0.782464
$690$	−3.69886	−0.140813
$691$	9.31713	0.354440	0.177220	−	0.984171i	$-0.443290\pi$
$691$	9.31713	0.354440	0.177220	+	0.984171i	$0.443290\pi$
$692$	−14.6980	−0.558732
$693$	−16.2574	−0.617566
$694$	−10.2349	−0.388512
$695$	−11.4704	−0.435098
$696$	−2.74101	−0.103898
$697$	−62.0447	−2.35011
$698$	−33.4660	−1.26671
$699$	−7.19550	−0.272159
$700$	3.30687	0.124988
$701$	10.3466	0.390785	0.195393	−	0.980725i	$-0.437402\pi$
$701$	10.3466	0.390785	0.195393	+	0.980725i	$0.437402\pi$
$702$	6.07236	0.229186
$703$	−6.15204	−0.232029
$704$	1.88207	0.0709331
$705$	−1.63135	−0.0614403
$706$	−9.58252	−0.360643
$707$	30.7072	1.15486
$708$	1.94850	0.0732290
$709$	−46.6145	−1.75064	−0.875321	−	0.483542i	$-0.839350\pi$
$709$	−46.6145	−1.75064	−0.875321	+	0.483542i	$0.839350\pi$
$710$	10.5321	0.395263
$711$	−16.0129	−0.600531
$712$	11.4350	0.428544
$713$	12.8990	0.483071
$714$	−12.2620	−0.458895
$715$	3.26988	0.122286
$716$	8.79442	0.328663
$717$	9.41059	0.351445
$718$	−30.4741	−1.13728
$719$	16.5340	0.616614	0.308307	−	0.951287i	$-0.400238\pi$
$719$	16.5340	0.616614	0.308307	+	0.951287i	$0.400238\pi$
$720$	−2.61215	−0.0973491
$721$	18.5193	0.689696
$722$	14.6261	0.544325
$723$	14.7191	0.547408
$724$	−9.10260	−0.338295
$725$	4.40127	0.163459
$726$	4.64456	0.172376
$727$	37.6835	1.39760	0.698802	−	0.715316i	$-0.253717\pi$
$727$	37.6835	1.39760	0.698802	+	0.715316i	$0.253717\pi$
$728$	−5.74530	−0.212935
$729$	−8.25415	−0.305709
$730$	−3.03116	−0.112188
$731$	−38.7706	−1.43398
$732$	−5.29769	−0.195808
$733$	40.9541	1.51267	0.756337	−	0.654182i	$-0.226987\pi$
$733$	40.9541	1.51267	0.756337	+	0.654182i	$0.226987\pi$
$734$	7.90669	0.291841
$735$	2.45086	0.0904014
$736$	−5.93930	−0.218926
$737$	11.1881	0.412118
$738$	−27.2201	−1.00199
$739$	−0.640476	−0.0235603	−0.0117802	−	0.999931i	$-0.503750\pi$
$739$	−0.640476	−0.0235603	−0.0117802	+	0.999931i	$0.503750\pi$
$740$	−2.94159	−0.108135
$741$	2.26290	0.0831296
$742$	−39.0926	−1.43514
$743$	38.3000	1.40509	0.702545	−	0.711639i	$-0.252047\pi$
$743$	38.3000	1.40509	0.702545	+	0.711639i	$0.252047\pi$
$744$	−1.35255	−0.0495868
$745$	6.06177	0.222086
$746$	−21.5711	−0.789773
$747$	25.7612	0.942552
$748$	11.2059	0.409730
$749$	−1.22840	−0.0448848
$750$	−0.622776	−0.0227406
$751$	−9.90176	−0.361320	−0.180660	−	0.983546i	$-0.557823\pi$
$751$	−9.90176	−0.361320	−0.180660	+	0.983546i	$0.557823\pi$
$752$	−2.61949	−0.0955228
$753$	6.95839	0.253578
$754$	−7.64670	−0.278476
$755$	15.4833	0.563495
$756$	−11.5579	−0.420356
$757$	25.3655	0.921925	0.460962	−	0.887420i	$-0.347504\pi$
$757$	25.3655	0.921925	0.460962	+	0.887420i	$0.347504\pi$
$758$	−3.72506	−0.135300
$759$	6.96150	0.252686
$760$	−2.09140	−0.0758630
$761$	−17.6439	−0.639590	−0.319795	−	0.947487i	$-0.603614\pi$
$761$	−17.6439	−0.639590	−0.319795	+	0.947487i	$0.603614\pi$
$762$	−1.43262	−0.0518983
$763$	33.8917	1.22696
$764$	−15.5253	−0.561686
$765$	−15.5529	−0.562316
$766$	25.2621	0.912757
$767$	5.43580	0.196275
$768$	0.622776	0.0224725
$769$	11.5858	0.417795	0.208897	−	0.977938i	$-0.433013\pi$
$769$	11.5858	0.417795	0.208897	+	0.977938i	$0.433013\pi$
$770$	−6.22375	−0.224288
$771$	−18.4611	−0.664860
$772$	22.4801	0.809077
$773$	−20.2231	−0.727376	−0.363688	−	0.931521i	$-0.618482\pi$
$773$	−20.2231	−0.727376	−0.363688	+	0.931521i	$0.618482\pi$
$774$	−17.0093	−0.611388
$775$	2.17180	0.0780135
$776$	−12.2492	−0.439721
$777$	−6.05803	−0.217331
$778$	−19.6944	−0.706080
$779$	−21.7936	−0.780836
$780$	1.08200	0.0387419
$781$	−19.8221	−0.709292
$782$	−35.3629	−1.26458
$783$	−15.3829	−0.549742
$784$	3.93538	0.140549
$785$	−5.89782	−0.210502
$786$	9.98929	0.356306
$787$	−16.3474	−0.582721	−0.291360	−	0.956613i	$-0.594108\pi$
$787$	−16.3474	−0.582721	−0.291360	+	0.956613i	$0.594108\pi$
$788$	7.12866	0.253948
$789$	−6.95231	−0.247509
$790$	−6.13016	−0.218101
$791$	0.718633	0.0255516
$792$	4.91624	0.174691
$793$	−14.7792	−0.524824
$794$	21.5556	0.764981
$795$	7.36224	0.261112
$796$	28.0040	0.992576
$797$	7.97845	0.282611	0.141306	−	0.989966i	$-0.454870\pi$
$797$	7.97845	0.282611	0.141306	+	0.989966i	$0.454870\pi$
$798$	−4.30711	−0.152470
$799$	−15.5966	−0.551767
$800$	−1.00000	−0.0353553
$801$	29.8698	1.05540
$802$	1.00000	0.0353112
$803$	5.70485	0.201320
$804$	3.70214	0.130564
$805$	19.6405	0.692236
$806$	−3.77326	−0.132907
$807$	8.73429	0.307461
$808$	−9.28587	−0.326676
$809$	5.32160	0.187098	0.0935488	−	0.995615i	$-0.470179\pi$
$809$	5.32160	0.187098	0.0935488	+	0.995615i	$0.470179\pi$
$810$	−5.65977	−0.198864
$811$	18.5201	0.650329	0.325164	−	0.945658i	$-0.394580\pi$
$811$	18.5201	0.650329	0.325164	+	0.945658i	$0.394580\pi$
$812$	14.5544	0.510760
$813$	1.20629	0.0423063
$814$	5.53627	0.194046
$815$	−3.57835	−0.125344
$816$	3.70805	0.129808
$817$	−13.6184	−0.476448
$818$	−11.0596	−0.386690
$819$	−15.0076	−0.524408
$820$	−10.4206	−0.363902
$821$	−44.1237	−1.53993	−0.769964	−	0.638088i	$-0.779726\pi$
$821$	−44.1237	−1.53993	−0.769964	+	0.638088i	$0.779726\pi$
$822$	−1.95409	−0.0681567
$823$	−23.1291	−0.806230	−0.403115	−	0.915149i	$-0.632072\pi$
$823$	−23.1291	−0.806230	−0.403115	+	0.915149i	$0.632072\pi$
$824$	−5.60026	−0.195094
$825$	1.17211	0.0408075
$826$	−10.3463	−0.359993
$827$	−11.7739	−0.409418	−0.204709	−	0.978823i	$-0.565625\pi$
$827$	−11.7739	−0.409418	−0.204709	+	0.978823i	$0.565625\pi$
$828$	−15.5143	−0.539161
$829$	−34.0135	−1.18134	−0.590668	−	0.806914i	$-0.701136\pi$
$829$	−34.0135	−1.18134	−0.590668	+	0.806914i	$0.701136\pi$
$830$	9.86206	0.342317
$831$	5.15133	0.178698
$832$	1.73738	0.0602330
$833$	23.4315	0.811853
$834$	7.14352	0.247360
$835$	−21.4986	−0.743988
$836$	3.93615	0.136135
$837$	−7.59070	−0.262373
$838$	−2.24274	−0.0774742
$839$	31.9759	1.10393	0.551966	−	0.833867i	$-0.313878\pi$
$839$	31.9759	1.10393	0.551966	+	0.833867i	$0.313878\pi$
$840$	−2.05944	−0.0710574
$841$	−9.62881	−0.332028
$842$	8.53014	0.293968
$843$	2.13354	0.0734831
$844$	12.6929	0.436906
$845$	−9.98149	−0.343374
$846$	−6.84249	−0.235250
$847$	−24.6620	−0.847398
$848$	11.8216	0.405957
$849$	−4.87757	−0.167398
$850$	−5.95406	−0.204222
$851$	−17.4710	−0.598898
$852$	−6.55915	−0.224713
$853$	−35.3277	−1.20960	−0.604798	−	0.796379i	$-0.706746\pi$
$853$	−35.3277	−1.20960	−0.604798	+	0.796379i	$0.706746\pi$
$854$	28.1301	0.962592
$855$	−5.46305	−0.186832
$856$	0.371470	0.0126966
$857$	2.01009	0.0686633	0.0343317	−	0.999410i	$-0.489070\pi$
$857$	2.01009	0.0686633	0.0343317	+	0.999410i	$0.489070\pi$
$858$	−2.03640	−0.0695216
$859$	13.8981	0.474196	0.237098	−	0.971486i	$-0.423804\pi$
$859$	13.8981	0.474196	0.237098	+	0.971486i	$0.423804\pi$
$860$	−6.51163	−0.222045
$861$	−21.4605	−0.731373
$862$	−5.56609	−0.189582
$863$	1.72457	0.0587049	0.0293525	−	0.999569i	$-0.490655\pi$
$863$	1.72457	0.0587049	0.0293525	+	0.999569i	$0.490655\pi$
$864$	3.49511	0.118906
$865$	−14.6980	−0.499745
$866$	13.5511	0.460486
$867$	11.4907	0.390245
$868$	7.18187	0.243769
$869$	11.5374	0.391379
$870$	−2.74101	−0.0929289
$871$	10.3280	0.349951
$872$	−10.2489	−0.347071
$873$	−31.9968	−1.08293
$874$	−12.4214	−0.420162
$875$	3.30687	0.111793
$876$	1.88774	0.0637807
$877$	−48.0203	−1.62153	−0.810764	−	0.585373i	$-0.800948\pi$
$877$	−48.0203	−1.62153	−0.810764	+	0.585373i	$0.800948\pi$
$878$	21.6716	0.731381
$879$	0.309949	0.0104543
$880$	1.88207	0.0634445
$881$	−27.3112	−0.920137	−0.460069	−	0.887883i	$-0.652175\pi$
$881$	−27.3112	−0.920137	−0.460069	+	0.887883i	$0.652175\pi$
$882$	10.2798	0.346139
$883$	−2.21603	−0.0745755	−0.0372877	−	0.999305i	$-0.511872\pi$
$883$	−2.21603	−0.0745755	−0.0372877	+	0.999305i	$0.511872\pi$
$884$	10.3445	0.347923
$885$	1.94850	0.0654980
$886$	−26.0198	−0.874154
$887$	11.2803	0.378755	0.189377	−	0.981904i	$-0.439353\pi$
$887$	11.2803	0.378755	0.189377	+	0.981904i	$0.439353\pi$
$888$	1.83195	0.0614763
$889$	7.60704	0.255132
$890$	11.4350	0.383301
$891$	10.6521	0.356858
$892$	13.7262	0.459588
$893$	−5.47839	−0.183327
$894$	−3.77513	−0.126259
$895$	8.79442	0.293965
$896$	−3.30687	−0.110475
$897$	6.42634	0.214569
$898$	14.1385	0.471808
$899$	9.55870	0.318800
$900$	−2.61215	−0.0870717
$901$	70.3867	2.34492
$902$	19.6122	0.653015
$903$	−13.4103	−0.446267
$904$	−0.217315	−0.00722779
$905$	−9.10260	−0.302581
$906$	−9.64264	−0.320355
$907$	14.7237	0.488891	0.244446	−	0.969663i	$-0.421394\pi$
$907$	14.7237	0.488891	0.244446	+	0.969663i	$0.421394\pi$
$908$	−27.3065	−0.906197
$909$	−24.2561	−0.804524
$910$	−5.74530	−0.190455
$911$	−3.51642	−0.116504	−0.0582521	−	0.998302i	$-0.518553\pi$
$911$	−3.51642	−0.116504	−0.0582521	+	0.998302i	$0.518553\pi$
$912$	1.30247	0.0431292
$913$	−18.5611	−0.614281
$914$	14.4250	0.477138
$915$	−5.29769	−0.175136
$916$	11.8029	0.389978
$917$	−53.0420	−1.75160
$918$	20.8101	0.686836
$919$	24.4619	0.806924	0.403462	−	0.914996i	$-0.367807\pi$
$919$	24.4619	0.806924	0.403462	+	0.914996i	$0.367807\pi$
$920$	−5.93930	−0.195813
$921$	6.95509	0.229178
$922$	−32.2623	−1.06250
$923$	−18.2983	−0.602297
$924$	3.87600	0.127511
$925$	−2.94159	−0.0967189
$926$	−28.6447	−0.941325
$927$	−14.6287	−0.480470
$928$	−4.40127	−0.144479
$929$	41.7820	1.37082	0.685411	−	0.728156i	$-0.259622\pi$
$929$	41.7820	1.37082	0.685411	+	0.728156i	$0.259622\pi$
$930$	−1.35255	−0.0443518
$931$	8.23045	0.269742
$932$	−11.5539	−0.378461
$933$	−8.60054	−0.281569
$934$	8.91661	0.291760
$935$	11.2059	0.366473
$936$	4.53831	0.148339
$937$	−36.4916	−1.19213	−0.596064	−	0.802937i	$-0.703270\pi$
$937$	−36.4916	−1.19213	−0.596064	+	0.802937i	$0.703270\pi$
$938$	−19.6579	−0.641854
$939$	−8.47497	−0.276570
$940$	−2.61949	−0.0854382
$941$	−41.3929	−1.34937	−0.674685	−	0.738106i	$-0.735720\pi$
$941$	−41.3929	−1.34937	−0.674685	+	0.738106i	$0.735720\pi$
$942$	3.67302	0.119674
$943$	−61.8909	−2.01545
$944$	3.12873	0.101831
$945$	−11.5579	−0.375978
$946$	12.2553	0.398455
$947$	8.47174	0.275295	0.137647	−	0.990481i	$-0.456046\pi$
$947$	8.47174	0.275295	0.137647	+	0.990481i	$0.456046\pi$
$948$	3.81772	0.123994
$949$	5.26630	0.170951
$950$	−2.09140	−0.0678539
$951$	−10.4681	−0.339450
$952$	−19.6893	−0.638133
$953$	−10.0552	−0.325721	−0.162860	−	0.986649i	$-0.552072\pi$
$953$	−10.0552	−0.325721	−0.162860	+	0.986649i	$0.552072\pi$
$954$	30.8799	0.999774
$955$	−15.5253	−0.502388
$956$	15.1107	0.488715
$957$	5.15876	0.166759
$958$	21.9610	0.709528
$959$	10.3760	0.335058
$960$	0.622776	0.0201000
$961$	−26.2833	−0.847847
$962$	5.11067	0.164775
$963$	0.970335	0.0312686
$964$	23.6346	0.761219
$965$	22.4801	0.723660
$966$	−12.2316	−0.393546
$967$	−3.83000	−0.123165	−0.0615823	−	0.998102i	$-0.519615\pi$
$967$	−3.83000	−0.123165	−0.0615823	+	0.998102i	$0.519615\pi$
$968$	7.45782	0.239703
$969$	7.75500	0.249127
$970$	−12.2492	−0.393299
$971$	15.2212	0.488472	0.244236	−	0.969716i	$-0.421463\pi$
$971$	15.2212	0.488472	0.244236	+	0.969716i	$0.421463\pi$
$972$	14.0101	0.449375
$973$	−37.9312	−1.21602
$974$	−8.87070	−0.284235
$975$	1.08200	0.0346518
$976$	−8.50657	−0.272289
$977$	22.8066	0.729647	0.364823	−	0.931077i	$-0.381129\pi$
$977$	22.8066	0.729647	0.364823	+	0.931077i	$0.381129\pi$
$978$	2.22851	0.0712600
$979$	−21.5214	−0.687826
$980$	3.93538	0.125711
$981$	−26.7716	−0.854752
$982$	16.3036	0.520269
$983$	−58.3016	−1.85953	−0.929766	−	0.368151i	$-0.879991\pi$
$983$	−58.3016	−1.85953	−0.929766	+	0.368151i	$0.879991\pi$
$984$	6.48969	0.206884
$985$	7.12866	0.227138
$986$	−26.2054	−0.834551
$987$	−5.39467	−0.171714
$988$	3.63356	0.115599
$989$	−38.6745	−1.22978
$990$	4.91624	0.156248
$991$	1.66417	0.0528642	0.0264321	−	0.999651i	$-0.491585\pi$
$991$	1.66417	0.0528642	0.0264321	+	0.999651i	$0.491585\pi$
$992$	−2.17180	−0.0689548
$993$	−5.41767	−0.171925
$994$	34.8283	1.10469
$995$	28.0040	0.887787
$996$	−6.14186	−0.194612
$997$	−11.0340	−0.349451	−0.174726	−	0.984617i	$-0.555904\pi$
$997$	−11.0340	−0.349451	−0.174726	+	0.984617i	$0.555904\pi$
$998$	18.0426	0.571129
$999$	10.2812	0.325283

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.m.1.11	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.m.1.11	✓	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.622776 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.622776 q^{6} +3.30687 q^{7} -1.00000 q^{8} -2.61215 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.622776 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.622776 q^{6} +3.30687 q^{7} -1.00000 q^{8} -2.61215 q^{9} -1.00000 q^{10} +1.88207 q^{11} +0.622776 q^{12} +1.73738 q^{13} -3.30687 q^{14} +0.622776 q^{15} +1.00000 q^{16} +5.95406 q^{17} +2.61215 q^{18} +2.09140 q^{19} +1.00000 q^{20} +2.05944 q^{21} -1.88207 q^{22} +5.93930 q^{23} -0.622776 q^{24} +1.00000 q^{25} -1.73738 q^{26} -3.49511 q^{27} +3.30687 q^{28} +4.40127 q^{29} -0.622776 q^{30} +2.17180 q^{31} -1.00000 q^{32} +1.17211 q^{33} -5.95406 q^{34} +3.30687 q^{35} -2.61215 q^{36} -2.94159 q^{37} -2.09140 q^{38} +1.08200 q^{39} -1.00000 q^{40} -10.4206 q^{41} -2.05944 q^{42} -6.51163 q^{43} +1.88207 q^{44} -2.61215 q^{45} -5.93930 q^{46} -2.61949 q^{47} +0.622776 q^{48} +3.93538 q^{49} -1.00000 q^{50} +3.70805 q^{51} +1.73738 q^{52} +11.8216 q^{53} +3.49511 q^{54} +1.88207 q^{55} -3.30687 q^{56} +1.30247 q^{57} -4.40127 q^{58} +3.12873 q^{59} +0.622776 q^{60} -8.50657 q^{61} -2.17180 q^{62} -8.63804 q^{63} +1.00000 q^{64} +1.73738 q^{65} -1.17211 q^{66} +5.94457 q^{67} +5.95406 q^{68} +3.69886 q^{69} -3.30687 q^{70} -10.5321 q^{71} +2.61215 q^{72} +3.03116 q^{73} +2.94159 q^{74} +0.622776 q^{75} +2.09140 q^{76} +6.22375 q^{77} -1.08200 q^{78} +6.13016 q^{79} +1.00000 q^{80} +5.65977 q^{81} +10.4206 q^{82} -9.86206 q^{83} +2.05944 q^{84} +5.95406 q^{85} +6.51163 q^{86} +2.74101 q^{87} -1.88207 q^{88} -11.4350 q^{89} +2.61215 q^{90} +5.74530 q^{91} +5.93930 q^{92} +1.35255 q^{93} +2.61949 q^{94} +2.09140 q^{95} -0.622776 q^{96} +12.2492 q^{97} -3.93538 q^{98} -4.91624 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 - 9 * q^13 - 11 * q^14 + 4 * q^15 + 20 * q^16 - 11 * q^17 - 26 * q^18 + 17 * q^19 + 20 * q^20 - 2 * q^21 - 10 * q^22 - 3 * q^23 - 4 * q^24 + 20 * q^25 + 9 * q^26 - 2 * q^27 + 11 * q^28 + 6 * q^29 - 4 * q^30 + 28 * q^31 - 20 * q^32 + 2 * q^33 + 11 * q^34 + 11 * q^35 + 26 * q^36 + 33 * q^37 - 17 * q^38 + 36 * q^39 - 20 * q^40 + 32 * q^41 + 2 * q^42 + 30 * q^43 + 10 * q^44 + 26 * q^45 + 3 * q^46 + 13 * q^47 + 4 * q^48 + 43 * q^49 - 20 * q^50 + 43 * q^51 - 9 * q^52 + 2 * q^54 + 10 * q^55 - 11 * q^56 + 19 * q^57 - 6 * q^58 + 52 * q^59 + 4 * q^60 + 25 * q^61 - 28 * q^62 + 16 * q^63 + 20 * q^64 - 9 * q^65 - 2 * q^66 + 40 * q^67 - 11 * q^68 + 39 * q^69 - 11 * q^70 + 25 * q^71 - 26 * q^72 - 5 * q^73 - 33 * q^74 + 4 * q^75 + 17 * q^76 - 9 * q^77 - 36 * q^78 + 40 * q^79 + 20 * q^80 + 48 * q^81 - 32 * q^82 + 10 * q^83 - 2 * q^84 - 11 * q^85 - 30 * q^86 + 10 * q^87 - 10 * q^88 + 17 * q^89 - 26 * q^90 + 88 * q^91 - 3 * q^92 - 4 * q^93 - 13 * q^94 + 17 * q^95 - 4 * q^96 - 2 * q^97 - 43 * q^98 + 50 * q^99

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