LMFDB - Embedded newform 4010.2.a.m.1.2

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			1.2
Root			$-2.99276$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.99276 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.99276 q^{6} +0.839081 q^{7} -1.00000 q^{8} +5.95662 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.99276 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.99276 q^{6} +0.839081 q^{7} -1.00000 q^{8} +5.95662 q^{9} -1.00000 q^{10} +4.32738 q^{11} -2.99276 q^{12} +0.593592 q^{13} -0.839081 q^{14} -2.99276 q^{15} +1.00000 q^{16} -0.302659 q^{17} -5.95662 q^{18} +3.41975 q^{19} +1.00000 q^{20} -2.51117 q^{21} -4.32738 q^{22} +1.14602 q^{23} +2.99276 q^{24} +1.00000 q^{25} -0.593592 q^{26} -8.84847 q^{27} +0.839081 q^{28} +9.67486 q^{29} +2.99276 q^{30} +4.06088 q^{31} -1.00000 q^{32} -12.9508 q^{33} +0.302659 q^{34} +0.839081 q^{35} +5.95662 q^{36} +1.09146 q^{37} -3.41975 q^{38} -1.77648 q^{39} -1.00000 q^{40} +11.6980 q^{41} +2.51117 q^{42} +8.97913 q^{43} +4.32738 q^{44} +5.95662 q^{45} -1.14602 q^{46} +12.2393 q^{47} -2.99276 q^{48} -6.29594 q^{49} -1.00000 q^{50} +0.905787 q^{51} +0.593592 q^{52} -1.48778 q^{53} +8.84847 q^{54} +4.32738 q^{55} -0.839081 q^{56} -10.2345 q^{57} -9.67486 q^{58} -9.58967 q^{59} -2.99276 q^{60} -9.64738 q^{61} -4.06088 q^{62} +4.99809 q^{63} +1.00000 q^{64} +0.593592 q^{65} +12.9508 q^{66} +1.19083 q^{67} -0.302659 q^{68} -3.42975 q^{69} -0.839081 q^{70} +10.8778 q^{71} -5.95662 q^{72} -2.30726 q^{73} -1.09146 q^{74} -2.99276 q^{75} +3.41975 q^{76} +3.63102 q^{77} +1.77648 q^{78} +6.33947 q^{79} +1.00000 q^{80} +8.61149 q^{81} -11.6980 q^{82} -15.2620 q^{83} -2.51117 q^{84} -0.302659 q^{85} -8.97913 q^{86} -28.9546 q^{87} -4.32738 q^{88} -17.4410 q^{89} -5.95662 q^{90} +0.498072 q^{91} +1.14602 q^{92} -12.1532 q^{93} -12.2393 q^{94} +3.41975 q^{95} +2.99276 q^{96} -15.7060 q^{97} +6.29594 q^{98} +25.7766 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.99276	−1.72787	−0.863936	−	0.503602i	$-0.832008\pi$
$3$	−2.99276	−1.72787	−0.863936	+	0.503602i	$0.832008\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.99276	1.22179
$7$	0.839081	0.317143	0.158571	−	0.987348i	$-0.449311\pi$
$7$	0.839081	0.317143	0.158571	+	0.987348i	$0.449311\pi$
$8$	−1.00000	−0.353553
$9$	5.95662	1.98554
$10$	−1.00000	−0.316228
$11$	4.32738	1.30475	0.652377	−	0.757895i	$-0.273772\pi$
$11$	4.32738	1.30475	0.652377	+	0.757895i	$0.273772\pi$
$12$	−2.99276	−0.863936
$13$	0.593592	0.164633	0.0823164	−	0.996606i	$-0.473768\pi$
$13$	0.593592	0.164633	0.0823164	+	0.996606i	$0.473768\pi$
$14$	−0.839081	−0.224254
$15$	−2.99276	−0.772728
$16$	1.00000	0.250000
$17$	−0.302659	−0.0734056	−0.0367028	−	0.999326i	$-0.511685\pi$
$17$	−0.302659	−0.0734056	−0.0367028	+	0.999326i	$0.511685\pi$
$18$	−5.95662	−1.40399
$19$	3.41975	0.784544	0.392272	−	0.919849i	$-0.371689\pi$
$19$	3.41975	0.784544	0.392272	+	0.919849i	$0.371689\pi$
$20$	1.00000	0.223607
$21$	−2.51117	−0.547982
$22$	−4.32738	−0.922600
$23$	1.14602	0.238961	0.119480	−	0.992837i	$-0.461877\pi$
$23$	1.14602	0.238961	0.119480	+	0.992837i	$0.461877\pi$
$24$	2.99276	0.610895
$25$	1.00000	0.200000
$26$	−0.593592	−0.116413
$27$	−8.84847	−1.70289
$28$	0.839081	0.158571
$29$	9.67486	1.79658	0.898288	−	0.439407i	$-0.144811\pi$
$29$	9.67486	1.79658	0.898288	+	0.439407i	$0.144811\pi$
$30$	2.99276	0.546401
$31$	4.06088	0.729355	0.364677	−	0.931134i	$-0.381179\pi$
$31$	4.06088	0.729355	0.364677	+	0.931134i	$0.381179\pi$
$32$	−1.00000	−0.176777
$33$	−12.9508	−2.25445
$34$	0.302659	0.0519056
$35$	0.839081	0.141831
$36$	5.95662	0.992771
$37$	1.09146	0.179435	0.0897174	−	0.995967i	$-0.471404\pi$
$37$	1.09146	0.179435	0.0897174	+	0.995967i	$0.471404\pi$
$38$	−3.41975	−0.554757
$39$	−1.77648	−0.284464
$40$	−1.00000	−0.158114
$41$	11.6980	1.82693	0.913463	−	0.406921i	$-0.133398\pi$
$41$	11.6980	1.82693	0.913463	+	0.406921i	$0.133398\pi$
$42$	2.51117	0.387482
$43$	8.97913	1.36930	0.684652	−	0.728870i	$-0.259954\pi$
$43$	8.97913	1.36930	0.684652	+	0.728870i	$0.259954\pi$
$44$	4.32738	0.652377
$45$	5.95662	0.887961
$46$	−1.14602	−0.168971
$47$	12.2393	1.78528	0.892640	−	0.450769i	$-0.148850\pi$
$47$	12.2393	1.78528	0.892640	+	0.450769i	$0.148850\pi$
$48$	−2.99276	−0.431968
$49$	−6.29594	−0.899420
$50$	−1.00000	−0.141421
$51$	0.905787	0.126836
$52$	0.593592	0.0823164
$53$	−1.48778	−0.204362	−0.102181	−	0.994766i	$-0.532582\pi$
$53$	−1.48778	−0.204362	−0.102181	+	0.994766i	$0.532582\pi$
$54$	8.84847	1.20412
$55$	4.32738	0.583503
$56$	−0.839081	−0.112127
$57$	−10.2345	−1.35559
$58$	−9.67486	−1.27037
$59$	−9.58967	−1.24847	−0.624235	−	0.781237i	$-0.714589\pi$
$59$	−9.58967	−1.24847	−0.624235	+	0.781237i	$0.714589\pi$
$60$	−2.99276	−0.386364
$61$	−9.64738	−1.23522	−0.617610	−	0.786485i	$-0.711899\pi$
$61$	−9.64738	−1.23522	−0.617610	+	0.786485i	$0.711899\pi$
$62$	−4.06088	−0.515732
$63$	4.99809	0.629700
$64$	1.00000	0.125000
$65$	0.593592	0.0736260
$66$	12.9508	1.59413
$67$	1.19083	0.145483	0.0727415	−	0.997351i	$-0.476825\pi$
$67$	1.19083	0.145483	0.0727415	+	0.997351i	$0.476825\pi$
$68$	−0.302659	−0.0367028
$69$	−3.42975	−0.412894
$70$	−0.839081	−0.100289
$71$	10.8778	1.29096	0.645479	−	0.763778i	$-0.276658\pi$
$71$	10.8778	1.29096	0.645479	+	0.763778i	$0.276658\pi$
$72$	−5.95662	−0.701995
$73$	−2.30726	−0.270044	−0.135022	−	0.990843i	$-0.543111\pi$
$73$	−2.30726	−0.270044	−0.135022	+	0.990843i	$0.543111\pi$
$74$	−1.09146	−0.126880
$75$	−2.99276	−0.345574
$76$	3.41975	0.392272
$77$	3.63102	0.413793
$78$	1.77648	0.201147
$79$	6.33947	0.713246	0.356623	−	0.934248i	$-0.383928\pi$
$79$	6.33947	0.713246	0.356623	+	0.934248i	$0.383928\pi$
$80$	1.00000	0.111803
$81$	8.61149	0.956833
$82$	−11.6980	−1.29183
$83$	−15.2620	−1.67522	−0.837611	−	0.546268i	$-0.816048\pi$
$83$	−15.2620	−1.67522	−0.837611	+	0.546268i	$0.816048\pi$
$84$	−2.51117	−0.273991
$85$	−0.302659	−0.0328280
$86$	−8.97913	−0.968244
$87$	−28.9546	−3.10425
$88$	−4.32738	−0.461300
$89$	−17.4410	−1.84874	−0.924369	−	0.381501i	$-0.875407\pi$
$89$	−17.4410	−1.84874	−0.924369	+	0.381501i	$0.875407\pi$
$90$	−5.95662	−0.627883
$91$	0.498072	0.0522121
$92$	1.14602	0.119480
$93$	−12.1532	−1.26023
$94$	−12.2393	−1.26238
$95$	3.41975	0.350859
$96$	2.99276	0.305447
$97$	−15.7060	−1.59470	−0.797350	−	0.603517i	$-0.793765\pi$
$97$	−15.7060	−1.59470	−0.797350	+	0.603517i	$0.793765\pi$
$98$	6.29594	0.635986
$99$	25.7766	2.59064
$100$	1.00000	0.100000
$101$	−7.20292	−0.716717	−0.358358	−	0.933584i	$-0.616663\pi$
$101$	−7.20292	−0.716717	−0.358358	+	0.933584i	$0.616663\pi$
$102$	−0.905787	−0.0896863
$103$	5.07200	0.499759	0.249880	−	0.968277i	$-0.419609\pi$
$103$	5.07200	0.499759	0.249880	+	0.968277i	$0.419609\pi$
$104$	−0.593592	−0.0582065
$105$	−2.51117	−0.245065
$106$	1.48778	0.144506
$107$	11.4193	1.10394	0.551972	−	0.833863i	$-0.313875\pi$
$107$	11.4193	1.10394	0.551972	+	0.833863i	$0.313875\pi$
$108$	−8.84847	−0.851444
$109$	17.5738	1.68326	0.841632	−	0.540052i	$-0.181595\pi$
$109$	17.5738	1.68326	0.841632	+	0.540052i	$0.181595\pi$
$110$	−4.32738	−0.412599
$111$	−3.26648	−0.310040
$112$	0.839081	0.0792857
$113$	−6.63988	−0.624627	−0.312314	−	0.949979i	$-0.601104\pi$
$113$	−6.63988	−0.624627	−0.312314	+	0.949979i	$0.601104\pi$
$114$	10.2345	0.958548
$115$	1.14602	0.106866
$116$	9.67486	0.898288
$117$	3.53580	0.326885
$118$	9.58967	0.882801
$119$	−0.253956	−0.0232801
$120$	2.99276	0.273201
$121$	7.72619	0.702381
$122$	9.64738	0.873432
$123$	−35.0094	−3.15670
$124$	4.06088	0.364677
$125$	1.00000	0.0894427
$126$	−4.99809	−0.445265
$127$	4.82348	0.428015	0.214007	−	0.976832i	$-0.431348\pi$
$127$	4.82348	0.428015	0.214007	+	0.976832i	$0.431348\pi$
$128$	−1.00000	−0.0883883
$129$	−26.8724	−2.36598
$130$	−0.593592	−0.0520615
$131$	−16.2753	−1.42198	−0.710989	−	0.703203i	$-0.751753\pi$
$131$	−16.2753	−1.42198	−0.710989	+	0.703203i	$0.751753\pi$
$132$	−12.9508	−1.12722
$133$	2.86945	0.248813
$134$	−1.19083	−0.102872
$135$	−8.84847	−0.761555
$136$	0.302659	0.0259528
$137$	−17.9345	−1.53224	−0.766122	−	0.642695i	$-0.777816\pi$
$137$	−17.9345	−1.53224	−0.766122	+	0.642695i	$0.777816\pi$
$138$	3.42975	0.291960
$139$	15.3836	1.30482	0.652409	−	0.757867i	$-0.273759\pi$
$139$	15.3836	1.30482	0.652409	+	0.757867i	$0.273759\pi$
$140$	0.839081	0.0709153
$141$	−36.6292	−3.08474
$142$	−10.8778	−0.912845
$143$	2.56870	0.214805
$144$	5.95662	0.496385
$145$	9.67486	0.803453
$146$	2.30726	0.190950
$147$	18.8423	1.55408
$148$	1.09146	0.0897174
$149$	18.7790	1.53843	0.769216	−	0.638988i	$-0.220647\pi$
$149$	18.7790	1.53843	0.769216	+	0.638988i	$0.220647\pi$
$150$	2.99276	0.244358
$151$	13.9045	1.13154	0.565768	−	0.824565i	$-0.308580\pi$
$151$	13.9045	1.13154	0.565768	+	0.824565i	$0.308580\pi$
$152$	−3.41975	−0.277378
$153$	−1.80283	−0.145750
$154$	−3.63102	−0.292596
$155$	4.06088	0.326177
$156$	−1.77648	−0.142232
$157$	20.2531	1.61638	0.808188	−	0.588925i	$-0.200449\pi$
$157$	20.2531	1.61638	0.808188	+	0.588925i	$0.200449\pi$
$158$	−6.33947	−0.504341
$159$	4.45257	0.353112
$160$	−1.00000	−0.0790569
$161$	0.961600	0.0757847
$162$	−8.61149	−0.676583
$163$	−9.22899	−0.722870	−0.361435	−	0.932397i	$-0.617713\pi$
$163$	−9.22899	−0.722870	−0.361435	+	0.932397i	$0.617713\pi$
$164$	11.6980	0.913463
$165$	−12.9508	−1.00822
$166$	15.2620	1.18456
$167$	0.633271	0.0490040	0.0245020	−	0.999700i	$-0.492200\pi$
$167$	0.633271	0.0490040	0.0245020	+	0.999700i	$0.492200\pi$
$168$	2.51117	0.193741
$169$	−12.6476	−0.972896
$170$	0.302659	0.0232129
$171$	20.3702	1.55775
$172$	8.97913	0.684652
$173$	−22.4477	−1.70667	−0.853333	−	0.521367i	$-0.825422\pi$
$173$	−22.4477	−1.70667	−0.853333	+	0.521367i	$0.825422\pi$
$174$	28.9546	2.19504
$175$	0.839081	0.0634286
$176$	4.32738	0.326188
$177$	28.6996	2.15719
$178$	17.4410	1.30725
$179$	−24.4322	−1.82615	−0.913075	−	0.407791i	$-0.866299\pi$
$179$	−24.4322	−1.82615	−0.913075	+	0.407791i	$0.866299\pi$
$180$	5.95662	0.443981
$181$	−11.2615	−0.837057	−0.418529	−	0.908204i	$-0.637454\pi$
$181$	−11.2615	−0.837057	−0.418529	+	0.908204i	$0.637454\pi$
$182$	−0.498072	−0.0369195
$183$	28.8723	2.13430
$184$	−1.14602	−0.0844854
$185$	1.09146	0.0802456
$186$	12.1532	0.891118
$187$	−1.30972	−0.0957763
$188$	12.2393	0.892640
$189$	−7.42458	−0.540059
$190$	−3.41975	−0.248095
$191$	−15.6002	−1.12879	−0.564394	−	0.825506i	$-0.690890\pi$
$191$	−15.6002	−1.12879	−0.564394	+	0.825506i	$0.690890\pi$
$192$	−2.99276	−0.215984
$193$	−16.4301	−1.18266	−0.591332	−	0.806428i	$-0.701398\pi$
$193$	−16.4301	−1.18266	−0.591332	+	0.806428i	$0.701398\pi$
$194$	15.7060	1.12762
$195$	−1.77648	−0.127216
$196$	−6.29594	−0.449710
$197$	10.5521	0.751807	0.375904	−	0.926659i	$-0.377332\pi$
$197$	10.5521	0.751807	0.375904	+	0.926659i	$0.377332\pi$
$198$	−25.7766	−1.83186
$199$	0.343775	0.0243695	0.0121848	−	0.999926i	$-0.496121\pi$
$199$	0.343775	0.0243695	0.0121848	+	0.999926i	$0.496121\pi$
$200$	−1.00000	−0.0707107
$201$	−3.56387	−0.251376
$202$	7.20292	0.506795
$203$	8.11799	0.569771
$204$	0.905787	0.0634178
$205$	11.6980	0.817027
$206$	−5.07200	−0.353383
$207$	6.82638	0.474466
$208$	0.593592	0.0411582
$209$	14.7985	1.02364
$210$	2.51117	0.173287
$211$	−6.23885	−0.429500	−0.214750	−	0.976669i	$-0.568894\pi$
$211$	−6.23885	−0.429500	−0.214750	+	0.976669i	$0.568894\pi$
$212$	−1.48778	−0.102181
$213$	−32.5547	−2.23061
$214$	−11.4193	−0.780607
$215$	8.97913	0.612371
$216$	8.84847	0.602062
$217$	3.40740	0.231310
$218$	−17.5738	−1.19025
$219$	6.90507	0.466601
$220$	4.32738	0.291752
$221$	−0.179656	−0.0120850
$222$	3.26648	0.219232
$223$	20.1101	1.34667	0.673337	−	0.739336i	$-0.264860\pi$
$223$	20.1101	1.34667	0.673337	+	0.739336i	$0.264860\pi$
$224$	−0.839081	−0.0560635
$225$	5.95662	0.397108
$226$	6.63988	0.441678
$227$	9.08979	0.603310	0.301655	−	0.953417i	$-0.402461\pi$
$227$	9.08979	0.603310	0.301655	+	0.953417i	$0.402461\pi$
$228$	−10.2345	−0.677796
$229$	−17.6824	−1.16849	−0.584244	−	0.811578i	$-0.698609\pi$
$229$	−17.6824	−1.16849	−0.584244	+	0.811578i	$0.698609\pi$
$230$	−1.14602	−0.0755660
$231$	−10.8668	−0.714981
$232$	−9.67486	−0.635186
$233$	−3.49531	−0.228986	−0.114493	−	0.993424i	$-0.536524\pi$
$233$	−3.49531	−0.228986	−0.114493	+	0.993424i	$0.536524\pi$
$234$	−3.53580	−0.231143
$235$	12.2393	0.798402
$236$	−9.58967	−0.624235
$237$	−18.9725	−1.23240
$238$	0.253956	0.0164615
$239$	−14.3780	−0.930039	−0.465019	−	0.885301i	$-0.653953\pi$
$239$	−14.3780	−0.930039	−0.465019	+	0.885301i	$0.653953\pi$
$240$	−2.99276	−0.193182
$241$	11.0365	0.710924	0.355462	−	0.934691i	$-0.384324\pi$
$241$	11.0365	0.710924	0.355462	+	0.934691i	$0.384324\pi$
$242$	−7.72619	−0.496658
$243$	0.773260	0.0496047
$244$	−9.64738	−0.617610
$245$	−6.29594	−0.402233
$246$	35.0094	2.23212
$247$	2.02994	0.129162
$248$	−4.06088	−0.257866
$249$	45.6755	2.89457
$250$	−1.00000	−0.0632456
$251$	−0.930554	−0.0587360	−0.0293680	−	0.999569i	$-0.509349\pi$
$251$	−0.930554	−0.0587360	−0.0293680	+	0.999569i	$0.509349\pi$
$252$	4.99809	0.314850
$253$	4.95924	0.311785
$254$	−4.82348	−0.302652
$255$	0.905787	0.0567226
$256$	1.00000	0.0625000
$257$	22.1262	1.38019	0.690097	−	0.723717i	$-0.257568\pi$
$257$	22.1262	1.38019	0.690097	+	0.723717i	$0.257568\pi$
$258$	26.8724	1.67300
$259$	0.915822	0.0569064
$260$	0.593592	0.0368130
$261$	57.6295	3.56718
$262$	16.2753	1.00549
$263$	12.6296	0.778773	0.389387	−	0.921074i	$-0.372687\pi$
$263$	12.6296	0.778773	0.389387	+	0.921074i	$0.372687\pi$
$264$	12.9508	0.797067
$265$	−1.48778	−0.0913937
$266$	−2.86945	−0.175937
$267$	52.1966	3.19438
$268$	1.19083	0.0727415
$269$	32.4147	1.97636	0.988181	−	0.153293i	$-0.0489877\pi$
$269$	32.4147	1.97636	0.988181	+	0.153293i	$0.0489877\pi$
$270$	8.84847	0.538501
$271$	−5.01228	−0.304474	−0.152237	−	0.988344i	$-0.548648\pi$
$271$	−5.01228	−0.304474	−0.152237	+	0.988344i	$0.548648\pi$
$272$	−0.302659	−0.0183514
$273$	−1.49061	−0.0902158
$274$	17.9345	1.08346
$275$	4.32738	0.260951
$276$	−3.42975	−0.206447
$277$	24.2890	1.45938	0.729692	−	0.683776i	$-0.239664\pi$
$277$	24.2890	1.45938	0.729692	+	0.683776i	$0.239664\pi$
$278$	−15.3836	−0.922645
$279$	24.1891	1.44816
$280$	−0.839081	−0.0501447
$281$	17.8597	1.06542	0.532709	−	0.846298i	$-0.321174\pi$
$281$	17.8597	1.06542	0.532709	+	0.846298i	$0.321174\pi$
$282$	36.6292	2.18124
$283$	8.25560	0.490744	0.245372	−	0.969429i	$-0.421090\pi$
$283$	8.25560	0.490744	0.245372	+	0.969429i	$0.421090\pi$
$284$	10.8778	0.645479
$285$	−10.2345	−0.606239
$286$	−2.56870	−0.151890
$287$	9.81560	0.579397
$288$	−5.95662	−0.350997
$289$	−16.9084	−0.994612
$290$	−9.67486	−0.568127
$291$	47.0042	2.75544
$292$	−2.30726	−0.135022
$293$	−1.25307	−0.0732052	−0.0366026	−	0.999330i	$-0.511654\pi$
$293$	−1.25307	−0.0732052	−0.0366026	+	0.999330i	$0.511654\pi$
$294$	−18.8423	−1.09890
$295$	−9.58967	−0.558332
$296$	−1.09146	−0.0634398
$297$	−38.2907	−2.22185
$298$	−18.7790	−1.08784
$299$	0.680266	0.0393408
$300$	−2.99276	−0.172787
$301$	7.53421	0.434265
$302$	−13.9045	−0.800117
$303$	21.5566	1.23839
$304$	3.41975	0.196136
$305$	−9.64738	−0.552407
$306$	1.80283	0.103061
$307$	−28.4969	−1.62640	−0.813201	−	0.581983i	$-0.802277\pi$
$307$	−28.4969	−1.62640	−0.813201	+	0.581983i	$0.802277\pi$
$308$	3.63102	0.206897
$309$	−15.1793	−0.863520
$310$	−4.06088	−0.230642
$311$	−13.9830	−0.792901	−0.396451	−	0.918056i	$-0.629758\pi$
$311$	−13.9830	−0.792901	−0.396451	+	0.918056i	$0.629758\pi$
$312$	1.77648	0.100573
$313$	−7.90339	−0.446726	−0.223363	−	0.974735i	$-0.571704\pi$
$313$	−7.90339	−0.446726	−0.223363	+	0.974735i	$0.571704\pi$
$314$	−20.2531	−1.14295
$315$	4.99809	0.281610
$316$	6.33947	0.356623
$317$	17.4503	0.980105	0.490053	−	0.871693i	$-0.336978\pi$
$317$	17.4503	0.980105	0.490053	+	0.871693i	$0.336978\pi$
$318$	−4.45257	−0.249688
$319$	41.8668	2.34409
$320$	1.00000	0.0559017
$321$	−34.1752	−1.90747
$322$	−0.961600	−0.0535879
$323$	−1.03502	−0.0575900
$324$	8.61149	0.478416
$325$	0.593592	0.0329266
$326$	9.22899	0.511146
$327$	−52.5941	−2.90846
$328$	−11.6980	−0.645916
$329$	10.2697	0.566189
$330$	12.9508	0.712919
$331$	−0.657717	−0.0361514	−0.0180757	−	0.999837i	$-0.505754\pi$
$331$	−0.657717	−0.0361514	−0.0180757	+	0.999837i	$0.505754\pi$
$332$	−15.2620	−0.837611
$333$	6.50141	0.356275
$334$	−0.633271	−0.0346510
$335$	1.19083	0.0650620
$336$	−2.51117	−0.136996
$337$	−0.378100	−0.0205964	−0.0102982	−	0.999947i	$-0.503278\pi$
$337$	−0.378100	−0.0205964	−0.0102982	+	0.999947i	$0.503278\pi$
$338$	12.6476	0.687941
$339$	19.8716	1.07928
$340$	−0.302659	−0.0164140
$341$	17.5729	0.951628
$342$	−20.3702	−1.10149
$343$	−11.1564	−0.602388
$344$	−8.97913	−0.484122
$345$	−3.42975	−0.184652
$346$	22.4477	1.20679
$347$	4.05906	0.217902	0.108951	−	0.994047i	$-0.465251\pi$
$347$	4.05906	0.217902	0.108951	+	0.994047i	$0.465251\pi$
$348$	−28.9546	−1.55213
$349$	30.9447	1.65643	0.828215	−	0.560410i	$-0.189356\pi$
$349$	30.9447	1.65643	0.828215	+	0.560410i	$0.189356\pi$
$350$	−0.839081	−0.0448508
$351$	−5.25238	−0.280351
$352$	−4.32738	−0.230650
$353$	−37.3739	−1.98921	−0.994607	−	0.103714i	$-0.966927\pi$
$353$	−37.3739	−1.98921	−0.994607	+	0.103714i	$0.966927\pi$
$354$	−28.6996	−1.52537
$355$	10.8778	0.577334
$356$	−17.4410	−0.924369
$357$	0.760029	0.0402250
$358$	24.4322	1.29128
$359$	−10.4399	−0.550999	−0.275499	−	0.961301i	$-0.588843\pi$
$359$	−10.4399	−0.550999	−0.275499	+	0.961301i	$0.588843\pi$
$360$	−5.95662	−0.313942
$361$	−7.30531	−0.384490
$362$	11.2615	0.591889
$363$	−23.1226	−1.21362
$364$	0.498072	0.0261061
$365$	−2.30726	−0.120767
$366$	−28.8723	−1.50918
$367$	−13.1737	−0.687663	−0.343831	−	0.939031i	$-0.611725\pi$
$367$	−13.1737	−0.687663	−0.343831	+	0.939031i	$0.611725\pi$
$368$	1.14602	0.0597402
$369$	69.6808	3.62744
$370$	−1.09146	−0.0567422
$371$	−1.24837	−0.0648121
$372$	−12.1532	−0.630116
$373$	−21.5557	−1.11611	−0.558057	−	0.829802i	$-0.688453\pi$
$373$	−21.5557	−1.11611	−0.558057	+	0.829802i	$0.688453\pi$
$374$	1.30972	0.0677240
$375$	−2.99276	−0.154546
$376$	−12.2393	−0.631192
$377$	5.74292	0.295775
$378$	7.42458	0.381879
$379$	34.0943	1.75131	0.875653	−	0.482941i	$-0.160431\pi$
$379$	34.0943	1.75131	0.875653	+	0.482941i	$0.160431\pi$
$380$	3.41975	0.175429
$381$	−14.4355	−0.739554
$382$	15.6002	0.798173
$383$	23.9191	1.22221	0.611105	−	0.791550i	$-0.290725\pi$
$383$	23.9191	1.22221	0.611105	+	0.791550i	$0.290725\pi$
$384$	2.99276	0.152724
$385$	3.63102	0.185054
$386$	16.4301	0.836270
$387$	53.4853	2.71881
$388$	−15.7060	−0.797350
$389$	−29.4422	−1.49278	−0.746388	−	0.665511i	$-0.768214\pi$
$389$	−29.4422	−1.49278	−0.746388	+	0.665511i	$0.768214\pi$
$390$	1.77648	0.0899555
$391$	−0.346852	−0.0175411
$392$	6.29594	0.317993
$393$	48.7081	2.45700
$394$	−10.5521	−0.531608
$395$	6.33947	0.318973
$396$	25.7766	1.29532
$397$	−0.482928	−0.0242375	−0.0121187	−	0.999927i	$-0.503858\pi$
$397$	−0.482928	−0.0242375	−0.0121187	+	0.999927i	$0.503858\pi$
$398$	−0.343775	−0.0172319
$399$	−8.58757	−0.429916
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	3.56387	0.177750
$403$	2.41050	0.120076
$404$	−7.20292	−0.358358
$405$	8.61149	0.427909
$406$	−8.11799	−0.402889
$407$	4.72315	0.234118
$408$	−0.905787	−0.0448431
$409$	31.6820	1.56658	0.783288	−	0.621659i	$-0.213541\pi$
$409$	31.6820	1.56658	0.783288	+	0.621659i	$0.213541\pi$
$410$	−11.6980	−0.577725
$411$	53.6735	2.64752
$412$	5.07200	0.249880
$413$	−8.04651	−0.395943
$414$	−6.82638	−0.335498
$415$	−15.2620	−0.749182
$416$	−0.593592	−0.0291032
$417$	−46.0394	−2.25456
$418$	−14.7985	−0.723821
$419$	16.7551	0.818538	0.409269	−	0.912414i	$-0.365784\pi$
$419$	16.7551	0.818538	0.409269	+	0.912414i	$0.365784\pi$
$420$	−2.51117	−0.122533
$421$	30.1762	1.47070	0.735349	−	0.677689i	$-0.237018\pi$
$421$	30.1762	1.47070	0.735349	+	0.677689i	$0.237018\pi$
$422$	6.23885	0.303703
$423$	72.9047	3.54475
$424$	1.48778	0.0722530
$425$	−0.302659	−0.0146811
$426$	32.5547	1.57728
$427$	−8.09493	−0.391741
$428$	11.4193	0.551972
$429$	−7.68750	−0.371156
$430$	−8.97913	−0.433012
$431$	−8.08564	−0.389472	−0.194736	−	0.980856i	$-0.562385\pi$
$431$	−8.08564	−0.389472	−0.194736	+	0.980856i	$0.562385\pi$
$432$	−8.84847	−0.425722
$433$	−15.2624	−0.733463	−0.366731	−	0.930327i	$-0.619523\pi$
$433$	−15.2624	−0.733463	−0.366731	+	0.930327i	$0.619523\pi$
$434$	−3.40740	−0.163561
$435$	−28.9546	−1.38826
$436$	17.5738	0.841632
$437$	3.91909	0.187475
$438$	−6.90507	−0.329937
$439$	28.3671	1.35389	0.676944	−	0.736034i	$-0.263304\pi$
$439$	28.3671	1.35389	0.676944	+	0.736034i	$0.263304\pi$
$440$	−4.32738	−0.206300
$441$	−37.5026	−1.78584
$442$	0.179656	0.00854537
$443$	−0.173228	−0.00823033	−0.00411517	−	0.999992i	$-0.501310\pi$
$443$	−0.173228	−0.00823033	−0.00411517	+	0.999992i	$0.501310\pi$
$444$	−3.26648	−0.155020
$445$	−17.4410	−0.826780
$446$	−20.1101	−0.952242
$447$	−56.2010	−2.65821
$448$	0.839081	0.0396429
$449$	−8.47088	−0.399765	−0.199883	−	0.979820i	$-0.564056\pi$
$449$	−8.47088	−0.399765	−0.199883	+	0.979820i	$0.564056\pi$
$450$	−5.95662	−0.280798
$451$	50.6218	2.38369
$452$	−6.63988	−0.312314
$453$	−41.6130	−1.95515
$454$	−9.08979	−0.426605
$455$	0.498072	0.0233500
$456$	10.2345	0.479274
$457$	1.81101	0.0847154	0.0423577	−	0.999103i	$-0.486513\pi$
$457$	1.81101	0.0847154	0.0423577	+	0.999103i	$0.486513\pi$
$458$	17.6824	0.826246
$459$	2.67807	0.125002
$460$	1.14602	0.0534332
$461$	10.4940	0.488754	0.244377	−	0.969680i	$-0.421417\pi$
$461$	10.4940	0.488754	0.244377	+	0.969680i	$0.421417\pi$
$462$	10.8668	0.505568
$463$	22.4969	1.04552	0.522761	−	0.852479i	$-0.324902\pi$
$463$	22.4969	1.04552	0.522761	+	0.852479i	$0.324902\pi$
$464$	9.67486	0.449144
$465$	−12.1532	−0.563593
$466$	3.49531	0.161917
$467$	0.478185	0.0221278	0.0110639	−	0.999939i	$-0.496478\pi$
$467$	0.478185	0.0221278	0.0110639	+	0.999939i	$0.496478\pi$
$468$	3.53580	0.163443
$469$	0.999203	0.0461389
$470$	−12.2393	−0.564555
$471$	−60.6128	−2.79289
$472$	9.58967	0.441401
$473$	38.8561	1.78660
$474$	18.9725	0.871436
$475$	3.41975	0.156909
$476$	−0.253956	−0.0116400
$477$	−8.86215	−0.405770
$478$	14.3780	0.657637
$479$	−12.4817	−0.570305	−0.285152	−	0.958482i	$-0.592044\pi$
$479$	−12.4817	−0.570305	−0.285152	+	0.958482i	$0.592044\pi$
$480$	2.99276	0.136600
$481$	0.647881	0.0295408
$482$	−11.0365	−0.502699
$483$	−2.87784	−0.130946
$484$	7.72619	0.351190
$485$	−15.7060	−0.713171
$486$	−0.773260	−0.0350758
$487$	23.9463	1.08511	0.542555	−	0.840020i	$-0.317457\pi$
$487$	23.9463	1.08511	0.542555	+	0.840020i	$0.317457\pi$
$488$	9.64738	0.436716
$489$	27.6202	1.24903
$490$	6.29594	0.284422
$491$	−5.14047	−0.231986	−0.115993	−	0.993250i	$-0.537005\pi$
$491$	−5.14047	−0.231986	−0.115993	+	0.993250i	$0.537005\pi$
$492$	−35.0094	−1.57835
$493$	−2.92819	−0.131879
$494$	−2.02994	−0.0913311
$495$	25.7766	1.15857
$496$	4.06088	0.182339
$497$	9.12736	0.409418
$498$	−45.6755	−2.04677
$499$	−34.0657	−1.52499	−0.762495	−	0.646994i	$-0.776026\pi$
$499$	−34.0657	−1.52499	−0.762495	+	0.646994i	$0.776026\pi$
$500$	1.00000	0.0447214
$501$	−1.89523	−0.0846726
$502$	0.930554	0.0415327
$503$	15.3721	0.685410	0.342705	−	0.939443i	$-0.388657\pi$
$503$	15.3721	0.685410	0.342705	+	0.939443i	$0.388657\pi$
$504$	−4.99809	−0.222633
$505$	−7.20292	−0.320526
$506$	−4.95924	−0.220465
$507$	37.8514	1.68104
$508$	4.82348	0.214007
$509$	2.63906	0.116974	0.0584872	−	0.998288i	$-0.481372\pi$
$509$	2.63906	0.116974	0.0584872	+	0.998288i	$0.481372\pi$
$510$	−0.905787	−0.0401089
$511$	−1.93598	−0.0856425
$512$	−1.00000	−0.0441942
$513$	−30.2596	−1.33599
$514$	−22.1262	−0.975944
$515$	5.07200	0.223499
$516$	−26.8724	−1.18299
$517$	52.9639	2.32935
$518$	−0.915822	−0.0402389
$519$	67.1805	2.94890
$520$	−0.593592	−0.0260307
$521$	24.1240	1.05689	0.528445	−	0.848967i	$-0.322775\pi$
$521$	24.1240	1.05689	0.528445	+	0.848967i	$0.322775\pi$
$522$	−57.6295	−2.52237
$523$	−13.1130	−0.573390	−0.286695	−	0.958022i	$-0.592557\pi$
$523$	−13.1130	−0.573390	−0.286695	+	0.958022i	$0.592557\pi$
$524$	−16.2753	−0.710989
$525$	−2.51117	−0.109596
$526$	−12.6296	−0.550676
$527$	−1.22906	−0.0535388
$528$	−12.9508	−0.563612
$529$	−21.6866	−0.942898
$530$	1.48778	0.0646251
$531$	−57.1221	−2.47889
$532$	2.86945	0.124406
$533$	6.94386	0.300772
$534$	−52.1966	−2.25877
$535$	11.4193	0.493699
$536$	−1.19083	−0.0514360
$537$	73.1199	3.15535
$538$	−32.4147	−1.39750
$539$	−27.2449	−1.17352
$540$	−8.84847	−0.380778
$541$	−26.2588	−1.12895	−0.564476	−	0.825449i	$-0.690922\pi$
$541$	−26.2588	−1.12895	−0.564476	+	0.825449i	$0.690922\pi$
$542$	5.01228	0.215296
$543$	33.7028	1.44633
$544$	0.302659	0.0129764
$545$	17.5738	0.752778
$546$	1.49061	0.0637922
$547$	17.5789	0.751619	0.375810	−	0.926697i	$-0.377365\pi$
$547$	17.5789	0.751619	0.375810	+	0.926697i	$0.377365\pi$
$548$	−17.9345	−0.766122
$549$	−57.4658	−2.45258
$550$	−4.32738	−0.184520
$551$	33.0856	1.40949
$552$	3.42975	0.145980
$553$	5.31933	0.226201
$554$	−24.2890	−1.03194
$555$	−3.26648	−0.138654
$556$	15.3836	0.652409
$557$	−16.6495	−0.705460	−0.352730	−	0.935725i	$-0.614747\pi$
$557$	−16.6495	−0.705460	−0.352730	+	0.935725i	$0.614747\pi$
$558$	−24.1891	−1.02401
$559$	5.32994	0.225432
$560$	0.839081	0.0354576
$561$	3.91968	0.165489
$562$	−17.8597	−0.753365
$563$	9.59816	0.404514	0.202257	−	0.979332i	$-0.435172\pi$
$563$	9.59816	0.404514	0.202257	+	0.979332i	$0.435172\pi$
$564$	−36.6292	−1.54237
$565$	−6.63988	−0.279342
$566$	−8.25560	−0.347009
$567$	7.22574	0.303453
$568$	−10.8778	−0.456422
$569$	−30.0475	−1.25966	−0.629828	−	0.776735i	$-0.716874\pi$
$569$	−30.0475	−1.25966	−0.629828	+	0.776735i	$0.716874\pi$
$570$	10.2345	0.428676
$571$	36.5916	1.53131	0.765655	−	0.643251i	$-0.222415\pi$
$571$	36.5916	1.53131	0.765655	+	0.643251i	$0.222415\pi$
$572$	2.56870	0.107403
$573$	46.6876	1.95040
$574$	−9.81560	−0.409695
$575$	1.14602	0.0477921
$576$	5.95662	0.248193
$577$	−7.83148	−0.326029	−0.163014	−	0.986624i	$-0.552122\pi$
$577$	−7.83148	−0.326029	−0.163014	+	0.986624i	$0.552122\pi$
$578$	16.9084	0.703297
$579$	49.1714	2.04349
$580$	9.67486	0.401727
$581$	−12.8060	−0.531284
$582$	−47.0042	−1.94839
$583$	−6.43819	−0.266643
$584$	2.30726	0.0954750
$585$	3.53580	0.146188
$586$	1.25307	0.0517639
$587$	−18.3332	−0.756693	−0.378347	−	0.925664i	$-0.623507\pi$
$587$	−18.3332	−0.756693	−0.378347	+	0.925664i	$0.623507\pi$
$588$	18.8423	0.777042
$589$	13.8872	0.572211
$590$	9.58967	0.394801
$591$	−31.5800	−1.29903
$592$	1.09146	0.0448587
$593$	−39.9287	−1.63967	−0.819837	−	0.572597i	$-0.805936\pi$
$593$	−39.9287	−1.63967	−0.819837	+	0.572597i	$0.805936\pi$
$594$	38.2907	1.57109
$595$	−0.253956	−0.0104112
$596$	18.7790	0.769216
$597$	−1.02884	−0.0421074
$598$	−0.680266	−0.0278181
$599$	−6.90810	−0.282257	−0.141129	−	0.989991i	$-0.545073\pi$
$599$	−6.90810	−0.282257	−0.141129	+	0.989991i	$0.545073\pi$
$600$	2.99276	0.122179
$601$	13.1916	0.538097	0.269048	−	0.963127i	$-0.413291\pi$
$601$	13.1916	0.538097	0.269048	+	0.963127i	$0.413291\pi$
$602$	−7.53421	−0.307072
$603$	7.09333	0.288863
$604$	13.9045	0.565768
$605$	7.72619	0.314114
$606$	−21.5566	−0.875677
$607$	13.2390	0.537353	0.268676	−	0.963231i	$-0.413414\pi$
$607$	13.2390	0.537353	0.268676	+	0.963231i	$0.413414\pi$
$608$	−3.41975	−0.138689
$609$	−24.2952	−0.984492
$610$	9.64738	0.390611
$611$	7.26513	0.293916
$612$	−1.80283	−0.0728750
$613$	−21.0730	−0.851131	−0.425565	−	0.904928i	$-0.639925\pi$
$613$	−21.0730	−0.851131	−0.425565	+	0.904928i	$0.639925\pi$
$614$	28.4969	1.15004
$615$	−35.0094	−1.41172
$616$	−3.63102	−0.146298
$617$	13.0182	0.524093	0.262047	−	0.965055i	$-0.415603\pi$
$617$	13.0182	0.524093	0.262047	+	0.965055i	$0.415603\pi$
$618$	15.1793	0.610601
$619$	47.6166	1.91387	0.956936	−	0.290298i	$-0.0937545\pi$
$619$	47.6166	1.91387	0.956936	+	0.290298i	$0.0937545\pi$
$620$	4.06088	0.163089
$621$	−10.1405	−0.406924
$622$	13.9830	0.560666
$623$	−14.6344	−0.586314
$624$	−1.77648	−0.0711161
$625$	1.00000	0.0400000
$626$	7.90339	0.315883
$627$	−44.2885	−1.76871
$628$	20.2531	0.808188
$629$	−0.330340	−0.0131715
$630$	−4.99809	−0.199129
$631$	−37.6167	−1.49750	−0.748750	−	0.662853i	$-0.769345\pi$
$631$	−37.6167	−1.49750	−0.748750	+	0.662853i	$0.769345\pi$
$632$	−6.33947	−0.252170
$633$	18.6714	0.742122
$634$	−17.4503	−0.693039
$635$	4.82348	0.191414
$636$	4.45257	0.176556
$637$	−3.73722	−0.148074
$638$	−41.8668	−1.65752
$639$	64.7950	2.56325
$640$	−1.00000	−0.0395285
$641$	−31.2138	−1.23287	−0.616435	−	0.787406i	$-0.711423\pi$
$641$	−31.2138	−1.23287	−0.616435	+	0.787406i	$0.711423\pi$
$642$	34.1752	1.34879
$643$	−14.6046	−0.575948	−0.287974	−	0.957638i	$-0.592982\pi$
$643$	−14.6046	−0.575948	−0.287974	+	0.957638i	$0.592982\pi$
$644$	0.961600	0.0378923
$645$	−26.8724	−1.05810
$646$	1.03502	0.0407223
$647$	−18.0115	−0.708106	−0.354053	−	0.935225i	$-0.615197\pi$
$647$	−18.0115	−0.708106	−0.354053	+	0.935225i	$0.615197\pi$
$648$	−8.61149	−0.338291
$649$	−41.4981	−1.62894
$650$	−0.593592	−0.0232826
$651$	−10.1975	−0.399673
$652$	−9.22899	−0.361435
$653$	15.2435	0.596526	0.298263	−	0.954484i	$-0.403593\pi$
$653$	15.2435	0.596526	0.298263	+	0.954484i	$0.403593\pi$
$654$	52.5941	2.05659
$655$	−16.2753	−0.635928
$656$	11.6980	0.456732
$657$	−13.7435	−0.536184
$658$	−10.2697	−0.400356
$659$	1.71721	0.0668930	0.0334465	−	0.999441i	$-0.489352\pi$
$659$	1.71721	0.0668930	0.0334465	+	0.999441i	$0.489352\pi$
$660$	−12.9508	−0.504110
$661$	−41.7721	−1.62474	−0.812372	−	0.583139i	$-0.801824\pi$
$661$	−41.7721	−1.62474	−0.812372	+	0.583139i	$0.801824\pi$
$662$	0.657717	0.0255629
$663$	0.537668	0.0208813
$664$	15.2620	0.592280
$665$	2.86945	0.111272
$666$	−6.50141	−0.251924
$667$	11.0875	0.429311
$668$	0.633271	0.0245020
$669$	−60.1848	−2.32688
$670$	−1.19083	−0.0460058
$671$	−41.7478	−1.61166
$672$	2.51117	0.0968705
$673$	32.5104	1.25318	0.626592	−	0.779347i	$-0.284449\pi$
$673$	32.5104	1.25318	0.626592	+	0.779347i	$0.284449\pi$
$674$	0.378100	0.0145639
$675$	−8.84847	−0.340578
$676$	−12.6476	−0.486448
$677$	−44.0696	−1.69373	−0.846865	−	0.531807i	$-0.821513\pi$
$677$	−44.0696	−1.69373	−0.846865	+	0.531807i	$0.821513\pi$
$678$	−19.8716	−0.763163
$679$	−13.1786	−0.505748
$680$	0.302659	0.0116065
$681$	−27.2036	−1.04244
$682$	−17.5729	−0.672903
$683$	2.36766	0.0905962	0.0452981	−	0.998974i	$-0.485576\pi$
$683$	2.36766	0.0905962	0.0452981	+	0.998974i	$0.485576\pi$
$684$	20.3702	0.778873
$685$	−17.9345	−0.685240
$686$	11.1564	0.425952
$687$	52.9193	2.01900
$688$	8.97913	0.342326
$689$	−0.883135	−0.0336448
$690$	3.42975	0.130568
$691$	−17.9148	−0.681513	−0.340756	−	0.940152i	$-0.610683\pi$
$691$	−17.9148	−0.681513	−0.340756	+	0.940152i	$0.610683\pi$
$692$	−22.4477	−0.853333
$693$	21.6286	0.821603
$694$	−4.05906	−0.154080
$695$	15.3836	0.583532
$696$	28.9546	1.09752
$697$	−3.54052	−0.134107
$698$	−30.9447	−1.17127
$699$	10.4606	0.395658
$700$	0.839081	0.0317143
$701$	−49.2924	−1.86175	−0.930874	−	0.365340i	$-0.880953\pi$
$701$	−49.2924	−1.86175	−0.930874	+	0.365340i	$0.880953\pi$
$702$	5.25238	0.198238
$703$	3.73252	0.140775
$704$	4.32738	0.163094
$705$	−36.6292	−1.37954
$706$	37.3739	1.40659
$707$	−6.04383	−0.227302
$708$	28.6996	1.07860
$709$	−20.9005	−0.784936	−0.392468	−	0.919766i	$-0.628379\pi$
$709$	−20.9005	−0.784936	−0.392468	+	0.919766i	$0.628379\pi$
$710$	−10.8778	−0.408237
$711$	37.7618	1.41618
$712$	17.4410	0.653627
$713$	4.65383	0.174287
$714$	−0.760029	−0.0284434
$715$	2.56870	0.0960638
$716$	−24.4322	−0.913075
$717$	43.0301	1.60699
$718$	10.4399	0.389615
$719$	−4.55660	−0.169933	−0.0849663	−	0.996384i	$-0.527078\pi$
$719$	−4.55660	−0.169933	−0.0849663	+	0.996384i	$0.527078\pi$
$720$	5.95662	0.221990
$721$	4.25582	0.158495
$722$	7.30531	0.271876
$723$	−33.0296	−1.22839
$724$	−11.2615	−0.418529
$725$	9.67486	0.359315
$726$	23.1226	0.858162
$727$	42.0995	1.56138	0.780692	−	0.624916i	$-0.214867\pi$
$727$	42.0995	1.56138	0.780692	+	0.624916i	$0.214867\pi$
$728$	−0.498072	−0.0184598
$729$	−28.1487	−1.04254
$730$	2.30726	0.0853954
$731$	−2.71762	−0.100515
$732$	28.8723	1.06715
$733$	32.3623	1.19533	0.597664	−	0.801747i	$-0.296096\pi$
$733$	32.3623	1.19533	0.597664	+	0.801747i	$0.296096\pi$
$734$	13.1737	0.486251
$735$	18.8423	0.695007
$736$	−1.14602	−0.0422427
$737$	5.15317	0.189820
$738$	−69.6808	−2.56499
$739$	49.0259	1.80345	0.901723	−	0.432315i	$-0.142303\pi$
$739$	49.0259	1.80345	0.901723	+	0.432315i	$0.142303\pi$
$740$	1.09146	0.0401228
$741$	−6.07512	−0.223175
$742$	1.24837	0.0458291
$743$	−39.4013	−1.44549	−0.722746	−	0.691114i	$-0.757120\pi$
$743$	−39.4013	−1.44549	−0.722746	+	0.691114i	$0.757120\pi$
$744$	12.1532	0.445559
$745$	18.7790	0.688008
$746$	21.5557	0.789212
$747$	−90.9100	−3.32622
$748$	−1.30972	−0.0478881
$749$	9.58171	0.350108
$750$	2.99276	0.109280
$751$	3.79636	0.138531	0.0692655	−	0.997598i	$-0.477934\pi$
$751$	3.79636	0.138531	0.0692655	+	0.997598i	$0.477934\pi$
$752$	12.2393	0.446320
$753$	2.78493	0.101488
$754$	−5.74292	−0.209145
$755$	13.9045	0.506038
$756$	−7.42458	−0.270029
$757$	−11.3485	−0.412468	−0.206234	−	0.978503i	$-0.566121\pi$
$757$	−11.3485	−0.412468	−0.206234	+	0.978503i	$0.566121\pi$
$758$	−34.0943	−1.23836
$759$	−14.8418	−0.538724
$760$	−3.41975	−0.124047
$761$	−6.62460	−0.240142	−0.120071	−	0.992765i	$-0.538312\pi$
$761$	−6.62460	−0.240142	−0.120071	+	0.992765i	$0.538312\pi$
$762$	14.4355	0.522944
$763$	14.7458	0.533835
$764$	−15.6002	−0.564394
$765$	−1.80283	−0.0651814
$766$	−23.9191	−0.864233
$767$	−5.69235	−0.205539
$768$	−2.99276	−0.107992
$769$	6.81088	0.245607	0.122803	−	0.992431i	$-0.460812\pi$
$769$	6.81088	0.245607	0.122803	+	0.992431i	$0.460812\pi$
$770$	−3.63102	−0.130853
$771$	−66.2184	−2.38480
$772$	−16.4301	−0.591332
$773$	−27.5928	−0.992445	−0.496222	−	0.868195i	$-0.665280\pi$
$773$	−27.5928	−0.992445	−0.496222	+	0.868195i	$0.665280\pi$
$774$	−53.4853	−1.92249
$775$	4.06088	0.145871
$776$	15.7060	0.563812
$777$	−2.74084	−0.0983270
$778$	29.4422	1.05555
$779$	40.0044	1.43331
$780$	−1.77648	−0.0636082
$781$	47.0723	1.68438
$782$	0.346852	0.0124034
$783$	−85.6077	−3.05937
$784$	−6.29594	−0.224855
$785$	20.2531	0.722865
$786$	−48.7081	−1.73736
$787$	33.5384	1.19552	0.597758	−	0.801677i	$-0.296058\pi$
$787$	33.5384	1.19552	0.597758	+	0.801677i	$0.296058\pi$
$788$	10.5521	0.375904
$789$	−37.7973	−1.34562
$790$	−6.33947	−0.225548
$791$	−5.57140	−0.198096
$792$	−25.7766	−0.915930
$793$	−5.72660	−0.203358
$794$	0.482928	0.0171385
$795$	4.45257	0.157917
$796$	0.343775	0.0121848
$797$	27.0153	0.956932	0.478466	−	0.878106i	$-0.341193\pi$
$797$	27.0153	0.956932	0.478466	+	0.878106i	$0.341193\pi$
$798$	8.58757	0.303997
$799$	−3.70433	−0.131050
$800$	−1.00000	−0.0353553
$801$	−103.889	−3.67074
$802$	1.00000	0.0353112
$803$	−9.98437	−0.352341
$804$	−3.56387	−0.125688
$805$	0.961600	0.0338919
$806$	−2.41050	−0.0849064
$807$	−97.0096	−3.41490
$808$	7.20292	0.253398
$809$	48.8777	1.71845	0.859225	−	0.511598i	$-0.170946\pi$
$809$	48.8777	1.71845	0.859225	+	0.511598i	$0.170946\pi$
$810$	−8.61149	−0.302577
$811$	−13.1227	−0.460802	−0.230401	−	0.973096i	$-0.574004\pi$
$811$	−13.1227	−0.460802	−0.230401	+	0.973096i	$0.574004\pi$
$812$	8.11799	0.284886
$813$	15.0006	0.526092
$814$	−4.72315	−0.165546
$815$	−9.22899	−0.323277
$816$	0.905787	0.0317089
$817$	30.7064	1.07428
$818$	−31.6820	−1.10774
$819$	2.96683	0.103669
$820$	11.6980	0.408513
$821$	−35.7394	−1.24732	−0.623658	−	0.781698i	$-0.714354\pi$
$821$	−35.7394	−1.24732	−0.623658	+	0.781698i	$0.714354\pi$
$822$	−53.6735	−1.87208
$823$	−42.8896	−1.49504	−0.747518	−	0.664242i	$-0.768755\pi$
$823$	−42.8896	−1.49504	−0.747518	+	0.664242i	$0.768755\pi$
$824$	−5.07200	−0.176692
$825$	−12.9508	−0.450889
$826$	8.04651	0.279974
$827$	53.8215	1.87156	0.935779	−	0.352588i	$-0.114698\pi$
$827$	53.8215	1.87156	0.935779	+	0.352588i	$0.114698\pi$
$828$	6.82638	0.237233
$829$	21.7082	0.753958	0.376979	−	0.926222i	$-0.376963\pi$
$829$	21.7082	0.753958	0.376979	+	0.926222i	$0.376963\pi$
$830$	15.2620	0.529752
$831$	−72.6911	−2.52163
$832$	0.593592	0.0205791
$833$	1.90553	0.0660225
$834$	46.0394	1.59421
$835$	0.633271	0.0219152
$836$	14.7985	0.511818
$837$	−35.9325	−1.24201
$838$	−16.7551	−0.578794
$839$	−35.7812	−1.23530	−0.617652	−	0.786452i	$-0.711916\pi$
$839$	−35.7812	−1.23530	−0.617652	+	0.786452i	$0.711916\pi$
$840$	2.51117	0.0866436
$841$	64.6029	2.22769
$842$	−30.1762	−1.03994
$843$	−53.4497	−1.84091
$844$	−6.23885	−0.214750
$845$	−12.6476	−0.435092
$846$	−72.9047	−2.50652
$847$	6.48290	0.222755
$848$	−1.48778	−0.0510906
$849$	−24.7070	−0.847943
$850$	0.302659	0.0103811
$851$	1.25083	0.0428778
$852$	−32.5547	−1.11530
$853$	43.8670	1.50198	0.750989	−	0.660314i	$-0.229577\pi$
$853$	43.8670	1.50198	0.750989	+	0.660314i	$0.229577\pi$
$854$	8.09493	0.277003
$855$	20.3702	0.696645
$856$	−11.4193	−0.390303
$857$	−49.5183	−1.69151	−0.845756	−	0.533570i	$-0.820850\pi$
$857$	−49.5183	−1.69151	−0.845756	+	0.533570i	$0.820850\pi$
$858$	7.68750	0.262447
$859$	43.8201	1.49512	0.747562	−	0.664192i	$-0.231224\pi$
$859$	43.8201	1.49512	0.747562	+	0.664192i	$0.231224\pi$
$860$	8.97913	0.306186
$861$	−29.3758	−1.00112
$862$	8.08564	0.275398
$863$	−52.1923	−1.77665	−0.888324	−	0.459217i	$-0.848130\pi$
$863$	−52.1923	−1.77665	−0.888324	+	0.459217i	$0.848130\pi$
$864$	8.84847	0.301031
$865$	−22.4477	−0.763244
$866$	15.2624	0.518636
$867$	50.6028	1.71856
$868$	3.40740	0.115655
$869$	27.4333	0.930610
$870$	28.9546	0.981651
$871$	0.706867	0.0239513
$872$	−17.5738	−0.595123
$873$	−93.5546	−3.16634
$874$	−3.91909	−0.132565
$875$	0.839081	0.0283661
$876$	6.90507	0.233301
$877$	−19.8742	−0.671103	−0.335552	−	0.942022i	$-0.608923\pi$
$877$	−19.8742	−0.671103	−0.335552	+	0.942022i	$0.608923\pi$
$878$	−28.3671	−0.957344
$879$	3.75015	0.126489
$880$	4.32738	0.145876
$881$	−19.0319	−0.641201	−0.320600	−	0.947215i	$-0.603885\pi$
$881$	−19.0319	−0.641201	−0.320600	+	0.947215i	$0.603885\pi$
$882$	37.5026	1.26278
$883$	20.4164	0.687067	0.343534	−	0.939140i	$-0.388376\pi$
$883$	20.4164	0.687067	0.343534	+	0.939140i	$0.388376\pi$
$884$	−0.179656	−0.00604249
$885$	28.6996	0.964727
$886$	0.173228	0.00581972
$887$	37.0330	1.24345	0.621723	−	0.783237i	$-0.286433\pi$
$887$	37.0330	1.24345	0.621723	+	0.783237i	$0.286433\pi$
$888$	3.26648	0.109616
$889$	4.04729	0.135742
$890$	17.4410	0.584622
$891$	37.2652	1.24843
$892$	20.1101	0.673337
$893$	41.8552	1.40063
$894$	56.2010	1.87964
$895$	−24.4322	−0.816680
$896$	−0.839081	−0.0280317
$897$	−2.03587	−0.0679758
$898$	8.47088	0.282677
$899$	39.2884	1.31034
$900$	5.95662	0.198554
$901$	0.450291	0.0150014
$902$	−50.6218	−1.68552
$903$	−22.5481	−0.750354
$904$	6.63988	0.220839
$905$	−11.2615	−0.374343
$906$	41.6130	1.38250
$907$	−45.0346	−1.49535	−0.747675	−	0.664065i	$-0.768830\pi$
$907$	−45.0346	−1.49535	−0.747675	+	0.664065i	$0.768830\pi$
$908$	9.08979	0.301655
$909$	−42.9051	−1.42307
$910$	−0.498072	−0.0165109
$911$	−3.85457	−0.127707	−0.0638537	−	0.997959i	$-0.520339\pi$
$911$	−3.85457	−0.127707	−0.0638537	+	0.997959i	$0.520339\pi$
$912$	−10.2345	−0.338898
$913$	−66.0444	−2.18575
$914$	−1.81101	−0.0599028
$915$	28.8723	0.954489
$916$	−17.6824	−0.584244
$917$	−13.6563	−0.450970
$918$	−2.67807	−0.0883895
$919$	−20.3674	−0.671860	−0.335930	−	0.941887i	$-0.609051\pi$
$919$	−20.3674	−0.671860	−0.335930	+	0.941887i	$0.609051\pi$
$920$	−1.14602	−0.0377830
$921$	85.2843	2.81021
$922$	−10.4940	−0.345601
$923$	6.45698	0.212534
$924$	−10.8668	−0.357491
$925$	1.09146	0.0358869
$926$	−22.4969	−0.739295
$927$	30.2120	0.992293
$928$	−9.67486	−0.317593
$929$	19.1067	0.626871	0.313435	−	0.949610i	$-0.398520\pi$
$929$	19.1067	0.626871	0.313435	+	0.949610i	$0.398520\pi$
$930$	12.1532	0.398520
$931$	−21.5306	−0.705635
$932$	−3.49531	−0.114493
$933$	41.8477	1.37003
$934$	−0.478185	−0.0156467
$935$	−1.30972	−0.0428324
$936$	−3.53580	−0.115571
$937$	−30.0046	−0.980206	−0.490103	−	0.871665i	$-0.663041\pi$
$937$	−30.0046	−0.980206	−0.490103	+	0.871665i	$0.663041\pi$
$938$	−0.999203	−0.0326251
$939$	23.6530	0.771885
$940$	12.2393	0.399201
$941$	−32.2150	−1.05018	−0.525090	−	0.851047i	$-0.675968\pi$
$941$	−32.2150	−1.05018	−0.525090	+	0.851047i	$0.675968\pi$
$942$	60.6128	1.97487
$943$	13.4061	0.436564
$944$	−9.58967	−0.312117
$945$	−7.42458	−0.241522
$946$	−38.8561	−1.26332
$947$	−13.2695	−0.431201	−0.215601	−	0.976482i	$-0.569171\pi$
$947$	−13.2695	−0.431201	−0.215601	+	0.976482i	$0.569171\pi$
$948$	−18.9725	−0.616199
$949$	−1.36957	−0.0444581
$950$	−3.41975	−0.110951
$951$	−52.2245	−1.69350
$952$	0.253956	0.00823075
$953$	23.1134	0.748716	0.374358	−	0.927284i	$-0.377863\pi$
$953$	23.1134	0.748716	0.374358	+	0.927284i	$0.377863\pi$
$954$	8.86215	0.286923
$955$	−15.6002	−0.504809
$956$	−14.3780	−0.465019
$957$	−125.297	−4.05029
$958$	12.4817	0.403267
$959$	−15.0485	−0.485940
$960$	−2.99276	−0.0965910
$961$	−14.5093	−0.468042
$962$	−0.647881	−0.0208885
$963$	68.0204	2.19193
$964$	11.0365	0.355462
$965$	−16.4301	−0.528904
$966$	2.87784	0.0925930
$967$	1.74945	0.0562585	0.0281293	−	0.999604i	$-0.491045\pi$
$967$	1.74945	0.0562585	0.0281293	+	0.999604i	$0.491045\pi$
$968$	−7.72619	−0.248329
$969$	3.09757	0.0995081
$970$	15.7060	0.504288
$971$	26.6753	0.856053	0.428026	−	0.903766i	$-0.359209\pi$
$971$	26.6753	0.856053	0.428026	+	0.903766i	$0.359209\pi$
$972$	0.773260	0.0248023
$973$	12.9081	0.413813
$974$	−23.9463	−0.767289
$975$	−1.77648	−0.0568929
$976$	−9.64738	−0.308805
$977$	21.2737	0.680607	0.340303	−	0.940316i	$-0.389470\pi$
$977$	21.2737	0.680607	0.340303	+	0.940316i	$0.389470\pi$
$978$	−27.6202	−0.883196
$979$	−75.4736	−2.41215
$980$	−6.29594	−0.201117
$981$	104.680	3.34219
$982$	5.14047	0.164039
$983$	−32.9808	−1.05192	−0.525962	−	0.850508i	$-0.676295\pi$
$983$	−32.9808	−1.05192	−0.525962	+	0.850508i	$0.676295\pi$
$984$	35.0094	1.11606
$985$	10.5521	0.336218
$986$	2.92819	0.0932524
$987$	−30.7349	−0.978302
$988$	2.02994	0.0645809
$989$	10.2902	0.327210
$990$	−25.7766	−0.819233
$991$	35.8833	1.13987	0.569935	−	0.821690i	$-0.306968\pi$
$991$	35.8833	1.13987	0.569935	+	0.821690i	$0.306968\pi$
$992$	−4.06088	−0.128933
$993$	1.96839	0.0624649
$994$	−9.12736	−0.289502
$995$	0.343775	0.0108984
$996$	45.6755	1.44728
$997$	8.69142	0.275260	0.137630	−	0.990484i	$-0.456051\pi$
$997$	8.69142	0.275260	0.137630	+	0.990484i	$0.456051\pi$
$998$	34.0657	1.07833
$999$	−9.65774	−0.305557

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.m.1.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.m.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.99276 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.99276 q^{6} +0.839081 q^{7} -1.00000 q^{8} +5.95662 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.99276 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.99276 q^{6} +0.839081 q^{7} -1.00000 q^{8} +5.95662 q^{9} -1.00000 q^{10} +4.32738 q^{11} -2.99276 q^{12} +0.593592 q^{13} -0.839081 q^{14} -2.99276 q^{15} +1.00000 q^{16} -0.302659 q^{17} -5.95662 q^{18} +3.41975 q^{19} +1.00000 q^{20} -2.51117 q^{21} -4.32738 q^{22} +1.14602 q^{23} +2.99276 q^{24} +1.00000 q^{25} -0.593592 q^{26} -8.84847 q^{27} +0.839081 q^{28} +9.67486 q^{29} +2.99276 q^{30} +4.06088 q^{31} -1.00000 q^{32} -12.9508 q^{33} +0.302659 q^{34} +0.839081 q^{35} +5.95662 q^{36} +1.09146 q^{37} -3.41975 q^{38} -1.77648 q^{39} -1.00000 q^{40} +11.6980 q^{41} +2.51117 q^{42} +8.97913 q^{43} +4.32738 q^{44} +5.95662 q^{45} -1.14602 q^{46} +12.2393 q^{47} -2.99276 q^{48} -6.29594 q^{49} -1.00000 q^{50} +0.905787 q^{51} +0.593592 q^{52} -1.48778 q^{53} +8.84847 q^{54} +4.32738 q^{55} -0.839081 q^{56} -10.2345 q^{57} -9.67486 q^{58} -9.58967 q^{59} -2.99276 q^{60} -9.64738 q^{61} -4.06088 q^{62} +4.99809 q^{63} +1.00000 q^{64} +0.593592 q^{65} +12.9508 q^{66} +1.19083 q^{67} -0.302659 q^{68} -3.42975 q^{69} -0.839081 q^{70} +10.8778 q^{71} -5.95662 q^{72} -2.30726 q^{73} -1.09146 q^{74} -2.99276 q^{75} +3.41975 q^{76} +3.63102 q^{77} +1.77648 q^{78} +6.33947 q^{79} +1.00000 q^{80} +8.61149 q^{81} -11.6980 q^{82} -15.2620 q^{83} -2.51117 q^{84} -0.302659 q^{85} -8.97913 q^{86} -28.9546 q^{87} -4.32738 q^{88} -17.4410 q^{89} -5.95662 q^{90} +0.498072 q^{91} +1.14602 q^{92} -12.1532 q^{93} -12.2393 q^{94} +3.41975 q^{95} +2.99276 q^{96} -15.7060 q^{97} +6.29594 q^{98} +25.7766 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) 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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