LMFDB - Embedded newform 4006.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.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:	$31.9880710497$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.94307 q^{3} +1.00000 q^{4} -2.92103 q^{5} -2.94307 q^{6} -1.64254 q^{7} +1.00000 q^{8} +5.66165 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.94307 q^{3} +1.00000 q^{4} -2.92103 q^{5} -2.94307 q^{6} -1.64254 q^{7} +1.00000 q^{8} +5.66165 q^{9} -2.92103 q^{10} +1.25720 q^{11} -2.94307 q^{12} -1.66284 q^{13} -1.64254 q^{14} +8.59679 q^{15} +1.00000 q^{16} +4.45720 q^{17} +5.66165 q^{18} -4.03054 q^{19} -2.92103 q^{20} +4.83412 q^{21} +1.25720 q^{22} -4.68796 q^{23} -2.94307 q^{24} +3.53241 q^{25} -1.66284 q^{26} -7.83342 q^{27} -1.64254 q^{28} +9.17058 q^{29} +8.59679 q^{30} -4.91974 q^{31} +1.00000 q^{32} -3.70002 q^{33} +4.45720 q^{34} +4.79792 q^{35} +5.66165 q^{36} +5.52361 q^{37} -4.03054 q^{38} +4.89384 q^{39} -2.92103 q^{40} +5.68787 q^{41} +4.83412 q^{42} +11.3508 q^{43} +1.25720 q^{44} -16.5378 q^{45} -4.68796 q^{46} -1.32437 q^{47} -2.94307 q^{48} -4.30205 q^{49} +3.53241 q^{50} -13.1179 q^{51} -1.66284 q^{52} +12.4934 q^{53} -7.83342 q^{54} -3.67231 q^{55} -1.64254 q^{56} +11.8622 q^{57} +9.17058 q^{58} +9.88554 q^{59} +8.59679 q^{60} -13.9052 q^{61} -4.91974 q^{62} -9.29951 q^{63} +1.00000 q^{64} +4.85720 q^{65} -3.70002 q^{66} -0.382778 q^{67} +4.45720 q^{68} +13.7970 q^{69} +4.79792 q^{70} +0.361868 q^{71} +5.66165 q^{72} -10.2001 q^{73} +5.52361 q^{74} -10.3961 q^{75} -4.03054 q^{76} -2.06500 q^{77} +4.89384 q^{78} -7.96455 q^{79} -2.92103 q^{80} +6.06935 q^{81} +5.68787 q^{82} -7.89601 q^{83} +4.83412 q^{84} -13.0196 q^{85} +11.3508 q^{86} -26.9897 q^{87} +1.25720 q^{88} -3.01688 q^{89} -16.5378 q^{90} +2.73128 q^{91} -4.68796 q^{92} +14.4791 q^{93} -1.32437 q^{94} +11.7733 q^{95} -2.94307 q^{96} +11.9998 q^{97} -4.30205 q^{98} +7.11781 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) $ 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9 - 23 * q^10 - 32 * q^11 - 13 * q^12 - 8 * q^13 - 18 * q^14 - 14 * q^15 + 31 * q^16 - 30 * q^17 + 20 * q^18 - 38 * q^19 - 23 * q^20 - 16 * q^21 - 32 * q^22 - 24 * q^23 - 13 * q^24 + 40 * q^25 - 8 * q^26 - 28 * q^27 - 18 * q^28 - 7 * q^29 - 14 * q^30 - 32 * q^31 + 31 * q^32 - 9 * q^33 - 30 * q^34 - 14 * q^35 + 20 * q^36 + 11 * q^37 - 38 * q^38 - 9 * q^39 - 23 * q^40 - 76 * q^41 - 16 * q^42 - 33 * q^43 - 32 * q^44 - 40 * q^45 - 24 * q^46 - 96 * q^47 - 13 * q^48 + 15 * q^49 + 40 * q^50 - 55 * q^51 - 8 * q^52 - 28 * q^53 - 28 * q^54 - 52 * q^55 - 18 * q^56 - 21 * q^57 - 7 * q^58 - 72 * q^59 - 14 * q^60 - 9 * q^61 - 32 * q^62 - 54 * q^63 + 31 * q^64 - 38 * q^65 - 9 * q^66 - 4 * q^67 - 30 * q^68 - 17 * q^69 - 14 * q^70 - 61 * q^71 + 20 * q^72 - 62 * q^73 + 11 * q^74 - 63 * q^75 - 38 * q^76 - 9 * q^77 - 9 * q^78 - 30 * q^79 - 23 * q^80 - 13 * q^81 - 76 * q^82 - 90 * q^83 - 16 * q^84 + 26 * q^85 - 33 * q^86 - 34 * q^87 - 32 * q^88 - 99 * q^89 - 40 * q^90 - 47 * q^91 - 24 * q^92 - 6 * q^93 - 96 * q^94 - 24 * q^95 - 13 * q^96 - 46 * q^97 + 15 * q^98 - 48 * 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
$3$	−2.94307	−1.69918	−0.849591	−	0.527443i	$-0.823151\pi$
$3$	−2.94307	−1.69918	−0.849591	+	0.527443i	$0.823151\pi$
$4$	1.00000	0.500000
$5$	−2.92103	−1.30632	−0.653162	−	0.757218i	$-0.726558\pi$
$5$	−2.92103	−1.30632	−0.653162	+	0.757218i	$0.726558\pi$
$6$	−2.94307	−1.20150
$7$	−1.64254	−0.620823	−0.310412	−	0.950602i	$-0.600467\pi$
$7$	−1.64254	−0.620823	−0.310412	+	0.950602i	$0.600467\pi$
$8$	1.00000	0.353553
$9$	5.66165	1.88722
$10$	−2.92103	−0.923710
$11$	1.25720	0.379059	0.189530	−	0.981875i	$-0.439304\pi$
$11$	1.25720	0.379059	0.189530	+	0.981875i	$0.439304\pi$
$12$	−2.94307	−0.849591
$13$	−1.66284	−0.461188	−0.230594	−	0.973050i	$-0.574067\pi$
$13$	−1.66284	−0.461188	−0.230594	+	0.973050i	$0.574067\pi$
$14$	−1.64254	−0.438988
$15$	8.59679	2.21968
$16$	1.00000	0.250000
$17$	4.45720	1.08103	0.540515	−	0.841334i	$-0.318229\pi$
$17$	4.45720	1.08103	0.540515	+	0.841334i	$0.318229\pi$
$18$	5.66165	1.33446
$19$	−4.03054	−0.924669	−0.462334	−	0.886706i	$-0.652988\pi$
$19$	−4.03054	−0.924669	−0.462334	+	0.886706i	$0.652988\pi$
$20$	−2.92103	−0.653162
$21$	4.83412	1.05489
$22$	1.25720	0.268035
$23$	−4.68796	−0.977508	−0.488754	−	0.872422i	$-0.662548\pi$
$23$	−4.68796	−0.977508	−0.488754	+	0.872422i	$0.662548\pi$
$24$	−2.94307	−0.600751
$25$	3.53241	0.706482
$26$	−1.66284	−0.326109
$27$	−7.83342	−1.50754
$28$	−1.64254	−0.310412
$29$	9.17058	1.70293	0.851467	−	0.524408i	$-0.175713\pi$
$29$	9.17058	1.70293	0.851467	+	0.524408i	$0.175713\pi$
$30$	8.59679	1.56955
$31$	−4.91974	−0.883612	−0.441806	−	0.897111i	$-0.645662\pi$
$31$	−4.91974	−0.883612	−0.441806	+	0.897111i	$0.645662\pi$
$32$	1.00000	0.176777
$33$	−3.70002	−0.644091
$34$	4.45720	0.764404
$35$	4.79792	0.810996
$36$	5.66165	0.943609
$37$	5.52361	0.908076	0.454038	−	0.890982i	$-0.349983\pi$
$37$	5.52361	0.908076	0.454038	+	0.890982i	$0.349983\pi$
$38$	−4.03054	−0.653840
$39$	4.89384	0.783642
$40$	−2.92103	−0.461855
$41$	5.68787	0.888296	0.444148	−	0.895953i	$-0.353506\pi$
$41$	5.68787	0.888296	0.444148	+	0.895953i	$0.353506\pi$
$42$	4.83412	0.745921
$43$	11.3508	1.73098	0.865492	−	0.500923i	$-0.167006\pi$
$43$	11.3508	1.73098	0.865492	+	0.500923i	$0.167006\pi$
$44$	1.25720	0.189530
$45$	−16.5378	−2.46532
$46$	−4.68796	−0.691203
$47$	−1.32437	−0.193180	−0.0965899	−	0.995324i	$-0.530794\pi$
$47$	−1.32437	−0.193180	−0.0965899	+	0.995324i	$0.530794\pi$
$48$	−2.94307	−0.424795
$49$	−4.30205	−0.614579
$50$	3.53241	0.499558
$51$	−13.1179	−1.83687
$52$	−1.66284	−0.230594
$53$	12.4934	1.71610	0.858051	−	0.513565i	$-0.171675\pi$
$53$	12.4934	1.71610	0.858051	+	0.513565i	$0.171675\pi$
$54$	−7.83342	−1.06599
$55$	−3.67231	−0.495174
$56$	−1.64254	−0.219494
$57$	11.8622	1.57118
$58$	9.17058	1.20416
$59$	9.88554	1.28699	0.643494	−	0.765451i	$-0.277484\pi$
$59$	9.88554	1.28699	0.643494	+	0.765451i	$0.277484\pi$
$60$	8.59679	1.10984
$61$	−13.9052	−1.78038	−0.890191	−	0.455587i	$-0.849429\pi$
$61$	−13.9052	−1.78038	−0.890191	+	0.455587i	$0.849429\pi$
$62$	−4.91974	−0.624808
$63$	−9.29951	−1.17163
$64$	1.00000	0.125000
$65$	4.85720	0.602461
$66$	−3.70002	−0.455441
$67$	−0.382778	−0.0467638	−0.0233819	−	0.999727i	$-0.507443\pi$
$67$	−0.382778	−0.0467638	−0.0233819	+	0.999727i	$0.507443\pi$
$68$	4.45720	0.540515
$69$	13.7970	1.66096
$70$	4.79792	0.573461
$71$	0.361868	0.0429458	0.0214729	−	0.999769i	$-0.493164\pi$
$71$	0.361868	0.0429458	0.0214729	+	0.999769i	$0.493164\pi$
$72$	5.66165	0.667232
$73$	−10.2001	−1.19383	−0.596917	−	0.802303i	$-0.703608\pi$
$73$	−10.2001	−1.19383	−0.596917	+	0.802303i	$0.703608\pi$
$74$	5.52361	0.642107
$75$	−10.3961	−1.20044
$76$	−4.03054	−0.462334
$77$	−2.06500	−0.235329
$78$	4.89384	0.554119
$79$	−7.96455	−0.896082	−0.448041	−	0.894013i	$-0.647878\pi$
$79$	−7.96455	−0.896082	−0.448041	+	0.894013i	$0.647878\pi$
$80$	−2.92103	−0.326581
$81$	6.06935	0.674372
$82$	5.68787	0.628120
$83$	−7.89601	−0.866700	−0.433350	−	0.901226i	$-0.642669\pi$
$83$	−7.89601	−0.866700	−0.433350	+	0.901226i	$0.642669\pi$
$84$	4.83412	0.527446
$85$	−13.0196	−1.41218
$86$	11.3508	1.22399
$87$	−26.9897	−2.89359
$88$	1.25720	0.134018
$89$	−3.01688	−0.319789	−0.159895	−	0.987134i	$-0.551115\pi$
$89$	−3.01688	−0.319789	−0.159895	+	0.987134i	$0.551115\pi$
$90$	−16.5378	−1.74324
$91$	2.73128	0.286316
$92$	−4.68796	−0.488754
$93$	14.4791	1.50142
$94$	−1.32437	−0.136599
$95$	11.7733	1.20792
$96$	−2.94307	−0.300376
$97$	11.9998	1.21839	0.609197	−	0.793019i	$-0.291492\pi$
$97$	11.9998	1.21839	0.609197	+	0.793019i	$0.291492\pi$
$98$	−4.30205	−0.434573
$99$	7.11781	0.715367
$100$	3.53241	0.353241
$101$	−2.88724	−0.287291	−0.143646	−	0.989629i	$-0.545883\pi$
$101$	−2.88724	−0.287291	−0.143646	+	0.989629i	$0.545883\pi$
$102$	−13.1179	−1.29886
$103$	16.3206	1.60812	0.804058	−	0.594551i	$-0.202670\pi$
$103$	16.3206	1.60812	0.804058	+	0.594551i	$0.202670\pi$
$104$	−1.66284	−0.163055
$105$	−14.1206	−1.37803
$106$	12.4934	1.21347
$107$	−11.0863	−1.07175	−0.535876	−	0.844297i	$-0.680019\pi$
$107$	−11.0863	−1.07175	−0.535876	+	0.844297i	$0.680019\pi$
$108$	−7.83342	−0.753772
$109$	4.44845	0.426085	0.213042	−	0.977043i	$-0.431663\pi$
$109$	4.44845	0.426085	0.213042	+	0.977043i	$0.431663\pi$
$110$	−3.67231	−0.350141
$111$	−16.2564	−1.54299
$112$	−1.64254	−0.155206
$113$	−2.79711	−0.263130	−0.131565	−	0.991308i	$-0.542000\pi$
$113$	−2.79711	−0.263130	−0.131565	+	0.991308i	$0.542000\pi$
$114$	11.8622	1.11099
$115$	13.6937	1.27694
$116$	9.17058	0.851467
$117$	−9.41441	−0.870362
$118$	9.88554	0.910038
$119$	−7.32115	−0.671129
$120$	8.59679	0.784776
$121$	−9.41945	−0.856314
$122$	−13.9052	−1.25892
$123$	−16.7398	−1.50938
$124$	−4.91974	−0.441806
$125$	4.28688	0.383430
$126$	−9.29951	−0.828466
$127$	−1.79939	−0.159670	−0.0798352	−	0.996808i	$-0.525439\pi$
$127$	−1.79939	−0.159670	−0.0798352	+	0.996808i	$0.525439\pi$
$128$	1.00000	0.0883883
$129$	−33.4062	−2.94125
$130$	4.85720	0.426004
$131$	−7.55304	−0.659912	−0.329956	−	0.943996i	$-0.607034\pi$
$131$	−7.55304	−0.659912	−0.329956	+	0.943996i	$0.607034\pi$
$132$	−3.70002	−0.322045
$133$	6.62034	0.574056
$134$	−0.382778	−0.0330670
$135$	22.8817	1.96934
$136$	4.45720	0.382202
$137$	−18.4869	−1.57945	−0.789723	−	0.613464i	$-0.789776\pi$
$137$	−18.4869	−1.57945	−0.789723	+	0.613464i	$0.789776\pi$
$138$	13.7970	1.17448
$139$	−19.8690	−1.68526	−0.842632	−	0.538490i	$-0.818995\pi$
$139$	−19.8690	−1.68526	−0.842632	+	0.538490i	$0.818995\pi$
$140$	4.79792	0.405498
$141$	3.89772	0.328247
$142$	0.361868	0.0303673
$143$	−2.09052	−0.174818
$144$	5.66165	0.471804
$145$	−26.7875	−2.22458
$146$	−10.2001	−0.844168
$147$	12.6612	1.04428
$148$	5.52361	0.454038
$149$	−16.4084	−1.34423	−0.672114	−	0.740448i	$-0.734613\pi$
$149$	−16.4084	−1.34423	−0.672114	+	0.740448i	$0.734613\pi$
$150$	−10.3961	−0.848840
$151$	−7.54365	−0.613894	−0.306947	−	0.951727i	$-0.599307\pi$
$151$	−7.54365	−0.613894	−0.306947	+	0.951727i	$0.599307\pi$
$152$	−4.03054	−0.326920
$153$	25.2351	2.04014
$154$	−2.06500	−0.166403
$155$	14.3707	1.15428
$156$	4.89384	0.391821
$157$	19.3011	1.54039	0.770196	−	0.637808i	$-0.220158\pi$
$157$	19.3011	1.54039	0.770196	+	0.637808i	$0.220158\pi$
$158$	−7.96455	−0.633626
$159$	−36.7690	−2.91597
$160$	−2.92103	−0.230928
$161$	7.70018	0.606860
$162$	6.06935	0.476853
$163$	−23.4058	−1.83328	−0.916640	−	0.399714i	$-0.869110\pi$
$163$	−23.4058	−1.83328	−0.916640	+	0.399714i	$0.869110\pi$
$164$	5.68787	0.444148
$165$	10.8079	0.841391
$166$	−7.89601	−0.612849
$167$	−13.3421	−1.03244	−0.516221	−	0.856455i	$-0.672662\pi$
$167$	−13.3421	−1.03244	−0.516221	+	0.856455i	$0.672662\pi$
$168$	4.83412	0.372960
$169$	−10.2350	−0.787305
$170$	−13.0196	−0.998559
$171$	−22.8195	−1.74505
$172$	11.3508	0.865492
$173$	−8.67377	−0.659454	−0.329727	−	0.944076i	$-0.606957\pi$
$173$	−8.67377	−0.659454	−0.329727	+	0.944076i	$0.606957\pi$
$174$	−26.9897	−2.04608
$175$	−5.80214	−0.438600
$176$	1.25720	0.0947648
$177$	−29.0938	−2.18683
$178$	−3.01688	−0.226125
$179$	−14.2629	−1.06606	−0.533028	−	0.846097i	$-0.678946\pi$
$179$	−14.2629	−1.06606	−0.533028	+	0.846097i	$0.678946\pi$
$180$	−16.5378	−1.23266
$181$	−11.7906	−0.876389	−0.438195	−	0.898880i	$-0.644382\pi$
$181$	−11.7906	−0.876389	−0.438195	+	0.898880i	$0.644382\pi$
$182$	2.73128	0.202456
$183$	40.9240	3.02519
$184$	−4.68796	−0.345601
$185$	−16.1346	−1.18624
$186$	14.4791	1.06166
$187$	5.60359	0.409775
$188$	−1.32437	−0.0965899
$189$	12.8667	0.935918
$190$	11.7733	0.854126
$191$	−15.3693	−1.11208	−0.556040	−	0.831155i	$-0.687680\pi$
$191$	−15.3693	−1.11208	−0.556040	+	0.831155i	$0.687680\pi$
$192$	−2.94307	−0.212398
$193$	1.91946	0.138166	0.0690828	−	0.997611i	$-0.477993\pi$
$193$	1.91946	0.138166	0.0690828	+	0.997611i	$0.477993\pi$
$194$	11.9998	0.861534
$195$	−14.2951	−1.02369
$196$	−4.30205	−0.307289
$197$	11.8963	0.847574	0.423787	−	0.905762i	$-0.360700\pi$
$197$	11.8963	0.847574	0.423787	+	0.905762i	$0.360700\pi$
$198$	7.11781	0.505841
$199$	0.978478	0.0693625	0.0346812	−	0.999398i	$-0.488958\pi$
$199$	0.978478	0.0693625	0.0346812	+	0.999398i	$0.488958\pi$
$200$	3.53241	0.249779
$201$	1.12654	0.0794601
$202$	−2.88724	−0.203146
$203$	−15.0631	−1.05722
$204$	−13.1179	−0.918434
$205$	−16.6144	−1.16040
$206$	16.3206	1.13711
$207$	−26.5416	−1.84477
$208$	−1.66284	−0.115297
$209$	−5.06718	−0.350504
$210$	−14.1206	−0.974414
$211$	15.9955	1.10117	0.550587	−	0.834778i	$-0.314404\pi$
$211$	15.9955	1.10117	0.550587	+	0.834778i	$0.314404\pi$
$212$	12.4934	0.858051
$213$	−1.06500	−0.0729727
$214$	−11.0863	−0.757843
$215$	−33.1561	−2.26122
$216$	−7.83342	−0.532997
$217$	8.08089	0.548567
$218$	4.44845	0.301287
$219$	30.0197	2.02854
$220$	−3.67231	−0.247587
$221$	−7.41161	−0.498559
$222$	−16.2564	−1.09106
$223$	19.6613	1.31662	0.658309	−	0.752748i	$-0.271272\pi$
$223$	19.6613	1.31662	0.658309	+	0.752748i	$0.271272\pi$
$224$	−1.64254	−0.109747
$225$	19.9993	1.33328
$226$	−2.79711	−0.186061
$227$	−1.11266	−0.0738498	−0.0369249	−	0.999318i	$-0.511756\pi$
$227$	−1.11266	−0.0738498	−0.0369249	+	0.999318i	$0.511756\pi$
$228$	11.8622	0.785590
$229$	25.7580	1.70214	0.851069	−	0.525053i	$-0.175955\pi$
$229$	25.7580	1.70214	0.851069	+	0.525053i	$0.175955\pi$
$230$	13.6937	0.902934
$231$	6.07744	0.399866
$232$	9.17058	0.602078
$233$	−8.68541	−0.569000	−0.284500	−	0.958676i	$-0.591828\pi$
$233$	−8.68541	−0.569000	−0.284500	+	0.958676i	$0.591828\pi$
$234$	−9.41441	−0.615439
$235$	3.86853	0.252355
$236$	9.88554	0.643494
$237$	23.4402	1.52261
$238$	−7.32115	−0.474560
$239$	16.9822	1.09849	0.549243	−	0.835663i	$-0.314916\pi$
$239$	16.9822	1.09849	0.549243	+	0.835663i	$0.314916\pi$
$240$	8.59679	0.554920
$241$	−4.11545	−0.265099	−0.132550	−	0.991176i	$-0.542316\pi$
$241$	−4.11545	−0.265099	−0.132550	+	0.991176i	$0.542316\pi$
$242$	−9.41945	−0.605505
$243$	5.63777	0.361663
$244$	−13.9052	−0.890191
$245$	12.5664	0.802839
$246$	−16.7398	−1.06729
$247$	6.70213	0.426446
$248$	−4.91974	−0.312404
$249$	23.2385	1.47268
$250$	4.28688	0.271126
$251$	10.6976	0.675224	0.337612	−	0.941285i	$-0.390381\pi$
$251$	10.6976	0.675224	0.337612	+	0.941285i	$0.390381\pi$
$252$	−9.29951	−0.585814
$253$	−5.89370	−0.370534
$254$	−1.79939	−0.112904
$255$	38.3176	2.39954
$256$	1.00000	0.0625000
$257$	3.05536	0.190588	0.0952940	−	0.995449i	$-0.469621\pi$
$257$	3.05536	0.190588	0.0952940	+	0.995449i	$0.469621\pi$
$258$	−33.4062	−2.07978
$259$	−9.07277	−0.563755
$260$	4.85720	0.301231
$261$	51.9206	3.21381
$262$	−7.55304	−0.466629
$263$	15.2516	0.940455	0.470227	−	0.882545i	$-0.344172\pi$
$263$	15.2516	0.940455	0.470227	+	0.882545i	$0.344172\pi$
$264$	−3.70002	−0.227720
$265$	−36.4936	−2.24178
$266$	6.62034	0.405919
$267$	8.87890	0.543380
$268$	−0.382778	−0.0233819
$269$	5.95741	0.363230	0.181615	−	0.983370i	$-0.441868\pi$
$269$	5.95741	0.363230	0.181615	+	0.983370i	$0.441868\pi$
$270$	22.8817	1.39253
$271$	−28.7736	−1.74787	−0.873936	−	0.486042i	$-0.838440\pi$
$271$	−28.7736	−1.74787	−0.873936	+	0.486042i	$0.838440\pi$
$272$	4.45720	0.270258
$273$	−8.03835	−0.486503
$274$	−18.4869	−1.11684
$275$	4.44094	0.267799
$276$	13.7970	0.830482
$277$	25.8473	1.55301	0.776507	−	0.630108i	$-0.216990\pi$
$277$	25.8473	1.55301	0.776507	+	0.630108i	$0.216990\pi$
$278$	−19.8690	−1.19166
$279$	−27.8539	−1.66757
$280$	4.79792	0.286730
$281$	−11.2305	−0.669956	−0.334978	−	0.942226i	$-0.608729\pi$
$281$	−11.2305	−0.669956	−0.334978	+	0.942226i	$0.608729\pi$
$282$	3.89772	0.232106
$283$	4.69252	0.278941	0.139471	−	0.990226i	$-0.455460\pi$
$283$	4.69252	0.278941	0.139471	+	0.990226i	$0.455460\pi$
$284$	0.361868	0.0214729
$285$	−34.6497	−2.05247
$286$	−2.09052	−0.123615
$287$	−9.34258	−0.551475
$288$	5.66165	0.333616
$289$	2.86666	0.168627
$290$	−26.7875	−1.57302
$291$	−35.3162	−2.07027
$292$	−10.2001	−0.596917
$293$	−11.6101	−0.678267	−0.339133	−	0.940738i	$-0.610134\pi$
$293$	−11.6101	−0.678267	−0.339133	+	0.940738i	$0.610134\pi$
$294$	12.6612	0.738418
$295$	−28.8760	−1.68122
$296$	5.52361	0.321053
$297$	−9.84816	−0.571448
$298$	−16.4084	−0.950512
$299$	7.79532	0.450815
$300$	−10.3961	−0.600220
$301$	−18.6442	−1.07463
$302$	−7.54365	−0.434089
$303$	8.49735	0.488160
$304$	−4.03054	−0.231167
$305$	40.6176	2.32576
$306$	25.2351	1.44260
$307$	−5.15456	−0.294186	−0.147093	−	0.989123i	$-0.546992\pi$
$307$	−5.15456	−0.294186	−0.147093	+	0.989123i	$0.546992\pi$
$308$	−2.06500	−0.117664
$309$	−48.0326	−2.73248
$310$	14.3707	0.816201
$311$	−9.20197	−0.521796	−0.260898	−	0.965366i	$-0.584019\pi$
$311$	−9.20197	−0.521796	−0.260898	+	0.965366i	$0.584019\pi$
$312$	4.89384	0.277059
$313$	9.26848	0.523885	0.261943	−	0.965083i	$-0.415637\pi$
$313$	9.26848	0.523885	0.261943	+	0.965083i	$0.415637\pi$
$314$	19.3011	1.08922
$315$	27.1641	1.53053
$316$	−7.96455	−0.448041
$317$	2.15885	0.121253	0.0606264	−	0.998161i	$-0.480690\pi$
$317$	2.15885	0.121253	0.0606264	+	0.998161i	$0.480690\pi$
$318$	−36.7690	−2.06190
$319$	11.5292	0.645513
$320$	−2.92103	−0.163290
$321$	32.6277	1.82110
$322$	7.70018	0.429115
$323$	−17.9649	−0.999595
$324$	6.06935	0.337186
$325$	−5.87382	−0.325821
$326$	−23.4058	−1.29632
$327$	−13.0921	−0.723995
$328$	5.68787	0.314060
$329$	2.17534	0.119930
$330$	10.8079	0.594953
$331$	28.2154	1.55086	0.775428	−	0.631436i	$-0.217534\pi$
$331$	28.2154	1.55086	0.775428	+	0.631436i	$0.217534\pi$
$332$	−7.89601	−0.433350
$333$	31.2728	1.71374
$334$	−13.3421	−0.730047
$335$	1.11811	0.0610886
$336$	4.83412	0.263723
$337$	0.753041	0.0410208	0.0205104	−	0.999790i	$-0.493471\pi$
$337$	0.753041	0.0410208	0.0205104	+	0.999790i	$0.493471\pi$
$338$	−10.2350	−0.556709
$339$	8.23208	0.447105
$340$	−13.0196	−0.706088
$341$	−6.18509	−0.334941
$342$	−22.8195	−1.23394
$343$	18.5641	1.00237
$344$	11.3508	0.611995
$345$	−40.3014	−2.16976
$346$	−8.67377	−0.466305
$347$	18.4799	0.992053	0.496026	−	0.868307i	$-0.334792\pi$
$347$	18.4799	0.992053	0.496026	+	0.868307i	$0.334792\pi$
$348$	−26.9897	−1.44680
$349$	4.61683	0.247133	0.123567	−	0.992336i	$-0.460567\pi$
$349$	4.61683	0.247133	0.123567	+	0.992336i	$0.460567\pi$
$350$	−5.80214	−0.310137
$351$	13.0257	0.695261
$352$	1.25720	0.0670089
$353$	−31.8262	−1.69394	−0.846970	−	0.531641i	$-0.821575\pi$
$353$	−31.8262	−1.69394	−0.846970	+	0.531641i	$0.821575\pi$
$354$	−29.0938	−1.54632
$355$	−1.05703	−0.0561011
$356$	−3.01688	−0.159895
$357$	21.5466	1.14037
$358$	−14.2629	−0.753816
$359$	1.41906	0.0748949	0.0374475	−	0.999299i	$-0.488077\pi$
$359$	1.41906	0.0748949	0.0374475	+	0.999299i	$0.488077\pi$
$360$	−16.5378	−0.871621
$361$	−2.75476	−0.144987
$362$	−11.7906	−0.619701
$363$	27.7221	1.45503
$364$	2.73128	0.143158
$365$	29.7949	1.55953
$366$	40.9240	2.13913
$367$	−29.0057	−1.51408	−0.757042	−	0.653367i	$-0.773356\pi$
$367$	−29.0057	−1.51408	−0.757042	+	0.653367i	$0.773356\pi$
$368$	−4.68796	−0.244377
$369$	32.2028	1.67641
$370$	−16.1346	−0.838799
$371$	−20.5210	−1.06540
$372$	14.4791	0.750708
$373$	0.970767	0.0502644	0.0251322	−	0.999684i	$-0.491999\pi$
$373$	0.970767	0.0502644	0.0251322	+	0.999684i	$0.491999\pi$
$374$	5.60359	0.289754
$375$	−12.6166	−0.651517
$376$	−1.32437	−0.0682993
$377$	−15.2492	−0.785373
$378$	12.8667	0.661794
$379$	−3.71977	−0.191072	−0.0955359	−	0.995426i	$-0.530456\pi$
$379$	−3.71977	−0.191072	−0.0955359	+	0.995426i	$0.530456\pi$
$380$	11.7733	0.603959
$381$	5.29574	0.271309
$382$	−15.3693	−0.786360
$383$	3.97141	0.202930	0.101465	−	0.994839i	$-0.467647\pi$
$383$	3.97141	0.202930	0.101465	+	0.994839i	$0.467647\pi$
$384$	−2.94307	−0.150188
$385$	6.03193	0.307416
$386$	1.91946	0.0976978
$387$	64.2644	3.26674
$388$	11.9998	0.609197
$389$	16.2835	0.825606	0.412803	−	0.910820i	$-0.364550\pi$
$389$	16.2835	0.825606	0.412803	+	0.910820i	$0.364550\pi$
$390$	−14.2951	−0.723859
$391$	−20.8952	−1.05672
$392$	−4.30205	−0.217286
$393$	22.2291	1.12131
$394$	11.8963	0.599325
$395$	23.2647	1.17057
$396$	7.11781	0.357684
$397$	−29.3946	−1.47527	−0.737636	−	0.675198i	$-0.764058\pi$
$397$	−29.3946	−1.47527	−0.737636	+	0.675198i	$0.764058\pi$
$398$	0.978478	0.0490467
$399$	−19.4841	−0.975425
$400$	3.53241	0.176620
$401$	−29.3304	−1.46469	−0.732346	−	0.680933i	$-0.761575\pi$
$401$	−29.3304	−1.46469	−0.732346	+	0.680933i	$0.761575\pi$
$402$	1.12654	0.0561868
$403$	8.18073	0.407511
$404$	−2.88724	−0.143646
$405$	−17.7287	−0.880948
$406$	−15.0631	−0.747568
$407$	6.94427	0.344215
$408$	−13.1179	−0.649431
$409$	−0.284469	−0.0140661	−0.00703305	−	0.999975i	$-0.502239\pi$
$409$	−0.284469	−0.0140661	−0.00703305	+	0.999975i	$0.502239\pi$
$410$	−16.6144	−0.820529
$411$	54.4083	2.68376
$412$	16.3206	0.804058
$413$	−16.2374	−0.798992
$414$	−26.5416	−1.30445
$415$	23.0645	1.13219
$416$	−1.66284	−0.0815273
$417$	58.4757	2.86357
$418$	−5.06718	−0.247844
$419$	−13.6721	−0.667926	−0.333963	−	0.942586i	$-0.608386\pi$
$419$	−13.6721	−0.667926	−0.333963	+	0.942586i	$0.608386\pi$
$420$	−14.1206	−0.689015
$421$	8.52537	0.415501	0.207751	−	0.978182i	$-0.433386\pi$
$421$	8.52537	0.415501	0.207751	+	0.978182i	$0.433386\pi$
$422$	15.9955	0.778648
$423$	−7.49814	−0.364572
$424$	12.4934	0.606734
$425$	15.7447	0.763728
$426$	−1.06500	−0.0515995
$427$	22.8399	1.10530
$428$	−11.0863	−0.535876
$429$	6.15253	0.297047
$430$	−33.1561	−1.59893
$431$	20.0998	0.968175	0.484088	−	0.875020i	$-0.339152\pi$
$431$	20.0998	0.968175	0.484088	+	0.875020i	$0.339152\pi$
$432$	−7.83342	−0.376886
$433$	1.80710	0.0868438	0.0434219	−	0.999057i	$-0.486174\pi$
$433$	1.80710	0.0868438	0.0434219	+	0.999057i	$0.486174\pi$
$434$	8.08089	0.387895
$435$	78.8376	3.77997
$436$	4.44845	0.213042
$437$	18.8950	0.903871
$438$	30.0197	1.43439
$439$	24.1446	1.15236	0.576179	−	0.817324i	$-0.304543\pi$
$439$	24.1446	1.15236	0.576179	+	0.817324i	$0.304543\pi$
$440$	−3.67231	−0.175071
$441$	−24.3567	−1.15984
$442$	−7.41161	−0.352534
$443$	−9.66836	−0.459358	−0.229679	−	0.973266i	$-0.573768\pi$
$443$	−9.66836	−0.459358	−0.229679	+	0.973266i	$0.573768\pi$
$444$	−16.2564	−0.771493
$445$	8.81240	0.417748
$446$	19.6613	0.930990
$447$	48.2910	2.28409
$448$	−1.64254	−0.0776029
$449$	−28.2517	−1.33328	−0.666639	−	0.745380i	$-0.732268\pi$
$449$	−28.2517	−1.33328	−0.666639	+	0.745380i	$0.732268\pi$
$450$	19.9993	0.942775
$451$	7.15078	0.336717
$452$	−2.79711	−0.131565
$453$	22.2015	1.04312
$454$	−1.11266	−0.0522197
$455$	−7.97816	−0.374022
$456$	11.8622	0.555496
$457$	23.0568	1.07855	0.539275	−	0.842130i	$-0.318698\pi$
$457$	23.0568	1.07855	0.539275	+	0.842130i	$0.318698\pi$
$458$	25.7580	1.20359
$459$	−34.9152	−1.62970
$460$	13.6937	0.638471
$461$	−27.3204	−1.27244	−0.636219	−	0.771509i	$-0.719502\pi$
$461$	−27.3204	−1.27244	−0.636219	+	0.771509i	$0.719502\pi$
$462$	6.07744	0.282748
$463$	12.0627	0.560601	0.280301	−	0.959912i	$-0.409566\pi$
$463$	12.0627	0.560601	0.280301	+	0.959912i	$0.409566\pi$
$464$	9.17058	0.425734
$465$	−42.2940	−1.96134
$466$	−8.68541	−0.402344
$467$	−28.8978	−1.33723	−0.668614	−	0.743609i	$-0.733112\pi$
$467$	−28.8978	−1.33723	−0.668614	+	0.743609i	$0.733112\pi$
$468$	−9.41441	−0.435181
$469$	0.628730	0.0290320
$470$	3.86853	0.178442
$471$	−56.8043	−2.61740
$472$	9.88554	0.455019
$473$	14.2702	0.656145
$474$	23.4402	1.07664
$475$	−14.2375	−0.653262
$476$	−7.32115	−0.335564
$477$	70.7333	3.23866
$478$	16.9822	0.776746
$479$	−34.2851	−1.56653	−0.783263	−	0.621691i	$-0.786446\pi$
$479$	−34.2851	−1.56653	−0.783263	+	0.621691i	$0.786446\pi$
$480$	8.59679	0.392388
$481$	−9.18487	−0.418794
$482$	−4.11545	−0.187454
$483$	−22.6622	−1.03116
$484$	−9.41945	−0.428157
$485$	−35.0517	−1.59162
$486$	5.63777	0.255734
$487$	−14.0500	−0.636665	−0.318332	−	0.947979i	$-0.603123\pi$
$487$	−14.0500	−0.636665	−0.318332	+	0.947979i	$0.603123\pi$
$488$	−13.9052	−0.629460
$489$	68.8847	3.11508
$490$	12.5664	0.567693
$491$	−33.3220	−1.50380	−0.751900	−	0.659277i	$-0.770863\pi$
$491$	−33.3220	−1.50380	−0.751900	+	0.659277i	$0.770863\pi$
$492$	−16.7398	−0.754688
$493$	40.8752	1.84092
$494$	6.70213	0.301543
$495$	−20.7913	−0.934501
$496$	−4.91974	−0.220903
$497$	−0.594384	−0.0266618
$498$	23.2385	1.04134
$499$	−32.1547	−1.43944	−0.719722	−	0.694263i	$-0.755731\pi$
$499$	−32.1547	−1.43944	−0.719722	+	0.694263i	$0.755731\pi$
$500$	4.28688	0.191715
$501$	39.2667	1.75431
$502$	10.6976	0.477455
$503$	16.1866	0.721725	0.360863	−	0.932619i	$-0.382482\pi$
$503$	16.1866	0.721725	0.360863	+	0.932619i	$0.382482\pi$
$504$	−9.29951	−0.414233
$505$	8.43371	0.375295
$506$	−5.89370	−0.262007
$507$	30.1222	1.33777
$508$	−1.79939	−0.0798352
$509$	41.9187	1.85801	0.929007	−	0.370061i	$-0.120663\pi$
$509$	41.9187	1.85801	0.929007	+	0.370061i	$0.120663\pi$
$510$	38.3176	1.69673
$511$	16.7541	0.741160
$512$	1.00000	0.0441942
$513$	31.5729	1.39398
$514$	3.05536	0.134766
$515$	−47.6729	−2.10072
$516$	−33.4062	−1.47063
$517$	−1.66500	−0.0732266
$518$	−9.07277	−0.398635
$519$	25.5275	1.12053
$520$	4.85720	0.213002
$521$	−21.8413	−0.956884	−0.478442	−	0.878119i	$-0.658798\pi$
$521$	−21.8413	−0.956884	−0.478442	+	0.878119i	$0.658798\pi$
$522$	51.9206	2.27251
$523$	4.48845	0.196266	0.0981332	−	0.995173i	$-0.468713\pi$
$523$	4.48845	0.196266	0.0981332	+	0.995173i	$0.468713\pi$
$524$	−7.55304	−0.329956
$525$	17.0761	0.745261
$526$	15.2516	0.665002
$527$	−21.9283	−0.955211
$528$	−3.70002	−0.161023
$529$	−1.02300	−0.0444781
$530$	−36.4936	−1.58518
$531$	55.9685	2.42883
$532$	6.62034	0.287028
$533$	−9.45801	−0.409672
$534$	8.87890	0.384227
$535$	32.3834	1.40006
$536$	−0.382778	−0.0165335
$537$	41.9766	1.81142
$538$	5.95741	0.256842
$539$	−5.40853	−0.232962
$540$	22.8817	0.984670
$541$	15.2830	0.657067	0.328533	−	0.944492i	$-0.393446\pi$
$541$	15.2830	0.657067	0.328533	+	0.944492i	$0.393446\pi$
$542$	−28.7736	−1.23593
$543$	34.7006	1.48914
$544$	4.45720	0.191101
$545$	−12.9941	−0.556605
$546$	−8.03835	−0.344010
$547$	−34.8820	−1.49145	−0.745723	−	0.666256i	$-0.767896\pi$
$547$	−34.8820	−1.49145	−0.745723	+	0.666256i	$0.767896\pi$
$548$	−18.4869	−0.789723
$549$	−78.7266	−3.35997
$550$	4.44094	0.189362
$551$	−36.9624	−1.57465
$552$	13.7970	0.587239
$553$	13.0821	0.556308
$554$	25.8473	1.09815
$555$	47.4853	2.01564
$556$	−19.8690	−0.842632
$557$	−37.0522	−1.56995	−0.784976	−	0.619526i	$-0.787325\pi$
$557$	−37.0522	−1.56995	−0.784976	+	0.619526i	$0.787325\pi$
$558$	−27.8539	−1.17915
$559$	−18.8746	−0.798309
$560$	4.79792	0.202749
$561$	−16.4917	−0.696282
$562$	−11.2305	−0.473730
$563$	23.6408	0.996341	0.498170	−	0.867079i	$-0.334005\pi$
$563$	23.6408	0.996341	0.498170	+	0.867079i	$0.334005\pi$
$564$	3.89772	0.164124
$565$	8.17043	0.343733
$566$	4.69252	0.197241
$567$	−9.96917	−0.418666
$568$	0.361868	0.0151836
$569$	17.5523	0.735832	0.367916	−	0.929859i	$-0.380071\pi$
$569$	17.5523	0.735832	0.367916	+	0.929859i	$0.380071\pi$
$570$	−34.6497	−1.45132
$571$	−34.8108	−1.45679	−0.728394	−	0.685159i	$-0.759733\pi$
$571$	−34.8108	−1.45679	−0.728394	+	0.685159i	$0.759733\pi$
$572$	−2.09052	−0.0874088
$573$	45.2328	1.88963
$574$	−9.34258	−0.389952
$575$	−16.5598	−0.690592
$576$	5.66165	0.235902
$577$	−7.64101	−0.318099	−0.159050	−	0.987271i	$-0.550843\pi$
$577$	−7.64101	−0.318099	−0.159050	+	0.987271i	$0.550843\pi$
$578$	2.86666	0.119237
$579$	−5.64909	−0.234768
$580$	−26.7875	−1.11229
$581$	12.9695	0.538067
$582$	−35.3162	−1.46390
$583$	15.7067	0.650504
$584$	−10.2001	−0.422084
$585$	27.4998	1.13697
$586$	−11.6101	−0.479607
$587$	31.7058	1.30864	0.654319	−	0.756219i	$-0.272955\pi$
$587$	31.7058	1.30864	0.654319	+	0.756219i	$0.272955\pi$
$588$	12.6612	0.522140
$589$	19.8292	0.817048
$590$	−28.8760	−1.18880
$591$	−35.0115	−1.44018
$592$	5.52361	0.227019
$593$	8.38932	0.344508	0.172254	−	0.985053i	$-0.444895\pi$
$593$	8.38932	0.344508	0.172254	+	0.985053i	$0.444895\pi$
$594$	−9.84816	−0.404075
$595$	21.3853	0.876711
$596$	−16.4084	−0.672114
$597$	−2.87973	−0.117859
$598$	7.79532	0.318774
$599$	−43.4791	−1.77651	−0.888254	−	0.459353i	$-0.848081\pi$
$599$	−43.4791	−1.77651	−0.888254	+	0.459353i	$0.848081\pi$
$600$	−10.3961	−0.424420
$601$	41.8333	1.70642	0.853209	−	0.521570i	$-0.174653\pi$
$601$	41.8333	1.70642	0.853209	+	0.521570i	$0.174653\pi$
$602$	−18.6442	−0.759881
$603$	−2.16716	−0.0882534
$604$	−7.54365	−0.306947
$605$	27.5145	1.11862
$606$	8.49735	0.345181
$607$	−6.50644	−0.264088	−0.132044	−	0.991244i	$-0.542154\pi$
$607$	−6.50644	−0.264088	−0.132044	+	0.991244i	$0.542154\pi$
$608$	−4.03054	−0.163460
$609$	44.3317	1.79641
$610$	40.6176	1.64456
$611$	2.20222	0.0890922
$612$	25.2351	1.02007
$613$	27.8676	1.12556	0.562781	−	0.826606i	$-0.309732\pi$
$613$	27.8676	1.12556	0.562781	+	0.826606i	$0.309732\pi$
$614$	−5.15456	−0.208021
$615$	48.8974	1.97173
$616$	−2.06500	−0.0832013
$617$	−17.0955	−0.688239	−0.344120	−	0.938926i	$-0.611823\pi$
$617$	−17.0955	−0.688239	−0.344120	+	0.938926i	$0.611823\pi$
$618$	−48.0326	−1.93215
$619$	−47.4462	−1.90702	−0.953511	−	0.301357i	$-0.902560\pi$
$619$	−47.4462	−1.90702	−0.953511	+	0.301357i	$0.902560\pi$
$620$	14.3707	0.577142
$621$	36.7228	1.47364
$622$	−9.20197	−0.368965
$623$	4.95536	0.198532
$624$	4.89384	0.195911
$625$	−30.1841	−1.20737
$626$	9.26848	0.370443
$627$	14.9131	0.595571
$628$	19.3011	0.770196
$629$	24.6199	0.981658
$630$	27.1641	1.08225
$631$	−9.81798	−0.390848	−0.195424	−	0.980719i	$-0.562608\pi$
$631$	−9.81798	−0.390848	−0.195424	+	0.980719i	$0.562608\pi$
$632$	−7.96455	−0.316813
$633$	−47.0758	−1.87109
$634$	2.15885	0.0857387
$635$	5.25608	0.208581
$636$	−36.7690	−1.45798
$637$	7.15361	0.283436
$638$	11.5292	0.456447
$639$	2.04877	0.0810481
$640$	−2.92103	−0.115464
$641$	35.9369	1.41942	0.709711	−	0.704493i	$-0.248826\pi$
$641$	35.9369	1.41942	0.709711	+	0.704493i	$0.248826\pi$
$642$	32.6277	1.28771
$643$	−10.5788	−0.417186	−0.208593	−	0.978003i	$-0.566888\pi$
$643$	−10.5788	−0.417186	−0.208593	+	0.978003i	$0.566888\pi$
$644$	7.70018	0.303430
$645$	97.5806	3.84223
$646$	−17.9649	−0.706821
$647$	−42.2732	−1.66193	−0.830966	−	0.556323i	$-0.812212\pi$
$647$	−42.2732	−1.66193	−0.830966	+	0.556323i	$0.812212\pi$
$648$	6.06935	0.238427
$649$	12.4281	0.487845
$650$	−5.87382	−0.230390
$651$	−23.7826	−0.932114
$652$	−23.4058	−0.916640
$653$	35.4039	1.38546	0.692731	−	0.721196i	$-0.256407\pi$
$653$	35.4039	1.38546	0.692731	+	0.721196i	$0.256407\pi$
$654$	−13.0921	−0.511942
$655$	22.0627	0.862059
$656$	5.68787	0.222074
$657$	−57.7496	−2.25302
$658$	2.17534	0.0848036
$659$	6.38951	0.248900	0.124450	−	0.992226i	$-0.460283\pi$
$659$	6.38951	0.248900	0.124450	+	0.992226i	$0.460283\pi$
$660$	10.8079	0.420695
$661$	−10.8833	−0.423312	−0.211656	−	0.977344i	$-0.567886\pi$
$661$	−10.8833	−0.423312	−0.211656	+	0.977344i	$0.567886\pi$
$662$	28.2154	1.09662
$663$	21.8129	0.847141
$664$	−7.89601	−0.306425
$665$	−19.3382	−0.749903
$666$	31.2728	1.21179
$667$	−42.9914	−1.66463
$668$	−13.3421	−0.516221
$669$	−57.8646	−2.23717
$670$	1.11811	0.0431962
$671$	−17.4816	−0.674871
$672$	4.83412	0.186480
$673$	−9.94617	−0.383397	−0.191698	−	0.981454i	$-0.561400\pi$
$673$	−9.94617	−0.383397	−0.191698	+	0.981454i	$0.561400\pi$
$674$	0.753041	0.0290061
$675$	−27.6709	−1.06505
$676$	−10.2350	−0.393653
$677$	30.8580	1.18597	0.592985	−	0.805214i	$-0.297949\pi$
$677$	30.8580	1.18597	0.592985	+	0.805214i	$0.297949\pi$
$678$	8.23208	0.316151
$679$	−19.7102	−0.756407
$680$	−13.0196	−0.499280
$681$	3.27463	0.125484
$682$	−6.18509	−0.236839
$683$	−17.9571	−0.687108	−0.343554	−	0.939133i	$-0.611631\pi$
$683$	−17.9571	−0.687108	−0.343554	+	0.939133i	$0.611631\pi$
$684$	−22.8195	−0.872526
$685$	54.0009	2.06327
$686$	18.5641	0.708781
$687$	−75.8077	−2.89224
$688$	11.3508	0.432746
$689$	−20.7745	−0.791446
$690$	−40.3014	−1.53425
$691$	−16.7605	−0.637599	−0.318800	−	0.947822i	$-0.603280\pi$
$691$	−16.7605	−0.637599	−0.318800	+	0.947822i	$0.603280\pi$
$692$	−8.67377	−0.329727
$693$	−11.6913	−0.444117
$694$	18.4799	0.701487
$695$	58.0378	2.20150
$696$	−26.9897	−1.02304
$697$	25.3520	0.960276
$698$	4.61683	0.174750
$699$	25.5618	0.966835
$700$	−5.80214	−0.219300
$701$	−23.3045	−0.880200	−0.440100	−	0.897949i	$-0.645057\pi$
$701$	−23.3045	−0.880200	−0.440100	+	0.897949i	$0.645057\pi$
$702$	13.0257	0.491624
$703$	−22.2631	−0.839670
$704$	1.25720	0.0473824
$705$	−11.3854	−0.428797
$706$	−31.8262	−1.19780
$707$	4.74242	0.178357
$708$	−29.0938	−1.09341
$709$	23.0973	0.867436	0.433718	−	0.901049i	$-0.357201\pi$
$709$	23.0973	0.867436	0.433718	+	0.901049i	$0.357201\pi$
$710$	−1.05703	−0.0396695
$711$	−45.0925	−1.69110
$712$	−3.01688	−0.113062
$713$	23.0636	0.863738
$714$	21.5466	0.806363
$715$	6.10646	0.228368
$716$	−14.2629	−0.533028
$717$	−49.9797	−1.86653
$718$	1.41906	0.0529587
$719$	−35.2542	−1.31476	−0.657380	−	0.753559i	$-0.728335\pi$
$719$	−35.2542	−1.31476	−0.657380	+	0.753559i	$0.728335\pi$
$720$	−16.5378	−0.616329
$721$	−26.8073	−0.998355
$722$	−2.75476	−0.102521
$723$	12.1120	0.450452
$724$	−11.7906	−0.438195
$725$	32.3942	1.20309
$726$	27.7221	1.02886
$727$	−14.2513	−0.528550	−0.264275	−	0.964447i	$-0.585133\pi$
$727$	−14.2513	−0.528550	−0.264275	+	0.964447i	$0.585133\pi$
$728$	2.73128	0.101228
$729$	−34.8004	−1.28890
$730$	29.7949	1.10276
$731$	50.5929	1.87125
$732$	40.9240	1.51260
$733$	22.2213	0.820761	0.410380	−	0.911914i	$-0.365396\pi$
$733$	22.2213	0.820761	0.410380	+	0.911914i	$0.365396\pi$
$734$	−29.0057	−1.07062
$735$	−36.9838	−1.36417
$736$	−4.68796	−0.172801
$737$	−0.481228	−0.0177262
$738$	32.2028	1.18540
$739$	−7.16325	−0.263505	−0.131752	−	0.991283i	$-0.542060\pi$
$739$	−7.16325	−0.263505	−0.131752	+	0.991283i	$0.542060\pi$
$740$	−16.1346	−0.593121
$741$	−19.7248	−0.724610
$742$	−20.5210	−0.753349
$743$	8.24461	0.302465	0.151233	−	0.988498i	$-0.451676\pi$
$743$	8.24461	0.302465	0.151233	+	0.988498i	$0.451676\pi$
$744$	14.4791	0.530831
$745$	47.9294	1.75600
$746$	0.970767	0.0355423
$747$	−44.7045	−1.63565
$748$	5.60359	0.204887
$749$	18.2097	0.665369
$750$	−12.6166	−0.460692
$751$	−6.10505	−0.222776	−0.111388	−	0.993777i	$-0.535530\pi$
$751$	−6.10505	−0.222776	−0.111388	+	0.993777i	$0.535530\pi$
$752$	−1.32437	−0.0482949
$753$	−31.4836	−1.14733
$754$	−15.2492	−0.555343
$755$	22.0352	0.801944
$756$	12.8667	0.467959
$757$	21.7492	0.790487	0.395243	−	0.918576i	$-0.370660\pi$
$757$	21.7492	0.790487	0.395243	+	0.918576i	$0.370660\pi$
$758$	−3.71977	−0.135108
$759$	17.3456	0.629604
$760$	11.7733	0.427063
$761$	38.3932	1.39175	0.695876	−	0.718161i	$-0.255016\pi$
$761$	38.3932	1.39175	0.695876	+	0.718161i	$0.255016\pi$
$762$	5.29574	0.191844
$763$	−7.30678	−0.264523
$764$	−15.3693	−0.556040
$765$	−73.7126	−2.66508
$766$	3.97141	0.143493
$767$	−16.4381	−0.593544
$768$	−2.94307	−0.106199
$769$	−29.6320	−1.06856	−0.534278	−	0.845309i	$-0.679417\pi$
$769$	−29.6320	−1.06856	−0.534278	+	0.845309i	$0.679417\pi$
$770$	6.03193	0.217376
$771$	−8.99213	−0.323844
$772$	1.91946	0.0690828
$773$	−27.4540	−0.987453	−0.493727	−	0.869617i	$-0.664366\pi$
$773$	−27.4540	−0.987453	−0.493727	+	0.869617i	$0.664366\pi$
$774$	64.2644	2.30994
$775$	−17.3785	−0.624256
$776$	11.9998	0.430767
$777$	26.7018	0.957921
$778$	16.2835	0.583792
$779$	−22.9252	−0.821380
$780$	−14.2951	−0.511845
$781$	0.454939	0.0162790
$782$	−20.8952	−0.747211
$783$	−71.8371	−2.56725
$784$	−4.30205	−0.153645
$785$	−56.3789	−2.01225
$786$	22.2291	0.792887
$787$	30.2089	1.07683	0.538415	−	0.842680i	$-0.319023\pi$
$787$	30.2089	1.07683	0.538415	+	0.842680i	$0.319023\pi$
$788$	11.8963	0.423787
$789$	−44.8865	−1.59800
$790$	23.2647	0.827720
$791$	4.59437	0.163357
$792$	7.11781	0.252921
$793$	23.1221	0.821091
$794$	−29.3946	−1.04318
$795$	107.403	3.80920
$796$	0.978478	0.0346812
$797$	−14.5053	−0.513803	−0.256902	−	0.966438i	$-0.582702\pi$
$797$	−14.5053	−0.513803	−0.256902	+	0.966438i	$0.582702\pi$
$798$	−19.4841	−0.689730
$799$	−5.90300	−0.208833
$800$	3.53241	0.124890
$801$	−17.0805	−0.603511
$802$	−29.3304	−1.03569
$803$	−12.8236	−0.452534
$804$	1.12654	0.0397301
$805$	−22.4925	−0.792755
$806$	8.18073	0.288154
$807$	−17.5331	−0.617194
$808$	−2.88724	−0.101573
$809$	−11.6841	−0.410790	−0.205395	−	0.978679i	$-0.565848\pi$
$809$	−11.6841	−0.410790	−0.205395	+	0.978679i	$0.565848\pi$
$810$	−17.7287	−0.622924
$811$	−1.13085	−0.0397097	−0.0198548	−	0.999803i	$-0.506320\pi$
$811$	−1.13085	−0.0397097	−0.0198548	+	0.999803i	$0.506320\pi$
$812$	−15.0631	−0.528611
$813$	84.6827	2.96995
$814$	6.94427	0.243396
$815$	68.3689	2.39486
$816$	−13.1179	−0.459217
$817$	−45.7499	−1.60059
$818$	−0.284469	−0.00994624
$819$	15.4636	0.540341
$820$	−16.6144	−0.580201
$821$	46.8313	1.63442	0.817212	−	0.576337i	$-0.195518\pi$
$821$	46.8313	1.63442	0.817212	+	0.576337i	$0.195518\pi$
$822$	54.4083	1.89771
$823$	23.7556	0.828069	0.414034	−	0.910261i	$-0.364119\pi$
$823$	23.7556	0.828069	0.414034	+	0.910261i	$0.364119\pi$
$824$	16.3206	0.568555
$825$	−13.0700	−0.455038
$826$	−16.2374	−0.564973
$827$	−53.5720	−1.86288	−0.931440	−	0.363896i	$-0.881446\pi$
$827$	−53.5720	−1.86288	−0.931440	+	0.363896i	$0.881446\pi$
$828$	−26.5416	−0.922385
$829$	26.2931	0.913198	0.456599	−	0.889673i	$-0.349067\pi$
$829$	26.2931	0.913198	0.456599	+	0.889673i	$0.349067\pi$
$830$	23.0645	0.800580
$831$	−76.0704	−2.63885
$832$	−1.66284	−0.0576485
$833$	−19.1751	−0.664378
$834$	58.4757	2.02485
$835$	38.9726	1.34870
$836$	−5.06718	−0.175252
$837$	38.5384	1.33208
$838$	−13.6721	−0.472295
$839$	31.9795	1.10405	0.552027	−	0.833826i	$-0.313854\pi$
$839$	31.9795	1.10405	0.552027	+	0.833826i	$0.313854\pi$
$840$	−14.1206	−0.487207
$841$	55.0996	1.89999
$842$	8.52537	0.293804
$843$	33.0521	1.13838
$844$	15.9955	0.550587
$845$	29.8966	1.02848
$846$	−7.49814	−0.257791
$847$	15.4719	0.531620
$848$	12.4934	0.429025
$849$	−13.8104	−0.473972
$850$	15.7447	0.540038
$851$	−25.8945	−0.887651
$852$	−1.06500	−0.0364864
$853$	2.61994	0.0897051	0.0448526	−	0.998994i	$-0.485718\pi$
$853$	2.61994	0.0897051	0.0448526	+	0.998994i	$0.485718\pi$
$854$	22.8399	0.781567
$855$	66.6564	2.27960
$856$	−11.0863	−0.378922
$857$	−46.3997	−1.58498	−0.792492	−	0.609883i	$-0.791217\pi$
$857$	−46.3997	−1.58498	−0.792492	+	0.609883i	$0.791217\pi$
$858$	6.15253	0.210044
$859$	17.8306	0.608372	0.304186	−	0.952613i	$-0.401616\pi$
$859$	17.8306	0.608372	0.304186	+	0.952613i	$0.401616\pi$
$860$	−33.1561	−1.13061
$861$	27.4958	0.937056
$862$	20.0998	0.684603
$863$	16.0405	0.546025	0.273013	−	0.962010i	$-0.411980\pi$
$863$	16.0405	0.546025	0.273013	+	0.962010i	$0.411980\pi$
$864$	−7.83342	−0.266499
$865$	25.3363	0.861461
$866$	1.80710	0.0614079
$867$	−8.43678	−0.286528
$868$	8.08089	0.274283
$869$	−10.0130	−0.339668
$870$	78.8376	2.67284
$871$	0.636498	0.0215669
$872$	4.44845	0.150644
$873$	67.9386	2.29937
$874$	18.8950	0.639134
$875$	−7.04138	−0.238042
$876$	30.0197	1.01427
$877$	−25.1395	−0.848901	−0.424451	−	0.905451i	$-0.639533\pi$
$877$	−25.1395	−0.848901	−0.424451	+	0.905451i	$0.639533\pi$
$878$	24.1446	0.814840
$879$	34.1692	1.15250
$880$	−3.67231	−0.123794
$881$	−53.6367	−1.80707	−0.903534	−	0.428517i	$-0.859036\pi$
$881$	−53.6367	−1.80707	−0.903534	+	0.428517i	$0.859036\pi$
$882$	−24.3567	−0.820133
$883$	−43.7560	−1.47251	−0.736254	−	0.676706i	$-0.763407\pi$
$883$	−43.7560	−1.47251	−0.736254	+	0.676706i	$0.763407\pi$
$884$	−7.41161	−0.249279
$885$	84.9839	2.85670
$886$	−9.66836	−0.324815
$887$	13.2588	0.445187	0.222593	−	0.974911i	$-0.428548\pi$
$887$	13.2588	0.445187	0.222593	+	0.974911i	$0.428548\pi$
$888$	−16.2564	−0.545528
$889$	2.95558	0.0991271
$890$	8.81240	0.295392
$891$	7.63037	0.255627
$892$	19.6613	0.658309
$893$	5.33794	0.178627
$894$	48.2910	1.61509
$895$	41.6622	1.39262
$896$	−1.64254	−0.0548735
$897$	−22.9422	−0.766017
$898$	−28.2517	−0.942770
$899$	−45.1169	−1.50473
$900$	19.9993	0.666642
$901$	55.6857	1.85516
$902$	7.15078	0.238095
$903$	54.8712	1.82600
$904$	−2.79711	−0.0930304
$905$	34.4407	1.14485
$906$	22.2015	0.737595
$907$	35.3770	1.17467	0.587337	−	0.809342i	$-0.300176\pi$
$907$	35.3770	1.17467	0.587337	+	0.809342i	$0.300176\pi$
$908$	−1.11266	−0.0369249
$909$	−16.3466	−0.542181
$910$	−7.97816	−0.264473
$911$	−33.7227	−1.11728	−0.558641	−	0.829409i	$-0.688677\pi$
$911$	−33.7227	−1.11728	−0.558641	+	0.829409i	$0.688677\pi$
$912$	11.8622	0.392795
$913$	−9.92685	−0.328531
$914$	23.0568	0.762650
$915$	−119.540	−3.95188
$916$	25.7580	0.851069
$917$	12.4062	0.409689
$918$	−34.9152	−1.15237
$919$	−40.6046	−1.33942	−0.669712	−	0.742621i	$-0.733582\pi$
$919$	−40.6046	−1.33942	−0.669712	+	0.742621i	$0.733582\pi$
$920$	13.6937	0.451467
$921$	15.1702	0.499876
$922$	−27.3204	−0.899749
$923$	−0.601727	−0.0198061
$924$	6.07744	0.199933
$925$	19.5116	0.641539
$926$	12.0627	0.396405
$927$	92.4015	3.03486
$928$	9.17058	0.301039
$929$	0.720619	0.0236427	0.0118214	−	0.999930i	$-0.496237\pi$
$929$	0.720619	0.0236427	0.0118214	+	0.999930i	$0.496237\pi$
$930$	−42.2940	−1.38687
$931$	17.3396	0.568282
$932$	−8.68541	−0.284500
$933$	27.0820	0.886626
$934$	−28.8978	−0.945563
$935$	−16.3682	−0.535298
$936$	−9.41441	−0.307720
$937$	5.17888	0.169187	0.0845934	−	0.996416i	$-0.473041\pi$
$937$	5.17888	0.169187	0.0845934	+	0.996416i	$0.473041\pi$
$938$	0.628730	0.0205288
$939$	−27.2778	−0.890176
$940$	3.86853	0.126178
$941$	38.1984	1.24523	0.622615	−	0.782528i	$-0.286070\pi$
$941$	38.1984	1.24523	0.622615	+	0.782528i	$0.286070\pi$
$942$	−56.8043	−1.85078
$943$	−26.6645	−0.868317
$944$	9.88554	0.321747
$945$	−37.5841	−1.22261
$946$	14.2702	0.463965
$947$	−43.4378	−1.41154	−0.705770	−	0.708441i	$-0.749399\pi$
$947$	−43.4378	−1.41154	−0.705770	+	0.708441i	$0.749399\pi$
$948$	23.4402	0.761303
$949$	16.9611	0.550582
$950$	−14.2375	−0.461926
$951$	−6.35363	−0.206031
$952$	−7.32115	−0.237280
$953$	−17.1930	−0.556935	−0.278467	−	0.960446i	$-0.589826\pi$
$953$	−17.1930	−0.556935	−0.278467	+	0.960446i	$0.589826\pi$
$954$	70.7333	2.29008
$955$	44.8941	1.45274
$956$	16.9822	0.549243
$957$	−33.9313	−1.09684
$958$	−34.2851	−1.10770
$959$	30.3656	0.980556
$960$	8.59679	0.277460
$961$	−6.79613	−0.219230
$962$	−9.18487	−0.296132
$963$	−62.7667	−2.02263
$964$	−4.11545	−0.132550
$965$	−5.60679	−0.180489
$966$	−22.6622	−0.729143
$967$	−36.9788	−1.18916	−0.594580	−	0.804037i	$-0.702682\pi$
$967$	−36.9788	−1.18916	−0.594580	+	0.804037i	$0.702682\pi$
$968$	−9.41945	−0.302753
$969$	52.8720	1.69849
$970$	−35.0517	−1.12544
$971$	10.8129	0.347004	0.173502	−	0.984834i	$-0.444492\pi$
$971$	10.8129	0.347004	0.173502	+	0.984834i	$0.444492\pi$
$972$	5.63777	0.180831
$973$	32.6356	1.04625
$974$	−14.0500	−0.450190
$975$	17.2871	0.553629
$976$	−13.9052	−0.445096
$977$	30.7771	0.984646	0.492323	−	0.870413i	$-0.336148\pi$
$977$	30.7771	0.984646	0.492323	+	0.870413i	$0.336148\pi$
$978$	68.8847	2.20269
$979$	−3.79282	−0.121219
$980$	12.5664	0.401419
$981$	25.1856	0.804114
$982$	−33.3220	−1.06335
$983$	55.8687	1.78193	0.890967	−	0.454069i	$-0.150028\pi$
$983$	55.8687	1.78193	0.890967	+	0.454069i	$0.150028\pi$
$984$	−16.7398	−0.533645
$985$	−34.7493	−1.10721
$986$	40.8752	1.30173
$987$	−6.40218	−0.203784
$988$	6.70213	0.213223
$989$	−53.2122	−1.69205
$990$	−20.7913	−0.660792
$991$	5.31519	0.168843	0.0844213	−	0.996430i	$-0.473096\pi$
$991$	5.31519	0.168843	0.0844213	+	0.996430i	$0.473096\pi$
$992$	−4.91974	−0.156202
$993$	−83.0397	−2.63519
$994$	−0.594384	−0.0188527
$995$	−2.85816	−0.0906099
$996$	23.2385	0.736340
$997$	42.7174	1.35287	0.676437	−	0.736501i	$-0.263523\pi$
$997$	42.7174	1.35287	0.676437	+	0.736501i	$0.263523\pi$
$998$	−32.1547	−1.01784
$999$	−43.2688	−1.36896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.f.1.3	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.f.1.3	✓	31	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.94307 q^{3} +1.00000 q^{4} -2.92103 q^{5} -2.94307 q^{6} -1.64254 q^{7} +1.00000 q^{8} +5.66165 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.94307 q^{3} +1.00000 q^{4} -2.92103 q^{5} -2.94307 q^{6} -1.64254 q^{7} +1.00000 q^{8} +5.66165 q^{9} -2.92103 q^{10} +1.25720 q^{11} -2.94307 q^{12} -1.66284 q^{13} -1.64254 q^{14} +8.59679 q^{15} +1.00000 q^{16} +4.45720 q^{17} +5.66165 q^{18} -4.03054 q^{19} -2.92103 q^{20} +4.83412 q^{21} +1.25720 q^{22} -4.68796 q^{23} -2.94307 q^{24} +3.53241 q^{25} -1.66284 q^{26} -7.83342 q^{27} -1.64254 q^{28} +9.17058 q^{29} +8.59679 q^{30} -4.91974 q^{31} +1.00000 q^{32} -3.70002 q^{33} +4.45720 q^{34} +4.79792 q^{35} +5.66165 q^{36} +5.52361 q^{37} -4.03054 q^{38} +4.89384 q^{39} -2.92103 q^{40} +5.68787 q^{41} +4.83412 q^{42} +11.3508 q^{43} +1.25720 q^{44} -16.5378 q^{45} -4.68796 q^{46} -1.32437 q^{47} -2.94307 q^{48} -4.30205 q^{49} +3.53241 q^{50} -13.1179 q^{51} -1.66284 q^{52} +12.4934 q^{53} -7.83342 q^{54} -3.67231 q^{55} -1.64254 q^{56} +11.8622 q^{57} +9.17058 q^{58} +9.88554 q^{59} +8.59679 q^{60} -13.9052 q^{61} -4.91974 q^{62} -9.29951 q^{63} +1.00000 q^{64} +4.85720 q^{65} -3.70002 q^{66} -0.382778 q^{67} +4.45720 q^{68} +13.7970 q^{69} +4.79792 q^{70} +0.361868 q^{71} +5.66165 q^{72} -10.2001 q^{73} +5.52361 q^{74} -10.3961 q^{75} -4.03054 q^{76} -2.06500 q^{77} +4.89384 q^{78} -7.96455 q^{79} -2.92103 q^{80} +6.06935 q^{81} +5.68787 q^{82} -7.89601 q^{83} +4.83412 q^{84} -13.0196 q^{85} +11.3508 q^{86} -26.9897 q^{87} +1.25720 q^{88} -3.01688 q^{89} -16.5378 q^{90} +2.73128 q^{91} -4.68796 q^{92} +14.4791 q^{93} -1.32437 q^{94} +11.7733 q^{95} -2.94307 q^{96} +11.9998 q^{97} -4.30205 q^{98} +7.11781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9 - 23 * q^10 - 32 * q^11 - 13 * q^12 - 8 * q^13 - 18 * q^14 - 14 * q^15 + 31 * q^16 - 30 * q^17 + 20 * q^18 - 38 * q^19 - 23 * q^20 - 16 * q^21 - 32 * q^22 - 24 * q^23 - 13 * q^24 + 40 * q^25 - 8 * q^26 - 28 * q^27 - 18 * q^28 - 7 * q^29 - 14 * q^30 - 32 * q^31 + 31 * q^32 - 9 * q^33 - 30 * q^34 - 14 * q^35 + 20 * q^36 + 11 * q^37 - 38 * q^38 - 9 * q^39 - 23 * q^40 - 76 * q^41 - 16 * q^42 - 33 * q^43 - 32 * q^44 - 40 * q^45 - 24 * q^46 - 96 * q^47 - 13 * q^48 + 15 * q^49 + 40 * q^50 - 55 * q^51 - 8 * q^52 - 28 * q^53 - 28 * q^54 - 52 * q^55 - 18 * q^56 - 21 * q^57 - 7 * q^58 - 72 * q^59 - 14 * q^60 - 9 * q^61 - 32 * q^62 - 54 * q^63 + 31 * q^64 - 38 * q^65 - 9 * q^66 - 4 * q^67 - 30 * q^68 - 17 * q^69 - 14 * q^70 - 61 * q^71 + 20 * q^72 - 62 * q^73 + 11 * q^74 - 63 * q^75 - 38 * q^76 - 9 * q^77 - 9 * q^78 - 30 * q^79 - 23 * q^80 - 13 * q^81 - 76 * q^82 - 90 * q^83 - 16 * q^84 + 26 * q^85 - 33 * q^86 - 34 * q^87 - 32 * q^88 - 99 * q^89 - 40 * q^90 - 47 * q^91 - 24 * q^92 - 6 * q^93 - 96 * q^94 - 24 * q^95 - 13 * q^96 - 46 * q^97 + 15 * q^98 - 48 * q^99

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