LMFDB - Embedded newform 4006.2.a.f.1.4

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			1.4
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.69988 q^{3} +1.00000 q^{4} +2.56179 q^{5} -2.69988 q^{6} -4.70163 q^{7} +1.00000 q^{8} +4.28934 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.69988 q^{3} +1.00000 q^{4} +2.56179 q^{5} -2.69988 q^{6} -4.70163 q^{7} +1.00000 q^{8} +4.28934 q^{9} +2.56179 q^{10} -3.35955 q^{11} -2.69988 q^{12} -1.77855 q^{13} -4.70163 q^{14} -6.91652 q^{15} +1.00000 q^{16} +7.86079 q^{17} +4.28934 q^{18} +2.18172 q^{19} +2.56179 q^{20} +12.6938 q^{21} -3.35955 q^{22} +1.70570 q^{23} -2.69988 q^{24} +1.56278 q^{25} -1.77855 q^{26} -3.48106 q^{27} -4.70163 q^{28} -1.59237 q^{29} -6.91652 q^{30} +0.397955 q^{31} +1.00000 q^{32} +9.07037 q^{33} +7.86079 q^{34} -12.0446 q^{35} +4.28934 q^{36} +0.252674 q^{37} +2.18172 q^{38} +4.80186 q^{39} +2.56179 q^{40} -6.51031 q^{41} +12.6938 q^{42} +9.23111 q^{43} -3.35955 q^{44} +10.9884 q^{45} +1.70570 q^{46} -0.749555 q^{47} -2.69988 q^{48} +15.1053 q^{49} +1.56278 q^{50} -21.2232 q^{51} -1.77855 q^{52} -4.43967 q^{53} -3.48106 q^{54} -8.60646 q^{55} -4.70163 q^{56} -5.89036 q^{57} -1.59237 q^{58} -2.84420 q^{59} -6.91652 q^{60} -0.902979 q^{61} +0.397955 q^{62} -20.1669 q^{63} +1.00000 q^{64} -4.55627 q^{65} +9.07037 q^{66} -13.9660 q^{67} +7.86079 q^{68} -4.60517 q^{69} -12.0446 q^{70} -11.7003 q^{71} +4.28934 q^{72} -14.1261 q^{73} +0.252674 q^{74} -4.21930 q^{75} +2.18172 q^{76} +15.7953 q^{77} +4.80186 q^{78} -6.82766 q^{79} +2.56179 q^{80} -3.46958 q^{81} -6.51031 q^{82} -10.4247 q^{83} +12.6938 q^{84} +20.1377 q^{85} +9.23111 q^{86} +4.29921 q^{87} -3.35955 q^{88} +11.3934 q^{89} +10.9884 q^{90} +8.36207 q^{91} +1.70570 q^{92} -1.07443 q^{93} -0.749555 q^{94} +5.58910 q^{95} -2.69988 q^{96} -3.25680 q^{97} +15.1053 q^{98} -14.4102 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.69988	−1.55878	−0.779388	−	0.626542i	$-0.784470\pi$
$3$	−2.69988	−1.55878	−0.779388	+	0.626542i	$0.784470\pi$
$4$	1.00000	0.500000
$5$	2.56179	1.14567	0.572834	−	0.819671i	$-0.305844\pi$
$5$	2.56179	1.14567	0.572834	+	0.819671i	$0.305844\pi$
$6$	−2.69988	−1.10222
$7$	−4.70163	−1.77705	−0.888524	−	0.458830i	$-0.848269\pi$
$7$	−4.70163	−1.77705	−0.888524	+	0.458830i	$0.848269\pi$
$8$	1.00000	0.353553
$9$	4.28934	1.42978
$10$	2.56179	0.810110
$11$	−3.35955	−1.01294	−0.506471	−	0.862257i	$-0.669050\pi$
$11$	−3.35955	−1.01294	−0.506471	+	0.862257i	$0.669050\pi$
$12$	−2.69988	−0.779388
$13$	−1.77855	−0.493280	−0.246640	−	0.969107i	$-0.579327\pi$
$13$	−1.77855	−0.493280	−0.246640	+	0.969107i	$0.579327\pi$
$14$	−4.70163	−1.25656
$15$	−6.91652	−1.78584
$16$	1.00000	0.250000
$17$	7.86079	1.90652	0.953261	−	0.302147i	$-0.0977033\pi$
$17$	7.86079	1.90652	0.953261	+	0.302147i	$0.0977033\pi$
$18$	4.28934	1.01101
$19$	2.18172	0.500520	0.250260	−	0.968179i	$-0.419484\pi$
$19$	2.18172	0.500520	0.250260	+	0.968179i	$0.419484\pi$
$20$	2.56179	0.572834
$21$	12.6938	2.77002
$22$	−3.35955	−0.716258
$23$	1.70570	0.355662	0.177831	−	0.984061i	$-0.443092\pi$
$23$	1.70570	0.355662	0.177831	+	0.984061i	$0.443092\pi$
$24$	−2.69988	−0.551110
$25$	1.56278	0.312555
$26$	−1.77855	−0.348802
$27$	−3.48106	−0.669931
$28$	−4.70163	−0.888524
$29$	−1.59237	−0.295696	−0.147848	−	0.989010i	$-0.547235\pi$
$29$	−1.59237	−0.295696	−0.147848	+	0.989010i	$0.547235\pi$
$30$	−6.91652	−1.26278
$31$	0.397955	0.0714749	0.0357374	−	0.999361i	$-0.488622\pi$
$31$	0.397955	0.0714749	0.0357374	+	0.999361i	$0.488622\pi$
$32$	1.00000	0.176777
$33$	9.07037	1.57895
$34$	7.86079	1.34812
$35$	−12.0446	−2.03591
$36$	4.28934	0.714890
$37$	0.252674	0.0415393	0.0207696	−	0.999784i	$-0.493388\pi$
$37$	0.252674	0.0415393	0.0207696	+	0.999784i	$0.493388\pi$
$38$	2.18172	0.353921
$39$	4.80186	0.768913
$40$	2.56179	0.405055
$41$	−6.51031	−1.01674	−0.508370	−	0.861139i	$-0.669752\pi$
$41$	−6.51031	−1.01674	−0.508370	+	0.861139i	$0.669752\pi$
$42$	12.6938	1.95870
$43$	9.23111	1.40773	0.703865	−	0.710334i	$-0.251456\pi$
$43$	9.23111	1.40773	0.703865	+	0.710334i	$0.251456\pi$
$44$	−3.35955	−0.506471
$45$	10.9884	1.63805
$46$	1.70570	0.251491
$47$	−0.749555	−0.109334	−0.0546669	−	0.998505i	$-0.517410\pi$
$47$	−0.749555	−0.109334	−0.0546669	+	0.998505i	$0.517410\pi$
$48$	−2.69988	−0.389694
$49$	15.1053	2.15790
$50$	1.56278	0.221010
$51$	−21.2232	−2.97184
$52$	−1.77855	−0.246640
$53$	−4.43967	−0.609835	−0.304918	−	0.952379i	$-0.598629\pi$
$53$	−4.43967	−0.609835	−0.304918	+	0.952379i	$0.598629\pi$
$54$	−3.48106	−0.473713
$55$	−8.60646	−1.16049
$56$	−4.70163	−0.628281
$57$	−5.89036	−0.780198
$58$	−1.59237	−0.209089
$59$	−2.84420	−0.370283	−0.185141	−	0.982712i	$-0.559274\pi$
$59$	−2.84420	−0.370283	−0.185141	+	0.982712i	$0.559274\pi$
$60$	−6.91652	−0.892919
$61$	−0.902979	−0.115615	−0.0578073	−	0.998328i	$-0.518411\pi$
$61$	−0.902979	−0.115615	−0.0578073	+	0.998328i	$0.518411\pi$
$62$	0.397955	0.0505404
$63$	−20.1669	−2.54079
$64$	1.00000	0.125000
$65$	−4.55627	−0.565135
$66$	9.07037	1.11648
$67$	−13.9660	−1.70622	−0.853110	−	0.521731i	$-0.825286\pi$
$67$	−13.9660	−1.70622	−0.853110	+	0.521731i	$0.825286\pi$
$68$	7.86079	0.953261
$69$	−4.60517	−0.554398
$70$	−12.0446	−1.43960
$71$	−11.7003	−1.38857	−0.694286	−	0.719699i	$-0.744280\pi$
$71$	−11.7003	−1.38857	−0.694286	+	0.719699i	$0.744280\pi$
$72$	4.28934	0.505504
$73$	−14.1261	−1.65334	−0.826669	−	0.562689i	$-0.809767\pi$
$73$	−14.1261	−1.65334	−0.826669	+	0.562689i	$0.809767\pi$
$74$	0.252674	0.0293727
$75$	−4.21930	−0.487203
$76$	2.18172	0.250260
$77$	15.7953	1.80005
$78$	4.80186	0.543704
$79$	−6.82766	−0.768172	−0.384086	−	0.923297i	$-0.625483\pi$
$79$	−6.82766	−0.768172	−0.384086	+	0.923297i	$0.625483\pi$
$80$	2.56179	0.286417
$81$	−3.46958	−0.385509
$82$	−6.51031	−0.718944
$83$	−10.4247	−1.14426	−0.572130	−	0.820163i	$-0.693883\pi$
$83$	−10.4247	−1.14426	−0.572130	+	0.820163i	$0.693883\pi$
$84$	12.6938	1.38501
$85$	20.1377	2.18424
$86$	9.23111	0.995416
$87$	4.29921	0.460924
$88$	−3.35955	−0.358129
$89$	11.3934	1.20770	0.603848	−	0.797099i	$-0.293633\pi$
$89$	11.3934	1.20770	0.603848	+	0.797099i	$0.293633\pi$
$90$	10.9884	1.15828
$91$	8.36207	0.876583
$92$	1.70570	0.177831
$93$	−1.07443	−0.111413
$94$	−0.749555	−0.0773107
$95$	5.58910	0.573430
$96$	−2.69988	−0.275555
$97$	−3.25680	−0.330678	−0.165339	−	0.986237i	$-0.552872\pi$
$97$	−3.25680	−0.330678	−0.165339	+	0.986237i	$0.552872\pi$
$98$	15.1053	1.52587
$99$	−14.4102	−1.44828
$100$	1.56278	0.156278
$101$	−1.11901	−0.111346	−0.0556730	−	0.998449i	$-0.517730\pi$
$101$	−1.11901	−0.111346	−0.0556730	+	0.998449i	$0.517730\pi$
$102$	−21.2232	−2.10141
$103$	8.46499	0.834081	0.417040	−	0.908888i	$-0.363067\pi$
$103$	8.46499	0.834081	0.417040	+	0.908888i	$0.363067\pi$
$104$	−1.77855	−0.174401
$105$	32.5189	3.17352
$106$	−4.43967	−0.431219
$107$	9.76485	0.944004	0.472002	−	0.881597i	$-0.343531\pi$
$107$	9.76485	0.944004	0.472002	+	0.881597i	$0.343531\pi$
$108$	−3.48106	−0.334965
$109$	−8.06094	−0.772098	−0.386049	−	0.922478i	$-0.626160\pi$
$109$	−8.06094	−0.772098	−0.386049	+	0.922478i	$0.626160\pi$
$110$	−8.60646	−0.820594
$111$	−0.682188	−0.0647504
$112$	−4.70163	−0.444262
$113$	−14.3324	−1.34828	−0.674141	−	0.738602i	$-0.735486\pi$
$113$	−14.3324	−1.34828	−0.674141	+	0.738602i	$0.735486\pi$
$114$	−5.89036	−0.551683
$115$	4.36964	0.407471
$116$	−1.59237	−0.147848
$117$	−7.62879	−0.705282
$118$	−2.84420	−0.261829
$119$	−36.9585	−3.38798
$120$	−6.91652	−0.631389
$121$	0.286554	0.0260503
$122$	−0.902979	−0.0817519
$123$	17.5770	1.58487
$124$	0.397955	0.0357374
$125$	−8.80545	−0.787584
$126$	−20.1669	−1.79661
$127$	−6.14118	−0.544942	−0.272471	−	0.962164i	$-0.587841\pi$
$127$	−6.14118	−0.544942	−0.272471	+	0.962164i	$0.587841\pi$
$128$	1.00000	0.0883883
$129$	−24.9229	−2.19434
$130$	−4.55627	−0.399611
$131$	9.33398	0.815514	0.407757	−	0.913091i	$-0.366311\pi$
$131$	9.33398	0.815514	0.407757	+	0.913091i	$0.366311\pi$
$132$	9.07037	0.789474
$133$	−10.2576	−0.889448
$134$	−13.9660	−1.20648
$135$	−8.91776	−0.767518
$136$	7.86079	0.674058
$137$	20.9553	1.79034	0.895168	−	0.445729i	$-0.147055\pi$
$137$	20.9553	1.79034	0.895168	+	0.445729i	$0.147055\pi$
$138$	−4.60517	−0.392018
$139$	1.54468	0.131018	0.0655088	−	0.997852i	$-0.479133\pi$
$139$	1.54468	0.131018	0.0655088	+	0.997852i	$0.479133\pi$
$140$	−12.0446	−1.01795
$141$	2.02371	0.170427
$142$	−11.7003	−0.981869
$143$	5.97511	0.499664
$144$	4.28934	0.357445
$145$	−4.07932	−0.338769
$146$	−14.1261	−1.16909
$147$	−40.7825	−3.36368
$148$	0.252674	0.0207696
$149$	−0.269120	−0.0220471	−0.0110236	−	0.999939i	$-0.503509\pi$
$149$	−0.269120	−0.0220471	−0.0110236	+	0.999939i	$0.503509\pi$
$150$	−4.21930	−0.344505
$151$	−11.9758	−0.974576	−0.487288	−	0.873241i	$-0.662014\pi$
$151$	−11.9758	−0.974576	−0.487288	+	0.873241i	$0.662014\pi$
$152$	2.18172	0.176960
$153$	33.7176	2.72591
$154$	15.7953	1.27282
$155$	1.01948	0.0818865
$156$	4.80186	0.384456
$157$	−23.4754	−1.87354	−0.936769	−	0.349948i	$-0.886199\pi$
$157$	−23.4754	−1.87354	−0.936769	+	0.349948i	$0.886199\pi$
$158$	−6.82766	−0.543179
$159$	11.9866	0.950596
$160$	2.56179	0.202527
$161$	−8.01955	−0.632029
$162$	−3.46958	−0.272596
$163$	−1.62522	−0.127297	−0.0636486	−	0.997972i	$-0.520274\pi$
$163$	−1.62522	−0.127297	−0.0636486	+	0.997972i	$0.520274\pi$
$164$	−6.51031	−0.508370
$165$	23.2364	1.80895
$166$	−10.4247	−0.809114
$167$	9.56941	0.740504	0.370252	−	0.928931i	$-0.379271\pi$
$167$	9.56941	0.740504	0.370252	+	0.928931i	$0.379271\pi$
$168$	12.6938	0.979350
$169$	−9.83677	−0.756675
$170$	20.1377	1.54449
$171$	9.35812	0.715633
$172$	9.23111	0.703865
$173$	18.7881	1.42843	0.714215	−	0.699926i	$-0.246784\pi$
$173$	18.7881	1.42843	0.714215	+	0.699926i	$0.246784\pi$
$174$	4.29921	0.325922
$175$	−7.34759	−0.555425
$176$	−3.35955	−0.253235
$177$	7.67898	0.577187
$178$	11.3934	0.853970
$179$	7.46164	0.557709	0.278855	−	0.960333i	$-0.410045\pi$
$179$	7.46164	0.557709	0.278855	+	0.960333i	$0.410045\pi$
$180$	10.9884	0.819027
$181$	12.1925	0.906264	0.453132	−	0.891444i	$-0.350307\pi$
$181$	12.1925	0.906264	0.453132	+	0.891444i	$0.350307\pi$
$182$	8.36207	0.619838
$183$	2.43793	0.180217
$184$	1.70570	0.125746
$185$	0.647297	0.0475902
$186$	−1.07443	−0.0787811
$187$	−26.4087	−1.93120
$188$	−0.749555	−0.0546669
$189$	16.3667	1.19050
$190$	5.58910	0.405476
$191$	−19.3248	−1.39829	−0.699145	−	0.714979i	$-0.746436\pi$
$191$	−19.3248	−1.39829	−0.699145	+	0.714979i	$0.746436\pi$
$192$	−2.69988	−0.194847
$193$	−19.6122	−1.41172	−0.705860	−	0.708351i	$-0.749439\pi$
$193$	−19.6122	−1.41172	−0.705860	+	0.708351i	$0.749439\pi$
$194$	−3.25680	−0.233824
$195$	12.3014	0.880919
$196$	15.1053	1.07895
$197$	18.7231	1.33396	0.666982	−	0.745074i	$-0.267586\pi$
$197$	18.7231	1.33396	0.666982	+	0.745074i	$0.267586\pi$
$198$	−14.4102	−1.02409
$199$	−9.88437	−0.700685	−0.350342	−	0.936622i	$-0.613935\pi$
$199$	−9.88437	−0.700685	−0.350342	+	0.936622i	$0.613935\pi$
$200$	1.56278	0.110505
$201$	37.7065	2.65961
$202$	−1.11901	−0.0787335
$203$	7.48674	0.525466
$204$	−21.2232	−1.48592
$205$	−16.6781	−1.16485
$206$	8.46499	0.589784
$207$	7.31631	0.508519
$208$	−1.77855	−0.123320
$209$	−7.32957	−0.506997
$210$	32.5189	2.24402
$211$	−22.1653	−1.52593	−0.762963	−	0.646443i	$-0.776256\pi$
$211$	−22.1653	−1.52593	−0.762963	+	0.646443i	$0.776256\pi$
$212$	−4.43967	−0.304918
$213$	31.5894	2.16447
$214$	9.76485	0.667512
$215$	23.6482	1.61279
$216$	−3.48106	−0.236856
$217$	−1.87104	−0.127014
$218$	−8.06094	−0.545956
$219$	38.1388	2.57718
$220$	−8.60646	−0.580247
$221$	−13.9808	−0.940450
$222$	−0.682188	−0.0457855
$223$	6.88751	0.461222	0.230611	−	0.973046i	$-0.425928\pi$
$223$	6.88751	0.461222	0.230611	+	0.973046i	$0.425928\pi$
$224$	−4.70163	−0.314141
$225$	6.70328	0.446885
$226$	−14.3324	−0.953380
$227$	−26.6524	−1.76898	−0.884492	−	0.466556i	$-0.845495\pi$
$227$	−26.6524	−1.76898	−0.884492	+	0.466556i	$0.845495\pi$
$228$	−5.89036	−0.390099
$229$	1.15984	0.0766443	0.0383221	−	0.999265i	$-0.487799\pi$
$229$	1.15984	0.0766443	0.0383221	+	0.999265i	$0.487799\pi$
$230$	4.36964	0.288125
$231$	−42.6455	−2.80587
$232$	−1.59237	−0.104544
$233$	17.9335	1.17486	0.587430	−	0.809275i	$-0.300140\pi$
$233$	17.9335	1.17486	0.587430	+	0.809275i	$0.300140\pi$
$234$	−7.62879	−0.498710
$235$	−1.92020	−0.125260
$236$	−2.84420	−0.185141
$237$	18.4338	1.19741
$238$	−36.9585	−2.39567
$239$	−17.8130	−1.15223	−0.576113	−	0.817370i	$-0.695431\pi$
$239$	−17.8130	−1.15223	−0.576113	+	0.817370i	$0.695431\pi$
$240$	−6.91652	−0.446460
$241$	−25.3142	−1.63063	−0.815317	−	0.579015i	$-0.803437\pi$
$241$	−25.3142	−1.63063	−0.815317	+	0.579015i	$0.803437\pi$
$242$	0.286554	0.0184204
$243$	19.8106	1.27085
$244$	−0.902979	−0.0578073
$245$	38.6966	2.47224
$246$	17.5770	1.12067
$247$	−3.88028	−0.246896
$248$	0.397955	0.0252702
$249$	28.1454	1.78364
$250$	−8.80545	−0.556906
$251$	−5.98580	−0.377821	−0.188910	−	0.981994i	$-0.560496\pi$
$251$	−5.98580	−0.377821	−0.188910	+	0.981994i	$0.560496\pi$
$252$	−20.1669	−1.27039
$253$	−5.73037	−0.360265
$254$	−6.14118	−0.385332
$255$	−54.3694	−3.40474
$256$	1.00000	0.0625000
$257$	1.56035	0.0973321	0.0486660	−	0.998815i	$-0.484503\pi$
$257$	1.56035	0.0973321	0.0486660	+	0.998815i	$0.484503\pi$
$258$	−24.9229	−1.55163
$259$	−1.18798	−0.0738173
$260$	−4.55627	−0.282568
$261$	−6.83022	−0.422780
$262$	9.33398	0.576655
$263$	13.3944	0.825934	0.412967	−	0.910746i	$-0.364492\pi$
$263$	13.3944	0.825934	0.412967	+	0.910746i	$0.364492\pi$
$264$	9.07037	0.558242
$265$	−11.3735	−0.698669
$266$	−10.2576	−0.628935
$267$	−30.7607	−1.88253
$268$	−13.9660	−0.853110
$269$	−6.59409	−0.402049	−0.201024	−	0.979586i	$-0.564427\pi$
$269$	−6.59409	−0.402049	−0.201024	+	0.979586i	$0.564427\pi$
$270$	−8.91776	−0.542717
$271$	−0.755657	−0.0459029	−0.0229514	−	0.999737i	$-0.507306\pi$
$271$	−0.755657	−0.0459029	−0.0229514	+	0.999737i	$0.507306\pi$
$272$	7.86079	0.476631
$273$	−22.5766	−1.36640
$274$	20.9553	1.26596
$275$	−5.25022	−0.316600
$276$	−4.60517	−0.277199
$277$	15.6450	0.940020	0.470010	−	0.882661i	$-0.344250\pi$
$277$	15.6450	0.940020	0.470010	+	0.882661i	$0.344250\pi$
$278$	1.54468	0.0926435
$279$	1.70697	0.102193
$280$	−12.0446	−0.719802
$281$	3.76053	0.224334	0.112167	−	0.993689i	$-0.464221\pi$
$281$	3.76053	0.224334	0.112167	+	0.993689i	$0.464221\pi$
$282$	2.02371	0.120510
$283$	−24.8214	−1.47548	−0.737739	−	0.675087i	$-0.764106\pi$
$283$	−24.8214	−1.47548	−0.737739	+	0.675087i	$0.764106\pi$
$284$	−11.7003	−0.694286
$285$	−15.0899	−0.893848
$286$	5.97511	0.353316
$287$	30.6091	1.80680
$288$	4.28934	0.252752
$289$	44.7921	2.63483
$290$	−4.07932	−0.239546
$291$	8.79296	0.515452
$292$	−14.1261	−0.826669
$293$	−14.4220	−0.842543	−0.421271	−	0.906935i	$-0.638416\pi$
$293$	−14.4220	−0.842543	−0.421271	+	0.906935i	$0.638416\pi$
$294$	−40.7825	−2.37848
$295$	−7.28624	−0.424221
$296$	0.252674	0.0146864
$297$	11.6948	0.678601
$298$	−0.269120	−0.0155897
$299$	−3.03366	−0.175441
$300$	−4.21930	−0.243602
$301$	−43.4012	−2.50160
$302$	−11.9758	−0.689129
$303$	3.02120	0.173563
$304$	2.18172	0.125130
$305$	−2.31324	−0.132456
$306$	33.7176	1.92751
$307$	−11.6719	−0.666150	−0.333075	−	0.942900i	$-0.608086\pi$
$307$	−11.6719	−0.666150	−0.333075	+	0.942900i	$0.608086\pi$
$308$	15.7953	0.900023
$309$	−22.8545	−1.30014
$310$	1.01948	0.0579025
$311$	−17.4229	−0.987963	−0.493981	−	0.869472i	$-0.664459\pi$
$311$	−17.4229	−0.987963	−0.493981	+	0.869472i	$0.664459\pi$
$312$	4.80186	0.271852
$313$	−26.3859	−1.49142	−0.745710	−	0.666271i	$-0.767889\pi$
$313$	−26.3859	−1.49142	−0.745710	+	0.666271i	$0.767889\pi$
$314$	−23.4754	−1.32479
$315$	−51.6634	−2.91090
$316$	−6.82766	−0.384086
$317$	−3.19053	−0.179198	−0.0895989	−	0.995978i	$-0.528559\pi$
$317$	−3.19053	−0.179198	−0.0895989	+	0.995978i	$0.528559\pi$
$318$	11.9866	0.672173
$319$	5.34965	0.299523
$320$	2.56179	0.143208
$321$	−26.3639	−1.47149
$322$	−8.01955	−0.446912
$323$	17.1500	0.954252
$324$	−3.46958	−0.192754
$325$	−2.77947	−0.154177
$326$	−1.62522	−0.0900127
$327$	21.7636	1.20353
$328$	−6.51031	−0.359472
$329$	3.52413	0.194291
$330$	23.2364	1.27912
$331$	−2.70034	−0.148424	−0.0742119	−	0.997242i	$-0.523644\pi$
$331$	−2.70034	−0.148424	−0.0742119	+	0.997242i	$0.523644\pi$
$332$	−10.4247	−0.572130
$333$	1.08380	0.0593921
$334$	9.56941	0.523615
$335$	−35.7780	−1.95476
$336$	12.6938	0.692505
$337$	−4.93622	−0.268893	−0.134447	−	0.990921i	$-0.542926\pi$
$337$	−4.93622	−0.268893	−0.134447	+	0.990921i	$0.542926\pi$
$338$	−9.83677	−0.535050
$339$	38.6958	2.10167
$340$	20.1377	1.09212
$341$	−1.33695	−0.0723999
$342$	9.35812	0.506029
$343$	−38.1081	−2.05765
$344$	9.23111	0.497708
$345$	−11.7975	−0.635156
$346$	18.7881	1.01005
$347$	−21.3827	−1.14789	−0.573943	−	0.818895i	$-0.694587\pi$
$347$	−21.3827	−1.14789	−0.573943	+	0.818895i	$0.694587\pi$
$348$	4.29921	0.230462
$349$	2.69881	0.144464	0.0722319	−	0.997388i	$-0.476988\pi$
$349$	2.69881	0.144464	0.0722319	+	0.997388i	$0.476988\pi$
$350$	−7.34759	−0.392745
$351$	6.19123	0.330464
$352$	−3.35955	−0.179064
$353$	−18.8719	−1.00445	−0.502226	−	0.864736i	$-0.667485\pi$
$353$	−18.8719	−1.00445	−0.502226	+	0.864736i	$0.667485\pi$
$354$	7.67898	0.408133
$355$	−29.9738	−1.59084
$356$	11.3934	0.603848
$357$	99.7835	5.28110
$358$	7.46164	0.394360
$359$	−2.27872	−0.120266	−0.0601331	−	0.998190i	$-0.519153\pi$
$359$	−2.27872	−0.120266	−0.0601331	+	0.998190i	$0.519153\pi$
$360$	10.9884	0.579139
$361$	−14.2401	−0.749480
$362$	12.1925	0.640825
$363$	−0.773660	−0.0406066
$364$	8.36207	0.438291
$365$	−36.1882	−1.89418
$366$	2.43793	0.127433
$367$	24.2854	1.26769	0.633843	−	0.773462i	$-0.281477\pi$
$367$	24.2854	1.26769	0.633843	+	0.773462i	$0.281477\pi$
$368$	1.70570	0.0889156
$369$	−27.9250	−1.45371
$370$	0.647297	0.0336514
$371$	20.8737	1.08371
$372$	−1.07443	−0.0557066
$373$	13.3098	0.689156	0.344578	−	0.938758i	$-0.388022\pi$
$373$	13.3098	0.689156	0.344578	+	0.938758i	$0.388022\pi$
$374$	−26.4087	−1.36556
$375$	23.7736	1.22767
$376$	−0.749555	−0.0386553
$377$	2.83211	0.145861
$378$	16.3667	0.841810
$379$	33.3308	1.71209	0.856045	−	0.516902i	$-0.172915\pi$
$379$	33.3308	1.71209	0.856045	+	0.516902i	$0.172915\pi$
$380$	5.58910	0.286715
$381$	16.5804	0.849442
$382$	−19.3248	−0.988741
$383$	2.32995	0.119055	0.0595273	−	0.998227i	$-0.481041\pi$
$383$	2.32995	0.119055	0.0595273	+	0.998227i	$0.481041\pi$
$384$	−2.69988	−0.137778
$385$	40.4644	2.06225
$386$	−19.6122	−0.998237
$387$	39.5954	2.01275
$388$	−3.25680	−0.165339
$389$	4.79289	0.243009	0.121505	−	0.992591i	$-0.461228\pi$
$389$	4.79289	0.243009	0.121505	+	0.992591i	$0.461228\pi$
$390$	12.3014	0.622904
$391$	13.4081	0.678078
$392$	15.1053	0.762933
$393$	−25.2006	−1.27120
$394$	18.7231	0.943254
$395$	−17.4910	−0.880070
$396$	−14.4102	−0.724142
$397$	29.7539	1.49331	0.746653	−	0.665214i	$-0.231660\pi$
$397$	29.7539	1.49331	0.746653	+	0.665214i	$0.231660\pi$
$398$	−9.88437	−0.495459
$399$	27.6943	1.38645
$400$	1.56278	0.0781388
$401$	18.9279	0.945214	0.472607	−	0.881273i	$-0.343313\pi$
$401$	18.9279	0.945214	0.472607	+	0.881273i	$0.343313\pi$
$402$	37.7065	1.88063
$403$	−0.707782	−0.0352571
$404$	−1.11901	−0.0556730
$405$	−8.88833	−0.441665
$406$	7.48674	0.371561
$407$	−0.848869	−0.0420769
$408$	−21.2232	−1.05070
$409$	1.18594	0.0586409	0.0293205	−	0.999570i	$-0.490666\pi$
$409$	1.18594	0.0586409	0.0293205	+	0.999570i	$0.490666\pi$
$410$	−16.6781	−0.823671
$411$	−56.5769	−2.79073
$412$	8.46499	0.417040
$413$	13.3723	0.658010
$414$	7.31631	0.359577
$415$	−26.7059	−1.31094
$416$	−1.77855	−0.0872004
$417$	−4.17044	−0.204227
$418$	−7.32957	−0.358501
$419$	0.337479	0.0164869	0.00824347	−	0.999966i	$-0.497376\pi$
$419$	0.337479	0.0164869	0.00824347	+	0.999966i	$0.497376\pi$
$420$	32.5189	1.58676
$421$	−6.59244	−0.321296	−0.160648	−	0.987012i	$-0.551358\pi$
$421$	−6.59244	−0.321296	−0.160648	+	0.987012i	$0.551358\pi$
$422$	−22.1653	−1.07899
$423$	−3.21510	−0.156323
$424$	−4.43967	−0.215609
$425$	12.2847	0.595893
$426$	31.5894	1.53051
$427$	4.24547	0.205453
$428$	9.76485	0.472002
$429$	−16.1321	−0.778864
$430$	23.6482	1.14042
$431$	7.14557	0.344190	0.172095	−	0.985080i	$-0.444946\pi$
$431$	7.14557	0.344190	0.172095	+	0.985080i	$0.444946\pi$
$432$	−3.48106	−0.167483
$433$	−1.28059	−0.0615411	−0.0307705	−	0.999526i	$-0.509796\pi$
$433$	−1.28059	−0.0615411	−0.0307705	+	0.999526i	$0.509796\pi$
$434$	−1.87104	−0.0898127
$435$	11.0137	0.528065
$436$	−8.06094	−0.386049
$437$	3.72134	0.178016
$438$	38.1388	1.82234
$439$	14.2595	0.680571	0.340285	−	0.940322i	$-0.389476\pi$
$439$	14.2595	0.680571	0.340285	+	0.940322i	$0.389476\pi$
$440$	−8.60646	−0.410297
$441$	64.7918	3.08532
$442$	−13.9808	−0.664998
$443$	20.9206	0.993968	0.496984	−	0.867760i	$-0.334441\pi$
$443$	20.9206	0.993968	0.496984	+	0.867760i	$0.334441\pi$
$444$	−0.682188	−0.0323752
$445$	29.1875	1.38362
$446$	6.88751	0.326133
$447$	0.726590	0.0343665
$448$	−4.70163	−0.222131
$449$	−34.0364	−1.60628	−0.803138	−	0.595793i	$-0.796838\pi$
$449$	−34.0364	−1.60628	−0.803138	+	0.595793i	$0.796838\pi$
$450$	6.70328	0.315995
$451$	21.8717	1.02990
$452$	−14.3324	−0.674141
$453$	32.3332	1.51915
$454$	−26.6524	−1.25086
$455$	21.4219	1.00427
$456$	−5.89036	−0.275842
$457$	−16.5841	−0.775773	−0.387886	−	0.921707i	$-0.626795\pi$
$457$	−16.5841	−0.775773	−0.387886	+	0.921707i	$0.626795\pi$
$458$	1.15984	0.0541957
$459$	−27.3639	−1.27724
$460$	4.36964	0.203735
$461$	−27.5089	−1.28122	−0.640608	−	0.767868i	$-0.721318\pi$
$461$	−27.5089	−1.28122	−0.640608	+	0.767868i	$0.721318\pi$
$462$	−42.6455	−1.98405
$463$	26.4623	1.22981	0.614904	−	0.788602i	$-0.289195\pi$
$463$	26.4623	1.22981	0.614904	+	0.788602i	$0.289195\pi$
$464$	−1.59237	−0.0739240
$465$	−2.75247	−0.127643
$466$	17.9335	0.830752
$467$	1.84003	0.0851464	0.0425732	−	0.999093i	$-0.486444\pi$
$467$	1.84003	0.0851464	0.0425732	+	0.999093i	$0.486444\pi$
$468$	−7.62879	−0.352641
$469$	65.6630	3.03203
$470$	−1.92020	−0.0885723
$471$	63.3806	2.92042
$472$	−2.84420	−0.130915
$473$	−31.0123	−1.42595
$474$	18.4338	0.846694
$475$	3.40953	0.156440
$476$	−36.9585	−1.69399
$477$	−19.0433	−0.871931
$478$	−17.8130	−0.814747
$479$	−3.53806	−0.161658	−0.0808290	−	0.996728i	$-0.525757\pi$
$479$	−3.53806	−0.161658	−0.0808290	+	0.996728i	$0.525757\pi$
$480$	−6.91652	−0.315695
$481$	−0.449392	−0.0204905
$482$	−25.3142	−1.15303
$483$	21.6518	0.985191
$484$	0.286554	0.0130252
$485$	−8.34324	−0.378847
$486$	19.8106	0.898628
$487$	−9.67518	−0.438424	−0.219212	−	0.975677i	$-0.570349\pi$
$487$	−9.67518	−0.438424	−0.219212	+	0.975677i	$0.570349\pi$
$488$	−0.902979	−0.0408759
$489$	4.38790	0.198428
$490$	38.6966	1.74814
$491$	33.6333	1.51785	0.758926	−	0.651177i	$-0.225725\pi$
$491$	33.6333	1.51785	0.758926	+	0.651177i	$0.225725\pi$
$492$	17.5770	0.792435
$493$	−12.5173	−0.563751
$494$	−3.88028	−0.174582
$495$	−36.9160	−1.65925
$496$	0.397955	0.0178687
$497$	55.0106	2.46756
$498$	28.1454	1.26123
$499$	−7.47281	−0.334529	−0.167264	−	0.985912i	$-0.553493\pi$
$499$	−7.47281	−0.334529	−0.167264	+	0.985912i	$0.553493\pi$
$500$	−8.80545	−0.393792
$501$	−25.8363	−1.15428
$502$	−5.98580	−0.267159
$503$	22.7769	1.01557	0.507786	−	0.861483i	$-0.330464\pi$
$503$	22.7769	1.01557	0.507786	+	0.861483i	$0.330464\pi$
$504$	−20.1669	−0.898304
$505$	−2.86668	−0.127566
$506$	−5.73037	−0.254746
$507$	26.5581	1.17949
$508$	−6.14118	−0.272471
$509$	8.86932	0.393126	0.196563	−	0.980491i	$-0.437022\pi$
$509$	8.86932	0.393126	0.196563	+	0.980491i	$0.437022\pi$
$510$	−54.3694	−2.40752
$511$	66.4158	2.93806
$512$	1.00000	0.0441942
$513$	−7.59469	−0.335314
$514$	1.56035	0.0688242
$515$	21.6855	0.955579
$516$	−24.9229	−1.09717
$517$	2.51816	0.110749
$518$	−1.18798	−0.0521967
$519$	−50.7255	−2.22660
$520$	−4.55627	−0.199805
$521$	−25.1324	−1.10107	−0.550535	−	0.834812i	$-0.685576\pi$
$521$	−25.1324	−1.10107	−0.550535	+	0.834812i	$0.685576\pi$
$522$	−6.83022	−0.298951
$523$	1.18632	0.0518742	0.0259371	−	0.999664i	$-0.491743\pi$
$523$	1.18632	0.0518742	0.0259371	+	0.999664i	$0.491743\pi$
$524$	9.33398	0.407757
$525$	19.8376	0.865783
$526$	13.3944	0.584024
$527$	3.12825	0.136268
$528$	9.07037	0.394737
$529$	−20.0906	−0.873504
$530$	−11.3735	−0.494033
$531$	−12.1997	−0.529423
$532$	−10.2576	−0.444724
$533$	11.5789	0.501538
$534$	−30.7607	−1.33115
$535$	25.0155	1.08152
$536$	−13.9660	−0.603240
$537$	−20.1455	−0.869343
$538$	−6.59409	−0.284291
$539$	−50.7470	−2.18583
$540$	−8.91776	−0.383759
$541$	−31.4425	−1.35182	−0.675910	−	0.736984i	$-0.736249\pi$
$541$	−31.4425	−1.35182	−0.675910	+	0.736984i	$0.736249\pi$
$542$	−0.755657	−0.0324582
$543$	−32.9183	−1.41266
$544$	7.86079	0.337029
$545$	−20.6505	−0.884568
$546$	−22.5766	−0.966187
$547$	23.2871	0.995685	0.497842	−	0.867268i	$-0.334126\pi$
$547$	23.2871	0.995685	0.497842	+	0.867268i	$0.334126\pi$
$548$	20.9553	0.895168
$549$	−3.87319	−0.165304
$550$	−5.25022	−0.223870
$551$	−3.47410	−0.148002
$552$	−4.60517	−0.196009
$553$	32.1011	1.36508
$554$	15.6450	0.664694
$555$	−1.74762	−0.0741825
$556$	1.54468	0.0655088
$557$	18.0456	0.764615	0.382308	−	0.924035i	$-0.375129\pi$
$557$	18.0456	0.764615	0.382308	+	0.924035i	$0.375129\pi$
$558$	1.70697	0.0722616
$559$	−16.4180	−0.694405
$560$	−12.0446	−0.508977
$561$	71.3003	3.01030
$562$	3.76053	0.158628
$563$	−17.4360	−0.734838	−0.367419	−	0.930056i	$-0.619759\pi$
$563$	−17.4360	−0.734838	−0.367419	+	0.930056i	$0.619759\pi$
$564$	2.02371	0.0852134
$565$	−36.7167	−1.54468
$566$	−24.8214	−1.04332
$567$	16.3127	0.685067
$568$	−11.7003	−0.490935
$569$	22.4157	0.939716	0.469858	−	0.882742i	$-0.344305\pi$
$569$	22.4157	0.939716	0.469858	+	0.882742i	$0.344305\pi$
$570$	−15.0899	−0.632046
$571$	4.71269	0.197220	0.0986100	−	0.995126i	$-0.468560\pi$
$571$	4.71269	0.197220	0.0986100	+	0.995126i	$0.468560\pi$
$572$	5.97511	0.249832
$573$	52.1745	2.17962
$574$	30.6091	1.27760
$575$	2.66562	0.111164
$576$	4.28934	0.178723
$577$	31.9478	1.33001	0.665003	−	0.746841i	$-0.268430\pi$
$577$	31.9478	1.33001	0.665003	+	0.746841i	$0.268430\pi$
$578$	44.7921	1.86311
$579$	52.9507	2.20055
$580$	−4.07932	−0.169385
$581$	49.0131	2.03341
$582$	8.79296	0.364480
$583$	14.9153	0.617727
$584$	−14.1261	−0.584543
$585$	−19.5434	−0.808019
$586$	−14.4220	−0.595768
$587$	−13.6855	−0.564861	−0.282430	−	0.959288i	$-0.591141\pi$
$587$	−13.6855	−0.564861	−0.282430	+	0.959288i	$0.591141\pi$
$588$	−40.7825	−1.68184
$589$	0.868225	0.0357746
$590$	−7.28624	−0.299970
$591$	−50.5500	−2.07935
$592$	0.252674	0.0103848
$593$	31.3841	1.28879	0.644396	−	0.764692i	$-0.277109\pi$
$593$	31.3841	1.28879	0.644396	+	0.764692i	$0.277109\pi$
$594$	11.6948	0.479843
$595$	−94.6801	−3.88150
$596$	−0.269120	−0.0110236
$597$	26.6866	1.09221
$598$	−3.03366	−0.124056
$599$	25.7071	1.05036	0.525182	−	0.850990i	$-0.323997\pi$
$599$	25.7071	1.05036	0.525182	+	0.850990i	$0.323997\pi$
$600$	−4.21930	−0.172252
$601$	−45.8689	−1.87103	−0.935516	−	0.353284i	$-0.885065\pi$
$601$	−45.8689	−1.87103	−0.935516	+	0.353284i	$0.885065\pi$
$602$	−43.4012	−1.76890
$603$	−59.9050	−2.43952
$604$	−11.9758	−0.487288
$605$	0.734091	0.0298450
$606$	3.02120	0.122728
$607$	18.8885	0.766659	0.383329	−	0.923612i	$-0.374777\pi$
$607$	18.8885	0.766659	0.383329	+	0.923612i	$0.374777\pi$
$608$	2.18172	0.0884802
$609$	−20.2133	−0.819083
$610$	−2.31324	−0.0936605
$611$	1.33312	0.0539322
$612$	33.7176	1.36295
$613$	2.82969	0.114290	0.0571451	−	0.998366i	$-0.481800\pi$
$613$	2.82969	0.114290	0.0571451	+	0.998366i	$0.481800\pi$
$614$	−11.6719	−0.471039
$615$	45.0287	1.81573
$616$	15.7953	0.636412
$617$	−3.27229	−0.131738	−0.0658688	−	0.997828i	$-0.520982\pi$
$617$	−3.27229	−0.131738	−0.0658688	+	0.997828i	$0.520982\pi$
$618$	−22.8545	−0.919341
$619$	−36.5683	−1.46980	−0.734901	−	0.678174i	$-0.762772\pi$
$619$	−36.5683	−1.46980	−0.734901	+	0.678174i	$0.762772\pi$
$620$	1.01948	0.0409432
$621$	−5.93764	−0.238269
$622$	−17.4229	−0.698595
$623$	−53.5674	−2.14613
$624$	4.80186	0.192228
$625$	−30.3716	−1.21486
$626$	−26.3859	−1.05459
$627$	19.7890	0.790295
$628$	−23.4754	−0.936769
$629$	1.98622	0.0791956
$630$	−51.6634	−2.05832
$631$	−34.6854	−1.38080	−0.690402	−	0.723426i	$-0.742566\pi$
$631$	−34.6854	−1.38080	−0.690402	+	0.723426i	$0.742566\pi$
$632$	−6.82766	−0.271590
$633$	59.8437	2.37857
$634$	−3.19053	−0.126712
$635$	−15.7324	−0.624322
$636$	11.9866	0.475298
$637$	−26.8655	−1.06445
$638$	5.34965	0.211795
$639$	−50.1867	−1.98535
$640$	2.56179	0.101264
$641$	−21.7407	−0.858705	−0.429352	−	0.903137i	$-0.641258\pi$
$641$	−21.7407	−0.858705	−0.429352	+	0.903137i	$0.641258\pi$
$642$	−26.3639	−1.04050
$643$	18.4249	0.726606	0.363303	−	0.931671i	$-0.381649\pi$
$643$	18.4249	0.726606	0.363303	+	0.931671i	$0.381649\pi$
$644$	−8.01955	−0.316015
$645$	−63.8472	−2.51398
$646$	17.1500	0.674758
$647$	49.1289	1.93146	0.965728	−	0.259558i	$-0.0835769\pi$
$647$	49.1289	1.93146	0.965728	+	0.259558i	$0.0835769\pi$
$648$	−3.46958	−0.136298
$649$	9.55521	0.375075
$650$	−2.77947	−0.109020
$651$	5.05157	0.197987
$652$	−1.62522	−0.0636486
$653$	22.9929	0.899781	0.449890	−	0.893084i	$-0.351463\pi$
$653$	22.9929	0.899781	0.449890	+	0.893084i	$0.351463\pi$
$654$	21.7636	0.851023
$655$	23.9117	0.934308
$656$	−6.51031	−0.254185
$657$	−60.5918	−2.36391
$658$	3.52413	0.137385
$659$	13.5270	0.526938	0.263469	−	0.964668i	$-0.415133\pi$
$659$	13.5270	0.526938	0.263469	+	0.964668i	$0.415133\pi$
$660$	23.2364	0.904475
$661$	−9.87253	−0.383997	−0.191998	−	0.981395i	$-0.561497\pi$
$661$	−9.87253	−0.383997	−0.191998	+	0.981395i	$0.561497\pi$
$662$	−2.70034	−0.104952
$663$	37.7464	1.46595
$664$	−10.4247	−0.404557
$665$	−26.2779	−1.01901
$666$	1.08380	0.0419965
$667$	−2.71610	−0.105168
$668$	9.56941	0.370252
$669$	−18.5954	−0.718941
$670$	−35.7780	−1.38222
$671$	3.03360	0.117111
$672$	12.6938	0.489675
$673$	−35.6070	−1.37255	−0.686275	−	0.727342i	$-0.740755\pi$
$673$	−35.6070	−1.37255	−0.686275	+	0.727342i	$0.740755\pi$
$674$	−4.93622	−0.190136
$675$	−5.44012	−0.209390
$676$	−9.83677	−0.378337
$677$	−28.5524	−1.09736	−0.548679	−	0.836033i	$-0.684869\pi$
$677$	−28.5524	−1.09736	−0.548679	+	0.836033i	$0.684869\pi$
$678$	38.6958	1.48610
$679$	15.3122	0.587630
$680$	20.1377	0.772246
$681$	71.9583	2.75745
$682$	−1.33695	−0.0511944
$683$	9.32897	0.356963	0.178482	−	0.983943i	$-0.442882\pi$
$683$	9.32897	0.356963	0.178482	+	0.983943i	$0.442882\pi$
$684$	9.35812	0.357817
$685$	53.6832	2.05113
$686$	−38.1081	−1.45498
$687$	−3.13142	−0.119471
$688$	9.23111	0.351933
$689$	7.89616	0.300820
$690$	−11.7975	−0.449123
$691$	−31.8466	−1.21150	−0.605751	−	0.795655i	$-0.707127\pi$
$691$	−31.8466	−1.21150	−0.605751	+	0.795655i	$0.707127\pi$
$692$	18.7881	0.714215
$693$	67.7516	2.57367
$694$	−21.3827	−0.811678
$695$	3.95714	0.150103
$696$	4.29921	0.162961
$697$	−51.1762	−1.93844
$698$	2.69881	0.102151
$699$	−48.4182	−1.83134
$700$	−7.34759	−0.277713
$701$	41.7993	1.57874	0.789369	−	0.613919i	$-0.210408\pi$
$701$	41.7993	1.57874	0.789369	+	0.613919i	$0.210408\pi$
$702$	6.19123	0.233673
$703$	0.551262	0.0207912
$704$	−3.35955	−0.126618
$705$	5.18431	0.195252
$706$	−18.8719	−0.710255
$707$	5.26119	0.197867
$708$	7.67898	0.288594
$709$	−21.6934	−0.814713	−0.407357	−	0.913269i	$-0.633549\pi$
$709$	−21.6934	−0.814713	−0.407357	+	0.913269i	$0.633549\pi$
$710$	−29.9738	−1.12490
$711$	−29.2862	−1.09832
$712$	11.3934	0.426985
$713$	0.678791	0.0254209
$714$	99.7835	3.73430
$715$	15.3070	0.572449
$716$	7.46164	0.278855
$717$	48.0929	1.79606
$718$	−2.27872	−0.0850411
$719$	7.33863	0.273685	0.136842	−	0.990593i	$-0.456305\pi$
$719$	7.33863	0.273685	0.136842	+	0.990593i	$0.456305\pi$
$720$	10.9884	0.409513
$721$	−39.7993	−1.48220
$722$	−14.2401	−0.529962
$723$	68.3454	2.54179
$724$	12.1925	0.453132
$725$	−2.48852	−0.0924213
$726$	−0.773660	−0.0287132
$727$	17.7080	0.656754	0.328377	−	0.944547i	$-0.393498\pi$
$727$	17.7080	0.656754	0.328377	+	0.944547i	$0.393498\pi$
$728$	8.36207	0.309919
$729$	−43.0775	−1.59546
$730$	−36.1882	−1.33938
$731$	72.5638	2.68387
$732$	2.43793	0.0901086
$733$	41.3047	1.52562	0.762812	−	0.646621i	$-0.223818\pi$
$733$	41.3047	1.52562	0.762812	+	0.646621i	$0.223818\pi$
$734$	24.2854	0.896389
$735$	−104.476	−3.85366
$736$	1.70570	0.0628728
$737$	46.9195	1.72830
$738$	−27.9250	−1.02793
$739$	23.0811	0.849051	0.424525	−	0.905416i	$-0.360441\pi$
$739$	23.0811	0.849051	0.424525	+	0.905416i	$0.360441\pi$
$740$	0.647297	0.0237951
$741$	10.4763	0.384856
$742$	20.8737	0.766296
$743$	36.0412	1.32222	0.661112	−	0.750287i	$-0.270085\pi$
$743$	36.0412	1.32222	0.661112	+	0.750287i	$0.270085\pi$
$744$	−1.07443	−0.0393905
$745$	−0.689429	−0.0252587
$746$	13.3098	0.487307
$747$	−44.7151	−1.63604
$748$	−26.4087	−0.965598
$749$	−45.9107	−1.67754
$750$	23.7736	0.868091
$751$	31.0980	1.13478	0.567391	−	0.823449i	$-0.307953\pi$
$751$	31.0980	1.13478	0.567391	+	0.823449i	$0.307953\pi$
$752$	−0.749555	−0.0273334
$753$	16.1609	0.588937
$754$	2.83211	0.103139
$755$	−30.6795	−1.11654
$756$	16.3667	0.595250
$757$	23.8594	0.867185	0.433592	−	0.901109i	$-0.357246\pi$
$757$	23.8594	0.867185	0.433592	+	0.901109i	$0.357246\pi$
$758$	33.3308	1.21063
$759$	15.4713	0.561572
$760$	5.58910	0.202738
$761$	30.3630	1.10066	0.550328	−	0.834949i	$-0.314503\pi$
$761$	30.3630	1.10066	0.550328	+	0.834949i	$0.314503\pi$
$762$	16.5804	0.600646
$763$	37.8996	1.37206
$764$	−19.3248	−0.699145
$765$	86.3775	3.12299
$766$	2.32995	0.0841844
$767$	5.05853	0.182653
$768$	−2.69988	−0.0974235
$769$	36.8709	1.32960	0.664799	−	0.747022i	$-0.268517\pi$
$769$	36.8709	1.32960	0.664799	+	0.747022i	$0.268517\pi$
$770$	40.4644	1.45823
$771$	−4.21276	−0.151719
$772$	−19.6122	−0.705860
$773$	14.2738	0.513394	0.256697	−	0.966492i	$-0.417366\pi$
$773$	14.2738	0.513394	0.256697	+	0.966492i	$0.417366\pi$
$774$	39.5954	1.42323
$775$	0.621915	0.0223398
$776$	−3.25680	−0.116912
$777$	3.20739	0.115065
$778$	4.79289	0.171833
$779$	−14.2036	−0.508899
$780$	12.3014	0.440459
$781$	39.3078	1.40654
$782$	13.4081	0.479474
$783$	5.54315	0.198096
$784$	15.1053	0.539475
$785$	−60.1390	−2.14645
$786$	−25.2006	−0.898876
$787$	19.0453	0.678893	0.339446	−	0.940625i	$-0.389760\pi$
$787$	19.0453	0.678893	0.339446	+	0.940625i	$0.389760\pi$
$788$	18.7231	0.666982
$789$	−36.1632	−1.28745
$790$	−17.4910	−0.622303
$791$	67.3858	2.39596
$792$	−14.4102	−0.512046
$793$	1.60599	0.0570304
$794$	29.7539	1.05593
$795$	30.7071	1.08907
$796$	−9.88437	−0.350342
$797$	0.0589370	0.00208765	0.00104383	−	0.999999i	$-0.499668\pi$
$797$	0.0589370	0.00208765	0.00104383	+	0.999999i	$0.499668\pi$
$798$	27.6943	0.980368
$799$	−5.89209	−0.208447
$800$	1.56278	0.0552524
$801$	48.8701	1.72674
$802$	18.9279	0.668367
$803$	47.4574	1.67473
$804$	37.7065	1.32981
$805$	−20.5444	−0.724096
$806$	−0.707782	−0.0249306
$807$	17.8032	0.626704
$808$	−1.11901	−0.0393668
$809$	−39.4627	−1.38744	−0.693718	−	0.720247i	$-0.744029\pi$
$809$	−39.4627	−1.38744	−0.693718	+	0.720247i	$0.744029\pi$
$810$	−8.88833	−0.312304
$811$	−0.963734	−0.0338413	−0.0169206	−	0.999857i	$-0.505386\pi$
$811$	−0.963734	−0.0338413	−0.0169206	+	0.999857i	$0.505386\pi$
$812$	7.48674	0.262733
$813$	2.04018	0.0715523
$814$	−0.848869	−0.0297528
$815$	−4.16348	−0.145840
$816$	−21.2232	−0.742960
$817$	20.1396	0.704597
$818$	1.18594	0.0414654
$819$	35.8678	1.25332
$820$	−16.6781	−0.582423
$821$	46.3542	1.61777	0.808887	−	0.587964i	$-0.200070\pi$
$821$	46.3542	1.61777	0.808887	+	0.587964i	$0.200070\pi$
$822$	−56.5769	−1.97335
$823$	57.1037	1.99051	0.995255	−	0.0973060i	$-0.0310225\pi$
$823$	57.1037	1.99051	0.995255	+	0.0973060i	$0.0310225\pi$
$824$	8.46499	0.294892
$825$	14.1749	0.493508
$826$	13.3723	0.465284
$827$	−43.7775	−1.52229	−0.761146	−	0.648581i	$-0.775363\pi$
$827$	−43.7775	−1.52229	−0.761146	+	0.648581i	$0.775363\pi$
$828$	7.31631	0.254259
$829$	−29.2482	−1.01583	−0.507916	−	0.861406i	$-0.669584\pi$
$829$	−29.2482	−1.01583	−0.507916	+	0.861406i	$0.669584\pi$
$830$	−26.7059	−0.926976
$831$	−42.2397	−1.46528
$832$	−1.77855	−0.0616600
$833$	118.740	4.11409
$834$	−4.17044	−0.144410
$835$	24.5148	0.848371
$836$	−7.32957	−0.253499
$837$	−1.38531	−0.0478832
$838$	0.337479	0.0116580
$839$	11.4574	0.395555	0.197777	−	0.980247i	$-0.436628\pi$
$839$	11.4574	0.395555	0.197777	+	0.980247i	$0.436628\pi$
$840$	32.5189	1.12201
$841$	−26.4644	−0.912564
$842$	−6.59244	−0.227190
$843$	−10.1530	−0.349687
$844$	−22.1653	−0.762963
$845$	−25.1998	−0.866898
$846$	−3.21510	−0.110537
$847$	−1.34727	−0.0462927
$848$	−4.43967	−0.152459
$849$	67.0147	2.29994
$850$	12.2847	0.421360
$851$	0.430985	0.0147740
$852$	31.5894	1.08224
$853$	−36.9586	−1.26544	−0.632719	−	0.774382i	$-0.718061\pi$
$853$	−36.9586	−1.26544	−0.632719	+	0.774382i	$0.718061\pi$
$854$	4.24547	0.145277
$855$	23.9736	0.819878
$856$	9.76485	0.333756
$857$	2.79425	0.0954497	0.0477249	−	0.998861i	$-0.484803\pi$
$857$	2.79425	0.0954497	0.0477249	+	0.998861i	$0.484803\pi$
$858$	−16.1321	−0.550740
$859$	−19.7069	−0.672391	−0.336195	−	0.941792i	$-0.609140\pi$
$859$	−19.7069	−0.672391	−0.336195	+	0.941792i	$0.609140\pi$
$860$	23.6482	0.806396
$861$	−82.6407	−2.81639
$862$	7.14557	0.243379
$863$	−38.0869	−1.29649	−0.648246	−	0.761431i	$-0.724497\pi$
$863$	−38.0869	−1.29649	−0.648246	+	0.761431i	$0.724497\pi$
$864$	−3.48106	−0.118428
$865$	48.1311	1.63651
$866$	−1.28059	−0.0435161
$867$	−120.933	−4.10711
$868$	−1.87104	−0.0635072
$869$	22.9378	0.778113
$870$	11.0137	0.373399
$871$	24.8392	0.841644
$872$	−8.06094	−0.272978
$873$	−13.9695	−0.472796
$874$	3.72134	0.125876
$875$	41.4000	1.39957
$876$	38.1388	1.28859
$877$	−37.2447	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$877$	−37.2447	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$878$	14.2595	0.481236
$879$	38.9377	1.31333
$880$	−8.60646	−0.290124
$881$	−43.3209	−1.45952	−0.729759	−	0.683705i	$-0.760368\pi$
$881$	−43.3209	−1.45952	−0.729759	+	0.683705i	$0.760368\pi$
$882$	64.7918	2.18165
$883$	9.99209	0.336261	0.168130	−	0.985765i	$-0.446227\pi$
$883$	9.99209	0.336261	0.168130	+	0.985765i	$0.446227\pi$
$884$	−13.9808	−0.470225
$885$	19.6719	0.661265
$886$	20.9206	0.702842
$887$	−19.8089	−0.665118	−0.332559	−	0.943082i	$-0.607912\pi$
$887$	−19.8089	−0.665118	−0.332559	+	0.943082i	$0.607912\pi$
$888$	−0.682188	−0.0228927
$889$	28.8735	0.968388
$890$	29.1875	0.978366
$891$	11.6562	0.390498
$892$	6.88751	0.230611
$893$	−1.63531	−0.0547237
$894$	0.726590	0.0243008
$895$	19.1152	0.638949
$896$	−4.70163	−0.157070
$897$	8.19052	0.273473
$898$	−34.0364	−1.13581
$899$	−0.633693	−0.0211348
$900$	6.70328	0.223443
$901$	−34.8993	−1.16266
$902$	21.8717	0.728248
$903$	117.178	3.89944
$904$	−14.3324	−0.476690
$905$	31.2347	1.03828
$906$	32.3332	1.07420
$907$	−53.2116	−1.76686	−0.883432	−	0.468560i	$-0.844773\pi$
$907$	−53.2116	−1.76686	−0.883432	+	0.468560i	$0.844773\pi$
$908$	−26.6524	−0.884492
$909$	−4.79983	−0.159200
$910$	21.4219	0.710128
$911$	0.539804	0.0178845	0.00894225	−	0.999960i	$-0.497154\pi$
$911$	0.539804	0.0178845	0.00894225	+	0.999960i	$0.497154\pi$
$912$	−5.89036	−0.195049
$913$	35.0223	1.15907
$914$	−16.5841	−0.548554
$915$	6.24548	0.206469
$916$	1.15984	0.0383221
$917$	−43.8849	−1.44921
$918$	−27.3639	−0.903144
$919$	22.1803	0.731661	0.365831	−	0.930681i	$-0.380785\pi$
$919$	22.1803	0.731661	0.365831	+	0.930681i	$0.380785\pi$
$920$	4.36964	0.144063
$921$	31.5127	1.03838
$922$	−27.5089	−0.905957
$923$	20.8096	0.684955
$924$	−42.6455	−1.40293
$925$	0.394872	0.0129833
$926$	26.4623	0.869605
$927$	36.3092	1.19255
$928$	−1.59237	−0.0522722
$929$	15.7738	0.517523	0.258761	−	0.965941i	$-0.416686\pi$
$929$	15.7738	0.517523	0.258761	+	0.965941i	$0.416686\pi$
$930$	−2.75247	−0.0902570
$931$	32.9555	1.08007
$932$	17.9335	0.587430
$933$	47.0397	1.54001
$934$	1.84003	0.0602076
$935$	−67.6536	−2.21251
$936$	−7.62879	−0.249355
$937$	34.2311	1.11828	0.559141	−	0.829072i	$-0.311131\pi$
$937$	34.2311	1.11828	0.559141	+	0.829072i	$0.311131\pi$
$938$	65.6630	2.14397
$939$	71.2387	2.32479
$940$	−1.92020	−0.0626301
$941$	−30.4831	−0.993722	−0.496861	−	0.867830i	$-0.665514\pi$
$941$	−30.4831	−0.993722	−0.496861	+	0.867830i	$0.665514\pi$
$942$	63.3806	2.06505
$943$	−11.1046	−0.361616
$944$	−2.84420	−0.0925707
$945$	41.9280	1.36392
$946$	−31.0123	−1.00830
$947$	−41.9181	−1.36216	−0.681078	−	0.732210i	$-0.738489\pi$
$947$	−41.9181	−1.36216	−0.681078	+	0.732210i	$0.738489\pi$
$948$	18.4338	0.598703
$949$	25.1240	0.815559
$950$	3.40953	0.110620
$951$	8.61403	0.279329
$952$	−36.9585	−1.19783
$953$	33.5677	1.08737	0.543683	−	0.839291i	$-0.317030\pi$
$953$	33.5677	1.08737	0.543683	+	0.839291i	$0.317030\pi$
$954$	−19.0433	−0.616548
$955$	−49.5060	−1.60198
$956$	−17.8130	−0.576113
$957$	−14.4434	−0.466889
$958$	−3.53806	−0.114309
$959$	−98.5242	−3.18151
$960$	−6.91652	−0.223230
$961$	−30.8416	−0.994891
$962$	−0.449392	−0.0144890
$963$	41.8848	1.34972
$964$	−25.3142	−0.815317
$965$	−50.2425	−1.61736
$966$	21.6518	0.696635
$967$	−18.2634	−0.587311	−0.293655	−	0.955911i	$-0.594872\pi$
$967$	−18.2634	−0.587311	−0.293655	+	0.955911i	$0.594872\pi$
$968$	0.286554	0.00921018
$969$	−46.3029	−1.48747
$970$	−8.34324	−0.267885
$971$	−33.5934	−1.07806	−0.539031	−	0.842286i	$-0.681209\pi$
$971$	−33.5934	−1.07806	−0.539031	+	0.842286i	$0.681209\pi$
$972$	19.8106	0.635426
$973$	−7.26249	−0.232825
$974$	−9.67518	−0.310013
$975$	7.50423	0.240328
$976$	−0.902979	−0.0289037
$977$	24.8653	0.795510	0.397755	−	0.917492i	$-0.369789\pi$
$977$	24.8653	0.795510	0.397755	+	0.917492i	$0.369789\pi$
$978$	4.38790	0.140310
$979$	−38.2766	−1.22333
$980$	38.6966	1.23612
$981$	−34.5761	−1.10393
$982$	33.6333	1.07328
$983$	−47.0709	−1.50133	−0.750665	−	0.660684i	$-0.770267\pi$
$983$	−47.0709	−1.50133	−0.750665	+	0.660684i	$0.770267\pi$
$984$	17.5770	0.560336
$985$	47.9646	1.52828
$986$	−12.5173	−0.398632
$987$	−9.51471	−0.302857
$988$	−3.88028	−0.123448
$989$	15.7455	0.500677
$990$	−36.9160	−1.17327
$991$	55.0040	1.74726	0.873630	−	0.486590i	$-0.161760\pi$
$991$	55.0040	1.74726	0.873630	+	0.486590i	$0.161760\pi$
$992$	0.397955	0.0126351
$993$	7.29058	0.231359
$994$	55.0106	1.74483
$995$	−25.3217	−0.802752
$996$	28.1454	0.891822
$997$	42.6399	1.35042	0.675210	−	0.737626i	$-0.264053\pi$
$997$	42.6399	1.35042	0.675210	+	0.737626i	$0.264053\pi$
$998$	−7.47281	−0.236548
$999$	−0.879573	−0.0278285

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.f.1.4	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.f.1.4	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.69988 q^{3} +1.00000 q^{4} +2.56179 q^{5} -2.69988 q^{6} -4.70163 q^{7} +1.00000 q^{8} +4.28934 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.69988 q^{3} +1.00000 q^{4} +2.56179 q^{5} -2.69988 q^{6} -4.70163 q^{7} +1.00000 q^{8} +4.28934 q^{9} +2.56179 q^{10} -3.35955 q^{11} -2.69988 q^{12} -1.77855 q^{13} -4.70163 q^{14} -6.91652 q^{15} +1.00000 q^{16} +7.86079 q^{17} +4.28934 q^{18} +2.18172 q^{19} +2.56179 q^{20} +12.6938 q^{21} -3.35955 q^{22} +1.70570 q^{23} -2.69988 q^{24} +1.56278 q^{25} -1.77855 q^{26} -3.48106 q^{27} -4.70163 q^{28} -1.59237 q^{29} -6.91652 q^{30} +0.397955 q^{31} +1.00000 q^{32} +9.07037 q^{33} +7.86079 q^{34} -12.0446 q^{35} +4.28934 q^{36} +0.252674 q^{37} +2.18172 q^{38} +4.80186 q^{39} +2.56179 q^{40} -6.51031 q^{41} +12.6938 q^{42} +9.23111 q^{43} -3.35955 q^{44} +10.9884 q^{45} +1.70570 q^{46} -0.749555 q^{47} -2.69988 q^{48} +15.1053 q^{49} +1.56278 q^{50} -21.2232 q^{51} -1.77855 q^{52} -4.43967 q^{53} -3.48106 q^{54} -8.60646 q^{55} -4.70163 q^{56} -5.89036 q^{57} -1.59237 q^{58} -2.84420 q^{59} -6.91652 q^{60} -0.902979 q^{61} +0.397955 q^{62} -20.1669 q^{63} +1.00000 q^{64} -4.55627 q^{65} +9.07037 q^{66} -13.9660 q^{67} +7.86079 q^{68} -4.60517 q^{69} -12.0446 q^{70} -11.7003 q^{71} +4.28934 q^{72} -14.1261 q^{73} +0.252674 q^{74} -4.21930 q^{75} +2.18172 q^{76} +15.7953 q^{77} +4.80186 q^{78} -6.82766 q^{79} +2.56179 q^{80} -3.46958 q^{81} -6.51031 q^{82} -10.4247 q^{83} +12.6938 q^{84} +20.1377 q^{85} +9.23111 q^{86} +4.29921 q^{87} -3.35955 q^{88} +11.3934 q^{89} +10.9884 q^{90} +8.36207 q^{91} +1.70570 q^{92} -1.07443 q^{93} -0.749555 q^{94} +5.58910 q^{95} -2.69988 q^{96} -3.25680 q^{97} +15.1053 q^{98} -14.4102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) \) 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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