LMFDB - Embedded newform 8043.2.a.n.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.09091 q^{2} +1.00000 q^{3} -0.809919 q^{4} -1.12988 q^{5} -1.09091 q^{6} +1.00000 q^{7} +3.06536 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.09091 q^{2} +1.00000 q^{3} -0.809919 q^{4} -1.12988 q^{5} -1.09091 q^{6} +1.00000 q^{7} +3.06536 q^{8} +1.00000 q^{9} +1.23260 q^{10} -1.68778 q^{11} -0.809919 q^{12} +5.87918 q^{13} -1.09091 q^{14} -1.12988 q^{15} -1.72419 q^{16} -5.56422 q^{17} -1.09091 q^{18} +6.84491 q^{19} +0.915111 q^{20} +1.00000 q^{21} +1.84121 q^{22} +5.37750 q^{23} +3.06536 q^{24} -3.72337 q^{25} -6.41364 q^{26} +1.00000 q^{27} -0.809919 q^{28} -2.65149 q^{29} +1.23260 q^{30} -7.74549 q^{31} -4.24979 q^{32} -1.68778 q^{33} +6.07006 q^{34} -1.12988 q^{35} -0.809919 q^{36} -1.41558 q^{37} -7.46717 q^{38} +5.87918 q^{39} -3.46349 q^{40} -9.04759 q^{41} -1.09091 q^{42} +0.379413 q^{43} +1.36696 q^{44} -1.12988 q^{45} -5.86636 q^{46} -5.21149 q^{47} -1.72419 q^{48} +1.00000 q^{49} +4.06186 q^{50} -5.56422 q^{51} -4.76166 q^{52} -9.76946 q^{53} -1.09091 q^{54} +1.90699 q^{55} +3.06536 q^{56} +6.84491 q^{57} +2.89254 q^{58} -1.50227 q^{59} +0.915111 q^{60} -3.41866 q^{61} +8.44962 q^{62} +1.00000 q^{63} +8.08452 q^{64} -6.64277 q^{65} +1.84121 q^{66} +5.33791 q^{67} +4.50657 q^{68} +5.37750 q^{69} +1.23260 q^{70} -11.6773 q^{71} +3.06536 q^{72} -1.24576 q^{73} +1.54427 q^{74} -3.72337 q^{75} -5.54383 q^{76} -1.68778 q^{77} -6.41364 q^{78} +0.907380 q^{79} +1.94813 q^{80} +1.00000 q^{81} +9.87009 q^{82} +3.77411 q^{83} -0.809919 q^{84} +6.28690 q^{85} -0.413905 q^{86} -2.65149 q^{87} -5.17365 q^{88} +2.34583 q^{89} +1.23260 q^{90} +5.87918 q^{91} -4.35534 q^{92} -7.74549 q^{93} +5.68526 q^{94} -7.73393 q^{95} -4.24979 q^{96} +6.88933 q^{97} -1.09091 q^{98} -1.68778 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * 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 $

$2$	−1.09091	−0.771389	−0.385694	−	0.922627i	$-0.626038\pi$
$2$	−1.09091	−0.771389	−0.385694	+	0.922627i	$0.626038\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.809919	−0.404960
$5$	−1.12988	−0.505298	−0.252649	−	0.967558i	$-0.581302\pi$
$5$	−1.12988	−0.505298	−0.252649	+	0.967558i	$0.581302\pi$
$6$	−1.09091	−0.445361
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.06536	1.08377
$9$	1.00000	0.333333
$10$	1.23260	0.389781
$11$	−1.68778	−0.508884	−0.254442	−	0.967088i	$-0.581892\pi$
$11$	−1.68778	−0.508884	−0.254442	+	0.967088i	$0.581892\pi$
$12$	−0.809919	−0.233804
$13$	5.87918	1.63059	0.815295	−	0.579045i	$-0.196575\pi$
$13$	5.87918	1.63059	0.815295	+	0.579045i	$0.196575\pi$
$14$	−1.09091	−0.291557
$15$	−1.12988	−0.291734
$16$	−1.72419	−0.431048
$17$	−5.56422	−1.34952	−0.674761	−	0.738036i	$-0.735753\pi$
$17$	−5.56422	−1.34952	−0.674761	+	0.738036i	$0.735753\pi$
$18$	−1.09091	−0.257130
$19$	6.84491	1.57033	0.785165	−	0.619286i	$-0.212578\pi$
$19$	6.84491	1.57033	0.785165	+	0.619286i	$0.212578\pi$
$20$	0.915111	0.204625
$21$	1.00000	0.218218
$22$	1.84121	0.392547
$23$	5.37750	1.12129	0.560643	−	0.828058i	$-0.310554\pi$
$23$	5.37750	1.12129	0.560643	+	0.828058i	$0.310554\pi$
$24$	3.06536	0.625715
$25$	−3.72337	−0.744674
$26$	−6.41364	−1.25782
$27$	1.00000	0.192450
$28$	−0.809919	−0.153060
$29$	−2.65149	−0.492370	−0.246185	−	0.969223i	$-0.579177\pi$
$29$	−2.65149	−0.492370	−0.246185	+	0.969223i	$0.579177\pi$
$30$	1.23260	0.225040
$31$	−7.74549	−1.39113	−0.695565	−	0.718463i	$-0.744846\pi$
$31$	−7.74549	−1.39113	−0.695565	+	0.718463i	$0.744846\pi$
$32$	−4.24979	−0.751264
$33$	−1.68778	−0.293804
$34$	6.07006	1.04101
$35$	−1.12988	−0.190985
$36$	−0.809919	−0.134987
$37$	−1.41558	−0.232720	−0.116360	−	0.993207i	$-0.537123\pi$
$37$	−1.41558	−0.232720	−0.116360	+	0.993207i	$0.537123\pi$
$38$	−7.46717	−1.21134
$39$	5.87918	0.941422
$40$	−3.46349	−0.547626
$41$	−9.04759	−1.41300	−0.706498	−	0.707715i	$-0.749726\pi$
$41$	−9.04759	−1.41300	−0.706498	+	0.707715i	$0.749726\pi$
$42$	−1.09091	−0.168331
$43$	0.379413	0.0578599	0.0289300	−	0.999581i	$-0.490790\pi$
$43$	0.379413	0.0578599	0.0289300	+	0.999581i	$0.490790\pi$
$44$	1.36696	0.206077
$45$	−1.12988	−0.168433
$46$	−5.86636	−0.864947
$47$	−5.21149	−0.760175	−0.380087	−	0.924951i	$-0.624106\pi$
$47$	−5.21149	−0.760175	−0.380087	+	0.924951i	$0.624106\pi$
$48$	−1.72419	−0.248866
$49$	1.00000	0.142857
$50$	4.06186	0.574433
$51$	−5.56422	−0.779147
$52$	−4.76166	−0.660323
$53$	−9.76946	−1.34194	−0.670969	−	0.741485i	$-0.734122\pi$
$53$	−9.76946	−1.34194	−0.670969	+	0.741485i	$0.734122\pi$
$54$	−1.09091	−0.148454
$55$	1.90699	0.257138
$56$	3.06536	0.409626
$57$	6.84491	0.906631
$58$	2.89254	0.379809
$59$	−1.50227	−0.195579	−0.0977895	−	0.995207i	$-0.531177\pi$
$59$	−1.50227	−0.195579	−0.0977895	+	0.995207i	$0.531177\pi$
$60$	0.915111	0.118140
$61$	−3.41866	−0.437714	−0.218857	−	0.975757i	$-0.570233\pi$
$61$	−3.41866	−0.437714	−0.218857	+	0.975757i	$0.570233\pi$
$62$	8.44962	1.07310
$63$	1.00000	0.125988
$64$	8.08452	1.01056
$65$	−6.64277	−0.823934
$66$	1.84121	0.226637
$67$	5.33791	0.652129	0.326065	−	0.945347i	$-0.394277\pi$
$67$	5.33791	0.652129	0.326065	+	0.945347i	$0.394277\pi$
$68$	4.50657	0.546502
$69$	5.37750	0.647375
$70$	1.23260	0.147323
$71$	−11.6773	−1.38584	−0.692921	−	0.721013i	$-0.743677\pi$
$71$	−11.6773	−1.38584	−0.692921	+	0.721013i	$0.743677\pi$
$72$	3.06536	0.361257
$73$	−1.24576	−0.145805	−0.0729023	−	0.997339i	$-0.523226\pi$
$73$	−1.24576	−0.145805	−0.0729023	+	0.997339i	$0.523226\pi$
$74$	1.54427	0.179518
$75$	−3.72337	−0.429938
$76$	−5.54383	−0.635920
$77$	−1.68778	−0.192340
$78$	−6.41364	−0.726202
$79$	0.907380	0.102088	0.0510441	−	0.998696i	$-0.483745\pi$
$79$	0.907380	0.102088	0.0510441	+	0.998696i	$0.483745\pi$
$80$	1.94813	0.217808
$81$	1.00000	0.111111
$82$	9.87009	1.08997
$83$	3.77411	0.414262	0.207131	−	0.978313i	$-0.433587\pi$
$83$	3.77411	0.414262	0.207131	+	0.978313i	$0.433587\pi$
$84$	−0.809919	−0.0883694
$85$	6.28690	0.681910
$86$	−0.413905	−0.0446325
$87$	−2.65149	−0.284270
$88$	−5.17365	−0.551513
$89$	2.34583	0.248658	0.124329	−	0.992241i	$-0.460322\pi$
$89$	2.34583	0.248658	0.124329	+	0.992241i	$0.460322\pi$
$90$	1.23260	0.129927
$91$	5.87918	0.616305
$92$	−4.35534	−0.454075
$93$	−7.74549	−0.803170
$94$	5.68526	0.586390
$95$	−7.73393	−0.793484
$96$	−4.24979	−0.433743
$97$	6.88933	0.699505	0.349753	−	0.936842i	$-0.386266\pi$
$97$	6.88933	0.699505	0.349753	+	0.936842i	$0.386266\pi$
$98$	−1.09091	−0.110198
$99$	−1.68778	−0.169628
$100$	3.01563	0.301563
$101$	6.61764	0.658480	0.329240	−	0.944246i	$-0.393207\pi$
$101$	6.61764	0.658480	0.329240	+	0.944246i	$0.393207\pi$
$102$	6.07006	0.601025
$103$	11.3394	1.11731	0.558654	−	0.829401i	$-0.311318\pi$
$103$	11.3394	1.11731	0.558654	+	0.829401i	$0.311318\pi$
$104$	18.0218	1.76719
$105$	−1.12988	−0.110265
$106$	10.6576	1.03516
$107$	10.4834	1.01346	0.506732	−	0.862103i	$-0.330853\pi$
$107$	10.4834	1.01346	0.506732	+	0.862103i	$0.330853\pi$
$108$	−0.809919	−0.0779345
$109$	2.10893	0.201999	0.101000	−	0.994886i	$-0.467796\pi$
$109$	2.10893	0.201999	0.101000	+	0.994886i	$0.467796\pi$
$110$	−2.08035	−0.198353
$111$	−1.41558	−0.134361
$112$	−1.72419	−0.162921
$113$	11.9443	1.12363	0.561814	−	0.827263i	$-0.310103\pi$
$113$	11.9443	1.12363	0.561814	+	0.827263i	$0.310103\pi$
$114$	−7.46717	−0.699365
$115$	−6.07593	−0.566583
$116$	2.14750	0.199390
$117$	5.87918	0.543530
$118$	1.63884	0.150867
$119$	−5.56422	−0.510071
$120$	−3.46349	−0.316172
$121$	−8.15141	−0.741037
$122$	3.72944	0.337648
$123$	−9.04759	−0.815794
$124$	6.27322	0.563352
$125$	9.85636	0.881580
$126$	−1.09091	−0.0971858
$127$	−18.1324	−1.60899	−0.804494	−	0.593961i	$-0.797563\pi$
$127$	−18.1324	−1.60899	−0.804494	+	0.593961i	$0.797563\pi$
$128$	−0.319884	−0.0282740
$129$	0.379413	0.0334054
$130$	7.24665	0.635573
$131$	−12.6935	−1.10903	−0.554517	−	0.832173i	$-0.687097\pi$
$131$	−12.6935	−1.10903	−0.554517	+	0.832173i	$0.687097\pi$
$132$	1.36696	0.118979
$133$	6.84491	0.593529
$134$	−5.82317	−0.503045
$135$	−1.12988	−0.0972446
$136$	−17.0564	−1.46257
$137$	2.87254	0.245418	0.122709	−	0.992443i	$-0.460842\pi$
$137$	2.87254	0.245418	0.122709	+	0.992443i	$0.460842\pi$
$138$	−5.86636	−0.499377
$139$	4.97779	0.422210	0.211105	−	0.977463i	$-0.432294\pi$
$139$	4.97779	0.422210	0.211105	+	0.977463i	$0.432294\pi$
$140$	0.915111	0.0773410
$141$	−5.21149	−0.438887
$142$	12.7389	1.06902
$143$	−9.92274	−0.829782
$144$	−1.72419	−0.143683
$145$	2.99587	0.248793
$146$	1.35901	0.112472
$147$	1.00000	0.0824786
$148$	1.14651	0.0942423
$149$	−7.67844	−0.629042	−0.314521	−	0.949250i	$-0.601844\pi$
$149$	−7.67844	−0.629042	−0.314521	+	0.949250i	$0.601844\pi$
$150$	4.06186	0.331649
$151$	14.5937	1.18762	0.593809	−	0.804606i	$-0.297623\pi$
$151$	14.5937	1.18762	0.593809	+	0.804606i	$0.297623\pi$
$152$	20.9821	1.70188
$153$	−5.56422	−0.449841
$154$	1.84121	0.148369
$155$	8.75147	0.702935
$156$	−4.76166	−0.381238
$157$	−14.3537	−1.14555	−0.572777	−	0.819711i	$-0.694134\pi$
$157$	−14.3537	−1.14555	−0.572777	+	0.819711i	$0.694134\pi$
$158$	−0.989869	−0.0787497
$159$	−9.76946	−0.774769
$160$	4.80175	0.379612
$161$	5.37750	0.423806
$162$	−1.09091	−0.0857098
$163$	−17.1103	−1.34019	−0.670093	−	0.742277i	$-0.733746\pi$
$163$	−17.1103	−1.34019	−0.670093	+	0.742277i	$0.733746\pi$
$164$	7.32782	0.572206
$165$	1.90699	0.148459
$166$	−4.11721	−0.319557
$167$	−22.2477	−1.72158	−0.860790	−	0.508960i	$-0.830030\pi$
$167$	−22.2477	−1.72158	−0.860790	+	0.508960i	$0.830030\pi$
$168$	3.06536	0.236498
$169$	21.5647	1.65883
$170$	−6.85843	−0.526018
$171$	6.84491	0.523444
$172$	−0.307294	−0.0234309
$173$	1.21562	0.0924219	0.0462109	−	0.998932i	$-0.485285\pi$
$173$	1.21562	0.0924219	0.0462109	+	0.998932i	$0.485285\pi$
$174$	2.89254	0.219283
$175$	−3.72337	−0.281460
$176$	2.91005	0.219354
$177$	−1.50227	−0.112918
$178$	−2.55909	−0.191812
$179$	−18.4070	−1.37580	−0.687901	−	0.725805i	$-0.741468\pi$
$179$	−18.4070	−1.37580	−0.687901	+	0.725805i	$0.741468\pi$
$180$	0.915111	0.0682084
$181$	7.77113	0.577624	0.288812	−	0.957386i	$-0.406740\pi$
$181$	7.77113	0.577624	0.288812	+	0.957386i	$0.406740\pi$
$182$	−6.41364	−0.475411
$183$	−3.41866	−0.252714
$184$	16.4840	1.21522
$185$	1.59944	0.117593
$186$	8.44962	0.619556
$187$	9.39117	0.686750
$188$	4.22089	0.307840
$189$	1.00000	0.0727393
$190$	8.43701	0.612085
$191$	−13.2457	−0.958423	−0.479211	−	0.877699i	$-0.659077\pi$
$191$	−13.2457	−0.958423	−0.479211	+	0.877699i	$0.659077\pi$
$192$	8.08452	0.583450
$193$	4.47200	0.321901	0.160951	−	0.986962i	$-0.448544\pi$
$193$	4.47200	0.321901	0.160951	+	0.986962i	$0.448544\pi$
$194$	−7.51562	−0.539590
$195$	−6.64277	−0.475698
$196$	−0.809919	−0.0578514
$197$	16.3180	1.16261	0.581303	−	0.813687i	$-0.302543\pi$
$197$	16.3180	1.16261	0.581303	+	0.813687i	$0.302543\pi$
$198$	1.84121	0.130849
$199$	−17.7794	−1.26035	−0.630173	−	0.776455i	$-0.717016\pi$
$199$	−17.7794	−1.26035	−0.630173	+	0.776455i	$0.717016\pi$
$200$	−11.4135	−0.807056
$201$	5.33791	0.376507
$202$	−7.21924	−0.507944
$203$	−2.65149	−0.186098
$204$	4.50657	0.315523
$205$	10.2227	0.713984
$206$	−12.3703	−0.861879
$207$	5.37750	0.373762
$208$	−10.1368	−0.702863
$209$	−11.5527	−0.799116
$210$	1.23260	0.0850572
$211$	−4.49884	−0.309713	−0.154856	−	0.987937i	$-0.549492\pi$
$211$	−4.49884	−0.309713	−0.154856	+	0.987937i	$0.549492\pi$
$212$	7.91247	0.543431
$213$	−11.6773	−0.800116
$214$	−11.4364	−0.781775
$215$	−0.428691	−0.0292365
$216$	3.06536	0.208572
$217$	−7.74549	−0.525798
$218$	−2.30065	−0.155820
$219$	−1.24576	−0.0841804
$220$	−1.54450	−0.104130
$221$	−32.7131	−2.20052
$222$	1.54427	0.103645
$223$	−21.5985	−1.44634	−0.723170	−	0.690670i	$-0.757316\pi$
$223$	−21.5985	−1.44634	−0.723170	+	0.690670i	$0.757316\pi$
$224$	−4.24979	−0.283951
$225$	−3.72337	−0.248225
$226$	−13.0302	−0.866755
$227$	29.8766	1.98298	0.991490	−	0.130184i	$-0.0415567\pi$
$227$	29.8766	1.98298	0.991490	+	0.130184i	$0.0415567\pi$
$228$	−5.54383	−0.367149
$229$	5.77762	0.381796	0.190898	−	0.981610i	$-0.438860\pi$
$229$	5.77762	0.381796	0.190898	+	0.981610i	$0.438860\pi$
$230$	6.62828	0.437056
$231$	−1.68778	−0.111048
$232$	−8.12779	−0.533616
$233$	−16.4801	−1.07965	−0.539824	−	0.841778i	$-0.681509\pi$
$233$	−16.4801	−1.07965	−0.539824	+	0.841778i	$0.681509\pi$
$234$	−6.41364	−0.419273
$235$	5.88836	0.384114
$236$	1.21672	0.0792016
$237$	0.907380	0.0589407
$238$	6.07006	0.393463
$239$	−6.10555	−0.394935	−0.197468	−	0.980309i	$-0.563272\pi$
$239$	−6.10555	−0.394935	−0.197468	+	0.980309i	$0.563272\pi$
$240$	1.94813	0.125751
$241$	−5.73679	−0.369539	−0.184769	−	0.982782i	$-0.559154\pi$
$241$	−5.73679	−0.369539	−0.184769	+	0.982782i	$0.559154\pi$
$242$	8.89244	0.571628
$243$	1.00000	0.0641500
$244$	2.76884	0.177257
$245$	−1.12988	−0.0721854
$246$	9.87009	0.629294
$247$	40.2425	2.56057
$248$	−23.7427	−1.50767
$249$	3.77411	0.239174
$250$	−10.7524	−0.680041
$251$	−19.5133	−1.23167	−0.615833	−	0.787876i	$-0.711181\pi$
$251$	−19.5133	−1.23167	−0.615833	+	0.787876i	$0.711181\pi$
$252$	−0.809919	−0.0510201
$253$	−9.07602	−0.570604
$254$	19.7808	1.24115
$255$	6.28690	0.393701
$256$	−15.8201	−0.988755
$257$	12.4109	0.774168	0.387084	−	0.922044i	$-0.373482\pi$
$257$	12.4109	0.774168	0.387084	+	0.922044i	$0.373482\pi$
$258$	−0.413905	−0.0257686
$259$	−1.41558	−0.0879600
$260$	5.38010	0.333660
$261$	−2.65149	−0.164123
$262$	13.8474	0.855496
$263$	17.4509	1.07607	0.538035	−	0.842922i	$-0.319167\pi$
$263$	17.4509	1.07607	0.538035	+	0.842922i	$0.319167\pi$
$264$	−5.17365	−0.318416
$265$	11.0383	0.678078
$266$	−7.46717	−0.457842
$267$	2.34583	0.143563
$268$	−4.32327	−0.264086
$269$	15.9257	0.971008	0.485504	−	0.874234i	$-0.338636\pi$
$269$	15.9257	0.971008	0.485504	+	0.874234i	$0.338636\pi$
$270$	1.23260	0.0750134
$271$	−27.4133	−1.66524	−0.832619	−	0.553846i	$-0.813160\pi$
$271$	−27.4133	−1.66524	−0.832619	+	0.553846i	$0.813160\pi$
$272$	9.59379	0.581709
$273$	5.87918	0.355824
$274$	−3.13368	−0.189313
$275$	6.28422	0.378953
$276$	−4.35534	−0.262161
$277$	−0.546370	−0.0328282	−0.0164141	−	0.999865i	$-0.505225\pi$
$277$	−0.546370	−0.0328282	−0.0164141	+	0.999865i	$0.505225\pi$
$278$	−5.43031	−0.325688
$279$	−7.74549	−0.463710
$280$	−3.46349	−0.206983
$281$	19.7703	1.17940	0.589700	−	0.807622i	$-0.299246\pi$
$281$	19.7703	1.17940	0.589700	+	0.807622i	$0.299246\pi$
$282$	5.68526	0.338552
$283$	8.49736	0.505115	0.252558	−	0.967582i	$-0.418728\pi$
$283$	8.49736	0.505115	0.252558	+	0.967582i	$0.418728\pi$
$284$	9.45768	0.561210
$285$	−7.73393	−0.458118
$286$	10.8248	0.640084
$287$	−9.04759	−0.534062
$288$	−4.24979	−0.250421
$289$	13.9606	0.821210
$290$	−3.26822	−0.191916
$291$	6.88933	0.403859
$292$	1.00896	0.0590450
$293$	14.2474	0.832343	0.416171	−	0.909286i	$-0.363372\pi$
$293$	14.2474	0.832343	0.416171	+	0.909286i	$0.363372\pi$
$294$	−1.09091	−0.0636231
$295$	1.69738	0.0988256
$296$	−4.33927	−0.252215
$297$	−1.68778	−0.0979348
$298$	8.37647	0.485236
$299$	31.6153	1.82836
$300$	3.01563	0.174107
$301$	0.379413	0.0218690
$302$	−15.9204	−0.916116
$303$	6.61764	0.380174
$304$	−11.8019	−0.676888
$305$	3.86267	0.221176
$306$	6.07006	0.347002
$307$	−32.1440	−1.83455	−0.917277	−	0.398250i	$-0.869618\pi$
$307$	−32.1440	−1.83455	−0.917277	+	0.398250i	$0.869618\pi$
$308$	1.36696	0.0778900
$309$	11.3394	0.645078
$310$	−9.54705	−0.542236
$311$	−13.0552	−0.740293	−0.370147	−	0.928973i	$-0.620693\pi$
$311$	−13.0552	−0.740293	−0.370147	+	0.928973i	$0.620693\pi$
$312$	18.0218	1.02028
$313$	−11.0071	−0.622159	−0.311080	−	0.950384i	$-0.600691\pi$
$313$	−11.0071	−0.622159	−0.311080	+	0.950384i	$0.600691\pi$
$314$	15.6586	0.883667
$315$	−1.12988	−0.0636615
$316$	−0.734905	−0.0413416
$317$	4.39807	0.247020	0.123510	−	0.992343i	$-0.460585\pi$
$317$	4.39807	0.247020	0.123510	+	0.992343i	$0.460585\pi$
$318$	10.6576	0.597648
$319$	4.47513	0.250559
$320$	−9.13453	−0.510636
$321$	10.4834	0.585124
$322$	−5.86636	−0.326919
$323$	−38.0866	−2.11920
$324$	−0.809919	−0.0449955
$325$	−21.8904	−1.21426
$326$	18.6658	1.03380
$327$	2.10893	0.116624
$328$	−27.7342	−1.53136
$329$	−5.21149	−0.287319
$330$	−2.08035	−0.114519
$331$	10.1132	0.555872	0.277936	−	0.960600i	$-0.410350\pi$
$331$	10.1132	0.555872	0.277936	+	0.960600i	$0.410350\pi$
$332$	−3.05672	−0.167759
$333$	−1.41558	−0.0775734
$334$	24.2702	1.32801
$335$	−6.03120	−0.329519
$336$	−1.72419	−0.0940624
$337$	−0.650054	−0.0354107	−0.0177054	−	0.999843i	$-0.505636\pi$
$337$	−0.650054	−0.0354107	−0.0177054	+	0.999843i	$0.505636\pi$
$338$	−23.5252	−1.27960
$339$	11.9443	0.648727
$340$	−5.09188	−0.276146
$341$	13.0727	0.707924
$342$	−7.46717	−0.403778
$343$	1.00000	0.0539949
$344$	1.16304	0.0627068
$345$	−6.07593	−0.327117
$346$	−1.32613	−0.0712932
$347$	−20.3942	−1.09482	−0.547410	−	0.836865i	$-0.684386\pi$
$347$	−20.3942	−1.09482	−0.547410	+	0.836865i	$0.684386\pi$
$348$	2.14750	0.115118
$349$	5.21595	0.279204	0.139602	−	0.990208i	$-0.455418\pi$
$349$	5.21595	0.279204	0.139602	+	0.990208i	$0.455418\pi$
$350$	4.06186	0.217115
$351$	5.87918	0.313807
$352$	7.17270	0.382306
$353$	−33.6780	−1.79250	−0.896248	−	0.443553i	$-0.853718\pi$
$353$	−33.6780	−1.79250	−0.896248	+	0.443553i	$0.853718\pi$
$354$	1.63884	0.0871033
$355$	13.1940	0.700263
$356$	−1.89993	−0.100696
$357$	−5.56422	−0.294490
$358$	20.0803	1.06128
$359$	−12.2346	−0.645719	−0.322859	−	0.946447i	$-0.604644\pi$
$359$	−12.2346	−0.645719	−0.322859	+	0.946447i	$0.604644\pi$
$360$	−3.46349	−0.182542
$361$	27.8528	1.46594
$362$	−8.47759	−0.445572
$363$	−8.15141	−0.427838
$364$	−4.76166	−0.249579
$365$	1.40755	0.0736748
$366$	3.72944	0.194941
$367$	−7.48552	−0.390741	−0.195370	−	0.980730i	$-0.562591\pi$
$367$	−7.48552	−0.390741	−0.195370	+	0.980730i	$0.562591\pi$
$368$	−9.27184	−0.483328
$369$	−9.04759	−0.470999
$370$	−1.74484	−0.0907099
$371$	−9.76946	−0.507205
$372$	6.27322	0.325251
$373$	5.70570	0.295430	0.147715	−	0.989030i	$-0.452808\pi$
$373$	5.70570	0.295430	0.147715	+	0.989030i	$0.452808\pi$
$374$	−10.2449	−0.529751
$375$	9.85636	0.508980
$376$	−15.9751	−0.823854
$377$	−15.5886	−0.802854
$378$	−1.09091	−0.0561103
$379$	12.8959	0.662416	0.331208	−	0.943558i	$-0.392544\pi$
$379$	12.8959	0.662416	0.331208	+	0.943558i	$0.392544\pi$
$380$	6.26386	0.321329
$381$	−18.1324	−0.928950
$382$	14.4498	0.739317
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−0.319884	−0.0163240
$385$	1.90699	0.0971890
$386$	−4.87854	−0.248311
$387$	0.379413	0.0192866
$388$	−5.57980	−0.283271
$389$	18.6585	0.946023	0.473011	−	0.881056i	$-0.343167\pi$
$389$	18.6585	0.946023	0.473011	+	0.881056i	$0.343167\pi$
$390$	7.24665	0.366948
$391$	−29.9216	−1.51320
$392$	3.06536	0.154824
$393$	−12.6935	−0.640301
$394$	−17.8014	−0.896821
$395$	−1.02523	−0.0515850
$396$	1.36696	0.0686925
$397$	11.5061	0.577472	0.288736	−	0.957409i	$-0.406765\pi$
$397$	11.5061	0.577472	0.288736	+	0.957409i	$0.406765\pi$
$398$	19.3957	0.972217
$399$	6.84491	0.342674
$400$	6.41981	0.320990
$401$	−27.9986	−1.39818	−0.699092	−	0.715032i	$-0.746412\pi$
$401$	−27.9986	−1.39818	−0.699092	+	0.715032i	$0.746412\pi$
$402$	−5.82317	−0.290433
$403$	−45.5371	−2.26837
$404$	−5.35976	−0.266658
$405$	−1.12988	−0.0561442
$406$	2.89254	0.143554
$407$	2.38919	0.118428
$408$	−17.0564	−0.844416
$409$	−32.1388	−1.58916	−0.794582	−	0.607157i	$-0.792310\pi$
$409$	−32.1388	−1.58916	−0.794582	+	0.607157i	$0.792310\pi$
$410$	−11.1520	−0.550759
$411$	2.87254	0.141692
$412$	−9.18403	−0.452465
$413$	−1.50227	−0.0739219
$414$	−5.86636	−0.288316
$415$	−4.26429	−0.209326
$416$	−24.9853	−1.22500
$417$	4.97779	0.243763
$418$	12.6029	0.616429
$419$	4.22558	0.206433	0.103217	−	0.994659i	$-0.467087\pi$
$419$	4.22558	0.206433	0.103217	+	0.994659i	$0.467087\pi$
$420$	0.915111	0.0446529
$421$	−1.45354	−0.0708412	−0.0354206	−	0.999372i	$-0.511277\pi$
$421$	−1.45354	−0.0708412	−0.0354206	+	0.999372i	$0.511277\pi$
$422$	4.90782	0.238909
$423$	−5.21149	−0.253392
$424$	−29.9470	−1.45435
$425$	20.7177	1.00495
$426$	12.7389	0.617201
$427$	−3.41866	−0.165440
$428$	−8.49068	−0.410412
$429$	−9.92274	−0.479075
$430$	0.467663	0.0225527
$431$	−10.6399	−0.512508	−0.256254	−	0.966609i	$-0.582488\pi$
$431$	−10.6399	−0.512508	−0.256254	+	0.966609i	$0.582488\pi$
$432$	−1.72419	−0.0829553
$433$	−7.88242	−0.378805	−0.189403	−	0.981900i	$-0.560655\pi$
$433$	−7.88242	−0.378805	−0.189403	+	0.981900i	$0.560655\pi$
$434$	8.44962	0.405595
$435$	2.99587	0.143641
$436$	−1.70807	−0.0818016
$437$	36.8085	1.76079
$438$	1.35901	0.0649358
$439$	22.6607	1.08154	0.540769	−	0.841171i	$-0.318133\pi$
$439$	22.6607	1.08154	0.540769	+	0.841171i	$0.318133\pi$
$440$	5.84561	0.278678
$441$	1.00000	0.0476190
$442$	35.6869	1.69746
$443$	−17.3764	−0.825578	−0.412789	−	0.910827i	$-0.635445\pi$
$443$	−17.3764	−0.825578	−0.412789	+	0.910827i	$0.635445\pi$
$444$	1.14651	0.0544108
$445$	−2.65051	−0.125646
$446$	23.5619	1.11569
$447$	−7.67844	−0.363178
$448$	8.08452	0.381958
$449$	−25.4150	−1.19941	−0.599703	−	0.800223i	$-0.704715\pi$
$449$	−25.4150	−1.19941	−0.599703	+	0.800223i	$0.704715\pi$
$450$	4.06186	0.191478
$451$	15.2703	0.719051
$452$	−9.67395	−0.455024
$453$	14.5937	0.685672
$454$	−32.5926	−1.52965
$455$	−6.64277	−0.311418
$456$	20.9821	0.982579
$457$	−24.5371	−1.14780	−0.573898	−	0.818927i	$-0.694569\pi$
$457$	−24.5371	−1.14780	−0.573898	+	0.818927i	$0.694569\pi$
$458$	−6.30285	−0.294513
$459$	−5.56422	−0.259716
$460$	4.92101	0.229443
$461$	−17.9423	−0.835654	−0.417827	−	0.908526i	$-0.637208\pi$
$461$	−17.9423	−0.835654	−0.417827	+	0.908526i	$0.637208\pi$
$462$	1.84121	0.0856609
$463$	27.2622	1.26698	0.633492	−	0.773750i	$-0.281621\pi$
$463$	27.2622	1.26698	0.633492	+	0.773750i	$0.281621\pi$
$464$	4.57168	0.212235
$465$	8.75147	0.405840
$466$	17.9783	0.832828
$467$	23.4632	1.08575	0.542875	−	0.839814i	$-0.317336\pi$
$467$	23.4632	1.08575	0.542875	+	0.839814i	$0.317336\pi$
$468$	−4.76166	−0.220108
$469$	5.33791	0.246482
$470$	−6.42366	−0.296302
$471$	−14.3537	−0.661386
$472$	−4.60500	−0.211963
$473$	−0.640365	−0.0294440
$474$	−0.989869	−0.0454662
$475$	−25.4862	−1.16938
$476$	4.50657	0.206558
$477$	−9.76946	−0.447313
$478$	6.66059	0.304648
$479$	−9.00231	−0.411326	−0.205663	−	0.978623i	$-0.565935\pi$
$479$	−9.00231	−0.411326	−0.205663	+	0.978623i	$0.565935\pi$
$480$	4.80175	0.219169
$481$	−8.32246	−0.379471
$482$	6.25831	0.285058
$483$	5.37750	0.244685
$484$	6.60198	0.300090
$485$	−7.78411	−0.353458
$486$	−1.09091	−0.0494846
$487$	−3.63000	−0.164491	−0.0822454	−	0.996612i	$-0.526209\pi$
$487$	−3.63000	−0.164491	−0.0822454	+	0.996612i	$0.526209\pi$
$488$	−10.4794	−0.474381
$489$	−17.1103	−0.773756
$490$	1.23260	0.0556830
$491$	−19.0420	−0.859352	−0.429676	−	0.902983i	$-0.641372\pi$
$491$	−19.0420	−0.859352	−0.429676	+	0.902983i	$0.641372\pi$
$492$	7.32782	0.330364
$493$	14.7535	0.664464
$494$	−43.9008	−1.97519
$495$	1.90699	0.0857126
$496$	13.3547	0.599644
$497$	−11.6773	−0.523799
$498$	−4.11721	−0.184496
$499$	28.6794	1.28386	0.641932	−	0.766761i	$-0.278133\pi$
$499$	28.6794	1.28386	0.641932	+	0.766761i	$0.278133\pi$
$500$	−7.98286	−0.357004
$501$	−22.2477	−0.993955
$502$	21.2872	0.950094
$503$	8.31236	0.370630	0.185315	−	0.982679i	$-0.440670\pi$
$503$	8.31236	0.370630	0.185315	+	0.982679i	$0.440670\pi$
$504$	3.06536	0.136542
$505$	−7.47714	−0.332729
$506$	9.90110	0.440158
$507$	21.5647	0.957724
$508$	14.6858	0.651575
$509$	−9.62687	−0.426703	−0.213352	−	0.976975i	$-0.568438\pi$
$509$	−9.62687	−0.426703	−0.213352	+	0.976975i	$0.568438\pi$
$510$	−6.85843	−0.303697
$511$	−1.24576	−0.0551090
$512$	17.8980	0.790988
$513$	6.84491	0.302210
$514$	−13.5391	−0.597184
$515$	−12.8122	−0.564573
$516$	−0.307294	−0.0135279
$517$	8.79584	0.386841
$518$	1.54427	0.0678513
$519$	1.21562	0.0533598
$520$	−20.3625	−0.892954
$521$	−17.8474	−0.781909	−0.390955	−	0.920410i	$-0.627855\pi$
$521$	−17.8474	−0.781909	−0.390955	+	0.920410i	$0.627855\pi$
$522$	2.89254	0.126603
$523$	33.1263	1.44851	0.724255	−	0.689532i	$-0.242184\pi$
$523$	33.1263	1.44851	0.724255	+	0.689532i	$0.242184\pi$
$524$	10.2807	0.449114
$525$	−3.72337	−0.162501
$526$	−19.0374	−0.830069
$527$	43.0976	1.87736
$528$	2.91005	0.126644
$529$	5.91748	0.257282
$530$	−12.0418	−0.523062
$531$	−1.50227	−0.0651930
$532$	−5.54383	−0.240355
$533$	−53.1924	−2.30402
$534$	−2.55909	−0.110743
$535$	−11.8449	−0.512101
$536$	16.3626	0.706758
$537$	−18.4070	−0.794319
$538$	−17.3735	−0.749025
$539$	−1.68778	−0.0726977
$540$	0.915111	0.0393801
$541$	−25.0832	−1.07841	−0.539206	−	0.842174i	$-0.681276\pi$
$541$	−25.0832	−1.07841	−0.539206	+	0.842174i	$0.681276\pi$
$542$	29.9054	1.28455
$543$	7.77113	0.333491
$544$	23.6468	1.01385
$545$	−2.38284	−0.102070
$546$	−6.41364	−0.274479
$547$	15.1791	0.649013	0.324507	−	0.945883i	$-0.394802\pi$
$547$	15.1791	0.649013	0.324507	+	0.945883i	$0.394802\pi$
$548$	−2.32653	−0.0993843
$549$	−3.41866	−0.145905
$550$	−6.85551	−0.292320
$551$	−18.1492	−0.773184
$552$	16.4840	0.701605
$553$	0.907380	0.0385857
$554$	0.596039	0.0253233
$555$	1.59944	0.0678923
$556$	−4.03160	−0.170978
$557$	8.05931	0.341484	0.170742	−	0.985316i	$-0.445384\pi$
$557$	8.05931	0.341484	0.170742	+	0.985316i	$0.445384\pi$
$558$	8.44962	0.357701
$559$	2.23064	0.0943459
$560$	1.94813	0.0823235
$561$	9.39117	0.396496
$562$	−21.5676	−0.909776
$563$	−17.1709	−0.723669	−0.361835	−	0.932242i	$-0.617849\pi$
$563$	−17.1709	−0.723669	−0.361835	+	0.932242i	$0.617849\pi$
$564$	4.22089	0.177732
$565$	−13.4957	−0.567767
$566$	−9.26984	−0.389640
$567$	1.00000	0.0419961
$568$	−35.7952	−1.50193
$569$	−17.7373	−0.743588	−0.371794	−	0.928315i	$-0.621257\pi$
$569$	−17.7373	−0.743588	−0.371794	+	0.928315i	$0.621257\pi$
$570$	8.43701	0.353387
$571$	−18.8947	−0.790719	−0.395359	−	0.918526i	$-0.629380\pi$
$571$	−18.8947	−0.790719	−0.395359	+	0.918526i	$0.629380\pi$
$572$	8.03662	0.336028
$573$	−13.2457	−0.553346
$574$	9.87009	0.411970
$575$	−20.0224	−0.834993
$576$	8.08452	0.336855
$577$	−40.9831	−1.70615	−0.853074	−	0.521790i	$-0.825264\pi$
$577$	−40.9831	−1.70615	−0.853074	+	0.521790i	$0.825264\pi$
$578$	−15.2297	−0.633472
$579$	4.47200	0.185850
$580$	−2.42641	−0.100751
$581$	3.77411	0.156576
$582$	−7.51562	−0.311533
$583$	16.4887	0.682891
$584$	−3.81870	−0.158019
$585$	−6.64277	−0.274645
$586$	−15.5426	−0.642060
$587$	−39.4874	−1.62982	−0.814909	−	0.579589i	$-0.803213\pi$
$587$	−39.4874	−1.62982	−0.814909	+	0.579589i	$0.803213\pi$
$588$	−0.809919	−0.0334005
$589$	−53.0172	−2.18454
$590$	−1.85169	−0.0762329
$591$	16.3180	0.671231
$592$	2.44074	0.100314
$593$	40.5499	1.66519	0.832593	−	0.553886i	$-0.186856\pi$
$593$	40.5499	1.66519	0.832593	+	0.553886i	$0.186856\pi$
$594$	1.84121	0.0755458
$595$	6.28690	0.257738
$596$	6.21891	0.254737
$597$	−17.7794	−0.727661
$598$	−34.4894	−1.41037
$599$	−40.2881	−1.64613	−0.823064	−	0.567948i	$-0.807737\pi$
$599$	−40.2881	−1.64613	−0.823064	+	0.567948i	$0.807737\pi$
$600$	−11.4135	−0.465954
$601$	−24.4405	−0.996949	−0.498474	−	0.866904i	$-0.666106\pi$
$601$	−24.4405	−0.996949	−0.498474	+	0.866904i	$0.666106\pi$
$602$	−0.413905	−0.0168695
$603$	5.33791	0.217376
$604$	−11.8197	−0.480938
$605$	9.21011	0.374444
$606$	−7.21924	−0.293262
$607$	26.7348	1.08513	0.542567	−	0.840013i	$-0.317453\pi$
$607$	26.7348	1.08513	0.542567	+	0.840013i	$0.317453\pi$
$608$	−29.0895	−1.17973
$609$	−2.65149	−0.107444
$610$	−4.21382	−0.170613
$611$	−30.6393	−1.23953
$612$	4.50657	0.182167
$613$	11.3060	0.456645	0.228323	−	0.973586i	$-0.426676\pi$
$613$	11.3060	0.456645	0.228323	+	0.973586i	$0.426676\pi$
$614$	35.0661	1.41515
$615$	10.2227	0.412219
$616$	−5.17365	−0.208452
$617$	15.6822	0.631344	0.315672	−	0.948868i	$-0.397770\pi$
$617$	15.6822	0.631344	0.315672	+	0.948868i	$0.397770\pi$
$618$	−12.3703	−0.497606
$619$	−9.55873	−0.384198	−0.192099	−	0.981376i	$-0.561529\pi$
$619$	−9.55873	−0.384198	−0.192099	+	0.981376i	$0.561529\pi$
$620$	−7.08798	−0.284660
$621$	5.37750	0.215792
$622$	14.2420	0.571054
$623$	2.34583	0.0939838
$624$	−10.1368	−0.405798
$625$	7.48035	0.299214
$626$	12.0078	0.479926
$627$	−11.5527	−0.461370
$628$	11.6254	0.463903
$629$	7.87661	0.314061
$630$	1.23260	0.0491078
$631$	21.9689	0.874569	0.437284	−	0.899323i	$-0.355940\pi$
$631$	21.9689	0.874569	0.437284	+	0.899323i	$0.355940\pi$
$632$	2.78145	0.110640
$633$	−4.49884	−0.178813
$634$	−4.79789	−0.190549
$635$	20.4874	0.813018
$636$	7.91247	0.313750
$637$	5.87918	0.232942
$638$	−4.88196	−0.193279
$639$	−11.6773	−0.461947
$640$	0.361430	0.0142868
$641$	−14.2149	−0.561456	−0.280728	−	0.959787i	$-0.590576\pi$
$641$	−14.2149	−0.561456	−0.280728	+	0.959787i	$0.590576\pi$
$642$	−11.4364	−0.451358
$643$	19.6764	0.775962	0.387981	−	0.921667i	$-0.373173\pi$
$643$	19.6764	0.775962	0.387981	+	0.921667i	$0.373173\pi$
$644$	−4.35534	−0.171624
$645$	−0.428691	−0.0168797
$646$	41.5490	1.63472
$647$	0.409640	0.0161046	0.00805230	−	0.999968i	$-0.497437\pi$
$647$	0.409640	0.0161046	0.00805230	+	0.999968i	$0.497437\pi$
$648$	3.06536	0.120419
$649$	2.53550	0.0995270
$650$	23.8804	0.936666
$651$	−7.74549	−0.303570
$652$	13.8580	0.542721
$653$	1.86209	0.0728692	0.0364346	−	0.999336i	$-0.488400\pi$
$653$	1.86209	0.0728692	0.0364346	+	0.999336i	$0.488400\pi$
$654$	−2.30065	−0.0899627
$655$	14.3421	0.560392
$656$	15.5998	0.609069
$657$	−1.24576	−0.0486016
$658$	5.68526	0.221635
$659$	−15.0998	−0.588204	−0.294102	−	0.955774i	$-0.595021\pi$
$659$	−15.0998	−0.588204	−0.294102	+	0.955774i	$0.595021\pi$
$660$	−1.54450	−0.0601197
$661$	38.7879	1.50867	0.754337	−	0.656488i	$-0.227959\pi$
$661$	38.7879	1.50867	0.754337	+	0.656488i	$0.227959\pi$
$662$	−11.0326	−0.428793
$663$	−32.7131	−1.27047
$664$	11.5690	0.448965
$665$	−7.73393	−0.299909
$666$	1.54427	0.0598392
$667$	−14.2584	−0.552087
$668$	18.0189	0.697170
$669$	−21.5985	−0.835045
$670$	6.57948	0.254188
$671$	5.76993	0.222746
$672$	−4.24979	−0.163939
$673$	18.0861	0.697168	0.348584	−	0.937278i	$-0.386663\pi$
$673$	18.0861	0.697168	0.348584	+	0.937278i	$0.386663\pi$
$674$	0.709150	0.0273154
$675$	−3.72337	−0.143313
$676$	−17.4657	−0.671758
$677$	20.7856	0.798857	0.399429	−	0.916764i	$-0.369209\pi$
$677$	20.7856	0.798857	0.399429	+	0.916764i	$0.369209\pi$
$678$	−13.0302	−0.500421
$679$	6.88933	0.264388
$680$	19.2716	0.739034
$681$	29.8766	1.14487
$682$	−14.2611	−0.546085
$683$	40.5438	1.55137	0.775683	−	0.631123i	$-0.217406\pi$
$683$	40.5438	1.55137	0.775683	+	0.631123i	$0.217406\pi$
$684$	−5.54383	−0.211973
$685$	−3.24563	−0.124009
$686$	−1.09091	−0.0416511
$687$	5.77762	0.220430
$688$	−0.654181	−0.0249404
$689$	−57.4364	−2.18815
$690$	6.62828	0.252334
$691$	−20.0760	−0.763727	−0.381863	−	0.924219i	$-0.624717\pi$
$691$	−20.0760	−0.763727	−0.381863	+	0.924219i	$0.624717\pi$
$692$	−0.984554	−0.0374271
$693$	−1.68778	−0.0641134
$694$	22.2482	0.844531
$695$	−5.62430	−0.213342
$696$	−8.12779	−0.308083
$697$	50.3428	1.90687
$698$	−5.69013	−0.215374
$699$	−16.4801	−0.623335
$700$	3.01563	0.113980
$701$	−7.39428	−0.279278	−0.139639	−	0.990202i	$-0.544594\pi$
$701$	−7.39428	−0.279278	−0.139639	+	0.990202i	$0.544594\pi$
$702$	−6.41364	−0.242067
$703$	−9.68953	−0.365448
$704$	−13.6449	−0.514260
$705$	5.88836	0.221769
$706$	36.7396	1.38271
$707$	6.61764	0.248882
$708$	1.21672	0.0457270
$709$	23.9644	0.900003	0.450002	−	0.893028i	$-0.351423\pi$
$709$	23.9644	0.900003	0.450002	+	0.893028i	$0.351423\pi$
$710$	−14.3934	−0.540175
$711$	0.907380	0.0340294
$712$	7.19083	0.269488
$713$	−41.6513	−1.55986
$714$	6.07006	0.227166
$715$	11.2115	0.419287
$716$	14.9082	0.557144
$717$	−6.10555	−0.228016
$718$	13.3469	0.498100
$719$	−21.7324	−0.810480	−0.405240	−	0.914210i	$-0.632812\pi$
$719$	−21.7324	−0.810480	−0.405240	+	0.914210i	$0.632812\pi$
$720$	1.94813	0.0726025
$721$	11.3394	0.422303
$722$	−30.3849	−1.13081
$723$	−5.73679	−0.213353
$724$	−6.29399	−0.233914
$725$	9.87249	0.366655
$726$	8.89244	0.330029
$727$	36.0030	1.33528	0.667639	−	0.744485i	$-0.267305\pi$
$727$	36.0030	1.33528	0.667639	+	0.744485i	$0.267305\pi$
$728$	18.0218	0.667933
$729$	1.00000	0.0370370
$730$	−1.53551	−0.0568319
$731$	−2.11114	−0.0780833
$732$	2.76884	0.102339
$733$	6.95749	0.256981	0.128490	−	0.991711i	$-0.458987\pi$
$733$	6.95749	0.256981	0.128490	+	0.991711i	$0.458987\pi$
$734$	8.16601	0.301413
$735$	−1.12988	−0.0416762
$736$	−22.8532	−0.842382
$737$	−9.00920	−0.331858
$738$	9.87009	0.363323
$739$	4.03660	0.148489	0.0742443	−	0.997240i	$-0.476346\pi$
$739$	4.03660	0.148489	0.0742443	+	0.997240i	$0.476346\pi$
$740$	−1.29542	−0.0476204
$741$	40.2425	1.47834
$742$	10.6576	0.391252
$743$	−38.4587	−1.41091	−0.705456	−	0.708754i	$-0.749258\pi$
$743$	−38.4587	−1.41091	−0.705456	+	0.708754i	$0.749258\pi$
$744$	−23.7427	−0.870451
$745$	8.67571	0.317854
$746$	−6.22440	−0.227891
$747$	3.77411	0.138087
$748$	−7.60609	−0.278106
$749$	10.4834	0.383054
$750$	−10.7524	−0.392622
$751$	−6.90406	−0.251933	−0.125966	−	0.992035i	$-0.540203\pi$
$751$	−6.90406	−0.251933	−0.125966	+	0.992035i	$0.540203\pi$
$752$	8.98562	0.327672
$753$	−19.5133	−0.711103
$754$	17.0057	0.619312
$755$	−16.4891	−0.600101
$756$	−0.809919	−0.0294565
$757$	31.7924	1.15551	0.577756	−	0.816209i	$-0.303928\pi$
$757$	31.7924	1.15551	0.577756	+	0.816209i	$0.303928\pi$
$758$	−14.0682	−0.510980
$759$	−9.07602	−0.329439
$760$	−23.7073	−0.859954
$761$	13.2570	0.480564	0.240282	−	0.970703i	$-0.422760\pi$
$761$	13.2570	0.480564	0.240282	+	0.970703i	$0.422760\pi$
$762$	19.7808	0.716581
$763$	2.10893	0.0763486
$764$	10.7279	0.388123
$765$	6.28690	0.227303
$766$	1.09091	0.0394161
$767$	−8.83211	−0.318909
$768$	−15.8201	−0.570858
$769$	−54.7042	−1.97268	−0.986342	−	0.164711i	$-0.947331\pi$
$769$	−54.7042	−1.97268	−0.986342	+	0.164711i	$0.947331\pi$
$770$	−2.08035	−0.0749705
$771$	12.4109	0.446966
$772$	−3.62196	−0.130357
$773$	36.2341	1.30325	0.651626	−	0.758541i	$-0.274087\pi$
$773$	36.2341	1.30325	0.651626	+	0.758541i	$0.274087\pi$
$774$	−0.413905	−0.0148775
$775$	28.8393	1.03594
$776$	21.1183	0.758102
$777$	−1.41558	−0.0507837
$778$	−20.3547	−0.729751
$779$	−61.9300	−2.21887
$780$	5.38010	0.192639
$781$	19.7087	0.705233
$782$	32.6417	1.16727
$783$	−2.65149	−0.0947566
$784$	−1.72419	−0.0615783
$785$	16.2180	0.578845
$786$	13.8474	0.493921
$787$	18.3237	0.653170	0.326585	−	0.945168i	$-0.394102\pi$
$787$	18.3237	0.653170	0.326585	+	0.945168i	$0.394102\pi$
$788$	−13.2162	−0.470809
$789$	17.4509	0.621270
$790$	1.11843	0.0397920
$791$	11.9443	0.424692
$792$	−5.17365	−0.183838
$793$	−20.0989	−0.713732
$794$	−12.5520	−0.445456
$795$	11.0383	0.391489
$796$	14.3999	0.510389
$797$	−49.4811	−1.75271	−0.876356	−	0.481664i	$-0.840032\pi$
$797$	−49.4811	−1.75271	−0.876356	+	0.481664i	$0.840032\pi$
$798$	−7.46717	−0.264335
$799$	28.9979	1.02587
$800$	15.8236	0.559447
$801$	2.34583	0.0828859
$802$	30.5439	1.07854
$803$	2.10256	0.0741977
$804$	−4.32327	−0.152470
$805$	−6.07593	−0.214148
$806$	49.6768	1.74979
$807$	15.9257	0.560612
$808$	20.2855	0.713641
$809$	24.5821	0.864260	0.432130	−	0.901811i	$-0.357762\pi$
$809$	24.5821	0.864260	0.432130	+	0.901811i	$0.357762\pi$
$810$	1.23260	0.0433090
$811$	4.63883	0.162891	0.0814457	−	0.996678i	$-0.474046\pi$
$811$	4.63883	0.162891	0.0814457	+	0.996678i	$0.474046\pi$
$812$	2.14750	0.0753623
$813$	−27.4133	−0.961426
$814$	−2.60638	−0.0913537
$815$	19.3326	0.677192
$816$	9.59379	0.335850
$817$	2.59705	0.0908592
$818$	35.0605	1.22586
$819$	5.87918	0.205435
$820$	−8.27955	−0.289135
$821$	31.4656	1.09816	0.549078	−	0.835771i	$-0.314979\pi$
$821$	31.4656	1.09816	0.549078	+	0.835771i	$0.314979\pi$
$822$	−3.13368	−0.109300
$823$	−29.1292	−1.01538	−0.507691	−	0.861539i	$-0.669501\pi$
$823$	−29.1292	−1.01538	−0.507691	+	0.861539i	$0.669501\pi$
$824$	34.7595	1.21090
$825$	6.28422	0.218789
$826$	1.63884	0.0570225
$827$	−31.4909	−1.09504	−0.547522	−	0.836791i	$-0.684429\pi$
$827$	−31.4909	−1.09504	−0.547522	+	0.836791i	$0.684429\pi$
$828$	−4.35534	−0.151358
$829$	26.2201	0.910663	0.455331	−	0.890322i	$-0.349521\pi$
$829$	26.2201	0.910663	0.455331	+	0.890322i	$0.349521\pi$
$830$	4.65195	0.161471
$831$	−0.546370	−0.0189534
$832$	47.5303	1.64782
$833$	−5.56422	−0.192789
$834$	−5.43031	−0.188036
$835$	25.1372	0.869910
$836$	9.35674	0.323610
$837$	−7.74549	−0.267723
$838$	−4.60972	−0.159240
$839$	−25.6259	−0.884704	−0.442352	−	0.896842i	$-0.645856\pi$
$839$	−25.6259	−0.884704	−0.442352	+	0.896842i	$0.645856\pi$
$840$	−3.46349	−0.119502
$841$	−21.9696	−0.757572
$842$	1.58568	0.0546461
$843$	19.7703	0.680927
$844$	3.64370	0.125421
$845$	−24.3656	−0.838201
$846$	5.68526	0.195463
$847$	−8.15141	−0.280086
$848$	16.8444	0.578440
$849$	8.49736	0.291629
$850$	−22.6011	−0.775211
$851$	−7.61229	−0.260946
$852$	9.45768	0.324015
$853$	41.8585	1.43321	0.716603	−	0.697481i	$-0.245696\pi$
$853$	41.8585	1.43321	0.716603	+	0.697481i	$0.245696\pi$
$854$	3.72944	0.127619
$855$	−7.73393	−0.264495
$856$	32.1353	1.09836
$857$	−55.2717	−1.88805	−0.944023	−	0.329879i	$-0.892992\pi$
$857$	−55.2717	−1.88805	−0.944023	+	0.329879i	$0.892992\pi$
$858$	10.8248	0.369553
$859$	−8.23416	−0.280946	−0.140473	−	0.990085i	$-0.544862\pi$
$859$	−8.23416	−0.280946	−0.140473	+	0.990085i	$0.544862\pi$
$860$	0.347205	0.0118396
$861$	−9.04759	−0.308341
$862$	11.6072	0.395343
$863$	−1.17270	−0.0399193	−0.0199596	−	0.999801i	$-0.506354\pi$
$863$	−1.17270	−0.0399193	−0.0199596	+	0.999801i	$0.506354\pi$
$864$	−4.24979	−0.144581
$865$	−1.37350	−0.0467006
$866$	8.59900	0.292206
$867$	13.9606	0.474126
$868$	6.27322	0.212927
$869$	−1.53146	−0.0519511
$870$	−3.26822	−0.110803
$871$	31.3825	1.06336
$872$	6.46465	0.218921
$873$	6.88933	0.233168
$874$	−40.1547	−1.35825
$875$	9.85636	0.333206
$876$	1.00896	0.0340897
$877$	−29.8215	−1.00700	−0.503500	−	0.863995i	$-0.667955\pi$
$877$	−29.8215	−1.00700	−0.503500	+	0.863995i	$0.667955\pi$
$878$	−24.7208	−0.834286
$879$	14.2474	0.480553
$880$	−3.28801	−0.110839
$881$	41.3211	1.39214	0.696071	−	0.717972i	$-0.254930\pi$
$881$	41.3211	1.39214	0.696071	+	0.717972i	$0.254930\pi$
$882$	−1.09091	−0.0367328
$883$	51.7159	1.74038	0.870189	−	0.492718i	$-0.163997\pi$
$883$	51.7159	1.74038	0.870189	+	0.492718i	$0.163997\pi$
$884$	26.4949	0.891121
$885$	1.69738	0.0570570
$886$	18.9561	0.636842
$887$	−6.08837	−0.204427	−0.102214	−	0.994762i	$-0.532593\pi$
$887$	−6.08837	−0.204427	−0.102214	+	0.994762i	$0.532593\pi$
$888$	−4.33927	−0.145616
$889$	−18.1324	−0.608140
$890$	2.89146	0.0969220
$891$	−1.68778	−0.0565427
$892$	17.4930	0.585709
$893$	−35.6722	−1.19373
$894$	8.37647	0.280151
$895$	20.7977	0.695189
$896$	−0.319884	−0.0106866
$897$	31.6153	1.05560
$898$	27.7254	0.925208
$899$	20.5371	0.684951
$900$	3.01563	0.100521
$901$	54.3595	1.81098
$902$	−16.6585	−0.554668
$903$	0.379413	0.0126261
$904$	36.6137	1.21776
$905$	−8.78045	−0.291872
$906$	−15.9204	−0.528920
$907$	51.4873	1.70961	0.854803	−	0.518952i	$-0.173678\pi$
$907$	51.4873	1.70961	0.854803	+	0.518952i	$0.173678\pi$
$908$	−24.1976	−0.803027
$909$	6.61764	0.219493
$910$	7.24665	0.240224
$911$	−18.5026	−0.613018	−0.306509	−	0.951868i	$-0.599161\pi$
$911$	−18.5026	−0.613018	−0.306509	+	0.951868i	$0.599161\pi$
$912$	−11.8019	−0.390802
$913$	−6.36986	−0.210811
$914$	26.7677	0.885396
$915$	3.86267	0.127696
$916$	−4.67940	−0.154612
$917$	−12.6935	−0.419175
$918$	6.07006	0.200342
$919$	−42.4377	−1.39989	−0.699946	−	0.714196i	$-0.746792\pi$
$919$	−42.4377	−1.39989	−0.699946	+	0.714196i	$0.746792\pi$
$920$	−18.6249	−0.614046
$921$	−32.1440	−1.05918
$922$	19.5734	0.644614
$923$	−68.6530	−2.25974
$924$	1.36696	0.0449698
$925$	5.27074	0.173301
$926$	−29.7406	−0.977336
$927$	11.3394	0.372436
$928$	11.2683	0.369900
$929$	24.0443	0.788867	0.394433	−	0.918925i	$-0.370941\pi$
$929$	24.0443	0.788867	0.394433	+	0.918925i	$0.370941\pi$
$930$	−9.54705	−0.313060
$931$	6.84491	0.224333
$932$	13.3476	0.437214
$933$	−13.0552	−0.427408
$934$	−25.5962	−0.837535
$935$	−10.6109	−0.347013
$936$	18.0218	0.589062
$937$	−28.3646	−0.926630	−0.463315	−	0.886194i	$-0.653340\pi$
$937$	−28.3646	−0.926630	−0.463315	+	0.886194i	$0.653340\pi$
$938$	−5.82317	−0.190133
$939$	−11.0071	−0.359204
$940$	−4.76910	−0.155551
$941$	−43.6910	−1.42429	−0.712143	−	0.702034i	$-0.752275\pi$
$941$	−43.6910	−1.42429	−0.712143	+	0.702034i	$0.752275\pi$
$942$	15.6586	0.510185
$943$	−48.6534	−1.58437
$944$	2.59020	0.0843039
$945$	−1.12988	−0.0367550
$946$	0.698579	0.0227128
$947$	−29.8643	−0.970461	−0.485230	−	0.874386i	$-0.661264\pi$
$947$	−29.8643	−0.970461	−0.485230	+	0.874386i	$0.661264\pi$
$948$	−0.734905	−0.0238686
$949$	−7.32402	−0.237748
$950$	27.8031	0.902050
$951$	4.39807	0.142617
$952$	−17.0564	−0.552800
$953$	−31.0132	−1.00462	−0.502308	−	0.864689i	$-0.667516\pi$
$953$	−31.0132	−1.00462	−0.502308	+	0.864689i	$0.667516\pi$
$954$	10.6576	0.345052
$955$	14.9660	0.484289
$956$	4.94500	0.159933
$957$	4.47513	0.144660
$958$	9.82069	0.317292
$959$	2.87254	0.0927592
$960$	−9.13453	−0.294816
$961$	28.9926	0.935245
$962$	9.07904	0.292720
$963$	10.4834	0.337822
$964$	4.64633	0.149648
$965$	−5.05282	−0.162656
$966$	−5.86636	−0.188747
$967$	−13.8455	−0.445241	−0.222620	−	0.974905i	$-0.571461\pi$
$967$	−13.8455	−0.445241	−0.222620	+	0.974905i	$0.571461\pi$
$968$	−24.9870	−0.803114
$969$	−38.0866	−1.22352
$970$	8.49175	0.272654
$971$	−12.2101	−0.391842	−0.195921	−	0.980620i	$-0.562770\pi$
$971$	−12.2101	−0.391842	−0.195921	+	0.980620i	$0.562770\pi$
$972$	−0.809919	−0.0259782
$973$	4.97779	0.159580
$974$	3.95999	0.126886
$975$	−21.8904	−0.701053
$976$	5.89442	0.188676
$977$	4.40923	0.141064	0.0705319	−	0.997510i	$-0.477530\pi$
$977$	4.40923	0.141064	0.0705319	+	0.997510i	$0.477530\pi$
$978$	18.6658	0.596867
$979$	−3.95924	−0.126538
$980$	0.915111	0.0292322
$981$	2.10893	0.0673331
$982$	20.7730	0.662894
$983$	29.6991	0.947255	0.473628	−	0.880725i	$-0.342944\pi$
$983$	29.6991	0.947255	0.473628	+	0.880725i	$0.342944\pi$
$984$	−27.7342	−0.884133
$985$	−18.4373	−0.587462
$986$	−16.0947	−0.512560
$987$	−5.21149	−0.165884
$988$	−32.5931	−1.03693
$989$	2.04029	0.0648775
$990$	−2.08035	−0.0661178
$991$	47.5611	1.51083	0.755415	−	0.655247i	$-0.227435\pi$
$991$	47.5611	1.51083	0.755415	+	0.655247i	$0.227435\pi$
$992$	32.9167	1.04511
$993$	10.1132	0.320933
$994$	12.7389	0.404053
$995$	20.0886	0.636850
$996$	−3.05672	−0.0968560
$997$	−11.3551	−0.359619	−0.179809	−	0.983701i	$-0.557548\pi$
$997$	−11.3551	−0.359619	−0.179809	+	0.983701i	$0.557548\pi$
$998$	−31.2866	−0.990359
$999$	−1.41558	−0.0447870

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.n.1.16	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.n.1.16	✓	40	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-1.09091 q^{2} +1.00000 q^{3} -0.809919 q^{4} -1.12988 q^{5} -1.09091 q^{6} +1.00000 q^{7} +3.06536 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.09091 q^{2} +1.00000 q^{3} -0.809919 q^{4} -1.12988 q^{5} -1.09091 q^{6} +1.00000 q^{7} +3.06536 q^{8} +1.00000 q^{9} +1.23260 q^{10} -1.68778 q^{11} -0.809919 q^{12} +5.87918 q^{13} -1.09091 q^{14} -1.12988 q^{15} -1.72419 q^{16} -5.56422 q^{17} -1.09091 q^{18} +6.84491 q^{19} +0.915111 q^{20} +1.00000 q^{21} +1.84121 q^{22} +5.37750 q^{23} +3.06536 q^{24} -3.72337 q^{25} -6.41364 q^{26} +1.00000 q^{27} -0.809919 q^{28} -2.65149 q^{29} +1.23260 q^{30} -7.74549 q^{31} -4.24979 q^{32} -1.68778 q^{33} +6.07006 q^{34} -1.12988 q^{35} -0.809919 q^{36} -1.41558 q^{37} -7.46717 q^{38} +5.87918 q^{39} -3.46349 q^{40} -9.04759 q^{41} -1.09091 q^{42} +0.379413 q^{43} +1.36696 q^{44} -1.12988 q^{45} -5.86636 q^{46} -5.21149 q^{47} -1.72419 q^{48} +1.00000 q^{49} +4.06186 q^{50} -5.56422 q^{51} -4.76166 q^{52} -9.76946 q^{53} -1.09091 q^{54} +1.90699 q^{55} +3.06536 q^{56} +6.84491 q^{57} +2.89254 q^{58} -1.50227 q^{59} +0.915111 q^{60} -3.41866 q^{61} +8.44962 q^{62} +1.00000 q^{63} +8.08452 q^{64} -6.64277 q^{65} +1.84121 q^{66} +5.33791 q^{67} +4.50657 q^{68} +5.37750 q^{69} +1.23260 q^{70} -11.6773 q^{71} +3.06536 q^{72} -1.24576 q^{73} +1.54427 q^{74} -3.72337 q^{75} -5.54383 q^{76} -1.68778 q^{77} -6.41364 q^{78} +0.907380 q^{79} +1.94813 q^{80} +1.00000 q^{81} +9.87009 q^{82} +3.77411 q^{83} -0.809919 q^{84} +6.28690 q^{85} -0.413905 q^{86} -2.65149 q^{87} -5.17365 q^{88} +2.34583 q^{89} +1.23260 q^{90} +5.87918 q^{91} -4.35534 q^{92} -7.74549 q^{93} +5.68526 q^{94} -7.73393 q^{95} -4.24979 q^{96} +6.88933 q^{97} -1.09091 q^{98} -1.68778 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * q^99

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