LMFDB - Embedded newform 8002.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.43715 q^{3} +1.00000 q^{4} -2.28285 q^{5} -3.43715 q^{6} -3.85963 q^{7} +1.00000 q^{8} +8.81400 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.43715 q^{3} +1.00000 q^{4} -2.28285 q^{5} -3.43715 q^{6} -3.85963 q^{7} +1.00000 q^{8} +8.81400 q^{9} -2.28285 q^{10} +0.361008 q^{11} -3.43715 q^{12} -4.38630 q^{13} -3.85963 q^{14} +7.84648 q^{15} +1.00000 q^{16} +1.54216 q^{17} +8.81400 q^{18} -7.18602 q^{19} -2.28285 q^{20} +13.2661 q^{21} +0.361008 q^{22} +6.47568 q^{23} -3.43715 q^{24} +0.211388 q^{25} -4.38630 q^{26} -19.9836 q^{27} -3.85963 q^{28} -4.60661 q^{29} +7.84648 q^{30} +9.01373 q^{31} +1.00000 q^{32} -1.24084 q^{33} +1.54216 q^{34} +8.81095 q^{35} +8.81400 q^{36} -0.704615 q^{37} -7.18602 q^{38} +15.0764 q^{39} -2.28285 q^{40} -1.54259 q^{41} +13.2661 q^{42} +6.07803 q^{43} +0.361008 q^{44} -20.1210 q^{45} +6.47568 q^{46} +6.06179 q^{47} -3.43715 q^{48} +7.89678 q^{49} +0.211388 q^{50} -5.30063 q^{51} -4.38630 q^{52} +2.49746 q^{53} -19.9836 q^{54} -0.824126 q^{55} -3.85963 q^{56} +24.6994 q^{57} -4.60661 q^{58} +9.64179 q^{59} +7.84648 q^{60} -5.89636 q^{61} +9.01373 q^{62} -34.0188 q^{63} +1.00000 q^{64} +10.0133 q^{65} -1.24084 q^{66} +9.47651 q^{67} +1.54216 q^{68} -22.2579 q^{69} +8.81095 q^{70} +10.4189 q^{71} +8.81400 q^{72} -13.9243 q^{73} -0.704615 q^{74} -0.726572 q^{75} -7.18602 q^{76} -1.39336 q^{77} +15.0764 q^{78} -16.0878 q^{79} -2.28285 q^{80} +42.2445 q^{81} -1.54259 q^{82} -10.7216 q^{83} +13.2661 q^{84} -3.52051 q^{85} +6.07803 q^{86} +15.8336 q^{87} +0.361008 q^{88} +18.3038 q^{89} -20.1210 q^{90} +16.9295 q^{91} +6.47568 q^{92} -30.9816 q^{93} +6.06179 q^{94} +16.4046 q^{95} -3.43715 q^{96} +1.80861 q^{97} +7.89678 q^{98} +3.18192 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * 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
$3$	−3.43715	−1.98444	−0.992220	−	0.124501i	$-0.960267\pi$
$3$	−3.43715	−1.98444	−0.992220	+	0.124501i	$0.960267\pi$
$4$	1.00000	0.500000
$5$	−2.28285	−1.02092	−0.510460	−	0.859902i	$-0.670525\pi$
$5$	−2.28285	−1.02092	−0.510460	+	0.859902i	$0.670525\pi$
$6$	−3.43715	−1.40321
$7$	−3.85963	−1.45880	−0.729402	−	0.684085i	$-0.760202\pi$
$7$	−3.85963	−1.45880	−0.729402	+	0.684085i	$0.760202\pi$
$8$	1.00000	0.353553
$9$	8.81400	2.93800
$10$	−2.28285	−0.721899
$11$	0.361008	0.108848	0.0544240	−	0.998518i	$-0.482668\pi$
$11$	0.361008	0.108848	0.0544240	+	0.998518i	$0.482668\pi$
$12$	−3.43715	−0.992220
$13$	−4.38630	−1.21654	−0.608271	−	0.793730i	$-0.708137\pi$
$13$	−4.38630	−1.21654	−0.608271	+	0.793730i	$0.708137\pi$
$14$	−3.85963	−1.03153
$15$	7.84648	2.02595
$16$	1.00000	0.250000
$17$	1.54216	0.374029	0.187014	−	0.982357i	$-0.440119\pi$
$17$	1.54216	0.374029	0.187014	+	0.982357i	$0.440119\pi$
$18$	8.81400	2.07748
$19$	−7.18602	−1.64859	−0.824293	−	0.566164i	$-0.808427\pi$
$19$	−7.18602	−1.64859	−0.824293	+	0.566164i	$0.808427\pi$
$20$	−2.28285	−0.510460
$21$	13.2661	2.89491
$22$	0.361008	0.0769672
$23$	6.47568	1.35027	0.675137	−	0.737693i	$-0.264085\pi$
$23$	6.47568	1.35027	0.675137	+	0.737693i	$0.264085\pi$
$24$	−3.43715	−0.701605
$25$	0.211388	0.0422776
$26$	−4.38630	−0.860225
$27$	−19.9836	−3.84584
$28$	−3.85963	−0.729402
$29$	−4.60661	−0.855426	−0.427713	−	0.903915i	$-0.640681\pi$
$29$	−4.60661	−0.855426	−0.427713	+	0.903915i	$0.640681\pi$
$30$	7.84648	1.43257
$31$	9.01373	1.61891	0.809457	−	0.587179i	$-0.199761\pi$
$31$	9.01373	1.61891	0.809457	+	0.587179i	$0.199761\pi$
$32$	1.00000	0.176777
$33$	−1.24084	−0.216002
$34$	1.54216	0.264478
$35$	8.81095	1.48932
$36$	8.81400	1.46900
$37$	−0.704615	−0.115838	−0.0579190	−	0.998321i	$-0.518447\pi$
$37$	−0.704615	−0.115838	−0.0579190	+	0.998321i	$0.518447\pi$
$38$	−7.18602	−1.16573
$39$	15.0764	2.41415
$40$	−2.28285	−0.360950
$41$	−1.54259	−0.240912	−0.120456	−	0.992719i	$-0.538436\pi$
$41$	−1.54259	−0.240912	−0.120456	+	0.992719i	$0.538436\pi$
$42$	13.2661	2.04701
$43$	6.07803	0.926891	0.463445	−	0.886126i	$-0.346613\pi$
$43$	6.07803	0.926891	0.463445	+	0.886126i	$0.346613\pi$
$44$	0.361008	0.0544240
$45$	−20.1210	−2.99946
$46$	6.47568	0.954787
$47$	6.06179	0.884203	0.442101	−	0.896965i	$-0.354233\pi$
$47$	6.06179	0.884203	0.442101	+	0.896965i	$0.354233\pi$
$48$	−3.43715	−0.496110
$49$	7.89678	1.12811
$50$	0.211388	0.0298948
$51$	−5.30063	−0.742237
$52$	−4.38630	−0.608271
$53$	2.49746	0.343052	0.171526	−	0.985180i	$-0.445130\pi$
$53$	2.49746	0.343052	0.171526	+	0.985180i	$0.445130\pi$
$54$	−19.9836	−2.71942
$55$	−0.824126	−0.111125
$56$	−3.85963	−0.515765
$57$	24.6994	3.27152
$58$	−4.60661	−0.604878
$59$	9.64179	1.25525	0.627627	−	0.778514i	$-0.284026\pi$
$59$	9.64179	1.25525	0.627627	+	0.778514i	$0.284026\pi$
$60$	7.84648	1.01298
$61$	−5.89636	−0.754951	−0.377476	−	0.926020i	$-0.623208\pi$
$61$	−5.89636	−0.754951	−0.377476	+	0.926020i	$0.623208\pi$
$62$	9.01373	1.14475
$63$	−34.0188	−4.28597
$64$	1.00000	0.125000
$65$	10.0133	1.24199
$66$	−1.24084	−0.152737
$67$	9.47651	1.15774	0.578870	−	0.815420i	$-0.303494\pi$
$67$	9.47651	1.15774	0.578870	+	0.815420i	$0.303494\pi$
$68$	1.54216	0.187014
$69$	−22.2579	−2.67953
$70$	8.81095	1.05311
$71$	10.4189	1.23649	0.618247	−	0.785984i	$-0.287843\pi$
$71$	10.4189	1.23649	0.618247	+	0.785984i	$0.287843\pi$
$72$	8.81400	1.03874
$73$	−13.9243	−1.62971	−0.814855	−	0.579664i	$-0.803184\pi$
$73$	−13.9243	−1.62971	−0.814855	+	0.579664i	$0.803184\pi$
$74$	−0.704615	−0.0819098
$75$	−0.726572	−0.0838973
$76$	−7.18602	−0.824293
$77$	−1.39336	−0.158788
$78$	15.0764	1.70706
$79$	−16.0878	−1.81002	−0.905011	−	0.425389i	$-0.860137\pi$
$79$	−16.0878	−1.81002	−0.905011	+	0.425389i	$0.860137\pi$
$80$	−2.28285	−0.255230
$81$	42.2445	4.69384
$82$	−1.54259	−0.170350
$83$	−10.7216	−1.17685	−0.588425	−	0.808552i	$-0.700252\pi$
$83$	−10.7216	−1.17685	−0.588425	+	0.808552i	$0.700252\pi$
$84$	13.2661	1.44745
$85$	−3.52051	−0.381853
$86$	6.07803	0.655411
$87$	15.8336	1.69754
$88$	0.361008	0.0384836
$89$	18.3038	1.94020	0.970098	−	0.242715i	$-0.0780378\pi$
$89$	18.3038	1.94020	0.970098	+	0.242715i	$0.0780378\pi$
$90$	−20.1210	−2.12094
$91$	16.9295	1.77470
$92$	6.47568	0.675137
$93$	−30.9816	−3.21264
$94$	6.06179	0.625226
$95$	16.4046	1.68307
$96$	−3.43715	−0.350803
$97$	1.80861	0.183636	0.0918180	−	0.995776i	$-0.470732\pi$
$97$	1.80861	0.183636	0.0918180	+	0.995776i	$0.470732\pi$
$98$	7.89678	0.797695
$99$	3.18192	0.319795
$100$	0.211388	0.0211388
$101$	20.0095	1.99102	0.995510	−	0.0946561i	$-0.0301752\pi$
$101$	20.0095	1.99102	0.995510	+	0.0946561i	$0.0301752\pi$
$102$	−5.30063	−0.524841
$103$	−1.48856	−0.146672	−0.0733361	−	0.997307i	$-0.523365\pi$
$103$	−1.48856	−0.146672	−0.0733361	+	0.997307i	$0.523365\pi$
$104$	−4.38630	−0.430112
$105$	−30.2846	−2.95547
$106$	2.49746	0.242574
$107$	4.37825	0.423261	0.211631	−	0.977350i	$-0.432123\pi$
$107$	4.37825	0.423261	0.211631	+	0.977350i	$0.432123\pi$
$108$	−19.9836	−1.92292
$109$	−13.2141	−1.26568	−0.632842	−	0.774281i	$-0.718112\pi$
$109$	−13.2141	−1.26568	−0.632842	+	0.774281i	$0.718112\pi$
$110$	−0.824126	−0.0785773
$111$	2.42187	0.229873
$112$	−3.85963	−0.364701
$113$	10.3975	0.978114	0.489057	−	0.872252i	$-0.337341\pi$
$113$	10.3975	0.978114	0.489057	+	0.872252i	$0.337341\pi$
$114$	24.6994	2.31331
$115$	−14.7830	−1.37852
$116$	−4.60661	−0.427713
$117$	−38.6609	−3.57420
$118$	9.64179	0.887598
$119$	−5.95217	−0.545635
$120$	7.84648	0.716283
$121$	−10.8697	−0.988152
$122$	−5.89636	−0.533831
$123$	5.30211	0.478075
$124$	9.01373	0.809457
$125$	10.9317	0.977758
$126$	−34.0188	−3.03064
$127$	4.60505	0.408632	0.204316	−	0.978905i	$-0.434503\pi$
$127$	4.60505	0.408632	0.204316	+	0.978905i	$0.434503\pi$
$128$	1.00000	0.0883883
$129$	−20.8911	−1.83936
$130$	10.0133	0.878221
$131$	15.2608	1.33335	0.666673	−	0.745351i	$-0.267718\pi$
$131$	15.2608	1.33335	0.666673	+	0.745351i	$0.267718\pi$
$132$	−1.24084	−0.108001
$133$	27.7354	2.40496
$134$	9.47651	0.818645
$135$	45.6194	3.92629
$136$	1.54216	0.132239
$137$	−1.15379	−0.0985748	−0.0492874	−	0.998785i	$-0.515695\pi$
$137$	−1.15379	−0.0985748	−0.0492874	+	0.998785i	$0.515695\pi$
$138$	−22.2579	−1.89472
$139$	−4.42923	−0.375682	−0.187841	−	0.982199i	$-0.560149\pi$
$139$	−4.42923	−0.375682	−0.187841	+	0.982199i	$0.560149\pi$
$140$	8.81095	0.744661
$141$	−20.8353	−1.75465
$142$	10.4189	0.874333
$143$	−1.58349	−0.132418
$144$	8.81400	0.734500
$145$	10.5162	0.873322
$146$	−13.9243	−1.15238
$147$	−27.1424	−2.23867
$148$	−0.704615	−0.0579190
$149$	−10.6811	−0.875028	−0.437514	−	0.899212i	$-0.644141\pi$
$149$	−10.6811	−0.875028	−0.437514	+	0.899212i	$0.644141\pi$
$150$	−0.726572	−0.0593244
$151$	17.9884	1.46387	0.731937	−	0.681372i	$-0.238616\pi$
$151$	17.9884	1.46387	0.731937	+	0.681372i	$0.238616\pi$
$152$	−7.18602	−0.582863
$153$	13.5926	1.09890
$154$	−1.39336	−0.112280
$155$	−20.5770	−1.65278
$156$	15.0764	1.20708
$157$	−7.37785	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$157$	−7.37785	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$158$	−16.0878	−1.27988
$159$	−8.58413	−0.680766
$160$	−2.28285	−0.180475
$161$	−24.9938	−1.96978
$162$	42.2445	3.31904
$163$	3.42250	0.268071	0.134036	−	0.990977i	$-0.457206\pi$
$163$	3.42250	0.268071	0.134036	+	0.990977i	$0.457206\pi$
$164$	−1.54259	−0.120456
$165$	2.83264	0.220521
$166$	−10.7216	−0.832158
$167$	11.9050	0.921234	0.460617	−	0.887599i	$-0.347628\pi$
$167$	11.9050	0.921234	0.460617	+	0.887599i	$0.347628\pi$
$168$	13.2661	1.02350
$169$	6.23966	0.479973
$170$	−3.52051	−0.270011
$171$	−63.3375	−4.84354
$172$	6.07803	0.463445
$173$	−21.8901	−1.66427	−0.832137	−	0.554570i	$-0.812883\pi$
$173$	−21.8901	−1.66427	−0.832137	+	0.554570i	$0.812883\pi$
$174$	15.8336	1.20034
$175$	−0.815880	−0.0616748
$176$	0.361008	0.0272120
$177$	−33.1403	−2.49097
$178$	18.3038	1.37193
$179$	4.15916	0.310870	0.155435	−	0.987846i	$-0.450322\pi$
$179$	4.15916	0.310870	0.155435	+	0.987846i	$0.450322\pi$
$180$	−20.1210	−1.49973
$181$	−1.08020	−0.0802908	−0.0401454	−	0.999194i	$-0.512782\pi$
$181$	−1.08020	−0.0802908	−0.0401454	+	0.999194i	$0.512782\pi$
$182$	16.9295	1.25490
$183$	20.2667	1.49815
$184$	6.47568	0.477394
$185$	1.60853	0.118261
$186$	−30.9816	−2.27168
$187$	0.556732	0.0407123
$188$	6.06179	0.442101
$189$	77.1293	5.61033
$190$	16.4046	1.19011
$191$	−0.505220	−0.0365565	−0.0182782	−	0.999833i	$-0.505818\pi$
$191$	−0.505220	−0.0365565	−0.0182782	+	0.999833i	$0.505818\pi$
$192$	−3.43715	−0.248055
$193$	−4.91064	−0.353475	−0.176738	−	0.984258i	$-0.556554\pi$
$193$	−4.91064	−0.353475	−0.176738	+	0.984258i	$0.556554\pi$
$194$	1.80861	0.129850
$195$	−34.4171	−2.46466
$196$	7.89678	0.564056
$197$	6.63410	0.472660	0.236330	−	0.971673i	$-0.424055\pi$
$197$	6.63410	0.472660	0.236330	+	0.971673i	$0.424055\pi$
$198$	3.18192	0.226129
$199$	12.0176	0.851903	0.425952	−	0.904746i	$-0.359939\pi$
$199$	12.0176	0.851903	0.425952	+	0.904746i	$0.359939\pi$
$200$	0.211388	0.0149474
$201$	−32.5722	−2.29746
$202$	20.0095	1.40786
$203$	17.7798	1.24790
$204$	−5.30063	−0.371119
$205$	3.52149	0.245952
$206$	−1.48856	−0.103713
$207$	57.0766	3.96710
$208$	−4.38630	−0.304135
$209$	−2.59421	−0.179445
$210$	−30.2846	−2.08983
$211$	8.77979	0.604426	0.302213	−	0.953240i	$-0.402275\pi$
$211$	8.77979	0.604426	0.302213	+	0.953240i	$0.402275\pi$
$212$	2.49746	0.171526
$213$	−35.8113	−2.45375
$214$	4.37825	0.299291
$215$	−13.8752	−0.946281
$216$	−19.9836	−1.35971
$217$	−34.7897	−2.36168
$218$	−13.2141	−0.894974
$219$	47.8597	3.23406
$220$	−0.824126	−0.0555625
$221$	−6.76438	−0.455022
$222$	2.42187	0.162545
$223$	−4.16238	−0.278734	−0.139367	−	0.990241i	$-0.544507\pi$
$223$	−4.16238	−0.278734	−0.139367	+	0.990241i	$0.544507\pi$
$224$	−3.85963	−0.257883
$225$	1.86317	0.124212
$226$	10.3975	0.691631
$227$	−10.5565	−0.700658	−0.350329	−	0.936627i	$-0.613930\pi$
$227$	−10.5565	−0.700658	−0.350329	+	0.936627i	$0.613930\pi$
$228$	24.6994	1.63576
$229$	5.99864	0.396401	0.198201	−	0.980161i	$-0.436490\pi$
$229$	5.99864	0.396401	0.198201	+	0.980161i	$0.436490\pi$
$230$	−14.7830	−0.974761
$231$	4.78918	0.315105
$232$	−4.60661	−0.302439
$233$	12.9067	0.845548	0.422774	−	0.906235i	$-0.361056\pi$
$233$	12.9067	0.845548	0.422774	+	0.906235i	$0.361056\pi$
$234$	−38.6609	−2.52734
$235$	−13.8381	−0.902700
$236$	9.64179	0.627627
$237$	55.2962	3.59188
$238$	−5.95217	−0.385822
$239$	−19.8478	−1.28384	−0.641922	−	0.766770i	$-0.721863\pi$
$239$	−19.8478	−1.28384	−0.641922	+	0.766770i	$0.721863\pi$
$240$	7.84648	0.506488
$241$	2.71243	0.174723	0.0873616	−	0.996177i	$-0.472156\pi$
$241$	2.71243	0.174723	0.0873616	+	0.996177i	$0.472156\pi$
$242$	−10.8697	−0.698729
$243$	−85.2500	−5.46879
$244$	−5.89636	−0.377476
$245$	−18.0271	−1.15171
$246$	5.30211	0.338050
$247$	31.5201	2.00557
$248$	9.01373	0.572373
$249$	36.8518	2.33539
$250$	10.9317	0.691379
$251$	0.460861	0.0290893	0.0145447	−	0.999894i	$-0.495370\pi$
$251$	0.460861	0.0290893	0.0145447	+	0.999894i	$0.495370\pi$
$252$	−34.0188	−2.14298
$253$	2.33777	0.146975
$254$	4.60505	0.288946
$255$	12.1005	0.757765
$256$	1.00000	0.0625000
$257$	−27.7877	−1.73335	−0.866676	−	0.498872i	$-0.833748\pi$
$257$	−27.7877	−1.73335	−0.866676	+	0.498872i	$0.833748\pi$
$258$	−20.8911	−1.30062
$259$	2.71956	0.168985
$260$	10.0133	0.620996
$261$	−40.6027	−2.51324
$262$	15.2608	0.942817
$263$	−15.5040	−0.956016	−0.478008	−	0.878356i	$-0.658641\pi$
$263$	−15.5040	−0.956016	−0.478008	+	0.878356i	$0.658641\pi$
$264$	−1.24084	−0.0763683
$265$	−5.70131	−0.350229
$266$	27.7354	1.70057
$267$	−62.9128	−3.85020
$268$	9.47651	0.578870
$269$	−3.74935	−0.228602	−0.114301	−	0.993446i	$-0.536463\pi$
$269$	−3.74935	−0.228602	−0.114301	+	0.993446i	$0.536463\pi$
$270$	45.6194	2.77631
$271$	−18.0585	−1.09698	−0.548489	−	0.836158i	$-0.684797\pi$
$271$	−18.0585	−1.09698	−0.548489	+	0.836158i	$0.684797\pi$
$272$	1.54216	0.0935072
$273$	−58.1893	−3.52178
$274$	−1.15379	−0.0697029
$275$	0.0763127	0.00460183
$276$	−22.2579	−1.33977
$277$	−17.6270	−1.05911	−0.529553	−	0.848277i	$-0.677640\pi$
$277$	−17.6270	−1.05911	−0.529553	+	0.848277i	$0.677640\pi$
$278$	−4.42923	−0.265647
$279$	79.4470	4.75637
$280$	8.81095	0.526555
$281$	4.89388	0.291944	0.145972	−	0.989289i	$-0.453369\pi$
$281$	4.89388	0.291944	0.145972	+	0.989289i	$0.453369\pi$
$282$	−20.8353	−1.24072
$283$	−12.7271	−0.756546	−0.378273	−	0.925694i	$-0.623482\pi$
$283$	−12.7271	−0.756546	−0.378273	+	0.925694i	$0.623482\pi$
$284$	10.4189	0.618247
$285$	−56.3850	−3.33996
$286$	−1.58349	−0.0936337
$287$	5.95383	0.351443
$288$	8.81400	0.519370
$289$	−14.6217	−0.860102
$290$	10.5162	0.617532
$291$	−6.21645	−0.364415
$292$	−13.9243	−0.814855
$293$	−25.5441	−1.49230	−0.746151	−	0.665776i	$-0.768101\pi$
$293$	−25.5441	−1.49230	−0.746151	+	0.665776i	$0.768101\pi$
$294$	−27.1424	−1.58298
$295$	−22.0107	−1.28151
$296$	−0.704615	−0.0409549
$297$	−7.21423	−0.418612
$298$	−10.6811	−0.618738
$299$	−28.4043	−1.64266
$300$	−0.726572	−0.0419487
$301$	−23.4590	−1.35215
$302$	17.9884	1.03512
$303$	−68.7757	−3.95106
$304$	−7.18602	−0.412146
$305$	13.4605	0.770745
$306$	13.5926	0.777037
$307$	−18.8491	−1.07578	−0.537888	−	0.843016i	$-0.680778\pi$
$307$	−18.8491	−1.07578	−0.537888	+	0.843016i	$0.680778\pi$
$308$	−1.39336	−0.0793940
$309$	5.11640	0.291062
$310$	−20.5770	−1.16869
$311$	23.3036	1.32143	0.660713	−	0.750639i	$-0.270254\pi$
$311$	23.3036	1.32143	0.660713	+	0.750639i	$0.270254\pi$
$312$	15.0764	0.853532
$313$	−34.6503	−1.95855	−0.979274	−	0.202540i	$-0.935080\pi$
$313$	−34.6503	−1.95855	−0.979274	+	0.202540i	$0.935080\pi$
$314$	−7.37785	−0.416356
$315$	77.6597	4.37563
$316$	−16.0878	−0.905011
$317$	−6.30150	−0.353928	−0.176964	−	0.984217i	$-0.556628\pi$
$317$	−6.30150	−0.353928	−0.176964	+	0.984217i	$0.556628\pi$
$318$	−8.58413	−0.481374
$319$	−1.66302	−0.0931114
$320$	−2.28285	−0.127615
$321$	−15.0487	−0.839936
$322$	−24.9938	−1.39285
$323$	−11.0820	−0.616618
$324$	42.2445	2.34692
$325$	−0.927212	−0.0514325
$326$	3.42250	0.189555
$327$	45.4189	2.51167
$328$	−1.54259	−0.0851752
$329$	−23.3963	−1.28988
$330$	2.83264	0.155932
$331$	−16.7890	−0.922804	−0.461402	−	0.887191i	$-0.652653\pi$
$331$	−16.7890	−0.922804	−0.461402	+	0.887191i	$0.652653\pi$
$332$	−10.7216	−0.588425
$333$	−6.21047	−0.340332
$334$	11.9050	0.651411
$335$	−21.6334	−1.18196
$336$	13.2661	0.723727
$337$	−8.49180	−0.462578	−0.231289	−	0.972885i	$-0.574294\pi$
$337$	−8.49180	−0.462578	−0.231289	+	0.972885i	$0.574294\pi$
$338$	6.23966	0.339393
$339$	−35.7378	−1.94101
$340$	−3.52051	−0.190927
$341$	3.25403	0.176216
$342$	−63.3375	−3.42490
$343$	−3.46124	−0.186889
$344$	6.07803	0.327705
$345$	50.8113	2.73559
$346$	−21.8901	−1.17682
$347$	−34.1733	−1.83452	−0.917258	−	0.398293i	$-0.869603\pi$
$347$	−34.1733	−1.83452	−0.917258	+	0.398293i	$0.869603\pi$
$348$	15.8336	0.848771
$349$	19.9392	1.06732	0.533661	−	0.845698i	$-0.320816\pi$
$349$	19.9392	1.06732	0.533661	+	0.845698i	$0.320816\pi$
$350$	−0.815880	−0.0436106
$351$	87.6540	4.67862
$352$	0.361008	0.0192418
$353$	−3.42769	−0.182437	−0.0912187	−	0.995831i	$-0.529076\pi$
$353$	−3.42769	−0.182437	−0.0912187	+	0.995831i	$0.529076\pi$
$354$	−33.1403	−1.76138
$355$	−23.7847	−1.26236
$356$	18.3038	0.970098
$357$	20.4585	1.08278
$358$	4.15916	0.219818
$359$	32.1213	1.69530	0.847649	−	0.530557i	$-0.178017\pi$
$359$	32.1213	1.69530	0.847649	+	0.530557i	$0.178017\pi$
$360$	−20.1210	−1.06047
$361$	32.6389	1.71783
$362$	−1.08020	−0.0567742
$363$	37.3607	1.96093
$364$	16.9295	0.887348
$365$	31.7869	1.66380
$366$	20.2667	1.05936
$367$	−31.1725	−1.62719	−0.813595	−	0.581432i	$-0.802492\pi$
$367$	−31.1725	−1.62719	−0.813595	+	0.581432i	$0.802492\pi$
$368$	6.47568	0.337568
$369$	−13.5964	−0.707799
$370$	1.60853	0.0836234
$371$	−9.63927	−0.500446
$372$	−30.9816	−1.60632
$373$	32.9584	1.70652	0.853259	−	0.521487i	$-0.174622\pi$
$373$	32.9584	1.70652	0.853259	+	0.521487i	$0.174622\pi$
$374$	0.556732	0.0287879
$375$	−37.5738	−1.94030
$376$	6.06179	0.312613
$377$	20.2060	1.04066
$378$	77.1293	3.96710
$379$	5.03123	0.258437	0.129218	−	0.991616i	$-0.458753\pi$
$379$	5.03123	0.258437	0.129218	+	0.991616i	$0.458753\pi$
$380$	16.4046	0.841537
$381$	−15.8282	−0.810905
$382$	−0.505220	−0.0258493
$383$	18.1170	0.925737	0.462868	−	0.886427i	$-0.346820\pi$
$383$	18.1170	0.925737	0.462868	+	0.886427i	$0.346820\pi$
$384$	−3.43715	−0.175401
$385$	3.18082	0.162110
$386$	−4.91064	−0.249945
$387$	53.5717	2.72320
$388$	1.80861	0.0918180
$389$	4.36529	0.221329	0.110664	−	0.993858i	$-0.464702\pi$
$389$	4.36529	0.221329	0.110664	+	0.993858i	$0.464702\pi$
$390$	−34.4171	−1.74278
$391$	9.98654	0.505041
$392$	7.89678	0.398848
$393$	−52.4538	−2.64594
$394$	6.63410	0.334221
$395$	36.7260	1.84789
$396$	3.18192	0.159898
$397$	−24.3276	−1.22097	−0.610485	−	0.792028i	$-0.709025\pi$
$397$	−24.3276	−1.22097	−0.610485	+	0.792028i	$0.709025\pi$
$398$	12.0176	0.602387
$399$	−95.3307	−4.77251
$400$	0.211388	0.0105694
$401$	−3.52602	−0.176081	−0.0880406	−	0.996117i	$-0.528061\pi$
$401$	−3.52602	−0.176081	−0.0880406	+	0.996117i	$0.528061\pi$
$402$	−32.5722	−1.62455
$403$	−39.5370	−1.96948
$404$	20.0095	0.995510
$405$	−96.4378	−4.79203
$406$	17.7798	0.882399
$407$	−0.254372	−0.0126087
$408$	−5.30063	−0.262421
$409$	−18.7840	−0.928807	−0.464403	−	0.885624i	$-0.653731\pi$
$409$	−18.7840	−0.928807	−0.464403	+	0.885624i	$0.653731\pi$
$410$	3.52149	0.173914
$411$	3.96574	0.195616
$412$	−1.48856	−0.0733361
$413$	−37.2138	−1.83117
$414$	57.0766	2.80516
$415$	24.4758	1.20147
$416$	−4.38630	−0.215056
$417$	15.2239	0.745518
$418$	−2.59421	−0.126887
$419$	24.8971	1.21630	0.608151	−	0.793822i	$-0.291912\pi$
$419$	24.8971	1.21630	0.608151	+	0.793822i	$0.291912\pi$
$420$	−30.2846	−1.47774
$421$	−3.33289	−0.162435	−0.0812175	−	0.996696i	$-0.525881\pi$
$421$	−3.33289	−0.162435	−0.0812175	+	0.996696i	$0.525881\pi$
$422$	8.77979	0.427393
$423$	53.4286	2.59779
$424$	2.49746	0.121287
$425$	0.325994	0.0158130
$426$	−35.8113	−1.73506
$427$	22.7578	1.10133
$428$	4.37825	0.211631
$429$	5.44269	0.262776
$430$	−13.8752	−0.669122
$431$	12.4942	0.601826	0.300913	−	0.953652i	$-0.402709\pi$
$431$	12.4942	0.601826	0.300913	+	0.953652i	$0.402709\pi$
$432$	−19.9836	−0.961460
$433$	−8.81737	−0.423736	−0.211868	−	0.977298i	$-0.567955\pi$
$433$	−8.81737	−0.423736	−0.211868	+	0.977298i	$0.567955\pi$
$434$	−34.7897	−1.66996
$435$	−36.1457	−1.73305
$436$	−13.2141	−0.632842
$437$	−46.5344	−2.22604
$438$	47.8597	2.28683
$439$	19.2818	0.920269	0.460135	−	0.887849i	$-0.347801\pi$
$439$	19.2818	0.920269	0.460135	+	0.887849i	$0.347801\pi$
$440$	−0.824126	−0.0392887
$441$	69.6022	3.31439
$442$	−6.76438	−0.321749
$443$	26.9186	1.27894	0.639471	−	0.768815i	$-0.279153\pi$
$443$	26.9186	1.27894	0.639471	+	0.768815i	$0.279153\pi$
$444$	2.42187	0.114937
$445$	−41.7847	−1.98078
$446$	−4.16238	−0.197095
$447$	36.7125	1.73644
$448$	−3.85963	−0.182351
$449$	−6.33393	−0.298916	−0.149458	−	0.988768i	$-0.547753\pi$
$449$	−6.33393	−0.298916	−0.149458	+	0.988768i	$0.547753\pi$
$450$	1.86317	0.0878308
$451$	−0.556887	−0.0262228
$452$	10.3975	0.489057
$453$	−61.8288	−2.90497
$454$	−10.5565	−0.495440
$455$	−38.6475	−1.81182
$456$	24.6994	1.15666
$457$	−8.02585	−0.375434	−0.187717	−	0.982223i	$-0.560109\pi$
$457$	−8.02585	−0.375434	−0.187717	+	0.982223i	$0.560109\pi$
$458$	5.99864	0.280298
$459$	−30.8179	−1.43845
$460$	−14.7830	−0.689260
$461$	27.5375	1.28255	0.641276	−	0.767311i	$-0.278406\pi$
$461$	27.5375	1.28255	0.641276	+	0.767311i	$0.278406\pi$
$462$	4.78918	0.222813
$463$	−40.8397	−1.89798	−0.948991	−	0.315302i	$-0.897894\pi$
$463$	−40.8397	−1.89798	−0.948991	+	0.315302i	$0.897894\pi$
$464$	−4.60661	−0.213857
$465$	70.7261	3.27985
$466$	12.9067	0.597892
$467$	12.2630	0.567463	0.283731	−	0.958904i	$-0.408428\pi$
$467$	12.2630	0.567463	0.283731	+	0.958904i	$0.408428\pi$
$468$	−38.6609	−1.78710
$469$	−36.5759	−1.68892
$470$	−13.8381	−0.638305
$471$	25.3588	1.16847
$472$	9.64179	0.443799
$473$	2.19422	0.100890
$474$	55.2962	2.53984
$475$	−1.51904	−0.0696982
$476$	−5.95217	−0.272817
$477$	22.0126	1.00789
$478$	−19.8478	−0.907815
$479$	29.2475	1.33635	0.668175	−	0.744004i	$-0.267076\pi$
$479$	29.2475	1.33635	0.668175	+	0.744004i	$0.267076\pi$
$480$	7.84648	0.358141
$481$	3.09066	0.140922
$482$	2.71243	0.123548
$483$	85.9073	3.90892
$484$	−10.8697	−0.494076
$485$	−4.12877	−0.187478
$486$	−85.2500	−3.86702
$487$	−14.5335	−0.658576	−0.329288	−	0.944230i	$-0.606809\pi$
$487$	−14.5335	−0.658576	−0.329288	+	0.944230i	$0.606809\pi$
$488$	−5.89636	−0.266916
$489$	−11.7637	−0.531971
$490$	−18.0271	−0.814383
$491$	−7.66250	−0.345804	−0.172902	−	0.984939i	$-0.555314\pi$
$491$	−7.66250	−0.345804	−0.172902	+	0.984939i	$0.555314\pi$
$492$	5.30211	0.239038
$493$	−7.10413	−0.319954
$494$	31.5201	1.41815
$495$	−7.26384	−0.326485
$496$	9.01373	0.404729
$497$	−40.2131	−1.80380
$498$	36.8518	1.65137
$499$	−31.1742	−1.39555	−0.697774	−	0.716318i	$-0.745826\pi$
$499$	−31.1742	−1.39555	−0.697774	+	0.716318i	$0.745826\pi$
$500$	10.9317	0.488879
$501$	−40.9192	−1.82813
$502$	0.460861	0.0205692
$503$	−21.0971	−0.940673	−0.470337	−	0.882487i	$-0.655867\pi$
$503$	−21.0971	−0.940673	−0.470337	+	0.882487i	$0.655867\pi$
$504$	−34.0188	−1.51532
$505$	−45.6786	−2.03267
$506$	2.33777	0.103927
$507$	−21.4466	−0.952478
$508$	4.60505	0.204316
$509$	12.4964	0.553893	0.276946	−	0.960885i	$-0.410678\pi$
$509$	12.4964	0.553893	0.276946	+	0.960885i	$0.410678\pi$
$510$	12.1005	0.535821
$511$	53.7425	2.37743
$512$	1.00000	0.0441942
$513$	143.602	6.34020
$514$	−27.7877	−1.22566
$515$	3.39815	0.149741
$516$	−20.8911	−0.919679
$517$	2.18835	0.0962437
$518$	2.71956	0.119490
$519$	75.2396	3.30265
$520$	10.0133	0.439110
$521$	−36.2419	−1.58778	−0.793892	−	0.608058i	$-0.791949\pi$
$521$	−36.2419	−1.58778	−0.793892	+	0.608058i	$0.791949\pi$
$522$	−40.6027	−1.77713
$523$	37.6417	1.64596	0.822978	−	0.568074i	$-0.192311\pi$
$523$	37.6417	1.64596	0.822978	+	0.568074i	$0.192311\pi$
$524$	15.2608	0.666673
$525$	2.80430	0.122390
$526$	−15.5040	−0.676005
$527$	13.9006	0.605521
$528$	−1.24084	−0.0540006
$529$	18.9345	0.823238
$530$	−5.70131	−0.247649
$531$	84.9827	3.68793
$532$	27.7354	1.20248
$533$	6.76626	0.293079
$534$	−62.9128	−2.72250
$535$	−9.99487	−0.432116
$536$	9.47651	0.409323
$537$	−14.2956	−0.616902
$538$	−3.74935	−0.161646
$539$	2.85080	0.122793
$540$	45.6194	1.96315
$541$	−8.58565	−0.369126	−0.184563	−	0.982821i	$-0.559087\pi$
$541$	−8.58565	−0.369126	−0.184563	+	0.982821i	$0.559087\pi$
$542$	−18.0585	−0.775680
$543$	3.71282	0.159332
$544$	1.54216	0.0661196
$545$	30.1658	1.29216
$546$	−58.1893	−2.49027
$547$	8.56072	0.366030	0.183015	−	0.983110i	$-0.441414\pi$
$547$	8.56072	0.366030	0.183015	+	0.983110i	$0.441414\pi$
$548$	−1.15379	−0.0492874
$549$	−51.9705	−2.21805
$550$	0.0763127	0.00325399
$551$	33.1032	1.41024
$552$	−22.2579	−0.947359
$553$	62.0931	2.64047
$554$	−17.6270	−0.748902
$555$	−5.52875	−0.234682
$556$	−4.42923	−0.187841
$557$	−7.87206	−0.333550	−0.166775	−	0.985995i	$-0.553335\pi$
$557$	−7.87206	−0.333550	−0.166775	+	0.985995i	$0.553335\pi$
$558$	79.4470	3.36326
$559$	−26.6601	−1.12760
$560$	8.81095	0.372331
$561$	−1.91357	−0.0807910
$562$	4.89388	0.206436
$563$	4.78943	0.201850	0.100925	−	0.994894i	$-0.467820\pi$
$563$	4.78943	0.201850	0.100925	+	0.994894i	$0.467820\pi$
$564$	−20.8353	−0.877323
$565$	−23.7359	−0.998577
$566$	−12.7271	−0.534959
$567$	−163.048	−6.84739
$568$	10.4189	0.437167
$569$	−34.3600	−1.44045	−0.720223	−	0.693742i	$-0.755961\pi$
$569$	−34.3600	−1.44045	−0.720223	+	0.693742i	$0.755961\pi$
$570$	−56.3850	−2.36171
$571$	6.62622	0.277299	0.138649	−	0.990342i	$-0.455724\pi$
$571$	6.62622	0.277299	0.138649	+	0.990342i	$0.455724\pi$
$572$	−1.58349	−0.0662091
$573$	1.73652	0.0725441
$574$	5.95383	0.248508
$575$	1.36888	0.0570863
$576$	8.81400	0.367250
$577$	18.4521	0.768170	0.384085	−	0.923298i	$-0.374517\pi$
$577$	18.4521	0.768170	0.384085	+	0.923298i	$0.374517\pi$
$578$	−14.6217	−0.608184
$579$	16.8786	0.701450
$580$	10.5162	0.436661
$581$	41.3815	1.71679
$582$	−6.21645	−0.257680
$583$	0.901601	0.0373405
$584$	−13.9243	−0.576190
$585$	88.2568	3.64897
$586$	−25.5441	−1.05522
$587$	−25.4131	−1.04891	−0.524455	−	0.851438i	$-0.675731\pi$
$587$	−25.4131	−1.04891	−0.524455	+	0.851438i	$0.675731\pi$
$588$	−27.1424	−1.11933
$589$	−64.7729	−2.66892
$590$	−22.0107	−0.906167
$591$	−22.8024	−0.937965
$592$	−0.704615	−0.0289595
$593$	−38.3779	−1.57599	−0.787996	−	0.615681i	$-0.788881\pi$
$593$	−38.3779	−1.57599	−0.787996	+	0.615681i	$0.788881\pi$
$594$	−7.21423	−0.296003
$595$	13.5879	0.557050
$596$	−10.6811	−0.437514
$597$	−41.3062	−1.69055
$598$	−28.4043	−1.16154
$599$	−20.7652	−0.848445	−0.424222	−	0.905558i	$-0.639452\pi$
$599$	−20.7652	−0.848445	−0.424222	+	0.905558i	$0.639452\pi$
$600$	−0.726572	−0.0296622
$601$	26.8648	1.09584	0.547919	−	0.836531i	$-0.315420\pi$
$601$	26.8648	1.09584	0.547919	+	0.836531i	$0.315420\pi$
$602$	−23.4590	−0.956116
$603$	83.5259	3.40144
$604$	17.9884	0.731937
$605$	24.8138	1.00882
$606$	−68.7757	−2.79382
$607$	2.21155	0.0897641	0.0448820	−	0.998992i	$-0.485709\pi$
$607$	2.21155	0.0897641	0.0448820	+	0.998992i	$0.485709\pi$
$608$	−7.18602	−0.291432
$609$	−61.1120	−2.47638
$610$	13.4605	0.544999
$611$	−26.5888	−1.07567
$612$	13.5926	0.549448
$613$	30.7580	1.24231	0.621153	−	0.783690i	$-0.286665\pi$
$613$	30.7580	1.24231	0.621153	+	0.783690i	$0.286665\pi$
$614$	−18.8491	−0.760689
$615$	−12.1039	−0.488076
$616$	−1.39336	−0.0561400
$617$	43.8140	1.76388	0.881942	−	0.471357i	$-0.156236\pi$
$617$	43.8140	1.76388	0.881942	+	0.471357i	$0.156236\pi$
$618$	5.11640	0.205812
$619$	18.2194	0.732301	0.366151	−	0.930556i	$-0.380676\pi$
$619$	18.2194	0.732301	0.366151	+	0.930556i	$0.380676\pi$
$620$	−20.5770	−0.826391
$621$	−129.407	−5.19293
$622$	23.3036	0.934389
$623$	−70.6459	−2.83037
$624$	15.0764	0.603538
$625$	−26.0123	−1.04049
$626$	−34.6503	−1.38490
$627$	8.91669	0.356098
$628$	−7.37785	−0.294408
$629$	−1.08663	−0.0433267
$630$	77.6597	3.09404
$631$	−3.30219	−0.131458	−0.0657292	−	0.997838i	$-0.520937\pi$
$631$	−3.30219	−0.131458	−0.0657292	+	0.997838i	$0.520937\pi$
$632$	−16.0878	−0.639939
$633$	−30.1774	−1.19945
$634$	−6.30150	−0.250265
$635$	−10.5126	−0.417180
$636$	−8.58413	−0.340383
$637$	−34.6377	−1.37239
$638$	−1.66302	−0.0658397
$639$	91.8320	3.63282
$640$	−2.28285	−0.0902374
$641$	14.1111	0.557354	0.278677	−	0.960385i	$-0.410104\pi$
$641$	14.1111	0.557354	0.278677	+	0.960385i	$0.410104\pi$
$642$	−15.0487	−0.593925
$643$	31.1308	1.22768	0.613840	−	0.789431i	$-0.289624\pi$
$643$	31.1308	1.22768	0.613840	+	0.789431i	$0.289624\pi$
$644$	−24.9938	−0.984892
$645$	47.6912	1.87784
$646$	−11.0820	−0.436015
$647$	19.4766	0.765703	0.382851	−	0.923810i	$-0.374942\pi$
$647$	19.4766	0.765703	0.382851	+	0.923810i	$0.374942\pi$
$648$	42.2445	1.65952
$649$	3.48076	0.136632
$650$	−0.927212	−0.0363682
$651$	119.577	4.68661
$652$	3.42250	0.134036
$653$	34.7814	1.36110	0.680550	−	0.732702i	$-0.261741\pi$
$653$	34.7814	1.36110	0.680550	+	0.732702i	$0.261741\pi$
$654$	45.4189	1.77602
$655$	−34.8381	−1.36124
$656$	−1.54259	−0.0602280
$657$	−122.728	−4.78809
$658$	−23.3963	−0.912082
$659$	−24.0845	−0.938200	−0.469100	−	0.883145i	$-0.655422\pi$
$659$	−24.0845	−0.938200	−0.469100	+	0.883145i	$0.655422\pi$
$660$	2.83264	0.110260
$661$	−18.6861	−0.726805	−0.363402	−	0.931632i	$-0.618385\pi$
$661$	−18.6861	−0.726805	−0.363402	+	0.931632i	$0.618385\pi$
$662$	−16.7890	−0.652521
$663$	23.2502	0.902963
$664$	−10.7216	−0.416079
$665$	−63.3157	−2.45528
$666$	−6.21047	−0.240651
$667$	−29.8310	−1.15506
$668$	11.9050	0.460617
$669$	14.3067	0.553130
$670$	−21.6334	−0.835772
$671$	−2.12863	−0.0821749
$672$	13.2661	0.511752
$673$	−39.9539	−1.54011	−0.770056	−	0.637977i	$-0.779772\pi$
$673$	−39.9539	−1.54011	−0.770056	+	0.637977i	$0.779772\pi$
$674$	−8.49180	−0.327092
$675$	−4.22429	−0.162593
$676$	6.23966	0.239987
$677$	−14.3836	−0.552805	−0.276402	−	0.961042i	$-0.589142\pi$
$677$	−14.3836	−0.552805	−0.276402	+	0.961042i	$0.589142\pi$
$678$	−35.7378	−1.37250
$679$	−6.98056	−0.267889
$680$	−3.52051	−0.135006
$681$	36.2842	1.39041
$682$	3.25403	0.124603
$683$	21.9011	0.838021	0.419011	−	0.907981i	$-0.362377\pi$
$683$	21.9011	0.838021	0.419011	+	0.907981i	$0.362377\pi$
$684$	−63.3375	−2.42177
$685$	2.63392	0.100637
$686$	−3.46124	−0.132151
$687$	−20.6182	−0.786634
$688$	6.07803	0.231723
$689$	−10.9546	−0.417337
$690$	50.8113	1.93435
$691$	−23.5784	−0.896965	−0.448483	−	0.893792i	$-0.648035\pi$
$691$	−23.5784	−0.896965	−0.448483	+	0.893792i	$0.648035\pi$
$692$	−21.8901	−0.832137
$693$	−12.2811	−0.466519
$694$	−34.1733	−1.29720
$695$	10.1112	0.383541
$696$	15.8336	0.600172
$697$	−2.37892	−0.0901080
$698$	19.9392	0.754711
$699$	−44.3623	−1.67794
$700$	−0.815880	−0.0308374
$701$	45.1051	1.70360	0.851798	−	0.523870i	$-0.175512\pi$
$701$	45.1051	1.70360	0.851798	+	0.523870i	$0.175512\pi$
$702$	87.6540	3.30829
$703$	5.06338	0.190969
$704$	0.361008	0.0136060
$705$	47.5637	1.79135
$706$	−3.42769	−0.129003
$707$	−77.2294	−2.90451
$708$	−33.1403	−1.24549
$709$	−10.0490	−0.377397	−0.188699	−	0.982035i	$-0.560427\pi$
$709$	−10.0490	−0.377397	−0.188699	+	0.982035i	$0.560427\pi$
$710$	−23.7847	−0.892625
$711$	−141.798	−5.31784
$712$	18.3038	0.685963
$713$	58.3701	2.18598
$714$	20.4585	0.765641
$715$	3.61487	0.135188
$716$	4.15916	0.155435
$717$	68.2197	2.54771
$718$	32.1213	1.19876
$719$	−7.74052	−0.288673	−0.144336	−	0.989529i	$-0.546105\pi$
$719$	−7.74052	−0.288673	−0.144336	+	0.989529i	$0.546105\pi$
$720$	−20.1210	−0.749865
$721$	5.74530	0.213966
$722$	32.6389	1.21469
$723$	−9.32304	−0.346727
$724$	−1.08020	−0.0401454
$725$	−0.973782	−0.0361654
$726$	37.3607	1.38659
$727$	−41.5408	−1.54066	−0.770332	−	0.637643i	$-0.779909\pi$
$727$	−41.5408	−1.54066	−0.770332	+	0.637643i	$0.779909\pi$
$728$	16.9295	0.627450
$729$	166.284	6.15865
$730$	31.7869	1.17649
$731$	9.37329	0.346684
$732$	20.2667	0.749077
$733$	21.0349	0.776943	0.388471	−	0.921461i	$-0.373003\pi$
$733$	21.0349	0.776943	0.388471	+	0.921461i	$0.373003\pi$
$734$	−31.1725	−1.15060
$735$	61.9620	2.28550
$736$	6.47568	0.238697
$737$	3.42109	0.126018
$738$	−13.5964	−0.500489
$739$	6.94372	0.255429	0.127714	−	0.991811i	$-0.459236\pi$
$739$	6.94372	0.255429	0.127714	+	0.991811i	$0.459236\pi$
$740$	1.60853	0.0591307
$741$	−108.339	−3.97994
$742$	−9.63927	−0.353869
$743$	−32.5720	−1.19495	−0.597475	−	0.801888i	$-0.703829\pi$
$743$	−32.5720	−1.19495	−0.597475	+	0.801888i	$0.703829\pi$
$744$	−30.9816	−1.13584
$745$	24.3833	0.893333
$746$	32.9584	1.20669
$747$	−94.5002	−3.45758
$748$	0.556732	0.0203561
$749$	−16.8984	−0.617455
$750$	−37.5738	−1.37200
$751$	−7.20426	−0.262887	−0.131444	−	0.991324i	$-0.541961\pi$
$751$	−7.20426	−0.262887	−0.131444	+	0.991324i	$0.541961\pi$
$752$	6.06179	0.221051
$753$	−1.58405	−0.0577260
$754$	20.2060	0.735859
$755$	−41.0647	−1.49450
$756$	77.1293	2.80516
$757$	1.28789	0.0468090	0.0234045	−	0.999726i	$-0.492549\pi$
$757$	1.28789	0.0468090	0.0234045	+	0.999726i	$0.492549\pi$
$758$	5.03123	0.182742
$759$	−8.03527	−0.291662
$760$	16.4046	0.595056
$761$	33.8984	1.22882	0.614408	−	0.788988i	$-0.289395\pi$
$761$	33.8984	1.22882	0.614408	+	0.788988i	$0.289395\pi$
$762$	−15.8282	−0.573396
$763$	51.0017	1.84639
$764$	−0.505220	−0.0182782
$765$	−31.0298	−1.12188
$766$	18.1170	0.654595
$767$	−42.2918	−1.52707
$768$	−3.43715	−0.124027
$769$	14.6411	0.527970	0.263985	−	0.964527i	$-0.414963\pi$
$769$	14.6411	0.527970	0.263985	+	0.964527i	$0.414963\pi$
$770$	3.18082	0.114629
$771$	95.5106	3.43973
$772$	−4.91064	−0.176738
$773$	16.1390	0.580480	0.290240	−	0.956954i	$-0.406265\pi$
$773$	16.1390	0.580480	0.290240	+	0.956954i	$0.406265\pi$
$774$	53.5717	1.92560
$775$	1.90540	0.0684438
$776$	1.80861	0.0649251
$777$	−9.34752	−0.335340
$778$	4.36529	0.156503
$779$	11.0851	0.397164
$780$	−34.4171	−1.23233
$781$	3.76130	0.134590
$782$	9.98654	0.357118
$783$	92.0566	3.28983
$784$	7.89678	0.282028
$785$	16.8425	0.601135
$786$	−52.4538	−1.87096
$787$	4.19515	0.149541	0.0747705	−	0.997201i	$-0.476178\pi$
$787$	4.19515	0.149541	0.0747705	+	0.997201i	$0.476178\pi$
$788$	6.63410	0.236330
$789$	53.2895	1.89716
$790$	36.7260	1.30665
$791$	−40.1305	−1.42688
$792$	3.18192	0.113065
$793$	25.8632	0.918430
$794$	−24.3276	−0.863356
$795$	19.5962	0.695007
$796$	12.0176	0.425952
$797$	45.9449	1.62745	0.813727	−	0.581248i	$-0.197435\pi$
$797$	45.9449	1.62745	0.813727	+	0.581248i	$0.197435\pi$
$798$	−95.3307	−3.37467
$799$	9.34825	0.330717
$800$	0.211388	0.00747369
$801$	161.329	5.70029
$802$	−3.52602	−0.124508
$803$	−5.02677	−0.177391
$804$	−32.5722	−1.14873
$805$	57.0569	2.01099
$806$	−39.5370	−1.39263
$807$	12.8871	0.453647
$808$	20.0095	0.703932
$809$	39.0275	1.37213	0.686067	−	0.727538i	$-0.259335\pi$
$809$	39.0275	1.37213	0.686067	+	0.727538i	$0.259335\pi$
$810$	−96.4378	−3.38848
$811$	−15.6739	−0.550384	−0.275192	−	0.961389i	$-0.588741\pi$
$811$	−15.6739	−0.550384	−0.275192	+	0.961389i	$0.588741\pi$
$812$	17.7798	0.623950
$813$	62.0699	2.17689
$814$	−0.254372	−0.00891572
$815$	−7.81305	−0.273679
$816$	−5.30063	−0.185559
$817$	−43.6768	−1.52806
$818$	−18.7840	−0.656766
$819$	149.217	5.21406
$820$	3.52149	0.122976
$821$	−23.4399	−0.818059	−0.409029	−	0.912521i	$-0.634133\pi$
$821$	−23.4399	−0.818059	−0.409029	+	0.912521i	$0.634133\pi$
$822$	3.96574	0.138321
$823$	−8.29980	−0.289313	−0.144656	−	0.989482i	$-0.546208\pi$
$823$	−8.29980	−0.289313	−0.144656	+	0.989482i	$0.546208\pi$
$824$	−1.48856	−0.0518564
$825$	−0.262298	−0.00913205
$826$	−37.2138	−1.29483
$827$	36.7606	1.27829	0.639146	−	0.769085i	$-0.279288\pi$
$827$	36.7606	1.27829	0.639146	+	0.769085i	$0.279288\pi$
$828$	57.0766	1.98355
$829$	−5.31370	−0.184553	−0.0922763	−	0.995733i	$-0.529414\pi$
$829$	−5.31370	−0.184553	−0.0922763	+	0.995733i	$0.529414\pi$
$830$	24.4758	0.849567
$831$	60.5868	2.10173
$832$	−4.38630	−0.152068
$833$	12.1781	0.421946
$834$	15.2239	0.527161
$835$	−27.1772	−0.940507
$836$	−2.59421	−0.0897226
$837$	−180.127	−6.22609
$838$	24.8971	0.860055
$839$	−10.9768	−0.378961	−0.189480	−	0.981885i	$-0.560680\pi$
$839$	−10.9768	−0.378961	−0.189480	+	0.981885i	$0.560680\pi$
$840$	−30.2846	−1.04492
$841$	−7.77913	−0.268246
$842$	−3.33289	−0.114859
$843$	−16.8210	−0.579346
$844$	8.77979	0.302213
$845$	−14.2442	−0.490015
$846$	53.4286	1.83691
$847$	41.9530	1.44152
$848$	2.49746	0.0857630
$849$	43.7448	1.50132
$850$	0.325994	0.0111815
$851$	−4.56286	−0.156413
$852$	−35.8113	−1.22687
$853$	7.04692	0.241282	0.120641	−	0.992696i	$-0.461505\pi$
$853$	7.04692	0.241282	0.120641	+	0.992696i	$0.461505\pi$
$854$	22.7578	0.778755
$855$	144.590	4.94487
$856$	4.37825	0.149645
$857$	−25.5924	−0.874218	−0.437109	−	0.899408i	$-0.643998\pi$
$857$	−25.5924	−0.874218	−0.437109	+	0.899408i	$0.643998\pi$
$858$	5.44269	0.185810
$859$	−25.6368	−0.874715	−0.437358	−	0.899288i	$-0.644086\pi$
$859$	−25.6368	−0.874715	−0.437358	+	0.899288i	$0.644086\pi$
$860$	−13.8752	−0.473141
$861$	−20.4642	−0.697418
$862$	12.4942	0.425555
$863$	−3.39448	−0.115549	−0.0577747	−	0.998330i	$-0.518400\pi$
$863$	−3.39448	−0.115549	−0.0577747	+	0.998330i	$0.518400\pi$
$864$	−19.9836	−0.679855
$865$	49.9717	1.69909
$866$	−8.81737	−0.299626
$867$	50.2571	1.70682
$868$	−34.7897	−1.18084
$869$	−5.80783	−0.197017
$870$	−36.1457	−1.22545
$871$	−41.5668	−1.40844
$872$	−13.2141	−0.447487
$873$	15.9410	0.539522
$874$	−46.5344	−1.57405
$875$	−42.1922	−1.42636
$876$	47.8597	1.61703
$877$	42.7031	1.44198	0.720991	−	0.692945i	$-0.243687\pi$
$877$	42.7031	1.44198	0.720991	+	0.692945i	$0.243687\pi$
$878$	19.2818	0.650729
$879$	87.7989	2.96138
$880$	−0.824126	−0.0277813
$881$	39.9187	1.34490	0.672448	−	0.740144i	$-0.265243\pi$
$881$	39.9187	1.34490	0.672448	+	0.740144i	$0.265243\pi$
$882$	69.6022	2.34363
$883$	42.1911	1.41984	0.709921	−	0.704281i	$-0.248730\pi$
$883$	42.1911	1.41984	0.709921	+	0.704281i	$0.248730\pi$
$884$	−6.76438	−0.227511
$885$	75.6541	2.54309
$886$	26.9186	0.904349
$887$	9.10136	0.305594	0.152797	−	0.988258i	$-0.451172\pi$
$887$	9.10136	0.305594	0.152797	+	0.988258i	$0.451172\pi$
$888$	2.42187	0.0812725
$889$	−17.7738	−0.596114
$890$	−41.7847	−1.40063
$891$	15.2506	0.510915
$892$	−4.16238	−0.139367
$893$	−43.5601	−1.45768
$894$	36.7125	1.22785
$895$	−9.49472	−0.317373
$896$	−3.85963	−0.128941
$897$	97.6298	3.25977
$898$	−6.33393	−0.211366
$899$	−41.5228	−1.38486
$900$	1.86317	0.0621058
$901$	3.85148	0.128311
$902$	−0.556887	−0.0185423
$903$	80.6320	2.68326
$904$	10.3975	0.345816
$905$	2.46594	0.0819705
$906$	−61.8288	−2.05412
$907$	28.9234	0.960386	0.480193	−	0.877163i	$-0.340567\pi$
$907$	28.9234	0.960386	0.480193	+	0.877163i	$0.340567\pi$
$908$	−10.5565	−0.350329
$909$	176.364	5.84961
$910$	−38.6475	−1.28115
$911$	16.0352	0.531271	0.265635	−	0.964074i	$-0.414418\pi$
$911$	16.0352	0.531271	0.265635	+	0.964074i	$0.414418\pi$
$912$	24.6994	0.817879
$913$	−3.87058	−0.128098
$914$	−8.02585	−0.265472
$915$	−46.2657	−1.52950
$916$	5.99864	0.198201
$917$	−58.9012	−1.94509
$918$	−30.8179	−1.01714
$919$	39.9042	1.31632	0.658158	−	0.752879i	$-0.271336\pi$
$919$	39.9042	1.31632	0.658158	+	0.752879i	$0.271336\pi$
$920$	−14.7830	−0.487381
$921$	64.7873	2.13481
$922$	27.5375	0.906901
$923$	−45.7004	−1.50425
$924$	4.78918	0.157553
$925$	−0.148947	−0.00489735
$926$	−40.8397	−1.34208
$927$	−13.1202	−0.430923
$928$	−4.60661	−0.151219
$929$	29.4339	0.965696	0.482848	−	0.875704i	$-0.339602\pi$
$929$	29.4339	0.965696	0.482848	+	0.875704i	$0.339602\pi$
$930$	70.7261	2.31920
$931$	−56.7464	−1.85979
$932$	12.9067	0.422774
$933$	−80.0979	−2.62229
$934$	12.2630	0.401257
$935$	−1.27093	−0.0415640
$936$	−38.6609	−1.26367
$937$	−38.8089	−1.26783	−0.633917	−	0.773401i	$-0.718554\pi$
$937$	−38.8089	−1.26783	−0.633917	+	0.773401i	$0.718554\pi$
$938$	−36.5759	−1.19424
$939$	119.098	3.88662
$940$	−13.8381	−0.451350
$941$	3.16882	0.103301	0.0516503	−	0.998665i	$-0.483552\pi$
$941$	3.16882	0.103301	0.0516503	+	0.998665i	$0.483552\pi$
$942$	25.3588	0.826234
$943$	−9.98932	−0.325297
$944$	9.64179	0.313813
$945$	−176.074	−5.72770
$946$	2.19422	0.0713401
$947$	−48.4182	−1.57338	−0.786690	−	0.617349i	$-0.788207\pi$
$947$	−48.4182	−1.57338	−0.786690	+	0.617349i	$0.788207\pi$
$948$	55.2962	1.79594
$949$	61.0760	1.98261
$950$	−1.51904	−0.0492841
$951$	21.6592	0.702348
$952$	−5.95217	−0.192911
$953$	22.4957	0.728706	0.364353	−	0.931261i	$-0.381290\pi$
$953$	22.4957	0.728706	0.364353	+	0.931261i	$0.381290\pi$
$954$	22.0126	0.712683
$955$	1.15334	0.0373212
$956$	−19.8478	−0.641922
$957$	5.71606	0.184774
$958$	29.2475	0.944942
$959$	4.45320	0.143801
$960$	7.84648	0.253244
$961$	50.2474	1.62088
$962$	3.09066	0.0996467
$963$	38.5899	1.24354
$964$	2.71243	0.0873616
$965$	11.2102	0.360870
$966$	85.9073	2.76402
$967$	−37.9318	−1.21980	−0.609902	−	0.792477i	$-0.708791\pi$
$967$	−37.9318	−1.21980	−0.609902	+	0.792477i	$0.708791\pi$
$968$	−10.8697	−0.349365
$969$	38.0905	1.22364
$970$	−4.12877	−0.132567
$971$	−42.8819	−1.37615	−0.688073	−	0.725642i	$-0.741543\pi$
$971$	−42.8819	−1.37615	−0.688073	+	0.725642i	$0.741543\pi$
$972$	−85.2500	−2.73440
$973$	17.0952	0.548047
$974$	−14.5335	−0.465683
$975$	3.18697	0.102065
$976$	−5.89636	−0.188738
$977$	37.7232	1.20687	0.603436	−	0.797411i	$-0.293798\pi$
$977$	37.7232	1.20687	0.603436	+	0.797411i	$0.293798\pi$
$978$	−11.7637	−0.376160
$979$	6.60781	0.211186
$980$	−18.0271	−0.575856
$981$	−116.469	−3.71858
$982$	−7.66250	−0.244520
$983$	−28.0142	−0.893514	−0.446757	−	0.894655i	$-0.647421\pi$
$983$	−28.0142	−0.893514	−0.446757	+	0.894655i	$0.647421\pi$
$984$	5.30211	0.169025
$985$	−15.1446	−0.482548
$986$	−7.10413	−0.226242
$987$	80.4165	2.55969
$988$	31.5201	1.00279
$989$	39.3594	1.25156
$990$	−7.26384	−0.230860
$991$	−38.3848	−1.21933	−0.609666	−	0.792658i	$-0.708696\pi$
$991$	−38.3848	−1.21933	−0.609666	+	0.792658i	$0.708696\pi$
$992$	9.01373	0.286186
$993$	57.7061	1.83125
$994$	−40.2131	−1.27548
$995$	−27.4343	−0.869725
$996$	36.8518	1.16769
$997$	−0.718145	−0.0227439	−0.0113719	−	0.999935i	$-0.503620\pi$
$997$	−0.718145	−0.0227439	−0.0113719	+	0.999935i	$0.503620\pi$
$998$	−31.1742	−0.986801
$999$	14.0807	0.445494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.1	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.1	✓	69	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.43715 q^{3} +1.00000 q^{4} -2.28285 q^{5} -3.43715 q^{6} -3.85963 q^{7} +1.00000 q^{8} +8.81400 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.43715 q^{3} +1.00000 q^{4} -2.28285 q^{5} -3.43715 q^{6} -3.85963 q^{7} +1.00000 q^{8} +8.81400 q^{9} -2.28285 q^{10} +0.361008 q^{11} -3.43715 q^{12} -4.38630 q^{13} -3.85963 q^{14} +7.84648 q^{15} +1.00000 q^{16} +1.54216 q^{17} +8.81400 q^{18} -7.18602 q^{19} -2.28285 q^{20} +13.2661 q^{21} +0.361008 q^{22} +6.47568 q^{23} -3.43715 q^{24} +0.211388 q^{25} -4.38630 q^{26} -19.9836 q^{27} -3.85963 q^{28} -4.60661 q^{29} +7.84648 q^{30} +9.01373 q^{31} +1.00000 q^{32} -1.24084 q^{33} +1.54216 q^{34} +8.81095 q^{35} +8.81400 q^{36} -0.704615 q^{37} -7.18602 q^{38} +15.0764 q^{39} -2.28285 q^{40} -1.54259 q^{41} +13.2661 q^{42} +6.07803 q^{43} +0.361008 q^{44} -20.1210 q^{45} +6.47568 q^{46} +6.06179 q^{47} -3.43715 q^{48} +7.89678 q^{49} +0.211388 q^{50} -5.30063 q^{51} -4.38630 q^{52} +2.49746 q^{53} -19.9836 q^{54} -0.824126 q^{55} -3.85963 q^{56} +24.6994 q^{57} -4.60661 q^{58} +9.64179 q^{59} +7.84648 q^{60} -5.89636 q^{61} +9.01373 q^{62} -34.0188 q^{63} +1.00000 q^{64} +10.0133 q^{65} -1.24084 q^{66} +9.47651 q^{67} +1.54216 q^{68} -22.2579 q^{69} +8.81095 q^{70} +10.4189 q^{71} +8.81400 q^{72} -13.9243 q^{73} -0.704615 q^{74} -0.726572 q^{75} -7.18602 q^{76} -1.39336 q^{77} +15.0764 q^{78} -16.0878 q^{79} -2.28285 q^{80} +42.2445 q^{81} -1.54259 q^{82} -10.7216 q^{83} +13.2661 q^{84} -3.52051 q^{85} +6.07803 q^{86} +15.8336 q^{87} +0.361008 q^{88} +18.3038 q^{89} -20.1210 q^{90} +16.9295 q^{91} +6.47568 q^{92} -30.9816 q^{93} +6.06179 q^{94} +16.4046 q^{95} -3.43715 q^{96} +1.80861 q^{97} +7.89678 q^{98} +3.18192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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