LMFDB - Embedded newform 8002.2.a.e.1.45

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:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.45
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)

$f(q)$	$=$	$q-1.00000 q^{2} +0.861874 q^{3} +1.00000 q^{4} -2.56007 q^{5} -0.861874 q^{6} -1.37195 q^{7} -1.00000 q^{8} -2.25717 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.861874 q^{3} +1.00000 q^{4} -2.56007 q^{5} -0.861874 q^{6} -1.37195 q^{7} -1.00000 q^{8} -2.25717 q^{9} +2.56007 q^{10} -3.71711 q^{11} +0.861874 q^{12} +4.81992 q^{13} +1.37195 q^{14} -2.20646 q^{15} +1.00000 q^{16} +4.22155 q^{17} +2.25717 q^{18} +0.679336 q^{19} -2.56007 q^{20} -1.18245 q^{21} +3.71711 q^{22} +7.08465 q^{23} -0.861874 q^{24} +1.55398 q^{25} -4.81992 q^{26} -4.53102 q^{27} -1.37195 q^{28} -5.18861 q^{29} +2.20646 q^{30} -8.85301 q^{31} -1.00000 q^{32} -3.20368 q^{33} -4.22155 q^{34} +3.51230 q^{35} -2.25717 q^{36} -1.14545 q^{37} -0.679336 q^{38} +4.15416 q^{39} +2.56007 q^{40} +7.40779 q^{41} +1.18245 q^{42} -9.93673 q^{43} -3.71711 q^{44} +5.77853 q^{45} -7.08465 q^{46} +0.701448 q^{47} +0.861874 q^{48} -5.11775 q^{49} -1.55398 q^{50} +3.63844 q^{51} +4.81992 q^{52} -3.73412 q^{53} +4.53102 q^{54} +9.51607 q^{55} +1.37195 q^{56} +0.585502 q^{57} +5.18861 q^{58} +6.36757 q^{59} -2.20646 q^{60} -2.54417 q^{61} +8.85301 q^{62} +3.09673 q^{63} +1.00000 q^{64} -12.3394 q^{65} +3.20368 q^{66} -6.70824 q^{67} +4.22155 q^{68} +6.10607 q^{69} -3.51230 q^{70} -7.17723 q^{71} +2.25717 q^{72} +2.02523 q^{73} +1.14545 q^{74} +1.33934 q^{75} +0.679336 q^{76} +5.09969 q^{77} -4.15416 q^{78} +15.6572 q^{79} -2.56007 q^{80} +2.86636 q^{81} -7.40779 q^{82} +1.56504 q^{83} -1.18245 q^{84} -10.8075 q^{85} +9.93673 q^{86} -4.47192 q^{87} +3.71711 q^{88} -3.15167 q^{89} -5.77853 q^{90} -6.61269 q^{91} +7.08465 q^{92} -7.63018 q^{93} -0.701448 q^{94} -1.73915 q^{95} -0.861874 q^{96} -0.959539 q^{97} +5.11775 q^{98} +8.39016 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * 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
$3$	0.861874	0.497603	0.248801	−	0.968554i	$-0.419963\pi$
$3$	0.861874	0.497603	0.248801	+	0.968554i	$0.419963\pi$
$4$	1.00000	0.500000
$5$	−2.56007	−1.14490	−0.572450	−	0.819940i	$-0.694007\pi$
$5$	−2.56007	−1.14490	−0.572450	+	0.819940i	$0.694007\pi$
$6$	−0.861874	−0.351858
$7$	−1.37195	−0.518549	−0.259274	−	0.965804i	$-0.583483\pi$
$7$	−1.37195	−0.518549	−0.259274	+	0.965804i	$0.583483\pi$
$8$	−1.00000	−0.353553
$9$	−2.25717	−0.752391
$10$	2.56007	0.809567
$11$	−3.71711	−1.12075	−0.560375	−	0.828239i	$-0.689343\pi$
$11$	−3.71711	−1.12075	−0.560375	+	0.828239i	$0.689343\pi$
$12$	0.861874	0.248801
$13$	4.81992	1.33680	0.668402	−	0.743800i	$-0.266978\pi$
$13$	4.81992	1.33680	0.668402	+	0.743800i	$0.266978\pi$
$14$	1.37195	0.366669
$15$	−2.20646	−0.569706
$16$	1.00000	0.250000
$17$	4.22155	1.02388	0.511938	−	0.859022i	$-0.328928\pi$
$17$	4.22155	1.02388	0.511938	+	0.859022i	$0.328928\pi$
$18$	2.25717	0.532021
$19$	0.679336	0.155850	0.0779252	−	0.996959i	$-0.475170\pi$
$19$	0.679336	0.155850	0.0779252	+	0.996959i	$0.475170\pi$
$20$	−2.56007	−0.572450
$21$	−1.18245	−0.258031
$22$	3.71711	0.792490
$23$	7.08465	1.47725	0.738625	−	0.674116i	$-0.235475\pi$
$23$	7.08465	1.47725	0.738625	+	0.674116i	$0.235475\pi$
$24$	−0.861874	−0.175929
$25$	1.55398	0.310797
$26$	−4.81992	−0.945264
$27$	−4.53102	−0.871995
$28$	−1.37195	−0.259274
$29$	−5.18861	−0.963500	−0.481750	−	0.876309i	$-0.659999\pi$
$29$	−5.18861	−0.963500	−0.481750	+	0.876309i	$0.659999\pi$
$30$	2.20646	0.402843
$31$	−8.85301	−1.59005	−0.795024	−	0.606578i	$-0.792542\pi$
$31$	−8.85301	−1.59005	−0.795024	+	0.606578i	$0.792542\pi$
$32$	−1.00000	−0.176777
$33$	−3.20368	−0.557688
$34$	−4.22155	−0.723990
$35$	3.51230	0.593687
$36$	−2.25717	−0.376196
$37$	−1.14545	−0.188310	−0.0941550	−	0.995558i	$-0.530015\pi$
$37$	−1.14545	−0.188310	−0.0941550	+	0.995558i	$0.530015\pi$
$38$	−0.679336	−0.110203
$39$	4.15416	0.665198
$40$	2.56007	0.404783
$41$	7.40779	1.15690	0.578451	−	0.815717i	$-0.303657\pi$
$41$	7.40779	1.15690	0.578451	+	0.815717i	$0.303657\pi$
$42$	1.18245	0.182456
$43$	−9.93673	−1.51534	−0.757668	−	0.652640i	$-0.773662\pi$
$43$	−9.93673	−1.51534	−0.757668	+	0.652640i	$0.773662\pi$
$44$	−3.71711	−0.560375
$45$	5.77853	0.861413
$46$	−7.08465	−1.04457
$47$	0.701448	0.102317	0.0511584	−	0.998691i	$-0.483709\pi$
$47$	0.701448	0.102317	0.0511584	+	0.998691i	$0.483709\pi$
$48$	0.861874	0.124401
$49$	−5.11775	−0.731107
$50$	−1.55398	−0.219766
$51$	3.63844	0.509484
$52$	4.81992	0.668402
$53$	−3.73412	−0.512920	−0.256460	−	0.966555i	$-0.582556\pi$
$53$	−3.73412	−0.512920	−0.256460	+	0.966555i	$0.582556\pi$
$54$	4.53102	0.616594
$55$	9.51607	1.28315
$56$	1.37195	0.183335
$57$	0.585502	0.0775516
$58$	5.18861	0.681298
$59$	6.36757	0.828987	0.414494	−	0.910052i	$-0.363959\pi$
$59$	6.36757	0.828987	0.414494	+	0.910052i	$0.363959\pi$
$60$	−2.20646	−0.284853
$61$	−2.54417	−0.325747	−0.162874	−	0.986647i	$-0.552076\pi$
$61$	−2.54417	−0.325747	−0.162874	+	0.986647i	$0.552076\pi$
$62$	8.85301	1.12433
$63$	3.09673	0.390152
$64$	1.00000	0.125000
$65$	−12.3394	−1.53051
$66$	3.20368	0.394345
$67$	−6.70824	−0.819542	−0.409771	−	0.912188i	$-0.634391\pi$
$67$	−6.70824	−0.819542	−0.409771	+	0.912188i	$0.634391\pi$
$68$	4.22155	0.511938
$69$	6.10607	0.735084
$70$	−3.51230	−0.419800
$71$	−7.17723	−0.851780	−0.425890	−	0.904775i	$-0.640039\pi$
$71$	−7.17723	−0.851780	−0.425890	+	0.904775i	$0.640039\pi$
$72$	2.25717	0.266010
$73$	2.02523	0.237035	0.118518	−	0.992952i	$-0.462186\pi$
$73$	2.02523	0.237035	0.118518	+	0.992952i	$0.462186\pi$
$74$	1.14545	0.133155
$75$	1.33934	0.154653
$76$	0.679336	0.0779252
$77$	5.09969	0.581163
$78$	−4.15416	−0.470366
$79$	15.6572	1.76157	0.880785	−	0.473516i	$-0.157015\pi$
$79$	15.6572	1.76157	0.880785	+	0.473516i	$0.157015\pi$
$80$	−2.56007	−0.286225
$81$	2.86636	0.318484
$82$	−7.40779	−0.818054
$83$	1.56504	0.171785	0.0858925	−	0.996304i	$-0.472626\pi$
$83$	1.56504	0.171785	0.0858925	+	0.996304i	$0.472626\pi$
$84$	−1.18245	−0.129016
$85$	−10.8075	−1.17224
$86$	9.93673	1.07150
$87$	−4.47192	−0.479441
$88$	3.71711	0.396245
$89$	−3.15167	−0.334077	−0.167038	−	0.985950i	$-0.553420\pi$
$89$	−3.15167	−0.334077	−0.167038	+	0.985950i	$0.553420\pi$
$90$	−5.77853	−0.609111
$91$	−6.61269	−0.693198
$92$	7.08465	0.738625
$93$	−7.63018	−0.791212
$94$	−0.701448	−0.0723488
$95$	−1.73915	−0.178433
$96$	−0.861874	−0.0879646
$97$	−0.959539	−0.0974264	−0.0487132	−	0.998813i	$-0.515512\pi$
$97$	−0.959539	−0.0974264	−0.0487132	+	0.998813i	$0.515512\pi$
$98$	5.11775	0.516971
$99$	8.39016	0.843242
$100$	1.55398	0.155398
$101$	−16.6446	−1.65620	−0.828102	−	0.560577i	$-0.810579\pi$
$101$	−16.6446	−1.65620	−0.828102	+	0.560577i	$0.810579\pi$
$102$	−3.63844	−0.360259
$103$	−11.2712	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$103$	−11.2712	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$104$	−4.81992	−0.472632
$105$	3.02716	0.295420
$106$	3.73412	0.362689
$107$	0.368055	0.0355812	0.0177906	−	0.999842i	$-0.494337\pi$
$107$	0.368055	0.0355812	0.0177906	+	0.999842i	$0.494337\pi$
$108$	−4.53102	−0.435998
$109$	−2.70211	−0.258815	−0.129408	−	0.991591i	$-0.541308\pi$
$109$	−2.70211	−0.258815	−0.129408	+	0.991591i	$0.541308\pi$
$110$	−9.51607	−0.907322
$111$	−0.987229	−0.0937036
$112$	−1.37195	−0.129637
$113$	1.28681	0.121053	0.0605265	−	0.998167i	$-0.480722\pi$
$113$	1.28681	0.121053	0.0605265	+	0.998167i	$0.480722\pi$
$114$	−0.585502	−0.0548373
$115$	−18.1372	−1.69130
$116$	−5.18861	−0.481750
$117$	−10.8794	−1.00580
$118$	−6.36757	−0.586183
$119$	−5.79176	−0.530930
$120$	2.20646	0.201421
$121$	2.81688	0.256080
$122$	2.54417	0.230338
$123$	6.38458	0.575678
$124$	−8.85301	−0.795024
$125$	8.82206	0.789069
$126$	−3.09673	−0.275879
$127$	−14.5218	−1.28860	−0.644299	−	0.764774i	$-0.722851\pi$
$127$	−14.5218	−1.28860	−0.644299	+	0.764774i	$0.722851\pi$
$128$	−1.00000	−0.0883883
$129$	−8.56420	−0.754036
$130$	12.3394	1.08223
$131$	−18.3605	−1.60417	−0.802084	−	0.597211i	$-0.796276\pi$
$131$	−18.3605	−1.60417	−0.802084	+	0.597211i	$0.796276\pi$
$132$	−3.20368	−0.278844
$133$	−0.932016	−0.0808160
$134$	6.70824	0.579504
$135$	11.5997	0.998347
$136$	−4.22155	−0.361995
$137$	7.39878	0.632120	0.316060	−	0.948739i	$-0.397640\pi$
$137$	7.39878	0.632120	0.316060	+	0.948739i	$0.397640\pi$
$138$	−6.10607	−0.519783
$139$	22.3562	1.89623	0.948115	−	0.317929i	$-0.102987\pi$
$139$	22.3562	1.89623	0.948115	+	0.317929i	$0.102987\pi$
$140$	3.51230	0.296843
$141$	0.604560	0.0509131
$142$	7.17723	0.602299
$143$	−17.9162	−1.49822
$144$	−2.25717	−0.188098
$145$	13.2832	1.10311
$146$	−2.02523	−0.167609
$147$	−4.41085	−0.363801
$148$	−1.14545	−0.0941550
$149$	16.4558	1.34811	0.674054	−	0.738682i	$-0.264552\pi$
$149$	16.4558	1.34811	0.674054	+	0.738682i	$0.264552\pi$
$150$	−1.33934	−0.109356
$151$	23.9475	1.94882	0.974410	−	0.224777i	$-0.0721654\pi$
$151$	23.9475	1.94882	0.974410	+	0.224777i	$0.0721654\pi$
$152$	−0.679336	−0.0551014
$153$	−9.52877	−0.770355
$154$	−5.09969	−0.410945
$155$	22.6644	1.82045
$156$	4.15416	0.332599
$157$	−14.0678	−1.12273	−0.561365	−	0.827568i	$-0.689724\pi$
$157$	−14.0678	−1.12273	−0.561365	+	0.827568i	$0.689724\pi$
$158$	−15.6572	−1.24562
$159$	−3.21834	−0.255231
$160$	2.56007	0.202392
$161$	−9.71979	−0.766026
$162$	−2.86636	−0.225202
$163$	8.17557	0.640360	0.320180	−	0.947357i	$-0.396257\pi$
$163$	8.17557	0.640360	0.320180	+	0.947357i	$0.396257\pi$
$164$	7.40779	0.578451
$165$	8.20165	0.638498
$166$	−1.56504	−0.121470
$167$	16.4335	1.27166	0.635830	−	0.771829i	$-0.280658\pi$
$167$	16.4335	1.27166	0.635830	+	0.771829i	$0.280658\pi$
$168$	1.18245	0.0912279
$169$	10.2316	0.787047
$170$	10.8075	0.828896
$171$	−1.53338	−0.117261
$172$	−9.93673	−0.757668
$173$	−1.75514	−0.133441	−0.0667203	−	0.997772i	$-0.521254\pi$
$173$	−1.75514	−0.133441	−0.0667203	+	0.997772i	$0.521254\pi$
$174$	4.47192	0.339016
$175$	−2.13199	−0.161163
$176$	−3.71711	−0.280187
$177$	5.48804	0.412507
$178$	3.15167	0.236228
$179$	25.8325	1.93081	0.965406	−	0.260751i	$-0.0839701\pi$
$179$	25.8325	1.93081	0.965406	+	0.260751i	$0.0839701\pi$
$180$	5.77853	0.430706
$181$	5.13134	0.381410	0.190705	−	0.981647i	$-0.438923\pi$
$181$	5.13134	0.381410	0.190705	+	0.981647i	$0.438923\pi$
$182$	6.61269	0.490165
$183$	−2.19275	−0.162093
$184$	−7.08465	−0.522287
$185$	2.93243	0.215596
$186$	7.63018	0.559472
$187$	−15.6919	−1.14751
$188$	0.701448	0.0511584
$189$	6.21634	0.452172
$190$	1.73915	0.126171
$191$	21.3470	1.54462	0.772308	−	0.635248i	$-0.219102\pi$
$191$	21.3470	1.54462	0.772308	+	0.635248i	$0.219102\pi$
$192$	0.861874	0.0622004
$193$	−17.4988	−1.25959	−0.629795	−	0.776762i	$-0.716861\pi$
$193$	−17.4988	−1.25959	−0.629795	+	0.776762i	$0.716861\pi$
$194$	0.959539	0.0688909
$195$	−10.6350	−0.761585
$196$	−5.11775	−0.365554
$197$	1.39390	0.0993109	0.0496555	−	0.998766i	$-0.484188\pi$
$197$	1.39390	0.0993109	0.0496555	+	0.998766i	$0.484188\pi$
$198$	−8.39016	−0.596262
$199$	5.98956	0.424589	0.212294	−	0.977206i	$-0.431906\pi$
$199$	5.98956	0.424589	0.212294	+	0.977206i	$0.431906\pi$
$200$	−1.55398	−0.109883
$201$	−5.78166	−0.407807
$202$	16.6446	1.17111
$203$	7.11852	0.499622
$204$	3.63844	0.254742
$205$	−18.9645	−1.32454
$206$	11.2712	0.785304
$207$	−15.9913	−1.11147
$208$	4.81992	0.334201
$209$	−2.52517	−0.174669
$210$	−3.02716	−0.208894
$211$	−10.2058	−0.702597	−0.351298	−	0.936264i	$-0.614260\pi$
$211$	−10.2058	−0.702597	−0.351298	+	0.936264i	$0.614260\pi$
$212$	−3.73412	−0.256460
$213$	−6.18586	−0.423848
$214$	−0.368055	−0.0251597
$215$	25.4388	1.73491
$216$	4.53102	0.308297
$217$	12.1459	0.824517
$218$	2.70211	0.183010
$219$	1.74549	0.117949
$220$	9.51607	0.641573
$221$	20.3475	1.36872
$222$	0.987229	0.0662585
$223$	25.1269	1.68262	0.841311	−	0.540551i	$-0.181784\pi$
$223$	25.1269	1.68262	0.841311	+	0.540551i	$0.181784\pi$
$224$	1.37195	0.0916673
$225$	−3.50761	−0.233841
$226$	−1.28681	−0.0855974
$227$	11.0071	0.730568	0.365284	−	0.930896i	$-0.380972\pi$
$227$	11.0071	0.730568	0.365284	+	0.930896i	$0.380972\pi$
$228$	0.585502	0.0387758
$229$	−3.88918	−0.257004	−0.128502	−	0.991709i	$-0.541017\pi$
$229$	−3.88918	−0.257004	−0.128502	+	0.991709i	$0.541017\pi$
$230$	18.1372	1.19593
$231$	4.39529	0.289189
$232$	5.18861	0.340649
$233$	−17.1979	−1.12667	−0.563335	−	0.826229i	$-0.690482\pi$
$233$	−17.1979	−1.12667	−0.563335	+	0.826229i	$0.690482\pi$
$234$	10.8794	0.711208
$235$	−1.79576	−0.117142
$236$	6.36757	0.414494
$237$	13.4945	0.876563
$238$	5.79176	0.375424
$239$	−19.0919	−1.23495	−0.617476	−	0.786590i	$-0.711845\pi$
$239$	−19.0919	−1.23495	−0.617476	+	0.786590i	$0.711845\pi$
$240$	−2.20646	−0.142426
$241$	15.0530	0.969651	0.484825	−	0.874611i	$-0.338883\pi$
$241$	15.0530	0.969651	0.484825	+	0.874611i	$0.338883\pi$
$242$	−2.81688	−0.181076
$243$	16.0635	1.03047
$244$	−2.54417	−0.162874
$245$	13.1018	0.837045
$246$	−6.38458	−0.407066
$247$	3.27435	0.208342
$248$	8.85301	0.562167
$249$	1.34886	0.0854808
$250$	−8.82206	−0.557956
$251$	−12.2476	−0.773063	−0.386532	−	0.922276i	$-0.626327\pi$
$251$	−12.2476	−0.773063	−0.386532	+	0.922276i	$0.626327\pi$
$252$	3.09673	0.195076
$253$	−26.3344	−1.65563
$254$	14.5218	0.911176
$255$	−9.31468	−0.583308
$256$	1.00000	0.0625000
$257$	29.8421	1.86150	0.930750	−	0.365656i	$-0.119155\pi$
$257$	29.8421	1.86150	0.930750	+	0.365656i	$0.119155\pi$
$258$	8.56420	0.533184
$259$	1.57149	0.0976479
$260$	−12.3394	−0.765254
$261$	11.7116	0.724929
$262$	18.3605	1.13432
$263$	22.9505	1.41519	0.707594	−	0.706620i	$-0.249781\pi$
$263$	22.9505	1.41519	0.707594	+	0.706620i	$0.249781\pi$
$264$	3.20368	0.197173
$265$	9.55962	0.587243
$266$	0.932016	0.0571456
$267$	−2.71635	−0.166238
$268$	−6.70824	−0.409771
$269$	11.0531	0.673921	0.336960	−	0.941519i	$-0.390601\pi$
$269$	11.0531	0.673921	0.336960	+	0.941519i	$0.390601\pi$
$270$	−11.5997	−0.705938
$271$	28.7940	1.74911	0.874557	−	0.484924i	$-0.161153\pi$
$271$	28.7940	1.74911	0.874557	+	0.484924i	$0.161153\pi$
$272$	4.22155	0.255969
$273$	−5.69930	−0.344938
$274$	−7.39878	−0.446977
$275$	−5.77632	−0.348325
$276$	6.10607	0.367542
$277$	−20.3399	−1.22210	−0.611052	−	0.791590i	$-0.709253\pi$
$277$	−20.3399	−1.22210	−0.611052	+	0.791590i	$0.709253\pi$
$278$	−22.3562	−1.34084
$279$	19.9828	1.19634
$280$	−3.51230	−0.209900
$281$	30.4326	1.81546	0.907729	−	0.419557i	$-0.137814\pi$
$281$	30.4326	1.81546	0.907729	+	0.419557i	$0.137814\pi$
$282$	−0.604560	−0.0360010
$283$	13.6464	0.811192	0.405596	−	0.914053i	$-0.367064\pi$
$283$	13.6464	0.811192	0.405596	+	0.914053i	$0.367064\pi$
$284$	−7.17723	−0.425890
$285$	−1.49893	−0.0887889
$286$	17.9162	1.05940
$287$	−10.1631	−0.599910
$288$	2.25717	0.133005
$289$	0.821475	0.0483220
$290$	−13.2832	−0.780018
$291$	−0.827001	−0.0484797
$292$	2.02523	0.118518
$293$	33.2919	1.94493	0.972466	−	0.233045i	$-0.0748690\pi$
$293$	33.2919	1.94493	0.972466	+	0.233045i	$0.0748690\pi$
$294$	4.41085	0.257246
$295$	−16.3015	−0.949108
$296$	1.14545	0.0665776
$297$	16.8423	0.977288
$298$	−16.4558	−0.953256
$299$	34.1474	1.97480
$300$	1.33934	0.0773267
$301$	13.6327	0.785776
$302$	−23.9475	−1.37802
$303$	−14.3456	−0.824132
$304$	0.679336	0.0389626
$305$	6.51326	0.372948
$306$	9.52877	0.544723
$307$	−30.0607	−1.71566	−0.857828	−	0.513937i	$-0.828186\pi$
$307$	−30.0607	−1.71566	−0.857828	+	0.513937i	$0.828186\pi$
$308$	5.09969	0.290582
$309$	−9.71438	−0.552632
$310$	−22.6644	−1.28725
$311$	−15.6599	−0.887990	−0.443995	−	0.896029i	$-0.646439\pi$
$311$	−15.6599	−0.887990	−0.443995	+	0.896029i	$0.646439\pi$
$312$	−4.15416	−0.235183
$313$	−7.98274	−0.451211	−0.225605	−	0.974219i	$-0.572436\pi$
$313$	−7.98274	−0.451211	−0.225605	+	0.974219i	$0.572436\pi$
$314$	14.0678	0.793890
$315$	−7.92786	−0.446685
$316$	15.6572	0.880785
$317$	−18.6256	−1.04612	−0.523058	−	0.852297i	$-0.675209\pi$
$317$	−18.6256	−1.04612	−0.523058	+	0.852297i	$0.675209\pi$
$318$	3.21834	0.180475
$319$	19.2866	1.07984
$320$	−2.56007	−0.143113
$321$	0.317217	0.0177053
$322$	9.71979	0.541662
$323$	2.86785	0.159572
$324$	2.86636	0.159242
$325$	7.49007	0.415474
$326$	−8.17557	−0.452803
$327$	−2.32888	−0.128787
$328$	−7.40779	−0.409027
$329$	−0.962352	−0.0530562
$330$	−8.20165	−0.451486
$331$	1.09940	0.0604285	0.0302143	−	0.999543i	$-0.490381\pi$
$331$	1.09940	0.0604285	0.0302143	+	0.999543i	$0.490381\pi$
$332$	1.56504	0.0858925
$333$	2.58547	0.141683
$334$	−16.4335	−0.899199
$335$	17.1736	0.938294
$336$	−1.18245	−0.0645078
$337$	−10.8781	−0.592566	−0.296283	−	0.955100i	$-0.595747\pi$
$337$	−10.8781	−0.592566	−0.296283	+	0.955100i	$0.595747\pi$
$338$	−10.2316	−0.556526
$339$	1.10907	0.0602364
$340$	−10.8075	−0.586118
$341$	32.9076	1.78205
$342$	1.53338	0.0829157
$343$	16.6250	0.897663
$344$	9.93673	0.535752
$345$	−15.6320	−0.841598
$346$	1.75514	0.0943568
$347$	31.1819	1.67393	0.836967	−	0.547254i	$-0.184327\pi$
$347$	31.1819	1.67393	0.836967	+	0.547254i	$0.184327\pi$
$348$	−4.47192	−0.239720
$349$	−24.5891	−1.31623	−0.658113	−	0.752919i	$-0.728645\pi$
$349$	−24.5891	−1.31623	−0.658113	+	0.752919i	$0.728645\pi$
$350$	2.13199	0.113960
$351$	−21.8391	−1.16569
$352$	3.71711	0.198122
$353$	20.0848	1.06900	0.534502	−	0.845167i	$-0.320499\pi$
$353$	20.0848	1.06900	0.534502	+	0.845167i	$0.320499\pi$
$354$	−5.48804	−0.291686
$355$	18.3742	0.975203
$356$	−3.15167	−0.167038
$357$	−4.99176	−0.264192
$358$	−25.8325	−1.36529
$359$	−4.84868	−0.255904	−0.127952	−	0.991780i	$-0.540840\pi$
$359$	−4.84868	−0.255904	−0.127952	+	0.991780i	$0.540840\pi$
$360$	−5.77853	−0.304555
$361$	−18.5385	−0.975711
$362$	−5.13134	−0.269697
$363$	2.42780	0.127426
$364$	−6.61269	−0.346599
$365$	−5.18474	−0.271382
$366$	2.19275	0.114617
$367$	15.7447	0.821868	0.410934	−	0.911665i	$-0.365203\pi$
$367$	15.7447	0.821868	0.410934	+	0.911665i	$0.365203\pi$
$368$	7.08465	0.369313
$369$	−16.7207	−0.870443
$370$	−2.93243	−0.152450
$371$	5.12302	0.265974
$372$	−7.63018	−0.395606
$373$	−26.4584	−1.36996	−0.684982	−	0.728560i	$-0.740190\pi$
$373$	−26.4584	−1.36996	−0.684982	+	0.728560i	$0.740190\pi$
$374$	15.6919	0.811411
$375$	7.60350	0.392643
$376$	−0.701448	−0.0361744
$377$	−25.0087	−1.28801
$378$	−6.21634	−0.319734
$379$	26.3062	1.35126	0.675629	−	0.737241i	$-0.263872\pi$
$379$	26.3062	1.35126	0.675629	+	0.737241i	$0.263872\pi$
$380$	−1.73915	−0.0892166
$381$	−12.5159	−0.641210
$382$	−21.3470	−1.09221
$383$	17.5661	0.897584	0.448792	−	0.893636i	$-0.351854\pi$
$383$	17.5661	0.897584	0.448792	+	0.893636i	$0.351854\pi$
$384$	−0.861874	−0.0439823
$385$	−13.0556	−0.665374
$386$	17.4988	0.890664
$387$	22.4289	1.14013
$388$	−0.959539	−0.0487132
$389$	32.9091	1.66856	0.834278	−	0.551344i	$-0.185885\pi$
$389$	32.9091	1.66856	0.834278	+	0.551344i	$0.185885\pi$
$390$	10.6350	0.538522
$391$	29.9082	1.51252
$392$	5.11775	0.258485
$393$	−15.8245	−0.798239
$394$	−1.39390	−0.0702234
$395$	−40.0836	−2.01682
$396$	8.39016	0.421621
$397$	−10.7565	−0.539854	−0.269927	−	0.962881i	$-0.587000\pi$
$397$	−10.7565	−0.539854	−0.269927	+	0.962881i	$0.587000\pi$
$398$	−5.98956	−0.300229
$399$	−0.803280	−0.0402143
$400$	1.55398	0.0776992
$401$	29.8151	1.48889	0.744447	−	0.667681i	$-0.232713\pi$
$401$	29.8151	1.48889	0.744447	+	0.667681i	$0.232713\pi$
$402$	5.78166	0.288363
$403$	−42.6708	−2.12558
$404$	−16.6446	−0.828102
$405$	−7.33808	−0.364632
$406$	−7.11852	−0.353286
$407$	4.25774	0.211048
$408$	−3.63844	−0.180130
$409$	−3.66982	−0.181461	−0.0907306	−	0.995875i	$-0.528920\pi$
$409$	−3.66982	−0.181461	−0.0907306	+	0.995875i	$0.528920\pi$
$410$	18.9645	0.936590
$411$	6.37681	0.314545
$412$	−11.2712	−0.555294
$413$	−8.73600	−0.429870
$414$	15.9913	0.785928
$415$	−4.00661	−0.196677
$416$	−4.81992	−0.236316
$417$	19.2682	0.943569
$418$	2.52517	0.123510
$419$	−15.0877	−0.737081	−0.368540	−	0.929612i	$-0.620142\pi$
$419$	−15.0877	−0.737081	−0.368540	+	0.929612i	$0.620142\pi$
$420$	3.02716	0.147710
$421$	−4.75992	−0.231984	−0.115992	−	0.993250i	$-0.537005\pi$
$421$	−4.75992	−0.231984	−0.115992	+	0.993250i	$0.537005\pi$
$422$	10.2058	0.496811
$423$	−1.58329	−0.0769822
$424$	3.73412	0.181345
$425$	6.56022	0.318217
$426$	6.18586	0.299706
$427$	3.49047	0.168916
$428$	0.368055	0.0177906
$429$	−15.4415	−0.745521
$430$	−25.4388	−1.22677
$431$	−22.8015	−1.09831	−0.549156	−	0.835720i	$-0.685051\pi$
$431$	−22.8015	−1.09831	−0.549156	+	0.835720i	$0.685051\pi$
$432$	−4.53102	−0.217999
$433$	28.1273	1.35171	0.675856	−	0.737034i	$-0.263774\pi$
$433$	28.1273	1.35171	0.675856	+	0.737034i	$0.263774\pi$
$434$	−12.1459	−0.583022
$435$	11.4485	0.548912
$436$	−2.70211	−0.129408
$437$	4.81286	0.230230
$438$	−1.74549	−0.0834028
$439$	−17.0850	−0.815423	−0.407711	−	0.913111i	$-0.633673\pi$
$439$	−17.0850	−0.815423	−0.407711	+	0.913111i	$0.633673\pi$
$440$	−9.51607	−0.453661
$441$	11.5517	0.550079
$442$	−20.3475	−0.967833
$443$	21.4448	1.01887	0.509437	−	0.860508i	$-0.329854\pi$
$443$	21.4448	1.01887	0.509437	+	0.860508i	$0.329854\pi$
$444$	−0.987229	−0.0468518
$445$	8.06852	0.382485
$446$	−25.1269	−1.18979
$447$	14.1828	0.670822
$448$	−1.37195	−0.0648186
$449$	−10.9327	−0.515946	−0.257973	−	0.966152i	$-0.583055\pi$
$449$	−10.9327	−0.515946	−0.257973	+	0.966152i	$0.583055\pi$
$450$	3.50761	0.165350
$451$	−27.5355	−1.29660
$452$	1.28681	0.0605265
$453$	20.6397	0.969739
$454$	−11.0071	−0.516590
$455$	16.9290	0.793643
$456$	−0.585502	−0.0274186
$457$	8.99175	0.420616	0.210308	−	0.977635i	$-0.432553\pi$
$457$	8.99175	0.420616	0.210308	+	0.977635i	$0.432553\pi$
$458$	3.88918	0.181730
$459$	−19.1279	−0.892815
$460$	−18.1372	−0.845652
$461$	29.2777	1.36360	0.681799	−	0.731540i	$-0.261198\pi$
$461$	29.2777	1.36360	0.681799	+	0.731540i	$0.261198\pi$
$462$	−4.39529	−0.204487
$463$	−10.5841	−0.491885	−0.245943	−	0.969284i	$-0.579098\pi$
$463$	−10.5841	−0.491885	−0.245943	+	0.969284i	$0.579098\pi$
$464$	−5.18861	−0.240875
$465$	19.5338	0.905859
$466$	17.1979	0.796675
$467$	9.53049	0.441018	0.220509	−	0.975385i	$-0.429228\pi$
$467$	9.53049	0.441018	0.220509	+	0.975385i	$0.429228\pi$
$468$	−10.8794	−0.502900
$469$	9.20338	0.424972
$470$	1.79576	0.0828322
$471$	−12.1246	−0.558674
$472$	−6.36757	−0.293091
$473$	36.9359	1.69831
$474$	−13.4945	−0.619823
$475$	1.05568	0.0484378
$476$	−5.79176	−0.265465
$477$	8.42855	0.385917
$478$	19.0919	0.873243
$479$	13.3680	0.610799	0.305400	−	0.952224i	$-0.401210\pi$
$479$	13.3680	0.610799	0.305400	+	0.952224i	$0.401210\pi$
$480$	2.20646	0.100711
$481$	−5.52095	−0.251734
$482$	−15.0530	−0.685647
$483$	−8.37723	−0.381177
$484$	2.81688	0.128040
$485$	2.45649	0.111544
$486$	−16.0635	−0.728655
$487$	−0.0444139	−0.00201259	−0.00100629	−	0.999999i	$-0.500320\pi$
$487$	−0.0444139	−0.00201259	−0.00100629	+	0.999999i	$0.500320\pi$
$488$	2.54417	0.115169
$489$	7.04631	0.318645
$490$	−13.1018	−0.591880
$491$	−32.7096	−1.47616	−0.738081	−	0.674712i	$-0.764268\pi$
$491$	−32.7096	−1.47616	−0.738081	+	0.674712i	$0.764268\pi$
$492$	6.38458	0.287839
$493$	−21.9040	−0.986505
$494$	−3.27435	−0.147320
$495$	−21.4794	−0.965429
$496$	−8.85301	−0.397512
$497$	9.84680	0.441689
$498$	−1.34886	−0.0604440
$499$	16.4175	0.734949	0.367474	−	0.930034i	$-0.380223\pi$
$499$	16.4175	0.734949	0.367474	+	0.930034i	$0.380223\pi$
$500$	8.82206	0.394535
$501$	14.1636	0.632782
$502$	12.2476	0.546638
$503$	−37.7581	−1.68355	−0.841776	−	0.539827i	$-0.818490\pi$
$503$	−37.7581	−1.68355	−0.841776	+	0.539827i	$0.818490\pi$
$504$	−3.09673	−0.137939
$505$	42.6115	1.89619
$506$	26.3344	1.17071
$507$	8.81835	0.391637
$508$	−14.5218	−0.644299
$509$	−12.1800	−0.539871	−0.269935	−	0.962878i	$-0.587002\pi$
$509$	−12.1800	−0.539871	−0.269935	+	0.962878i	$0.587002\pi$
$510$	9.31468	0.412461
$511$	−2.77852	−0.122914
$512$	−1.00000	−0.0441942
$513$	−3.07809	−0.135901
$514$	−29.8421	−1.31628
$515$	28.8552	1.27151
$516$	−8.56420	−0.377018
$517$	−2.60736	−0.114671
$518$	−1.57149	−0.0690475
$519$	−1.51271	−0.0664004
$520$	12.3394	0.541116
$521$	−11.2777	−0.494087	−0.247043	−	0.969004i	$-0.579459\pi$
$521$	−11.2777	−0.494087	−0.247043	+	0.969004i	$0.579459\pi$
$522$	−11.7116	−0.512602
$523$	−2.82030	−0.123323	−0.0616616	−	0.998097i	$-0.519640\pi$
$523$	−2.82030	−0.123323	−0.0616616	+	0.998097i	$0.519640\pi$
$524$	−18.3605	−0.802084
$525$	−1.83750	−0.0801953
$526$	−22.9505	−1.00069
$527$	−37.3734	−1.62801
$528$	−3.20368	−0.139422
$529$	27.1922	1.18227
$530$	−9.55962	−0.415243
$531$	−14.3727	−0.623723
$532$	−0.932016	−0.0404080
$533$	35.7049	1.54655
$534$	2.71635	0.117548
$535$	−0.942249	−0.0407370
$536$	6.70824	0.289752
$537$	22.2644	0.960778
$538$	−11.0531	−0.476534
$539$	19.0232	0.819388
$540$	11.5997	0.499174
$541$	16.6001	0.713694	0.356847	−	0.934163i	$-0.383852\pi$
$541$	16.6001	0.713694	0.356847	+	0.934163i	$0.383852\pi$
$542$	−28.7940	−1.23681
$543$	4.42257	0.189791
$544$	−4.22155	−0.180997
$545$	6.91761	0.296318
$546$	5.69930	0.243908
$547$	28.1103	1.20191	0.600954	−	0.799283i	$-0.294787\pi$
$547$	28.1103	1.20191	0.600954	+	0.799283i	$0.294787\pi$
$548$	7.39878	0.316060
$549$	5.74263	0.245089
$550$	5.77632	0.246303
$551$	−3.52481	−0.150162
$552$	−6.10607	−0.259892
$553$	−21.4809	−0.913460
$554$	20.3399	0.864158
$555$	2.52738	0.107281
$556$	22.3562	0.948115
$557$	−24.6791	−1.04569	−0.522844	−	0.852429i	$-0.675129\pi$
$557$	−24.6791	−1.04569	−0.522844	+	0.852429i	$0.675129\pi$
$558$	−19.9828	−0.845939
$559$	−47.8942	−2.02571
$560$	3.51230	0.148422
$561$	−13.5245	−0.571004
$562$	−30.4326	−1.28372
$563$	−38.3025	−1.61426	−0.807129	−	0.590376i	$-0.798980\pi$
$563$	−38.3025	−1.61426	−0.807129	+	0.590376i	$0.798980\pi$
$564$	0.604560	0.0254565
$565$	−3.29433	−0.138594
$566$	−13.6464	−0.573599
$567$	−3.93250	−0.165149
$568$	7.17723	0.301150
$569$	20.2864	0.850450	0.425225	−	0.905088i	$-0.360195\pi$
$569$	20.2864	0.850450	0.425225	+	0.905088i	$0.360195\pi$
$570$	1.49893	0.0627832
$571$	22.4107	0.937860	0.468930	−	0.883235i	$-0.344640\pi$
$571$	22.4107	0.937860	0.468930	+	0.883235i	$0.344640\pi$
$572$	−17.9162	−0.749112
$573$	18.3984	0.768606
$574$	10.1631	0.424201
$575$	11.0094	0.459125
$576$	−2.25717	−0.0940489
$577$	35.8276	1.49152	0.745760	−	0.666214i	$-0.232086\pi$
$577$	35.8276	1.49152	0.745760	+	0.666214i	$0.232086\pi$
$578$	−0.821475	−0.0341688
$579$	−15.0817	−0.626775
$580$	13.2832	0.551556
$581$	−2.14715	−0.0890789
$582$	0.827001	0.0342803
$583$	13.8801	0.574855
$584$	−2.02523	−0.0838046
$585$	27.8521	1.15154
$586$	−33.2919	−1.37527
$587$	−15.5770	−0.642932	−0.321466	−	0.946921i	$-0.604176\pi$
$587$	−15.5770	−0.642932	−0.321466	+	0.946921i	$0.604176\pi$
$588$	−4.41085	−0.181901
$589$	−6.01417	−0.247810
$590$	16.3015	0.671121
$591$	1.20136	0.0494174
$592$	−1.14545	−0.0470775
$593$	−19.9351	−0.818637	−0.409318	−	0.912392i	$-0.634233\pi$
$593$	−19.9351	−0.818637	−0.409318	+	0.912392i	$0.634233\pi$
$594$	−16.8423	−0.691047
$595$	14.8273	0.607861
$596$	16.4558	0.674054
$597$	5.16224	0.211277
$598$	−34.1474	−1.39639
$599$	26.7597	1.09337	0.546687	−	0.837337i	$-0.315889\pi$
$599$	26.7597	1.09337	0.546687	+	0.837337i	$0.315889\pi$
$600$	−1.33934	−0.0546782
$601$	12.9656	0.528876	0.264438	−	0.964403i	$-0.414813\pi$
$601$	12.9656	0.528876	0.264438	+	0.964403i	$0.414813\pi$
$602$	−13.6327	−0.555627
$603$	15.1417	0.616616
$604$	23.9475	0.974410
$605$	−7.21143	−0.293186
$606$	14.3456	0.582749
$607$	31.1276	1.26343	0.631715	−	0.775201i	$-0.282351\pi$
$607$	31.1276	1.26343	0.631715	+	0.775201i	$0.282351\pi$
$608$	−0.679336	−0.0275507
$609$	6.13526	0.248613
$610$	−6.51326	−0.263714
$611$	3.38092	0.136777
$612$	−9.52877	−0.385178
$613$	−17.6590	−0.713242	−0.356621	−	0.934249i	$-0.616071\pi$
$613$	−17.6590	−0.713242	−0.356621	+	0.934249i	$0.616071\pi$
$614$	30.0607	1.21315
$615$	−16.3450	−0.659094
$616$	−5.09969	−0.205472
$617$	−14.8652	−0.598450	−0.299225	−	0.954183i	$-0.596728\pi$
$617$	−14.8652	−0.598450	−0.299225	+	0.954183i	$0.596728\pi$
$618$	9.71438	0.390770
$619$	21.9947	0.884040	0.442020	−	0.897005i	$-0.354262\pi$
$619$	21.9947	0.884040	0.442020	+	0.897005i	$0.354262\pi$
$620$	22.6644	0.910223
$621$	−32.1007	−1.28816
$622$	15.6599	0.627904
$623$	4.32394	0.173235
$624$	4.15416	0.166300
$625$	−30.3551	−1.21420
$626$	7.98274	0.319054
$627$	−2.17637	−0.0869160
$628$	−14.0678	−0.561365
$629$	−4.83555	−0.192806
$630$	7.92786	0.315854
$631$	0.269671	0.0107354	0.00536771	−	0.999986i	$-0.498291\pi$
$631$	0.269671	0.0107354	0.00536771	+	0.999986i	$0.498291\pi$
$632$	−15.6572	−0.622809
$633$	−8.79612	−0.349614
$634$	18.6256	0.739716
$635$	37.1768	1.47532
$636$	−3.21834	−0.127615
$637$	−24.6671	−0.977348
$638$	−19.2866	−0.763564
$639$	16.2002	0.640872
$640$	2.56007	0.101196
$641$	29.6258	1.17015	0.585074	−	0.810980i	$-0.301066\pi$
$641$	29.6258	1.17015	0.585074	+	0.810980i	$0.301066\pi$
$642$	−0.317217	−0.0125196
$643$	−32.4837	−1.28103	−0.640516	−	0.767944i	$-0.721280\pi$
$643$	−32.4837	−1.28103	−0.640516	+	0.767944i	$0.721280\pi$
$644$	−9.71979	−0.383013
$645$	21.9250	0.863296
$646$	−2.86785	−0.112834
$647$	44.0628	1.73229	0.866145	−	0.499793i	$-0.166591\pi$
$647$	44.0628	1.73229	0.866145	+	0.499793i	$0.166591\pi$
$648$	−2.86636	−0.112601
$649$	−23.6689	−0.929087
$650$	−7.49007	−0.293785
$651$	10.4682	0.410282
$652$	8.17557	0.320180
$653$	40.2835	1.57642	0.788208	−	0.615409i	$-0.211009\pi$
$653$	40.2835	1.57642	0.788208	+	0.615409i	$0.211009\pi$
$654$	2.32888	0.0910664
$655$	47.0044	1.83661
$656$	7.40779	0.289226
$657$	−4.57130	−0.178343
$658$	0.962352	0.0375164
$659$	23.7677	0.925858	0.462929	−	0.886395i	$-0.346798\pi$
$659$	23.7677	0.925858	0.462929	+	0.886395i	$0.346798\pi$
$660$	8.20165	0.319249
$661$	34.9769	1.36044	0.680221	−	0.733007i	$-0.261884\pi$
$661$	34.9769	1.36044	0.680221	+	0.733007i	$0.261884\pi$
$662$	−1.09940	−0.0427294
$663$	17.5370	0.681080
$664$	−1.56504	−0.0607352
$665$	2.38603	0.0925263
$666$	−2.58547	−0.100185
$667$	−36.7595	−1.42333
$668$	16.4335	0.635830
$669$	21.6562	0.837278
$670$	−17.1736	−0.663474
$671$	9.45694	0.365081
$672$	1.18245	0.0456139
$673$	−41.4974	−1.59961	−0.799803	−	0.600263i	$-0.795063\pi$
$673$	−41.4974	−1.59961	−0.799803	+	0.600263i	$0.795063\pi$
$674$	10.8781	0.419007
$675$	−7.04113	−0.271013
$676$	10.2316	0.393523
$677$	13.0669	0.502202	0.251101	−	0.967961i	$-0.419207\pi$
$677$	13.0669	0.502202	0.251101	+	0.967961i	$0.419207\pi$
$678$	−1.10907	−0.0425935
$679$	1.31644	0.0505203
$680$	10.8075	0.414448
$681$	9.48674	0.363533
$682$	−32.9076	−1.26010
$683$	−25.9046	−0.991213	−0.495607	−	0.868547i	$-0.665054\pi$
$683$	−25.9046	−0.991213	−0.495607	+	0.868547i	$0.665054\pi$
$684$	−1.53338	−0.0586303
$685$	−18.9414	−0.723715
$686$	−16.6250	−0.634744
$687$	−3.35198	−0.127886
$688$	−9.93673	−0.378834
$689$	−17.9981	−0.685674
$690$	15.6320	0.595100
$691$	−21.6967	−0.825380	−0.412690	−	0.910872i	$-0.635411\pi$
$691$	−21.6967	−0.825380	−0.412690	+	0.910872i	$0.635411\pi$
$692$	−1.75514	−0.0667203
$693$	−11.5109	−0.437262
$694$	−31.1819	−1.18365
$695$	−57.2336	−2.17099
$696$	4.47192	0.169508
$697$	31.2723	1.18452
$698$	24.5891	0.930712
$699$	−14.8224	−0.560634
$700$	−2.13199	−0.0805816
$701$	−35.7234	−1.34925	−0.674627	−	0.738159i	$-0.735696\pi$
$701$	−35.7234	−1.34925	−0.674627	+	0.738159i	$0.735696\pi$
$702$	21.8391	0.824265
$703$	−0.778142	−0.0293482
$704$	−3.71711	−0.140094
$705$	−1.54772	−0.0582904
$706$	−20.0848	−0.755900
$707$	22.8356	0.858823
$708$	5.48804	0.206253
$709$	6.10051	0.229110	0.114555	−	0.993417i	$-0.463456\pi$
$709$	6.10051	0.229110	0.114555	+	0.993417i	$0.463456\pi$
$710$	−18.3742	−0.689573
$711$	−35.3410	−1.32539
$712$	3.15167	0.118114
$713$	−62.7204	−2.34890
$714$	4.99176	0.186812
$715$	45.8667	1.71532
$716$	25.8325	0.965406
$717$	−16.4548	−0.614516
$718$	4.84868	0.180951
$719$	15.4881	0.577609	0.288805	−	0.957388i	$-0.406742\pi$
$719$	15.4881	0.577609	0.288805	+	0.957388i	$0.406742\pi$
$720$	5.77853	0.215353
$721$	15.4636	0.575894
$722$	18.5385	0.689932
$723$	12.9738	0.482501
$724$	5.13134	0.190705
$725$	−8.06301	−0.299453
$726$	−2.42780	−0.0901040
$727$	−38.1309	−1.41420	−0.707098	−	0.707115i	$-0.749996\pi$
$727$	−38.1309	−1.41420	−0.707098	+	0.707115i	$0.749996\pi$
$728$	6.61269	0.245083
$729$	5.24564	0.194283
$730$	5.18474	0.191896
$731$	−41.9484	−1.55152
$732$	−2.19275	−0.0810464
$733$	−28.6733	−1.05907	−0.529536	−	0.848288i	$-0.677634\pi$
$733$	−28.6733	−1.05907	−0.529536	+	0.848288i	$0.677634\pi$
$734$	−15.7447	−0.581148
$735$	11.2921	0.416516
$736$	−7.08465	−0.261143
$737$	24.9352	0.918502
$738$	16.7207	0.615496
$739$	5.63381	0.207243	0.103622	−	0.994617i	$-0.466957\pi$
$739$	5.63381	0.207243	0.103622	+	0.994617i	$0.466957\pi$
$740$	2.93243	0.107798
$741$	2.82207	0.103671
$742$	−5.12302	−0.188072
$743$	45.3908	1.66523	0.832614	−	0.553854i	$-0.186844\pi$
$743$	45.3908	1.66523	0.832614	+	0.553854i	$0.186844\pi$
$744$	7.63018	0.279736
$745$	−42.1280	−1.54345
$746$	26.4584	0.968710
$747$	−3.53256	−0.129250
$748$	−15.6919	−0.573754
$749$	−0.504954	−0.0184506
$750$	−7.60350	−0.277641
$751$	40.5174	1.47850	0.739250	−	0.673432i	$-0.235180\pi$
$751$	40.5174	1.47850	0.739250	+	0.673432i	$0.235180\pi$
$752$	0.701448	0.0255792
$753$	−10.5559	−0.384679
$754$	25.0087	0.910762
$755$	−61.3074	−2.23121
$756$	6.21634	0.226086
$757$	−17.1920	−0.624852	−0.312426	−	0.949942i	$-0.601142\pi$
$757$	−17.1920	−0.624852	−0.312426	+	0.949942i	$0.601142\pi$
$758$	−26.3062	−0.955484
$759$	−22.6969	−0.823846
$760$	1.73915	0.0630857
$761$	−1.34850	−0.0488830	−0.0244415	−	0.999701i	$-0.507781\pi$
$761$	−1.34850	−0.0488830	−0.0244415	+	0.999701i	$0.507781\pi$
$762$	12.5159	0.453404
$763$	3.70716	0.134208
$764$	21.3470	0.772308
$765$	24.3944	0.881980
$766$	−17.5661	−0.634687
$767$	30.6912	1.10819
$768$	0.861874	0.0311002
$769$	−14.1012	−0.508504	−0.254252	−	0.967138i	$-0.581829\pi$
$769$	−14.1012	−0.508504	−0.254252	+	0.967138i	$0.581829\pi$
$770$	13.0556	0.470491
$771$	25.7201	0.926288
$772$	−17.4988	−0.629795
$773$	8.63777	0.310679	0.155340	−	0.987861i	$-0.450353\pi$
$773$	8.63777	0.310679	0.155340	+	0.987861i	$0.450353\pi$
$774$	−22.4289	−0.806191
$775$	−13.7574	−0.494181
$776$	0.959539	0.0344454
$777$	1.35443	0.0485899
$778$	−32.9091	−1.17985
$779$	5.03238	0.180304
$780$	−10.6350	−0.380793
$781$	26.6785	0.954632
$782$	−29.9082	−1.06951
$783$	23.5097	0.840168
$784$	−5.11775	−0.182777
$785$	36.0145	1.28541
$786$	15.8245	0.564440
$787$	−24.5653	−0.875658	−0.437829	−	0.899058i	$-0.644252\pi$
$787$	−24.5653	−0.875658	−0.437829	+	0.899058i	$0.644252\pi$
$788$	1.39390	0.0496555
$789$	19.7804	0.704201
$790$	40.0836	1.42611
$791$	−1.76544	−0.0627719
$792$	−8.39016	−0.298131
$793$	−12.2627	−0.435460
$794$	10.7565	0.381735
$795$	8.23918	0.292214
$796$	5.98956	0.212294
$797$	13.3368	0.472415	0.236208	−	0.971703i	$-0.424095\pi$
$797$	13.3368	0.472415	0.236208	+	0.971703i	$0.424095\pi$
$798$	0.803280	0.0284358
$799$	2.96120	0.104760
$800$	−1.55398	−0.0549416
$801$	7.11388	0.251357
$802$	−29.8151	−1.05281
$803$	−7.52799	−0.265657
$804$	−5.78166	−0.203903
$805$	24.8834	0.877024
$806$	42.6708	1.50301
$807$	9.52639	0.335345
$808$	16.6446	0.585557
$809$	−37.5862	−1.32146	−0.660730	−	0.750624i	$-0.729753\pi$
$809$	−37.5862	−1.32146	−0.660730	+	0.750624i	$0.729753\pi$
$810$	7.33808	0.257834
$811$	13.4985	0.473998	0.236999	−	0.971510i	$-0.423836\pi$
$811$	13.4985	0.473998	0.236999	+	0.971510i	$0.423836\pi$
$812$	7.11852	0.249811
$813$	24.8168	0.870364
$814$	−4.25774	−0.149234
$815$	−20.9301	−0.733148
$816$	3.63844	0.127371
$817$	−6.75038	−0.236166
$818$	3.66982	0.128312
$819$	14.9260	0.521556
$820$	−18.9645	−0.662269
$821$	9.67558	0.337680	0.168840	−	0.985643i	$-0.445998\pi$
$821$	9.67558	0.337680	0.168840	+	0.985643i	$0.445998\pi$
$822$	−6.37681	−0.222417
$823$	6.45474	0.224998	0.112499	−	0.993652i	$-0.464114\pi$
$823$	6.45474	0.224998	0.112499	+	0.993652i	$0.464114\pi$
$824$	11.2712	0.392652
$825$	−4.97846	−0.173328
$826$	8.73600	0.303964
$827$	−5.63035	−0.195786	−0.0978932	−	0.995197i	$-0.531210\pi$
$827$	−5.63035	−0.195786	−0.0978932	+	0.995197i	$0.531210\pi$
$828$	−15.9913	−0.555735
$829$	−28.3352	−0.984121	−0.492061	−	0.870561i	$-0.663756\pi$
$829$	−28.3352	−0.984121	−0.492061	+	0.870561i	$0.663756\pi$
$830$	4.00661	0.139072
$831$	−17.5304	−0.608123
$832$	4.81992	0.167101
$833$	−21.6048	−0.748563
$834$	−19.2682	−0.667204
$835$	−42.0709	−1.45592
$836$	−2.52517	−0.0873347
$837$	40.1132	1.38651
$838$	15.0877	0.521195
$839$	−42.8480	−1.47928	−0.739639	−	0.673004i	$-0.765004\pi$
$839$	−42.8480	−1.47928	−0.739639	+	0.673004i	$0.765004\pi$
$840$	−3.02716	−0.104447
$841$	−2.07834	−0.0716671
$842$	4.75992	0.164038
$843$	26.2291	0.903377
$844$	−10.2058	−0.351298
$845$	−26.1937	−0.901090
$846$	1.58329	0.0544346
$847$	−3.86463	−0.132790
$848$	−3.73412	−0.128230
$849$	11.7614	0.403651
$850$	−6.56022	−0.225014
$851$	−8.11507	−0.278181
$852$	−6.18586	−0.211924
$853$	−3.75087	−0.128427	−0.0642136	−	0.997936i	$-0.520454\pi$
$853$	−3.75087	−0.128427	−0.0642136	+	0.997936i	$0.520454\pi$
$854$	−3.49047	−0.119442
$855$	3.92557	0.134252
$856$	−0.368055	−0.0125799
$857$	12.8434	0.438721	0.219361	−	0.975644i	$-0.429603\pi$
$857$	12.8434	0.438721	0.219361	+	0.975644i	$0.429603\pi$
$858$	15.4415	0.527163
$859$	−54.9330	−1.87429	−0.937144	−	0.348943i	$-0.886541\pi$
$859$	−54.9330	−1.87429	−0.937144	+	0.348943i	$0.886541\pi$
$860$	25.4388	0.867455
$861$	−8.75933	−0.298517
$862$	22.8015	0.776623
$863$	−25.5190	−0.868678	−0.434339	−	0.900749i	$-0.643018\pi$
$863$	−25.5190	−0.868678	−0.434339	+	0.900749i	$0.643018\pi$
$864$	4.53102	0.154148
$865$	4.49328	0.152776
$866$	−28.1273	−0.955804
$867$	0.708007	0.0240452
$868$	12.1459	0.412259
$869$	−58.1994	−1.97428
$870$	−11.4485	−0.388139
$871$	−32.3332	−1.09557
$872$	2.70211	0.0915050
$873$	2.16585	0.0733028
$874$	−4.81286	−0.162797
$875$	−12.1034	−0.409171
$876$	1.74549	0.0589747
$877$	26.6448	0.899731	0.449866	−	0.893096i	$-0.351472\pi$
$877$	26.6448	0.899731	0.449866	+	0.893096i	$0.351472\pi$
$878$	17.0850	0.576591
$879$	28.6934	0.967804
$880$	9.51607	0.320787
$881$	−43.0716	−1.45112	−0.725559	−	0.688160i	$-0.758419\pi$
$881$	−43.0716	−1.45112	−0.725559	+	0.688160i	$0.758419\pi$
$882$	−11.5517	−0.388964
$883$	−47.7327	−1.60633	−0.803167	−	0.595754i	$-0.796853\pi$
$883$	−47.7327	−1.60633	−0.803167	+	0.595754i	$0.796853\pi$
$884$	20.3475	0.684361
$885$	−14.0498	−0.472279
$886$	−21.4448	−0.720453
$887$	−4.45013	−0.149421	−0.0747104	−	0.997205i	$-0.523803\pi$
$887$	−4.45013	−0.149421	−0.0747104	+	0.997205i	$0.523803\pi$
$888$	0.987229	0.0331292
$889$	19.9231	0.668201
$890$	−8.06852	−0.270458
$891$	−10.6545	−0.356941
$892$	25.1269	0.841311
$893$	0.476519	0.0159461
$894$	−14.1828	−0.474343
$895$	−66.1332	−2.21059
$896$	1.37195	0.0458337
$897$	29.4308	0.982664
$898$	10.9327	0.364829
$899$	45.9348	1.53201
$900$	−3.50761	−0.116920
$901$	−15.7638	−0.525167
$902$	27.5355	0.916834
$903$	11.7497	0.391004
$904$	−1.28681	−0.0427987
$905$	−13.1366	−0.436676
$906$	−20.6397	−0.685709
$907$	46.9376	1.55854	0.779269	−	0.626689i	$-0.215590\pi$
$907$	46.9376	1.55854	0.779269	+	0.626689i	$0.215590\pi$
$908$	11.0071	0.365284
$909$	37.5699	1.24611
$910$	−16.9290	−0.561190
$911$	−52.3114	−1.73315	−0.866577	−	0.499044i	$-0.833685\pi$
$911$	−52.3114	−1.73315	−0.866577	+	0.499044i	$0.833685\pi$
$912$	0.585502	0.0193879
$913$	−5.81741	−0.192528
$914$	−8.99175	−0.297421
$915$	5.61361	0.185580
$916$	−3.88918	−0.128502
$917$	25.1898	0.831839
$918$	19.1279	0.631315
$919$	14.9032	0.491611	0.245806	−	0.969319i	$-0.420948\pi$
$919$	14.9032	0.491611	0.245806	+	0.969319i	$0.420948\pi$
$920$	18.1372	0.597967
$921$	−25.9085	−0.853715
$922$	−29.2777	−0.964209
$923$	−34.5936	−1.13866
$924$	4.39529	0.144594
$925$	−1.78000	−0.0585261
$926$	10.5841	0.347816
$927$	25.4411	0.835596
$928$	5.18861	0.170324
$929$	3.17037	0.104016	0.0520082	−	0.998647i	$-0.483438\pi$
$929$	3.17037	0.104016	0.0520082	+	0.998647i	$0.483438\pi$
$930$	−19.5338	−0.640539
$931$	−3.47667	−0.113943
$932$	−17.1979	−0.563335
$933$	−13.4968	−0.441866
$934$	−9.53049	−0.311847
$935$	40.1726	1.31378
$936$	10.8794	0.355604
$937$	19.1601	0.625932	0.312966	−	0.949764i	$-0.398677\pi$
$937$	19.1601	0.625932	0.312966	+	0.949764i	$0.398677\pi$
$938$	−9.20338	−0.300501
$939$	−6.88011	−0.224524
$940$	−1.79576	−0.0585712
$941$	−4.13124	−0.134675	−0.0673373	−	0.997730i	$-0.521450\pi$
$941$	−4.13124	−0.134675	−0.0673373	+	0.997730i	$0.521450\pi$
$942$	12.1246	0.395042
$943$	52.4816	1.70904
$944$	6.36757	0.207247
$945$	−15.9143	−0.517692
$946$	−36.9359	−1.20089
$947$	54.0990	1.75798	0.878990	−	0.476840i	$-0.158218\pi$
$947$	54.0990	1.75798	0.878990	+	0.476840i	$0.158218\pi$
$948$	13.4945	0.438281
$949$	9.76144	0.316870
$950$	−1.05568	−0.0342507
$951$	−16.0529	−0.520550
$952$	5.79176	0.187712
$953$	25.2515	0.817977	0.408989	−	0.912539i	$-0.365882\pi$
$953$	25.2515	0.817977	0.408989	+	0.912539i	$0.365882\pi$
$954$	−8.42855	−0.272884
$955$	−54.6500	−1.76843
$956$	−19.0919	−0.617476
$957$	16.6226	0.537333
$958$	−13.3680	−0.431900
$959$	−10.1508	−0.327785
$960$	−2.20646	−0.0712132
$961$	47.3758	1.52825
$962$	5.52095	0.178003
$963$	−0.830765	−0.0267710
$964$	15.0530	0.484825
$965$	44.7982	1.44210
$966$	8.37723	0.269533
$967$	52.5703	1.69055	0.845274	−	0.534334i	$-0.179437\pi$
$967$	52.5703	1.69055	0.845274	+	0.534334i	$0.179437\pi$
$968$	−2.81688	−0.0905381
$969$	2.47173	0.0794033
$970$	−2.45649	−0.0788732
$971$	17.8557	0.573018	0.286509	−	0.958078i	$-0.407505\pi$
$971$	17.8557	0.573018	0.286509	+	0.958078i	$0.407505\pi$
$972$	16.0635	0.515237
$973$	−30.6716	−0.983287
$974$	0.0444139	0.00142311
$975$	6.45550	0.206741
$976$	−2.54417	−0.0814368
$977$	−19.3812	−0.620059	−0.310030	−	0.950727i	$-0.600339\pi$
$977$	−19.3812	−0.620059	−0.310030	+	0.950727i	$0.600339\pi$
$978$	−7.04631	−0.225316
$979$	11.7151	0.374417
$980$	13.1018	0.418522
$981$	6.09913	0.194730
$982$	32.7096	1.04380
$983$	−36.2978	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$983$	−36.2978	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$984$	−6.38458	−0.203533
$985$	−3.56848	−0.113701
$986$	21.9040	0.697564
$987$	−0.829426	−0.0264009
$988$	3.27435	0.104171
$989$	−70.3982	−2.23853
$990$	21.4794	0.682661
$991$	21.5554	0.684729	0.342364	−	0.939567i	$-0.388772\pi$
$991$	21.5554	0.684729	0.342364	+	0.939567i	$0.388772\pi$
$992$	8.85301	0.281083
$993$	0.947544	0.0300694
$994$	−9.84680	−0.312322
$995$	−15.3337	−0.486112
$996$	1.34886	0.0427404
$997$	18.6960	0.592109	0.296055	−	0.955171i	$-0.404329\pi$
$997$	18.6960	0.592109	0.296055	+	0.955171i	$0.404329\pi$
$998$	−16.4175	−0.519687
$999$	5.19003	0.164205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.45	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.45	✓	77	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.861874 q^{3} +1.00000 q^{4} -2.56007 q^{5} -0.861874 q^{6} -1.37195 q^{7} -1.00000 q^{8} -2.25717 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.861874 q^{3} +1.00000 q^{4} -2.56007 q^{5} -0.861874 q^{6} -1.37195 q^{7} -1.00000 q^{8} -2.25717 q^{9} +2.56007 q^{10} -3.71711 q^{11} +0.861874 q^{12} +4.81992 q^{13} +1.37195 q^{14} -2.20646 q^{15} +1.00000 q^{16} +4.22155 q^{17} +2.25717 q^{18} +0.679336 q^{19} -2.56007 q^{20} -1.18245 q^{21} +3.71711 q^{22} +7.08465 q^{23} -0.861874 q^{24} +1.55398 q^{25} -4.81992 q^{26} -4.53102 q^{27} -1.37195 q^{28} -5.18861 q^{29} +2.20646 q^{30} -8.85301 q^{31} -1.00000 q^{32} -3.20368 q^{33} -4.22155 q^{34} +3.51230 q^{35} -2.25717 q^{36} -1.14545 q^{37} -0.679336 q^{38} +4.15416 q^{39} +2.56007 q^{40} +7.40779 q^{41} +1.18245 q^{42} -9.93673 q^{43} -3.71711 q^{44} +5.77853 q^{45} -7.08465 q^{46} +0.701448 q^{47} +0.861874 q^{48} -5.11775 q^{49} -1.55398 q^{50} +3.63844 q^{51} +4.81992 q^{52} -3.73412 q^{53} +4.53102 q^{54} +9.51607 q^{55} +1.37195 q^{56} +0.585502 q^{57} +5.18861 q^{58} +6.36757 q^{59} -2.20646 q^{60} -2.54417 q^{61} +8.85301 q^{62} +3.09673 q^{63} +1.00000 q^{64} -12.3394 q^{65} +3.20368 q^{66} -6.70824 q^{67} +4.22155 q^{68} +6.10607 q^{69} -3.51230 q^{70} -7.17723 q^{71} +2.25717 q^{72} +2.02523 q^{73} +1.14545 q^{74} +1.33934 q^{75} +0.679336 q^{76} +5.09969 q^{77} -4.15416 q^{78} +15.6572 q^{79} -2.56007 q^{80} +2.86636 q^{81} -7.40779 q^{82} +1.56504 q^{83} -1.18245 q^{84} -10.8075 q^{85} +9.93673 q^{86} -4.47192 q^{87} +3.71711 q^{88} -3.15167 q^{89} -5.77853 q^{90} -6.61269 q^{91} +7.08465 q^{92} -7.63018 q^{93} -0.701448 q^{94} -1.73915 q^{95} -0.861874 q^{96} -0.959539 q^{97} +5.11775 q^{98} +8.39016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

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