LMFDB - Embedded newform 8042.2.a.b.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.27252 q^{3} +1.00000 q^{4} +1.18592 q^{5} +2.27252 q^{6} +3.11806 q^{7} -1.00000 q^{8} +2.16434 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.27252 q^{3} +1.00000 q^{4} +1.18592 q^{5} +2.27252 q^{6} +3.11806 q^{7} -1.00000 q^{8} +2.16434 q^{9} -1.18592 q^{10} -3.30500 q^{11} -2.27252 q^{12} -2.07784 q^{13} -3.11806 q^{14} -2.69503 q^{15} +1.00000 q^{16} -1.14302 q^{17} -2.16434 q^{18} -6.68822 q^{19} +1.18592 q^{20} -7.08585 q^{21} +3.30500 q^{22} +0.381268 q^{23} +2.27252 q^{24} -3.59359 q^{25} +2.07784 q^{26} +1.89905 q^{27} +3.11806 q^{28} +2.54569 q^{29} +2.69503 q^{30} +7.10661 q^{31} -1.00000 q^{32} +7.51068 q^{33} +1.14302 q^{34} +3.69777 q^{35} +2.16434 q^{36} +7.62559 q^{37} +6.68822 q^{38} +4.72193 q^{39} -1.18592 q^{40} +9.76257 q^{41} +7.08585 q^{42} -8.92914 q^{43} -3.30500 q^{44} +2.56674 q^{45} -0.381268 q^{46} +1.77454 q^{47} -2.27252 q^{48} +2.72228 q^{49} +3.59359 q^{50} +2.59754 q^{51} -2.07784 q^{52} +4.63002 q^{53} -1.89905 q^{54} -3.91947 q^{55} -3.11806 q^{56} +15.1991 q^{57} -2.54569 q^{58} -1.77952 q^{59} -2.69503 q^{60} -1.61034 q^{61} -7.10661 q^{62} +6.74854 q^{63} +1.00000 q^{64} -2.46415 q^{65} -7.51068 q^{66} +4.69517 q^{67} -1.14302 q^{68} -0.866440 q^{69} -3.69777 q^{70} +7.63111 q^{71} -2.16434 q^{72} +1.19400 q^{73} -7.62559 q^{74} +8.16650 q^{75} -6.68822 q^{76} -10.3052 q^{77} -4.72193 q^{78} +3.30984 q^{79} +1.18592 q^{80} -10.8086 q^{81} -9.76257 q^{82} -13.8355 q^{83} -7.08585 q^{84} -1.35553 q^{85} +8.92914 q^{86} -5.78512 q^{87} +3.30500 q^{88} -6.24467 q^{89} -2.56674 q^{90} -6.47882 q^{91} +0.381268 q^{92} -16.1499 q^{93} -1.77454 q^{94} -7.93170 q^{95} +2.27252 q^{96} -2.35973 q^{97} -2.72228 q^{98} -7.15316 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) $ 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.27252	−1.31204	−0.656020	−	0.754744i	$-0.727761\pi$
$3$	−2.27252	−1.31204	−0.656020	+	0.754744i	$0.727761\pi$
$4$	1.00000	0.500000
$5$	1.18592	0.530360	0.265180	−	0.964199i	$-0.414569\pi$
$5$	1.18592	0.530360	0.265180	+	0.964199i	$0.414569\pi$
$6$	2.27252	0.927752
$7$	3.11806	1.17851	0.589257	−	0.807945i	$-0.299420\pi$
$7$	3.11806	1.17851	0.589257	+	0.807945i	$0.299420\pi$
$8$	−1.00000	−0.353553
$9$	2.16434	0.721448
$10$	−1.18592	−0.375021
$11$	−3.30500	−0.996496	−0.498248	−	0.867035i	$-0.666023\pi$
$11$	−3.30500	−0.996496	−0.498248	+	0.867035i	$0.666023\pi$
$12$	−2.27252	−0.656020
$13$	−2.07784	−0.576289	−0.288145	−	0.957587i	$-0.593038\pi$
$13$	−2.07784	−0.576289	−0.288145	+	0.957587i	$0.593038\pi$
$14$	−3.11806	−0.833336
$15$	−2.69503	−0.695853
$16$	1.00000	0.250000
$17$	−1.14302	−0.277224	−0.138612	−	0.990347i	$-0.544264\pi$
$17$	−1.14302	−0.277224	−0.138612	+	0.990347i	$0.544264\pi$
$18$	−2.16434	−0.510140
$19$	−6.68822	−1.53438	−0.767191	−	0.641419i	$-0.778346\pi$
$19$	−6.68822	−1.53438	−0.767191	+	0.641419i	$0.778346\pi$
$20$	1.18592	0.265180
$21$	−7.08585	−1.54626
$22$	3.30500	0.704629
$23$	0.381268	0.0795000	0.0397500	−	0.999210i	$-0.487344\pi$
$23$	0.381268	0.0795000	0.0397500	+	0.999210i	$0.487344\pi$
$24$	2.27252	0.463876
$25$	−3.59359	−0.718718
$26$	2.07784	0.407498
$27$	1.89905	0.365472
$28$	3.11806	0.589257
$29$	2.54569	0.472722	0.236361	−	0.971665i	$-0.424045\pi$
$29$	2.54569	0.472722	0.236361	+	0.971665i	$0.424045\pi$
$30$	2.69503	0.492043
$31$	7.10661	1.27638	0.638192	−	0.769877i	$-0.279682\pi$
$31$	7.10661	1.27638	0.638192	+	0.769877i	$0.279682\pi$
$32$	−1.00000	−0.176777
$33$	7.51068	1.30744
$34$	1.14302	0.196027
$35$	3.69777	0.625037
$36$	2.16434	0.360724
$37$	7.62559	1.25364	0.626819	−	0.779165i	$-0.284356\pi$
$37$	7.62559	1.25364	0.626819	+	0.779165i	$0.284356\pi$
$38$	6.68822	1.08497
$39$	4.72193	0.756114
$40$	−1.18592	−0.187511
$41$	9.76257	1.52466	0.762328	−	0.647191i	$-0.224056\pi$
$41$	9.76257	1.52466	0.762328	+	0.647191i	$0.224056\pi$
$42$	7.08585	1.09337
$43$	−8.92914	−1.36168	−0.680840	−	0.732432i	$-0.738385\pi$
$43$	−8.92914	−1.36168	−0.680840	+	0.732432i	$0.738385\pi$
$44$	−3.30500	−0.498248
$45$	2.56674	0.382627
$46$	−0.381268	−0.0562150
$47$	1.77454	0.258843	0.129421	−	0.991590i	$-0.458688\pi$
$47$	1.77454	0.258843	0.129421	+	0.991590i	$0.458688\pi$
$48$	−2.27252	−0.328010
$49$	2.72228	0.388898
$50$	3.59359	0.508210
$51$	2.59754	0.363728
$52$	−2.07784	−0.288145
$53$	4.63002	0.635982	0.317991	−	0.948094i	$-0.396992\pi$
$53$	4.63002	0.635982	0.317991	+	0.948094i	$0.396992\pi$
$54$	−1.89905	−0.258428
$55$	−3.91947	−0.528502
$56$	−3.11806	−0.416668
$57$	15.1991	2.01317
$58$	−2.54569	−0.334265
$59$	−1.77952	−0.231674	−0.115837	−	0.993268i	$-0.536955\pi$
$59$	−1.77952	−0.231674	−0.115837	+	0.993268i	$0.536955\pi$
$60$	−2.69503	−0.347927
$61$	−1.61034	−0.206182	−0.103091	−	0.994672i	$-0.532873\pi$
$61$	−1.61034	−0.206182	−0.103091	+	0.994672i	$0.532873\pi$
$62$	−7.10661	−0.902540
$63$	6.74854	0.850237
$64$	1.00000	0.125000
$65$	−2.46415	−0.305641
$66$	−7.51068	−0.924501
$67$	4.69517	0.573606	0.286803	−	0.957990i	$-0.407407\pi$
$67$	4.69517	0.573606	0.286803	+	0.957990i	$0.407407\pi$
$68$	−1.14302	−0.138612
$69$	−0.866440	−0.104307
$70$	−3.69777	−0.441968
$71$	7.63111	0.905646	0.452823	−	0.891600i	$-0.350417\pi$
$71$	7.63111	0.905646	0.452823	+	0.891600i	$0.350417\pi$
$72$	−2.16434	−0.255070
$73$	1.19400	0.139747	0.0698737	−	0.997556i	$-0.477740\pi$
$73$	1.19400	0.139747	0.0698737	+	0.997556i	$0.477740\pi$
$74$	−7.62559	−0.886456
$75$	8.16650	0.942987
$76$	−6.68822	−0.767191
$77$	−10.3052	−1.17438
$78$	−4.72193	−0.534653
$79$	3.30984	0.372386	0.186193	−	0.982513i	$-0.440385\pi$
$79$	3.30984	0.372386	0.186193	+	0.982513i	$0.440385\pi$
$80$	1.18592	0.132590
$81$	−10.8086	−1.20096
$82$	−9.76257	−1.07810
$83$	−13.8355	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$83$	−13.8355	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$84$	−7.08585	−0.773129
$85$	−1.35553	−0.147028
$86$	8.92914	0.962854
$87$	−5.78512	−0.620230
$88$	3.30500	0.352314
$89$	−6.24467	−0.661934	−0.330967	−	0.943642i	$-0.607375\pi$
$89$	−6.24467	−0.661934	−0.330967	+	0.943642i	$0.607375\pi$
$90$	−2.56674	−0.270558
$91$	−6.47882	−0.679165
$92$	0.381268	0.0397500
$93$	−16.1499	−1.67467
$94$	−1.77454	−0.183030
$95$	−7.93170	−0.813775
$96$	2.27252	0.231938
$97$	−2.35973	−0.239594	−0.119797	−	0.992798i	$-0.538224\pi$
$97$	−2.35973	−0.239594	−0.119797	+	0.992798i	$0.538224\pi$
$98$	−2.72228	−0.274992
$99$	−7.15316	−0.718919
$100$	−3.59359	−0.359359
$101$	1.39959	0.139264	0.0696320	−	0.997573i	$-0.477817\pi$
$101$	1.39959	0.139264	0.0696320	+	0.997573i	$0.477817\pi$
$102$	−2.59754	−0.257195
$103$	15.9024	1.56691	0.783456	−	0.621447i	$-0.213455\pi$
$103$	15.9024	1.56691	0.783456	+	0.621447i	$0.213455\pi$
$104$	2.07784	0.203749
$105$	−8.40325	−0.820074
$106$	−4.63002	−0.449707
$107$	−4.28671	−0.414412	−0.207206	−	0.978297i	$-0.566437\pi$
$107$	−4.28671	−0.414412	−0.207206	+	0.978297i	$0.566437\pi$
$108$	1.89905	0.182736
$109$	2.93400	0.281026	0.140513	−	0.990079i	$-0.455125\pi$
$109$	2.93400	0.281026	0.140513	+	0.990079i	$0.455125\pi$
$110$	3.91947	0.373707
$111$	−17.3293	−1.64482
$112$	3.11806	0.294629
$113$	−6.07148	−0.571156	−0.285578	−	0.958355i	$-0.592186\pi$
$113$	−6.07148	−0.571156	−0.285578	+	0.958355i	$0.592186\pi$
$114$	−15.1991	−1.42353
$115$	0.452154	0.0421636
$116$	2.54569	0.236361
$117$	−4.49716	−0.415762
$118$	1.77952	0.163818
$119$	−3.56401	−0.326712
$120$	2.69503	0.246021
$121$	−0.0769632	−0.00699665
$122$	1.61034	0.145793
$123$	−22.1856	−2.00041
$124$	7.10661	0.638192
$125$	−10.1913	−0.911540
$126$	−6.74854	−0.601208
$127$	−3.29739	−0.292596	−0.146298	−	0.989241i	$-0.546736\pi$
$127$	−3.29739	−0.292596	−0.146298	+	0.989241i	$0.546736\pi$
$128$	−1.00000	−0.0883883
$129$	20.2916	1.78658
$130$	2.46415	0.216121
$131$	6.01135	0.525214	0.262607	−	0.964903i	$-0.415418\pi$
$131$	6.01135	0.525214	0.262607	+	0.964903i	$0.415418\pi$
$132$	7.51068	0.653721
$133$	−20.8542	−1.80829
$134$	−4.69517	−0.405601
$135$	2.25212	0.193832
$136$	1.14302	0.0980134
$137$	12.0101	1.02609	0.513044	−	0.858362i	$-0.328518\pi$
$137$	12.0101	1.02609	0.513044	+	0.858362i	$0.328518\pi$
$138$	0.866440	0.0737563
$139$	−6.59961	−0.559771	−0.279886	−	0.960033i	$-0.590297\pi$
$139$	−6.59961	−0.559771	−0.279886	+	0.960033i	$0.590297\pi$
$140$	3.69777	0.312519
$141$	−4.03267	−0.339612
$142$	−7.63111	−0.640389
$143$	6.86726	0.574269
$144$	2.16434	0.180362
$145$	3.01898	0.250713
$146$	−1.19400	−0.0988164
$147$	−6.18644	−0.510249
$148$	7.62559	0.626819
$149$	−0.0639852	−0.00524187	−0.00262093	−	0.999997i	$-0.500834\pi$
$149$	−0.0639852	−0.00524187	−0.00262093	+	0.999997i	$0.500834\pi$
$150$	−8.16650	−0.666792
$151$	−9.26977	−0.754363	−0.377181	−	0.926139i	$-0.623107\pi$
$151$	−9.26977	−0.754363	−0.377181	+	0.926139i	$0.623107\pi$
$152$	6.68822	0.542486
$153$	−2.47389	−0.200002
$154$	10.3052	0.830416
$155$	8.42788	0.676944
$156$	4.72193	0.378057
$157$	−7.68202	−0.613092	−0.306546	−	0.951856i	$-0.599173\pi$
$157$	−7.68202	−0.613092	−0.306546	+	0.951856i	$0.599173\pi$
$158$	−3.30984	−0.263317
$159$	−10.5218	−0.834434
$160$	−1.18592	−0.0937553
$161$	1.18882	0.0936919
$162$	10.8086	0.849208
$163$	−5.85674	−0.458735	−0.229367	−	0.973340i	$-0.573666\pi$
$163$	−5.85674	−0.458735	−0.229367	+	0.973340i	$0.573666\pi$
$164$	9.76257	0.762328
$165$	8.90707	0.693415
$166$	13.8355	1.07384
$167$	−11.6025	−0.897827	−0.448913	−	0.893575i	$-0.648189\pi$
$167$	−11.6025	−0.897827	−0.448913	+	0.893575i	$0.648189\pi$
$168$	7.08585	0.546685
$169$	−8.68258	−0.667891
$170$	1.35553	0.103965
$171$	−14.4756	−1.10698
$172$	−8.92914	−0.680840
$173$	3.23943	0.246289	0.123145	−	0.992389i	$-0.460702\pi$
$173$	3.23943	0.246289	0.123145	+	0.992389i	$0.460702\pi$
$174$	5.78512	0.438569
$175$	−11.2050	−0.847020
$176$	−3.30500	−0.249124
$177$	4.04399	0.303965
$178$	6.24467	0.468058
$179$	−13.8310	−1.03378	−0.516890	−	0.856052i	$-0.672910\pi$
$179$	−13.8310	−1.03378	−0.516890	+	0.856052i	$0.672910\pi$
$180$	2.56674	0.191314
$181$	10.9823	0.816308	0.408154	−	0.912913i	$-0.366173\pi$
$181$	10.9823	0.816308	0.408154	+	0.912913i	$0.366173\pi$
$182$	6.47882	0.480242
$183$	3.65952	0.270519
$184$	−0.381268	−0.0281075
$185$	9.04335	0.664880
$186$	16.1499	1.18417
$187$	3.77769	0.276252
$188$	1.77454	0.129421
$189$	5.92134	0.430714
$190$	7.93170	0.575426
$191$	2.13863	0.154746	0.0773728	−	0.997002i	$-0.475347\pi$
$191$	2.13863	0.154746	0.0773728	+	0.997002i	$0.475347\pi$
$192$	−2.27252	−0.164005
$193$	24.1264	1.73666	0.868329	−	0.495989i	$-0.165194\pi$
$193$	24.1264	1.73666	0.868329	+	0.495989i	$0.165194\pi$
$194$	2.35973	0.169419
$195$	5.59984	0.401013
$196$	2.72228	0.194449
$197$	20.0149	1.42600	0.713001	−	0.701163i	$-0.247336\pi$
$197$	20.0149	1.42600	0.713001	+	0.701163i	$0.247336\pi$
$198$	7.15316	0.508353
$199$	−1.62026	−0.114857	−0.0574287	−	0.998350i	$-0.518290\pi$
$199$	−1.62026	−0.114857	−0.0574287	+	0.998350i	$0.518290\pi$
$200$	3.59359	0.254105
$201$	−10.6699	−0.752594
$202$	−1.39959	−0.0984746
$203$	7.93760	0.557110
$204$	2.59754	0.181864
$205$	11.5776	0.808617
$206$	−15.9024	−1.10797
$207$	0.825196	0.0573551
$208$	−2.07784	−0.144072
$209$	22.1046	1.52900
$210$	8.40325	0.579880
$211$	−25.9492	−1.78642	−0.893208	−	0.449644i	$-0.851551\pi$
$211$	−25.9492	−1.78642	−0.893208	+	0.449644i	$0.851551\pi$
$212$	4.63002	0.317991
$213$	−17.3418	−1.18824
$214$	4.28671	0.293034
$215$	−10.5893	−0.722181
$216$	−1.89905	−0.129214
$217$	22.1588	1.50424
$218$	−2.93400	−0.198715
$219$	−2.71339	−0.183354
$220$	−3.91947	−0.264251
$221$	2.37502	0.159761
$222$	17.3293	1.16307
$223$	−6.03849	−0.404367	−0.202184	−	0.979348i	$-0.564804\pi$
$223$	−6.03849	−0.404367	−0.202184	+	0.979348i	$0.564804\pi$
$224$	−3.11806	−0.208334
$225$	−7.77776	−0.518517
$226$	6.07148	0.403869
$227$	−12.5895	−0.835593	−0.417797	−	0.908541i	$-0.637198\pi$
$227$	−12.5895	−0.835593	−0.417797	+	0.908541i	$0.637198\pi$
$228$	15.1991	1.00658
$229$	2.24373	0.148270	0.0741349	−	0.997248i	$-0.476380\pi$
$229$	2.24373	0.148270	0.0741349	+	0.997248i	$0.476380\pi$
$230$	−0.452154	−0.0298142
$231$	23.4187	1.54084
$232$	−2.54569	−0.167132
$233$	−12.5640	−0.823095	−0.411548	−	0.911388i	$-0.635012\pi$
$233$	−12.5640	−0.823095	−0.411548	+	0.911388i	$0.635012\pi$
$234$	4.49716	0.293988
$235$	2.10446	0.137280
$236$	−1.77952	−0.115837
$237$	−7.52167	−0.488585
$238$	3.56401	0.231020
$239$	9.37918	0.606689	0.303344	−	0.952881i	$-0.401897\pi$
$239$	9.37918	0.606689	0.303344	+	0.952881i	$0.401897\pi$
$240$	−2.69503	−0.173963
$241$	−19.6524	−1.26593	−0.632963	−	0.774182i	$-0.718161\pi$
$241$	−19.6524	−1.26593	−0.632963	+	0.774182i	$0.718161\pi$
$242$	0.0769632	0.00494738
$243$	18.8657	1.21024
$244$	−1.61034	−0.103091
$245$	3.22841	0.206256
$246$	22.1856	1.41450
$247$	13.8970	0.884247
$248$	−7.10661	−0.451270
$249$	31.4414	1.99252
$250$	10.1913	0.644556
$251$	23.1624	1.46200	0.730998	−	0.682380i	$-0.239055\pi$
$251$	23.1624	1.46200	0.730998	+	0.682380i	$0.239055\pi$
$252$	6.74854	0.425118
$253$	−1.26009	−0.0792214
$254$	3.29739	0.206897
$255$	3.08048	0.192907
$256$	1.00000	0.0625000
$257$	17.2468	1.07583	0.537913	−	0.843000i	$-0.319213\pi$
$257$	17.2468	1.07583	0.537913	+	0.843000i	$0.319213\pi$
$258$	−20.2916	−1.26330
$259$	23.7770	1.47743
$260$	−2.46415	−0.152820
$261$	5.50974	0.341044
$262$	−6.01135	−0.371383
$263$	−17.0777	−1.05306	−0.526529	−	0.850157i	$-0.676507\pi$
$263$	−17.0777	−1.05306	−0.526529	+	0.850157i	$0.676507\pi$
$264$	−7.51068	−0.462250
$265$	5.49084	0.337300
$266$	20.8542	1.27866
$267$	14.1911	0.868484
$268$	4.69517	0.286803
$269$	−15.3821	−0.937865	−0.468933	−	0.883234i	$-0.655361\pi$
$269$	−15.3821	−0.937865	−0.468933	+	0.883234i	$0.655361\pi$
$270$	−2.25212	−0.137060
$271$	−2.18931	−0.132991	−0.0664955	−	0.997787i	$-0.521182\pi$
$271$	−2.18931	−0.132991	−0.0664955	+	0.997787i	$0.521182\pi$
$272$	−1.14302	−0.0693059
$273$	14.7233	0.891092
$274$	−12.0101	−0.725554
$275$	11.8768	0.716199
$276$	−0.866440	−0.0521536
$277$	2.00336	0.120370	0.0601850	−	0.998187i	$-0.480831\pi$
$277$	2.00336	0.120370	0.0601850	+	0.998187i	$0.480831\pi$
$278$	6.59961	0.395818
$279$	15.3811	0.920845
$280$	−3.69777	−0.220984
$281$	9.24534	0.551531	0.275765	−	0.961225i	$-0.411069\pi$
$281$	9.24534	0.551531	0.275765	+	0.961225i	$0.411069\pi$
$282$	4.03267	0.240142
$283$	−20.5038	−1.21883	−0.609413	−	0.792853i	$-0.708595\pi$
$283$	−20.5038	−1.21883	−0.609413	+	0.792853i	$0.708595\pi$
$284$	7.63111	0.452823
$285$	18.0249	1.06770
$286$	−6.86726	−0.406070
$287$	30.4402	1.79683
$288$	−2.16434	−0.127535
$289$	−15.6935	−0.923147
$290$	−3.01898	−0.177281
$291$	5.36252	0.314357
$292$	1.19400	0.0698737
$293$	16.2517	0.949432	0.474716	−	0.880139i	$-0.342551\pi$
$293$	16.2517	0.949432	0.474716	+	0.880139i	$0.342551\pi$
$294$	6.18644	0.360800
$295$	−2.11037	−0.122871
$296$	−7.62559	−0.443228
$297$	−6.27635	−0.364191
$298$	0.0639852	0.00370656
$299$	−0.792215	−0.0458150
$300$	8.16650	0.471493
$301$	−27.8416	−1.60476
$302$	9.26977	0.533415
$303$	−3.18059	−0.182720
$304$	−6.68822	−0.383595
$305$	−1.90973	−0.109351
$306$	2.47389	0.141423
$307$	−15.8714	−0.905829	−0.452915	−	0.891554i	$-0.649616\pi$
$307$	−15.8714	−0.905829	−0.452915	+	0.891554i	$0.649616\pi$
$308$	−10.3052	−0.587192
$309$	−36.1385	−2.05585
$310$	−8.42788	−0.478671
$311$	10.4187	0.590791	0.295395	−	0.955375i	$-0.404549\pi$
$311$	10.4187	0.590791	0.295395	+	0.955375i	$0.404549\pi$
$312$	−4.72193	−0.267327
$313$	−1.70660	−0.0964625	−0.0482312	−	0.998836i	$-0.515358\pi$
$313$	−1.70660	−0.0964625	−0.0482312	+	0.998836i	$0.515358\pi$
$314$	7.68202	0.433521
$315$	8.00324	0.450932
$316$	3.30984	0.186193
$317$	−18.8641	−1.05952	−0.529758	−	0.848149i	$-0.677717\pi$
$317$	−18.8641	−1.05952	−0.529758	+	0.848149i	$0.677717\pi$
$318$	10.5218	0.590034
$319$	−8.41350	−0.471065
$320$	1.18592	0.0662950
$321$	9.74163	0.543725
$322$	−1.18882	−0.0662502
$323$	7.64478	0.425367
$324$	−10.8086	−0.600480
$325$	7.46691	0.414189
$326$	5.85674	0.324375
$327$	−6.66756	−0.368717
$328$	−9.76257	−0.539048
$329$	5.53311	0.305050
$330$	−8.90707	−0.490318
$331$	−26.5708	−1.46047	−0.730233	−	0.683198i	$-0.760588\pi$
$331$	−26.5708	−1.46047	−0.730233	+	0.683198i	$0.760588\pi$
$332$	−13.8355	−0.759321
$333$	16.5044	0.904435
$334$	11.6025	0.634859
$335$	5.56810	0.304218
$336$	−7.08585	−0.386565
$337$	2.79079	0.152024	0.0760121	−	0.997107i	$-0.475781\pi$
$337$	2.79079	0.152024	0.0760121	+	0.997107i	$0.475781\pi$
$338$	8.68258	0.472270
$339$	13.7975	0.749380
$340$	−1.35553	−0.0735142
$341$	−23.4874	−1.27191
$342$	14.4756	0.782750
$343$	−13.3382	−0.720193
$344$	8.92914	0.481427
$345$	−1.02753	−0.0553203
$346$	−3.23943	−0.174153
$347$	−4.87842	−0.261887	−0.130944	−	0.991390i	$-0.541801\pi$
$347$	−4.87842	−0.261887	−0.130944	+	0.991390i	$0.541801\pi$
$348$	−5.78512	−0.310115
$349$	−20.7284	−1.10956	−0.554782	−	0.831995i	$-0.687198\pi$
$349$	−20.7284	−1.10956	−0.554782	+	0.831995i	$0.687198\pi$
$350$	11.2050	0.598934
$351$	−3.94592	−0.210617
$352$	3.30500	0.176157
$353$	−24.3912	−1.29821	−0.649105	−	0.760699i	$-0.724857\pi$
$353$	−24.3912	−1.29821	−0.649105	+	0.760699i	$0.724857\pi$
$354$	−4.04399	−0.214936
$355$	9.04990	0.480319
$356$	−6.24467	−0.330967
$357$	8.09928	0.428659
$358$	13.8310	0.730993
$359$	−20.9230	−1.10427	−0.552136	−	0.833754i	$-0.686187\pi$
$359$	−20.9230	−1.10427	−0.552136	+	0.833754i	$0.686187\pi$
$360$	−2.56674	−0.135279
$361$	25.7322	1.35433
$362$	−10.9823	−0.577217
$363$	0.174900	0.00917988
$364$	−6.47882	−0.339583
$365$	1.41599	0.0741165
$366$	−3.65952	−0.191286
$367$	28.1195	1.46783	0.733913	−	0.679243i	$-0.237692\pi$
$367$	28.1195	1.46783	0.733913	+	0.679243i	$0.237692\pi$
$368$	0.381268	0.0198750
$369$	21.1295	1.09996
$370$	−9.04335	−0.470141
$371$	14.4367	0.749515
$372$	−16.1499	−0.837334
$373$	−4.92940	−0.255234	−0.127617	−	0.991823i	$-0.540733\pi$
$373$	−4.92940	−0.255234	−0.127617	+	0.991823i	$0.540733\pi$
$374$	−3.77769	−0.195340
$375$	23.1600	1.19598
$376$	−1.77454	−0.0915148
$377$	−5.28953	−0.272425
$378$	−5.92134	−0.304561
$379$	−4.84984	−0.249120	−0.124560	−	0.992212i	$-0.539752\pi$
$379$	−4.84984	−0.249120	−0.124560	+	0.992212i	$0.539752\pi$
$380$	−7.93170	−0.406887
$381$	7.49338	0.383897
$382$	−2.13863	−0.109422
$383$	3.77804	0.193049	0.0965243	−	0.995331i	$-0.469227\pi$
$383$	3.77804	0.193049	0.0965243	+	0.995331i	$0.469227\pi$
$384$	2.27252	0.115969
$385$	−12.2211	−0.622847
$386$	−24.1264	−1.22800
$387$	−19.3257	−0.982381
$388$	−2.35973	−0.119797
$389$	−2.19245	−0.111162	−0.0555808	−	0.998454i	$-0.517701\pi$
$389$	−2.19245	−0.111162	−0.0555808	+	0.998454i	$0.517701\pi$
$390$	−5.59984	−0.283559
$391$	−0.435798	−0.0220393
$392$	−2.72228	−0.137496
$393$	−13.6609	−0.689102
$394$	−20.0149	−1.00834
$395$	3.92521	0.197499
$396$	−7.15316	−0.359460
$397$	4.78192	0.239998	0.119999	−	0.992774i	$-0.461711\pi$
$397$	4.78192	0.239998	0.119999	+	0.992774i	$0.461711\pi$
$398$	1.62026	0.0812165
$399$	47.3917	2.37255
$400$	−3.59359	−0.179680
$401$	18.4255	0.920127	0.460064	−	0.887886i	$-0.347827\pi$
$401$	18.4255	0.920127	0.460064	+	0.887886i	$0.347827\pi$
$402$	10.6699	0.532164
$403$	−14.7664	−0.735567
$404$	1.39959	0.0696320
$405$	−12.8182	−0.636942
$406$	−7.93760	−0.393936
$407$	−25.2026	−1.24925
$408$	−2.59754	−0.128597
$409$	−26.6829	−1.31939	−0.659693	−	0.751535i	$-0.729314\pi$
$409$	−26.6829	−1.31939	−0.659693	+	0.751535i	$0.729314\pi$
$410$	−11.5776	−0.571779
$411$	−27.2931	−1.34627
$412$	15.9024	0.783456
$413$	−5.54865	−0.273031
$414$	−0.825196	−0.0405562
$415$	−16.4078	−0.805427
$416$	2.07784	0.101874
$417$	14.9977	0.734442
$418$	−22.1046	−1.08117
$419$	−1.11478	−0.0544605	−0.0272303	−	0.999629i	$-0.508669\pi$
$419$	−1.11478	−0.0544605	−0.0272303	+	0.999629i	$0.508669\pi$
$420$	−8.40325	−0.410037
$421$	−16.8296	−0.820227	−0.410113	−	0.912035i	$-0.634511\pi$
$421$	−16.8296	−0.820227	−0.410113	+	0.912035i	$0.634511\pi$
$422$	25.9492	1.26319
$423$	3.84071	0.186742
$424$	−4.63002	−0.224854
$425$	4.10755	0.199246
$426$	17.3418	0.840215
$427$	−5.02112	−0.242989
$428$	−4.28671	−0.207206
$429$	−15.6060	−0.753464
$430$	10.5893	0.510659
$431$	6.82633	0.328813	0.164406	−	0.986393i	$-0.447429\pi$
$431$	6.82633	0.328813	0.164406	+	0.986393i	$0.447429\pi$
$432$	1.89905	0.0913680
$433$	−30.6831	−1.47454	−0.737268	−	0.675601i	$-0.763884\pi$
$433$	−30.6831	−1.47454	−0.737268	+	0.675601i	$0.763884\pi$
$434$	−22.1588	−1.06366
$435$	−6.86070	−0.328945
$436$	2.93400	0.140513
$437$	−2.55001	−0.121983
$438$	2.71339	0.129651
$439$	−24.8620	−1.18660	−0.593299	−	0.804982i	$-0.702175\pi$
$439$	−24.8620	−1.18660	−0.593299	+	0.804982i	$0.702175\pi$
$440$	3.91947	0.186853
$441$	5.89195	0.280569
$442$	−2.37502	−0.112968
$443$	18.4032	0.874364	0.437182	−	0.899373i	$-0.355976\pi$
$443$	18.4032	0.874364	0.437182	+	0.899373i	$0.355976\pi$
$444$	−17.3293	−0.822412
$445$	−7.40569	−0.351063
$446$	6.03849	0.285931
$447$	0.145407	0.00687754
$448$	3.11806	0.147314
$449$	34.9356	1.64871	0.824357	−	0.566070i	$-0.191537\pi$
$449$	34.9356	1.64871	0.824357	+	0.566070i	$0.191537\pi$
$450$	7.77776	0.366647
$451$	−32.2653	−1.51931
$452$	−6.07148	−0.285578
$453$	21.0657	0.989754
$454$	12.5895	0.590854
$455$	−7.68338	−0.360202
$456$	−15.1991	−0.711763
$457$	20.5308	0.960391	0.480196	−	0.877161i	$-0.340566\pi$
$457$	20.5308	0.960391	0.480196	+	0.877161i	$0.340566\pi$
$458$	−2.24373	−0.104843
$459$	−2.17065	−0.101317
$460$	0.452154	0.0210818
$461$	−28.8729	−1.34474	−0.672371	−	0.740214i	$-0.734724\pi$
$461$	−28.8729	−1.34474	−0.672371	+	0.740214i	$0.734724\pi$
$462$	−23.4187	−1.08954
$463$	0.541309	0.0251568	0.0125784	−	0.999921i	$-0.495996\pi$
$463$	0.541309	0.0251568	0.0125784	+	0.999921i	$0.495996\pi$
$464$	2.54569	0.118181
$465$	−19.1525	−0.888177
$466$	12.5640	0.582016
$467$	−26.9876	−1.24884	−0.624418	−	0.781090i	$-0.714664\pi$
$467$	−26.9876	−1.24884	−0.624418	+	0.781090i	$0.714664\pi$
$468$	−4.49716	−0.207881
$469$	14.6398	0.676004
$470$	−2.10446	−0.0970716
$471$	17.4575	0.804400
$472$	1.77952	0.0819091
$473$	29.5108	1.35691
$474$	7.52167	0.345482
$475$	24.0347	1.10279
$476$	−3.56401	−0.163356
$477$	10.0210	0.458828
$478$	−9.37918	−0.428994
$479$	27.1549	1.24074	0.620369	−	0.784310i	$-0.286983\pi$
$479$	27.1549	1.24074	0.620369	+	0.784310i	$0.286983\pi$
$480$	2.69503	0.123011
$481$	−15.8447	−0.722458
$482$	19.6524	0.895144
$483$	−2.70161	−0.122927
$484$	−0.0769632	−0.00349833
$485$	−2.79845	−0.127071
$486$	−18.8657	−0.855766
$487$	−20.0393	−0.908068	−0.454034	−	0.890984i	$-0.650016\pi$
$487$	−20.0393	−0.908068	−0.454034	+	0.890984i	$0.650016\pi$
$488$	1.61034	0.0728965
$489$	13.3095	0.601878
$490$	−3.22841	−0.145845
$491$	35.6506	1.60889	0.804445	−	0.594027i	$-0.202463\pi$
$491$	35.6506	1.60889	0.804445	+	0.594027i	$0.202463\pi$
$492$	−22.1856	−1.00020
$493$	−2.90978	−0.131050
$494$	−13.8970	−0.625257
$495$	−8.48308	−0.381286
$496$	7.10661	0.319096
$497$	23.7942	1.06732
$498$	−31.4414	−1.40892
$499$	−21.9554	−0.982859	−0.491430	−	0.870917i	$-0.663525\pi$
$499$	−21.9554	−0.982859	−0.491430	+	0.870917i	$0.663525\pi$
$500$	−10.1913	−0.455770
$501$	26.3668	1.17798
$502$	−23.1624	−1.03379
$503$	−10.6522	−0.474960	−0.237480	−	0.971392i	$-0.576321\pi$
$503$	−10.6522	−0.474960	−0.237480	+	0.971392i	$0.576321\pi$
$504$	−6.74854	−0.300604
$505$	1.65980	0.0738601
$506$	1.26009	0.0560180
$507$	19.7313	0.876299
$508$	−3.29739	−0.146298
$509$	26.7150	1.18412	0.592062	−	0.805893i	$-0.298314\pi$
$509$	26.7150	1.18412	0.592062	+	0.805893i	$0.298314\pi$
$510$	−3.08048	−0.136406
$511$	3.72297	0.164695
$512$	−1.00000	−0.0441942
$513$	−12.7012	−0.560773
$514$	−17.2468	−0.760724
$515$	18.8590	0.831027
$516$	20.2916	0.893290
$517$	−5.86485	−0.257936
$518$	−23.7770	−1.04470
$519$	−7.36166	−0.323141
$520$	2.46415	0.108060
$521$	26.3691	1.15525	0.577626	−	0.816301i	$-0.303979\pi$
$521$	26.3691	1.15525	0.577626	+	0.816301i	$0.303979\pi$
$522$	−5.50974	−0.241155
$523$	−27.4437	−1.20003	−0.600014	−	0.799989i	$-0.704838\pi$
$523$	−27.4437	−1.20003	−0.600014	+	0.799989i	$0.704838\pi$
$524$	6.01135	0.262607
$525$	25.4636	1.11132
$526$	17.0777	0.744625
$527$	−8.12301	−0.353844
$528$	7.51068	0.326860
$529$	−22.8546	−0.993680
$530$	−5.49084	−0.238507
$531$	−3.85149	−0.167141
$532$	−20.8542	−0.904146
$533$	−20.2850	−0.878643
$534$	−14.1911	−0.614111
$535$	−5.08370	−0.219788
$536$	−4.69517	−0.202800
$537$	31.4313	1.35636
$538$	15.3821	0.663171
$539$	−8.99715	−0.387535
$540$	2.25212	0.0969158
$541$	30.4414	1.30878	0.654389	−	0.756158i	$-0.272926\pi$
$541$	30.4414	1.30878	0.654389	+	0.756158i	$0.272926\pi$
$542$	2.18931	0.0940389
$543$	−24.9575	−1.07103
$544$	1.14302	0.0490067
$545$	3.47949	0.149045
$546$	−14.7233	−0.630097
$547$	1.32319	0.0565754	0.0282877	−	0.999600i	$-0.490995\pi$
$547$	1.32319	0.0565754	0.0282877	+	0.999600i	$0.490995\pi$
$548$	12.0101	0.513044
$549$	−3.48532	−0.148750
$550$	−11.8768	−0.506429
$551$	−17.0261	−0.725336
$552$	0.866440	0.0368781
$553$	10.3203	0.438863
$554$	−2.00336	−0.0851145
$555$	−20.5512	−0.872349
$556$	−6.59961	−0.279886
$557$	11.2017	0.474632	0.237316	−	0.971432i	$-0.423732\pi$
$557$	11.2017	0.474632	0.237316	+	0.971432i	$0.423732\pi$
$558$	−15.3811	−0.651136
$559$	18.5533	0.784722
$560$	3.69777	0.156259
$561$	−8.58488	−0.362454
$562$	−9.24534	−0.389991
$563$	22.5720	0.951296	0.475648	−	0.879636i	$-0.342214\pi$
$563$	22.5720	0.951296	0.475648	+	0.879636i	$0.342214\pi$
$564$	−4.03267	−0.169806
$565$	−7.20029	−0.302919
$566$	20.5038	0.861841
$567$	−33.7020	−1.41535
$568$	−7.63111	−0.320194
$569$	8.41932	0.352956	0.176478	−	0.984305i	$-0.443530\pi$
$569$	8.41932	0.352956	0.176478	+	0.984305i	$0.443530\pi$
$570$	−18.0249	−0.754981
$571$	−2.11148	−0.0883627	−0.0441813	−	0.999024i	$-0.514068\pi$
$571$	−2.11148	−0.0883627	−0.0441813	+	0.999024i	$0.514068\pi$
$572$	6.86726	0.287135
$573$	−4.86007	−0.203032
$574$	−30.4402	−1.27055
$575$	−1.37012	−0.0571381
$576$	2.16434	0.0901809
$577$	−8.64381	−0.359847	−0.179923	−	0.983681i	$-0.557585\pi$
$577$	−8.64381	−0.359847	−0.179923	+	0.983681i	$0.557585\pi$
$578$	15.6935	0.652764
$579$	−54.8277	−2.27856
$580$	3.01898	0.125356
$581$	−43.1399	−1.78974
$582$	−5.36252	−0.222284
$583$	−15.3022	−0.633753
$584$	−1.19400	−0.0494082
$585$	−5.33327	−0.220504
$586$	−16.2517	−0.671350
$587$	−34.6554	−1.43038	−0.715191	−	0.698929i	$-0.753660\pi$
$587$	−34.6554	−1.43038	−0.715191	+	0.698929i	$0.753660\pi$
$588$	−6.18644	−0.255124
$589$	−47.5305	−1.95846
$590$	2.11037	0.0868826
$591$	−45.4842	−1.87097
$592$	7.62559	0.313410
$593$	−17.8470	−0.732889	−0.366444	−	0.930440i	$-0.619425\pi$
$593$	−17.8470	−0.732889	−0.366444	+	0.930440i	$0.619425\pi$
$594$	6.27635	0.257522
$595$	−4.22664	−0.173275
$596$	−0.0639852	−0.00262093
$597$	3.68208	0.150697
$598$	0.792215	0.0323961
$599$	−3.85445	−0.157489	−0.0787443	−	0.996895i	$-0.525091\pi$
$599$	−3.85445	−0.157489	−0.0787443	+	0.996895i	$0.525091\pi$
$600$	−8.16650	−0.333396
$601$	−34.1868	−1.39451	−0.697254	−	0.716824i	$-0.745595\pi$
$601$	−34.1868	−1.39451	−0.697254	+	0.716824i	$0.745595\pi$
$602$	27.8416	1.13474
$603$	10.1620	0.413827
$604$	−9.26977	−0.377181
$605$	−0.0912723	−0.00371075
$606$	3.18059	0.129203
$607$	5.60396	0.227458	0.113729	−	0.993512i	$-0.463720\pi$
$607$	5.60396	0.227458	0.113729	+	0.993512i	$0.463720\pi$
$608$	6.68822	0.271243
$609$	−18.0383	−0.730950
$610$	1.90973	0.0773228
$611$	−3.68720	−0.149168
$612$	−2.47389	−0.100001
$613$	27.2301	1.09981	0.549907	−	0.835226i	$-0.314663\pi$
$613$	27.2301	1.09981	0.549907	+	0.835226i	$0.314663\pi$
$614$	15.8714	0.640518
$615$	−26.3104	−1.06094
$616$	10.3052	0.415208
$617$	−1.82993	−0.0736703	−0.0368351	−	0.999321i	$-0.511728\pi$
$617$	−1.82993	−0.0736703	−0.0368351	+	0.999321i	$0.511728\pi$
$618$	36.1385	1.45371
$619$	45.4855	1.82822	0.914108	−	0.405471i	$-0.132892\pi$
$619$	45.4855	1.82822	0.914108	+	0.405471i	$0.132892\pi$
$620$	8.42788	0.338472
$621$	0.724047	0.0290550
$622$	−10.4187	−0.417752
$623$	−19.4713	−0.780099
$624$	4.72193	0.189028
$625$	5.88185	0.235274
$626$	1.70660	0.0682093
$627$	−50.2330	−2.00611
$628$	−7.68202	−0.306546
$629$	−8.71622	−0.347538
$630$	−8.00324	−0.318857
$631$	−27.6179	−1.09945	−0.549726	−	0.835345i	$-0.685268\pi$
$631$	−27.6179	−1.09945	−0.549726	+	0.835345i	$0.685268\pi$
$632$	−3.30984	−0.131658
$633$	58.9700	2.34385
$634$	18.8641	0.749191
$635$	−3.91044	−0.155181
$636$	−10.5218	−0.417217
$637$	−5.65647	−0.224117
$638$	8.41350	0.333094
$639$	16.5163	0.653376
$640$	−1.18592	−0.0468777
$641$	0.629091	0.0248476	0.0124238	−	0.999923i	$-0.496045\pi$
$641$	0.629091	0.0248476	0.0124238	+	0.999923i	$0.496045\pi$
$642$	−9.74163	−0.384472
$643$	−26.3846	−1.04051	−0.520254	−	0.854012i	$-0.674163\pi$
$643$	−26.3846	−1.04051	−0.520254	+	0.854012i	$0.674163\pi$
$644$	1.18882	0.0468460
$645$	24.0643	0.947530
$646$	−7.64478	−0.300780
$647$	26.1128	1.02660	0.513300	−	0.858209i	$-0.328423\pi$
$647$	26.1128	1.02660	0.513300	+	0.858209i	$0.328423\pi$
$648$	10.8086	0.424604
$649$	5.88132	0.230862
$650$	−7.46691	−0.292876
$651$	−50.3563	−1.97362
$652$	−5.85674	−0.229367
$653$	−46.4451	−1.81754	−0.908768	−	0.417301i	$-0.862976\pi$
$653$	−46.4451	−1.81754	−0.908768	+	0.417301i	$0.862976\pi$
$654$	6.66756	0.260722
$655$	7.12899	0.278553
$656$	9.76257	0.381164
$657$	2.58423	0.100820
$658$	−5.53311	−0.215703
$659$	−48.1884	−1.87716	−0.938578	−	0.345068i	$-0.887856\pi$
$659$	−48.1884	−1.87716	−0.938578	+	0.345068i	$0.887856\pi$
$660$	8.90707	0.346707
$661$	25.5691	0.994524	0.497262	−	0.867600i	$-0.334339\pi$
$661$	25.5691	0.994524	0.497262	+	0.867600i	$0.334339\pi$
$662$	26.5708	1.03271
$663$	−5.39727	−0.209613
$664$	13.8355	0.536921
$665$	−24.7315	−0.959046
$666$	−16.5044	−0.639532
$667$	0.970590	0.0375814
$668$	−11.6025	−0.448913
$669$	13.7226	0.530546
$670$	−5.56810	−0.215115
$671$	5.32216	0.205460
$672$	7.08585	0.273342
$673$	−38.7247	−1.49273	−0.746365	−	0.665537i	$-0.768202\pi$
$673$	−38.7247	−1.49273	−0.746365	+	0.665537i	$0.768202\pi$
$674$	−2.79079	−0.107497
$675$	−6.82440	−0.262671
$676$	−8.68258	−0.333945
$677$	−38.6436	−1.48519	−0.742597	−	0.669739i	$-0.766406\pi$
$677$	−38.6436	−1.48519	−0.742597	+	0.669739i	$0.766406\pi$
$678$	−13.7975	−0.529892
$679$	−7.35776	−0.282365
$680$	1.35553	0.0519824
$681$	28.6098	1.09633
$682$	23.4874	0.899377
$683$	−40.7183	−1.55804	−0.779021	−	0.626998i	$-0.784283\pi$
$683$	−40.7183	−1.55804	−0.779021	+	0.626998i	$0.784283\pi$
$684$	−14.4756	−0.553488
$685$	14.2430	0.544197
$686$	13.3382	0.509254
$687$	−5.09892	−0.194536
$688$	−8.92914	−0.340420
$689$	−9.62044	−0.366510
$690$	1.02753	0.0391174
$691$	25.3226	0.963316	0.481658	−	0.876359i	$-0.340035\pi$
$691$	25.3226	0.963316	0.481658	+	0.876359i	$0.340035\pi$
$692$	3.23943	0.123145
$693$	−22.3040	−0.847257
$694$	4.87842	0.185182
$695$	−7.82662	−0.296880
$696$	5.78512	0.219284
$697$	−11.1588	−0.422671
$698$	20.7284	0.784581
$699$	28.5519	1.07993
$700$	−11.2050	−0.423510
$701$	25.3130	0.956060	0.478030	−	0.878344i	$-0.341351\pi$
$701$	25.3130	0.956060	0.478030	+	0.878344i	$0.341351\pi$
$702$	3.94592	0.148929
$703$	−51.0016	−1.92356
$704$	−3.30500	−0.124562
$705$	−4.78243	−0.180117
$706$	24.3912	0.917974
$707$	4.36399	0.164125
$708$	4.04399	0.151983
$709$	7.39972	0.277902	0.138951	−	0.990299i	$-0.455627\pi$
$709$	7.39972	0.277902	0.138951	+	0.990299i	$0.455627\pi$
$710$	−9.04990	−0.339637
$711$	7.16363	0.268657
$712$	6.24467	0.234029
$713$	2.70953	0.101473
$714$	−8.09928	−0.303108
$715$	8.14403	0.304570
$716$	−13.8310	−0.516890
$717$	−21.3144	−0.796000
$718$	20.9230	0.780838
$719$	−11.7876	−0.439602	−0.219801	−	0.975545i	$-0.570541\pi$
$719$	−11.7876	−0.439602	−0.219801	+	0.975545i	$0.570541\pi$
$720$	2.56674	0.0956568
$721$	49.5846	1.84663
$722$	−25.7322	−0.957654
$723$	44.6605	1.66094
$724$	10.9823	0.408154
$725$	−9.14815	−0.339754
$726$	−0.174900	−0.00649116
$727$	−37.0993	−1.37594	−0.687969	−	0.725741i	$-0.741497\pi$
$727$	−37.0993	−1.37594	−0.687969	+	0.725741i	$0.741497\pi$
$728$	6.47882	0.240121
$729$	−10.4468	−0.386917
$730$	−1.41599	−0.0524083
$731$	10.2062	0.377490
$732$	3.65952	0.135260
$733$	−19.0596	−0.703984	−0.351992	−	0.936003i	$-0.614496\pi$
$733$	−19.0596	−0.703984	−0.351992	+	0.936003i	$0.614496\pi$
$734$	−28.1195	−1.03791
$735$	−7.33663	−0.270616
$736$	−0.381268	−0.0140537
$737$	−15.5175	−0.571596
$738$	−21.1295	−0.777789
$739$	−26.3286	−0.968515	−0.484257	−	0.874926i	$-0.660910\pi$
$739$	−26.3286	−0.968515	−0.484257	+	0.874926i	$0.660910\pi$
$740$	9.04335	0.332440
$741$	−31.5813	−1.16017
$742$	−14.4367	−0.529987
$743$	6.87100	0.252072	0.126036	−	0.992026i	$-0.459774\pi$
$743$	6.87100	0.252072	0.126036	+	0.992026i	$0.459774\pi$
$744$	16.1499	0.592084
$745$	−0.0758814	−0.00278008
$746$	4.92940	0.180478
$747$	−29.9447	−1.09562
$748$	3.77769	0.138126
$749$	−13.3662	−0.488391
$750$	−23.1600	−0.845683
$751$	−5.09592	−0.185953	−0.0929765	−	0.995668i	$-0.529638\pi$
$751$	−5.09592	−0.185953	−0.0929765	+	0.995668i	$0.529638\pi$
$752$	1.77454	0.0647107
$753$	−52.6369	−1.91820
$754$	5.28953	0.192633
$755$	−10.9932	−0.400084
$756$	5.92134	0.215357
$757$	11.3449	0.412336	0.206168	−	0.978517i	$-0.433901\pi$
$757$	11.3449	0.412336	0.206168	+	0.978517i	$0.433901\pi$
$758$	4.84984	0.176154
$759$	2.86359	0.103942
$760$	7.93170	0.287713
$761$	31.0031	1.12386	0.561931	−	0.827184i	$-0.310059\pi$
$761$	31.0031	1.12386	0.561931	+	0.827184i	$0.310059\pi$
$762$	−7.49338	−0.271456
$763$	9.14837	0.331193
$764$	2.13863	0.0773728
$765$	−2.93384	−0.106073
$766$	−3.77804	−0.136506
$767$	3.69756	0.133511
$768$	−2.27252	−0.0820025
$769$	29.4504	1.06201	0.531005	−	0.847369i	$-0.321814\pi$
$769$	29.4504	1.06201	0.531005	+	0.847369i	$0.321814\pi$
$770$	12.2211	0.440419
$771$	−39.1937	−1.41153
$772$	24.1264	0.868329
$773$	−17.2744	−0.621317	−0.310659	−	0.950522i	$-0.600550\pi$
$773$	−17.2744	−0.621317	−0.310659	+	0.950522i	$0.600550\pi$
$774$	19.3257	0.694648
$775$	−25.5382	−0.917361
$776$	2.35973	0.0847093
$777$	−54.0337	−1.93845
$778$	2.19245	0.0786032
$779$	−65.2941	−2.33941
$780$	5.59984	0.200506
$781$	−25.2208	−0.902473
$782$	0.435798	0.0155841
$783$	4.83438	0.172767
$784$	2.72228	0.0972244
$785$	−9.11027	−0.325159
$786$	13.6609	0.487269
$787$	6.84992	0.244173	0.122087	−	0.992519i	$-0.461041\pi$
$787$	6.84992	0.244173	0.122087	+	0.992519i	$0.461041\pi$
$788$	20.0149	0.713001
$789$	38.8095	1.38165
$790$	−3.92521	−0.139653
$791$	−18.9312	−0.673116
$792$	7.15316	0.254176
$793$	3.34602	0.118821
$794$	−4.78192	−0.169704
$795$	−12.4780	−0.442550
$796$	−1.62026	−0.0574287
$797$	−9.48315	−0.335910	−0.167955	−	0.985795i	$-0.553716\pi$
$797$	−9.48315	−0.335910	−0.167955	+	0.985795i	$0.553716\pi$
$798$	−47.3917	−1.67765
$799$	−2.02834	−0.0717574
$800$	3.59359	0.127053
$801$	−13.5156	−0.477551
$802$	−18.4255	−0.650628
$803$	−3.94618	−0.139258
$804$	−10.6699	−0.376297
$805$	1.40984	0.0496905
$806$	14.7664	0.520124
$807$	34.9562	1.23052
$808$	−1.39959	−0.0492373
$809$	−12.9631	−0.455757	−0.227879	−	0.973690i	$-0.573179\pi$
$809$	−12.9631	−0.455757	−0.227879	+	0.973690i	$0.573179\pi$
$810$	12.8182	0.450386
$811$	−20.3665	−0.715165	−0.357583	−	0.933881i	$-0.616399\pi$
$811$	−20.3665	−0.715165	−0.357583	+	0.933881i	$0.616399\pi$
$812$	7.93760	0.278555
$813$	4.97525	0.174489
$814$	25.2026	0.883350
$815$	−6.94563	−0.243295
$816$	2.59754	0.0909321
$817$	59.7200	2.08934
$818$	26.6829	0.932947
$819$	−14.0224	−0.489982
$820$	11.5776	0.404309
$821$	48.8981	1.70656	0.853278	−	0.521456i	$-0.174611\pi$
$821$	48.8981	1.70656	0.853278	+	0.521456i	$0.174611\pi$
$822$	27.2931	0.951956
$823$	19.8076	0.690449	0.345225	−	0.938520i	$-0.387803\pi$
$823$	19.8076	0.690449	0.345225	+	0.938520i	$0.387803\pi$
$824$	−15.9024	−0.553987
$825$	−26.9903	−0.939682
$826$	5.54865	0.193062
$827$	−8.38482	−0.291569	−0.145784	−	0.989316i	$-0.546571\pi$
$827$	−8.38482	−0.291569	−0.145784	+	0.989316i	$0.546571\pi$
$828$	0.825196	0.0286775
$829$	39.5112	1.37228	0.686140	−	0.727470i	$-0.259304\pi$
$829$	39.5112	1.37228	0.686140	+	0.727470i	$0.259304\pi$
$830$	16.4078	0.569523
$831$	−4.55267	−0.157930
$832$	−2.07784	−0.0720361
$833$	−3.11163	−0.107812
$834$	−14.9977	−0.519329
$835$	−13.7596	−0.476171
$836$	22.1046	0.764502
$837$	13.4958	0.466483
$838$	1.11478	0.0385094
$839$	−4.01385	−0.138573	−0.0692867	−	0.997597i	$-0.522072\pi$
$839$	−4.01385	−0.138573	−0.0692867	+	0.997597i	$0.522072\pi$
$840$	8.40325	0.289940
$841$	−22.5195	−0.776534
$842$	16.8296	0.579988
$843$	−21.0102	−0.723630
$844$	−25.9492	−0.893208
$845$	−10.2969	−0.354223
$846$	−3.84071	−0.132046
$847$	−0.239976	−0.00824566
$848$	4.63002	0.158996
$849$	46.5954	1.59915
$850$	−4.10755	−0.140888
$851$	2.90740	0.0996642
$852$	−17.3418	−0.594122
$853$	−46.0245	−1.57585	−0.787925	−	0.615771i	$-0.788845\pi$
$853$	−46.0245	−1.57585	−0.787925	+	0.615771i	$0.788845\pi$
$854$	5.02112	0.171819
$855$	−17.1669	−0.587096
$856$	4.28671	0.146517
$857$	−35.0302	−1.19661	−0.598304	−	0.801269i	$-0.704159\pi$
$857$	−35.0302	−1.19661	−0.598304	+	0.801269i	$0.704159\pi$
$858$	15.6060	0.532780
$859$	19.1909	0.654787	0.327393	−	0.944888i	$-0.393830\pi$
$859$	19.1909	0.654787	0.327393	+	0.944888i	$0.393830\pi$
$860$	−10.5893	−0.361091
$861$	−69.1760	−2.35751
$862$	−6.82633	−0.232506
$863$	−28.7753	−0.979522	−0.489761	−	0.871857i	$-0.662916\pi$
$863$	−28.7753	−0.979522	−0.489761	+	0.871857i	$0.662916\pi$
$864$	−1.89905	−0.0646069
$865$	3.84171	0.130622
$866$	30.6831	1.04265
$867$	35.6638	1.21121
$868$	22.1588	0.752119
$869$	−10.9390	−0.371081
$870$	6.86070	0.232599
$871$	−9.75581	−0.330563
$872$	−2.93400	−0.0993577
$873$	−5.10726	−0.172854
$874$	2.55001	0.0862552
$875$	−31.7771	−1.07426
$876$	−2.71339	−0.0916771
$877$	17.5597	0.592948	0.296474	−	0.955041i	$-0.404189\pi$
$877$	17.5597	0.592948	0.296474	+	0.955041i	$0.404189\pi$
$878$	24.8620	0.839052
$879$	−36.9322	−1.24569
$880$	−3.91947	−0.132125
$881$	54.9231	1.85041	0.925203	−	0.379472i	$-0.123894\pi$
$881$	54.9231	1.85041	0.925203	+	0.379472i	$0.123894\pi$
$882$	−5.89195	−0.198392
$883$	−22.3278	−0.751389	−0.375695	−	0.926743i	$-0.622596\pi$
$883$	−22.3278	−0.751389	−0.375695	+	0.926743i	$0.622596\pi$
$884$	2.37502	0.0798805
$885$	4.79586	0.161211
$886$	−18.4032	−0.618269
$887$	−26.7278	−0.897432	−0.448716	−	0.893674i	$-0.648119\pi$
$887$	−26.7278	−0.897432	−0.448716	+	0.893674i	$0.648119\pi$
$888$	17.3293	0.581533
$889$	−10.2814	−0.344829
$890$	7.40569	0.248239
$891$	35.7226	1.19675
$892$	−6.03849	−0.202184
$893$	−11.8685	−0.397164
$894$	−0.145407	−0.00486315
$895$	−16.4025	−0.548276
$896$	−3.11806	−0.104167
$897$	1.80032	0.0601110
$898$	−34.9356	−1.16582
$899$	18.0912	0.603375
$900$	−7.77776	−0.259259
$901$	−5.29222	−0.176309
$902$	32.2653	1.07432
$903$	63.2705	2.10551
$904$	6.07148	0.201934
$905$	13.0241	0.432937
$906$	−21.0657	−0.699862
$907$	9.35460	0.310614	0.155307	−	0.987866i	$-0.450363\pi$
$907$	9.35460	0.310614	0.155307	+	0.987866i	$0.450363\pi$
$908$	−12.5895	−0.417797
$909$	3.02918	0.100472
$910$	7.68338	0.254701
$911$	−43.4068	−1.43813	−0.719066	−	0.694941i	$-0.755430\pi$
$911$	−43.4068	−1.43813	−0.719066	+	0.694941i	$0.755430\pi$
$912$	15.1991	0.503292
$913$	45.7263	1.51332
$914$	−20.5308	−0.679099
$915$	4.33990	0.143473
$916$	2.24373	0.0741349
$917$	18.7437	0.618973
$918$	2.17065	0.0716422
$919$	−43.5352	−1.43609	−0.718046	−	0.695995i	$-0.754963\pi$
$919$	−43.5352	−1.43609	−0.718046	+	0.695995i	$0.754963\pi$
$920$	−0.452154	−0.0149071
$921$	36.0681	1.18848
$922$	28.8729	0.950877
$923$	−15.8562	−0.521914
$924$	23.4187	0.770420
$925$	−27.4032	−0.901013
$926$	−0.541309	−0.0177885
$927$	34.4183	1.13044
$928$	−2.54569	−0.0835662
$929$	34.7916	1.14147	0.570737	−	0.821133i	$-0.306657\pi$
$929$	34.7916	1.14147	0.570737	+	0.821133i	$0.306657\pi$
$930$	19.1525	0.628036
$931$	−18.2072	−0.596717
$932$	−12.5640	−0.411548
$933$	−23.6767	−0.775141
$934$	26.9876	0.883060
$935$	4.48004	0.146513
$936$	4.49716	0.146994
$937$	−31.5074	−1.02930	−0.514650	−	0.857400i	$-0.672078\pi$
$937$	−31.5074	−1.02930	−0.514650	+	0.857400i	$0.672078\pi$
$938$	−14.6398	−0.478007
$939$	3.87827	0.126563
$940$	2.10446	0.0686400
$941$	17.6385	0.574998	0.287499	−	0.957781i	$-0.407176\pi$
$941$	17.6385	0.574998	0.287499	+	0.957781i	$0.407176\pi$
$942$	−17.4575	−0.568797
$943$	3.72216	0.121210
$944$	−1.77952	−0.0579185
$945$	7.02224	0.228434
$946$	−29.5108	−0.959479
$947$	−32.7012	−1.06265	−0.531324	−	0.847169i	$-0.678305\pi$
$947$	−32.7012	−1.06265	−0.531324	+	0.847169i	$0.678305\pi$
$948$	−7.52167	−0.244293
$949$	−2.48095	−0.0805349
$950$	−24.0347	−0.779789
$951$	42.8691	1.39013
$952$	3.56401	0.115510
$953$	−1.05582	−0.0342015	−0.0171007	−	0.999854i	$-0.505444\pi$
$953$	−1.05582	−0.0342015	−0.0171007	+	0.999854i	$0.505444\pi$
$954$	−10.0210	−0.324440
$955$	2.53624	0.0820709
$956$	9.37918	0.303344
$957$	19.1198	0.618056
$958$	−27.1549	−0.877334
$959$	37.4481	1.20926
$960$	−2.69503	−0.0869817
$961$	19.5039	0.629158
$962$	15.8447	0.510855
$963$	−9.27791	−0.298977
$964$	−19.6524	−0.632963
$965$	28.6120	0.921054
$966$	2.70161	0.0869229
$967$	−56.3158	−1.81099	−0.905496	−	0.424354i	$-0.860501\pi$
$967$	−56.3158	−1.81099	−0.905496	+	0.424354i	$0.860501\pi$
$968$	0.0769632	0.00247369
$969$	−17.3729	−0.558098
$970$	2.79845	0.0898528
$971$	18.2290	0.584997	0.292498	−	0.956266i	$-0.405513\pi$
$971$	18.2290	0.584997	0.292498	+	0.956266i	$0.405513\pi$
$972$	18.8657	0.605118
$973$	−20.5780	−0.659699
$974$	20.0393	0.642101
$975$	−16.9687	−0.543433
$976$	−1.61034	−0.0515456
$977$	31.8478	1.01890	0.509451	−	0.860500i	$-0.329848\pi$
$977$	31.8478	1.01890	0.509451	+	0.860500i	$0.329848\pi$
$978$	−13.3095	−0.425592
$979$	20.6387	0.659614
$980$	3.22841	0.103128
$981$	6.35018	0.202745
$982$	−35.6506	−1.13766
$983$	46.7864	1.49225	0.746127	−	0.665804i	$-0.231911\pi$
$983$	46.7864	1.49225	0.746127	+	0.665804i	$0.231911\pi$
$984$	22.1856	0.707252
$985$	23.7361	0.756294
$986$	2.90978	0.0926662
$987$	−12.5741	−0.400238
$988$	13.8970	0.442124
$989$	−3.40440	−0.108254
$990$	8.48308	0.269610
$991$	20.2720	0.643961	0.321981	−	0.946746i	$-0.395651\pi$
$991$	20.2720	0.643961	0.321981	+	0.946746i	$0.395651\pi$
$992$	−7.10661	−0.225635
$993$	60.3827	1.91619
$994$	−23.7942	−0.754708
$995$	−1.92151	−0.0609158
$996$	31.4414	0.996259
$997$	−7.11500	−0.225334	−0.112667	−	0.993633i	$-0.535939\pi$
$997$	−7.11500	−0.225334	−0.112667	+	0.993633i	$0.535939\pi$
$998$	21.9554	0.694986
$999$	14.4813	0.458170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.b.1.17	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.b.1.17	✓	82	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.27252 q^{3} +1.00000 q^{4} +1.18592 q^{5} +2.27252 q^{6} +3.11806 q^{7} -1.00000 q^{8} +2.16434 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.27252 q^{3} +1.00000 q^{4} +1.18592 q^{5} +2.27252 q^{6} +3.11806 q^{7} -1.00000 q^{8} +2.16434 q^{9} -1.18592 q^{10} -3.30500 q^{11} -2.27252 q^{12} -2.07784 q^{13} -3.11806 q^{14} -2.69503 q^{15} +1.00000 q^{16} -1.14302 q^{17} -2.16434 q^{18} -6.68822 q^{19} +1.18592 q^{20} -7.08585 q^{21} +3.30500 q^{22} +0.381268 q^{23} +2.27252 q^{24} -3.59359 q^{25} +2.07784 q^{26} +1.89905 q^{27} +3.11806 q^{28} +2.54569 q^{29} +2.69503 q^{30} +7.10661 q^{31} -1.00000 q^{32} +7.51068 q^{33} +1.14302 q^{34} +3.69777 q^{35} +2.16434 q^{36} +7.62559 q^{37} +6.68822 q^{38} +4.72193 q^{39} -1.18592 q^{40} +9.76257 q^{41} +7.08585 q^{42} -8.92914 q^{43} -3.30500 q^{44} +2.56674 q^{45} -0.381268 q^{46} +1.77454 q^{47} -2.27252 q^{48} +2.72228 q^{49} +3.59359 q^{50} +2.59754 q^{51} -2.07784 q^{52} +4.63002 q^{53} -1.89905 q^{54} -3.91947 q^{55} -3.11806 q^{56} +15.1991 q^{57} -2.54569 q^{58} -1.77952 q^{59} -2.69503 q^{60} -1.61034 q^{61} -7.10661 q^{62} +6.74854 q^{63} +1.00000 q^{64} -2.46415 q^{65} -7.51068 q^{66} +4.69517 q^{67} -1.14302 q^{68} -0.866440 q^{69} -3.69777 q^{70} +7.63111 q^{71} -2.16434 q^{72} +1.19400 q^{73} -7.62559 q^{74} +8.16650 q^{75} -6.68822 q^{76} -10.3052 q^{77} -4.72193 q^{78} +3.30984 q^{79} +1.18592 q^{80} -10.8086 q^{81} -9.76257 q^{82} -13.8355 q^{83} -7.08585 q^{84} -1.35553 q^{85} +8.92914 q^{86} -5.78512 q^{87} +3.30500 q^{88} -6.24467 q^{89} -2.56674 q^{90} -6.47882 q^{91} +0.381268 q^{92} -16.1499 q^{93} -1.77454 q^{94} -7.93170 q^{95} +2.27252 q^{96} -2.35973 q^{97} -2.72228 q^{98} -7.15316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * q^99

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