LMFDB - Embedded newform 8042.2.a.a.1.53

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:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.53
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.59153 q^{3} +1.00000 q^{4} -1.20895 q^{5} +1.59153 q^{6} +0.467535 q^{7} +1.00000 q^{8} -0.467036 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.59153 q^{3} +1.00000 q^{4} -1.20895 q^{5} +1.59153 q^{6} +0.467535 q^{7} +1.00000 q^{8} -0.467036 q^{9} -1.20895 q^{10} -3.54567 q^{11} +1.59153 q^{12} +1.50984 q^{13} +0.467535 q^{14} -1.92408 q^{15} +1.00000 q^{16} +3.82933 q^{17} -0.467036 q^{18} -4.53504 q^{19} -1.20895 q^{20} +0.744096 q^{21} -3.54567 q^{22} -1.59168 q^{23} +1.59153 q^{24} -3.53844 q^{25} +1.50984 q^{26} -5.51789 q^{27} +0.467535 q^{28} +1.13991 q^{29} -1.92408 q^{30} -1.51691 q^{31} +1.00000 q^{32} -5.64304 q^{33} +3.82933 q^{34} -0.565226 q^{35} -0.467036 q^{36} +8.67298 q^{37} -4.53504 q^{38} +2.40295 q^{39} -1.20895 q^{40} +2.66861 q^{41} +0.744096 q^{42} -10.1113 q^{43} -3.54567 q^{44} +0.564622 q^{45} -1.59168 q^{46} -7.70855 q^{47} +1.59153 q^{48} -6.78141 q^{49} -3.53844 q^{50} +6.09449 q^{51} +1.50984 q^{52} -9.11813 q^{53} -5.51789 q^{54} +4.28653 q^{55} +0.467535 q^{56} -7.21765 q^{57} +1.13991 q^{58} +8.65938 q^{59} -1.92408 q^{60} +0.572026 q^{61} -1.51691 q^{62} -0.218356 q^{63} +1.00000 q^{64} -1.82531 q^{65} -5.64304 q^{66} +0.363374 q^{67} +3.82933 q^{68} -2.53321 q^{69} -0.565226 q^{70} -3.76741 q^{71} -0.467036 q^{72} +3.81409 q^{73} +8.67298 q^{74} -5.63154 q^{75} -4.53504 q^{76} -1.65773 q^{77} +2.40295 q^{78} -1.67985 q^{79} -1.20895 q^{80} -7.38077 q^{81} +2.66861 q^{82} +2.98527 q^{83} +0.744096 q^{84} -4.62947 q^{85} -10.1113 q^{86} +1.81420 q^{87} -3.54567 q^{88} -9.06120 q^{89} +0.564622 q^{90} +0.705902 q^{91} -1.59168 q^{92} -2.41420 q^{93} -7.70855 q^{94} +5.48263 q^{95} +1.59153 q^{96} -15.3940 q^{97} -6.78141 q^{98} +1.65596 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	1.59153	0.918870	0.459435	−	0.888212i	$-0.348052\pi$
$3$	1.59153	0.918870	0.459435	+	0.888212i	$0.348052\pi$
$4$	1.00000	0.500000
$5$	−1.20895	−0.540658	−0.270329	−	0.962768i	$-0.587132\pi$
$5$	−1.20895	−0.540658	−0.270329	+	0.962768i	$0.587132\pi$
$6$	1.59153	0.649739
$7$	0.467535	0.176712	0.0883559	−	0.996089i	$-0.471839\pi$
$7$	0.467535	0.176712	0.0883559	+	0.996089i	$0.471839\pi$
$8$	1.00000	0.353553
$9$	−0.467036	−0.155679
$10$	−1.20895	−0.382303
$11$	−3.54567	−1.06906	−0.534530	−	0.845150i	$-0.679511\pi$
$11$	−3.54567	−1.06906	−0.534530	+	0.845150i	$0.679511\pi$
$12$	1.59153	0.459435
$13$	1.50984	0.418753	0.209377	−	0.977835i	$-0.432857\pi$
$13$	1.50984	0.418753	0.209377	+	0.977835i	$0.432857\pi$
$14$	0.467535	0.124954
$15$	−1.92408	−0.496794
$16$	1.00000	0.250000
$17$	3.82933	0.928750	0.464375	−	0.885639i	$-0.346279\pi$
$17$	3.82933	0.928750	0.464375	+	0.885639i	$0.346279\pi$
$18$	−0.467036	−0.110081
$19$	−4.53504	−1.04041	−0.520205	−	0.854042i	$-0.674144\pi$
$19$	−4.53504	−1.04041	−0.520205	+	0.854042i	$0.674144\pi$
$20$	−1.20895	−0.270329
$21$	0.744096	0.162375
$22$	−3.54567	−0.755939
$23$	−1.59168	−0.331889	−0.165944	−	0.986135i	$-0.553067\pi$
$23$	−1.59168	−0.331889	−0.165944	+	0.986135i	$0.553067\pi$
$24$	1.59153	0.324869
$25$	−3.53844	−0.707689
$26$	1.50984	0.296103
$27$	−5.51789	−1.06192
$28$	0.467535	0.0883559
$29$	1.13991	0.211676	0.105838	−	0.994383i	$-0.466247\pi$
$29$	1.13991	0.211676	0.105838	+	0.994383i	$0.466247\pi$
$30$	−1.92408	−0.351287
$31$	−1.51691	−0.272445	−0.136222	−	0.990678i	$-0.543496\pi$
$31$	−1.51691	−0.272445	−0.136222	+	0.990678i	$0.543496\pi$
$32$	1.00000	0.176777
$33$	−5.64304	−0.982327
$34$	3.82933	0.656725
$35$	−0.565226	−0.0955406
$36$	−0.467036	−0.0778393
$37$	8.67298	1.42583	0.712914	−	0.701251i	$-0.247375\pi$
$37$	8.67298	1.42583	0.712914	+	0.701251i	$0.247375\pi$
$38$	−4.53504	−0.735681
$39$	2.40295	0.384780
$40$	−1.20895	−0.191151
$41$	2.66861	0.416767	0.208383	−	0.978047i	$-0.433180\pi$
$41$	2.66861	0.416767	0.208383	+	0.978047i	$0.433180\pi$
$42$	0.744096	0.114817
$43$	−10.1113	−1.54196	−0.770981	−	0.636858i	$-0.780234\pi$
$43$	−10.1113	−1.54196	−0.770981	+	0.636858i	$0.780234\pi$
$44$	−3.54567	−0.534530
$45$	0.564622	0.0841689
$46$	−1.59168	−0.234681
$47$	−7.70855	−1.12441	−0.562204	−	0.826999i	$-0.690046\pi$
$47$	−7.70855	−1.12441	−0.562204	+	0.826999i	$0.690046\pi$
$48$	1.59153	0.229717
$49$	−6.78141	−0.968773
$50$	−3.53844	−0.500412
$51$	6.09449	0.853400
$52$	1.50984	0.209377
$53$	−9.11813	−1.25247	−0.626236	−	0.779634i	$-0.715405\pi$
$53$	−9.11813	−1.25247	−0.626236	+	0.779634i	$0.715405\pi$
$54$	−5.51789	−0.750889
$55$	4.28653	0.577996
$56$	0.467535	0.0624770
$57$	−7.21765	−0.956001
$58$	1.13991	0.149678
$59$	8.65938	1.12736	0.563678	−	0.825995i	$-0.309386\pi$
$59$	8.65938	1.12736	0.563678	+	0.825995i	$0.309386\pi$
$60$	−1.92408	−0.248397
$61$	0.572026	0.0732405	0.0366202	−	0.999329i	$-0.488341\pi$
$61$	0.572026	0.0732405	0.0366202	+	0.999329i	$0.488341\pi$
$62$	−1.51691	−0.192647
$63$	−0.218356	−0.0275103
$64$	1.00000	0.125000
$65$	−1.82531	−0.226402
$66$	−5.64304	−0.694610
$67$	0.363374	0.0443932	0.0221966	−	0.999754i	$-0.492934\pi$
$67$	0.363374	0.0443932	0.0221966	+	0.999754i	$0.492934\pi$
$68$	3.82933	0.464375
$69$	−2.53321	−0.304962
$70$	−0.565226	−0.0675574
$71$	−3.76741	−0.447109	−0.223555	−	0.974691i	$-0.571766\pi$
$71$	−3.76741	−0.447109	−0.223555	+	0.974691i	$0.571766\pi$
$72$	−0.467036	−0.0550407
$73$	3.81409	0.446405	0.223203	−	0.974772i	$-0.428349\pi$
$73$	3.81409	0.446405	0.223203	+	0.974772i	$0.428349\pi$
$74$	8.67298	1.00821
$75$	−5.63154	−0.650274
$76$	−4.53504	−0.520205
$77$	−1.65773	−0.188915
$78$	2.40295	0.272080
$79$	−1.67985	−0.188998	−0.0944990	−	0.995525i	$-0.530125\pi$
$79$	−1.67985	−0.188998	−0.0944990	+	0.995525i	$0.530125\pi$
$80$	−1.20895	−0.135165
$81$	−7.38077	−0.820085
$82$	2.66861	0.294698
$83$	2.98527	0.327676	0.163838	−	0.986487i	$-0.447613\pi$
$83$	2.98527	0.327676	0.163838	+	0.986487i	$0.447613\pi$
$84$	0.744096	0.0811875
$85$	−4.62947	−0.502136
$86$	−10.1113	−1.09033
$87$	1.81420	0.194503
$88$	−3.54567	−0.377970
$89$	−9.06120	−0.960486	−0.480243	−	0.877136i	$-0.659451\pi$
$89$	−9.06120	−0.960486	−0.480243	+	0.877136i	$0.659451\pi$
$90$	0.564622	0.0595164
$91$	0.705902	0.0739986
$92$	−1.59168	−0.165944
$93$	−2.41420	−0.250341
$94$	−7.70855	−0.795077
$95$	5.48263	0.562506
$96$	1.59153	0.162435
$97$	−15.3940	−1.56303	−0.781513	−	0.623889i	$-0.785552\pi$
$97$	−15.3940	−1.56303	−0.781513	+	0.623889i	$0.785552\pi$
$98$	−6.78141	−0.685026
$99$	1.65596	0.166430
$100$	−3.53844	−0.353844
$101$	−5.04864	−0.502358	−0.251179	−	0.967941i	$-0.580818\pi$
$101$	−5.04864	−0.502358	−0.251179	+	0.967941i	$0.580818\pi$
$102$	6.09449	0.603445
$103$	−7.55411	−0.744329	−0.372165	−	0.928167i	$-0.621384\pi$
$103$	−7.55411	−0.744329	−0.372165	+	0.928167i	$0.621384\pi$
$104$	1.50984	0.148052
$105$	−0.899573	−0.0877894
$106$	−9.11813	−0.885631
$107$	2.40147	0.232159	0.116080	−	0.993240i	$-0.462967\pi$
$107$	2.40147	0.232159	0.116080	+	0.993240i	$0.462967\pi$
$108$	−5.51789	−0.530959
$109$	−2.69590	−0.258221	−0.129110	−	0.991630i	$-0.541212\pi$
$109$	−2.69590	−0.258221	−0.129110	+	0.991630i	$0.541212\pi$
$110$	4.28653	0.408705
$111$	13.8033	1.31015
$112$	0.467535	0.0441779
$113$	−17.7134	−1.66634	−0.833170	−	0.553017i	$-0.813477\pi$
$113$	−17.7134	−1.66634	−0.833170	+	0.553017i	$0.813477\pi$
$114$	−7.21765	−0.675994
$115$	1.92426	0.179438
$116$	1.13991	0.105838
$117$	−0.705148	−0.0651909
$118$	8.65938	0.797160
$119$	1.79035	0.164121
$120$	−1.92408	−0.175643
$121$	1.57178	0.142889
$122$	0.572026	0.0517888
$123$	4.24717	0.382954
$124$	−1.51691	−0.136222
$125$	10.3225	0.923276
$126$	−0.218356	−0.0194527
$127$	22.0230	1.95423	0.977114	−	0.212718i	$-0.0682315\pi$
$127$	22.0230	1.95423	0.977114	+	0.212718i	$0.0682315\pi$
$128$	1.00000	0.0883883
$129$	−16.0925	−1.41686
$130$	−1.82531	−0.160091
$131$	3.01623	0.263529	0.131765	−	0.991281i	$-0.457936\pi$
$131$	3.01623	0.263529	0.131765	+	0.991281i	$0.457936\pi$
$132$	−5.64304	−0.491163
$133$	−2.12029	−0.183853
$134$	0.363374	0.0313907
$135$	6.67084	0.574135
$136$	3.82933	0.328363
$137$	5.25994	0.449387	0.224693	−	0.974430i	$-0.427862\pi$
$137$	5.25994	0.449387	0.224693	+	0.974430i	$0.427862\pi$
$138$	−2.53321	−0.215641
$139$	−1.33622	−0.113337	−0.0566685	−	0.998393i	$-0.518048\pi$
$139$	−1.33622	−0.113337	−0.0566685	+	0.998393i	$0.518048\pi$
$140$	−0.565226	−0.0477703
$141$	−12.2684	−1.03318
$142$	−3.76741	−0.316154
$143$	−5.35338	−0.447672
$144$	−0.467036	−0.0389197
$145$	−1.37810	−0.114445
$146$	3.81409	0.315656
$147$	−10.7928	−0.890176
$148$	8.67298	0.712914
$149$	4.99843	0.409487	0.204744	−	0.978816i	$-0.434364\pi$
$149$	4.99843	0.409487	0.204744	+	0.978816i	$0.434364\pi$
$150$	−5.63154	−0.459813
$151$	−2.40241	−0.195505	−0.0977527	−	0.995211i	$-0.531165\pi$
$151$	−2.40241	−0.195505	−0.0977527	+	0.995211i	$0.531165\pi$
$152$	−4.53504	−0.367840
$153$	−1.78844	−0.144587
$154$	−1.65773	−0.133583
$155$	1.83386	0.147299
$156$	2.40295	0.192390
$157$	−4.96397	−0.396168	−0.198084	−	0.980185i	$-0.563472\pi$
$157$	−4.96397	−0.396168	−0.198084	+	0.980185i	$0.563472\pi$
$158$	−1.67985	−0.133642
$159$	−14.5118	−1.15086
$160$	−1.20895	−0.0955757
$161$	−0.744168	−0.0586486
$162$	−7.38077	−0.579888
$163$	0.471025	0.0368935	0.0184467	−	0.999830i	$-0.494128\pi$
$163$	0.471025	0.0368935	0.0184467	+	0.999830i	$0.494128\pi$
$164$	2.66861	0.208383
$165$	6.82214	0.531103
$166$	2.98527	0.231702
$167$	−0.160711	−0.0124362	−0.00621811	−	0.999981i	$-0.501979\pi$
$167$	−0.160711	−0.0124362	−0.00621811	+	0.999981i	$0.501979\pi$
$168$	0.744096	0.0574083
$169$	−10.7204	−0.824646
$170$	−4.62947	−0.355064
$171$	2.11803	0.161970
$172$	−10.1113	−0.770981
$173$	−8.13822	−0.618738	−0.309369	−	0.950942i	$-0.600118\pi$
$173$	−8.13822	−0.618738	−0.309369	+	0.950942i	$0.600118\pi$
$174$	1.81420	0.137534
$175$	−1.65435	−0.125057
$176$	−3.54567	−0.267265
$177$	13.7816	1.03589
$178$	−9.06120	−0.679166
$179$	−21.0409	−1.57267	−0.786335	−	0.617800i	$-0.788024\pi$
$179$	−21.0409	−1.57267	−0.786335	+	0.617800i	$0.788024\pi$
$180$	0.564622	0.0420845
$181$	5.90698	0.439062	0.219531	−	0.975605i	$-0.429547\pi$
$181$	5.90698	0.439062	0.219531	+	0.975605i	$0.429547\pi$
$182$	0.705902	0.0523249
$183$	0.910396	0.0672984
$184$	−1.59168	−0.117340
$185$	−10.4852	−0.770886
$186$	−2.41420	−0.177018
$187$	−13.5776	−0.992889
$188$	−7.70855	−0.562204
$189$	−2.57981	−0.187653
$190$	5.48263	0.397752
$191$	19.2673	1.39414	0.697068	−	0.717005i	$-0.254487\pi$
$191$	19.2673	1.39414	0.697068	+	0.717005i	$0.254487\pi$
$192$	1.59153	0.114859
$193$	7.26344	0.522834	0.261417	−	0.965226i	$-0.415810\pi$
$193$	7.26344	0.522834	0.261417	+	0.965226i	$0.415810\pi$
$194$	−15.3940	−1.10523
$195$	−2.90504	−0.208034
$196$	−6.78141	−0.484386
$197$	−15.0292	−1.07079	−0.535393	−	0.844603i	$-0.679837\pi$
$197$	−15.0292	−1.07079	−0.535393	+	0.844603i	$0.679837\pi$
$198$	1.65596	0.117684
$199$	25.1850	1.78532	0.892658	−	0.450734i	$-0.148838\pi$
$199$	25.1850	1.78532	0.892658	+	0.450734i	$0.148838\pi$
$200$	−3.53844	−0.250206
$201$	0.578320	0.0407916
$202$	−5.04864	−0.355221
$203$	0.532950	0.0374057
$204$	6.09449	0.426700
$205$	−3.22621	−0.225328
$206$	−7.55411	−0.526320
$207$	0.743373	0.0516680
$208$	1.50984	0.104688
$209$	16.0798	1.11226
$210$	−0.899573	−0.0620765
$211$	−19.9823	−1.37564	−0.687818	−	0.725883i	$-0.741431\pi$
$211$	−19.9823	−1.37564	−0.687818	+	0.725883i	$0.741431\pi$
$212$	−9.11813	−0.626236
$213$	−5.99594	−0.410835
$214$	2.40147	0.164161
$215$	12.2241	0.833674
$216$	−5.51789	−0.375445
$217$	−0.709208	−0.0481442
$218$	−2.69590	−0.182590
$219$	6.07023	0.410188
$220$	4.28653	0.288998
$221$	5.78167	0.388917
$222$	13.8033	0.926416
$223$	16.3252	1.09322	0.546609	−	0.837388i	$-0.315919\pi$
$223$	16.3252	1.09322	0.546609	+	0.837388i	$0.315919\pi$
$224$	0.467535	0.0312385
$225$	1.65258	0.110172
$226$	−17.7134	−1.17828
$227$	3.15936	0.209694	0.104847	−	0.994488i	$-0.466565\pi$
$227$	3.15936	0.209694	0.104847	+	0.994488i	$0.466565\pi$
$228$	−7.21765	−0.478000
$229$	−25.8453	−1.70790	−0.853951	−	0.520353i	$-0.825800\pi$
$229$	−25.8453	−1.70790	−0.853951	+	0.520353i	$0.825800\pi$
$230$	1.92426	0.126882
$231$	−2.63832	−0.173589
$232$	1.13991	0.0748389
$233$	−24.0451	−1.57525	−0.787625	−	0.616155i	$-0.788689\pi$
$233$	−24.0451	−1.57525	−0.787625	+	0.616155i	$0.788689\pi$
$234$	−0.705148	−0.0460970
$235$	9.31924	0.607920
$236$	8.65938	0.563678
$237$	−2.67353	−0.173664
$238$	1.79035	0.116051
$239$	30.7966	1.99207	0.996034	−	0.0889682i	$-0.0283570\pi$
$239$	30.7966	1.99207	0.996034	+	0.0889682i	$0.0283570\pi$
$240$	−1.92408	−0.124199
$241$	2.48117	0.159826	0.0799130	−	0.996802i	$-0.474536\pi$
$241$	2.48117	0.159826	0.0799130	+	0.996802i	$0.474536\pi$
$242$	1.57178	0.101038
$243$	4.80696	0.308366
$244$	0.572026	0.0366202
$245$	8.19837	0.523775
$246$	4.24717	0.270789
$247$	−6.84717	−0.435675
$248$	−1.51691	−0.0963237
$249$	4.75115	0.301092
$250$	10.3225	0.652855
$251$	−4.51563	−0.285024	−0.142512	−	0.989793i	$-0.545518\pi$
$251$	−4.51563	−0.285024	−0.142512	+	0.989793i	$0.545518\pi$
$252$	−0.218356	−0.0137551
$253$	5.64358	0.354809
$254$	22.0230	1.38185
$255$	−7.36793	−0.461398
$256$	1.00000	0.0625000
$257$	−14.4262	−0.899883	−0.449941	−	0.893058i	$-0.648555\pi$
$257$	−14.4262	−0.899883	−0.449941	+	0.893058i	$0.648555\pi$
$258$	−16.0925	−1.00187
$259$	4.05492	0.251961
$260$	−1.82531	−0.113201
$261$	−0.532380	−0.0329535
$262$	3.01623	0.186343
$263$	18.3434	1.13110	0.565550	−	0.824714i	$-0.308664\pi$
$263$	18.3434	1.13110	0.565550	+	0.824714i	$0.308664\pi$
$264$	−5.64304	−0.347305
$265$	11.0233	0.677159
$266$	−2.12029	−0.130003
$267$	−14.4212	−0.882561
$268$	0.363374	0.0221966
$269$	−17.2814	−1.05367	−0.526833	−	0.849969i	$-0.676621\pi$
$269$	−17.2814	−1.05367	−0.526833	+	0.849969i	$0.676621\pi$
$270$	6.67084	0.405974
$271$	−18.9665	−1.15214	−0.576068	−	0.817402i	$-0.695414\pi$
$271$	−18.9665	−1.15214	−0.576068	+	0.817402i	$0.695414\pi$
$272$	3.82933	0.232187
$273$	1.12346	0.0679951
$274$	5.25994	0.317764
$275$	12.5462	0.756562
$276$	−2.53321	−0.152481
$277$	0.897548	0.0539285	0.0269642	−	0.999636i	$-0.491416\pi$
$277$	0.897548	0.0539285	0.0269642	+	0.999636i	$0.491416\pi$
$278$	−1.33622	−0.0801414
$279$	0.708450	0.0424138
$280$	−0.565226	−0.0337787
$281$	27.0321	1.61260	0.806301	−	0.591505i	$-0.201466\pi$
$281$	27.0321	1.61260	0.806301	+	0.591505i	$0.201466\pi$
$282$	−12.2684	−0.730572
$283$	−7.92214	−0.470922	−0.235461	−	0.971884i	$-0.575660\pi$
$283$	−7.92214	−0.470922	−0.235461	+	0.971884i	$0.575660\pi$
$284$	−3.76741	−0.223555
$285$	8.72576	0.516869
$286$	−5.35338	−0.316552
$287$	1.24767	0.0736476
$288$	−0.467036	−0.0275204
$289$	−2.33620	−0.137424
$290$	−1.37810	−0.0809246
$291$	−24.5000	−1.43622
$292$	3.81409	0.223203
$293$	−15.6766	−0.915835	−0.457917	−	0.888995i	$-0.651404\pi$
$293$	−15.6766	−0.915835	−0.457917	+	0.888995i	$0.651404\pi$
$294$	−10.7928	−0.629449
$295$	−10.4687	−0.609514
$296$	8.67298	0.504107
$297$	19.5646	1.13525
$298$	4.99843	0.289551
$299$	−2.40318	−0.138979
$300$	−5.63154	−0.325137
$301$	−4.72740	−0.272483
$302$	−2.40241	−0.138243
$303$	−8.03505	−0.461602
$304$	−4.53504	−0.260102
$305$	−0.691550	−0.0395980
$306$	−1.78844	−0.102238
$307$	11.4925	0.655914	0.327957	−	0.944693i	$-0.393640\pi$
$307$	11.4925	0.655914	0.327957	+	0.944693i	$0.393640\pi$
$308$	−1.65773	−0.0944577
$309$	−12.0226	−0.683941
$310$	1.83386	0.104156
$311$	−26.9016	−1.52545	−0.762724	−	0.646724i	$-0.776139\pi$
$311$	−26.9016	−1.52545	−0.762724	+	0.646724i	$0.776139\pi$
$312$	2.40295	0.136040
$313$	−8.00290	−0.452350	−0.226175	−	0.974087i	$-0.572622\pi$
$313$	−8.00290	−0.452350	−0.226175	+	0.974087i	$0.572622\pi$
$314$	−4.96397	−0.280133
$315$	0.263981	0.0148736
$316$	−1.67985	−0.0944990
$317$	9.63069	0.540914	0.270457	−	0.962732i	$-0.412825\pi$
$317$	9.63069	0.540914	0.270457	+	0.962732i	$0.412825\pi$
$318$	−14.5118	−0.813779
$319$	−4.04176	−0.226295
$320$	−1.20895	−0.0675823
$321$	3.82201	0.213324
$322$	−0.744168	−0.0414708
$323$	−17.3662	−0.966280
$324$	−7.38077	−0.410043
$325$	−5.34247	−0.296347
$326$	0.471025	0.0260876
$327$	−4.29061	−0.237271
$328$	2.66861	0.147349
$329$	−3.60402	−0.198696
$330$	6.82214	0.375546
$331$	27.9303	1.53519	0.767593	−	0.640937i	$-0.221454\pi$
$331$	27.9303	1.53519	0.767593	+	0.640937i	$0.221454\pi$
$332$	2.98527	0.163838
$333$	−4.05059	−0.221971
$334$	−0.160711	−0.00879373
$335$	−0.439300	−0.0240015
$336$	0.744096	0.0405938
$337$	20.2874	1.10513	0.552563	−	0.833471i	$-0.313650\pi$
$337$	20.2874	1.10513	0.552563	+	0.833471i	$0.313650\pi$
$338$	−10.7204	−0.583113
$339$	−28.1915	−1.53115
$340$	−4.62947	−0.251068
$341$	5.37845	0.291260
$342$	2.11803	0.114530
$343$	−6.44330	−0.347905
$344$	−10.1113	−0.545166
$345$	3.06252	0.164880
$346$	−8.13822	−0.437514
$347$	2.60711	0.139957	0.0699784	−	0.997549i	$-0.477707\pi$
$347$	2.60711	0.139957	0.0699784	+	0.997549i	$0.477707\pi$
$348$	1.81420	0.0972515
$349$	7.67867	0.411030	0.205515	−	0.978654i	$-0.434113\pi$
$349$	7.67867	0.411030	0.205515	+	0.978654i	$0.434113\pi$
$350$	−1.65435	−0.0884286
$351$	−8.33111	−0.444682
$352$	−3.54567	−0.188985
$353$	−12.3417	−0.656881	−0.328441	−	0.944525i	$-0.606523\pi$
$353$	−12.3417	−0.656881	−0.328441	+	0.944525i	$0.606523\pi$
$354$	13.7816	0.732486
$355$	4.55460	0.241733
$356$	−9.06120	−0.480243
$357$	2.84939	0.150806
$358$	−21.0409	−1.11205
$359$	−24.7367	−1.30555	−0.652777	−	0.757550i	$-0.726396\pi$
$359$	−24.7367	−1.30555	−0.652777	+	0.757550i	$0.726396\pi$
$360$	0.564622	0.0297582
$361$	1.56658	0.0824518
$362$	5.90698	0.310464
$363$	2.50153	0.131296
$364$	0.705902	0.0369993
$365$	−4.61103	−0.241352
$366$	0.910396	0.0475872
$367$	5.82849	0.304245	0.152122	−	0.988362i	$-0.451389\pi$
$367$	5.82849	0.304245	0.152122	+	0.988362i	$0.451389\pi$
$368$	−1.59168	−0.0829722
$369$	−1.24634	−0.0648817
$370$	−10.4852	−0.545099
$371$	−4.26305	−0.221326
$372$	−2.41420	−0.125171
$373$	−6.45818	−0.334392	−0.167196	−	0.985924i	$-0.553471\pi$
$373$	−6.45818	−0.334392	−0.167196	+	0.985924i	$0.553471\pi$
$374$	−13.5776	−0.702079
$375$	16.4286	0.848370
$376$	−7.70855	−0.397538
$377$	1.72108	0.0886402
$378$	−2.57981	−0.132691
$379$	−4.12216	−0.211741	−0.105871	−	0.994380i	$-0.533763\pi$
$379$	−4.12216	−0.211741	−0.105871	+	0.994380i	$0.533763\pi$
$380$	5.48263	0.281253
$381$	35.0503	1.79568
$382$	19.2673	0.985803
$383$	11.9602	0.611138	0.305569	−	0.952170i	$-0.401153\pi$
$383$	11.9602	0.611138	0.305569	+	0.952170i	$0.401153\pi$
$384$	1.59153	0.0812174
$385$	2.00411	0.102139
$386$	7.26344	0.369699
$387$	4.72235	0.240051
$388$	−15.3940	−0.781513
$389$	27.0701	1.37251	0.686254	−	0.727362i	$-0.259254\pi$
$389$	27.0701	1.37251	0.686254	+	0.727362i	$0.259254\pi$
$390$	−2.90504	−0.147102
$391$	−6.09508	−0.308242
$392$	−6.78141	−0.342513
$393$	4.80041	0.242149
$394$	−15.0292	−0.757161
$395$	2.03085	0.102183
$396$	1.65596	0.0832149
$397$	−11.6477	−0.584581	−0.292290	−	0.956330i	$-0.594417\pi$
$397$	−11.6477	−0.584581	−0.292290	+	0.956330i	$0.594417\pi$
$398$	25.1850	1.26241
$399$	−3.37450	−0.168937
$400$	−3.53844	−0.176922
$401$	12.3210	0.615282	0.307641	−	0.951502i	$-0.400460\pi$
$401$	12.3210	0.615282	0.307641	+	0.951502i	$0.400460\pi$
$402$	0.578320	0.0288440
$403$	−2.29028	−0.114087
$404$	−5.04864	−0.251179
$405$	8.92297	0.443386
$406$	0.532950	0.0264498
$407$	−30.7515	−1.52430
$408$	6.09449	0.301722
$409$	16.2604	0.804025	0.402013	−	0.915634i	$-0.368311\pi$
$409$	16.2604	0.804025	0.402013	+	0.915634i	$0.368311\pi$
$410$	−3.22621	−0.159331
$411$	8.37134	0.412928
$412$	−7.55411	−0.372165
$413$	4.04857	0.199217
$414$	0.743373	0.0365348
$415$	−3.60904	−0.177161
$416$	1.50984	0.0740258
$417$	−2.12664	−0.104142
$418$	16.0798	0.786487
$419$	33.1673	1.62033	0.810165	−	0.586202i	$-0.199378\pi$
$419$	33.1673	1.62033	0.810165	+	0.586202i	$0.199378\pi$
$420$	−0.899573	−0.0438947
$421$	8.37947	0.408391	0.204195	−	0.978930i	$-0.434542\pi$
$421$	8.37947	0.408391	0.204195	+	0.978930i	$0.434542\pi$
$422$	−19.9823	−0.972722
$423$	3.60017	0.175046
$424$	−9.11813	−0.442815
$425$	−13.5499	−0.657266
$426$	−5.99594	−0.290504
$427$	0.267443	0.0129425
$428$	2.40147	0.116080
$429$	−8.52006	−0.411352
$430$	12.2241	0.589497
$431$	15.6043	0.751633	0.375817	−	0.926694i	$-0.377362\pi$
$431$	15.6043	0.751633	0.375817	+	0.926694i	$0.377362\pi$
$432$	−5.51789	−0.265479
$433$	−9.36151	−0.449886	−0.224943	−	0.974372i	$-0.572220\pi$
$433$	−9.36151	−0.449886	−0.224943	+	0.974372i	$0.572220\pi$
$434$	−0.709208	−0.0340431
$435$	−2.19328	−0.105160
$436$	−2.69590	−0.129110
$437$	7.21834	0.345300
$438$	6.07023	0.290047
$439$	29.9385	1.42889	0.714444	−	0.699692i	$-0.246680\pi$
$439$	29.9385	1.42889	0.714444	+	0.699692i	$0.246680\pi$
$440$	4.28653	0.204352
$441$	3.16716	0.150817
$442$	5.78167	0.275006
$443$	−1.35466	−0.0643619	−0.0321809	−	0.999482i	$-0.510245\pi$
$443$	−1.35466	−0.0643619	−0.0321809	+	0.999482i	$0.510245\pi$
$444$	13.8033	0.655075
$445$	10.9545	0.519294
$446$	16.3252	0.773021
$447$	7.95515	0.376265
$448$	0.467535	0.0220890
$449$	−13.0979	−0.618130	−0.309065	−	0.951041i	$-0.600016\pi$
$449$	−13.0979	−0.618130	−0.309065	+	0.951041i	$0.600016\pi$
$450$	1.65258	0.0779034
$451$	−9.46200	−0.445548
$452$	−17.7134	−0.833170
$453$	−3.82351	−0.179644
$454$	3.15936	0.148276
$455$	−0.853399	−0.0400079
$456$	−7.21765	−0.337997
$457$	15.2413	0.712957	0.356479	−	0.934303i	$-0.383977\pi$
$457$	15.2413	0.712957	0.356479	+	0.934303i	$0.383977\pi$
$458$	−25.8453	−1.20767
$459$	−21.1298	−0.986256
$460$	1.92426	0.0897191
$461$	20.2112	0.941331	0.470666	−	0.882312i	$-0.344014\pi$
$461$	20.2112	0.941331	0.470666	+	0.882312i	$0.344014\pi$
$462$	−2.63832	−0.122746
$463$	−34.2594	−1.59217	−0.796086	−	0.605184i	$-0.793100\pi$
$463$	−34.2594	−1.59217	−0.796086	+	0.605184i	$0.793100\pi$
$464$	1.13991	0.0529191
$465$	2.91864	0.135349
$466$	−24.0451	−1.11387
$467$	−14.9612	−0.692324	−0.346162	−	0.938175i	$-0.612515\pi$
$467$	−14.9612	−0.692324	−0.346162	+	0.938175i	$0.612515\pi$
$468$	−0.705148	−0.0325955
$469$	0.169890	0.00784480
$470$	9.31924	0.429865
$471$	−7.90030	−0.364027
$472$	8.65938	0.398580
$473$	35.8514	1.64845
$474$	−2.67353	−0.122799
$475$	16.0470	0.736286
$476$	1.79035	0.0820605
$477$	4.25849	0.194983
$478$	30.7966	1.40861
$479$	1.08841	0.0497305	0.0248653	−	0.999691i	$-0.492084\pi$
$479$	1.08841	0.0497305	0.0248653	+	0.999691i	$0.492084\pi$
$480$	−1.92408	−0.0878216
$481$	13.0948	0.597070
$482$	2.48117	0.113014
$483$	−1.18436	−0.0538904
$484$	1.57178	0.0714445
$485$	18.6106	0.845063
$486$	4.80696	0.218048
$487$	−18.4840	−0.837591	−0.418796	−	0.908080i	$-0.637548\pi$
$487$	−18.4840	−0.837591	−0.418796	+	0.908080i	$0.637548\pi$
$488$	0.572026	0.0258944
$489$	0.749649	0.0339003
$490$	8.19837	0.370365
$491$	14.8352	0.669502	0.334751	−	0.942307i	$-0.391348\pi$
$491$	14.8352	0.669502	0.334751	+	0.942307i	$0.391348\pi$
$492$	4.24717	0.191477
$493$	4.36511	0.196595
$494$	−6.84717	−0.308069
$495$	−2.00196	−0.0899816
$496$	−1.51691	−0.0681111
$497$	−1.76140	−0.0790095
$498$	4.75115	0.212904
$499$	10.1880	0.456077	0.228039	−	0.973652i	$-0.426769\pi$
$499$	10.1880	0.456077	0.228039	+	0.973652i	$0.426769\pi$
$500$	10.3225	0.461638
$501$	−0.255777	−0.0114273
$502$	−4.51563	−0.201542
$503$	11.4225	0.509305	0.254652	−	0.967033i	$-0.418039\pi$
$503$	11.4225	0.509305	0.254652	+	0.967033i	$0.418039\pi$
$504$	−0.218356	−0.00972634
$505$	6.10354	0.271604
$506$	5.64358	0.250888
$507$	−17.0618	−0.757742
$508$	22.0230	0.977114
$509$	−15.1092	−0.669705	−0.334852	−	0.942271i	$-0.608686\pi$
$509$	−15.1092	−0.669705	−0.334852	+	0.942271i	$0.608686\pi$
$510$	−7.36793	−0.326257
$511$	1.78322	0.0788850
$512$	1.00000	0.0441942
$513$	25.0238	1.10483
$514$	−14.4262	−0.636313
$515$	9.13253	0.402427
$516$	−16.0925	−0.708431
$517$	27.3320	1.20206
$518$	4.05492	0.178163
$519$	−12.9522	−0.568539
$520$	−1.82531	−0.0800453
$521$	9.36682	0.410368	0.205184	−	0.978723i	$-0.434221\pi$
$521$	9.36682	0.410368	0.205184	+	0.978723i	$0.434221\pi$
$522$	−0.532380	−0.0233017
$523$	−7.48581	−0.327332	−0.163666	−	0.986516i	$-0.552332\pi$
$523$	−7.48581	−0.327332	−0.163666	+	0.986516i	$0.552332\pi$
$524$	3.01623	0.131765
$525$	−2.63294	−0.114911
$526$	18.3434	0.799809
$527$	−5.80874	−0.253033
$528$	−5.64304	−0.245582
$529$	−20.4665	−0.889850
$530$	11.0233	0.478823
$531$	−4.04424	−0.175505
$532$	−2.12029	−0.0919263
$533$	4.02916	0.174522
$534$	−14.4212	−0.624065
$535$	−2.90326	−0.125519
$536$	0.363374	0.0156954
$537$	−33.4872	−1.44508
$538$	−17.2814	−0.745054
$539$	24.0446	1.03568
$540$	6.67084	0.287067
$541$	−20.7651	−0.892761	−0.446380	−	0.894843i	$-0.647287\pi$
$541$	−20.7651	−0.892761	−0.446380	+	0.894843i	$0.647287\pi$
$542$	−18.9665	−0.814683
$543$	9.40113	0.403441
$544$	3.82933	0.164181
$545$	3.25921	0.139609
$546$	1.12346	0.0480798
$547$	46.2532	1.97764	0.988822	−	0.149100i	$-0.0476377\pi$
$547$	46.2532	1.97764	0.988822	+	0.149100i	$0.0476377\pi$
$548$	5.25994	0.224693
$549$	−0.267157	−0.0114020
$550$	12.5462	0.534970
$551$	−5.16955	−0.220230
$552$	−2.53321	−0.107820
$553$	−0.785389	−0.0333982
$554$	0.897548	0.0381332
$555$	−16.6875	−0.708344
$556$	−1.33622	−0.0566685
$557$	−21.2022	−0.898364	−0.449182	−	0.893440i	$-0.648285\pi$
$557$	−21.2022	−0.898364	−0.449182	+	0.893440i	$0.648285\pi$
$558$	0.708450	0.0299911
$559$	−15.2664	−0.645701
$560$	−0.565226	−0.0238852
$561$	−21.6091	−0.912336
$562$	27.0321	1.14028
$563$	3.90334	0.164506	0.0822531	−	0.996611i	$-0.473788\pi$
$563$	3.90334	0.164506	0.0822531	+	0.996611i	$0.473788\pi$
$564$	−12.2684	−0.516592
$565$	21.4146	0.900920
$566$	−7.92214	−0.332992
$567$	−3.45077	−0.144919
$568$	−3.76741	−0.158077
$569$	23.1461	0.970335	0.485168	−	0.874421i	$-0.338759\pi$
$569$	23.1461	0.970335	0.485168	+	0.874421i	$0.338759\pi$
$570$	8.72576	0.365482
$571$	9.72475	0.406968	0.203484	−	0.979078i	$-0.434773\pi$
$571$	9.72475	0.406968	0.203484	+	0.979078i	$0.434773\pi$
$572$	−5.35338	−0.223836
$573$	30.6645	1.28103
$574$	1.24767	0.0520767
$575$	5.63208	0.234874
$576$	−0.467036	−0.0194598
$577$	−45.6542	−1.90061	−0.950304	−	0.311323i	$-0.899228\pi$
$577$	−45.6542	−1.90061	−0.950304	+	0.311323i	$0.899228\pi$
$578$	−2.33620	−0.0971732
$579$	11.5600	0.480416
$580$	−1.37810	−0.0572223
$581$	1.39572	0.0579043
$582$	−24.5000	−1.01556
$583$	32.3299	1.33897
$584$	3.81409	0.157828
$585$	0.852487	0.0352460
$586$	−15.6766	−0.647593
$587$	43.6250	1.80060	0.900299	−	0.435273i	$-0.143348\pi$
$587$	43.6250	1.80060	0.900299	+	0.435273i	$0.143348\pi$
$588$	−10.7928	−0.445088
$589$	6.87923	0.283454
$590$	−10.4687	−0.430991
$591$	−23.9194	−0.983913
$592$	8.67298	0.356457
$593$	40.5145	1.66373	0.831865	−	0.554978i	$-0.187273\pi$
$593$	40.5145	1.66373	0.831865	+	0.554978i	$0.187273\pi$
$594$	19.5646	0.802746
$595$	−2.16444	−0.0887334
$596$	4.99843	0.204744
$597$	40.0826	1.64047
$598$	−2.40318	−0.0982733
$599$	28.7765	1.17578	0.587889	−	0.808942i	$-0.299959\pi$
$599$	28.7765	1.17578	0.587889	+	0.808942i	$0.299959\pi$
$600$	−5.63154	−0.229906
$601$	−17.8561	−0.728365	−0.364182	−	0.931328i	$-0.618651\pi$
$601$	−17.8561	−0.728365	−0.364182	+	0.931328i	$0.618651\pi$
$602$	−4.72740	−0.192674
$603$	−0.169709	−0.00691108
$604$	−2.40241	−0.0977527
$605$	−1.90020	−0.0772541
$606$	−8.03505	−0.326402
$607$	−28.4535	−1.15489	−0.577446	−	0.816429i	$-0.695951\pi$
$607$	−28.4535	−1.15489	−0.577446	+	0.816429i	$0.695951\pi$
$608$	−4.53504	−0.183920
$609$	0.848205	0.0343710
$610$	−0.691550	−0.0280000
$611$	−11.6387	−0.470849
$612$	−1.78844	−0.0722933
$613$	−20.9347	−0.845546	−0.422773	−	0.906236i	$-0.638943\pi$
$613$	−20.9347	−0.845546	−0.422773	+	0.906236i	$0.638943\pi$
$614$	11.4925	0.463801
$615$	−5.13460	−0.207047
$616$	−1.65773	−0.0667917
$617$	−17.6868	−0.712045	−0.356023	−	0.934477i	$-0.615867\pi$
$617$	−17.6868	−0.712045	−0.356023	+	0.934477i	$0.615867\pi$
$618$	−12.0226	−0.483620
$619$	1.30770	0.0525608	0.0262804	−	0.999655i	$-0.491634\pi$
$619$	1.30770	0.0525608	0.0262804	+	0.999655i	$0.491634\pi$
$620$	1.83386	0.0736497
$621$	8.78272	0.352439
$622$	−26.9016	−1.07865
$623$	−4.23643	−0.169729
$624$	2.40295	0.0961949
$625$	5.21281	0.208512
$626$	−8.00290	−0.319860
$627$	25.5914	1.02202
$628$	−4.96397	−0.198084
$629$	33.2117	1.32424
$630$	0.263981	0.0105173
$631$	25.2442	1.00495	0.502477	−	0.864590i	$-0.332422\pi$
$631$	25.2442	1.00495	0.502477	+	0.864590i	$0.332422\pi$
$632$	−1.67985	−0.0668209
$633$	−31.8024	−1.26403
$634$	9.63069	0.382484
$635$	−26.6247	−1.05657
$636$	−14.5118	−0.575429
$637$	−10.2388	−0.405677
$638$	−4.04176	−0.160015
$639$	1.75952	0.0696054
$640$	−1.20895	−0.0477879
$641$	−39.7479	−1.56995	−0.784973	−	0.619530i	$-0.787323\pi$
$641$	−39.7479	−1.56995	−0.784973	+	0.619530i	$0.787323\pi$
$642$	3.82201	0.150843
$643$	−48.5042	−1.91282	−0.956409	−	0.292031i	$-0.905669\pi$
$643$	−48.5042	−1.91282	−0.956409	+	0.292031i	$0.905669\pi$
$644$	−0.744168	−0.0293243
$645$	19.4549	0.766038
$646$	−17.3662	−0.683263
$647$	29.0541	1.14223	0.571117	−	0.820868i	$-0.306510\pi$
$647$	29.0541	1.14223	0.571117	+	0.820868i	$0.306510\pi$
$648$	−7.38077	−0.289944
$649$	−30.7033	−1.20521
$650$	−5.34247	−0.209549
$651$	−1.12872	−0.0442382
$652$	0.471025	0.0184467
$653$	9.78151	0.382780	0.191390	−	0.981514i	$-0.438700\pi$
$653$	9.78151	0.382780	0.191390	+	0.981514i	$0.438700\pi$
$654$	−4.29061	−0.167776
$655$	−3.64646	−0.142479
$656$	2.66861	0.104192
$657$	−1.78132	−0.0694958
$658$	−3.60402	−0.140499
$659$	25.5450	0.995093	0.497546	−	0.867437i	$-0.334234\pi$
$659$	25.5450	0.995093	0.497546	+	0.867437i	$0.334234\pi$
$660$	6.82214	0.265551
$661$	11.2692	0.438321	0.219161	−	0.975689i	$-0.429668\pi$
$661$	11.2692	0.438321	0.219161	+	0.975689i	$0.429668\pi$
$662$	27.9303	1.08554
$663$	9.20169	0.357364
$664$	2.98527	0.115851
$665$	2.56332	0.0994014
$666$	−4.05059	−0.156957
$667$	−1.81438	−0.0702530
$668$	−0.160711	−0.00621811
$669$	25.9820	1.00452
$670$	−0.439300	−0.0169717
$671$	−2.02822	−0.0782984
$672$	0.744096	0.0287041
$673$	1.89280	0.0729620	0.0364810	−	0.999334i	$-0.488385\pi$
$673$	1.89280	0.0729620	0.0364810	+	0.999334i	$0.488385\pi$
$674$	20.2874	0.781442
$675$	19.5247	0.751508
$676$	−10.7204	−0.412323
$677$	−28.9683	−1.11334	−0.556672	−	0.830732i	$-0.687922\pi$
$677$	−28.9683	−1.11334	−0.556672	+	0.830732i	$0.687922\pi$
$678$	−28.1915	−1.08269
$679$	−7.19725	−0.276205
$680$	−4.62947	−0.177532
$681$	5.02822	0.192682
$682$	5.37845	0.205952
$683$	42.9233	1.64241	0.821206	−	0.570631i	$-0.193301\pi$
$683$	42.9233	1.64241	0.821206	+	0.570631i	$0.193301\pi$
$684$	2.11803	0.0809848
$685$	−6.35899	−0.242965
$686$	−6.44330	−0.246006
$687$	−41.1335	−1.56934
$688$	−10.1113	−0.385490
$689$	−13.7669	−0.524476
$690$	3.06252	0.116588
$691$	−13.4517	−0.511725	−0.255863	−	0.966713i	$-0.582360\pi$
$691$	−13.4517	−0.511725	−0.255863	+	0.966713i	$0.582360\pi$
$692$	−8.13822	−0.309369
$693$	0.774218	0.0294101
$694$	2.60711	0.0989644
$695$	1.61542	0.0612766
$696$	1.81420	0.0687672
$697$	10.2190	0.387072
$698$	7.67867	0.290642
$699$	−38.2685	−1.44745
$700$	−1.65435	−0.0625285
$701$	7.13215	0.269378	0.134689	−	0.990888i	$-0.456997\pi$
$701$	7.13215	0.269378	0.134689	+	0.990888i	$0.456997\pi$
$702$	−8.33111	−0.314437
$703$	−39.3323	−1.48345
$704$	−3.54567	−0.133632
$705$	14.8318	0.558599
$706$	−12.3417	−0.464485
$707$	−2.36042	−0.0887726
$708$	13.7816	0.517946
$709$	−48.4571	−1.81985	−0.909923	−	0.414778i	$-0.863859\pi$
$709$	−48.4571	−1.81985	−0.909923	+	0.414778i	$0.863859\pi$
$710$	4.55460	0.170931
$711$	0.784551	0.0294229
$712$	−9.06120	−0.339583
$713$	2.41443	0.0904213
$714$	2.84939	0.106636
$715$	6.47196	0.242038
$716$	−21.0409	−0.786335
$717$	49.0137	1.83045
$718$	−24.7367	−0.923167
$719$	−9.28097	−0.346122	−0.173061	−	0.984911i	$-0.555366\pi$
$719$	−9.28097	−0.346122	−0.173061	+	0.984911i	$0.555366\pi$
$720$	0.564622	0.0210422
$721$	−3.53182	−0.131532
$722$	1.56658	0.0583022
$723$	3.94885	0.146859
$724$	5.90698	0.219531
$725$	−4.03352	−0.149801
$726$	2.50153	0.0928405
$727$	12.8416	0.476267	0.238133	−	0.971232i	$-0.423464\pi$
$727$	12.8416	0.476267	0.238133	+	0.971232i	$0.423464\pi$
$728$	0.705902	0.0261625
$729$	29.7927	1.10343
$730$	−4.61103	−0.170662
$731$	−38.7196	−1.43210
$732$	0.910396	0.0336492
$733$	10.1049	0.373234	0.186617	−	0.982433i	$-0.440248\pi$
$733$	10.1049	0.373234	0.186617	+	0.982433i	$0.440248\pi$
$734$	5.82849	0.215133
$735$	13.0479	0.481281
$736$	−1.59168	−0.0586702
$737$	−1.28840	−0.0474590
$738$	−1.24634	−0.0458783
$739$	−11.4484	−0.421137	−0.210569	−	0.977579i	$-0.567532\pi$
$739$	−11.4484	−0.421137	−0.210569	+	0.977579i	$0.567532\pi$
$740$	−10.4852	−0.385443
$741$	−10.8975	−0.400328
$742$	−4.26305	−0.156501
$743$	35.3755	1.29780	0.648900	−	0.760873i	$-0.275229\pi$
$743$	35.3755	1.29780	0.648900	+	0.760873i	$0.275229\pi$
$744$	−2.41420	−0.0885089
$745$	−6.04284	−0.221393
$746$	−6.45818	−0.236451
$747$	−1.39423	−0.0510122
$748$	−13.5776	−0.496445
$749$	1.12277	0.0410253
$750$	16.4286	0.599888
$751$	−11.8175	−0.431225	−0.215613	−	0.976479i	$-0.569175\pi$
$751$	−11.8175	−0.431225	−0.215613	+	0.976479i	$0.569175\pi$
$752$	−7.70855	−0.281102
$753$	−7.18676	−0.261900
$754$	1.72108	0.0626781
$755$	2.90439	0.105702
$756$	−2.57981	−0.0938267
$757$	1.73088	0.0629098	0.0314549	−	0.999505i	$-0.489986\pi$
$757$	1.73088	0.0629098	0.0314549	+	0.999505i	$0.489986\pi$
$758$	−4.12216	−0.149724
$759$	8.98192	0.326023
$760$	5.48263	0.198876
$761$	4.22127	0.153021	0.0765105	−	0.997069i	$-0.475622\pi$
$761$	4.22127	0.153021	0.0765105	+	0.997069i	$0.475622\pi$
$762$	35.0503	1.26974
$763$	−1.26043	−0.0456307
$764$	19.2673	0.697068
$765$	2.16213	0.0781719
$766$	11.9602	0.432140
$767$	13.0742	0.472083
$768$	1.59153	0.0574293
$769$	11.8992	0.429097	0.214548	−	0.976713i	$-0.431172\pi$
$769$	11.8992	0.429097	0.214548	+	0.976713i	$0.431172\pi$
$770$	2.00411	0.0722229
$771$	−22.9597	−0.826875
$772$	7.26344	0.261417
$773$	9.69596	0.348739	0.174370	−	0.984680i	$-0.444211\pi$
$773$	9.69596	0.348739	0.174370	+	0.984680i	$0.444211\pi$
$774$	4.72235	0.169741
$775$	5.36749	0.192806
$776$	−15.3940	−0.552613
$777$	6.45353	0.231519
$778$	27.0701	0.970510
$779$	−12.1022	−0.433608
$780$	−2.90504	−0.104017
$781$	13.3580	0.477987
$782$	−6.09508	−0.217960
$783$	−6.28991	−0.224783
$784$	−6.78141	−0.242193
$785$	6.00118	0.214191
$786$	4.80041	0.171225
$787$	−29.6467	−1.05679	−0.528396	−	0.848998i	$-0.677206\pi$
$787$	−29.6467	−1.05679	−0.528396	+	0.848998i	$0.677206\pi$
$788$	−15.0292	−0.535393
$789$	29.1940	1.03933
$790$	2.03085	0.0722545
$791$	−8.28166	−0.294462
$792$	1.65596	0.0588418
$793$	0.863666	0.0306697
$794$	−11.6477	−0.413361
$795$	17.5440	0.622220
$796$	25.1850	0.892658
$797$	38.7158	1.37139	0.685693	−	0.727891i	$-0.259499\pi$
$797$	38.7158	1.37139	0.685693	+	0.727891i	$0.259499\pi$
$798$	−3.37450	−0.119456
$799$	−29.5186	−1.04429
$800$	−3.53844	−0.125103
$801$	4.23191	0.149527
$802$	12.3210	0.435070
$803$	−13.5235	−0.477234
$804$	0.578320	0.0203958
$805$	0.899660	0.0317089
$806$	−2.29028	−0.0806717
$807$	−27.5039	−0.968182
$808$	−5.04864	−0.177611
$809$	27.6274	0.971326	0.485663	−	0.874146i	$-0.338578\pi$
$809$	27.6274	0.971326	0.485663	+	0.874146i	$0.338578\pi$
$810$	8.92297	0.313521
$811$	−7.38398	−0.259287	−0.129643	−	0.991561i	$-0.541383\pi$
$811$	−7.38398	−0.259287	−0.129643	+	0.991561i	$0.541383\pi$
$812$	0.532950	0.0187029
$813$	−30.1858	−1.05866
$814$	−30.7515	−1.07784
$815$	−0.569444	−0.0199468
$816$	6.09449	0.213350
$817$	45.8552	1.60427
$818$	16.2604	0.568532
$819$	−0.329682	−0.0115200
$820$	−3.22621	−0.112664
$821$	15.0873	0.526549	0.263274	−	0.964721i	$-0.415198\pi$
$821$	15.0873	0.526549	0.263274	+	0.964721i	$0.415198\pi$
$822$	8.37134	0.291984
$823$	−18.7500	−0.653585	−0.326792	−	0.945096i	$-0.605968\pi$
$823$	−18.7500	−0.653585	−0.326792	+	0.945096i	$0.605968\pi$
$824$	−7.55411	−0.263160
$825$	19.9676	0.695182
$826$	4.04857	0.140868
$827$	−12.3165	−0.428285	−0.214143	−	0.976802i	$-0.568696\pi$
$827$	−12.3165	−0.428285	−0.214143	+	0.976802i	$0.568696\pi$
$828$	0.743373	0.0258340
$829$	3.00997	0.104540	0.0522702	−	0.998633i	$-0.483354\pi$
$829$	3.00997	0.104540	0.0522702	+	0.998633i	$0.483354\pi$
$830$	−3.60904	−0.125272
$831$	1.42847	0.0495532
$832$	1.50984	0.0523441
$833$	−25.9683	−0.899748
$834$	−2.12664	−0.0736395
$835$	0.194292	0.00672374
$836$	16.0798	0.556130
$837$	8.37012	0.289314
$838$	33.1673	1.14575
$839$	−13.5435	−0.467573	−0.233787	−	0.972288i	$-0.575112\pi$
$839$	−13.5435	−0.467573	−0.233787	+	0.972288i	$0.575112\pi$
$840$	−0.899573	−0.0310382
$841$	−27.7006	−0.955193
$842$	8.37947	0.288776
$843$	43.0224	1.48177
$844$	−19.9823	−0.687818
$845$	12.9604	0.445851
$846$	3.60017	0.123776
$847$	0.734862	0.0252502
$848$	−9.11813	−0.313118
$849$	−12.6083	−0.432716
$850$	−13.5499	−0.464757
$851$	−13.8046	−0.473216
$852$	−5.99594	−0.205418
$853$	32.1264	1.09999	0.549993	−	0.835169i	$-0.314630\pi$
$853$	32.1264	1.09999	0.549993	+	0.835169i	$0.314630\pi$
$854$	0.267443	0.00915169
$855$	−2.56058	−0.0875702
$856$	2.40147	0.0820807
$857$	4.47068	0.152715	0.0763577	−	0.997080i	$-0.475671\pi$
$857$	4.47068	0.152715	0.0763577	+	0.997080i	$0.475671\pi$
$858$	−8.52006	−0.290870
$859$	8.24010	0.281149	0.140574	−	0.990070i	$-0.455105\pi$
$859$	8.24010	0.281149	0.140574	+	0.990070i	$0.455105\pi$
$860$	12.2241	0.416837
$861$	1.98570	0.0676725
$862$	15.6043	0.531485
$863$	−49.2516	−1.67654	−0.838272	−	0.545253i	$-0.816434\pi$
$863$	−49.2516	−1.67654	−0.838272	+	0.545253i	$0.816434\pi$
$864$	−5.51789	−0.187722
$865$	9.83869	0.334526
$866$	−9.36151	−0.318117
$867$	−3.71813	−0.126274
$868$	−0.709208	−0.0240721
$869$	5.95619	0.202050
$870$	−2.19328	−0.0743591
$871$	0.548635	0.0185898
$872$	−2.69590	−0.0912949
$873$	7.18956	0.243330
$874$	7.21834	0.244164
$875$	4.82615	0.163154
$876$	6.07023	0.205094
$877$	22.1331	0.747382	0.373691	−	0.927553i	$-0.378092\pi$
$877$	22.1331	0.747382	0.373691	+	0.927553i	$0.378092\pi$
$878$	29.9385	1.01038
$879$	−24.9497	−0.841533
$880$	4.28653	0.144499
$881$	−18.6481	−0.628271	−0.314136	−	0.949378i	$-0.601715\pi$
$881$	−18.6481	−0.628271	−0.314136	+	0.949378i	$0.601715\pi$
$882$	3.16716	0.106644
$883$	0.353495	0.0118961	0.00594803	−	0.999982i	$-0.498107\pi$
$883$	0.353495	0.0118961	0.00594803	+	0.999982i	$0.498107\pi$
$884$	5.78167	0.194458
$885$	−16.6613	−0.560063
$886$	−1.35466	−0.0455107
$887$	48.5510	1.63018	0.815091	−	0.579333i	$-0.196687\pi$
$887$	48.5510	1.63018	0.815091	+	0.579333i	$0.196687\pi$
$888$	13.8033	0.463208
$889$	10.2965	0.345335
$890$	10.9545	0.367196
$891$	26.1698	0.876720
$892$	16.3252	0.546609
$893$	34.9586	1.16984
$894$	7.95515	0.266060
$895$	25.4373	0.850277
$896$	0.467535	0.0156193
$897$	−3.82473	−0.127704
$898$	−13.0979	−0.437084
$899$	−1.72914	−0.0576701
$900$	1.65258	0.0550860
$901$	−34.9164	−1.16323
$902$	−9.46200	−0.315050
$903$	−7.52379	−0.250376
$904$	−17.7134	−0.589140
$905$	−7.14123	−0.237383
$906$	−3.82351	−0.127027
$907$	−0.395100	−0.0131191	−0.00655954	−	0.999978i	$-0.502088\pi$
$907$	−0.395100	−0.0131191	−0.00655954	+	0.999978i	$0.502088\pi$
$908$	3.15936	0.104847
$909$	2.35790	0.0782065
$910$	−0.853399	−0.0282899
$911$	−34.1729	−1.13220	−0.566099	−	0.824337i	$-0.691548\pi$
$911$	−34.1729	−1.13220	−0.566099	+	0.824337i	$0.691548\pi$
$912$	−7.21765	−0.239000
$913$	−10.5848	−0.350306
$914$	15.2413	0.504137
$915$	−1.10062	−0.0363854
$916$	−25.8453	−0.853951
$917$	1.41019	0.0465687
$918$	−21.1298	−0.697388
$919$	−7.37481	−0.243273	−0.121636	−	0.992575i	$-0.538814\pi$
$919$	−7.37481	−0.243273	−0.121636	+	0.992575i	$0.538814\pi$
$920$	1.92426	0.0634410
$921$	18.2907	0.602699
$922$	20.2112	0.665622
$923$	−5.68817	−0.187228
$924$	−2.63832	−0.0867943
$925$	−30.6888	−1.00904
$926$	−34.2594	−1.12584
$927$	3.52804	0.115876
$928$	1.13991	0.0374195
$929$	19.9183	0.653500	0.326750	−	0.945111i	$-0.394047\pi$
$929$	19.9183	0.653500	0.326750	+	0.945111i	$0.394047\pi$
$930$	2.91864	0.0957061
$931$	30.7540	1.00792
$932$	−24.0451	−0.787625
$933$	−42.8146	−1.40169
$934$	−14.9612	−0.489547
$935$	16.4146	0.536814
$936$	−0.705148	−0.0230485
$937$	13.4662	0.439922	0.219961	−	0.975509i	$-0.429407\pi$
$937$	13.4662	0.439922	0.219961	+	0.975509i	$0.429407\pi$
$938$	0.169890	0.00554711
$939$	−12.7368	−0.415651
$940$	9.31924	0.303960
$941$	25.1164	0.818771	0.409386	−	0.912361i	$-0.365743\pi$
$941$	25.1164	0.818771	0.409386	+	0.912361i	$0.365743\pi$
$942$	−7.90030	−0.257406
$943$	−4.24758	−0.138320
$944$	8.65938	0.281839
$945$	3.11885	0.101456
$946$	35.8514	1.16563
$947$	14.5285	0.472113	0.236056	−	0.971739i	$-0.424145\pi$
$947$	14.5285	0.472113	0.236056	+	0.971739i	$0.424145\pi$
$948$	−2.67353	−0.0868322
$949$	5.75864	0.186934
$950$	16.0470	0.520633
$951$	15.3275	0.497029
$952$	1.79035	0.0580255
$953$	14.1096	0.457053	0.228527	−	0.973538i	$-0.426609\pi$
$953$	14.1096	0.457053	0.228527	+	0.973538i	$0.426609\pi$
$954$	4.25849	0.137874
$955$	−23.2932	−0.753751
$956$	30.7966	0.996034
$957$	−6.43257	−0.207935
$958$	1.08841	0.0351648
$959$	2.45921	0.0794119
$960$	−1.92408	−0.0620993
$961$	−28.6990	−0.925774
$962$	13.0948	0.422192
$963$	−1.12157	−0.0361422
$964$	2.48117	0.0799130
$965$	−8.78112	−0.282674
$966$	−1.18436	−0.0381063
$967$	−10.0797	−0.324140	−0.162070	−	0.986779i	$-0.551817\pi$
$967$	−10.0797	−0.324140	−0.162070	+	0.986779i	$0.551817\pi$
$968$	1.57178	0.0505189
$969$	−27.6388	−0.887885
$970$	18.6106	0.597549
$971$	−35.5105	−1.13959	−0.569793	−	0.821788i	$-0.692977\pi$
$971$	−35.5105	−1.13959	−0.569793	+	0.821788i	$0.692977\pi$
$972$	4.80696	0.154183
$973$	−0.624732	−0.0200280
$974$	−18.4840	−0.592267
$975$	−8.50270	−0.272304
$976$	0.572026	0.0183101
$977$	−6.06581	−0.194062	−0.0970312	−	0.995281i	$-0.530935\pi$
$977$	−6.06581	−0.194062	−0.0970312	+	0.995281i	$0.530935\pi$
$978$	0.749649	0.0239711
$979$	32.1280	1.02682
$980$	8.19837	0.261887
$981$	1.25908	0.0401995
$982$	14.8352	0.473409
$983$	9.70309	0.309480	0.154740	−	0.987955i	$-0.450546\pi$
$983$	9.70309	0.309480	0.154740	+	0.987955i	$0.450546\pi$
$984$	4.24717	0.135395
$985$	18.1695	0.578929
$986$	4.36511	0.139013
$987$	−5.73590	−0.182576
$988$	−6.84717	−0.217837
$989$	16.0940	0.511760
$990$	−2.00196	−0.0636266
$991$	−18.5596	−0.589564	−0.294782	−	0.955564i	$-0.595247\pi$
$991$	−18.5596	−0.589564	−0.294782	+	0.955564i	$0.595247\pi$
$992$	−1.51691	−0.0481619
$993$	44.4518	1.41064
$994$	−1.76140	−0.0558681
$995$	−30.4473	−0.965246
$996$	4.75115	0.150546
$997$	6.23500	0.197465	0.0987323	−	0.995114i	$-0.468521\pi$
$997$	6.23500	0.197465	0.0987323	+	0.995114i	$0.468521\pi$
$998$	10.1880	0.322495
$999$	−47.8565	−1.51411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.53	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.53	✓	67	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.59153 q^{3} +1.00000 q^{4} -1.20895 q^{5} +1.59153 q^{6} +0.467535 q^{7} +1.00000 q^{8} -0.467036 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.59153 q^{3} +1.00000 q^{4} -1.20895 q^{5} +1.59153 q^{6} +0.467535 q^{7} +1.00000 q^{8} -0.467036 q^{9} -1.20895 q^{10} -3.54567 q^{11} +1.59153 q^{12} +1.50984 q^{13} +0.467535 q^{14} -1.92408 q^{15} +1.00000 q^{16} +3.82933 q^{17} -0.467036 q^{18} -4.53504 q^{19} -1.20895 q^{20} +0.744096 q^{21} -3.54567 q^{22} -1.59168 q^{23} +1.59153 q^{24} -3.53844 q^{25} +1.50984 q^{26} -5.51789 q^{27} +0.467535 q^{28} +1.13991 q^{29} -1.92408 q^{30} -1.51691 q^{31} +1.00000 q^{32} -5.64304 q^{33} +3.82933 q^{34} -0.565226 q^{35} -0.467036 q^{36} +8.67298 q^{37} -4.53504 q^{38} +2.40295 q^{39} -1.20895 q^{40} +2.66861 q^{41} +0.744096 q^{42} -10.1113 q^{43} -3.54567 q^{44} +0.564622 q^{45} -1.59168 q^{46} -7.70855 q^{47} +1.59153 q^{48} -6.78141 q^{49} -3.53844 q^{50} +6.09449 q^{51} +1.50984 q^{52} -9.11813 q^{53} -5.51789 q^{54} +4.28653 q^{55} +0.467535 q^{56} -7.21765 q^{57} +1.13991 q^{58} +8.65938 q^{59} -1.92408 q^{60} +0.572026 q^{61} -1.51691 q^{62} -0.218356 q^{63} +1.00000 q^{64} -1.82531 q^{65} -5.64304 q^{66} +0.363374 q^{67} +3.82933 q^{68} -2.53321 q^{69} -0.565226 q^{70} -3.76741 q^{71} -0.467036 q^{72} +3.81409 q^{73} +8.67298 q^{74} -5.63154 q^{75} -4.53504 q^{76} -1.65773 q^{77} +2.40295 q^{78} -1.67985 q^{79} -1.20895 q^{80} -7.38077 q^{81} +2.66861 q^{82} +2.98527 q^{83} +0.744096 q^{84} -4.62947 q^{85} -10.1113 q^{86} +1.81420 q^{87} -3.54567 q^{88} -9.06120 q^{89} +0.564622 q^{90} +0.705902 q^{91} -1.59168 q^{92} -2.41420 q^{93} -7.70855 q^{94} +5.48263 q^{95} +1.59153 q^{96} -15.3940 q^{97} -6.78141 q^{98} +1.65596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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