LMFDB - Embedded newform 8041.2.a.i.1.55

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$78$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.55
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35148 q^{2} -3.16834 q^{3} -0.173514 q^{4} +0.946723 q^{5} -4.28194 q^{6} -1.01609 q^{7} -2.93745 q^{8} +7.03839 q^{9} +O(q^{10})$	\(q+1.35148 q^{2} -3.16834 q^{3} -0.173514 q^{4} +0.946723 q^{5} -4.28194 q^{6} -1.01609 q^{7} -2.93745 q^{8} +7.03839 q^{9} +1.27947 q^{10} +1.00000 q^{11} +0.549751 q^{12} -0.670329 q^{13} -1.37322 q^{14} -2.99954 q^{15} -3.62287 q^{16} -1.00000 q^{17} +9.51222 q^{18} +7.01059 q^{19} -0.164270 q^{20} +3.21931 q^{21} +1.35148 q^{22} +5.30650 q^{23} +9.30685 q^{24} -4.10372 q^{25} -0.905933 q^{26} -12.7950 q^{27} +0.176305 q^{28} +7.70636 q^{29} -4.05381 q^{30} -1.08361 q^{31} +0.978688 q^{32} -3.16834 q^{33} -1.35148 q^{34} -0.961953 q^{35} -1.22126 q^{36} -8.33194 q^{37} +9.47465 q^{38} +2.12383 q^{39} -2.78095 q^{40} -3.46047 q^{41} +4.35082 q^{42} -1.00000 q^{43} -0.173514 q^{44} +6.66341 q^{45} +7.17160 q^{46} -8.42060 q^{47} +11.4785 q^{48} -5.96757 q^{49} -5.54607 q^{50} +3.16834 q^{51} +0.116311 q^{52} +7.09260 q^{53} -17.2921 q^{54} +0.946723 q^{55} +2.98471 q^{56} -22.2120 q^{57} +10.4150 q^{58} +4.60654 q^{59} +0.520462 q^{60} +4.42971 q^{61} -1.46447 q^{62} -7.15162 q^{63} +8.56840 q^{64} -0.634616 q^{65} -4.28194 q^{66} -7.54042 q^{67} +0.173514 q^{68} -16.8128 q^{69} -1.30006 q^{70} -7.27703 q^{71} -20.6749 q^{72} +0.284343 q^{73} -11.2604 q^{74} +13.0020 q^{75} -1.21644 q^{76} -1.01609 q^{77} +2.87031 q^{78} -2.46064 q^{79} -3.42985 q^{80} +19.4238 q^{81} -4.67674 q^{82} -0.746569 q^{83} -0.558595 q^{84} -0.946723 q^{85} -1.35148 q^{86} -24.4164 q^{87} -2.93745 q^{88} -6.32785 q^{89} +9.00543 q^{90} +0.681113 q^{91} -0.920751 q^{92} +3.43325 q^{93} -11.3802 q^{94} +6.63709 q^{95} -3.10082 q^{96} +6.02995 q^{97} -8.06502 q^{98} +7.03839 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) $ 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * 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 $

$2$	1.35148	0.955638	0.477819	−	0.878458i	$-0.341428\pi$
$2$	1.35148	0.955638	0.477819	+	0.878458i	$0.341428\pi$
$3$	−3.16834	−1.82924	−0.914622	−	0.404311i	$-0.867511\pi$
$3$	−3.16834	−1.82924	−0.914622	+	0.404311i	$0.867511\pi$
$4$	−0.173514	−0.0867569
$5$	0.946723	0.423387	0.211694	−	0.977336i	$-0.432102\pi$
$5$	0.946723	0.423387	0.211694	+	0.977336i	$0.432102\pi$
$6$	−4.28194	−1.74809
$7$	−1.01609	−0.384045	−0.192023	−	0.981391i	$-0.561505\pi$
$7$	−1.01609	−0.384045	−0.192023	+	0.981391i	$0.561505\pi$
$8$	−2.93745	−1.03855
$9$	7.03839	2.34613
$10$	1.27947	0.404605
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.549751	0.158700
$13$	−0.670329	−0.185916	−0.0929579	−	0.995670i	$-0.529632\pi$
$13$	−0.670329	−0.185916	−0.0929579	+	0.995670i	$0.529632\pi$
$14$	−1.37322	−0.367008
$15$	−2.99954	−0.774478
$16$	−3.62287	−0.905716
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	9.51222	2.24205
$19$	7.01059	1.60834	0.804170	−	0.594399i	$-0.202610\pi$
$19$	7.01059	1.60834	0.804170	+	0.594399i	$0.202610\pi$
$20$	−0.164270	−0.0367318
$21$	3.21931	0.702512
$22$	1.35148	0.288136
$23$	5.30650	1.10648	0.553241	−	0.833021i	$-0.313391\pi$
$23$	5.30650	1.10648	0.553241	+	0.833021i	$0.313391\pi$
$24$	9.30685	1.89975
$25$	−4.10372	−0.820743
$26$	−0.905933	−0.177668
$27$	−12.7950	−2.46240
$28$	0.176305	0.0333186
$29$	7.70636	1.43103	0.715517	−	0.698595i	$-0.246191\pi$
$29$	7.70636	1.43103	0.715517	+	0.698595i	$0.246191\pi$
$30$	−4.05381	−0.740121
$31$	−1.08361	−0.194622	−0.0973110	−	0.995254i	$-0.531024\pi$
$31$	−1.08361	−0.194622	−0.0973110	+	0.995254i	$0.531024\pi$
$32$	0.978688	0.173009
$33$	−3.16834	−0.551538
$34$	−1.35148	−0.231776
$35$	−0.961953	−0.162600
$36$	−1.22126	−0.203543
$37$	−8.33194	−1.36976	−0.684882	−	0.728654i	$-0.740146\pi$
$37$	−8.33194	−1.36976	−0.684882	+	0.728654i	$0.740146\pi$
$38$	9.47465	1.53699
$39$	2.12383	0.340085
$40$	−2.78095	−0.439707
$41$	−3.46047	−0.540435	−0.270217	−	0.962799i	$-0.587096\pi$
$41$	−3.46047	−0.540435	−0.270217	+	0.962799i	$0.587096\pi$
$42$	4.35082	0.671347
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−0.173514	−0.0261582
$45$	6.66341	0.993322
$46$	7.17160	1.05740
$47$	−8.42060	−1.22827	−0.614135	−	0.789201i	$-0.710495\pi$
$47$	−8.42060	−1.22827	−0.614135	+	0.789201i	$0.710495\pi$
$48$	11.4785	1.65678
$49$	−5.96757	−0.852509
$50$	−5.54607	−0.784333
$51$	3.16834	0.443657
$52$	0.116311	0.0161295
$53$	7.09260	0.974243	0.487121	−	0.873334i	$-0.338047\pi$
$53$	7.09260	0.974243	0.487121	+	0.873334i	$0.338047\pi$
$54$	−17.2921	−2.35316
$55$	0.946723	0.127656
$56$	2.98471	0.398848
$57$	−22.2120	−2.94205
$58$	10.4150	1.36755
$59$	4.60654	0.599720	0.299860	−	0.953983i	$-0.403060\pi$
$59$	4.60654	0.599720	0.299860	+	0.953983i	$0.403060\pi$
$60$	0.520462	0.0671914
$61$	4.42971	0.567167	0.283583	−	0.958948i	$-0.408477\pi$
$61$	4.42971	0.567167	0.283583	+	0.958948i	$0.408477\pi$
$62$	−1.46447	−0.185988
$63$	−7.15162	−0.901020
$64$	8.56840	1.07105
$65$	−0.634616	−0.0787144
$66$	−4.28194	−0.527070
$67$	−7.54042	−0.921209	−0.460605	−	0.887605i	$-0.652367\pi$
$67$	−7.54042	−0.921209	−0.460605	+	0.887605i	$0.652367\pi$
$68$	0.173514	0.0210416
$69$	−16.8128	−2.02402
$70$	−1.30006	−0.155386
$71$	−7.27703	−0.863625	−0.431812	−	0.901963i	$-0.642126\pi$
$71$	−7.27703	−0.863625	−0.431812	+	0.901963i	$0.642126\pi$
$72$	−20.6749	−2.43656
$73$	0.284343	0.0332799	0.0166399	−	0.999862i	$-0.494703\pi$
$73$	0.284343	0.0332799	0.0166399	+	0.999862i	$0.494703\pi$
$74$	−11.2604	−1.30900
$75$	13.0020	1.50134
$76$	−1.21644	−0.139535
$77$	−1.01609	−0.115794
$78$	2.87031	0.324998
$79$	−2.46064	−0.276844	−0.138422	−	0.990373i	$-0.544203\pi$
$79$	−2.46064	−0.276844	−0.138422	+	0.990373i	$0.544203\pi$
$80$	−3.42985	−0.383469
$81$	19.4238	2.15820
$82$	−4.67674	−0.516460
$83$	−0.746569	−0.0819466	−0.0409733	−	0.999160i	$-0.513046\pi$
$83$	−0.746569	−0.0819466	−0.0409733	+	0.999160i	$0.513046\pi$
$84$	−0.558595	−0.0609478
$85$	−0.946723	−0.102686
$86$	−1.35148	−0.145733
$87$	−24.4164	−2.61771
$88$	−2.93745	−0.313133
$89$	−6.32785	−0.670751	−0.335375	−	0.942085i	$-0.608863\pi$
$89$	−6.32785	−0.670751	−0.335375	+	0.942085i	$0.608863\pi$
$90$	9.00543	0.949256
$91$	0.681113	0.0714001
$92$	−0.920751	−0.0959949
$93$	3.43325	0.356011
$94$	−11.3802	−1.17378
$95$	6.63709	0.680951
$96$	−3.10082	−0.316476
$97$	6.02995	0.612249	0.306124	−	0.951992i	$-0.400968\pi$
$97$	6.02995	0.612249	0.306124	+	0.951992i	$0.400968\pi$
$98$	−8.06502	−0.814690
$99$	7.03839	0.707385
$100$	0.712052	0.0712052
$101$	1.00935	0.100434	0.0502172	−	0.998738i	$-0.484009\pi$
$101$	1.00935	0.100434	0.0502172	+	0.998738i	$0.484009\pi$
$102$	4.28194	0.423975
$103$	15.7432	1.55123	0.775613	−	0.631209i	$-0.217441\pi$
$103$	15.7432	1.55123	0.775613	+	0.631209i	$0.217441\pi$
$104$	1.96906	0.193082
$105$	3.04780	0.297435
$106$	9.58547	0.931023
$107$	12.6326	1.22124	0.610618	−	0.791926i	$-0.290921\pi$
$107$	12.6326	1.22124	0.610618	+	0.791926i	$0.290921\pi$
$108$	2.22011	0.213630
$109$	0.749602	0.0717989	0.0358994	−	0.999355i	$-0.488570\pi$
$109$	0.749602	0.0717989	0.0358994	+	0.999355i	$0.488570\pi$
$110$	1.27947	0.121993
$111$	26.3984	2.50563
$112$	3.68115	0.347836
$113$	−4.13836	−0.389304	−0.194652	−	0.980872i	$-0.562358\pi$
$113$	−4.13836	−0.389304	−0.194652	+	0.980872i	$0.562358\pi$
$114$	−30.0189	−2.81153
$115$	5.02378	0.468470
$116$	−1.33716	−0.124152
$117$	−4.71804	−0.436183
$118$	6.22562	0.573115
$119$	1.01609	0.0931446
$120$	8.81101	0.804331
$121$	1.00000	0.0909091
$122$	5.98665	0.542006
$123$	10.9640	0.988586
$124$	0.188021	0.0168848
$125$	−8.61870	−0.770880
$126$	−9.66524	−0.861048
$127$	−8.42307	−0.747427	−0.373713	−	0.927544i	$-0.621916\pi$
$127$	−8.42307	−0.747427	−0.373713	+	0.927544i	$0.621916\pi$
$128$	9.62261	0.850527
$129$	3.16834	0.278957
$130$	−0.857668	−0.0752224
$131$	−14.7837	−1.29166	−0.645829	−	0.763482i	$-0.723488\pi$
$131$	−14.7837	−1.29166	−0.645829	+	0.763482i	$0.723488\pi$
$132$	0.549751	0.0478497
$133$	−7.12338	−0.617675
$134$	−10.1907	−0.880342
$135$	−12.1133	−1.04255
$136$	2.93745	0.251884
$137$	−9.56333	−0.817050	−0.408525	−	0.912747i	$-0.633957\pi$
$137$	−9.56333	−0.817050	−0.408525	+	0.912747i	$0.633957\pi$
$138$	−22.7221	−1.93423
$139$	11.1229	0.943432	0.471716	−	0.881750i	$-0.343635\pi$
$139$	11.1229	0.943432	0.471716	+	0.881750i	$0.343635\pi$
$140$	0.166912	0.0141067
$141$	26.6793	2.24681
$142$	−9.83473	−0.825312
$143$	−0.670329	−0.0560557
$144$	−25.4991	−2.12493
$145$	7.29579	0.605882
$146$	0.384283	0.0318035
$147$	18.9073	1.55945
$148$	1.44571	0.118836
$149$	6.51152	0.533444	0.266722	−	0.963773i	$-0.414059\pi$
$149$	6.51152	0.533444	0.266722	+	0.963773i	$0.414059\pi$
$150$	17.5719	1.43474
$151$	1.94298	0.158117	0.0790585	−	0.996870i	$-0.474809\pi$
$151$	1.94298	0.158117	0.0790585	+	0.996870i	$0.474809\pi$
$152$	−20.5933	−1.67034
$153$	−7.03839	−0.569020
$154$	−1.37322	−0.110657
$155$	−1.02588	−0.0824005
$156$	−0.368514	−0.0295048
$157$	14.1422	1.12867	0.564334	−	0.825547i	$-0.309133\pi$
$157$	14.1422	1.12867	0.564334	+	0.825547i	$0.309133\pi$
$158$	−3.32550	−0.264562
$159$	−22.4718	−1.78213
$160$	0.926546	0.0732499
$161$	−5.39187	−0.424939
$162$	26.2508	2.06246
$163$	10.1434	0.794492	0.397246	−	0.917712i	$-0.369966\pi$
$163$	10.1434	0.794492	0.397246	+	0.917712i	$0.369966\pi$
$164$	0.600439	0.0468864
$165$	−2.99954	−0.233514
$166$	−1.00897	−0.0783113
$167$	0.0934925	0.00723467	0.00361733	−	0.999993i	$-0.498849\pi$
$167$	0.0934925	0.00723467	0.00361733	+	0.999993i	$0.498849\pi$
$168$	−9.45657	−0.729591
$169$	−12.5507	−0.965435
$170$	−1.27947	−0.0981311
$171$	49.3433	3.77338
$172$	0.173514	0.0132303
$173$	−15.7950	−1.20087	−0.600434	−	0.799674i	$-0.705005\pi$
$173$	−15.7950	−1.20087	−0.600434	+	0.799674i	$0.705005\pi$
$174$	−32.9981	−2.50158
$175$	4.16974	0.315202
$176$	−3.62287	−0.273084
$177$	−14.5951	−1.09703
$178$	−8.55193	−0.640995
$179$	1.07728	0.0805193	0.0402597	−	0.999189i	$-0.487181\pi$
$179$	1.07728	0.0805193	0.0402597	+	0.999189i	$0.487181\pi$
$180$	−1.15619	−0.0861776
$181$	2.92469	0.217391	0.108695	−	0.994075i	$-0.465333\pi$
$181$	2.92469	0.217391	0.108695	+	0.994075i	$0.465333\pi$
$182$	0.920508	0.0682326
$183$	−14.0348	−1.03749
$184$	−15.5876	−1.14913
$185$	−7.88804	−0.579940
$186$	4.63995	0.340217
$187$	−1.00000	−0.0731272
$188$	1.46109	0.106561
$189$	13.0009	0.945673
$190$	8.96986	0.650742
$191$	−8.08955	−0.585339	−0.292670	−	0.956214i	$-0.594544\pi$
$191$	−8.08955	−0.585339	−0.292670	+	0.956214i	$0.594544\pi$
$192$	−27.1476	−1.95921
$193$	14.0550	1.01170	0.505850	−	0.862622i	$-0.331179\pi$
$193$	14.0550	1.01170	0.505850	+	0.862622i	$0.331179\pi$
$194$	8.14933	0.585088
$195$	2.01068	0.143988
$196$	1.03546	0.0739611
$197$	19.9826	1.42370	0.711851	−	0.702330i	$-0.247857\pi$
$197$	19.9826	1.42370	0.711851	+	0.702330i	$0.247857\pi$
$198$	9.51222	0.676004
$199$	8.70032	0.616749	0.308375	−	0.951265i	$-0.400215\pi$
$199$	8.70032	0.616749	0.308375	+	0.951265i	$0.400215\pi$
$200$	12.0545	0.852379
$201$	23.8906	1.68512
$202$	1.36412	0.0959789
$203$	−7.83034	−0.549582
$204$	−0.549751	−0.0384903
$205$	−3.27611	−0.228813
$206$	21.2766	1.48241
$207$	37.3492	2.59595
$208$	2.42851	0.168387
$209$	7.01059	0.484933
$210$	4.11902	0.284240
$211$	18.4725	1.27170	0.635848	−	0.771814i	$-0.280650\pi$
$211$	18.4725	1.27170	0.635848	+	0.771814i	$0.280650\pi$
$212$	−1.23066	−0.0845223
$213$	23.0561	1.57978
$214$	17.0726	1.16706
$215$	−0.946723	−0.0645660
$216$	37.5847	2.55732
$217$	1.10104	0.0747436
$218$	1.01307	0.0686137
$219$	−0.900897	−0.0608770
$220$	−0.164270	−0.0110750
$221$	0.670329	0.0450912
$222$	35.6769	2.39447
$223$	−10.0912	−0.675755	−0.337878	−	0.941190i	$-0.609709\pi$
$223$	−10.0912	−0.675755	−0.337878	+	0.941190i	$0.609709\pi$
$224$	−0.994433	−0.0664433
$225$	−28.8836	−1.92557
$226$	−5.59290	−0.372034
$227$	−2.23054	−0.148046	−0.0740230	−	0.997257i	$-0.523584\pi$
$227$	−2.23054	−0.148046	−0.0740230	+	0.997257i	$0.523584\pi$
$228$	3.85408	0.255243
$229$	−28.6133	−1.89082	−0.945412	−	0.325879i	$-0.894340\pi$
$229$	−28.6133	−1.89082	−0.945412	+	0.325879i	$0.894340\pi$
$230$	6.78952	0.447688
$231$	3.21931	0.211815
$232$	−22.6371	−1.48620
$233$	−5.16959	−0.338671	−0.169335	−	0.985558i	$-0.554162\pi$
$233$	−5.16959	−0.338671	−0.169335	+	0.985558i	$0.554162\pi$
$234$	−6.37632	−0.416833
$235$	−7.97197	−0.520034
$236$	−0.799298	−0.0520299
$237$	7.79615	0.506414
$238$	1.37322	0.0890125
$239$	25.0746	1.62194	0.810972	−	0.585085i	$-0.198939\pi$
$239$	25.0746	1.62194	0.810972	+	0.585085i	$0.198939\pi$
$240$	10.8669	0.701458
$241$	17.2454	1.11087	0.555436	−	0.831559i	$-0.312551\pi$
$241$	17.2454	1.11087	0.555436	+	0.831559i	$0.312551\pi$
$242$	1.35148	0.0868761
$243$	−23.1562	−1.48547
$244$	−0.768616	−0.0492056
$245$	−5.64963	−0.360942
$246$	14.8175	0.944730
$247$	−4.69941	−0.299016
$248$	3.18305	0.202124
$249$	2.36539	0.149900
$250$	−11.6480	−0.736681
$251$	9.63565	0.608197	0.304098	−	0.952641i	$-0.401645\pi$
$251$	9.63565	0.608197	0.304098	+	0.952641i	$0.401645\pi$
$252$	1.24091	0.0781697
$253$	5.30650	0.333617
$254$	−11.3836	−0.714269
$255$	2.99954	0.187839
$256$	−4.13208	−0.258255
$257$	−8.12386	−0.506752	−0.253376	−	0.967368i	$-0.581541\pi$
$257$	−8.12386	−0.506752	−0.253376	+	0.967368i	$0.581541\pi$
$258$	4.28194	0.266582
$259$	8.46598	0.526051
$260$	0.110115	0.00682902
$261$	54.2404	3.35740
$262$	−19.9798	−1.23436
$263$	17.8541	1.10093	0.550466	−	0.834857i	$-0.314450\pi$
$263$	17.8541	1.10093	0.550466	+	0.834857i	$0.314450\pi$
$264$	9.30685	0.572797
$265$	6.71472	0.412482
$266$	−9.62707	−0.590274
$267$	20.0488	1.22697
$268$	1.30837	0.0799213
$269$	10.0791	0.614535	0.307268	−	0.951623i	$-0.400585\pi$
$269$	10.0791	0.614535	0.307268	+	0.951623i	$0.400585\pi$
$270$	−16.3709	−0.996299
$271$	26.1978	1.59141	0.795703	−	0.605687i	$-0.207102\pi$
$271$	26.1978	1.59141	0.795703	+	0.605687i	$0.207102\pi$
$272$	3.62287	0.219668
$273$	−2.15800	−0.130608
$274$	−12.9246	−0.780804
$275$	−4.10372	−0.247463
$276$	2.91725	0.175598
$277$	25.9252	1.55770	0.778848	−	0.627212i	$-0.215804\pi$
$277$	25.9252	1.55770	0.778848	+	0.627212i	$0.215804\pi$
$278$	15.0323	0.901579
$279$	−7.62687	−0.456609
$280$	2.82569	0.168867
$281$	−2.17068	−0.129492	−0.0647458	−	0.997902i	$-0.520624\pi$
$281$	−2.17068	−0.129492	−0.0647458	+	0.997902i	$0.520624\pi$
$282$	36.0565	2.14713
$283$	−10.5490	−0.627074	−0.313537	−	0.949576i	$-0.601514\pi$
$283$	−10.5490	−0.627074	−0.313537	+	0.949576i	$0.601514\pi$
$284$	1.26267	0.0749254
$285$	−21.0286	−1.24562
$286$	−0.905933	−0.0535690
$287$	3.51614	0.207551
$288$	6.88839	0.405902
$289$	1.00000	0.0588235
$290$	9.86008	0.579004
$291$	−19.1049	−1.11995
$292$	−0.0493375	−0.00288726
$293$	−27.4832	−1.60558	−0.802792	−	0.596259i	$-0.796653\pi$
$293$	−27.4832	−1.60558	−0.802792	+	0.596259i	$0.796653\pi$
$294$	25.5527	1.49027
$295$	4.36111	0.253914
$296$	24.4747	1.42256
$297$	−12.7950	−0.742442
$298$	8.80016	0.509779
$299$	−3.55710	−0.205712
$300$	−2.25602	−0.130252
$301$	1.01609	0.0585663
$302$	2.62588	0.151103
$303$	−3.19798	−0.183719
$304$	−25.3984	−1.45670
$305$	4.19371	0.240131
$306$	−9.51222	−0.543777
$307$	13.9681	0.797202	0.398601	−	0.917124i	$-0.369496\pi$
$307$	13.9681	0.797202	0.398601	+	0.917124i	$0.369496\pi$
$308$	0.176305	0.0100459
$309$	−49.8799	−2.83757
$310$	−1.38645	−0.0787450
$311$	29.2927	1.66104	0.830518	−	0.556992i	$-0.188044\pi$
$311$	29.2927	1.66104	0.830518	+	0.556992i	$0.188044\pi$
$312$	−6.23865	−0.353194
$313$	−5.34851	−0.302316	−0.151158	−	0.988510i	$-0.548300\pi$
$313$	−5.34851	−0.302316	−0.151158	+	0.988510i	$0.548300\pi$
$314$	19.1128	1.07860
$315$	−6.77060	−0.381480
$316$	0.426955	0.0240181
$317$	13.2317	0.743167	0.371583	−	0.928400i	$-0.378815\pi$
$317$	13.2317	0.743167	0.371583	+	0.928400i	$0.378815\pi$
$318$	−30.3700	−1.70307
$319$	7.70636	0.431473
$320$	8.11190	0.453469
$321$	−40.0243	−2.23394
$322$	−7.28698	−0.406087
$323$	−7.01059	−0.390080
$324$	−3.37030	−0.187239
$325$	2.75084	0.152589
$326$	13.7085	0.759246
$327$	−2.37500	−0.131338
$328$	10.1650	0.561266
$329$	8.55607	0.471711
$330$	−4.05381	−0.223155
$331$	22.0201	1.21034	0.605168	−	0.796098i	$-0.293106\pi$
$331$	22.0201	1.21034	0.605168	+	0.796098i	$0.293106\pi$
$332$	0.129540	0.00710944
$333$	−58.6435	−3.21364
$334$	0.126353	0.00691372
$335$	−7.13869	−0.390028
$336$	−11.6631	−0.636276
$337$	3.32257	0.180992	0.0904958	−	0.995897i	$-0.471155\pi$
$337$	3.32257	0.180992	0.0904958	+	0.995897i	$0.471155\pi$
$338$	−16.9619	−0.922606
$339$	13.1117	0.712132
$340$	0.164270	0.00890876
$341$	−1.08361	−0.0586807
$342$	66.6863	3.60598
$343$	13.1762	0.711447
$344$	2.93745	0.158377
$345$	−15.9171	−0.856946
$346$	−21.3465	−1.14759
$347$	14.1460	0.759398	0.379699	−	0.925110i	$-0.376028\pi$
$347$	14.1460	0.759398	0.379699	+	0.925110i	$0.376028\pi$
$348$	4.23658	0.227105
$349$	10.1998	0.545981	0.272990	−	0.962017i	$-0.411987\pi$
$349$	10.1998	0.545981	0.272990	+	0.962017i	$0.411987\pi$
$350$	5.63530	0.301219
$351$	8.57687	0.457799
$352$	0.978688	0.0521642
$353$	−0.801671	−0.0426686	−0.0213343	−	0.999772i	$-0.506791\pi$
$353$	−0.801671	−0.0426686	−0.0213343	+	0.999772i	$0.506791\pi$
$354$	−19.7249	−1.04837
$355$	−6.88933	−0.365648
$356$	1.09797	0.0581923
$357$	−3.21931	−0.170384
$358$	1.45591	0.0769473
$359$	−7.14554	−0.377127	−0.188564	−	0.982061i	$-0.560383\pi$
$359$	−7.14554	−0.377127	−0.188564	+	0.982061i	$0.560383\pi$
$360$	−19.5734	−1.03161
$361$	30.1484	1.58676
$362$	3.95265	0.207747
$363$	−3.16834	−0.166295
$364$	−0.118183	−0.00619445
$365$	0.269194	0.0140903
$366$	−18.9677	−0.991460
$367$	−15.9914	−0.834743	−0.417371	−	0.908736i	$-0.637049\pi$
$367$	−15.9914	−0.834743	−0.417371	+	0.908736i	$0.637049\pi$
$368$	−19.2247	−1.00216
$369$	−24.3561	−1.26793
$370$	−10.6605	−0.554213
$371$	−7.20670	−0.374153
$372$	−0.595716	−0.0308864
$373$	−28.9969	−1.50140	−0.750700	−	0.660643i	$-0.770284\pi$
$373$	−28.9969	−1.50140	−0.750700	+	0.660643i	$0.770284\pi$
$374$	−1.35148	−0.0698831
$375$	27.3070	1.41013
$376$	24.7351	1.27561
$377$	−5.16580	−0.266052
$378$	17.5703	0.903720
$379$	20.4472	1.05030	0.525152	−	0.851008i	$-0.324008\pi$
$379$	20.4472	1.05030	0.525152	+	0.851008i	$0.324008\pi$
$380$	−1.15163	−0.0590772
$381$	26.6872	1.36723
$382$	−10.9328	−0.559372
$383$	2.88581	0.147458	0.0737289	−	0.997278i	$-0.476510\pi$
$383$	2.88581	0.147458	0.0737289	+	0.997278i	$0.476510\pi$
$384$	−30.4877	−1.55582
$385$	−0.961953	−0.0490257
$386$	18.9950	0.966818
$387$	−7.03839	−0.357782
$388$	−1.04628	−0.0531168
$389$	30.0664	1.52443	0.762213	−	0.647326i	$-0.224113\pi$
$389$	30.0664	1.52443	0.762213	+	0.647326i	$0.224113\pi$
$390$	2.71739	0.137600
$391$	−5.30650	−0.268361
$392$	17.5294	0.885370
$393$	46.8398	2.36276
$394$	27.0060	1.36054
$395$	−2.32954	−0.117212
$396$	−1.22126	−0.0613706
$397$	−0.508663	−0.0255291	−0.0127645	−	0.999919i	$-0.504063\pi$
$397$	−0.508663	−0.0255291	−0.0127645	+	0.999919i	$0.504063\pi$
$398$	11.7583	0.589389
$399$	22.5693	1.12988
$400$	14.8672	0.743361
$401$	−28.3650	−1.41648	−0.708241	−	0.705971i	$-0.750511\pi$
$401$	−28.3650	−1.41648	−0.708241	+	0.705971i	$0.750511\pi$
$402$	32.2876	1.61036
$403$	0.726375	0.0361833
$404$	−0.175137	−0.00871338
$405$	18.3889	0.913754
$406$	−10.5825	−0.525201
$407$	−8.33194	−0.412999
$408$	−9.30685	−0.460758
$409$	29.3564	1.45158	0.725791	−	0.687915i	$-0.241474\pi$
$409$	29.3564	1.45158	0.725791	+	0.687915i	$0.241474\pi$
$410$	−4.42758	−0.218662
$411$	30.2999	1.49458
$412$	−2.73167	−0.134580
$413$	−4.68065	−0.230320
$414$	50.4766	2.48079
$415$	−0.706794	−0.0346952
$416$	−0.656043	−0.0321652
$417$	−35.2412	−1.72577
$418$	9.47465	0.463420
$419$	−16.7368	−0.817646	−0.408823	−	0.912614i	$-0.634061\pi$
$419$	−16.7368	−0.817646	−0.408823	+	0.912614i	$0.634061\pi$
$420$	−0.528835	−0.0258045
$421$	−29.8240	−1.45353	−0.726766	−	0.686885i	$-0.758978\pi$
$421$	−29.8240	−1.45353	−0.726766	+	0.686885i	$0.758978\pi$
$422$	24.9651	1.21528
$423$	−59.2675	−2.88168
$424$	−20.8342	−1.01180
$425$	4.10372	0.199059
$426$	31.1598	1.50970
$427$	−4.50098	−0.217817
$428$	−2.19192	−0.105951
$429$	2.12383	0.102540
$430$	−1.27947	−0.0617016
$431$	32.5302	1.56693	0.783463	−	0.621439i	$-0.213452\pi$
$431$	32.5302	1.56693	0.783463	+	0.621439i	$0.213452\pi$
$432$	46.3546	2.23024
$433$	−3.46246	−0.166395	−0.0831976	−	0.996533i	$-0.526513\pi$
$433$	−3.46246	−0.166395	−0.0831976	+	0.996533i	$0.526513\pi$
$434$	1.48803	0.0714278
$435$	−23.1155	−1.10831
$436$	−0.130066	−0.00622905
$437$	37.2017	1.77960
$438$	−1.21754	−0.0581763
$439$	11.4920	0.548485	0.274242	−	0.961661i	$-0.411573\pi$
$439$	11.4920	0.548485	0.274242	+	0.961661i	$0.411573\pi$
$440$	−2.78095	−0.132577
$441$	−42.0021	−2.00010
$442$	0.905933	0.0430909
$443$	16.5437	0.786016	0.393008	−	0.919535i	$-0.371434\pi$
$443$	16.5437	0.786016	0.393008	+	0.919535i	$0.371434\pi$
$444$	−4.58050	−0.217381
$445$	−5.99072	−0.283987
$446$	−13.6380	−0.645777
$447$	−20.6307	−0.975799
$448$	−8.70625	−0.411332
$449$	−7.87338	−0.371568	−0.185784	−	0.982591i	$-0.559482\pi$
$449$	−7.87338	−0.371568	−0.185784	+	0.982591i	$0.559482\pi$
$450$	−39.0354	−1.84015
$451$	−3.46047	−0.162947
$452$	0.718063	0.0337748
$453$	−6.15601	−0.289235
$454$	−3.01452	−0.141478
$455$	0.644825	0.0302299
$456$	65.2465	3.05545
$457$	−8.95666	−0.418975	−0.209487	−	0.977811i	$-0.567180\pi$
$457$	−8.95666	−0.418975	−0.209487	+	0.977811i	$0.567180\pi$
$458$	−38.6702	−1.80694
$459$	12.7950	0.597220
$460$	−0.871696	−0.0406430
$461$	3.28809	0.153142	0.0765709	−	0.997064i	$-0.475603\pi$
$461$	3.28809	0.153142	0.0765709	+	0.997064i	$0.475603\pi$
$462$	4.35082	0.202419
$463$	31.2377	1.45174	0.725870	−	0.687832i	$-0.241437\pi$
$463$	31.2377	1.45174	0.725870	+	0.687832i	$0.241437\pi$
$464$	−27.9191	−1.29611
$465$	3.25033	0.150730
$466$	−6.98657	−0.323647
$467$	−42.7423	−1.97788	−0.988939	−	0.148321i	$-0.952613\pi$
$467$	−42.7423	−1.97788	−0.988939	+	0.148321i	$0.952613\pi$
$468$	0.818645	0.0378419
$469$	7.66173	0.353786
$470$	−10.7739	−0.496964
$471$	−44.8072	−2.06461
$472$	−13.5315	−0.622837
$473$	−1.00000	−0.0459800
$474$	10.5363	0.483949
$475$	−28.7695	−1.32003
$476$	−0.176305	−0.00808094
$477$	49.9205	2.28570
$478$	33.8878	1.54999
$479$	21.3471	0.975375	0.487688	−	0.873018i	$-0.337840\pi$
$479$	21.3471	0.975375	0.487688	+	0.873018i	$0.337840\pi$
$480$	−2.93561	−0.133992
$481$	5.58514	0.254661
$482$	23.3067	1.06159
$483$	17.0833	0.777316
$484$	−0.173514	−0.00788699
$485$	5.70869	0.259218
$486$	−31.2950	−1.41957
$487$	−26.2563	−1.18979	−0.594894	−	0.803804i	$-0.702806\pi$
$487$	−26.2563	−1.18979	−0.594894	+	0.803804i	$0.702806\pi$
$488$	−13.0121	−0.589028
$489$	−32.1377	−1.45332
$490$	−7.63534	−0.344929
$491$	19.4056	0.875764	0.437882	−	0.899032i	$-0.355729\pi$
$491$	19.4056	0.875764	0.437882	+	0.899032i	$0.355729\pi$
$492$	−1.90240	−0.0857667
$493$	−7.70636	−0.347077
$494$	−6.35113	−0.285751
$495$	6.66341	0.299498
$496$	3.92577	0.176272
$497$	7.39410	0.331671
$498$	3.19676	0.143250
$499$	28.8579	1.29186	0.645929	−	0.763397i	$-0.276470\pi$
$499$	28.8579	1.29186	0.645929	+	0.763397i	$0.276470\pi$
$500$	1.49546	0.0668791
$501$	−0.296216	−0.0132340
$502$	13.0223	0.581216
$503$	0.462296	0.0206128	0.0103064	−	0.999947i	$-0.496719\pi$
$503$	0.462296	0.0206128	0.0103064	+	0.999947i	$0.496719\pi$
$504$	21.0075	0.935750
$505$	0.955578	0.0425226
$506$	7.17160	0.318817
$507$	39.7648	1.76602
$508$	1.46152	0.0648444
$509$	18.0697	0.800925	0.400463	−	0.916313i	$-0.368849\pi$
$509$	18.0697	0.800925	0.400463	+	0.916313i	$0.368849\pi$
$510$	4.05381	0.179506
$511$	−0.288918	−0.0127810
$512$	−24.8296	−1.09733
$513$	−89.7006	−3.96038
$514$	−10.9792	−0.484271
$515$	14.9045	0.656769
$516$	−0.549751	−0.0242014
$517$	−8.42060	−0.370337
$518$	11.4416	0.502714
$519$	50.0438	2.19668
$520$	1.86415	0.0817485
$521$	27.2859	1.19542	0.597708	−	0.801714i	$-0.296078\pi$
$521$	27.2859	1.19542	0.597708	+	0.801714i	$0.296078\pi$
$522$	73.3045	3.20845
$523$	42.2018	1.84535	0.922677	−	0.385573i	$-0.125996\pi$
$523$	42.2018	1.84535	0.922677	+	0.385573i	$0.125996\pi$
$524$	2.56518	0.112060
$525$	−13.2111	−0.576582
$526$	24.1294	1.05209
$527$	1.08361	0.0472028
$528$	11.4785	0.499537
$529$	5.15893	0.224301
$530$	9.07478	0.394183
$531$	32.4226	1.40702
$532$	1.23600	0.0535876
$533$	2.31965	0.100475
$534$	27.0955	1.17254
$535$	11.9595	0.517055
$536$	22.1496	0.956718
$537$	−3.41318	−0.147289
$538$	13.6217	0.587273
$539$	−5.96757	−0.257041
$540$	2.10183	0.0904484
$541$	25.6463	1.10262	0.551310	−	0.834300i	$-0.314128\pi$
$541$	25.6463	1.10262	0.551310	+	0.834300i	$0.314128\pi$
$542$	35.4058	1.52081
$543$	−9.26643	−0.397661
$544$	−0.978688	−0.0419609
$545$	0.709666	0.0303987
$546$	−2.91648	−0.124814
$547$	6.10861	0.261185	0.130593	−	0.991436i	$-0.458312\pi$
$547$	6.10861	0.261185	0.130593	+	0.991436i	$0.458312\pi$
$548$	1.65937	0.0708848
$549$	31.1781	1.33065
$550$	−5.54607	−0.236485
$551$	54.0262	2.30159
$552$	49.3868	2.10204
$553$	2.50023	0.106320
$554$	35.0373	1.48859
$555$	24.9920	1.06085
$556$	−1.92998	−0.0818493
$557$	−38.8236	−1.64501	−0.822504	−	0.568760i	$-0.807423\pi$
$557$	−38.8236	−1.64501	−0.822504	+	0.568760i	$0.807423\pi$
$558$	−10.3075	−0.436352
$559$	0.670329	0.0283519
$560$	3.48503	0.147269
$561$	3.16834	0.133768
$562$	−2.93362	−0.123747
$563$	−13.0579	−0.550324	−0.275162	−	0.961398i	$-0.588732\pi$
$563$	−13.0579	−0.550324	−0.275162	+	0.961398i	$0.588732\pi$
$564$	−4.62923	−0.194926
$565$	−3.91788	−0.164827
$566$	−14.2567	−0.599255
$567$	−19.7363	−0.828846
$568$	21.3759	0.896914
$569$	28.7226	1.20412	0.602058	−	0.798452i	$-0.294348\pi$
$569$	28.7226	1.20412	0.602058	+	0.798452i	$0.294348\pi$
$570$	−28.4196	−1.19037
$571$	−3.74251	−0.156619	−0.0783096	−	0.996929i	$-0.524952\pi$
$571$	−3.74251	−0.156619	−0.0783096	+	0.996929i	$0.524952\pi$
$572$	0.116311	0.00486322
$573$	25.6305	1.07073
$574$	4.75198	0.198344
$575$	−21.7764	−0.908137
$576$	60.3078	2.51282
$577$	−34.6012	−1.44047	−0.720234	−	0.693732i	$-0.755965\pi$
$577$	−34.6012	−1.44047	−0.720234	+	0.693732i	$0.755965\pi$
$578$	1.35148	0.0562140
$579$	−44.5310	−1.85064
$580$	−1.26592	−0.0525645
$581$	0.758580	0.0314712
$582$	−25.8199	−1.07027
$583$	7.09260	0.293745
$584$	−0.835245	−0.0345627
$585$	−4.46668	−0.184674
$586$	−37.1428	−1.53436
$587$	31.7969	1.31240	0.656200	−	0.754587i	$-0.272163\pi$
$587$	31.7969	1.31240	0.656200	+	0.754587i	$0.272163\pi$
$588$	−3.28068	−0.135293
$589$	−7.59675	−0.313018
$590$	5.89394	0.242650
$591$	−63.3118	−2.60430
$592$	30.1855	1.24062
$593$	0.0645581	0.00265108	0.00132554	−	0.999999i	$-0.499578\pi$
$593$	0.0645581	0.00265108	0.00132554	+	0.999999i	$0.499578\pi$
$594$	−17.2921	−0.709505
$595$	0.961953	0.0394362
$596$	−1.12984	−0.0462800
$597$	−27.5656	−1.12818
$598$	−4.80733	−0.196587
$599$	16.4947	0.673954	0.336977	−	0.941513i	$-0.390595\pi$
$599$	16.4947	0.673954	0.336977	+	0.941513i	$0.390595\pi$
$600$	−38.1927	−1.55921
$601$	45.0492	1.83760	0.918798	−	0.394728i	$-0.129161\pi$
$601$	45.0492	1.83760	0.918798	+	0.394728i	$0.129161\pi$
$602$	1.37322	0.0559682
$603$	−53.0724	−2.16128
$604$	−0.337133	−0.0137178
$605$	0.946723	0.0384898
$606$	−4.32199	−0.175569
$607$	34.6634	1.40695	0.703473	−	0.710722i	$-0.251632\pi$
$607$	34.6634	1.40695	0.703473	+	0.710722i	$0.251632\pi$
$608$	6.86118	0.278258
$609$	24.8092	1.00532
$610$	5.66770	0.229478
$611$	5.64457	0.228355
$612$	1.22126	0.0493665
$613$	−9.57191	−0.386606	−0.193303	−	0.981139i	$-0.561920\pi$
$613$	−9.57191	−0.386606	−0.193303	+	0.981139i	$0.561920\pi$
$614$	18.8776	0.761836
$615$	10.3798	0.418555
$616$	2.98471	0.120257
$617$	45.3798	1.82692	0.913460	−	0.406928i	$-0.133400\pi$
$617$	45.3798	1.82692	0.913460	+	0.406928i	$0.133400\pi$
$618$	−67.4115	−2.71169
$619$	−28.8787	−1.16073	−0.580366	−	0.814355i	$-0.697091\pi$
$619$	−28.8787	−1.16073	−0.580366	+	0.814355i	$0.697091\pi$
$620$	0.178004	0.00714881
$621$	−67.8967	−2.72460
$622$	39.5884	1.58735
$623$	6.42965	0.257598
$624$	−7.69436	−0.308021
$625$	12.3591	0.494363
$626$	−7.22838	−0.288904
$627$	−22.2120	−0.887060
$628$	−2.45386	−0.0979197
$629$	8.33194	0.332216
$630$	−9.15031	−0.364557
$631$	−19.9707	−0.795019	−0.397510	−	0.917598i	$-0.630125\pi$
$631$	−19.9707	−0.795019	−0.397510	+	0.917598i	$0.630125\pi$
$632$	7.22801	0.287515
$633$	−58.5271	−2.32624
$634$	17.8823	0.710198
$635$	−7.97431	−0.316451
$636$	3.89916	0.154612
$637$	4.00023	0.158495
$638$	10.4150	0.412332
$639$	−51.2186	−2.02618
$640$	9.10995	0.360102
$641$	29.8076	1.17733	0.588664	−	0.808378i	$-0.299654\pi$
$641$	29.8076	1.17733	0.588664	+	0.808378i	$0.299654\pi$
$642$	−54.0918	−2.13483
$643$	32.9459	1.29926	0.649629	−	0.760251i	$-0.274924\pi$
$643$	32.9459	1.29926	0.649629	+	0.760251i	$0.274924\pi$
$644$	0.935564	0.0368664
$645$	2.99954	0.118107
$646$	−9.47465	−0.372775
$647$	−28.7455	−1.13010	−0.565051	−	0.825056i	$-0.691143\pi$
$647$	−28.7455	−1.13010	−0.565051	+	0.825056i	$0.691143\pi$
$648$	−57.0564	−2.24139
$649$	4.60654	0.180822
$650$	3.71769	0.145820
$651$	−3.48848	−0.136724
$652$	−1.76002	−0.0689276
$653$	33.5796	1.31407	0.657035	−	0.753860i	$-0.271810\pi$
$653$	33.5796	1.31407	0.657035	+	0.753860i	$0.271810\pi$
$654$	−3.20975	−0.125511
$655$	−13.9961	−0.546871
$656$	12.5368	0.489480
$657$	2.00132	0.0780790
$658$	11.5633	0.450785
$659$	20.9729	0.816988	0.408494	−	0.912761i	$-0.366054\pi$
$659$	20.9729	0.816988	0.408494	+	0.912761i	$0.366054\pi$
$660$	0.520462	0.0202590
$661$	−36.4895	−1.41928	−0.709638	−	0.704566i	$-0.751142\pi$
$661$	−36.4895	−1.41928	−0.709638	+	0.704566i	$0.751142\pi$
$662$	29.7597	1.15664
$663$	−2.12383	−0.0824828
$664$	2.19301	0.0851053
$665$	−6.74386	−0.261516
$666$	−79.2552	−3.07108
$667$	40.8938	1.58341
$668$	−0.0162222	−0.000627658 0
$669$	31.9723	1.23612
$670$	−9.64776	−0.372726
$671$	4.42971	0.171007
$672$	3.15070	0.121541
$673$	8.49583	0.327490	0.163745	−	0.986503i	$-0.447643\pi$
$673$	8.49583	0.327490	0.163745	+	0.986503i	$0.447643\pi$
$674$	4.49037	0.172962
$675$	52.5071	2.02100
$676$	2.17771	0.0837582
$677$	−17.9339	−0.689255	−0.344627	−	0.938740i	$-0.611995\pi$
$677$	−17.9339	−0.689255	−0.344627	+	0.938740i	$0.611995\pi$
$678$	17.7202	0.680540
$679$	−6.12696	−0.235131
$680$	2.78095	0.106645
$681$	7.06711	0.270812
$682$	−1.46447	−0.0560775
$683$	−1.86771	−0.0714659	−0.0357330	−	0.999361i	$-0.511377\pi$
$683$	−1.86771	−0.0714659	−0.0357330	+	0.999361i	$0.511377\pi$
$684$	−8.56175	−0.327367
$685$	−9.05382	−0.345929
$686$	17.8073	0.679885
$687$	90.6569	3.45878
$688$	3.62287	0.138120
$689$	−4.75437	−0.181127
$690$	−21.5115	−0.818930
$691$	0.143237	0.00544900	0.00272450	−	0.999996i	$-0.499133\pi$
$691$	0.143237	0.00544900	0.00272450	+	0.999996i	$0.499133\pi$
$692$	2.74064	0.104184
$693$	−7.15162	−0.271668
$694$	19.1180	0.725709
$695$	10.5303	0.399437
$696$	71.7219	2.71861
$697$	3.46047	0.131075
$698$	13.7847	0.521760
$699$	16.3790	0.619511
$700$	−0.723507	−0.0273460
$701$	−49.4153	−1.86639	−0.933196	−	0.359368i	$-0.882992\pi$
$701$	−49.4153	−1.86639	−0.933196	+	0.359368i	$0.882992\pi$
$702$	11.5914	0.437490
$703$	−58.4119	−2.20305
$704$	8.56840	0.322934
$705$	25.2579	0.951269
$706$	−1.08344	−0.0407757
$707$	−1.02559	−0.0385713
$708$	2.53245	0.0951753
$709$	−28.7705	−1.08050	−0.540249	−	0.841505i	$-0.681670\pi$
$709$	−28.7705	−1.08050	−0.540249	+	0.841505i	$0.681670\pi$
$710$	−9.31076	−0.349427
$711$	−17.3190	−0.649512
$712$	18.5877	0.696605
$713$	−5.75017	−0.215346
$714$	−4.35082	−0.162825
$715$	−0.634616	−0.0237333
$716$	−0.186922	−0.00698561
$717$	−79.4451	−2.96693
$718$	−9.65703	−0.360397
$719$	−8.06788	−0.300881	−0.150441	−	0.988619i	$-0.548069\pi$
$719$	−8.06788	−0.300881	−0.150441	+	0.988619i	$0.548069\pi$
$720$	−24.1406	−0.899668
$721$	−15.9965	−0.595740
$722$	40.7449	1.51637
$723$	−54.6393	−2.03206
$724$	−0.507475	−0.0188602
$725$	−31.6247	−1.17451
$726$	−4.28194	−0.158918
$727$	−4.02286	−0.149200	−0.0745999	−	0.997214i	$-0.523768\pi$
$727$	−4.02286	−0.149200	−0.0745999	+	0.997214i	$0.523768\pi$
$728$	−2.00074	−0.0741522
$729$	15.0954	0.559087
$730$	0.363810	0.0134652
$731$	1.00000	0.0369863
$732$	2.43524	0.0900091
$733$	−39.1590	−1.44637	−0.723185	−	0.690654i	$-0.757323\pi$
$733$	−39.1590	−1.44637	−0.723185	+	0.690654i	$0.757323\pi$
$734$	−21.6119	−0.797712
$735$	17.9000	0.660250
$736$	5.19341	0.191431
$737$	−7.54042	−0.277755
$738$	−32.9167	−1.21168
$739$	45.0590	1.65752	0.828760	−	0.559604i	$-0.189047\pi$
$739$	45.0590	1.65752	0.828760	+	0.559604i	$0.189047\pi$
$740$	1.36868	0.0503138
$741$	14.8893	0.546973
$742$	−9.73968	−0.357555
$743$	−16.2373	−0.595689	−0.297845	−	0.954614i	$-0.596268\pi$
$743$	−16.2373	−0.595689	−0.297845	+	0.954614i	$0.596268\pi$
$744$	−10.0850	−0.369734
$745$	6.16460	0.225854
$746$	−39.1885	−1.43479
$747$	−5.25465	−0.192258
$748$	0.173514	0.00634429
$749$	−12.8358	−0.469009
$750$	36.9047	1.34757
$751$	−50.7579	−1.85218	−0.926090	−	0.377302i	$-0.876852\pi$
$751$	−50.7579	−1.85218	−0.926090	+	0.377302i	$0.876852\pi$
$752$	30.5067	1.11246
$753$	−30.5290	−1.11254
$754$	−6.98145	−0.254249
$755$	1.83946	0.0669448
$756$	−2.25583	−0.0820437
$757$	−3.88366	−0.141154	−0.0705770	−	0.997506i	$-0.522484\pi$
$757$	−3.88366	−0.141154	−0.0705770	+	0.997506i	$0.522484\pi$
$758$	27.6340	1.00371
$759$	−16.8128	−0.610266
$760$	−19.4961	−0.707199
$761$	−16.4849	−0.597576	−0.298788	−	0.954320i	$-0.596582\pi$
$761$	−16.4849	−0.597576	−0.298788	+	0.954320i	$0.596582\pi$
$762$	36.0671	1.30657
$763$	−0.761662	−0.0275740
$764$	1.40365	0.0507822
$765$	−6.66341	−0.240916
$766$	3.90010	0.140916
$767$	−3.08790	−0.111497
$768$	13.0918	0.472411
$769$	−1.77733	−0.0640922	−0.0320461	−	0.999486i	$-0.510202\pi$
$769$	−1.77733	−0.0640922	−0.0320461	+	0.999486i	$0.510202\pi$
$770$	−1.30006	−0.0468508
$771$	25.7392	0.926973
$772$	−2.43873	−0.0877719
$773$	−22.9889	−0.826855	−0.413427	−	0.910537i	$-0.635668\pi$
$773$	−22.9889	−0.826855	−0.413427	+	0.910537i	$0.635668\pi$
$774$	−9.51222	−0.341910
$775$	4.44682	0.159735
$776$	−17.7127	−0.635848
$777$	−26.8231	−0.962275
$778$	40.6340	1.45680
$779$	−24.2599	−0.869203
$780$	−0.348881	−0.0124919
$781$	−7.27703	−0.260393
$782$	−7.17160	−0.256456
$783$	−98.6029	−3.52378
$784$	21.6197	0.772132
$785$	13.3887	0.477864
$786$	63.3028	2.25794
$787$	20.4405	0.728627	0.364313	−	0.931276i	$-0.381304\pi$
$787$	20.4405	0.728627	0.364313	+	0.931276i	$0.381304\pi$
$788$	−3.46726	−0.123516
$789$	−56.5680	−2.01387
$790$	−3.14832	−0.112012
$791$	4.20494	0.149510
$792$	−20.6749	−0.734652
$793$	−2.96937	−0.105445
$794$	−0.687446	−0.0243965
$795$	−21.2745	−0.754530
$796$	−1.50963	−0.0535073
$797$	25.8017	0.913942	0.456971	−	0.889482i	$-0.348934\pi$
$797$	25.8017	0.913942	0.456971	+	0.889482i	$0.348934\pi$
$798$	30.5019	1.07975
$799$	8.42060	0.297899
$800$	−4.01626	−0.141996
$801$	−44.5379	−1.57367
$802$	−38.3347	−1.35364
$803$	0.284343	0.0100343
$804$	−4.14536	−0.146195
$805$	−5.10460	−0.179914
$806$	0.981678	0.0345781
$807$	−31.9341	−1.12413
$808$	−2.96493	−0.104306
$809$	14.0008	0.492243	0.246122	−	0.969239i	$-0.420844\pi$
$809$	14.0008	0.492243	0.246122	+	0.969239i	$0.420844\pi$
$810$	24.8522	0.873218
$811$	13.4188	0.471198	0.235599	−	0.971850i	$-0.424295\pi$
$811$	13.4188	0.471198	0.235599	+	0.971850i	$0.424295\pi$
$812$	1.35867	0.0476800
$813$	−83.0038	−2.91107
$814$	−11.2604	−0.394677
$815$	9.60298	0.336378
$816$	−11.4785	−0.401827
$817$	−7.01059	−0.245270
$818$	39.6745	1.38719
$819$	4.79394	0.167514
$820$	0.568450	0.0198511
$821$	−11.5651	−0.403626	−0.201813	−	0.979424i	$-0.564683\pi$
$821$	−11.5651	−0.403626	−0.201813	+	0.979424i	$0.564683\pi$
$822$	40.9496	1.42828
$823$	9.03067	0.314789	0.157395	−	0.987536i	$-0.449691\pi$
$823$	9.03067	0.314789	0.157395	+	0.987536i	$0.449691\pi$
$824$	−46.2449	−1.61102
$825$	13.0020	0.452671
$826$	−6.32578	−0.220102
$827$	−18.2075	−0.633138	−0.316569	−	0.948569i	$-0.602531\pi$
$827$	−18.2075	−0.633138	−0.316569	+	0.948569i	$0.602531\pi$
$828$	−6.48061	−0.225217
$829$	10.2849	0.357210	0.178605	−	0.983921i	$-0.442842\pi$
$829$	10.2849	0.357210	0.178605	+	0.983921i	$0.442842\pi$
$830$	−0.955215	−0.0331560
$831$	−82.1400	−2.84941
$832$	−5.74365	−0.199125
$833$	5.96757	0.206764
$834$	−47.6276	−1.64921
$835$	0.0885115	0.00306307
$836$	−1.21644	−0.0420713
$837$	13.8648	0.479237
$838$	−22.6194	−0.781373
$839$	30.9437	1.06830	0.534148	−	0.845391i	$-0.320632\pi$
$839$	30.9437	1.06830	0.534148	+	0.845391i	$0.320632\pi$
$840$	−8.95275	−0.308899
$841$	30.3880	1.04786
$842$	−40.3064	−1.38905
$843$	6.87744	0.236872
$844$	−3.20523	−0.110328
$845$	−11.8820	−0.408753
$846$	−80.0985	−2.75384
$847$	−1.01609	−0.0349132
$848$	−25.6955	−0.882388
$849$	33.4229	1.14707
$850$	5.54607	0.190229
$851$	−44.2134	−1.51562
$852$	−4.00056	−0.137057
$853$	−47.7841	−1.63610	−0.818048	−	0.575149i	$-0.804944\pi$
$853$	−47.7841	−1.63610	−0.818048	+	0.575149i	$0.804944\pi$
$854$	−6.08296	−0.208155
$855$	46.7144	1.59760
$856$	−37.1075	−1.26831
$857$	−51.8896	−1.77251	−0.886257	−	0.463193i	$-0.846704\pi$
$857$	−51.8896	−1.77251	−0.886257	+	0.463193i	$0.846704\pi$
$858$	2.87031	0.0979907
$859$	−15.7979	−0.539016	−0.269508	−	0.962998i	$-0.586861\pi$
$859$	−15.7979	−0.539016	−0.269508	+	0.962998i	$0.586861\pi$
$860$	0.164270	0.00560154
$861$	−11.1403	−0.379662
$862$	43.9638	1.49741
$863$	24.3109	0.827553	0.413776	−	0.910379i	$-0.364210\pi$
$863$	24.3109	0.827553	0.413776	+	0.910379i	$0.364210\pi$
$864$	−12.5223	−0.426018
$865$	−14.9534	−0.508432
$866$	−4.67943	−0.159014
$867$	−3.16834	−0.107603
$868$	−0.191046	−0.00648452
$869$	−2.46064	−0.0834715
$870$	−31.2401	−1.05914
$871$	5.05456	0.171267
$872$	−2.20192	−0.0745664
$873$	42.4412	1.43642
$874$	50.2772	1.70065
$875$	8.75735	0.296052
$876$	0.156318	0.00528150
$877$	−2.29470	−0.0774864	−0.0387432	−	0.999249i	$-0.512335\pi$
$877$	−2.29470	−0.0774864	−0.0387432	+	0.999249i	$0.512335\pi$
$878$	15.5312	0.524152
$879$	87.0761	2.93700
$880$	−3.42985	−0.115620
$881$	34.7615	1.17114	0.585572	−	0.810621i	$-0.300870\pi$
$881$	34.7615	1.17114	0.585572	+	0.810621i	$0.300870\pi$
$882$	−56.7648	−1.91137
$883$	4.95087	0.166610	0.0833051	−	0.996524i	$-0.473452\pi$
$883$	4.95087	0.166610	0.0833051	+	0.996524i	$0.473452\pi$
$884$	−0.116311	−0.00391198
$885$	−13.8175	−0.464470
$886$	22.3584	0.751146
$887$	55.1690	1.85240	0.926198	−	0.377038i	$-0.123058\pi$
$887$	55.1690	1.85240	0.926198	+	0.377038i	$0.123058\pi$
$888$	−77.5441	−2.60221
$889$	8.55858	0.287046
$890$	−8.09631	−0.271389
$891$	19.4238	0.650721
$892$	1.75096	0.0586264
$893$	−59.0334	−1.97548
$894$	−27.8819	−0.932511
$895$	1.01988	0.0340909
$896$	−9.77742	−0.326641
$897$	11.2701	0.376298
$898$	−10.6407	−0.355084
$899$	−8.35068	−0.278511
$900$	5.01170	0.167057
$901$	−7.09260	−0.236289
$902$	−4.67674	−0.155718
$903$	−3.21931	−0.107132
$904$	12.1562	0.404310
$905$	2.76887	0.0920405
$906$	−8.31970	−0.276403
$907$	−4.87713	−0.161942	−0.0809712	−	0.996716i	$-0.525802\pi$
$907$	−4.87713	−0.161942	−0.0809712	+	0.996716i	$0.525802\pi$
$908$	0.387029	0.0128440
$909$	7.10422	0.235632
$910$	0.871466	0.0288888
$911$	−22.2472	−0.737083	−0.368542	−	0.929611i	$-0.620143\pi$
$911$	−22.2472	−0.737083	−0.368542	+	0.929611i	$0.620143\pi$
$912$	80.4709	2.66466
$913$	−0.746569	−0.0247078
$914$	−12.1047	−0.400388
$915$	−13.2871	−0.439258
$916$	4.96481	0.164042
$917$	15.0215	0.496055
$918$	17.2921	0.570726
$919$	33.5575	1.10696	0.553480	−	0.832863i	$-0.313300\pi$
$919$	33.5575	1.10696	0.553480	+	0.832863i	$0.313300\pi$
$920$	−14.7571	−0.486528
$921$	−44.2558	−1.45828
$922$	4.44378	0.146348
$923$	4.87801	0.160562
$924$	−0.558595	−0.0183764
$925$	34.1919	1.12422
$926$	42.2170	1.38734
$927$	110.807	3.63938
$928$	7.54212	0.247582
$929$	−32.3751	−1.06219	−0.531096	−	0.847312i	$-0.678220\pi$
$929$	−32.3751	−1.06219	−0.531096	+	0.847312i	$0.678220\pi$
$930$	4.39274	0.144044
$931$	−41.8362	−1.37113
$932$	0.896995	0.0293820
$933$	−92.8093	−3.03844
$934$	−57.7652	−1.89013
$935$	−0.946723	−0.0309611
$936$	13.8590	0.452996
$937$	−18.4102	−0.601437	−0.300718	−	0.953713i	$-0.597226\pi$
$937$	−18.4102	−0.601437	−0.300718	+	0.953713i	$0.597226\pi$
$938$	10.3546	0.338091
$939$	16.9459	0.553009
$940$	1.38325	0.0451166
$941$	−45.6768	−1.48902	−0.744510	−	0.667611i	$-0.767317\pi$
$941$	−45.6768	−1.48902	−0.744510	+	0.667611i	$0.767317\pi$
$942$	−60.5559	−1.97302
$943$	−18.3630	−0.597981
$944$	−16.6889	−0.543176
$945$	12.3082	0.400386
$946$	−1.35148	−0.0439403
$947$	−30.6270	−0.995242	−0.497621	−	0.867395i	$-0.665793\pi$
$947$	−30.6270	−0.995242	−0.497621	+	0.867395i	$0.665793\pi$
$948$	−1.35274	−0.0439350
$949$	−0.190604	−0.00618726
$950$	−38.8813	−1.26147
$951$	−41.9226	−1.35943
$952$	−2.98471	−0.0967349
$953$	−44.4776	−1.44077	−0.720385	−	0.693574i	$-0.756035\pi$
$953$	−44.4776	−1.44077	−0.720385	+	0.693574i	$0.756035\pi$
$954$	67.4663	2.18430
$955$	−7.65856	−0.247825
$956$	−4.35080	−0.140715
$957$	−24.4164	−0.789270
$958$	28.8501	0.932105
$959$	9.71718	0.313784
$960$	−25.7013	−0.829505
$961$	−29.8258	−0.962122
$962$	7.54819	0.243363
$963$	88.9129	2.86518
$964$	−2.99231	−0.0963759
$965$	13.3062	0.428341
$966$	23.0876	0.742833
$967$	−6.87083	−0.220951	−0.110475	−	0.993879i	$-0.535237\pi$
$967$	−6.87083	−0.220951	−0.110475	+	0.993879i	$0.535237\pi$
$968$	−2.93745	−0.0944132
$969$	22.2120	0.713551
$970$	7.71516	0.247719
$971$	−11.8000	−0.378679	−0.189340	−	0.981912i	$-0.560635\pi$
$971$	−11.8000	−0.378679	−0.189340	+	0.981912i	$0.560635\pi$
$972$	4.01792	0.128875
$973$	−11.3018	−0.362320
$974$	−35.4848	−1.13701
$975$	−8.71560	−0.279123
$976$	−16.0483	−0.513692
$977$	24.7867	0.792998	0.396499	−	0.918035i	$-0.370225\pi$
$977$	24.7867	0.792998	0.396499	+	0.918035i	$0.370225\pi$
$978$	−43.4334	−1.38885
$979$	−6.32785	−0.202239
$980$	0.980289	0.0313142
$981$	5.27600	0.168450
$982$	26.2262	0.836913
$983$	−54.3337	−1.73298	−0.866489	−	0.499197i	$-0.833628\pi$
$983$	−54.3337	−1.73298	−0.866489	+	0.499197i	$0.833628\pi$
$984$	−32.2061	−1.02669
$985$	18.9180	0.602778
$986$	−10.4150	−0.331680
$987$	−27.1085	−0.862874
$988$	0.815412	0.0259417
$989$	−5.30650	−0.168737
$990$	9.00543	0.286211
$991$	46.1164	1.46494	0.732468	−	0.680801i	$-0.238368\pi$
$991$	46.1164	1.46494	0.732468	+	0.680801i	$0.238368\pi$
$992$	−1.06052	−0.0336714
$993$	−69.7673	−2.21400
$994$	9.99295	0.316957
$995$	8.23679	0.261124
$996$	−0.410427	−0.0130049
$997$	5.50800	0.174440	0.0872200	−	0.996189i	$-0.472202\pi$
$997$	5.50800	0.174440	0.0872200	+	0.996189i	$0.472202\pi$
$998$	39.0008	1.23455
$999$	106.607	3.37291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.i.1.55	✓	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.i.1.55	✓	78	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35148 q^{2} -3.16834 q^{3} -0.173514 q^{4} +0.946723 q^{5} -4.28194 q^{6} -1.01609 q^{7} -2.93745 q^{8} +7.03839 q^{9} +O(q^{10})\)	\(q+1.35148 q^{2} -3.16834 q^{3} -0.173514 q^{4} +0.946723 q^{5} -4.28194 q^{6} -1.01609 q^{7} -2.93745 q^{8} +7.03839 q^{9} +1.27947 q^{10} +1.00000 q^{11} +0.549751 q^{12} -0.670329 q^{13} -1.37322 q^{14} -2.99954 q^{15} -3.62287 q^{16} -1.00000 q^{17} +9.51222 q^{18} +7.01059 q^{19} -0.164270 q^{20} +3.21931 q^{21} +1.35148 q^{22} +5.30650 q^{23} +9.30685 q^{24} -4.10372 q^{25} -0.905933 q^{26} -12.7950 q^{27} +0.176305 q^{28} +7.70636 q^{29} -4.05381 q^{30} -1.08361 q^{31} +0.978688 q^{32} -3.16834 q^{33} -1.35148 q^{34} -0.961953 q^{35} -1.22126 q^{36} -8.33194 q^{37} +9.47465 q^{38} +2.12383 q^{39} -2.78095 q^{40} -3.46047 q^{41} +4.35082 q^{42} -1.00000 q^{43} -0.173514 q^{44} +6.66341 q^{45} +7.17160 q^{46} -8.42060 q^{47} +11.4785 q^{48} -5.96757 q^{49} -5.54607 q^{50} +3.16834 q^{51} +0.116311 q^{52} +7.09260 q^{53} -17.2921 q^{54} +0.946723 q^{55} +2.98471 q^{56} -22.2120 q^{57} +10.4150 q^{58} +4.60654 q^{59} +0.520462 q^{60} +4.42971 q^{61} -1.46447 q^{62} -7.15162 q^{63} +8.56840 q^{64} -0.634616 q^{65} -4.28194 q^{66} -7.54042 q^{67} +0.173514 q^{68} -16.8128 q^{69} -1.30006 q^{70} -7.27703 q^{71} -20.6749 q^{72} +0.284343 q^{73} -11.2604 q^{74} +13.0020 q^{75} -1.21644 q^{76} -1.01609 q^{77} +2.87031 q^{78} -2.46064 q^{79} -3.42985 q^{80} +19.4238 q^{81} -4.67674 q^{82} -0.746569 q^{83} -0.558595 q^{84} -0.946723 q^{85} -1.35148 q^{86} -24.4164 q^{87} -2.93745 q^{88} -6.32785 q^{89} +9.00543 q^{90} +0.681113 q^{91} -0.920751 q^{92} +3.43325 q^{93} -11.3802 q^{94} +6.63709 q^{95} -3.10082 q^{96} +6.02995 q^{97} -8.06502 q^{98} +7.03839 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * q^99

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