LMFDB - Embedded newform 8043.2.a.t.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.890431 q^{2} -1.00000 q^{3} -1.20713 q^{4} -2.94587 q^{5} +0.890431 q^{6} +1.00000 q^{7} +2.85573 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.890431 q^{2} -1.00000 q^{3} -1.20713 q^{4} -2.94587 q^{5} +0.890431 q^{6} +1.00000 q^{7} +2.85573 q^{8} +1.00000 q^{9} +2.62309 q^{10} +0.179614 q^{11} +1.20713 q^{12} -1.10516 q^{13} -0.890431 q^{14} +2.94587 q^{15} -0.128564 q^{16} +5.29902 q^{17} -0.890431 q^{18} +7.81335 q^{19} +3.55606 q^{20} -1.00000 q^{21} -0.159933 q^{22} -0.108668 q^{23} -2.85573 q^{24} +3.67816 q^{25} +0.984070 q^{26} -1.00000 q^{27} -1.20713 q^{28} +8.19715 q^{29} -2.62309 q^{30} +7.92895 q^{31} -5.59698 q^{32} -0.179614 q^{33} -4.71841 q^{34} -2.94587 q^{35} -1.20713 q^{36} +6.33542 q^{37} -6.95724 q^{38} +1.10516 q^{39} -8.41261 q^{40} +9.17700 q^{41} +0.890431 q^{42} -3.90929 q^{43} -0.216817 q^{44} -2.94587 q^{45} +0.0967615 q^{46} +5.06383 q^{47} +0.128564 q^{48} +1.00000 q^{49} -3.27514 q^{50} -5.29902 q^{51} +1.33408 q^{52} +3.78868 q^{53} +0.890431 q^{54} -0.529118 q^{55} +2.85573 q^{56} -7.81335 q^{57} -7.29899 q^{58} -2.54639 q^{59} -3.55606 q^{60} -3.05804 q^{61} -7.06018 q^{62} +1.00000 q^{63} +5.24085 q^{64} +3.25566 q^{65} +0.159933 q^{66} +12.8695 q^{67} -6.39663 q^{68} +0.108668 q^{69} +2.62309 q^{70} +4.82286 q^{71} +2.85573 q^{72} +11.1792 q^{73} -5.64125 q^{74} -3.67816 q^{75} -9.43175 q^{76} +0.179614 q^{77} -0.984070 q^{78} +8.14904 q^{79} +0.378732 q^{80} +1.00000 q^{81} -8.17148 q^{82} -2.18994 q^{83} +1.20713 q^{84} -15.6102 q^{85} +3.48095 q^{86} -8.19715 q^{87} +0.512928 q^{88} -18.0733 q^{89} +2.62309 q^{90} -1.10516 q^{91} +0.131177 q^{92} -7.92895 q^{93} -4.50899 q^{94} -23.0171 q^{95} +5.59698 q^{96} +3.58909 q^{97} -0.890431 q^{98} +0.179614 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	−0.890431	−0.629630	−0.314815	−	0.949153i	$-0.601942\pi$
$2$	−0.890431	−0.629630	−0.314815	+	0.949153i	$0.601942\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.20713	−0.603567
$5$	−2.94587	−1.31743	−0.658717	−	0.752391i	$-0.728900\pi$
$5$	−2.94587	−1.31743	−0.658717	+	0.752391i	$0.728900\pi$
$6$	0.890431	0.363517
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.85573	1.00965
$9$	1.00000	0.333333
$10$	2.62309	0.829495
$11$	0.179614	0.0541555	0.0270778	−	0.999633i	$-0.491380\pi$
$11$	0.179614	0.0541555	0.0270778	+	0.999633i	$0.491380\pi$
$12$	1.20713	0.348469
$13$	−1.10516	−0.306517	−0.153258	−	0.988186i	$-0.548977\pi$
$13$	−1.10516	−0.306517	−0.153258	+	0.988186i	$0.548977\pi$
$14$	−0.890431	−0.237978
$15$	2.94587	0.760621
$16$	−0.128564	−0.0321409
$17$	5.29902	1.28520	0.642601	−	0.766201i	$-0.277855\pi$
$17$	5.29902	1.28520	0.642601	+	0.766201i	$0.277855\pi$
$18$	−0.890431	−0.209877
$19$	7.81335	1.79250	0.896252	−	0.443545i	$-0.146279\pi$
$19$	7.81335	1.79250	0.896252	+	0.443545i	$0.146279\pi$
$20$	3.55606	0.795159
$21$	−1.00000	−0.218218
$22$	−0.159933	−0.0340979
$23$	−0.108668	−0.0226589	−0.0113294	−	0.999936i	$-0.503606\pi$
$23$	−0.108668	−0.0226589	−0.0113294	+	0.999936i	$0.503606\pi$
$24$	−2.85573	−0.582923
$25$	3.67816	0.735632
$26$	0.984070	0.192992
$27$	−1.00000	−0.192450
$28$	−1.20713	−0.228127
$29$	8.19715	1.52217	0.761086	−	0.648651i	$-0.224667\pi$
$29$	8.19715	1.52217	0.761086	+	0.648651i	$0.224667\pi$
$30$	−2.62309	−0.478909
$31$	7.92895	1.42408	0.712040	−	0.702139i	$-0.247771\pi$
$31$	7.92895	1.42408	0.712040	+	0.702139i	$0.247771\pi$
$32$	−5.59698	−0.989416
$33$	−0.179614	−0.0312667
$34$	−4.71841	−0.809201
$35$	−2.94587	−0.497943
$36$	−1.20713	−0.201189
$37$	6.33542	1.04154	0.520768	−	0.853698i	$-0.325646\pi$
$37$	6.33542	1.04154	0.520768	+	0.853698i	$0.325646\pi$
$38$	−6.95724	−1.12861
$39$	1.10516	0.176968
$40$	−8.41261	−1.33015
$41$	9.17700	1.43321	0.716603	−	0.697481i	$-0.245696\pi$
$41$	9.17700	1.43321	0.716603	+	0.697481i	$0.245696\pi$
$42$	0.890431	0.137396
$43$	−3.90929	−0.596161	−0.298081	−	0.954541i	$-0.596346\pi$
$43$	−3.90929	−0.596161	−0.298081	+	0.954541i	$0.596346\pi$
$44$	−0.216817	−0.0326865
$45$	−2.94587	−0.439145
$46$	0.0967615	0.0142667
$47$	5.06383	0.738636	0.369318	−	0.929303i	$-0.379591\pi$
$47$	5.06383	0.738636	0.369318	+	0.929303i	$0.379591\pi$
$48$	0.128564	0.0185565
$49$	1.00000	0.142857
$50$	−3.27514	−0.463175
$51$	−5.29902	−0.742012
$52$	1.33408	0.185003
$53$	3.78868	0.520415	0.260207	−	0.965553i	$-0.416209\pi$
$53$	3.78868	0.520415	0.260207	+	0.965553i	$0.416209\pi$
$54$	0.890431	0.121172
$55$	−0.529118	−0.0713463
$56$	2.85573	0.381613
$57$	−7.81335	−1.03490
$58$	−7.29899	−0.958405
$59$	−2.54639	−0.331512	−0.165756	−	0.986167i	$-0.553006\pi$
$59$	−2.54639	−0.331512	−0.165756	+	0.986167i	$0.553006\pi$
$60$	−3.55606	−0.459085
$61$	−3.05804	−0.391542	−0.195771	−	0.980650i	$-0.562721\pi$
$61$	−3.05804	−0.391542	−0.195771	+	0.980650i	$0.562721\pi$
$62$	−7.06018	−0.896643
$63$	1.00000	0.125988
$64$	5.24085	0.655107
$65$	3.25566	0.403815
$66$	0.159933	0.0196864
$67$	12.8695	1.57226	0.786130	−	0.618061i	$-0.212081\pi$
$67$	12.8695	1.57226	0.786130	+	0.618061i	$0.212081\pi$
$68$	−6.39663	−0.775705
$69$	0.108668	0.0130821
$70$	2.62309	0.313520
$71$	4.82286	0.572368	0.286184	−	0.958175i	$-0.407613\pi$
$71$	4.82286	0.572368	0.286184	+	0.958175i	$0.407613\pi$
$72$	2.85573	0.336551
$73$	11.1792	1.30843	0.654215	−	0.756308i	$-0.272999\pi$
$73$	11.1792	1.30843	0.654215	+	0.756308i	$0.272999\pi$
$74$	−5.64125	−0.655782
$75$	−3.67816	−0.424717
$76$	−9.43175	−1.08190
$77$	0.179614	0.0204689
$78$	−0.984070	−0.111424
$79$	8.14904	0.916838	0.458419	−	0.888736i	$-0.348416\pi$
$79$	8.14904	0.916838	0.458419	+	0.888736i	$0.348416\pi$
$80$	0.378732	0.0423435
$81$	1.00000	0.111111
$82$	−8.17148	−0.902389
$83$	−2.18994	−0.240377	−0.120189	−	0.992751i	$-0.538350\pi$
$83$	−2.18994	−0.240377	−0.120189	+	0.992751i	$0.538350\pi$
$84$	1.20713	0.131709
$85$	−15.6102	−1.69317
$86$	3.48095	0.375361
$87$	−8.19715	−0.878826
$88$	0.512928	0.0546783
$89$	−18.0733	−1.91576	−0.957882	−	0.287161i	$-0.907289\pi$
$89$	−18.0733	−1.91576	−0.957882	+	0.287161i	$0.907289\pi$
$90$	2.62309	0.276498
$91$	−1.10516	−0.115852
$92$	0.131177	0.0136761
$93$	−7.92895	−0.822193
$94$	−4.50899	−0.465067
$95$	−23.0171	−2.36151
$96$	5.59698	0.571240
$97$	3.58909	0.364416	0.182208	−	0.983260i	$-0.441675\pi$
$97$	3.58909	0.364416	0.182208	+	0.983260i	$0.441675\pi$
$98$	−0.890431	−0.0899471
$99$	0.179614	0.0180518
$100$	−4.44003	−0.444003
$101$	−6.96029	−0.692575	−0.346287	−	0.938128i	$-0.612558\pi$
$101$	−6.96029	−0.692575	−0.346287	+	0.938128i	$0.612558\pi$
$102$	4.71841	0.467192
$103$	−8.75017	−0.862180	−0.431090	−	0.902309i	$-0.641871\pi$
$103$	−8.75017	−0.862180	−0.431090	+	0.902309i	$0.641871\pi$
$104$	−3.15604	−0.309476
$105$	2.94587	0.287488
$106$	−3.37355	−0.327668
$107$	17.6296	1.70432	0.852158	−	0.523285i	$-0.175294\pi$
$107$	17.6296	1.70432	0.852158	+	0.523285i	$0.175294\pi$
$108$	1.20713	0.116156
$109$	18.1317	1.73670	0.868349	−	0.495954i	$-0.165182\pi$
$109$	18.1317	1.73670	0.868349	+	0.495954i	$0.165182\pi$
$110$	0.471143	0.0449217
$111$	−6.33542	−0.601331
$112$	−0.128564	−0.0121481
$113$	−18.2232	−1.71430	−0.857149	−	0.515069i	$-0.827766\pi$
$113$	−18.2232	−1.71430	−0.857149	+	0.515069i	$0.827766\pi$
$114$	6.95724	0.651605
$115$	0.320123	0.0298516
$116$	−9.89505	−0.918732
$117$	−1.10516	−0.102172
$118$	2.26738	0.208730
$119$	5.29902	0.485761
$120$	8.41261	0.767963
$121$	−10.9677	−0.997067
$122$	2.72297	0.246527
$123$	−9.17700	−0.827462
$124$	−9.57129	−0.859527
$125$	3.89398	0.348288
$126$	−0.890431	−0.0793259
$127$	−17.4437	−1.54788	−0.773941	−	0.633258i	$-0.781717\pi$
$127$	−17.4437	−1.54788	−0.773941	+	0.633258i	$0.781717\pi$
$128$	6.52735	0.576942
$129$	3.90929	0.344194
$130$	−2.89894	−0.254254
$131$	0.203233	0.0177566	0.00887829	−	0.999961i	$-0.497174\pi$
$131$	0.203233	0.0177566	0.00887829	+	0.999961i	$0.497174\pi$
$132$	0.216817	0.0188715
$133$	7.81335	0.677503
$134$	−11.4594	−0.989942
$135$	2.94587	0.253540
$136$	15.1326	1.29761
$137$	17.0426	1.45604	0.728022	−	0.685554i	$-0.240440\pi$
$137$	17.0426	1.45604	0.728022	+	0.685554i	$0.240440\pi$
$138$	−0.0967615	−0.00823689
$139$	−10.6387	−0.902359	−0.451179	−	0.892433i	$-0.648997\pi$
$139$	−10.6387	−0.902359	−0.451179	+	0.892433i	$0.648997\pi$
$140$	3.55606	0.300542
$141$	−5.06383	−0.426452
$142$	−4.29442	−0.360380
$143$	−0.198502	−0.0165996
$144$	−0.128564	−0.0107136
$145$	−24.1477	−2.00536
$146$	−9.95433	−0.823827
$147$	−1.00000	−0.0824786
$148$	−7.64769	−0.628637
$149$	−23.8580	−1.95453	−0.977264	−	0.212027i	$-0.931993\pi$
$149$	−23.8580	−1.95453	−0.977264	+	0.212027i	$0.931993\pi$
$150$	3.27514	0.267414
$151$	−2.24687	−0.182848	−0.0914238	−	0.995812i	$-0.529142\pi$
$151$	−2.24687	−0.182848	−0.0914238	+	0.995812i	$0.529142\pi$
$152$	22.3128	1.80981
$153$	5.29902	0.428401
$154$	−0.159933	−0.0128878
$155$	−23.3577	−1.87613
$156$	−1.33408	−0.106812
$157$	17.3498	1.38467	0.692333	−	0.721578i	$-0.256583\pi$
$157$	17.3498	1.38467	0.692333	+	0.721578i	$0.256583\pi$
$158$	−7.25615	−0.577269
$159$	−3.78868	−0.300462
$160$	16.4880	1.30349
$161$	−0.108668	−0.00856425
$162$	−0.890431	−0.0699588
$163$	−1.99087	−0.155937	−0.0779685	−	0.996956i	$-0.524843\pi$
$163$	−1.99087	−0.155937	−0.0779685	+	0.996956i	$0.524843\pi$
$164$	−11.0779	−0.865035
$165$	0.529118	0.0411918
$166$	1.94999	0.151349
$167$	6.68332	0.517171	0.258586	−	0.965988i	$-0.416744\pi$
$167$	6.68332	0.517171	0.258586	+	0.965988i	$0.416744\pi$
$168$	−2.85573	−0.220324
$169$	−11.7786	−0.906047
$170$	13.8998	1.06607
$171$	7.81335	0.597501
$172$	4.71904	0.359823
$173$	−4.49817	−0.341989	−0.170995	−	0.985272i	$-0.554698\pi$
$173$	−4.49817	−0.341989	−0.170995	+	0.985272i	$0.554698\pi$
$174$	7.29899	0.553335
$175$	3.67816	0.278043
$176$	−0.0230918	−0.00174061
$177$	2.54639	0.191398
$178$	16.0930	1.20622
$179$	−19.2849	−1.44142	−0.720710	−	0.693237i	$-0.756184\pi$
$179$	−19.2849	−1.44142	−0.720710	+	0.693237i	$0.756184\pi$
$180$	3.55606	0.265053
$181$	22.3462	1.66098	0.830490	−	0.557034i	$-0.188061\pi$
$181$	22.3462	1.66098	0.830490	+	0.557034i	$0.188061\pi$
$182$	0.984070	0.0729441
$183$	3.05804	0.226057
$184$	−0.310327	−0.0228776
$185$	−18.6633	−1.37216
$186$	7.06018	0.517677
$187$	0.951776	0.0696008
$188$	−6.11272	−0.445816
$189$	−1.00000	−0.0727393
$190$	20.4951	1.48687
$191$	3.26969	0.236586	0.118293	−	0.992979i	$-0.462258\pi$
$191$	3.26969	0.236586	0.118293	+	0.992979i	$0.462258\pi$
$192$	−5.24085	−0.378226
$193$	23.0322	1.65790	0.828948	−	0.559325i	$-0.188940\pi$
$193$	23.0322	1.65790	0.828948	+	0.559325i	$0.188940\pi$
$194$	−3.19583	−0.229447
$195$	−3.25566	−0.233143
$196$	−1.20713	−0.0862238
$197$	1.86990	0.133225	0.0666123	−	0.997779i	$-0.478781\pi$
$197$	1.86990	0.133225	0.0666123	+	0.997779i	$0.478781\pi$
$198$	−0.159933	−0.0113660
$199$	5.49704	0.389675	0.194837	−	0.980836i	$-0.437582\pi$
$199$	5.49704	0.389675	0.194837	+	0.980836i	$0.437582\pi$
$200$	10.5038	0.742733
$201$	−12.8695	−0.907745
$202$	6.19766	0.436066
$203$	8.19715	0.575327
$204$	6.39663	0.447853
$205$	−27.0342	−1.88815
$206$	7.79142	0.542854
$207$	−0.108668	−0.00755296
$208$	0.142084	0.00985172
$209$	1.40338	0.0970740
$210$	−2.62309	−0.181011
$211$	21.6086	1.48759	0.743797	−	0.668405i	$-0.233023\pi$
$211$	21.6086	1.48759	0.743797	+	0.668405i	$0.233023\pi$
$212$	−4.57344	−0.314105
$213$	−4.82286	−0.330457
$214$	−15.6979	−1.07309
$215$	11.5163	0.785403
$216$	−2.85573	−0.194308
$217$	7.92895	0.538252
$218$	−16.1450	−1.09348
$219$	−11.1792	−0.755423
$220$	0.638716	0.0430622
$221$	−5.85628	−0.393936
$222$	5.64125	0.378616
$223$	−9.50590	−0.636562	−0.318281	−	0.947996i	$-0.603106\pi$
$223$	−9.50590	−0.636562	−0.318281	+	0.947996i	$0.603106\pi$
$224$	−5.59698	−0.373964
$225$	3.67816	0.245211
$226$	16.2265	1.07937
$227$	8.04296	0.533830	0.266915	−	0.963720i	$-0.413996\pi$
$227$	8.04296	0.533830	0.266915	+	0.963720i	$0.413996\pi$
$228$	9.43175	0.624633
$229$	25.4483	1.68167	0.840834	−	0.541293i	$-0.182065\pi$
$229$	25.4483	1.68167	0.840834	+	0.541293i	$0.182065\pi$
$230$	−0.285047	−0.0187954
$231$	−0.179614	−0.0118177
$232$	23.4088	1.53687
$233$	6.54227	0.428598	0.214299	−	0.976768i	$-0.431253\pi$
$233$	6.54227	0.428598	0.214299	+	0.976768i	$0.431253\pi$
$234$	0.984070	0.0643307
$235$	−14.9174	−0.973104
$236$	3.07383	0.200089
$237$	−8.14904	−0.529337
$238$	−4.71841	−0.305849
$239$	−4.76519	−0.308235	−0.154117	−	0.988053i	$-0.549253\pi$
$239$	−4.76519	−0.308235	−0.154117	+	0.988053i	$0.549253\pi$
$240$	−0.378732	−0.0244470
$241$	−23.1446	−1.49087	−0.745437	−	0.666576i	$-0.767759\pi$
$241$	−23.1446	−1.49087	−0.745437	+	0.666576i	$0.767759\pi$
$242$	9.76601	0.627783
$243$	−1.00000	−0.0641500
$244$	3.69146	0.236322
$245$	−2.94587	−0.188205
$246$	8.17148	0.520994
$247$	−8.63501	−0.549433
$248$	22.6429	1.43783
$249$	2.18994	0.138782
$250$	−3.46732	−0.219292
$251$	−17.9686	−1.13417	−0.567084	−	0.823660i	$-0.691928\pi$
$251$	−17.9686	−1.13417	−0.567084	+	0.823660i	$0.691928\pi$
$252$	−1.20713	−0.0760422
$253$	−0.0195183	−0.00122710
$254$	15.5324	0.974592
$255$	15.6102	0.977551
$256$	−16.2939	−1.01837
$257$	−20.7887	−1.29676	−0.648381	−	0.761316i	$-0.724554\pi$
$257$	−20.7887	−1.29676	−0.648381	+	0.761316i	$0.724554\pi$
$258$	−3.48095	−0.216715
$259$	6.33542	0.393664
$260$	−3.93002	−0.243730
$261$	8.19715	0.507391
$262$	−0.180965	−0.0111801
$263$	−1.97070	−0.121519	−0.0607593	−	0.998152i	$-0.519352\pi$
$263$	−1.97070	−0.121519	−0.0607593	+	0.998152i	$0.519352\pi$
$264$	−0.512928	−0.0315685
$265$	−11.1610	−0.685612
$266$	−6.95724	−0.426576
$267$	18.0733	1.10607
$268$	−15.5352	−0.948964
$269$	−0.0240334	−0.00146534	−0.000732670	−	1.00000i	$-0.500233\pi$
$269$	−0.0240334	−0.00146534	−0.000732670		1.00000i	$0.500233\pi$
$270$	−2.62309	−0.159636
$271$	−31.5550	−1.91683	−0.958416	−	0.285374i	$-0.907882\pi$
$271$	−31.5550	−1.91683	−0.958416	+	0.285374i	$0.907882\pi$
$272$	−0.681261	−0.0413075
$273$	1.10516	0.0668874
$274$	−15.1752	−0.916768
$275$	0.660647	0.0398385
$276$	−0.131177	−0.00789593
$277$	−21.0207	−1.26301	−0.631506	−	0.775371i	$-0.717563\pi$
$277$	−21.0207	−1.26301	−0.631506	+	0.775371i	$0.717563\pi$
$278$	9.47299	0.568152
$279$	7.92895	0.474694
$280$	−8.41261	−0.502750
$281$	−2.03809	−0.121582	−0.0607910	−	0.998151i	$-0.519362\pi$
$281$	−2.03809	−0.121582	−0.0607910	+	0.998151i	$0.519362\pi$
$282$	4.50899	0.268507
$283$	2.37671	0.141281	0.0706404	−	0.997502i	$-0.477496\pi$
$283$	2.37671	0.141281	0.0706404	+	0.997502i	$0.477496\pi$
$284$	−5.82183	−0.345462
$285$	23.0171	1.36342
$286$	0.176752	0.0104516
$287$	9.17700	0.541701
$288$	−5.59698	−0.329805
$289$	11.0796	0.651744
$290$	21.5019	1.26263
$291$	−3.58909	−0.210396
$292$	−13.4948	−0.789725
$293$	−1.11171	−0.0649466	−0.0324733	−	0.999473i	$-0.510338\pi$
$293$	−1.11171	−0.0649466	−0.0324733	+	0.999473i	$0.510338\pi$
$294$	0.890431	0.0519310
$295$	7.50133	0.436745
$296$	18.0922	1.05159
$297$	−0.179614	−0.0104222
$298$	21.2439	1.23063
$299$	0.120096	0.00694533
$300$	4.44003	0.256345
$301$	−3.90929	−0.225328
$302$	2.00068	0.115126
$303$	6.96029	0.399858
$304$	−1.00451	−0.0576127
$305$	9.00860	0.515831
$306$	−4.71841	−0.269734
$307$	9.80791	0.559767	0.279883	−	0.960034i	$-0.409704\pi$
$307$	9.80791	0.559767	0.279883	+	0.960034i	$0.409704\pi$
$308$	−0.216817	−0.0123543
$309$	8.75017	0.497780
$310$	20.7984	1.18127
$311$	−2.01734	−0.114393	−0.0571965	−	0.998363i	$-0.518216\pi$
$311$	−2.01734	−0.114393	−0.0571965	+	0.998363i	$0.518216\pi$
$312$	3.15604	0.178676
$313$	5.85414	0.330896	0.165448	−	0.986219i	$-0.447093\pi$
$313$	5.85414	0.330896	0.165448	+	0.986219i	$0.447093\pi$
$314$	−15.4488	−0.871827
$315$	−2.94587	−0.165981
$316$	−9.83697	−0.553373
$317$	−4.04197	−0.227020	−0.113510	−	0.993537i	$-0.536209\pi$
$317$	−4.04197	−0.227020	−0.113510	+	0.993537i	$0.536209\pi$
$318$	3.37355	0.189179
$319$	1.47232	0.0824340
$320$	−15.4389	−0.863059
$321$	−17.6296	−0.983987
$322$	0.0967615	0.00539231
$323$	41.4031	2.30373
$324$	−1.20713	−0.0670629
$325$	−4.06496	−0.225483
$326$	1.77273	0.0981826
$327$	−18.1317	−1.00268
$328$	26.2070	1.44704
$329$	5.06383	0.279178
$330$	−0.471143	−0.0259356
$331$	32.8278	1.80438	0.902189	−	0.431341i	$-0.141959\pi$
$331$	32.8278	1.80438	0.902189	+	0.431341i	$0.141959\pi$
$332$	2.64355	0.145084
$333$	6.33542	0.347179
$334$	−5.95104	−0.325626
$335$	−37.9119	−2.07135
$336$	0.128564	0.00701372
$337$	−3.12536	−0.170249	−0.0851246	−	0.996370i	$-0.527129\pi$
$337$	−3.12536	−0.170249	−0.0851246	+	0.996370i	$0.527129\pi$
$338$	10.4880	0.570474
$339$	18.2232	0.989750
$340$	18.8436	1.02194
$341$	1.42415	0.0771218
$342$	−6.95724	−0.376205
$343$	1.00000	0.0539949
$344$	−11.1639	−0.601916
$345$	−0.320123	−0.0172348
$346$	4.00530	0.215327
$347$	−5.18110	−0.278136	−0.139068	−	0.990283i	$-0.544411\pi$
$347$	−5.18110	−0.278136	−0.139068	+	0.990283i	$0.544411\pi$
$348$	9.89505	0.530430
$349$	6.95601	0.372347	0.186173	−	0.982517i	$-0.440391\pi$
$349$	6.95601	0.372347	0.186173	+	0.982517i	$0.440391\pi$
$350$	−3.27514	−0.175064
$351$	1.10516	0.0589892
$352$	−1.00529	−0.0535823
$353$	23.4941	1.25047	0.625233	−	0.780438i	$-0.285004\pi$
$353$	23.4941	1.25047	0.625233	+	0.780438i	$0.285004\pi$
$354$	−2.26738	−0.120510
$355$	−14.2075	−0.754056
$356$	21.8169	1.15629
$357$	−5.29902	−0.280454
$358$	17.1719	0.907561
$359$	−2.70150	−0.142580	−0.0712898	−	0.997456i	$-0.522712\pi$
$359$	−2.70150	−0.142580	−0.0712898	+	0.997456i	$0.522712\pi$
$360$	−8.41261	−0.443384
$361$	42.0484	2.21307
$362$	−19.8977	−1.04580
$363$	10.9677	0.575657
$364$	1.33408	0.0699247
$365$	−32.9326	−1.72377
$366$	−2.72297	−0.142332
$367$	22.5269	1.17590	0.587948	−	0.808899i	$-0.299936\pi$
$367$	22.5269	1.17590	0.587948	+	0.808899i	$0.299936\pi$
$368$	0.0139708	0.000728277 0
$369$	9.17700	0.477735
$370$	16.6184	0.863950
$371$	3.78868	0.196698
$372$	9.57129	0.496248
$373$	18.5352	0.959714	0.479857	−	0.877347i	$-0.340689\pi$
$373$	18.5352	0.959714	0.479857	+	0.877347i	$0.340689\pi$
$374$	−0.847491	−0.0438227
$375$	−3.89398	−0.201084
$376$	14.4609	0.745766
$377$	−9.05917	−0.466571
$378$	0.890431	0.0457988
$379$	29.6595	1.52351	0.761754	−	0.647866i	$-0.224339\pi$
$379$	29.6595	1.52351	0.761754	+	0.647866i	$0.224339\pi$
$380$	27.7847	1.42533
$381$	17.4437	0.893670
$382$	−2.91143	−0.148962
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−6.52735	−0.333097
$385$	−0.529118	−0.0269664
$386$	−20.5086	−1.04386
$387$	−3.90929	−0.198720
$388$	−4.33250	−0.219950
$389$	15.9439	0.808389	0.404194	−	0.914673i	$-0.367552\pi$
$389$	15.9439	0.808389	0.404194	+	0.914673i	$0.367552\pi$
$390$	2.89894	0.146794
$391$	−0.575835	−0.0291212
$392$	2.85573	0.144236
$393$	−0.203233	−0.0102518
$394$	−1.66501	−0.0838821
$395$	−24.0060	−1.20787
$396$	−0.216817	−0.0108955
$397$	−23.5523	−1.18206	−0.591028	−	0.806651i	$-0.701278\pi$
$397$	−23.5523	−1.18206	−0.591028	+	0.806651i	$0.701278\pi$
$398$	−4.89473	−0.245351
$399$	−7.81335	−0.391157
$400$	−0.472877	−0.0236438
$401$	−23.9425	−1.19563	−0.597815	−	0.801634i	$-0.703964\pi$
$401$	−23.9425	−1.19563	−0.597815	+	0.801634i	$0.703964\pi$
$402$	11.4594	0.571543
$403$	−8.76277	−0.436505
$404$	8.40200	0.418015
$405$	−2.94587	−0.146382
$406$	−7.29899	−0.362243
$407$	1.13793	0.0564050
$408$	−15.1326	−0.749174
$409$	−27.8489	−1.37704	−0.688521	−	0.725217i	$-0.741740\pi$
$409$	−27.8489	−1.37704	−0.688521	+	0.725217i	$0.741740\pi$
$410$	24.0721	1.18884
$411$	−17.0426	−0.840647
$412$	10.5626	0.520383
$413$	−2.54639	−0.125300
$414$	0.0967615	0.00475557
$415$	6.45129	0.316681
$416$	6.18557	0.303273
$417$	10.6387	0.520977
$418$	−1.24961	−0.0611207
$419$	−14.5369	−0.710176	−0.355088	−	0.934833i	$-0.615549\pi$
$419$	−14.5369	−0.710176	−0.355088	+	0.934833i	$0.615549\pi$
$420$	−3.55606	−0.173518
$421$	2.89161	0.140928	0.0704641	−	0.997514i	$-0.477552\pi$
$421$	2.89161	0.140928	0.0704641	+	0.997514i	$0.477552\pi$
$422$	−19.2409	−0.936634
$423$	5.06383	0.246212
$424$	10.8194	0.525438
$425$	19.4906	0.945435
$426$	4.29442	0.208065
$427$	−3.05804	−0.147989
$428$	−21.2813	−1.02867
$429$	0.198502	0.00958377
$430$	−10.2544	−0.494513
$431$	−20.4085	−0.983043	−0.491521	−	0.870865i	$-0.663559\pi$
$431$	−20.4085	−0.983043	−0.491521	+	0.870865i	$0.663559\pi$
$432$	0.128564	0.00618552
$433$	−13.3664	−0.642350	−0.321175	−	0.947020i	$-0.604078\pi$
$433$	−13.3664	−0.642350	−0.321175	+	0.947020i	$0.604078\pi$
$434$	−7.06018	−0.338899
$435$	24.1477	1.15780
$436$	−21.8873	−1.04821
$437$	−0.849062	−0.0406162
$438$	9.95433	0.475636
$439$	−3.69398	−0.176304	−0.0881520	−	0.996107i	$-0.528096\pi$
$439$	−3.69398	−0.176304	−0.0881520	+	0.996107i	$0.528096\pi$
$440$	−1.51102	−0.0720350
$441$	1.00000	0.0476190
$442$	5.21461	0.248034
$443$	1.01285	0.0481222	0.0240611	−	0.999710i	$-0.492340\pi$
$443$	1.01285	0.0481222	0.0240611	+	0.999710i	$0.492340\pi$
$444$	7.64769	0.362944
$445$	53.2416	2.52389
$446$	8.46435	0.400798
$447$	23.8580	1.12845
$448$	5.24085	0.247607
$449$	10.7969	0.509537	0.254769	−	0.967002i	$-0.418001\pi$
$449$	10.7969	0.509537	0.254769	+	0.967002i	$0.418001\pi$
$450$	−3.27514	−0.154392
$451$	1.64831	0.0776160
$452$	21.9979	1.03469
$453$	2.24687	0.105567
$454$	−7.16170	−0.336115
$455$	3.25566	0.152628
$456$	−22.3128	−1.04489
$457$	14.5257	0.679482	0.339741	−	0.940519i	$-0.389661\pi$
$457$	14.5257	0.679482	0.339741	+	0.940519i	$0.389661\pi$
$458$	−22.6599	−1.05883
$459$	−5.29902	−0.247337
$460$	−0.386430	−0.0180174
$461$	−5.73860	−0.267273	−0.133637	−	0.991030i	$-0.542666\pi$
$461$	−5.73860	−0.267273	−0.133637	+	0.991030i	$0.542666\pi$
$462$	0.159933	0.00744078
$463$	−12.6109	−0.586080	−0.293040	−	0.956100i	$-0.594667\pi$
$463$	−12.6109	−0.586080	−0.293040	+	0.956100i	$0.594667\pi$
$464$	−1.05385	−0.0489240
$465$	23.3577	1.08319
$466$	−5.82544	−0.269858
$467$	−24.1168	−1.11599	−0.557997	−	0.829843i	$-0.688430\pi$
$467$	−24.1168	−1.11599	−0.557997	+	0.829843i	$0.688430\pi$
$468$	1.33408	0.0616677
$469$	12.8695	0.594259
$470$	13.2829	0.612695
$471$	−17.3498	−0.799438
$472$	−7.27180	−0.334712
$473$	−0.702162	−0.0322854
$474$	7.25615	0.333286
$475$	28.7387	1.31862
$476$	−6.39663	−0.293189
$477$	3.78868	0.173472
$478$	4.24307	0.194074
$479$	27.9106	1.27527	0.637635	−	0.770339i	$-0.279913\pi$
$479$	27.9106	1.27527	0.637635	+	0.770339i	$0.279913\pi$
$480$	−16.4880	−0.752570
$481$	−7.00166	−0.319248
$482$	20.6087	0.938698
$483$	0.108668	0.00494457
$484$	13.2395	0.601796
$485$	−10.5730	−0.480095
$486$	0.890431	0.0403908
$487$	21.1250	0.957263	0.478632	−	0.878016i	$-0.341133\pi$
$487$	21.1250	0.957263	0.478632	+	0.878016i	$0.341133\pi$
$488$	−8.73294	−0.395322
$489$	1.99087	0.0900303
$490$	2.62309	0.118499
$491$	24.7918	1.11884	0.559420	−	0.828885i	$-0.311024\pi$
$491$	24.7918	1.11884	0.559420	+	0.828885i	$0.311024\pi$
$492$	11.0779	0.499428
$493$	43.4369	1.95630
$494$	7.68888	0.345939
$495$	−0.529118	−0.0237821
$496$	−1.01937	−0.0457712
$497$	4.82286	0.216335
$498$	−1.94999	−0.0873812
$499$	5.84699	0.261747	0.130874	−	0.991399i	$-0.458222\pi$
$499$	5.84699	0.261747	0.130874	+	0.991399i	$0.458222\pi$
$500$	−4.70055	−0.210215
$501$	−6.68332	−0.298589
$502$	15.9998	0.714106
$503$	2.96347	0.132134	0.0660672	−	0.997815i	$-0.478955\pi$
$503$	2.96347	0.132134	0.0660672	+	0.997815i	$0.478955\pi$
$504$	2.85573	0.127204
$505$	20.5041	0.912422
$506$	0.0173797	0.000772621 0
$507$	11.7786	0.523107
$508$	21.0569	0.934249
$509$	−15.6701	−0.694566	−0.347283	−	0.937760i	$-0.612896\pi$
$509$	−15.6701	−0.694566	−0.347283	+	0.937760i	$0.612896\pi$
$510$	−13.8998	−0.615495
$511$	11.1792	0.494540
$512$	1.45385	0.0642519
$513$	−7.81335	−0.344968
$514$	18.5109	0.816480
$515$	25.7769	1.13586
$516$	−4.71904	−0.207744
$517$	0.909533	0.0400012
$518$	−5.64125	−0.247862
$519$	4.49817	0.197448
$520$	9.29730	0.407713
$521$	16.4535	0.720842	0.360421	−	0.932790i	$-0.382633\pi$
$521$	16.4535	0.720842	0.360421	+	0.932790i	$0.382633\pi$
$522$	−7.29899	−0.319468
$523$	−2.50744	−0.109643	−0.0548213	−	0.998496i	$-0.517459\pi$
$523$	−2.50744	−0.109643	−0.0548213	+	0.998496i	$0.517459\pi$
$524$	−0.245330	−0.0107173
$525$	−3.67816	−0.160528
$526$	1.75477	0.0765118
$527$	42.0157	1.83023
$528$	0.0230918	0.00100494
$529$	−22.9882	−0.999487
$530$	9.93805	0.431681
$531$	−2.54639	−0.110504
$532$	−9.43175	−0.408918
$533$	−10.1421	−0.439302
$534$	−16.0930	−0.696413
$535$	−51.9345	−2.24532
$536$	36.7518	1.58744
$537$	19.2849	0.832204
$538$	0.0214000	0.000922621 0
$539$	0.179614	0.00773650
$540$	−3.55606	−0.153028
$541$	30.3724	1.30581	0.652905	−	0.757439i	$-0.273550\pi$
$541$	30.3724	1.30581	0.652905	+	0.757439i	$0.273550\pi$
$542$	28.0976	1.20689
$543$	−22.3462	−0.958967
$544$	−29.6585	−1.27160
$545$	−53.4135	−2.28798
$546$	−0.984070	−0.0421143
$547$	1.19088	0.0509183	0.0254591	−	0.999676i	$-0.491895\pi$
$547$	1.19088	0.0509183	0.0254591	+	0.999676i	$0.491895\pi$
$548$	−20.5726	−0.878819
$549$	−3.05804	−0.130514
$550$	−0.588260	−0.0250835
$551$	64.0471	2.72850
$552$	0.310327	0.0132084
$553$	8.14904	0.346532
$554$	18.7175	0.795230
$555$	18.6633	0.792214
$556$	12.8423	0.544634
$557$	−27.8643	−1.18065	−0.590324	−	0.807166i	$-0.701000\pi$
$557$	−27.8643	−1.18065	−0.590324	+	0.807166i	$0.701000\pi$
$558$	−7.06018	−0.298881
$559$	4.32040	0.182733
$560$	0.378732	0.0160043
$561$	−0.951776	−0.0401840
$562$	1.81477	0.0765516
$563$	−16.8300	−0.709299	−0.354649	−	0.934999i	$-0.615400\pi$
$563$	−16.8300	−0.709299	−0.354649	+	0.934999i	$0.615400\pi$
$564$	6.11272	0.257392
$565$	53.6833	2.25847
$566$	−2.11630	−0.0889546
$567$	1.00000	0.0419961
$568$	13.7728	0.577893
$569$	−33.8841	−1.42050	−0.710248	−	0.703952i	$-0.751417\pi$
$569$	−33.8841	−1.42050	−0.710248	+	0.703952i	$0.751417\pi$
$570$	−20.4951	−0.858447
$571$	4.55793	0.190743	0.0953717	−	0.995442i	$-0.469596\pi$
$571$	4.55793	0.190743	0.0953717	+	0.995442i	$0.469596\pi$
$572$	0.239618	0.0100189
$573$	−3.26969	−0.136593
$574$	−8.17148	−0.341071
$575$	−0.399699	−0.0166686
$576$	5.24085	0.218369
$577$	44.1403	1.83758	0.918792	−	0.394742i	$-0.129166\pi$
$577$	44.1403	1.83758	0.918792	+	0.394742i	$0.129166\pi$
$578$	−9.86566	−0.410357
$579$	−23.0322	−0.957187
$580$	29.1495	1.21037
$581$	−2.18994	−0.0908541
$582$	3.19583	0.132472
$583$	0.680497	0.0281833
$584$	31.9249	1.32106
$585$	3.25566	0.134605
$586$	0.989898	0.0408923
$587$	41.3913	1.70840	0.854201	−	0.519943i	$-0.174047\pi$
$587$	41.3913	1.70840	0.854201	+	0.519943i	$0.174047\pi$
$588$	1.20713	0.0497813
$589$	61.9516	2.55267
$590$	−6.67942	−0.274987
$591$	−1.86990	−0.0769172
$592$	−0.814504	−0.0334759
$593$	−16.9202	−0.694831	−0.347416	−	0.937711i	$-0.612941\pi$
$593$	−16.9202	−0.694831	−0.347416	+	0.937711i	$0.612941\pi$
$594$	0.159933	0.00656215
$595$	−15.6102	−0.639957
$596$	28.7998	1.17969
$597$	−5.49704	−0.224979
$598$	−0.106937	−0.00437298
$599$	30.7165	1.25504	0.627522	−	0.778599i	$-0.284069\pi$
$599$	30.7165	1.25504	0.627522	+	0.778599i	$0.284069\pi$
$600$	−10.5038	−0.428817
$601$	9.87946	0.402991	0.201496	−	0.979489i	$-0.435420\pi$
$601$	9.87946	0.402991	0.201496	+	0.979489i	$0.435420\pi$
$602$	3.48095	0.141873
$603$	12.8695	0.524087
$604$	2.71227	0.110361
$605$	32.3095	1.31357
$606$	−6.19766	−0.251763
$607$	11.3103	0.459072	0.229536	−	0.973300i	$-0.426279\pi$
$607$	11.3103	0.459072	0.229536	+	0.973300i	$0.426279\pi$
$608$	−43.7312	−1.77353
$609$	−8.19715	−0.332165
$610$	−8.02153	−0.324782
$611$	−5.59635	−0.226404
$612$	−6.39663	−0.258568
$613$	17.9127	0.723489	0.361744	−	0.932277i	$-0.382181\pi$
$613$	17.9127	0.723489	0.361744	+	0.932277i	$0.382181\pi$
$614$	−8.73326	−0.352446
$615$	27.0342	1.09013
$616$	0.512928	0.0206664
$617$	35.7533	1.43938	0.719688	−	0.694298i	$-0.244285\pi$
$617$	35.7533	1.43938	0.719688	+	0.694298i	$0.244285\pi$
$618$	−7.79142	−0.313417
$619$	−4.52029	−0.181686	−0.0908429	−	0.995865i	$-0.528956\pi$
$619$	−4.52029	−0.181686	−0.0908429	+	0.995865i	$0.528956\pi$
$620$	28.1958	1.13237
$621$	0.108668	0.00436070
$622$	1.79630	0.0720252
$623$	−18.0733	−0.724091
$624$	−0.142084	−0.00568789
$625$	−29.8619	−1.19448
$626$	−5.21271	−0.208342
$627$	−1.40338	−0.0560457
$628$	−20.9435	−0.835739
$629$	33.5715	1.33858
$630$	2.62309	0.104507
$631$	−20.4559	−0.814337	−0.407168	−	0.913353i	$-0.633484\pi$
$631$	−20.4559	−0.814337	−0.407168	+	0.913353i	$0.633484\pi$
$632$	23.2715	0.925689
$633$	−21.6086	−0.858863
$634$	3.59910	0.142938
$635$	51.3870	2.03923
$636$	4.57344	0.181349
$637$	−1.10516	−0.0437881
$638$	−1.31100	−0.0519029
$639$	4.82286	0.190789
$640$	−19.2287	−0.760082
$641$	−34.5673	−1.36532	−0.682662	−	0.730734i	$-0.739178\pi$
$641$	−34.5673	−1.36532	−0.682662	+	0.730734i	$0.739178\pi$
$642$	15.6979	0.619548
$643$	4.81828	0.190014	0.0950071	−	0.995477i	$-0.469713\pi$
$643$	4.81828	0.190014	0.0950071	+	0.995477i	$0.469713\pi$
$644$	0.131177	0.00516910
$645$	−11.5163	−0.453453
$646$	−36.8666	−1.45050
$647$	−24.1169	−0.948132	−0.474066	−	0.880489i	$-0.657214\pi$
$647$	−24.1169	−0.948132	−0.474066	+	0.880489i	$0.657214\pi$
$648$	2.85573	0.112184
$649$	−0.457366	−0.0179532
$650$	3.61956	0.141971
$651$	−7.92895	−0.310760
$652$	2.40325	0.0941184
$653$	−33.1468	−1.29714	−0.648568	−	0.761157i	$-0.724632\pi$
$653$	−33.1468	−1.29714	−0.648568	+	0.761157i	$0.724632\pi$
$654$	16.1450	0.631319
$655$	−0.598699	−0.0233931
$656$	−1.17983	−0.0460645
$657$	11.1792	0.436143
$658$	−4.50899	−0.175779
$659$	−12.3931	−0.482768	−0.241384	−	0.970430i	$-0.577601\pi$
$659$	−12.3931	−0.482768	−0.241384	+	0.970430i	$0.577601\pi$
$660$	−0.638716	−0.0248620
$661$	10.8871	0.423460	0.211730	−	0.977328i	$-0.432090\pi$
$661$	10.8871	0.423460	0.211730	+	0.977328i	$0.432090\pi$
$662$	−29.2309	−1.13609
$663$	5.85628	0.227439
$664$	−6.25388	−0.242698
$665$	−23.0171	−0.892565
$666$	−5.64125	−0.218594
$667$	−0.890769	−0.0344907
$668$	−8.06766	−0.312147
$669$	9.50590	0.367519
$670$	33.7579	1.30418
$671$	−0.549266	−0.0212042
$672$	5.59698	0.215908
$673$	−42.7063	−1.64621	−0.823104	−	0.567891i	$-0.807759\pi$
$673$	−42.7063	−1.64621	−0.823104	+	0.567891i	$0.807759\pi$
$674$	2.78292	0.107194
$675$	−3.67816	−0.141572
$676$	14.2184	0.546860
$677$	−50.2974	−1.93309	−0.966544	−	0.256501i	$-0.917430\pi$
$677$	−50.2974	−1.93309	−0.966544	+	0.256501i	$0.917430\pi$
$678$	−16.2265	−0.623176
$679$	3.58909	0.137736
$680$	−44.5786	−1.70951
$681$	−8.04296	−0.308207
$682$	−1.26810	−0.0485582
$683$	11.5889	0.443437	0.221718	−	0.975111i	$-0.428833\pi$
$683$	11.5889	0.443437	0.221718	+	0.975111i	$0.428833\pi$
$684$	−9.43175	−0.360632
$685$	−50.2052	−1.91824
$686$	−0.890431	−0.0339968
$687$	−25.4483	−0.970912
$688$	0.502592	0.0191612
$689$	−4.18710	−0.159516
$690$	0.285047	0.0108516
$691$	22.0525	0.838918	0.419459	−	0.907774i	$-0.362220\pi$
$691$	22.0525	0.838918	0.419459	+	0.907774i	$0.362220\pi$
$692$	5.42988	0.206413
$693$	0.179614	0.00682295
$694$	4.61341	0.175123
$695$	31.3401	1.18880
$696$	−23.4088	−0.887310
$697$	48.6291	1.84196
$698$	−6.19384	−0.234440
$699$	−6.54227	−0.247451
$700$	−4.44003	−0.167817
$701$	11.4967	0.434225	0.217113	−	0.976147i	$-0.430336\pi$
$701$	11.4967	0.434225	0.217113	+	0.976147i	$0.430336\pi$
$702$	−0.984070	−0.0371413
$703$	49.5008	1.86696
$704$	0.941328	0.0354776
$705$	14.9174	0.561822
$706$	−20.9199	−0.787330
$707$	−6.96029	−0.261769
$708$	−3.07383	−0.115522
$709$	25.9943	0.976237	0.488119	−	0.872777i	$-0.337683\pi$
$709$	25.9943	0.976237	0.488119	+	0.872777i	$0.337683\pi$
$710$	12.6508	0.474776
$711$	8.14904	0.305613
$712$	−51.6124	−1.93426
$713$	−0.861624	−0.0322681
$714$	4.71841	0.176582
$715$	0.584761	0.0218688
$716$	23.2794	0.869993
$717$	4.76519	0.177959
$718$	2.40550	0.0897723
$719$	−15.1295	−0.564234	−0.282117	−	0.959380i	$-0.591037\pi$
$719$	−15.1295	−0.564234	−0.282117	+	0.959380i	$0.591037\pi$
$720$	0.378732	0.0141145
$721$	−8.75017	−0.325873
$722$	−37.4412	−1.39342
$723$	23.1446	0.860756
$724$	−26.9748	−1.00251
$725$	30.1504	1.11976
$726$	−9.76601	−0.362451
$727$	0.394234	0.0146213	0.00731067	−	0.999973i	$-0.497673\pi$
$727$	0.394234	0.0146213	0.00731067	+	0.999973i	$0.497673\pi$
$728$	−3.15604	−0.116971
$729$	1.00000	0.0370370
$730$	29.3242	1.08534
$731$	−20.7154	−0.766188
$732$	−3.69146	−0.136440
$733$	−30.2905	−1.11881	−0.559403	−	0.828896i	$-0.688970\pi$
$733$	−30.2905	−1.11881	−0.559403	+	0.828896i	$0.688970\pi$
$734$	−20.0587	−0.740379
$735$	2.94587	0.108660
$736$	0.608214	0.0224191
$737$	2.31154	0.0851466
$738$	−8.17148	−0.300796
$739$	50.5886	1.86093	0.930466	−	0.366378i	$-0.119402\pi$
$739$	50.5886	1.86093	0.930466	+	0.366378i	$0.119402\pi$
$740$	22.5291	0.828187
$741$	8.63501	0.317215
$742$	−3.37355	−0.123847
$743$	−5.07292	−0.186107	−0.0930536	−	0.995661i	$-0.529663\pi$
$743$	−5.07292	−0.186107	−0.0930536	+	0.995661i	$0.529663\pi$
$744$	−22.6429	−0.830130
$745$	70.2827	2.57496
$746$	−16.5043	−0.604264
$747$	−2.18994	−0.0801258
$748$	−1.14892	−0.0420087
$749$	17.6296	0.644171
$750$	3.46732	0.126609
$751$	−5.11461	−0.186635	−0.0933174	−	0.995636i	$-0.529747\pi$
$751$	−5.11461	−0.186635	−0.0933174	+	0.995636i	$0.529747\pi$
$752$	−0.651024	−0.0237404
$753$	17.9686	0.654812
$754$	8.06657	0.293767
$755$	6.61899	0.240890
$756$	1.20713	0.0439030
$757$	19.7492	0.717797	0.358898	−	0.933377i	$-0.383152\pi$
$757$	19.7492	0.717797	0.358898	+	0.933377i	$0.383152\pi$
$758$	−26.4098	−0.959246
$759$	0.0195183	0.000708469 0
$760$	−65.7306	−2.38430
$761$	−25.1720	−0.912485	−0.456242	−	0.889856i	$-0.650805\pi$
$761$	−25.1720	−0.912485	−0.456242	+	0.889856i	$0.650805\pi$
$762$	−15.5324	−0.562681
$763$	18.1317	0.656410
$764$	−3.94695	−0.142796
$765$	−15.6102	−0.564389
$766$	0.890431	0.0321726
$767$	2.81417	0.101614
$768$	16.2939	0.587954
$769$	−38.0079	−1.37060	−0.685300	−	0.728261i	$-0.740329\pi$
$769$	−38.0079	−1.37060	−0.685300	+	0.728261i	$0.740329\pi$
$770$	0.471143	0.0169788
$771$	20.7887	0.748686
$772$	−27.8030	−1.00065
$773$	50.9564	1.83278	0.916388	−	0.400292i	$-0.131091\pi$
$773$	50.9564	1.83278	0.916388	+	0.400292i	$0.131091\pi$
$774$	3.48095	0.125120
$775$	29.1639	1.04760
$776$	10.2495	0.367934
$777$	−6.33542	−0.227282
$778$	−14.1970	−0.508986
$779$	71.7030	2.56903
$780$	3.93002	0.140717
$781$	0.866250	0.0309969
$782$	0.512741	0.0183356
$783$	−8.19715	−0.292942
$784$	−0.128564	−0.00459156
$785$	−51.1104	−1.82421
$786$	0.180965	0.00645482
$787$	−43.2607	−1.54208	−0.771039	−	0.636787i	$-0.780263\pi$
$787$	−43.2607	−1.54208	−0.771039	+	0.636787i	$0.780263\pi$
$788$	−2.25721	−0.0804099
$789$	1.97070	0.0701589
$790$	21.3757	0.760513
$791$	−18.2232	−0.647943
$792$	0.512928	0.0182261
$793$	3.37963	0.120014
$794$	20.9717	0.744257
$795$	11.1610	0.395838
$796$	−6.63566	−0.235195
$797$	−37.5932	−1.33162	−0.665810	−	0.746121i	$-0.731914\pi$
$797$	−37.5932	−1.33162	−0.665810	+	0.746121i	$0.731914\pi$
$798$	6.95724	0.246284
$799$	26.8334	0.949296
$800$	−20.5866	−0.727846
$801$	−18.0733	−0.638588
$802$	21.3191	0.752804
$803$	2.00794	0.0708587
$804$	15.5352	0.547885
$805$	0.320123	0.0112828
$806$	7.80264	0.274836
$807$	0.0240334	0.000846014 0
$808$	−19.8767	−0.699260
$809$	23.0303	0.809702	0.404851	−	0.914383i	$-0.367323\pi$
$809$	23.0303	0.809702	0.404851	+	0.914383i	$0.367323\pi$
$810$	2.62309	0.0921661
$811$	−39.7311	−1.39515	−0.697573	−	0.716513i	$-0.745737\pi$
$811$	−39.7311	−1.39515	−0.697573	+	0.716513i	$0.745737\pi$
$812$	−9.89505	−0.347248
$813$	31.5550	1.10668
$814$	−1.01325	−0.0355142
$815$	5.86485	0.205437
$816$	0.681261	0.0238489
$817$	−30.5446	−1.06862
$818$	24.7976	0.867026
$819$	−1.10516	−0.0386175
$820$	32.6339	1.13963
$821$	33.8058	1.17983	0.589914	−	0.807466i	$-0.299161\pi$
$821$	33.8058	1.17983	0.589914	+	0.807466i	$0.299161\pi$
$822$	15.1752	0.529296
$823$	22.3721	0.779841	0.389921	−	0.920848i	$-0.372502\pi$
$823$	22.3721	0.779841	0.389921	+	0.920848i	$0.372502\pi$
$824$	−24.9881	−0.870502
$825$	−0.660647	−0.0230008
$826$	2.26738	0.0788924
$827$	37.8507	1.31620	0.658099	−	0.752931i	$-0.271361\pi$
$827$	37.8507	1.31620	0.658099	+	0.752931i	$0.271361\pi$
$828$	0.131177	0.00455872
$829$	−33.4541	−1.16191	−0.580955	−	0.813936i	$-0.697321\pi$
$829$	−33.4541	−1.16191	−0.580955	+	0.813936i	$0.697321\pi$
$830$	−5.74442	−0.199392
$831$	21.0207	0.729200
$832$	−5.79199	−0.200801
$833$	5.29902	0.183600
$834$	−9.47299	−0.328023
$835$	−19.6882	−0.681339
$836$	−1.69407	−0.0585906
$837$	−7.92895	−0.274064
$838$	12.9441	0.447148
$839$	−17.3285	−0.598247	−0.299123	−	0.954214i	$-0.596694\pi$
$839$	−17.3285	−0.598247	−0.299123	+	0.954214i	$0.596694\pi$
$840$	8.41261	0.290263
$841$	38.1932	1.31701
$842$	−2.57477	−0.0887326
$843$	2.03809	0.0701954
$844$	−26.0844	−0.897862
$845$	34.6983	1.19366
$846$	−4.50899	−0.155022
$847$	−10.9677	−0.376856
$848$	−0.487086	−0.0167266
$849$	−2.37671	−0.0815685
$850$	−17.3551	−0.595274
$851$	−0.688459	−0.0236001
$852$	5.82183	0.199453
$853$	−4.52297	−0.154864	−0.0774318	−	0.996998i	$-0.524672\pi$
$853$	−4.52297	−0.154864	−0.0774318	+	0.996998i	$0.524672\pi$
$854$	2.72297	0.0931783
$855$	−23.0171	−0.787169
$856$	50.3453	1.72077
$857$	−6.56825	−0.224367	−0.112184	−	0.993688i	$-0.535784\pi$
$857$	−6.56825	−0.224367	−0.112184	+	0.993688i	$0.535784\pi$
$858$	−0.176752	−0.00603422
$859$	−41.9273	−1.43054	−0.715270	−	0.698849i	$-0.753696\pi$
$859$	−41.9273	−1.43054	−0.715270	+	0.698849i	$0.753696\pi$
$860$	−13.9017	−0.474043
$861$	−9.17700	−0.312751
$862$	18.1724	0.618953
$863$	48.3610	1.64623	0.823114	−	0.567876i	$-0.192235\pi$
$863$	48.3610	1.64623	0.823114	+	0.567876i	$0.192235\pi$
$864$	5.59698	0.190413
$865$	13.2510	0.450548
$866$	11.9019	0.404443
$867$	−11.0796	−0.376284
$868$	−9.57129	−0.324871
$869$	1.46368	0.0496519
$870$	−21.5019	−0.728982
$871$	−14.2229	−0.481924
$872$	51.7791	1.75346
$873$	3.58909	0.121472
$874$	0.756031	0.0255731
$875$	3.89398	0.131640
$876$	13.4948	0.455948
$877$	−54.0861	−1.82636	−0.913179	−	0.407559i	$-0.866380\pi$
$877$	−54.0861	−1.82636	−0.913179	+	0.407559i	$0.866380\pi$
$878$	3.28923	0.111006
$879$	1.11171	0.0374970
$880$	0.0680253	0.00229313
$881$	−49.8600	−1.67983	−0.839913	−	0.542721i	$-0.817394\pi$
$881$	−49.8600	−1.67983	−0.839913	+	0.542721i	$0.817394\pi$
$882$	−0.890431	−0.0299824
$883$	−36.6056	−1.23188	−0.615938	−	0.787795i	$-0.711223\pi$
$883$	−36.6056	−1.23188	−0.615938	+	0.787795i	$0.711223\pi$
$884$	7.06931	0.237767
$885$	−7.50133	−0.252155
$886$	−0.901877	−0.0302991
$887$	−51.4721	−1.72827	−0.864133	−	0.503264i	$-0.832132\pi$
$887$	−51.4721	−1.72827	−0.864133	+	0.503264i	$0.832132\pi$
$888$	−18.0922	−0.607136
$889$	−17.4437	−0.585044
$890$	−47.4079	−1.58912
$891$	0.179614	0.00601728
$892$	11.4749	0.384208
$893$	39.5655	1.32401
$894$	−21.2439	−0.710504
$895$	56.8108	1.89898
$896$	6.52735	0.218063
$897$	−0.120096	−0.00400989
$898$	−9.61390	−0.320820
$899$	64.9947	2.16770
$900$	−4.44003	−0.148001
$901$	20.0763	0.668838
$902$	−1.46771	−0.0488693
$903$	3.90929	0.130093
$904$	−52.0406	−1.73085
$905$	−65.8290	−2.18823
$906$	−2.00068	−0.0664682
$907$	50.7450	1.68496	0.842481	−	0.538726i	$-0.181094\pi$
$907$	50.7450	1.68496	0.842481	+	0.538726i	$0.181094\pi$
$908$	−9.70893	−0.322202
$909$	−6.96029	−0.230858
$910$	−2.89894	−0.0960990
$911$	−50.7678	−1.68201	−0.841006	−	0.541025i	$-0.818036\pi$
$911$	−50.7678	−1.68201	−0.841006	+	0.541025i	$0.818036\pi$
$912$	1.00451	0.0332627
$913$	−0.393343	−0.0130178
$914$	−12.9341	−0.427822
$915$	−9.00860	−0.297815
$916$	−30.7194	−1.01500
$917$	0.203233	0.00671136
$918$	4.71841	0.155731
$919$	−37.7250	−1.24443	−0.622217	−	0.782845i	$-0.713768\pi$
$919$	−37.7250	−1.24443	−0.622217	+	0.782845i	$0.713768\pi$
$920$	0.914183	0.0301397
$921$	−9.80791	−0.323182
$922$	5.10983	0.168283
$923$	−5.33004	−0.175440
$924$	0.216817	0.00713277
$925$	23.3027	0.766187
$926$	11.2292	0.369013
$927$	−8.75017	−0.287393
$928$	−45.8793	−1.50606
$929$	19.1943	0.629744	0.314872	−	0.949134i	$-0.398038\pi$
$929$	19.1943	0.629744	0.314872	+	0.949134i	$0.398038\pi$
$930$	−20.7984	−0.682006
$931$	7.81335	0.256072
$932$	−7.89739	−0.258688
$933$	2.01734	0.0660448
$934$	21.4744	0.702662
$935$	−2.80381	−0.0916944
$936$	−3.15604	−0.103159
$937$	−45.7399	−1.49426	−0.747128	−	0.664680i	$-0.768568\pi$
$937$	−45.7399	−1.49426	−0.747128	+	0.664680i	$0.768568\pi$
$938$	−11.4594	−0.374163
$939$	−5.85414	−0.191043
$940$	18.0073	0.587333
$941$	26.7344	0.871516	0.435758	−	0.900064i	$-0.356480\pi$
$941$	26.7344	0.871516	0.435758	+	0.900064i	$0.356480\pi$
$942$	15.4488	0.503350
$943$	−0.997248	−0.0324748
$944$	0.327373	0.0106551
$945$	2.94587	0.0958292
$946$	0.625226	0.0203279
$947$	−11.0398	−0.358745	−0.179372	−	0.983781i	$-0.557407\pi$
$947$	−11.0398	−0.358745	−0.179372	+	0.983781i	$0.557407\pi$
$948$	9.83697	0.319490
$949$	−12.3549	−0.401056
$950$	−25.5898	−0.830244
$951$	4.04197	0.131070
$952$	15.1326	0.490450
$953$	−47.0511	−1.52413	−0.762067	−	0.647498i	$-0.775816\pi$
$953$	−47.0511	−1.52413	−0.762067	+	0.647498i	$0.775816\pi$
$954$	−3.37355	−0.109223
$955$	−9.63208	−0.311687
$956$	5.75222	0.186040
$957$	−1.47232	−0.0475933
$958$	−24.8525	−0.802948
$959$	17.0426	0.550333
$960$	15.4389	0.498288
$961$	31.8682	1.02801
$962$	6.23450	0.201008
$963$	17.6296	0.568105
$964$	27.9386	0.899842
$965$	−67.8500	−2.18417
$966$	−0.0967615	−0.00311325
$967$	−4.99050	−0.160484	−0.0802418	−	0.996775i	$-0.525569\pi$
$967$	−4.99050	−0.160484	−0.0802418	+	0.996775i	$0.525569\pi$
$968$	−31.3209	−1.00669
$969$	−41.4031	−1.33006
$970$	9.41451	0.302282
$971$	−39.9038	−1.28057	−0.640286	−	0.768136i	$-0.721184\pi$
$971$	−39.9038	−1.28057	−0.640286	+	0.768136i	$0.721184\pi$
$972$	1.20713	0.0387188
$973$	−10.6387	−0.341060
$974$	−18.8103	−0.602721
$975$	4.06496	0.130183
$976$	0.393153	0.0125845
$977$	19.6317	0.628074	0.314037	−	0.949411i	$-0.398318\pi$
$977$	19.6317	0.628074	0.314037	+	0.949411i	$0.398318\pi$
$978$	−1.77273	−0.0566857
$979$	−3.24621	−0.103749
$980$	3.55606	0.113594
$981$	18.1317	0.578899
$982$	−22.0754	−0.704454
$983$	−8.59545	−0.274152	−0.137076	−	0.990561i	$-0.543771\pi$
$983$	−8.59545	−0.274152	−0.137076	+	0.990561i	$0.543771\pi$
$984$	−26.2070	−0.835449
$985$	−5.50847	−0.175514
$986$	−38.6775	−1.23174
$987$	−5.06383	−0.161184
$988$	10.4236	0.331619
$989$	0.424816	0.0135084
$990$	0.471143	0.0149739
$991$	39.1777	1.24452	0.622261	−	0.782810i	$-0.286214\pi$
$991$	39.1777	1.24452	0.622261	+	0.782810i	$0.286214\pi$
$992$	−44.3782	−1.40901
$993$	−32.8278	−1.04176
$994$	−4.29442	−0.136211
$995$	−16.1936	−0.513370
$996$	−2.64355	−0.0837641
$997$	−51.4864	−1.63059	−0.815296	−	0.579044i	$-0.803426\pi$
$997$	−51.4864	−1.63059	−0.815296	+	0.579044i	$0.803426\pi$
$998$	−5.20634	−0.164804
$999$	−6.33542	−0.200444

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.18	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.18	✓	52	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.890431 q^{2} -1.00000 q^{3} -1.20713 q^{4} -2.94587 q^{5} +0.890431 q^{6} +1.00000 q^{7} +2.85573 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.890431 q^{2} -1.00000 q^{3} -1.20713 q^{4} -2.94587 q^{5} +0.890431 q^{6} +1.00000 q^{7} +2.85573 q^{8} +1.00000 q^{9} +2.62309 q^{10} +0.179614 q^{11} +1.20713 q^{12} -1.10516 q^{13} -0.890431 q^{14} +2.94587 q^{15} -0.128564 q^{16} +5.29902 q^{17} -0.890431 q^{18} +7.81335 q^{19} +3.55606 q^{20} -1.00000 q^{21} -0.159933 q^{22} -0.108668 q^{23} -2.85573 q^{24} +3.67816 q^{25} +0.984070 q^{26} -1.00000 q^{27} -1.20713 q^{28} +8.19715 q^{29} -2.62309 q^{30} +7.92895 q^{31} -5.59698 q^{32} -0.179614 q^{33} -4.71841 q^{34} -2.94587 q^{35} -1.20713 q^{36} +6.33542 q^{37} -6.95724 q^{38} +1.10516 q^{39} -8.41261 q^{40} +9.17700 q^{41} +0.890431 q^{42} -3.90929 q^{43} -0.216817 q^{44} -2.94587 q^{45} +0.0967615 q^{46} +5.06383 q^{47} +0.128564 q^{48} +1.00000 q^{49} -3.27514 q^{50} -5.29902 q^{51} +1.33408 q^{52} +3.78868 q^{53} +0.890431 q^{54} -0.529118 q^{55} +2.85573 q^{56} -7.81335 q^{57} -7.29899 q^{58} -2.54639 q^{59} -3.55606 q^{60} -3.05804 q^{61} -7.06018 q^{62} +1.00000 q^{63} +5.24085 q^{64} +3.25566 q^{65} +0.159933 q^{66} +12.8695 q^{67} -6.39663 q^{68} +0.108668 q^{69} +2.62309 q^{70} +4.82286 q^{71} +2.85573 q^{72} +11.1792 q^{73} -5.64125 q^{74} -3.67816 q^{75} -9.43175 q^{76} +0.179614 q^{77} -0.984070 q^{78} +8.14904 q^{79} +0.378732 q^{80} +1.00000 q^{81} -8.17148 q^{82} -2.18994 q^{83} +1.20713 q^{84} -15.6102 q^{85} +3.48095 q^{86} -8.19715 q^{87} +0.512928 q^{88} -18.0733 q^{89} +2.62309 q^{90} -1.10516 q^{91} +0.131177 q^{92} -7.92895 q^{93} -4.50899 q^{94} -23.0171 q^{95} +5.59698 q^{96} +3.58909 q^{97} -0.890431 q^{98} +0.179614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

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