LMFDB - Embedded newform 8043.2.a.t.1.8

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			1.8
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)

$f(q)$	$=$	$q-1.95494 q^{2} -1.00000 q^{3} +1.82179 q^{4} -1.46647 q^{5} +1.95494 q^{6} +1.00000 q^{7} +0.348396 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.95494 q^{2} -1.00000 q^{3} +1.82179 q^{4} -1.46647 q^{5} +1.95494 q^{6} +1.00000 q^{7} +0.348396 q^{8} +1.00000 q^{9} +2.86686 q^{10} +0.940724 q^{11} -1.82179 q^{12} -5.06092 q^{13} -1.95494 q^{14} +1.46647 q^{15} -4.32467 q^{16} -2.26100 q^{17} -1.95494 q^{18} +0.999722 q^{19} -2.67160 q^{20} -1.00000 q^{21} -1.83906 q^{22} -6.50961 q^{23} -0.348396 q^{24} -2.84946 q^{25} +9.89378 q^{26} -1.00000 q^{27} +1.82179 q^{28} -3.46685 q^{29} -2.86686 q^{30} -6.78497 q^{31} +7.75767 q^{32} -0.940724 q^{33} +4.42012 q^{34} -1.46647 q^{35} +1.82179 q^{36} -2.02507 q^{37} -1.95440 q^{38} +5.06092 q^{39} -0.510913 q^{40} -11.1501 q^{41} +1.95494 q^{42} -3.65442 q^{43} +1.71380 q^{44} -1.46647 q^{45} +12.7259 q^{46} +3.37434 q^{47} +4.32467 q^{48} +1.00000 q^{49} +5.57053 q^{50} +2.26100 q^{51} -9.21991 q^{52} -4.27583 q^{53} +1.95494 q^{54} -1.37954 q^{55} +0.348396 q^{56} -0.999722 q^{57} +6.77749 q^{58} -12.2811 q^{59} +2.67160 q^{60} +1.16065 q^{61} +13.2642 q^{62} +1.00000 q^{63} -6.51643 q^{64} +7.42169 q^{65} +1.83906 q^{66} +10.2723 q^{67} -4.11906 q^{68} +6.50961 q^{69} +2.86686 q^{70} +2.38139 q^{71} +0.348396 q^{72} -13.8312 q^{73} +3.95889 q^{74} +2.84946 q^{75} +1.82128 q^{76} +0.940724 q^{77} -9.89378 q^{78} +8.72222 q^{79} +6.34200 q^{80} +1.00000 q^{81} +21.7977 q^{82} -2.92312 q^{83} -1.82179 q^{84} +3.31569 q^{85} +7.14416 q^{86} +3.46685 q^{87} +0.327745 q^{88} -14.5404 q^{89} +2.86686 q^{90} -5.06092 q^{91} -11.8591 q^{92} +6.78497 q^{93} -6.59663 q^{94} -1.46606 q^{95} -7.75767 q^{96} +11.4502 q^{97} -1.95494 q^{98} +0.940724 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.95494	−1.38235	−0.691175	−	0.722687i	$-0.742907\pi$
$2$	−1.95494	−1.38235	−0.691175	+	0.722687i	$0.742907\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.82179	0.910893
$5$	−1.46647	−0.655826	−0.327913	−	0.944708i	$-0.606345\pi$
$5$	−1.46647	−0.655826	−0.327913	+	0.944708i	$0.606345\pi$
$6$	1.95494	0.798101
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.348396	0.123177
$9$	1.00000	0.333333
$10$	2.86686	0.906581
$11$	0.940724	0.283639	0.141819	−	0.989893i	$-0.454705\pi$
$11$	0.940724	0.283639	0.141819	+	0.989893i	$0.454705\pi$
$12$	−1.82179	−0.525904
$13$	−5.06092	−1.40365	−0.701823	−	0.712351i	$-0.747630\pi$
$13$	−5.06092	−1.40365	−0.701823	+	0.712351i	$0.747630\pi$
$14$	−1.95494	−0.522479
$15$	1.46647	0.378641
$16$	−4.32467	−1.08117
$17$	−2.26100	−0.548373	−0.274187	−	0.961677i	$-0.588409\pi$
$17$	−2.26100	−0.548373	−0.274187	+	0.961677i	$0.588409\pi$
$18$	−1.95494	−0.460784
$19$	0.999722	0.229352	0.114676	−	0.993403i	$-0.463417\pi$
$19$	0.999722	0.229352	0.114676	+	0.993403i	$0.463417\pi$
$20$	−2.67160	−0.597387
$21$	−1.00000	−0.218218
$22$	−1.83906	−0.392088
$23$	−6.50961	−1.35735	−0.678674	−	0.734440i	$-0.737445\pi$
$23$	−6.50961	−1.35735	−0.678674	+	0.734440i	$0.737445\pi$
$24$	−0.348396	−0.0711161
$25$	−2.84946	−0.569892
$26$	9.89378	1.94033
$27$	−1.00000	−0.192450
$28$	1.82179	0.344285
$29$	−3.46685	−0.643779	−0.321889	−	0.946777i	$-0.604318\pi$
$29$	−3.46685	−0.643779	−0.321889	+	0.946777i	$0.604318\pi$
$30$	−2.86686	−0.523415
$31$	−6.78497	−1.21862	−0.609308	−	0.792934i	$-0.708553\pi$
$31$	−6.78497	−1.21862	−0.609308	+	0.792934i	$0.708553\pi$
$32$	7.75767	1.37137
$33$	−0.940724	−0.163759
$34$	4.42012	0.758044
$35$	−1.46647	−0.247879
$36$	1.82179	0.303631
$37$	−2.02507	−0.332920	−0.166460	−	0.986048i	$-0.553234\pi$
$37$	−2.02507	−0.332920	−0.166460	+	0.986048i	$0.553234\pi$
$38$	−1.95440	−0.317045
$39$	5.06092	0.810395
$40$	−0.510913	−0.0807825
$41$	−11.1501	−1.74135	−0.870673	−	0.491862i	$-0.836316\pi$
$41$	−11.1501	−1.74135	−0.870673	+	0.491862i	$0.836316\pi$
$42$	1.95494	0.301654
$43$	−3.65442	−0.557293	−0.278647	−	0.960394i	$-0.589886\pi$
$43$	−3.65442	−0.557293	−0.278647	+	0.960394i	$0.589886\pi$
$44$	1.71380	0.258365
$45$	−1.46647	−0.218609
$46$	12.7259	1.87633
$47$	3.37434	0.492198	0.246099	−	0.969245i	$-0.420851\pi$
$47$	3.37434	0.492198	0.246099	+	0.969245i	$0.420851\pi$
$48$	4.32467	0.624212
$49$	1.00000	0.142857
$50$	5.57053	0.787791
$51$	2.26100	0.316603
$52$	−9.21991	−1.27857
$53$	−4.27583	−0.587331	−0.293665	−	0.955908i	$-0.594875\pi$
$53$	−4.27583	−0.587331	−0.293665	+	0.955908i	$0.594875\pi$
$54$	1.95494	0.266034
$55$	−1.37954	−0.186018
$56$	0.348396	0.0465564
$57$	−0.999722	−0.132416
$58$	6.77749	0.889928
$59$	−12.2811	−1.59887	−0.799434	−	0.600754i	$-0.794867\pi$
$59$	−12.2811	−1.59887	−0.799434	+	0.600754i	$0.794867\pi$
$60$	2.67160	0.344902
$61$	1.16065	0.148607	0.0743033	−	0.997236i	$-0.476327\pi$
$61$	1.16065	0.148607	0.0743033	+	0.997236i	$0.476327\pi$
$62$	13.2642	1.68456
$63$	1.00000	0.125988
$64$	−6.51643	−0.814554
$65$	7.42169	0.920547
$66$	1.83906	0.226372
$67$	10.2723	1.25496	0.627481	−	0.778632i	$-0.284086\pi$
$67$	10.2723	1.25496	0.627481	+	0.778632i	$0.284086\pi$
$68$	−4.11906	−0.499509
$69$	6.50961	0.783665
$70$	2.86686	0.342656
$71$	2.38139	0.282619	0.141309	−	0.989965i	$-0.454869\pi$
$71$	2.38139	0.282619	0.141309	+	0.989965i	$0.454869\pi$
$72$	0.348396	0.0410589
$73$	−13.8312	−1.61882	−0.809410	−	0.587244i	$-0.800213\pi$
$73$	−13.8312	−1.61882	−0.809410	+	0.587244i	$0.800213\pi$
$74$	3.95889	0.460212
$75$	2.84946	0.329028
$76$	1.82128	0.208915
$77$	0.940724	0.107205
$78$	−9.89378	−1.12025
$79$	8.72222	0.981326	0.490663	−	0.871349i	$-0.336755\pi$
$79$	8.72222	0.981326	0.490663	+	0.871349i	$0.336755\pi$
$80$	6.34200	0.709057
$81$	1.00000	0.111111
$82$	21.7977	2.40715
$83$	−2.92312	−0.320854	−0.160427	−	0.987048i	$-0.551287\pi$
$83$	−2.92312	−0.320854	−0.160427	+	0.987048i	$0.551287\pi$
$84$	−1.82179	−0.198773
$85$	3.31569	0.359637
$86$	7.14416	0.770375
$87$	3.46685	0.371686
$88$	0.327745	0.0349377
$89$	−14.5404	−1.54127	−0.770637	−	0.637274i	$-0.780062\pi$
$89$	−14.5404	−1.54127	−0.770637	+	0.637274i	$0.780062\pi$
$90$	2.86686	0.302194
$91$	−5.06092	−0.530528
$92$	−11.8591	−1.23640
$93$	6.78497	0.703568
$94$	−6.59663	−0.680391
$95$	−1.46606	−0.150415
$96$	−7.75767	−0.791764
$97$	11.4502	1.16260	0.581298	−	0.813691i	$-0.302545\pi$
$97$	11.4502	1.16260	0.581298	+	0.813691i	$0.302545\pi$
$98$	−1.95494	−0.197479
$99$	0.940724	0.0945463
$100$	−5.19111	−0.519111
$101$	−1.17288	−0.116706	−0.0583530	−	0.998296i	$-0.518585\pi$
$101$	−1.17288	−0.116706	−0.0583530	+	0.998296i	$0.518585\pi$
$102$	−4.42012	−0.437657
$103$	−4.96745	−0.489457	−0.244729	−	0.969592i	$-0.578699\pi$
$103$	−4.96745	−0.489457	−0.244729	+	0.969592i	$0.578699\pi$
$104$	−1.76321	−0.172897
$105$	1.46647	0.143113
$106$	8.35899	0.811897
$107$	−2.93347	−0.283589	−0.141795	−	0.989896i	$-0.545287\pi$
$107$	−2.93347	−0.283589	−0.141795	+	0.989896i	$0.545287\pi$
$108$	−1.82179	−0.175301
$109$	12.0323	1.15248	0.576242	−	0.817279i	$-0.304519\pi$
$109$	12.0323	1.15248	0.576242	+	0.817279i	$0.304519\pi$
$110$	2.69692	0.257142
$111$	2.02507	0.192211
$112$	−4.32467	−0.408643
$113$	6.24653	0.587624	0.293812	−	0.955863i	$-0.405076\pi$
$113$	6.24653	0.587624	0.293812	+	0.955863i	$0.405076\pi$
$114$	1.95440	0.183046
$115$	9.54615	0.890183
$116$	−6.31587	−0.586414
$117$	−5.06092	−0.467882
$118$	24.0089	2.21020
$119$	−2.26100	−0.207266
$120$	0.510913	0.0466398
$121$	−10.1150	−0.919549
$122$	−2.26901	−0.205426
$123$	11.1501	1.00537
$124$	−12.3608	−1.11003
$125$	11.5110	1.02958
$126$	−1.95494	−0.174160
$127$	18.1860	1.61375	0.806875	−	0.590723i	$-0.201157\pi$
$127$	18.1860	1.61375	0.806875	+	0.590723i	$0.201157\pi$
$128$	−2.77611	−0.245375
$129$	3.65442	0.321754
$130$	−14.5089	−1.27252
$131$	−16.4153	−1.43421	−0.717104	−	0.696966i	$-0.754533\pi$
$131$	−16.4153	−1.43421	−0.717104	+	0.696966i	$0.754533\pi$
$132$	−1.71380	−0.149167
$133$	0.999722	0.0866869
$134$	−20.0817	−1.73480
$135$	1.46647	0.126214
$136$	−0.787724	−0.0675468
$137$	8.89871	0.760268	0.380134	−	0.924931i	$-0.375878\pi$
$137$	8.89871	0.760268	0.380134	+	0.924931i	$0.375878\pi$
$138$	−12.7259	−1.08330
$139$	12.1427	1.02993	0.514964	−	0.857212i	$-0.327805\pi$
$139$	12.1427	1.02993	0.514964	+	0.857212i	$0.327805\pi$
$140$	−2.67160	−0.225791
$141$	−3.37434	−0.284171
$142$	−4.65546	−0.390678
$143$	−4.76092	−0.398128
$144$	−4.32467	−0.360389
$145$	5.08404	0.422207
$146$	27.0392	2.23778
$147$	−1.00000	−0.0824786
$148$	−3.68925	−0.303254
$149$	−8.52571	−0.698453	−0.349227	−	0.937038i	$-0.613556\pi$
$149$	−8.52571	−0.698453	−0.349227	+	0.937038i	$0.613556\pi$
$150$	−5.57053	−0.454831
$151$	−0.344333	−0.0280214	−0.0140107	−	0.999902i	$-0.504460\pi$
$151$	−0.344333	−0.0280214	−0.0140107	+	0.999902i	$0.504460\pi$
$152$	0.348300	0.0282508
$153$	−2.26100	−0.182791
$154$	−1.83906	−0.148195
$155$	9.94996	0.799200
$156$	9.21991	0.738184
$157$	11.2990	0.901756	0.450878	−	0.892586i	$-0.351111\pi$
$157$	11.2990	0.901756	0.450878	+	0.892586i	$0.351111\pi$
$158$	−17.0514	−1.35654
$159$	4.27583	0.339095
$160$	−11.3764	−0.899383
$161$	−6.50961	−0.513029
$162$	−1.95494	−0.153595
$163$	−11.9852	−0.938753	−0.469377	−	0.882998i	$-0.655521\pi$
$163$	−11.9852	−0.938753	−0.469377	+	0.882998i	$0.655521\pi$
$164$	−20.3130	−1.58618
$165$	1.37954	0.107397
$166$	5.71452	0.443533
$167$	5.74616	0.444651	0.222326	−	0.974972i	$-0.428635\pi$
$167$	5.74616	0.444651	0.222326	+	0.974972i	$0.428635\pi$
$168$	−0.348396	−0.0268794
$169$	12.6129	0.970222
$170$	−6.48197	−0.497145
$171$	0.999722	0.0764507
$172$	−6.65757	−0.507635
$173$	−8.70102	−0.661526	−0.330763	−	0.943714i	$-0.607306\pi$
$173$	−8.70102	−0.661526	−0.330763	+	0.943714i	$0.607306\pi$
$174$	−6.77749	−0.513800
$175$	−2.84946	−0.215399
$176$	−4.06832	−0.306661
$177$	12.2811	0.923106
$178$	28.4255	2.13058
$179$	−4.99206	−0.373124	−0.186562	−	0.982443i	$-0.559735\pi$
$179$	−4.99206	−0.373124	−0.186562	+	0.982443i	$0.559735\pi$
$180$	−2.67160	−0.199129
$181$	−16.3303	−1.21382	−0.606910	−	0.794771i	$-0.707591\pi$
$181$	−16.3303	−1.21382	−0.606910	+	0.794771i	$0.707591\pi$
$182$	9.89378	0.733376
$183$	−1.16065	−0.0857980
$184$	−2.26792	−0.167194
$185$	2.96971	0.218337
$186$	−13.2642	−0.972578
$187$	−2.12698	−0.155540
$188$	6.14733	0.448340
$189$	−1.00000	−0.0727393
$190$	2.86607	0.207926
$191$	6.47927	0.468823	0.234412	−	0.972137i	$-0.424684\pi$
$191$	6.47927	0.468823	0.234412	+	0.972137i	$0.424684\pi$
$192$	6.51643	0.470283
$193$	1.34244	0.0966310	0.0483155	−	0.998832i	$-0.484615\pi$
$193$	1.34244	0.0966310	0.0483155	+	0.998832i	$0.484615\pi$
$194$	−22.3845	−1.60712
$195$	−7.42169	−0.531478
$196$	1.82179	0.130128
$197$	−24.8526	−1.77067	−0.885336	−	0.464952i	$-0.846071\pi$
$197$	−24.8526	−1.77067	−0.885336	+	0.464952i	$0.846071\pi$
$198$	−1.83906	−0.130696
$199$	13.5146	0.958025	0.479013	−	0.877808i	$-0.340995\pi$
$199$	13.5146	0.958025	0.479013	+	0.877808i	$0.340995\pi$
$200$	−0.992743	−0.0701975
$201$	−10.2723	−0.724553
$202$	2.29291	0.161329
$203$	−3.46685	−0.243325
$204$	4.11906	0.288392
$205$	16.3512	1.14202
$206$	9.71106	0.676601
$207$	−6.50961	−0.452449
$208$	21.8868	1.51758
$209$	0.940462	0.0650532
$210$	−2.86686	−0.197832
$211$	−11.3459	−0.781085	−0.390543	−	0.920585i	$-0.627713\pi$
$211$	−11.3459	−0.781085	−0.390543	+	0.920585i	$0.627713\pi$
$212$	−7.78965	−0.534996
$213$	−2.38139	−0.163170
$214$	5.73476	0.392020
$215$	5.35910	0.365487
$216$	−0.348396	−0.0237054
$217$	−6.78497	−0.460594
$218$	−23.5224	−1.59314
$219$	13.8312	0.934626
$220$	−2.51323	−0.169442
$221$	11.4427	0.769722
$222$	−3.95889	−0.265703
$223$	−3.47833	−0.232926	−0.116463	−	0.993195i	$-0.537156\pi$
$223$	−3.47833	−0.232926	−0.116463	+	0.993195i	$0.537156\pi$
$224$	7.75767	0.518331
$225$	−2.84946	−0.189964
$226$	−12.2116	−0.812303
$227$	17.5596	1.16547	0.582737	−	0.812661i	$-0.301982\pi$
$227$	17.5596	1.16547	0.582737	+	0.812661i	$0.301982\pi$
$228$	−1.82128	−0.120617
$229$	−9.29845	−0.614459	−0.307230	−	0.951635i	$-0.599402\pi$
$229$	−9.29845	−0.614459	−0.307230	+	0.951635i	$0.599402\pi$
$230$	−18.6621	−1.23055
$231$	−0.940724	−0.0618951
$232$	−1.20784	−0.0792986
$233$	−22.8398	−1.49629	−0.748143	−	0.663538i	$-0.769054\pi$
$233$	−22.8398	−1.49629	−0.748143	+	0.663538i	$0.769054\pi$
$234$	9.89378	0.646777
$235$	−4.94837	−0.322796
$236$	−22.3736	−1.45640
$237$	−8.72222	−0.566569
$238$	4.42012	0.286514
$239$	0.882131	0.0570603	0.0285302	−	0.999593i	$-0.490917\pi$
$239$	0.882131	0.0570603	0.0285302	+	0.999593i	$0.490917\pi$
$240$	−6.34200	−0.409374
$241$	−12.1698	−0.783928	−0.391964	−	0.919981i	$-0.628204\pi$
$241$	−12.1698	−0.783928	−0.391964	+	0.919981i	$0.628204\pi$
$242$	19.7743	1.27114
$243$	−1.00000	−0.0641500
$244$	2.11446	0.135365
$245$	−1.46647	−0.0936894
$246$	−21.7977	−1.38977
$247$	−5.05951	−0.321929
$248$	−2.36386	−0.150105
$249$	2.92312	0.185245
$250$	−22.5033	−1.42324
$251$	11.5041	0.726133	0.363067	−	0.931763i	$-0.381730\pi$
$251$	11.5041	0.726133	0.363067	+	0.931763i	$0.381730\pi$
$252$	1.82179	0.114762
$253$	−6.12374	−0.384996
$254$	−35.5526	−2.23077
$255$	−3.31569	−0.207637
$256$	18.4600	1.15375
$257$	−13.9902	−0.872682	−0.436341	−	0.899781i	$-0.643726\pi$
$257$	−13.9902	−0.872682	−0.436341	+	0.899781i	$0.643726\pi$
$258$	−7.14416	−0.444776
$259$	−2.02507	−0.125832
$260$	13.5207	0.838520
$261$	−3.46685	−0.214593
$262$	32.0908	1.98258
$263$	−3.47715	−0.214410	−0.107205	−	0.994237i	$-0.534190\pi$
$263$	−3.47715	−0.214410	−0.107205	+	0.994237i	$0.534190\pi$
$264$	−0.327745	−0.0201713
$265$	6.27038	0.385187
$266$	−1.95440	−0.119832
$267$	14.5404	0.889855
$268$	18.7139	1.14314
$269$	−20.4025	−1.24396	−0.621980	−	0.783033i	$-0.713671\pi$
$269$	−20.4025	−1.24396	−0.621980	+	0.783033i	$0.713671\pi$
$270$	−2.86686	−0.174472
$271$	−28.2484	−1.71597	−0.857983	−	0.513679i	$-0.828282\pi$
$271$	−28.2484	−1.71597	−0.857983	+	0.513679i	$0.828282\pi$
$272$	9.77807	0.592883
$273$	5.06092	0.306301
$274$	−17.3964	−1.05096
$275$	−2.68056	−0.161644
$276$	11.8591	0.713835
$277$	−8.01304	−0.481457	−0.240728	−	0.970593i	$-0.577386\pi$
$277$	−8.01304	−0.481457	−0.240728	+	0.970593i	$0.577386\pi$
$278$	−23.7382	−1.42372
$279$	−6.78497	−0.406205
$280$	−0.510913	−0.0305329
$281$	4.18120	0.249430	0.124715	−	0.992193i	$-0.460198\pi$
$281$	4.18120	0.249430	0.124715	+	0.992193i	$0.460198\pi$
$282$	6.59663	0.392824
$283$	23.6767	1.40743	0.703716	−	0.710481i	$-0.251523\pi$
$283$	23.6767	1.40743	0.703716	+	0.710481i	$0.251523\pi$
$284$	4.33838	0.257435
$285$	1.46606	0.0868421
$286$	9.30732	0.550353
$287$	−11.1501	−0.658167
$288$	7.75767	0.457125
$289$	−11.8879	−0.699287
$290$	−9.93899	−0.583638
$291$	−11.4502	−0.671225
$292$	−25.1975	−1.47457
$293$	−4.96299	−0.289941	−0.144971	−	0.989436i	$-0.546309\pi$
$293$	−4.96299	−0.289941	−0.144971	+	0.989436i	$0.546309\pi$
$294$	1.95494	0.114014
$295$	18.0099	1.04858
$296$	−0.705528	−0.0410080
$297$	−0.940724	−0.0545863
$298$	16.6672	0.965507
$299$	32.9446	1.90523
$300$	5.19111	0.299709
$301$	−3.65442	−0.210637
$302$	0.673151	0.0387355
$303$	1.17288	0.0673803
$304$	−4.32347	−0.247968
$305$	−1.70207	−0.0974600
$306$	4.42012	0.252681
$307$	20.5415	1.17236	0.586181	−	0.810180i	$-0.300631\pi$
$307$	20.5415	1.17236	0.586181	+	0.810180i	$0.300631\pi$
$308$	1.71380	0.0976527
$309$	4.96745	0.282588
$310$	−19.4516	−1.10477
$311$	−14.1106	−0.800136	−0.400068	−	0.916485i	$-0.631013\pi$
$311$	−14.1106	−0.800136	−0.400068	+	0.916485i	$0.631013\pi$
$312$	1.76321	0.0998219
$313$	33.3060	1.88257	0.941283	−	0.337618i	$-0.109621\pi$
$313$	33.3060	1.88257	0.941283	+	0.337618i	$0.109621\pi$
$314$	−22.0888	−1.24654
$315$	−1.46647	−0.0826263
$316$	15.8900	0.893884
$317$	−21.8091	−1.22492	−0.612459	−	0.790502i	$-0.709820\pi$
$317$	−21.8091	−1.22492	−0.612459	+	0.790502i	$0.709820\pi$
$318$	−8.35899	−0.468749
$319$	−3.26135	−0.182601
$320$	9.55616	0.534206
$321$	2.93347	0.163730
$322$	12.7259	0.709186
$323$	−2.26037	−0.125770
$324$	1.82179	0.101210
$325$	14.4209	0.799927
$326$	23.4303	1.29769
$327$	−12.0323	−0.665387
$328$	−3.88464	−0.214493
$329$	3.37434	0.186033
$330$	−2.69692	−0.148461
$331$	13.4444	0.738973	0.369486	−	0.929236i	$-0.379534\pi$
$331$	13.4444	0.738973	0.369486	+	0.929236i	$0.379534\pi$
$332$	−5.32530	−0.292264
$333$	−2.02507	−0.110973
$334$	−11.2334	−0.614664
$335$	−15.0640	−0.823036
$336$	4.32467	0.235930
$337$	19.6861	1.07237	0.536186	−	0.844100i	$-0.319864\pi$
$337$	19.6861	1.07237	0.536186	+	0.844100i	$0.319864\pi$
$338$	−24.6574	−1.34119
$339$	−6.24653	−0.339265
$340$	6.04048	0.327591
$341$	−6.38278	−0.345647
$342$	−1.95440	−0.105682
$343$	1.00000	0.0539949
$344$	−1.27319	−0.0686456
$345$	−9.54615	−0.513948
$346$	17.0100	0.914461
$347$	12.8226	0.688354	0.344177	−	0.938905i	$-0.388158\pi$
$347$	12.8226	0.688354	0.344177	+	0.938905i	$0.388158\pi$
$348$	6.31587	0.338566
$349$	5.75324	0.307964	0.153982	−	0.988074i	$-0.450790\pi$
$349$	5.75324	0.307964	0.153982	+	0.988074i	$0.450790\pi$
$350$	5.57053	0.297757
$351$	5.06092	0.270132
$352$	7.29782	0.388975
$353$	16.0578	0.854669	0.427335	−	0.904093i	$-0.359453\pi$
$353$	16.0578	0.854669	0.427335	+	0.904093i	$0.359453\pi$
$354$	−24.0089	−1.27606
$355$	−3.49223	−0.185349
$356$	−26.4894	−1.40394
$357$	2.26100	0.119665
$358$	9.75917	0.515788
$359$	3.14721	0.166103	0.0830517	−	0.996545i	$-0.473533\pi$
$359$	3.14721	0.166103	0.0830517	+	0.996545i	$0.473533\pi$
$360$	−0.510913	−0.0269275
$361$	−18.0006	−0.947398
$362$	31.9247	1.67792
$363$	10.1150	0.530902
$364$	−9.21991	−0.483255
$365$	20.2831	1.06166
$366$	2.26901	0.118603
$367$	−8.21567	−0.428855	−0.214427	−	0.976740i	$-0.568788\pi$
$367$	−8.21567	−0.428855	−0.214427	+	0.976740i	$0.568788\pi$
$368$	28.1519	1.46752
$369$	−11.1501	−0.580449
$370$	−5.80560	−0.301819
$371$	−4.27583	−0.221990
$372$	12.3608	0.640876
$373$	−13.0096	−0.673612	−0.336806	−	0.941574i	$-0.609347\pi$
$373$	−13.0096	−0.673612	−0.336806	+	0.941574i	$0.609347\pi$
$374$	4.15811	0.215011
$375$	−11.5110	−0.594426
$376$	1.17561	0.0606274
$377$	17.5455	0.903637
$378$	1.95494	0.100551
$379$	12.1158	0.622347	0.311173	−	0.950353i	$-0.399278\pi$
$379$	12.1158	0.622347	0.311173	+	0.950353i	$0.399278\pi$
$380$	−2.67086	−0.137012
$381$	−18.1860	−0.931699
$382$	−12.6666	−0.648078
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	2.77611	0.141668
$385$	−1.37954	−0.0703081
$386$	−2.62439	−0.133578
$387$	−3.65442	−0.185764
$388$	20.8599	1.05900
$389$	−15.4354	−0.782604	−0.391302	−	0.920262i	$-0.627975\pi$
$389$	−15.4354	−0.782604	−0.391302	+	0.920262i	$0.627975\pi$
$390$	14.5089	0.734689
$391$	14.7182	0.744333
$392$	0.348396	0.0175967
$393$	16.4153	0.828040
$394$	48.5852	2.44769
$395$	−12.7909	−0.643579
$396$	1.71380	0.0861216
$397$	14.5802	0.731759	0.365880	−	0.930662i	$-0.380768\pi$
$397$	14.5802	0.731759	0.365880	+	0.930662i	$0.380768\pi$
$398$	−26.4202	−1.32433
$399$	−0.999722	−0.0500487
$400$	12.3230	0.616149
$401$	−8.42292	−0.420621	−0.210310	−	0.977635i	$-0.567447\pi$
$401$	−8.42292	−0.420621	−0.210310	+	0.977635i	$0.567447\pi$
$402$	20.0817	1.00159
$403$	34.3382	1.71051
$404$	−2.13674	−0.106307
$405$	−1.46647	−0.0728695
$406$	6.77749	0.336361
$407$	−1.90503	−0.0944290
$408$	0.787724	0.0389982
$409$	2.32799	0.115112	0.0575559	−	0.998342i	$-0.481669\pi$
$409$	2.32799	0.115112	0.0575559	+	0.998342i	$0.481669\pi$
$410$	−31.9657	−1.57867
$411$	−8.89871	−0.438941
$412$	−9.04963	−0.445843
$413$	−12.2811	−0.604315
$414$	12.7259	0.625443
$415$	4.28667	0.210424
$416$	−39.2609	−1.92492
$417$	−12.1427	−0.594629
$418$	−1.83855	−0.0899263
$419$	−24.7529	−1.20926	−0.604630	−	0.796507i	$-0.706679\pi$
$419$	−24.7529	−1.20926	−0.604630	+	0.796507i	$0.706679\pi$
$420$	2.67160	0.130361
$421$	18.9269	0.922442	0.461221	−	0.887285i	$-0.347412\pi$
$421$	18.9269	0.922442	0.461221	+	0.887285i	$0.347412\pi$
$422$	22.1806	1.07973
$423$	3.37434	0.164066
$424$	−1.48968	−0.0723455
$425$	6.44263	0.312514
$426$	4.65546	0.225558
$427$	1.16065	0.0561680
$428$	−5.34416	−0.258320
$429$	4.76092	0.229860
$430$	−10.4767	−0.505232
$431$	−0.733052	−0.0353099	−0.0176549	−	0.999844i	$-0.505620\pi$
$431$	−0.733052	−0.0353099	−0.0176549	+	0.999844i	$0.505620\pi$
$432$	4.32467	0.208071
$433$	−1.90369	−0.0914856	−0.0457428	−	0.998953i	$-0.514565\pi$
$433$	−1.90369	−0.0914856	−0.0457428	+	0.998953i	$0.514565\pi$
$434$	13.2642	0.636702
$435$	−5.08404	−0.243761
$436$	21.9202	1.04979
$437$	−6.50780	−0.311310
$438$	−27.0392	−1.29198
$439$	−15.0644	−0.718985	−0.359492	−	0.933148i	$-0.617050\pi$
$439$	−15.0644	−0.718985	−0.359492	+	0.933148i	$0.617050\pi$
$440$	−0.480628	−0.0229131
$441$	1.00000	0.0476190
$442$	−22.3698	−1.06403
$443$	28.4508	1.35174	0.675869	−	0.737021i	$-0.263768\pi$
$443$	28.4508	1.35174	0.675869	+	0.737021i	$0.263768\pi$
$444$	3.68925	0.175084
$445$	21.3230	1.01081
$446$	6.79992	0.321986
$447$	8.52571	0.403252
$448$	−6.51643	−0.307872
$449$	7.79538	0.367887	0.183943	−	0.982937i	$-0.441114\pi$
$449$	7.79538	0.367887	0.183943	+	0.982937i	$0.441114\pi$
$450$	5.57053	0.262597
$451$	−10.4891	−0.493913
$452$	11.3798	0.535263
$453$	0.344333	0.0161782
$454$	−34.3280	−1.61109
$455$	7.42169	0.347934
$456$	−0.348300	−0.0163106
$457$	9.99146	0.467381	0.233691	−	0.972311i	$-0.424920\pi$
$457$	9.99146	0.467381	0.233691	+	0.972311i	$0.424920\pi$
$458$	18.1779	0.849398
$459$	2.26100	0.105534
$460$	17.3911	0.810862
$461$	13.2834	0.618670	0.309335	−	0.950953i	$-0.399894\pi$
$461$	13.2834	0.618670	0.309335	+	0.950953i	$0.399894\pi$
$462$	1.83906	0.0855607
$463$	−24.4910	−1.13819	−0.569097	−	0.822270i	$-0.692707\pi$
$463$	−24.4910	−1.13819	−0.569097	+	0.822270i	$0.692707\pi$
$464$	14.9930	0.696032
$465$	−9.94996	−0.461418
$466$	44.6504	2.06839
$467$	−0.0512726	−0.00237261	−0.00118631	−	0.999999i	$-0.500378\pi$
$467$	−0.0512726	−0.00237261	−0.00118631	+	0.999999i	$0.500378\pi$
$468$	−9.21991	−0.426191
$469$	10.2723	0.474331
$470$	9.67377	0.446218
$471$	−11.2990	−0.520629
$472$	−4.27870	−0.196943
$473$	−3.43780	−0.158070
$474$	17.0514	0.783197
$475$	−2.84867	−0.130706
$476$	−4.11906	−0.188797
$477$	−4.27583	−0.195777
$478$	−1.72451	−0.0788774
$479$	3.22870	0.147523	0.0737615	−	0.997276i	$-0.476500\pi$
$479$	3.22870	0.147523	0.0737615	+	0.997276i	$0.476500\pi$
$480$	11.3764	0.519259
$481$	10.2487	0.467301
$482$	23.7913	1.08366
$483$	6.50961	0.296197
$484$	−18.4274	−0.837611
$485$	−16.7914	−0.762460
$486$	1.95494	0.0886778
$487$	3.28653	0.148927	0.0744636	−	0.997224i	$-0.476276\pi$
$487$	3.28653	0.148927	0.0744636	+	0.997224i	$0.476276\pi$
$488$	0.404368	0.0183049
$489$	11.9852	0.541990
$490$	2.86686	0.129512
$491$	6.63849	0.299591	0.149796	−	0.988717i	$-0.452138\pi$
$491$	6.63849	0.299591	0.149796	+	0.988717i	$0.452138\pi$
$492$	20.3130	0.915782
$493$	7.83856	0.353031
$494$	9.89104	0.445019
$495$	−1.37954	−0.0620059
$496$	29.3427	1.31753
$497$	2.38139	0.106820
$498$	−5.71452	−0.256074
$499$	−9.04986	−0.405127	−0.202564	−	0.979269i	$-0.564927\pi$
$499$	−9.04986	−0.405127	−0.202564	+	0.979269i	$0.564927\pi$
$500$	20.9706	0.937834
$501$	−5.74616	−0.256719
$502$	−22.4898	−1.00377
$503$	18.9139	0.843331	0.421665	−	0.906751i	$-0.361446\pi$
$503$	18.9139	0.843331	0.421665	+	0.906751i	$0.361446\pi$
$504$	0.348396	0.0155188
$505$	1.72000	0.0765388
$506$	11.9715	0.532200
$507$	−12.6129	−0.560158
$508$	33.1311	1.46995
$509$	7.79853	0.345664	0.172832	−	0.984951i	$-0.444708\pi$
$509$	7.79853	0.345664	0.172832	+	0.984951i	$0.444708\pi$
$510$	6.48197	0.287027
$511$	−13.8312	−0.611857
$512$	−30.5359	−1.34951
$513$	−0.999722	−0.0441388
$514$	27.3499	1.20635
$515$	7.28462	0.320999
$516$	6.65757	0.293083
$517$	3.17432	0.139607
$518$	3.95889	0.173944
$519$	8.70102	0.381932
$520$	2.58569	0.113390
$521$	−27.1107	−1.18774	−0.593871	−	0.804560i	$-0.702401\pi$
$521$	−27.1107	−1.18774	−0.593871	+	0.804560i	$0.702401\pi$
$522$	6.77749	0.296643
$523$	−6.00622	−0.262634	−0.131317	−	0.991340i	$-0.541921\pi$
$523$	−6.00622	−0.262634	−0.131317	+	0.991340i	$0.541921\pi$
$524$	−29.9051	−1.30641
$525$	2.84946	0.124361
$526$	6.79762	0.296390
$527$	15.3408	0.668256
$528$	4.06832	0.177051
$529$	19.3750	0.842391
$530$	−12.2582	−0.532463
$531$	−12.2811	−0.532956
$532$	1.82128	0.0789625
$533$	56.4295	2.44423
$534$	−28.4255	−1.23009
$535$	4.30185	0.185985
$536$	3.57883	0.154582
$537$	4.99206	0.215423
$538$	39.8856	1.71959
$539$	0.940724	0.0405198
$540$	2.67160	0.114967
$541$	−12.8455	−0.552273	−0.276136	−	0.961118i	$-0.589054\pi$
$541$	−12.8455	−0.552273	−0.276136	+	0.961118i	$0.589054\pi$
$542$	55.2238	2.37207
$543$	16.3303	0.700799
$544$	−17.5401	−0.752025
$545$	−17.6450	−0.755828
$546$	−9.89378	−0.423415
$547$	−20.6363	−0.882345	−0.441172	−	0.897422i	$-0.645437\pi$
$547$	−20.6363	−0.882345	−0.441172	+	0.897422i	$0.645437\pi$
$548$	16.2116	0.692523
$549$	1.16065	0.0495355
$550$	5.24032	0.223448
$551$	−3.46589	−0.147652
$552$	2.26792	0.0965293
$553$	8.72222	0.370906
$554$	15.6650	0.665542
$555$	−2.96971	−0.126057
$556$	22.1214	0.938154
$557$	30.6316	1.29790	0.648951	−	0.760830i	$-0.275208\pi$
$557$	30.6316	1.29790	0.648951	+	0.760830i	$0.275208\pi$
$558$	13.2642	0.561518
$559$	18.4947	0.782243
$560$	6.34200	0.267998
$561$	2.12698	0.0898010
$562$	−8.17400	−0.344799
$563$	7.88912	0.332487	0.166243	−	0.986085i	$-0.446836\pi$
$563$	7.88912	0.332487	0.166243	+	0.986085i	$0.446836\pi$
$564$	−6.14733	−0.258849
$565$	−9.16036	−0.385379
$566$	−46.2864	−1.94556
$567$	1.00000	0.0419961
$568$	0.829667	0.0348120
$569$	−27.1127	−1.13662	−0.568312	−	0.822813i	$-0.692403\pi$
$569$	−27.1127	−1.13662	−0.568312	+	0.822813i	$0.692403\pi$
$570$	−2.86607	−0.120046
$571$	43.5057	1.82066	0.910329	−	0.413885i	$-0.135829\pi$
$571$	43.5057	1.82066	0.910329	+	0.413885i	$0.135829\pi$
$572$	−8.67339	−0.362653
$573$	−6.47927	−0.270675
$574$	21.7977	0.909817
$575$	18.5489	0.773542
$576$	−6.51643	−0.271518
$577$	−6.44412	−0.268272	−0.134136	−	0.990963i	$-0.542826\pi$
$577$	−6.44412	−0.268272	−0.134136	+	0.990963i	$0.542826\pi$
$578$	23.2401	0.966660
$579$	−1.34244	−0.0557900
$580$	9.26204	0.384585
$581$	−2.92312	−0.121271
$582$	22.3845	0.927868
$583$	−4.02238	−0.166590
$584$	−4.81874	−0.199401
$585$	7.42169	0.306849
$586$	9.70234	0.400800
$587$	−21.3449	−0.881000	−0.440500	−	0.897753i	$-0.645199\pi$
$587$	−21.3449	−0.881000	−0.440500	+	0.897753i	$0.645199\pi$
$588$	−1.82179	−0.0751292
$589$	−6.78309	−0.279492
$590$	−35.2083	−1.44950
$591$	24.8526	1.02230
$592$	8.75776	0.359942
$593$	30.0206	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$593$	30.0206	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$594$	1.83906	0.0754574
$595$	3.31569	0.135930
$596$	−15.5320	−0.636216
$597$	−13.5146	−0.553116
$598$	−64.4047	−2.63370
$599$	−9.35266	−0.382139	−0.191070	−	0.981576i	$-0.561196\pi$
$599$	−9.35266	−0.382139	−0.191070	+	0.981576i	$0.561196\pi$
$600$	0.992743	0.0405285
$601$	−34.8271	−1.42063	−0.710314	−	0.703885i	$-0.751447\pi$
$601$	−34.8271	−1.42063	−0.710314	+	0.703885i	$0.751447\pi$
$602$	7.14416	0.291174
$603$	10.2723	0.418321
$604$	−0.627302	−0.0255245
$605$	14.8334	0.603064
$606$	−2.29291	−0.0931432
$607$	−11.0556	−0.448735	−0.224367	−	0.974505i	$-0.572032\pi$
$607$	−11.0556	−0.448735	−0.224367	+	0.974505i	$0.572032\pi$
$608$	7.75551	0.314528
$609$	3.46685	0.140484
$610$	3.32744	0.134724
$611$	−17.0773	−0.690872
$612$	−4.11906	−0.166503
$613$	−21.4290	−0.865509	−0.432755	−	0.901512i	$-0.642458\pi$
$613$	−21.4290	−0.865509	−0.432755	+	0.901512i	$0.642458\pi$
$614$	−40.1573	−1.62062
$615$	−16.3512	−0.659345
$616$	0.327745	0.0132052
$617$	31.1684	1.25479	0.627396	−	0.778700i	$-0.284121\pi$
$617$	31.1684	1.25479	0.627396	+	0.778700i	$0.284121\pi$
$618$	−9.71106	−0.390636
$619$	0.395176	0.0158835	0.00794173	−	0.999968i	$-0.497472\pi$
$619$	0.395176	0.0158835	0.00794173	+	0.999968i	$0.497472\pi$
$620$	18.1267	0.727986
$621$	6.50961	0.261222
$622$	27.5853	1.10607
$623$	−14.5404	−0.582547
$624$	−21.8868	−0.876172
$625$	−2.63325	−0.105330
$626$	−65.1112	−2.60237
$627$	−0.940462	−0.0375585
$628$	20.5843	0.821404
$629$	4.57869	0.182564
$630$	2.86686	0.114219
$631$	15.7391	0.626565	0.313282	−	0.949660i	$-0.398571\pi$
$631$	15.7391	0.626565	0.313282	+	0.949660i	$0.398571\pi$
$632$	3.03879	0.120877
$633$	11.3459	0.450960
$634$	42.6354	1.69327
$635$	−26.6693	−1.05834
$636$	7.78965	0.308880
$637$	−5.06092	−0.200521
$638$	6.37574	0.252418
$639$	2.38139	0.0942062
$640$	4.07108	0.160924
$641$	26.6973	1.05448	0.527240	−	0.849716i	$-0.323227\pi$
$641$	26.6973	1.05448	0.527240	+	0.849716i	$0.323227\pi$
$642$	−5.73476	−0.226333
$643$	0.959406	0.0378353	0.0189176	−	0.999821i	$-0.493978\pi$
$643$	0.959406	0.0378353	0.0189176	+	0.999821i	$0.493978\pi$
$644$	−11.8591	−0.467315
$645$	−5.35910	−0.211014
$646$	4.41889	0.173859
$647$	−25.0747	−0.985790	−0.492895	−	0.870089i	$-0.664061\pi$
$647$	−25.0747	−0.985790	−0.492895	+	0.870089i	$0.664061\pi$
$648$	0.348396	0.0136863
$649$	−11.5532	−0.453501
$650$	−28.1920	−1.10578
$651$	6.78497	0.265924
$652$	−21.8345	−0.855104
$653$	−1.12924	−0.0441907	−0.0220953	−	0.999756i	$-0.507034\pi$
$653$	−1.12924	−0.0441907	−0.0220953	+	0.999756i	$0.507034\pi$
$654$	23.5224	0.919798
$655$	24.0725	0.940591
$656$	48.2203	1.88269
$657$	−13.8312	−0.539607
$658$	−6.59663	−0.257164
$659$	−22.1447	−0.862636	−0.431318	−	0.902200i	$-0.641951\pi$
$659$	−22.1447	−0.862636	−0.431318	+	0.902200i	$0.641951\pi$
$660$	2.51323	0.0978275
$661$	16.1186	0.626941	0.313470	−	0.949598i	$-0.398508\pi$
$661$	16.1186	0.626941	0.313470	+	0.949598i	$0.398508\pi$
$662$	−26.2830	−1.02152
$663$	−11.4427	−0.444399
$664$	−1.01840	−0.0395217
$665$	−1.46606	−0.0568515
$666$	3.95889	0.153404
$667$	22.5679	0.873831
$668$	10.4683	0.405030
$669$	3.47833	0.134480
$670$	29.4493	1.13772
$671$	1.09185	0.0421506
$672$	−7.75767	−0.299259
$673$	13.1733	0.507792	0.253896	−	0.967231i	$-0.418288\pi$
$673$	13.1733	0.507792	0.253896	+	0.967231i	$0.418288\pi$
$674$	−38.4852	−1.48239
$675$	2.84946	0.109676
$676$	22.9780	0.883768
$677$	−6.89888	−0.265145	−0.132573	−	0.991173i	$-0.542324\pi$
$677$	−6.89888	−0.265145	−0.132573	+	0.991173i	$0.542324\pi$
$678$	12.2116	0.468983
$679$	11.4502	0.439420
$680$	1.15518	0.0442989
$681$	−17.5596	−0.672887
$682$	12.4779	0.477805
$683$	17.7170	0.677924	0.338962	−	0.940800i	$-0.389924\pi$
$683$	17.7170	0.677924	0.338962	+	0.940800i	$0.389924\pi$
$684$	1.82128	0.0696384
$685$	−13.0497	−0.498603
$686$	−1.95494	−0.0746399
$687$	9.29845	0.354758
$688$	15.8041	0.602527
$689$	21.6396	0.824404
$690$	18.6621	0.710456
$691$	−42.7887	−1.62776	−0.813879	−	0.581034i	$-0.802648\pi$
$691$	−42.7887	−1.62776	−0.813879	+	0.581034i	$0.802648\pi$
$692$	−15.8514	−0.602580
$693$	0.940724	0.0357351
$694$	−25.0674	−0.951547
$695$	−17.8069	−0.675453
$696$	1.20784	0.0457830
$697$	25.2103	0.954907
$698$	−11.2472	−0.425714
$699$	22.8398	0.863881
$700$	−5.19111	−0.196206
$701$	−1.75483	−0.0662789	−0.0331394	−	0.999451i	$-0.510551\pi$
$701$	−1.75483	−0.0662789	−0.0331394	+	0.999451i	$0.510551\pi$
$702$	−9.89378	−0.373417
$703$	−2.02451	−0.0763558
$704$	−6.13016	−0.231039
$705$	4.94837	0.186367
$706$	−31.3920	−1.18145
$707$	−1.17288	−0.0441107
$708$	22.3736	0.840851
$709$	41.5674	1.56110	0.780548	−	0.625095i	$-0.214940\pi$
$709$	41.5674	1.56110	0.780548	+	0.625095i	$0.214940\pi$
$710$	6.82710	0.256217
$711$	8.72222	0.327109
$712$	−5.06581	−0.189849
$713$	44.1675	1.65409
$714$	−4.42012	−0.165419
$715$	6.98176	0.261103
$716$	−9.09446	−0.339876
$717$	−0.882131	−0.0329438
$718$	−6.15261	−0.229613
$719$	11.3608	0.423685	0.211842	−	0.977304i	$-0.432054\pi$
$719$	11.3608	0.423685	0.211842	+	0.977304i	$0.432054\pi$
$720$	6.34200	0.236352
$721$	−4.96745	−0.184997
$722$	35.1900	1.30964
$723$	12.1698	0.452601
$724$	−29.7503	−1.10566
$725$	9.87867	0.366885
$726$	−19.7743	−0.733893
$727$	36.6164	1.35803	0.679014	−	0.734125i	$-0.262407\pi$
$727$	36.6164	1.35803	0.679014	+	0.734125i	$0.262407\pi$
$728$	−1.76321	−0.0653487
$729$	1.00000	0.0370370
$730$	−39.6522	−1.46759
$731$	8.26264	0.305605
$732$	−2.11446	−0.0781529
$733$	−43.2343	−1.59690	−0.798448	−	0.602063i	$-0.794345\pi$
$733$	−43.2343	−1.59690	−0.798448	+	0.602063i	$0.794345\pi$
$734$	16.0611	0.592827
$735$	1.46647	0.0540916
$736$	−50.4994	−1.86143
$737$	9.66340	0.355956
$738$	21.7977	0.802383
$739$	4.29526	0.158004	0.0790018	−	0.996874i	$-0.474827\pi$
$739$	4.29526	0.158004	0.0790018	+	0.996874i	$0.474827\pi$
$740$	5.41018	0.198882
$741$	5.05951	0.185866
$742$	8.35899	0.306868
$743$	10.9100	0.400248	0.200124	−	0.979771i	$-0.435866\pi$
$743$	10.9100	0.400248	0.200124	+	0.979771i	$0.435866\pi$
$744$	2.36386	0.0866633
$745$	12.5027	0.458064
$746$	25.4330	0.931167
$747$	−2.92312	−0.106951
$748$	−3.87490	−0.141680
$749$	−2.93347	−0.107187
$750$	22.5033	0.821705
$751$	4.57642	0.166996	0.0834980	−	0.996508i	$-0.473391\pi$
$751$	4.57642	0.166996	0.0834980	+	0.996508i	$0.473391\pi$
$752$	−14.5929	−0.532148
$753$	−11.5041	−0.419233
$754$	−34.3003	−1.24914
$755$	0.504955	0.0183772
$756$	−1.82179	−0.0662577
$757$	−3.68657	−0.133991	−0.0669953	−	0.997753i	$-0.521341\pi$
$757$	−3.68657	−0.133991	−0.0669953	+	0.997753i	$0.521341\pi$
$758$	−23.6857	−0.860302
$759$	6.12374	0.222278
$760$	−0.510772	−0.0185276
$761$	−16.0754	−0.582734	−0.291367	−	0.956611i	$-0.594110\pi$
$761$	−16.0754	−0.582734	−0.291367	+	0.956611i	$0.594110\pi$
$762$	35.5526	1.28793
$763$	12.0323	0.435598
$764$	11.8038	0.427048
$765$	3.31569	0.119879
$766$	1.95494	0.0706348
$767$	62.1538	2.24424
$768$	−18.4600	−0.666117
$769$	25.5334	0.920758	0.460379	−	0.887722i	$-0.347714\pi$
$769$	25.5334	0.920758	0.460379	+	0.887722i	$0.347714\pi$
$770$	2.69692	0.0971904
$771$	13.9902	0.503843
$772$	2.44564	0.0880206
$773$	6.45187	0.232057	0.116029	−	0.993246i	$-0.462984\pi$
$773$	6.45187	0.232057	0.116029	+	0.993246i	$0.462984\pi$
$774$	7.14416	0.256792
$775$	19.3335	0.694480
$776$	3.98922	0.143205
$777$	2.02507	0.0726491
$778$	30.1752	1.08183
$779$	−11.1470	−0.399381
$780$	−13.5207	−0.484120
$781$	2.24023	0.0801616
$782$	−28.7732	−1.02893
$783$	3.46685	0.123895
$784$	−4.32467	−0.154452
$785$	−16.5696	−0.591395
$786$	−32.0908	−1.14464
$787$	4.02494	0.143474	0.0717368	−	0.997424i	$-0.477146\pi$
$787$	4.02494	0.143474	0.0717368	+	0.997424i	$0.477146\pi$
$788$	−45.2761	−1.61289
$789$	3.47715	0.123790
$790$	25.0054	0.889652
$791$	6.24653	0.222101
$792$	0.327745	0.0116459
$793$	−5.87398	−0.208591
$794$	−28.5034	−1.01155
$795$	−6.27038	−0.222388
$796$	24.6207	0.872659
$797$	−25.8606	−0.916028	−0.458014	−	0.888945i	$-0.651439\pi$
$797$	−25.8606	−0.916028	−0.458014	+	0.888945i	$0.651439\pi$
$798$	1.95440	0.0691849
$799$	−7.62939	−0.269908
$800$	−22.1052	−0.781536
$801$	−14.5404	−0.513758
$802$	16.4663	0.581445
$803$	−13.0113	−0.459160
$804$	−18.7139	−0.659990
$805$	9.54615	0.336458
$806$	−67.1290	−2.36452
$807$	20.4025	0.718200
$808$	−0.408628	−0.0143755
$809$	11.0174	0.387353	0.193676	−	0.981065i	$-0.437959\pi$
$809$	11.0174	0.387353	0.193676	+	0.981065i	$0.437959\pi$
$810$	2.86686	0.100731
$811$	46.7015	1.63991	0.819956	−	0.572426i	$-0.193998\pi$
$811$	46.7015	1.63991	0.819956	+	0.572426i	$0.193998\pi$
$812$	−6.31587	−0.221644
$813$	28.2484	0.990713
$814$	3.72422	0.130534
$815$	17.5760	0.615659
$816$	−9.77807	−0.342301
$817$	−3.65340	−0.127816
$818$	−4.55108	−0.159125
$819$	−5.06092	−0.176843
$820$	29.7885	1.04026
$821$	34.6546	1.20945	0.604727	−	0.796433i	$-0.293282\pi$
$821$	34.6546	1.20945	0.604727	+	0.796433i	$0.293282\pi$
$822$	17.3964	0.606770
$823$	5.59300	0.194960	0.0974798	−	0.995238i	$-0.468922\pi$
$823$	5.59300	0.194960	0.0974798	+	0.995238i	$0.468922\pi$
$824$	−1.73064	−0.0602897
$825$	2.68056	0.0933250
$826$	24.0089	0.835375
$827$	−24.8420	−0.863842	−0.431921	−	0.901911i	$-0.642164\pi$
$827$	−24.8420	−0.863842	−0.431921	+	0.901911i	$0.642164\pi$
$828$	−11.8591	−0.412133
$829$	41.6683	1.44720	0.723600	−	0.690220i	$-0.242486\pi$
$829$	41.6683	1.44720	0.723600	+	0.690220i	$0.242486\pi$
$830$	−8.38018	−0.290880
$831$	8.01304	0.277969
$832$	32.9791	1.14335
$833$	−2.26100	−0.0783390
$834$	23.7382	0.821986
$835$	−8.42657	−0.291614
$836$	1.71332	0.0592565
$837$	6.78497	0.234523
$838$	48.3905	1.67162
$839$	−15.4387	−0.533002	−0.266501	−	0.963835i	$-0.585868\pi$
$839$	−15.4387	−0.533002	−0.266501	+	0.963835i	$0.585868\pi$
$840$	0.510913	0.0176282
$841$	−16.9809	−0.585549
$842$	−37.0010	−1.27514
$843$	−4.18120	−0.144008
$844$	−20.6698	−0.711485
$845$	−18.4964	−0.636296
$846$	−6.59663	−0.226797
$847$	−10.1150	−0.347557
$848$	18.4915	0.635002
$849$	−23.6767	−0.812581
$850$	−12.5950	−0.432004
$851$	13.1824	0.451888
$852$	−4.33838	−0.148630
$853$	−53.1353	−1.81932	−0.909659	−	0.415357i	$-0.863657\pi$
$853$	−53.1353	−1.81932	−0.909659	+	0.415357i	$0.863657\pi$
$854$	−2.26901	−0.0776439
$855$	−1.46606	−0.0501383
$856$	−1.02201	−0.0349316
$857$	−54.1981	−1.85137	−0.925685	−	0.378295i	$-0.876511\pi$
$857$	−54.1981	−1.85137	−0.925685	+	0.378295i	$0.876511\pi$
$858$	−9.30732	−0.317747
$859$	−4.42347	−0.150927	−0.0754633	−	0.997149i	$-0.524044\pi$
$859$	−4.42347	−0.150927	−0.0754633	+	0.997149i	$0.524044\pi$
$860$	9.76313	0.332920
$861$	11.1501	0.379993
$862$	1.43307	0.0488106
$863$	−3.73404	−0.127108	−0.0635541	−	0.997978i	$-0.520244\pi$
$863$	−3.73404	−0.127108	−0.0635541	+	0.997978i	$0.520244\pi$
$864$	−7.75767	−0.263921
$865$	12.7598	0.433846
$866$	3.72160	0.126465
$867$	11.8879	0.403734
$868$	−12.3608	−0.419552
$869$	8.20520	0.278342
$870$	9.93899	0.336963
$871$	−51.9873	−1.76152
$872$	4.19200	0.141959
$873$	11.4502	0.387532
$874$	12.7224	0.430340
$875$	11.5110	0.389143
$876$	25.1975	0.851345
$877$	34.0569	1.15002	0.575010	−	0.818147i	$-0.304998\pi$
$877$	34.0569	1.15002	0.575010	+	0.818147i	$0.304998\pi$
$878$	29.4500	0.993889
$879$	4.96299	0.167398
$880$	5.96607	0.201116
$881$	31.5757	1.06381	0.531906	−	0.846803i	$-0.321476\pi$
$881$	31.5757	1.06381	0.531906	+	0.846803i	$0.321476\pi$
$882$	−1.95494	−0.0658262
$883$	−37.6272	−1.26626	−0.633128	−	0.774047i	$-0.718229\pi$
$883$	−37.6272	−1.26626	−0.633128	+	0.774047i	$0.718229\pi$
$884$	20.8462	0.701134
$885$	−18.0099	−0.605397
$886$	−55.6196	−1.86858
$887$	29.9572	1.00587	0.502933	−	0.864325i	$-0.332254\pi$
$887$	29.9572	1.00587	0.502933	+	0.864325i	$0.332254\pi$
$888$	0.705528	0.0236760
$889$	18.1860	0.609940
$890$	−41.6852	−1.39729
$891$	0.940724	0.0315154
$892$	−6.33678	−0.212171
$893$	3.37341	0.112887
$894$	−16.6672	−0.557436
$895$	7.32071	0.244704
$896$	−2.77611	−0.0927432
$897$	−32.9446	−1.09999
$898$	−15.2395	−0.508549
$899$	23.5225	0.784519
$900$	−5.19111	−0.173037
$901$	9.66766	0.322076
$902$	20.5056	0.682761
$903$	3.65442	0.121611
$904$	2.17627	0.0723817
$905$	23.9479	0.796054
$906$	−0.673151	−0.0223639
$907$	−1.89647	−0.0629711	−0.0314856	−	0.999504i	$-0.510024\pi$
$907$	−1.89647	−0.0629711	−0.0314856	+	0.999504i	$0.510024\pi$
$908$	31.9899	1.06162
$909$	−1.17288	−0.0389020
$910$	−14.5089	−0.480967
$911$	−13.5165	−0.447820	−0.223910	−	0.974610i	$-0.571882\pi$
$911$	−13.5165	−0.447820	−0.223910	+	0.974610i	$0.571882\pi$
$912$	4.32347	0.143164
$913$	−2.74985	−0.0910066
$914$	−19.5327	−0.646084
$915$	1.70207	0.0562686
$916$	−16.9398	−0.559707
$917$	−16.4153	−0.542080
$918$	−4.42012	−0.145886
$919$	−11.4446	−0.377523	−0.188762	−	0.982023i	$-0.560447\pi$
$919$	−11.4446	−0.377523	−0.188762	+	0.982023i	$0.560447\pi$
$920$	3.32585	0.109650
$921$	−20.5415	−0.676864
$922$	−25.9683	−0.855219
$923$	−12.0520	−0.396696
$924$	−1.71380	−0.0563798
$925$	5.77037	0.189728
$926$	47.8784	1.57338
$927$	−4.96745	−0.163152
$928$	−26.8947	−0.882862
$929$	−5.49232	−0.180197	−0.0900986	−	0.995933i	$-0.528718\pi$
$929$	−5.49232	−0.180197	−0.0900986	+	0.995933i	$0.528718\pi$
$930$	19.4516	0.637842
$931$	0.999722	0.0327646
$932$	−41.6093	−1.36296
$933$	14.1106	0.461959
$934$	0.100235	0.00327978
$935$	3.11915	0.102007
$936$	−1.76321	−0.0576322
$937$	−17.5025	−0.571780	−0.285890	−	0.958262i	$-0.592289\pi$
$937$	−17.5025	−0.571780	−0.285890	+	0.958262i	$0.592289\pi$
$938$	−20.0817	−0.655692
$939$	−33.3060	−1.08690
$940$	−9.01488	−0.294033
$941$	0.517014	0.0168542	0.00842709	−	0.999964i	$-0.497318\pi$
$941$	0.517014	0.0168542	0.00842709	+	0.999964i	$0.497318\pi$
$942$	22.0888	0.719692
$943$	72.5825	2.36361
$944$	53.1118	1.72864
$945$	1.46647	0.0477043
$946$	6.72068	0.218508
$947$	26.0157	0.845396	0.422698	−	0.906271i	$-0.361083\pi$
$947$	26.0157	0.845396	0.422698	+	0.906271i	$0.361083\pi$
$948$	−15.8900	−0.516084
$949$	69.9986	2.27225
$950$	5.56898	0.180682
$951$	21.8091	0.707207
$952$	−0.787724	−0.0255303
$953$	11.7992	0.382215	0.191108	−	0.981569i	$-0.438792\pi$
$953$	11.7992	0.382215	0.191108	+	0.981569i	$0.438792\pi$
$954$	8.35899	0.270632
$955$	−9.50166	−0.307466
$956$	1.60706	0.0519759
$957$	3.26135	0.105425
$958$	−6.31191	−0.203929
$959$	8.89871	0.287354
$960$	−9.55616	−0.308424
$961$	15.0358	0.485026
$962$	−20.0356	−0.645974
$963$	−2.93347	−0.0945298
$964$	−22.1708	−0.714074
$965$	−1.96865	−0.0633731
$966$	−12.7259	−0.409449
$967$	−15.9799	−0.513877	−0.256939	−	0.966428i	$-0.582714\pi$
$967$	−15.9799	−0.513877	−0.256939	+	0.966428i	$0.582714\pi$
$968$	−3.52404	−0.113267
$969$	2.26037	0.0726136
$970$	32.8263	1.05399
$971$	−7.48091	−0.240074	−0.120037	−	0.992769i	$-0.538301\pi$
$971$	−7.48091	−0.240074	−0.120037	+	0.992769i	$0.538301\pi$
$972$	−1.82179	−0.0584338
$973$	12.1427	0.389276
$974$	−6.42498	−0.205869
$975$	−14.4209	−0.461838
$976$	−5.01944	−0.160668
$977$	−22.1403	−0.708330	−0.354165	−	0.935183i	$-0.615235\pi$
$977$	−22.1403	−0.708330	−0.354165	+	0.935183i	$0.615235\pi$
$978$	−23.4303	−0.749220
$979$	−13.6785	−0.437165
$980$	−2.67160	−0.0853410
$981$	12.0323	0.384161
$982$	−12.9779	−0.414140
$983$	9.42302	0.300548	0.150274	−	0.988644i	$-0.451984\pi$
$983$	9.42302	0.300548	0.150274	+	0.988644i	$0.451984\pi$
$984$	3.88464	0.123838
$985$	36.4456	1.16125
$986$	−15.3239	−0.488012
$987$	−3.37434	−0.107406
$988$	−9.21735	−0.293243
$989$	23.7888	0.756441
$990$	2.69692	0.0857139
$991$	55.5757	1.76542	0.882710	−	0.469917i	$-0.155716\pi$
$991$	55.5757	1.76542	0.882710	+	0.469917i	$0.155716\pi$
$992$	−52.6355	−1.67118
$993$	−13.4444	−0.426646
$994$	−4.65546	−0.147662
$995$	−19.8188	−0.628298
$996$	5.32530	0.168739
$997$	33.9626	1.07561	0.537804	−	0.843070i	$-0.319254\pi$
$997$	33.9626	1.07561	0.537804	+	0.843070i	$0.319254\pi$
$998$	17.6919	0.560028
$999$	2.02507	0.0640704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.8	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-1.95494 q^{2} -1.00000 q^{3} +1.82179 q^{4} -1.46647 q^{5} +1.95494 q^{6} +1.00000 q^{7} +0.348396 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.95494 q^{2} -1.00000 q^{3} +1.82179 q^{4} -1.46647 q^{5} +1.95494 q^{6} +1.00000 q^{7} +0.348396 q^{8} +1.00000 q^{9} +2.86686 q^{10} +0.940724 q^{11} -1.82179 q^{12} -5.06092 q^{13} -1.95494 q^{14} +1.46647 q^{15} -4.32467 q^{16} -2.26100 q^{17} -1.95494 q^{18} +0.999722 q^{19} -2.67160 q^{20} -1.00000 q^{21} -1.83906 q^{22} -6.50961 q^{23} -0.348396 q^{24} -2.84946 q^{25} +9.89378 q^{26} -1.00000 q^{27} +1.82179 q^{28} -3.46685 q^{29} -2.86686 q^{30} -6.78497 q^{31} +7.75767 q^{32} -0.940724 q^{33} +4.42012 q^{34} -1.46647 q^{35} +1.82179 q^{36} -2.02507 q^{37} -1.95440 q^{38} +5.06092 q^{39} -0.510913 q^{40} -11.1501 q^{41} +1.95494 q^{42} -3.65442 q^{43} +1.71380 q^{44} -1.46647 q^{45} +12.7259 q^{46} +3.37434 q^{47} +4.32467 q^{48} +1.00000 q^{49} +5.57053 q^{50} +2.26100 q^{51} -9.21991 q^{52} -4.27583 q^{53} +1.95494 q^{54} -1.37954 q^{55} +0.348396 q^{56} -0.999722 q^{57} +6.77749 q^{58} -12.2811 q^{59} +2.67160 q^{60} +1.16065 q^{61} +13.2642 q^{62} +1.00000 q^{63} -6.51643 q^{64} +7.42169 q^{65} +1.83906 q^{66} +10.2723 q^{67} -4.11906 q^{68} +6.50961 q^{69} +2.86686 q^{70} +2.38139 q^{71} +0.348396 q^{72} -13.8312 q^{73} +3.95889 q^{74} +2.84946 q^{75} +1.82128 q^{76} +0.940724 q^{77} -9.89378 q^{78} +8.72222 q^{79} +6.34200 q^{80} +1.00000 q^{81} +21.7977 q^{82} -2.92312 q^{83} -1.82179 q^{84} +3.31569 q^{85} +7.14416 q^{86} +3.46685 q^{87} +0.327745 q^{88} -14.5404 q^{89} +2.86686 q^{90} -5.06092 q^{91} -11.8591 q^{92} +6.78497 q^{93} -6.59663 q^{94} -1.46606 q^{95} -7.75767 q^{96} +11.4502 q^{97} -1.95494 q^{98} +0.940724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 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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