LMFDB - Embedded newform 4006.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.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:	$31.9880710497$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.67899 q^{3} +1.00000 q^{4} +3.78338 q^{5} -2.67899 q^{6} +0.625122 q^{7} +1.00000 q^{8} +4.17700 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.67899 q^{3} +1.00000 q^{4} +3.78338 q^{5} -2.67899 q^{6} +0.625122 q^{7} +1.00000 q^{8} +4.17700 q^{9} +3.78338 q^{10} +0.853817 q^{11} -2.67899 q^{12} +2.70785 q^{13} +0.625122 q^{14} -10.1356 q^{15} +1.00000 q^{16} -5.14733 q^{17} +4.17700 q^{18} +4.43495 q^{19} +3.78338 q^{20} -1.67470 q^{21} +0.853817 q^{22} +3.96804 q^{23} -2.67899 q^{24} +9.31393 q^{25} +2.70785 q^{26} -3.15317 q^{27} +0.625122 q^{28} -1.84441 q^{29} -10.1356 q^{30} +7.54094 q^{31} +1.00000 q^{32} -2.28737 q^{33} -5.14733 q^{34} +2.36507 q^{35} +4.17700 q^{36} -7.26788 q^{37} +4.43495 q^{38} -7.25431 q^{39} +3.78338 q^{40} -11.9980 q^{41} -1.67470 q^{42} +8.82565 q^{43} +0.853817 q^{44} +15.8032 q^{45} +3.96804 q^{46} +7.32582 q^{47} -2.67899 q^{48} -6.60922 q^{49} +9.31393 q^{50} +13.7897 q^{51} +2.70785 q^{52} +10.6840 q^{53} -3.15317 q^{54} +3.23031 q^{55} +0.625122 q^{56} -11.8812 q^{57} -1.84441 q^{58} +7.78467 q^{59} -10.1356 q^{60} -12.3736 q^{61} +7.54094 q^{62} +2.61113 q^{63} +1.00000 q^{64} +10.2448 q^{65} -2.28737 q^{66} -1.89719 q^{67} -5.14733 q^{68} -10.6304 q^{69} +2.36507 q^{70} -5.74352 q^{71} +4.17700 q^{72} +11.7188 q^{73} -7.26788 q^{74} -24.9519 q^{75} +4.43495 q^{76} +0.533740 q^{77} -7.25431 q^{78} -7.72379 q^{79} +3.78338 q^{80} -4.08368 q^{81} -11.9980 q^{82} -0.592104 q^{83} -1.67470 q^{84} -19.4743 q^{85} +8.82565 q^{86} +4.94115 q^{87} +0.853817 q^{88} +13.2052 q^{89} +15.8032 q^{90} +1.69274 q^{91} +3.96804 q^{92} -20.2021 q^{93} +7.32582 q^{94} +16.7791 q^{95} -2.67899 q^{96} +3.18485 q^{97} -6.60922 q^{98} +3.56639 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9}+O(q^{10}) $ 46 * q + 46 * q^2 + 21 * q^3 + 46 * q^4 + 23 * q^5 + 21 * q^6 + 26 * q^7 + 46 * q^8 + 59 * q^9	\( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9} + 23 q^{10} + 39 q^{11} + 21 q^{12} + 8 q^{13} + 26 q^{14} + 14 q^{15} + 46 q^{16} + 36 q^{17} + 59 q^{18} + 37 q^{19} + 23 q^{20} + 20 q^{21} + 39 q^{22} + 38 q^{23} + 21 q^{24} + 57 q^{25} + 8 q^{26} + 63 q^{27} + 26 q^{28} + 23 q^{29} + 14 q^{30} + 44 q^{31} + 46 q^{32} + 25 q^{33} + 36 q^{34} + 26 q^{35} + 59 q^{36} + 9 q^{37} + 37 q^{38} - 2 q^{39} + 23 q^{40} + 50 q^{41} + 20 q^{42} + 46 q^{43} + 39 q^{44} + 30 q^{45} + 38 q^{46} + 57 q^{47} + 21 q^{48} + 62 q^{49} + 57 q^{50} + 5 q^{51} + 8 q^{52} + 21 q^{53} + 63 q^{54} + 40 q^{55} + 26 q^{56} + 3 q^{57} + 23 q^{58} + 68 q^{59} + 14 q^{60} - q^{61} + 44 q^{62} + 40 q^{63} + 46 q^{64} + 18 q^{65} + 25 q^{66} + 42 q^{67} + 36 q^{68} - 7 q^{69} + 26 q^{70} + 67 q^{71} + 59 q^{72} + 48 q^{73} + 9 q^{74} + 71 q^{75} + 37 q^{76} - 5 q^{77} - 2 q^{78} + 40 q^{79} + 23 q^{80} + 82 q^{81} + 50 q^{82} + 48 q^{83} + 20 q^{84} - 68 q^{85} + 46 q^{86} + 18 q^{87} + 39 q^{88} + 90 q^{89} + 30 q^{90} + 9 q^{91} + 38 q^{92} - 42 q^{93} + 57 q^{94} + 6 q^{95} + 21 q^{96} + 46 q^{97} + 62 q^{98} + 61 q^{99}+O(q^{100}) \) 46 * q + 46 * q^2 + 21 * q^3 + 46 * q^4 + 23 * q^5 + 21 * q^6 + 26 * q^7 + 46 * q^8 + 59 * q^9 + 23 * q^10 + 39 * q^11 + 21 * q^12 + 8 * q^13 + 26 * q^14 + 14 * q^15 + 46 * q^16 + 36 * q^17 + 59 * q^18 + 37 * q^19 + 23 * q^20 + 20 * q^21 + 39 * q^22 + 38 * q^23 + 21 * q^24 + 57 * q^25 + 8 * q^26 + 63 * q^27 + 26 * q^28 + 23 * q^29 + 14 * q^30 + 44 * q^31 + 46 * q^32 + 25 * q^33 + 36 * q^34 + 26 * q^35 + 59 * q^36 + 9 * q^37 + 37 * q^38 - 2 * q^39 + 23 * q^40 + 50 * q^41 + 20 * q^42 + 46 * q^43 + 39 * q^44 + 30 * q^45 + 38 * q^46 + 57 * q^47 + 21 * q^48 + 62 * q^49 + 57 * q^50 + 5 * q^51 + 8 * q^52 + 21 * q^53 + 63 * q^54 + 40 * q^55 + 26 * q^56 + 3 * q^57 + 23 * q^58 + 68 * q^59 + 14 * q^60 - q^61 + 44 * q^62 + 40 * q^63 + 46 * q^64 + 18 * q^65 + 25 * q^66 + 42 * q^67 + 36 * q^68 - 7 * q^69 + 26 * q^70 + 67 * q^71 + 59 * q^72 + 48 * q^73 + 9 * q^74 + 71 * q^75 + 37 * q^76 - 5 * q^77 - 2 * q^78 + 40 * q^79 + 23 * q^80 + 82 * q^81 + 50 * q^82 + 48 * q^83 + 20 * q^84 - 68 * q^85 + 46 * q^86 + 18 * q^87 + 39 * q^88 + 90 * q^89 + 30 * q^90 + 9 * q^91 + 38 * q^92 - 42 * q^93 + 57 * q^94 + 6 * q^95 + 21 * q^96 + 46 * q^97 + 62 * q^98 + 61 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.67899	−1.54672	−0.773358	−	0.633969i	$-0.781425\pi$
$3$	−2.67899	−1.54672	−0.773358	+	0.633969i	$0.781425\pi$
$4$	1.00000	0.500000
$5$	3.78338	1.69198	0.845988	−	0.533201i	$-0.179011\pi$
$5$	3.78338	1.69198	0.845988	+	0.533201i	$0.179011\pi$
$6$	−2.67899	−1.09369
$7$	0.625122	0.236274	0.118137	−	0.992997i	$-0.462308\pi$
$7$	0.625122	0.236274	0.118137	+	0.992997i	$0.462308\pi$
$8$	1.00000	0.353553
$9$	4.17700	1.39233
$10$	3.78338	1.19641
$11$	0.853817	0.257436	0.128718	−	0.991681i	$-0.458914\pi$
$11$	0.853817	0.257436	0.128718	+	0.991681i	$0.458914\pi$
$12$	−2.67899	−0.773358
$13$	2.70785	0.751022	0.375511	−	0.926818i	$-0.377467\pi$
$13$	2.70785	0.751022	0.375511	+	0.926818i	$0.377467\pi$
$14$	0.625122	0.167071
$15$	−10.1356	−2.61701
$16$	1.00000	0.250000
$17$	−5.14733	−1.24841	−0.624205	−	0.781260i	$-0.714577\pi$
$17$	−5.14733	−1.24841	−0.624205	+	0.781260i	$0.714577\pi$
$18$	4.17700	0.984528
$19$	4.43495	1.01745	0.508723	−	0.860930i	$-0.330118\pi$
$19$	4.43495	1.01745	0.508723	+	0.860930i	$0.330118\pi$
$20$	3.78338	0.845988
$21$	−1.67470	−0.365449
$22$	0.853817	0.182034
$23$	3.96804	0.827394	0.413697	−	0.910415i	$-0.364237\pi$
$23$	3.96804	0.827394	0.413697	+	0.910415i	$0.364237\pi$
$24$	−2.67899	−0.546847
$25$	9.31393	1.86279
$26$	2.70785	0.531053
$27$	−3.15317	−0.606828
$28$	0.625122	0.118137
$29$	−1.84441	−0.342498	−0.171249	−	0.985228i	$-0.554780\pi$
$29$	−1.84441	−0.342498	−0.171249	+	0.985228i	$0.554780\pi$
$30$	−10.1356	−1.85050
$31$	7.54094	1.35439	0.677196	−	0.735802i	$-0.263195\pi$
$31$	7.54094	1.35439	0.677196	+	0.735802i	$0.263195\pi$
$32$	1.00000	0.176777
$33$	−2.28737	−0.398180
$34$	−5.14733	−0.882760
$35$	2.36507	0.399770
$36$	4.17700	0.696166
$37$	−7.26788	−1.19483	−0.597416	−	0.801931i	$-0.703806\pi$
$37$	−7.26788	−1.19483	−0.597416	+	0.801931i	$0.703806\pi$
$38$	4.43495	0.719443
$39$	−7.25431	−1.16162
$40$	3.78338	0.598204
$41$	−11.9980	−1.87377	−0.936885	−	0.349639i	$-0.886304\pi$
$41$	−11.9980	−1.87377	−0.936885	+	0.349639i	$0.886304\pi$
$42$	−1.67470	−0.258411
$43$	8.82565	1.34590	0.672949	−	0.739688i	$-0.265027\pi$
$43$	8.82565	1.34590	0.672949	+	0.739688i	$0.265027\pi$
$44$	0.853817	0.128718
$45$	15.8032	2.35579
$46$	3.96804	0.585056
$47$	7.32582	1.06858	0.534290	−	0.845301i	$-0.320579\pi$
$47$	7.32582	1.06858	0.534290	+	0.845301i	$0.320579\pi$
$48$	−2.67899	−0.386679
$49$	−6.60922	−0.944175
$50$	9.31393	1.31719
$51$	13.7897	1.93094
$52$	2.70785	0.375511
$53$	10.6840	1.46756	0.733778	−	0.679389i	$-0.237755\pi$
$53$	10.6840	1.46756	0.733778	+	0.679389i	$0.237755\pi$
$54$	−3.15317	−0.429092
$55$	3.23031	0.435575
$56$	0.625122	0.0835354
$57$	−11.8812	−1.57370
$58$	−1.84441	−0.242183
$59$	7.78467	1.01348	0.506739	−	0.862100i	$-0.330851\pi$
$59$	7.78467	1.01348	0.506739	+	0.862100i	$0.330851\pi$
$60$	−10.1356	−1.30850
$61$	−12.3736	−1.58428	−0.792141	−	0.610338i	$-0.791034\pi$
$61$	−12.3736	−1.58428	−0.792141	+	0.610338i	$0.791034\pi$
$62$	7.54094	0.957700
$63$	2.61113	0.328972
$64$	1.00000	0.125000
$65$	10.2448	1.27071
$66$	−2.28737	−0.281556
$67$	−1.89719	−0.231779	−0.115889	−	0.993262i	$-0.536972\pi$
$67$	−1.89719	−0.231779	−0.115889	+	0.993262i	$0.536972\pi$
$68$	−5.14733	−0.624205
$69$	−10.6304	−1.27974
$70$	2.36507	0.282680
$71$	−5.74352	−0.681630	−0.340815	−	0.940130i	$-0.610703\pi$
$71$	−5.74352	−0.681630	−0.340815	+	0.940130i	$0.610703\pi$
$72$	4.17700	0.492264
$73$	11.7188	1.37158	0.685790	−	0.727799i	$-0.259457\pi$
$73$	11.7188	1.37158	0.685790	+	0.727799i	$0.259457\pi$
$74$	−7.26788	−0.844874
$75$	−24.9519	−2.88120
$76$	4.43495	0.508723
$77$	0.533740	0.0608253
$78$	−7.25431	−0.821389
$79$	−7.72379	−0.868995	−0.434497	−	0.900673i	$-0.643074\pi$
$79$	−7.72379	−0.868995	−0.434497	+	0.900673i	$0.643074\pi$
$80$	3.78338	0.422994
$81$	−4.08368	−0.453742
$82$	−11.9980	−1.32496
$83$	−0.592104	−0.0649919	−0.0324959	−	0.999472i	$-0.510346\pi$
$83$	−0.592104	−0.0649919	−0.0324959	+	0.999472i	$0.510346\pi$
$84$	−1.67470	−0.182724
$85$	−19.4743	−2.11228
$86$	8.82565	0.951694
$87$	4.94115	0.529747
$88$	0.853817	0.0910172
$89$	13.2052	1.39975	0.699873	−	0.714267i	$-0.253240\pi$
$89$	13.2052	1.39975	0.699873	+	0.714267i	$0.253240\pi$
$90$	15.8032	1.66580
$91$	1.69274	0.177447
$92$	3.96804	0.413697
$93$	−20.2021	−2.09486
$94$	7.32582	0.755600
$95$	16.7791	1.72150
$96$	−2.67899	−0.273423
$97$	3.18485	0.323373	0.161686	−	0.986842i	$-0.448307\pi$
$97$	3.18485	0.323373	0.161686	+	0.986842i	$0.448307\pi$
$98$	−6.60922	−0.667632
$99$	3.56639	0.358436
$100$	9.31393	0.931393
$101$	−13.8439	−1.37752	−0.688762	−	0.724988i	$-0.741845\pi$
$101$	−13.8439	−1.37752	−0.688762	+	0.724988i	$0.741845\pi$
$102$	13.7897	1.36538
$103$	−16.8107	−1.65641	−0.828203	−	0.560428i	$-0.810637\pi$
$103$	−16.8107	−1.65641	−0.828203	+	0.560428i	$0.810637\pi$
$104$	2.70785	0.265526
$105$	−6.33600	−0.618331
$106$	10.6840	1.03772
$107$	−10.9565	−1.05920	−0.529601	−	0.848247i	$-0.677658\pi$
$107$	−10.9565	−1.05920	−0.529601	+	0.848247i	$0.677658\pi$
$108$	−3.15317	−0.303414
$109$	16.3671	1.56768	0.783840	−	0.620963i	$-0.213258\pi$
$109$	16.3671	1.56768	0.783840	+	0.620963i	$0.213258\pi$
$110$	3.23031	0.307998
$111$	19.4706	1.84807
$112$	0.625122	0.0590684
$113$	9.94394	0.935447	0.467723	−	0.883875i	$-0.345074\pi$
$113$	9.94394	0.935447	0.467723	+	0.883875i	$0.345074\pi$
$114$	−11.8812	−1.11277
$115$	15.0126	1.39993
$116$	−1.84441	−0.171249
$117$	11.3107	1.04567
$118$	7.78467	0.716637
$119$	−3.21771	−0.294967
$120$	−10.1356	−0.925252
$121$	−10.2710	−0.933727
$122$	−12.3736	−1.12026
$123$	32.1425	2.89819
$124$	7.54094	0.677196
$125$	16.3212	1.45981
$126$	2.61113	0.232618
$127$	−13.8425	−1.22832	−0.614161	−	0.789181i	$-0.710506\pi$
$127$	−13.8425	−1.22832	−0.614161	+	0.789181i	$0.710506\pi$
$128$	1.00000	0.0883883
$129$	−23.6438	−2.08172
$130$	10.2448	0.898529
$131$	14.5137	1.26807	0.634036	−	0.773304i	$-0.281397\pi$
$131$	14.5137	1.26807	0.634036	+	0.773304i	$0.281397\pi$
$132$	−2.28737	−0.199090
$133$	2.77238	0.240396
$134$	−1.89719	−0.163892
$135$	−11.9296	−1.02674
$136$	−5.14733	−0.441380
$137$	−10.2842	−0.878636	−0.439318	−	0.898332i	$-0.644780\pi$
$137$	−10.2842	−0.878636	−0.439318	+	0.898332i	$0.644780\pi$
$138$	−10.6304	−0.904916
$139$	−0.770491	−0.0653522	−0.0326761	−	0.999466i	$-0.510403\pi$
$139$	−0.770491	−0.0653522	−0.0326761	+	0.999466i	$0.510403\pi$
$140$	2.36507	0.199885
$141$	−19.6258	−1.65279
$142$	−5.74352	−0.481986
$143$	2.31201	0.193340
$144$	4.17700	0.348083
$145$	−6.97809	−0.579499
$146$	11.7188	0.969854
$147$	17.7061	1.46037
$148$	−7.26788	−0.597416
$149$	17.0667	1.39816	0.699080	−	0.715043i	$-0.253593\pi$
$149$	17.0667	1.39816	0.699080	+	0.715043i	$0.253593\pi$
$150$	−24.9519	−2.03732
$151$	13.0753	1.06406	0.532028	−	0.846727i	$-0.321430\pi$
$151$	13.0753	1.06406	0.532028	+	0.846727i	$0.321430\pi$
$152$	4.43495	0.359722
$153$	−21.5004	−1.73820
$154$	0.533740	0.0430100
$155$	28.5302	2.29160
$156$	−7.25431	−0.580809
$157$	−24.0097	−1.91618	−0.958092	−	0.286459i	$-0.907522\pi$
$157$	−24.0097	−1.91618	−0.958092	+	0.286459i	$0.907522\pi$
$158$	−7.72379	−0.614472
$159$	−28.6223	−2.26989
$160$	3.78338	0.299102
$161$	2.48051	0.195492
$162$	−4.08368	−0.320844
$163$	−3.12481	−0.244754	−0.122377	−	0.992484i	$-0.539052\pi$
$163$	−3.12481	−0.244754	−0.122377	+	0.992484i	$0.539052\pi$
$164$	−11.9980	−0.936885
$165$	−8.65398	−0.673711
$166$	−0.592104	−0.0459562
$167$	7.65047	0.592011	0.296005	−	0.955186i	$-0.404345\pi$
$167$	7.65047	0.592011	0.296005	+	0.955186i	$0.404345\pi$
$168$	−1.67470	−0.129206
$169$	−5.66755	−0.435965
$170$	−19.4743	−1.49361
$171$	18.5248	1.41662
$172$	8.82565	0.672949
$173$	−2.56069	−0.194686	−0.0973428	−	0.995251i	$-0.531034\pi$
$173$	−2.56069	−0.194686	−0.0973428	+	0.995251i	$0.531034\pi$
$174$	4.94115	0.374588
$175$	5.82234	0.440127
$176$	0.853817	0.0643589
$177$	−20.8551	−1.56756
$178$	13.2052	0.989770
$179$	−0.700096	−0.0523277	−0.0261638	−	0.999658i	$-0.508329\pi$
$179$	−0.700096	−0.0523277	−0.0261638	+	0.999658i	$0.508329\pi$
$180$	15.8032	1.17790
$181$	24.0626	1.78856	0.894278	−	0.447512i	$-0.147690\pi$
$181$	24.0626	1.78856	0.894278	+	0.447512i	$0.147690\pi$
$182$	1.69274	0.125474
$183$	33.1489	2.45044
$184$	3.96804	0.292528
$185$	−27.4971	−2.02163
$186$	−20.2021	−1.48129
$187$	−4.39488	−0.321385
$188$	7.32582	0.534290
$189$	−1.97111	−0.143377
$190$	16.7791	1.21728
$191$	12.5631	0.909034	0.454517	−	0.890738i	$-0.349812\pi$
$191$	12.5631	0.909034	0.454517	+	0.890738i	$0.349812\pi$
$192$	−2.67899	−0.193340
$193$	−4.94984	−0.356297	−0.178149	−	0.984004i	$-0.557011\pi$
$193$	−4.94984	−0.356297	−0.178149	+	0.984004i	$0.557011\pi$
$194$	3.18485	0.228659
$195$	−27.4458	−1.96543
$196$	−6.60922	−0.472087
$197$	13.8157	0.984325	0.492162	−	0.870503i	$-0.336207\pi$
$197$	13.8157	0.984325	0.492162	+	0.870503i	$0.336207\pi$
$198$	3.56639	0.253453
$199$	−5.51912	−0.391240	−0.195620	−	0.980680i	$-0.562672\pi$
$199$	−5.51912	−0.391240	−0.195620	+	0.980680i	$0.562672\pi$
$200$	9.31393	0.658594
$201$	5.08256	0.358496
$202$	−13.8439	−0.974056
$203$	−1.15298	−0.0809233
$204$	13.7897	0.965469
$205$	−45.3929	−3.17037
$206$	−16.8107	−1.17126
$207$	16.5745	1.15201
$208$	2.70785	0.187756
$209$	3.78663	0.261927
$210$	−6.33600	−0.437226
$211$	18.2425	1.25587	0.627933	−	0.778267i	$-0.283901\pi$
$211$	18.2425	1.25587	0.627933	+	0.778267i	$0.283901\pi$
$212$	10.6840	0.733778
$213$	15.3868	1.05429
$214$	−10.9565	−0.748969
$215$	33.3907	2.27723
$216$	−3.15317	−0.214546
$217$	4.71400	0.320007
$218$	16.3671	1.10852
$219$	−31.3945	−2.12145
$220$	3.23031	0.217788
$221$	−13.9382	−0.937584
$222$	19.4706	1.30678
$223$	18.4946	1.23849	0.619244	−	0.785199i	$-0.287439\pi$
$223$	18.4946	1.23849	0.619244	+	0.785199i	$0.287439\pi$
$224$	0.625122	0.0417677
$225$	38.9043	2.59362
$226$	9.94394	0.661461
$227$	14.7419	0.978456	0.489228	−	0.872156i	$-0.337279\pi$
$227$	14.7419	0.978456	0.489228	+	0.872156i	$0.337279\pi$
$228$	−11.8812	−0.786851
$229$	10.6790	0.705688	0.352844	−	0.935682i	$-0.385215\pi$
$229$	10.6790	0.705688	0.352844	+	0.935682i	$0.385215\pi$
$230$	15.0126	0.989902
$231$	−1.42988	−0.0940795
$232$	−1.84441	−0.121091
$233$	−2.76191	−0.180939	−0.0904694	−	0.995899i	$-0.528837\pi$
$233$	−2.76191	−0.180939	−0.0904694	+	0.995899i	$0.528837\pi$
$234$	11.3107	0.739402
$235$	27.7163	1.80801
$236$	7.78467	0.506739
$237$	20.6920	1.34409
$238$	−3.21771	−0.208573
$239$	−3.17798	−0.205566	−0.102783	−	0.994704i	$-0.532775\pi$
$239$	−3.17798	−0.205566	−0.102783	+	0.994704i	$0.532775\pi$
$240$	−10.1356	−0.654252
$241$	9.67007	0.622904	0.311452	−	0.950262i	$-0.399185\pi$
$241$	9.67007	0.622904	0.311452	+	0.950262i	$0.399185\pi$
$242$	−10.2710	−0.660245
$243$	20.3997	1.30864
$244$	−12.3736	−0.792141
$245$	−25.0052	−1.59752
$246$	32.1425	2.04933
$247$	12.0092	0.764125
$248$	7.54094	0.478850
$249$	1.58624	0.100524
$250$	16.3212	1.03224
$251$	18.2156	1.14976	0.574879	−	0.818238i	$-0.305049\pi$
$251$	18.2156	1.14976	0.574879	+	0.818238i	$0.305049\pi$
$252$	2.61113	0.164486
$253$	3.38798	0.213001
$254$	−13.8425	−0.868555
$255$	52.1714	3.26710
$256$	1.00000	0.0625000
$257$	−15.0497	−0.938774	−0.469387	−	0.882992i	$-0.655525\pi$
$257$	−15.0497	−0.938774	−0.469387	+	0.882992i	$0.655525\pi$
$258$	−23.6438	−1.47200
$259$	−4.54331	−0.282308
$260$	10.2448	0.635356
$261$	−7.70409	−0.476871
$262$	14.5137	0.896662
$263$	−19.4309	−1.19816	−0.599080	−	0.800689i	$-0.704467\pi$
$263$	−19.4309	−1.19816	−0.599080	+	0.800689i	$0.704467\pi$
$264$	−2.28737	−0.140778
$265$	40.4215	2.48307
$266$	2.77238	0.169986
$267$	−35.3766	−2.16501
$268$	−1.89719	−0.115889
$269$	19.9200	1.21455	0.607273	−	0.794493i	$-0.292263\pi$
$269$	19.9200	1.21455	0.607273	+	0.794493i	$0.292263\pi$
$270$	−11.9296	−0.726014
$271$	−3.49171	−0.212107	−0.106053	−	0.994360i	$-0.533821\pi$
$271$	−3.49171	−0.212107	−0.106053	+	0.994360i	$0.533821\pi$
$272$	−5.14733	−0.312103
$273$	−4.53482	−0.274460
$274$	−10.2842	−0.621290
$275$	7.95239	0.479547
$276$	−10.6304	−0.639872
$277$	0.467841	0.0281098	0.0140549	−	0.999901i	$-0.495526\pi$
$277$	0.467841	0.0281098	0.0140549	+	0.999901i	$0.495526\pi$
$278$	−0.770491	−0.0462110
$279$	31.4985	1.88577
$280$	2.36507	0.141340
$281$	29.6775	1.77041	0.885207	−	0.465197i	$-0.154017\pi$
$281$	29.6775	1.77041	0.885207	+	0.465197i	$0.154017\pi$
$282$	−19.6258	−1.16870
$283$	17.2030	1.02261	0.511306	−	0.859399i	$-0.329162\pi$
$283$	17.2030	1.02261	0.511306	+	0.859399i	$0.329162\pi$
$284$	−5.74352	−0.340815
$285$	−44.9510	−2.66267
$286$	2.31201	0.136712
$287$	−7.50020	−0.442723
$288$	4.17700	0.246132
$289$	9.49500	0.558529
$290$	−6.97809	−0.409767
$291$	−8.53219	−0.500166
$292$	11.7188	0.685790
$293$	−25.8802	−1.51194	−0.755969	−	0.654607i	$-0.772834\pi$
$293$	−25.8802	−1.51194	−0.755969	+	0.654607i	$0.772834\pi$
$294$	17.7061	1.03264
$295$	29.4523	1.71478
$296$	−7.26788	−0.422437
$297$	−2.69223	−0.156219
$298$	17.0667	0.988649
$299$	10.7449	0.621392
$300$	−24.9519	−1.44060
$301$	5.51710	0.318001
$302$	13.0753	0.752402
$303$	37.0878	2.13064
$304$	4.43495	0.254362
$305$	−46.8141	−2.68057
$306$	−21.5004	−1.22910
$307$	11.5079	0.656790	0.328395	−	0.944540i	$-0.393492\pi$
$307$	11.5079	0.656790	0.328395	+	0.944540i	$0.393492\pi$
$308$	0.533740	0.0304126
$309$	45.0357	2.56199
$310$	28.5302	1.62041
$311$	4.03199	0.228633	0.114317	−	0.993444i	$-0.463532\pi$
$311$	4.03199	0.228633	0.114317	+	0.993444i	$0.463532\pi$
$312$	−7.25431	−0.410694
$313$	21.6141	1.22170	0.610851	−	0.791746i	$-0.290827\pi$
$313$	21.6141	1.22170	0.610851	+	0.791746i	$0.290827\pi$
$314$	−24.0097	−1.35495
$315$	9.87889	0.556612
$316$	−7.72379	−0.434497
$317$	−14.4195	−0.809879	−0.404939	−	0.914343i	$-0.632707\pi$
$317$	−14.4195	−0.809879	−0.404939	+	0.914343i	$0.632707\pi$
$318$	−28.6223	−1.60506
$319$	−1.57479	−0.0881712
$320$	3.78338	0.211497
$321$	29.3523	1.63829
$322$	2.48051	0.138233
$323$	−22.8281	−1.27019
$324$	−4.08368	−0.226871
$325$	25.2207	1.39899
$326$	−3.12481	−0.173067
$327$	−43.8472	−2.42476
$328$	−11.9980	−0.662478
$329$	4.57953	0.252478
$330$	−8.65398	−0.476386
$331$	−10.3424	−0.568470	−0.284235	−	0.958755i	$-0.591740\pi$
$331$	−10.3424	−0.568470	−0.284235	+	0.958755i	$0.591740\pi$
$332$	−0.592104	−0.0324959
$333$	−30.3579	−1.66360
$334$	7.65047	0.418615
$335$	−7.17779	−0.392165
$336$	−1.67470	−0.0913621
$337$	32.2666	1.75767	0.878837	−	0.477122i	$-0.158320\pi$
$337$	32.2666	1.75767	0.878837	+	0.477122i	$0.158320\pi$
$338$	−5.66755	−0.308274
$339$	−26.6397	−1.44687
$340$	−19.4743	−1.05614
$341$	6.43858	0.348669
$342$	18.5248	1.00170
$343$	−8.50742	−0.459357
$344$	8.82565	0.475847
$345$	−40.2186	−2.16530
$346$	−2.56069	−0.137663
$347$	−4.17535	−0.224145	−0.112072	−	0.993700i	$-0.535749\pi$
$347$	−4.17535	−0.224145	−0.112072	+	0.993700i	$0.535749\pi$
$348$	4.94115	0.264874
$349$	−30.5979	−1.63787	−0.818935	−	0.573886i	$-0.805435\pi$
$349$	−30.5979	−1.63787	−0.818935	+	0.573886i	$0.805435\pi$
$350$	5.82234	0.311217
$351$	−8.53831	−0.455741
$352$	0.853817	0.0455086
$353$	−6.28193	−0.334353	−0.167177	−	0.985927i	$-0.553465\pi$
$353$	−6.28193	−0.334353	−0.167177	+	0.985927i	$0.553465\pi$
$354$	−20.8551	−1.10843
$355$	−21.7299	−1.15330
$356$	13.2052	0.699873
$357$	8.62021	0.456230
$358$	−0.700096	−0.0370012
$359$	24.9096	1.31468	0.657340	−	0.753594i	$-0.271682\pi$
$359$	24.9096	1.31468	0.657340	+	0.753594i	$0.271682\pi$
$360$	15.8032	0.832899
$361$	0.668744	0.0351970
$362$	24.0626	1.26470
$363$	27.5159	1.44421
$364$	1.69274	0.0887234
$365$	44.3366	2.32068
$366$	33.1489	1.73272
$367$	−34.4751	−1.79959	−0.899793	−	0.436317i	$-0.856283\pi$
$367$	−34.4751	−1.79959	−0.899793	+	0.436317i	$0.856283\pi$
$368$	3.96804	0.206849
$369$	−50.1155	−2.60891
$370$	−27.4971	−1.42951
$371$	6.67878	0.346745
$372$	−20.2021	−1.04743
$373$	−11.6571	−0.603580	−0.301790	−	0.953374i	$-0.597584\pi$
$373$	−11.6571	−0.603580	−0.301790	+	0.953374i	$0.597584\pi$
$374$	−4.39488	−0.227254
$375$	−43.7244	−2.25792
$376$	7.32582	0.377800
$377$	−4.99438	−0.257224
$378$	−1.97111	−0.101383
$379$	−18.0348	−0.926384	−0.463192	−	0.886258i	$-0.653296\pi$
$379$	−18.0348	−0.926384	−0.463192	+	0.886258i	$0.653296\pi$
$380$	16.7791	0.860748
$381$	37.0839	1.89987
$382$	12.5631	0.642784
$383$	−30.5418	−1.56061	−0.780306	−	0.625398i	$-0.784937\pi$
$383$	−30.5418	−1.56061	−0.780306	+	0.625398i	$0.784937\pi$
$384$	−2.67899	−0.136712
$385$	2.01934	0.102915
$386$	−4.94984	−0.251940
$387$	36.8647	1.87394
$388$	3.18485	0.161686
$389$	31.8591	1.61532	0.807659	−	0.589649i	$-0.200734\pi$
$389$	31.8591	1.61532	0.807659	+	0.589649i	$0.200734\pi$
$390$	−27.4458	−1.38977
$391$	−20.4248	−1.03293
$392$	−6.60922	−0.333816
$393$	−38.8822	−1.96135
$394$	13.8157	0.696023
$395$	−29.2220	−1.47032
$396$	3.56639	0.179218
$397$	0.169172	0.00849052	0.00424526	−	0.999991i	$-0.498649\pi$
$397$	0.169172	0.00849052	0.00424526	+	0.999991i	$0.498649\pi$
$398$	−5.51912	−0.276649
$399$	−7.42719	−0.371824
$400$	9.31393	0.465696
$401$	−10.7299	−0.535824	−0.267912	−	0.963443i	$-0.586334\pi$
$401$	−10.7299	−0.535824	−0.267912	+	0.963443i	$0.586334\pi$
$402$	5.08256	0.253495
$403$	20.4197	1.01718
$404$	−13.8439	−0.688762
$405$	−15.4501	−0.767721
$406$	−1.15298	−0.0572214
$407$	−6.20545	−0.307593
$408$	13.7897	0.682690
$409$	8.29473	0.410148	0.205074	−	0.978746i	$-0.434256\pi$
$409$	8.29473	0.410148	0.205074	+	0.978746i	$0.434256\pi$
$410$	−45.3929	−2.24179
$411$	27.5512	1.35900
$412$	−16.8107	−0.828203
$413$	4.86636	0.239458
$414$	16.5745	0.814593
$415$	−2.24015	−0.109965
$416$	2.70785	0.132763
$417$	2.06414	0.101081
$418$	3.78663	0.185210
$419$	28.1068	1.37311	0.686553	−	0.727080i	$-0.259123\pi$
$419$	28.1068	1.37311	0.686553	+	0.727080i	$0.259123\pi$
$420$	−6.33600	−0.309165
$421$	−28.3848	−1.38339	−0.691694	−	0.722191i	$-0.743135\pi$
$421$	−28.3848	−1.38339	−0.691694	+	0.722191i	$0.743135\pi$
$422$	18.2425	0.888032
$423$	30.5999	1.48782
$424$	10.6840	0.518859
$425$	−47.9419	−2.32552
$426$	15.3868	0.745495
$427$	−7.73503	−0.374324
$428$	−10.9565	−0.529601
$429$	−6.19385	−0.299042
$430$	33.3907	1.61024
$431$	−3.20773	−0.154511	−0.0772555	−	0.997011i	$-0.524616\pi$
$431$	−3.20773	−0.154511	−0.0772555	+	0.997011i	$0.524616\pi$
$432$	−3.15317	−0.151707
$433$	−36.7333	−1.76529	−0.882646	−	0.470039i	$-0.844240\pi$
$433$	−36.7333	−1.76529	−0.882646	+	0.470039i	$0.844240\pi$
$434$	4.71400	0.226279
$435$	18.6942	0.896320
$436$	16.3671	0.783840
$437$	17.5981	0.841829
$438$	−31.3945	−1.50009
$439$	16.7964	0.801650	0.400825	−	0.916155i	$-0.368724\pi$
$439$	16.7964	0.801650	0.400825	+	0.916155i	$0.368724\pi$
$440$	3.23031	0.153999
$441$	−27.6067	−1.31461
$442$	−13.9382	−0.662972
$443$	−28.2986	−1.34451	−0.672253	−	0.740321i	$-0.734673\pi$
$443$	−28.2986	−1.34451	−0.672253	+	0.740321i	$0.734673\pi$
$444$	19.4706	0.924034
$445$	49.9601	2.36834
$446$	18.4946	0.875743
$447$	−45.7216	−2.16256
$448$	0.625122	0.0295342
$449$	−18.9450	−0.894072	−0.447036	−	0.894516i	$-0.647520\pi$
$449$	−18.9450	−0.894072	−0.447036	+	0.894516i	$0.647520\pi$
$450$	38.9043	1.83396
$451$	−10.2441	−0.482375
$452$	9.94394	0.467723
$453$	−35.0287	−1.64579
$454$	14.7419	0.691873
$455$	6.40425	0.300236
$456$	−11.8812	−0.556387
$457$	−0.231900	−0.0108478	−0.00542392	−	0.999985i	$-0.501726\pi$
$457$	−0.231900	−0.0108478	−0.00542392	+	0.999985i	$0.501726\pi$
$458$	10.6790	0.498997
$459$	16.2304	0.757570
$460$	15.0126	0.699966
$461$	−30.8774	−1.43810	−0.719052	−	0.694956i	$-0.755424\pi$
$461$	−30.8774	−1.43810	−0.719052	+	0.694956i	$0.755424\pi$
$462$	−1.42988	−0.0665242
$463$	−38.0320	−1.76750	−0.883748	−	0.467963i	$-0.844988\pi$
$463$	−38.0320	−1.76750	−0.883748	+	0.467963i	$0.844988\pi$
$464$	−1.84441	−0.0856245
$465$	−76.4322	−3.54446
$466$	−2.76191	−0.127943
$467$	−29.3096	−1.35629	−0.678143	−	0.734930i	$-0.737215\pi$
$467$	−29.3096	−1.35629	−0.678143	+	0.734930i	$0.737215\pi$
$468$	11.3107	0.522836
$469$	−1.18598	−0.0547633
$470$	27.7163	1.27846
$471$	64.3219	2.96380
$472$	7.78467	0.358318
$473$	7.53549	0.346482
$474$	20.6920	0.950414
$475$	41.3068	1.89528
$476$	−3.21771	−0.147483
$477$	44.6269	2.04333
$478$	−3.17798	−0.145357
$479$	−43.2030	−1.97399	−0.986997	−	0.160736i	$-0.948613\pi$
$479$	−43.2030	−1.97399	−0.986997	+	0.160736i	$0.948613\pi$
$480$	−10.1356	−0.462626
$481$	−19.6803	−0.897346
$482$	9.67007	0.440460
$483$	−6.64527	−0.302370
$484$	−10.2710	−0.466863
$485$	12.0495	0.547139
$486$	20.3997	0.925347
$487$	3.82789	0.173458	0.0867291	−	0.996232i	$-0.472359\pi$
$487$	3.82789	0.173458	0.0867291	+	0.996232i	$0.472359\pi$
$488$	−12.3736	−0.560128
$489$	8.37133	0.378565
$490$	−25.0052	−1.12962
$491$	−9.75524	−0.440248	−0.220124	−	0.975472i	$-0.570646\pi$
$491$	−9.75524	−0.440248	−0.220124	+	0.975472i	$0.570646\pi$
$492$	32.1425	1.44910
$493$	9.49377	0.427578
$494$	12.0092	0.540318
$495$	13.4930	0.606465
$496$	7.54094	0.338598
$497$	−3.59040	−0.161051
$498$	1.58624	0.0710812
$499$	17.2108	0.770462	0.385231	−	0.922820i	$-0.374122\pi$
$499$	17.2108	0.770462	0.385231	+	0.922820i	$0.374122\pi$
$500$	16.3212	0.729907
$501$	−20.4955	−0.915673
$502$	18.2156	0.813002
$503$	−40.3061	−1.79716	−0.898581	−	0.438808i	$-0.855401\pi$
$503$	−40.3061	−1.79716	−0.898581	+	0.438808i	$0.855401\pi$
$504$	2.61113	0.116309
$505$	−52.3768	−2.33074
$506$	3.38798	0.150614
$507$	15.1833	0.674315
$508$	−13.8425	−0.614161
$509$	41.2776	1.82960	0.914798	−	0.403911i	$-0.132349\pi$
$509$	41.2776	1.82960	0.914798	+	0.403911i	$0.132349\pi$
$510$	52.1714	2.31019
$511$	7.32567	0.324069
$512$	1.00000	0.0441942
$513$	−13.9841	−0.617415
$514$	−15.0497	−0.663814
$515$	−63.6012	−2.80260
$516$	−23.6438	−1.04086
$517$	6.25491	0.275091
$518$	−4.54331	−0.199622
$519$	6.86006	0.301123
$520$	10.2448	0.449265
$521$	11.9888	0.525238	0.262619	−	0.964900i	$-0.415414\pi$
$521$	11.9888	0.525238	0.262619	+	0.964900i	$0.415414\pi$
$522$	−7.70409	−0.337199
$523$	3.18603	0.139315	0.0696576	−	0.997571i	$-0.477809\pi$
$523$	3.18603	0.139315	0.0696576	+	0.997571i	$0.477809\pi$
$524$	14.5137	0.634036
$525$	−15.5980	−0.680752
$526$	−19.4309	−0.847227
$527$	−38.8157	−1.69084
$528$	−2.28737	−0.0995450
$529$	−7.25463	−0.315418
$530$	40.4215	1.75580
$531$	32.5165	1.41110
$532$	2.77238	0.120198
$533$	−32.4887	−1.40724
$534$	−35.3766	−1.53089
$535$	−41.4524	−1.79214
$536$	−1.89719	−0.0819462
$537$	1.87555	0.0809361
$538$	19.9200	0.858814
$539$	−5.64307	−0.243064
$540$	−11.9296	−0.513369
$541$	−33.7281	−1.45008	−0.725041	−	0.688706i	$-0.758179\pi$
$541$	−33.7281	−1.45008	−0.725041	+	0.688706i	$0.758179\pi$
$542$	−3.49171	−0.149982
$543$	−64.4634	−2.76639
$544$	−5.14733	−0.220690
$545$	61.9227	2.65248
$546$	−4.53482	−0.194073
$547$	−44.3088	−1.89451	−0.947253	−	0.320487i	$-0.896153\pi$
$547$	−44.3088	−1.89451	−0.947253	+	0.320487i	$0.896153\pi$
$548$	−10.2842	−0.439318
$549$	−51.6847	−2.20585
$550$	7.95239	0.339091
$551$	−8.17985	−0.348473
$552$	−10.6304	−0.452458
$553$	−4.82831	−0.205321
$554$	0.467841	0.0198766
$555$	73.6646	3.12689
$556$	−0.770491	−0.0326761
$557$	5.88813	0.249488	0.124744	−	0.992189i	$-0.460189\pi$
$557$	5.88813	0.249488	0.124744	+	0.992189i	$0.460189\pi$
$558$	31.4985	1.33344
$559$	23.8985	1.01080
$560$	2.36507	0.0999424
$561$	11.7738	0.497092
$562$	29.6775	1.25187
$563$	−35.8928	−1.51270	−0.756351	−	0.654166i	$-0.773020\pi$
$563$	−35.8928	−1.51270	−0.756351	+	0.654166i	$0.773020\pi$
$564$	−19.6258	−0.826396
$565$	37.6216	1.58275
$566$	17.2030	0.723095
$567$	−2.55280	−0.107207
$568$	−5.74352	−0.240993
$569$	−2.30607	−0.0966756	−0.0483378	−	0.998831i	$-0.515392\pi$
$569$	−2.30607	−0.0966756	−0.0483378	+	0.998831i	$0.515392\pi$
$570$	−44.9510	−1.88279
$571$	26.7724	1.12039	0.560196	−	0.828360i	$-0.310726\pi$
$571$	26.7724	1.12039	0.560196	+	0.828360i	$0.310726\pi$
$572$	2.31201	0.0966699
$573$	−33.6564	−1.40602
$574$	−7.50020	−0.313052
$575$	36.9581	1.54126
$576$	4.17700	0.174042
$577$	25.1829	1.04838	0.524189	−	0.851602i	$-0.324369\pi$
$577$	25.1829	1.04838	0.524189	+	0.851602i	$0.324369\pi$
$578$	9.49500	0.394940
$579$	13.2606	0.551091
$580$	−6.97809	−0.289749
$581$	−0.370137	−0.0153559
$582$	−8.53219	−0.353671
$583$	9.12216	0.377801
$584$	11.7188	0.484927
$585$	42.7926	1.76925
$586$	−25.8802	−1.06910
$587$	43.2441	1.78488	0.892438	−	0.451171i	$-0.148993\pi$
$587$	43.2441	1.78488	0.892438	+	0.451171i	$0.148993\pi$
$588$	17.7061	0.730185
$589$	33.4437	1.37802
$590$	29.4523	1.21253
$591$	−37.0120	−1.52247
$592$	−7.26788	−0.298708
$593$	−6.30170	−0.258780	−0.129390	−	0.991594i	$-0.541302\pi$
$593$	−6.30170	−0.258780	−0.129390	+	0.991594i	$0.541302\pi$
$594$	−2.69223	−0.110464
$595$	−12.1738	−0.499077
$596$	17.0667	0.699080
$597$	14.7857	0.605138
$598$	10.7449	0.439390
$599$	10.0159	0.409239	0.204619	−	0.978842i	$-0.434404\pi$
$599$	10.0159	0.409239	0.204619	+	0.978842i	$0.434404\pi$
$600$	−24.9519	−1.01866
$601$	35.3513	1.44201	0.721006	−	0.692929i	$-0.243680\pi$
$601$	35.3513	1.44201	0.721006	+	0.692929i	$0.243680\pi$
$602$	5.51710	0.224860
$603$	−7.92457	−0.322713
$604$	13.0753	0.532028
$605$	−38.8590	−1.57984
$606$	37.0878	1.50659
$607$	22.9878	0.933048	0.466524	−	0.884509i	$-0.345506\pi$
$607$	22.9878	0.933048	0.466524	+	0.884509i	$0.345506\pi$
$608$	4.43495	0.179861
$609$	3.08882	0.125165
$610$	−46.8141	−1.89545
$611$	19.8372	0.802528
$612$	−21.5004	−0.869102
$613$	−29.0920	−1.17502	−0.587508	−	0.809219i	$-0.699891\pi$
$613$	−29.0920	−1.17502	−0.587508	+	0.809219i	$0.699891\pi$
$614$	11.5079	0.464421
$615$	121.607	4.90367
$616$	0.533740	0.0215050
$617$	26.1334	1.05209	0.526046	−	0.850456i	$-0.323674\pi$
$617$	26.1334	1.05209	0.526046	+	0.850456i	$0.323674\pi$
$618$	45.0357	1.81160
$619$	24.5729	0.987670	0.493835	−	0.869556i	$-0.335595\pi$
$619$	24.5729	0.987670	0.493835	+	0.869556i	$0.335595\pi$
$620$	28.5302	1.14580
$621$	−12.5119	−0.502086
$622$	4.03199	0.161668
$623$	8.25484	0.330723
$624$	−7.25431	−0.290405
$625$	15.1796	0.607185
$626$	21.6141	0.863874
$627$	−10.1444	−0.405127
$628$	−24.0097	−0.958092
$629$	37.4102	1.49164
$630$	9.87889	0.393584
$631$	4.36407	0.173731	0.0868655	−	0.996220i	$-0.472315\pi$
$631$	4.36407	0.173731	0.0868655	+	0.996220i	$0.472315\pi$
$632$	−7.72379	−0.307236
$633$	−48.8716	−1.94247
$634$	−14.4195	−0.572671
$635$	−52.3713	−2.07829
$636$	−28.6223	−1.13495
$637$	−17.8968	−0.709096
$638$	−1.57479	−0.0623464
$639$	−23.9907	−0.949056
$640$	3.78338	0.149551
$641$	19.2381	0.759858	0.379929	−	0.925016i	$-0.375948\pi$
$641$	19.2381	0.759858	0.379929	+	0.925016i	$0.375948\pi$
$642$	29.3523	1.15844
$643$	−34.2608	−1.35111	−0.675557	−	0.737308i	$-0.736097\pi$
$643$	−34.2608	−1.35111	−0.675557	+	0.737308i	$0.736097\pi$
$644$	2.48051	0.0977458
$645$	−89.4535	−3.52223
$646$	−22.8281	−0.898161
$647$	9.27218	0.364527	0.182263	−	0.983250i	$-0.441658\pi$
$647$	9.27218	0.364527	0.182263	+	0.983250i	$0.441658\pi$
$648$	−4.08368	−0.160422
$649$	6.64668	0.260905
$650$	25.2207	0.989238
$651$	−12.6288	−0.494961
$652$	−3.12481	−0.122377
$653$	−16.5695	−0.648413	−0.324207	−	0.945986i	$-0.605097\pi$
$653$	−16.5695	−0.648413	−0.324207	+	0.945986i	$0.605097\pi$
$654$	−43.8472	−1.71456
$655$	54.9109	2.14555
$656$	−11.9980	−0.468442
$657$	48.9494	1.90970
$658$	4.57953	0.178529
$659$	23.0129	0.896457	0.448228	−	0.893919i	$-0.352055\pi$
$659$	23.0129	0.896457	0.448228	+	0.893919i	$0.352055\pi$
$660$	−8.65398	−0.336856
$661$	−23.4725	−0.912973	−0.456486	−	0.889730i	$-0.650892\pi$
$661$	−23.4725	−0.912973	−0.456486	+	0.889730i	$0.650892\pi$
$662$	−10.3424	−0.401969
$663$	37.3403	1.45018
$664$	−0.592104	−0.0229781
$665$	10.4890	0.406744
$666$	−30.3579	−1.17635
$667$	−7.31869	−0.283381
$668$	7.65047	0.296005
$669$	−49.5468	−1.91559
$670$	−7.17779	−0.277302
$671$	−10.5648	−0.407851
$672$	−1.67470	−0.0646028
$673$	24.2799	0.935922	0.467961	−	0.883749i	$-0.344989\pi$
$673$	24.2799	0.935922	0.467961	+	0.883749i	$0.344989\pi$
$674$	32.2666	1.24286
$675$	−29.3684	−1.13039
$676$	−5.66755	−0.217983
$677$	−31.5851	−1.21391	−0.606957	−	0.794735i	$-0.707610\pi$
$677$	−31.5851	−1.21391	−0.606957	+	0.794735i	$0.707610\pi$
$678$	−26.6397	−1.02309
$679$	1.99092	0.0764045
$680$	−19.4743	−0.746804
$681$	−39.4935	−1.51339
$682$	6.43858	0.246546
$683$	−10.4444	−0.399645	−0.199823	−	0.979832i	$-0.564037\pi$
$683$	−10.4444	−0.399645	−0.199823	+	0.979832i	$0.564037\pi$
$684$	18.5248	0.708312
$685$	−38.9089	−1.48663
$686$	−8.50742	−0.324815
$687$	−28.6090	−1.09150
$688$	8.82565	0.336475
$689$	28.9306	1.10217
$690$	−40.2186	−1.53110
$691$	1.96895	0.0749022	0.0374511	−	0.999298i	$-0.488076\pi$
$691$	1.96895	0.0749022	0.0374511	+	0.999298i	$0.488076\pi$
$692$	−2.56069	−0.0973428
$693$	2.22943	0.0846890
$694$	−4.17535	−0.158494
$695$	−2.91505	−0.110574
$696$	4.94115	0.187294
$697$	61.7575	2.33923
$698$	−30.5979	−1.15815
$699$	7.39913	0.279861
$700$	5.82234	0.220064
$701$	−43.8413	−1.65586	−0.827931	−	0.560830i	$-0.810482\pi$
$701$	−43.8413	−1.65586	−0.827931	+	0.560830i	$0.810482\pi$
$702$	−8.53831	−0.322258
$703$	−32.2327	−1.21568
$704$	0.853817	0.0321795
$705$	−74.2518	−2.79648
$706$	−6.28193	−0.236423
$707$	−8.65415	−0.325473
$708$	−20.8551	−0.783781
$709$	−12.3883	−0.465252	−0.232626	−	0.972566i	$-0.574732\pi$
$709$	−12.3883	−0.465252	−0.232626	+	0.972566i	$0.574732\pi$
$710$	−21.7299	−0.815508
$711$	−32.2623	−1.20993
$712$	13.2052	0.494885
$713$	29.9228	1.12062
$714$	8.62021	0.322603
$715$	8.74720	0.327127
$716$	−0.700096	−0.0261638
$717$	8.51378	0.317953
$718$	24.9096	0.929619
$719$	−9.31049	−0.347223	−0.173611	−	0.984814i	$-0.555544\pi$
$719$	−9.31049	−0.347223	−0.173611	+	0.984814i	$0.555544\pi$
$720$	15.8032	0.588949
$721$	−10.5087	−0.391365
$722$	0.668744	0.0248881
$723$	−25.9060	−0.963456
$724$	24.0626	0.894278
$725$	−17.1787	−0.638000
$726$	27.5159	1.02121
$727$	−22.8989	−0.849272	−0.424636	−	0.905364i	$-0.639598\pi$
$727$	−22.8989	−0.849272	−0.424636	+	0.905364i	$0.639598\pi$
$728$	1.69274	0.0627369
$729$	−42.3995	−1.57035
$730$	44.3366	1.64097
$731$	−45.4285	−1.68023
$732$	33.1489	1.22522
$733$	5.05945	0.186875	0.0934376	−	0.995625i	$-0.470214\pi$
$733$	5.05945	0.186875	0.0934376	+	0.995625i	$0.470214\pi$
$734$	−34.4751	−1.27250
$735$	66.9887	2.47091
$736$	3.96804	0.146264
$737$	−1.61986	−0.0596682
$738$	−50.1155	−1.84478
$739$	−4.12013	−0.151562	−0.0757808	−	0.997125i	$-0.524145\pi$
$739$	−4.12013	−0.151562	−0.0757808	+	0.997125i	$0.524145\pi$
$740$	−27.4971	−1.01081
$741$	−32.1725	−1.18188
$742$	6.67878	0.245186
$743$	−22.0816	−0.810096	−0.405048	−	0.914295i	$-0.632745\pi$
$743$	−22.0816	−0.810096	−0.405048	+	0.914295i	$0.632745\pi$
$744$	−20.2021	−0.740645
$745$	64.5698	2.36566
$746$	−11.6571	−0.426796
$747$	−2.47322	−0.0904903
$748$	−4.39488	−0.160693
$749$	−6.84912	−0.250262
$750$	−43.7244	−1.59659
$751$	−14.5880	−0.532325	−0.266163	−	0.963928i	$-0.585756\pi$
$751$	−14.5880	−0.532325	−0.266163	+	0.963928i	$0.585756\pi$
$752$	7.32582	0.267145
$753$	−48.7994	−1.77835
$754$	−4.99438	−0.181885
$755$	49.4689	1.80036
$756$	−1.97111	−0.0716887
$757$	−20.9401	−0.761080	−0.380540	−	0.924764i	$-0.624262\pi$
$757$	−20.9401	−0.761080	−0.380540	+	0.924764i	$0.624262\pi$
$758$	−18.0348	−0.655053
$759$	−9.07638	−0.329452
$760$	16.7791	0.608641
$761$	−10.3357	−0.374670	−0.187335	−	0.982296i	$-0.559985\pi$
$761$	−10.3357	−0.374670	−0.187335	+	0.982296i	$0.559985\pi$
$762$	37.0839	1.34341
$763$	10.2314	0.370402
$764$	12.5631	0.454517
$765$	−81.3440	−2.94100
$766$	−30.5418	−1.10352
$767$	21.0797	0.761144
$768$	−2.67899	−0.0966698
$769$	9.24647	0.333436	0.166718	−	0.986005i	$-0.446683\pi$
$769$	9.24647	0.333436	0.166718	+	0.986005i	$0.446683\pi$
$770$	2.01934	0.0727719
$771$	40.3180	1.45202
$772$	−4.94984	−0.178149
$773$	22.3093	0.802412	0.401206	−	0.915988i	$-0.368591\pi$
$773$	22.3093	0.802412	0.401206	+	0.915988i	$0.368591\pi$
$774$	36.8647	1.32508
$775$	70.2358	2.52294
$776$	3.18485	0.114330
$777$	12.1715	0.436650
$778$	31.8591	1.14220
$779$	−53.2104	−1.90646
$780$	−27.4458	−0.982716
$781$	−4.90392	−0.175476
$782$	−20.4248	−0.730390
$783$	5.81573	0.207837
$784$	−6.60922	−0.236044
$785$	−90.8378	−3.24214
$786$	−38.8822	−1.38688
$787$	−28.5171	−1.01652	−0.508262	−	0.861202i	$-0.669712\pi$
$787$	−28.5171	−1.01652	−0.508262	+	0.861202i	$0.669712\pi$
$788$	13.8157	0.492162
$789$	52.0552	1.85321
$790$	−29.2220	−1.03967
$791$	6.21617	0.221022
$792$	3.56639	0.126726
$793$	−33.5060	−1.18983
$794$	0.169172	0.00600370
$795$	−108.289	−3.84061
$796$	−5.51912	−0.195620
$797$	27.6621	0.979840	0.489920	−	0.871767i	$-0.337026\pi$
$797$	27.6621	0.979840	0.489920	+	0.871767i	$0.337026\pi$
$798$	−7.42719	−0.262920
$799$	−37.7084	−1.33403
$800$	9.31393	0.329297
$801$	55.1580	1.94891
$802$	−10.7299	−0.378885
$803$	10.0057	0.353094
$804$	5.08256	0.179248
$805$	9.38470	0.330767
$806$	20.4197	0.719254
$807$	−53.3656	−1.87856
$808$	−13.8439	−0.487028
$809$	−0.471779	−0.0165869	−0.00829344	−	0.999966i	$-0.502640\pi$
$809$	−0.471779	−0.0165869	−0.00829344	+	0.999966i	$0.502640\pi$
$810$	−15.4501	−0.542861
$811$	−11.8233	−0.415171	−0.207585	−	0.978217i	$-0.566560\pi$
$811$	−11.8233	−0.415171	−0.207585	+	0.978217i	$0.566560\pi$
$812$	−1.15298	−0.0404616
$813$	9.35427	0.328069
$814$	−6.20545	−0.217501
$815$	−11.8223	−0.414118
$816$	13.7897	0.482734
$817$	39.1413	1.36938
$818$	8.29473	0.290018
$819$	7.07055	0.247065
$820$	−45.3929	−1.58519
$821$	28.4267	0.992099	0.496049	−	0.868294i	$-0.334783\pi$
$821$	28.4267	0.992099	0.496049	+	0.868294i	$0.334783\pi$
$822$	27.5512	0.960959
$823$	3.97963	0.138721	0.0693606	−	0.997592i	$-0.477904\pi$
$823$	3.97963	0.138721	0.0693606	+	0.997592i	$0.477904\pi$
$824$	−16.8107	−0.585628
$825$	−21.3044	−0.741724
$826$	4.86636	0.169322
$827$	−11.7918	−0.410042	−0.205021	−	0.978758i	$-0.565726\pi$
$827$	−11.7918	−0.410042	−0.205021	+	0.978758i	$0.565726\pi$
$828$	16.5745	0.576004
$829$	15.3492	0.533098	0.266549	−	0.963821i	$-0.414117\pi$
$829$	15.3492	0.533098	0.266549	+	0.963821i	$0.414117\pi$
$830$	−2.24015	−0.0777568
$831$	−1.25334	−0.0434779
$832$	2.70785	0.0938778
$833$	34.0198	1.17872
$834$	2.06414	0.0714753
$835$	28.9446	1.00167
$836$	3.78663	0.130963
$837$	−23.7779	−0.821883
$838$	28.1068	0.970933
$839$	49.8012	1.71933	0.859664	−	0.510860i	$-0.170673\pi$
$839$	49.8012	1.71933	0.859664	+	0.510860i	$0.170673\pi$
$840$	−6.33600	−0.218613
$841$	−25.5982	−0.882695
$842$	−28.3848	−0.978203
$843$	−79.5059	−2.73833
$844$	18.2425	0.627933
$845$	−21.4425	−0.737644
$846$	30.5999	1.05205
$847$	−6.42062	−0.220615
$848$	10.6840	0.366889
$849$	−46.0867	−1.58169
$850$	−47.9419	−1.64439
$851$	−28.8393	−0.988598
$852$	15.3868	0.527145
$853$	41.6601	1.42641	0.713207	−	0.700953i	$-0.247242\pi$
$853$	41.6601	1.42641	0.713207	+	0.700953i	$0.247242\pi$
$854$	−7.73503	−0.264687
$855$	70.0861	2.39689
$856$	−10.9565	−0.374484
$857$	−38.3734	−1.31081	−0.655405	−	0.755278i	$-0.727502\pi$
$857$	−38.3734	−1.31081	−0.655405	+	0.755278i	$0.727502\pi$
$858$	−6.19385	−0.211455
$859$	−29.6282	−1.01090	−0.505450	−	0.862856i	$-0.668673\pi$
$859$	−29.6282	−1.01090	−0.505450	+	0.862856i	$0.668673\pi$
$860$	33.3907	1.13861
$861$	20.0930	0.684766
$862$	−3.20773	−0.109256
$863$	−16.0664	−0.546907	−0.273454	−	0.961885i	$-0.588166\pi$
$863$	−16.0664	−0.546907	−0.273454	+	0.961885i	$0.588166\pi$
$864$	−3.15317	−0.107273
$865$	−9.68805	−0.329403
$866$	−36.7333	−1.24825
$867$	−25.4370	−0.863887
$868$	4.71400	0.160004
$869$	−6.59471	−0.223710
$870$	18.6942	0.633794
$871$	−5.13731	−0.174071
$872$	16.3671	0.554258
$873$	13.3031	0.450242
$874$	17.5981	0.595263
$875$	10.2027	0.344916
$876$	−31.3945	−1.06072
$877$	6.49547	0.219337	0.109668	−	0.993968i	$-0.465021\pi$
$877$	6.49547	0.219337	0.109668	+	0.993968i	$0.465021\pi$
$878$	16.7964	0.566852
$879$	69.3329	2.33854
$880$	3.23031	0.108894
$881$	−9.08845	−0.306198	−0.153099	−	0.988211i	$-0.548925\pi$
$881$	−9.08845	−0.306198	−0.153099	+	0.988211i	$0.548925\pi$
$882$	−27.6067	−0.929566
$883$	−18.9624	−0.638135	−0.319068	−	0.947732i	$-0.603370\pi$
$883$	−18.9624	−0.638135	−0.319068	+	0.947732i	$0.603370\pi$
$884$	−13.9382	−0.468792
$885$	−78.9025	−2.65228
$886$	−28.2986	−0.950710
$887$	37.9925	1.27566	0.637831	−	0.770176i	$-0.279832\pi$
$887$	37.9925	1.27566	0.637831	+	0.770176i	$0.279832\pi$
$888$	19.4706	0.653391
$889$	−8.65324	−0.290220
$890$	49.9601	1.67467
$891$	−3.48672	−0.116809
$892$	18.4946	0.619244
$893$	32.4896	1.08722
$894$	−45.7216	−1.52916
$895$	−2.64873	−0.0885372
$896$	0.625122	0.0208838
$897$	−28.7854	−0.961117
$898$	−18.9450	−0.632204
$899$	−13.9086	−0.463877
$900$	38.9043	1.29681
$901$	−54.9939	−1.83211
$902$	−10.2441	−0.341091
$903$	−14.7803	−0.491857
$904$	9.94394	0.330730
$905$	91.0377	3.02619
$906$	−35.0287	−1.16375
$907$	−9.29501	−0.308636	−0.154318	−	0.988021i	$-0.549318\pi$
$907$	−9.29501	−0.308636	−0.154318	+	0.988021i	$0.549318\pi$
$908$	14.7419	0.489228
$909$	−57.8261	−1.91797
$910$	6.40425	0.212299
$911$	31.3453	1.03852	0.519258	−	0.854618i	$-0.326208\pi$
$911$	31.3453	1.03852	0.519258	+	0.854618i	$0.326208\pi$
$912$	−11.8812	−0.393425
$913$	−0.505549	−0.0167312
$914$	−0.231900	−0.00767058
$915$	125.415	4.14608
$916$	10.6790	0.352844
$917$	9.07285	0.299612
$918$	16.2304	0.535683
$919$	−24.9485	−0.822974	−0.411487	−	0.911416i	$-0.634990\pi$
$919$	−24.9485	−0.822974	−0.411487	+	0.911416i	$0.634990\pi$
$920$	15.0126	0.494951
$921$	−30.8296	−1.01587
$922$	−30.8774	−1.01689
$923$	−15.5526	−0.511920
$924$	−1.42988	−0.0470397
$925$	−67.6926	−2.22572
$926$	−38.0320	−1.24981
$927$	−70.2182	−2.30627
$928$	−1.84441	−0.0605457
$929$	24.9856	0.819750	0.409875	−	0.912142i	$-0.365572\pi$
$929$	24.9856	0.819750	0.409875	+	0.912142i	$0.365572\pi$
$930$	−76.4322	−2.50631
$931$	−29.3115	−0.960647
$932$	−2.76191	−0.0904694
$933$	−10.8017	−0.353631
$934$	−29.3096	−0.959039
$935$	−16.6275	−0.543777
$936$	11.3107	0.369701
$937$	6.08571	0.198812	0.0994058	−	0.995047i	$-0.468306\pi$
$937$	6.08571	0.198812	0.0994058	+	0.995047i	$0.468306\pi$
$938$	−1.18598	−0.0387235
$939$	−57.9040	−1.88963
$940$	27.7163	0.904007
$941$	−8.94802	−0.291697	−0.145849	−	0.989307i	$-0.546591\pi$
$941$	−8.94802	−0.291697	−0.145849	+	0.989307i	$0.546591\pi$
$942$	64.3219	2.09572
$943$	−47.6085	−1.55035
$944$	7.78467	0.253369
$945$	−7.45746	−0.242591
$946$	7.53549	0.245000
$947$	23.7197	0.770786	0.385393	−	0.922752i	$-0.374066\pi$
$947$	23.7197	0.770786	0.385393	+	0.922752i	$0.374066\pi$
$948$	20.6920	0.672044
$949$	31.7327	1.03009
$950$	41.3068	1.34017
$951$	38.6297	1.25265
$952$	−3.21771	−0.104286
$953$	−24.6827	−0.799552	−0.399776	−	0.916613i	$-0.630912\pi$
$953$	−24.6827	−0.799552	−0.399776	+	0.916613i	$0.630912\pi$
$954$	44.6269	1.44485
$955$	47.5309	1.53806
$956$	−3.17798	−0.102783
$957$	4.21884	0.136376
$958$	−43.2030	−1.39583
$959$	−6.42886	−0.207599
$960$	−10.1356	−0.327126
$961$	25.8658	0.834379
$962$	−19.6803	−0.634520
$963$	−45.7651	−1.47476
$964$	9.67007	0.311452
$965$	−18.7271	−0.602847
$966$	−6.64527	−0.213808
$967$	−46.3578	−1.49077	−0.745383	−	0.666636i	$-0.767733\pi$
$967$	−46.3578	−1.49077	−0.745383	+	0.666636i	$0.767733\pi$
$968$	−10.2710	−0.330122
$969$	61.1564	1.96463
$970$	12.0495	0.386886
$971$	−48.0239	−1.54116	−0.770580	−	0.637343i	$-0.780034\pi$
$971$	−48.0239	−1.54116	−0.770580	+	0.637343i	$0.780034\pi$
$972$	20.3997	0.654319
$973$	−0.481650	−0.0154410
$974$	3.82789	0.122653
$975$	−67.5661	−2.16385
$976$	−12.3736	−0.396071
$977$	1.07506	0.0343943	0.0171972	−	0.999852i	$-0.494526\pi$
$977$	1.07506	0.0343943	0.0171972	+	0.999852i	$0.494526\pi$
$978$	8.37133	0.267686
$979$	11.2748	0.360344
$980$	−25.0052	−0.798761
$981$	68.3652	2.18273
$982$	−9.75524	−0.311302
$983$	−4.11113	−0.131125	−0.0655624	−	0.997848i	$-0.520884\pi$
$983$	−4.11113	−0.131125	−0.0655624	+	0.997848i	$0.520884\pi$
$984$	32.1425	1.02467
$985$	52.2698	1.66545
$986$	9.49377	0.302343
$987$	−12.2685	−0.390511
$988$	12.0092	0.382062
$989$	35.0206	1.11359
$990$	13.4930	0.428836
$991$	35.9808	1.14297	0.571484	−	0.820613i	$-0.306368\pi$
$991$	35.9808	1.14297	0.571484	+	0.820613i	$0.306368\pi$
$992$	7.54094	0.239425
$993$	27.7072	0.879262
$994$	−3.59040	−0.113881
$995$	−20.8809	−0.661970
$996$	1.58624	0.0502620
$997$	−49.8509	−1.57879	−0.789396	−	0.613884i	$-0.789606\pi$
$997$	−49.8509	−1.57879	−0.789396	+	0.613884i	$0.789606\pi$
$998$	17.2108	0.544799
$999$	22.9169	0.725058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.i.1.4	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.i.1.4	✓	46	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.67899 q^{3} +1.00000 q^{4} +3.78338 q^{5} -2.67899 q^{6} +0.625122 q^{7} +1.00000 q^{8} +4.17700 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.67899 q^{3} +1.00000 q^{4} +3.78338 q^{5} -2.67899 q^{6} +0.625122 q^{7} +1.00000 q^{8} +4.17700 q^{9} +3.78338 q^{10} +0.853817 q^{11} -2.67899 q^{12} +2.70785 q^{13} +0.625122 q^{14} -10.1356 q^{15} +1.00000 q^{16} -5.14733 q^{17} +4.17700 q^{18} +4.43495 q^{19} +3.78338 q^{20} -1.67470 q^{21} +0.853817 q^{22} +3.96804 q^{23} -2.67899 q^{24} +9.31393 q^{25} +2.70785 q^{26} -3.15317 q^{27} +0.625122 q^{28} -1.84441 q^{29} -10.1356 q^{30} +7.54094 q^{31} +1.00000 q^{32} -2.28737 q^{33} -5.14733 q^{34} +2.36507 q^{35} +4.17700 q^{36} -7.26788 q^{37} +4.43495 q^{38} -7.25431 q^{39} +3.78338 q^{40} -11.9980 q^{41} -1.67470 q^{42} +8.82565 q^{43} +0.853817 q^{44} +15.8032 q^{45} +3.96804 q^{46} +7.32582 q^{47} -2.67899 q^{48} -6.60922 q^{49} +9.31393 q^{50} +13.7897 q^{51} +2.70785 q^{52} +10.6840 q^{53} -3.15317 q^{54} +3.23031 q^{55} +0.625122 q^{56} -11.8812 q^{57} -1.84441 q^{58} +7.78467 q^{59} -10.1356 q^{60} -12.3736 q^{61} +7.54094 q^{62} +2.61113 q^{63} +1.00000 q^{64} +10.2448 q^{65} -2.28737 q^{66} -1.89719 q^{67} -5.14733 q^{68} -10.6304 q^{69} +2.36507 q^{70} -5.74352 q^{71} +4.17700 q^{72} +11.7188 q^{73} -7.26788 q^{74} -24.9519 q^{75} +4.43495 q^{76} +0.533740 q^{77} -7.25431 q^{78} -7.72379 q^{79} +3.78338 q^{80} -4.08368 q^{81} -11.9980 q^{82} -0.592104 q^{83} -1.67470 q^{84} -19.4743 q^{85} +8.82565 q^{86} +4.94115 q^{87} +0.853817 q^{88} +13.2052 q^{89} +15.8032 q^{90} +1.69274 q^{91} +3.96804 q^{92} -20.2021 q^{93} +7.32582 q^{94} +16.7791 q^{95} -2.67899 q^{96} +3.18485 q^{97} -6.60922 q^{98} +3.56639 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9}+O(q^{10}) \) 46 * q + 46 * q^2 + 21 * q^3 + 46 * q^4 + 23 * q^5 + 21 * q^6 + 26 * q^7 + 46 * q^8 + 59 * q^9	\( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9} + 23 q^{10} + 39 q^{11} + 21 q^{12} + 8 q^{13} + 26 q^{14} + 14 q^{15} + 46 q^{16} + 36 q^{17} + 59 q^{18} + 37 q^{19} + 23 q^{20} + 20 q^{21} + 39 q^{22} + 38 q^{23} + 21 q^{24} + 57 q^{25} + 8 q^{26} + 63 q^{27} + 26 q^{28} + 23 q^{29} + 14 q^{30} + 44 q^{31} + 46 q^{32} + 25 q^{33} + 36 q^{34} + 26 q^{35} + 59 q^{36} + 9 q^{37} + 37 q^{38} - 2 q^{39} + 23 q^{40} + 50 q^{41} + 20 q^{42} + 46 q^{43} + 39 q^{44} + 30 q^{45} + 38 q^{46} + 57 q^{47} + 21 q^{48} + 62 q^{49} + 57 q^{50} + 5 q^{51} + 8 q^{52} + 21 q^{53} + 63 q^{54} + 40 q^{55} + 26 q^{56} + 3 q^{57} + 23 q^{58} + 68 q^{59} + 14 q^{60} - q^{61} + 44 q^{62} + 40 q^{63} + 46 q^{64} + 18 q^{65} + 25 q^{66} + 42 q^{67} + 36 q^{68} - 7 q^{69} + 26 q^{70} + 67 q^{71} + 59 q^{72} + 48 q^{73} + 9 q^{74} + 71 q^{75} + 37 q^{76} - 5 q^{77} - 2 q^{78} + 40 q^{79} + 23 q^{80} + 82 q^{81} + 50 q^{82} + 48 q^{83} + 20 q^{84} - 68 q^{85} + 46 q^{86} + 18 q^{87} + 39 q^{88} + 90 q^{89} + 30 q^{90} + 9 q^{91} + 38 q^{92} - 42 q^{93} + 57 q^{94} + 6 q^{95} + 21 q^{96} + 46 q^{97} + 62 q^{98} + 61 q^{99}+O(q^{100}) \) 46 * q + 46 * q^2 + 21 * q^3 + 46 * q^4 + 23 * q^5 + 21 * q^6 + 26 * q^7 + 46 * q^8 + 59 * q^9 + 23 * q^10 + 39 * q^11 + 21 * q^12 + 8 * q^13 + 26 * q^14 + 14 * q^15 + 46 * q^16 + 36 * q^17 + 59 * q^18 + 37 * q^19 + 23 * q^20 + 20 * q^21 + 39 * q^22 + 38 * q^23 + 21 * q^24 + 57 * q^25 + 8 * q^26 + 63 * q^27 + 26 * q^28 + 23 * q^29 + 14 * q^30 + 44 * q^31 + 46 * q^32 + 25 * q^33 + 36 * q^34 + 26 * q^35 + 59 * q^36 + 9 * q^37 + 37 * q^38 - 2 * q^39 + 23 * q^40 + 50 * q^41 + 20 * q^42 + 46 * q^43 + 39 * q^44 + 30 * q^45 + 38 * q^46 + 57 * q^47 + 21 * q^48 + 62 * q^49 + 57 * q^50 + 5 * q^51 + 8 * q^52 + 21 * q^53 + 63 * q^54 + 40 * q^55 + 26 * q^56 + 3 * q^57 + 23 * q^58 + 68 * q^59 + 14 * q^60 - q^61 + 44 * q^62 + 40 * q^63 + 46 * q^64 + 18 * q^65 + 25 * q^66 + 42 * q^67 + 36 * q^68 - 7 * q^69 + 26 * q^70 + 67 * q^71 + 59 * q^72 + 48 * q^73 + 9 * q^74 + 71 * q^75 + 37 * q^76 - 5 * q^77 - 2 * q^78 + 40 * q^79 + 23 * q^80 + 82 * q^81 + 50 * q^82 + 48 * q^83 + 20 * q^84 - 68 * q^85 + 46 * q^86 + 18 * q^87 + 39 * q^88 + 90 * q^89 + 30 * q^90 + 9 * q^91 + 38 * q^92 - 42 * q^93 + 57 * q^94 + 6 * q^95 + 21 * q^96 + 46 * q^97 + 62 * q^98 + 61 * q^99

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