LMFDB - Embedded newform 8007.2.a.f.1.41

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.41
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.99297 q^{2} -1.00000 q^{3} +1.97192 q^{4} +1.60110 q^{5} -1.99297 q^{6} -0.432857 q^{7} -0.0559561 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.99297 q^{2} -1.00000 q^{3} +1.97192 q^{4} +1.60110 q^{5} -1.99297 q^{6} -0.432857 q^{7} -0.0559561 q^{8} +1.00000 q^{9} +3.19095 q^{10} +1.01167 q^{11} -1.97192 q^{12} -6.62498 q^{13} -0.862669 q^{14} -1.60110 q^{15} -4.05537 q^{16} -1.00000 q^{17} +1.99297 q^{18} +1.52546 q^{19} +3.15726 q^{20} +0.432857 q^{21} +2.01622 q^{22} +4.61601 q^{23} +0.0559561 q^{24} -2.43646 q^{25} -13.2034 q^{26} -1.00000 q^{27} -0.853560 q^{28} +10.2590 q^{29} -3.19095 q^{30} +1.72586 q^{31} -7.97030 q^{32} -1.01167 q^{33} -1.99297 q^{34} -0.693049 q^{35} +1.97192 q^{36} -2.02527 q^{37} +3.04019 q^{38} +6.62498 q^{39} -0.0895916 q^{40} -0.00303029 q^{41} +0.862669 q^{42} +2.14892 q^{43} +1.99493 q^{44} +1.60110 q^{45} +9.19956 q^{46} -3.31533 q^{47} +4.05537 q^{48} -6.81264 q^{49} -4.85580 q^{50} +1.00000 q^{51} -13.0640 q^{52} -4.49136 q^{53} -1.99297 q^{54} +1.61979 q^{55} +0.0242210 q^{56} -1.52546 q^{57} +20.4460 q^{58} -12.6510 q^{59} -3.15726 q^{60} -1.11802 q^{61} +3.43959 q^{62} -0.432857 q^{63} -7.77383 q^{64} -10.6073 q^{65} -2.01622 q^{66} -13.6575 q^{67} -1.97192 q^{68} -4.61601 q^{69} -1.38122 q^{70} -7.62182 q^{71} -0.0559561 q^{72} +8.54418 q^{73} -4.03629 q^{74} +2.43646 q^{75} +3.00808 q^{76} -0.437907 q^{77} +13.2034 q^{78} +4.23499 q^{79} -6.49306 q^{80} +1.00000 q^{81} -0.00603926 q^{82} -11.5256 q^{83} +0.853560 q^{84} -1.60110 q^{85} +4.28272 q^{86} -10.2590 q^{87} -0.0566090 q^{88} +9.19145 q^{89} +3.19095 q^{90} +2.86767 q^{91} +9.10241 q^{92} -1.72586 q^{93} -6.60735 q^{94} +2.44242 q^{95} +7.97030 q^{96} +0.643877 q^{97} -13.5774 q^{98} +1.01167 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	1.99297	1.40924	0.704621	−	0.709584i	$-0.251117\pi$
$2$	1.99297	1.40924	0.704621	+	0.709584i	$0.251117\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.97192	0.985962
$5$	1.60110	0.716036	0.358018	−	0.933715i	$-0.383453\pi$
$5$	1.60110	0.716036	0.358018	+	0.933715i	$0.383453\pi$
$6$	−1.99297	−0.813626
$7$	−0.432857	−0.163604	−0.0818022	−	0.996649i	$-0.526068\pi$
$7$	−0.432857	−0.163604	−0.0818022	+	0.996649i	$0.526068\pi$
$8$	−0.0559561	−0.0197835
$9$	1.00000	0.333333
$10$	3.19095	1.00907
$11$	1.01167	0.305029	0.152515	−	0.988301i	$-0.451263\pi$
$11$	1.01167	0.305029	0.152515	+	0.988301i	$0.451263\pi$
$12$	−1.97192	−0.569245
$13$	−6.62498	−1.83744	−0.918720	−	0.394910i	$-0.870776\pi$
$13$	−6.62498	−1.83744	−0.918720	+	0.394910i	$0.870776\pi$
$14$	−0.862669	−0.230558
$15$	−1.60110	−0.413403
$16$	−4.05537	−1.01384
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.99297	0.469747
$19$	1.52546	0.349964	0.174982	−	0.984572i	$-0.444013\pi$
$19$	1.52546	0.349964	0.174982	+	0.984572i	$0.444013\pi$
$20$	3.15726	0.705984
$21$	0.432857	0.0944570
$22$	2.01622	0.429860
$23$	4.61601	0.962504	0.481252	−	0.876582i	$-0.340182\pi$
$23$	4.61601	0.962504	0.481252	+	0.876582i	$0.340182\pi$
$24$	0.0559561	0.0114220
$25$	−2.43646	−0.487293
$26$	−13.2034	−2.58940
$27$	−1.00000	−0.192450
$28$	−0.853560	−0.161308
$29$	10.2590	1.90506	0.952529	−	0.304449i	$-0.0984723\pi$
$29$	10.2590	1.90506	0.952529	+	0.304449i	$0.0984723\pi$
$30$	−3.19095	−0.582585
$31$	1.72586	0.309974	0.154987	−	0.987917i	$-0.450466\pi$
$31$	1.72586	0.309974	0.154987	+	0.987917i	$0.450466\pi$
$32$	−7.97030	−1.40896
$33$	−1.01167	−0.176109
$34$	−1.99297	−0.341791
$35$	−0.693049	−0.117147
$36$	1.97192	0.328654
$37$	−2.02527	−0.332952	−0.166476	−	0.986046i	$-0.553239\pi$
$37$	−2.02527	−0.332952	−0.166476	+	0.986046i	$0.553239\pi$
$38$	3.04019	0.493183
$39$	6.62498	1.06085
$40$	−0.0895916	−0.0141657
$41$	−0.00303029	−0.000473251 0	−0.000236626	−	1.00000i	$-0.500075\pi$
$41$	−0.00303029	−0.000473251 0	−0.000236626		1.00000i	$0.500075\pi$
$42$	0.862669	0.133113
$43$	2.14892	0.327707	0.163853	−	0.986485i	$-0.447608\pi$
$43$	2.14892	0.327707	0.163853	+	0.986485i	$0.447608\pi$
$44$	1.99493	0.300747
$45$	1.60110	0.238679
$46$	9.19956	1.35640
$47$	−3.31533	−0.483591	−0.241795	−	0.970327i	$-0.577736\pi$
$47$	−3.31533	−0.483591	−0.241795	+	0.970327i	$0.577736\pi$
$48$	4.05537	0.585342
$49$	−6.81264	−0.973234
$50$	−4.85580	−0.686713
$51$	1.00000	0.140028
$52$	−13.0640	−1.81165
$53$	−4.49136	−0.616935	−0.308468	−	0.951235i	$-0.599816\pi$
$53$	−4.49136	−0.616935	−0.308468	+	0.951235i	$0.599816\pi$
$54$	−1.99297	−0.271209
$55$	1.61979	0.218412
$56$	0.0242210	0.00323666
$57$	−1.52546	−0.202052
$58$	20.4460	2.68469
$59$	−12.6510	−1.64702	−0.823511	−	0.567300i	$-0.807988\pi$
$59$	−12.6510	−1.64702	−0.823511	+	0.567300i	$0.807988\pi$
$60$	−3.15726	−0.407600
$61$	−1.11802	−0.143148	−0.0715738	−	0.997435i	$-0.522802\pi$
$61$	−1.11802	−0.143148	−0.0715738	+	0.997435i	$0.522802\pi$
$62$	3.43959	0.436828
$63$	−0.432857	−0.0545348
$64$	−7.77383	−0.971729
$65$	−10.6073	−1.31567
$66$	−2.01622	−0.248180
$67$	−13.6575	−1.66853	−0.834266	−	0.551362i	$-0.814108\pi$
$67$	−13.6575	−1.66853	−0.834266	+	0.551362i	$0.814108\pi$
$68$	−1.97192	−0.239131
$69$	−4.61601	−0.555702
$70$	−1.38122	−0.165088
$71$	−7.62182	−0.904544	−0.452272	−	0.891880i	$-0.649386\pi$
$71$	−7.62182	−0.904544	−0.452272	+	0.891880i	$0.649386\pi$
$72$	−0.0559561	−0.00659449
$73$	8.54418	1.00002	0.500010	−	0.866019i	$-0.333330\pi$
$73$	8.54418	1.00002	0.500010	+	0.866019i	$0.333330\pi$
$74$	−4.03629	−0.469209
$75$	2.43646	0.281339
$76$	3.00808	0.345051
$77$	−0.437907	−0.0499041
$78$	13.2034	1.49499
$79$	4.23499	0.476474	0.238237	−	0.971207i	$-0.423431\pi$
$79$	4.23499	0.476474	0.238237	+	0.971207i	$0.423431\pi$
$80$	−6.49306	−0.725947
$81$	1.00000	0.111111
$82$	−0.00603926	−0.000666925 0
$83$	−11.5256	−1.26510	−0.632552	−	0.774518i	$-0.717993\pi$
$83$	−11.5256	−1.26510	−0.632552	+	0.774518i	$0.717993\pi$
$84$	0.853560	0.0931310
$85$	−1.60110	−0.173664
$86$	4.28272	0.461818
$87$	−10.2590	−1.09989
$88$	−0.0566090	−0.00603454
$89$	9.19145	0.974291	0.487146	−	0.873321i	$-0.338038\pi$
$89$	9.19145	0.974291	0.487146	+	0.873321i	$0.338038\pi$
$90$	3.19095	0.336356
$91$	2.86767	0.300613
$92$	9.10241	0.948992
$93$	−1.72586	−0.178964
$94$	−6.60735	−0.681496
$95$	2.44242	0.250587
$96$	7.97030	0.813466
$97$	0.643877	0.0653758	0.0326879	−	0.999466i	$-0.489593\pi$
$97$	0.643877	0.0653758	0.0326879	+	0.999466i	$0.489593\pi$
$98$	−13.5774	−1.37152
$99$	1.01167	0.101676
$100$	−4.80452	−0.480452
$101$	−16.9792	−1.68949	−0.844744	−	0.535170i	$-0.820248\pi$
$101$	−16.9792	−1.68949	−0.844744	+	0.535170i	$0.820248\pi$
$102$	1.99297	0.197333
$103$	0.0962444	0.00948325	0.00474162	−	0.999989i	$-0.498491\pi$
$103$	0.0962444	0.00948325	0.00474162	+	0.999989i	$0.498491\pi$
$104$	0.370708	0.0363509
$105$	0.693049	0.0676346
$106$	−8.95113	−0.869411
$107$	−19.1244	−1.84883	−0.924414	−	0.381390i	$-0.875445\pi$
$107$	−19.1244	−1.84883	−0.924414	+	0.381390i	$0.875445\pi$
$108$	−1.97192	−0.189748
$109$	10.2719	0.983871	0.491935	−	0.870632i	$-0.336290\pi$
$109$	10.2719	0.983871	0.491935	+	0.870632i	$0.336290\pi$
$110$	3.22818	0.307795
$111$	2.02527	0.192230
$112$	1.75539	0.165869
$113$	−14.4236	−1.35686	−0.678429	−	0.734666i	$-0.737339\pi$
$113$	−14.4236	−1.35686	−0.678429	+	0.734666i	$0.737339\pi$
$114$	−3.04019	−0.284740
$115$	7.39071	0.689187
$116$	20.2301	1.87831
$117$	−6.62498	−0.612480
$118$	−25.2131	−2.32105
$119$	0.432857	0.0396799
$120$	0.0895916	0.00817855
$121$	−9.97653	−0.906957
$122$	−2.22818	−0.201730
$123$	0.00303029	0.000273232 0
$124$	3.40327	0.305623
$125$	−11.9066	−1.06495
$126$	−0.862669	−0.0768527
$127$	−8.54721	−0.758442	−0.379221	−	0.925306i	$-0.623808\pi$
$127$	−8.54721	−0.758442	−0.379221	+	0.925306i	$0.623808\pi$
$128$	0.447605	0.0395630
$129$	−2.14892	−0.189202
$130$	−21.1400	−1.85410
$131$	8.69462	0.759653	0.379826	−	0.925058i	$-0.375984\pi$
$131$	8.69462	0.759653	0.379826	+	0.925058i	$0.375984\pi$
$132$	−1.99493	−0.173636
$133$	−0.660304	−0.0572556
$134$	−27.2190	−2.35136
$135$	−1.60110	−0.137801
$136$	0.0559561	0.00479820
$137$	−8.40565	−0.718143	−0.359072	−	0.933310i	$-0.616907\pi$
$137$	−8.40565	−0.718143	−0.359072	+	0.933310i	$0.616907\pi$
$138$	−9.19956	−0.783118
$139$	9.53648	0.808874	0.404437	−	0.914566i	$-0.367468\pi$
$139$	9.53648	0.808874	0.404437	+	0.914566i	$0.367468\pi$
$140$	−1.36664	−0.115502
$141$	3.31533	0.279201
$142$	−15.1900	−1.27472
$143$	−6.70228	−0.560473
$144$	−4.05537	−0.337947
$145$	16.4258	1.36409
$146$	17.0283	1.40927
$147$	6.81264	0.561897
$148$	−3.99367	−0.328278
$149$	0.518737	0.0424966	0.0212483	−	0.999774i	$-0.493236\pi$
$149$	0.518737	0.0424966	0.0212483	+	0.999774i	$0.493236\pi$
$150$	4.85580	0.396474
$151$	−10.0706	−0.819532	−0.409766	−	0.912191i	$-0.634390\pi$
$151$	−10.0706	−0.819532	−0.409766	+	0.912191i	$0.634390\pi$
$152$	−0.0853586	−0.00692350
$153$	−1.00000	−0.0808452
$154$	−0.872735	−0.0703270
$155$	2.76329	0.221953
$156$	13.0640	1.04595
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	8.44020	0.671467
$159$	4.49136	0.356188
$160$	−12.7613	−1.00887
$161$	−1.99807	−0.157470
$162$	1.99297	0.156582
$163$	−19.6205	−1.53679	−0.768397	−	0.639973i	$-0.778946\pi$
$163$	−19.6205	−1.53679	−0.768397	+	0.639973i	$0.778946\pi$
$164$	−0.00597549	−0.000466607 0
$165$	−1.61979	−0.126100
$166$	−22.9702	−1.78284
$167$	−19.6883	−1.52353	−0.761763	−	0.647856i	$-0.775666\pi$
$167$	−19.6883	−1.52353	−0.761763	+	0.647856i	$0.775666\pi$
$168$	−0.0242210	−0.00186869
$169$	30.8904	2.37619
$170$	−3.19095	−0.244735
$171$	1.52546	0.116655
$172$	4.23750	0.323106
$173$	4.81636	0.366181	0.183090	−	0.983096i	$-0.441390\pi$
$173$	4.81636	0.366181	0.183090	+	0.983096i	$0.441390\pi$
$174$	−20.4460	−1.55000
$175$	1.05464	0.0797233
$176$	−4.10268	−0.309251
$177$	12.6510	0.950909
$178$	18.3183	1.37301
$179$	15.0731	1.12662	0.563308	−	0.826247i	$-0.309529\pi$
$179$	15.0731	1.12662	0.563308	+	0.826247i	$0.309529\pi$
$180$	3.15726	0.235328
$181$	10.6825	0.794023	0.397012	−	0.917814i	$-0.370047\pi$
$181$	10.6825	0.794023	0.397012	+	0.917814i	$0.370047\pi$
$182$	5.71517	0.423637
$183$	1.11802	0.0826463
$184$	−0.258294	−0.0190417
$185$	−3.24266	−0.238405
$186$	−3.43959	−0.252203
$187$	−1.01167	−0.0739805
$188$	−6.53758	−0.476802
$189$	0.432857	0.0314857
$190$	4.86766	0.353137
$191$	−14.9692	−1.08314	−0.541568	−	0.840657i	$-0.682169\pi$
$191$	−14.9692	−1.08314	−0.541568	+	0.840657i	$0.682169\pi$
$192$	7.77383	0.561028
$193$	−15.9861	−1.15070	−0.575352	−	0.817906i	$-0.695135\pi$
$193$	−15.9861	−1.15070	−0.575352	+	0.817906i	$0.695135\pi$
$194$	1.28323	0.0921303
$195$	10.6073	0.759604
$196$	−13.4340	−0.959571
$197$	−1.46441	−0.104335	−0.0521676	−	0.998638i	$-0.516613\pi$
$197$	−1.46441	−0.104335	−0.0521676	+	0.998638i	$0.516613\pi$
$198$	2.01622	0.143287
$199$	25.2045	1.78670	0.893350	−	0.449361i	$-0.148348\pi$
$199$	25.2045	1.78670	0.893350	+	0.449361i	$0.148348\pi$
$200$	0.136335	0.00964034
$201$	13.6575	0.963327
$202$	−33.8389	−2.38090
$203$	−4.44070	−0.311676
$204$	1.97192	0.138062
$205$	−0.00485180	−0.000338865 0
$206$	0.191812	0.0133642
$207$	4.61601	0.320835
$208$	26.8667	1.86287
$209$	1.54326	0.106749
$210$	1.38122	0.0953135
$211$	−16.0553	−1.10529	−0.552645	−	0.833416i	$-0.686382\pi$
$211$	−16.0553	−1.10529	−0.552645	+	0.833416i	$0.686382\pi$
$212$	−8.85661	−0.608274
$213$	7.62182	0.522238
$214$	−38.1144	−2.60545
$215$	3.44064	0.234650
$216$	0.0559561	0.00380733
$217$	−0.747051	−0.0507131
$218$	20.4716	1.38651
$219$	−8.54418	−0.577362
$220$	3.19409	0.215346
$221$	6.62498	0.445645
$222$	4.03629	0.270898
$223$	16.6067	1.11206	0.556032	−	0.831161i	$-0.312323\pi$
$223$	16.6067	1.11206	0.556032	+	0.831161i	$0.312323\pi$
$224$	3.45000	0.230513
$225$	−2.43646	−0.162431
$226$	−28.7458	−1.91214
$227$	13.4179	0.890575	0.445288	−	0.895388i	$-0.353101\pi$
$227$	13.4179	0.890575	0.445288	+	0.895388i	$0.353101\pi$
$228$	−3.00808	−0.199215
$229$	13.1180	0.866862	0.433431	−	0.901187i	$-0.357303\pi$
$229$	13.1180	0.866862	0.433431	+	0.901187i	$0.357303\pi$
$230$	14.7295	0.971231
$231$	0.437907	0.0288122
$232$	−0.574056	−0.0376886
$233$	3.59337	0.235409	0.117705	−	0.993049i	$-0.462446\pi$
$233$	3.59337	0.235409	0.117705	+	0.993049i	$0.462446\pi$
$234$	−13.2034	−0.863132
$235$	−5.30819	−0.346268
$236$	−24.9468	−1.62390
$237$	−4.23499	−0.275092
$238$	0.862669	0.0559186
$239$	1.63998	0.106081	0.0530407	−	0.998592i	$-0.483109\pi$
$239$	1.63998	0.106081	0.0530407	+	0.998592i	$0.483109\pi$
$240$	6.49306	0.419125
$241$	−7.66528	−0.493764	−0.246882	−	0.969046i	$-0.579406\pi$
$241$	−7.66528	−0.493764	−0.246882	+	0.969046i	$0.579406\pi$
$242$	−19.8829	−1.27812
$243$	−1.00000	−0.0641500
$244$	−2.20465	−0.141138
$245$	−10.9077	−0.696870
$246$	0.00603926	0.000385049 0
$247$	−10.1061	−0.643037
$248$	−0.0965726	−0.00613236
$249$	11.5256	0.730408
$250$	−23.7294	−1.50078
$251$	27.9627	1.76499	0.882496	−	0.470319i	$-0.155861\pi$
$251$	27.9627	1.76499	0.882496	+	0.470319i	$0.155861\pi$
$252$	−0.853560	−0.0537692
$253$	4.66987	0.293592
$254$	−17.0343	−1.06883
$255$	1.60110	0.100265
$256$	16.4397	1.02748
$257$	−4.96860	−0.309933	−0.154966	−	0.987920i	$-0.549527\pi$
$257$	−4.96860	−0.309933	−0.154966	+	0.987920i	$0.549527\pi$
$258$	−4.28272	−0.266631
$259$	0.876650	0.0544724
$260$	−20.9168	−1.29720
$261$	10.2590	0.635019
$262$	17.3281	1.07053
$263$	−28.1225	−1.73411	−0.867055	−	0.498213i	$-0.833990\pi$
$263$	−28.1225	−1.73411	−0.867055	+	0.498213i	$0.833990\pi$
$264$	0.0566090	0.00348404
$265$	−7.19113	−0.441748
$266$	−1.31596	−0.0806870
$267$	−9.19145	−0.562507
$268$	−26.9316	−1.64511
$269$	−4.56721	−0.278468	−0.139234	−	0.990260i	$-0.544464\pi$
$269$	−4.56721	−0.278468	−0.139234	+	0.990260i	$0.544464\pi$
$270$	−3.19095	−0.194195
$271$	7.03561	0.427383	0.213692	−	0.976901i	$-0.431451\pi$
$271$	7.03561	0.427383	0.213692	+	0.976901i	$0.431451\pi$
$272$	4.05537	0.245893
$273$	−2.86767	−0.173559
$274$	−16.7522	−1.01204
$275$	−2.46489	−0.148639
$276$	−9.10241	−0.547901
$277$	−19.6517	−1.18075	−0.590377	−	0.807128i	$-0.701021\pi$
$277$	−19.6517	−1.18075	−0.590377	+	0.807128i	$0.701021\pi$
$278$	19.0059	1.13990
$279$	1.72586	0.103325
$280$	0.0387803	0.00231757
$281$	−5.73467	−0.342101	−0.171051	−	0.985262i	$-0.554716\pi$
$281$	−5.73467	−0.342101	−0.171051	+	0.985262i	$0.554716\pi$
$282$	6.60735	0.393462
$283$	11.4583	0.681123	0.340561	−	0.940222i	$-0.389383\pi$
$283$	11.4583	0.681123	0.340561	+	0.940222i	$0.389383\pi$
$284$	−15.0296	−0.891845
$285$	−2.44242	−0.144676
$286$	−13.3574	−0.789842
$287$	0.00131168	7.74260e−5 0
$288$	−7.97030	−0.469655
$289$	1.00000	0.0588235
$290$	32.7361	1.92233
$291$	−0.643877	−0.0377447
$292$	16.8485	0.985982
$293$	28.0591	1.63923	0.819616	−	0.572913i	$-0.194187\pi$
$293$	28.0591	1.63923	0.819616	+	0.572913i	$0.194187\pi$
$294$	13.5774	0.791848
$295$	−20.2556	−1.17933
$296$	0.113326	0.00658694
$297$	−1.01167	−0.0587029
$298$	1.03383	0.0598879
$299$	−30.5810	−1.76854
$300$	4.80452	0.277389
$301$	−0.930173	−0.0536143
$302$	−20.0703	−1.15492
$303$	16.9792	0.975427
$304$	−6.18628	−0.354808
$305$	−1.79006	−0.102499
$306$	−1.99297	−0.113930
$307$	25.2277	1.43982	0.719912	−	0.694066i	$-0.244182\pi$
$307$	25.2277	1.43982	0.719912	+	0.694066i	$0.244182\pi$
$308$	−0.863519	−0.0492036
$309$	−0.0962444	−0.00547515
$310$	5.50714	0.312785
$311$	−30.4752	−1.72809	−0.864044	−	0.503417i	$-0.832076\pi$
$311$	−30.4752	−1.72809	−0.864044	+	0.503417i	$0.832076\pi$
$312$	−0.370708	−0.0209872
$313$	−8.90078	−0.503102	−0.251551	−	0.967844i	$-0.580941\pi$
$313$	−8.90078	−0.503102	−0.251551	+	0.967844i	$0.580941\pi$
$314$	−1.99297	−0.112470
$315$	−0.693049	−0.0390489
$316$	8.35108	0.469785
$317$	21.2670	1.19447	0.597237	−	0.802065i	$-0.296265\pi$
$317$	21.2670	1.19447	0.597237	+	0.802065i	$0.296265\pi$
$318$	8.95113	0.501954
$319$	10.3787	0.581098
$320$	−12.4467	−0.695793
$321$	19.1244	1.06742
$322$	−3.98209	−0.221913
$323$	−1.52546	−0.0848787
$324$	1.97192	0.109551
$325$	16.1415	0.895371
$326$	−39.1030	−2.16571
$327$	−10.2719	−0.568038
$328$	0.000169563 0	9.36255e−6 0
$329$	1.43506	0.0791175
$330$	−3.22818	−0.177706
$331$	26.6616	1.46546	0.732728	−	0.680522i	$-0.238247\pi$
$331$	26.6616	1.46546	0.732728	+	0.680522i	$0.238247\pi$
$332$	−22.7277	−1.24734
$333$	−2.02527	−0.110984
$334$	−39.2381	−2.14702
$335$	−21.8671	−1.19473
$336$	−1.75539	−0.0957644
$337$	23.3113	1.26985	0.634925	−	0.772574i	$-0.281031\pi$
$337$	23.3113	1.26985	0.634925	+	0.772574i	$0.281031\pi$
$338$	61.5636	3.34862
$339$	14.4236	0.783382
$340$	−3.15726	−0.171226
$341$	1.74600	0.0945512
$342$	3.04019	0.164394
$343$	5.97889	0.322830
$344$	−0.120245	−0.00648318
$345$	−7.39071	−0.397902
$346$	9.59885	0.516037
$347$	24.8691	1.33504	0.667521	−	0.744591i	$-0.267356\pi$
$347$	24.8691	1.33504	0.667521	+	0.744591i	$0.267356\pi$
$348$	−20.2301	−1.08444
$349$	−24.0955	−1.28980	−0.644901	−	0.764266i	$-0.723101\pi$
$349$	−24.0955	−1.28980	−0.644901	+	0.764266i	$0.723101\pi$
$350$	2.10186	0.112349
$351$	6.62498	0.353615
$352$	−8.06330	−0.429775
$353$	8.21102	0.437029	0.218514	−	0.975834i	$-0.429879\pi$
$353$	8.21102	0.437029	0.218514	+	0.975834i	$0.429879\pi$
$354$	25.2131	1.34006
$355$	−12.2033	−0.647686
$356$	18.1248	0.960614
$357$	−0.432857	−0.0229092
$358$	30.0402	1.58767
$359$	0.725163	0.0382727	0.0191363	−	0.999817i	$-0.493908\pi$
$359$	0.725163	0.0382727	0.0191363	+	0.999817i	$0.493908\pi$
$360$	−0.0895916	−0.00472189
$361$	−16.6730	−0.877525
$362$	21.2899	1.11897
$363$	9.97653	0.523632
$364$	5.65482	0.296393
$365$	13.6801	0.716050
$366$	2.22818	0.116469
$367$	11.9680	0.624723	0.312362	−	0.949963i	$-0.398880\pi$
$367$	11.9680	0.624723	0.312362	+	0.949963i	$0.398880\pi$
$368$	−18.7196	−0.975826
$369$	−0.00303029	−0.000157750 0
$370$	−6.46252	−0.335971
$371$	1.94411	0.100933
$372$	−3.40327	−0.176451
$373$	−24.0803	−1.24683	−0.623415	−	0.781891i	$-0.714255\pi$
$373$	−24.0803	−1.24683	−0.623415	+	0.781891i	$0.714255\pi$
$374$	−2.01622	−0.104256
$375$	11.9066	0.614852
$376$	0.185513	0.00956710
$377$	−67.9660	−3.50043
$378$	0.862669	0.0443709
$379$	8.38669	0.430795	0.215398	−	0.976526i	$-0.430895\pi$
$379$	8.38669	0.430795	0.215398	+	0.976526i	$0.430895\pi$
$380$	4.81626	0.247069
$381$	8.54721	0.437887
$382$	−29.8332	−1.52640
$383$	32.2643	1.64863	0.824314	−	0.566133i	$-0.191561\pi$
$383$	32.2643	1.64863	0.824314	+	0.566133i	$0.191561\pi$
$384$	−0.447605	−0.0228417
$385$	−0.701135	−0.0357331
$386$	−31.8597	−1.62162
$387$	2.14892	0.109236
$388$	1.26968	0.0644580
$389$	7.64544	0.387639	0.193820	−	0.981037i	$-0.437912\pi$
$389$	7.64544	0.387639	0.193820	+	0.981037i	$0.437912\pi$
$390$	21.1400	1.07047
$391$	−4.61601	−0.233442
$392$	0.381209	0.0192539
$393$	−8.69462	−0.438586
$394$	−2.91853	−0.147034
$395$	6.78066	0.341172
$396$	1.99493	0.100249
$397$	12.7191	0.638354	0.319177	−	0.947695i	$-0.396594\pi$
$397$	12.7191	0.638354	0.319177	+	0.947695i	$0.396594\pi$
$398$	50.2318	2.51789
$399$	0.660304	0.0330565
$400$	9.88075	0.494038
$401$	36.2024	1.80786	0.903930	−	0.427680i	$-0.140669\pi$
$401$	36.2024	1.80786	0.903930	+	0.427680i	$0.140669\pi$
$402$	27.2190	1.35756
$403$	−11.4338	−0.569559
$404$	−33.4816	−1.66577
$405$	1.60110	0.0795595
$406$	−8.85017	−0.439226
$407$	−2.04890	−0.101560
$408$	−0.0559561	−0.00277024
$409$	−38.0524	−1.88157	−0.940785	−	0.339004i	$-0.889910\pi$
$409$	−38.0524	−1.88157	−0.940785	+	0.339004i	$0.889910\pi$
$410$	−0.00966949	−0.000477542 0
$411$	8.40565	0.414620
$412$	0.189787	0.00935012
$413$	5.47608	0.269460
$414$	9.19956	0.452134
$415$	−18.4538	−0.905860
$416$	52.8031	2.58889
$417$	−9.53648	−0.467004
$418$	3.07566	0.150435
$419$	17.6411	0.861824	0.430912	−	0.902394i	$-0.358192\pi$
$419$	17.6411	0.861824	0.430912	+	0.902394i	$0.358192\pi$
$420$	1.36664	0.0666851
$421$	−24.1281	−1.17593	−0.587967	−	0.808885i	$-0.700071\pi$
$421$	−24.1281	−1.17593	−0.587967	+	0.808885i	$0.700071\pi$
$422$	−31.9977	−1.55762
$423$	−3.31533	−0.161197
$424$	0.251319	0.0122051
$425$	2.43646	0.118186
$426$	15.1900	0.735960
$427$	0.483942	0.0234196
$428$	−37.7119	−1.82287
$429$	6.70228	0.323589
$430$	6.85709	0.330678
$431$	−26.7262	−1.28736	−0.643678	−	0.765296i	$-0.722592\pi$
$431$	−26.7262	−1.28736	−0.643678	+	0.765296i	$0.722592\pi$
$432$	4.05537	0.195114
$433$	8.13961	0.391165	0.195582	−	0.980687i	$-0.437340\pi$
$433$	8.13961	0.391165	0.195582	+	0.980687i	$0.437340\pi$
$434$	−1.48885	−0.0714670
$435$	−16.4258	−0.787557
$436$	20.2554	0.970059
$437$	7.04152	0.336841
$438$	−17.0283	−0.813643
$439$	7.27620	0.347274	0.173637	−	0.984810i	$-0.444448\pi$
$439$	7.27620	0.347274	0.173637	+	0.984810i	$0.444448\pi$
$440$	−0.0906369	−0.00432095
$441$	−6.81264	−0.324411
$442$	13.2034	0.628021
$443$	5.49941	0.261285	0.130642	−	0.991430i	$-0.458296\pi$
$443$	5.49941	0.261285	0.130642	+	0.991430i	$0.458296\pi$
$444$	3.99367	0.189531
$445$	14.7165	0.697627
$446$	33.0966	1.56717
$447$	−0.518737	−0.0245354
$448$	3.36495	0.158979
$449$	−5.84793	−0.275981	−0.137990	−	0.990434i	$-0.544064\pi$
$449$	−5.84793	−0.275981	−0.137990	+	0.990434i	$0.544064\pi$
$450$	−4.85580	−0.228904
$451$	−0.00306564	−0.000144355 0
$452$	−28.4422	−1.33781
$453$	10.0706	0.473157
$454$	26.7414	1.25504
$455$	4.59144	0.215250
$456$	0.0853586	0.00399728
$457$	−21.2467	−0.993877	−0.496939	−	0.867786i	$-0.665543\pi$
$457$	−21.2467	−0.993877	−0.496939	+	0.867786i	$0.665543\pi$
$458$	26.1438	1.22162
$459$	1.00000	0.0466760
$460$	14.5739	0.679512
$461$	−3.33124	−0.155151	−0.0775756	−	0.996986i	$-0.524718\pi$
$461$	−3.33124	−0.155151	−0.0775756	+	0.996986i	$0.524718\pi$
$462$	0.872735	0.0406033
$463$	−36.8487	−1.71251	−0.856253	−	0.516557i	$-0.827213\pi$
$463$	−36.8487	−1.71251	−0.856253	+	0.516557i	$0.827213\pi$
$464$	−41.6042	−1.93143
$465$	−2.76329	−0.128144
$466$	7.16147	0.331749
$467$	−26.9362	−1.24646	−0.623229	−	0.782040i	$-0.714179\pi$
$467$	−26.9362	−1.24646	−0.623229	+	0.782040i	$0.714179\pi$
$468$	−13.0640	−0.603882
$469$	5.91175	0.272979
$470$	−10.5791	−0.487975
$471$	1.00000	0.0460776
$472$	0.707902	0.0325838
$473$	2.17399	0.0999602
$474$	−8.44020	−0.387671
$475$	−3.71672	−0.170535
$476$	0.853560	0.0391229
$477$	−4.49136	−0.205645
$478$	3.26843	0.149494
$479$	−19.7511	−0.902450	−0.451225	−	0.892410i	$-0.649013\pi$
$479$	−19.7511	−0.902450	−0.451225	+	0.892410i	$0.649013\pi$
$480$	12.7613	0.582470
$481$	13.4174	0.611779
$482$	−15.2767	−0.695832
$483$	1.99807	0.0909153
$484$	−19.6729	−0.894225
$485$	1.03091	0.0468114
$486$	−1.99297	−0.0904029
$487$	22.2900	1.01006	0.505028	−	0.863103i	$-0.331482\pi$
$487$	22.2900	1.01006	0.505028	+	0.863103i	$0.331482\pi$
$488$	0.0625600	0.00283196
$489$	19.6205	0.887269
$490$	−21.7388	−0.982058
$491$	5.41662	0.244449	0.122224	−	0.992502i	$-0.460997\pi$
$491$	5.41662	0.244449	0.122224	+	0.992502i	$0.460997\pi$
$492$	0.00597549	0.000269396 0
$493$	−10.2590	−0.462044
$494$	−20.1412	−0.906195
$495$	1.61979	0.0728040
$496$	−6.99900	−0.314265
$497$	3.29915	0.147987
$498$	22.9702	1.02932
$499$	−29.6564	−1.32760	−0.663801	−	0.747910i	$-0.731058\pi$
$499$	−29.6564	−1.32760	−0.663801	+	0.747910i	$0.731058\pi$
$500$	−23.4788	−1.05000
$501$	19.6883	0.879608
$502$	55.7289	2.48730
$503$	−16.8329	−0.750542	−0.375271	−	0.926915i	$-0.622450\pi$
$503$	−16.8329	−0.750542	−0.375271	+	0.926915i	$0.622450\pi$
$504$	0.0242210	0.00107889
$505$	−27.1854	−1.20973
$506$	9.30690	0.413742
$507$	−30.8904	−1.37189
$508$	−16.8544	−0.747795
$509$	−34.0036	−1.50718	−0.753592	−	0.657342i	$-0.771681\pi$
$509$	−34.0036	−1.50718	−0.753592	+	0.657342i	$0.771681\pi$
$510$	3.19095	0.141298
$511$	−3.69840	−0.163608
$512$	31.8686	1.40841
$513$	−1.52546	−0.0673505
$514$	−9.90226	−0.436770
$515$	0.154097	0.00679034
$516$	−4.23750	−0.186546
$517$	−3.35401	−0.147509
$518$	1.74714	0.0767647
$519$	−4.81636	−0.211415
$520$	0.593543	0.0260286
$521$	24.7635	1.08491	0.542454	−	0.840086i	$-0.317495\pi$
$521$	24.7635	1.08491	0.542454	+	0.840086i	$0.317495\pi$
$522$	20.4460	0.894895
$523$	24.0515	1.05170	0.525849	−	0.850578i	$-0.323748\pi$
$523$	24.0515	1.05170	0.525849	+	0.850578i	$0.323748\pi$
$524$	17.1451	0.748988
$525$	−1.05464	−0.0460282
$526$	−56.0473	−2.44378
$527$	−1.72586	−0.0751798
$528$	4.10268	0.178546
$529$	−1.69248	−0.0735861
$530$	−14.3317	−0.622529
$531$	−12.6510	−0.549008
$532$	−1.30207	−0.0564518
$533$	0.0200756	0.000869571 0
$534$	−18.3183	−0.792709
$535$	−30.6202	−1.32383
$536$	0.764222	0.0330093
$537$	−15.0731	−0.650452
$538$	−9.10231	−0.392429
$539$	−6.89212	−0.296865
$540$	−3.15726	−0.135867
$541$	−39.8999	−1.71543	−0.857715	−	0.514125i	$-0.828117\pi$
$541$	−39.8999	−1.71543	−0.857715	+	0.514125i	$0.828117\pi$
$542$	14.0218	0.602286
$543$	−10.6825	−0.458430
$544$	7.97030	0.341724
$545$	16.4464	0.704486
$546$	−5.71517	−0.244587
$547$	19.2275	0.822110	0.411055	−	0.911611i	$-0.365160\pi$
$547$	19.2275	0.822110	0.411055	+	0.911611i	$0.365160\pi$
$548$	−16.5753	−0.708062
$549$	−1.11802	−0.0477159
$550$	−4.91245	−0.209468
$551$	15.6497	0.666701
$552$	0.258294	0.0109937
$553$	−1.83314	−0.0779532
$554$	−39.1651	−1.66397
$555$	3.24266	0.137643
$556$	18.8052	0.797519
$557$	42.0447	1.78149	0.890745	−	0.454503i	$-0.150183\pi$
$557$	42.0447	1.78149	0.890745	+	0.454503i	$0.150183\pi$
$558$	3.43959	0.145609
$559$	−14.2365	−0.602142
$560$	2.81056	0.118768
$561$	1.01167	0.0427127
$562$	−11.4290	−0.482104
$563$	−36.1407	−1.52315	−0.761574	−	0.648078i	$-0.775573\pi$
$563$	−36.1407	−1.52315	−0.761574	+	0.648078i	$0.775573\pi$
$564$	6.53758	0.275282
$565$	−23.0937	−0.971559
$566$	22.8359	0.959866
$567$	−0.432857	−0.0181783
$568$	0.426487	0.0178950
$569$	12.3688	0.518529	0.259264	−	0.965806i	$-0.416520\pi$
$569$	12.3688	0.518529	0.259264	+	0.965806i	$0.416520\pi$
$570$	−4.86766	−0.203884
$571$	26.1997	1.09642	0.548212	−	0.836340i	$-0.315309\pi$
$571$	26.1997	1.09642	0.548212	+	0.836340i	$0.315309\pi$
$572$	−13.2164	−0.552605
$573$	14.9692	0.625349
$574$	0.00261413	0.000109112 0
$575$	−11.2467	−0.469021
$576$	−7.77383	−0.323910
$577$	14.0642	0.585500	0.292750	−	0.956189i	$-0.405430\pi$
$577$	14.0642	0.585500	0.292750	+	0.956189i	$0.405430\pi$
$578$	1.99297	0.0828966
$579$	15.9861	0.664359
$580$	32.3904	1.34494
$581$	4.98895	0.206977
$582$	−1.28323	−0.0531914
$583$	−4.54376	−0.188183
$584$	−0.478099	−0.0197839
$585$	−10.6073	−0.438558
$586$	55.9210	2.31007
$587$	−24.1041	−0.994884	−0.497442	−	0.867497i	$-0.665727\pi$
$587$	−24.1041	−0.994884	−0.497442	+	0.867497i	$0.665727\pi$
$588$	13.4340	0.554009
$589$	2.63273	0.108480
$590$	−40.3688	−1.66196
$591$	1.46441	0.0602380
$592$	8.21319	0.337560
$593$	−18.0093	−0.739552	−0.369776	−	0.929121i	$-0.620566\pi$
$593$	−18.0093	−0.739552	−0.369776	+	0.929121i	$0.620566\pi$
$594$	−2.01622	−0.0827266
$595$	0.693049	0.0284122
$596$	1.02291	0.0419000
$597$	−25.2045	−1.03155
$598$	−60.9469	−2.49230
$599$	−4.37223	−0.178645	−0.0893223	−	0.996003i	$-0.528470\pi$
$599$	−4.37223	−0.178645	−0.0893223	+	0.996003i	$0.528470\pi$
$600$	−0.136335	−0.00556586
$601$	26.2338	1.07010	0.535049	−	0.844821i	$-0.320293\pi$
$601$	26.2338	1.07010	0.535049	+	0.844821i	$0.320293\pi$
$602$	−1.85380	−0.0755554
$603$	−13.6575	−0.556177
$604$	−19.8584	−0.808027
$605$	−15.9735	−0.649414
$606$	33.8389	1.37461
$607$	−20.8752	−0.847297	−0.423649	−	0.905827i	$-0.639251\pi$
$607$	−20.8752	−0.847297	−0.423649	+	0.905827i	$0.639251\pi$
$608$	−12.1583	−0.493086
$609$	4.44070	0.179946
$610$	−3.56754	−0.144446
$611$	21.9640	0.888569
$612$	−1.97192	−0.0797103
$613$	−29.3060	−1.18366	−0.591829	−	0.806064i	$-0.701594\pi$
$613$	−29.3060	−1.18366	−0.591829	+	0.806064i	$0.701594\pi$
$614$	50.2781	2.02906
$615$	0.00485180	0.000195644 0
$616$	0.0245036	0.000987277 0
$617$	−0.294244	−0.0118458	−0.00592291	−	0.999982i	$-0.501885\pi$
$617$	−0.294244	−0.0118458	−0.00592291	+	0.999982i	$0.501885\pi$
$618$	−0.191812	−0.00771581
$619$	−26.2966	−1.05695	−0.528476	−	0.848948i	$-0.677236\pi$
$619$	−26.2966	−1.05695	−0.528476	+	0.848948i	$0.677236\pi$
$620$	5.44899	0.218837
$621$	−4.61601	−0.185234
$622$	−60.7360	−2.43529
$623$	−3.97858	−0.159398
$624$	−26.8667	−1.07553
$625$	−6.88132	−0.275253
$626$	−17.7390	−0.708992
$627$	−1.54326	−0.0616317
$628$	−1.97192	−0.0786883
$629$	2.02527	0.0807526
$630$	−1.38122	−0.0550293
$631$	31.8860	1.26936	0.634681	−	0.772774i	$-0.281132\pi$
$631$	31.8860	1.26936	0.634681	+	0.772774i	$0.281132\pi$
$632$	−0.236974	−0.00942630
$633$	16.0553	0.638140
$634$	42.3845	1.68330
$635$	−13.6850	−0.543072
$636$	8.85661	0.351187
$637$	45.1336	1.78826
$638$	20.6845	0.818908
$639$	−7.62182	−0.301515
$640$	0.716662	0.0283286
$641$	24.5313	0.968929	0.484465	−	0.874811i	$-0.339014\pi$
$641$	24.5313	0.968929	0.484465	+	0.874811i	$0.339014\pi$
$642$	38.1144	1.50425
$643$	37.7961	1.49053	0.745267	−	0.666766i	$-0.232322\pi$
$643$	37.7961	1.49053	0.745267	+	0.666766i	$0.232322\pi$
$644$	−3.94004	−0.155259
$645$	−3.44064	−0.135475
$646$	−3.04019	−0.119615
$647$	−26.1212	−1.02693	−0.513465	−	0.858111i	$-0.671638\pi$
$647$	−26.1212	−1.02693	−0.513465	+	0.858111i	$0.671638\pi$
$648$	−0.0559561	−0.00219816
$649$	−12.7986	−0.502390
$650$	32.1696	1.26179
$651$	0.747051	0.0292792
$652$	−38.6901	−1.51522
$653$	−34.4860	−1.34954	−0.674772	−	0.738027i	$-0.735758\pi$
$653$	−34.4860	−1.34954	−0.674772	+	0.738027i	$0.735758\pi$
$654$	−20.4716	−0.800503
$655$	13.9210	0.543938
$656$	0.0122889	0.000479802 0
$657$	8.54418	0.333340
$658$	2.86003	0.111496
$659$	−26.2577	−1.02285	−0.511427	−	0.859327i	$-0.670883\pi$
$659$	−26.2577	−1.02285	−0.511427	+	0.859327i	$0.670883\pi$
$660$	−3.19409	−0.124330
$661$	−26.2617	−1.02146	−0.510731	−	0.859741i	$-0.670625\pi$
$661$	−26.2617	−1.02146	−0.510731	+	0.859741i	$0.670625\pi$
$662$	53.1358	2.06518
$663$	−6.62498	−0.257293
$664$	0.644930	0.0250281
$665$	−1.05722	−0.0409971
$666$	−4.03629	−0.156403
$667$	47.3558	1.83363
$668$	−38.8238	−1.50214
$669$	−16.6067	−0.642051
$670$	−43.5805	−1.68366
$671$	−1.13106	−0.0436642
$672$	−3.45000	−0.133087
$673$	−12.2140	−0.470816	−0.235408	−	0.971897i	$-0.575643\pi$
$673$	−12.2140	−0.470816	−0.235408	+	0.971897i	$0.575643\pi$
$674$	46.4588	1.78952
$675$	2.43646	0.0937796
$676$	60.9135	2.34283
$677$	−29.4272	−1.13098	−0.565490	−	0.824755i	$-0.691313\pi$
$677$	−29.4272	−1.13098	−0.565490	+	0.824755i	$0.691313\pi$
$678$	28.7458	1.10397
$679$	−0.278706	−0.0106958
$680$	0.0895916	0.00343568
$681$	−13.4179	−0.514174
$682$	3.47972	0.133245
$683$	39.4492	1.50948	0.754741	−	0.656023i	$-0.227762\pi$
$683$	39.4492	1.50948	0.754741	+	0.656023i	$0.227762\pi$
$684$	3.00808	0.115017
$685$	−13.4583	−0.514216
$686$	11.9157	0.454945
$687$	−13.1180	−0.500483
$688$	−8.71464	−0.332243
$689$	29.7552	1.13358
$690$	−14.7295	−0.560741
$691$	12.5047	0.475702	0.237851	−	0.971302i	$-0.423557\pi$
$691$	12.5047	0.475702	0.237851	+	0.971302i	$0.423557\pi$
$692$	9.49749	0.361040
$693$	−0.437907	−0.0166347
$694$	49.5633	1.88140
$695$	15.2689	0.579183
$696$	0.574056	0.0217596
$697$	0.00303029	0.000114780 0
$698$	−48.0215	−1.81764
$699$	−3.59337	−0.135914
$700$	2.07967	0.0786041
$701$	−4.30313	−0.162527	−0.0812636	−	0.996693i	$-0.525896\pi$
$701$	−4.30313	−0.162527	−0.0812636	+	0.996693i	$0.525896\pi$
$702$	13.2034	0.498330
$703$	−3.08946	−0.116521
$704$	−7.86454	−0.296406
$705$	5.30819	0.199918
$706$	16.3643	0.615879
$707$	7.34954	0.276408
$708$	24.9468	0.937560
$709$	−40.0643	−1.50465	−0.752323	−	0.658794i	$-0.771067\pi$
$709$	−40.0643	−1.50465	−0.752323	+	0.658794i	$0.771067\pi$
$710$	−24.3209	−0.912745
$711$	4.23499	0.158825
$712$	−0.514318	−0.0192749
$713$	7.96659	0.298351
$714$	−0.862669	−0.0322846
$715$	−10.7311	−0.401319
$716$	29.7230	1.11080
$717$	−1.63998	−0.0612462
$718$	1.44523	0.0539354
$719$	35.2408	1.31426	0.657129	−	0.753778i	$-0.271771\pi$
$719$	35.2408	1.31426	0.657129	+	0.753778i	$0.271771\pi$
$720$	−6.49306	−0.241982
$721$	−0.0416600	−0.00155150
$722$	−33.2287	−1.23665
$723$	7.66528	0.285075
$724$	21.0651	0.782876
$725$	−24.9958	−0.928321
$726$	19.8829	0.737924
$727$	0.701989	0.0260353	0.0130177	−	0.999915i	$-0.495856\pi$
$727$	0.701989	0.0260353	0.0130177	+	0.999915i	$0.495856\pi$
$728$	−0.160464	−0.00594717
$729$	1.00000	0.0370370
$730$	27.2641	1.00909
$731$	−2.14892	−0.0794806
$732$	2.20465	0.0814861
$733$	−16.8186	−0.621208	−0.310604	−	0.950539i	$-0.600531\pi$
$733$	−16.8186	−0.621208	−0.310604	+	0.950539i	$0.600531\pi$
$734$	23.8518	0.880386
$735$	10.9077	0.402338
$736$	−36.7910	−1.35613
$737$	−13.8169	−0.508951
$738$	−0.00603926	−0.000222308 0
$739$	45.2138	1.66322	0.831609	−	0.555361i	$-0.187420\pi$
$739$	45.2138	1.66322	0.831609	+	0.555361i	$0.187420\pi$
$740$	−6.39428	−0.235058
$741$	10.1061	0.371258
$742$	3.87455	0.142239
$743$	25.6625	0.941466	0.470733	−	0.882276i	$-0.343989\pi$
$743$	25.6625	0.941466	0.470733	+	0.882276i	$0.343989\pi$
$744$	0.0965726	0.00354052
$745$	0.830552	0.0304291
$746$	−47.9913	−1.75709
$747$	−11.5256	−0.421701
$748$	−1.99493	−0.0729419
$749$	8.27813	0.302476
$750$	23.7294	0.866475
$751$	14.2218	0.518961	0.259481	−	0.965748i	$-0.416449\pi$
$751$	14.2218	0.518961	0.259481	+	0.965748i	$0.416449\pi$
$752$	13.4449	0.490284
$753$	−27.9627	−1.01902
$754$	−135.454	−4.93295
$755$	−16.1240	−0.586814
$756$	0.853560	0.0310437
$757$	−11.8283	−0.429906	−0.214953	−	0.976624i	$-0.568960\pi$
$757$	−11.8283	−0.429906	−0.214953	+	0.976624i	$0.568960\pi$
$758$	16.7144	0.607094
$759$	−4.66987	−0.169505
$760$	−0.136668	−0.00495747
$761$	8.85396	0.320956	0.160478	−	0.987039i	$-0.448696\pi$
$761$	8.85396	0.320956	0.160478	+	0.987039i	$0.448696\pi$
$762$	17.0343	0.617088
$763$	−4.44626	−0.160966
$764$	−29.5182	−1.06793
$765$	−1.60110	−0.0578881
$766$	64.3017	2.32331
$767$	83.8128	3.02631
$768$	−16.4397	−0.593217
$769$	13.8857	0.500731	0.250365	−	0.968151i	$-0.419449\pi$
$769$	13.8857	0.500731	0.250365	+	0.968151i	$0.419449\pi$
$770$	−1.39734	−0.0503566
$771$	4.96860	0.178940
$772$	−31.5233	−1.13455
$773$	−4.70018	−0.169054	−0.0845268	−	0.996421i	$-0.526938\pi$
$773$	−4.70018	−0.169054	−0.0845268	+	0.996421i	$0.526938\pi$
$774$	4.28272	0.153939
$775$	−4.20500	−0.151048
$776$	−0.0360288	−0.00129336
$777$	−0.876650	−0.0314496
$778$	15.2371	0.546277
$779$	−0.00462257	−0.000165621 0
$780$	20.9168	0.748940
$781$	−7.71075	−0.275912
$782$	−9.19956	−0.328975
$783$	−10.2590	−0.366628
$784$	27.6277	0.986704
$785$	−1.60110	−0.0571459
$786$	−17.3281	−0.618073
$787$	−24.1514	−0.860903	−0.430451	−	0.902614i	$-0.641646\pi$
$787$	−24.1514	−0.860903	−0.430451	+	0.902614i	$0.641646\pi$
$788$	−2.88771	−0.102871
$789$	28.1225	1.00119
$790$	13.5136	0.480794
$791$	6.24335	0.221988
$792$	−0.0566090	−0.00201151
$793$	7.40686	0.263025
$794$	25.3488	0.899595
$795$	7.19113	0.255043
$796$	49.7014	1.76162
$797$	−6.01909	−0.213207	−0.106604	−	0.994302i	$-0.533998\pi$
$797$	−6.01909	−0.213207	−0.106604	+	0.994302i	$0.533998\pi$
$798$	1.31596	0.0465846
$799$	3.31533	0.117288
$800$	19.4194	0.686578
$801$	9.19145	0.324764
$802$	72.1502	2.54771
$803$	8.64387	0.305036
$804$	26.9316	0.949804
$805$	−3.19912	−0.112754
$806$	−22.7872	−0.802646
$807$	4.56721	0.160774
$808$	0.950087	0.0334240
$809$	29.4135	1.03412	0.517062	−	0.855948i	$-0.327026\pi$
$809$	29.4135	1.03412	0.517062	+	0.855948i	$0.327026\pi$
$810$	3.19095	0.112119
$811$	49.2940	1.73095	0.865473	−	0.500956i	$-0.167018\pi$
$811$	49.2940	1.73095	0.865473	+	0.500956i	$0.167018\pi$
$812$	−8.75671	−0.307300
$813$	−7.03561	−0.246750
$814$	−4.08339	−0.143123
$815$	−31.4144	−1.10040
$816$	−4.05537	−0.141966
$817$	3.27808	0.114685
$818$	−75.8372	−2.65159
$819$	2.86767	0.100204
$820$	−0.00956739	−0.000334108 0
$821$	40.0959	1.39936	0.699678	−	0.714458i	$-0.253327\pi$
$821$	40.0959	1.39936	0.699678	+	0.714458i	$0.253327\pi$
$822$	16.7522	0.584300
$823$	37.8978	1.32103	0.660517	−	0.750811i	$-0.270337\pi$
$823$	37.8978	1.32103	0.660517	+	0.750811i	$0.270337\pi$
$824$	−0.00538546	−0.000187612 0
$825$	2.46489	0.0858165
$826$	10.9137	0.379734
$827$	25.3479	0.881432	0.440716	−	0.897647i	$-0.354725\pi$
$827$	25.3479	0.881432	0.440716	+	0.897647i	$0.354725\pi$
$828$	9.10241	0.316331
$829$	−2.63384	−0.0914772	−0.0457386	−	0.998953i	$-0.514564\pi$
$829$	−2.63384	−0.0914772	−0.0457386	+	0.998953i	$0.514564\pi$
$830$	−36.7778	−1.27657
$831$	19.6517	0.681708
$832$	51.5015	1.78549
$833$	6.81264	0.236044
$834$	−19.0059	−0.658121
$835$	−31.5230	−1.09090
$836$	3.04318	0.105251
$837$	−1.72586	−0.0596545
$838$	35.1581	1.21452
$839$	16.2186	0.559929	0.279964	−	0.960010i	$-0.409677\pi$
$839$	16.2186	0.559929	0.279964	+	0.960010i	$0.409677\pi$
$840$	−0.0387803	−0.00133805
$841$	76.2481	2.62924
$842$	−48.0866	−1.65717
$843$	5.73467	0.197512
$844$	−31.6598	−1.08977
$845$	49.4588	1.70143
$846$	−6.60735	−0.227165
$847$	4.31841	0.148382
$848$	18.2141	0.625474
$849$	−11.4583	−0.393246
$850$	4.85580	0.166552
$851$	−9.34864	−0.320467
$852$	15.0296	0.514907
$853$	−8.32339	−0.284988	−0.142494	−	0.989796i	$-0.545512\pi$
$853$	−8.32339	−0.284988	−0.142494	+	0.989796i	$0.545512\pi$
$854$	0.964480	0.0330038
$855$	2.44242	0.0835288
$856$	1.07013	0.0365762
$857$	−6.64789	−0.227088	−0.113544	−	0.993533i	$-0.536220\pi$
$857$	−6.64789	−0.227088	−0.113544	+	0.993533i	$0.536220\pi$
$858$	13.3574	0.456015
$859$	48.1724	1.64362	0.821811	−	0.569760i	$-0.192964\pi$
$859$	48.1724	1.64362	0.821811	+	0.569760i	$0.192964\pi$
$860$	6.78468	0.231356
$861$	−0.00131168	−4.47019e−5 0
$862$	−53.2645	−1.81420
$863$	−48.0557	−1.63583	−0.817917	−	0.575336i	$-0.804871\pi$
$863$	−48.0557	−1.63583	−0.817917	+	0.575336i	$0.804871\pi$
$864$	7.97030	0.271155
$865$	7.71149	0.262199
$866$	16.2220	0.551245
$867$	−1.00000	−0.0339618
$868$	−1.47313	−0.0500012
$869$	4.28440	0.145338
$870$	−32.7361	−1.10986
$871$	90.4809	3.06583
$872$	−0.574776	−0.0194644
$873$	0.643877	0.0217919
$874$	14.0335	0.474691
$875$	5.15383	0.174231
$876$	−16.8485	−0.569257
$877$	−3.43556	−0.116011	−0.0580053	−	0.998316i	$-0.518474\pi$
$877$	−3.43556	−0.116011	−0.0580053	+	0.998316i	$0.518474\pi$
$878$	14.5012	0.489393
$879$	−28.0591	−0.946412
$880$	−6.56882	−0.221435
$881$	51.0698	1.72058	0.860292	−	0.509802i	$-0.170281\pi$
$881$	51.0698	1.72058	0.860292	+	0.509802i	$0.170281\pi$
$882$	−13.5774	−0.457174
$883$	−35.5021	−1.19474	−0.597371	−	0.801965i	$-0.703788\pi$
$883$	−35.5021	−1.19474	−0.597371	+	0.801965i	$0.703788\pi$
$884$	13.0640	0.439389
$885$	20.2556	0.680885
$886$	10.9602	0.368214
$887$	37.8314	1.27025	0.635126	−	0.772408i	$-0.280948\pi$
$887$	37.8314	1.27025	0.635126	+	0.772408i	$0.280948\pi$
$888$	−0.113326	−0.00380297
$889$	3.69971	0.124084
$890$	29.3295	0.983126
$891$	1.01167	0.0338921
$892$	32.7471	1.09645
$893$	−5.05739	−0.169239
$894$	−1.03383	−0.0345763
$895$	24.1336	0.806697
$896$	−0.193749	−0.00647269
$897$	30.5810	1.02107
$898$	−11.6547	−0.388924
$899$	17.7057	0.590518
$900$	−4.80452	−0.160151
$901$	4.49136	0.149629
$902$	−0.00610973	−0.000203432 0
$903$	0.930173	0.0309542
$904$	0.807088	0.0268434
$905$	17.1038	0.568549
$906$	20.0703	0.666792
$907$	0.918220	0.0304890	0.0152445	−	0.999884i	$-0.495147\pi$
$907$	0.918220	0.0304890	0.0152445	+	0.999884i	$0.495147\pi$
$908$	26.4590	0.878073
$909$	−16.9792	−0.563163
$910$	9.15059	0.303339
$911$	−38.2580	−1.26754	−0.633772	−	0.773520i	$-0.718494\pi$
$911$	−38.2580	−1.26754	−0.633772	+	0.773520i	$0.718494\pi$
$912$	6.18628	0.204848
$913$	−11.6601	−0.385894
$914$	−42.3439	−1.40061
$915$	1.79006	0.0591777
$916$	25.8677	0.854692
$917$	−3.76352	−0.124282
$918$	1.99297	0.0657778
$919$	19.5757	0.645743	0.322871	−	0.946443i	$-0.395352\pi$
$919$	19.5757	0.645743	0.322871	+	0.946443i	$0.395352\pi$
$920$	−0.413555	−0.0136345
$921$	−25.2277	−0.831282
$922$	−6.63905	−0.218645
$923$	50.4944	1.66204
$924$	0.863519	0.0284077
$925$	4.93449	0.162245
$926$	−73.4383	−2.41333
$927$	0.0962444	0.00316108
$928$	−81.7677	−2.68416
$929$	−17.2376	−0.565548	−0.282774	−	0.959187i	$-0.591255\pi$
$929$	−17.2376	−0.565548	−0.282774	+	0.959187i	$0.591255\pi$
$930$	−5.50714	−0.180586
$931$	−10.3924	−0.340596
$932$	7.08585	0.232105
$933$	30.4752	0.997712
$934$	−53.6829	−1.75656
$935$	−1.61979	−0.0529727
$936$	0.370708	0.0121170
$937$	48.9576	1.59938	0.799688	−	0.600416i	$-0.204998\pi$
$937$	48.9576	1.59938	0.799688	+	0.600416i	$0.204998\pi$
$938$	11.7819	0.384693
$939$	8.90078	0.290466
$940$	−10.4673	−0.341407
$941$	−14.1271	−0.460532	−0.230266	−	0.973128i	$-0.573960\pi$
$941$	−14.1271	−0.460532	−0.230266	+	0.973128i	$0.573960\pi$
$942$	1.99297	0.0649344
$943$	−0.0139878	−0.000455506 0
$944$	51.3045	1.66982
$945$	0.693049	0.0225449
$946$	4.33269	0.140868
$947$	−22.5368	−0.732347	−0.366173	−	0.930547i	$-0.619332\pi$
$947$	−22.5368	−0.732347	−0.366173	+	0.930547i	$0.619332\pi$
$948$	−8.35108	−0.271230
$949$	−56.6050	−1.83748
$950$	−7.40731	−0.240325
$951$	−21.2670	−0.689630
$952$	−0.0242210	−0.000785006 0
$953$	−16.0109	−0.518644	−0.259322	−	0.965791i	$-0.583499\pi$
$953$	−16.0109	−0.518644	−0.259322	+	0.965791i	$0.583499\pi$
$954$	−8.95113	−0.289804
$955$	−23.9673	−0.775564
$956$	3.23391	0.104592
$957$	−10.3787	−0.335497
$958$	−39.3633	−1.27177
$959$	3.63844	0.117491
$960$	12.4467	0.401716
$961$	−28.0214	−0.903916
$962$	26.7404	0.862144
$963$	−19.1244	−0.616276
$964$	−15.1153	−0.486832
$965$	−25.5954	−0.823944
$966$	3.98209	0.128122
$967$	43.0782	1.38530	0.692651	−	0.721273i	$-0.256443\pi$
$967$	43.0782	1.38530	0.692651	+	0.721273i	$0.256443\pi$
$968$	0.558248	0.0179428
$969$	1.52546	0.0490047
$970$	2.05458	0.0659686
$971$	48.0368	1.54157	0.770787	−	0.637093i	$-0.219863\pi$
$971$	48.0368	1.54157	0.770787	+	0.637093i	$0.219863\pi$
$972$	−1.97192	−0.0632495
$973$	−4.12793	−0.132335
$974$	44.4233	1.42341
$975$	−16.1415	−0.516943
$976$	4.53397	0.145129
$977$	−10.8729	−0.347856	−0.173928	−	0.984758i	$-0.555646\pi$
$977$	−10.8729	−0.347856	−0.173928	+	0.984758i	$0.555646\pi$
$978$	39.1030	1.25038
$979$	9.29869	0.297187
$980$	−21.5092	−0.687087
$981$	10.2719	0.327957
$982$	10.7952	0.344487
$983$	−26.8200	−0.855425	−0.427712	−	0.903915i	$-0.640680\pi$
$983$	−26.8200	−0.855425	−0.427712	+	0.903915i	$0.640680\pi$
$984$	−0.000169563 0	−5.40547e−6 0
$985$	−2.34468	−0.0747078
$986$	−20.4460	−0.651132
$987$	−1.43506	−0.0456785
$988$	−19.9285	−0.634010
$989$	9.91942	0.315419
$990$	3.22818	0.102598
$991$	7.75001	0.246187	0.123094	−	0.992395i	$-0.460718\pi$
$991$	7.75001	0.246187	0.123094	+	0.992395i	$0.460718\pi$
$992$	−13.7556	−0.436742
$993$	−26.6616	−0.846081
$994$	6.57511	0.208550
$995$	40.3550	1.27934
$996$	22.7277	0.720154
$997$	37.1197	1.17559	0.587797	−	0.809009i	$-0.299996\pi$
$997$	37.1197	1.17559	0.587797	+	0.809009i	$0.299996\pi$
$998$	−59.1042	−1.87091
$999$	2.02527	0.0640766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.41	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.41	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+1.99297 q^{2} -1.00000 q^{3} +1.97192 q^{4} +1.60110 q^{5} -1.99297 q^{6} -0.432857 q^{7} -0.0559561 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.99297 q^{2} -1.00000 q^{3} +1.97192 q^{4} +1.60110 q^{5} -1.99297 q^{6} -0.432857 q^{7} -0.0559561 q^{8} +1.00000 q^{9} +3.19095 q^{10} +1.01167 q^{11} -1.97192 q^{12} -6.62498 q^{13} -0.862669 q^{14} -1.60110 q^{15} -4.05537 q^{16} -1.00000 q^{17} +1.99297 q^{18} +1.52546 q^{19} +3.15726 q^{20} +0.432857 q^{21} +2.01622 q^{22} +4.61601 q^{23} +0.0559561 q^{24} -2.43646 q^{25} -13.2034 q^{26} -1.00000 q^{27} -0.853560 q^{28} +10.2590 q^{29} -3.19095 q^{30} +1.72586 q^{31} -7.97030 q^{32} -1.01167 q^{33} -1.99297 q^{34} -0.693049 q^{35} +1.97192 q^{36} -2.02527 q^{37} +3.04019 q^{38} +6.62498 q^{39} -0.0895916 q^{40} -0.00303029 q^{41} +0.862669 q^{42} +2.14892 q^{43} +1.99493 q^{44} +1.60110 q^{45} +9.19956 q^{46} -3.31533 q^{47} +4.05537 q^{48} -6.81264 q^{49} -4.85580 q^{50} +1.00000 q^{51} -13.0640 q^{52} -4.49136 q^{53} -1.99297 q^{54} +1.61979 q^{55} +0.0242210 q^{56} -1.52546 q^{57} +20.4460 q^{58} -12.6510 q^{59} -3.15726 q^{60} -1.11802 q^{61} +3.43959 q^{62} -0.432857 q^{63} -7.77383 q^{64} -10.6073 q^{65} -2.01622 q^{66} -13.6575 q^{67} -1.97192 q^{68} -4.61601 q^{69} -1.38122 q^{70} -7.62182 q^{71} -0.0559561 q^{72} +8.54418 q^{73} -4.03629 q^{74} +2.43646 q^{75} +3.00808 q^{76} -0.437907 q^{77} +13.2034 q^{78} +4.23499 q^{79} -6.49306 q^{80} +1.00000 q^{81} -0.00603926 q^{82} -11.5256 q^{83} +0.853560 q^{84} -1.60110 q^{85} +4.28272 q^{86} -10.2590 q^{87} -0.0566090 q^{88} +9.19145 q^{89} +3.19095 q^{90} +2.86767 q^{91} +9.10241 q^{92} -1.72586 q^{93} -6.60735 q^{94} +2.44242 q^{95} +7.97030 q^{96} +0.643877 q^{97} -13.5774 q^{98} +1.01167 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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