LMFDB - Embedded newform 8042.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.08937 q^{3} +1.00000 q^{4} -1.29634 q^{5} -3.08937 q^{6} +4.06006 q^{7} +1.00000 q^{8} +6.54422 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.08937 q^{3} +1.00000 q^{4} -1.29634 q^{5} -3.08937 q^{6} +4.06006 q^{7} +1.00000 q^{8} +6.54422 q^{9} -1.29634 q^{10} +2.11310 q^{11} -3.08937 q^{12} +1.62348 q^{13} +4.06006 q^{14} +4.00488 q^{15} +1.00000 q^{16} -2.05899 q^{17} +6.54422 q^{18} +1.37650 q^{19} -1.29634 q^{20} -12.5430 q^{21} +2.11310 q^{22} -2.66845 q^{23} -3.08937 q^{24} -3.31950 q^{25} +1.62348 q^{26} -10.9494 q^{27} +4.06006 q^{28} -1.01589 q^{29} +4.00488 q^{30} -3.05425 q^{31} +1.00000 q^{32} -6.52814 q^{33} -2.05899 q^{34} -5.26323 q^{35} +6.54422 q^{36} -7.60060 q^{37} +1.37650 q^{38} -5.01553 q^{39} -1.29634 q^{40} +5.22545 q^{41} -12.5430 q^{42} -9.75420 q^{43} +2.11310 q^{44} -8.48354 q^{45} -2.66845 q^{46} -11.0245 q^{47} -3.08937 q^{48} +9.48407 q^{49} -3.31950 q^{50} +6.36097 q^{51} +1.62348 q^{52} -6.35741 q^{53} -10.9494 q^{54} -2.73930 q^{55} +4.06006 q^{56} -4.25252 q^{57} -1.01589 q^{58} +10.0360 q^{59} +4.00488 q^{60} +3.96562 q^{61} -3.05425 q^{62} +26.5699 q^{63} +1.00000 q^{64} -2.10458 q^{65} -6.52814 q^{66} -8.49862 q^{67} -2.05899 q^{68} +8.24383 q^{69} -5.26323 q^{70} -12.7609 q^{71} +6.54422 q^{72} -14.3477 q^{73} -7.60060 q^{74} +10.2552 q^{75} +1.37650 q^{76} +8.57930 q^{77} -5.01553 q^{78} +4.45222 q^{79} -1.29634 q^{80} +14.1941 q^{81} +5.22545 q^{82} +13.3840 q^{83} -12.5430 q^{84} +2.66915 q^{85} -9.75420 q^{86} +3.13847 q^{87} +2.11310 q^{88} +1.76396 q^{89} -8.48354 q^{90} +6.59141 q^{91} -2.66845 q^{92} +9.43570 q^{93} -11.0245 q^{94} -1.78442 q^{95} -3.08937 q^{96} -6.28206 q^{97} +9.48407 q^{98} +13.8286 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−3.08937	−1.78365	−0.891825	−	0.452381i	$-0.850575\pi$
$3$	−3.08937	−1.78365	−0.891825	+	0.452381i	$0.850575\pi$
$4$	1.00000	0.500000
$5$	−1.29634	−0.579742	−0.289871	−	0.957066i	$-0.593612\pi$
$5$	−1.29634	−0.579742	−0.289871	+	0.957066i	$0.593612\pi$
$6$	−3.08937	−1.26123
$7$	4.06006	1.53456	0.767279	−	0.641314i	$-0.221610\pi$
$7$	4.06006	1.53456	0.767279	+	0.641314i	$0.221610\pi$
$8$	1.00000	0.353553
$9$	6.54422	2.18141
$10$	−1.29634	−0.409939
$11$	2.11310	0.637123	0.318561	−	0.947902i	$-0.396800\pi$
$11$	2.11310	0.637123	0.318561	+	0.947902i	$0.396800\pi$
$12$	−3.08937	−0.891825
$13$	1.62348	0.450272	0.225136	−	0.974327i	$-0.427717\pi$
$13$	1.62348	0.450272	0.225136	+	0.974327i	$0.427717\pi$
$14$	4.06006	1.08510
$15$	4.00488	1.03406
$16$	1.00000	0.250000
$17$	−2.05899	−0.499377	−0.249689	−	0.968326i	$-0.580328\pi$
$17$	−2.05899	−0.499377	−0.249689	+	0.968326i	$0.580328\pi$
$18$	6.54422	1.54249
$19$	1.37650	0.315791	0.157896	−	0.987456i	$-0.449529\pi$
$19$	1.37650	0.315791	0.157896	+	0.987456i	$0.449529\pi$
$20$	−1.29634	−0.289871
$21$	−12.5430	−2.73711
$22$	2.11310	0.450514
$23$	−2.66845	−0.556410	−0.278205	−	0.960522i	$-0.589740\pi$
$23$	−2.66845	−0.556410	−0.278205	+	0.960522i	$0.589740\pi$
$24$	−3.08937	−0.630615
$25$	−3.31950	−0.663899
$26$	1.62348	0.318390
$27$	−10.9494	−2.10721
$28$	4.06006	0.767279
$29$	−1.01589	−0.188647	−0.0943234	−	0.995542i	$-0.530069\pi$
$29$	−1.01589	−0.188647	−0.0943234	+	0.995542i	$0.530069\pi$
$30$	4.00488	0.731188
$31$	−3.05425	−0.548559	−0.274280	−	0.961650i	$-0.588439\pi$
$31$	−3.05425	−0.548559	−0.274280	+	0.961650i	$0.588439\pi$
$32$	1.00000	0.176777
$33$	−6.52814	−1.13640
$34$	−2.05899	−0.353113
$35$	−5.26323	−0.889647
$36$	6.54422	1.09070
$37$	−7.60060	−1.24953	−0.624765	−	0.780813i	$-0.714805\pi$
$37$	−7.60060	−1.24953	−0.624765	+	0.780813i	$0.714805\pi$
$38$	1.37650	0.223298
$39$	−5.01553	−0.803127
$40$	−1.29634	−0.204970
$41$	5.22545	0.816079	0.408039	−	0.912964i	$-0.366213\pi$
$41$	5.22545	0.816079	0.408039	+	0.912964i	$0.366213\pi$
$42$	−12.5430	−1.93543
$43$	−9.75420	−1.48750	−0.743751	−	0.668457i	$-0.766955\pi$
$43$	−9.75420	−1.48750	−0.743751	+	0.668457i	$0.766955\pi$
$44$	2.11310	0.318561
$45$	−8.48354	−1.26465
$46$	−2.66845	−0.393441
$47$	−11.0245	−1.60809	−0.804047	−	0.594565i	$-0.797324\pi$
$47$	−11.0245	−1.60809	−0.804047	+	0.594565i	$0.797324\pi$
$48$	−3.08937	−0.445912
$49$	9.48407	1.35487
$50$	−3.31950	−0.469448
$51$	6.36097	0.890714
$52$	1.62348	0.225136
$53$	−6.35741	−0.873257	−0.436629	−	0.899642i	$-0.643828\pi$
$53$	−6.35741	−0.873257	−0.436629	+	0.899642i	$0.643828\pi$
$54$	−10.9494	−1.49002
$55$	−2.73930	−0.369367
$56$	4.06006	0.542548
$57$	−4.25252	−0.563261
$58$	−1.01589	−0.133393
$59$	10.0360	1.30658	0.653289	−	0.757108i	$-0.273388\pi$
$59$	10.0360	1.30658	0.653289	+	0.757108i	$0.273388\pi$
$60$	4.00488	0.517028
$61$	3.96562	0.507745	0.253873	−	0.967238i	$-0.418296\pi$
$61$	3.96562	0.507745	0.253873	+	0.967238i	$0.418296\pi$
$62$	−3.05425	−0.387890
$63$	26.5699	3.34749
$64$	1.00000	0.125000
$65$	−2.10458	−0.261041
$66$	−6.52814	−0.803559
$67$	−8.49862	−1.03827	−0.519136	−	0.854692i	$-0.673746\pi$
$67$	−8.49862	−1.03827	−0.519136	+	0.854692i	$0.673746\pi$
$68$	−2.05899	−0.249689
$69$	8.24383	0.992441
$70$	−5.26323	−0.629076
$71$	−12.7609	−1.51444	−0.757218	−	0.653162i	$-0.773442\pi$
$71$	−12.7609	−1.51444	−0.757218	+	0.653162i	$0.773442\pi$
$72$	6.54422	0.771243
$73$	−14.3477	−1.67928	−0.839638	−	0.543146i	$-0.817233\pi$
$73$	−14.3477	−1.67928	−0.839638	+	0.543146i	$0.817233\pi$
$74$	−7.60060	−0.883552
$75$	10.2552	1.18416
$76$	1.37650	0.157896
$77$	8.57930	0.977702
$78$	−5.01553	−0.567896
$79$	4.45222	0.500914	0.250457	−	0.968128i	$-0.419419\pi$
$79$	4.45222	0.500914	0.250457	+	0.968128i	$0.419419\pi$
$80$	−1.29634	−0.144935
$81$	14.1941	1.57712
$82$	5.22545	0.577055
$83$	13.3840	1.46909	0.734545	−	0.678560i	$-0.237396\pi$
$83$	13.3840	1.46909	0.734545	+	0.678560i	$0.237396\pi$
$84$	−12.5430	−1.36856
$85$	2.66915	0.289510
$86$	−9.75420	−1.05182
$87$	3.13847	0.336480
$88$	2.11310	0.225257
$89$	1.76396	0.186980	0.0934899	−	0.995620i	$-0.470198\pi$
$89$	1.76396	0.186980	0.0934899	+	0.995620i	$0.470198\pi$
$90$	−8.48354	−0.894244
$91$	6.59141	0.690968
$92$	−2.66845	−0.278205
$93$	9.43570	0.978437
$94$	−11.0245	−1.13709
$95$	−1.78442	−0.183077
$96$	−3.08937	−0.315308
$97$	−6.28206	−0.637846	−0.318923	−	0.947781i	$-0.603321\pi$
$97$	−6.28206	−0.637846	−0.318923	+	0.947781i	$0.603321\pi$
$98$	9.48407	0.958036
$99$	13.8286	1.38982
$100$	−3.31950	−0.331950
$101$	−3.10164	−0.308625	−0.154312	−	0.988022i	$-0.549316\pi$
$101$	−3.10164	−0.308625	−0.154312	+	0.988022i	$0.549316\pi$
$102$	6.36097	0.629830
$103$	−15.1890	−1.49661	−0.748307	−	0.663352i	$-0.769133\pi$
$103$	−15.1890	−1.49661	−0.748307	+	0.663352i	$0.769133\pi$
$104$	1.62348	0.159195
$105$	16.2601	1.58682
$106$	−6.35741	−0.617486
$107$	−10.0387	−0.970476	−0.485238	−	0.874382i	$-0.661267\pi$
$107$	−10.0387	−0.970476	−0.485238	+	0.874382i	$0.661267\pi$
$108$	−10.9494	−1.05361
$109$	20.2167	1.93641	0.968207	−	0.250152i	$-0.0804805\pi$
$109$	20.2167	1.93641	0.968207	+	0.250152i	$0.0804805\pi$
$110$	−2.73930	−0.261182
$111$	23.4811	2.22872
$112$	4.06006	0.383639
$113$	−5.84705	−0.550044	−0.275022	−	0.961438i	$-0.588685\pi$
$113$	−5.84705	−0.550044	−0.275022	+	0.961438i	$0.588685\pi$
$114$	−4.25252	−0.398285
$115$	3.45922	0.322574
$116$	−1.01589	−0.0943234
$117$	10.6244	0.982225
$118$	10.0360	0.923891
$119$	−8.35960	−0.766323
$120$	4.00488	0.365594
$121$	−6.53482	−0.594074
$122$	3.96562	0.359030
$123$	−16.1434	−1.45560
$124$	−3.05425	−0.274280
$125$	10.7849	0.964632
$126$	26.5699	2.36703
$127$	−3.85113	−0.341732	−0.170866	−	0.985294i	$-0.554657\pi$
$127$	−3.85113	−0.341732	−0.170866	+	0.985294i	$0.554657\pi$
$128$	1.00000	0.0883883
$129$	30.1344	2.65318
$130$	−2.10458	−0.184584
$131$	10.6395	0.929581	0.464791	−	0.885421i	$-0.346130\pi$
$131$	10.6395	0.929581	0.464791	+	0.885421i	$0.346130\pi$
$132$	−6.52814	−0.568202
$133$	5.58868	0.484600
$134$	−8.49862	−0.734169
$135$	14.1942	1.22164
$136$	−2.05899	−0.176557
$137$	21.6958	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$137$	21.6958	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$138$	8.24383	0.701762
$139$	4.00382	0.339599	0.169800	−	0.985479i	$-0.445688\pi$
$139$	4.00382	0.339599	0.169800	+	0.985479i	$0.445688\pi$
$140$	−5.26323	−0.444824
$141$	34.0589	2.86828
$142$	−12.7609	−1.07087
$143$	3.43057	0.286878
$144$	6.54422	0.545351
$145$	1.31695	0.109366
$146$	−14.3477	−1.18743
$147$	−29.2998	−2.41661
$148$	−7.60060	−0.624765
$149$	−3.27870	−0.268602	−0.134301	−	0.990941i	$-0.542879\pi$
$149$	−3.27870	−0.268602	−0.134301	+	0.990941i	$0.542879\pi$
$150$	10.2552	0.837330
$151$	−7.08842	−0.576848	−0.288424	−	0.957503i	$-0.593131\pi$
$151$	−7.08842	−0.576848	−0.288424	+	0.957503i	$0.593131\pi$
$152$	1.37650	0.111649
$153$	−13.4744	−1.08934
$154$	8.57930	0.691340
$155$	3.95935	0.318023
$156$	−5.01553	−0.401563
$157$	2.27988	0.181954	0.0909770	−	0.995853i	$-0.471001\pi$
$157$	2.27988	0.181954	0.0909770	+	0.995853i	$0.471001\pi$
$158$	4.45222	0.354200
$159$	19.6404	1.55758
$160$	−1.29634	−0.102485
$161$	−10.8341	−0.853844
$162$	14.1941	1.11519
$163$	6.05792	0.474493	0.237247	−	0.971449i	$-0.423755\pi$
$163$	6.05792	0.474493	0.237247	+	0.971449i	$0.423755\pi$
$164$	5.22545	0.408039
$165$	8.46271	0.658821
$166$	13.3840	1.03880
$167$	−5.10081	−0.394713	−0.197356	−	0.980332i	$-0.563236\pi$
$167$	−5.10081	−0.394713	−0.197356	+	0.980332i	$0.563236\pi$
$168$	−12.5430	−0.967716
$169$	−10.3643	−0.797255
$170$	2.66915	0.204714
$171$	9.00812	0.688868
$172$	−9.75420	−0.743751
$173$	−1.16851	−0.0888403	−0.0444202	−	0.999013i	$-0.514144\pi$
$173$	−1.16851	−0.0888403	−0.0444202	+	0.999013i	$0.514144\pi$
$174$	3.13847	0.237927
$175$	−13.4774	−1.01879
$176$	2.11310	0.159281
$177$	−31.0050	−2.33048
$178$	1.76396	0.132215
$179$	−7.18255	−0.536849	−0.268424	−	0.963301i	$-0.586503\pi$
$179$	−7.18255	−0.536849	−0.268424	+	0.963301i	$0.586503\pi$
$180$	−8.48354	−0.632326
$181$	−8.04614	−0.598065	−0.299033	−	0.954243i	$-0.596664\pi$
$181$	−8.04614	−0.598065	−0.299033	+	0.954243i	$0.596664\pi$
$182$	6.59141	0.488588
$183$	−12.2513	−0.905639
$184$	−2.66845	−0.196721
$185$	9.85298	0.724405
$186$	9.43570	0.691859
$187$	−4.35084	−0.318165
$188$	−11.0245	−0.804047
$189$	−44.4552	−3.23364
$190$	−1.78442	−0.129455
$191$	−0.946898	−0.0685151	−0.0342576	−	0.999413i	$-0.510907\pi$
$191$	−0.946898	−0.0685151	−0.0342576	+	0.999413i	$0.510907\pi$
$192$	−3.08937	−0.222956
$193$	−8.90872	−0.641263	−0.320632	−	0.947204i	$-0.603895\pi$
$193$	−8.90872	−0.641263	−0.320632	+	0.947204i	$0.603895\pi$
$194$	−6.28206	−0.451025
$195$	6.50184	0.465606
$196$	9.48407	0.677434
$197$	−15.2659	−1.08765	−0.543826	−	0.839198i	$-0.683025\pi$
$197$	−15.2659	−1.08765	−0.543826	+	0.839198i	$0.683025\pi$
$198$	13.8286	0.982753
$199$	13.1743	0.933900	0.466950	−	0.884284i	$-0.345353\pi$
$199$	13.1743	0.933900	0.466950	+	0.884284i	$0.345353\pi$
$200$	−3.31950	−0.234724
$201$	26.2554	1.85191
$202$	−3.10164	−0.218231
$203$	−4.12459	−0.289489
$204$	6.36097	0.445357
$205$	−6.77398	−0.473115
$206$	−15.1890	−1.05827
$207$	−17.4629	−1.21376
$208$	1.62348	0.112568
$209$	2.90868	0.201198
$210$	16.2601	1.12205
$211$	−0.619013	−0.0426146	−0.0213073	−	0.999773i	$-0.506783\pi$
$211$	−0.619013	−0.0426146	−0.0213073	+	0.999773i	$0.506783\pi$
$212$	−6.35741	−0.436629
$213$	39.4231	2.70122
$214$	−10.0387	−0.686230
$215$	12.6448	0.862367
$216$	−10.9494	−0.745012
$217$	−12.4004	−0.841796
$218$	20.2167	1.36925
$219$	44.3255	2.99524
$220$	−2.73930	−0.184683
$221$	−3.34272	−0.224855
$222$	23.4811	1.57595
$223$	−0.204315	−0.0136820	−0.00684098	−	0.999977i	$-0.502178\pi$
$223$	−0.204315	−0.0136820	−0.00684098	+	0.999977i	$0.502178\pi$
$224$	4.06006	0.271274
$225$	−21.7235	−1.44823
$226$	−5.84705	−0.388940
$227$	0.393365	0.0261086	0.0130543	−	0.999915i	$-0.495845\pi$
$227$	0.393365	0.0261086	0.0130543	+	0.999915i	$0.495845\pi$
$228$	−4.25252	−0.281630
$229$	13.1248	0.867311	0.433655	−	0.901079i	$-0.357224\pi$
$229$	13.1248	0.867311	0.433655	+	0.901079i	$0.357224\pi$
$230$	3.45922	0.228094
$231$	−26.5046	−1.74388
$232$	−1.01589	−0.0666967
$233$	5.68219	0.372253	0.186126	−	0.982526i	$-0.440407\pi$
$233$	5.68219	0.372253	0.186126	+	0.982526i	$0.440407\pi$
$234$	10.6244	0.694538
$235$	14.2916	0.932280
$236$	10.0360	0.653289
$237$	−13.7546	−0.893456
$238$	−8.35960	−0.541872
$239$	−6.22273	−0.402515	−0.201257	−	0.979538i	$-0.564503\pi$
$239$	−6.22273	−0.402515	−0.201257	+	0.979538i	$0.564503\pi$
$240$	4.00488	0.258514
$241$	−0.356819	−0.0229847	−0.0114924	−	0.999934i	$-0.503658\pi$
$241$	−0.356819	−0.0229847	−0.0114924	+	0.999934i	$0.503658\pi$
$242$	−6.53482	−0.420074
$243$	−11.0027	−0.705822
$244$	3.96562	0.253873
$245$	−12.2946	−0.785474
$246$	−16.1434	−1.02926
$247$	2.23472	0.142192
$248$	−3.05425	−0.193945
$249$	−41.3483	−2.62034
$250$	10.7849	0.682098
$251$	24.3385	1.53623	0.768115	−	0.640312i	$-0.221195\pi$
$251$	24.3385	1.53623	0.768115	+	0.640312i	$0.221195\pi$
$252$	26.5699	1.67375
$253$	−5.63869	−0.354502
$254$	−3.85113	−0.241641
$255$	−8.24599	−0.516384
$256$	1.00000	0.0625000
$257$	−8.64281	−0.539124	−0.269562	−	0.962983i	$-0.586879\pi$
$257$	−8.64281	−0.539124	−0.269562	+	0.962983i	$0.586879\pi$
$258$	30.1344	1.87608
$259$	−30.8589	−1.91748
$260$	−2.10458	−0.130521
$261$	−6.64823	−0.411515
$262$	10.6395	0.657313
$263$	−1.29572	−0.0798972	−0.0399486	−	0.999202i	$-0.512719\pi$
$263$	−1.29572	−0.0798972	−0.0399486	+	0.999202i	$0.512719\pi$
$264$	−6.52814	−0.401779
$265$	8.24138	0.506264
$266$	5.58868	0.342664
$267$	−5.44954	−0.333506
$268$	−8.49862	−0.519136
$269$	−26.0035	−1.58546	−0.792731	−	0.609571i	$-0.791342\pi$
$269$	−26.0035	−1.58546	−0.792731	+	0.609571i	$0.791342\pi$
$270$	14.1942	0.863829
$271$	−21.8067	−1.32466	−0.662332	−	0.749210i	$-0.730433\pi$
$271$	−21.8067	−1.32466	−0.662332	+	0.749210i	$0.730433\pi$
$272$	−2.05899	−0.124844
$273$	−20.3633	−1.23244
$274$	21.6958	1.31069
$275$	−7.01442	−0.422985
$276$	8.24383	0.496220
$277$	−16.3304	−0.981200	−0.490600	−	0.871385i	$-0.663222\pi$
$277$	−16.3304	−0.981200	−0.490600	+	0.871385i	$0.663222\pi$
$278$	4.00382	0.240133
$279$	−19.9877	−1.19663
$280$	−5.26323	−0.314538
$281$	0.876484	0.0522866	0.0261433	−	0.999658i	$-0.491677\pi$
$281$	0.876484	0.0522866	0.0261433	+	0.999658i	$0.491677\pi$
$282$	34.0589	2.02818
$283$	7.00305	0.416288	0.208144	−	0.978098i	$-0.433258\pi$
$283$	7.00305	0.416288	0.208144	+	0.978098i	$0.433258\pi$
$284$	−12.7609	−0.757218
$285$	5.51273	0.326546
$286$	3.43057	0.202854
$287$	21.2156	1.25232
$288$	6.54422	0.385622
$289$	−12.7606	−0.750622
$290$	1.31695	0.0773338
$291$	19.4076	1.13769
$292$	−14.3477	−0.839638
$293$	5.56377	0.325039	0.162520	−	0.986705i	$-0.448038\pi$
$293$	5.56377	0.325039	0.162520	+	0.986705i	$0.448038\pi$
$294$	−29.2998	−1.70880
$295$	−13.0101	−0.757478
$296$	−7.60060	−0.441776
$297$	−23.1371	−1.34255
$298$	−3.27870	−0.189930
$299$	−4.33217	−0.250536
$300$	10.2552	0.592082
$301$	−39.6026	−2.28266
$302$	−7.08842	−0.407893
$303$	9.58212	0.550478
$304$	1.37650	0.0789478
$305$	−5.14080	−0.294361
$306$	−13.4744	−0.770283
$307$	−19.8160	−1.13096	−0.565479	−	0.824762i	$-0.691309\pi$
$307$	−19.8160	−1.13096	−0.565479	+	0.824762i	$0.691309\pi$
$308$	8.57930	0.488851
$309$	46.9244	2.66944
$310$	3.95935	0.224876
$311$	−13.8156	−0.783412	−0.391706	−	0.920090i	$-0.628115\pi$
$311$	−13.8156	−0.783412	−0.391706	+	0.920090i	$0.628115\pi$
$312$	−5.01553	−0.283948
$313$	−3.51493	−0.198676	−0.0993379	−	0.995054i	$-0.531672\pi$
$313$	−3.51493	−0.198676	−0.0993379	+	0.995054i	$0.531672\pi$
$314$	2.27988	0.128661
$315$	−34.4437	−1.94068
$316$	4.45222	0.250457
$317$	8.03903	0.451517	0.225758	−	0.974183i	$-0.427514\pi$
$317$	8.03903	0.451517	0.225758	+	0.974183i	$0.427514\pi$
$318$	19.6404	1.10138
$319$	−2.14668	−0.120191
$320$	−1.29634	−0.0724677
$321$	31.0132	1.73099
$322$	−10.8341	−0.603759
$323$	−2.83420	−0.157699
$324$	14.1941	0.788561
$325$	−5.38913	−0.298935
$326$	6.05792	0.335517
$327$	−62.4570	−3.45388
$328$	5.22545	0.288527
$329$	−44.7603	−2.46771
$330$	8.46271	0.465857
$331$	33.8157	1.85868	0.929340	−	0.369224i	$-0.120376\pi$
$331$	33.8157	1.85868	0.929340	+	0.369224i	$0.120376\pi$
$332$	13.3840	0.734545
$333$	−49.7400	−2.72573
$334$	−5.10081	−0.279104
$335$	11.0171	0.601929
$336$	−12.5430	−0.684278
$337$	−18.9839	−1.03412	−0.517060	−	0.855949i	$-0.672974\pi$
$337$	−18.9839	−1.03412	−0.517060	+	0.855949i	$0.672974\pi$
$338$	−10.3643	−0.563745
$339$	18.0637	0.981085
$340$	2.66915	0.144755
$341$	−6.45392	−0.349500
$342$	9.00812	0.487103
$343$	10.0855	0.544565
$344$	−9.75420	−0.525911
$345$	−10.6868	−0.575359
$346$	−1.16851	−0.0628196
$347$	31.9817	1.71687	0.858434	−	0.512924i	$-0.171438\pi$
$347$	31.9817	1.71687	0.858434	+	0.512924i	$0.171438\pi$
$348$	3.13847	0.168240
$349$	−23.9454	−1.28177	−0.640884	−	0.767638i	$-0.721432\pi$
$349$	−23.9454	−1.28177	−0.640884	+	0.767638i	$0.721432\pi$
$350$	−13.4774	−0.720395
$351$	−17.7761	−0.948818
$352$	2.11310	0.112628
$353$	5.51931	0.293763	0.146882	−	0.989154i	$-0.453076\pi$
$353$	5.51931	0.293763	0.146882	+	0.989154i	$0.453076\pi$
$354$	−31.0050	−1.64790
$355$	16.5425	0.877982
$356$	1.76396	0.0934899
$357$	25.8259	1.36685
$358$	−7.18255	−0.379610
$359$	−5.60049	−0.295583	−0.147791	−	0.989019i	$-0.547216\pi$
$359$	−5.60049	−0.295583	−0.147791	+	0.989019i	$0.547216\pi$
$360$	−8.48354	−0.447122
$361$	−17.1052	−0.900276
$362$	−8.04614	−0.422896
$363$	20.1885	1.05962
$364$	6.59141	0.345484
$365$	18.5996	0.973547
$366$	−12.2513	−0.640384
$367$	5.33401	0.278433	0.139216	−	0.990262i	$-0.455542\pi$
$367$	5.33401	0.278433	0.139216	+	0.990262i	$0.455542\pi$
$368$	−2.66845	−0.139103
$369$	34.1965	1.78020
$370$	9.85298	0.512232
$371$	−25.8114	−1.34006
$372$	9.43570	0.489219
$373$	11.3224	0.586251	0.293125	−	0.956074i	$-0.405305\pi$
$373$	11.3224	0.586251	0.293125	+	0.956074i	$0.405305\pi$
$374$	−4.35084	−0.224976
$375$	−33.3186	−1.72057
$376$	−11.0245	−0.568547
$377$	−1.64928	−0.0849423
$378$	−44.4552	−2.28653
$379$	−1.48057	−0.0760519	−0.0380260	−	0.999277i	$-0.512107\pi$
$379$	−1.48057	−0.0760519	−0.0380260	+	0.999277i	$0.512107\pi$
$380$	−1.78442	−0.0915387
$381$	11.8976	0.609531
$382$	−0.946898	−0.0484475
$383$	32.5825	1.66489	0.832443	−	0.554110i	$-0.186941\pi$
$383$	32.5825	1.66489	0.832443	+	0.554110i	$0.186941\pi$
$384$	−3.08937	−0.157654
$385$	−11.1217	−0.566815
$386$	−8.90872	−0.453442
$387$	−63.8336	−3.24484
$388$	−6.28206	−0.318923
$389$	−26.5123	−1.34423	−0.672114	−	0.740448i	$-0.734614\pi$
$389$	−26.5123	−1.34423	−0.672114	+	0.740448i	$0.734614\pi$
$390$	6.50184	0.329233
$391$	5.49430	0.277859
$392$	9.48407	0.479018
$393$	−32.8695	−1.65805
$394$	−15.2659	−0.769086
$395$	−5.77161	−0.290401
$396$	13.8286	0.694911
$397$	−17.5556	−0.881092	−0.440546	−	0.897730i	$-0.645215\pi$
$397$	−17.5556	−0.881092	−0.440546	+	0.897730i	$0.645215\pi$
$398$	13.1743	0.660367
$399$	−17.2655	−0.864356
$400$	−3.31950	−0.165975
$401$	−2.24267	−0.111993	−0.0559967	−	0.998431i	$-0.517834\pi$
$401$	−2.24267	−0.111993	−0.0559967	+	0.998431i	$0.517834\pi$
$402$	26.2554	1.30950
$403$	−4.95850	−0.247001
$404$	−3.10164	−0.154312
$405$	−18.4004	−0.914324
$406$	−4.12459	−0.204700
$407$	−16.0608	−0.796105
$408$	6.36097	0.314915
$409$	11.0297	0.545385	0.272693	−	0.962101i	$-0.412086\pi$
$409$	11.0297	0.545385	0.272693	+	0.962101i	$0.412086\pi$
$410$	−6.77398	−0.334543
$411$	−67.0265	−3.30617
$412$	−15.1890	−0.748307
$413$	40.7468	2.00502
$414$	−17.4629	−0.858255
$415$	−17.3503	−0.851693
$416$	1.62348	0.0795975
$417$	−12.3693	−0.605726
$418$	2.90868	0.142268
$419$	4.30389	0.210259	0.105129	−	0.994459i	$-0.466474\pi$
$419$	4.30389	0.210259	0.105129	+	0.994459i	$0.466474\pi$
$420$	16.2601	0.793410
$421$	−22.5563	−1.09933	−0.549663	−	0.835387i	$-0.685244\pi$
$421$	−22.5563	−1.09933	−0.549663	+	0.835387i	$0.685244\pi$
$422$	−0.619013	−0.0301331
$423$	−72.1470	−3.50791
$424$	−6.35741	−0.308743
$425$	6.83480	0.331536
$426$	39.4231	1.91005
$427$	16.1006	0.779164
$428$	−10.0387	−0.485238
$429$	−10.5983	−0.511690
$430$	12.6448	0.609786
$431$	−39.0845	−1.88263	−0.941316	−	0.337527i	$-0.890410\pi$
$431$	−39.0845	−1.88263	−0.941316	+	0.337527i	$0.890410\pi$
$432$	−10.9494	−0.526803
$433$	24.8409	1.19378	0.596889	−	0.802323i	$-0.296403\pi$
$433$	24.8409	1.19378	0.596889	+	0.802323i	$0.296403\pi$
$434$	−12.4004	−0.595239
$435$	−4.06854	−0.195071
$436$	20.2167	0.968207
$437$	−3.67313	−0.175709
$438$	44.3255	2.11795
$439$	6.42377	0.306590	0.153295	−	0.988180i	$-0.451012\pi$
$439$	6.42377	0.306590	0.153295	+	0.988180i	$0.451012\pi$
$440$	−2.73930	−0.130591
$441$	62.0658	2.95552
$442$	−3.34272	−0.158997
$443$	31.7437	1.50819	0.754096	−	0.656765i	$-0.228076\pi$
$443$	31.7437	1.50819	0.754096	+	0.656765i	$0.228076\pi$
$444$	23.4811	1.11436
$445$	−2.28670	−0.108400
$446$	−0.204315	−0.00967461
$447$	10.1291	0.479091
$448$	4.06006	0.191820
$449$	20.8885	0.985790	0.492895	−	0.870089i	$-0.335939\pi$
$449$	20.8885	0.985790	0.492895	+	0.870089i	$0.335939\pi$
$450$	−21.7235	−1.02406
$451$	11.0419	0.519942
$452$	−5.84705	−0.275022
$453$	21.8988	1.02889
$454$	0.393365	0.0184615
$455$	−8.54473	−0.400583
$456$	−4.25252	−0.199143
$457$	−5.10176	−0.238650	−0.119325	−	0.992855i	$-0.538073\pi$
$457$	−5.10176	−0.238650	−0.119325	+	0.992855i	$0.538073\pi$
$458$	13.1248	0.613281
$459$	22.5446	1.05229
$460$	3.45922	0.161287
$461$	−11.3630	−0.529227	−0.264614	−	0.964354i	$-0.585244\pi$
$461$	−11.3630	−0.529227	−0.264614	+	0.964354i	$0.585244\pi$
$462$	−26.5046	−1.23311
$463$	−24.4595	−1.13673	−0.568365	−	0.822777i	$-0.692424\pi$
$463$	−24.4595	−1.13673	−0.568365	+	0.822777i	$0.692424\pi$
$464$	−1.01589	−0.0471617
$465$	−12.2319	−0.567241
$466$	5.68219	0.263222
$467$	2.83365	0.131126	0.0655628	−	0.997848i	$-0.479116\pi$
$467$	2.83365	0.131126	0.0655628	+	0.997848i	$0.479116\pi$
$468$	10.6244	0.491112
$469$	−34.5049	−1.59329
$470$	14.2916	0.659221
$471$	−7.04339	−0.324542
$472$	10.0360	0.461945
$473$	−20.6116	−0.947722
$474$	−13.7546	−0.631769
$475$	−4.56929	−0.209653
$476$	−8.35960	−0.383162
$477$	−41.6042	−1.90493
$478$	−6.22273	−0.284621
$479$	23.1850	1.05935	0.529674	−	0.848201i	$-0.322314\pi$
$479$	23.1850	1.05935	0.529674	+	0.848201i	$0.322314\pi$
$480$	4.00488	0.182797
$481$	−12.3394	−0.562628
$482$	−0.356819	−0.0162527
$483$	33.4704	1.52296
$484$	−6.53482	−0.297037
$485$	8.14370	0.369786
$486$	−11.0027	−0.499091
$487$	17.6224	0.798549	0.399274	−	0.916831i	$-0.369262\pi$
$487$	17.6224	0.798549	0.399274	+	0.916831i	$0.369262\pi$
$488$	3.96562	0.179515
$489$	−18.7152	−0.846330
$490$	−12.2946	−0.555414
$491$	11.0642	0.499319	0.249660	−	0.968334i	$-0.419681\pi$
$491$	11.0642	0.499319	0.249660	+	0.968334i	$0.419681\pi$
$492$	−16.1434	−0.727799
$493$	2.09171	0.0942060
$494$	2.23472	0.100545
$495$	−17.9266	−0.805739
$496$	−3.05425	−0.137140
$497$	−51.8099	−2.32399
$498$	−41.3483	−1.85286
$499$	−0.501950	−0.0224704	−0.0112352	−	0.999937i	$-0.503576\pi$
$499$	−0.501950	−0.0224704	−0.0112352	+	0.999937i	$0.503576\pi$
$500$	10.7849	0.482316
$501$	15.7583	0.704029
$502$	24.3385	1.08628
$503$	−9.01821	−0.402102	−0.201051	−	0.979581i	$-0.564436\pi$
$503$	−9.01821	−0.402102	−0.201051	+	0.979581i	$0.564436\pi$
$504$	26.5699	1.18352
$505$	4.02079	0.178923
$506$	−5.63869	−0.250671
$507$	32.0192	1.42202
$508$	−3.85113	−0.170866
$509$	12.9461	0.573828	0.286914	−	0.957956i	$-0.407371\pi$
$509$	12.9461	0.573828	0.286914	+	0.957956i	$0.407371\pi$
$510$	−8.24599	−0.365139
$511$	−58.2527	−2.57695
$512$	1.00000	0.0441942
$513$	−15.0719	−0.665439
$514$	−8.64281	−0.381218
$515$	19.6901	0.867650
$516$	30.1344	1.32659
$517$	−23.2959	−1.02455
$518$	−30.8589	−1.35586
$519$	3.60997	0.158460
$520$	−2.10458	−0.0922921
$521$	8.71704	0.381900	0.190950	−	0.981600i	$-0.438843\pi$
$521$	8.71704	0.381900	0.190950	+	0.981600i	$0.438843\pi$
$522$	−6.64823	−0.290985
$523$	−0.781995	−0.0341943	−0.0170971	−	0.999854i	$-0.505442\pi$
$523$	−0.781995	−0.0341943	−0.0170971	+	0.999854i	$0.505442\pi$
$524$	10.6395	0.464791
$525$	41.6365	1.81717
$526$	−1.29572	−0.0564959
$527$	6.28865	0.273938
$528$	−6.52814	−0.284101
$529$	−15.8794	−0.690408
$530$	8.24138	0.357983
$531$	65.6779	2.85018
$532$	5.58868	0.242300
$533$	8.48341	0.367457
$534$	−5.44954	−0.235825
$535$	13.0136	0.562626
$536$	−8.49862	−0.367084
$537$	22.1896	0.957550
$538$	−26.0035	−1.12109
$539$	20.0408	0.863217
$540$	14.1942	0.610820
$541$	31.8528	1.36946	0.684730	−	0.728797i	$-0.259920\pi$
$541$	31.8528	1.36946	0.684730	+	0.728797i	$0.259920\pi$
$542$	−21.8067	−0.936680
$543$	24.8575	1.06674
$544$	−2.05899	−0.0882783
$545$	−26.2078	−1.12262
$546$	−20.3633	−0.871470
$547$	−31.8632	−1.36237	−0.681185	−	0.732111i	$-0.738535\pi$
$547$	−31.8632	−1.36237	−0.681185	+	0.732111i	$0.738535\pi$
$548$	21.6958	0.926800
$549$	25.9518	1.10760
$550$	−7.01442	−0.299096
$551$	−1.39838	−0.0595730
$552$	8.24383	0.350881
$553$	18.0763	0.768682
$554$	−16.3304	−0.693813
$555$	−30.4395	−1.29208
$556$	4.00382	0.169800
$557$	7.89836	0.334664	0.167332	−	0.985901i	$-0.446485\pi$
$557$	7.89836	0.334664	0.167332	+	0.985901i	$0.446485\pi$
$558$	−19.9877	−0.846145
$559$	−15.8357	−0.669780
$560$	−5.26323	−0.222412
$561$	13.4414	0.567494
$562$	0.876484	0.0369722
$563$	−20.3205	−0.856406	−0.428203	−	0.903683i	$-0.640853\pi$
$563$	−20.3205	−0.856406	−0.428203	+	0.903683i	$0.640853\pi$
$564$	34.0589	1.43414
$565$	7.57977	0.318883
$566$	7.00305	0.294360
$567$	57.6289	2.42019
$568$	−12.7609	−0.535434
$569$	12.4138	0.520412	0.260206	−	0.965553i	$-0.416209\pi$
$569$	12.4138	0.520412	0.260206	+	0.965553i	$0.416209\pi$
$570$	5.51273	0.230903
$571$	−1.87974	−0.0786648	−0.0393324	−	0.999226i	$-0.512523\pi$
$571$	−1.87974	−0.0786648	−0.0393324	+	0.999226i	$0.512523\pi$
$572$	3.43057	0.143439
$573$	2.92532	0.122207
$574$	21.2156	0.885524
$575$	8.85791	0.369400
$576$	6.54422	0.272676
$577$	−9.01851	−0.375445	−0.187723	−	0.982222i	$-0.560111\pi$
$577$	−9.01851	−0.375445	−0.187723	+	0.982222i	$0.560111\pi$
$578$	−12.7606	−0.530770
$579$	27.5223	1.14379
$580$	1.31695	0.0546832
$581$	54.3400	2.25440
$582$	19.4076	0.804471
$583$	−13.4338	−0.556372
$584$	−14.3477	−0.593714
$585$	−13.7728	−0.569437
$586$	5.56377	0.229837
$587$	31.0377	1.28106	0.640532	−	0.767931i	$-0.278714\pi$
$587$	31.0377	1.28106	0.640532	+	0.767931i	$0.278714\pi$
$588$	−29.2998	−1.20830
$589$	−4.20418	−0.173230
$590$	−13.0101	−0.535618
$591$	47.1621	1.93999
$592$	−7.60060	−0.312383
$593$	−33.6345	−1.38120	−0.690602	−	0.723235i	$-0.742654\pi$
$593$	−33.6345	−1.38120	−0.690602	+	0.723235i	$0.742654\pi$
$594$	−23.1371	−0.949328
$595$	10.8369	0.444270
$596$	−3.27870	−0.134301
$597$	−40.7003	−1.66575
$598$	−4.33217	−0.177156
$599$	7.06471	0.288656	0.144328	−	0.989530i	$-0.453898\pi$
$599$	7.06471	0.288656	0.144328	+	0.989530i	$0.453898\pi$
$600$	10.2552	0.418665
$601$	10.6039	0.432540	0.216270	−	0.976334i	$-0.430611\pi$
$601$	10.6039	0.432540	0.216270	+	0.976334i	$0.430611\pi$
$602$	−39.6026	−1.61408
$603$	−55.6168	−2.26489
$604$	−7.08842	−0.288424
$605$	8.47136	0.344410
$606$	9.58212	0.389247
$607$	−2.86115	−0.116130	−0.0580652	−	0.998313i	$-0.518493\pi$
$607$	−2.86115	−0.116130	−0.0580652	+	0.998313i	$0.518493\pi$
$608$	1.37650	0.0558245
$609$	12.7424	0.516348
$610$	−5.14080	−0.208145
$611$	−17.8981	−0.724080
$612$	−13.4744	−0.544672
$613$	31.7484	1.28231	0.641153	−	0.767413i	$-0.278456\pi$
$613$	31.7484	1.28231	0.641153	+	0.767413i	$0.278456\pi$
$614$	−19.8160	−0.799709
$615$	20.9273	0.843871
$616$	8.57930	0.345670
$617$	−21.9552	−0.883883	−0.441942	−	0.897044i	$-0.645710\pi$
$617$	−21.9552	−0.883883	−0.441942	+	0.897044i	$0.645710\pi$
$618$	46.9244	1.88758
$619$	−11.2808	−0.453416	−0.226708	−	0.973963i	$-0.572796\pi$
$619$	−11.2808	−0.453416	−0.226708	+	0.973963i	$0.572796\pi$
$620$	3.95935	0.159011
$621$	29.2179	1.17247
$622$	−13.8156	−0.553956
$623$	7.16180	0.286931
$624$	−5.01553	−0.200782
$625$	2.61654	0.104662
$626$	−3.51493	−0.140485
$627$	−8.98600	−0.358866
$628$	2.27988	0.0909770
$629$	15.6495	0.623987
$630$	−34.4437	−1.37227
$631$	−33.2046	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$631$	−33.2046	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$632$	4.45222	0.177100
$633$	1.91236	0.0760095
$634$	8.03903	0.319271
$635$	4.99238	0.198117
$636$	19.6404	0.778792
$637$	15.3972	0.610059
$638$	−2.14668	−0.0849880
$639$	−83.5099	−3.30360
$640$	−1.29634	−0.0512424
$641$	22.4413	0.886378	0.443189	−	0.896428i	$-0.353847\pi$
$641$	22.4413	0.886378	0.443189	+	0.896428i	$0.353847\pi$
$642$	31.0132	1.22399
$643$	34.6292	1.36564	0.682821	−	0.730586i	$-0.260753\pi$
$643$	34.6292	1.36564	0.682821	+	0.730586i	$0.260753\pi$
$644$	−10.8341	−0.426922
$645$	−39.0644	−1.53816
$646$	−2.83420	−0.111510
$647$	24.1715	0.950282	0.475141	−	0.879910i	$-0.342397\pi$
$647$	24.1715	0.950282	0.475141	+	0.879910i	$0.342397\pi$
$648$	14.1941	0.557597
$649$	21.2071	0.832451
$650$	−5.38913	−0.211379
$651$	38.3095	1.50147
$652$	6.05792	0.237247
$653$	18.3575	0.718386	0.359193	−	0.933263i	$-0.383052\pi$
$653$	18.3575	0.718386	0.359193	+	0.933263i	$0.383052\pi$
$654$	−62.4570	−2.44226
$655$	−13.7925	−0.538917
$656$	5.22545	0.204020
$657$	−93.8947	−3.66318
$658$	−44.7603	−1.74494
$659$	−46.7204	−1.81997	−0.909985	−	0.414641i	$-0.863907\pi$
$659$	−46.7204	−1.81997	−0.909985	+	0.414641i	$0.863907\pi$
$660$	8.46271	0.329410
$661$	−13.2024	−0.513514	−0.256757	−	0.966476i	$-0.582654\pi$
$661$	−13.2024	−0.513514	−0.256757	+	0.966476i	$0.582654\pi$
$662$	33.8157	1.31429
$663$	10.3269	0.401063
$664$	13.3840	0.519402
$665$	−7.24484	−0.280943
$666$	−49.7400	−1.92738
$667$	2.71086	0.104965
$668$	−5.10081	−0.197356
$669$	0.631206	0.0244038
$670$	11.0171	0.425628
$671$	8.37973	0.323496
$672$	−12.5430	−0.483858
$673$	−17.1502	−0.661093	−0.330547	−	0.943790i	$-0.607233\pi$
$673$	−17.1502	−0.661093	−0.330547	+	0.943790i	$0.607233\pi$
$674$	−18.9839	−0.731234
$675$	36.3465	1.39898
$676$	−10.3643	−0.398628
$677$	−19.0405	−0.731786	−0.365893	−	0.930657i	$-0.619236\pi$
$677$	−19.0405	−0.731786	−0.365893	+	0.930657i	$0.619236\pi$
$678$	18.0637	0.693732
$679$	−25.5055	−0.978812
$680$	2.66915	0.102357
$681$	−1.21525	−0.0465685
$682$	−6.45392	−0.247133
$683$	1.16481	0.0445704	0.0222852	−	0.999752i	$-0.492906\pi$
$683$	1.16481	0.0445704	0.0222852	+	0.999752i	$0.492906\pi$
$684$	9.00812	0.344434
$685$	−28.1252	−1.07461
$686$	10.0855	0.385066
$687$	−40.5474	−1.54698
$688$	−9.75420	−0.371876
$689$	−10.3211	−0.393203
$690$	−10.6868	−0.406841
$691$	30.9579	1.17769	0.588847	−	0.808244i	$-0.299582\pi$
$691$	30.9579	1.17769	0.588847	+	0.808244i	$0.299582\pi$
$692$	−1.16851	−0.0444202
$693$	56.1448	2.13276
$694$	31.9817	1.21401
$695$	−5.19032	−0.196880
$696$	3.13847	0.118964
$697$	−10.7591	−0.407531
$698$	−23.9454	−0.906346
$699$	−17.5544	−0.663968
$700$	−13.4774	−0.509396
$701$	−51.8275	−1.95750	−0.978748	−	0.205066i	$-0.934259\pi$
$701$	−51.8275	−1.95750	−0.978748	+	0.205066i	$0.934259\pi$
$702$	−17.7761	−0.670916
$703$	−10.4622	−0.394591
$704$	2.11310	0.0796404
$705$	−44.1520	−1.66286
$706$	5.51931	0.207722
$707$	−12.5928	−0.473602
$708$	−31.0050	−1.16524
$709$	−39.9155	−1.49906	−0.749529	−	0.661971i	$-0.769720\pi$
$709$	−39.9155	−1.49906	−0.749529	+	0.661971i	$0.769720\pi$
$710$	16.5425	0.620827
$711$	29.1363	1.09270
$712$	1.76396	0.0661073
$713$	8.15011	0.305224
$714$	25.8259	0.966510
$715$	−4.44719	−0.166315
$716$	−7.18255	−0.268424
$717$	19.2243	0.717946
$718$	−5.60049	−0.209009
$719$	10.3363	0.385478	0.192739	−	0.981250i	$-0.438263\pi$
$719$	10.3363	0.385478	0.192739	+	0.981250i	$0.438263\pi$
$720$	−8.48354	−0.316163
$721$	−61.6681	−2.29664
$722$	−17.1052	−0.636591
$723$	1.10235	0.0409967
$724$	−8.04614	−0.299033
$725$	3.37226	0.125243
$726$	20.1885	0.749265
$727$	8.75877	0.324845	0.162422	−	0.986721i	$-0.448069\pi$
$727$	8.75877	0.324845	0.162422	+	0.986721i	$0.448069\pi$
$728$	6.59141	0.244294
$729$	−8.59097	−0.318184
$730$	18.5996	0.688402
$731$	20.0838	0.742825
$732$	−12.2513	−0.452820
$733$	−46.2230	−1.70729	−0.853643	−	0.520858i	$-0.825612\pi$
$733$	−46.2230	−1.70729	−0.853643	+	0.520858i	$0.825612\pi$
$734$	5.33401	0.196882
$735$	37.9826	1.40101
$736$	−2.66845	−0.0983604
$737$	−17.9584	−0.661506
$738$	34.1965	1.25879
$739$	35.6242	1.31046	0.655229	−	0.755430i	$-0.272572\pi$
$739$	35.6242	1.31046	0.655229	+	0.755430i	$0.272572\pi$
$740$	9.85298	0.362203
$741$	−6.90388	−0.253620
$742$	−25.8114	−0.947568
$743$	50.3343	1.84658	0.923292	−	0.384098i	$-0.125488\pi$
$743$	50.3343	1.84658	0.923292	+	0.384098i	$0.125488\pi$
$744$	9.43570	0.345930
$745$	4.25032	0.155720
$746$	11.3224	0.414542
$747$	87.5880	3.20468
$748$	−4.35084	−0.159082
$749$	−40.7576	−1.48925
$750$	−33.3186	−1.21662
$751$	34.6928	1.26596	0.632979	−	0.774169i	$-0.281832\pi$
$751$	34.6928	1.26596	0.632979	+	0.774169i	$0.281832\pi$
$752$	−11.0245	−0.402024
$753$	−75.1905	−2.74010
$754$	−1.64928	−0.0600633
$755$	9.18902	0.334423
$756$	−44.4552	−1.61682
$757$	−22.6917	−0.824743	−0.412372	−	0.911016i	$-0.635300\pi$
$757$	−22.6917	−0.824743	−0.412372	+	0.911016i	$0.635300\pi$
$758$	−1.48057	−0.0537768
$759$	17.4200	0.632307
$760$	−1.78442	−0.0647276
$761$	38.3794	1.39125	0.695627	−	0.718403i	$-0.255127\pi$
$761$	38.3794	1.39125	0.695627	+	0.718403i	$0.255127\pi$
$762$	11.8976	0.431003
$763$	82.0812	2.97154
$764$	−0.946898	−0.0342576
$765$	17.4675	0.631538
$766$	32.5825	1.17725
$767$	16.2933	0.588315
$768$	−3.08937	−0.111478
$769$	47.1485	1.70022	0.850109	−	0.526607i	$-0.176536\pi$
$769$	47.1485	1.70022	0.850109	+	0.526607i	$0.176536\pi$
$770$	−11.1217	−0.400799
$771$	26.7009	0.961608
$772$	−8.90872	−0.320632
$773$	−13.7478	−0.494474	−0.247237	−	0.968955i	$-0.579523\pi$
$773$	−13.7478	−0.494474	−0.247237	+	0.968955i	$0.579523\pi$
$774$	−63.8336	−2.29445
$775$	10.1386	0.364188
$776$	−6.28206	−0.225513
$777$	95.3345	3.42011
$778$	−26.5123	−0.950513
$779$	7.19284	0.257710
$780$	6.50184	0.232803
$781$	−26.9650	−0.964882
$782$	5.49430	0.196476
$783$	11.1234	0.397519
$784$	9.48407	0.338717
$785$	−2.95550	−0.105486
$786$	−32.8695	−1.17242
$787$	−32.2917	−1.15107	−0.575537	−	0.817776i	$-0.695207\pi$
$787$	−32.2917	−1.15107	−0.575537	+	0.817776i	$0.695207\pi$
$788$	−15.2659	−0.543826
$789$	4.00295	0.142509
$790$	−5.77161	−0.205345
$791$	−23.7394	−0.844074
$792$	13.8286	0.491377
$793$	6.43809	0.228623
$794$	−17.5556	−0.623026
$795$	−25.4607	−0.902997
$796$	13.1743	0.466950
$797$	2.86355	0.101432	0.0507161	−	0.998713i	$-0.483850\pi$
$797$	2.86355	0.101432	0.0507161	+	0.998713i	$0.483850\pi$
$798$	−17.2655	−0.611192
$799$	22.6994	0.803046
$800$	−3.31950	−0.117362
$801$	11.5438	0.407879
$802$	−2.24267	−0.0791913
$803$	−30.3182	−1.06991
$804$	26.2554	0.925956
$805$	14.0447	0.495009
$806$	−4.95850	−0.174656
$807$	80.3345	2.82791
$808$	−3.10164	−0.109115
$809$	−48.6359	−1.70995	−0.854974	−	0.518671i	$-0.826427\pi$
$809$	−48.6359	−1.70995	−0.854974	+	0.518671i	$0.826427\pi$
$810$	−18.4004	−0.646525
$811$	28.8391	1.01268	0.506339	−	0.862335i	$-0.330999\pi$
$811$	28.8391	1.01268	0.506339	+	0.862335i	$0.330999\pi$
$812$	−4.12459	−0.144745
$813$	67.3691	2.36274
$814$	−16.0608	−0.562931
$815$	−7.85314	−0.275084
$816$	6.36097	0.222679
$817$	−13.4267	−0.469740
$818$	11.0297	0.385645
$819$	43.1356	1.50728
$820$	−6.77398	−0.236557
$821$	23.0737	0.805277	0.402638	−	0.915359i	$-0.368093\pi$
$821$	23.0737	0.805277	0.402638	+	0.915359i	$0.368093\pi$
$822$	−67.0265	−2.33782
$823$	−15.4743	−0.539399	−0.269699	−	0.962945i	$-0.586924\pi$
$823$	−15.4743	−0.539399	−0.269699	+	0.962945i	$0.586924\pi$
$824$	−15.1890	−0.529133
$825$	21.6701	0.754458
$826$	40.7468	1.41776
$827$	13.2760	0.461650	0.230825	−	0.972995i	$-0.425857\pi$
$827$	13.2760	0.461650	0.230825	+	0.972995i	$0.425857\pi$
$828$	−17.4629	−0.606878
$829$	−12.0588	−0.418821	−0.209410	−	0.977828i	$-0.567154\pi$
$829$	−12.0588	−0.418821	−0.209410	+	0.977828i	$0.567154\pi$
$830$	−17.3503	−0.602238
$831$	50.4507	1.75012
$832$	1.62348	0.0562840
$833$	−19.5276	−0.676590
$834$	−12.3693	−0.428313
$835$	6.61240	0.228831
$836$	2.90868	0.100599
$837$	33.4422	1.15593
$838$	4.30389	0.148675
$839$	−24.5569	−0.847798	−0.423899	−	0.905710i	$-0.639339\pi$
$839$	−24.5569	−0.847798	−0.423899	+	0.905710i	$0.639339\pi$
$840$	16.2601	0.561025
$841$	−27.9680	−0.964412
$842$	−22.5563	−0.777340
$843$	−2.70778	−0.0932610
$844$	−0.619013	−0.0213073
$845$	13.4357	0.462202
$846$	−72.1470	−2.48046
$847$	−26.5317	−0.911642
$848$	−6.35741	−0.218314
$849$	−21.6350	−0.742512
$850$	6.83480	0.234432
$851$	20.2818	0.695252
$852$	39.4231	1.35061
$853$	−24.9643	−0.854760	−0.427380	−	0.904072i	$-0.640563\pi$
$853$	−24.9643	−0.854760	−0.427380	+	0.904072i	$0.640563\pi$
$854$	16.1006	0.550952
$855$	−11.6776	−0.399366
$856$	−10.0387	−0.343115
$857$	−10.4137	−0.355726	−0.177863	−	0.984055i	$-0.556918\pi$
$857$	−10.4137	−0.355726	−0.177863	+	0.984055i	$0.556918\pi$
$858$	−10.5983	−0.361820
$859$	25.0221	0.853743	0.426872	−	0.904312i	$-0.359616\pi$
$859$	25.0221	0.853743	0.426872	+	0.904312i	$0.359616\pi$
$860$	12.6448	0.431184
$861$	−65.5430	−2.23370
$862$	−39.0845	−1.33122
$863$	30.7530	1.04685	0.523423	−	0.852073i	$-0.324655\pi$
$863$	30.7530	1.04685	0.523423	+	0.852073i	$0.324655\pi$
$864$	−10.9494	−0.372506
$865$	1.51479	0.0515045
$866$	24.8409	0.844129
$867$	39.4222	1.33885
$868$	−12.4004	−0.420898
$869$	9.40798	0.319144
$870$	−4.06854	−0.137936
$871$	−13.7973	−0.467504
$872$	20.2167	0.684626
$873$	−41.1111	−1.39140
$874$	−3.67313	−0.124245
$875$	43.7874	1.48028
$876$	44.3255	1.49762
$877$	−9.00643	−0.304126	−0.152063	−	0.988371i	$-0.548592\pi$
$877$	−9.00643	−0.304126	−0.152063	+	0.988371i	$0.548592\pi$
$878$	6.42377	0.216792
$879$	−17.1886	−0.579756
$880$	−2.73930	−0.0923417
$881$	−8.55682	−0.288287	−0.144143	−	0.989557i	$-0.546043\pi$
$881$	−8.55682	−0.288287	−0.144143	+	0.989557i	$0.546043\pi$
$882$	62.0658	2.08986
$883$	−15.3471	−0.516470	−0.258235	−	0.966082i	$-0.583141\pi$
$883$	−15.3471	−0.516470	−0.258235	+	0.966082i	$0.583141\pi$
$884$	−3.34272	−0.112428
$885$	40.1931	1.35108
$886$	31.7437	1.06645
$887$	44.9107	1.50795	0.753977	−	0.656901i	$-0.228133\pi$
$887$	44.9107	1.50795	0.753977	+	0.656901i	$0.228133\pi$
$888$	23.4811	0.787973
$889$	−15.6358	−0.524408
$890$	−2.28670	−0.0766504
$891$	29.9935	1.00482
$892$	−0.204315	−0.00684098
$893$	−15.1753	−0.507822
$894$	10.1291	0.338769
$895$	9.31104	0.311234
$896$	4.06006	0.135637
$897$	13.3837	0.446868
$898$	20.8885	0.697059
$899$	3.10279	0.103484
$900$	−21.7235	−0.724117
$901$	13.0898	0.436085
$902$	11.0419	0.367655
$903$	122.347	4.07146
$904$	−5.84705	−0.194470
$905$	10.4306	0.346723
$906$	21.8988	0.727538
$907$	12.3194	0.409059	0.204529	−	0.978860i	$-0.434434\pi$
$907$	12.3194	0.409059	0.204529	+	0.978860i	$0.434434\pi$
$908$	0.393365	0.0130543
$909$	−20.2978	−0.673235
$910$	−8.54473	−0.283255
$911$	−39.3485	−1.30368	−0.651838	−	0.758358i	$-0.726002\pi$
$911$	−39.3485	−1.30368	−0.651838	+	0.758358i	$0.726002\pi$
$912$	−4.25252	−0.140815
$913$	28.2818	0.935990
$914$	−5.10176	−0.168751
$915$	15.8818	0.525037
$916$	13.1248	0.433655
$917$	43.1972	1.42650
$918$	22.5446	0.744084
$919$	−39.7912	−1.31259	−0.656295	−	0.754505i	$-0.727877\pi$
$919$	−39.7912	−1.31259	−0.656295	+	0.754505i	$0.727877\pi$
$920$	3.45922	0.114047
$921$	61.2190	2.01723
$922$	−11.3630	−0.374220
$923$	−20.7170	−0.681908
$924$	−26.5046	−0.871939
$925$	25.2302	0.829563
$926$	−24.4595	−0.803789
$927$	−99.3999	−3.26472
$928$	−1.01589	−0.0333484
$929$	−20.0133	−0.656616	−0.328308	−	0.944571i	$-0.606478\pi$
$929$	−20.0133	−0.656616	−0.328308	+	0.944571i	$0.606478\pi$
$930$	−12.2319	−0.401100
$931$	13.0548	0.427855
$932$	5.68219	0.186126
$933$	42.6816	1.39733
$934$	2.83365	0.0927197
$935$	5.64017	0.184453
$936$	10.6244	0.347269
$937$	−3.20578	−0.104728	−0.0523642	−	0.998628i	$-0.516676\pi$
$937$	−3.20578	−0.104728	−0.0523642	+	0.998628i	$0.516676\pi$
$938$	−34.5049	−1.12662
$939$	10.8589	0.354368
$940$	14.2916	0.466140
$941$	29.5916	0.964659	0.482330	−	0.875990i	$-0.339791\pi$
$941$	29.5916	0.964659	0.482330	+	0.875990i	$0.339791\pi$
$942$	−7.04339	−0.229486
$943$	−13.9439	−0.454075
$944$	10.0360	0.326645
$945$	57.6291	1.87468
$946$	−20.6116	−0.670140
$947$	−4.28243	−0.139160	−0.0695802	−	0.997576i	$-0.522166\pi$
$947$	−4.28243	−0.139160	−0.0695802	+	0.997576i	$0.522166\pi$
$948$	−13.7546	−0.446728
$949$	−23.2932	−0.756131
$950$	−4.56929	−0.148247
$951$	−24.8356	−0.805348
$952$	−8.35960	−0.270936
$953$	32.7192	1.05988	0.529940	−	0.848035i	$-0.322214\pi$
$953$	32.7192	1.05988	0.529940	+	0.848035i	$0.322214\pi$
$954$	−41.6042	−1.34699
$955$	1.22750	0.0397211
$956$	−6.22273	−0.201257
$957$	6.63190	0.214379
$958$	23.1850	0.749072
$959$	88.0864	2.84446
$960$	4.00488	0.129257
$961$	−21.6716	−0.699083
$962$	−12.3394	−0.397838
$963$	−65.6953	−2.11700
$964$	−0.356819	−0.0114924
$965$	11.5487	0.371767
$966$	33.4704	1.07689
$967$	36.4107	1.17089	0.585444	−	0.810713i	$-0.300920\pi$
$967$	36.4107	1.17089	0.585444	+	0.810713i	$0.300920\pi$
$968$	−6.53482	−0.210037
$969$	8.75589	0.281280
$970$	8.14370	0.261478
$971$	15.4501	0.495817	0.247909	−	0.968783i	$-0.420257\pi$
$971$	15.4501	0.495817	0.247909	+	0.968783i	$0.420257\pi$
$972$	−11.0027	−0.352911
$973$	16.2557	0.521135
$974$	17.6224	0.564659
$975$	16.6490	0.533195
$976$	3.96562	0.126936
$977$	35.9023	1.14862	0.574309	−	0.818639i	$-0.305271\pi$
$977$	35.9023	1.14862	0.574309	+	0.818639i	$0.305271\pi$
$978$	−18.7152	−0.598445
$979$	3.72743	0.119129
$980$	−12.2946	−0.392737
$981$	132.303	4.22410
$982$	11.0642	0.353072
$983$	−57.4177	−1.83134	−0.915670	−	0.401930i	$-0.868340\pi$
$983$	−57.4177	−1.83134	−0.915670	+	0.401930i	$0.868340\pi$
$984$	−16.1434	−0.514632
$985$	19.7899	0.630557
$986$	2.09171	0.0666137
$987$	138.281	4.40154
$988$	2.23472	0.0710959
$989$	26.0286	0.827662
$990$	−17.9266	−0.569743
$991$	−45.0499	−1.43106	−0.715529	−	0.698583i	$-0.753814\pi$
$991$	−45.0499	−1.43106	−0.715529	+	0.698583i	$0.753814\pi$
$992$	−3.05425	−0.0969725
$993$	−104.469	−3.31524
$994$	−51.8099	−1.64331
$995$	−17.0784	−0.541421
$996$	−41.3483	−1.31017
$997$	42.1523	1.33498	0.667488	−	0.744621i	$-0.267370\pi$
$997$	42.1523	1.33498	0.667488	+	0.744621i	$0.267370\pi$
$998$	−0.501950	−0.0158890
$999$	83.2220	2.63303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.2	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.2	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.08937 q^{3} +1.00000 q^{4} -1.29634 q^{5} -3.08937 q^{6} +4.06006 q^{7} +1.00000 q^{8} +6.54422 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.08937 q^{3} +1.00000 q^{4} -1.29634 q^{5} -3.08937 q^{6} +4.06006 q^{7} +1.00000 q^{8} +6.54422 q^{9} -1.29634 q^{10} +2.11310 q^{11} -3.08937 q^{12} +1.62348 q^{13} +4.06006 q^{14} +4.00488 q^{15} +1.00000 q^{16} -2.05899 q^{17} +6.54422 q^{18} +1.37650 q^{19} -1.29634 q^{20} -12.5430 q^{21} +2.11310 q^{22} -2.66845 q^{23} -3.08937 q^{24} -3.31950 q^{25} +1.62348 q^{26} -10.9494 q^{27} +4.06006 q^{28} -1.01589 q^{29} +4.00488 q^{30} -3.05425 q^{31} +1.00000 q^{32} -6.52814 q^{33} -2.05899 q^{34} -5.26323 q^{35} +6.54422 q^{36} -7.60060 q^{37} +1.37650 q^{38} -5.01553 q^{39} -1.29634 q^{40} +5.22545 q^{41} -12.5430 q^{42} -9.75420 q^{43} +2.11310 q^{44} -8.48354 q^{45} -2.66845 q^{46} -11.0245 q^{47} -3.08937 q^{48} +9.48407 q^{49} -3.31950 q^{50} +6.36097 q^{51} +1.62348 q^{52} -6.35741 q^{53} -10.9494 q^{54} -2.73930 q^{55} +4.06006 q^{56} -4.25252 q^{57} -1.01589 q^{58} +10.0360 q^{59} +4.00488 q^{60} +3.96562 q^{61} -3.05425 q^{62} +26.5699 q^{63} +1.00000 q^{64} -2.10458 q^{65} -6.52814 q^{66} -8.49862 q^{67} -2.05899 q^{68} +8.24383 q^{69} -5.26323 q^{70} -12.7609 q^{71} +6.54422 q^{72} -14.3477 q^{73} -7.60060 q^{74} +10.2552 q^{75} +1.37650 q^{76} +8.57930 q^{77} -5.01553 q^{78} +4.45222 q^{79} -1.29634 q^{80} +14.1941 q^{81} +5.22545 q^{82} +13.3840 q^{83} -12.5430 q^{84} +2.66915 q^{85} -9.75420 q^{86} +3.13847 q^{87} +2.11310 q^{88} +1.76396 q^{89} -8.48354 q^{90} +6.59141 q^{91} -2.66845 q^{92} +9.43570 q^{93} -11.0245 q^{94} -1.78442 q^{95} -3.08937 q^{96} -6.28206 q^{97} +9.48407 q^{98} +13.8286 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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