LMFDB - Embedded newform 6038.2.a.d.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.58255 q^{3} +1.00000 q^{4} -3.72234 q^{5} +1.58255 q^{6} -1.18949 q^{7} -1.00000 q^{8} -0.495522 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.58255 q^{3} +1.00000 q^{4} -3.72234 q^{5} +1.58255 q^{6} -1.18949 q^{7} -1.00000 q^{8} -0.495522 q^{9} +3.72234 q^{10} -1.91466 q^{11} -1.58255 q^{12} +6.23531 q^{13} +1.18949 q^{14} +5.89081 q^{15} +1.00000 q^{16} -3.63211 q^{17} +0.495522 q^{18} -0.859459 q^{19} -3.72234 q^{20} +1.88243 q^{21} +1.91466 q^{22} -1.23295 q^{23} +1.58255 q^{24} +8.85585 q^{25} -6.23531 q^{26} +5.53185 q^{27} -1.18949 q^{28} +0.548342 q^{29} -5.89081 q^{30} -1.85637 q^{31} -1.00000 q^{32} +3.03005 q^{33} +3.63211 q^{34} +4.42768 q^{35} -0.495522 q^{36} -8.13928 q^{37} +0.859459 q^{38} -9.86772 q^{39} +3.72234 q^{40} +4.04899 q^{41} -1.88243 q^{42} -4.58447 q^{43} -1.91466 q^{44} +1.84450 q^{45} +1.23295 q^{46} -5.93418 q^{47} -1.58255 q^{48} -5.58512 q^{49} -8.85585 q^{50} +5.74801 q^{51} +6.23531 q^{52} -4.81228 q^{53} -5.53185 q^{54} +7.12701 q^{55} +1.18949 q^{56} +1.36014 q^{57} -0.548342 q^{58} -14.1120 q^{59} +5.89081 q^{60} +3.31907 q^{61} +1.85637 q^{62} +0.589418 q^{63} +1.00000 q^{64} -23.2100 q^{65} -3.03005 q^{66} -0.743541 q^{67} -3.63211 q^{68} +1.95121 q^{69} -4.42768 q^{70} -3.46188 q^{71} +0.495522 q^{72} -2.17574 q^{73} +8.13928 q^{74} -14.0149 q^{75} -0.859459 q^{76} +2.27746 q^{77} +9.86772 q^{78} -3.04863 q^{79} -3.72234 q^{80} -7.26789 q^{81} -4.04899 q^{82} -5.30972 q^{83} +1.88243 q^{84} +13.5200 q^{85} +4.58447 q^{86} -0.867780 q^{87} +1.91466 q^{88} +0.338672 q^{89} -1.84450 q^{90} -7.41683 q^{91} -1.23295 q^{92} +2.93781 q^{93} +5.93418 q^{94} +3.19920 q^{95} +1.58255 q^{96} -1.62639 q^{97} +5.58512 q^{98} +0.948754 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * 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
$3$	−1.58255	−0.913688	−0.456844	−	0.889547i	$-0.651020\pi$
$3$	−1.58255	−0.913688	−0.456844	+	0.889547i	$0.651020\pi$
$4$	1.00000	0.500000
$5$	−3.72234	−1.66468	−0.832342	−	0.554263i	$-0.813000\pi$
$5$	−3.72234	−1.66468	−0.832342	+	0.554263i	$0.813000\pi$
$6$	1.58255	0.646075
$7$	−1.18949	−0.449584	−0.224792	−	0.974407i	$-0.572170\pi$
$7$	−1.18949	−0.449584	−0.224792	+	0.974407i	$0.572170\pi$
$8$	−1.00000	−0.353553
$9$	−0.495522	−0.165174
$10$	3.72234	1.17711
$11$	−1.91466	−0.577290	−0.288645	−	0.957436i	$-0.593205\pi$
$11$	−1.91466	−0.577290	−0.288645	+	0.957436i	$0.593205\pi$
$12$	−1.58255	−0.456844
$13$	6.23531	1.72937	0.864683	−	0.502319i	$-0.167520\pi$
$13$	6.23531	1.72937	0.864683	+	0.502319i	$0.167520\pi$
$14$	1.18949	0.317904
$15$	5.89081	1.52100
$16$	1.00000	0.250000
$17$	−3.63211	−0.880915	−0.440458	−	0.897773i	$-0.645184\pi$
$17$	−3.63211	−0.880915	−0.440458	+	0.897773i	$0.645184\pi$
$18$	0.495522	0.116796
$19$	−0.859459	−0.197173	−0.0985867	−	0.995128i	$-0.531432\pi$
$19$	−0.859459	−0.197173	−0.0985867	+	0.995128i	$0.531432\pi$
$20$	−3.72234	−0.832342
$21$	1.88243	0.410780
$22$	1.91466	0.408206
$23$	−1.23295	−0.257088	−0.128544	−	0.991704i	$-0.541030\pi$
$23$	−1.23295	−0.257088	−0.128544	+	0.991704i	$0.541030\pi$
$24$	1.58255	0.323038
$25$	8.85585	1.77117
$26$	−6.23531	−1.22285
$27$	5.53185	1.06461
$28$	−1.18949	−0.224792
$29$	0.548342	0.101824	0.0509122	−	0.998703i	$-0.483787\pi$
$29$	0.548342	0.101824	0.0509122	+	0.998703i	$0.483787\pi$
$30$	−5.89081	−1.07551
$31$	−1.85637	−0.333414	−0.166707	−	0.986006i	$-0.553313\pi$
$31$	−1.85637	−0.333414	−0.166707	+	0.986006i	$0.553313\pi$
$32$	−1.00000	−0.176777
$33$	3.03005	0.527463
$34$	3.63211	0.622901
$35$	4.42768	0.748415
$36$	−0.495522	−0.0825870
$37$	−8.13928	−1.33809	−0.669045	−	0.743222i	$-0.733296\pi$
$37$	−8.13928	−1.33809	−0.669045	+	0.743222i	$0.733296\pi$
$38$	0.859459	0.139423
$39$	−9.86772	−1.58010
$40$	3.72234	0.588554
$41$	4.04899	0.632346	0.316173	−	0.948702i	$-0.397602\pi$
$41$	4.04899	0.632346	0.316173	+	0.948702i	$0.397602\pi$
$42$	−1.88243	−0.290465
$43$	−4.58447	−0.699125	−0.349563	−	0.936913i	$-0.613670\pi$
$43$	−4.58447	−0.699125	−0.349563	+	0.936913i	$0.613670\pi$
$44$	−1.91466	−0.288645
$45$	1.84450	0.274962
$46$	1.23295	0.181789
$47$	−5.93418	−0.865590	−0.432795	−	0.901492i	$-0.642473\pi$
$47$	−5.93418	−0.865590	−0.432795	+	0.901492i	$0.642473\pi$
$48$	−1.58255	−0.228422
$49$	−5.58512	−0.797874
$50$	−8.85585	−1.25241
$51$	5.74801	0.804882
$52$	6.23531	0.864683
$53$	−4.81228	−0.661018	−0.330509	−	0.943803i	$-0.607220\pi$
$53$	−4.81228	−0.661018	−0.330509	+	0.943803i	$0.607220\pi$
$54$	−5.53185	−0.752790
$55$	7.12701	0.961006
$56$	1.18949	0.158952
$57$	1.36014	0.180155
$58$	−0.548342	−0.0720008
$59$	−14.1120	−1.83723	−0.918616	−	0.395151i	$-0.870692\pi$
$59$	−14.1120	−1.83723	−0.918616	+	0.395151i	$0.870692\pi$
$60$	5.89081	0.760501
$61$	3.31907	0.424964	0.212482	−	0.977165i	$-0.431845\pi$
$61$	3.31907	0.424964	0.212482	+	0.977165i	$0.431845\pi$
$62$	1.85637	0.235759
$63$	0.589418	0.0742596
$64$	1.00000	0.125000
$65$	−23.2100	−2.87885
$66$	−3.03005	−0.372973
$67$	−0.743541	−0.0908380	−0.0454190	−	0.998968i	$-0.514462\pi$
$67$	−0.743541	−0.0908380	−0.0454190	+	0.998968i	$0.514462\pi$
$68$	−3.63211	−0.440458
$69$	1.95121	0.234898
$70$	−4.42768	−0.529209
$71$	−3.46188	−0.410850	−0.205425	−	0.978673i	$-0.565858\pi$
$71$	−3.46188	−0.410850	−0.205425	+	0.978673i	$0.565858\pi$
$72$	0.495522	0.0583978
$73$	−2.17574	−0.254652	−0.127326	−	0.991861i	$-0.540639\pi$
$73$	−2.17574	−0.254652	−0.127326	+	0.991861i	$0.540639\pi$
$74$	8.13928	0.946172
$75$	−14.0149	−1.61830
$76$	−0.859459	−0.0985867
$77$	2.27746	0.259541
$78$	9.86772	1.11730
$79$	−3.04863	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$79$	−3.04863	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$80$	−3.72234	−0.416171
$81$	−7.26789	−0.807543
$82$	−4.04899	−0.447136
$83$	−5.30972	−0.582817	−0.291409	−	0.956599i	$-0.594124\pi$
$83$	−5.30972	−0.582817	−0.291409	+	0.956599i	$0.594124\pi$
$84$	1.88243	0.205390
$85$	13.5200	1.46644
$86$	4.58447	0.494356
$87$	−0.867780	−0.0930358
$88$	1.91466	0.204103
$89$	0.338672	0.0358992	0.0179496	−	0.999839i	$-0.494286\pi$
$89$	0.338672	0.0358992	0.0179496	+	0.999839i	$0.494286\pi$
$90$	−1.84450	−0.194428
$91$	−7.41683	−0.777495
$92$	−1.23295	−0.128544
$93$	2.93781	0.304636
$94$	5.93418	0.612064
$95$	3.19920	0.328231
$96$	1.58255	0.161519
$97$	−1.62639	−0.165135	−0.0825676	−	0.996585i	$-0.526312\pi$
$97$	−1.62639	−0.165135	−0.0825676	+	0.996585i	$0.526312\pi$
$98$	5.58512	0.564182
$99$	0.948754	0.0953534
$100$	8.85585	0.885585
$101$	−14.9948	−1.49203	−0.746017	−	0.665927i	$-0.768036\pi$
$101$	−14.9948	−1.49203	−0.746017	+	0.665927i	$0.768036\pi$
$102$	−5.74801	−0.569137
$103$	9.26461	0.912869	0.456434	−	0.889757i	$-0.349126\pi$
$103$	9.26461	0.912869	0.456434	+	0.889757i	$0.349126\pi$
$104$	−6.23531	−0.611423
$105$	−7.00705	−0.683818
$106$	4.81228	0.467410
$107$	−7.19902	−0.695956	−0.347978	−	0.937503i	$-0.613132\pi$
$107$	−7.19902	−0.695956	−0.347978	+	0.937503i	$0.613132\pi$
$108$	5.53185	0.532303
$109$	−2.23163	−0.213751	−0.106875	−	0.994272i	$-0.534085\pi$
$109$	−2.23163	−0.213751	−0.106875	+	0.994272i	$0.534085\pi$
$110$	−7.12701	−0.679534
$111$	12.8808	1.22260
$112$	−1.18949	−0.112396
$113$	5.64316	0.530864	0.265432	−	0.964130i	$-0.414485\pi$
$113$	5.64316	0.530864	0.265432	+	0.964130i	$0.414485\pi$
$114$	−1.36014	−0.127389
$115$	4.58947	0.427970
$116$	0.548342	0.0509122
$117$	−3.08974	−0.285646
$118$	14.1120	1.29912
$119$	4.32035	0.396045
$120$	−5.89081	−0.537755
$121$	−7.33409	−0.666736
$122$	−3.31907	−0.300495
$123$	−6.40775	−0.577767
$124$	−1.85637	−0.166707
$125$	−14.3528	−1.28375
$126$	−0.589418	−0.0525095
$127$	−10.4571	−0.927914	−0.463957	−	0.885858i	$-0.653571\pi$
$127$	−10.4571	−0.927914	−0.463957	+	0.885858i	$0.653571\pi$
$128$	−1.00000	−0.0883883
$129$	7.25517	0.638782
$130$	23.2100	2.03565
$131$	−13.0291	−1.13835	−0.569177	−	0.822215i	$-0.692738\pi$
$131$	−13.0291	−1.13835	−0.569177	+	0.822215i	$0.692738\pi$
$132$	3.03005	0.263732
$133$	1.02232	0.0886460
$134$	0.743541	0.0642322
$135$	−20.5915	−1.77223
$136$	3.63211	0.311451
$137$	−4.24174	−0.362396	−0.181198	−	0.983447i	$-0.557997\pi$
$137$	−4.24174	−0.362396	−0.181198	+	0.983447i	$0.557997\pi$
$138$	−1.95121	−0.166098
$139$	−0.521028	−0.0441930	−0.0220965	−	0.999756i	$-0.507034\pi$
$139$	−0.521028	−0.0441930	−0.0220965	+	0.999756i	$0.507034\pi$
$140$	4.42768	0.374208
$141$	9.39117	0.790879
$142$	3.46188	0.290515
$143$	−11.9385	−0.998346
$144$	−0.495522	−0.0412935
$145$	−2.04112	−0.169506
$146$	2.17574	0.180066
$147$	8.83875	0.729008
$148$	−8.13928	−0.669045
$149$	20.1874	1.65382	0.826910	−	0.562335i	$-0.190097\pi$
$149$	20.1874	1.65382	0.826910	+	0.562335i	$0.190097\pi$
$150$	14.0149	1.14431
$151$	−6.48866	−0.528040	−0.264020	−	0.964517i	$-0.585048\pi$
$151$	−6.48866	−0.528040	−0.264020	+	0.964517i	$0.585048\pi$
$152$	0.859459	0.0697113
$153$	1.79979	0.145504
$154$	−2.27746	−0.183523
$155$	6.91005	0.555028
$156$	−9.86772	−0.790050
$157$	−12.7537	−1.01786	−0.508928	−	0.860809i	$-0.669958\pi$
$157$	−12.7537	−1.01786	−0.508928	+	0.860809i	$0.669958\pi$
$158$	3.04863	0.242536
$159$	7.61570	0.603964
$160$	3.72234	0.294277
$161$	1.46658	0.115583
$162$	7.26789	0.571019
$163$	8.95061	0.701066	0.350533	−	0.936550i	$-0.386001\pi$
$163$	8.95061	0.701066	0.350533	+	0.936550i	$0.386001\pi$
$164$	4.04899	0.316173
$165$	−11.2789	−0.878059
$166$	5.30972	0.412114
$167$	−9.87995	−0.764534	−0.382267	−	0.924052i	$-0.624856\pi$
$167$	−9.87995	−0.764534	−0.382267	+	0.924052i	$0.624856\pi$
$168$	−1.88243	−0.145233
$169$	25.8792	1.99070
$170$	−13.5200	−1.03693
$171$	0.425881	0.0325679
$172$	−4.58447	−0.349563
$173$	−10.4601	−0.795269	−0.397634	−	0.917544i	$-0.630169\pi$
$173$	−10.4601	−0.795269	−0.397634	+	0.917544i	$0.630169\pi$
$174$	0.867780	0.0657863
$175$	−10.5339	−0.796290
$176$	−1.91466	−0.144323
$177$	22.3331	1.67866
$178$	−0.338672	−0.0253845
$179$	17.0888	1.27727	0.638637	−	0.769508i	$-0.279499\pi$
$179$	17.0888	1.27727	0.638637	+	0.769508i	$0.279499\pi$
$180$	1.84450	0.137481
$181$	−14.7622	−1.09727	−0.548635	−	0.836062i	$-0.684852\pi$
$181$	−14.7622	−1.09727	−0.548635	+	0.836062i	$0.684852\pi$
$182$	7.41683	0.549772
$183$	−5.25261	−0.388284
$184$	1.23295	0.0908943
$185$	30.2972	2.22749
$186$	−2.93781	−0.215410
$187$	6.95423	0.508544
$188$	−5.93418	−0.432795
$189$	−6.58007	−0.478630
$190$	−3.19920	−0.232095
$191$	−19.1531	−1.38587	−0.692935	−	0.721000i	$-0.743683\pi$
$191$	−19.1531	−1.38587	−0.692935	+	0.721000i	$0.743683\pi$
$192$	−1.58255	−0.114211
$193$	−2.27302	−0.163615	−0.0818077	−	0.996648i	$-0.526069\pi$
$193$	−2.27302	−0.163615	−0.0818077	+	0.996648i	$0.526069\pi$
$194$	1.62639	0.116768
$195$	36.7311	2.63037
$196$	−5.58512	−0.398937
$197$	24.9497	1.77759	0.888795	−	0.458306i	$-0.151544\pi$
$197$	24.9497	1.77759	0.888795	+	0.458306i	$0.151544\pi$
$198$	−0.948754	−0.0674250
$199$	8.23209	0.583557	0.291779	−	0.956486i	$-0.405753\pi$
$199$	8.23209	0.583557	0.291779	+	0.956486i	$0.405753\pi$
$200$	−8.85585	−0.626203
$201$	1.17669	0.0829976
$202$	14.9948	1.05503
$203$	−0.652246	−0.0457787
$204$	5.74801	0.402441
$205$	−15.0717	−1.05266
$206$	−9.26461	−0.645496
$207$	0.610954	0.0424643
$208$	6.23531	0.432341
$209$	1.64557	0.113826
$210$	7.00705	0.483532
$211$	14.6009	1.00516	0.502582	−	0.864529i	$-0.332383\pi$
$211$	14.6009	1.00516	0.502582	+	0.864529i	$0.332383\pi$
$212$	−4.81228	−0.330509
$213$	5.47862	0.375389
$214$	7.19902	0.492115
$215$	17.0650	1.16382
$216$	−5.53185	−0.376395
$217$	2.20813	0.149898
$218$	2.23163	0.151145
$219$	3.44323	0.232672
$220$	7.12701	0.480503
$221$	−22.6473	−1.52342
$222$	−12.8808	−0.864506
$223$	−15.6811	−1.05008	−0.525040	−	0.851077i	$-0.675950\pi$
$223$	−15.6811	−1.05008	−0.525040	+	0.851077i	$0.675950\pi$
$224$	1.18949	0.0794760
$225$	−4.38827	−0.292551
$226$	−5.64316	−0.375378
$227$	−5.26107	−0.349190	−0.174595	−	0.984640i	$-0.555862\pi$
$227$	−5.26107	−0.349190	−0.174595	+	0.984640i	$0.555862\pi$
$228$	1.36014	0.0900775
$229$	1.85948	0.122878	0.0614390	−	0.998111i	$-0.480431\pi$
$229$	1.85948	0.122878	0.0614390	+	0.998111i	$0.480431\pi$
$230$	−4.58947	−0.302620
$231$	−3.60420	−0.237139
$232$	−0.548342	−0.0360004
$233$	−13.1315	−0.860276	−0.430138	−	0.902763i	$-0.641535\pi$
$233$	−13.1315	−0.860276	−0.430138	+	0.902763i	$0.641535\pi$
$234$	3.08974	0.201982
$235$	22.0891	1.44093
$236$	−14.1120	−0.918616
$237$	4.82461	0.313392
$238$	−4.32035	−0.280046
$239$	25.8536	1.67233	0.836166	−	0.548477i	$-0.184792\pi$
$239$	25.8536	1.67233	0.836166	+	0.548477i	$0.184792\pi$
$240$	5.89081	0.380250
$241$	12.3439	0.795138	0.397569	−	0.917572i	$-0.369854\pi$
$241$	12.3439	0.795138	0.397569	+	0.917572i	$0.369854\pi$
$242$	7.33409	0.471453
$243$	−5.09373	−0.326763
$244$	3.31907	0.212482
$245$	20.7897	1.32821
$246$	6.40775	0.408543
$247$	−5.35900	−0.340985
$248$	1.85637	0.117880
$249$	8.40292	0.532513
$250$	14.3528	0.907751
$251$	−9.59188	−0.605434	−0.302717	−	0.953080i	$-0.597894\pi$
$251$	−9.59188	−0.605434	−0.302717	+	0.953080i	$0.597894\pi$
$252$	0.589418	0.0371298
$253$	2.36068	0.148414
$254$	10.4571	0.656135
$255$	−21.3961	−1.33987
$256$	1.00000	0.0625000
$257$	−5.49006	−0.342461	−0.171230	−	0.985231i	$-0.554774\pi$
$257$	−5.49006	−0.342461	−0.171230	+	0.985231i	$0.554774\pi$
$258$	−7.25517	−0.451687
$259$	9.68157	0.601584
$260$	−23.2100	−1.43942
$261$	−0.271715	−0.0168188
$262$	13.0291	0.804938
$263$	−11.5067	−0.709536	−0.354768	−	0.934954i	$-0.615440\pi$
$263$	−11.5067	−0.709536	−0.354768	+	0.934954i	$0.615440\pi$
$264$	−3.03005	−0.186486
$265$	17.9130	1.10039
$266$	−1.02232	−0.0626822
$267$	−0.535967	−0.0328006
$268$	−0.743541	−0.0454190
$269$	−23.7062	−1.44539	−0.722697	−	0.691165i	$-0.757098\pi$
$269$	−23.7062	−1.44539	−0.722697	+	0.691165i	$0.757098\pi$
$270$	20.5915	1.25316
$271$	4.08268	0.248005	0.124003	−	0.992282i	$-0.460427\pi$
$271$	4.08268	0.248005	0.124003	+	0.992282i	$0.460427\pi$
$272$	−3.63211	−0.220229
$273$	11.7375	0.710388
$274$	4.24174	0.256253
$275$	−16.9559	−1.02248
$276$	1.95121	0.117449
$277$	30.9596	1.86018	0.930090	−	0.367331i	$-0.119728\pi$
$277$	30.9596	1.86018	0.930090	+	0.367331i	$0.119728\pi$
$278$	0.521028	0.0312492
$279$	0.919872	0.0550713
$280$	−4.42768	−0.264605
$281$	−10.1232	−0.603897	−0.301948	−	0.953324i	$-0.597637\pi$
$281$	−10.1232	−0.603897	−0.301948	+	0.953324i	$0.597637\pi$
$282$	−9.39117	−0.559236
$283$	27.5821	1.63959	0.819794	−	0.572659i	$-0.194088\pi$
$283$	27.5821	1.63959	0.819794	+	0.572659i	$0.194088\pi$
$284$	−3.46188	−0.205425
$285$	−5.06291	−0.299901
$286$	11.9385	0.705937
$287$	−4.81622	−0.284293
$288$	0.495522	0.0291989
$289$	−3.80780	−0.223988
$290$	2.04112	0.119859
$291$	2.57386	0.150882
$292$	−2.17574	−0.127326
$293$	23.2008	1.35540	0.677701	−	0.735337i	$-0.262976\pi$
$293$	23.2008	1.35540	0.677701	+	0.735337i	$0.262976\pi$
$294$	−8.83875	−0.515487
$295$	52.5299	3.05841
$296$	8.13928	0.473086
$297$	−10.5916	−0.614587
$298$	−20.1874	−1.16943
$299$	−7.68783	−0.444599
$300$	−14.0149	−0.809149
$301$	5.45317	0.314316
$302$	6.48866	0.373381
$303$	23.7300	1.36325
$304$	−0.859459	−0.0492934
$305$	−12.3547	−0.707430
$306$	−1.79979	−0.102887
$307$	−3.73575	−0.213211	−0.106605	−	0.994301i	$-0.533998\pi$
$307$	−3.73575	−0.213211	−0.106605	+	0.994301i	$0.533998\pi$
$308$	2.27746	0.129770
$309$	−14.6617	−0.834077
$310$	−6.91005	−0.392464
$311$	4.34034	0.246118	0.123059	−	0.992399i	$-0.460730\pi$
$311$	4.34034	0.246118	0.123059	+	0.992399i	$0.460730\pi$
$312$	9.86772	0.558650
$313$	30.0712	1.69972	0.849862	−	0.527006i	$-0.176685\pi$
$313$	30.0712	1.69972	0.849862	+	0.527006i	$0.176685\pi$
$314$	12.7537	0.719734
$315$	−2.19402	−0.123619
$316$	−3.04863	−0.171499
$317$	4.82810	0.271173	0.135587	−	0.990766i	$-0.456708\pi$
$317$	4.82810	0.271173	0.135587	+	0.990766i	$0.456708\pi$
$318$	−7.61570	−0.427067
$319$	−1.04989	−0.0587823
$320$	−3.72234	−0.208085
$321$	11.3928	0.635886
$322$	−1.46658	−0.0817293
$323$	3.12165	0.173693
$324$	−7.26789	−0.403772
$325$	55.2190	3.06300
$326$	−8.95061	−0.495728
$327$	3.53167	0.195302
$328$	−4.04899	−0.223568
$329$	7.05864	0.389155
$330$	11.2789	0.620882
$331$	14.0657	0.773119	0.386559	−	0.922265i	$-0.373663\pi$
$331$	14.0657	0.773119	0.386559	+	0.922265i	$0.373663\pi$
$332$	−5.30972	−0.291409
$333$	4.03319	0.221018
$334$	9.87995	0.540607
$335$	2.76772	0.151217
$336$	1.88243	0.102695
$337$	−17.3269	−0.943855	−0.471927	−	0.881637i	$-0.656442\pi$
$337$	−17.3269	−0.943855	−0.471927	+	0.881637i	$0.656442\pi$
$338$	−25.8792	−1.40764
$339$	−8.93061	−0.485044
$340$	13.5200	0.733222
$341$	3.55431	0.192477
$342$	−0.425881	−0.0230290
$343$	14.9698	0.808296
$344$	4.58447	0.247178
$345$	−7.26308	−0.391031
$346$	10.4601	0.562340
$347$	−21.0947	−1.13243	−0.566213	−	0.824259i	$-0.691592\pi$
$347$	−21.0947	−1.13243	−0.566213	+	0.824259i	$0.691592\pi$
$348$	−0.867780	−0.0465179
$349$	−22.5311	−1.20606	−0.603031	−	0.797718i	$-0.706041\pi$
$349$	−22.5311	−1.20606	−0.603031	+	0.797718i	$0.706041\pi$
$350$	10.5339	0.563062
$351$	34.4928	1.84109
$352$	1.91466	0.102052
$353$	23.3357	1.24203	0.621017	−	0.783797i	$-0.286720\pi$
$353$	23.3357	1.24203	0.621017	+	0.783797i	$0.286720\pi$
$354$	−22.3331	−1.18699
$355$	12.8863	0.683935
$356$	0.338672	0.0179496
$357$	−6.83718	−0.361862
$358$	−17.0888	−0.903169
$359$	−30.3387	−1.60121	−0.800607	−	0.599190i	$-0.795489\pi$
$359$	−30.3387	−1.60121	−0.800607	+	0.599190i	$0.795489\pi$
$360$	−1.84450	−0.0972139
$361$	−18.2613	−0.961123
$362$	14.7622	0.775887
$363$	11.6066	0.609188
$364$	−7.41683	−0.388748
$365$	8.09887	0.423914
$366$	5.25261	0.274558
$367$	−1.18981	−0.0621075	−0.0310537	−	0.999518i	$-0.509886\pi$
$367$	−1.18981	−0.0621075	−0.0310537	+	0.999518i	$0.509886\pi$
$368$	−1.23295	−0.0642720
$369$	−2.00636	−0.104447
$370$	−30.2972	−1.57508
$371$	5.72415	0.297183
$372$	2.93781	0.152318
$373$	2.89681	0.149991	0.0749955	−	0.997184i	$-0.476106\pi$
$373$	2.89681	0.149991	0.0749955	+	0.997184i	$0.476106\pi$
$374$	−6.95423	−0.359595
$375$	22.7141	1.17295
$376$	5.93418	0.306032
$377$	3.41908	0.176092
$378$	6.58007	0.338442
$379$	−3.20154	−0.164452	−0.0822260	−	0.996614i	$-0.526203\pi$
$379$	−3.20154	−0.164452	−0.0822260	+	0.996614i	$0.526203\pi$
$380$	3.19920	0.164116
$381$	16.5489	0.847824
$382$	19.1531	0.979958
$383$	−10.6798	−0.545710	−0.272855	−	0.962055i	$-0.587968\pi$
$383$	−10.6798	−0.545710	−0.272855	+	0.962055i	$0.587968\pi$
$384$	1.58255	0.0807594
$385$	−8.47749	−0.432053
$386$	2.27302	0.115694
$387$	2.27171	0.115477
$388$	−1.62639	−0.0825676
$389$	−32.3732	−1.64139	−0.820693	−	0.571369i	$-0.806412\pi$
$389$	−32.3732	−1.64139	−0.820693	+	0.571369i	$0.806412\pi$
$390$	−36.7311	−1.85995
$391$	4.47821	0.226473
$392$	5.58512	0.282091
$393$	20.6192	1.04010
$394$	−24.9497	−1.25695
$395$	11.3480	0.570982
$396$	0.948754	0.0476767
$397$	−18.0958	−0.908202	−0.454101	−	0.890950i	$-0.650040\pi$
$397$	−18.0958	−0.908202	−0.454101	+	0.890950i	$0.650040\pi$
$398$	−8.23209	−0.412637
$399$	−1.61787	−0.0809948
$400$	8.85585	0.442793
$401$	10.0364	0.501193	0.250596	−	0.968092i	$-0.419373\pi$
$401$	10.0364	0.501193	0.250596	+	0.968092i	$0.419373\pi$
$402$	−1.17669	−0.0586882
$403$	−11.5750	−0.576594
$404$	−14.9948	−0.746017
$405$	27.0536	1.34430
$406$	0.652246	0.0323704
$407$	15.5839	0.772466
$408$	−5.74801	−0.284569
$409$	−23.5160	−1.16279	−0.581396	−	0.813621i	$-0.697493\pi$
$409$	−23.5160	−1.16279	−0.581396	+	0.813621i	$0.697493\pi$
$410$	15.0717	0.744340
$411$	6.71278	0.331117
$412$	9.26461	0.456434
$413$	16.7861	0.825990
$414$	−0.610954	−0.0300268
$415$	19.7646	0.970206
$416$	−6.23531	−0.305711
$417$	0.824555	0.0403786
$418$	−1.64557	−0.0804874
$419$	−0.398234	−0.0194550	−0.00972749	−	0.999953i	$-0.503096\pi$
$419$	−0.398234	−0.0194550	−0.00972749	+	0.999953i	$0.503096\pi$
$420$	−7.00705	−0.341909
$421$	7.77374	0.378869	0.189434	−	0.981893i	$-0.439335\pi$
$421$	7.77374	0.378869	0.189434	+	0.981893i	$0.439335\pi$
$422$	−14.6009	−0.710759
$423$	2.94052	0.142973
$424$	4.81228	0.233705
$425$	−32.1654	−1.56025
$426$	−5.47862	−0.265440
$427$	−3.94800	−0.191057
$428$	−7.19902	−0.347978
$429$	18.8933	0.912177
$430$	−17.0650	−0.822947
$431$	−17.1754	−0.827309	−0.413654	−	0.910434i	$-0.635748\pi$
$431$	−17.1754	−0.827309	−0.413654	+	0.910434i	$0.635748\pi$
$432$	5.53185	0.266151
$433$	13.3710	0.642567	0.321283	−	0.946983i	$-0.395886\pi$
$433$	13.3710	0.642567	0.321283	+	0.946983i	$0.395886\pi$
$434$	−2.20813	−0.105994
$435$	3.23018	0.154875
$436$	−2.23163	−0.106875
$437$	1.05967	0.0506909
$438$	−3.44323	−0.164524
$439$	−28.4069	−1.35579	−0.677894	−	0.735160i	$-0.737107\pi$
$439$	−28.4069	−1.35579	−0.677894	+	0.735160i	$0.737107\pi$
$440$	−7.12701	−0.339767
$441$	2.76755	0.131788
$442$	22.6473	1.07722
$443$	7.88929	0.374832	0.187416	−	0.982281i	$-0.439989\pi$
$443$	7.88929	0.374832	0.187416	+	0.982281i	$0.439989\pi$
$444$	12.8808	0.611298
$445$	−1.26065	−0.0597607
$446$	15.6811	0.742519
$447$	−31.9477	−1.51108
$448$	−1.18949	−0.0561980
$449$	9.95348	0.469734	0.234867	−	0.972028i	$-0.424535\pi$
$449$	9.95348	0.469734	0.234867	+	0.972028i	$0.424535\pi$
$450$	4.38827	0.206865
$451$	−7.75242	−0.365047
$452$	5.64316	0.265432
$453$	10.2687	0.482464
$454$	5.26107	0.246914
$455$	27.6080	1.29428
$456$	−1.36014	−0.0636944
$457$	6.28517	0.294008	0.147004	−	0.989136i	$-0.453037\pi$
$457$	6.28517	0.294008	0.147004	+	0.989136i	$0.453037\pi$
$458$	−1.85948	−0.0868879
$459$	−20.0923	−0.937827
$460$	4.58947	0.213985
$461$	10.2891	0.479212	0.239606	−	0.970870i	$-0.422982\pi$
$461$	10.2891	0.479212	0.239606	+	0.970870i	$0.422982\pi$
$462$	3.60420	0.167683
$463$	−20.5322	−0.954212	−0.477106	−	0.878846i	$-0.658314\pi$
$463$	−20.5322	−0.954212	−0.477106	+	0.878846i	$0.658314\pi$
$464$	0.548342	0.0254561
$465$	−10.9355	−0.507123
$466$	13.1315	0.608307
$467$	−25.7547	−1.19178	−0.595892	−	0.803064i	$-0.703201\pi$
$467$	−25.7547	−1.19178	−0.595892	+	0.803064i	$0.703201\pi$
$468$	−3.08974	−0.142823
$469$	0.884433	0.0408393
$470$	−22.0891	−1.01889
$471$	20.1834	0.930004
$472$	14.1120	0.649560
$473$	8.77768	0.403598
$474$	−4.82461	−0.221602
$475$	−7.61124	−0.349228
$476$	4.32035	0.198023
$477$	2.38459	0.109183
$478$	−25.8536	−1.18252
$479$	5.14750	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$479$	5.14750	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$480$	−5.89081	−0.268878
$481$	−50.7510	−2.31404
$482$	−12.3439	−0.562248
$483$	−2.32094	−0.105606
$484$	−7.33409	−0.333368
$485$	6.05400	0.274898
$486$	5.09373	0.231056
$487$	5.88744	0.266785	0.133393	−	0.991063i	$-0.457413\pi$
$487$	5.88744	0.266785	0.133393	+	0.991063i	$0.457413\pi$
$488$	−3.31907	−0.150247
$489$	−14.1648	−0.640555
$490$	−20.7897	−0.939185
$491$	−15.9104	−0.718024	−0.359012	−	0.933333i	$-0.616886\pi$
$491$	−15.9104	−0.718024	−0.359012	+	0.933333i	$0.616886\pi$
$492$	−6.40775	−0.288884
$493$	−1.99164	−0.0896987
$494$	5.35900	0.241113
$495$	−3.53159	−0.158733
$496$	−1.85637	−0.0833535
$497$	4.11787	0.184712
$498$	−8.40292	−0.376544
$499$	7.30822	0.327161	0.163580	−	0.986530i	$-0.447696\pi$
$499$	7.30822	0.327161	0.163580	+	0.986530i	$0.447696\pi$
$500$	−14.3528	−0.641877
$501$	15.6356	0.698545
$502$	9.59188	0.428107
$503$	35.9670	1.60369	0.801845	−	0.597532i	$-0.203852\pi$
$503$	35.9670	1.60369	0.801845	+	0.597532i	$0.203852\pi$
$504$	−0.589418	−0.0262547
$505$	55.8157	2.48376
$506$	−2.36068	−0.104945
$507$	−40.9552	−1.81888
$508$	−10.4571	−0.463957
$509$	29.9960	1.32955	0.664774	−	0.747045i	$-0.268528\pi$
$509$	29.9960	1.32955	0.664774	+	0.747045i	$0.268528\pi$
$510$	21.3961	0.947433
$511$	2.58802	0.114487
$512$	−1.00000	−0.0441942
$513$	−4.75440	−0.209912
$514$	5.49006	0.242156
$515$	−34.4861	−1.51964
$516$	7.25517	0.319391
$517$	11.3619	0.499697
$518$	−9.68157	−0.425384
$519$	16.5537	0.726627
$520$	23.2100	1.01783
$521$	−15.3707	−0.673404	−0.336702	−	0.941611i	$-0.609311\pi$
$521$	−15.3707	−0.673404	−0.336702	+	0.941611i	$0.609311\pi$
$522$	0.271715	0.0118927
$523$	26.3849	1.15373	0.576866	−	0.816839i	$-0.304275\pi$
$523$	26.3849	1.15373	0.576866	+	0.816839i	$0.304275\pi$
$524$	−13.0291	−0.569177
$525$	16.6705	0.727561
$526$	11.5067	0.501718
$527$	6.74253	0.293709
$528$	3.03005	0.131866
$529$	−21.4798	−0.933906
$530$	−17.9130	−0.778090
$531$	6.99283	0.303463
$532$	1.02232	0.0443230
$533$	25.2467	1.09356
$534$	0.535967	0.0231936
$535$	26.7972	1.15855
$536$	0.743541	0.0321161
$537$	−27.0439	−1.16703
$538$	23.7062	1.02205
$539$	10.6936	0.460605
$540$	−20.5915	−0.886116
$541$	6.70169	0.288128	0.144064	−	0.989568i	$-0.453983\pi$
$541$	6.70169	0.288128	0.144064	+	0.989568i	$0.453983\pi$
$542$	−4.08268	−0.175366
$543$	23.3621	1.00256
$544$	3.63211	0.155725
$545$	8.30688	0.355828
$546$	−11.7375	−0.502320
$547$	44.6496	1.90908	0.954541	−	0.298081i	$-0.0963466\pi$
$547$	44.6496	1.90908	0.954541	+	0.298081i	$0.0963466\pi$
$548$	−4.24174	−0.181198
$549$	−1.64467	−0.0701930
$550$	16.9559	0.723002
$551$	−0.471277	−0.0200771
$552$	−1.95121	−0.0830491
$553$	3.62630	0.154206
$554$	−30.9596	−1.31535
$555$	−47.9470	−2.03524
$556$	−0.521028	−0.0220965
$557$	11.7332	0.497150	0.248575	−	0.968613i	$-0.420038\pi$
$557$	11.7332	0.497150	0.248575	+	0.968613i	$0.420038\pi$
$558$	−0.919872	−0.0389413
$559$	−28.5856	−1.20904
$560$	4.42768	0.187104
$561$	−11.0055	−0.464651
$562$	10.1232	0.427020
$563$	26.1504	1.10211	0.551053	−	0.834470i	$-0.314226\pi$
$563$	26.1504	1.10211	0.551053	+	0.834470i	$0.314226\pi$
$564$	9.39117	0.395439
$565$	−21.0058	−0.883721
$566$	−27.5821	−1.15936
$567$	8.64507	0.363059
$568$	3.46188	0.145257
$569$	37.0036	1.55127	0.775636	−	0.631181i	$-0.217430\pi$
$569$	37.0036	1.55127	0.775636	+	0.631181i	$0.217430\pi$
$570$	5.06291	0.212062
$571$	34.2160	1.43190	0.715948	−	0.698154i	$-0.245995\pi$
$571$	34.2160	1.43190	0.715948	+	0.698154i	$0.245995\pi$
$572$	−11.9385	−0.499173
$573$	30.3108	1.26625
$574$	4.81622	0.201025
$575$	−10.9188	−0.455347
$576$	−0.495522	−0.0206468
$577$	4.29935	0.178984	0.0894921	−	0.995988i	$-0.471476\pi$
$577$	4.29935	0.178984	0.0894921	+	0.995988i	$0.471476\pi$
$578$	3.80780	0.158384
$579$	3.59717	0.149493
$580$	−2.04112	−0.0847528
$581$	6.31585	0.262025
$582$	−2.57386	−0.106690
$583$	9.21387	0.381599
$584$	2.17574	0.0900329
$585$	11.5011	0.475511
$586$	−23.2008	−0.958414
$587$	28.0225	1.15661	0.578305	−	0.815820i	$-0.303714\pi$
$587$	28.0225	1.15661	0.578305	+	0.815820i	$0.303714\pi$
$588$	8.83875	0.364504
$589$	1.59547	0.0657404
$590$	−52.5299	−2.16262
$591$	−39.4842	−1.62416
$592$	−8.13928	−0.334522
$593$	6.46289	0.265399	0.132700	−	0.991156i	$-0.457635\pi$
$593$	6.46289	0.265399	0.132700	+	0.991156i	$0.457635\pi$
$594$	10.5916	0.434578
$595$	−16.0818	−0.659290
$596$	20.1874	0.826910
$597$	−13.0277	−0.533189
$598$	7.68783	0.314379
$599$	18.8534	0.770330	0.385165	−	0.922848i	$-0.374144\pi$
$599$	18.8534	0.770330	0.385165	+	0.922848i	$0.374144\pi$
$600$	14.0149	0.572154
$601$	−6.61401	−0.269791	−0.134896	−	0.990860i	$-0.543070\pi$
$601$	−6.61401	−0.269791	−0.134896	+	0.990860i	$0.543070\pi$
$602$	−5.45317	−0.222255
$603$	0.368441	0.0150041
$604$	−6.48866	−0.264020
$605$	27.3000	1.10990
$606$	−23.7300	−0.963966
$607$	−2.69896	−0.109547	−0.0547737	−	0.998499i	$-0.517444\pi$
$607$	−2.69896	−0.109547	−0.0547737	+	0.998499i	$0.517444\pi$
$608$	0.859459	0.0348557
$609$	1.03221	0.0418274
$610$	12.3547	0.500229
$611$	−37.0015	−1.49692
$612$	1.79979	0.0727522
$613$	9.77126	0.394658	0.197329	−	0.980337i	$-0.436773\pi$
$613$	9.77126	0.394658	0.197329	+	0.980337i	$0.436773\pi$
$614$	3.73575	0.150763
$615$	23.8518	0.961799
$616$	−2.27746	−0.0917615
$617$	−4.16208	−0.167559	−0.0837796	−	0.996484i	$-0.526699\pi$
$617$	−4.16208	−0.167559	−0.0837796	+	0.996484i	$0.526699\pi$
$618$	14.6617	0.589782
$619$	−12.5204	−0.503237	−0.251619	−	0.967826i	$-0.580963\pi$
$619$	−12.5204	−0.503237	−0.251619	+	0.967826i	$0.580963\pi$
$620$	6.91005	0.277514
$621$	−6.82050	−0.273697
$622$	−4.34034	−0.174032
$623$	−0.402846	−0.0161397
$624$	−9.86772	−0.395025
$625$	9.14683	0.365873
$626$	−30.0712	−1.20189
$627$	−2.60420	−0.104002
$628$	−12.7537	−0.508928
$629$	29.5627	1.17874
$630$	2.19402	0.0874117
$631$	2.17780	0.0866970	0.0433485	−	0.999060i	$-0.486197\pi$
$631$	2.17780	0.0866970	0.0433485	+	0.999060i	$0.486197\pi$
$632$	3.04863	0.121268
$633$	−23.1067	−0.918407
$634$	−4.82810	−0.191748
$635$	38.9248	1.54468
$636$	7.61570	0.301982
$637$	−34.8250	−1.37982
$638$	1.04989	0.0415654
$639$	1.71544	0.0678618
$640$	3.72234	0.147139
$641$	−8.36550	−0.330417	−0.165209	−	0.986259i	$-0.552830\pi$
$641$	−8.36550	−0.330417	−0.165209	+	0.986259i	$0.552830\pi$
$642$	−11.3928	−0.449640
$643$	−32.4373	−1.27920	−0.639600	−	0.768708i	$-0.720900\pi$
$643$	−32.4373	−1.27920	−0.639600	+	0.768708i	$0.720900\pi$
$644$	1.46658	0.0577913
$645$	−27.0063	−1.06337
$646$	−3.12165	−0.122820
$647$	42.5404	1.67244	0.836218	−	0.548397i	$-0.184762\pi$
$647$	42.5404	1.67244	0.836218	+	0.548397i	$0.184762\pi$
$648$	7.26789	0.285510
$649$	27.0197	1.06062
$650$	−55.2190	−2.16587
$651$	−3.49448	−0.136960
$652$	8.95061	0.350533
$653$	−35.3015	−1.38145	−0.690727	−	0.723115i	$-0.742709\pi$
$653$	−35.3015	−1.38145	−0.690727	+	0.723115i	$0.742709\pi$
$654$	−3.53167	−0.138099
$655$	48.4987	1.89500
$656$	4.04899	0.158087
$657$	1.07813	0.0420618
$658$	−7.05864	−0.275174
$659$	−49.0530	−1.91083	−0.955416	−	0.295263i	$-0.904593\pi$
$659$	−49.0530	−1.91083	−0.955416	+	0.295263i	$0.904593\pi$
$660$	−11.2789	−0.439030
$661$	31.9759	1.24372	0.621860	−	0.783128i	$-0.286377\pi$
$661$	31.9759	1.24372	0.621860	+	0.783128i	$0.286377\pi$
$662$	−14.0657	−0.546677
$663$	35.8406	1.39193
$664$	5.30972	0.206057
$665$	−3.80541	−0.147568
$666$	−4.03319	−0.156283
$667$	−0.676078	−0.0261779
$668$	−9.87995	−0.382267
$669$	24.8161	0.959446
$670$	−2.76772	−0.106926
$671$	−6.35488	−0.245328
$672$	−1.88243	−0.0726163
$673$	−7.88437	−0.303920	−0.151960	−	0.988387i	$-0.548559\pi$
$673$	−7.88437	−0.303920	−0.151960	+	0.988387i	$0.548559\pi$
$674$	17.3269	0.667406
$675$	48.9893	1.88560
$676$	25.8792	0.995352
$677$	36.9167	1.41882	0.709412	−	0.704794i	$-0.248961\pi$
$677$	36.9167	1.41882	0.709412	+	0.704794i	$0.248961\pi$
$678$	8.93061	0.342978
$679$	1.93457	0.0742422
$680$	−13.5200	−0.518466
$681$	8.32593	0.319050
$682$	−3.55431	−0.136102
$683$	6.13856	0.234885	0.117443	−	0.993080i	$-0.462530\pi$
$683$	6.13856	0.234885	0.117443	+	0.993080i	$0.462530\pi$
$684$	0.425881	0.0162840
$685$	15.7892	0.603274
$686$	−14.9698	−0.571551
$687$	−2.94273	−0.112272
$688$	−4.58447	−0.174781
$689$	−30.0061	−1.14314
$690$	7.26308	0.276501
$691$	−6.97563	−0.265365	−0.132683	−	0.991159i	$-0.542359\pi$
$691$	−6.97563	−0.265365	−0.132683	+	0.991159i	$0.542359\pi$
$692$	−10.4601	−0.397634
$693$	−1.12853	−0.0428694
$694$	21.0947	0.800746
$695$	1.93945	0.0735674
$696$	0.867780	0.0328931
$697$	−14.7064	−0.557043
$698$	22.5311	0.852815
$699$	20.7814	0.786024
$700$	−10.5339	−0.398145
$701$	21.9180	0.827831	0.413915	−	0.910315i	$-0.364161\pi$
$701$	21.9180	0.827831	0.413915	+	0.910315i	$0.364161\pi$
$702$	−34.4928	−1.30185
$703$	6.99538	0.263836
$704$	−1.91466	−0.0721613
$705$	−34.9572	−1.31656
$706$	−23.3357	−0.878250
$707$	17.8361	0.670795
$708$	22.3331	0.839329
$709$	47.6318	1.78885	0.894426	−	0.447217i	$-0.147585\pi$
$709$	47.6318	1.78885	0.894426	+	0.447217i	$0.147585\pi$
$710$	−12.8863	−0.483615
$711$	1.51066	0.0566542
$712$	−0.338672	−0.0126923
$713$	2.28881	0.0857167
$714$	6.83718	0.255875
$715$	44.4391	1.66193
$716$	17.0888	0.638637
$717$	−40.9147	−1.52799
$718$	30.3387	1.13223
$719$	9.95231	0.371159	0.185579	−	0.982629i	$-0.440584\pi$
$719$	9.95231	0.371159	0.185579	+	0.982629i	$0.440584\pi$
$720$	1.84450	0.0687406
$721$	−11.0201	−0.410411
$722$	18.2613	0.679616
$723$	−19.5348	−0.726508
$724$	−14.7622	−0.548635
$725$	4.85603	0.180349
$726$	−11.6066	−0.430761
$727$	19.1351	0.709683	0.354842	−	0.934926i	$-0.384535\pi$
$727$	19.1351	0.709683	0.354842	+	0.934926i	$0.384535\pi$
$728$	7.41683	0.274886
$729$	29.8648	1.10610
$730$	−8.09887	−0.299753
$731$	16.6513	0.615870
$732$	−5.25261	−0.194142
$733$	8.94809	0.330505	0.165253	−	0.986251i	$-0.447156\pi$
$733$	8.94809	0.330505	0.165253	+	0.986251i	$0.447156\pi$
$734$	1.18981	0.0439166
$735$	−32.9009	−1.21357
$736$	1.23295	0.0454472
$737$	1.42363	0.0524399
$738$	2.00636	0.0738553
$739$	−36.8160	−1.35430	−0.677149	−	0.735846i	$-0.736785\pi$
$739$	−36.8160	−1.35430	−0.677149	+	0.735846i	$0.736785\pi$
$740$	30.2972	1.11375
$741$	8.48090	0.311554
$742$	−5.72415	−0.210140
$743$	41.0516	1.50604	0.753018	−	0.658000i	$-0.228597\pi$
$743$	41.0516	1.50604	0.753018	+	0.658000i	$0.228597\pi$
$744$	−2.93781	−0.107705
$745$	−75.1446	−2.75309
$746$	−2.89681	−0.106060
$747$	2.63108	0.0962663
$748$	6.95423	0.254272
$749$	8.56315	0.312891
$750$	−22.7141	−0.829401
$751$	−27.3030	−0.996300	−0.498150	−	0.867091i	$-0.665987\pi$
$751$	−27.3030	−0.996300	−0.498150	+	0.867091i	$0.665987\pi$
$752$	−5.93418	−0.216397
$753$	15.1797	0.553178
$754$	−3.41908	−0.124516
$755$	24.1530	0.879019
$756$	−6.58007	−0.239315
$757$	−15.9698	−0.580432	−0.290216	−	0.956961i	$-0.593727\pi$
$757$	−15.9698	−0.580432	−0.290216	+	0.956961i	$0.593727\pi$
$758$	3.20154	0.116285
$759$	−3.73590	−0.135604
$760$	−3.19920	−0.116047
$761$	12.0103	0.435374	0.217687	−	0.976019i	$-0.430149\pi$
$761$	12.0103	0.435374	0.217687	+	0.976019i	$0.430149\pi$
$762$	−16.5489	−0.599502
$763$	2.65449	0.0960990
$764$	−19.1531	−0.692935
$765$	−6.69944	−0.242219
$766$	10.6798	0.385875
$767$	−87.9931	−3.17725
$768$	−1.58255	−0.0571055
$769$	42.6007	1.53622	0.768111	−	0.640317i	$-0.221197\pi$
$769$	42.6007	1.53622	0.768111	+	0.640317i	$0.221197\pi$
$770$	8.47749	0.305508
$771$	8.68832	0.312902
$772$	−2.27302	−0.0818077
$773$	19.5831	0.704355	0.352177	−	0.935933i	$-0.385441\pi$
$773$	19.5831	0.704355	0.352177	+	0.935933i	$0.385441\pi$
$774$	−2.27171	−0.0816548
$775$	−16.4397	−0.590533
$776$	1.62639	0.0583841
$777$	−15.3216	−0.549660
$778$	32.3732	1.16064
$779$	−3.47994	−0.124682
$780$	36.7311	1.31518
$781$	6.62832	0.237180
$782$	−4.47821	−0.160140
$783$	3.03335	0.108403
$784$	−5.58512	−0.199469
$785$	47.4737	1.69441
$786$	−20.6192	−0.735463
$787$	−4.04209	−0.144085	−0.0720424	−	0.997402i	$-0.522952\pi$
$787$	−4.04209	−0.144085	−0.0720424	+	0.997402i	$0.522952\pi$
$788$	24.9497	0.888795
$789$	18.2101	0.648295
$790$	−11.3480	−0.403745
$791$	−6.71247	−0.238668
$792$	−0.948754	−0.0337125
$793$	20.6955	0.734918
$794$	18.0958	0.642196
$795$	−28.3483	−1.00541
$796$	8.23209	0.291779
$797$	23.4187	0.829532	0.414766	−	0.909928i	$-0.363863\pi$
$797$	23.4187	0.829532	0.414766	+	0.909928i	$0.363863\pi$
$798$	1.61787	0.0572720
$799$	21.5536	0.762511
$800$	−8.85585	−0.313102
$801$	−0.167820	−0.00592961
$802$	−10.0364	−0.354397
$803$	4.16580	0.147008
$804$	1.17669	0.0414988
$805$	−5.45911	−0.192408
$806$	11.5750	0.407714
$807$	37.5164	1.32064
$808$	14.9948	0.527514
$809$	8.86066	0.311524	0.155762	−	0.987795i	$-0.450217\pi$
$809$	8.86066	0.311524	0.155762	+	0.987795i	$0.450217\pi$
$810$	−27.0536	−0.950566
$811$	−32.2847	−1.13367	−0.566835	−	0.823831i	$-0.691832\pi$
$811$	−32.2847	−1.13367	−0.566835	+	0.823831i	$0.691832\pi$
$812$	−0.652246	−0.0228893
$813$	−6.46106	−0.226599
$814$	−15.5839	−0.546216
$815$	−33.3172	−1.16705
$816$	5.74801	0.201220
$817$	3.94017	0.137849
$818$	23.5160	0.822218
$819$	3.67520	0.128422
$820$	−15.0717	−0.526328
$821$	13.9378	0.486431	0.243215	−	0.969972i	$-0.421798\pi$
$821$	13.9378	0.486431	0.243215	+	0.969972i	$0.421798\pi$
$822$	−6.71278	−0.234135
$823$	−29.5897	−1.03143	−0.515715	−	0.856760i	$-0.672474\pi$
$823$	−29.5897	−1.03143	−0.515715	+	0.856760i	$0.672474\pi$
$824$	−9.26461	−0.322748
$825$	26.8336	0.934227
$826$	−16.7861	−0.584063
$827$	53.1046	1.84663	0.923314	−	0.384047i	$-0.125470\pi$
$827$	53.1046	1.84663	0.923314	+	0.384047i	$0.125470\pi$
$828$	0.610954	0.0212321
$829$	7.92822	0.275358	0.137679	−	0.990477i	$-0.456036\pi$
$829$	7.92822	0.275358	0.137679	+	0.990477i	$0.456036\pi$
$830$	−19.7646	−0.686040
$831$	−48.9952	−1.69963
$832$	6.23531	0.216171
$833$	20.2857	0.702859
$834$	−0.824555	−0.0285520
$835$	36.7766	1.27271
$836$	1.64557	0.0569132
$837$	−10.2692	−0.354954
$838$	0.398234	0.0137568
$839$	−35.5575	−1.22758	−0.613791	−	0.789469i	$-0.710356\pi$
$839$	−35.5575	−1.22758	−0.613791	+	0.789469i	$0.710356\pi$
$840$	7.00705	0.241766
$841$	−28.6993	−0.989632
$842$	−7.77374	−0.267901
$843$	16.0204	0.551773
$844$	14.6009	0.502582
$845$	−96.3311	−3.31389
$846$	−2.94052	−0.101097
$847$	8.72381	0.299754
$848$	−4.81228	−0.165255
$849$	−43.6502	−1.49807
$850$	32.1654	1.10326
$851$	10.0353	0.344007
$852$	5.47862	0.187694
$853$	−40.9903	−1.40348	−0.701741	−	0.712432i	$-0.747593\pi$
$853$	−40.9903	−1.40348	−0.701741	+	0.712432i	$0.747593\pi$
$854$	3.94800	0.135098
$855$	−1.58528	−0.0542153
$856$	7.19902	0.246057
$857$	−6.15374	−0.210208	−0.105104	−	0.994461i	$-0.533517\pi$
$857$	−6.15374	−0.210208	−0.105104	+	0.994461i	$0.533517\pi$
$858$	−18.8933	−0.645006
$859$	−1.32114	−0.0450766	−0.0225383	−	0.999746i	$-0.507175\pi$
$859$	−1.32114	−0.0450766	−0.0225383	+	0.999746i	$0.507175\pi$
$860$	17.0650	0.581911
$861$	7.62194	0.259755
$862$	17.1754	0.584996
$863$	−38.2670	−1.30262	−0.651312	−	0.758810i	$-0.725781\pi$
$863$	−38.2670	−1.30262	−0.651312	+	0.758810i	$0.725781\pi$
$864$	−5.53185	−0.188197
$865$	38.9362	1.32387
$866$	−13.3710	−0.454363
$867$	6.02606	0.204656
$868$	2.20813	0.0749488
$869$	5.83707	0.198009
$870$	−3.23018	−0.109513
$871$	−4.63622	−0.157092
$872$	2.23163	0.0755724
$873$	0.805914	0.0272761
$874$	−1.05967	−0.0358439
$875$	17.0725	0.577155
$876$	3.44323	0.116336
$877$	49.8288	1.68260	0.841299	−	0.540570i	$-0.181791\pi$
$877$	49.8288	1.68260	0.841299	+	0.540570i	$0.181791\pi$
$878$	28.4069	0.958687
$879$	−36.7165	−1.23842
$880$	7.12701	0.240251
$881$	13.6029	0.458293	0.229146	−	0.973392i	$-0.426407\pi$
$881$	13.6029	0.458293	0.229146	+	0.973392i	$0.426407\pi$
$882$	−2.76755	−0.0931883
$883$	−25.8379	−0.869515	−0.434758	−	0.900548i	$-0.643166\pi$
$883$	−25.8379	−0.869515	−0.434758	+	0.900548i	$0.643166\pi$
$884$	−22.6473	−0.761712
$885$	−83.1314	−2.79443
$886$	−7.88929	−0.265046
$887$	−47.3649	−1.59036	−0.795179	−	0.606375i	$-0.792623\pi$
$887$	−47.3649	−1.59036	−0.795179	+	0.606375i	$0.792623\pi$
$888$	−12.8808	−0.432253
$889$	12.4385	0.417176
$890$	1.26065	0.0422572
$891$	13.9155	0.466187
$892$	−15.6811	−0.525040
$893$	5.10019	0.170671
$894$	31.9477	1.06849
$895$	−63.6102	−2.12626
$896$	1.18949	0.0397380
$897$	12.1664	0.406225
$898$	−9.95348	−0.332152
$899$	−1.01792	−0.0339497
$900$	−4.38827	−0.146276
$901$	17.4787	0.582301
$902$	7.75242	0.258128
$903$	−8.62994	−0.287186
$904$	−5.64316	−0.187689
$905$	54.9502	1.82661
$906$	−10.2687	−0.341153
$907$	−52.3137	−1.73705	−0.868524	−	0.495647i	$-0.834931\pi$
$907$	−52.3137	−1.73705	−0.868524	+	0.495647i	$0.834931\pi$
$908$	−5.26107	−0.174595
$909$	7.43024	0.246445
$910$	−27.6080	−0.915196
$911$	−21.7883	−0.721877	−0.360939	−	0.932590i	$-0.617544\pi$
$911$	−21.7883	−0.721877	−0.360939	+	0.932590i	$0.617544\pi$
$912$	1.36014	0.0450388
$913$	10.1663	0.336455
$914$	−6.28517	−0.207895
$915$	19.5520	0.646370
$916$	1.85948	0.0614390
$917$	15.4979	0.511786
$918$	20.0923	0.663144
$919$	−22.2981	−0.735545	−0.367773	−	0.929916i	$-0.619880\pi$
$919$	−22.2981	−0.735545	−0.367773	+	0.929916i	$0.619880\pi$
$920$	−4.58947	−0.151310
$921$	5.91203	0.194808
$922$	−10.2891	−0.338854
$923$	−21.5859	−0.710510
$924$	−3.60420	−0.118570
$925$	−72.0802	−2.36998
$926$	20.5322	0.674730
$927$	−4.59082	−0.150782
$928$	−0.548342	−0.0180002
$929$	−18.7337	−0.614633	−0.307316	−	0.951607i	$-0.599431\pi$
$929$	−18.7337	−0.614633	−0.307316	+	0.951607i	$0.599431\pi$
$930$	10.9355	0.358590
$931$	4.80018	0.157320
$932$	−13.1315	−0.430138
$933$	−6.86883	−0.224875
$934$	25.7547	0.842719
$935$	−25.8861	−0.846565
$936$	3.08974	0.100991
$937$	9.81062	0.320499	0.160249	−	0.987077i	$-0.448770\pi$
$937$	9.81062	0.320499	0.160249	+	0.987077i	$0.448770\pi$
$938$	−0.884433	−0.0288778
$939$	−47.5893	−1.55302
$940$	22.0891	0.720466
$941$	24.1778	0.788174	0.394087	−	0.919073i	$-0.371061\pi$
$941$	24.1778	0.788174	0.394087	+	0.919073i	$0.371061\pi$
$942$	−20.1834	−0.657612
$943$	−4.99221	−0.162569
$944$	−14.1120	−0.459308
$945$	24.4933	0.796767
$946$	−8.77768	−0.285387
$947$	−17.0741	−0.554832	−0.277416	−	0.960750i	$-0.589478\pi$
$947$	−17.0741	−0.554832	−0.277416	+	0.960750i	$0.589478\pi$
$948$	4.82461	0.156696
$949$	−13.5664	−0.440386
$950$	7.61124	0.246941
$951$	−7.64073	−0.247768
$952$	−4.32035	−0.140023
$953$	−26.4799	−0.857768	−0.428884	−	0.903360i	$-0.641093\pi$
$953$	−26.4799	−0.857768	−0.428884	+	0.903360i	$0.641093\pi$
$954$	−2.38459	−0.0772041
$955$	71.2945	2.30703
$956$	25.8536	0.836166
$957$	1.66150	0.0537087
$958$	−5.14750	−0.166308
$959$	5.04549	0.162927
$960$	5.89081	0.190125
$961$	−27.5539	−0.888835
$962$	50.7510	1.63628
$963$	3.56727	0.114954
$964$	12.3439	0.397569
$965$	8.46096	0.272368
$966$	2.32094	0.0746751
$967$	19.6328	0.631349	0.315674	−	0.948868i	$-0.397769\pi$
$967$	19.6328	0.631349	0.315674	+	0.948868i	$0.397769\pi$
$968$	7.33409	0.235727
$969$	−4.94018	−0.158701
$970$	−6.05400	−0.194382
$971$	3.39179	0.108848	0.0544239	−	0.998518i	$-0.482668\pi$
$971$	3.39179	0.108848	0.0544239	+	0.998518i	$0.482668\pi$
$972$	−5.09373	−0.163381
$973$	0.619756	0.0198685
$974$	−5.88744	−0.188646
$975$	−87.3871	−2.79863
$976$	3.31907	0.106241
$977$	−8.24147	−0.263668	−0.131834	−	0.991272i	$-0.542087\pi$
$977$	−8.24147	−0.263668	−0.131834	+	0.991272i	$0.542087\pi$
$978$	14.1648	0.452941
$979$	−0.648441	−0.0207242
$980$	20.7897	0.664104
$981$	1.10582	0.0353061
$982$	15.9104	0.507720
$983$	29.3933	0.937502	0.468751	−	0.883330i	$-0.344704\pi$
$983$	29.3933	0.937502	0.468751	+	0.883330i	$0.344704\pi$
$984$	6.40775	0.204272
$985$	−92.8712	−2.95912
$986$	1.99164	0.0634266
$987$	−11.1707	−0.355567
$988$	−5.35900	−0.170492
$989$	5.65243	0.179737
$990$	3.53159	0.112241
$991$	−15.6913	−0.498450	−0.249225	−	0.968446i	$-0.580176\pi$
$991$	−15.6913	−0.498450	−0.249225	+	0.968446i	$0.580176\pi$
$992$	1.85637	0.0589398
$993$	−22.2597	−0.706389
$994$	−4.11787	−0.130611
$995$	−30.6427	−0.971438
$996$	8.40292	0.266257
$997$	−33.4797	−1.06031	−0.530156	−	0.847900i	$-0.677867\pi$
$997$	−33.4797	−1.06031	−0.530156	+	0.847900i	$0.677867\pi$
$998$	−7.30822	−0.231338
$999$	−45.0253	−1.42454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.19	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.19	✓	69	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.58255 q^{3} +1.00000 q^{4} -3.72234 q^{5} +1.58255 q^{6} -1.18949 q^{7} -1.00000 q^{8} -0.495522 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.58255 q^{3} +1.00000 q^{4} -3.72234 q^{5} +1.58255 q^{6} -1.18949 q^{7} -1.00000 q^{8} -0.495522 q^{9} +3.72234 q^{10} -1.91466 q^{11} -1.58255 q^{12} +6.23531 q^{13} +1.18949 q^{14} +5.89081 q^{15} +1.00000 q^{16} -3.63211 q^{17} +0.495522 q^{18} -0.859459 q^{19} -3.72234 q^{20} +1.88243 q^{21} +1.91466 q^{22} -1.23295 q^{23} +1.58255 q^{24} +8.85585 q^{25} -6.23531 q^{26} +5.53185 q^{27} -1.18949 q^{28} +0.548342 q^{29} -5.89081 q^{30} -1.85637 q^{31} -1.00000 q^{32} +3.03005 q^{33} +3.63211 q^{34} +4.42768 q^{35} -0.495522 q^{36} -8.13928 q^{37} +0.859459 q^{38} -9.86772 q^{39} +3.72234 q^{40} +4.04899 q^{41} -1.88243 q^{42} -4.58447 q^{43} -1.91466 q^{44} +1.84450 q^{45} +1.23295 q^{46} -5.93418 q^{47} -1.58255 q^{48} -5.58512 q^{49} -8.85585 q^{50} +5.74801 q^{51} +6.23531 q^{52} -4.81228 q^{53} -5.53185 q^{54} +7.12701 q^{55} +1.18949 q^{56} +1.36014 q^{57} -0.548342 q^{58} -14.1120 q^{59} +5.89081 q^{60} +3.31907 q^{61} +1.85637 q^{62} +0.589418 q^{63} +1.00000 q^{64} -23.2100 q^{65} -3.03005 q^{66} -0.743541 q^{67} -3.63211 q^{68} +1.95121 q^{69} -4.42768 q^{70} -3.46188 q^{71} +0.495522 q^{72} -2.17574 q^{73} +8.13928 q^{74} -14.0149 q^{75} -0.859459 q^{76} +2.27746 q^{77} +9.86772 q^{78} -3.04863 q^{79} -3.72234 q^{80} -7.26789 q^{81} -4.04899 q^{82} -5.30972 q^{83} +1.88243 q^{84} +13.5200 q^{85} +4.58447 q^{86} -0.867780 q^{87} +1.91466 q^{88} +0.338672 q^{89} -1.84450 q^{90} -7.41683 q^{91} -1.23295 q^{92} +2.93781 q^{93} +5.93418 q^{94} +3.19920 q^{95} +1.58255 q^{96} -1.62639 q^{97} +5.58512 q^{98} +0.948754 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

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