LMFDB - Embedded newform 6038.2.a.d.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.22022 q^{3} +1.00000 q^{4} +2.70863 q^{5} +3.22022 q^{6} +4.47236 q^{7} -1.00000 q^{8} +7.36980 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.22022 q^{3} +1.00000 q^{4} +2.70863 q^{5} +3.22022 q^{6} +4.47236 q^{7} -1.00000 q^{8} +7.36980 q^{9} -2.70863 q^{10} +5.27121 q^{11} -3.22022 q^{12} +1.35627 q^{13} -4.47236 q^{14} -8.72238 q^{15} +1.00000 q^{16} +4.29095 q^{17} -7.36980 q^{18} -8.43362 q^{19} +2.70863 q^{20} -14.4020 q^{21} -5.27121 q^{22} -4.66909 q^{23} +3.22022 q^{24} +2.33669 q^{25} -1.35627 q^{26} -14.0717 q^{27} +4.47236 q^{28} +1.90252 q^{29} +8.72238 q^{30} +3.33464 q^{31} -1.00000 q^{32} -16.9744 q^{33} -4.29095 q^{34} +12.1140 q^{35} +7.36980 q^{36} -2.69141 q^{37} +8.43362 q^{38} -4.36750 q^{39} -2.70863 q^{40} +7.88850 q^{41} +14.4020 q^{42} +2.67530 q^{43} +5.27121 q^{44} +19.9621 q^{45} +4.66909 q^{46} +3.75850 q^{47} -3.22022 q^{48} +13.0020 q^{49} -2.33669 q^{50} -13.8178 q^{51} +1.35627 q^{52} +8.05316 q^{53} +14.0717 q^{54} +14.2778 q^{55} -4.47236 q^{56} +27.1581 q^{57} -1.90252 q^{58} -8.71857 q^{59} -8.72238 q^{60} +1.40068 q^{61} -3.33464 q^{62} +32.9604 q^{63} +1.00000 q^{64} +3.67365 q^{65} +16.9744 q^{66} +11.8701 q^{67} +4.29095 q^{68} +15.0355 q^{69} -12.1140 q^{70} -7.97200 q^{71} -7.36980 q^{72} -7.69457 q^{73} +2.69141 q^{74} -7.52464 q^{75} -8.43362 q^{76} +23.5747 q^{77} +4.36750 q^{78} -9.85183 q^{79} +2.70863 q^{80} +23.2045 q^{81} -7.88850 q^{82} +10.9166 q^{83} -14.4020 q^{84} +11.6226 q^{85} -2.67530 q^{86} -6.12653 q^{87} -5.27121 q^{88} -16.7401 q^{89} -19.9621 q^{90} +6.06574 q^{91} -4.66909 q^{92} -10.7383 q^{93} -3.75850 q^{94} -22.8436 q^{95} +3.22022 q^{96} +2.63120 q^{97} -13.0020 q^{98} +38.8477 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.22022	−1.85919	−0.929596	−	0.368579i	$-0.879844\pi$
$3$	−3.22022	−1.85919	−0.929596	+	0.368579i	$0.879844\pi$
$4$	1.00000	0.500000
$5$	2.70863	1.21134	0.605669	−	0.795717i	$-0.292906\pi$
$5$	2.70863	1.21134	0.605669	+	0.795717i	$0.292906\pi$
$6$	3.22022	1.31465
$7$	4.47236	1.69039	0.845196	−	0.534456i	$-0.179484\pi$
$7$	4.47236	1.69039	0.845196	+	0.534456i	$0.179484\pi$
$8$	−1.00000	−0.353553
$9$	7.36980	2.45660
$10$	−2.70863	−0.856545
$11$	5.27121	1.58933	0.794665	−	0.607048i	$-0.207647\pi$
$11$	5.27121	1.58933	0.794665	+	0.607048i	$0.207647\pi$
$12$	−3.22022	−0.929596
$13$	1.35627	0.376163	0.188081	−	0.982153i	$-0.439773\pi$
$13$	1.35627	0.376163	0.188081	+	0.982153i	$0.439773\pi$
$14$	−4.47236	−1.19529
$15$	−8.72238	−2.25211
$16$	1.00000	0.250000
$17$	4.29095	1.04071	0.520355	−	0.853950i	$-0.325800\pi$
$17$	4.29095	1.04071	0.520355	+	0.853950i	$0.325800\pi$
$18$	−7.36980	−1.73708
$19$	−8.43362	−1.93480	−0.967402	−	0.253246i	$-0.918502\pi$
$19$	−8.43362	−1.93480	−0.967402	+	0.253246i	$0.918502\pi$
$20$	2.70863	0.605669
$21$	−14.4020	−3.14277
$22$	−5.27121	−1.12383
$23$	−4.66909	−0.973573	−0.486787	−	0.873521i	$-0.661831\pi$
$23$	−4.66909	−0.973573	−0.486787	+	0.873521i	$0.661831\pi$
$24$	3.22022	0.657324
$25$	2.33669	0.467338
$26$	−1.35627	−0.265987
$27$	−14.0717	−2.70810
$28$	4.47236	0.845196
$29$	1.90252	0.353289	0.176645	−	0.984275i	$-0.443476\pi$
$29$	1.90252	0.353289	0.176645	+	0.984275i	$0.443476\pi$
$30$	8.72238	1.59248
$31$	3.33464	0.598919	0.299460	−	0.954109i	$-0.403194\pi$
$31$	3.33464	0.598919	0.299460	+	0.954109i	$0.403194\pi$
$32$	−1.00000	−0.176777
$33$	−16.9744	−2.95487
$34$	−4.29095	−0.735892
$35$	12.1140	2.04764
$36$	7.36980	1.22830
$37$	−2.69141	−0.442466	−0.221233	−	0.975221i	$-0.571008\pi$
$37$	−2.69141	−0.442466	−0.221233	+	0.975221i	$0.571008\pi$
$38$	8.43362	1.36811
$39$	−4.36750	−0.699359
$40$	−2.70863	−0.428272
$41$	7.88850	1.23198	0.615989	−	0.787755i	$-0.288757\pi$
$41$	7.88850	1.23198	0.615989	+	0.787755i	$0.288757\pi$
$42$	14.4020	2.22227
$43$	2.67530	0.407979	0.203990	−	0.978973i	$-0.434609\pi$
$43$	2.67530	0.407979	0.203990	+	0.978973i	$0.434609\pi$
$44$	5.27121	0.794665
$45$	19.9621	2.97577
$46$	4.66909	0.688420
$47$	3.75850	0.548234	0.274117	−	0.961696i	$-0.411614\pi$
$47$	3.75850	0.548234	0.274117	+	0.961696i	$0.411614\pi$
$48$	−3.22022	−0.464798
$49$	13.0020	1.85743
$50$	−2.33669	−0.330458
$51$	−13.8178	−1.93488
$52$	1.35627	0.188081
$53$	8.05316	1.10619	0.553093	−	0.833119i	$-0.313447\pi$
$53$	8.05316	1.10619	0.553093	+	0.833119i	$0.313447\pi$
$54$	14.0717	1.91491
$55$	14.2778	1.92521
$56$	−4.47236	−0.597644
$57$	27.1581	3.59717
$58$	−1.90252	−0.249813
$59$	−8.71857	−1.13506	−0.567530	−	0.823353i	$-0.692101\pi$
$59$	−8.71857	−1.13506	−0.567530	+	0.823353i	$0.692101\pi$
$60$	−8.72238	−1.12605
$61$	1.40068	0.179338	0.0896691	−	0.995972i	$-0.471419\pi$
$61$	1.40068	0.179338	0.0896691	+	0.995972i	$0.471419\pi$
$62$	−3.33464	−0.423500
$63$	32.9604	4.15262
$64$	1.00000	0.125000
$65$	3.67365	0.455660
$66$	16.9744	2.08941
$67$	11.8701	1.45016	0.725082	−	0.688662i	$-0.241802\pi$
$67$	11.8701	1.45016	0.725082	+	0.688662i	$0.241802\pi$
$68$	4.29095	0.520355
$69$	15.0355	1.81006
$70$	−12.1140	−1.44790
$71$	−7.97200	−0.946102	−0.473051	−	0.881035i	$-0.656847\pi$
$71$	−7.97200	−0.946102	−0.473051	+	0.881035i	$0.656847\pi$
$72$	−7.36980	−0.868539
$73$	−7.69457	−0.900581	−0.450290	−	0.892882i	$-0.648680\pi$
$73$	−7.69457	−0.900581	−0.450290	+	0.892882i	$0.648680\pi$
$74$	2.69141	0.312871
$75$	−7.52464	−0.868871
$76$	−8.43362	−0.967402
$77$	23.5747	2.68659
$78$	4.36750	0.494522
$79$	−9.85183	−1.10842	−0.554209	−	0.832378i	$-0.686979\pi$
$79$	−9.85183	−1.10842	−0.554209	+	0.832378i	$0.686979\pi$
$80$	2.70863	0.302834
$81$	23.2045	2.57828
$82$	−7.88850	−0.871140
$83$	10.9166	1.19826	0.599129	−	0.800653i	$-0.295514\pi$
$83$	10.9166	1.19826	0.599129	+	0.800653i	$0.295514\pi$
$84$	−14.4020	−1.57138
$85$	11.6226	1.26065
$86$	−2.67530	−0.288485
$87$	−6.12653	−0.656833
$88$	−5.27121	−0.561913
$89$	−16.7401	−1.77445	−0.887223	−	0.461340i	$-0.847369\pi$
$89$	−16.7401	−1.77445	−0.887223	+	0.461340i	$0.847369\pi$
$90$	−19.9621	−2.10419
$91$	6.06574	0.635863
$92$	−4.66909	−0.486787
$93$	−10.7383	−1.11351
$94$	−3.75850	−0.387660
$95$	−22.8436	−2.34370
$96$	3.22022	0.328662
$97$	2.63120	0.267158	0.133579	−	0.991038i	$-0.457353\pi$
$97$	2.63120	0.267158	0.133579	+	0.991038i	$0.457353\pi$
$98$	−13.0020	−1.31340
$99$	38.8477	3.90435
$100$	2.33669	0.233669
$101$	11.1256	1.10704	0.553521	−	0.832835i	$-0.313284\pi$
$101$	11.1256	1.10704	0.553521	+	0.832835i	$0.313284\pi$
$102$	13.8178	1.36817
$103$	−1.34578	−0.132604	−0.0663020	−	0.997800i	$-0.521120\pi$
$103$	−1.34578	−0.132604	−0.0663020	+	0.997800i	$0.521120\pi$
$104$	−1.35627	−0.132994
$105$	−39.0096	−3.80695
$106$	−8.05316	−0.782192
$107$	0.962311	0.0930301	0.0465150	−	0.998918i	$-0.485188\pi$
$107$	0.962311	0.0930301	0.0465150	+	0.998918i	$0.485188\pi$
$108$	−14.0717	−1.35405
$109$	−4.58186	−0.438863	−0.219432	−	0.975628i	$-0.570420\pi$
$109$	−4.58186	−0.438863	−0.219432	+	0.975628i	$0.570420\pi$
$110$	−14.2778	−1.36133
$111$	8.66694	0.822629
$112$	4.47236	0.422598
$113$	12.1024	1.13849	0.569247	−	0.822166i	$-0.307235\pi$
$113$	12.1024	1.13849	0.569247	+	0.822166i	$0.307235\pi$
$114$	−27.1581	−2.54359
$115$	−12.6469	−1.17933
$116$	1.90252	0.176645
$117$	9.99546	0.924081
$118$	8.71857	0.802609
$119$	19.1907	1.75921
$120$	8.72238	0.796241
$121$	16.7857	1.52597
$122$	−1.40068	−0.126811
$123$	−25.4027	−2.29048
$124$	3.33464	0.299460
$125$	−7.21393	−0.645234
$126$	−32.9604	−2.93634
$127$	−13.8631	−1.23015	−0.615076	−	0.788468i	$-0.710875\pi$
$127$	−13.8631	−1.23015	−0.615076	+	0.788468i	$0.710875\pi$
$128$	−1.00000	−0.0883883
$129$	−8.61504	−0.758512
$130$	−3.67365	−0.322200
$131$	7.93819	0.693563	0.346782	−	0.937946i	$-0.387275\pi$
$131$	7.93819	0.693563	0.346782	+	0.937946i	$0.387275\pi$
$132$	−16.9744	−1.47744
$133$	−37.7181	−3.27058
$134$	−11.8701	−1.02542
$135$	−38.1150	−3.28042
$136$	−4.29095	−0.367946
$137$	−6.46343	−0.552208	−0.276104	−	0.961128i	$-0.589043\pi$
$137$	−6.46343	−0.552208	−0.276104	+	0.961128i	$0.589043\pi$
$138$	−15.0355	−1.27991
$139$	16.3026	1.38276	0.691382	−	0.722489i	$-0.257002\pi$
$139$	16.3026	1.38276	0.691382	+	0.722489i	$0.257002\pi$
$140$	12.1140	1.02382
$141$	−12.1032	−1.01927
$142$	7.97200	0.668995
$143$	7.14921	0.597847
$144$	7.36980	0.614150
$145$	5.15323	0.427953
$146$	7.69457	0.636807
$147$	−41.8692	−3.45331
$148$	−2.69141	−0.221233
$149$	22.2770	1.82501	0.912503	−	0.409070i	$-0.134147\pi$
$149$	22.2770	1.82501	0.912503	+	0.409070i	$0.134147\pi$
$150$	7.52464	0.614384
$151$	−2.02255	−0.164593	−0.0822964	−	0.996608i	$-0.526225\pi$
$151$	−2.02255	−0.164593	−0.0822964	+	0.996608i	$0.526225\pi$
$152$	8.43362	0.684057
$153$	31.6235	2.55660
$154$	−23.5747	−1.89971
$155$	9.03232	0.725493
$156$	−4.36750	−0.349680
$157$	20.1691	1.60967	0.804834	−	0.593501i	$-0.202255\pi$
$157$	20.1691	1.60967	0.804834	+	0.593501i	$0.202255\pi$
$158$	9.85183	0.783769
$159$	−25.9329	−2.05661
$160$	−2.70863	−0.214136
$161$	−20.8819	−1.64572
$162$	−23.2045	−1.82312
$163$	16.4494	1.28842	0.644208	−	0.764851i	$-0.277187\pi$
$163$	16.4494	1.28842	0.644208	+	0.764851i	$0.277187\pi$
$164$	7.88850	0.615989
$165$	−45.9775	−3.57934
$166$	−10.9166	−0.847296
$167$	8.41094	0.650858	0.325429	−	0.945566i	$-0.394491\pi$
$167$	8.41094	0.650858	0.325429	+	0.945566i	$0.394491\pi$
$168$	14.4020	1.11114
$169$	−11.1605	−0.858502
$170$	−11.6226	−0.891414
$171$	−62.1540	−4.75304
$172$	2.67530	0.203990
$173$	−6.47837	−0.492541	−0.246271	−	0.969201i	$-0.579205\pi$
$173$	−6.47837	−0.492541	−0.246271	+	0.969201i	$0.579205\pi$
$174$	6.12653	0.464451
$175$	10.4505	0.789984
$176$	5.27121	0.397332
$177$	28.0757	2.11030
$178$	16.7401	1.25472
$179$	−6.00397	−0.448758	−0.224379	−	0.974502i	$-0.572035\pi$
$179$	−6.00397	−0.448758	−0.224379	+	0.974502i	$0.572035\pi$
$180$	19.9621	1.48788
$181$	25.5447	1.89872	0.949360	−	0.314190i	$-0.101733\pi$
$181$	25.5447	1.89872	0.949360	+	0.314190i	$0.101733\pi$
$182$	−6.06574	−0.449623
$183$	−4.51048	−0.333424
$184$	4.66909	0.344210
$185$	−7.29005	−0.535975
$186$	10.7383	0.787368
$187$	22.6185	1.65403
$188$	3.75850	0.274117
$189$	−62.9336	−4.57775
$190$	22.8436	1.65725
$191$	−12.5070	−0.904972	−0.452486	−	0.891772i	$-0.649463\pi$
$191$	−12.5070	−0.904972	−0.452486	+	0.891772i	$0.649463\pi$
$192$	−3.22022	−0.232399
$193$	−7.66398	−0.551666	−0.275833	−	0.961206i	$-0.588954\pi$
$193$	−7.66398	−0.551666	−0.275833	+	0.961206i	$0.588954\pi$
$194$	−2.63120	−0.188909
$195$	−11.8299	−0.847160
$196$	13.0020	0.928713
$197$	−6.96091	−0.495944	−0.247972	−	0.968767i	$-0.579764\pi$
$197$	−6.96091	−0.495944	−0.247972	+	0.968767i	$0.579764\pi$
$198$	−38.8477	−2.76079
$199$	0.613801	0.0435112	0.0217556	−	0.999763i	$-0.493074\pi$
$199$	0.613801	0.0435112	0.0217556	+	0.999763i	$0.493074\pi$
$200$	−2.33669	−0.165229
$201$	−38.2243	−2.69614
$202$	−11.1256	−0.782797
$203$	8.50876	0.597198
$204$	−13.8178	−0.967440
$205$	21.3671	1.49234
$206$	1.34578	0.0937652
$207$	−34.4103	−2.39168
$208$	1.35627	0.0940407
$209$	−44.4554	−3.07504
$210$	39.0096	2.69192
$211$	−6.97139	−0.479930	−0.239965	−	0.970782i	$-0.577136\pi$
$211$	−6.97139	−0.479930	−0.239965	+	0.970782i	$0.577136\pi$
$212$	8.05316	0.553093
$213$	25.6716	1.75899
$214$	−0.962311	−0.0657822
$215$	7.24640	0.494200
$216$	14.0717	0.957457
$217$	14.9137	1.01241
$218$	4.58186	0.310323
$219$	24.7782	1.67435
$220$	14.2778	0.962607
$221$	5.81971	0.391476
$222$	−8.66694	−0.581687
$223$	6.72630	0.450426	0.225213	−	0.974310i	$-0.427692\pi$
$223$	6.72630	0.450426	0.225213	+	0.974310i	$0.427692\pi$
$224$	−4.47236	−0.298822
$225$	17.2209	1.14806
$226$	−12.1024	−0.805037
$227$	13.4458	0.892427	0.446213	−	0.894927i	$-0.352772\pi$
$227$	13.4458	0.892427	0.446213	+	0.894927i	$0.352772\pi$
$228$	27.1581	1.79859
$229$	−17.0138	−1.12431	−0.562153	−	0.827033i	$-0.690027\pi$
$229$	−17.0138	−1.12431	−0.562153	+	0.827033i	$0.690027\pi$
$230$	12.6469	0.833909
$231$	−75.9158	−4.99489
$232$	−1.90252	−0.124907
$233$	−6.69008	−0.438282	−0.219141	−	0.975693i	$-0.570325\pi$
$233$	−6.69008	−0.438282	−0.219141	+	0.975693i	$0.570325\pi$
$234$	−9.99546	−0.653424
$235$	10.1804	0.664097
$236$	−8.71857	−0.567530
$237$	31.7250	2.06076
$238$	−19.1907	−1.24395
$239$	−12.0507	−0.779493	−0.389746	−	0.920922i	$-0.627437\pi$
$239$	−12.0507	−0.779493	−0.389746	+	0.920922i	$0.627437\pi$
$240$	−8.72238	−0.563027
$241$	−0.371642	−0.0239396	−0.0119698	−	0.999928i	$-0.503810\pi$
$241$	−0.371642	−0.0239396	−0.0119698	+	0.999928i	$0.503810\pi$
$242$	−16.7857	−1.07902
$243$	−32.5085	−2.08542
$244$	1.40068	0.0896691
$245$	35.2176	2.24997
$246$	25.4027	1.61962
$247$	−11.4383	−0.727801
$248$	−3.33464	−0.211750
$249$	−35.1540	−2.22779
$250$	7.21393	0.456249
$251$	30.5249	1.92671	0.963357	−	0.268221i	$-0.0864358\pi$
$251$	30.5249	1.92671	0.963357	+	0.268221i	$0.0864358\pi$
$252$	32.9604	2.07631
$253$	−24.6118	−1.54733
$254$	13.8631	0.869849
$255$	−37.4273	−2.34379
$256$	1.00000	0.0625000
$257$	−26.6734	−1.66384	−0.831919	−	0.554897i	$-0.812758\pi$
$257$	−26.6734	−1.66384	−0.831919	+	0.554897i	$0.812758\pi$
$258$	8.61504	0.536349
$259$	−12.0370	−0.747941
$260$	3.67365	0.227830
$261$	14.0212	0.867890
$262$	−7.93819	−0.490423
$263$	5.29395	0.326439	0.163220	−	0.986590i	$-0.447812\pi$
$263$	5.29395	0.326439	0.163220	+	0.986590i	$0.447812\pi$
$264$	16.9744	1.04470
$265$	21.8130	1.33996
$266$	37.7181	2.31265
$267$	53.9067	3.29904
$268$	11.8701	0.725082
$269$	1.11650	0.0680744	0.0340372	−	0.999421i	$-0.489164\pi$
$269$	1.11650	0.0680744	0.0340372	+	0.999421i	$0.489164\pi$
$270$	38.1150	2.31961
$271$	−4.15237	−0.252238	−0.126119	−	0.992015i	$-0.540252\pi$
$271$	−4.15237	−0.252238	−0.126119	+	0.992015i	$0.540252\pi$
$272$	4.29095	0.260177
$273$	−19.5330	−1.18219
$274$	6.46343	0.390470
$275$	12.3172	0.742753
$276$	15.0355	0.905030
$277$	−9.75276	−0.585987	−0.292993	−	0.956114i	$-0.594651\pi$
$277$	−9.75276	−0.585987	−0.292993	+	0.956114i	$0.594651\pi$
$278$	−16.3026	−0.977762
$279$	24.5756	1.47130
$280$	−12.1140	−0.723948
$281$	0.670679	0.0400093	0.0200047	−	0.999800i	$-0.493632\pi$
$281$	0.670679	0.0400093	0.0200047	+	0.999800i	$0.493632\pi$
$282$	12.1032	0.720735
$283$	7.53352	0.447821	0.223911	−	0.974610i	$-0.428118\pi$
$283$	7.53352	0.447821	0.223911	+	0.974610i	$0.428118\pi$
$284$	−7.97200	−0.473051
$285$	73.5612	4.35739
$286$	−7.14921	−0.422741
$287$	35.2802	2.08253
$288$	−7.36980	−0.434269
$289$	1.41228	0.0830755
$290$	−5.15323	−0.302608
$291$	−8.47303	−0.496698
$292$	−7.69457	−0.450290
$293$	14.4351	0.843310	0.421655	−	0.906756i	$-0.361449\pi$
$293$	14.4351	0.843310	0.421655	+	0.906756i	$0.361449\pi$
$294$	41.8692	2.44186
$295$	−23.6154	−1.37494
$296$	2.69141	0.156435
$297$	−74.1748	−4.30406
$298$	−22.2770	−1.29047
$299$	−6.33257	−0.366222
$300$	−7.52464	−0.434435
$301$	11.9649	0.689645
$302$	2.02255	0.116385
$303$	−35.8270	−2.05821
$304$	−8.43362	−0.483701
$305$	3.79392	0.217239
$306$	−31.6235	−1.80779
$307$	−12.1748	−0.694853	−0.347426	−	0.937707i	$-0.612944\pi$
$307$	−12.1748	−0.694853	−0.347426	+	0.937707i	$0.612944\pi$
$308$	23.5747	1.34330
$309$	4.33371	0.246536
$310$	−9.03232	−0.513001
$311$	−29.9743	−1.69969	−0.849844	−	0.527034i	$-0.823304\pi$
$311$	−29.9743	−1.69969	−0.849844	+	0.527034i	$0.823304\pi$
$312$	4.36750	0.247261
$313$	23.3820	1.32163	0.660814	−	0.750550i	$-0.270211\pi$
$313$	23.3820	1.32163	0.660814	+	0.750550i	$0.270211\pi$
$314$	−20.1691	−1.13821
$315$	89.2775	5.03022
$316$	−9.85183	−0.554209
$317$	27.4486	1.54166	0.770832	−	0.637038i	$-0.219841\pi$
$317$	27.4486	1.54166	0.770832	+	0.637038i	$0.219841\pi$
$318$	25.9329	1.45425
$319$	10.0286	0.561493
$320$	2.70863	0.151417
$321$	−3.09885	−0.172961
$322$	20.8819	1.16370
$323$	−36.1883	−2.01357
$324$	23.2045	1.28914
$325$	3.16919	0.175795
$326$	−16.4494	−0.911047
$327$	14.7546	0.815931
$328$	−7.88850	−0.435570
$329$	16.8094	0.926731
$330$	45.9775	2.53098
$331$	−19.0581	−1.04753	−0.523763	−	0.851864i	$-0.675472\pi$
$331$	−19.0581	−1.04753	−0.523763	+	0.851864i	$0.675472\pi$
$332$	10.9166	0.599129
$333$	−19.8352	−1.08696
$334$	−8.41094	−0.460226
$335$	32.1518	1.75664
$336$	−14.4020	−0.785691
$337$	36.2018	1.97204	0.986018	−	0.166642i	$-0.0532923\pi$
$337$	36.2018	1.97204	0.986018	+	0.166642i	$0.0532923\pi$
$338$	11.1605	0.607052
$339$	−38.9722	−2.11668
$340$	11.6226	0.630325
$341$	17.5776	0.951880
$342$	62.1540	3.36090
$343$	26.8430	1.44939
$344$	−2.67530	−0.144242
$345$	40.7256	2.19259
$346$	6.47837	0.348279
$347$	−11.6913	−0.627622	−0.313811	−	0.949485i	$-0.601606\pi$
$347$	−11.6913	−0.627622	−0.313811	+	0.949485i	$0.601606\pi$
$348$	−6.12653	−0.328417
$349$	−33.9689	−1.81831	−0.909156	−	0.416455i	$-0.863272\pi$
$349$	−33.9689	−1.81831	−0.909156	+	0.416455i	$0.863272\pi$
$350$	−10.4505	−0.558603
$351$	−19.0851	−1.01869
$352$	−5.27121	−0.280956
$353$	−35.9126	−1.91143	−0.955717	−	0.294288i	$-0.904917\pi$
$353$	−35.9126	−1.91143	−0.955717	+	0.294288i	$0.904917\pi$
$354$	−28.0757	−1.49221
$355$	−21.5932	−1.14605
$356$	−16.7401	−0.887223
$357$	−61.7982	−3.27071
$358$	6.00397	0.317319
$359$	−14.7814	−0.780133	−0.390067	−	0.920787i	$-0.627548\pi$
$359$	−14.7814	−0.780133	−0.390067	+	0.920787i	$0.627548\pi$
$360$	−19.9621	−1.05209
$361$	52.1259	2.74347
$362$	−25.5447	−1.34260
$363$	−54.0535	−2.83707
$364$	6.06574	0.317931
$365$	−20.8417	−1.09091
$366$	4.51048	0.235767
$367$	−27.7343	−1.44772	−0.723860	−	0.689947i	$-0.757634\pi$
$367$	−27.7343	−1.44772	−0.723860	+	0.689947i	$0.757634\pi$
$368$	−4.66909	−0.243393
$369$	58.1367	3.02647
$370$	7.29005	0.378992
$371$	36.0166	1.86989
$372$	−10.7383	−0.556753
$373$	18.0713	0.935699	0.467849	−	0.883808i	$-0.345029\pi$
$373$	18.0713	0.935699	0.467849	+	0.883808i	$0.345029\pi$
$374$	−22.6185	−1.16958
$375$	23.2304	1.19961
$376$	−3.75850	−0.193830
$377$	2.58034	0.132894
$378$	62.9336	3.23696
$379$	−4.01458	−0.206215	−0.103107	−	0.994670i	$-0.532879\pi$
$379$	−4.01458	−0.206215	−0.103107	+	0.994670i	$0.532879\pi$
$380$	−22.8436	−1.17185
$381$	44.6422	2.28709
$382$	12.5070	0.639912
$383$	−13.7815	−0.704201	−0.352101	−	0.935962i	$-0.614533\pi$
$383$	−13.7815	−0.704201	−0.352101	+	0.935962i	$0.614533\pi$
$384$	3.22022	0.164331
$385$	63.8553	3.25437
$386$	7.66398	0.390086
$387$	19.7164	1.00224
$388$	2.63120	0.133579
$389$	4.48620	0.227459	0.113730	−	0.993512i	$-0.463720\pi$
$389$	4.48620	0.227459	0.113730	+	0.993512i	$0.463720\pi$
$390$	11.8299	0.599032
$391$	−20.0349	−1.01321
$392$	−13.0020	−0.656700
$393$	−25.5627	−1.28947
$394$	6.96091	0.350686
$395$	−26.6850	−1.34267
$396$	38.8477	1.95217
$397$	30.2396	1.51768	0.758841	−	0.651275i	$-0.225766\pi$
$397$	30.2396	1.51768	0.758841	+	0.651275i	$0.225766\pi$
$398$	−0.613801	−0.0307671
$399$	121.461	6.08064
$400$	2.33669	0.116834
$401$	−17.5790	−0.877855	−0.438928	−	0.898522i	$-0.644642\pi$
$401$	−17.5790	−0.877855	−0.438928	+	0.898522i	$0.644642\pi$
$402$	38.2243	1.90646
$403$	4.52269	0.225291
$404$	11.1256	0.553521
$405$	62.8525	3.12316
$406$	−8.50876	−0.422283
$407$	−14.1870	−0.703224
$408$	13.8178	0.684083
$409$	26.8012	1.32524	0.662618	−	0.748958i	$-0.269445\pi$
$409$	26.8012	1.32524	0.662618	+	0.748958i	$0.269445\pi$
$410$	−21.3671	−1.05524
$411$	20.8136	1.02666
$412$	−1.34578	−0.0663020
$413$	−38.9925	−1.91870
$414$	34.4103	1.69117
$415$	29.5692	1.45149
$416$	−1.35627	−0.0664968
$417$	−52.4978	−2.57083
$418$	44.4554	2.17438
$419$	8.51366	0.415920	0.207960	−	0.978137i	$-0.433318\pi$
$419$	8.51366	0.415920	0.207960	+	0.978137i	$0.433318\pi$
$420$	−39.0096	−1.90347
$421$	−18.0140	−0.877950	−0.438975	−	0.898499i	$-0.644658\pi$
$421$	−18.0140	−0.877950	−0.438975	+	0.898499i	$0.644658\pi$
$422$	6.97139	0.339362
$423$	27.6994	1.34679
$424$	−8.05316	−0.391096
$425$	10.0266	0.486362
$426$	−25.6716	−1.24379
$427$	6.26433	0.303152
$428$	0.962311	0.0465150
$429$	−23.0220	−1.11151
$430$	−7.24640	−0.349452
$431$	−3.04963	−0.146896	−0.0734478	−	0.997299i	$-0.523400\pi$
$431$	−3.04963	−0.146896	−0.0734478	+	0.997299i	$0.523400\pi$
$432$	−14.0717	−0.677024
$433$	40.9454	1.96771	0.983854	−	0.178971i	$-0.0572768\pi$
$433$	40.9454	1.96771	0.983854	+	0.178971i	$0.0572768\pi$
$434$	−14.9137	−0.715881
$435$	−16.5945	−0.795646
$436$	−4.58186	−0.219432
$437$	39.3773	1.88367
$438$	−24.7782	−1.18395
$439$	30.1880	1.44080	0.720398	−	0.693561i	$-0.243959\pi$
$439$	30.1880	1.44080	0.720398	+	0.693561i	$0.243959\pi$
$440$	−14.2778	−0.680666
$441$	95.8220	4.56295
$442$	−5.81971	−0.276815
$443$	−22.9994	−1.09273	−0.546367	−	0.837546i	$-0.683990\pi$
$443$	−22.9994	−1.09273	−0.546367	+	0.837546i	$0.683990\pi$
$444$	8.66694	0.411315
$445$	−45.3428	−2.14945
$446$	−6.72630	−0.318500
$447$	−71.7369	−3.39304
$448$	4.47236	0.211299
$449$	−18.9251	−0.893131	−0.446566	−	0.894751i	$-0.647353\pi$
$449$	−18.9251	−0.893131	−0.446566	+	0.894751i	$0.647353\pi$
$450$	−17.2209	−0.811801
$451$	41.5820	1.95802
$452$	12.1024	0.569247
$453$	6.51305	0.306010
$454$	−13.4458	−0.631041
$455$	16.4299	0.770244
$456$	−27.1581	−1.27179
$457$	12.6548	0.591967	0.295983	−	0.955193i	$-0.404353\pi$
$457$	12.6548	0.591967	0.295983	+	0.955193i	$0.404353\pi$
$458$	17.0138	0.795004
$459$	−60.3810	−2.81834
$460$	−12.6469	−0.589663
$461$	1.23235	0.0573964	0.0286982	−	0.999588i	$-0.490864\pi$
$461$	1.23235	0.0573964	0.0286982	+	0.999588i	$0.490864\pi$
$462$	75.9158	3.53192
$463$	2.86881	0.133325	0.0666624	−	0.997776i	$-0.478765\pi$
$463$	2.86881	0.133325	0.0666624	+	0.997776i	$0.478765\pi$
$464$	1.90252	0.0883224
$465$	−29.0860	−1.34883
$466$	6.69008	0.309912
$467$	−33.7132	−1.56006	−0.780030	−	0.625742i	$-0.784796\pi$
$467$	−33.7132	−1.56006	−0.780030	+	0.625742i	$0.784796\pi$
$468$	9.99546	0.462040
$469$	53.0874	2.45135
$470$	−10.1804	−0.469587
$471$	−64.9488	−2.99268
$472$	8.71857	0.401305
$473$	14.1021	0.648413
$474$	−31.7250	−1.45718
$475$	−19.7067	−0.904206
$476$	19.1907	0.879603
$477$	59.3501	2.71746
$478$	12.0507	0.551185
$479$	−24.0891	−1.10066	−0.550331	−	0.834947i	$-0.685498\pi$
$479$	−24.0891	−1.10066	−0.550331	+	0.834947i	$0.685498\pi$
$480$	8.72238	0.398120
$481$	−3.65029	−0.166439
$482$	0.371642	0.0169279
$483$	67.2441	3.05971
$484$	16.7857	0.762985
$485$	7.12695	0.323618
$486$	32.5085	1.47461
$487$	−3.00528	−0.136182	−0.0680911	−	0.997679i	$-0.521691\pi$
$487$	−3.00528	−0.136182	−0.0680911	+	0.997679i	$0.521691\pi$
$488$	−1.40068	−0.0634056
$489$	−52.9706	−2.39541
$490$	−35.2176	−1.59097
$491$	−37.6648	−1.69979	−0.849894	−	0.526954i	$-0.823334\pi$
$491$	−37.6648	−1.69979	−0.849894	+	0.526954i	$0.823334\pi$
$492$	−25.4027	−1.14524
$493$	8.16363	0.367672
$494$	11.4383	0.514633
$495$	105.224	4.72948
$496$	3.33464	0.149730
$497$	−35.6536	−1.59928
$498$	35.1540	1.57529
$499$	−14.8086	−0.662924	−0.331462	−	0.943468i	$-0.607542\pi$
$499$	−14.8086	−0.662924	−0.331462	+	0.943468i	$0.607542\pi$
$500$	−7.21393	−0.322617
$501$	−27.0851	−1.21007
$502$	−30.5249	−1.36239
$503$	27.2322	1.21423	0.607113	−	0.794616i	$-0.292328\pi$
$503$	27.2322	1.21423	0.607113	+	0.794616i	$0.292328\pi$
$504$	−32.9604	−1.46817
$505$	30.1353	1.34100
$506$	24.6118	1.09413
$507$	35.9393	1.59612
$508$	−13.8631	−0.615076
$509$	7.33009	0.324900	0.162450	−	0.986717i	$-0.448060\pi$
$509$	7.33009	0.324900	0.162450	+	0.986717i	$0.448060\pi$
$510$	37.4273	1.65731
$511$	−34.4129	−1.52233
$512$	−1.00000	−0.0441942
$513$	118.675	5.23964
$514$	26.6734	1.17651
$515$	−3.64523	−0.160628
$516$	−8.61504	−0.379256
$517$	19.8119	0.871325
$518$	12.0370	0.528874
$519$	20.8617	0.915729
$520$	−3.67365	−0.161100
$521$	24.6872	1.08156	0.540782	−	0.841163i	$-0.318128\pi$
$521$	24.6872	1.08156	0.540782	+	0.841163i	$0.318128\pi$
$522$	−14.0212	−0.613691
$523$	−0.585554	−0.0256045	−0.0128023	−	0.999918i	$-0.504075\pi$
$523$	−0.585554	−0.0256045	−0.0128023	+	0.999918i	$0.504075\pi$
$524$	7.93819	0.346782
$525$	−33.6529	−1.46873
$526$	−5.29395	−0.230827
$527$	14.3088	0.623301
$528$	−16.9744	−0.738718
$529$	−1.19958	−0.0521555
$530$	−21.8130	−0.947498
$531$	−64.2540	−2.78839
$532$	−37.7181	−1.63529
$533$	10.6990	0.463424
$534$	−53.9067	−2.33277
$535$	2.60655	0.112691
$536$	−11.8701	−0.512711
$537$	19.3341	0.834327
$538$	−1.11650	−0.0481359
$539$	68.5362	2.95206
$540$	−38.1150	−1.64021
$541$	0.553653	0.0238034	0.0119017	−	0.999929i	$-0.496211\pi$
$541$	0.553653	0.0238034	0.0119017	+	0.999929i	$0.496211\pi$
$542$	4.15237	0.178359
$543$	−82.2594	−3.53009
$544$	−4.29095	−0.183973
$545$	−12.4106	−0.531611
$546$	19.5330	0.835936
$547$	−15.1433	−0.647479	−0.323740	−	0.946146i	$-0.604940\pi$
$547$	−15.1433	−0.647479	−0.323740	+	0.946146i	$0.604940\pi$
$548$	−6.46343	−0.276104
$549$	10.3227	0.440562
$550$	−12.3172	−0.525206
$551$	−16.0451	−0.683546
$552$	−15.0355	−0.639953
$553$	−44.0609	−1.87366
$554$	9.75276	0.414355
$555$	23.4755	0.996481
$556$	16.3026	0.691382
$557$	4.96361	0.210315	0.105157	−	0.994456i	$-0.466465\pi$
$557$	4.96361	0.210315	0.105157	+	0.994456i	$0.466465\pi$
$558$	−24.5756	−1.04037
$559$	3.62844	0.153467
$560$	12.1140	0.511909
$561$	−72.8365	−3.07516
$562$	−0.670679	−0.0282909
$563$	−19.5324	−0.823194	−0.411597	−	0.911366i	$-0.635029\pi$
$563$	−19.5324	−0.823194	−0.411597	+	0.911366i	$0.635029\pi$
$564$	−12.1032	−0.509637
$565$	32.7809	1.37910
$566$	−7.53352	−0.316657
$567$	103.779	4.35830
$568$	7.97200	0.334498
$569$	32.7035	1.37100	0.685500	−	0.728072i	$-0.259584\pi$
$569$	32.7035	1.37100	0.685500	+	0.728072i	$0.259584\pi$
$570$	−73.5612	−3.08114
$571$	−5.43789	−0.227569	−0.113784	−	0.993505i	$-0.536297\pi$
$571$	−5.43789	−0.227569	−0.113784	+	0.993505i	$0.536297\pi$
$572$	7.14921	0.298923
$573$	40.2751	1.68252
$574$	−35.2802	−1.47257
$575$	−10.9102	−0.454987
$576$	7.36980	0.307075
$577$	20.8181	0.866670	0.433335	−	0.901233i	$-0.357337\pi$
$577$	20.8181	0.866670	0.433335	+	0.901233i	$0.357337\pi$
$578$	−1.41228	−0.0587432
$579$	24.6797	1.02565
$580$	5.15323	0.213976
$581$	48.8231	2.02552
$582$	8.47303	0.351219
$583$	42.4499	1.75809
$584$	7.69457	0.318403
$585$	27.0740	1.11937
$586$	−14.4351	−0.596310
$587$	10.8339	0.447162	0.223581	−	0.974685i	$-0.428225\pi$
$587$	10.8339	0.447162	0.223581	+	0.974685i	$0.428225\pi$
$588$	−41.8692	−1.72666
$589$	−28.1231	−1.15879
$590$	23.6154	0.972230
$591$	22.4156	0.922056
$592$	−2.69141	−0.110616
$593$	−23.1100	−0.949014	−0.474507	−	0.880252i	$-0.657374\pi$
$593$	−23.1100	−0.949014	−0.474507	+	0.880252i	$0.657374\pi$
$594$	74.1748	3.04343
$595$	51.9805	2.13099
$596$	22.2770	0.912503
$597$	−1.97657	−0.0808957
$598$	6.33257	0.258958
$599$	−5.75408	−0.235106	−0.117553	−	0.993067i	$-0.537505\pi$
$599$	−5.75408	−0.235106	−0.117553	+	0.993067i	$0.537505\pi$
$600$	7.52464	0.307192
$601$	−48.0578	−1.96032	−0.980160	−	0.198207i	$-0.936488\pi$
$601$	−48.0578	−1.96032	−0.980160	+	0.198207i	$0.936488\pi$
$602$	−11.9649	−0.487653
$603$	87.4803	3.56247
$604$	−2.02255	−0.0822964
$605$	45.4662	1.84846
$606$	35.8270	1.45537
$607$	−22.6532	−0.919466	−0.459733	−	0.888057i	$-0.652055\pi$
$607$	−22.6532	−0.919466	−0.459733	+	0.888057i	$0.652055\pi$
$608$	8.43362	0.342028
$609$	−27.4000	−1.11031
$610$	−3.79392	−0.153611
$611$	5.09756	0.206225
$612$	31.6235	1.27830
$613$	42.2989	1.70844	0.854219	−	0.519913i	$-0.174036\pi$
$613$	42.2989	1.70844	0.854219	+	0.519913i	$0.174036\pi$
$614$	12.1748	0.491335
$615$	−68.8066	−2.77455
$616$	−23.5747	−0.949853
$617$	12.0080	0.483423	0.241712	−	0.970348i	$-0.422291\pi$
$617$	12.0080	0.483423	0.241712	+	0.970348i	$0.422291\pi$
$618$	−4.33371	−0.174328
$619$	−6.44800	−0.259167	−0.129583	−	0.991569i	$-0.541364\pi$
$619$	−6.44800	−0.259167	−0.129583	+	0.991569i	$0.541364\pi$
$620$	9.03232	0.362747
$621$	65.7020	2.63653
$622$	29.9743	1.20186
$623$	−74.8677	−2.99951
$624$	−4.36750	−0.174840
$625$	−31.2233	−1.24893
$626$	−23.3820	−0.934532
$627$	143.156	5.71710
$628$	20.1691	0.804834
$629$	−11.5487	−0.460478
$630$	−89.2775	−3.55690
$631$	33.8544	1.34772	0.673862	−	0.738857i	$-0.264634\pi$
$631$	33.8544	1.34772	0.673862	+	0.738857i	$0.264634\pi$
$632$	9.85183	0.391885
$633$	22.4494	0.892283
$634$	−27.4486	−1.09012
$635$	−37.5501	−1.49013
$636$	−25.9329	−1.02831
$637$	17.6343	0.698695
$638$	−10.0286	−0.397036
$639$	−58.7520	−2.32419
$640$	−2.70863	−0.107068
$641$	−32.6875	−1.29108	−0.645540	−	0.763726i	$-0.723368\pi$
$641$	−32.6875	−1.29108	−0.645540	+	0.763726i	$0.723368\pi$
$642$	3.09885	0.122302
$643$	7.25077	0.285943	0.142971	−	0.989727i	$-0.454334\pi$
$643$	7.25077	0.285943	0.142971	+	0.989727i	$0.454334\pi$
$644$	−20.8819	−0.822860
$645$	−23.3350	−0.918814
$646$	36.1883	1.42381
$647$	47.9375	1.88462	0.942308	−	0.334747i	$-0.108651\pi$
$647$	47.9375	1.88462	0.942308	+	0.334747i	$0.108651\pi$
$648$	−23.2045	−0.911559
$649$	−45.9574	−1.80399
$650$	−3.16919	−0.124306
$651$	−48.0254	−1.88226
$652$	16.4494	0.644208
$653$	−23.0320	−0.901312	−0.450656	−	0.892698i	$-0.648810\pi$
$653$	−23.0320	−0.901312	−0.450656	+	0.892698i	$0.648810\pi$
$654$	−14.7546	−0.576950
$655$	21.5016	0.840139
$656$	7.88850	0.307994
$657$	−56.7074	−2.21237
$658$	−16.8094	−0.655298
$659$	−11.3318	−0.441425	−0.220713	−	0.975339i	$-0.570838\pi$
$659$	−11.3318	−0.441425	−0.220713	+	0.975339i	$0.570838\pi$
$660$	−45.9775	−1.78967
$661$	31.8968	1.24064	0.620322	−	0.784348i	$-0.287002\pi$
$661$	31.8968	1.24064	0.620322	+	0.784348i	$0.287002\pi$
$662$	19.0581	0.740712
$663$	−18.7407	−0.727829
$664$	−10.9166	−0.423648
$665$	−102.165	−3.96177
$666$	19.8352	0.768597
$667$	−8.88305	−0.343953
$668$	8.41094	0.325429
$669$	−21.6601	−0.837430
$670$	−32.1518	−1.24213
$671$	7.38326	0.285028
$672$	14.4020	0.555568
$673$	26.0084	1.00255	0.501276	−	0.865287i	$-0.332864\pi$
$673$	26.0084	1.00255	0.501276	+	0.865287i	$0.332864\pi$
$674$	−36.2018	−1.39444
$675$	−32.8811	−1.26560
$676$	−11.1605	−0.429251
$677$	20.6256	0.792705	0.396352	−	0.918098i	$-0.370276\pi$
$677$	20.6256	0.792705	0.396352	+	0.918098i	$0.370276\pi$
$678$	38.9722	1.49672
$679$	11.7677	0.451602
$680$	−11.6226	−0.445707
$681$	−43.2983	−1.65919
$682$	−17.5776	−0.673081
$683$	24.7685	0.947741	0.473871	−	0.880594i	$-0.342856\pi$
$683$	24.7685	0.947741	0.473871	+	0.880594i	$0.342856\pi$
$684$	−62.1540	−2.37652
$685$	−17.5070	−0.668910
$686$	−26.8430	−1.02487
$687$	54.7882	2.09030
$688$	2.67530	0.101995
$689$	10.9223	0.416106
$690$	−40.7256	−1.55040
$691$	−42.5685	−1.61938	−0.809691	−	0.586857i	$-0.800365\pi$
$691$	−42.5685	−1.61938	−0.809691	+	0.586857i	$0.800365\pi$
$692$	−6.47837	−0.246271
$693$	173.741	6.59988
$694$	11.6913	0.443796
$695$	44.1576	1.67499
$696$	6.12653	0.232226
$697$	33.8492	1.28213
$698$	33.9689	1.28574
$699$	21.5435	0.814850
$700$	10.4505	0.394992
$701$	35.3691	1.33587	0.667936	−	0.744219i	$-0.267178\pi$
$701$	35.3691	1.33587	0.667936	+	0.744219i	$0.267178\pi$
$702$	19.0851	0.720319
$703$	22.6983	0.856084
$704$	5.27121	0.198666
$705$	−32.7831	−1.23468
$706$	35.9126	1.35159
$707$	49.7579	1.87134
$708$	28.0757	1.05515
$709$	−0.673330	−0.0252875	−0.0126437	−	0.999920i	$-0.504025\pi$
$709$	−0.673330	−0.0252875	−0.0126437	+	0.999920i	$0.504025\pi$
$710$	21.5932	0.810379
$711$	−72.6060	−2.72294
$712$	16.7401	0.627362
$713$	−15.5697	−0.583092
$714$	61.7982	2.31274
$715$	19.3646	0.724194
$716$	−6.00397	−0.224379
$717$	38.8058	1.44923
$718$	14.7814	0.551638
$719$	3.70087	0.138019	0.0690095	−	0.997616i	$-0.478016\pi$
$719$	3.70087	0.138019	0.0690095	+	0.997616i	$0.478016\pi$
$720$	19.9621	0.743942
$721$	−6.01882	−0.224153
$722$	−52.1259	−1.93992
$723$	1.19677	0.0445083
$724$	25.5447	0.949360
$725$	4.44560	0.165105
$726$	54.0535	2.00611
$727$	49.0757	1.82012	0.910058	−	0.414482i	$-0.136037\pi$
$727$	49.0757	1.82012	0.910058	+	0.414482i	$0.136037\pi$
$728$	−6.06574	−0.224811
$729$	35.0708	1.29892
$730$	20.8417	0.771388
$731$	11.4796	0.424588
$732$	−4.51048	−0.166712
$733$	−7.68596	−0.283887	−0.141944	−	0.989875i	$-0.545335\pi$
$733$	−7.68596	−0.283887	−0.141944	+	0.989875i	$0.545335\pi$
$734$	27.7343	1.02369
$735$	−113.408	−4.18313
$736$	4.66909	0.172105
$737$	62.5699	2.30479
$738$	−58.1367	−2.14004
$739$	−28.9762	−1.06591	−0.532953	−	0.846145i	$-0.678918\pi$
$739$	−28.9762	−1.06591	−0.532953	+	0.846145i	$0.678918\pi$
$740$	−7.29005	−0.267988
$741$	36.8338	1.35312
$742$	−36.0166	−1.32221
$743$	−14.6035	−0.535750	−0.267875	−	0.963454i	$-0.586321\pi$
$743$	−14.6035	−0.535750	−0.267875	+	0.963454i	$0.586321\pi$
$744$	10.7383	0.393684
$745$	60.3403	2.21070
$746$	−18.0713	−0.661639
$747$	80.4534	2.94364
$748$	22.6185	0.827015
$749$	4.30380	0.157257
$750$	−23.2304	−0.848255
$751$	15.0966	0.550883	0.275441	−	0.961318i	$-0.411176\pi$
$751$	15.0966	0.550883	0.275441	+	0.961318i	$0.411176\pi$
$752$	3.75850	0.137059
$753$	−98.2968	−3.58213
$754$	−2.58034	−0.0939705
$755$	−5.47834	−0.199377
$756$	−62.9336	−2.28887
$757$	−37.4889	−1.36256	−0.681278	−	0.732025i	$-0.738576\pi$
$757$	−37.4889	−1.36256	−0.681278	+	0.732025i	$0.738576\pi$
$758$	4.01458	0.145816
$759$	79.2552	2.87678
$760$	22.8436	0.828623
$761$	11.3459	0.411290	0.205645	−	0.978627i	$-0.434071\pi$
$761$	11.3459	0.411290	0.205645	+	0.978627i	$0.434071\pi$
$762$	−44.6422	−1.61722
$763$	−20.4917	−0.741851
$764$	−12.5070	−0.452486
$765$	85.6563	3.09691
$766$	13.7815	0.497946
$767$	−11.8248	−0.426968
$768$	−3.22022	−0.116200
$769$	−51.3414	−1.85142	−0.925709	−	0.378236i	$-0.876531\pi$
$769$	−51.3414	−1.85142	−0.925709	+	0.378236i	$0.876531\pi$
$770$	−63.8553	−2.30119
$771$	85.8940	3.09340
$772$	−7.66398	−0.275833
$773$	−43.1690	−1.55268	−0.776340	−	0.630314i	$-0.782926\pi$
$773$	−43.1690	−1.55268	−0.776340	+	0.630314i	$0.782926\pi$
$774$	−19.7164	−0.708691
$775$	7.79201	0.279897
$776$	−2.63120	−0.0944546
$777$	38.7616	1.39057
$778$	−4.48620	−0.160838
$779$	−66.5286	−2.38363
$780$	−11.8299	−0.423580
$781$	−42.0221	−1.50367
$782$	20.0349	0.716445
$783$	−26.7717	−0.956742
$784$	13.0020	0.464357
$785$	54.6306	1.94985
$786$	25.5627	0.911791
$787$	35.8506	1.27794	0.638968	−	0.769233i	$-0.279362\pi$
$787$	35.8506	1.27794	0.638968	+	0.769233i	$0.279362\pi$
$788$	−6.96091	−0.247972
$789$	−17.0477	−0.606913
$790$	26.6850	0.949409
$791$	54.1261	1.92450
$792$	−38.8477	−1.38039
$793$	1.89970	0.0674604
$794$	−30.2396	−1.07316
$795$	−70.2427	−2.49125
$796$	0.613801	0.0217556
$797$	−51.8556	−1.83682	−0.918410	−	0.395631i	$-0.870526\pi$
$797$	−51.8556	−1.83682	−0.918410	+	0.395631i	$0.870526\pi$
$798$	−121.461	−4.29966
$799$	16.1276	0.570552
$800$	−2.33669	−0.0826144
$801$	−123.371	−4.35910
$802$	17.5790	0.620737
$803$	−40.5597	−1.43132
$804$	−38.2243	−1.34807
$805$	−56.5613	−1.99352
$806$	−4.52269	−0.159305
$807$	−3.59538	−0.126563
$808$	−11.1256	−0.391399
$809$	−17.2680	−0.607110	−0.303555	−	0.952814i	$-0.598174\pi$
$809$	−17.2680	−0.607110	−0.303555	+	0.952814i	$0.598174\pi$
$810$	−62.8525	−2.20841
$811$	7.99784	0.280842	0.140421	−	0.990092i	$-0.455154\pi$
$811$	7.99784	0.280842	0.140421	+	0.990092i	$0.455154\pi$
$812$	8.50876	0.298599
$813$	13.3715	0.468960
$814$	14.1870	0.497254
$815$	44.5553	1.56071
$816$	−13.8178	−0.483720
$817$	−22.5624	−0.789360
$818$	−26.8012	−0.937083
$819$	44.7033	1.56206
$820$	21.3671	0.746170
$821$	−14.3358	−0.500323	−0.250161	−	0.968204i	$-0.580484\pi$
$821$	−14.3358	−0.500323	−0.250161	+	0.968204i	$0.580484\pi$
$822$	−20.8136	−0.725959
$823$	−48.1283	−1.67765	−0.838823	−	0.544404i	$-0.816756\pi$
$823$	−48.1283	−1.67765	−0.838823	+	0.544404i	$0.816756\pi$
$824$	1.34578	0.0468826
$825$	−39.6640	−1.38092
$826$	38.9925	1.35672
$827$	3.21283	0.111721	0.0558605	−	0.998439i	$-0.482210\pi$
$827$	3.21283	0.111721	0.0558605	+	0.998439i	$0.482210\pi$
$828$	−34.4103	−1.19584
$829$	29.6107	1.02842	0.514211	−	0.857664i	$-0.328085\pi$
$829$	29.6107	1.02842	0.514211	+	0.857664i	$0.328085\pi$
$830$	−29.5692	−1.02636
$831$	31.4060	1.08946
$832$	1.35627	0.0470203
$833$	55.7909	1.93304
$834$	52.4978	1.81785
$835$	22.7821	0.788409
$836$	−44.4554	−1.53752
$837$	−46.9240	−1.62193
$838$	−8.51366	−0.294100
$839$	−19.4676	−0.672097	−0.336049	−	0.941845i	$-0.609091\pi$
$839$	−19.4676	−0.672097	−0.336049	+	0.941845i	$0.609091\pi$
$840$	39.0096	1.34596
$841$	−25.3804	−0.875187
$842$	18.0140	0.620804
$843$	−2.15973	−0.0743851
$844$	−6.97139	−0.239965
$845$	−30.2297	−1.03993
$846$	−27.6994	−0.952325
$847$	75.0715	2.57949
$848$	8.05316	0.276547
$849$	−24.2596	−0.832586
$850$	−10.0266	−0.343910
$851$	12.5665	0.430773
$852$	25.6716	0.879493
$853$	35.5710	1.21793	0.608964	−	0.793198i	$-0.291585\pi$
$853$	35.5710	1.21793	0.608964	+	0.793198i	$0.291585\pi$
$854$	−6.26433	−0.214361
$855$	−168.352	−5.75753
$856$	−0.962311	−0.0328911
$857$	−10.9339	−0.373496	−0.186748	−	0.982408i	$-0.559795\pi$
$857$	−10.9339	−0.373496	−0.186748	+	0.982408i	$0.559795\pi$
$858$	23.0220	0.785958
$859$	−22.5032	−0.767801	−0.383900	−	0.923374i	$-0.625419\pi$
$859$	−22.5032	−0.767801	−0.383900	+	0.923374i	$0.625419\pi$
$860$	7.24640	0.247100
$861$	−113.610	−3.87182
$862$	3.04963	0.103871
$863$	−8.05723	−0.274271	−0.137136	−	0.990552i	$-0.543790\pi$
$863$	−8.05723	−0.274271	−0.137136	+	0.990552i	$0.543790\pi$
$864$	14.0717	0.478729
$865$	−17.5475	−0.596633
$866$	−40.9454	−1.39138
$867$	−4.54786	−0.154453
$868$	14.9137	0.506204
$869$	−51.9311	−1.76164
$870$	16.5945	0.562607
$871$	16.0991	0.545498
$872$	4.58186	0.155162
$873$	19.3914	0.656300
$874$	−39.3773	−1.33196
$875$	−32.2633	−1.09070
$876$	24.7782	0.837177
$877$	−5.09471	−0.172036	−0.0860180	−	0.996294i	$-0.527414\pi$
$877$	−5.09471	−0.172036	−0.0860180	+	0.996294i	$0.527414\pi$
$878$	−30.1880	−1.01880
$879$	−46.4843	−1.56788
$880$	14.2778	0.481304
$881$	0.780401	0.0262924	0.0131462	−	0.999914i	$-0.495815\pi$
$881$	0.780401	0.0262924	0.0131462	+	0.999914i	$0.495815\pi$
$882$	−95.8220	−3.22649
$883$	−41.8731	−1.40914	−0.704571	−	0.709633i	$-0.748861\pi$
$883$	−41.8731	−1.40914	−0.704571	+	0.709633i	$0.748861\pi$
$884$	5.81971	0.195738
$885$	76.0467	2.55628
$886$	22.9994	0.772679
$887$	28.5145	0.957423	0.478712	−	0.877972i	$-0.341104\pi$
$887$	28.5145	0.957423	0.478712	+	0.877972i	$0.341104\pi$
$888$	−8.66694	−0.290843
$889$	−62.0008	−2.07944
$890$	45.3428	1.51989
$891$	122.316	4.09773
$892$	6.72630	0.225213
$893$	−31.6978	−1.06073
$894$	71.7369	2.39924
$895$	−16.2625	−0.543597
$896$	−4.47236	−0.149411
$897$	20.3922	0.680877
$898$	18.9251	0.631539
$899$	6.34423	0.211592
$900$	17.2209	0.574030
$901$	34.5557	1.15122
$902$	−41.5820	−1.38453
$903$	−38.5295	−1.28218
$904$	−12.1024	−0.402519
$905$	69.1911	2.29999
$906$	−6.51305	−0.216382
$907$	47.3692	1.57287	0.786435	−	0.617673i	$-0.211925\pi$
$907$	47.3692	1.57287	0.786435	+	0.617673i	$0.211925\pi$
$908$	13.4458	0.446213
$909$	81.9937	2.71956
$910$	−16.4299	−0.544645
$911$	−1.00262	−0.0332184	−0.0166092	−	0.999862i	$-0.505287\pi$
$911$	−1.00262	−0.0332184	−0.0166092	+	0.999862i	$0.505287\pi$
$912$	27.1581	0.899294
$913$	57.5439	1.90443
$914$	−12.6548	−0.418584
$915$	−12.2172	−0.403889
$916$	−17.0138	−0.562153
$917$	35.5024	1.17239
$918$	60.3810	1.99287
$919$	56.3140	1.85763	0.928813	−	0.370549i	$-0.120830\pi$
$919$	56.3140	1.85763	0.928813	+	0.370549i	$0.120830\pi$
$920$	12.6469	0.416954
$921$	39.2055	1.29187
$922$	−1.23235	−0.0405854
$923$	−10.8122	−0.355888
$924$	−75.9158	−2.49745
$925$	−6.28899	−0.206781
$926$	−2.86881	−0.0942748
$927$	−9.91815	−0.325755
$928$	−1.90252	−0.0624533
$929$	9.61789	0.315553	0.157776	−	0.987475i	$-0.449567\pi$
$929$	9.61789	0.315553	0.157776	+	0.987475i	$0.449567\pi$
$930$	29.0860	0.953768
$931$	−109.654	−3.59376
$932$	−6.69008	−0.219141
$933$	96.5238	3.16005
$934$	33.7132	1.10313
$935$	61.2652	2.00359
$936$	−9.99546	−0.326712
$937$	16.4083	0.536037	0.268018	−	0.963414i	$-0.413631\pi$
$937$	16.4083	0.536037	0.268018	+	0.963414i	$0.413631\pi$
$938$	−53.0874	−1.73336
$939$	−75.2950	−2.45716
$940$	10.1804	0.332048
$941$	2.88007	0.0938877	0.0469438	−	0.998898i	$-0.485052\pi$
$941$	2.88007	0.0938877	0.0469438	+	0.998898i	$0.485052\pi$
$942$	64.9488	2.11615
$943$	−36.8322	−1.19942
$944$	−8.71857	−0.283765
$945$	−170.464	−5.54520
$946$	−14.1021	−0.458498
$947$	−53.0265	−1.72313	−0.861566	−	0.507646i	$-0.830516\pi$
$947$	−53.0265	−1.72313	−0.861566	+	0.507646i	$0.830516\pi$
$948$	31.7250	1.03038
$949$	−10.4359	−0.338765
$950$	19.7067	0.639371
$951$	−88.3903	−2.86625
$952$	−19.1907	−0.621974
$953$	−54.1430	−1.75386	−0.876932	−	0.480615i	$-0.840414\pi$
$953$	−54.1430	−1.75386	−0.876932	+	0.480615i	$0.840414\pi$
$954$	−59.3501	−1.92153
$955$	−33.8768	−1.09623
$956$	−12.0507	−0.389746
$957$	−32.2942	−1.04392
$958$	24.0891	0.778285
$959$	−28.9068	−0.933448
$960$	−8.72238	−0.281514
$961$	−19.8802	−0.641296
$962$	3.65029	0.117690
$963$	7.09203	0.228538
$964$	−0.371642	−0.0119698
$965$	−20.7589	−0.668253
$966$	−67.2441	−2.16354
$967$	27.4727	0.883462	0.441731	−	0.897148i	$-0.354365\pi$
$967$	27.4727	0.883462	0.441731	+	0.897148i	$0.354365\pi$
$968$	−16.7857	−0.539512
$969$	116.534	3.74361
$970$	−7.12695	−0.228833
$971$	51.2526	1.64477	0.822387	−	0.568929i	$-0.192642\pi$
$971$	51.2526	1.64477	0.822387	+	0.568929i	$0.192642\pi$
$972$	−32.5085	−1.04271
$973$	72.9109	2.33742
$974$	3.00528	0.0962954
$975$	−10.2055	−0.326837
$976$	1.40068	0.0448346
$977$	45.7576	1.46392	0.731958	−	0.681350i	$-0.238607\pi$
$977$	45.7576	1.46392	0.731958	+	0.681350i	$0.238607\pi$
$978$	52.9706	1.69381
$979$	−88.2406	−2.82018
$980$	35.2176	1.12499
$981$	−33.7674	−1.07811
$982$	37.6648	1.20193
$983$	30.6722	0.978290	0.489145	−	0.872202i	$-0.337309\pi$
$983$	30.6722	0.978290	0.489145	+	0.872202i	$0.337309\pi$
$984$	25.4027	0.809808
$985$	−18.8545	−0.600756
$986$	−8.16363	−0.259983
$987$	−54.1298	−1.72297
$988$	−11.4383	−0.363901
$989$	−12.4912	−0.397197
$990$	−105.224	−3.34425
$991$	54.7054	1.73777	0.868887	−	0.495010i	$-0.164835\pi$
$991$	54.7054	1.73777	0.868887	+	0.495010i	$0.164835\pi$
$992$	−3.33464	−0.105875
$993$	61.3711	1.94755
$994$	35.6536	1.13086
$995$	1.66256	0.0527067
$996$	−35.1540	−1.11390
$997$	−11.2506	−0.356309	−0.178154	−	0.984003i	$-0.557013\pi$
$997$	−11.2506	−0.356309	−0.178154	+	0.984003i	$0.557013\pi$
$998$	14.8086	0.468758
$999$	37.8727	1.19824

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.2	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.22022 q^{3} +1.00000 q^{4} +2.70863 q^{5} +3.22022 q^{6} +4.47236 q^{7} -1.00000 q^{8} +7.36980 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.22022 q^{3} +1.00000 q^{4} +2.70863 q^{5} +3.22022 q^{6} +4.47236 q^{7} -1.00000 q^{8} +7.36980 q^{9} -2.70863 q^{10} +5.27121 q^{11} -3.22022 q^{12} +1.35627 q^{13} -4.47236 q^{14} -8.72238 q^{15} +1.00000 q^{16} +4.29095 q^{17} -7.36980 q^{18} -8.43362 q^{19} +2.70863 q^{20} -14.4020 q^{21} -5.27121 q^{22} -4.66909 q^{23} +3.22022 q^{24} +2.33669 q^{25} -1.35627 q^{26} -14.0717 q^{27} +4.47236 q^{28} +1.90252 q^{29} +8.72238 q^{30} +3.33464 q^{31} -1.00000 q^{32} -16.9744 q^{33} -4.29095 q^{34} +12.1140 q^{35} +7.36980 q^{36} -2.69141 q^{37} +8.43362 q^{38} -4.36750 q^{39} -2.70863 q^{40} +7.88850 q^{41} +14.4020 q^{42} +2.67530 q^{43} +5.27121 q^{44} +19.9621 q^{45} +4.66909 q^{46} +3.75850 q^{47} -3.22022 q^{48} +13.0020 q^{49} -2.33669 q^{50} -13.8178 q^{51} +1.35627 q^{52} +8.05316 q^{53} +14.0717 q^{54} +14.2778 q^{55} -4.47236 q^{56} +27.1581 q^{57} -1.90252 q^{58} -8.71857 q^{59} -8.72238 q^{60} +1.40068 q^{61} -3.33464 q^{62} +32.9604 q^{63} +1.00000 q^{64} +3.67365 q^{65} +16.9744 q^{66} +11.8701 q^{67} +4.29095 q^{68} +15.0355 q^{69} -12.1140 q^{70} -7.97200 q^{71} -7.36980 q^{72} -7.69457 q^{73} +2.69141 q^{74} -7.52464 q^{75} -8.43362 q^{76} +23.5747 q^{77} +4.36750 q^{78} -9.85183 q^{79} +2.70863 q^{80} +23.2045 q^{81} -7.88850 q^{82} +10.9166 q^{83} -14.4020 q^{84} +11.6226 q^{85} -2.67530 q^{86} -6.12653 q^{87} -5.27121 q^{88} -16.7401 q^{89} -19.9621 q^{90} +6.06574 q^{91} -4.66909 q^{92} -10.7383 q^{93} -3.75850 q^{94} -22.8436 q^{95} +3.22022 q^{96} +2.63120 q^{97} -13.0020 q^{98} +38.8477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 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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