LMFDB - Embedded newform 8043.2.a.r.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90871 q^{2} -1.00000 q^{3} +1.64317 q^{4} +2.98743 q^{5} +1.90871 q^{6} -1.00000 q^{7} +0.681087 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.90871 q^{2} -1.00000 q^{3} +1.64317 q^{4} +2.98743 q^{5} +1.90871 q^{6} -1.00000 q^{7} +0.681087 q^{8} +1.00000 q^{9} -5.70214 q^{10} +0.0579542 q^{11} -1.64317 q^{12} -2.15097 q^{13} +1.90871 q^{14} -2.98743 q^{15} -4.58633 q^{16} +0.787689 q^{17} -1.90871 q^{18} +5.09093 q^{19} +4.90885 q^{20} +1.00000 q^{21} -0.110618 q^{22} -4.55088 q^{23} -0.681087 q^{24} +3.92475 q^{25} +4.10558 q^{26} -1.00000 q^{27} -1.64317 q^{28} -2.45471 q^{29} +5.70214 q^{30} +4.20126 q^{31} +7.39180 q^{32} -0.0579542 q^{33} -1.50347 q^{34} -2.98743 q^{35} +1.64317 q^{36} -1.61258 q^{37} -9.71709 q^{38} +2.15097 q^{39} +2.03470 q^{40} -1.19436 q^{41} -1.90871 q^{42} -7.27102 q^{43} +0.0952285 q^{44} +2.98743 q^{45} +8.68630 q^{46} +2.26126 q^{47} +4.58633 q^{48} +1.00000 q^{49} -7.49120 q^{50} -0.787689 q^{51} -3.53441 q^{52} -8.28090 q^{53} +1.90871 q^{54} +0.173134 q^{55} -0.681087 q^{56} -5.09093 q^{57} +4.68533 q^{58} -9.80775 q^{59} -4.90885 q^{60} +8.02840 q^{61} -8.01898 q^{62} -1.00000 q^{63} -4.93613 q^{64} -6.42588 q^{65} +0.110618 q^{66} -4.14642 q^{67} +1.29431 q^{68} +4.55088 q^{69} +5.70214 q^{70} +13.6486 q^{71} +0.681087 q^{72} +10.6181 q^{73} +3.07794 q^{74} -3.92475 q^{75} +8.36525 q^{76} -0.0579542 q^{77} -4.10558 q^{78} +3.17309 q^{79} -13.7014 q^{80} +1.00000 q^{81} +2.27968 q^{82} +11.7530 q^{83} +1.64317 q^{84} +2.35317 q^{85} +13.8783 q^{86} +2.45471 q^{87} +0.0394718 q^{88} +11.5677 q^{89} -5.70214 q^{90} +2.15097 q^{91} -7.47786 q^{92} -4.20126 q^{93} -4.31609 q^{94} +15.2088 q^{95} -7.39180 q^{96} -7.83010 q^{97} -1.90871 q^{98} +0.0579542 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 3 q^{2} - 46 q^{3} + 45 q^{4} - 9 q^{5} - 3 q^{6} - 46 q^{7} + 6 q^{8} + 46 q^{9}+O(q^{10}) $ 46 * q + 3 * q^2 - 46 * q^3 + 45 * q^4 - 9 * q^5 - 3 * q^6 - 46 * q^7 + 6 * q^8 + 46 * q^9	\( 46 q + 3 q^{2} - 46 q^{3} + 45 q^{4} - 9 q^{5} - 3 q^{6} - 46 q^{7} + 6 q^{8} + 46 q^{9} - 10 q^{10} + 31 q^{11} - 45 q^{12} - 32 q^{13} - 3 q^{14} + 9 q^{15} + 43 q^{16} - 36 q^{17} + 3 q^{18} - 13 q^{19} - 19 q^{20} + 46 q^{21} - 13 q^{22} + 24 q^{23} - 6 q^{24} + 35 q^{25} - 11 q^{26} - 46 q^{27} - 45 q^{28} + 11 q^{29} + 10 q^{30} - 23 q^{31} + 5 q^{32} - 31 q^{33} - 35 q^{34} + 9 q^{35} + 45 q^{36} - 37 q^{37} - 32 q^{38} + 32 q^{39} - 28 q^{40} - 27 q^{41} + 3 q^{42} - 7 q^{43} + 46 q^{44} - 9 q^{45} + 16 q^{46} - 18 q^{47} - 43 q^{48} + 46 q^{49} + 10 q^{50} + 36 q^{51} - 62 q^{52} - 62 q^{53} - 3 q^{54} - 28 q^{55} - 6 q^{56} + 13 q^{57} - 36 q^{58} - 3 q^{59} + 19 q^{60} - 31 q^{61} - 41 q^{62} - 46 q^{63} + 42 q^{64} + 2 q^{65} + 13 q^{66} - 9 q^{67} - 70 q^{68} - 24 q^{69} + 10 q^{70} + 77 q^{71} + 6 q^{72} - 38 q^{73} + 14 q^{74} - 35 q^{75} - 41 q^{76} - 31 q^{77} + 11 q^{78} + 8 q^{79} - 59 q^{80} + 46 q^{81} - 53 q^{82} - 38 q^{83} + 45 q^{84} - 26 q^{85} + 37 q^{86} - 11 q^{87} - 26 q^{88} - 39 q^{89} - 10 q^{90} + 32 q^{91} + 2 q^{92} + 23 q^{93} - 55 q^{94} + 35 q^{95} - 5 q^{96} - 61 q^{97} + 3 q^{98} + 31 q^{99}+O(q^{100}) \) 46 * q + 3 * q^2 - 46 * q^3 + 45 * q^4 - 9 * q^5 - 3 * q^6 - 46 * q^7 + 6 * q^8 + 46 * q^9 - 10 * q^10 + 31 * q^11 - 45 * q^12 - 32 * q^13 - 3 * q^14 + 9 * q^15 + 43 * q^16 - 36 * q^17 + 3 * q^18 - 13 * q^19 - 19 * q^20 + 46 * q^21 - 13 * q^22 + 24 * q^23 - 6 * q^24 + 35 * q^25 - 11 * q^26 - 46 * q^27 - 45 * q^28 + 11 * q^29 + 10 * q^30 - 23 * q^31 + 5 * q^32 - 31 * q^33 - 35 * q^34 + 9 * q^35 + 45 * q^36 - 37 * q^37 - 32 * q^38 + 32 * q^39 - 28 * q^40 - 27 * q^41 + 3 * q^42 - 7 * q^43 + 46 * q^44 - 9 * q^45 + 16 * q^46 - 18 * q^47 - 43 * q^48 + 46 * q^49 + 10 * q^50 + 36 * q^51 - 62 * q^52 - 62 * q^53 - 3 * q^54 - 28 * q^55 - 6 * q^56 + 13 * q^57 - 36 * q^58 - 3 * q^59 + 19 * q^60 - 31 * q^61 - 41 * q^62 - 46 * q^63 + 42 * q^64 + 2 * q^65 + 13 * q^66 - 9 * q^67 - 70 * q^68 - 24 * q^69 + 10 * q^70 + 77 * q^71 + 6 * q^72 - 38 * q^73 + 14 * q^74 - 35 * q^75 - 41 * q^76 - 31 * q^77 + 11 * q^78 + 8 * q^79 - 59 * q^80 + 46 * q^81 - 53 * q^82 - 38 * q^83 + 45 * q^84 - 26 * q^85 + 37 * q^86 - 11 * q^87 - 26 * q^88 - 39 * q^89 - 10 * q^90 + 32 * q^91 + 2 * q^92 + 23 * q^93 - 55 * q^94 + 35 * q^95 - 5 * q^96 - 61 * q^97 + 3 * q^98 + 31 * 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 $

$2$	−1.90871	−1.34966	−0.674830	−	0.737973i	$-0.735783\pi$
$2$	−1.90871	−1.34966	−0.674830	+	0.737973i	$0.735783\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.64317	0.821584
$5$	2.98743	1.33602	0.668010	−	0.744152i	$-0.267146\pi$
$5$	2.98743	1.33602	0.668010	+	0.744152i	$0.267146\pi$
$6$	1.90871	0.779227
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.681087	0.240801
$9$	1.00000	0.333333
$10$	−5.70214	−1.80317
$11$	0.0579542	0.0174738	0.00873692	−	0.999962i	$-0.497219\pi$
$11$	0.0579542	0.0174738	0.00873692	+	0.999962i	$0.497219\pi$
$12$	−1.64317	−0.474342
$13$	−2.15097	−0.596572	−0.298286	−	0.954476i	$-0.596415\pi$
$13$	−2.15097	−0.596572	−0.298286	+	0.954476i	$0.596415\pi$
$14$	1.90871	0.510124
$15$	−2.98743	−0.771352
$16$	−4.58633	−1.14658
$17$	0.787689	0.191043	0.0955213	−	0.995427i	$-0.469548\pi$
$17$	0.787689	0.191043	0.0955213	+	0.995427i	$0.469548\pi$
$18$	−1.90871	−0.449887
$19$	5.09093	1.16794	0.583969	−	0.811776i	$-0.301499\pi$
$19$	5.09093	1.16794	0.583969	+	0.811776i	$0.301499\pi$
$20$	4.90885	1.09765
$21$	1.00000	0.218218
$22$	−0.110618	−0.0235838
$23$	−4.55088	−0.948924	−0.474462	−	0.880276i	$-0.657357\pi$
$23$	−4.55088	−0.948924	−0.474462	+	0.880276i	$0.657357\pi$
$24$	−0.681087	−0.139026
$25$	3.92475	0.784949
$26$	4.10558	0.805170
$27$	−1.00000	−0.192450
$28$	−1.64317	−0.310530
$29$	−2.45471	−0.455828	−0.227914	−	0.973681i	$-0.573191\pi$
$29$	−2.45471	−0.455828	−0.227914	+	0.973681i	$0.573191\pi$
$30$	5.70214	1.04106
$31$	4.20126	0.754569	0.377284	−	0.926097i	$-0.376858\pi$
$31$	4.20126	0.754569	0.377284	+	0.926097i	$0.376858\pi$
$32$	7.39180	1.30670
$33$	−0.0579542	−0.0100885
$34$	−1.50347	−0.257843
$35$	−2.98743	−0.504968
$36$	1.64317	0.273861
$37$	−1.61258	−0.265106	−0.132553	−	0.991176i	$-0.542317\pi$
$37$	−1.61258	−0.265106	−0.132553	+	0.991176i	$0.542317\pi$
$38$	−9.71709	−1.57632
$39$	2.15097	0.344431
$40$	2.03470	0.321714
$41$	−1.19436	−0.186528	−0.0932638	−	0.995641i	$-0.529730\pi$
$41$	−1.19436	−0.186528	−0.0932638	+	0.995641i	$0.529730\pi$
$42$	−1.90871	−0.294520
$43$	−7.27102	−1.10882	−0.554410	−	0.832244i	$-0.687056\pi$
$43$	−7.27102	−1.10882	−0.554410	+	0.832244i	$0.687056\pi$
$44$	0.0952285	0.0143562
$45$	2.98743	0.445340
$46$	8.68630	1.28072
$47$	2.26126	0.329839	0.164919	−	0.986307i	$-0.447264\pi$
$47$	2.26126	0.329839	0.164919	+	0.986307i	$0.447264\pi$
$48$	4.58633	0.661980
$49$	1.00000	0.142857
$50$	−7.49120	−1.05942
$51$	−0.787689	−0.110299
$52$	−3.53441	−0.490134
$53$	−8.28090	−1.13747	−0.568735	−	0.822521i	$-0.692567\pi$
$53$	−8.28090	−1.13747	−0.568735	+	0.822521i	$0.692567\pi$
$54$	1.90871	0.259742
$55$	0.173134	0.0233454
$56$	−0.681087	−0.0910141
$57$	−5.09093	−0.674310
$58$	4.68533	0.615213
$59$	−9.80775	−1.27686	−0.638430	−	0.769680i	$-0.720416\pi$
$59$	−9.80775	−1.27686	−0.638430	+	0.769680i	$0.720416\pi$
$60$	−4.90885	−0.633730
$61$	8.02840	1.02793	0.513965	−	0.857811i	$-0.328176\pi$
$61$	8.02840	1.02793	0.513965	+	0.857811i	$0.328176\pi$
$62$	−8.01898	−1.01841
$63$	−1.00000	−0.125988
$64$	−4.93613	−0.617016
$65$	−6.42588	−0.797033
$66$	0.110618	0.0136161
$67$	−4.14642	−0.506566	−0.253283	−	0.967392i	$-0.581510\pi$
$67$	−4.14642	−0.506566	−0.253283	+	0.967392i	$0.581510\pi$
$68$	1.29431	0.156958
$69$	4.55088	0.547861
$70$	5.70214	0.681536
$71$	13.6486	1.61979	0.809895	−	0.586575i	$-0.199524\pi$
$71$	13.6486	1.61979	0.809895	+	0.586575i	$0.199524\pi$
$72$	0.681087	0.0802669
$73$	10.6181	1.24276	0.621379	−	0.783510i	$-0.286573\pi$
$73$	10.6181	1.24276	0.621379	+	0.783510i	$0.286573\pi$
$74$	3.07794	0.357803
$75$	−3.92475	−0.453191
$76$	8.36525	0.959560
$77$	−0.0579542	−0.00660449
$78$	−4.10558	−0.464865
$79$	3.17309	0.357001	0.178500	−	0.983940i	$-0.442875\pi$
$79$	3.17309	0.357001	0.178500	+	0.983940i	$0.442875\pi$
$80$	−13.7014	−1.53186
$81$	1.00000	0.111111
$82$	2.27968	0.251749
$83$	11.7530	1.29006	0.645030	−	0.764158i	$-0.276845\pi$
$83$	11.7530	1.29006	0.645030	+	0.764158i	$0.276845\pi$
$84$	1.64317	0.179284
$85$	2.35317	0.255237
$86$	13.8783	1.49653
$87$	2.45471	0.263173
$88$	0.0394718	0.00420771
$89$	11.5677	1.22617	0.613085	−	0.790017i	$-0.289928\pi$
$89$	11.5677	1.22617	0.613085	+	0.790017i	$0.289928\pi$
$90$	−5.70214	−0.601058
$91$	2.15097	0.225483
$92$	−7.47786	−0.779621
$93$	−4.20126	−0.435650
$94$	−4.31609	−0.445171
$95$	15.2088	1.56039
$96$	−7.39180	−0.754423
$97$	−7.83010	−0.795026	−0.397513	−	0.917597i	$-0.630127\pi$
$97$	−7.83010	−0.795026	−0.397513	+	0.917597i	$0.630127\pi$
$98$	−1.90871	−0.192809
$99$	0.0579542	0.00582461
$100$	6.44902	0.644902
$101$	−13.0062	−1.29416	−0.647081	−	0.762421i	$-0.724010\pi$
$101$	−13.0062	−1.29416	−0.647081	+	0.762421i	$0.724010\pi$
$102$	1.50347	0.148866
$103$	−8.83571	−0.870608	−0.435304	−	0.900284i	$-0.643359\pi$
$103$	−8.83571	−0.870608	−0.435304	+	0.900284i	$0.643359\pi$
$104$	−1.46500	−0.143655
$105$	2.98743	0.291543
$106$	15.8058	1.53520
$107$	−19.1036	−1.84681	−0.923406	−	0.383824i	$-0.874607\pi$
$107$	−19.1036	−1.84681	−0.923406	+	0.383824i	$0.874607\pi$
$108$	−1.64317	−0.158114
$109$	−1.07761	−0.103217	−0.0516083	−	0.998667i	$-0.516435\pi$
$109$	−1.07761	−0.103217	−0.0516083	+	0.998667i	$0.516435\pi$
$110$	−0.330463	−0.0315084
$111$	1.61258	0.153059
$112$	4.58633	0.433368
$113$	−18.5734	−1.74724	−0.873619	−	0.486611i	$-0.838233\pi$
$113$	−18.5734	−1.74724	−0.873619	+	0.486611i	$0.838233\pi$
$114$	9.71709	0.910089
$115$	−13.5954	−1.26778
$116$	−4.03350	−0.374501
$117$	−2.15097	−0.198857
$118$	18.7201	1.72333
$119$	−0.787689	−0.0722073
$120$	−2.03470	−0.185742
$121$	−10.9966	−0.999695
$122$	−15.3239	−1.38736
$123$	1.19436	0.107692
$124$	6.90338	0.619942
$125$	−3.21224	−0.287312
$126$	1.90871	0.170041
$127$	−11.7413	−1.04187	−0.520935	−	0.853596i	$-0.674417\pi$
$127$	−11.7413	−1.04187	−0.520935	+	0.853596i	$0.674417\pi$
$128$	−5.36197	−0.473936
$129$	7.27102	0.640178
$130$	12.2651	1.07572
$131$	15.0942	1.31879	0.659395	−	0.751797i	$-0.270813\pi$
$131$	15.0942	1.31879	0.659395	+	0.751797i	$0.270813\pi$
$132$	−0.0952285	−0.00828858
$133$	−5.09093	−0.441439
$134$	7.91431	0.683692
$135$	−2.98743	−0.257117
$136$	0.536485	0.0460032
$137$	−14.6225	−1.24928	−0.624642	−	0.780911i	$-0.714755\pi$
$137$	−14.6225	−1.24928	−0.624642	+	0.780911i	$0.714755\pi$
$138$	−8.68630	−0.739427
$139$	−23.2143	−1.96901	−0.984506	−	0.175352i	$-0.943894\pi$
$139$	−23.2143	−1.96901	−0.984506	+	0.175352i	$0.943894\pi$
$140$	−4.90885	−0.414874
$141$	−2.26126	−0.190433
$142$	−26.0512	−2.18617
$143$	−0.124658	−0.0104244
$144$	−4.58633	−0.382195
$145$	−7.33328	−0.608996
$146$	−20.2669	−1.67730
$147$	−1.00000	−0.0824786
$148$	−2.64974	−0.217807
$149$	−20.7826	−1.70258	−0.851288	−	0.524699i	$-0.824178\pi$
$149$	−20.7826	−1.70258	−0.851288	+	0.524699i	$0.824178\pi$
$150$	7.49120	0.611654
$151$	5.71619	0.465177	0.232588	−	0.972575i	$-0.425281\pi$
$151$	5.71619	0.465177	0.232588	+	0.972575i	$0.425281\pi$
$152$	3.46736	0.281240
$153$	0.787689	0.0636809
$154$	0.110618	0.00891382
$155$	12.5510	1.00812
$156$	3.53441	0.282979
$157$	16.7482	1.33665	0.668325	−	0.743869i	$-0.267011\pi$
$157$	16.7482	1.33665	0.668325	+	0.743869i	$0.267011\pi$
$158$	−6.05651	−0.481830
$159$	8.28090	0.656718
$160$	22.0825	1.74577
$161$	4.55088	0.358659
$162$	−1.90871	−0.149962
$163$	−0.935824	−0.0732994	−0.0366497	−	0.999328i	$-0.511669\pi$
$163$	−0.935824	−0.0732994	−0.0366497	+	0.999328i	$0.511669\pi$
$164$	−1.96253	−0.153248
$165$	−0.173134	−0.0134785
$166$	−22.4330	−1.74114
$167$	16.1962	1.25330	0.626650	−	0.779301i	$-0.284425\pi$
$167$	16.1962	1.25330	0.626650	+	0.779301i	$0.284425\pi$
$168$	0.681087	0.0525470
$169$	−8.37332	−0.644101
$170$	−4.49151	−0.344483
$171$	5.09093	0.389313
$172$	−11.9475	−0.910989
$173$	−9.29842	−0.706945	−0.353473	−	0.935445i	$-0.614999\pi$
$173$	−9.29842	−0.706945	−0.353473	+	0.935445i	$0.614999\pi$
$174$	−4.68533	−0.355194
$175$	−3.92475	−0.296683
$176$	−0.265797	−0.0200352
$177$	9.80775	0.737196
$178$	−22.0793	−1.65491
$179$	12.0435	0.900174	0.450087	−	0.892985i	$-0.351393\pi$
$179$	12.0435	0.900174	0.450087	+	0.892985i	$0.351393\pi$
$180$	4.90885	0.365884
$181$	−7.53008	−0.559706	−0.279853	−	0.960043i	$-0.590286\pi$
$181$	−7.53008	−0.559706	−0.279853	+	0.960043i	$0.590286\pi$
$182$	−4.10558	−0.304326
$183$	−8.02840	−0.593476
$184$	−3.09954	−0.228501
$185$	−4.81746	−0.354187
$186$	8.01898	0.587980
$187$	0.0456499	0.00333825
$188$	3.71563	0.270990
$189$	1.00000	0.0727393
$190$	−29.0292	−2.10600
$191$	12.7168	0.920154	0.460077	−	0.887879i	$-0.347822\pi$
$191$	12.7168	0.920154	0.460077	+	0.887879i	$0.347822\pi$
$192$	4.93613	0.356234
$193$	−5.82472	−0.419273	−0.209636	−	0.977779i	$-0.567228\pi$
$193$	−5.82472	−0.419273	−0.209636	+	0.977779i	$0.567228\pi$
$194$	14.9454	1.07302
$195$	6.42588	0.460167
$196$	1.64317	0.117369
$197$	−0.991567	−0.0706463	−0.0353231	−	0.999376i	$-0.511246\pi$
$197$	−0.991567	−0.0706463	−0.0353231	+	0.999376i	$0.511246\pi$
$198$	−0.110618	−0.00786125
$199$	26.9275	1.90884	0.954419	−	0.298469i	$-0.0964760\pi$
$199$	26.9275	1.90884	0.954419	+	0.298469i	$0.0964760\pi$
$200$	2.67309	0.189016
$201$	4.14642	0.292466
$202$	24.8250	1.74668
$203$	2.45471	0.172287
$204$	−1.29431	−0.0906195
$205$	−3.56807	−0.249205
$206$	16.8648	1.17503
$207$	−4.55088	−0.316308
$208$	9.86508	0.684020
$209$	0.295040	0.0204084
$210$	−5.70214	−0.393485
$211$	−10.2491	−0.705574	−0.352787	−	0.935704i	$-0.614766\pi$
$211$	−10.2491	−0.705574	−0.352787	+	0.935704i	$0.614766\pi$
$212$	−13.6069	−0.934527
$213$	−13.6486	−0.935186
$214$	36.4632	2.49257
$215$	−21.7217	−1.48141
$216$	−0.681087	−0.0463421
$217$	−4.20126	−0.285200
$218$	2.05685	0.139307
$219$	−10.6181	−0.717507
$220$	0.284489	0.0191802
$221$	−1.69430	−0.113971
$222$	−3.07794	−0.206578
$223$	12.9754	0.868897	0.434448	−	0.900697i	$-0.356943\pi$
$223$	12.9754	0.868897	0.434448	+	0.900697i	$0.356943\pi$
$224$	−7.39180	−0.493886
$225$	3.92475	0.261650
$226$	35.4512	2.35818
$227$	14.1897	0.941805	0.470903	−	0.882185i	$-0.343928\pi$
$227$	14.1897	0.941805	0.470903	+	0.882185i	$0.343928\pi$
$228$	−8.36525	−0.554002
$229$	−11.2725	−0.744905	−0.372452	−	0.928051i	$-0.621483\pi$
$229$	−11.2725	−0.744905	−0.372452	+	0.928051i	$0.621483\pi$
$230$	25.9497	1.71107
$231$	0.0579542	0.00381310
$232$	−1.67187	−0.109764
$233$	20.2955	1.32960	0.664801	−	0.747021i	$-0.268516\pi$
$233$	20.2955	1.32960	0.664801	+	0.747021i	$0.268516\pi$
$234$	4.10558	0.268390
$235$	6.75536	0.440671
$236$	−16.1158	−1.04905
$237$	−3.17309	−0.206115
$238$	1.50347	0.0974554
$239$	1.85496	0.119987	0.0599937	−	0.998199i	$-0.480892\pi$
$239$	1.85496	0.119987	0.0599937	+	0.998199i	$0.480892\pi$
$240$	13.7014	0.884419
$241$	−10.8234	−0.697193	−0.348597	−	0.937273i	$-0.613342\pi$
$241$	−10.8234	−0.697193	−0.348597	+	0.937273i	$0.613342\pi$
$242$	20.9894	1.34925
$243$	−1.00000	−0.0641500
$244$	13.1920	0.844532
$245$	2.98743	0.190860
$246$	−2.27968	−0.145347
$247$	−10.9504	−0.696760
$248$	2.86142	0.181701
$249$	−11.7530	−0.744816
$250$	6.13124	0.387774
$251$	6.00483	0.379022	0.189511	−	0.981879i	$-0.439310\pi$
$251$	6.00483	0.379022	0.189511	+	0.981879i	$0.439310\pi$
$252$	−1.64317	−0.103510
$253$	−0.263742	−0.0165813
$254$	22.4107	1.40617
$255$	−2.35317	−0.147361
$256$	20.1067	1.25667
$257$	−14.1743	−0.884171	−0.442086	−	0.896973i	$-0.645761\pi$
$257$	−14.1743	−0.884171	−0.442086	+	0.896973i	$0.645761\pi$
$258$	−13.8783	−0.864023
$259$	1.61258	0.100201
$260$	−10.5588	−0.654829
$261$	−2.45471	−0.151943
$262$	−28.8105	−1.77992
$263$	26.4817	1.63293	0.816466	−	0.577393i	$-0.195930\pi$
$263$	26.4817	1.63293	0.816466	+	0.577393i	$0.195930\pi$
$264$	−0.0394718	−0.00242932
$265$	−24.7386	−1.51968
$266$	9.71709	0.595793
$267$	−11.5677	−0.707929
$268$	−6.81327	−0.416187
$269$	6.96464	0.424642	0.212321	−	0.977200i	$-0.431898\pi$
$269$	6.96464	0.424642	0.212321	+	0.977200i	$0.431898\pi$
$270$	5.70214	0.347021
$271$	−11.8871	−0.722088	−0.361044	−	0.932549i	$-0.617580\pi$
$271$	−11.8871	−0.722088	−0.361044	+	0.932549i	$0.617580\pi$
$272$	−3.61260	−0.219046
$273$	−2.15097	−0.130183
$274$	27.9101	1.68611
$275$	0.227455	0.0137161
$276$	7.47786	0.450114
$277$	−17.2343	−1.03551	−0.517755	−	0.855529i	$-0.673232\pi$
$277$	−17.2343	−1.03551	−0.517755	+	0.855529i	$0.673232\pi$
$278$	44.3093	2.65750
$279$	4.20126	0.251523
$280$	−2.03470	−0.121597
$281$	−14.3026	−0.853219	−0.426609	−	0.904436i	$-0.640292\pi$
$281$	−14.3026	−0.853219	−0.426609	+	0.904436i	$0.640292\pi$
$282$	4.31609	0.257019
$283$	−22.5514	−1.34054	−0.670271	−	0.742117i	$-0.733822\pi$
$283$	−22.5514	−1.34054	−0.670271	+	0.742117i	$0.733822\pi$
$284$	22.4269	1.33079
$285$	−15.2088	−0.900891
$286$	0.237935	0.0140694
$287$	1.19436	0.0705008
$288$	7.39180	0.435566
$289$	−16.3795	−0.963503
$290$	13.9971	0.821937
$291$	7.83010	0.459008
$292$	17.4474	1.02103
$293$	16.6292	0.971488	0.485744	−	0.874101i	$-0.338549\pi$
$293$	16.6292	0.971488	0.485744	+	0.874101i	$0.338549\pi$
$294$	1.90871	0.111318
$295$	−29.3000	−1.70591
$296$	−1.09830	−0.0638377
$297$	−0.0579542	−0.00336284
$298$	39.6679	2.29790
$299$	9.78881	0.566102
$300$	−6.44902	−0.372334
$301$	7.27102	0.419095
$302$	−10.9105	−0.627831
$303$	13.0062	0.747185
$304$	−23.3487	−1.33914
$305$	23.9843	1.37334
$306$	−1.50347	−0.0859476
$307$	−9.34402	−0.533291	−0.266646	−	0.963795i	$-0.585915\pi$
$307$	−9.34402	−0.533291	−0.266646	+	0.963795i	$0.585915\pi$
$308$	−0.0952285	−0.00542615
$309$	8.83571	0.502646
$310$	−23.9562	−1.36062
$311$	0.594707	0.0337228	0.0168614	−	0.999858i	$-0.494633\pi$
$311$	0.594707	0.0337228	0.0168614	+	0.999858i	$0.494633\pi$
$312$	1.46500	0.0829392
$313$	33.8418	1.91285	0.956427	−	0.291972i	$-0.0943116\pi$
$313$	33.8418	1.91285	0.956427	+	0.291972i	$0.0943116\pi$
$314$	−31.9674	−1.80402
$315$	−2.98743	−0.168323
$316$	5.21393	0.293306
$317$	23.1111	1.29805	0.649026	−	0.760766i	$-0.275177\pi$
$317$	23.1111	1.29805	0.649026	+	0.760766i	$0.275177\pi$
$318$	−15.8058	−0.886347
$319$	−0.142261	−0.00796507
$320$	−14.7463	−0.824346
$321$	19.1036	1.06626
$322$	−8.68630	−0.484069
$323$	4.01007	0.223126
$324$	1.64317	0.0912871
$325$	−8.44202	−0.468279
$326$	1.78621	0.0989293
$327$	1.07761	0.0595921
$328$	−0.813463	−0.0449160
$329$	−2.26126	−0.124667
$330$	0.330463	0.0181914
$331$	−26.2981	−1.44548	−0.722738	−	0.691122i	$-0.757117\pi$
$331$	−26.2981	−1.44548	−0.722738	+	0.691122i	$0.757117\pi$
$332$	19.3122	1.05989
$333$	−1.61258	−0.0883686
$334$	−30.9139	−1.69153
$335$	−12.3871	−0.676782
$336$	−4.58633	−0.250205
$337$	17.1837	0.936056	0.468028	−	0.883714i	$-0.344965\pi$
$337$	17.1837	0.936056	0.468028	+	0.883714i	$0.344965\pi$
$338$	15.9822	0.869319
$339$	18.5734	1.00877
$340$	3.86665	0.209699
$341$	0.243481	0.0131852
$342$	−9.71709	−0.525440
$343$	−1.00000	−0.0539949
$344$	−4.95220	−0.267005
$345$	13.5954	0.731954
$346$	17.7480	0.954137
$347$	−9.71580	−0.521571	−0.260786	−	0.965397i	$-0.583982\pi$
$347$	−9.71580	−0.521571	−0.260786	+	0.965397i	$0.583982\pi$
$348$	4.03350	0.216218
$349$	−15.1132	−0.808993	−0.404496	−	0.914540i	$-0.632553\pi$
$349$	−15.1132	−0.808993	−0.404496	+	0.914540i	$0.632553\pi$
$350$	7.49120	0.400421
$351$	2.15097	0.114810
$352$	0.428386	0.0228330
$353$	−6.34326	−0.337618	−0.168809	−	0.985649i	$-0.553992\pi$
$353$	−6.34326	−0.337618	−0.168809	+	0.985649i	$0.553992\pi$
$354$	−18.7201	−0.994965
$355$	40.7743	2.16407
$356$	19.0076	1.00740
$357$	0.787689	0.0416889
$358$	−22.9876	−1.21493
$359$	−8.75524	−0.462084	−0.231042	−	0.972944i	$-0.574213\pi$
$359$	−8.75524	−0.462084	−0.231042	+	0.972944i	$0.574213\pi$
$360$	2.03470	0.107238
$361$	6.91753	0.364080
$362$	14.3727	0.755414
$363$	10.9966	0.577174
$364$	3.53441	0.185253
$365$	31.7209	1.66035
$366$	15.3239	0.800992
$367$	31.7236	1.65596	0.827979	−	0.560759i	$-0.189490\pi$
$367$	31.7236	1.65596	0.827979	+	0.560759i	$0.189490\pi$
$368$	20.8718	1.08802
$369$	−1.19436	−0.0621759
$370$	9.19513	0.478032
$371$	8.28090	0.429923
$372$	−6.90338	−0.357924
$373$	−29.4108	−1.52284	−0.761418	−	0.648261i	$-0.775496\pi$
$373$	−29.4108	−1.52284	−0.761418	+	0.648261i	$0.775496\pi$
$374$	−0.0871323	−0.00450550
$375$	3.21224	0.165880
$376$	1.54012	0.0794254
$377$	5.28001	0.271934
$378$	−1.90871	−0.0981734
$379$	−11.2270	−0.576694	−0.288347	−	0.957526i	$-0.593106\pi$
$379$	−11.2270	−0.576694	−0.288347	+	0.957526i	$0.593106\pi$
$380$	24.9906	1.28199
$381$	11.7413	0.601524
$382$	−24.2726	−1.24190
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	5.36197	0.273627
$385$	−0.173134	−0.00882373
$386$	11.1177	0.565876
$387$	−7.27102	−0.369607
$388$	−12.8662	−0.653181
$389$	15.6751	0.794760	0.397380	−	0.917654i	$-0.369919\pi$
$389$	15.6751	0.794760	0.397380	+	0.917654i	$0.369919\pi$
$390$	−12.2651	−0.621069
$391$	−3.58468	−0.181285
$392$	0.681087	0.0344001
$393$	−15.0942	−0.761403
$394$	1.89261	0.0953485
$395$	9.47940	0.476960
$396$	0.0952285	0.00478541
$397$	−5.75224	−0.288697	−0.144348	−	0.989527i	$-0.546109\pi$
$397$	−5.75224	−0.288697	−0.144348	+	0.989527i	$0.546109\pi$
$398$	−51.3967	−2.57628
$399$	5.09093	0.254865
$400$	−18.0002	−0.900010
$401$	−2.41250	−0.120475	−0.0602373	−	0.998184i	$-0.519186\pi$
$401$	−2.41250	−0.120475	−0.0602373	+	0.998184i	$0.519186\pi$
$402$	−7.91431	−0.394730
$403$	−9.03679	−0.450155
$404$	−21.3713	−1.06326
$405$	2.98743	0.148447
$406$	−4.68533	−0.232529
$407$	−0.0934555	−0.00463242
$408$	−0.536485	−0.0265600
$409$	−0.951722	−0.0470596	−0.0235298	−	0.999723i	$-0.507490\pi$
$409$	−0.951722	−0.0470596	−0.0235298	+	0.999723i	$0.507490\pi$
$410$	6.81040	0.336342
$411$	14.6225	0.721275
$412$	−14.5186	−0.715278
$413$	9.80775	0.482608
$414$	8.68630	0.426908
$415$	35.1113	1.72354
$416$	−15.8996	−0.779540
$417$	23.2143	1.13681
$418$	−0.563146	−0.0275444
$419$	−20.4150	−0.997336	−0.498668	−	0.866793i	$-0.666177\pi$
$419$	−20.4150	−0.997336	−0.498668	+	0.866793i	$0.666177\pi$
$420$	4.90885	0.239528
$421$	34.5886	1.68574	0.842872	−	0.538115i	$-0.180863\pi$
$421$	34.5886	1.68574	0.842872	+	0.538115i	$0.180863\pi$
$422$	19.5625	0.952286
$423$	2.26126	0.109946
$424$	−5.64002	−0.273903
$425$	3.09148	0.149959
$426$	26.0512	1.26218
$427$	−8.02840	−0.388521
$428$	−31.3904	−1.51731
$429$	0.124658	0.00601854
$430$	41.4603	1.99940
$431$	14.1623	0.682176	0.341088	−	0.940031i	$-0.389205\pi$
$431$	14.1623	0.682176	0.341088	+	0.940031i	$0.389205\pi$
$432$	4.58633	0.220660
$433$	6.23885	0.299820	0.149910	−	0.988700i	$-0.452102\pi$
$433$	6.23885	0.299820	0.149910	+	0.988700i	$0.452102\pi$
$434$	8.01898	0.384923
$435$	7.33328	0.351604
$436$	−1.77070	−0.0848012
$437$	−23.1682	−1.10828
$438$	20.2669	0.968391
$439$	22.6693	1.08195	0.540973	−	0.841040i	$-0.318056\pi$
$439$	22.6693	1.08195	0.540973	+	0.841040i	$0.318056\pi$
$440$	0.117919	0.00562159
$441$	1.00000	0.0476190
$442$	3.23392	0.153822
$443$	7.95055	0.377742	0.188871	−	0.982002i	$-0.439517\pi$
$443$	7.95055	0.377742	0.188871	+	0.982002i	$0.439517\pi$
$444$	2.64974	0.125751
$445$	34.5576	1.63819
$446$	−24.7662	−1.17272
$447$	20.7826	0.982982
$448$	4.93613	0.233210
$449$	31.1558	1.47033	0.735167	−	0.677886i	$-0.237104\pi$
$449$	31.1558	1.47033	0.735167	+	0.677886i	$0.237104\pi$
$450$	−7.49120	−0.353138
$451$	−0.0692181	−0.00325935
$452$	−30.5192	−1.43550
$453$	−5.71619	−0.268570
$454$	−27.0841	−1.27112
$455$	6.42588	0.301250
$456$	−3.46736	−0.162374
$457$	−37.5195	−1.75509	−0.877545	−	0.479495i	$-0.840820\pi$
$457$	−37.5195	−1.75509	−0.877545	+	0.479495i	$0.840820\pi$
$458$	21.5158	1.00537
$459$	−0.787689	−0.0367662
$460$	−22.3396	−1.04159
$461$	−36.0666	−1.67979	−0.839896	−	0.542748i	$-0.817384\pi$
$461$	−36.0666	−1.67979	−0.839896	+	0.542748i	$0.817384\pi$
$462$	−0.110618	−0.00514640
$463$	−9.57498	−0.444987	−0.222494	−	0.974934i	$-0.571420\pi$
$463$	−9.57498	−0.444987	−0.222494	+	0.974934i	$0.571420\pi$
$464$	11.2581	0.522645
$465$	−12.5510	−0.582038
$466$	−38.7382	−1.79451
$467$	−1.40576	−0.0650509	−0.0325255	−	0.999471i	$-0.510355\pi$
$467$	−1.40576	−0.0650509	−0.0325255	+	0.999471i	$0.510355\pi$
$468$	−3.53441	−0.163378
$469$	4.14642	0.191464
$470$	−12.8940	−0.594757
$471$	−16.7482	−0.771715
$472$	−6.67993	−0.307469
$473$	−0.421386	−0.0193753
$474$	6.05651	0.278185
$475$	19.9806	0.916773
$476$	−1.29431	−0.0593244
$477$	−8.28090	−0.379156
$478$	−3.54058	−0.161942
$479$	28.9120	1.32102	0.660512	−	0.750816i	$-0.270339\pi$
$479$	28.9120	1.32102	0.660512	+	0.750816i	$0.270339\pi$
$480$	−22.0825	−1.00792
$481$	3.46861	0.158155
$482$	20.6586	0.940974
$483$	−4.55088	−0.207072
$484$	−18.0693	−0.821333
$485$	−23.3919	−1.06217
$486$	1.90871	0.0865808
$487$	−24.3928	−1.10534	−0.552672	−	0.833399i	$-0.686392\pi$
$487$	−24.3928	−1.10534	−0.552672	+	0.833399i	$0.686392\pi$
$488$	5.46804	0.247526
$489$	0.935824	0.0423194
$490$	−5.70214	−0.257596
$491$	−29.5905	−1.33540	−0.667700	−	0.744431i	$-0.732721\pi$
$491$	−29.5905	−1.33540	−0.667700	+	0.744431i	$0.732721\pi$
$492$	1.96253	0.0884778
$493$	−1.93355	−0.0870826
$494$	20.9012	0.940389
$495$	0.173134	0.00778180
$496$	−19.2684	−0.865176
$497$	−13.6486	−0.612223
$498$	22.4330	1.00525
$499$	−32.0458	−1.43457	−0.717283	−	0.696782i	$-0.754614\pi$
$499$	−32.0458	−1.43457	−0.717283	+	0.696782i	$0.754614\pi$
$500$	−5.27826	−0.236051
$501$	−16.1962	−0.723593
$502$	−11.4615	−0.511551
$503$	−0.327403	−0.0145982	−0.00729909	−	0.999973i	$-0.502323\pi$
$503$	−0.327403	−0.0145982	−0.00729909	+	0.999973i	$0.502323\pi$
$504$	−0.681087	−0.0303380
$505$	−38.8550	−1.72903
$506$	0.503407	0.0223792
$507$	8.37332	0.371872
$508$	−19.2929	−0.855984
$509$	0.839304	0.0372015	0.0186007	−	0.999827i	$-0.494079\pi$
$509$	0.839304	0.0372015	0.0186007	+	0.999827i	$0.494079\pi$
$510$	4.49151	0.198887
$511$	−10.6181	−0.469718
$512$	−27.6539	−1.22214
$513$	−5.09093	−0.224770
$514$	27.0547	1.19333
$515$	−26.3961	−1.16315
$516$	11.9475	0.525960
$517$	0.131050	0.00576355
$518$	−3.07794	−0.135237
$519$	9.29842	0.408155
$520$	−4.37658	−0.191926
$521$	−34.1545	−1.49634	−0.748169	−	0.663508i	$-0.769067\pi$
$521$	−34.1545	−1.49634	−0.748169	+	0.663508i	$0.769067\pi$
$522$	4.68533	0.205071
$523$	6.03200	0.263761	0.131881	−	0.991266i	$-0.457898\pi$
$523$	6.03200	0.263761	0.131881	+	0.991266i	$0.457898\pi$
$524$	24.8024	1.08350
$525$	3.92475	0.171290
$526$	−50.5459	−2.20391
$527$	3.30929	0.144155
$528$	0.265797	0.0115673
$529$	−2.28951	−0.0995441
$530$	47.2188	2.05106
$531$	−9.80775	−0.425620
$532$	−8.36525	−0.362680
$533$	2.56903	0.111277
$534$	22.0793	0.955464
$535$	−57.0706	−2.46738
$536$	−2.82407	−0.121981
$537$	−12.0435	−0.519716
$538$	−13.2935	−0.573122
$539$	0.0579542	0.00249626
$540$	−4.90885	−0.211243
$541$	−22.4791	−0.966453	−0.483226	−	0.875495i	$-0.660535\pi$
$541$	−22.4791	−0.966453	−0.483226	+	0.875495i	$0.660535\pi$
$542$	22.6890	0.974575
$543$	7.53008	0.323147
$544$	5.82244	0.249635
$545$	−3.21930	−0.137899
$546$	4.10558	0.175703
$547$	−36.2682	−1.55072	−0.775358	−	0.631522i	$-0.782431\pi$
$547$	−36.2682	−1.55072	−0.775358	+	0.631522i	$0.782431\pi$
$548$	−24.0272	−1.02639
$549$	8.02840	0.342644
$550$	−0.434146	−0.0185121
$551$	−12.4967	−0.532379
$552$	3.09954	0.131925
$553$	−3.17309	−0.134934
$554$	32.8953	1.39759
$555$	4.81746	0.204490
$556$	−38.1450	−1.61771
$557$	19.7614	0.837317	0.418659	−	0.908144i	$-0.362500\pi$
$557$	19.7614	0.837317	0.418659	+	0.908144i	$0.362500\pi$
$558$	−8.01898	−0.339471
$559$	15.6398	0.661491
$560$	13.7014	0.578988
$561$	−0.0456499	−0.00192734
$562$	27.2994	1.15156
$563$	−35.8764	−1.51201	−0.756006	−	0.654565i	$-0.772852\pi$
$563$	−35.8764	−1.51201	−0.756006	+	0.654565i	$0.772852\pi$
$564$	−3.71563	−0.156456
$565$	−55.4867	−2.33434
$566$	43.0440	1.80928
$567$	−1.00000	−0.0419961
$568$	9.29588	0.390047
$569$	47.0700	1.97328	0.986639	−	0.162924i	$-0.0520924\pi$
$569$	47.0700	1.97328	0.986639	+	0.162924i	$0.0520924\pi$
$570$	29.0292	1.21590
$571$	31.7906	1.33039	0.665197	−	0.746667i	$-0.268347\pi$
$571$	31.7906	1.33039	0.665197	+	0.746667i	$0.268347\pi$
$572$	−0.204834	−0.00856453
$573$	−12.7168	−0.531251
$574$	−2.27968	−0.0951522
$575$	−17.8610	−0.744857
$576$	−4.93613	−0.205672
$577$	−4.23892	−0.176468	−0.0882342	−	0.996100i	$-0.528122\pi$
$577$	−4.23892	−0.176468	−0.0882342	+	0.996100i	$0.528122\pi$
$578$	31.2638	1.30040
$579$	5.82472	0.242067
$580$	−12.0498	−0.500341
$581$	−11.7530	−0.487597
$582$	−14.9454	−0.619506
$583$	−0.479913	−0.0198760
$584$	7.23187	0.299257
$585$	−6.42588	−0.265678
$586$	−31.7403	−1.31118
$587$	31.9478	1.31862	0.659312	−	0.751869i	$-0.270847\pi$
$587$	31.9478	1.31862	0.659312	+	0.751869i	$0.270847\pi$
$588$	−1.64317	−0.0677631
$589$	21.3883	0.881290
$590$	55.9252	2.30240
$591$	0.991567	0.0407876
$592$	7.39581	0.303966
$593$	−33.3405	−1.36913	−0.684566	−	0.728951i	$-0.740008\pi$
$593$	−33.3405	−1.36913	−0.684566	+	0.728951i	$0.740008\pi$
$594$	0.110618	0.00453870
$595$	−2.35317	−0.0964704
$596$	−34.1493	−1.39881
$597$	−26.9275	−1.10207
$598$	−18.6840	−0.764045
$599$	46.4976	1.89984	0.949920	−	0.312492i	$-0.101164\pi$
$599$	46.4976	1.89984	0.949920	+	0.312492i	$0.101164\pi$
$600$	−2.67309	−0.109129
$601$	−11.8090	−0.481699	−0.240849	−	0.970563i	$-0.577426\pi$
$601$	−11.8090	−0.481699	−0.240849	+	0.970563i	$0.577426\pi$
$602$	−13.8783	−0.565636
$603$	−4.14642	−0.168855
$604$	9.39266	0.382182
$605$	−32.8517	−1.33561
$606$	−24.8250	−1.00845
$607$	−16.3972	−0.665543	−0.332771	−	0.943008i	$-0.607984\pi$
$607$	−16.3972	−0.665543	−0.332771	+	0.943008i	$0.607984\pi$
$608$	37.6311	1.52614
$609$	−2.45471	−0.0994699
$610$	−45.7790	−1.85354
$611$	−4.86391	−0.196773
$612$	1.29431	0.0523192
$613$	−28.1093	−1.13532	−0.567662	−	0.823262i	$-0.692152\pi$
$613$	−28.1093	−1.13532	−0.567662	+	0.823262i	$0.692152\pi$
$614$	17.8350	0.719762
$615$	3.56807	0.143878
$616$	−0.0394718	−0.00159037
$617$	−9.18394	−0.369732	−0.184866	−	0.982764i	$-0.559185\pi$
$617$	−9.18394	−0.369732	−0.184866	+	0.982764i	$0.559185\pi$
$618$	−16.8648	−0.678401
$619$	14.9505	0.600911	0.300456	−	0.953796i	$-0.402861\pi$
$619$	14.9505	0.600911	0.300456	+	0.953796i	$0.402861\pi$
$620$	20.6234	0.828255
$621$	4.55088	0.182620
$622$	−1.13512	−0.0455143
$623$	−11.5677	−0.463448
$624$	−9.86508	−0.394919
$625$	−29.2201	−1.16880
$626$	−64.5942	−2.58170
$627$	−0.295040	−0.0117828
$628$	27.5201	1.09817
$629$	−1.27021	−0.0506465
$630$	5.70214	0.227179
$631$	29.1985	1.16238	0.581188	−	0.813770i	$-0.302588\pi$
$631$	29.1985	1.16238	0.581188	+	0.813770i	$0.302588\pi$
$632$	2.16115	0.0859660
$633$	10.2491	0.407363
$634$	−44.1124	−1.75193
$635$	−35.0763	−1.39196
$636$	13.6069	0.539549
$637$	−2.15097	−0.0852246
$638$	0.271534	0.0107501
$639$	13.6486	0.539930
$640$	−16.0185	−0.633188
$641$	19.3237	0.763239	0.381620	−	0.924320i	$-0.375366\pi$
$641$	19.3237	0.763239	0.381620	+	0.924320i	$0.375366\pi$
$642$	−36.4632	−1.43909
$643$	−0.936187	−0.0369196	−0.0184598	−	0.999830i	$-0.505876\pi$
$643$	−0.936187	−0.0369196	−0.0184598	+	0.999830i	$0.505876\pi$
$644$	7.47786	0.294669
$645$	21.7217	0.855290
$646$	−7.65405	−0.301144
$647$	17.0152	0.668938	0.334469	−	0.942407i	$-0.391443\pi$
$647$	17.0152	0.668938	0.334469	+	0.942407i	$0.391443\pi$
$648$	0.681087	0.0267556
$649$	−0.568400	−0.0223117
$650$	16.1134	0.632018
$651$	4.20126	0.164660
$652$	−1.53772	−0.0602216
$653$	12.8426	0.502571	0.251285	−	0.967913i	$-0.419147\pi$
$653$	12.8426	0.502571	0.251285	+	0.967913i	$0.419147\pi$
$654$	−2.05685	−0.0804292
$655$	45.0930	1.76193
$656$	5.47773	0.213869
$657$	10.6181	0.414253
$658$	4.31609	0.168259
$659$	13.9090	0.541818	0.270909	−	0.962605i	$-0.412676\pi$
$659$	13.9090	0.541818	0.270909	+	0.962605i	$0.412676\pi$
$660$	−0.284489	−0.0110737
$661$	2.57156	0.100022	0.0500110	−	0.998749i	$-0.484074\pi$
$661$	2.57156	0.100022	0.0500110	+	0.998749i	$0.484074\pi$
$662$	50.1955	1.95090
$663$	1.69430	0.0658010
$664$	8.00482	0.310647
$665$	−15.2088	−0.589772
$666$	3.07794	0.119268
$667$	11.1711	0.432546
$668$	26.6131	1.02969
$669$	−12.9754	−0.501658
$670$	23.6435	0.913426
$671$	0.465279	0.0179619
$672$	7.39180	0.285145
$673$	−27.9744	−1.07833	−0.539167	−	0.842199i	$-0.681261\pi$
$673$	−27.9744	−1.07833	−0.539167	+	0.842199i	$0.681261\pi$
$674$	−32.7987	−1.26336
$675$	−3.92475	−0.151064
$676$	−13.7588	−0.529184
$677$	−28.9757	−1.11362	−0.556812	−	0.830638i	$-0.687976\pi$
$677$	−28.9757	−1.11362	−0.556812	+	0.830638i	$0.687976\pi$
$678$	−35.4512	−1.36149
$679$	7.83010	0.300492
$680$	1.60271	0.0614612
$681$	−14.1897	−0.543751
$682$	−0.464733	−0.0177956
$683$	−43.8218	−1.67679	−0.838397	−	0.545060i	$-0.816507\pi$
$683$	−43.8218	−1.67679	−0.838397	+	0.545060i	$0.816507\pi$
$684$	8.36525	0.319853
$685$	−43.6837	−1.66907
$686$	1.90871	0.0728748
$687$	11.2725	0.430071
$688$	33.3473	1.27135
$689$	17.8120	0.678583
$690$	−25.9497	−0.987889
$691$	−12.0872	−0.459820	−0.229910	−	0.973212i	$-0.573843\pi$
$691$	−12.0872	−0.459820	−0.229910	+	0.973212i	$0.573843\pi$
$692$	−15.2789	−0.580815
$693$	−0.0579542	−0.00220150
$694$	18.5446	0.703945
$695$	−69.3511	−2.63064
$696$	1.67187	0.0633721
$697$	−0.940783	−0.0356347
$698$	28.8468	1.09187
$699$	−20.2955	−0.767646
$700$	−6.44902	−0.243750
$701$	43.6315	1.64794	0.823969	−	0.566635i	$-0.191755\pi$
$701$	43.6315	1.64794	0.823969	+	0.566635i	$0.191755\pi$
$702$	−4.10558	−0.154955
$703$	−8.20951	−0.309627
$704$	−0.286069	−0.0107816
$705$	−6.75536	−0.254422
$706$	12.1074	0.455670
$707$	13.0062	0.489147
$708$	16.1158	0.605669
$709$	−33.2311	−1.24802	−0.624010	−	0.781417i	$-0.714497\pi$
$709$	−33.2311	−1.24802	−0.624010	+	0.781417i	$0.714497\pi$
$710$	−77.8262	−2.92076
$711$	3.17309	0.119000
$712$	7.87858	0.295262
$713$	−19.1194	−0.716028
$714$	−1.50347	−0.0562659
$715$	−0.372407	−0.0139272
$716$	19.7895	0.739569
$717$	−1.85496	−0.0692748
$718$	16.7112	0.623657
$719$	−44.7627	−1.66937	−0.834684	−	0.550729i	$-0.814350\pi$
$719$	−44.7627	−1.66937	−0.834684	+	0.550729i	$0.814350\pi$
$720$	−13.7014	−0.510620
$721$	8.83571	0.329059
$722$	−13.2035	−0.491385
$723$	10.8234	0.402525
$724$	−12.3732	−0.459846
$725$	−9.63412	−0.357802
$726$	−20.9894	−0.778989
$727$	−26.3521	−0.977344	−0.488672	−	0.872467i	$-0.662519\pi$
$727$	−26.3521	−0.977344	−0.488672	+	0.872467i	$0.662519\pi$
$728$	1.46500	0.0542965
$729$	1.00000	0.0370370
$730$	−60.5460	−2.24091
$731$	−5.72730	−0.211832
$732$	−13.1920	−0.487591
$733$	8.27067	0.305484	0.152742	−	0.988266i	$-0.451190\pi$
$733$	8.27067	0.305484	0.152742	+	0.988266i	$0.451190\pi$
$734$	−60.5511	−2.23498
$735$	−2.98743	−0.110193
$736$	−33.6392	−1.23996
$737$	−0.240302	−0.00885165
$738$	2.27968	0.0839163
$739$	1.86550	0.0686236	0.0343118	−	0.999411i	$-0.489076\pi$
$739$	1.86550	0.0686236	0.0343118	+	0.999411i	$0.489076\pi$
$740$	−7.91590	−0.290994
$741$	10.9504	0.402274
$742$	−15.8058	−0.580250
$743$	−51.5196	−1.89007	−0.945036	−	0.326965i	$-0.893974\pi$
$743$	−51.5196	−1.89007	−0.945036	+	0.326965i	$0.893974\pi$
$744$	−2.86142	−0.104905
$745$	−62.0865	−2.27467
$746$	56.1367	2.05531
$747$	11.7530	0.430020
$748$	0.0750104	0.00274265
$749$	19.1036	0.698029
$750$	−6.13124	−0.223881
$751$	−13.2528	−0.483600	−0.241800	−	0.970326i	$-0.577738\pi$
$751$	−13.2528	−0.483600	−0.241800	+	0.970326i	$0.577738\pi$
$752$	−10.3709	−0.378188
$753$	−6.00483	−0.218828
$754$	−10.0780	−0.367019
$755$	17.0767	0.621485
$756$	1.64317	0.0597615
$757$	−33.7820	−1.22783	−0.613913	−	0.789374i	$-0.710405\pi$
$757$	−33.7820	−1.22783	−0.613913	+	0.789374i	$0.710405\pi$
$758$	21.4291	0.778341
$759$	0.263742	0.00957324
$760$	10.3585	0.375743
$761$	4.13458	0.149878	0.0749392	−	0.997188i	$-0.476124\pi$
$761$	4.13458	0.149878	0.0749392	+	0.997188i	$0.476124\pi$
$762$	−22.4107	−0.811853
$763$	1.07761	0.0390122
$764$	20.8958	0.755984
$765$	2.35317	0.0850789
$766$	1.90871	0.0689644
$767$	21.0962	0.761740
$768$	−20.1067	−0.725538
$769$	35.7451	1.28900	0.644501	−	0.764603i	$-0.277065\pi$
$769$	35.7451	1.28900	0.644501	+	0.764603i	$0.277065\pi$
$770$	0.330463	0.0119090
$771$	14.1743	0.510476
$772$	−9.57100	−0.344468
$773$	26.3240	0.946810	0.473405	−	0.880845i	$-0.343025\pi$
$773$	26.3240	0.946810	0.473405	+	0.880845i	$0.343025\pi$
$774$	13.8783	0.498844
$775$	16.4889	0.592298
$776$	−5.33298	−0.191443
$777$	−1.61258	−0.0578509
$778$	−29.9192	−1.07266
$779$	−6.08039	−0.217853
$780$	10.5588	0.378066
$781$	0.790993	0.0283040
$782$	6.84210	0.244673
$783$	2.45471	0.0877242
$784$	−4.58633	−0.163798
$785$	50.0340	1.78579
$786$	28.8105	1.02764
$787$	−51.5530	−1.83767	−0.918833	−	0.394647i	$-0.870867\pi$
$787$	−51.5530	−1.83767	−0.918833	+	0.394647i	$0.870867\pi$
$788$	−1.62931	−0.0580419
$789$	−26.4817	−0.942774
$790$	−18.0934	−0.643735
$791$	18.5734	0.660394
$792$	0.0394718	0.00140257
$793$	−17.2689	−0.613235
$794$	10.9794	0.389643
$795$	24.7386	0.877389
$796$	44.2464	1.56827
$797$	3.99809	0.141620	0.0708098	−	0.997490i	$-0.477442\pi$
$797$	3.99809	0.141620	0.0708098	+	0.997490i	$0.477442\pi$
$798$	−9.71709	−0.343981
$799$	1.78117	0.0630133
$800$	29.0110	1.02569
$801$	11.5677	0.408723
$802$	4.60476	0.162600
$803$	0.615365	0.0217158
$804$	6.81327	0.240285
$805$	13.5954	0.479176
$806$	17.2486	0.607556
$807$	−6.96464	−0.245167
$808$	−8.85833	−0.311635
$809$	−15.7449	−0.553562	−0.276781	−	0.960933i	$-0.589268\pi$
$809$	−15.7449	−0.553562	−0.276781	+	0.960933i	$0.589268\pi$
$810$	−5.70214	−0.200353
$811$	55.4365	1.94664	0.973320	−	0.229453i	$-0.0736938\pi$
$811$	55.4365	1.94664	0.973320	+	0.229453i	$0.0736938\pi$
$812$	4.03350	0.141548
$813$	11.8871	0.416898
$814$	0.178379	0.00625219
$815$	−2.79571	−0.0979294
$816$	3.61260	0.126466
$817$	−37.0162	−1.29503
$818$	1.81656	0.0635145
$819$	2.15097	0.0751610
$820$	−5.86293	−0.204743
$821$	−11.2355	−0.392122	−0.196061	−	0.980592i	$-0.562815\pi$
$821$	−11.2355	−0.392122	−0.196061	+	0.980592i	$0.562815\pi$
$822$	−27.9101	−0.973476
$823$	−13.2636	−0.462341	−0.231171	−	0.972913i	$-0.574256\pi$
$823$	−13.2636	−0.462341	−0.231171	+	0.972913i	$0.574256\pi$
$824$	−6.01789	−0.209643
$825$	−0.227455	−0.00791898
$826$	−18.7201	−0.651357
$827$	4.05103	0.140868	0.0704340	−	0.997516i	$-0.477562\pi$
$827$	4.05103	0.140868	0.0704340	+	0.997516i	$0.477562\pi$
$828$	−7.47786	−0.259874
$829$	−29.2525	−1.01598	−0.507991	−	0.861362i	$-0.669612\pi$
$829$	−29.2525	−1.01598	−0.507991	+	0.861362i	$0.669612\pi$
$830$	−67.0172	−2.32620
$831$	17.2343	0.597853
$832$	10.6175	0.368095
$833$	0.787689	0.0272918
$834$	−44.3093	−1.53431
$835$	48.3851	1.67443
$836$	0.484801	0.0167672
$837$	−4.20126	−0.145217
$838$	38.9662	1.34607
$839$	−45.7837	−1.58063	−0.790315	−	0.612701i	$-0.790083\pi$
$839$	−45.7837	−1.58063	−0.790315	+	0.612701i	$0.790083\pi$
$840$	2.03470	0.0702039
$841$	−22.9744	−0.792221
$842$	−66.0195	−2.27518
$843$	14.3026	0.492606
$844$	−16.8409	−0.579689
$845$	−25.0147	−0.860532
$846$	−4.31609	−0.148390
$847$	10.9966	0.377849
$848$	37.9790	1.30420
$849$	22.5514	0.773962
$850$	−5.90073	−0.202394
$851$	7.33864	0.251565
$852$	−22.4269	−0.768335
$853$	−3.34491	−0.114528	−0.0572638	−	0.998359i	$-0.518238\pi$
$853$	−3.34491	−0.114528	−0.0572638	+	0.998359i	$0.518238\pi$
$854$	15.3239	0.524372
$855$	15.2088	0.520130
$856$	−13.0112	−0.444714
$857$	13.1186	0.448122	0.224061	−	0.974575i	$-0.428068\pi$
$857$	13.1186	0.448122	0.224061	+	0.974575i	$0.428068\pi$
$858$	−0.237935	−0.00812298
$859$	−20.9327	−0.714213	−0.357106	−	0.934064i	$-0.616237\pi$
$859$	−20.9327	−0.714213	−0.357106	+	0.934064i	$0.616237\pi$
$860$	−35.6924	−1.21710
$861$	−1.19436	−0.0407037
$862$	−27.0318	−0.920707
$863$	−55.4770	−1.88846	−0.944230	−	0.329288i	$-0.893191\pi$
$863$	−55.4770	−1.88846	−0.944230	+	0.329288i	$0.893191\pi$
$864$	−7.39180	−0.251474
$865$	−27.7784	−0.944493
$866$	−11.9081	−0.404655
$867$	16.3795	0.556279
$868$	−6.90338	−0.234316
$869$	0.183894	0.00623818
$870$	−13.9971	−0.474546
$871$	8.91884	0.302203
$872$	−0.733948	−0.0248546
$873$	−7.83010	−0.265009
$874$	44.2213	1.49581
$875$	3.21224	0.108594
$876$	−17.4474	−0.589492
$877$	−16.0783	−0.542926	−0.271463	−	0.962449i	$-0.587507\pi$
$877$	−16.0783	−0.542926	−0.271463	+	0.962449i	$0.587507\pi$
$878$	−43.2691	−1.46026
$879$	−16.6292	−0.560889
$880$	−0.794051	−0.0267675
$881$	34.0933	1.14863	0.574316	−	0.818634i	$-0.305268\pi$
$881$	34.0933	1.14863	0.574316	+	0.818634i	$0.305268\pi$
$882$	−1.90871	−0.0642696
$883$	11.8741	0.399595	0.199798	−	0.979837i	$-0.435971\pi$
$883$	11.8741	0.399595	0.199798	+	0.979837i	$0.435971\pi$
$884$	−2.78402	−0.0936366
$885$	29.3000	0.984909
$886$	−15.1753	−0.509823
$887$	−16.2388	−0.545247	−0.272623	−	0.962121i	$-0.587891\pi$
$887$	−16.2388	−0.545247	−0.272623	+	0.962121i	$0.587891\pi$
$888$	1.09830	0.0368567
$889$	11.7413	0.393790
$890$	−65.9604	−2.21100
$891$	0.0579542	0.00194154
$892$	21.3208	0.713872
$893$	11.5119	0.385232
$894$	−39.6679	−1.32669
$895$	35.9792	1.20265
$896$	5.36197	0.179131
$897$	−9.78881	−0.326839
$898$	−59.4674	−1.98445
$899$	−10.3129	−0.343954
$900$	6.44902	0.214967
$901$	−6.52277	−0.217305
$902$	0.132117	0.00439902
$903$	−7.27102	−0.241964
$904$	−12.6501	−0.420736
$905$	−22.4956	−0.747779
$906$	10.9105	0.362478
$907$	16.5732	0.550304	0.275152	−	0.961401i	$-0.411272\pi$
$907$	16.5732	0.550304	0.275152	+	0.961401i	$0.411272\pi$
$908$	23.3161	0.773772
$909$	−13.0062	−0.431387
$910$	−12.2651	−0.406585
$911$	−4.67204	−0.154792	−0.0773958	−	0.997000i	$-0.524660\pi$
$911$	−4.67204	−0.154792	−0.0773958	+	0.997000i	$0.524660\pi$
$912$	23.3487	0.773152
$913$	0.681135	0.0225423
$914$	71.6138	2.36878
$915$	−23.9843	−0.792896
$916$	−18.5225	−0.612002
$917$	−15.0942	−0.498456
$918$	1.50347	0.0496219
$919$	8.30210	0.273861	0.136931	−	0.990581i	$-0.456276\pi$
$919$	8.30210	0.273861	0.136931	+	0.990581i	$0.456276\pi$
$920$	−9.25967	−0.305282
$921$	9.34402	0.307896
$922$	68.8407	2.26715
$923$	−29.3578	−0.966322
$924$	0.0952285	0.00313279
$925$	−6.32895	−0.208095
$926$	18.2758	0.600582
$927$	−8.83571	−0.290203
$928$	−18.1447	−0.595630
$929$	−59.3002	−1.94558	−0.972788	−	0.231697i	$-0.925572\pi$
$929$	−59.3002	−1.94558	−0.972788	+	0.231697i	$0.925572\pi$
$930$	23.9562	0.785553
$931$	5.09093	0.166848
$932$	33.3489	1.09238
$933$	−0.594707	−0.0194698
$934$	2.68319	0.0877967
$935$	0.136376	0.00445997
$936$	−1.46500	−0.0478850
$937$	32.1097	1.04898	0.524489	−	0.851417i	$-0.324257\pi$
$937$	32.1097	1.04898	0.524489	+	0.851417i	$0.324257\pi$
$938$	−7.91431	−0.258411
$939$	−33.8418	−1.10439
$940$	11.1002	0.362049
$941$	−44.0544	−1.43613	−0.718066	−	0.695975i	$-0.754973\pi$
$941$	−44.0544	−1.43613	−0.718066	+	0.695975i	$0.754973\pi$
$942$	31.9674	1.04155
$943$	5.43538	0.177000
$944$	44.9816	1.46403
$945$	2.98743	0.0971812
$946$	0.804303	0.0261501
$947$	−29.0470	−0.943902	−0.471951	−	0.881625i	$-0.656450\pi$
$947$	−29.0470	−0.943902	−0.471951	+	0.881625i	$0.656450\pi$
$948$	−5.21393	−0.169340
$949$	−22.8393	−0.741395
$950$	−38.1371	−1.23733
$951$	−23.1111	−0.749430
$952$	−0.536485	−0.0173876
$953$	−50.6270	−1.63997	−0.819985	−	0.572385i	$-0.806018\pi$
$953$	−50.6270	−1.63997	−0.819985	+	0.572385i	$0.806018\pi$
$954$	15.8058	0.511733
$955$	37.9905	1.22934
$956$	3.04801	0.0985798
$957$	0.142261	0.00459864
$958$	−55.1846	−1.78293
$959$	14.6225	0.472185
$960$	14.7463	0.475936
$961$	−13.3494	−0.430626
$962$	−6.62056	−0.213455
$963$	−19.1036	−0.615604
$964$	−17.7846	−0.572803
$965$	−17.4010	−0.560157
$966$	8.68630	0.279477
$967$	11.4538	0.368330	0.184165	−	0.982895i	$-0.441042\pi$
$967$	11.4538	0.368330	0.184165	+	0.982895i	$0.441042\pi$
$968$	−7.48967	−0.240727
$969$	−4.01007	−0.128822
$970$	44.6483	1.43357
$971$	23.4898	0.753822	0.376911	−	0.926249i	$-0.376986\pi$
$971$	23.4898	0.753822	0.376911	+	0.926249i	$0.376986\pi$
$972$	−1.64317	−0.0527047
$973$	23.2143	0.744216
$974$	46.5588	1.49184
$975$	8.44202	0.270361
$976$	−36.8209	−1.17861
$977$	29.3504	0.939002	0.469501	−	0.882932i	$-0.344434\pi$
$977$	29.3504	0.939002	0.469501	+	0.882932i	$0.344434\pi$
$978$	−1.78621	−0.0571169
$979$	0.670394	0.0214259
$980$	4.90885	0.156808
$981$	−1.07761	−0.0344055
$982$	56.4796	1.80234
$983$	−4.93733	−0.157476	−0.0787381	−	0.996895i	$-0.525089\pi$
$983$	−4.93733	−0.157476	−0.0787381	+	0.996895i	$0.525089\pi$
$984$	0.813463	0.0259322
$985$	−2.96224	−0.0943848
$986$	3.69058	0.117532
$987$	2.26126	0.0719767
$988$	−17.9934	−0.572447
$989$	33.0895	1.05219
$990$	−0.330463	−0.0105028
$991$	−39.9154	−1.26796	−0.633978	−	0.773351i	$-0.718579\pi$
$991$	−39.9154	−1.26796	−0.633978	+	0.773351i	$0.718579\pi$
$992$	31.0549	0.985993
$993$	26.2981	0.834546
$994$	26.0512	0.826294
$995$	80.4440	2.55025
$996$	−19.3122	−0.611929
$997$	53.4088	1.69147	0.845737	−	0.533600i	$-0.179161\pi$
$997$	53.4088	1.69147	0.845737	+	0.533600i	$0.179161\pi$
$998$	61.1660	1.93618
$999$	1.61258	0.0510197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.r.1.8	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.r.1.8	✓	46	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-1.90871 q^{2} -1.00000 q^{3} +1.64317 q^{4} +2.98743 q^{5} +1.90871 q^{6} -1.00000 q^{7} +0.681087 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.90871 q^{2} -1.00000 q^{3} +1.64317 q^{4} +2.98743 q^{5} +1.90871 q^{6} -1.00000 q^{7} +0.681087 q^{8} +1.00000 q^{9} -5.70214 q^{10} +0.0579542 q^{11} -1.64317 q^{12} -2.15097 q^{13} +1.90871 q^{14} -2.98743 q^{15} -4.58633 q^{16} +0.787689 q^{17} -1.90871 q^{18} +5.09093 q^{19} +4.90885 q^{20} +1.00000 q^{21} -0.110618 q^{22} -4.55088 q^{23} -0.681087 q^{24} +3.92475 q^{25} +4.10558 q^{26} -1.00000 q^{27} -1.64317 q^{28} -2.45471 q^{29} +5.70214 q^{30} +4.20126 q^{31} +7.39180 q^{32} -0.0579542 q^{33} -1.50347 q^{34} -2.98743 q^{35} +1.64317 q^{36} -1.61258 q^{37} -9.71709 q^{38} +2.15097 q^{39} +2.03470 q^{40} -1.19436 q^{41} -1.90871 q^{42} -7.27102 q^{43} +0.0952285 q^{44} +2.98743 q^{45} +8.68630 q^{46} +2.26126 q^{47} +4.58633 q^{48} +1.00000 q^{49} -7.49120 q^{50} -0.787689 q^{51} -3.53441 q^{52} -8.28090 q^{53} +1.90871 q^{54} +0.173134 q^{55} -0.681087 q^{56} -5.09093 q^{57} +4.68533 q^{58} -9.80775 q^{59} -4.90885 q^{60} +8.02840 q^{61} -8.01898 q^{62} -1.00000 q^{63} -4.93613 q^{64} -6.42588 q^{65} +0.110618 q^{66} -4.14642 q^{67} +1.29431 q^{68} +4.55088 q^{69} +5.70214 q^{70} +13.6486 q^{71} +0.681087 q^{72} +10.6181 q^{73} +3.07794 q^{74} -3.92475 q^{75} +8.36525 q^{76} -0.0579542 q^{77} -4.10558 q^{78} +3.17309 q^{79} -13.7014 q^{80} +1.00000 q^{81} +2.27968 q^{82} +11.7530 q^{83} +1.64317 q^{84} +2.35317 q^{85} +13.8783 q^{86} +2.45471 q^{87} +0.0394718 q^{88} +11.5677 q^{89} -5.70214 q^{90} +2.15097 q^{91} -7.47786 q^{92} -4.20126 q^{93} -4.31609 q^{94} +15.2088 q^{95} -7.39180 q^{96} -7.83010 q^{97} -1.90871 q^{98} +0.0579542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 3 q^{2} - 46 q^{3} + 45 q^{4} - 9 q^{5} - 3 q^{6} - 46 q^{7} + 6 q^{8} + 46 q^{9}+O(q^{10}) \) 46 * q + 3 * q^2 - 46 * q^3 + 45 * q^4 - 9 * q^5 - 3 * q^6 - 46 * q^7 + 6 * q^8 + 46 * q^9	\( 46 q + 3 q^{2} - 46 q^{3} + 45 q^{4} - 9 q^{5} - 3 q^{6} - 46 q^{7} + 6 q^{8} + 46 q^{9} - 10 q^{10} + 31 q^{11} - 45 q^{12} - 32 q^{13} - 3 q^{14} + 9 q^{15} + 43 q^{16} - 36 q^{17} + 3 q^{18} - 13 q^{19} - 19 q^{20} + 46 q^{21} - 13 q^{22} + 24 q^{23} - 6 q^{24} + 35 q^{25} - 11 q^{26} - 46 q^{27} - 45 q^{28} + 11 q^{29} + 10 q^{30} - 23 q^{31} + 5 q^{32} - 31 q^{33} - 35 q^{34} + 9 q^{35} + 45 q^{36} - 37 q^{37} - 32 q^{38} + 32 q^{39} - 28 q^{40} - 27 q^{41} + 3 q^{42} - 7 q^{43} + 46 q^{44} - 9 q^{45} + 16 q^{46} - 18 q^{47} - 43 q^{48} + 46 q^{49} + 10 q^{50} + 36 q^{51} - 62 q^{52} - 62 q^{53} - 3 q^{54} - 28 q^{55} - 6 q^{56} + 13 q^{57} - 36 q^{58} - 3 q^{59} + 19 q^{60} - 31 q^{61} - 41 q^{62} - 46 q^{63} + 42 q^{64} + 2 q^{65} + 13 q^{66} - 9 q^{67} - 70 q^{68} - 24 q^{69} + 10 q^{70} + 77 q^{71} + 6 q^{72} - 38 q^{73} + 14 q^{74} - 35 q^{75} - 41 q^{76} - 31 q^{77} + 11 q^{78} + 8 q^{79} - 59 q^{80} + 46 q^{81} - 53 q^{82} - 38 q^{83} + 45 q^{84} - 26 q^{85} + 37 q^{86} - 11 q^{87} - 26 q^{88} - 39 q^{89} - 10 q^{90} + 32 q^{91} + 2 q^{92} + 23 q^{93} - 55 q^{94} + 35 q^{95} - 5 q^{96} - 61 q^{97} + 3 q^{98} + 31 q^{99}+O(q^{100}) \) 46 * q + 3 * q^2 - 46 * q^3 + 45 * q^4 - 9 * q^5 - 3 * q^6 - 46 * q^7 + 6 * q^8 + 46 * q^9 - 10 * q^10 + 31 * q^11 - 45 * q^12 - 32 * q^13 - 3 * q^14 + 9 * q^15 + 43 * q^16 - 36 * q^17 + 3 * q^18 - 13 * q^19 - 19 * q^20 + 46 * q^21 - 13 * q^22 + 24 * q^23 - 6 * q^24 + 35 * q^25 - 11 * q^26 - 46 * q^27 - 45 * q^28 + 11 * q^29 + 10 * q^30 - 23 * q^31 + 5 * q^32 - 31 * q^33 - 35 * q^34 + 9 * q^35 + 45 * q^36 - 37 * q^37 - 32 * q^38 + 32 * q^39 - 28 * q^40 - 27 * q^41 + 3 * q^42 - 7 * q^43 + 46 * q^44 - 9 * q^45 + 16 * q^46 - 18 * q^47 - 43 * q^48 + 46 * q^49 + 10 * q^50 + 36 * q^51 - 62 * q^52 - 62 * q^53 - 3 * q^54 - 28 * q^55 - 6 * q^56 + 13 * q^57 - 36 * q^58 - 3 * q^59 + 19 * q^60 - 31 * q^61 - 41 * q^62 - 46 * q^63 + 42 * q^64 + 2 * q^65 + 13 * q^66 - 9 * q^67 - 70 * q^68 - 24 * q^69 + 10 * q^70 + 77 * q^71 + 6 * q^72 - 38 * q^73 + 14 * q^74 - 35 * q^75 - 41 * q^76 - 31 * q^77 + 11 * q^78 + 8 * q^79 - 59 * q^80 + 46 * q^81 - 53 * q^82 - 38 * q^83 + 45 * q^84 - 26 * q^85 + 37 * q^86 - 11 * q^87 - 26 * q^88 - 39 * q^89 - 10 * q^90 + 32 * q^91 + 2 * q^92 + 23 * q^93 - 55 * q^94 + 35 * q^95 - 5 * q^96 - 61 * q^97 + 3 * q^98 + 31 * q^99

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