LMFDB - Embedded newform 8042.2.a.c.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.17249 q^{3} +1.00000 q^{4} -0.488966 q^{5} +2.17249 q^{6} -0.203617 q^{7} -1.00000 q^{8} +1.71971 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.17249 q^{3} +1.00000 q^{4} -0.488966 q^{5} +2.17249 q^{6} -0.203617 q^{7} -1.00000 q^{8} +1.71971 q^{9} +0.488966 q^{10} -2.88551 q^{11} -2.17249 q^{12} +1.03871 q^{13} +0.203617 q^{14} +1.06227 q^{15} +1.00000 q^{16} -1.76208 q^{17} -1.71971 q^{18} +3.22380 q^{19} -0.488966 q^{20} +0.442356 q^{21} +2.88551 q^{22} -9.36265 q^{23} +2.17249 q^{24} -4.76091 q^{25} -1.03871 q^{26} +2.78142 q^{27} -0.203617 q^{28} -5.34763 q^{29} -1.06227 q^{30} -0.0460050 q^{31} -1.00000 q^{32} +6.26874 q^{33} +1.76208 q^{34} +0.0995620 q^{35} +1.71971 q^{36} -5.50737 q^{37} -3.22380 q^{38} -2.25660 q^{39} +0.488966 q^{40} -6.91171 q^{41} -0.442356 q^{42} +2.67630 q^{43} -2.88551 q^{44} -0.840880 q^{45} +9.36265 q^{46} +12.3295 q^{47} -2.17249 q^{48} -6.95854 q^{49} +4.76091 q^{50} +3.82809 q^{51} +1.03871 q^{52} +11.3286 q^{53} -2.78142 q^{54} +1.41092 q^{55} +0.203617 q^{56} -7.00367 q^{57} +5.34763 q^{58} -14.6626 q^{59} +1.06227 q^{60} +1.54765 q^{61} +0.0460050 q^{62} -0.350163 q^{63} +1.00000 q^{64} -0.507897 q^{65} -6.26874 q^{66} -10.4059 q^{67} -1.76208 q^{68} +20.3403 q^{69} -0.0995620 q^{70} +0.421760 q^{71} -1.71971 q^{72} +7.89700 q^{73} +5.50737 q^{74} +10.3430 q^{75} +3.22380 q^{76} +0.587540 q^{77} +2.25660 q^{78} +7.66659 q^{79} -0.488966 q^{80} -11.2017 q^{81} +6.91171 q^{82} -14.2667 q^{83} +0.442356 q^{84} +0.861596 q^{85} -2.67630 q^{86} +11.6177 q^{87} +2.88551 q^{88} +13.3220 q^{89} +0.840880 q^{90} -0.211500 q^{91} -9.36265 q^{92} +0.0999453 q^{93} -12.3295 q^{94} -1.57633 q^{95} +2.17249 q^{96} -12.4930 q^{97} +6.95854 q^{98} -4.96223 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * 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
$3$	−2.17249	−1.25429	−0.627144	−	0.778904i	$-0.715776\pi$
$3$	−2.17249	−1.25429	−0.627144	+	0.778904i	$0.715776\pi$
$4$	1.00000	0.500000
$5$	−0.488966	−0.218672	−0.109336	−	0.994005i	$-0.534873\pi$
$5$	−0.488966	−0.218672	−0.109336	+	0.994005i	$0.534873\pi$
$6$	2.17249	0.886915
$7$	−0.203617	−0.0769601	−0.0384801	−	0.999259i	$-0.512252\pi$
$7$	−0.203617	−0.0769601	−0.0384801	+	0.999259i	$0.512252\pi$
$8$	−1.00000	−0.353553
$9$	1.71971	0.573236
$10$	0.488966	0.154625
$11$	−2.88551	−0.870014	−0.435007	−	0.900427i	$-0.643254\pi$
$11$	−2.88551	−0.870014	−0.435007	+	0.900427i	$0.643254\pi$
$12$	−2.17249	−0.627144
$13$	1.03871	0.288088	0.144044	−	0.989571i	$-0.453989\pi$
$13$	1.03871	0.288088	0.144044	+	0.989571i	$0.453989\pi$
$14$	0.203617	0.0544190
$15$	1.06227	0.274278
$16$	1.00000	0.250000
$17$	−1.76208	−0.427366	−0.213683	−	0.976903i	$-0.568546\pi$
$17$	−1.76208	−0.427366	−0.213683	+	0.976903i	$0.568546\pi$
$18$	−1.71971	−0.405339
$19$	3.22380	0.739590	0.369795	−	0.929113i	$-0.379428\pi$
$19$	3.22380	0.739590	0.369795	+	0.929113i	$0.379428\pi$
$20$	−0.488966	−0.109336
$21$	0.442356	0.0965301
$22$	2.88551	0.615193
$23$	−9.36265	−1.95225	−0.976124	−	0.217215i	$-0.930303\pi$
$23$	−9.36265	−1.95225	−0.976124	+	0.217215i	$0.930303\pi$
$24$	2.17249	0.443457
$25$	−4.76091	−0.952182
$26$	−1.03871	−0.203709
$27$	2.78142	0.535284
$28$	−0.203617	−0.0384801
$29$	−5.34763	−0.993030	−0.496515	−	0.868028i	$-0.665387\pi$
$29$	−5.34763	−0.993030	−0.496515	+	0.868028i	$0.665387\pi$
$30$	−1.06227	−0.193944
$31$	−0.0460050	−0.00826274	−0.00413137	−	0.999991i	$-0.501315\pi$
$31$	−0.0460050	−0.00826274	−0.00413137	+	0.999991i	$0.501315\pi$
$32$	−1.00000	−0.176777
$33$	6.26874	1.09125
$34$	1.76208	0.302194
$35$	0.0995620	0.0168291
$36$	1.71971	0.286618
$37$	−5.50737	−0.905406	−0.452703	−	0.891661i	$-0.649540\pi$
$37$	−5.50737	−0.905406	−0.452703	+	0.891661i	$0.649540\pi$
$38$	−3.22380	−0.522969
$39$	−2.25660	−0.361345
$40$	0.488966	0.0773124
$41$	−6.91171	−1.07943	−0.539714	−	0.841849i	$-0.681467\pi$
$41$	−6.91171	−1.07943	−0.539714	+	0.841849i	$0.681467\pi$
$42$	−0.442356	−0.0682571
$43$	2.67630	0.408132	0.204066	−	0.978957i	$-0.434584\pi$
$43$	2.67630	0.408132	0.204066	+	0.978957i	$0.434584\pi$
$44$	−2.88551	−0.435007
$45$	−0.840880	−0.125351
$46$	9.36265	1.38045
$47$	12.3295	1.79844	0.899220	−	0.437497i	$-0.144135\pi$
$47$	12.3295	1.79844	0.899220	+	0.437497i	$0.144135\pi$
$48$	−2.17249	−0.313572
$49$	−6.95854	−0.994077
$50$	4.76091	0.673295
$51$	3.82809	0.536040
$52$	1.03871	0.144044
$53$	11.3286	1.55610	0.778052	−	0.628200i	$-0.216208\pi$
$53$	11.3286	1.55610	0.778052	+	0.628200i	$0.216208\pi$
$54$	−2.78142	−0.378503
$55$	1.41092	0.190248
$56$	0.203617	0.0272095
$57$	−7.00367	−0.927658
$58$	5.34763	0.702178
$59$	−14.6626	−1.90891	−0.954455	−	0.298355i	$-0.903562\pi$
$59$	−14.6626	−1.90891	−0.954455	+	0.298355i	$0.903562\pi$
$60$	1.06227	0.137139
$61$	1.54765	0.198157	0.0990783	−	0.995080i	$-0.468411\pi$
$61$	1.54765	0.198157	0.0990783	+	0.995080i	$0.468411\pi$
$62$	0.0460050	0.00584264
$63$	−0.350163	−0.0441163
$64$	1.00000	0.125000
$65$	−0.507897	−0.0629968
$66$	−6.26874	−0.771628
$67$	−10.4059	−1.27128	−0.635641	−	0.771985i	$-0.719264\pi$
$67$	−10.4059	−1.27128	−0.635641	+	0.771985i	$0.719264\pi$
$68$	−1.76208	−0.213683
$69$	20.3403	2.44868
$70$	−0.0995620	−0.0118999
$71$	0.421760	0.0500537	0.0250268	−	0.999687i	$-0.492033\pi$
$71$	0.421760	0.0500537	0.0250268	+	0.999687i	$0.492033\pi$
$72$	−1.71971	−0.202670
$73$	7.89700	0.924274	0.462137	−	0.886809i	$-0.347083\pi$
$73$	7.89700	0.924274	0.462137	+	0.886809i	$0.347083\pi$
$74$	5.50737	0.640219
$75$	10.3430	1.19431
$76$	3.22380	0.369795
$77$	0.587540	0.0669564
$78$	2.25660	0.255509
$79$	7.66659	0.862559	0.431280	−	0.902218i	$-0.358062\pi$
$79$	7.66659	0.862559	0.431280	+	0.902218i	$0.358062\pi$
$80$	−0.488966	−0.0546681
$81$	−11.2017	−1.24464
$82$	6.91171	0.763270
$83$	−14.2667	−1.56597	−0.782986	−	0.622039i	$-0.786304\pi$
$83$	−14.2667	−1.56597	−0.782986	+	0.622039i	$0.786304\pi$
$84$	0.442356	0.0482650
$85$	0.861596	0.0934532
$86$	−2.67630	−0.288593
$87$	11.6177	1.24555
$88$	2.88551	0.307596
$89$	13.3220	1.41213	0.706063	−	0.708149i	$-0.250470\pi$
$89$	13.3220	1.41213	0.706063	+	0.708149i	$0.250470\pi$
$90$	0.840880	0.0886365
$91$	−0.211500	−0.0221713
$92$	−9.36265	−0.976124
$93$	0.0999453	0.0103638
$94$	−12.3295	−1.27169
$95$	−1.57633	−0.161728
$96$	2.17249	0.221729
$97$	−12.4930	−1.26847	−0.634234	−	0.773141i	$-0.718685\pi$
$97$	−12.4930	−1.26847	−0.634234	+	0.773141i	$0.718685\pi$
$98$	6.95854	0.702919
$99$	−4.96223	−0.498723
$100$	−4.76091	−0.476091
$101$	3.05855	0.304337	0.152169	−	0.988355i	$-0.451374\pi$
$101$	3.05855	0.304337	0.152169	+	0.988355i	$0.451374\pi$
$102$	−3.82809	−0.379038
$103$	−10.4018	−1.02492	−0.512458	−	0.858712i	$-0.671265\pi$
$103$	−10.4018	−1.02492	−0.512458	+	0.858712i	$0.671265\pi$
$104$	−1.03871	−0.101854
$105$	−0.216297	−0.0211085
$106$	−11.3286	−1.10033
$107$	−19.5104	−1.88614	−0.943069	−	0.332597i	$-0.892075\pi$
$107$	−19.5104	−1.88614	−0.943069	+	0.332597i	$0.892075\pi$
$108$	2.78142	0.267642
$109$	−0.995382	−0.0953403	−0.0476702	−	0.998863i	$-0.515180\pi$
$109$	−0.995382	−0.0953403	−0.0476702	+	0.998863i	$0.515180\pi$
$110$	−1.41092	−0.134526
$111$	11.9647	1.13564
$112$	−0.203617	−0.0192400
$113$	−7.34669	−0.691118	−0.345559	−	0.938397i	$-0.612311\pi$
$113$	−7.34669	−0.691118	−0.345559	+	0.938397i	$0.612311\pi$
$114$	7.00367	0.655954
$115$	4.57802	0.426903
$116$	−5.34763	−0.496515
$117$	1.78629	0.165142
$118$	14.6626	1.34980
$119$	0.358789	0.0328902
$120$	−1.06227	−0.0969719
$121$	−2.67384	−0.243076
$122$	−1.54765	−0.140118
$123$	15.0156	1.35391
$124$	−0.0460050	−0.00413137
$125$	4.77276	0.426888
$126$	0.350163	0.0311950
$127$	10.5809	0.938899	0.469450	−	0.882959i	$-0.344452\pi$
$127$	10.5809	0.938899	0.469450	+	0.882959i	$0.344452\pi$
$128$	−1.00000	−0.0883883
$129$	−5.81424	−0.511915
$130$	0.507897	0.0445455
$131$	−4.80680	−0.419972	−0.209986	−	0.977704i	$-0.567342\pi$
$131$	−4.80680	−0.419972	−0.209986	+	0.977704i	$0.567342\pi$
$132$	6.26874	0.545623
$133$	−0.656421	−0.0569189
$134$	10.4059	0.898932
$135$	−1.36002	−0.117052
$136$	1.76208	0.151097
$137$	−7.66777	−0.655101	−0.327551	−	0.944834i	$-0.606223\pi$
$137$	−7.66777	−0.655101	−0.327551	+	0.944834i	$0.606223\pi$
$138$	−20.3403	−1.73148
$139$	15.1655	1.28632	0.643161	−	0.765731i	$-0.277623\pi$
$139$	15.1655	1.28632	0.643161	+	0.765731i	$0.277623\pi$
$140$	0.0995620	0.00841453
$141$	−26.7857	−2.25576
$142$	−0.421760	−0.0353933
$143$	−2.99722	−0.250640
$144$	1.71971	0.143309
$145$	2.61481	0.217148
$146$	−7.89700	−0.653560
$147$	15.1174	1.24686
$148$	−5.50737	−0.452703
$149$	−19.4470	−1.59316	−0.796580	−	0.604533i	$-0.793360\pi$
$149$	−19.4470	−1.59316	−0.796580	+	0.604533i	$0.793360\pi$
$150$	−10.3430	−0.844505
$151$	20.6951	1.68414	0.842070	−	0.539369i	$-0.181337\pi$
$151$	20.6951	1.68414	0.842070	+	0.539369i	$0.181337\pi$
$152$	−3.22380	−0.261485
$153$	−3.03026	−0.244982
$154$	−0.587540	−0.0473453
$155$	0.0224949	0.00180683
$156$	−2.25660	−0.180672
$157$	14.0367	1.12025	0.560127	−	0.828407i	$-0.310752\pi$
$157$	14.0367	1.12025	0.560127	+	0.828407i	$0.310752\pi$
$158$	−7.66659	−0.609921
$159$	−24.6113	−1.95180
$160$	0.488966	0.0386562
$161$	1.90640	0.150245
$162$	11.2017	0.880091
$163$	−10.7706	−0.843615	−0.421808	−	0.906685i	$-0.638604\pi$
$163$	−10.7706	−0.843615	−0.421808	+	0.906685i	$0.638604\pi$
$164$	−6.91171	−0.539714
$165$	−3.06520	−0.238626
$166$	14.2667	1.10731
$167$	18.9859	1.46917	0.734586	−	0.678516i	$-0.237377\pi$
$167$	18.9859	1.46917	0.734586	+	0.678516i	$0.237377\pi$
$168$	−0.442356	−0.0341285
$169$	−11.9211	−0.917006
$170$	−0.861596	−0.0660814
$171$	5.54400	0.423960
$172$	2.67630	0.204066
$173$	6.10393	0.464073	0.232037	−	0.972707i	$-0.425461\pi$
$173$	6.10393	0.464073	0.232037	+	0.972707i	$0.425461\pi$
$174$	−11.6177	−0.880733
$175$	0.969404	0.0732801
$176$	−2.88551	−0.217503
$177$	31.8544	2.39432
$178$	−13.3220	−0.998524
$179$	−13.2117	−0.987486	−0.493743	−	0.869608i	$-0.664372\pi$
$179$	−13.2117	−0.987486	−0.493743	+	0.869608i	$0.664372\pi$
$180$	−0.840880	−0.0626755
$181$	−10.2089	−0.758825	−0.379413	−	0.925228i	$-0.623874\pi$
$181$	−10.2089	−0.758825	−0.379413	+	0.925228i	$0.623874\pi$
$182$	0.211500	0.0156774
$183$	−3.36226	−0.248545
$184$	9.36265	0.690224
$185$	2.69292	0.197987
$186$	−0.0999453	−0.00732835
$187$	5.08449	0.371815
$188$	12.3295	0.899220
$189$	−0.566345	−0.0411955
$190$	1.57633	0.114359
$191$	−26.4907	−1.91680	−0.958398	−	0.285435i	$-0.907862\pi$
$191$	−26.4907	−1.91680	−0.958398	+	0.285435i	$0.907862\pi$
$192$	−2.17249	−0.156786
$193$	−0.470343	−0.0338560	−0.0169280	−	0.999857i	$-0.505389\pi$
$193$	−0.470343	−0.0338560	−0.0169280	+	0.999857i	$0.505389\pi$
$194$	12.4930	0.896943
$195$	1.10340	0.0790161
$196$	−6.95854	−0.497039
$197$	3.51558	0.250475	0.125238	−	0.992127i	$-0.460031\pi$
$197$	3.51558	0.250475	0.125238	+	0.992127i	$0.460031\pi$
$198$	4.96223	0.352651
$199$	13.1428	0.931671	0.465835	−	0.884871i	$-0.345754\pi$
$199$	13.1428	0.931671	0.465835	+	0.884871i	$0.345754\pi$
$200$	4.76091	0.336647
$201$	22.6067	1.59455
$202$	−3.05855	−0.215199
$203$	1.08887	0.0764237
$204$	3.82809	0.268020
$205$	3.37959	0.236041
$206$	10.4018	0.724725
$207$	−16.1010	−1.11910
$208$	1.03871	0.0720219
$209$	−9.30230	−0.643453
$210$	0.216297	0.0149259
$211$	3.40461	0.234383	0.117192	−	0.993109i	$-0.462611\pi$
$211$	3.40461	0.234383	0.117192	+	0.993109i	$0.462611\pi$
$212$	11.3286	0.778052
$213$	−0.916269	−0.0627817
$214$	19.5104	1.33370
$215$	−1.30862	−0.0892472
$216$	−2.78142	−0.189252
$217$	0.00936741	0.000635901 0
$218$	0.995382	0.0674158
$219$	−17.1561	−1.15930
$220$	1.41092	0.0951240
$221$	−1.83030	−0.123119
$222$	−11.9647	−0.803018
$223$	20.3123	1.36021	0.680106	−	0.733114i	$-0.261934\pi$
$223$	20.3123	1.36021	0.680106	+	0.733114i	$0.261934\pi$
$224$	0.203617	0.0136048
$225$	−8.18738	−0.545825
$226$	7.34669	0.488694
$227$	−24.7343	−1.64168	−0.820838	−	0.571161i	$-0.806493\pi$
$227$	−24.7343	−1.64168	−0.820838	+	0.571161i	$0.806493\pi$
$228$	−7.00367	−0.463829
$229$	8.57570	0.566698	0.283349	−	0.959017i	$-0.408555\pi$
$229$	8.57570	0.566698	0.283349	+	0.959017i	$0.408555\pi$
$230$	−4.57802	−0.301866
$231$	−1.27642	−0.0839825
$232$	5.34763	0.351089
$233$	−26.9900	−1.76817	−0.884087	−	0.467322i	$-0.845219\pi$
$233$	−26.9900	−1.76817	−0.884087	+	0.467322i	$0.845219\pi$
$234$	−1.78629	−0.116773
$235$	−6.02870	−0.393269
$236$	−14.6626	−0.954455
$237$	−16.6556	−1.08190
$238$	−0.358789	−0.0232569
$239$	−2.11151	−0.136583	−0.0682913	−	0.997665i	$-0.521755\pi$
$239$	−2.11151	−0.136583	−0.0682913	+	0.997665i	$0.521755\pi$
$240$	1.06227	0.0685695
$241$	−25.8562	−1.66555	−0.832773	−	0.553614i	$-0.813248\pi$
$241$	−25.8562	−1.66555	−0.832773	+	0.553614i	$0.813248\pi$
$242$	2.67384	0.171881
$243$	15.9914	1.02585
$244$	1.54765	0.0990783
$245$	3.40249	0.217377
$246$	−15.0156	−0.957360
$247$	3.34861	0.213067
$248$	0.0460050	0.00292132
$249$	30.9942	1.96418
$250$	−4.77276	−0.301856
$251$	5.48143	0.345985	0.172993	−	0.984923i	$-0.444656\pi$
$251$	5.48143	0.345985	0.172993	+	0.984923i	$0.444656\pi$
$252$	−0.350163	−0.0220582
$253$	27.0160	1.69848
$254$	−10.5809	−0.663902
$255$	−1.87181	−0.117217
$256$	1.00000	0.0625000
$257$	15.4009	0.960681	0.480341	−	0.877082i	$-0.340513\pi$
$257$	15.4009	0.960681	0.480341	+	0.877082i	$0.340513\pi$
$258$	5.81424	0.361979
$259$	1.12140	0.0696802
$260$	−0.507897	−0.0314984
$261$	−9.19637	−0.569241
$262$	4.80680	0.296965
$263$	−3.22332	−0.198758	−0.0993791	−	0.995050i	$-0.531686\pi$
$263$	−3.22332	−0.198758	−0.0993791	+	0.995050i	$0.531686\pi$
$264$	−6.26874	−0.385814
$265$	−5.53931	−0.340277
$266$	0.656421	0.0402478
$267$	−28.9418	−1.77121
$268$	−10.4059	−0.635641
$269$	6.47628	0.394866	0.197433	−	0.980316i	$-0.436740\pi$
$269$	6.47628	0.394866	0.197433	+	0.980316i	$0.436740\pi$
$270$	1.36002	0.0827682
$271$	−1.08767	−0.0660711	−0.0330355	−	0.999454i	$-0.510517\pi$
$271$	−1.08767	−0.0660711	−0.0330355	+	0.999454i	$0.510517\pi$
$272$	−1.76208	−0.106842
$273$	0.459482	0.0278091
$274$	7.66777	0.463227
$275$	13.7377	0.828412
$276$	20.3403	1.22434
$277$	−4.48401	−0.269418	−0.134709	−	0.990885i	$-0.543010\pi$
$277$	−4.48401	−0.269418	−0.134709	+	0.990885i	$0.543010\pi$
$278$	−15.1655	−0.909567
$279$	−0.0791152	−0.00473650
$280$	−0.0995620	−0.00594997
$281$	−10.8553	−0.647571	−0.323785	−	0.946131i	$-0.604956\pi$
$281$	−10.8553	−0.647571	−0.323785	+	0.946131i	$0.604956\pi$
$282$	26.7857	1.59506
$283$	6.81521	0.405122	0.202561	−	0.979270i	$-0.435074\pi$
$283$	6.81521	0.405122	0.202561	+	0.979270i	$0.435074\pi$
$284$	0.421760	0.0250268
$285$	3.42456	0.202853
$286$	2.99722	0.177229
$287$	1.40734	0.0830728
$288$	−1.71971	−0.101335
$289$	−13.8951	−0.817358
$290$	−2.61481	−0.153547
$291$	27.1408	1.59102
$292$	7.89700	0.462137
$293$	−1.98542	−0.115989	−0.0579947	−	0.998317i	$-0.518471\pi$
$293$	−1.98542	−0.115989	−0.0579947	+	0.998317i	$0.518471\pi$
$294$	−15.1174	−0.881662
$295$	7.16952	0.417426
$296$	5.50737	0.320109
$297$	−8.02581	−0.465705
$298$	19.4470	1.12653
$299$	−9.72512	−0.562418
$300$	10.3430	0.597155
$301$	−0.544941	−0.0314099
$302$	−20.6951	−1.19087
$303$	−6.64467	−0.381727
$304$	3.22380	0.184898
$305$	−0.756750	−0.0433314
$306$	3.03026	0.173228
$307$	17.8245	1.01730	0.508648	−	0.860975i	$-0.330145\pi$
$307$	17.8245	1.01730	0.508648	+	0.860975i	$0.330145\pi$
$308$	0.587540	0.0334782
$309$	22.5977	1.28554
$310$	−0.0224949	−0.00127762
$311$	28.4071	1.61082	0.805409	−	0.592719i	$-0.201946\pi$
$311$	28.4071	1.61082	0.805409	+	0.592719i	$0.201946\pi$
$312$	2.25660	0.127755
$313$	−10.3805	−0.586742	−0.293371	−	0.955999i	$-0.594777\pi$
$313$	−10.3805	−0.586742	−0.293371	+	0.955999i	$0.594777\pi$
$314$	−14.0367	−0.792139
$315$	0.171218	0.00964702
$316$	7.66659	0.431280
$317$	18.5391	1.04126	0.520629	−	0.853783i	$-0.325698\pi$
$317$	18.5391	1.04126	0.520629	+	0.853783i	$0.325698\pi$
$318$	24.6113	1.38013
$319$	15.4306	0.863950
$320$	−0.488966	−0.0273340
$321$	42.3860	2.36576
$322$	−1.90640	−0.106239
$323$	−5.68058	−0.316076
$324$	−11.2017	−0.622318
$325$	−4.94523	−0.274312
$326$	10.7706	0.596526
$327$	2.16246	0.119584
$328$	6.91171	0.381635
$329$	−2.51050	−0.138408
$330$	3.06520	0.168734
$331$	−18.6176	−1.02331	−0.511657	−	0.859190i	$-0.670968\pi$
$331$	−18.6176	−1.02331	−0.511657	+	0.859190i	$0.670968\pi$
$332$	−14.2667	−0.782986
$333$	−9.47108	−0.519012
$334$	−18.9859	−1.03886
$335$	5.08813	0.277994
$336$	0.442356	0.0241325
$337$	−12.2109	−0.665168	−0.332584	−	0.943074i	$-0.607920\pi$
$337$	−12.2109	−0.665168	−0.332584	+	0.943074i	$0.607920\pi$
$338$	11.9211	0.648421
$339$	15.9606	0.866861
$340$	0.861596	0.0467266
$341$	0.132748	0.00718869
$342$	−5.54400	−0.299785
$343$	2.84220	0.153464
$344$	−2.67630	−0.144297
$345$	−9.94570	−0.535459
$346$	−6.10393	−0.328149
$347$	4.14405	0.222464	0.111232	−	0.993794i	$-0.464520\pi$
$347$	4.14405	0.222464	0.111232	+	0.993794i	$0.464520\pi$
$348$	11.6177	0.622773
$349$	−24.4945	−1.31116	−0.655580	−	0.755126i	$-0.727576\pi$
$349$	−24.4945	−1.31116	−0.655580	+	0.755126i	$0.727576\pi$
$350$	−0.969404	−0.0518168
$351$	2.88910	0.154209
$352$	2.88551	0.153798
$353$	−9.39438	−0.500012	−0.250006	−	0.968244i	$-0.580433\pi$
$353$	−9.39438	−0.500012	−0.250006	+	0.968244i	$0.580433\pi$
$354$	−31.8544	−1.69304
$355$	−0.206226	−0.0109454
$356$	13.3220	0.706063
$357$	−0.779466	−0.0412537
$358$	13.2117	0.698258
$359$	−18.2266	−0.961965	−0.480983	−	0.876730i	$-0.659720\pi$
$359$	−18.2266	−0.961965	−0.480983	+	0.876730i	$0.659720\pi$
$360$	0.840880	0.0443183
$361$	−8.60712	−0.453006
$362$	10.2089	0.536570
$363$	5.80889	0.304888
$364$	−0.211500	−0.0110856
$365$	−3.86137	−0.202113
$366$	3.36226	0.175748
$367$	9.46509	0.494074	0.247037	−	0.969006i	$-0.420543\pi$
$367$	9.46509	0.494074	0.247037	+	0.969006i	$0.420543\pi$
$368$	−9.36265	−0.488062
$369$	−11.8861	−0.618767
$370$	−2.69292	−0.139998
$371$	−2.30670	−0.119758
$372$	0.0999453	0.00518192
$373$	8.81810	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$373$	8.81810	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$374$	−5.08449	−0.262913
$375$	−10.3688	−0.535441
$376$	−12.3295	−0.635845
$377$	−5.55466	−0.286080
$378$	0.566345	0.0291296
$379$	8.70651	0.447224	0.223612	−	0.974678i	$-0.428215\pi$
$379$	8.70651	0.447224	0.223612	+	0.974678i	$0.428215\pi$
$380$	−1.57633	−0.0808640
$381$	−22.9868	−1.17765
$382$	26.4907	1.35538
$383$	18.7321	0.957166	0.478583	−	0.878042i	$-0.341151\pi$
$383$	18.7321	0.957166	0.478583	+	0.878042i	$0.341151\pi$
$384$	2.17249	0.110864
$385$	−0.287287	−0.0146415
$386$	0.470343	0.0239398
$387$	4.60246	0.233956
$388$	−12.4930	−0.634234
$389$	−38.9238	−1.97351	−0.986757	−	0.162208i	$-0.948138\pi$
$389$	−38.9238	−1.97351	−0.986757	+	0.162208i	$0.948138\pi$
$390$	−1.10340	−0.0558728
$391$	16.4977	0.834325
$392$	6.95854	0.351459
$393$	10.4427	0.526766
$394$	−3.51558	−0.177113
$395$	−3.74871	−0.188618
$396$	−4.96223	−0.249362
$397$	−0.502089	−0.0251991	−0.0125996	−	0.999921i	$-0.504011\pi$
$397$	−0.502089	−0.0251991	−0.0125996	+	0.999921i	$0.504011\pi$
$398$	−13.1428	−0.658791
$399$	1.42607	0.0713927
$400$	−4.76091	−0.238046
$401$	−22.2487	−1.11105	−0.555523	−	0.831501i	$-0.687482\pi$
$401$	−22.2487	−1.11105	−0.555523	+	0.831501i	$0.687482\pi$
$402$	−22.6067	−1.12752
$403$	−0.0477860	−0.00238039
$404$	3.05855	0.152169
$405$	5.47727	0.272168
$406$	−1.08887	−0.0540397
$407$	15.8916	0.787716
$408$	−3.82809	−0.189519
$409$	−22.2002	−1.09773	−0.548863	−	0.835912i	$-0.684939\pi$
$409$	−22.2002	−1.09773	−0.548863	+	0.835912i	$0.684939\pi$
$410$	−3.37959	−0.166906
$411$	16.6581	0.821685
$412$	−10.4018	−0.512458
$413$	2.98556	0.146910
$414$	16.1010	0.791323
$415$	6.97593	0.342435
$416$	−1.03871	−0.0509272
$417$	−32.9469	−1.61342
$418$	9.30230	0.454990
$419$	7.90202	0.386039	0.193020	−	0.981195i	$-0.438172\pi$
$419$	7.90202	0.386039	0.193020	+	0.981195i	$0.438172\pi$
$420$	−0.216297	−0.0105542
$421$	−15.7001	−0.765174	−0.382587	−	0.923919i	$-0.624967\pi$
$421$	−15.7001	−0.765174	−0.382587	+	0.923919i	$0.624967\pi$
$422$	−3.40461	−0.165734
$423$	21.2031	1.03093
$424$	−11.3286	−0.550166
$425$	8.38909	0.406931
$426$	0.916269	0.0443934
$427$	−0.315129	−0.0152502
$428$	−19.5104	−0.943069
$429$	6.51143	0.314375
$430$	1.30862	0.0631073
$431$	−3.49340	−0.168271	−0.0841356	−	0.996454i	$-0.526813\pi$
$431$	−3.49340	−0.168271	−0.0841356	+	0.996454i	$0.526813\pi$
$432$	2.78142	0.133821
$433$	34.6022	1.66288	0.831439	−	0.555616i	$-0.187518\pi$
$433$	34.6022	1.66288	0.831439	+	0.555616i	$0.187518\pi$
$434$	−0.00936741	−0.000449650 0
$435$	−5.68065	−0.272366
$436$	−0.995382	−0.0476702
$437$	−30.1833	−1.44386
$438$	17.1561	0.819752
$439$	−39.1443	−1.86826	−0.934129	−	0.356937i	$-0.883821\pi$
$439$	−39.1443	−1.86826	−0.934129	+	0.356937i	$0.883821\pi$
$440$	−1.41092	−0.0672628
$441$	−11.9667	−0.569841
$442$	1.83030	0.0870583
$443$	27.6015	1.31139	0.655693	−	0.755028i	$-0.272377\pi$
$443$	27.6015	1.31139	0.655693	+	0.755028i	$0.272377\pi$
$444$	11.9647	0.567820
$445$	−6.51399	−0.308793
$446$	−20.3123	−0.961816
$447$	42.2484	1.99828
$448$	−0.203617	−0.00962002
$449$	39.9154	1.88372	0.941862	−	0.336000i	$-0.109074\pi$
$449$	39.9154	1.88372	0.941862	+	0.336000i	$0.109074\pi$
$450$	8.18738	0.385957
$451$	19.9438	0.939116
$452$	−7.34669	−0.345559
$453$	−44.9598	−2.11239
$454$	24.7343	1.16084
$455$	0.103417	0.00484824
$456$	7.00367	0.327977
$457$	−14.9312	−0.698452	−0.349226	−	0.937039i	$-0.613555\pi$
$457$	−14.9312	−0.698452	−0.349226	+	0.937039i	$0.613555\pi$
$458$	−8.57570	−0.400716
$459$	−4.90107	−0.228763
$460$	4.57802	0.213451
$461$	39.2288	1.82707	0.913534	−	0.406762i	$-0.133342\pi$
$461$	39.2288	1.82707	0.913534	+	0.406762i	$0.133342\pi$
$462$	1.27642	0.0593846
$463$	42.9339	1.99531	0.997653	−	0.0684661i	$-0.0218105\pi$
$463$	42.9339	1.99531	0.997653	+	0.0684661i	$0.0218105\pi$
$464$	−5.34763	−0.248258
$465$	−0.0488699	−0.00226629
$466$	26.9900	1.25029
$467$	16.0978	0.744918	0.372459	−	0.928049i	$-0.378515\pi$
$467$	16.0978	0.744918	0.372459	+	0.928049i	$0.378515\pi$
$468$	1.78629	0.0825711
$469$	2.11882	0.0978380
$470$	6.02870	0.278083
$471$	−30.4947	−1.40512
$472$	14.6626	0.674902
$473$	−7.72249	−0.355081
$474$	16.6556	0.765017
$475$	−15.3482	−0.704225
$476$	0.358789	0.0164451
$477$	19.4819	0.892015
$478$	2.11151	0.0965784
$479$	4.32721	0.197715	0.0988576	−	0.995102i	$-0.468481\pi$
$479$	4.32721	0.197715	0.0988576	+	0.995102i	$0.468481\pi$
$480$	−1.06227	−0.0484860
$481$	−5.72059	−0.260836
$482$	25.8562	1.17772
$483$	−4.14163	−0.188451
$484$	−2.67384	−0.121538
$485$	6.10864	0.277379
$486$	−15.9914	−0.725384
$487$	30.5083	1.38246	0.691231	−	0.722634i	$-0.257069\pi$
$487$	30.5083	1.38246	0.691231	+	0.722634i	$0.257069\pi$
$488$	−1.54765	−0.0700590
$489$	23.3989	1.05814
$490$	−3.40249	−0.153709
$491$	25.7415	1.16170	0.580848	−	0.814012i	$-0.302721\pi$
$491$	25.7415	1.16170	0.580848	+	0.814012i	$0.302721\pi$
$492$	15.0156	0.676956
$493$	9.42294	0.424388
$494$	−3.34861	−0.150661
$495$	2.42637	0.109057
$496$	−0.0460050	−0.00206568
$497$	−0.0858776	−0.00385214
$498$	−30.9942	−1.38888
$499$	27.5830	1.23478	0.617392	−	0.786655i	$-0.288189\pi$
$499$	27.5830	1.23478	0.617392	+	0.786655i	$0.288189\pi$
$500$	4.77276	0.213444
$501$	−41.2466	−1.84276
$502$	−5.48143	−0.244648
$503$	28.3287	1.26312	0.631558	−	0.775328i	$-0.282416\pi$
$503$	28.3287	1.26312	0.631558	+	0.775328i	$0.282416\pi$
$504$	0.350163	0.0155975
$505$	−1.49553	−0.0665502
$506$	−27.0160	−1.20101
$507$	25.8984	1.15019
$508$	10.5809	0.469450
$509$	17.7260	0.785692	0.392846	−	0.919604i	$-0.371491\pi$
$509$	17.7260	0.785692	0.392846	+	0.919604i	$0.371491\pi$
$510$	1.87181	0.0828851
$511$	−1.60797	−0.0711322
$512$	−1.00000	−0.0441942
$513$	8.96673	0.395891
$514$	−15.4009	−0.679304
$515$	5.08611	0.224121
$516$	−5.81424	−0.255957
$517$	−35.5768	−1.56467
$518$	−1.12140	−0.0492713
$519$	−13.2607	−0.582081
$520$	0.507897	0.0222727
$521$	−20.7207	−0.907789	−0.453894	−	0.891056i	$-0.649966\pi$
$521$	−20.7207	−0.907789	−0.453894	+	0.891056i	$0.649966\pi$
$522$	9.19637	0.402514
$523$	9.25301	0.404606	0.202303	−	0.979323i	$-0.435157\pi$
$523$	9.25301	0.404606	0.202303	+	0.979323i	$0.435157\pi$
$524$	−4.80680	−0.209986
$525$	−2.10602	−0.0919143
$526$	3.22332	0.140543
$527$	0.0810643	0.00353122
$528$	6.26874	0.272812
$529$	64.6592	2.81127
$530$	5.53931	0.240612
$531$	−25.2154	−1.09426
$532$	−0.656421	−0.0284595
$533$	−7.17929	−0.310970
$534$	28.9418	1.25244
$535$	9.53991	0.412446
$536$	10.4059	0.449466
$537$	28.7022	1.23859
$538$	−6.47628	−0.279212
$539$	20.0789	0.864861
$540$	−1.36002	−0.0585259
$541$	18.9714	0.815643	0.407821	−	0.913062i	$-0.366289\pi$
$541$	18.9714	0.815643	0.407821	+	0.913062i	$0.366289\pi$
$542$	1.08767	0.0467193
$543$	22.1788	0.951785
$544$	1.76208	0.0755484
$545$	0.486708	0.0208483
$546$	−0.459482	−0.0196640
$547$	12.2319	0.522999	0.261500	−	0.965204i	$-0.415783\pi$
$547$	12.2319	0.522999	0.261500	+	0.965204i	$0.415783\pi$
$548$	−7.66777	−0.327551
$549$	2.66151	0.113591
$550$	−13.7377	−0.585775
$551$	−17.2397	−0.734435
$552$	−20.3403	−0.865739
$553$	−1.56105	−0.0663827
$554$	4.48401	0.190507
$555$	−5.85034	−0.248333
$556$	15.1655	0.643161
$557$	−14.4338	−0.611578	−0.305789	−	0.952099i	$-0.598920\pi$
$557$	−14.4338	−0.611578	−0.305789	+	0.952099i	$0.598920\pi$
$558$	0.0791152	0.00334921
$559$	2.77991	0.117578
$560$	0.0995620	0.00420726
$561$	−11.0460	−0.466362
$562$	10.8553	0.457902
$563$	9.26653	0.390538	0.195269	−	0.980750i	$-0.437442\pi$
$563$	9.26653	0.390538	0.195269	+	0.980750i	$0.437442\pi$
$564$	−26.7857	−1.12788
$565$	3.59228	0.151128
$566$	−6.81521	−0.286465
$567$	2.28087	0.0957874
$568$	−0.421760	−0.0176966
$569$	23.7731	0.996622	0.498311	−	0.866998i	$-0.333954\pi$
$569$	23.7731	0.996622	0.498311	+	0.866998i	$0.333954\pi$
$570$	−3.42456	−0.143439
$571$	−9.49436	−0.397327	−0.198663	−	0.980068i	$-0.563660\pi$
$571$	−9.49436	−0.397327	−0.198663	+	0.980068i	$0.563660\pi$
$572$	−2.99722	−0.125320
$573$	57.5507	2.40421
$574$	−1.40734	−0.0587414
$575$	44.5748	1.85890
$576$	1.71971	0.0716545
$577$	−42.0049	−1.74869	−0.874343	−	0.485308i	$-0.838707\pi$
$577$	−42.0049	−1.74869	−0.874343	+	0.485308i	$0.838707\pi$
$578$	13.8951	0.577959
$579$	1.02181	0.0424652
$580$	2.61481	0.108574
$581$	2.90494	0.120517
$582$	−27.1408	−1.12502
$583$	−32.6888	−1.35383
$584$	−7.89700	−0.326780
$585$	−0.873434	−0.0361121
$586$	1.98542	0.0820169
$587$	2.38132	0.0982877	0.0491438	−	0.998792i	$-0.484351\pi$
$587$	2.38132	0.0982877	0.0491438	+	0.998792i	$0.484351\pi$
$588$	15.1174	0.623429
$589$	−0.148311	−0.00611104
$590$	−7.16952	−0.295165
$591$	−7.63757	−0.314168
$592$	−5.50737	−0.226352
$593$	43.6316	1.79173	0.895867	−	0.444323i	$-0.146556\pi$
$593$	43.6316	1.79173	0.895867	+	0.444323i	$0.146556\pi$
$594$	8.02581	0.329303
$595$	−0.175436	−0.00719217
$596$	−19.4470	−0.796580
$597$	−28.5527	−1.16858
$598$	9.72512	0.397690
$599$	2.90427	0.118665	0.0593327	−	0.998238i	$-0.481103\pi$
$599$	2.90427	0.118665	0.0593327	+	0.998238i	$0.481103\pi$
$600$	−10.3430	−0.422252
$601$	−22.1328	−0.902817	−0.451409	−	0.892317i	$-0.649078\pi$
$601$	−22.1328	−0.902817	−0.451409	+	0.892317i	$0.649078\pi$
$602$	0.544941	0.0222102
$603$	−17.8951	−0.728745
$604$	20.6951	0.842070
$605$	1.30742	0.0531541
$606$	6.64467	0.269921
$607$	9.56319	0.388158	0.194079	−	0.980986i	$-0.437828\pi$
$607$	9.56319	0.388158	0.194079	+	0.980986i	$0.437828\pi$
$608$	−3.22380	−0.130742
$609$	−2.36556	−0.0958573
$610$	0.756750	0.0306399
$611$	12.8068	0.518108
$612$	−3.03026	−0.122491
$613$	−32.0874	−1.29600	−0.647999	−	0.761641i	$-0.724394\pi$
$613$	−32.0874	−1.29600	−0.647999	+	0.761641i	$0.724394\pi$
$614$	−17.8245	−0.719337
$615$	−7.34213	−0.296063
$616$	−0.587540	−0.0236726
$617$	−18.6881	−0.752355	−0.376177	−	0.926548i	$-0.622762\pi$
$617$	−18.6881	−0.752355	−0.376177	+	0.926548i	$0.622762\pi$
$618$	−22.5977	−0.909013
$619$	−3.67169	−0.147578	−0.0737888	−	0.997274i	$-0.523509\pi$
$619$	−3.67169	−0.147578	−0.0737888	+	0.997274i	$0.523509\pi$
$620$	0.0224949	0.000903416 0
$621$	−26.0415	−1.04501
$622$	−28.4071	−1.13902
$623$	−2.71258	−0.108677
$624$	−2.25660	−0.0903362
$625$	21.4708	0.858834
$626$	10.3805	0.414889
$627$	20.2091	0.807075
$628$	14.0367	0.560127
$629$	9.70441	0.386940
$630$	−0.171218	−0.00682148
$631$	8.42177	0.335266	0.167633	−	0.985850i	$-0.446388\pi$
$631$	8.42177	0.335266	0.167633	+	0.985850i	$0.446388\pi$
$632$	−7.66659	−0.304961
$633$	−7.39649	−0.293984
$634$	−18.5391	−0.736280
$635$	−5.17368	−0.205311
$636$	−24.6113	−0.975901
$637$	−7.22794	−0.286381
$638$	−15.4306	−0.610905
$639$	0.725304	0.0286926
$640$	0.488966	0.0193281
$641$	−9.44964	−0.373238	−0.186619	−	0.982432i	$-0.559753\pi$
$641$	−9.44964	−0.373238	−0.186619	+	0.982432i	$0.559753\pi$
$642$	−42.3860	−1.67284
$643$	41.9318	1.65363	0.826815	−	0.562474i	$-0.190150\pi$
$643$	41.9318	1.65363	0.826815	+	0.562474i	$0.190150\pi$
$644$	1.90640	0.0751226
$645$	2.84297	0.111942
$646$	5.68058	0.223499
$647$	−1.64765	−0.0647759	−0.0323880	−	0.999475i	$-0.510311\pi$
$647$	−1.64765	−0.0647759	−0.0323880	+	0.999475i	$0.510311\pi$
$648$	11.2017	0.440045
$649$	42.3091	1.66078
$650$	4.94523	0.193968
$651$	−0.0203506	−0.000797603 0
$652$	−10.7706	−0.421808
$653$	16.6303	0.650793	0.325396	−	0.945578i	$-0.394502\pi$
$653$	16.6303	0.650793	0.325396	+	0.945578i	$0.394502\pi$
$654$	−2.16246	−0.0845588
$655$	2.35036	0.0918363
$656$	−6.91171	−0.269857
$657$	13.5805	0.529827
$658$	2.51050	0.0978694
$659$	5.97780	0.232862	0.116431	−	0.993199i	$-0.462855\pi$
$659$	5.97780	0.232862	0.116431	+	0.993199i	$0.462855\pi$
$660$	−3.06520	−0.119313
$661$	42.6240	1.65788	0.828940	−	0.559337i	$-0.188944\pi$
$661$	42.6240	1.65788	0.828940	+	0.559337i	$0.188944\pi$
$662$	18.6176	0.723593
$663$	3.97630	0.154427
$664$	14.2667	0.553655
$665$	0.320968	0.0124466
$666$	9.47108	0.366997
$667$	50.0680	1.93864
$668$	18.9859	0.734586
$669$	−44.1283	−1.70610
$670$	−5.08813	−0.196572
$671$	−4.46577	−0.172399
$672$	−0.442356	−0.0170643
$673$	−11.7469	−0.452811	−0.226406	−	0.974033i	$-0.572697\pi$
$673$	−11.7469	−0.452811	−0.226406	+	0.974033i	$0.572697\pi$
$674$	12.2109	0.470345
$675$	−13.2421	−0.509688
$676$	−11.9211	−0.458503
$677$	30.6925	1.17961	0.589803	−	0.807547i	$-0.299205\pi$
$677$	30.6925	1.17961	0.589803	+	0.807547i	$0.299205\pi$
$678$	−15.9606	−0.612963
$679$	2.54379	0.0976215
$680$	−0.861596	−0.0330407
$681$	53.7351	2.05913
$682$	−0.132748	−0.00508317
$683$	34.3476	1.31428	0.657138	−	0.753770i	$-0.271767\pi$
$683$	34.3476	1.31428	0.657138	+	0.753770i	$0.271767\pi$
$684$	5.54400	0.211980
$685$	3.74928	0.143253
$686$	−2.84220	−0.108516
$687$	−18.6306	−0.710802
$688$	2.67630	0.102033
$689$	11.7672	0.448294
$690$	9.94570	0.378626
$691$	−9.71358	−0.369522	−0.184761	−	0.982783i	$-0.559151\pi$
$691$	−9.71358	−0.369522	−0.184761	+	0.982783i	$0.559151\pi$
$692$	6.10393	0.232037
$693$	1.01040	0.0383818
$694$	−4.14405	−0.157306
$695$	−7.41542	−0.281283
$696$	−11.6177	−0.440367
$697$	12.1790	0.461311
$698$	24.4945	0.927130
$699$	58.6355	2.21780
$700$	0.969404	0.0366400
$701$	−0.375372	−0.0141776	−0.00708879	−	0.999975i	$-0.502256\pi$
$701$	−0.375372	−0.0141776	−0.00708879	+	0.999975i	$0.502256\pi$
$702$	−2.88910	−0.109042
$703$	−17.7547	−0.669630
$704$	−2.88551	−0.108752
$705$	13.0973	0.493272
$706$	9.39438	0.353562
$707$	−0.622775	−0.0234218
$708$	31.8544	1.19716
$709$	12.4631	0.468062	0.234031	−	0.972229i	$-0.424808\pi$
$709$	12.4631	0.468062	0.234031	+	0.972229i	$0.424808\pi$
$710$	0.206226	0.00773954
$711$	13.1843	0.494450
$712$	−13.3220	−0.499262
$713$	0.430729	0.0161309
$714$	0.779466	0.0291708
$715$	1.46554	0.0548081
$716$	−13.2117	−0.493743
$717$	4.58724	0.171314
$718$	18.2266	0.680212
$719$	8.56040	0.319249	0.159625	−	0.987178i	$-0.448972\pi$
$719$	8.56040	0.319249	0.159625	+	0.987178i	$0.448972\pi$
$720$	−0.840880	−0.0313377
$721$	2.11798	0.0788777
$722$	8.60712	0.320324
$723$	56.1724	2.08907
$724$	−10.2089	−0.379413
$725$	25.4596	0.945546
$726$	−5.80889	−0.215588
$727$	−25.9133	−0.961069	−0.480535	−	0.876976i	$-0.659557\pi$
$727$	−25.9133	−0.961069	−0.480535	+	0.876976i	$0.659557\pi$
$728$	0.211500	0.00783872
$729$	−1.13591	−0.0420706
$730$	3.86137	0.142916
$731$	−4.71585	−0.174422
$732$	−3.36226	−0.124273
$733$	49.2678	1.81975	0.909873	−	0.414887i	$-0.136179\pi$
$733$	49.2678	1.81975	0.909873	+	0.414887i	$0.136179\pi$
$734$	−9.46509	−0.349363
$735$	−7.39188	−0.272653
$736$	9.36265	0.345112
$737$	30.0263	1.10603
$738$	11.8861	0.437534
$739$	−11.3619	−0.417953	−0.208977	−	0.977921i	$-0.567013\pi$
$739$	−11.3619	−0.417953	−0.208977	+	0.977921i	$0.567013\pi$
$740$	2.69292	0.0989937
$741$	−7.27481	−0.267247
$742$	2.30670	0.0846817
$743$	−16.9122	−0.620450	−0.310225	−	0.950663i	$-0.600404\pi$
$743$	−16.9122	−0.620450	−0.310225	+	0.950663i	$0.600404\pi$
$744$	−0.0999453	−0.00366417
$745$	9.50893	0.348380
$746$	−8.81810	−0.322854
$747$	−24.5345	−0.897672
$748$	5.08449	0.185907
$749$	3.97265	0.145157
$750$	10.3688	0.378614
$751$	5.03224	0.183629	0.0918145	−	0.995776i	$-0.470733\pi$
$751$	5.03224	0.183629	0.0918145	+	0.995776i	$0.470733\pi$
$752$	12.3295	0.449610
$753$	−11.9084	−0.433965
$754$	5.55466	0.202289
$755$	−10.1192	−0.368275
$756$	−0.566345	−0.0205978
$757$	25.4817	0.926148	0.463074	−	0.886320i	$-0.346746\pi$
$757$	25.4817	0.926148	0.463074	+	0.886320i	$0.346746\pi$
$758$	−8.70651	−0.316235
$759$	−58.6920	−2.13038
$760$	1.57633	0.0571795
$761$	47.4129	1.71872	0.859359	−	0.511373i	$-0.170863\pi$
$761$	47.4129	1.71872	0.859359	+	0.511373i	$0.170863\pi$
$762$	22.9868	0.832724
$763$	0.202677	0.00733740
$764$	−26.4907	−0.958398
$765$	1.48169	0.0535708
$766$	−18.7321	−0.676819
$767$	−15.2303	−0.549933
$768$	−2.17249	−0.0783929
$769$	41.9112	1.51136	0.755678	−	0.654944i	$-0.227308\pi$
$769$	41.9112	1.51136	0.755678	+	0.654944i	$0.227308\pi$
$770$	0.287287	0.0103531
$771$	−33.4583	−1.20497
$772$	−0.470343	−0.0169280
$773$	−16.7872	−0.603793	−0.301897	−	0.953341i	$-0.597620\pi$
$773$	−16.7872	−0.603793	−0.301897	+	0.953341i	$0.597620\pi$
$774$	−4.60246	−0.165432
$775$	0.219026	0.00786763
$776$	12.4930	0.448471
$777$	−2.43622	−0.0873990
$778$	38.9238	1.39548
$779$	−22.2819	−0.798334
$780$	1.10340	0.0395080
$781$	−1.21699	−0.0435474
$782$	−16.4977	−0.589957
$783$	−14.8740	−0.531553
$784$	−6.95854	−0.248519
$785$	−6.86349	−0.244969
$786$	−10.4427	−0.372480
$787$	26.5389	0.946010	0.473005	−	0.881060i	$-0.343169\pi$
$787$	26.5389	0.946010	0.473005	+	0.881060i	$0.343169\pi$
$788$	3.51558	0.125238
$789$	7.00262	0.249300
$790$	3.74871	0.133373
$791$	1.49591	0.0531885
$792$	4.96223	0.176325
$793$	1.60757	0.0570865
$794$	0.502089	0.0178185
$795$	12.0341	0.426805
$796$	13.1428	0.465835
$797$	−28.9689	−1.02613	−0.513066	−	0.858349i	$-0.671490\pi$
$797$	−28.9689	−1.02613	−0.513066	+	0.858349i	$0.671490\pi$
$798$	−1.42607	−0.0504823
$799$	−21.7255	−0.768593
$800$	4.76091	0.168324
$801$	22.9099	0.809482
$802$	22.2487	0.785628
$803$	−22.7869	−0.804131
$804$	22.6067	0.797276
$805$	−0.932165	−0.0328545
$806$	0.0477860	0.00168319
$807$	−14.0697	−0.495275
$808$	−3.05855	−0.107600
$809$	1.07223	0.0376976	0.0188488	−	0.999822i	$-0.494000\pi$
$809$	1.07223	0.0376976	0.0188488	+	0.999822i	$0.494000\pi$
$810$	−5.47727	−0.192452
$811$	18.8385	0.661508	0.330754	−	0.943717i	$-0.392697\pi$
$811$	18.8385	0.661508	0.330754	+	0.943717i	$0.392697\pi$
$812$	1.08887	0.0382119
$813$	2.36294	0.0828721
$814$	−15.8916	−0.556999
$815$	5.26644	0.184475
$816$	3.82809	0.134010
$817$	8.62786	0.301851
$818$	22.2002	0.776210
$819$	−0.363719	−0.0127094
$820$	3.37959	0.118020
$821$	26.4016	0.921421	0.460710	−	0.887551i	$-0.347595\pi$
$821$	26.4016	0.921421	0.460710	+	0.887551i	$0.347595\pi$
$822$	−16.6581	−0.581019
$823$	12.6884	0.442290	0.221145	−	0.975241i	$-0.429021\pi$
$823$	12.6884	0.442290	0.221145	+	0.975241i	$0.429021\pi$
$824$	10.4018	0.362363
$825$	−29.8449	−1.03907
$826$	−2.98556	−0.103881
$827$	21.8703	0.760503	0.380252	−	0.924883i	$-0.375837\pi$
$827$	21.8703	0.760503	0.380252	+	0.924883i	$0.375837\pi$
$828$	−16.1010	−0.559550
$829$	1.79540	0.0623570	0.0311785	−	0.999514i	$-0.490074\pi$
$829$	1.79540	0.0623570	0.0311785	+	0.999514i	$0.490074\pi$
$830$	−6.97593	−0.242138
$831$	9.74145	0.337927
$832$	1.03871	0.0360110
$833$	12.2615	0.424835
$834$	32.9469	1.14086
$835$	−9.28345	−0.321267
$836$	−9.30230	−0.321727
$837$	−0.127959	−0.00442291
$838$	−7.90202	−0.272971
$839$	16.8864	0.582983	0.291491	−	0.956573i	$-0.405849\pi$
$839$	16.8864	0.582983	0.291491	+	0.956573i	$0.405849\pi$
$840$	0.216297	0.00746297
$841$	−0.402835	−0.0138909
$842$	15.7001	0.541060
$843$	23.5829	0.812240
$844$	3.40461	0.117192
$845$	5.82900	0.200524
$846$	−21.2031	−0.728978
$847$	0.544440	0.0187072
$848$	11.3286	0.389026
$849$	−14.8060	−0.508139
$850$	−8.38909	−0.287744
$851$	51.5636	1.76758
$852$	−0.916269	−0.0313908
$853$	12.2733	0.420231	0.210116	−	0.977677i	$-0.432616\pi$
$853$	12.2733	0.420231	0.210116	+	0.977677i	$0.432616\pi$
$854$	0.315129	0.0107835
$855$	−2.71083	−0.0927083
$856$	19.5104	0.666850
$857$	2.66241	0.0909461	0.0454731	−	0.998966i	$-0.485520\pi$
$857$	2.66241	0.0909461	0.0454731	+	0.998966i	$0.485520\pi$
$858$	−6.51143	−0.222297
$859$	17.3821	0.593068	0.296534	−	0.955022i	$-0.404169\pi$
$859$	17.3821	0.593068	0.296534	+	0.955022i	$0.404169\pi$
$860$	−1.30862	−0.0446236
$861$	−3.05744	−0.104197
$862$	3.49340	0.118986
$863$	31.0369	1.05651	0.528254	−	0.849086i	$-0.322847\pi$
$863$	31.0369	1.05651	0.528254	+	0.849086i	$0.322847\pi$
$864$	−2.78142	−0.0946258
$865$	−2.98462	−0.101480
$866$	−34.6022	−1.17583
$867$	30.1869	1.02520
$868$	0.00936741	0.000317951 0
$869$	−22.1220	−0.750438
$870$	5.68065	0.192592
$871$	−10.8087	−0.366240
$872$	0.995382	0.0337079
$873$	−21.4843	−0.727132
$874$	30.1833	1.02097
$875$	−0.971816	−0.0328534
$876$	−17.1561	−0.579652
$877$	1.99987	0.0675308	0.0337654	−	0.999430i	$-0.489250\pi$
$877$	1.99987	0.0675308	0.0337654	+	0.999430i	$0.489250\pi$
$878$	39.1443	1.32106
$879$	4.31330	0.145484
$880$	1.41092	0.0475620
$881$	−39.7793	−1.34020	−0.670100	−	0.742271i	$-0.733749\pi$
$881$	−39.7793	−1.34020	−0.670100	+	0.742271i	$0.733749\pi$
$882$	11.9667	0.402938
$883$	−5.18990	−0.174654	−0.0873271	−	0.996180i	$-0.527833\pi$
$883$	−5.18990	−0.174654	−0.0873271	+	0.996180i	$0.527833\pi$
$884$	−1.83030	−0.0615595
$885$	−15.5757	−0.523572
$886$	−27.6015	−0.927290
$887$	−32.8075	−1.10157	−0.550784	−	0.834648i	$-0.685671\pi$
$887$	−32.8075	−1.10157	−0.550784	+	0.834648i	$0.685671\pi$
$888$	−11.9647	−0.401509
$889$	−2.15445	−0.0722578
$890$	6.51399	0.218350
$891$	32.3227	1.08285
$892$	20.3123	0.680106
$893$	39.7478	1.33011
$894$	−42.2484	−1.41300
$895$	6.46006	0.215936
$896$	0.203617	0.00680238
$897$	21.1277	0.705434
$898$	−39.9154	−1.33199
$899$	0.246018	0.00820515
$900$	−8.18738	−0.272913
$901$	−19.9619	−0.665027
$902$	−19.9438	−0.664055
$903$	1.18388	0.0393970
$904$	7.34669	0.244347
$905$	4.99183	0.165934
$906$	44.9598	1.49369
$907$	32.2211	1.06988	0.534942	−	0.844889i	$-0.320334\pi$
$907$	32.2211	1.06988	0.534942	+	0.844889i	$0.320334\pi$
$908$	−24.7343	−0.820838
$909$	5.25982	0.174457
$910$	−0.103417	−0.00342823
$911$	46.4288	1.53826	0.769128	−	0.639095i	$-0.220691\pi$
$911$	46.4288	1.53826	0.769128	+	0.639095i	$0.220691\pi$
$912$	−7.00367	−0.231915
$913$	41.1666	1.36242
$914$	14.9312	0.493880
$915$	1.64403	0.0543500
$916$	8.57570	0.283349
$917$	0.978748	0.0323211
$918$	4.90107	0.161760
$919$	−25.4097	−0.838190	−0.419095	−	0.907942i	$-0.637653\pi$
$919$	−25.4097	−0.838190	−0.419095	+	0.907942i	$0.637653\pi$
$920$	−4.57802	−0.150933
$921$	−38.7235	−1.27598
$922$	−39.2288	−1.29193
$923$	0.438088	0.0144198
$924$	−1.27642	−0.0419912
$925$	26.2201	0.862112
$926$	−42.9339	−1.41090
$927$	−17.8880	−0.587519
$928$	5.34763	0.175545
$929$	−7.04364	−0.231094	−0.115547	−	0.993302i	$-0.536862\pi$
$929$	−7.04364	−0.231094	−0.115547	+	0.993302i	$0.536862\pi$
$930$	0.0488699	0.00160251
$931$	−22.4329	−0.735210
$932$	−26.9900	−0.884087
$933$	−61.7141	−2.02043
$934$	−16.0978	−0.526737
$935$	−2.48614	−0.0813056
$936$	−1.78629	−0.0583866
$937$	−2.67022	−0.0872322	−0.0436161	−	0.999048i	$-0.513888\pi$
$937$	−2.67022	−0.0872322	−0.0436161	+	0.999048i	$0.513888\pi$
$938$	−2.11882	−0.0691819
$939$	22.5516	0.735943
$940$	−6.02870	−0.196635
$941$	−34.6547	−1.12971	−0.564856	−	0.825190i	$-0.691068\pi$
$941$	−34.6547	−1.12971	−0.564856	+	0.825190i	$0.691068\pi$
$942$	30.4947	0.993570
$943$	64.7119	2.10731
$944$	−14.6626	−0.477227
$945$	0.276924	0.00900833
$946$	7.72249	0.251080
$947$	−28.2273	−0.917263	−0.458632	−	0.888626i	$-0.651660\pi$
$947$	−28.2273	−0.917263	−0.458632	+	0.888626i	$0.651660\pi$
$948$	−16.6556	−0.540948
$949$	8.20273	0.266272
$950$	15.3482	0.497962
$951$	−40.2759	−1.30604
$952$	−0.358789	−0.0116284
$953$	18.3980	0.595970	0.297985	−	0.954571i	$-0.403685\pi$
$953$	18.3980	0.595970	0.297985	+	0.954571i	$0.403685\pi$
$954$	−19.4819	−0.630750
$955$	12.9530	0.419150
$956$	−2.11151	−0.0682913
$957$	−33.5229	−1.08364
$958$	−4.32721	−0.139806
$959$	1.56129	0.0504167
$960$	1.06227	0.0342847
$961$	−30.9979	−0.999932
$962$	5.72059	0.184439
$963$	−33.5521	−1.08120
$964$	−25.8562	−0.832773
$965$	0.229982	0.00740337
$966$	4.14163	0.133255
$967$	−27.8954	−0.897056	−0.448528	−	0.893769i	$-0.648052\pi$
$967$	−27.8954	−0.897056	−0.448528	+	0.893769i	$0.648052\pi$
$968$	2.67384	0.0859405
$969$	12.3410	0.396450
$970$	−6.10864	−0.196137
$971$	17.5049	0.561760	0.280880	−	0.959743i	$-0.409374\pi$
$971$	17.5049	0.561760	0.280880	+	0.959743i	$0.409374\pi$
$972$	15.9914	0.512924
$973$	−3.08796	−0.0989955
$974$	−30.5083	−0.977548
$975$	10.7435	0.344066
$976$	1.54765	0.0495392
$977$	11.8848	0.380229	0.190114	−	0.981762i	$-0.439114\pi$
$977$	11.8848	0.380229	0.190114	+	0.981762i	$0.439114\pi$
$978$	−23.3989	−0.748215
$979$	−38.4407	−1.22857
$980$	3.40249	0.108689
$981$	−1.71177	−0.0546525
$982$	−25.7415	−0.821443
$983$	−41.4259	−1.32128	−0.660640	−	0.750703i	$-0.729715\pi$
$983$	−41.4259	−1.32128	−0.660640	+	0.750703i	$0.729715\pi$
$984$	−15.0156	−0.478680
$985$	−1.71900	−0.0547720
$986$	−9.42294	−0.300087
$987$	5.45403	0.173604
$988$	3.34861	0.106533
$989$	−25.0573	−0.796775
$990$	−2.42637	−0.0771150
$991$	−15.3591	−0.487899	−0.243949	−	0.969788i	$-0.578443\pi$
$991$	−15.3591	−0.487899	−0.243949	+	0.969788i	$0.578443\pi$
$992$	0.0460050	0.00146066
$993$	40.4465	1.28353
$994$	0.0858776	0.00272387
$995$	−6.42640	−0.203731
$996$	30.9942	0.982089
$997$	24.7752	0.784638	0.392319	−	0.919829i	$-0.371673\pi$
$997$	24.7752	0.784638	0.392319	+	0.919829i	$0.371673\pi$
$998$	−27.5830	−0.873125
$999$	−15.3183	−0.484650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.14	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.14	✓	86	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.17249 q^{3} +1.00000 q^{4} -0.488966 q^{5} +2.17249 q^{6} -0.203617 q^{7} -1.00000 q^{8} +1.71971 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.17249 q^{3} +1.00000 q^{4} -0.488966 q^{5} +2.17249 q^{6} -0.203617 q^{7} -1.00000 q^{8} +1.71971 q^{9} +0.488966 q^{10} -2.88551 q^{11} -2.17249 q^{12} +1.03871 q^{13} +0.203617 q^{14} +1.06227 q^{15} +1.00000 q^{16} -1.76208 q^{17} -1.71971 q^{18} +3.22380 q^{19} -0.488966 q^{20} +0.442356 q^{21} +2.88551 q^{22} -9.36265 q^{23} +2.17249 q^{24} -4.76091 q^{25} -1.03871 q^{26} +2.78142 q^{27} -0.203617 q^{28} -5.34763 q^{29} -1.06227 q^{30} -0.0460050 q^{31} -1.00000 q^{32} +6.26874 q^{33} +1.76208 q^{34} +0.0995620 q^{35} +1.71971 q^{36} -5.50737 q^{37} -3.22380 q^{38} -2.25660 q^{39} +0.488966 q^{40} -6.91171 q^{41} -0.442356 q^{42} +2.67630 q^{43} -2.88551 q^{44} -0.840880 q^{45} +9.36265 q^{46} +12.3295 q^{47} -2.17249 q^{48} -6.95854 q^{49} +4.76091 q^{50} +3.82809 q^{51} +1.03871 q^{52} +11.3286 q^{53} -2.78142 q^{54} +1.41092 q^{55} +0.203617 q^{56} -7.00367 q^{57} +5.34763 q^{58} -14.6626 q^{59} +1.06227 q^{60} +1.54765 q^{61} +0.0460050 q^{62} -0.350163 q^{63} +1.00000 q^{64} -0.507897 q^{65} -6.26874 q^{66} -10.4059 q^{67} -1.76208 q^{68} +20.3403 q^{69} -0.0995620 q^{70} +0.421760 q^{71} -1.71971 q^{72} +7.89700 q^{73} +5.50737 q^{74} +10.3430 q^{75} +3.22380 q^{76} +0.587540 q^{77} +2.25660 q^{78} +7.66659 q^{79} -0.488966 q^{80} -11.2017 q^{81} +6.91171 q^{82} -14.2667 q^{83} +0.442356 q^{84} +0.861596 q^{85} -2.67630 q^{86} +11.6177 q^{87} +2.88551 q^{88} +13.3220 q^{89} +0.840880 q^{90} -0.211500 q^{91} -9.36265 q^{92} +0.0999453 q^{93} -12.3295 q^{94} -1.57633 q^{95} +2.17249 q^{96} -12.4930 q^{97} +6.95854 q^{98} -4.96223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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