LMFDB - Embedded newform 4022.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, 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("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.17886 q^{3} +1.00000 q^{4} +2.94122 q^{5} -3.17886 q^{6} -2.37061 q^{7} +1.00000 q^{8} +7.10517 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.17886 q^{3} +1.00000 q^{4} +2.94122 q^{5} -3.17886 q^{6} -2.37061 q^{7} +1.00000 q^{8} +7.10517 q^{9} +2.94122 q^{10} +0.0808043 q^{11} -3.17886 q^{12} +4.21634 q^{13} -2.37061 q^{14} -9.34974 q^{15} +1.00000 q^{16} -1.48787 q^{17} +7.10517 q^{18} -5.97005 q^{19} +2.94122 q^{20} +7.53585 q^{21} +0.0808043 q^{22} -7.06428 q^{23} -3.17886 q^{24} +3.65079 q^{25} +4.21634 q^{26} -13.0498 q^{27} -2.37061 q^{28} -6.65920 q^{29} -9.34974 q^{30} -4.37680 q^{31} +1.00000 q^{32} -0.256866 q^{33} -1.48787 q^{34} -6.97250 q^{35} +7.10517 q^{36} -3.06628 q^{37} -5.97005 q^{38} -13.4032 q^{39} +2.94122 q^{40} +8.70098 q^{41} +7.53585 q^{42} +8.55822 q^{43} +0.0808043 q^{44} +20.8979 q^{45} -7.06428 q^{46} +4.15060 q^{47} -3.17886 q^{48} -1.38019 q^{49} +3.65079 q^{50} +4.72974 q^{51} +4.21634 q^{52} -9.62784 q^{53} -13.0498 q^{54} +0.237664 q^{55} -2.37061 q^{56} +18.9780 q^{57} -6.65920 q^{58} +3.70188 q^{59} -9.34974 q^{60} -5.39464 q^{61} -4.37680 q^{62} -16.8436 q^{63} +1.00000 q^{64} +12.4012 q^{65} -0.256866 q^{66} +5.55648 q^{67} -1.48787 q^{68} +22.4564 q^{69} -6.97250 q^{70} -0.130112 q^{71} +7.10517 q^{72} -4.04440 q^{73} -3.06628 q^{74} -11.6054 q^{75} -5.97005 q^{76} -0.191556 q^{77} -13.4032 q^{78} -0.158725 q^{79} +2.94122 q^{80} +20.1679 q^{81} +8.70098 q^{82} +3.40851 q^{83} +7.53585 q^{84} -4.37617 q^{85} +8.55822 q^{86} +21.1687 q^{87} +0.0808043 q^{88} +7.82888 q^{89} +20.8979 q^{90} -9.99532 q^{91} -7.06428 q^{92} +13.9133 q^{93} +4.15060 q^{94} -17.5593 q^{95} -3.17886 q^{96} -16.9582 q^{97} -1.38019 q^{98} +0.574128 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9}+O(q^{10}) $ 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9} - 13 q^{10} - 29 q^{11} - 14 q^{12} - 23 q^{13} - 29 q^{14} - 14 q^{15} + 31 q^{16} - 19 q^{17} + 27 q^{18} - 36 q^{19} - 13 q^{20} - 29 q^{22} - 24 q^{23} - 14 q^{24} + 4 q^{25} - 23 q^{26} - 65 q^{27} - 29 q^{28} + 18 q^{29} - 14 q^{30} - 41 q^{31} + 31 q^{32} - 18 q^{33} - 19 q^{34} - 28 q^{35} + 27 q^{36} - 31 q^{37} - 36 q^{38} - 31 q^{39} - 13 q^{40} - 51 q^{41} - 14 q^{43} - 29 q^{44} - 34 q^{45} - 24 q^{46} - 64 q^{47} - 14 q^{48} + 8 q^{49} + 4 q^{50} - 17 q^{51} - 23 q^{52} + 3 q^{53} - 65 q^{54} - 50 q^{55} - 29 q^{56} - 19 q^{57} + 18 q^{58} - 58 q^{59} - 14 q^{60} - 14 q^{61} - 41 q^{62} - 66 q^{63} + 31 q^{64} + 6 q^{65} - 18 q^{66} - 55 q^{67} - 19 q^{68} + q^{69} - 28 q^{70} - 42 q^{71} + 27 q^{72} - 83 q^{73} - 31 q^{74} - 49 q^{75} - 36 q^{76} + 18 q^{77} - 31 q^{78} - 56 q^{79} - 13 q^{80} + 3 q^{81} - 51 q^{82} - 43 q^{83} + 4 q^{85} - 14 q^{86} - 76 q^{87} - 29 q^{88} + 17 q^{89} - 34 q^{90} - 49 q^{91} - 24 q^{92} - 24 q^{93} - 64 q^{94} - 43 q^{95} - 14 q^{96} - 98 q^{97} + 8 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9 - 13 * q^10 - 29 * q^11 - 14 * q^12 - 23 * q^13 - 29 * q^14 - 14 * q^15 + 31 * q^16 - 19 * q^17 + 27 * q^18 - 36 * q^19 - 13 * q^20 - 29 * q^22 - 24 * q^23 - 14 * q^24 + 4 * q^25 - 23 * q^26 - 65 * q^27 - 29 * q^28 + 18 * q^29 - 14 * q^30 - 41 * q^31 + 31 * q^32 - 18 * q^33 - 19 * q^34 - 28 * q^35 + 27 * q^36 - 31 * q^37 - 36 * q^38 - 31 * q^39 - 13 * q^40 - 51 * q^41 - 14 * q^43 - 29 * q^44 - 34 * q^45 - 24 * q^46 - 64 * q^47 - 14 * q^48 + 8 * q^49 + 4 * q^50 - 17 * q^51 - 23 * q^52 + 3 * q^53 - 65 * q^54 - 50 * q^55 - 29 * q^56 - 19 * q^57 + 18 * q^58 - 58 * q^59 - 14 * q^60 - 14 * q^61 - 41 * q^62 - 66 * q^63 + 31 * q^64 + 6 * q^65 - 18 * q^66 - 55 * q^67 - 19 * q^68 + q^69 - 28 * q^70 - 42 * q^71 + 27 * q^72 - 83 * q^73 - 31 * q^74 - 49 * q^75 - 36 * q^76 + 18 * q^77 - 31 * q^78 - 56 * q^79 - 13 * q^80 + 3 * q^81 - 51 * q^82 - 43 * q^83 + 4 * q^85 - 14 * q^86 - 76 * q^87 - 29 * q^88 + 17 * q^89 - 34 * q^90 - 49 * q^91 - 24 * q^92 - 24 * q^93 - 64 * q^94 - 43 * q^95 - 14 * q^96 - 98 * q^97 + 8 * q^98 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.17886	−1.83532	−0.917659	−	0.397370i	$-0.869923\pi$
$3$	−3.17886	−1.83532	−0.917659	+	0.397370i	$0.869923\pi$
$4$	1.00000	0.500000
$5$	2.94122	1.31535	0.657677	−	0.753300i	$-0.271539\pi$
$5$	2.94122	1.31535	0.657677	+	0.753300i	$0.271539\pi$
$6$	−3.17886	−1.29777
$7$	−2.37061	−0.896008	−0.448004	−	0.894032i	$-0.647865\pi$
$7$	−2.37061	−0.896008	−0.448004	+	0.894032i	$0.647865\pi$
$8$	1.00000	0.353553
$9$	7.10517	2.36839
$10$	2.94122	0.930096
$11$	0.0808043	0.0243634	0.0121817	−	0.999926i	$-0.496122\pi$
$11$	0.0808043	0.0243634	0.0121817	+	0.999926i	$0.496122\pi$
$12$	−3.17886	−0.917659
$13$	4.21634	1.16940	0.584702	−	0.811248i	$-0.301212\pi$
$13$	4.21634	1.16940	0.584702	+	0.811248i	$0.301212\pi$
$14$	−2.37061	−0.633573
$15$	−9.34974	−2.41409
$16$	1.00000	0.250000
$17$	−1.48787	−0.360862	−0.180431	−	0.983588i	$-0.557749\pi$
$17$	−1.48787	−0.360862	−0.180431	+	0.983588i	$0.557749\pi$
$18$	7.10517	1.67470
$19$	−5.97005	−1.36962	−0.684812	−	0.728720i	$-0.740116\pi$
$19$	−5.97005	−1.36962	−0.684812	+	0.728720i	$0.740116\pi$
$20$	2.94122	0.657677
$21$	7.53585	1.64446
$22$	0.0808043	0.0172275
$23$	−7.06428	−1.47300	−0.736502	−	0.676436i	$-0.763524\pi$
$23$	−7.06428	−1.47300	−0.736502	+	0.676436i	$0.763524\pi$
$24$	−3.17886	−0.648883
$25$	3.65079	0.730158
$26$	4.21634	0.826893
$27$	−13.0498	−2.51143
$28$	−2.37061	−0.448004
$29$	−6.65920	−1.23658	−0.618291	−	0.785949i	$-0.712175\pi$
$29$	−6.65920	−1.23658	−0.618291	+	0.785949i	$0.712175\pi$
$30$	−9.34974	−1.70702
$31$	−4.37680	−0.786097	−0.393048	−	0.919518i	$-0.628580\pi$
$31$	−4.37680	−0.786097	−0.393048	+	0.919518i	$0.628580\pi$
$32$	1.00000	0.176777
$33$	−0.256866	−0.0447146
$34$	−1.48787	−0.255168
$35$	−6.97250	−1.17857
$36$	7.10517	1.18419
$37$	−3.06628	−0.504094	−0.252047	−	0.967715i	$-0.581104\pi$
$37$	−3.06628	−0.504094	−0.252047	+	0.967715i	$0.581104\pi$
$38$	−5.97005	−0.968471
$39$	−13.4032	−2.14623
$40$	2.94122	0.465048
$41$	8.70098	1.35887	0.679433	−	0.733738i	$-0.262226\pi$
$41$	8.70098	1.35887	0.679433	+	0.733738i	$0.262226\pi$
$42$	7.53585	1.16281
$43$	8.55822	1.30512	0.652558	−	0.757739i	$-0.273696\pi$
$43$	8.55822	1.30512	0.652558	+	0.757739i	$0.273696\pi$
$44$	0.0808043	0.0121817
$45$	20.8979	3.11527
$46$	−7.06428	−1.04157
$47$	4.15060	0.605427	0.302714	−	0.953082i	$-0.402107\pi$
$47$	4.15060	0.605427	0.302714	+	0.953082i	$0.402107\pi$
$48$	−3.17886	−0.458829
$49$	−1.38019	−0.197170
$50$	3.65079	0.516300
$51$	4.72974	0.662297
$52$	4.21634	0.584702
$53$	−9.62784	−1.32248	−0.661242	−	0.750172i	$-0.729971\pi$
$53$	−9.62784	−1.32248	−0.661242	+	0.750172i	$0.729971\pi$
$54$	−13.0498	−1.77585
$55$	0.237664	0.0320465
$56$	−2.37061	−0.316787
$57$	18.9780	2.51369
$58$	−6.65920	−0.874395
$59$	3.70188	0.481944	0.240972	−	0.970532i	$-0.422534\pi$
$59$	3.70188	0.481944	0.240972	+	0.970532i	$0.422534\pi$
$60$	−9.34974	−1.20705
$61$	−5.39464	−0.690713	−0.345356	−	0.938472i	$-0.612242\pi$
$61$	−5.39464	−0.690713	−0.345356	+	0.938472i	$0.612242\pi$
$62$	−4.37680	−0.555854
$63$	−16.8436	−2.12209
$64$	1.00000	0.125000
$65$	12.4012	1.53818
$66$	−0.256866	−0.0316180
$67$	5.55648	0.678832	0.339416	−	0.940636i	$-0.389771\pi$
$67$	5.55648	0.678832	0.339416	+	0.940636i	$0.389771\pi$
$68$	−1.48787	−0.180431
$69$	22.4564	2.70343
$70$	−6.97250	−0.833373
$71$	−0.130112	−0.0154415	−0.00772073	−	0.999970i	$-0.502458\pi$
$71$	−0.130112	−0.0154415	−0.00772073	+	0.999970i	$0.502458\pi$
$72$	7.10517	0.837352
$73$	−4.04440	−0.473361	−0.236681	−	0.971588i	$-0.576059\pi$
$73$	−4.04440	−0.473361	−0.236681	+	0.971588i	$0.576059\pi$
$74$	−3.06628	−0.356448
$75$	−11.6054	−1.34007
$76$	−5.97005	−0.684812
$77$	−0.191556	−0.0218298
$78$	−13.4032	−1.51761
$79$	−0.158725	−0.0178579	−0.00892897	−	0.999960i	$-0.502842\pi$
$79$	−0.158725	−0.0178579	−0.00892897	+	0.999960i	$0.502842\pi$
$80$	2.94122	0.328839
$81$	20.1679	2.24088
$82$	8.70098	0.960863
$83$	3.40851	0.374133	0.187066	−	0.982347i	$-0.440102\pi$
$83$	3.40851	0.374133	0.187066	+	0.982347i	$0.440102\pi$
$84$	7.53585	0.822229
$85$	−4.37617	−0.474662
$86$	8.55822	0.922856
$87$	21.1687	2.26952
$88$	0.0808043	0.00861377
$89$	7.82888	0.829860	0.414930	−	0.909853i	$-0.363806\pi$
$89$	7.82888	0.829860	0.414930	+	0.909853i	$0.363806\pi$
$90$	20.8979	2.20283
$91$	−9.99532	−1.04779
$92$	−7.06428	−0.736502
$93$	13.9133	1.44274
$94$	4.15060	0.428102
$95$	−17.5593	−1.80154
$96$	−3.17886	−0.324441
$97$	−16.9582	−1.72185	−0.860923	−	0.508735i	$-0.830113\pi$
$97$	−16.9582	−1.72185	−0.860923	+	0.508735i	$0.830113\pi$
$98$	−1.38019	−0.139421
$99$	0.574128	0.0577020
$100$	3.65079	0.365079
$101$	−6.03778	−0.600782	−0.300391	−	0.953816i	$-0.597117\pi$
$101$	−6.03778	−0.600782	−0.300391	+	0.953816i	$0.597117\pi$
$102$	4.72974	0.468314
$103$	−3.59063	−0.353796	−0.176898	−	0.984229i	$-0.556606\pi$
$103$	−3.59063	−0.353796	−0.176898	+	0.984229i	$0.556606\pi$
$104$	4.21634	0.413447
$105$	22.1646	2.16305
$106$	−9.62784	−0.935138
$107$	−9.75388	−0.942943	−0.471472	−	0.881881i	$-0.656277\pi$
$107$	−9.75388	−0.942943	−0.471472	+	0.881881i	$0.656277\pi$
$108$	−13.0498	−1.25571
$109$	3.05824	0.292927	0.146463	−	0.989216i	$-0.453211\pi$
$109$	3.05824	0.292927	0.146463	+	0.989216i	$0.453211\pi$
$110$	0.237664	0.0226603
$111$	9.74729	0.925172
$112$	−2.37061	−0.224002
$113$	−10.9700	−1.03197	−0.515987	−	0.856596i	$-0.672575\pi$
$113$	−10.9700	−1.03197	−0.515987	+	0.856596i	$0.672575\pi$
$114$	18.9780	1.77745
$115$	−20.7776	−1.93752
$116$	−6.65920	−0.618291
$117$	29.9578	2.76960
$118$	3.70188	0.340786
$119$	3.52717	0.323335
$120$	−9.34974	−0.853511
$121$	−10.9935	−0.999406
$122$	−5.39464	−0.488408
$123$	−27.6592	−2.49395
$124$	−4.37680	−0.393048
$125$	−3.96832	−0.354938
$126$	−16.8436	−1.50055
$127$	−5.78824	−0.513623	−0.256812	−	0.966461i	$-0.582672\pi$
$127$	−5.78824	−0.513623	−0.256812	+	0.966461i	$0.582672\pi$
$128$	1.00000	0.0883883
$129$	−27.2054	−2.39530
$130$	12.4012	1.08766
$131$	−13.4612	−1.17611	−0.588057	−	0.808820i	$-0.700107\pi$
$131$	−13.4612	−1.17611	−0.588057	+	0.808820i	$0.700107\pi$
$132$	−0.256866	−0.0223573
$133$	14.1527	1.22719
$134$	5.55648	0.480007
$135$	−38.3822	−3.30342
$136$	−1.48787	−0.127584
$137$	9.55982	0.816751	0.408375	−	0.912814i	$-0.366095\pi$
$137$	9.55982	0.816751	0.408375	+	0.912814i	$0.366095\pi$
$138$	22.4564	1.91161
$139$	1.87971	0.159435	0.0797175	−	0.996817i	$-0.474598\pi$
$139$	1.87971	0.159435	0.0797175	+	0.996817i	$0.474598\pi$
$140$	−6.97250	−0.589284
$141$	−13.1942	−1.11115
$142$	−0.130112	−0.0109188
$143$	0.340699	0.0284907
$144$	7.10517	0.592097
$145$	−19.5862	−1.62654
$146$	−4.04440	−0.334717
$147$	4.38744	0.361870
$148$	−3.06628	−0.252047
$149$	8.11336	0.664673	0.332336	−	0.943161i	$-0.392163\pi$
$149$	8.11336	0.664673	0.332336	+	0.943161i	$0.392163\pi$
$150$	−11.6054	−0.947574
$151$	1.65047	0.134313	0.0671565	−	0.997742i	$-0.478607\pi$
$151$	1.65047	0.134313	0.0671565	+	0.997742i	$0.478607\pi$
$152$	−5.97005	−0.484235
$153$	−10.5716	−0.854662
$154$	−0.191556	−0.0154360
$155$	−12.8732	−1.03400
$156$	−13.4032	−1.07311
$157$	0.115118	0.00918741	0.00459370	−	0.999989i	$-0.498538\pi$
$157$	0.115118	0.00918741	0.00459370	+	0.999989i	$0.498538\pi$
$158$	−0.158725	−0.0126275
$159$	30.6056	2.42718
$160$	2.94122	0.232524
$161$	16.7467	1.31982
$162$	20.1679	1.58454
$163$	−16.4268	−1.28664	−0.643321	−	0.765596i	$-0.722444\pi$
$163$	−16.4268	−1.28664	−0.643321	+	0.765596i	$0.722444\pi$
$164$	8.70098	0.679433
$165$	−0.755500	−0.0588156
$166$	3.40851	0.264552
$167$	3.08466	0.238698	0.119349	−	0.992852i	$-0.461919\pi$
$167$	3.08466	0.238698	0.119349	+	0.992852i	$0.461919\pi$
$168$	7.53585	0.581404
$169$	4.77756	0.367505
$170$	−4.37617	−0.335637
$171$	−42.4182	−3.24380
$172$	8.55822	0.652558
$173$	−20.6344	−1.56881	−0.784404	−	0.620251i	$-0.787031\pi$
$173$	−20.6344	−1.56881	−0.784404	+	0.620251i	$0.787031\pi$
$174$	21.1687	1.60479
$175$	−8.65461	−0.654227
$176$	0.0808043	0.00609085
$177$	−11.7678	−0.884519
$178$	7.82888	0.586800
$179$	19.7648	1.47729	0.738645	−	0.674094i	$-0.235466\pi$
$179$	19.7648	1.47729	0.738645	+	0.674094i	$0.235466\pi$
$180$	20.8979	1.55764
$181$	14.9645	1.11230	0.556150	−	0.831082i	$-0.312278\pi$
$181$	14.9645	1.11230	0.556150	+	0.831082i	$0.312278\pi$
$182$	−9.99532	−0.740903
$183$	17.1488	1.26768
$184$	−7.06428	−0.520785
$185$	−9.01862	−0.663062
$186$	13.9133	1.02017
$187$	−0.120227	−0.00879184
$188$	4.15060	0.302714
$189$	30.9359	2.25026
$190$	−17.5593	−1.27388
$191$	11.4721	0.830089	0.415044	−	0.909801i	$-0.363766\pi$
$191$	11.4721	0.830089	0.415044	+	0.909801i	$0.363766\pi$
$192$	−3.17886	−0.229415
$193$	21.1602	1.52315	0.761573	−	0.648079i	$-0.224427\pi$
$193$	21.1602	1.52315	0.761573	+	0.648079i	$0.224427\pi$
$194$	−16.9582	−1.21753
$195$	−39.4217	−2.82305
$196$	−1.38019	−0.0985852
$197$	−10.4956	−0.747783	−0.373891	−	0.927472i	$-0.621977\pi$
$197$	−10.4956	−0.747783	−0.373891	+	0.927472i	$0.621977\pi$
$198$	0.574128	0.0408015
$199$	−1.85627	−0.131588	−0.0657939	−	0.997833i	$-0.520958\pi$
$199$	−1.85627	−0.131588	−0.0657939	+	0.997833i	$0.520958\pi$
$200$	3.65079	0.258150
$201$	−17.6633	−1.24587
$202$	−6.03778	−0.424817
$203$	15.7864	1.10799
$204$	4.72974	0.331148
$205$	25.5915	1.78739
$206$	−3.59063	−0.250171
$207$	−50.1929	−3.48864
$208$	4.21634	0.292351
$209$	−0.482406	−0.0333687
$210$	22.1646	1.52950
$211$	−1.02646	−0.0706641	−0.0353321	−	0.999376i	$-0.511249\pi$
$211$	−1.02646	−0.0706641	−0.0353321	+	0.999376i	$0.511249\pi$
$212$	−9.62784	−0.661242
$213$	0.413608	0.0283400
$214$	−9.75388	−0.666762
$215$	25.1716	1.71669
$216$	−13.0498	−0.887924
$217$	10.3757	0.704349
$218$	3.05824	0.207130
$219$	12.8566	0.868768
$220$	0.237664	0.0160233
$221$	−6.27339	−0.421994
$222$	9.74729	0.654195
$223$	3.96287	0.265373	0.132687	−	0.991158i	$-0.457640\pi$
$223$	3.96287	0.265373	0.132687	+	0.991158i	$0.457640\pi$
$224$	−2.37061	−0.158393
$225$	25.9395	1.72930
$226$	−10.9700	−0.729716
$227$	−24.3236	−1.61441	−0.807206	−	0.590270i	$-0.799021\pi$
$227$	−24.3236	−1.61441	−0.807206	+	0.590270i	$0.799021\pi$
$228$	18.9780	1.25685
$229$	3.65978	0.241845	0.120923	−	0.992662i	$-0.461415\pi$
$229$	3.65978	0.241845	0.120923	+	0.992662i	$0.461415\pi$
$230$	−20.7776	−1.37003
$231$	0.608930	0.0400646
$232$	−6.65920	−0.437198
$233$	−10.8548	−0.711120	−0.355560	−	0.934653i	$-0.615710\pi$
$233$	−10.8548	−0.711120	−0.355560	+	0.934653i	$0.615710\pi$
$234$	29.9578	1.95840
$235$	12.2078	0.796351
$236$	3.70188	0.240972
$237$	0.504565	0.0327750
$238$	3.52717	0.228633
$239$	−23.3707	−1.51173	−0.755863	−	0.654730i	$-0.772782\pi$
$239$	−23.3707	−1.51173	−0.755863	+	0.654730i	$0.772782\pi$
$240$	−9.34974	−0.603523
$241$	−18.3300	−1.18074	−0.590369	−	0.807134i	$-0.701018\pi$
$241$	−18.3300	−1.18074	−0.590369	+	0.807134i	$0.701018\pi$
$242$	−10.9935	−0.706687
$243$	−24.9617	−1.60129
$244$	−5.39464	−0.345356
$245$	−4.05945	−0.259349
$246$	−27.6592	−1.76349
$247$	−25.1718	−1.60164
$248$	−4.37680	−0.277927
$249$	−10.8352	−0.686652
$250$	−3.96832	−0.250979
$251$	−13.0129	−0.821370	−0.410685	−	0.911777i	$-0.634710\pi$
$251$	−13.0129	−0.821370	−0.410685	+	0.911777i	$0.634710\pi$
$252$	−16.8436	−1.06105
$253$	−0.570824	−0.0358874
$254$	−5.78824	−0.363186
$255$	13.9112	0.871155
$256$	1.00000	0.0625000
$257$	−19.0457	−1.18804	−0.594020	−	0.804450i	$-0.702460\pi$
$257$	−19.0457	−1.18804	−0.594020	+	0.804450i	$0.702460\pi$
$258$	−27.2054	−1.69373
$259$	7.26897	0.451672
$260$	12.4012	0.769090
$261$	−47.3147	−2.92871
$262$	−13.4612	−0.831638
$263$	20.9668	1.29287	0.646436	−	0.762969i	$-0.276259\pi$
$263$	20.9668	1.29287	0.646436	+	0.762969i	$0.276259\pi$
$264$	−0.256866	−0.0158090
$265$	−28.3176	−1.73954
$266$	14.1527	0.867757
$267$	−24.8869	−1.52306
$268$	5.55648	0.339416
$269$	9.70137	0.591503	0.295752	−	0.955265i	$-0.404430\pi$
$269$	9.70137	0.591503	0.295752	+	0.955265i	$0.404430\pi$
$270$	−38.3822	−2.33587
$271$	15.0767	0.915843	0.457922	−	0.888993i	$-0.348594\pi$
$271$	15.0767	0.915843	0.457922	+	0.888993i	$0.348594\pi$
$272$	−1.48787	−0.0902156
$273$	31.7738	1.92304
$274$	9.55982	0.577530
$275$	0.295000	0.0177892
$276$	22.4564	1.35171
$277$	0.333418	0.0200332	0.0100166	−	0.999950i	$-0.496812\pi$
$277$	0.333418	0.0200332	0.0100166	+	0.999950i	$0.496812\pi$
$278$	1.87971	0.112738
$279$	−31.0979	−1.86178
$280$	−6.97250	−0.416687
$281$	17.4003	1.03801	0.519006	−	0.854770i	$-0.326302\pi$
$281$	17.4003	1.03801	0.519006	+	0.854770i	$0.326302\pi$
$282$	−13.1942	−0.785702
$283$	−21.7111	−1.29059	−0.645295	−	0.763934i	$-0.723265\pi$
$283$	−21.7111	−1.29059	−0.645295	+	0.763934i	$0.723265\pi$
$284$	−0.130112	−0.00772073
$285$	55.8185	3.30640
$286$	0.340699	0.0201459
$287$	−20.6267	−1.21755
$288$	7.10517	0.418676
$289$	−14.7862	−0.869778
$290$	−19.5862	−1.15014
$291$	53.9079	3.16013
$292$	−4.04440	−0.236681
$293$	−30.0173	−1.75363	−0.876814	−	0.480829i	$-0.840336\pi$
$293$	−30.0173	−1.75363	−0.876814	+	0.480829i	$0.840336\pi$
$294$	4.38744	0.255881
$295$	10.8880	0.633927
$296$	−3.06628	−0.178224
$297$	−1.05448	−0.0611870
$298$	8.11336	0.469995
$299$	−29.7854	−1.72254
$300$	−11.6054	−0.670036
$301$	−20.2882	−1.16939
$302$	1.65047	0.0949736
$303$	19.1933	1.10263
$304$	−5.97005	−0.342406
$305$	−15.8668	−0.908532
$306$	−10.5716	−0.604337
$307$	19.2956	1.10126	0.550630	−	0.834749i	$-0.314387\pi$
$307$	19.2956	1.10126	0.550630	+	0.834749i	$0.314387\pi$
$308$	−0.191556	−0.0109149
$309$	11.4141	0.649327
$310$	−12.8732	−0.731146
$311$	1.81829	0.103106	0.0515529	−	0.998670i	$-0.483583\pi$
$311$	1.81829	0.103106	0.0515529	+	0.998670i	$0.483583\pi$
$312$	−13.4032	−0.758806
$313$	−28.7168	−1.62317	−0.811584	−	0.584236i	$-0.801394\pi$
$313$	−28.7168	−1.62317	−0.811584	+	0.584236i	$0.801394\pi$
$314$	0.115118	0.00649648
$315$	−49.5408	−2.79131
$316$	−0.158725	−0.00892897
$317$	−25.3000	−1.42099	−0.710496	−	0.703701i	$-0.751529\pi$
$317$	−25.3000	−1.42099	−0.710496	+	0.703701i	$0.751529\pi$
$318$	30.6056	1.71627
$319$	−0.538092	−0.0301274
$320$	2.94122	0.164419
$321$	31.0062	1.73060
$322$	16.7467	0.933255
$323$	8.88268	0.494246
$324$	20.1679	1.12044
$325$	15.3930	0.853850
$326$	−16.4268	−0.909794
$327$	−9.72173	−0.537613
$328$	8.70098	0.480431
$329$	−9.83946	−0.542467
$330$	−0.755500	−0.0415889
$331$	8.21971	0.451796	0.225898	−	0.974151i	$-0.427468\pi$
$331$	8.21971	0.451796	0.225898	+	0.974151i	$0.427468\pi$
$332$	3.40851	0.187066
$333$	−21.7865	−1.19389
$334$	3.08466	0.168785
$335$	16.3429	0.892905
$336$	7.53585	0.411115
$337$	10.5182	0.572965	0.286483	−	0.958085i	$-0.407514\pi$
$337$	10.5182	0.572965	0.286483	+	0.958085i	$0.407514\pi$
$338$	4.77756	0.259865
$339$	34.8722	1.89400
$340$	−4.37617	−0.237331
$341$	−0.353665	−0.0191520
$342$	−42.4182	−2.29371
$343$	19.8662	1.07267
$344$	8.55822	0.461428
$345$	66.0492	3.55597
$346$	−20.6344	−1.10931
$347$	12.1766	0.653676	0.326838	−	0.945080i	$-0.394017\pi$
$347$	12.1766	0.653676	0.326838	+	0.945080i	$0.394017\pi$
$348$	21.1687	1.13476
$349$	−21.3305	−1.14180	−0.570898	−	0.821021i	$-0.693405\pi$
$349$	−21.3305	−1.14180	−0.570898	+	0.821021i	$0.693405\pi$
$350$	−8.65461	−0.462609
$351$	−55.0223	−2.93687
$352$	0.0808043	0.00430688
$353$	2.69742	0.143569	0.0717846	−	0.997420i	$-0.477131\pi$
$353$	2.69742	0.143569	0.0717846	+	0.997420i	$0.477131\pi$
$354$	−11.7678	−0.625450
$355$	−0.382689	−0.0203110
$356$	7.82888	0.414930
$357$	−11.2124	−0.593423
$358$	19.7648	1.04460
$359$	−4.72010	−0.249117	−0.124559	−	0.992212i	$-0.539751\pi$
$359$	−4.72010	−0.249117	−0.124559	+	0.992212i	$0.539751\pi$
$360$	20.8979	1.10141
$361$	16.6415	0.875870
$362$	14.9645	0.786515
$363$	34.9467	1.83423
$364$	−9.99532	−0.523897
$365$	−11.8955	−0.622638
$366$	17.1488	0.896383
$367$	9.17930	0.479156	0.239578	−	0.970877i	$-0.422991\pi$
$367$	9.17930	0.479156	0.239578	+	0.970877i	$0.422991\pi$
$368$	−7.06428	−0.368251
$369$	61.8219	3.21832
$370$	−9.01862	−0.468856
$371$	22.8239	1.18496
$372$	13.9133	0.721369
$373$	−23.8818	−1.23655	−0.618276	−	0.785961i	$-0.712169\pi$
$373$	−23.8818	−1.23655	−0.618276	+	0.785961i	$0.712169\pi$
$374$	−0.120227	−0.00621677
$375$	12.6148	0.651423
$376$	4.15060	0.214051
$377$	−28.0775	−1.44606
$378$	30.9359	1.59117
$379$	0.740745	0.0380495	0.0190247	−	0.999819i	$-0.493944\pi$
$379$	0.740745	0.0380495	0.0190247	+	0.999819i	$0.493944\pi$
$380$	−17.5593	−0.900771
$381$	18.4000	0.942661
$382$	11.4721	0.586962
$383$	−30.6903	−1.56820	−0.784099	−	0.620635i	$-0.786875\pi$
$383$	−30.6903	−1.56820	−0.784099	+	0.620635i	$0.786875\pi$
$384$	−3.17886	−0.162221
$385$	−0.563408	−0.0287139
$386$	21.1602	1.07703
$387$	60.8076	3.09102
$388$	−16.9582	−0.860923
$389$	10.4284	0.528740	0.264370	−	0.964421i	$-0.414836\pi$
$389$	10.4284	0.528740	0.264370	+	0.964421i	$0.414836\pi$
$390$	−39.4217	−1.99620
$391$	10.5107	0.531551
$392$	−1.38019	−0.0697103
$393$	42.7914	2.15854
$394$	−10.4956	−0.528762
$395$	−0.466845	−0.0234895
$396$	0.574128	0.0288510
$397$	24.0230	1.20568	0.602839	−	0.797863i	$-0.294036\pi$
$397$	24.0230	1.20568	0.602839	+	0.797863i	$0.294036\pi$
$398$	−1.85627	−0.0930466
$399$	−44.9894	−2.25229
$400$	3.65079	0.182540
$401$	−15.1554	−0.756826	−0.378413	−	0.925637i	$-0.623530\pi$
$401$	−15.1554	−0.756826	−0.378413	+	0.925637i	$0.623530\pi$
$402$	−17.6633	−0.880965
$403$	−18.4541	−0.919264
$404$	−6.03778	−0.300391
$405$	59.3183	2.94755
$406$	15.7864	0.783465
$407$	−0.247769	−0.0122814
$408$	4.72974	0.234157
$409$	−35.1690	−1.73900	−0.869498	−	0.493937i	$-0.835557\pi$
$409$	−35.1690	−1.73900	−0.869498	+	0.493937i	$0.835557\pi$
$410$	25.5915	1.26388
$411$	−30.3894	−1.49900
$412$	−3.59063	−0.176898
$413$	−8.77572	−0.431825
$414$	−50.1929	−2.46684
$415$	10.0252	0.492117
$416$	4.21634	0.206723
$417$	−5.97534	−0.292614
$418$	−0.482406	−0.0235953
$419$	3.14952	0.153864	0.0769319	−	0.997036i	$-0.475488\pi$
$419$	3.14952	0.153864	0.0769319	+	0.997036i	$0.475488\pi$
$420$	22.1646	1.08152
$421$	37.0440	1.80541	0.902707	−	0.430255i	$-0.141576\pi$
$421$	37.0440	1.80541	0.902707	+	0.430255i	$0.141576\pi$
$422$	−1.02646	−0.0499671
$423$	29.4907	1.43389
$424$	−9.62784	−0.467569
$425$	−5.43191	−0.263487
$426$	0.413608	0.0200394
$427$	12.7886	0.618884
$428$	−9.75388	−0.471472
$429$	−1.08303	−0.0522894
$430$	25.1716	1.21388
$431$	31.1320	1.49957	0.749787	−	0.661679i	$-0.230156\pi$
$431$	31.1320	1.49957	0.749787	+	0.661679i	$0.230156\pi$
$432$	−13.0498	−0.627857
$433$	17.5725	0.844479	0.422239	−	0.906484i	$-0.361244\pi$
$433$	17.5725	0.844479	0.422239	+	0.906484i	$0.361244\pi$
$434$	10.3757	0.498050
$435$	62.2618	2.98522
$436$	3.05824	0.146463
$437$	42.1741	2.01746
$438$	12.8566	0.614312
$439$	−0.980355	−0.0467898	−0.0233949	−	0.999726i	$-0.507448\pi$
$439$	−0.980355	−0.0467898	−0.0233949	+	0.999726i	$0.507448\pi$
$440$	0.237664	0.0113302
$441$	−9.80650	−0.466976
$442$	−6.27339	−0.298395
$443$	−3.04011	−0.144440	−0.0722200	−	0.997389i	$-0.523008\pi$
$443$	−3.04011	−0.144440	−0.0722200	+	0.997389i	$0.523008\pi$
$444$	9.74729	0.462586
$445$	23.0265	1.09156
$446$	3.96287	0.187647
$447$	−25.7913	−1.21988
$448$	−2.37061	−0.112001
$449$	1.28937	0.0608491	0.0304245	−	0.999537i	$-0.490314\pi$
$449$	1.28937	0.0608491	0.0304245	+	0.999537i	$0.490314\pi$
$450$	25.9395	1.22280
$451$	0.703077	0.0331066
$452$	−10.9700	−0.515987
$453$	−5.24660	−0.246507
$454$	−24.3236	−1.14156
$455$	−29.3985	−1.37822
$456$	18.9780	0.888725
$457$	1.91348	0.0895088	0.0447544	−	0.998998i	$-0.485749\pi$
$457$	1.91348	0.0895088	0.0447544	+	0.998998i	$0.485749\pi$
$458$	3.65978	0.171010
$459$	19.4164	0.906279
$460$	−20.7776	−0.968761
$461$	−22.0841	−1.02856	−0.514279	−	0.857623i	$-0.671940\pi$
$461$	−22.0841	−1.02856	−0.514279	+	0.857623i	$0.671940\pi$
$462$	0.608930	0.0283300
$463$	−30.4642	−1.41579	−0.707895	−	0.706318i	$-0.750355\pi$
$463$	−30.4642	−1.41579	−0.707895	+	0.706318i	$0.750355\pi$
$464$	−6.65920	−0.309145
$465$	40.9220	1.89771
$466$	−10.8548	−0.502838
$467$	−19.4938	−0.902067	−0.451033	−	0.892507i	$-0.648944\pi$
$467$	−19.4938	−0.902067	−0.451033	+	0.892507i	$0.648944\pi$
$468$	29.9578	1.38480
$469$	−13.1723	−0.608239
$470$	12.2078	0.563105
$471$	−0.365944	−0.0168618
$472$	3.70188	0.170393
$473$	0.691541	0.0317971
$474$	0.504565	0.0231754
$475$	−21.7954	−1.00004
$476$	3.52717	0.161668
$477$	−68.4074	−3.13216
$478$	−23.3707	−1.06895
$479$	3.48055	0.159030	0.0795152	−	0.996834i	$-0.474663\pi$
$479$	3.48055	0.159030	0.0795152	+	0.996834i	$0.474663\pi$
$480$	−9.34974	−0.426755
$481$	−12.9285	−0.589489
$482$	−18.3300	−0.834907
$483$	−53.2353	−2.42229
$484$	−10.9935	−0.499703
$485$	−49.8779	−2.26484
$486$	−24.9617	−1.13228
$487$	42.1274	1.90898	0.954488	−	0.298250i	$-0.0964029\pi$
$487$	42.1274	1.90898	0.954488	+	0.298250i	$0.0964029\pi$
$488$	−5.39464	−0.244204
$489$	52.2184	2.36140
$490$	−4.05945	−0.183387
$491$	16.5010	0.744678	0.372339	−	0.928097i	$-0.378556\pi$
$491$	16.5010	0.744678	0.372339	+	0.928097i	$0.378556\pi$
$492$	−27.6592	−1.24697
$493$	9.90804	0.446236
$494$	−25.1718	−1.13253
$495$	1.68864	0.0758987
$496$	−4.37680	−0.196524
$497$	0.308445	0.0138357
$498$	−10.8352	−0.485537
$499$	28.7145	1.28544	0.642720	−	0.766101i	$-0.277806\pi$
$499$	28.7145	1.28544	0.642720	+	0.766101i	$0.277806\pi$
$500$	−3.96832	−0.177469
$501$	−9.80572	−0.438087
$502$	−13.0129	−0.580796
$503$	26.5886	1.18553	0.592764	−	0.805376i	$-0.298037\pi$
$503$	26.5886	1.18553	0.592764	+	0.805376i	$0.298037\pi$
$504$	−16.8436	−0.750274
$505$	−17.7585	−0.790241
$506$	−0.570824	−0.0253762
$507$	−15.1872	−0.674488
$508$	−5.78824	−0.256812
$509$	32.9251	1.45938	0.729691	−	0.683777i	$-0.239664\pi$
$509$	32.9251	1.45938	0.729691	+	0.683777i	$0.239664\pi$
$510$	13.9112	0.616000
$511$	9.58771	0.424135
$512$	1.00000	0.0441942
$513$	77.9078	3.43971
$514$	−19.0457	−0.840072
$515$	−10.5609	−0.465367
$516$	−27.2054	−1.19765
$517$	0.335386	0.0147503
$518$	7.26897	0.319380
$519$	65.5940	2.87926
$520$	12.4012	0.543829
$521$	29.8128	1.30612	0.653061	−	0.757305i	$-0.273484\pi$
$521$	29.8128	1.30612	0.653061	+	0.757305i	$0.273484\pi$
$522$	−47.3147	−2.07091
$523$	−40.9650	−1.79127	−0.895637	−	0.444785i	$-0.853280\pi$
$523$	−40.9650	−1.79127	−0.895637	+	0.444785i	$0.853280\pi$
$524$	−13.4612	−0.588057
$525$	27.5118	1.20071
$526$	20.9668	0.914198
$527$	6.51213	0.283673
$528$	−0.256866	−0.0111786
$529$	26.9040	1.16974
$530$	−28.3176	−1.23004
$531$	26.3025	1.14143
$532$	14.1527	0.613597
$533$	36.6863	1.58906
$534$	−24.8869	−1.07696
$535$	−28.6883	−1.24030
$536$	5.55648	0.240003
$537$	−62.8296	−2.71130
$538$	9.70137	0.418256
$539$	−0.111526	−0.00480375
$540$	−38.3822	−1.65171
$541$	21.4426	0.921890	0.460945	−	0.887429i	$-0.347511\pi$
$541$	21.4426	0.921890	0.460945	+	0.887429i	$0.347511\pi$
$542$	15.0767	0.647599
$543$	−47.5700	−2.04142
$544$	−1.48787	−0.0637920
$545$	8.99497	0.385302
$546$	31.7738	1.35979
$547$	32.5843	1.39321	0.696603	−	0.717457i	$-0.254694\pi$
$547$	32.5843	1.39321	0.696603	+	0.717457i	$0.254694\pi$
$548$	9.55982	0.408375
$549$	−38.3298	−1.63588
$550$	0.295000	0.0125788
$551$	39.7558	1.69365
$552$	22.4564	0.955806
$553$	0.376275	0.0160009
$554$	0.333418	0.0141656
$555$	28.6690	1.21693
$556$	1.87971	0.0797175
$557$	14.7738	0.625986	0.312993	−	0.949755i	$-0.398668\pi$
$557$	14.7738	0.625986	0.312993	+	0.949755i	$0.398668\pi$
$558$	−31.0979	−1.31648
$559$	36.0844	1.52621
$560$	−6.97250	−0.294642
$561$	0.382184	0.0161358
$562$	17.4003	0.733986
$563$	12.1948	0.513949	0.256974	−	0.966418i	$-0.417274\pi$
$563$	12.1948	0.513949	0.256974	+	0.966418i	$0.417274\pi$
$564$	−13.1942	−0.555575
$565$	−32.2653	−1.35741
$566$	−21.7111	−0.912584
$567$	−47.8103	−2.00784
$568$	−0.130112	−0.00545938
$569$	16.2345	0.680585	0.340292	−	0.940320i	$-0.389474\pi$
$569$	16.2345	0.680585	0.340292	+	0.940320i	$0.389474\pi$
$570$	55.8185	2.33798
$571$	47.6507	1.99412	0.997060	−	0.0766285i	$-0.0244156\pi$
$571$	47.6507	1.99412	0.997060	+	0.0766285i	$0.0244156\pi$
$572$	0.340699	0.0142453
$573$	−36.4681	−1.52348
$574$	−20.6267	−0.860940
$575$	−25.7902	−1.07553
$576$	7.10517	0.296049
$577$	−21.8012	−0.907596	−0.453798	−	0.891105i	$-0.649931\pi$
$577$	−21.8012	−0.907596	−0.453798	+	0.891105i	$0.649931\pi$
$578$	−14.7862	−0.615026
$579$	−67.2654	−2.79546
$580$	−19.5862	−0.813272
$581$	−8.08027	−0.335226
$582$	53.9079	2.23455
$583$	−0.777971	−0.0322203
$584$	−4.04440	−0.167358
$585$	88.1126	3.64301
$586$	−30.0173	−1.24000
$587$	−28.4445	−1.17403	−0.587014	−	0.809577i	$-0.699697\pi$
$587$	−28.4445	−1.17403	−0.587014	+	0.809577i	$0.699697\pi$
$588$	4.38744	0.180935
$589$	26.1297	1.07666
$590$	10.8880	0.448254
$591$	33.3642	1.37242
$592$	−3.06628	−0.126023
$593$	2.35284	0.0966194	0.0483097	−	0.998832i	$-0.484617\pi$
$593$	2.35284	0.0966194	0.0483097	+	0.998832i	$0.484617\pi$
$594$	−1.05448	−0.0432657
$595$	10.3742	0.425301
$596$	8.11336	0.332336
$597$	5.90084	0.241505
$598$	−29.7854	−1.21802
$599$	6.21772	0.254049	0.127025	−	0.991900i	$-0.459457\pi$
$599$	6.21772	0.254049	0.127025	+	0.991900i	$0.459457\pi$
$600$	−11.6054	−0.473787
$601$	−1.69087	−0.0689718	−0.0344859	−	0.999405i	$-0.510979\pi$
$601$	−1.69087	−0.0689718	−0.0344859	+	0.999405i	$0.510979\pi$
$602$	−20.2882	−0.826886
$603$	39.4797	1.60774
$604$	1.65047	0.0671565
$605$	−32.3342	−1.31457
$606$	19.1933	0.779674
$607$	9.66294	0.392207	0.196103	−	0.980583i	$-0.437171\pi$
$607$	9.66294	0.392207	0.196103	+	0.980583i	$0.437171\pi$
$608$	−5.97005	−0.242118
$609$	−50.1827	−2.03351
$610$	−15.8668	−0.642429
$611$	17.5004	0.707989
$612$	−10.5716	−0.427331
$613$	−22.2750	−0.899677	−0.449839	−	0.893110i	$-0.648518\pi$
$613$	−22.2750	−0.899677	−0.449839	+	0.893110i	$0.648518\pi$
$614$	19.2956	0.778709
$615$	−81.3519	−3.28043
$616$	−0.191556	−0.00771800
$617$	−31.1966	−1.25593	−0.627964	−	0.778242i	$-0.716112\pi$
$617$	−31.1966	−1.25593	−0.627964	+	0.778242i	$0.716112\pi$
$618$	11.4141	0.459144
$619$	16.4123	0.659664	0.329832	−	0.944040i	$-0.393008\pi$
$619$	16.4123	0.659664	0.329832	+	0.944040i	$0.393008\pi$
$620$	−12.8732	−0.516998
$621$	92.1871	3.69934
$622$	1.81829	0.0729068
$623$	−18.5593	−0.743561
$624$	−13.4032	−0.536557
$625$	−29.9257	−1.19703
$626$	−28.7168	−1.14775
$627$	1.53350	0.0612422
$628$	0.115118	0.00459370
$629$	4.56224	0.181908
$630$	−49.5408	−1.97375
$631$	−29.8819	−1.18958	−0.594790	−	0.803881i	$-0.702765\pi$
$631$	−29.8819	−1.18958	−0.594790	+	0.803881i	$0.702765\pi$
$632$	−0.158725	−0.00631374
$633$	3.26296	0.129691
$634$	−25.3000	−1.00479
$635$	−17.0245	−0.675597
$636$	30.6056	1.21359
$637$	−5.81937	−0.230572
$638$	−0.538092	−0.0213033
$639$	−0.924468	−0.0365714
$640$	2.94122	0.116262
$641$	−14.8023	−0.584656	−0.292328	−	0.956318i	$-0.594430\pi$
$641$	−14.8023	−0.584656	−0.292328	+	0.956318i	$0.594430\pi$
$642$	31.0062	1.22372
$643$	27.4207	1.08137	0.540683	−	0.841227i	$-0.318166\pi$
$643$	27.4207	1.08137	0.540683	+	0.841227i	$0.318166\pi$
$644$	16.7467	0.659911
$645$	−80.0171	−3.15067
$646$	8.88268	0.349484
$647$	−39.4609	−1.55137	−0.775685	−	0.631121i	$-0.782595\pi$
$647$	−39.4609	−1.55137	−0.775685	+	0.631121i	$0.782595\pi$
$648$	20.1679	0.792270
$649$	0.299128	0.0117418
$650$	15.3930	0.603763
$651$	−32.9829	−1.29270
$652$	−16.4268	−0.643321
$653$	7.52918	0.294640	0.147320	−	0.989089i	$-0.452935\pi$
$653$	7.52918	0.294640	0.147320	+	0.989089i	$0.452935\pi$
$654$	−9.72173	−0.380150
$655$	−39.5925	−1.54701
$656$	8.70098	0.339716
$657$	−28.7361	−1.12110
$658$	−9.83946	−0.383582
$659$	−36.3509	−1.41603	−0.708016	−	0.706197i	$-0.750409\pi$
$659$	−36.3509	−1.41603	−0.708016	+	0.706197i	$0.750409\pi$
$660$	−0.755500	−0.0294078
$661$	32.4920	1.26379	0.631896	−	0.775053i	$-0.282277\pi$
$661$	32.4920	1.26379	0.631896	+	0.775053i	$0.282277\pi$
$662$	8.21971	0.319468
$663$	19.9422	0.774492
$664$	3.40851	0.132276
$665$	41.6262	1.61420
$666$	−21.7865	−0.844208
$667$	47.0424	1.82149
$668$	3.08466	0.119349
$669$	−12.5974	−0.487044
$670$	16.3429	0.631379
$671$	−0.435910	−0.0168281
$672$	7.53585	0.290702
$673$	−39.5308	−1.52380	−0.761899	−	0.647695i	$-0.775733\pi$
$673$	−39.5308	−1.52380	−0.761899	+	0.647695i	$0.775733\pi$
$674$	10.5182	0.405148
$675$	−47.6419	−1.83374
$676$	4.77756	0.183752
$677$	−11.8195	−0.454262	−0.227131	−	0.973864i	$-0.572935\pi$
$677$	−11.8195	−0.454262	−0.227131	+	0.973864i	$0.572935\pi$
$678$	34.8722	1.33926
$679$	40.2014	1.54279
$680$	−4.37617	−0.167818
$681$	77.3213	2.96296
$682$	−0.353665	−0.0135425
$683$	−13.1971	−0.504974	−0.252487	−	0.967600i	$-0.581249\pi$
$683$	−13.1971	−0.504974	−0.252487	+	0.967600i	$0.581249\pi$
$684$	−42.4182	−1.62190
$685$	28.1176	1.07432
$686$	19.8662	0.758495
$687$	−11.6339	−0.443863
$688$	8.55822	0.326279
$689$	−40.5943	−1.54652
$690$	66.0492	2.51445
$691$	−40.8408	−1.55366	−0.776828	−	0.629713i	$-0.783173\pi$
$691$	−40.8408	−1.55366	−0.776828	+	0.629713i	$0.783173\pi$
$692$	−20.6344	−0.784404
$693$	−1.36104	−0.0517015
$694$	12.1766	0.462219
$695$	5.52865	0.209714
$696$	21.1687	0.802396
$697$	−12.9460	−0.490363
$698$	−21.3305	−0.807372
$699$	34.5058	1.30513
$700$	−8.65461	−0.327114
$701$	11.9385	0.450911	0.225456	−	0.974253i	$-0.427613\pi$
$701$	11.9385	0.450911	0.225456	+	0.974253i	$0.427613\pi$
$702$	−55.0223	−2.07668
$703$	18.3059	0.690419
$704$	0.0808043	0.00304543
$705$	−38.8070	−1.46156
$706$	2.69742	0.101519
$707$	14.3132	0.538305
$708$	−11.7678	−0.442260
$709$	36.1580	1.35794	0.678971	−	0.734165i	$-0.262426\pi$
$709$	36.1580	1.35794	0.678971	+	0.734165i	$0.262426\pi$
$710$	−0.382689	−0.0143620
$711$	−1.12777	−0.0422946
$712$	7.82888	0.293400
$713$	30.9189	1.15792
$714$	−11.2124	−0.419613
$715$	1.00207	0.0374753
$716$	19.7648	0.738645
$717$	74.2923	2.77450
$718$	−4.72010	−0.176152
$719$	2.24320	0.0836571	0.0418285	−	0.999125i	$-0.486682\pi$
$719$	2.24320	0.0836571	0.0418285	+	0.999125i	$0.486682\pi$
$720$	20.8979	0.778818
$721$	8.51200	0.317004
$722$	16.6415	0.619334
$723$	58.2685	2.16703
$724$	14.9645	0.556150
$725$	−24.3113	−0.902900
$726$	34.9467	1.29699
$727$	45.5297	1.68860	0.844302	−	0.535867i	$-0.180015\pi$
$727$	45.5297	1.68860	0.844302	+	0.535867i	$0.180015\pi$
$728$	−9.99532	−0.370451
$729$	18.8460	0.698002
$730$	−11.8955	−0.440271
$731$	−12.7335	−0.470967
$732$	17.1488	0.633839
$733$	−27.0160	−0.997858	−0.498929	−	0.866643i	$-0.666273\pi$
$733$	−27.0160	−0.997858	−0.498929	+	0.866643i	$0.666273\pi$
$734$	9.17930	0.338814
$735$	12.9044	0.475988
$736$	−7.06428	−0.260393
$737$	0.448988	0.0165387
$738$	61.8219	2.27570
$739$	−16.6875	−0.613860	−0.306930	−	0.951732i	$-0.599302\pi$
$739$	−16.6875	−0.613860	−0.306930	+	0.951732i	$0.599302\pi$
$740$	−9.01862	−0.331531
$741$	80.0177	2.93952
$742$	22.8239	0.837891
$743$	−22.1686	−0.813285	−0.406643	−	0.913587i	$-0.633301\pi$
$743$	−22.1686	−0.813285	−0.406643	+	0.913587i	$0.633301\pi$
$744$	13.9133	0.510085
$745$	23.8632	0.874280
$746$	−23.8818	−0.874375
$747$	24.2180	0.886092
$748$	−0.120227	−0.00439592
$749$	23.1227	0.844884
$750$	12.6148	0.460626
$751$	−45.4351	−1.65795	−0.828976	−	0.559284i	$-0.811076\pi$
$751$	−45.4351	−1.65795	−0.828976	+	0.559284i	$0.811076\pi$
$752$	4.15060	0.151357
$753$	41.3664	1.50747
$754$	−28.0775	−1.02252
$755$	4.85439	0.176669
$756$	30.9359	1.12513
$757$	33.2358	1.20798	0.603988	−	0.796994i	$-0.293578\pi$
$757$	33.2358	1.20798	0.603988	+	0.796994i	$0.293578\pi$
$758$	0.740745	0.0269051
$759$	1.81457	0.0658647
$760$	−17.5593	−0.636941
$761$	25.1588	0.912005	0.456002	−	0.889979i	$-0.349281\pi$
$761$	25.1588	0.912005	0.456002	+	0.889979i	$0.349281\pi$
$762$	18.4000	0.666562
$763$	−7.24991	−0.262464
$764$	11.4721	0.415044
$765$	−31.0934	−1.12418
$766$	−30.6903	−1.10888
$767$	15.6084	0.563586
$768$	−3.17886	−0.114707
$769$	34.2477	1.23500	0.617502	−	0.786569i	$-0.288145\pi$
$769$	34.2477	1.23500	0.617502	+	0.786569i	$0.288145\pi$
$770$	−0.563408	−0.0203038
$771$	60.5438	2.18043
$772$	21.1602	0.761573
$773$	34.0865	1.22601	0.613003	−	0.790080i	$-0.289961\pi$
$773$	34.0865	1.22601	0.613003	+	0.790080i	$0.289961\pi$
$774$	60.8076	2.18568
$775$	−15.9788	−0.573975
$776$	−16.9582	−0.608765
$777$	−23.1071	−0.828961
$778$	10.4284	0.373876
$779$	−51.9453	−1.86113
$780$	−39.4217	−1.41152
$781$	−0.0105136	−0.000376207 0
$782$	10.5107	0.375864
$783$	86.9009	3.10559
$784$	−1.38019	−0.0492926
$785$	0.338587	0.0120847
$786$	42.7914	1.52632
$787$	−25.1539	−0.896641	−0.448320	−	0.893873i	$-0.647978\pi$
$787$	−25.1539	−0.896641	−0.448320	+	0.893873i	$0.647978\pi$
$788$	−10.4956	−0.373891
$789$	−66.6507	−2.37283
$790$	−0.466845	−0.0166096
$791$	26.0057	0.924656
$792$	0.574128	0.0204008
$793$	−22.7457	−0.807722
$794$	24.0230	0.852543
$795$	90.0178	3.19260
$796$	−1.85627	−0.0657939
$797$	23.7365	0.840791	0.420395	−	0.907341i	$-0.361891\pi$
$797$	23.7365	0.840791	0.420395	+	0.907341i	$0.361891\pi$
$798$	−44.9894	−1.59261
$799$	−6.17556	−0.218476
$800$	3.65079	0.129075
$801$	55.6255	1.96543
$802$	−15.1554	−0.535157
$803$	−0.326805	−0.0115327
$804$	−17.6633	−0.622936
$805$	49.2557	1.73603
$806$	−18.4541	−0.650018
$807$	−30.8393	−1.08560
$808$	−6.03778	−0.212408
$809$	−2.74440	−0.0964878	−0.0482439	−	0.998836i	$-0.515362\pi$
$809$	−2.74440	−0.0964878	−0.0482439	+	0.998836i	$0.515362\pi$
$810$	59.3183	2.08423
$811$	−0.465285	−0.0163384	−0.00816918	−	0.999967i	$-0.502600\pi$
$811$	−0.465285	−0.0163384	−0.00816918	+	0.999967i	$0.502600\pi$
$812$	15.7864	0.553993
$813$	−47.9267	−1.68086
$814$	−0.247769	−0.00868430
$815$	−48.3147	−1.69239
$816$	4.72974	0.165574
$817$	−51.0930	−1.78752
$818$	−35.1690	−1.22966
$819$	−71.0184	−2.48158
$820$	25.5915	0.893695
$821$	26.0879	0.910474	0.455237	−	0.890370i	$-0.349554\pi$
$821$	26.0879	0.910474	0.455237	+	0.890370i	$0.349554\pi$
$822$	−30.3894	−1.05995
$823$	32.1607	1.12105	0.560526	−	0.828137i	$-0.310599\pi$
$823$	32.1607	1.12105	0.560526	+	0.828137i	$0.310599\pi$
$824$	−3.59063	−0.125086
$825$	−0.937763	−0.0326487
$826$	−8.77572	−0.305346
$827$	13.0132	0.452512	0.226256	−	0.974068i	$-0.427351\pi$
$827$	13.0132	0.452512	0.226256	+	0.974068i	$0.427351\pi$
$828$	−50.1929	−1.74432
$829$	−40.5956	−1.40994	−0.704972	−	0.709236i	$-0.749040\pi$
$829$	−40.5956	−1.40994	−0.704972	+	0.709236i	$0.749040\pi$
$830$	10.0252	0.347980
$831$	−1.05989	−0.0367672
$832$	4.21634	0.146175
$833$	2.05355	0.0711514
$834$	−5.97534	−0.206909
$835$	9.07268	0.313973
$836$	−0.482406	−0.0166844
$837$	57.1162	1.97423
$838$	3.14952	0.108798
$839$	−50.0246	−1.72704	−0.863520	−	0.504315i	$-0.831745\pi$
$839$	−50.0246	−1.72704	−0.863520	+	0.504315i	$0.831745\pi$
$840$	22.1646	0.764752
$841$	15.3449	0.529135
$842$	37.0440	1.27662
$843$	−55.3130	−1.90508
$844$	−1.02646	−0.0353321
$845$	14.0519	0.483399
$846$	29.4907	1.01391
$847$	26.0613	0.895476
$848$	−9.62784	−0.330621
$849$	69.0165	2.36864
$850$	−5.43191	−0.186313
$851$	21.6611	0.742532
$852$	0.413608	0.0141700
$853$	49.0350	1.67893	0.839463	−	0.543417i	$-0.182870\pi$
$853$	49.0350	1.67893	0.839463	+	0.543417i	$0.182870\pi$
$854$	12.7886	0.437617
$855$	−124.761	−4.26675
$856$	−9.75388	−0.333381
$857$	52.6466	1.79837	0.899187	−	0.437565i	$-0.144159\pi$
$857$	52.6466	1.79837	0.899187	+	0.437565i	$0.144159\pi$
$858$	−1.08303	−0.0369742
$859$	−39.2918	−1.34062	−0.670310	−	0.742081i	$-0.733839\pi$
$859$	−39.2918	−1.34062	−0.670310	+	0.742081i	$0.733839\pi$
$860$	25.1716	0.858345
$861$	65.5693	2.23460
$862$	31.1320	1.06036
$863$	12.9200	0.439802	0.219901	−	0.975522i	$-0.429426\pi$
$863$	12.9200	0.439802	0.219901	+	0.975522i	$0.429426\pi$
$864$	−13.0498	−0.443962
$865$	−60.6905	−2.06354
$866$	17.5725	0.597136
$867$	47.0034	1.59632
$868$	10.3757	0.352174
$869$	−0.0128257	−0.000435081 0
$870$	62.2618	2.11087
$871$	23.4280	0.793829
$872$	3.05824	0.103565
$873$	−120.491	−4.07800
$874$	42.1741	1.42656
$875$	9.40736	0.318027
$876$	12.8566	0.434384
$877$	−35.3492	−1.19366	−0.596829	−	0.802368i	$-0.703573\pi$
$877$	−35.3492	−1.19366	−0.596829	+	0.802368i	$0.703573\pi$
$878$	−0.980355	−0.0330854
$879$	95.4208	3.21846
$880$	0.237664	0.00801163
$881$	25.7831	0.868654	0.434327	−	0.900755i	$-0.356986\pi$
$881$	25.7831	0.868654	0.434327	+	0.900755i	$0.356986\pi$
$882$	−9.80650	−0.330202
$883$	9.71635	0.326981	0.163491	−	0.986545i	$-0.447725\pi$
$883$	9.71635	0.326981	0.163491	+	0.986545i	$0.447725\pi$
$884$	−6.27339	−0.210997
$885$	−34.6116	−1.16346
$886$	−3.04011	−0.102134
$887$	−53.8270	−1.80733	−0.903666	−	0.428238i	$-0.859135\pi$
$887$	−53.8270	−1.80733	−0.903666	+	0.428238i	$0.859135\pi$
$888$	9.74729	0.327098
$889$	13.7217	0.460210
$890$	23.0265	0.771850
$891$	1.62965	0.0545954
$892$	3.96287	0.132687
$893$	−24.7793	−0.829208
$894$	−25.7913	−0.862589
$895$	58.1327	1.94316
$896$	−2.37061	−0.0791966
$897$	94.6837	3.16140
$898$	1.28937	0.0430268
$899$	29.1460	0.972073
$900$	25.9395	0.864649
$901$	14.3250	0.477235
$902$	0.703077	0.0234099
$903$	64.4935	2.14621
$904$	−10.9700	−0.364858
$905$	44.0138	1.46307
$906$	−5.24660	−0.174307
$907$	17.6808	0.587082	0.293541	−	0.955946i	$-0.405166\pi$
$907$	17.6808	0.587082	0.293541	+	0.955946i	$0.405166\pi$
$908$	−24.3236	−0.807206
$909$	−42.8994	−1.42288
$910$	−29.3985	−0.974550
$911$	16.0011	0.530141	0.265070	−	0.964229i	$-0.414605\pi$
$911$	16.0011	0.530141	0.265070	+	0.964229i	$0.414605\pi$
$912$	18.9780	0.628424
$913$	0.275423	0.00911516
$914$	1.91348	0.0632923
$915$	50.4385	1.66745
$916$	3.65978	0.120923
$917$	31.9114	1.05381
$918$	19.4164	0.640836
$919$	−41.4585	−1.36759	−0.683794	−	0.729675i	$-0.739671\pi$
$919$	−41.4585	−1.36759	−0.683794	+	0.729675i	$0.739671\pi$
$920$	−20.7776	−0.685017
$921$	−61.3382	−2.02116
$922$	−22.0841	−0.727300
$923$	−0.548597	−0.0180573
$924$	0.608930	0.0200323
$925$	−11.1944	−0.368068
$926$	−30.4642	−1.00111
$927$	−25.5121	−0.837926
$928$	−6.65920	−0.218599
$929$	10.0055	0.328270	0.164135	−	0.986438i	$-0.447517\pi$
$929$	10.0055	0.328270	0.164135	+	0.986438i	$0.447517\pi$
$930$	40.9220	1.34188
$931$	8.23983	0.270049
$932$	−10.8548	−0.355560
$933$	−5.78009	−0.189232
$934$	−19.4938	−0.637858
$935$	−0.353613	−0.0115644
$936$	29.9578	0.979202
$937$	−4.13317	−0.135025	−0.0675123	−	0.997718i	$-0.521506\pi$
$937$	−4.13317	−0.135025	−0.0675123	+	0.997718i	$0.521506\pi$
$938$	−13.1723	−0.430090
$939$	91.2867	2.97903
$940$	12.2078	0.398176
$941$	61.1781	1.99435	0.997175	−	0.0751127i	$-0.0239317\pi$
$941$	61.1781	1.99435	0.997175	+	0.0751127i	$0.0239317\pi$
$942$	−0.365944	−0.0119231
$943$	−61.4661	−2.00161
$944$	3.70188	0.120486
$945$	90.9895	2.95989
$946$	0.691541	0.0224839
$947$	3.97061	0.129028	0.0645138	−	0.997917i	$-0.479450\pi$
$947$	3.97061	0.129028	0.0645138	+	0.997917i	$0.479450\pi$
$948$	0.504565	0.0163875
$949$	−17.0526	−0.553550
$950$	−21.7954	−0.707137
$951$	80.4253	2.60797
$952$	3.52717	0.114316
$953$	13.3127	0.431240	0.215620	−	0.976477i	$-0.430823\pi$
$953$	13.3127	0.431240	0.215620	+	0.976477i	$0.430823\pi$
$954$	−68.4074	−2.21477
$955$	33.7419	1.09186
$956$	−23.3707	−0.755863
$957$	1.71052	0.0552933
$958$	3.48055	0.112451
$959$	−22.6626	−0.731815
$960$	−9.34974	−0.301762
$961$	−11.8436	−0.382052
$962$	−12.9285	−0.416832
$963$	−69.3029	−2.23326
$964$	−18.3300	−0.590369
$965$	62.2369	2.00348
$966$	−53.2353	−1.71282
$967$	52.0315	1.67322	0.836611	−	0.547798i	$-0.184534\pi$
$967$	52.0315	1.67322	0.836611	+	0.547798i	$0.184534\pi$
$968$	−10.9935	−0.353344
$969$	−28.2368	−0.907098
$970$	−49.8779	−1.60148
$971$	−11.7289	−0.376397	−0.188199	−	0.982131i	$-0.560265\pi$
$971$	−11.7289	−0.376397	−0.188199	+	0.982131i	$0.560265\pi$
$972$	−24.9617	−0.800646
$973$	−4.45607	−0.142855
$974$	42.1274	1.34985
$975$	−48.9322	−1.56708
$976$	−5.39464	−0.172678
$977$	−53.5207	−1.71228	−0.856140	−	0.516745i	$-0.827144\pi$
$977$	−53.5207	−1.71228	−0.856140	+	0.516745i	$0.827144\pi$
$978$	52.2184	1.66976
$979$	0.632608	0.0202182
$980$	−4.05945	−0.129675
$981$	21.7293	0.693764
$982$	16.5010	0.526567
$983$	8.03816	0.256378	0.128189	−	0.991750i	$-0.459084\pi$
$983$	8.03816	0.256378	0.128189	+	0.991750i	$0.459084\pi$
$984$	−27.6592	−0.881744
$985$	−30.8700	−0.983600
$986$	9.90804	0.315536
$987$	31.2783	0.995599
$988$	−25.1718	−0.800822
$989$	−60.4576	−1.92244
$990$	1.68864	0.0536685
$991$	47.3799	1.50507	0.752536	−	0.658551i	$-0.228830\pi$
$991$	47.3799	1.50507	0.752536	+	0.658551i	$0.228830\pi$
$992$	−4.37680	−0.138964
$993$	−26.1293	−0.829189
$994$	0.308445	0.00978329
$995$	−5.45971	−0.173085
$996$	−10.8352	−0.343326
$997$	−36.2349	−1.14757	−0.573785	−	0.819006i	$-0.694526\pi$
$997$	−36.2349	−1.14757	−0.573785	+	0.819006i	$0.694526\pi$
$998$	28.7145	0.908943
$999$	40.0143	1.26600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.c.1.2	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.c.1.2	✓	31	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.17886 q^{3} +1.00000 q^{4} +2.94122 q^{5} -3.17886 q^{6} -2.37061 q^{7} +1.00000 q^{8} +7.10517 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.17886 q^{3} +1.00000 q^{4} +2.94122 q^{5} -3.17886 q^{6} -2.37061 q^{7} +1.00000 q^{8} +7.10517 q^{9} +2.94122 q^{10} +0.0808043 q^{11} -3.17886 q^{12} +4.21634 q^{13} -2.37061 q^{14} -9.34974 q^{15} +1.00000 q^{16} -1.48787 q^{17} +7.10517 q^{18} -5.97005 q^{19} +2.94122 q^{20} +7.53585 q^{21} +0.0808043 q^{22} -7.06428 q^{23} -3.17886 q^{24} +3.65079 q^{25} +4.21634 q^{26} -13.0498 q^{27} -2.37061 q^{28} -6.65920 q^{29} -9.34974 q^{30} -4.37680 q^{31} +1.00000 q^{32} -0.256866 q^{33} -1.48787 q^{34} -6.97250 q^{35} +7.10517 q^{36} -3.06628 q^{37} -5.97005 q^{38} -13.4032 q^{39} +2.94122 q^{40} +8.70098 q^{41} +7.53585 q^{42} +8.55822 q^{43} +0.0808043 q^{44} +20.8979 q^{45} -7.06428 q^{46} +4.15060 q^{47} -3.17886 q^{48} -1.38019 q^{49} +3.65079 q^{50} +4.72974 q^{51} +4.21634 q^{52} -9.62784 q^{53} -13.0498 q^{54} +0.237664 q^{55} -2.37061 q^{56} +18.9780 q^{57} -6.65920 q^{58} +3.70188 q^{59} -9.34974 q^{60} -5.39464 q^{61} -4.37680 q^{62} -16.8436 q^{63} +1.00000 q^{64} +12.4012 q^{65} -0.256866 q^{66} +5.55648 q^{67} -1.48787 q^{68} +22.4564 q^{69} -6.97250 q^{70} -0.130112 q^{71} +7.10517 q^{72} -4.04440 q^{73} -3.06628 q^{74} -11.6054 q^{75} -5.97005 q^{76} -0.191556 q^{77} -13.4032 q^{78} -0.158725 q^{79} +2.94122 q^{80} +20.1679 q^{81} +8.70098 q^{82} +3.40851 q^{83} +7.53585 q^{84} -4.37617 q^{85} +8.55822 q^{86} +21.1687 q^{87} +0.0808043 q^{88} +7.82888 q^{89} +20.8979 q^{90} -9.99532 q^{91} -7.06428 q^{92} +13.9133 q^{93} +4.15060 q^{94} -17.5593 q^{95} -3.17886 q^{96} -16.9582 q^{97} -1.38019 q^{98} +0.574128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9}+O(q^{10}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9} - 13 q^{10} - 29 q^{11} - 14 q^{12} - 23 q^{13} - 29 q^{14} - 14 q^{15} + 31 q^{16} - 19 q^{17} + 27 q^{18} - 36 q^{19} - 13 q^{20} - 29 q^{22} - 24 q^{23} - 14 q^{24} + 4 q^{25} - 23 q^{26} - 65 q^{27} - 29 q^{28} + 18 q^{29} - 14 q^{30} - 41 q^{31} + 31 q^{32} - 18 q^{33} - 19 q^{34} - 28 q^{35} + 27 q^{36} - 31 q^{37} - 36 q^{38} - 31 q^{39} - 13 q^{40} - 51 q^{41} - 14 q^{43} - 29 q^{44} - 34 q^{45} - 24 q^{46} - 64 q^{47} - 14 q^{48} + 8 q^{49} + 4 q^{50} - 17 q^{51} - 23 q^{52} + 3 q^{53} - 65 q^{54} - 50 q^{55} - 29 q^{56} - 19 q^{57} + 18 q^{58} - 58 q^{59} - 14 q^{60} - 14 q^{61} - 41 q^{62} - 66 q^{63} + 31 q^{64} + 6 q^{65} - 18 q^{66} - 55 q^{67} - 19 q^{68} + q^{69} - 28 q^{70} - 42 q^{71} + 27 q^{72} - 83 q^{73} - 31 q^{74} - 49 q^{75} - 36 q^{76} + 18 q^{77} - 31 q^{78} - 56 q^{79} - 13 q^{80} + 3 q^{81} - 51 q^{82} - 43 q^{83} + 4 q^{85} - 14 q^{86} - 76 q^{87} - 29 q^{88} + 17 q^{89} - 34 q^{90} - 49 q^{91} - 24 q^{92} - 24 q^{93} - 64 q^{94} - 43 q^{95} - 14 q^{96} - 98 q^{97} + 8 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9 - 13 * q^10 - 29 * q^11 - 14 * q^12 - 23 * q^13 - 29 * q^14 - 14 * q^15 + 31 * q^16 - 19 * q^17 + 27 * q^18 - 36 * q^19 - 13 * q^20 - 29 * q^22 - 24 * q^23 - 14 * q^24 + 4 * q^25 - 23 * q^26 - 65 * q^27 - 29 * q^28 + 18 * q^29 - 14 * q^30 - 41 * q^31 + 31 * q^32 - 18 * q^33 - 19 * q^34 - 28 * q^35 + 27 * q^36 - 31 * q^37 - 36 * q^38 - 31 * q^39 - 13 * q^40 - 51 * q^41 - 14 * q^43 - 29 * q^44 - 34 * q^45 - 24 * q^46 - 64 * q^47 - 14 * q^48 + 8 * q^49 + 4 * q^50 - 17 * q^51 - 23 * q^52 + 3 * q^53 - 65 * q^54 - 50 * q^55 - 29 * q^56 - 19 * q^57 + 18 * q^58 - 58 * q^59 - 14 * q^60 - 14 * q^61 - 41 * q^62 - 66 * q^63 + 31 * q^64 + 6 * q^65 - 18 * q^66 - 55 * q^67 - 19 * q^68 + q^69 - 28 * q^70 - 42 * q^71 + 27 * q^72 - 83 * q^73 - 31 * q^74 - 49 * q^75 - 36 * q^76 + 18 * q^77 - 31 * q^78 - 56 * q^79 - 13 * q^80 + 3 * q^81 - 51 * q^82 - 43 * q^83 + 4 * q^85 - 14 * q^86 - 76 * q^87 - 29 * q^88 + 17 * q^89 - 34 * q^90 - 49 * q^91 - 24 * q^92 - 24 * q^93 - 64 * q^94 - 43 * q^95 - 14 * q^96 - 98 * q^97 + 8 * q^98 - q^99

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