LMFDB - Embedded newform 8013.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8013,2,Mod(1,8013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8013 = 3 \cdot 2671 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8013.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:	$63.9841271397$
Analytic rank:	$0$
Dimension:	$106$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35891 q^{2} -1.00000 q^{3} +3.56446 q^{4} -3.22536 q^{5} +2.35891 q^{6} -2.62511 q^{7} -3.69041 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.35891 q^{2} -1.00000 q^{3} +3.56446 q^{4} -3.22536 q^{5} +2.35891 q^{6} -2.62511 q^{7} -3.69041 q^{8} +1.00000 q^{9} +7.60832 q^{10} -0.0148698 q^{11} -3.56446 q^{12} -1.39949 q^{13} +6.19239 q^{14} +3.22536 q^{15} +1.57644 q^{16} +5.76402 q^{17} -2.35891 q^{18} -0.0520376 q^{19} -11.4966 q^{20} +2.62511 q^{21} +0.0350765 q^{22} -0.276238 q^{23} +3.69041 q^{24} +5.40292 q^{25} +3.30127 q^{26} -1.00000 q^{27} -9.35708 q^{28} -2.65781 q^{29} -7.60832 q^{30} -8.40824 q^{31} +3.66215 q^{32} +0.0148698 q^{33} -13.5968 q^{34} +8.46690 q^{35} +3.56446 q^{36} +1.93189 q^{37} +0.122752 q^{38} +1.39949 q^{39} +11.9029 q^{40} +7.85726 q^{41} -6.19239 q^{42} +5.09649 q^{43} -0.0530027 q^{44} -3.22536 q^{45} +0.651621 q^{46} -4.89010 q^{47} -1.57644 q^{48} -0.108818 q^{49} -12.7450 q^{50} -5.76402 q^{51} -4.98843 q^{52} -3.27263 q^{53} +2.35891 q^{54} +0.0479603 q^{55} +9.68772 q^{56} +0.0520376 q^{57} +6.26954 q^{58} +11.7482 q^{59} +11.4966 q^{60} +2.23208 q^{61} +19.8343 q^{62} -2.62511 q^{63} -11.7916 q^{64} +4.51386 q^{65} -0.0350765 q^{66} +10.4804 q^{67} +20.5456 q^{68} +0.276238 q^{69} -19.9727 q^{70} +2.18180 q^{71} -3.69041 q^{72} -11.6348 q^{73} -4.55716 q^{74} -5.40292 q^{75} -0.185486 q^{76} +0.0390348 q^{77} -3.30127 q^{78} -4.61768 q^{79} -5.08457 q^{80} +1.00000 q^{81} -18.5346 q^{82} +5.61334 q^{83} +9.35708 q^{84} -18.5910 q^{85} -12.0222 q^{86} +2.65781 q^{87} +0.0548756 q^{88} +2.48143 q^{89} +7.60832 q^{90} +3.67381 q^{91} -0.984638 q^{92} +8.40824 q^{93} +11.5353 q^{94} +0.167840 q^{95} -3.66215 q^{96} +2.44363 q^{97} +0.256691 q^{98} -0.0148698 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9}+O(q^{10}) $ 106 * q + 15 * q^2 - 106 * q^3 + 109 * q^4 + 16 * q^5 - 15 * q^6 + 35 * q^7 + 48 * q^8 + 106 * q^9	\( 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9} - 3 q^{10} + 55 q^{11} - 109 q^{12} - 8 q^{13} + 27 q^{14} - 16 q^{15} + 111 q^{16} + 28 q^{17} + 15 q^{18} + q^{19} + 54 q^{20} - 35 q^{21} + 20 q^{22} + 62 q^{23} - 48 q^{24} + 102 q^{25} + 21 q^{26} - 106 q^{27} + 79 q^{28} + 36 q^{29} + 3 q^{30} + q^{31} + 111 q^{32} - 55 q^{33} - 27 q^{34} + 72 q^{35} + 109 q^{36} + 31 q^{37} + 43 q^{38} + 8 q^{39} - 13 q^{40} + 35 q^{41} - 27 q^{42} + 98 q^{43} + 121 q^{44} + 16 q^{45} + 8 q^{46} + 75 q^{47} - 111 q^{48} + 49 q^{49} + 83 q^{50} - 28 q^{51} - 18 q^{52} + 60 q^{53} - 15 q^{54} + 14 q^{55} + 85 q^{56} - q^{57} + 65 q^{58} + 77 q^{59} - 54 q^{60} - 55 q^{61} + 83 q^{62} + 35 q^{63} + 122 q^{64} + 86 q^{65} - 20 q^{66} + 121 q^{67} + 80 q^{68} - 62 q^{69} - 11 q^{70} + 79 q^{71} + 48 q^{72} - 29 q^{73} + 91 q^{74} - 102 q^{75} - 10 q^{76} + 87 q^{77} - 21 q^{78} + 15 q^{79} + 108 q^{80} + 106 q^{81} + 21 q^{82} + 196 q^{83} - 79 q^{84} - 5 q^{85} + 65 q^{86} - 36 q^{87} + 84 q^{88} + 34 q^{89} - 3 q^{90} + 17 q^{91} + 162 q^{92} - q^{93} - 35 q^{94} + 113 q^{95} - 111 q^{96} - 63 q^{97} + 112 q^{98} + 55 q^{99}+O(q^{100}) \) 106 * q + 15 * q^2 - 106 * q^3 + 109 * q^4 + 16 * q^5 - 15 * q^6 + 35 * q^7 + 48 * q^8 + 106 * q^9 - 3 * q^10 + 55 * q^11 - 109 * q^12 - 8 * q^13 + 27 * q^14 - 16 * q^15 + 111 * q^16 + 28 * q^17 + 15 * q^18 + q^19 + 54 * q^20 - 35 * q^21 + 20 * q^22 + 62 * q^23 - 48 * q^24 + 102 * q^25 + 21 * q^26 - 106 * q^27 + 79 * q^28 + 36 * q^29 + 3 * q^30 + q^31 + 111 * q^32 - 55 * q^33 - 27 * q^34 + 72 * q^35 + 109 * q^36 + 31 * q^37 + 43 * q^38 + 8 * q^39 - 13 * q^40 + 35 * q^41 - 27 * q^42 + 98 * q^43 + 121 * q^44 + 16 * q^45 + 8 * q^46 + 75 * q^47 - 111 * q^48 + 49 * q^49 + 83 * q^50 - 28 * q^51 - 18 * q^52 + 60 * q^53 - 15 * q^54 + 14 * q^55 + 85 * q^56 - q^57 + 65 * q^58 + 77 * q^59 - 54 * q^60 - 55 * q^61 + 83 * q^62 + 35 * q^63 + 122 * q^64 + 86 * q^65 - 20 * q^66 + 121 * q^67 + 80 * q^68 - 62 * q^69 - 11 * q^70 + 79 * q^71 + 48 * q^72 - 29 * q^73 + 91 * q^74 - 102 * q^75 - 10 * q^76 + 87 * q^77 - 21 * q^78 + 15 * q^79 + 108 * q^80 + 106 * q^81 + 21 * q^82 + 196 * q^83 - 79 * q^84 - 5 * q^85 + 65 * q^86 - 36 * q^87 + 84 * q^88 + 34 * q^89 - 3 * q^90 + 17 * q^91 + 162 * q^92 - q^93 - 35 * q^94 + 113 * q^95 - 111 * q^96 - 63 * q^97 + 112 * q^98 + 55 * 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 $

$2$	−2.35891	−1.66800	−0.834001	−	0.551763i	$-0.813955\pi$
$2$	−2.35891	−1.66800	−0.834001	+	0.551763i	$0.813955\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.56446	1.78223
$5$	−3.22536	−1.44242	−0.721211	−	0.692715i	$-0.756414\pi$
$5$	−3.22536	−1.44242	−0.721211	+	0.692715i	$0.756414\pi$
$6$	2.35891	0.963021
$7$	−2.62511	−0.992197	−0.496098	−	0.868266i	$-0.665234\pi$
$7$	−2.62511	−0.992197	−0.496098	+	0.868266i	$0.665234\pi$
$8$	−3.69041	−1.30476
$9$	1.00000	0.333333
$10$	7.60832	2.40596
$11$	−0.0148698	−0.00448341	−0.00224170	−	0.999997i	$-0.500714\pi$
$11$	−0.0148698	−0.00448341	−0.00224170	+	0.999997i	$0.500714\pi$
$12$	−3.56446	−1.02897
$13$	−1.39949	−0.388149	−0.194075	−	0.980987i	$-0.562170\pi$
$13$	−1.39949	−0.388149	−0.194075	+	0.980987i	$0.562170\pi$
$14$	6.19239	1.65499
$15$	3.22536	0.832783
$16$	1.57644	0.394109
$17$	5.76402	1.39798	0.698990	−	0.715131i	$-0.253633\pi$
$17$	5.76402	1.39798	0.698990	+	0.715131i	$0.253633\pi$
$18$	−2.35891	−0.556000
$19$	−0.0520376	−0.0119382	−0.00596912	−	0.999982i	$-0.501900\pi$
$19$	−0.0520376	−0.0119382	−0.00596912	+	0.999982i	$0.501900\pi$
$20$	−11.4966	−2.57073
$21$	2.62511	0.572845
$22$	0.0350765	0.00747833
$23$	−0.276238	−0.0575996	−0.0287998	−	0.999585i	$-0.509169\pi$
$23$	−0.276238	−0.0575996	−0.0287998	+	0.999585i	$0.509169\pi$
$24$	3.69041	0.753302
$25$	5.40292	1.08058
$26$	3.30127	0.647433
$27$	−1.00000	−0.192450
$28$	−9.35708	−1.76832
$29$	−2.65781	−0.493543	−0.246772	−	0.969074i	$-0.579370\pi$
$29$	−2.65781	−0.493543	−0.246772	+	0.969074i	$0.579370\pi$
$30$	−7.60832	−1.38908
$31$	−8.40824	−1.51017	−0.755083	−	0.655630i	$-0.772403\pi$
$31$	−8.40824	−1.51017	−0.755083	+	0.655630i	$0.772403\pi$
$32$	3.66215	0.647383
$33$	0.0148698	0.00258850
$34$	−13.5968	−2.33183
$35$	8.46690	1.43117
$36$	3.56446	0.594076
$37$	1.93189	0.317601	0.158801	−	0.987311i	$-0.449237\pi$
$37$	1.93189	0.317601	0.158801	+	0.987311i	$0.449237\pi$
$38$	0.122752	0.0199130
$39$	1.39949	0.224098
$40$	11.9029	1.88201
$41$	7.85726	1.22710	0.613549	−	0.789657i	$-0.289741\pi$
$41$	7.85726	1.22710	0.613549	+	0.789657i	$0.289741\pi$
$42$	−6.19239	−0.955506
$43$	5.09649	0.777207	0.388604	−	0.921405i	$-0.372958\pi$
$43$	5.09649	0.777207	0.388604	+	0.921405i	$0.372958\pi$
$44$	−0.0530027	−0.00799046
$45$	−3.22536	−0.480808
$46$	0.651621	0.0960762
$47$	−4.89010	−0.713295	−0.356647	−	0.934239i	$-0.616080\pi$
$47$	−4.89010	−0.713295	−0.356647	+	0.934239i	$0.616080\pi$
$48$	−1.57644	−0.227539
$49$	−0.108818	−0.0155454
$50$	−12.7450	−1.80241
$51$	−5.76402	−0.807125
$52$	−4.98843	−0.691770
$53$	−3.27263	−0.449530	−0.224765	−	0.974413i	$-0.572161\pi$
$53$	−3.27263	−0.449530	−0.224765	+	0.974413i	$0.572161\pi$
$54$	2.35891	0.321007
$55$	0.0479603	0.00646697
$56$	9.68772	1.29458
$57$	0.0520376	0.00689255
$58$	6.26954	0.823231
$59$	11.7482	1.52948	0.764742	−	0.644337i	$-0.222866\pi$
$59$	11.7482	1.52948	0.764742	+	0.644337i	$0.222866\pi$
$60$	11.4966	1.48421
$61$	2.23208	0.285789	0.142895	−	0.989738i	$-0.454359\pi$
$61$	2.23208	0.285789	0.142895	+	0.989738i	$0.454359\pi$
$62$	19.8343	2.51896
$63$	−2.62511	−0.330732
$64$	−11.7916	−1.47394
$65$	4.51386	0.559875
$66$	−0.0350765	−0.00431762
$67$	10.4804	1.28039	0.640194	−	0.768214i	$-0.278854\pi$
$67$	10.4804	1.28039	0.640194	+	0.768214i	$0.278854\pi$
$68$	20.5456	2.49152
$69$	0.276238	0.0332551
$70$	−19.9727	−2.38719
$71$	2.18180	0.258932	0.129466	−	0.991584i	$-0.458674\pi$
$71$	2.18180	0.258932	0.129466	+	0.991584i	$0.458674\pi$
$72$	−3.69041	−0.434919
$73$	−11.6348	−1.36175	−0.680877	−	0.732398i	$-0.738401\pi$
$73$	−11.6348	−1.36175	−0.680877	+	0.732398i	$0.738401\pi$
$74$	−4.55716	−0.529759
$75$	−5.40292	−0.623875
$76$	−0.185486	−0.0212767
$77$	0.0390348	0.00444842
$78$	−3.30127	−0.373796
$79$	−4.61768	−0.519530	−0.259765	−	0.965672i	$-0.583645\pi$
$79$	−4.61768	−0.519530	−0.259765	+	0.965672i	$0.583645\pi$
$80$	−5.08457	−0.568472
$81$	1.00000	0.111111
$82$	−18.5346	−2.04680
$83$	5.61334	0.616145	0.308072	−	0.951363i	$-0.400316\pi$
$83$	5.61334	0.616145	0.308072	+	0.951363i	$0.400316\pi$
$84$	9.35708	1.02094
$85$	−18.5910	−2.01648
$86$	−12.0222	−1.29638
$87$	2.65781	0.284947
$88$	0.0548756	0.00584976
$89$	2.48143	0.263031	0.131515	−	0.991314i	$-0.458016\pi$
$89$	2.48143	0.263031	0.131515	+	0.991314i	$0.458016\pi$
$90$	7.60832	0.801988
$91$	3.67381	0.385120
$92$	−0.984638	−0.102656
$93$	8.40824	0.871894
$94$	11.5353	1.18978
$95$	0.167840	0.0172200
$96$	−3.66215	−0.373767
$97$	2.44363	0.248113	0.124057	−	0.992275i	$-0.460410\pi$
$97$	2.44363	0.248113	0.124057	+	0.992275i	$0.460410\pi$
$98$	0.256691	0.0259297
$99$	−0.0148698	−0.00149447
$100$	19.2585	1.92585
$101$	−13.8126	−1.37441	−0.687205	−	0.726464i	$-0.741162\pi$
$101$	−13.8126	−1.37441	−0.687205	+	0.726464i	$0.741162\pi$
$102$	13.5968	1.34628
$103$	−15.7283	−1.54975	−0.774877	−	0.632112i	$-0.782188\pi$
$103$	−15.7283	−1.54975	−0.774877	+	0.632112i	$0.782188\pi$
$104$	5.16470	0.506440
$105$	−8.46690	−0.826285
$106$	7.71984	0.749817
$107$	−1.99078	−0.192456	−0.0962278	−	0.995359i	$-0.530678\pi$
$107$	−1.99078	−0.192456	−0.0962278	+	0.995359i	$0.530678\pi$
$108$	−3.56446	−0.342990
$109$	4.82255	0.461917	0.230958	−	0.972964i	$-0.425814\pi$
$109$	4.82255	0.461917	0.230958	+	0.972964i	$0.425814\pi$
$110$	−0.113134	−0.0107869
$111$	−1.93189	−0.183367
$112$	−4.13831	−0.391034
$113$	17.1025	1.60887	0.804434	−	0.594042i	$-0.202469\pi$
$113$	17.1025	1.60887	0.804434	+	0.594042i	$0.202469\pi$
$114$	−0.122752	−0.0114968
$115$	0.890966	0.0830830
$116$	−9.47365	−0.879606
$117$	−1.39949	−0.129383
$118$	−27.7129	−2.55118
$119$	−15.1312	−1.38707
$120$	−11.9029	−1.08658
$121$	−10.9998	−0.999980
$122$	−5.26529	−0.476697
$123$	−7.85726	−0.708465
$124$	−29.9708	−2.69146
$125$	−1.29955	−0.116236
$126$	6.19239	0.551662
$127$	7.73226	0.686128	0.343064	−	0.939312i	$-0.388535\pi$
$127$	7.73226	0.686128	0.343064	+	0.939312i	$0.388535\pi$
$128$	20.4909	1.81116
$129$	−5.09649	−0.448721
$130$	−10.6478	−0.933872
$131$	−15.6925	−1.37106	−0.685529	−	0.728045i	$-0.740429\pi$
$131$	−15.6925	−1.37106	−0.685529	+	0.728045i	$0.740429\pi$
$132$	0.0530027	0.00461329
$133$	0.136604	0.0118451
$134$	−24.7224	−2.13569
$135$	3.22536	0.277594
$136$	−21.2716	−1.82403
$137$	−12.4310	−1.06205	−0.531026	−	0.847355i	$-0.678194\pi$
$137$	−12.4310	−1.06205	−0.531026	+	0.847355i	$0.678194\pi$
$138$	−0.651621	−0.0554696
$139$	−16.0584	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$139$	−16.0584	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$140$	30.1799	2.55067
$141$	4.89010	0.411821
$142$	−5.14667	−0.431899
$143$	0.0208101	0.00174023
$144$	1.57644	0.131370
$145$	8.57238	0.711898
$146$	27.4455	2.27141
$147$	0.108818	0.00897512
$148$	6.88615	0.566038
$149$	−3.53636	−0.289710	−0.144855	−	0.989453i	$-0.546272\pi$
$149$	−3.53636	−0.289710	−0.144855	+	0.989453i	$0.546272\pi$
$150$	12.7450	1.04062
$151$	−8.65328	−0.704194	−0.352097	−	0.935964i	$-0.614531\pi$
$151$	−8.65328	−0.704194	−0.352097	+	0.935964i	$0.614531\pi$
$152$	0.192040	0.0155765
$153$	5.76402	0.465994
$154$	−0.0920795	−0.00741998
$155$	27.1196	2.17830
$156$	4.98843	0.399394
$157$	−13.0204	−1.03914	−0.519572	−	0.854427i	$-0.673909\pi$
$157$	−13.0204	−1.03914	−0.519572	+	0.854427i	$0.673909\pi$
$158$	10.8927	0.866576
$159$	3.27263	0.259536
$160$	−11.8117	−0.933800
$161$	0.725154	0.0571501
$162$	−2.35891	−0.185333
$163$	−6.64074	−0.520143	−0.260071	−	0.965589i	$-0.583746\pi$
$163$	−6.64074	−0.520143	−0.260071	+	0.965589i	$0.583746\pi$
$164$	28.0069	2.18697
$165$	−0.0479603	−0.00373371
$166$	−13.2414	−1.02773
$167$	24.2637	1.87758	0.938791	−	0.344487i	$-0.111947\pi$
$167$	24.2637	1.87758	0.938791	+	0.344487i	$0.111947\pi$
$168$	−9.68772	−0.747424
$169$	−11.0414	−0.849340
$170$	43.8545	3.36349
$171$	−0.0520376	−0.00397942
$172$	18.1662	1.38516
$173$	−8.14585	−0.619317	−0.309659	−	0.950848i	$-0.600215\pi$
$173$	−8.14585	−0.619317	−0.309659	+	0.950848i	$0.600215\pi$
$174$	−6.26954	−0.475292
$175$	−14.1832	−1.07215
$176$	−0.0234413	−0.00176695
$177$	−11.7482	−0.883048
$178$	−5.85346	−0.438735
$179$	−1.79724	−0.134332	−0.0671659	−	0.997742i	$-0.521396\pi$
$179$	−1.79724	−0.134332	−0.0671659	+	0.997742i	$0.521396\pi$
$180$	−11.4966	−0.856909
$181$	−12.5435	−0.932350	−0.466175	−	0.884693i	$-0.654368\pi$
$181$	−12.5435	−0.932350	−0.466175	+	0.884693i	$0.654368\pi$
$182$	−8.66620	−0.642381
$183$	−2.23208	−0.165000
$184$	1.01943	0.0751535
$185$	−6.23104	−0.458115
$186$	−19.8343	−1.45432
$187$	−0.0857098	−0.00626772
$188$	−17.4306	−1.27125
$189$	2.62511	0.190948
$190$	−0.395919	−0.0287230
$191$	−23.1689	−1.67644	−0.838220	−	0.545332i	$-0.816403\pi$
$191$	−23.1689	−1.67644	−0.838220	+	0.545332i	$0.816403\pi$
$192$	11.7916	0.850983
$193$	−19.4382	−1.39919	−0.699595	−	0.714539i	$-0.746636\pi$
$193$	−19.4382	−1.39919	−0.699595	+	0.714539i	$0.746636\pi$
$194$	−5.76431	−0.413854
$195$	−4.51386	−0.323244
$196$	−0.387875	−0.0277054
$197$	−14.9454	−1.06482	−0.532409	−	0.846487i	$-0.678713\pi$
$197$	−14.9454	−1.06482	−0.532409	+	0.846487i	$0.678713\pi$
$198$	0.0350765	0.00249278
$199$	19.2275	1.36300	0.681500	−	0.731818i	$-0.261328\pi$
$199$	19.2275	1.36300	0.681500	+	0.731818i	$0.261328\pi$
$200$	−19.9390	−1.40990
$201$	−10.4804	−0.739232
$202$	32.5828	2.29252
$203$	6.97704	0.489692
$204$	−20.5456	−1.43848
$205$	−25.3425	−1.76999
$206$	37.1016	2.58499
$207$	−0.276238	−0.0191999
$208$	−2.20621	−0.152973
$209$	0.000773788 0	5.35240e−5 0
$210$	19.9727	1.37824
$211$	26.0962	1.79654	0.898270	−	0.439445i	$-0.144825\pi$
$211$	26.0962	1.79654	0.898270	+	0.439445i	$0.144825\pi$
$212$	−11.6651	−0.801165
$213$	−2.18180	−0.149494
$214$	4.69606	0.321016
$215$	−16.4380	−1.12106
$216$	3.69041	0.251101
$217$	22.0725	1.49838
$218$	−11.3760	−0.770478
$219$	11.6348	0.786209
$220$	0.170952	0.0115256
$221$	−8.06670	−0.542625
$222$	4.55716	0.305857
$223$	8.74117	0.585352	0.292676	−	0.956212i	$-0.405454\pi$
$223$	8.74117	0.585352	0.292676	+	0.956212i	$0.405454\pi$
$224$	−9.61354	−0.642332
$225$	5.40292	0.360195
$226$	−40.3433	−2.68359
$227$	13.6654	0.907008	0.453504	−	0.891254i	$-0.350174\pi$
$227$	13.6654	0.907008	0.453504	+	0.891254i	$0.350174\pi$
$228$	0.185486	0.0122841
$229$	−2.56541	−0.169527	−0.0847635	−	0.996401i	$-0.527013\pi$
$229$	−2.56541	−0.169527	−0.0847635	+	0.996401i	$0.527013\pi$
$230$	−2.10171	−0.138583
$231$	−0.0390348	−0.00256830
$232$	9.80842	0.643954
$233$	7.72098	0.505818	0.252909	−	0.967490i	$-0.418613\pi$
$233$	7.72098	0.505818	0.252909	+	0.967490i	$0.418613\pi$
$234$	3.30127	0.215811
$235$	15.7723	1.02887
$236$	41.8759	2.72589
$237$	4.61768	0.299951
$238$	35.6931	2.31364
$239$	22.3975	1.44877	0.724386	−	0.689394i	$-0.242123\pi$
$239$	22.3975	1.44877	0.724386	+	0.689394i	$0.242123\pi$
$240$	5.08457	0.328207
$241$	−13.8147	−0.889886	−0.444943	−	0.895559i	$-0.646776\pi$
$241$	−13.8147	−0.889886	−0.444943	+	0.895559i	$0.646776\pi$
$242$	25.9475	1.66797
$243$	−1.00000	−0.0641500
$244$	7.95617	0.509341
$245$	0.350975	0.0224230
$246$	18.5346	1.18172
$247$	0.0728262	0.00463382
$248$	31.0299	1.97040
$249$	−5.61334	−0.355731
$250$	3.06553	0.193881
$251$	−15.6635	−0.988671	−0.494336	−	0.869271i	$-0.664589\pi$
$251$	−15.6635	−0.988671	−0.494336	+	0.869271i	$0.664589\pi$
$252$	−9.35708	−0.589440
$253$	0.00410760	0.000258242 0
$254$	−18.2397	−1.14446
$255$	18.5910	1.16422
$256$	−24.7531	−1.54707
$257$	28.2674	1.76327	0.881637	−	0.471927i	$-0.156442\pi$
$257$	28.2674	1.76327	0.881637	+	0.471927i	$0.156442\pi$
$258$	12.0222	0.748467
$259$	−5.07143	−0.315123
$260$	16.0894	0.997825
$261$	−2.65781	−0.164514
$262$	37.0171	2.28693
$263$	−14.8573	−0.916141	−0.458070	−	0.888916i	$-0.651459\pi$
$263$	−14.8573	−0.916141	−0.458070	+	0.888916i	$0.651459\pi$
$264$	−0.0548756	−0.00337736
$265$	10.5554	0.648412
$266$	−0.322237	−0.0197576
$267$	−2.48143	−0.151861
$268$	37.3570	2.28194
$269$	−23.8883	−1.45649	−0.728246	−	0.685315i	$-0.759664\pi$
$269$	−23.8883	−1.45649	−0.728246	+	0.685315i	$0.759664\pi$
$270$	−7.60832	−0.463028
$271$	−18.4761	−1.12234	−0.561172	−	0.827699i	$-0.689649\pi$
$271$	−18.4761	−1.12234	−0.561172	+	0.827699i	$0.689649\pi$
$272$	9.08661	0.550957
$273$	−3.67381	−0.222349
$274$	29.3236	1.77151
$275$	−0.0803402	−0.00484470
$276$	0.984638	0.0592683
$277$	−16.9207	−1.01666	−0.508332	−	0.861161i	$-0.669738\pi$
$277$	−16.9207	−1.01666	−0.508332	+	0.861161i	$0.669738\pi$
$278$	37.8803	2.27191
$279$	−8.40824	−0.503388
$280$	−31.2463	−1.86733
$281$	−5.33830	−0.318456	−0.159228	−	0.987242i	$-0.550900\pi$
$281$	−5.33830	−0.318456	−0.159228	+	0.987242i	$0.550900\pi$
$282$	−11.5353	−0.686918
$283$	−16.4446	−0.977527	−0.488764	−	0.872416i	$-0.662552\pi$
$283$	−16.4446	−0.977527	−0.488764	+	0.872416i	$0.662552\pi$
$284$	7.77693	0.461476
$285$	−0.167840	−0.00994197
$286$	−0.0490892	−0.00290271
$287$	−20.6261	−1.21752
$288$	3.66215	0.215794
$289$	16.2240	0.954351
$290$	−20.2215	−1.18745
$291$	−2.44363	−0.143248
$292$	−41.4719	−2.42696
$293$	5.48938	0.320693	0.160347	−	0.987061i	$-0.448739\pi$
$293$	5.48938	0.320693	0.160347	+	0.987061i	$0.448739\pi$
$294$	−0.256691	−0.0149705
$295$	−37.8921	−2.20616
$296$	−7.12948	−0.414393
$297$	0.0148698	0.000862832 0
$298$	8.34195	0.483236
$299$	0.386593	0.0223572
$300$	−19.2585	−1.11189
$301$	−13.3788	−0.771142
$302$	20.4123	1.17460
$303$	13.8126	0.793516
$304$	−0.0820340	−0.00470497
$305$	−7.19927	−0.412229
$306$	−13.5968	−0.777278
$307$	−6.23178	−0.355666	−0.177833	−	0.984061i	$-0.556909\pi$
$307$	−6.23178	−0.355666	−0.177833	+	0.984061i	$0.556909\pi$
$308$	0.139138	0.00792810
$309$	15.7283	0.894751
$310$	−63.9726	−3.63340
$311$	27.6942	1.57039	0.785196	−	0.619247i	$-0.212562\pi$
$311$	27.6942	1.57039	0.785196	+	0.619247i	$0.212562\pi$
$312$	−5.16470	−0.292394
$313$	−8.19762	−0.463357	−0.231679	−	0.972792i	$-0.574422\pi$
$313$	−8.19762	−0.463357	−0.231679	+	0.972792i	$0.574422\pi$
$314$	30.7140	1.73329
$315$	8.46690	0.477056
$316$	−16.4595	−0.925920
$317$	22.4479	1.26080	0.630400	−	0.776270i	$-0.282891\pi$
$317$	22.4479	1.26080	0.630400	+	0.776270i	$0.282891\pi$
$318$	−7.71984	−0.432907
$319$	0.0395211	0.00221275
$320$	38.0320	2.12605
$321$	1.99078	0.111114
$322$	−1.71057	−0.0953265
$323$	−0.299946	−0.0166894
$324$	3.56446	0.198025
$325$	−7.56134	−0.419428
$326$	15.6649	0.867599
$327$	−4.82255	−0.266688
$328$	−28.9965	−1.60106
$329$	12.8370	0.707729
$330$	0.113134	0.00622783
$331$	6.75301	0.371179	0.185589	−	0.982627i	$-0.440581\pi$
$331$	6.75301	0.371179	0.185589	+	0.982627i	$0.440581\pi$
$332$	20.0085	1.09811
$333$	1.93189	0.105867
$334$	−57.2359	−3.13181
$335$	−33.8031	−1.84686
$336$	4.13831	0.225763
$337$	25.5838	1.39364	0.696820	−	0.717246i	$-0.254598\pi$
$337$	25.5838	1.39364	0.696820	+	0.717246i	$0.254598\pi$
$338$	26.0457	1.41670
$339$	−17.1025	−0.928881
$340$	−66.2669	−3.59383
$341$	0.125029	0.00677069
$342$	0.122752	0.00663767
$343$	18.6614	1.00762
$344$	−18.8081	−1.01407
$345$	−0.890966	−0.0479680
$346$	19.2153	1.03302
$347$	−26.6338	−1.42978	−0.714888	−	0.699239i	$-0.753522\pi$
$347$	−26.6338	−1.42978	−0.714888	+	0.699239i	$0.753522\pi$
$348$	9.47365	0.507841
$349$	−2.33216	−0.124838	−0.0624190	−	0.998050i	$-0.519881\pi$
$349$	−2.33216	−0.124838	−0.0624190	+	0.998050i	$0.519881\pi$
$350$	33.4570	1.78835
$351$	1.39949	0.0746993
$352$	−0.0544554	−0.00290248
$353$	−16.7528	−0.891661	−0.445830	−	0.895117i	$-0.647092\pi$
$353$	−16.7528	−0.891661	−0.445830	+	0.895117i	$0.647092\pi$
$354$	27.7129	1.47293
$355$	−7.03708	−0.373489
$356$	8.84493	0.468781
$357$	15.1312	0.800827
$358$	4.23952	0.224066
$359$	−11.6383	−0.614246	−0.307123	−	0.951670i	$-0.599366\pi$
$359$	−11.6383	−0.614246	−0.307123	+	0.951670i	$0.599366\pi$
$360$	11.9029	0.627337
$361$	−18.9973	−0.999857
$362$	29.5889	1.55516
$363$	10.9998	0.577339
$364$	13.0951	0.686372
$365$	37.5265	1.96423
$366$	5.26529	0.275221
$367$	−6.71971	−0.350766	−0.175383	−	0.984500i	$-0.556116\pi$
$367$	−6.71971	−0.350766	−0.175383	+	0.984500i	$0.556116\pi$
$368$	−0.435472	−0.0227005
$369$	7.85726	0.409033
$370$	14.6985	0.764137
$371$	8.59100	0.446022
$372$	29.9708	1.55391
$373$	27.9135	1.44531	0.722653	−	0.691211i	$-0.242922\pi$
$373$	27.9135	1.44531	0.722653	+	0.691211i	$0.242922\pi$
$374$	0.202182	0.0104546
$375$	1.29955	0.0671086
$376$	18.0465	0.930677
$377$	3.71958	0.191568
$378$	−6.19239	−0.318502
$379$	30.8134	1.58278	0.791390	−	0.611312i	$-0.209358\pi$
$379$	30.8134	1.58278	0.791390	+	0.611312i	$0.209358\pi$
$380$	0.598258	0.0306900
$381$	−7.73226	−0.396136
$382$	54.6533	2.79630
$383$	19.6893	1.00608	0.503039	−	0.864264i	$-0.332215\pi$
$383$	19.6893	1.00608	0.503039	+	0.864264i	$0.332215\pi$
$384$	−20.4909	−1.04567
$385$	−0.125901	−0.00641651
$386$	45.8529	2.33385
$387$	5.09649	0.259069
$388$	8.71023	0.442195
$389$	−1.30817	−0.0663270	−0.0331635	−	0.999450i	$-0.510558\pi$
$389$	−1.30817	−0.0663270	−0.0331635	+	0.999450i	$0.510558\pi$
$390$	10.6478	0.539171
$391$	−1.59224	−0.0805231
$392$	0.401581	0.0202829
$393$	15.6925	0.791581
$394$	35.2549	1.77612
$395$	14.8937	0.749381
$396$	−0.0530027	−0.00266349
$397$	−15.4837	−0.777107	−0.388553	−	0.921426i	$-0.627025\pi$
$397$	−15.4837	−0.777107	−0.388553	+	0.921426i	$0.627025\pi$
$398$	−45.3559	−2.27349
$399$	−0.136604	−0.00683877
$400$	8.51736	0.425868
$401$	−36.2794	−1.81170	−0.905852	−	0.423594i	$-0.860768\pi$
$401$	−36.2794	−1.81170	−0.905852	+	0.423594i	$0.860768\pi$
$402$	24.7224	1.23304
$403$	11.7673	0.586169
$404$	−49.2346	−2.44951
$405$	−3.22536	−0.160269
$406$	−16.4582	−0.816807
$407$	−0.0287268	−0.00142394
$408$	21.2716	1.05310
$409$	−23.0021	−1.13738	−0.568689	−	0.822552i	$-0.692549\pi$
$409$	−23.0021	−1.13738	−0.568689	+	0.822552i	$0.692549\pi$
$410$	59.7806	2.95235
$411$	12.4310	0.613176
$412$	−56.0628	−2.76202
$413$	−30.8402	−1.51755
$414$	0.651621	0.0320254
$415$	−18.1050	−0.888741
$416$	−5.12515	−0.251281
$417$	16.0584	0.786384
$418$	−0.00182530	−8.92781e−5 0
$419$	23.9656	1.17080	0.585398	−	0.810746i	$-0.300938\pi$
$419$	23.9656	1.17080	0.585398	+	0.810746i	$0.300938\pi$
$420$	−30.1799	−1.47263
$421$	−10.6253	−0.517845	−0.258922	−	0.965898i	$-0.583367\pi$
$421$	−10.6253	−0.517845	−0.258922	+	0.965898i	$0.583367\pi$
$422$	−61.5587	−2.99663
$423$	−4.89010	−0.237765
$424$	12.0773	0.586528
$425$	31.1425	1.51064
$426$	5.14667	0.249357
$427$	−5.85946	−0.283559
$428$	−7.09604	−0.343000
$429$	−0.0208101	−0.00100472
$430$	38.7757	1.86993
$431$	33.2104	1.59969	0.799845	−	0.600207i	$-0.204915\pi$
$431$	33.2104	1.59969	0.799845	+	0.600207i	$0.204915\pi$
$432$	−1.57644	−0.0758463
$433$	25.6670	1.23348	0.616738	−	0.787169i	$-0.288454\pi$
$433$	25.6670	1.23348	0.616738	+	0.787169i	$0.288454\pi$
$434$	−52.0671	−2.49930
$435$	−8.57238	−0.411014
$436$	17.1898	0.823241
$437$	0.0143748	0.000687638 0
$438$	−27.4455	−1.31140
$439$	−26.6152	−1.27027	−0.635137	−	0.772400i	$-0.719056\pi$
$439$	−26.6152	−1.27027	−0.635137	+	0.772400i	$0.719056\pi$
$440$	−0.176993	−0.00843783
$441$	−0.108818	−0.00518179
$442$	19.0286	0.905099
$443$	−1.94319	−0.0923236	−0.0461618	−	0.998934i	$-0.514699\pi$
$443$	−1.94319	−0.0923236	−0.0461618	+	0.998934i	$0.514699\pi$
$444$	−6.88615	−0.326802
$445$	−8.00348	−0.379401
$446$	−20.6196	−0.976368
$447$	3.53636	0.167264
$448$	30.9541	1.46244
$449$	−19.3862	−0.914891	−0.457445	−	0.889238i	$-0.651235\pi$
$449$	−19.3862	−0.914891	−0.457445	+	0.889238i	$0.651235\pi$
$450$	−12.7450	−0.600805
$451$	−0.116836	−0.00550158
$452$	60.9611	2.86737
$453$	8.65328	0.406567
$454$	−32.2356	−1.51289
$455$	−11.8494	−0.555506
$456$	−0.192040	−0.00899311
$457$	0.837265	0.0391656	0.0195828	−	0.999808i	$-0.493766\pi$
$457$	0.837265	0.0391656	0.0195828	+	0.999808i	$0.493766\pi$
$458$	6.05157	0.282771
$459$	−5.76402	−0.269042
$460$	3.17581	0.148073
$461$	7.14765	0.332899	0.166450	−	0.986050i	$-0.446770\pi$
$461$	7.14765	0.332899	0.166450	+	0.986050i	$0.446770\pi$
$462$	0.0920795	0.00428392
$463$	11.4292	0.531162	0.265581	−	0.964089i	$-0.414436\pi$
$463$	11.4292	0.531162	0.265581	+	0.964089i	$0.414436\pi$
$464$	−4.18987	−0.194510
$465$	−27.1196	−1.25764
$466$	−18.2131	−0.843705
$467$	−21.1159	−0.977128	−0.488564	−	0.872528i	$-0.662479\pi$
$467$	−21.1159	−0.977128	−0.488564	+	0.872528i	$0.662479\pi$
$468$	−4.98843	−0.230590
$469$	−27.5122	−1.27040
$470$	−37.2055	−1.71616
$471$	13.0204	0.599950
$472$	−43.3557	−1.99561
$473$	−0.0757837	−0.00348454
$474$	−10.8927	−0.500318
$475$	−0.281155	−0.0129003
$476$	−53.9344	−2.47208
$477$	−3.27263	−0.149843
$478$	−52.8337	−2.41656
$479$	−34.6003	−1.58093	−0.790464	−	0.612508i	$-0.790161\pi$
$479$	−34.6003	−1.58093	−0.790464	+	0.612508i	$0.790161\pi$
$480$	11.8117	0.539130
$481$	−2.70367	−0.123277
$482$	32.5877	1.48433
$483$	−0.725154	−0.0329957
$484$	−39.2082	−1.78219
$485$	−7.88159	−0.357884
$486$	2.35891	0.107002
$487$	17.7777	0.805583	0.402792	−	0.915292i	$-0.368040\pi$
$487$	17.7777	0.805583	0.402792	+	0.915292i	$0.368040\pi$
$488$	−8.23731	−0.372885
$489$	6.64074	0.300305
$490$	−0.827919	−0.0374016
$491$	−6.58380	−0.297123	−0.148561	−	0.988903i	$-0.547464\pi$
$491$	−6.58380	−0.297123	−0.148561	+	0.988903i	$0.547464\pi$
$492$	−28.0069	−1.26265
$493$	−15.3197	−0.689964
$494$	−0.171790	−0.00772922
$495$	0.0479603	0.00215566
$496$	−13.2551	−0.595170
$497$	−5.72746	−0.256912
$498$	13.2414	0.593360
$499$	31.9385	1.42976	0.714881	−	0.699246i	$-0.246481\pi$
$499$	31.9385	1.42976	0.714881	+	0.699246i	$0.246481\pi$
$500$	−4.63220	−0.207158
$501$	−24.2637	−1.08402
$502$	36.9488	1.64910
$503$	−26.7489	−1.19268	−0.596338	−	0.802734i	$-0.703378\pi$
$503$	−26.7489	−1.19268	−0.596338	+	0.802734i	$0.703378\pi$
$504$	9.68772	0.431525
$505$	44.5507	1.98248
$506$	−0.00968945	−0.000430749 0
$507$	11.0414	0.490367
$508$	27.5613	1.22284
$509$	8.26632	0.366398	0.183199	−	0.983076i	$-0.441355\pi$
$509$	8.26632	0.366398	0.183199	+	0.983076i	$0.441355\pi$
$510$	−43.8545	−1.94191
$511$	30.5427	1.35113
$512$	17.4085	0.769356
$513$	0.0520376	0.00229752
$514$	−66.6804	−2.94114
$515$	50.7293	2.23540
$516$	−18.1662	−0.799723
$517$	0.0727148	0.00319799
$518$	11.9630	0.525626
$519$	8.14585	0.357563
$520$	−16.6580	−0.730501
$521$	−11.0883	−0.485786	−0.242893	−	0.970053i	$-0.578096\pi$
$521$	−11.0883	−0.485786	−0.242893	+	0.970053i	$0.578096\pi$
$522$	6.26954	0.274410
$523$	1.28325	0.0561124	0.0280562	−	0.999606i	$-0.491068\pi$
$523$	1.28325	0.0561124	0.0280562	+	0.999606i	$0.491068\pi$
$524$	−55.9352	−2.44354
$525$	14.1832	0.619007
$526$	35.0471	1.52812
$527$	−48.4653	−2.11118
$528$	0.0234413	0.00102015
$529$	−22.9237	−0.996682
$530$	−24.8992	−1.08155
$531$	11.7482	0.509828
$532$	0.486920	0.0211107
$533$	−10.9962	−0.476297
$534$	5.85346	0.253304
$535$	6.42096	0.277602
$536$	−38.6771	−1.67059
$537$	1.79724	0.0775565
$538$	56.3503	2.42943
$539$	0.00161809	6.96962e−5 0
$540$	11.4966	0.494737
$541$	20.9850	0.902214	0.451107	−	0.892470i	$-0.351029\pi$
$541$	20.9850	0.902214	0.451107	+	0.892470i	$0.351029\pi$
$542$	43.5835	1.87207
$543$	12.5435	0.538292
$544$	21.1087	0.905029
$545$	−15.5544	−0.666279
$546$	8.66620	0.370879
$547$	7.76800	0.332136	0.166068	−	0.986114i	$-0.446893\pi$
$547$	7.76800	0.332136	0.166068	+	0.986114i	$0.446893\pi$
$548$	−44.3098	−1.89282
$549$	2.23208	0.0952630
$550$	0.189515	0.00808096
$551$	0.138306	0.00589204
$552$	−1.01943	−0.0433899
$553$	12.1219	0.515476
$554$	39.9143	1.69580
$555$	6.23104	0.264493
$556$	−57.2395	−2.42749
$557$	−7.44215	−0.315334	−0.157667	−	0.987492i	$-0.550397\pi$
$557$	−7.44215	−0.315334	−0.157667	+	0.987492i	$0.550397\pi$
$558$	19.8343	0.839652
$559$	−7.13249	−0.301672
$560$	13.3475	0.564036
$561$	0.0857098	0.00361867
$562$	12.5926	0.531185
$563$	−21.6757	−0.913524	−0.456762	−	0.889589i	$-0.650991\pi$
$563$	−21.6757	−0.913524	−0.456762	+	0.889589i	$0.650991\pi$
$564$	17.4306	0.733959
$565$	−55.1617	−2.32067
$566$	38.7912	1.63052
$567$	−2.62511	−0.110244
$568$	−8.05174	−0.337844
$569$	6.56668	0.275290	0.137645	−	0.990482i	$-0.456047\pi$
$569$	6.56668	0.275290	0.137645	+	0.990482i	$0.456047\pi$
$570$	0.395919	0.0165832
$571$	−16.8093	−0.703449	−0.351724	−	0.936104i	$-0.614405\pi$
$571$	−16.8093	−0.703449	−0.351724	+	0.936104i	$0.614405\pi$
$572$	0.0741768	0.00310149
$573$	23.1689	0.967893
$574$	48.6552	2.03083
$575$	−1.49249	−0.0622412
$576$	−11.7916	−0.491315
$577$	29.4287	1.22513	0.612566	−	0.790419i	$-0.290137\pi$
$577$	29.4287	1.22513	0.612566	+	0.790419i	$0.290137\pi$
$578$	−38.2709	−1.59186
$579$	19.4382	0.807823
$580$	30.5559	1.26876
$581$	−14.7356	−0.611337
$582$	5.76431	0.238938
$583$	0.0486633	0.00201543
$584$	42.9373	1.77676
$585$	4.51386	0.186625
$586$	−12.9490	−0.534917
$587$	44.3551	1.83073	0.915366	−	0.402623i	$-0.131901\pi$
$587$	44.3551	1.83073	0.915366	+	0.402623i	$0.131901\pi$
$588$	0.387875	0.0159957
$589$	0.437545	0.0180287
$590$	89.3840	3.67988
$591$	14.9454	0.614773
$592$	3.04551	0.125170
$593$	19.7369	0.810498	0.405249	−	0.914206i	$-0.367185\pi$
$593$	19.7369	0.810498	0.405249	+	0.914206i	$0.367185\pi$
$594$	−0.0350765	−0.00143921
$595$	48.8034	2.00074
$596$	−12.6052	−0.516329
$597$	−19.2275	−0.786929
$598$	−0.911937	−0.0372919
$599$	6.34985	0.259448	0.129724	−	0.991550i	$-0.458591\pi$
$599$	6.34985	0.259448	0.129724	+	0.991550i	$0.458591\pi$
$600$	19.9390	0.814006
$601$	45.7396	1.86576	0.932879	−	0.360191i	$-0.117288\pi$
$601$	45.7396	1.86576	0.932879	+	0.360191i	$0.117288\pi$
$602$	31.5594	1.28627
$603$	10.4804	0.426796
$604$	−30.8442	−1.25503
$605$	35.4782	1.44239
$606$	−32.5828	−1.32359
$607$	−4.40443	−0.178770	−0.0893852	−	0.995997i	$-0.528490\pi$
$607$	−4.40443	−0.178770	−0.0893852	+	0.995997i	$0.528490\pi$
$608$	−0.190570	−0.00772862
$609$	−6.97704	−0.282724
$610$	16.9824	0.687598
$611$	6.84366	0.276865
$612$	20.5456	0.830507
$613$	−3.62493	−0.146410	−0.0732048	−	0.997317i	$-0.523323\pi$
$613$	−3.62493	−0.146410	−0.0732048	+	0.997317i	$0.523323\pi$
$614$	14.7002	0.593252
$615$	25.3425	1.02191
$616$	−0.144054	−0.00580411
$617$	2.41492	0.0972211	0.0486105	−	0.998818i	$-0.484521\pi$
$617$	2.41492	0.0972211	0.0486105	+	0.998818i	$0.484521\pi$
$618$	−37.1016	−1.49245
$619$	−4.10502	−0.164995	−0.0824974	−	0.996591i	$-0.526290\pi$
$619$	−4.10502	−0.164995	−0.0824974	+	0.996591i	$0.526290\pi$
$620$	96.6665	3.88222
$621$	0.276238	0.0110850
$622$	−65.3280	−2.61942
$623$	−6.51401	−0.260978
$624$	2.20621	0.0883190
$625$	−22.8231	−0.912923
$626$	19.3375	0.772880
$627$	−0.000773788 0	−3.09021e−5 0
$628$	−46.4108	−1.85199
$629$	11.1355	0.444001
$630$	−19.9727	−0.795730
$631$	12.8413	0.511203	0.255602	−	0.966782i	$-0.417726\pi$
$631$	12.8413	0.511203	0.255602	+	0.966782i	$0.417726\pi$
$632$	17.0411	0.677860
$633$	−26.0962	−1.03723
$634$	−52.9526	−2.10302
$635$	−24.9393	−0.989686
$636$	11.6651	0.462553
$637$	0.152289	0.00603392
$638$	−0.0932266	−0.00369088
$639$	2.18180	0.0863107
$640$	−66.0905	−2.61246
$641$	−22.0979	−0.872813	−0.436406	−	0.899750i	$-0.643749\pi$
$641$	−22.0979	−0.872813	−0.436406	+	0.899750i	$0.643749\pi$
$642$	−4.69606	−0.185339
$643$	40.9150	1.61353	0.806765	−	0.590873i	$-0.201217\pi$
$643$	40.9150	1.61353	0.806765	+	0.590873i	$0.201217\pi$
$644$	2.58478	0.101855
$645$	16.4380	0.647245
$646$	0.707545	0.0278380
$647$	37.0522	1.45667	0.728335	−	0.685221i	$-0.240294\pi$
$647$	37.0522	1.45667	0.728335	+	0.685221i	$0.240294\pi$
$648$	−3.69041	−0.144973
$649$	−0.174693	−0.00685730
$650$	17.8365	0.699606
$651$	−22.0725	−0.865091
$652$	−23.6706	−0.927013
$653$	12.5487	0.491069	0.245535	−	0.969388i	$-0.421036\pi$
$653$	12.5487	0.491069	0.245535	+	0.969388i	$0.421036\pi$
$654$	11.3760	0.444836
$655$	50.6138	1.97765
$656$	12.3865	0.483610
$657$	−11.6348	−0.453918
$658$	−30.2814	−1.18049
$659$	−38.8876	−1.51484	−0.757422	−	0.652925i	$-0.773542\pi$
$659$	−38.8876	−1.51484	−0.757422	+	0.652925i	$0.773542\pi$
$660$	−0.170952	−0.00665432
$661$	−46.9237	−1.82512	−0.912561	−	0.408941i	$-0.865898\pi$
$661$	−46.9237	−1.82512	−0.912561	+	0.408941i	$0.865898\pi$
$662$	−15.9297	−0.619127
$663$	8.06670	0.313285
$664$	−20.7155	−0.803919
$665$	−0.440597	−0.0170856
$666$	−4.55716	−0.176586
$667$	0.734188	0.0284279
$668$	86.4869	3.34628
$669$	−8.74117	−0.337953
$670$	79.7384	3.08056
$671$	−0.0331906	−0.00128131
$672$	9.61354	0.370850
$673$	21.2510	0.819165	0.409583	−	0.912273i	$-0.365674\pi$
$673$	21.2510	0.819165	0.409583	+	0.912273i	$0.365674\pi$
$674$	−60.3499	−2.32459
$675$	−5.40292	−0.207958
$676$	−39.3567	−1.51372
$677$	−8.05610	−0.309621	−0.154811	−	0.987944i	$-0.549477\pi$
$677$	−8.05610	−0.309621	−0.154811	+	0.987944i	$0.549477\pi$
$678$	40.3433	1.54937
$679$	−6.41480	−0.246177
$680$	68.6085	2.63102
$681$	−13.6654	−0.523661
$682$	−0.294932	−0.0112935
$683$	37.4825	1.43423	0.717114	−	0.696955i	$-0.245462\pi$
$683$	37.4825	1.43423	0.717114	+	0.696955i	$0.245462\pi$
$684$	−0.185486	−0.00709223
$685$	40.0944	1.53193
$686$	−44.0206	−1.68071
$687$	2.56541	0.0978765
$688$	8.03429	0.306304
$689$	4.58002	0.174485
$690$	2.10171	0.0800106
$691$	25.8455	0.983208	0.491604	−	0.870819i	$-0.336411\pi$
$691$	25.8455	0.983208	0.491604	+	0.870819i	$0.336411\pi$
$692$	−29.0355	−1.10376
$693$	0.0390348	0.00148281
$694$	62.8267	2.38487
$695$	51.7941	1.96466
$696$	−9.80842	−0.371787
$697$	45.2894	1.71546
$698$	5.50137	0.208230
$699$	−7.72098	−0.292034
$700$	−50.5555	−1.91082
$701$	49.2101	1.85864	0.929320	−	0.369275i	$-0.120394\pi$
$701$	49.2101	1.85864	0.929320	+	0.369275i	$0.120394\pi$
$702$	−3.30127	−0.124599
$703$	−0.100531	−0.00379160
$704$	0.175338	0.00660830
$705$	−15.7723	−0.594020
$706$	39.5183	1.48729
$707$	36.2597	1.36368
$708$	−41.8759	−1.57379
$709$	−12.7418	−0.478528	−0.239264	−	0.970955i	$-0.576906\pi$
$709$	−12.7418	−0.478528	−0.239264	+	0.970955i	$0.576906\pi$
$710$	16.5998	0.622981
$711$	−4.61768	−0.173177
$712$	−9.15748	−0.343191
$713$	2.32268	0.0869849
$714$	−35.6931	−1.33578
$715$	−0.0671201	−0.00251015
$716$	−6.40617	−0.239410
$717$	−22.3975	−0.836449
$718$	27.4537	1.02456
$719$	15.7751	0.588312	0.294156	−	0.955757i	$-0.404962\pi$
$719$	15.7751	0.588312	0.294156	+	0.955757i	$0.404962\pi$
$720$	−5.08457	−0.189491
$721$	41.2884	1.53766
$722$	44.8129	1.66776
$723$	13.8147	0.513776
$724$	−44.7107	−1.66166
$725$	−14.3599	−0.533315
$726$	−25.9475	−0.963002
$727$	36.7189	1.36183	0.680915	−	0.732363i	$-0.261583\pi$
$727$	36.7189	1.36183	0.680915	+	0.732363i	$0.261583\pi$
$728$	−13.5579	−0.502489
$729$	1.00000	0.0370370
$730$	−88.5216	−3.27633
$731$	29.3763	1.08652
$732$	−7.95617	−0.294068
$733$	−24.7722	−0.914983	−0.457492	−	0.889214i	$-0.651252\pi$
$733$	−24.7722	−0.914983	−0.457492	+	0.889214i	$0.651252\pi$
$734$	15.8512	0.585078
$735$	−0.350975	−0.0129459
$736$	−1.01163	−0.0372890
$737$	−0.155842	−0.00574050
$738$	−18.5346	−0.682267
$739$	−10.3153	−0.379455	−0.189727	−	0.981837i	$-0.560760\pi$
$739$	−10.3153	−0.379455	−0.189727	+	0.981837i	$0.560760\pi$
$740$	−22.2103	−0.816466
$741$	−0.0728262	−0.00267534
$742$	−20.2654	−0.743966
$743$	−51.9331	−1.90524	−0.952621	−	0.304159i	$-0.901625\pi$
$743$	−51.9331	−1.90524	−0.952621	+	0.304159i	$0.901625\pi$
$744$	−31.0299	−1.13761
$745$	11.4060	0.417884
$746$	−65.8455	−2.41077
$747$	5.61334	0.205382
$748$	−0.305509	−0.0111705
$749$	5.22600	0.190954
$750$	−3.06553	−0.111937
$751$	−9.19563	−0.335553	−0.167777	−	0.985825i	$-0.553659\pi$
$751$	−9.19563	−0.335553	−0.167777	+	0.985825i	$0.553659\pi$
$752$	−7.70894	−0.281116
$753$	15.6635	0.570809
$754$	−8.77416	−0.319536
$755$	27.9099	1.01575
$756$	9.35708	0.340314
$757$	13.1629	0.478413	0.239207	−	0.970969i	$-0.423113\pi$
$757$	13.1629	0.478413	0.239207	+	0.970969i	$0.423113\pi$
$758$	−72.6861	−2.64008
$759$	−0.00410760	−0.000149096 0
$760$	−0.619398	−0.0224679
$761$	9.56260	0.346644	0.173322	−	0.984865i	$-0.444550\pi$
$761$	9.56260	0.346644	0.173322	+	0.984865i	$0.444550\pi$
$762$	18.2397	0.660755
$763$	−12.6597	−0.458312
$764$	−82.5844	−2.98780
$765$	−18.5910	−0.672160
$766$	−46.4454	−1.67814
$767$	−16.4415	−0.593668
$768$	24.7531	0.893201
$769$	20.8610	0.752269	0.376134	−	0.926565i	$-0.377253\pi$
$769$	20.8610	0.752269	0.376134	+	0.926565i	$0.377253\pi$
$770$	0.296989	0.0107027
$771$	−28.2674	−1.01803
$772$	−69.2865	−2.49368
$773$	9.42169	0.338875	0.169437	−	0.985541i	$-0.445805\pi$
$773$	9.42169	0.338875	0.169437	+	0.985541i	$0.445805\pi$
$774$	−12.0222	−0.432127
$775$	−45.4291	−1.63186
$776$	−9.01801	−0.323728
$777$	5.07143	0.181936
$778$	3.08586	0.110634
$779$	−0.408873	−0.0146494
$780$	−16.0894	−0.576095
$781$	−0.0324429	−0.00116090
$782$	3.75596	0.134313
$783$	2.65781	0.0949824
$784$	−0.171544	−0.00612657
$785$	41.9955	1.49888
$786$	−37.0171	−1.32036
$787$	−6.62041	−0.235992	−0.117996	−	0.993014i	$-0.537647\pi$
$787$	−6.62041	−0.235992	−0.117996	+	0.993014i	$0.537647\pi$
$788$	−53.2723	−1.89775
$789$	14.8573	0.528934
$790$	−35.1328	−1.24997
$791$	−44.8959	−1.59631
$792$	0.0548756	0.00194992
$793$	−3.12378	−0.110929
$794$	36.5248	1.29622
$795$	−10.5554	−0.374361
$796$	68.5355	2.42918
$797$	−16.1146	−0.570809	−0.285404	−	0.958407i	$-0.592128\pi$
$797$	−16.1146	−0.570809	−0.285404	+	0.958407i	$0.592128\pi$
$798$	0.322237	0.0114071
$799$	−28.1867	−0.997173
$800$	19.7863	0.699552
$801$	2.48143	0.0876769
$802$	85.5797	3.02193
$803$	0.173007	0.00610530
$804$	−37.3570	−1.31748
$805$	−2.33888	−0.0824347
$806$	−27.7579	−0.977731
$807$	23.8883	0.840907
$808$	50.9743	1.79327
$809$	44.8140	1.57558	0.787788	−	0.615947i	$-0.211226\pi$
$809$	44.8140	1.57558	0.787788	+	0.615947i	$0.211226\pi$
$810$	7.60832	0.267329
$811$	1.18334	0.0415526	0.0207763	−	0.999784i	$-0.493386\pi$
$811$	1.18334	0.0415526	0.0207763	+	0.999784i	$0.493386\pi$
$812$	24.8693	0.872743
$813$	18.4761	0.647986
$814$	0.0677640	0.00237513
$815$	21.4187	0.750266
$816$	−9.08661	−0.318095
$817$	−0.265209	−0.00927849
$818$	54.2598	1.89715
$819$	3.67381	0.128373
$820$	−90.3321	−3.15453
$821$	−0.0586438	−0.00204668	−0.00102334	−	0.999999i	$-0.500326\pi$
$821$	−0.0586438	−0.00204668	−0.00102334	+	0.999999i	$0.500326\pi$
$822$	−29.3236	−1.02278
$823$	14.5233	0.506252	0.253126	−	0.967433i	$-0.418541\pi$
$823$	14.5233	0.506252	0.253126	+	0.967433i	$0.418541\pi$
$824$	58.0439	2.02205
$825$	0.0803402	0.00279709
$826$	72.7494	2.53127
$827$	9.29703	0.323289	0.161645	−	0.986849i	$-0.448320\pi$
$827$	9.29703	0.323289	0.161645	+	0.986849i	$0.448320\pi$
$828$	−0.984638	−0.0342185
$829$	−32.3142	−1.12232	−0.561158	−	0.827708i	$-0.689644\pi$
$829$	−32.3142	−1.12232	−0.561158	+	0.827708i	$0.689644\pi$
$830$	42.7081	1.48242
$831$	16.9207	0.586971
$832$	16.5022	0.572110
$833$	−0.627227	−0.0217321
$834$	−37.8803	−1.31169
$835$	−78.2591	−2.70827
$836$	0.00275813	9.53920e−5 0
$837$	8.40824	0.290631
$838$	−56.5327	−1.95289
$839$	−29.9881	−1.03531	−0.517653	−	0.855591i	$-0.673194\pi$
$839$	−29.9881	−1.03531	−0.517653	+	0.855591i	$0.673194\pi$
$840$	31.2463	1.07810
$841$	−21.9360	−0.756415
$842$	25.0641	0.863766
$843$	5.33830	0.183861
$844$	93.0189	3.20184
$845$	35.6125	1.22511
$846$	11.5353	0.396592
$847$	28.8756	0.992177
$848$	−5.15909	−0.177164
$849$	16.4446	0.564376
$850$	−73.4625	−2.51974
$851$	−0.533662	−0.0182937
$852$	−7.77693	−0.266433
$853$	−22.2337	−0.761267	−0.380634	−	0.924726i	$-0.624294\pi$
$853$	−22.2337	−0.761267	−0.380634	+	0.924726i	$0.624294\pi$
$854$	13.8219	0.472977
$855$	0.167840	0.00574000
$856$	7.34678	0.251108
$857$	−6.35544	−0.217098	−0.108549	−	0.994091i	$-0.534620\pi$
$857$	−6.35544	−0.217098	−0.108549	+	0.994091i	$0.534620\pi$
$858$	0.0490892	0.00167588
$859$	37.5515	1.28124	0.640620	−	0.767858i	$-0.278678\pi$
$859$	37.5515	1.28124	0.640620	+	0.767858i	$0.278678\pi$
$860$	−58.5925	−1.99799
$861$	20.6261	0.702937
$862$	−78.3404	−2.66828
$863$	9.79000	0.333256	0.166628	−	0.986020i	$-0.446712\pi$
$863$	9.79000	0.333256	0.166628	+	0.986020i	$0.446712\pi$
$864$	−3.66215	−0.124589
$865$	26.2733	0.893318
$866$	−60.5460	−2.05744
$867$	−16.2240	−0.550995
$868$	78.6766	2.67046
$869$	0.0686639	0.00232926
$870$	20.2215	0.685573
$871$	−14.6673	−0.496981
$872$	−17.7972	−0.602689
$873$	2.44363	0.0827045
$874$	−0.0339088	−0.00114698
$875$	3.41146	0.115329
$876$	41.4719	1.40120
$877$	21.1290	0.713477	0.356739	−	0.934204i	$-0.383889\pi$
$877$	21.1290	0.713477	0.356739	+	0.934204i	$0.383889\pi$
$878$	62.7828	2.11882
$879$	−5.48938	−0.185152
$880$	0.0756064	0.00254869
$881$	21.1220	0.711619	0.355810	−	0.934558i	$-0.384205\pi$
$881$	21.1220	0.711619	0.355810	+	0.934558i	$0.384205\pi$
$882$	0.256691	0.00864323
$883$	−24.1423	−0.812453	−0.406227	−	0.913772i	$-0.633156\pi$
$883$	−24.1423	−0.812453	−0.406227	+	0.913772i	$0.633156\pi$
$884$	−28.7534	−0.967082
$885$	37.8921	1.27373
$886$	4.58380	0.153996
$887$	22.6158	0.759366	0.379683	−	0.925117i	$-0.376033\pi$
$887$	22.6158	0.759366	0.379683	+	0.925117i	$0.376033\pi$
$888$	7.12948	0.239250
$889$	−20.2980	−0.680774
$890$	18.8795	0.632842
$891$	−0.0148698	−0.000498156 0
$892$	31.1575	1.04323
$893$	0.254469	0.00851549
$894$	−8.34195	−0.278996
$895$	5.79673	0.193763
$896$	−53.7909	−1.79703
$897$	−0.386593	−0.0129080
$898$	45.7303	1.52604
$899$	22.3475	0.745332
$900$	19.2585	0.641949
$901$	−18.8635	−0.628435
$902$	0.275605	0.00917664
$903$	13.3788	0.445219
$904$	−63.1153	−2.09918
$905$	40.4572	1.34484
$906$	−20.4123	−0.678153
$907$	45.6705	1.51646	0.758231	−	0.651985i	$-0.226064\pi$
$907$	45.6705	1.51646	0.758231	+	0.651985i	$0.226064\pi$
$908$	48.7099	1.61649
$909$	−13.8126	−0.458136
$910$	27.9516	0.926585
$911$	−54.6868	−1.81185	−0.905927	−	0.423435i	$-0.860824\pi$
$911$	−54.6868	−1.81185	−0.905927	+	0.423435i	$0.860824\pi$
$912$	0.0820340	0.00271642
$913$	−0.0834692	−0.00276243
$914$	−1.97503	−0.0653282
$915$	7.19927	0.238000
$916$	−9.14429	−0.302136
$917$	41.1944	1.36036
$918$	13.5968	0.448762
$919$	−24.5918	−0.811209	−0.405604	−	0.914049i	$-0.632939\pi$
$919$	−24.5918	−0.811209	−0.405604	+	0.914049i	$0.632939\pi$
$920$	−3.28803	−0.108403
$921$	6.23178	0.205344
$922$	−16.8607	−0.555276
$923$	−3.05341	−0.100504
$924$	−0.139138	−0.00457729
$925$	10.4379	0.343195
$926$	−26.9606	−0.885979
$927$	−15.7283	−0.516585
$928$	−9.73331	−0.319511
$929$	−11.0232	−0.361659	−0.180830	−	0.983514i	$-0.557878\pi$
$929$	−11.0232	−0.361659	−0.180830	+	0.983514i	$0.557878\pi$
$930$	63.9726	2.09775
$931$	0.00566260	0.000185584 0
$932$	27.5211	0.901483
$933$	−27.6942	−0.906666
$934$	49.8106	1.62985
$935$	0.276444	0.00904070
$936$	5.16470	0.168813
$937$	−23.1615	−0.756652	−0.378326	−	0.925672i	$-0.623500\pi$
$937$	−23.1615	−0.756652	−0.378326	+	0.925672i	$0.623500\pi$
$938$	64.8988	2.11902
$939$	8.19762	0.267519
$940$	56.2198	1.83369
$941$	16.8556	0.549476	0.274738	−	0.961519i	$-0.411409\pi$
$941$	16.8556	0.549476	0.274738	+	0.961519i	$0.411409\pi$
$942$	−30.7140	−1.00072
$943$	−2.17047	−0.0706803
$944$	18.5203	0.602784
$945$	−8.46690	−0.275428
$946$	0.178767	0.00581221
$947$	−16.8145	−0.546397	−0.273199	−	0.961958i	$-0.588082\pi$
$947$	−16.8145	−0.546397	−0.273199	+	0.961958i	$0.588082\pi$
$948$	16.4595	0.534580
$949$	16.2829	0.528564
$950$	0.663219	0.0215177
$951$	−22.4479	−0.727924
$952$	55.8403	1.80979
$953$	38.0584	1.23283	0.616417	−	0.787420i	$-0.288584\pi$
$953$	38.0584	1.23283	0.616417	+	0.787420i	$0.288584\pi$
$954$	7.71984	0.249939
$955$	74.7278	2.41814
$956$	79.8349	2.58204
$957$	−0.0395211	−0.00127753
$958$	81.6190	2.63699
$959$	32.6327	1.05377
$960$	−38.0320	−1.22748
$961$	39.6986	1.28060
$962$	6.37771	0.205626
$963$	−1.99078	−0.0641519
$964$	−49.2420	−1.58598
$965$	62.6950	2.01822
$966$	1.71057	0.0550368
$967$	−13.9541	−0.448735	−0.224367	−	0.974505i	$-0.572032\pi$
$967$	−13.9541	−0.448735	−0.224367	+	0.974505i	$0.572032\pi$
$968$	40.5937	1.30473
$969$	0.299946	0.00963565
$970$	18.5920	0.596952
$971$	0.536100	0.0172043	0.00860214	−	0.999963i	$-0.497262\pi$
$971$	0.536100	0.0172043	0.00860214	+	0.999963i	$0.497262\pi$
$972$	−3.56446	−0.114330
$973$	42.1550	1.35143
$974$	−41.9359	−1.34371
$975$	7.56134	0.242157
$976$	3.51874	0.112632
$977$	−18.6144	−0.595529	−0.297764	−	0.954639i	$-0.596241\pi$
$977$	−18.6144	−0.595529	−0.297764	+	0.954639i	$0.596241\pi$
$978$	−15.6649	−0.500908
$979$	−0.0368983	−0.00117927
$980$	1.25104	0.0399629
$981$	4.82255	0.153972
$982$	15.5306	0.495601
$983$	−23.5363	−0.750690	−0.375345	−	0.926885i	$-0.622476\pi$
$983$	−23.5363	−0.750690	−0.375345	+	0.926885i	$0.622476\pi$
$984$	28.9965	0.924375
$985$	48.2043	1.53592
$986$	36.1378	1.15086
$987$	−12.8370	−0.408608
$988$	0.259586	0.00825852
$989$	−1.40784	−0.0447668
$990$	−0.113134	−0.00359564
$991$	15.5925	0.495312	0.247656	−	0.968848i	$-0.420340\pi$
$991$	15.5925	0.495312	0.247656	+	0.968848i	$0.420340\pi$
$992$	−30.7923	−0.977656
$993$	−6.75301	−0.214300
$994$	13.5106	0.428529
$995$	−62.0155	−1.96602
$996$	−20.0085	−0.633994
$997$	48.5543	1.53773	0.768865	−	0.639411i	$-0.220822\pi$
$997$	48.5543	1.53773	0.768865	+	0.639411i	$0.220822\pi$
$998$	−75.3400	−2.38485
$999$	−1.93189	−0.0611224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8013.2.a.b.1.9	✓	106

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8013.2.a.b.1.9	✓	106	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:	\( N \)	\(=\)	\( 8013 = 3 \cdot 2671 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35891 q^{2} -1.00000 q^{3} +3.56446 q^{4} -3.22536 q^{5} +2.35891 q^{6} -2.62511 q^{7} -3.69041 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.35891 q^{2} -1.00000 q^{3} +3.56446 q^{4} -3.22536 q^{5} +2.35891 q^{6} -2.62511 q^{7} -3.69041 q^{8} +1.00000 q^{9} +7.60832 q^{10} -0.0148698 q^{11} -3.56446 q^{12} -1.39949 q^{13} +6.19239 q^{14} +3.22536 q^{15} +1.57644 q^{16} +5.76402 q^{17} -2.35891 q^{18} -0.0520376 q^{19} -11.4966 q^{20} +2.62511 q^{21} +0.0350765 q^{22} -0.276238 q^{23} +3.69041 q^{24} +5.40292 q^{25} +3.30127 q^{26} -1.00000 q^{27} -9.35708 q^{28} -2.65781 q^{29} -7.60832 q^{30} -8.40824 q^{31} +3.66215 q^{32} +0.0148698 q^{33} -13.5968 q^{34} +8.46690 q^{35} +3.56446 q^{36} +1.93189 q^{37} +0.122752 q^{38} +1.39949 q^{39} +11.9029 q^{40} +7.85726 q^{41} -6.19239 q^{42} +5.09649 q^{43} -0.0530027 q^{44} -3.22536 q^{45} +0.651621 q^{46} -4.89010 q^{47} -1.57644 q^{48} -0.108818 q^{49} -12.7450 q^{50} -5.76402 q^{51} -4.98843 q^{52} -3.27263 q^{53} +2.35891 q^{54} +0.0479603 q^{55} +9.68772 q^{56} +0.0520376 q^{57} +6.26954 q^{58} +11.7482 q^{59} +11.4966 q^{60} +2.23208 q^{61} +19.8343 q^{62} -2.62511 q^{63} -11.7916 q^{64} +4.51386 q^{65} -0.0350765 q^{66} +10.4804 q^{67} +20.5456 q^{68} +0.276238 q^{69} -19.9727 q^{70} +2.18180 q^{71} -3.69041 q^{72} -11.6348 q^{73} -4.55716 q^{74} -5.40292 q^{75} -0.185486 q^{76} +0.0390348 q^{77} -3.30127 q^{78} -4.61768 q^{79} -5.08457 q^{80} +1.00000 q^{81} -18.5346 q^{82} +5.61334 q^{83} +9.35708 q^{84} -18.5910 q^{85} -12.0222 q^{86} +2.65781 q^{87} +0.0548756 q^{88} +2.48143 q^{89} +7.60832 q^{90} +3.67381 q^{91} -0.984638 q^{92} +8.40824 q^{93} +11.5353 q^{94} +0.167840 q^{95} -3.66215 q^{96} +2.44363 q^{97} +0.256691 q^{98} -0.0148698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9}+O(q^{10}) \) 106 * q + 15 * q^2 - 106 * q^3 + 109 * q^4 + 16 * q^5 - 15 * q^6 + 35 * q^7 + 48 * q^8 + 106 * q^9	\( 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9} - 3 q^{10} + 55 q^{11} - 109 q^{12} - 8 q^{13} + 27 q^{14} - 16 q^{15} + 111 q^{16} + 28 q^{17} + 15 q^{18} + q^{19} + 54 q^{20} - 35 q^{21} + 20 q^{22} + 62 q^{23} - 48 q^{24} + 102 q^{25} + 21 q^{26} - 106 q^{27} + 79 q^{28} + 36 q^{29} + 3 q^{30} + q^{31} + 111 q^{32} - 55 q^{33} - 27 q^{34} + 72 q^{35} + 109 q^{36} + 31 q^{37} + 43 q^{38} + 8 q^{39} - 13 q^{40} + 35 q^{41} - 27 q^{42} + 98 q^{43} + 121 q^{44} + 16 q^{45} + 8 q^{46} + 75 q^{47} - 111 q^{48} + 49 q^{49} + 83 q^{50} - 28 q^{51} - 18 q^{52} + 60 q^{53} - 15 q^{54} + 14 q^{55} + 85 q^{56} - q^{57} + 65 q^{58} + 77 q^{59} - 54 q^{60} - 55 q^{61} + 83 q^{62} + 35 q^{63} + 122 q^{64} + 86 q^{65} - 20 q^{66} + 121 q^{67} + 80 q^{68} - 62 q^{69} - 11 q^{70} + 79 q^{71} + 48 q^{72} - 29 q^{73} + 91 q^{74} - 102 q^{75} - 10 q^{76} + 87 q^{77} - 21 q^{78} + 15 q^{79} + 108 q^{80} + 106 q^{81} + 21 q^{82} + 196 q^{83} - 79 q^{84} - 5 q^{85} + 65 q^{86} - 36 q^{87} + 84 q^{88} + 34 q^{89} - 3 q^{90} + 17 q^{91} + 162 q^{92} - q^{93} - 35 q^{94} + 113 q^{95} - 111 q^{96} - 63 q^{97} + 112 q^{98} + 55 q^{99}+O(q^{100}) \) 106 * q + 15 * q^2 - 106 * q^3 + 109 * q^4 + 16 * q^5 - 15 * q^6 + 35 * q^7 + 48 * q^8 + 106 * q^9 - 3 * q^10 + 55 * q^11 - 109 * q^12 - 8 * q^13 + 27 * q^14 - 16 * q^15 + 111 * q^16 + 28 * q^17 + 15 * q^18 + q^19 + 54 * q^20 - 35 * q^21 + 20 * q^22 + 62 * q^23 - 48 * q^24 + 102 * q^25 + 21 * q^26 - 106 * q^27 + 79 * q^28 + 36 * q^29 + 3 * q^30 + q^31 + 111 * q^32 - 55 * q^33 - 27 * q^34 + 72 * q^35 + 109 * q^36 + 31 * q^37 + 43 * q^38 + 8 * q^39 - 13 * q^40 + 35 * q^41 - 27 * q^42 + 98 * q^43 + 121 * q^44 + 16 * q^45 + 8 * q^46 + 75 * q^47 - 111 * q^48 + 49 * q^49 + 83 * q^50 - 28 * q^51 - 18 * q^52 + 60 * q^53 - 15 * q^54 + 14 * q^55 + 85 * q^56 - q^57 + 65 * q^58 + 77 * q^59 - 54 * q^60 - 55 * q^61 + 83 * q^62 + 35 * q^63 + 122 * q^64 + 86 * q^65 - 20 * q^66 + 121 * q^67 + 80 * q^68 - 62 * q^69 - 11 * q^70 + 79 * q^71 + 48 * q^72 - 29 * q^73 + 91 * q^74 - 102 * q^75 - 10 * q^76 + 87 * q^77 - 21 * q^78 + 15 * q^79 + 108 * q^80 + 106 * q^81 + 21 * q^82 + 196 * q^83 - 79 * q^84 - 5 * q^85 + 65 * q^86 - 36 * q^87 + 84 * q^88 + 34 * q^89 - 3 * q^90 + 17 * q^91 + 162 * q^92 - q^93 - 35 * q^94 + 113 * q^95 - 111 * q^96 - 63 * q^97 + 112 * q^98 + 55 * q^99

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