LMFDB - Embedded newform 8002.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.32456 q^{3} +1.00000 q^{4} +2.37351 q^{5} +3.32456 q^{6} -1.54900 q^{7} -1.00000 q^{8} +8.05269 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.32456 q^{3} +1.00000 q^{4} +2.37351 q^{5} +3.32456 q^{6} -1.54900 q^{7} -1.00000 q^{8} +8.05269 q^{9} -2.37351 q^{10} -3.72614 q^{11} -3.32456 q^{12} +5.29209 q^{13} +1.54900 q^{14} -7.89089 q^{15} +1.00000 q^{16} +0.656701 q^{17} -8.05269 q^{18} +2.19741 q^{19} +2.37351 q^{20} +5.14973 q^{21} +3.72614 q^{22} +0.664257 q^{23} +3.32456 q^{24} +0.633570 q^{25} -5.29209 q^{26} -16.7980 q^{27} -1.54900 q^{28} +9.52073 q^{29} +7.89089 q^{30} -8.87603 q^{31} -1.00000 q^{32} +12.3878 q^{33} -0.656701 q^{34} -3.67656 q^{35} +8.05269 q^{36} -4.68196 q^{37} -2.19741 q^{38} -17.5939 q^{39} -2.37351 q^{40} +3.30142 q^{41} -5.14973 q^{42} -2.72676 q^{43} -3.72614 q^{44} +19.1132 q^{45} -0.664257 q^{46} +10.9786 q^{47} -3.32456 q^{48} -4.60061 q^{49} -0.633570 q^{50} -2.18324 q^{51} +5.29209 q^{52} -12.1771 q^{53} +16.7980 q^{54} -8.84405 q^{55} +1.54900 q^{56} -7.30543 q^{57} -9.52073 q^{58} -4.63655 q^{59} -7.89089 q^{60} -2.71078 q^{61} +8.87603 q^{62} -12.4736 q^{63} +1.00000 q^{64} +12.5608 q^{65} -12.3878 q^{66} +9.75477 q^{67} +0.656701 q^{68} -2.20836 q^{69} +3.67656 q^{70} +2.93701 q^{71} -8.05269 q^{72} -5.43208 q^{73} +4.68196 q^{74} -2.10634 q^{75} +2.19741 q^{76} +5.77177 q^{77} +17.5939 q^{78} -2.67589 q^{79} +2.37351 q^{80} +31.6878 q^{81} -3.30142 q^{82} -4.14703 q^{83} +5.14973 q^{84} +1.55869 q^{85} +2.72676 q^{86} -31.6522 q^{87} +3.72614 q^{88} +8.42081 q^{89} -19.1132 q^{90} -8.19742 q^{91} +0.664257 q^{92} +29.5089 q^{93} -10.9786 q^{94} +5.21559 q^{95} +3.32456 q^{96} -14.4293 q^{97} +4.60061 q^{98} -30.0055 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * 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
$3$	−3.32456	−1.91944	−0.959718	−	0.280967i	$-0.909345\pi$
$3$	−3.32456	−1.91944	−0.959718	+	0.280967i	$0.909345\pi$
$4$	1.00000	0.500000
$5$	2.37351	1.06147	0.530734	−	0.847538i	$-0.321916\pi$
$5$	2.37351	1.06147	0.530734	+	0.847538i	$0.321916\pi$
$6$	3.32456	1.35725
$7$	−1.54900	−0.585465	−0.292733	−	0.956194i	$-0.594565\pi$
$7$	−1.54900	−0.585465	−0.292733	+	0.956194i	$0.594565\pi$
$8$	−1.00000	−0.353553
$9$	8.05269	2.68423
$10$	−2.37351	−0.750571
$11$	−3.72614	−1.12347	−0.561737	−	0.827316i	$-0.689867\pi$
$11$	−3.72614	−1.12347	−0.561737	+	0.827316i	$0.689867\pi$
$12$	−3.32456	−0.959718
$13$	5.29209	1.46776	0.733880	−	0.679279i	$-0.237707\pi$
$13$	5.29209	1.46776	0.733880	+	0.679279i	$0.237707\pi$
$14$	1.54900	0.413986
$15$	−7.89089	−2.03742
$16$	1.00000	0.250000
$17$	0.656701	0.159273	0.0796367	−	0.996824i	$-0.474624\pi$
$17$	0.656701	0.159273	0.0796367	+	0.996824i	$0.474624\pi$
$18$	−8.05269	−1.89804
$19$	2.19741	0.504121	0.252061	−	0.967711i	$-0.418892\pi$
$19$	2.19741	0.504121	0.252061	+	0.967711i	$0.418892\pi$
$20$	2.37351	0.530734
$21$	5.14973	1.12376
$22$	3.72614	0.794416
$23$	0.664257	0.138507	0.0692535	−	0.997599i	$-0.477938\pi$
$23$	0.664257	0.138507	0.0692535	+	0.997599i	$0.477938\pi$
$24$	3.32456	0.678623
$25$	0.633570	0.126714
$26$	−5.29209	−1.03786
$27$	−16.7980	−3.23277
$28$	−1.54900	−0.292733
$29$	9.52073	1.76795	0.883977	−	0.467530i	$-0.154856\pi$
$29$	9.52073	1.76795	0.883977	+	0.467530i	$0.154856\pi$
$30$	7.89089	1.44067
$31$	−8.87603	−1.59418	−0.797091	−	0.603859i	$-0.793629\pi$
$31$	−8.87603	−1.59418	−0.797091	+	0.603859i	$0.793629\pi$
$32$	−1.00000	−0.176777
$33$	12.3878	2.15643
$34$	−0.656701	−0.112623
$35$	−3.67656	−0.621452
$36$	8.05269	1.34212
$37$	−4.68196	−0.769710	−0.384855	−	0.922977i	$-0.625749\pi$
$37$	−4.68196	−0.769710	−0.384855	+	0.922977i	$0.625749\pi$
$38$	−2.19741	−0.356467
$39$	−17.5939	−2.81727
$40$	−2.37351	−0.375286
$41$	3.30142	0.515595	0.257797	−	0.966199i	$-0.417003\pi$
$41$	3.30142	0.515595	0.257797	+	0.966199i	$0.417003\pi$
$42$	−5.14973	−0.794620
$43$	−2.72676	−0.415827	−0.207913	−	0.978147i	$-0.566667\pi$
$43$	−2.72676	−0.415827	−0.207913	+	0.978147i	$0.566667\pi$
$44$	−3.72614	−0.561737
$45$	19.1132	2.84923
$46$	−0.664257	−0.0979393
$47$	10.9786	1.60139	0.800694	−	0.599074i	$-0.204464\pi$
$47$	10.9786	1.60139	0.800694	+	0.599074i	$0.204464\pi$
$48$	−3.32456	−0.479859
$49$	−4.60061	−0.657231
$50$	−0.633570	−0.0896003
$51$	−2.18324	−0.305715
$52$	5.29209	0.733880
$53$	−12.1771	−1.67265	−0.836326	−	0.548233i	$-0.815301\pi$
$53$	−12.1771	−1.67265	−0.836326	+	0.548233i	$0.815301\pi$
$54$	16.7980	2.28592
$55$	−8.84405	−1.19253
$56$	1.54900	0.206993
$57$	−7.30543	−0.967628
$58$	−9.52073	−1.25013
$59$	−4.63655	−0.603627	−0.301814	−	0.953367i	$-0.597592\pi$
$59$	−4.63655	−0.603627	−0.301814	+	0.953367i	$0.597592\pi$
$60$	−7.89089	−1.01871
$61$	−2.71078	−0.347080	−0.173540	−	0.984827i	$-0.555521\pi$
$61$	−2.71078	−0.347080	−0.173540	+	0.984827i	$0.555521\pi$
$62$	8.87603	1.12726
$63$	−12.4736	−1.57152
$64$	1.00000	0.125000
$65$	12.5608	1.55798
$66$	−12.3878	−1.52483
$67$	9.75477	1.19173	0.595867	−	0.803083i	$-0.296808\pi$
$67$	9.75477	1.19173	0.595867	+	0.803083i	$0.296808\pi$
$68$	0.656701	0.0796367
$69$	−2.20836	−0.265855
$70$	3.67656	0.439433
$71$	2.93701	0.348558	0.174279	−	0.984696i	$-0.444240\pi$
$71$	2.93701	0.348558	0.174279	+	0.984696i	$0.444240\pi$
$72$	−8.05269	−0.949019
$73$	−5.43208	−0.635777	−0.317888	−	0.948128i	$-0.602974\pi$
$73$	−5.43208	−0.635777	−0.317888	+	0.948128i	$0.602974\pi$
$74$	4.68196	0.544267
$75$	−2.10634	−0.243219
$76$	2.19741	0.252061
$77$	5.77177	0.657754
$78$	17.5939	1.99211
$79$	−2.67589	−0.301062	−0.150531	−	0.988605i	$-0.548098\pi$
$79$	−2.67589	−0.301062	−0.150531	+	0.988605i	$0.548098\pi$
$80$	2.37351	0.265367
$81$	31.6878	3.52087
$82$	−3.30142	−0.364581
$83$	−4.14703	−0.455196	−0.227598	−	0.973755i	$-0.573087\pi$
$83$	−4.14703	−0.455196	−0.227598	+	0.973755i	$0.573087\pi$
$84$	5.14973	0.561881
$85$	1.55869	0.169064
$86$	2.72676	0.294034
$87$	−31.6522	−3.39347
$88$	3.72614	0.397208
$89$	8.42081	0.892604	0.446302	−	0.894883i	$-0.352741\pi$
$89$	8.42081	0.892604	0.446302	+	0.894883i	$0.352741\pi$
$90$	−19.1132	−2.01471
$91$	−8.19742	−0.859323
$92$	0.664257	0.0692535
$93$	29.5089	3.05993
$94$	−10.9786	−1.13235
$95$	5.21559	0.535108
$96$	3.32456	0.339311
$97$	−14.4293	−1.46507	−0.732535	−	0.680730i	$-0.761663\pi$
$97$	−14.4293	−1.46507	−0.732535	+	0.680730i	$0.761663\pi$
$98$	4.60061	0.464732
$99$	−30.0055	−3.01566
$100$	0.633570	0.0633570
$101$	5.02596	0.500102	0.250051	−	0.968233i	$-0.419553\pi$
$101$	5.02596	0.500102	0.250051	+	0.968233i	$0.419553\pi$
$102$	2.18324	0.216173
$103$	10.1311	0.998249	0.499124	−	0.866530i	$-0.333655\pi$
$103$	10.1311	0.998249	0.499124	+	0.866530i	$0.333655\pi$
$104$	−5.29209	−0.518932
$105$	12.2229	1.19284
$106$	12.1771	1.18274
$107$	−9.08797	−0.878567	−0.439283	−	0.898348i	$-0.644768\pi$
$107$	−9.08797	−0.878567	−0.439283	+	0.898348i	$0.644768\pi$
$108$	−16.7980	−1.61639
$109$	−9.45755	−0.905869	−0.452935	−	0.891544i	$-0.649623\pi$
$109$	−9.45755	−0.905869	−0.452935	+	0.891544i	$0.649623\pi$
$110$	8.84405	0.843247
$111$	15.5655	1.47741
$112$	−1.54900	−0.146366
$113$	−15.7655	−1.48309	−0.741547	−	0.670901i	$-0.765908\pi$
$113$	−15.7655	−1.48309	−0.741547	+	0.670901i	$0.765908\pi$
$114$	7.30543	0.684216
$115$	1.57662	0.147021
$116$	9.52073	0.883977
$117$	42.6156	3.93981
$118$	4.63655	0.426829
$119$	−1.01723	−0.0932490
$120$	7.89089	0.720336
$121$	2.88412	0.262192
$122$	2.71078	0.245423
$123$	−10.9758	−0.989651
$124$	−8.87603	−0.797091
$125$	−10.3638	−0.926965
$126$	12.4736	1.11124
$127$	−6.20055	−0.550210	−0.275105	−	0.961414i	$-0.588713\pi$
$127$	−6.20055	−0.550210	−0.275105	+	0.961414i	$0.588713\pi$
$128$	−1.00000	−0.0883883
$129$	9.06527	0.798152
$130$	−12.5608	−1.10166
$131$	−21.1384	−1.84687	−0.923437	−	0.383750i	$-0.874632\pi$
$131$	−21.1384	−1.84687	−0.923437	+	0.383750i	$0.874632\pi$
$132$	12.3878	1.07822
$133$	−3.40378	−0.295145
$134$	−9.75477	−0.842684
$135$	−39.8702	−3.43148
$136$	−0.656701	−0.0563117
$137$	−17.6345	−1.50661	−0.753307	−	0.657669i	$-0.771543\pi$
$137$	−17.6345	−1.50661	−0.753307	+	0.657669i	$0.771543\pi$
$138$	2.20836	0.187988
$139$	9.67879	0.820945	0.410472	−	0.911873i	$-0.365364\pi$
$139$	9.67879	0.820945	0.410472	+	0.911873i	$0.365364\pi$
$140$	−3.67656	−0.310726
$141$	−36.4989	−3.07376
$142$	−2.93701	−0.246468
$143$	−19.7191	−1.64899
$144$	8.05269	0.671058
$145$	22.5976	1.87663
$146$	5.43208	0.449562
$147$	15.2950	1.26151
$148$	−4.68196	−0.384855
$149$	20.2181	1.65633	0.828164	−	0.560486i	$-0.189386\pi$
$149$	20.2181	1.65633	0.828164	+	0.560486i	$0.189386\pi$
$150$	2.10634	0.171982
$151$	13.0434	1.06146	0.530730	−	0.847541i	$-0.321918\pi$
$151$	13.0434	1.06146	0.530730	+	0.847541i	$0.321918\pi$
$152$	−2.19741	−0.178234
$153$	5.28821	0.427527
$154$	−5.77177	−0.465103
$155$	−21.0674	−1.69217
$156$	−17.5939	−1.40864
$157$	−20.6171	−1.64542	−0.822712	−	0.568458i	$-0.807540\pi$
$157$	−20.6171	−1.64542	−0.822712	+	0.568458i	$0.807540\pi$
$158$	2.67589	0.212883
$159$	40.4834	3.21055
$160$	−2.37351	−0.187643
$161$	−1.02893	−0.0810911
$162$	−31.6878	−2.48963
$163$	6.20364	0.485907	0.242953	−	0.970038i	$-0.421884\pi$
$163$	6.20364	0.485907	0.242953	+	0.970038i	$0.421884\pi$
$164$	3.30142	0.257797
$165$	29.4026	2.28899
$166$	4.14703	0.321872
$167$	−1.20670	−0.0933775	−0.0466887	−	0.998909i	$-0.514867\pi$
$167$	−1.20670	−0.0933775	−0.0466887	+	0.998909i	$0.514867\pi$
$168$	−5.14973	−0.397310
$169$	15.0062	1.15432
$170$	−1.55869	−0.119546
$171$	17.6951	1.35318
$172$	−2.72676	−0.207913
$173$	16.0783	1.22241	0.611204	−	0.791473i	$-0.290685\pi$
$173$	16.0783	1.22241	0.611204	+	0.791473i	$0.290685\pi$
$174$	31.6522	2.39955
$175$	−0.981397	−0.0741866
$176$	−3.72614	−0.280868
$177$	15.4145	1.15862
$178$	−8.42081	−0.631166
$179$	−3.21919	−0.240613	−0.120307	−	0.992737i	$-0.538388\pi$
$179$	−3.21919	−0.240613	−0.120307	+	0.992737i	$0.538388\pi$
$180$	19.1132	1.42461
$181$	9.22008	0.685323	0.342661	−	0.939459i	$-0.388672\pi$
$181$	9.22008	0.685323	0.342661	+	0.939459i	$0.388672\pi$
$182$	8.19742	0.607633
$183$	9.01216	0.666198
$184$	−0.664257	−0.0489696
$185$	−11.1127	−0.817022
$186$	−29.5089	−2.16370
$187$	−2.44696	−0.178939
$188$	10.9786	0.800694
$189$	26.0200	1.89268
$190$	−5.21559	−0.378379
$191$	−12.2639	−0.887384	−0.443692	−	0.896179i	$-0.646332\pi$
$191$	−12.2639	−0.887384	−0.443692	+	0.896179i	$0.646332\pi$
$192$	−3.32456	−0.239929
$193$	3.76442	0.270969	0.135484	−	0.990779i	$-0.456741\pi$
$193$	3.76442	0.270969	0.135484	+	0.990779i	$0.456741\pi$
$194$	14.4293	1.03596
$195$	−41.7593	−2.99044
$196$	−4.60061	−0.328615
$197$	−14.3107	−1.01960	−0.509798	−	0.860294i	$-0.670280\pi$
$197$	−14.3107	−1.01960	−0.509798	+	0.860294i	$0.670280\pi$
$198$	30.0055	2.13240
$199$	24.6485	1.74729	0.873643	−	0.486567i	$-0.161751\pi$
$199$	24.6485	1.74729	0.873643	+	0.486567i	$0.161751\pi$
$200$	−0.633570	−0.0448002
$201$	−32.4303	−2.28746
$202$	−5.02596	−0.353625
$203$	−14.7476	−1.03508
$204$	−2.18324	−0.152857
$205$	7.83596	0.547287
$206$	−10.1311	−0.705869
$207$	5.34906	0.371785
$208$	5.29209	0.366940
$209$	−8.18787	−0.566367
$210$	−12.2229	−0.843463
$211$	25.8050	1.77649	0.888245	−	0.459370i	$-0.151925\pi$
$211$	25.8050	1.77649	0.888245	+	0.459370i	$0.151925\pi$
$212$	−12.1771	−0.836326
$213$	−9.76425	−0.669035
$214$	9.08797	0.621241
$215$	−6.47200	−0.441387
$216$	16.7980	1.14296
$217$	13.7489	0.933338
$218$	9.45755	0.640546
$219$	18.0593	1.22033
$220$	−8.84405	−0.596265
$221$	3.47532	0.233775
$222$	−15.5655	−1.04469
$223$	−1.74963	−0.117164	−0.0585819	−	0.998283i	$-0.518658\pi$
$223$	−1.74963	−0.117164	−0.0585819	+	0.998283i	$0.518658\pi$
$224$	1.54900	0.103497
$225$	5.10195	0.340130
$226$	15.7655	1.04871
$227$	−10.5074	−0.697401	−0.348701	−	0.937234i	$-0.613377\pi$
$227$	−10.5074	−0.697401	−0.348701	+	0.937234i	$0.613377\pi$
$228$	−7.30543	−0.483814
$229$	14.4544	0.955176	0.477588	−	0.878584i	$-0.341511\pi$
$229$	14.4544	0.955176	0.477588	+	0.878584i	$0.341511\pi$
$230$	−1.57662	−0.103959
$231$	−19.1886	−1.26252
$232$	−9.52073	−0.625066
$233$	28.5701	1.87169	0.935846	−	0.352409i	$-0.114637\pi$
$233$	28.5701	1.87169	0.935846	+	0.352409i	$0.114637\pi$
$234$	−42.6156	−2.78587
$235$	26.0578	1.69982
$236$	−4.63655	−0.301814
$237$	8.89617	0.577868
$238$	1.01723	0.0659370
$239$	−0.659452	−0.0426564	−0.0213282	−	0.999773i	$-0.506789\pi$
$239$	−0.659452	−0.0426564	−0.0213282	+	0.999773i	$0.506789\pi$
$240$	−7.89089	−0.509355
$241$	22.4029	1.44310	0.721549	−	0.692363i	$-0.243430\pi$
$241$	22.4029	1.44310	0.721549	+	0.692363i	$0.243430\pi$
$242$	−2.88412	−0.185398
$243$	−54.9540	−3.52530
$244$	−2.71078	−0.173540
$245$	−10.9196	−0.697629
$246$	10.9758	0.699789
$247$	11.6289	0.739929
$248$	8.87603	0.563629
$249$	13.7871	0.873719
$250$	10.3638	0.655463
$251$	10.1826	0.642721	0.321361	−	0.946957i	$-0.395860\pi$
$251$	10.1826	0.642721	0.321361	+	0.946957i	$0.395860\pi$
$252$	−12.4736	−0.785762
$253$	−2.47511	−0.155609
$254$	6.20055	0.389057
$255$	−5.18196	−0.324507
$256$	1.00000	0.0625000
$257$	−9.54125	−0.595167	−0.297583	−	0.954696i	$-0.596181\pi$
$257$	−9.54125	−0.595167	−0.297583	+	0.954696i	$0.596181\pi$
$258$	−9.06527	−0.564379
$259$	7.25234	0.450638
$260$	12.5608	0.778991
$261$	76.6675	4.74560
$262$	21.1384	1.30594
$263$	25.9333	1.59912	0.799558	−	0.600589i	$-0.205067\pi$
$263$	25.9333	1.59912	0.799558	+	0.600589i	$0.205067\pi$
$264$	−12.3878	−0.762415
$265$	−28.9025	−1.77547
$266$	3.40378	0.208699
$267$	−27.9955	−1.71329
$268$	9.75477	0.595867
$269$	−30.9772	−1.88872	−0.944358	−	0.328920i	$-0.893315\pi$
$269$	−30.9772	−1.88872	−0.944358	+	0.328920i	$0.893315\pi$
$270$	39.8702	2.42643
$271$	−22.9523	−1.39425	−0.697126	−	0.716949i	$-0.745538\pi$
$271$	−22.9523	−1.39425	−0.697126	+	0.716949i	$0.745538\pi$
$272$	0.656701	0.0398184
$273$	27.2528	1.64941
$274$	17.6345	1.06534
$275$	−2.36077	−0.142360
$276$	−2.20836	−0.132928
$277$	−1.54038	−0.0925523	−0.0462762	−	0.998929i	$-0.514735\pi$
$277$	−1.54038	−0.0925523	−0.0462762	+	0.998929i	$0.514735\pi$
$278$	−9.67879	−0.580495
$279$	−71.4760	−4.27915
$280$	3.67656	0.219717
$281$	11.9143	0.710747	0.355373	−	0.934724i	$-0.384354\pi$
$281$	11.9143	0.710747	0.355373	+	0.934724i	$0.384354\pi$
$282$	36.4989	2.17348
$283$	31.7330	1.88633	0.943165	−	0.332324i	$-0.107833\pi$
$283$	31.7330	1.88633	0.943165	+	0.332324i	$0.107833\pi$
$284$	2.93701	0.174279
$285$	−17.3395	−1.02711
$286$	19.7191	1.16601
$287$	−5.11388	−0.301863
$288$	−8.05269	−0.474510
$289$	−16.5687	−0.974632
$290$	−22.5976	−1.32698
$291$	47.9709	2.81211
$292$	−5.43208	−0.317888
$293$	14.1815	0.828490	0.414245	−	0.910165i	$-0.364046\pi$
$293$	14.1815	0.828490	0.414245	+	0.910165i	$0.364046\pi$
$294$	−15.2950	−0.892023
$295$	−11.0049	−0.640731
$296$	4.68196	0.272134
$297$	62.5916	3.63193
$298$	−20.2181	−1.17120
$299$	3.51530	0.203295
$300$	−2.10634	−0.121610
$301$	4.22373	0.243452
$302$	−13.0434	−0.750566
$303$	−16.7091	−0.959913
$304$	2.19741	0.126030
$305$	−6.43408	−0.368414
$306$	−5.28821	−0.302307
$307$	−6.84866	−0.390873	−0.195437	−	0.980716i	$-0.562612\pi$
$307$	−6.84866	−0.390873	−0.195437	+	0.980716i	$0.562612\pi$
$308$	5.77177	0.328877
$309$	−33.6815	−1.91607
$310$	21.0674	1.19655
$311$	−1.30641	−0.0740798	−0.0370399	−	0.999314i	$-0.511793\pi$
$311$	−1.30641	−0.0740798	−0.0370399	+	0.999314i	$0.511793\pi$
$312$	17.5939	0.996056
$313$	10.9405	0.618393	0.309197	−	0.950998i	$-0.399940\pi$
$313$	10.9405	0.618393	0.309197	+	0.950998i	$0.399940\pi$
$314$	20.6171	1.16349
$315$	−29.6062	−1.66812
$316$	−2.67589	−0.150531
$317$	34.1638	1.91883	0.959416	−	0.281994i	$-0.0909958\pi$
$317$	34.1638	1.91883	0.959416	+	0.281994i	$0.0909958\pi$
$318$	−40.4834	−2.27020
$319$	−35.4756	−1.98625
$320$	2.37351	0.132683
$321$	30.2135	1.68635
$322$	1.02893	0.0573400
$323$	1.44304	0.0802931
$324$	31.6878	1.76043
$325$	3.35291	0.185986
$326$	−6.20364	−0.343588
$327$	31.4422	1.73876
$328$	−3.30142	−0.182290
$329$	−17.0057	−0.937557
$330$	−29.4026	−1.61856
$331$	−11.0770	−0.608848	−0.304424	−	0.952537i	$-0.598464\pi$
$331$	−11.0770	−0.608848	−0.304424	+	0.952537i	$0.598464\pi$
$332$	−4.14703	−0.227598
$333$	−37.7024	−2.06608
$334$	1.20670	0.0660279
$335$	23.1531	1.26499
$336$	5.14973	0.280941
$337$	17.1643	0.934997	0.467499	−	0.883994i	$-0.345155\pi$
$337$	17.1643	0.934997	0.467499	+	0.883994i	$0.345155\pi$
$338$	−15.0062	−0.816229
$339$	52.4134	2.84670
$340$	1.55869	0.0845318
$341$	33.0733	1.79102
$342$	−17.6951	−0.956841
$343$	17.9693	0.970251
$344$	2.72676	0.147017
$345$	−5.24158	−0.282197
$346$	−16.0783	−0.864373
$347$	−20.6108	−1.10645	−0.553223	−	0.833033i	$-0.686602\pi$
$347$	−20.6108	−1.10645	−0.553223	+	0.833033i	$0.686602\pi$
$348$	−31.6522	−1.69674
$349$	−5.50037	−0.294428	−0.147214	−	0.989105i	$-0.547031\pi$
$349$	−5.50037	−0.294428	−0.147214	+	0.989105i	$0.547031\pi$
$350$	0.981397	0.0524579
$351$	−88.8964	−4.74494
$352$	3.72614	0.198604
$353$	−31.4752	−1.67525	−0.837627	−	0.546243i	$-0.816058\pi$
$353$	−31.4752	−1.67525	−0.837627	+	0.546243i	$0.816058\pi$
$354$	−15.4145	−0.819270
$355$	6.97102	0.369984
$356$	8.42081	0.446302
$357$	3.38183	0.178985
$358$	3.21919	0.170139
$359$	−19.1175	−1.00898	−0.504491	−	0.863417i	$-0.668320\pi$
$359$	−19.1175	−1.00898	−0.504491	+	0.863417i	$0.668320\pi$
$360$	−19.1132	−1.00735
$361$	−14.1714	−0.745862
$362$	−9.22008	−0.484597
$363$	−9.58841	−0.503261
$364$	−8.19742	−0.429661
$365$	−12.8931	−0.674857
$366$	−9.01216	−0.471073
$367$	0.437869	0.0228566	0.0114283	−	0.999935i	$-0.496362\pi$
$367$	0.437869	0.0228566	0.0114283	+	0.999935i	$0.496362\pi$
$368$	0.664257	0.0346268
$369$	26.5853	1.38398
$370$	11.1127	0.577722
$371$	18.8622	0.979279
$372$	29.5089	1.52997
$373$	−32.6681	−1.69149	−0.845745	−	0.533588i	$-0.820843\pi$
$373$	−32.6681	−1.69149	−0.845745	+	0.533588i	$0.820843\pi$
$374$	2.44696	0.126529
$375$	34.4550	1.77925
$376$	−10.9786	−0.566176
$377$	50.3845	2.59494
$378$	−26.0200	−1.33832
$379$	−11.3540	−0.583216	−0.291608	−	0.956538i	$-0.594190\pi$
$379$	−11.3540	−0.583216	−0.291608	+	0.956538i	$0.594190\pi$
$380$	5.21559	0.267554
$381$	20.6141	1.05609
$382$	12.2639	0.627475
$383$	−2.24894	−0.114915	−0.0574576	−	0.998348i	$-0.518299\pi$
$383$	−2.24894	−0.114915	−0.0574576	+	0.998348i	$0.518299\pi$
$384$	3.32456	0.169656
$385$	13.6994	0.698185
$386$	−3.76442	−0.191604
$387$	−21.9577	−1.11617
$388$	−14.4293	−0.732535
$389$	−5.66524	−0.287239	−0.143620	−	0.989633i	$-0.545874\pi$
$389$	−5.66524	−0.287239	−0.143620	+	0.989633i	$0.545874\pi$
$390$	41.7593	2.11456
$391$	0.436218	0.0220605
$392$	4.60061	0.232366
$393$	70.2760	3.54495
$394$	14.3107	0.720963
$395$	−6.35127	−0.319567
$396$	−30.0055	−1.50783
$397$	−27.1393	−1.36208	−0.681041	−	0.732245i	$-0.738473\pi$
$397$	−27.1393	−1.36208	−0.681041	+	0.732245i	$0.738473\pi$
$398$	−24.6485	−1.23552
$399$	11.3161	0.566512
$400$	0.633570	0.0316785
$401$	−33.1607	−1.65597	−0.827983	−	0.560753i	$-0.810512\pi$
$401$	−33.1607	−1.65597	−0.827983	+	0.560753i	$0.810512\pi$
$402$	32.4303	1.61748
$403$	−46.9727	−2.33988
$404$	5.02596	0.250051
$405$	75.2114	3.73729
$406$	14.7476	0.731909
$407$	17.4456	0.864749
$408$	2.18324	0.108087
$409$	17.7516	0.877759	0.438880	−	0.898546i	$-0.355375\pi$
$409$	17.7516	0.877759	0.438880	+	0.898546i	$0.355375\pi$
$410$	−7.83596	−0.386991
$411$	58.6268	2.89185
$412$	10.1311	0.499124
$413$	7.18199	0.353403
$414$	−5.34906	−0.262892
$415$	−9.84304	−0.483176
$416$	−5.29209	−0.259466
$417$	−32.1777	−1.57575
$418$	8.18787	0.400482
$419$	−8.07923	−0.394696	−0.197348	−	0.980333i	$-0.563233\pi$
$419$	−8.07923	−0.394696	−0.197348	+	0.980333i	$0.563233\pi$
$420$	12.2229	0.596419
$421$	15.5078	0.755805	0.377902	−	0.925845i	$-0.376645\pi$
$421$	15.5078	0.755805	0.377902	+	0.925845i	$0.376645\pi$
$422$	−25.8050	−1.25617
$423$	88.4070	4.29850
$424$	12.1771	0.591372
$425$	0.416066	0.0201822
$426$	9.76425	0.473079
$427$	4.19899	0.203203
$428$	−9.08797	−0.439283
$429$	65.5572	3.16513
$430$	6.47200	0.312107
$431$	−3.27709	−0.157852	−0.0789259	−	0.996880i	$-0.525149\pi$
$431$	−3.27709	−0.157852	−0.0789259	+	0.996880i	$0.525149\pi$
$432$	−16.7980	−0.808193
$433$	−27.4552	−1.31941	−0.659707	−	0.751523i	$-0.729320\pi$
$433$	−27.4552	−1.31941	−0.659707	+	0.751523i	$0.729320\pi$
$434$	−13.7489	−0.659970
$435$	−75.1270	−3.60206
$436$	−9.45755	−0.452935
$437$	1.45965	0.0698243
$438$	−18.0593	−0.862905
$439$	18.7295	0.893912	0.446956	−	0.894556i	$-0.352508\pi$
$439$	18.7295	0.893912	0.446956	+	0.894556i	$0.352508\pi$
$440$	8.84405	0.421623
$441$	−37.0473	−1.76416
$442$	−3.47532	−0.165304
$443$	−2.23604	−0.106238	−0.0531188	−	0.998588i	$-0.516916\pi$
$443$	−2.23604	−0.106238	−0.0531188	+	0.998588i	$0.516916\pi$
$444$	15.5655	0.738704
$445$	19.9869	0.947470
$446$	1.74963	0.0828473
$447$	−67.2161	−3.17921
$448$	−1.54900	−0.0731831
$449$	−36.1923	−1.70802	−0.854009	−	0.520258i	$-0.825836\pi$
$449$	−36.1923	−1.70802	−0.854009	+	0.520258i	$0.825836\pi$
$450$	−5.10195	−0.240508
$451$	−12.3015	−0.579257
$452$	−15.7655	−0.741547
$453$	−43.3637	−2.03740
$454$	10.5074	0.493137
$455$	−19.4567	−0.912144
$456$	7.30543	0.342108
$457$	−41.1830	−1.92646	−0.963230	−	0.268680i	$-0.913413\pi$
$457$	−41.1830	−1.92646	−0.963230	+	0.268680i	$0.913413\pi$
$458$	−14.4544	−0.675411
$459$	−11.0313	−0.514895
$460$	1.57662	0.0735104
$461$	7.36302	0.342930	0.171465	−	0.985190i	$-0.445150\pi$
$461$	7.36302	0.342930	0.171465	+	0.985190i	$0.445150\pi$
$462$	19.1886	0.892734
$463$	13.9817	0.649784	0.324892	−	0.945751i	$-0.394672\pi$
$463$	13.9817	0.649784	0.324892	+	0.945751i	$0.394672\pi$
$464$	9.52073	0.441989
$465$	70.0398	3.24802
$466$	−28.5701	−1.32349
$467$	−29.3165	−1.35661	−0.678303	−	0.734782i	$-0.737284\pi$
$467$	−29.3165	−1.35661	−0.678303	+	0.734782i	$0.737284\pi$
$468$	42.6156	1.96990
$469$	−15.1101	−0.697719
$470$	−26.0578	−1.20196
$471$	68.5428	3.15828
$472$	4.63655	0.213414
$473$	10.1603	0.467170
$474$	−8.89617	−0.408615
$475$	1.39221	0.0638792
$476$	−1.01723	−0.0466245
$477$	−98.0583	−4.48978
$478$	0.659452	0.0301626
$479$	27.3079	1.24773	0.623865	−	0.781532i	$-0.285562\pi$
$479$	27.3079	1.24773	0.623865	+	0.781532i	$0.285562\pi$
$480$	7.89089	0.360168
$481$	−24.7774	−1.12975
$482$	−22.4029	−1.02042
$483$	3.42074	0.155649
$484$	2.88412	0.131096
$485$	−34.2481	−1.55512
$486$	54.9540	2.49276
$487$	−13.7250	−0.621937	−0.310969	−	0.950420i	$-0.600653\pi$
$487$	−13.7250	−0.621937	−0.310969	+	0.950420i	$0.600653\pi$
$488$	2.71078	0.122711
$489$	−20.6244	−0.932667
$490$	10.9196	0.493298
$491$	27.8718	1.25784	0.628918	−	0.777472i	$-0.283498\pi$
$491$	27.8718	1.25784	0.628918	+	0.777472i	$0.283498\pi$
$492$	−10.9758	−0.494825
$493$	6.25227	0.281588
$494$	−11.6289	−0.523209
$495$	−71.2184	−3.20103
$496$	−8.87603	−0.398546
$497$	−4.54941	−0.204069
$498$	−13.7871	−0.617813
$499$	−28.8904	−1.29331	−0.646656	−	0.762782i	$-0.723833\pi$
$499$	−28.8904	−1.29331	−0.646656	+	0.762782i	$0.723833\pi$
$500$	−10.3638	−0.463483
$501$	4.01176	0.179232
$502$	−10.1826	−0.454472
$503$	−35.5763	−1.58627	−0.793134	−	0.609047i	$-0.791552\pi$
$503$	−35.5763	−1.58627	−0.793134	+	0.609047i	$0.791552\pi$
$504$	12.4736	0.555618
$505$	11.9292	0.530842
$506$	2.47511	0.110032
$507$	−49.8890	−2.21565
$508$	−6.20055	−0.275105
$509$	−21.6988	−0.961784	−0.480892	−	0.876780i	$-0.659687\pi$
$509$	−21.6988	−0.961784	−0.480892	+	0.876780i	$0.659687\pi$
$510$	5.18196	0.229461
$511$	8.41427	0.372225
$512$	−1.00000	−0.0441942
$513$	−36.9121	−1.62971
$514$	9.54125	0.420846
$515$	24.0464	1.05961
$516$	9.06527	0.399076
$517$	−40.9077	−1.79912
$518$	−7.25234	−0.318649
$519$	−53.4532	−2.34633
$520$	−12.5608	−0.550829
$521$	−0.0939307	−0.00411518	−0.00205759	−	0.999998i	$-0.500655\pi$
$521$	−0.0939307	−0.00411518	−0.00205759	+	0.999998i	$0.500655\pi$
$522$	−76.6675	−3.35565
$523$	27.0037	1.18079	0.590395	−	0.807114i	$-0.298972\pi$
$523$	27.0037	1.18079	0.590395	+	0.807114i	$0.298972\pi$
$524$	−21.1384	−0.923437
$525$	3.26271	0.142396
$526$	−25.9333	−1.13075
$527$	−5.82890	−0.253911
$528$	12.3878	0.539109
$529$	−22.5588	−0.980816
$530$	28.9025	1.25544
$531$	−37.3367	−1.62027
$532$	−3.40378	−0.147573
$533$	17.4714	0.756770
$534$	27.9955	1.21148
$535$	−21.5704	−0.932571
$536$	−9.75477	−0.421342
$537$	10.7024	0.461841
$538$	30.9772	1.33552
$539$	17.1425	0.738381
$540$	−39.8702	−1.71574
$541$	−13.3951	−0.575899	−0.287949	−	0.957646i	$-0.592973\pi$
$541$	−13.3951	−0.575899	−0.287949	+	0.957646i	$0.592973\pi$
$542$	22.9523	0.985885
$543$	−30.6527	−1.31543
$544$	−0.656701	−0.0281558
$545$	−22.4476	−0.961551
$546$	−27.2528	−1.16631
$547$	0.788407	0.0337099	0.0168549	−	0.999858i	$-0.494635\pi$
$547$	0.788407	0.0337099	0.0168549	+	0.999858i	$0.494635\pi$
$548$	−17.6345	−0.753307
$549$	−21.8291	−0.931644
$550$	2.36077	0.100664
$551$	20.9210	0.891263
$552$	2.20836	0.0939941
$553$	4.14495	0.176261
$554$	1.54038	0.0654444
$555$	36.9448	1.56822
$556$	9.67879	0.410472
$557$	−26.2326	−1.11151	−0.555756	−	0.831345i	$-0.687571\pi$
$557$	−26.2326	−1.11151	−0.555756	+	0.831345i	$0.687571\pi$
$558$	71.4760	3.02582
$559$	−14.4302	−0.610334
$560$	−3.67656	−0.155363
$561$	8.13506	0.343463
$562$	−11.9143	−0.502574
$563$	6.23149	0.262626	0.131313	−	0.991341i	$-0.458081\pi$
$563$	6.23149	0.262626	0.131313	+	0.991341i	$0.458081\pi$
$564$	−36.4989	−1.53688
$565$	−37.4197	−1.57426
$566$	−31.7330	−1.33384
$567$	−49.0842	−2.06134
$568$	−2.93701	−0.123234
$569$	−36.8888	−1.54646	−0.773229	−	0.634127i	$-0.781360\pi$
$569$	−36.8888	−1.54646	−0.773229	+	0.634127i	$0.781360\pi$
$570$	17.3395	0.726273
$571$	−34.3202	−1.43625	−0.718127	−	0.695912i	$-0.755000\pi$
$571$	−34.3202	−1.43625	−0.718127	+	0.695912i	$0.755000\pi$
$572$	−19.7191	−0.824495
$573$	40.7720	1.70328
$574$	5.11388	0.213449
$575$	0.420853	0.0175508
$576$	8.05269	0.335529
$577$	−2.04227	−0.0850210	−0.0425105	−	0.999096i	$-0.513536\pi$
$577$	−2.04227	−0.0850210	−0.0425105	+	0.999096i	$0.513536\pi$
$578$	16.5687	0.689169
$579$	−12.5150	−0.520107
$580$	22.5976	0.938314
$581$	6.42373	0.266501
$582$	−47.9709	−1.98846
$583$	45.3735	1.87918
$584$	5.43208	0.224781
$585$	101.149	4.18198
$586$	−14.1815	−0.585831
$587$	−9.44733	−0.389933	−0.194966	−	0.980810i	$-0.562460\pi$
$587$	−9.44733	−0.389933	−0.194966	+	0.980810i	$0.562460\pi$
$588$	15.2950	0.630756
$589$	−19.5043	−0.803661
$590$	11.0049	0.453065
$591$	47.5768	1.95705
$592$	−4.68196	−0.192427
$593$	16.8399	0.691534	0.345767	−	0.938320i	$-0.387619\pi$
$593$	16.8399	0.691534	0.345767	+	0.938320i	$0.387619\pi$
$594$	−62.5916	−2.56817
$595$	−2.41440	−0.0989808
$596$	20.2181	0.828164
$597$	−81.9454	−3.35380
$598$	−3.51530	−0.143751
$599$	−3.32419	−0.135823	−0.0679113	−	0.997691i	$-0.521633\pi$
$599$	−3.32419	−0.135823	−0.0679113	+	0.997691i	$0.521633\pi$
$600$	2.10634	0.0859910
$601$	37.4243	1.52657	0.763285	−	0.646062i	$-0.223585\pi$
$601$	37.4243	1.52657	0.763285	+	0.646062i	$0.223585\pi$
$602$	−4.22373	−0.172147
$603$	78.5522	3.19889
$604$	13.0434	0.530730
$605$	6.84549	0.278309
$606$	16.7091	0.678761
$607$	11.5406	0.468419	0.234209	−	0.972186i	$-0.424750\pi$
$607$	11.5406	0.468419	0.234209	+	0.972186i	$0.424750\pi$
$608$	−2.19741	−0.0891168
$609$	49.0291	1.98676
$610$	6.43408	0.260508
$611$	58.0995	2.35045
$612$	5.28821	0.213763
$613$	8.78959	0.355008	0.177504	−	0.984120i	$-0.443198\pi$
$613$	8.78959	0.355008	0.177504	+	0.984120i	$0.443198\pi$
$614$	6.84866	0.276389
$615$	−26.0511	−1.05048
$616$	−5.77177	−0.232551
$617$	−36.3180	−1.46211	−0.731053	−	0.682321i	$-0.760971\pi$
$617$	−36.3180	−1.46211	−0.731053	+	0.682321i	$0.760971\pi$
$618$	33.6815	1.35487
$619$	35.8825	1.44224	0.721121	−	0.692809i	$-0.243627\pi$
$619$	35.8825	1.44224	0.721121	+	0.692809i	$0.243627\pi$
$620$	−21.0674	−0.846087
$621$	−11.1582	−0.447762
$622$	1.30641	0.0523823
$623$	−13.0438	−0.522588
$624$	−17.5939	−0.704318
$625$	−27.7664	−1.11066
$626$	−10.9405	−0.437270
$627$	27.2210	1.08710
$628$	−20.6171	−0.822712
$629$	−3.07465	−0.122594
$630$	29.6062	1.17954
$631$	30.1247	1.19925	0.599623	−	0.800283i	$-0.295317\pi$
$631$	30.1247	1.19925	0.599623	+	0.800283i	$0.295317\pi$
$632$	2.67589	0.106441
$633$	−85.7903	−3.40986
$634$	−34.1638	−1.35682
$635$	−14.7171	−0.584030
$636$	40.4834	1.60527
$637$	−24.3469	−0.964657
$638$	35.4756	1.40449
$639$	23.6508	0.935611
$640$	−2.37351	−0.0938214
$641$	25.7345	1.01645	0.508227	−	0.861223i	$-0.330301\pi$
$641$	25.7345	1.01645	0.508227	+	0.861223i	$0.330301\pi$
$642$	−30.2135	−1.19243
$643$	−10.4869	−0.413563	−0.206782	−	0.978387i	$-0.566299\pi$
$643$	−10.4869	−0.413563	−0.206782	+	0.978387i	$0.566299\pi$
$644$	−1.02893	−0.0405455
$645$	21.5165	0.847213
$646$	−1.44304	−0.0567758
$647$	−28.1274	−1.10580	−0.552901	−	0.833247i	$-0.686479\pi$
$647$	−28.1274	−1.10580	−0.552901	+	0.833247i	$0.686479\pi$
$648$	−31.6878	−1.24481
$649$	17.2764	0.678159
$650$	−3.35291	−0.131512
$651$	−45.7091	−1.79148
$652$	6.20364	0.242953
$653$	8.02500	0.314043	0.157021	−	0.987595i	$-0.449811\pi$
$653$	8.02500	0.314043	0.157021	+	0.987595i	$0.449811\pi$
$654$	−31.4422	−1.22949
$655$	−50.1724	−1.96040
$656$	3.30142	0.128899
$657$	−43.7429	−1.70657
$658$	17.0057	0.662953
$659$	24.0531	0.936976	0.468488	−	0.883470i	$-0.344799\pi$
$659$	24.0531	0.936976	0.468488	+	0.883470i	$0.344799\pi$
$660$	29.4026	1.14449
$661$	14.0682	0.547190	0.273595	−	0.961845i	$-0.411787\pi$
$661$	14.0682	0.547190	0.273595	+	0.961845i	$0.411787\pi$
$662$	11.0770	0.430520
$663$	−11.5539	−0.448717
$664$	4.14703	0.160936
$665$	−8.07892	−0.313287
$666$	37.7024	1.46094
$667$	6.32421	0.244874
$668$	−1.20670	−0.0466887
$669$	5.81674	0.224888
$670$	−23.1531	−0.894482
$671$	10.1008	0.389935
$672$	−5.14973	−0.198655
$673$	−31.5491	−1.21613	−0.608064	−	0.793888i	$-0.708054\pi$
$673$	−31.5491	−1.21613	−0.608064	+	0.793888i	$0.708054\pi$
$674$	−17.1643	−0.661143
$675$	−10.6427	−0.409638
$676$	15.0062	0.577161
$677$	−2.17341	−0.0835309	−0.0417654	−	0.999127i	$-0.513298\pi$
$677$	−2.17341	−0.0835309	−0.0417654	+	0.999127i	$0.513298\pi$
$678$	−52.4134	−2.01292
$679$	22.3508	0.857747
$680$	−1.55869	−0.0597730
$681$	34.9325	1.33862
$682$	−33.0733	−1.26644
$683$	−41.3641	−1.58275	−0.791376	−	0.611330i	$-0.790635\pi$
$683$	−41.3641	−1.58275	−0.791376	+	0.611330i	$0.790635\pi$
$684$	17.6951	0.676589
$685$	−41.8557	−1.59922
$686$	−17.9693	−0.686071
$687$	−48.0546	−1.83340
$688$	−2.72676	−0.103957
$689$	−64.4422	−2.45505
$690$	5.24158	0.199543
$691$	−8.36994	−0.318407	−0.159204	−	0.987246i	$-0.550893\pi$
$691$	−8.36994	−0.318407	−0.159204	+	0.987246i	$0.550893\pi$
$692$	16.0783	0.611204
$693$	46.4783	1.76556
$694$	20.6108	0.782375
$695$	22.9728	0.871406
$696$	31.6522	1.19977
$697$	2.16804	0.0821206
$698$	5.50037	0.208192
$699$	−94.9831	−3.59259
$700$	−0.981397	−0.0370933
$701$	25.5006	0.963145	0.481572	−	0.876406i	$-0.340066\pi$
$701$	25.5006	0.963145	0.481572	+	0.876406i	$0.340066\pi$
$702$	88.8964	3.35518
$703$	−10.2882	−0.388027
$704$	−3.72614	−0.140434
$705$	−86.6306	−3.26270
$706$	31.4752	1.18458
$707$	−7.78519	−0.292792
$708$	15.4145	0.579312
$709$	45.5582	1.71097	0.855487	−	0.517824i	$-0.173258\pi$
$709$	45.5582	1.71097	0.855487	+	0.517824i	$0.173258\pi$
$710$	−6.97102	−0.261618
$711$	−21.5482	−0.808119
$712$	−8.42081	−0.315583
$713$	−5.89596	−0.220806
$714$	−3.38183	−0.126562
$715$	−46.8035	−1.75035
$716$	−3.21919	−0.120307
$717$	2.19239	0.0818762
$718$	19.1175	0.713458
$719$	7.64761	0.285208	0.142604	−	0.989780i	$-0.454453\pi$
$719$	7.64761	0.285208	0.142604	+	0.989780i	$0.454453\pi$
$720$	19.1132	0.712306
$721$	−15.6931	−0.584440
$722$	14.1714	0.527404
$723$	−74.4798	−2.76993
$724$	9.22008	0.342661
$725$	6.03205	0.224025
$726$	9.58841	0.355859
$727$	−7.30522	−0.270935	−0.135468	−	0.990782i	$-0.543254\pi$
$727$	−7.30522	−0.270935	−0.135468	+	0.990782i	$0.543254\pi$
$728$	8.19742	0.303816
$729$	87.6345	3.24572
$730$	12.8931	0.477196
$731$	−1.79066	−0.0662301
$732$	9.01216	0.333099
$733$	−2.04879	−0.0756736	−0.0378368	−	0.999284i	$-0.512047\pi$
$733$	−2.04879	−0.0756736	−0.0378368	+	0.999284i	$0.512047\pi$
$734$	−0.437869	−0.0161620
$735$	36.3029	1.33905
$736$	−0.664257	−0.0244848
$737$	−36.3476	−1.33888
$738$	−26.5853	−0.978619
$739$	−9.61229	−0.353594	−0.176797	−	0.984247i	$-0.556574\pi$
$739$	−9.61229	−0.353594	−0.176797	+	0.984247i	$0.556574\pi$
$740$	−11.1127	−0.408511
$741$	−38.6610	−1.42025
$742$	−18.8622	−0.692455
$743$	−10.4724	−0.384196	−0.192098	−	0.981376i	$-0.561529\pi$
$743$	−10.4724	−0.384196	−0.192098	+	0.981376i	$0.561529\pi$
$744$	−29.5089	−1.08185
$745$	47.9879	1.75814
$746$	32.6681	1.19606
$747$	−33.3948	−1.22185
$748$	−2.44696	−0.0894697
$749$	14.0772	0.514370
$750$	−34.4550	−1.25812
$751$	−11.4780	−0.418840	−0.209420	−	0.977826i	$-0.567158\pi$
$751$	−11.4780	−0.418840	−0.209420	+	0.977826i	$0.567158\pi$
$752$	10.9786	0.400347
$753$	−33.8527	−1.23366
$754$	−50.3845	−1.83490
$755$	30.9588	1.12671
$756$	26.0200	0.946338
$757$	−24.1425	−0.877472	−0.438736	−	0.898616i	$-0.644574\pi$
$757$	−24.1425	−0.877472	−0.438736	+	0.898616i	$0.644574\pi$
$758$	11.3540	0.412396
$759$	8.22866	0.298681
$760$	−5.21559	−0.189189
$761$	15.0396	0.545184	0.272592	−	0.962130i	$-0.412119\pi$
$761$	15.0396	0.545184	0.272592	+	0.962130i	$0.412119\pi$
$762$	−20.6141	−0.746770
$763$	14.6497	0.530355
$764$	−12.2639	−0.443692
$765$	12.5516	0.453806
$766$	2.24894	0.0812573
$767$	−24.5370	−0.885980
$768$	−3.32456	−0.119965
$769$	−9.29786	−0.335290	−0.167645	−	0.985847i	$-0.553616\pi$
$769$	−9.29786	−0.335290	−0.167645	+	0.985847i	$0.553616\pi$
$770$	−13.6994	−0.493691
$771$	31.7204	1.14238
$772$	3.76442	0.135484
$773$	−16.8056	−0.604457	−0.302229	−	0.953235i	$-0.597731\pi$
$773$	−16.8056	−0.604457	−0.302229	+	0.953235i	$0.597731\pi$
$774$	21.9577	0.789255
$775$	−5.62359	−0.202005
$776$	14.4293	0.517980
$777$	−24.1108	−0.864971
$778$	5.66524	0.203109
$779$	7.25458	0.259922
$780$	−41.7593	−1.49522
$781$	−10.9437	−0.391596
$782$	−0.436218	−0.0155991
$783$	−159.929	−5.71540
$784$	−4.60061	−0.164308
$785$	−48.9350	−1.74656
$786$	−70.2760	−2.50666
$787$	22.3736	0.797532	0.398766	−	0.917053i	$-0.369439\pi$
$787$	22.3736	0.797532	0.398766	+	0.917053i	$0.369439\pi$
$788$	−14.3107	−0.509798
$789$	−86.2168	−3.06940
$790$	6.35127	0.225968
$791$	24.4207	0.868300
$792$	30.0055	1.06620
$793$	−14.3457	−0.509431
$794$	27.1393	0.963138
$795$	96.0880	3.40789
$796$	24.6485	0.873643
$797$	12.4217	0.439999	0.219999	−	0.975500i	$-0.429394\pi$
$797$	12.4217	0.439999	0.219999	+	0.975500i	$0.429394\pi$
$798$	−11.3161	−0.400585
$799$	7.20963	0.255059
$800$	−0.633570	−0.0224001
$801$	67.8102	2.39595
$802$	33.1607	1.17095
$803$	20.2407	0.714278
$804$	−32.4303	−1.14373
$805$	−2.44218	−0.0860756
$806$	46.9727	1.65454
$807$	102.986	3.62527
$808$	−5.02596	−0.176813
$809$	−19.2161	−0.675602	−0.337801	−	0.941218i	$-0.609683\pi$
$809$	−19.2161	−0.675602	−0.337801	+	0.941218i	$0.609683\pi$
$810$	−75.2114	−2.64266
$811$	−12.4782	−0.438169	−0.219085	−	0.975706i	$-0.570307\pi$
$811$	−12.4782	−0.438169	−0.219085	+	0.975706i	$0.570307\pi$
$812$	−14.7476	−0.517538
$813$	76.3062	2.67618
$814$	−17.4456	−0.611470
$815$	14.7244	0.515775
$816$	−2.18324	−0.0764287
$817$	−5.99181	−0.209627
$818$	−17.7516	−0.620669
$819$	−66.0113	−2.30662
$820$	7.83596	0.273644
$821$	−7.95099	−0.277491	−0.138746	−	0.990328i	$-0.544307\pi$
$821$	−7.95099	−0.277491	−0.138746	+	0.990328i	$0.544307\pi$
$822$	−58.6268	−2.04485
$823$	−8.20828	−0.286123	−0.143061	−	0.989714i	$-0.545695\pi$
$823$	−8.20828	−0.286123	−0.143061	+	0.989714i	$0.545695\pi$
$824$	−10.1311	−0.352934
$825$	7.84852	0.273250
$826$	−7.18199	−0.249893
$827$	−49.4249	−1.71867	−0.859337	−	0.511410i	$-0.829123\pi$
$827$	−49.4249	−1.71867	−0.859337	+	0.511410i	$0.829123\pi$
$828$	5.34906	0.185893
$829$	35.2495	1.22426	0.612132	−	0.790755i	$-0.290312\pi$
$829$	35.2495	1.22426	0.612132	+	0.790755i	$0.290312\pi$
$830$	9.84304	0.341657
$831$	5.12108	0.177648
$832$	5.29209	0.183470
$833$	−3.02123	−0.104679
$834$	32.1777	1.11422
$835$	−2.86413	−0.0991172
$836$	−8.18787	−0.283183
$837$	149.099	5.15363
$838$	8.07923	0.279092
$839$	−20.8392	−0.719448	−0.359724	−	0.933059i	$-0.617129\pi$
$839$	−20.8392	−0.719448	−0.359724	+	0.933059i	$0.617129\pi$
$840$	−12.2229	−0.421732
$841$	61.6443	2.12566
$842$	−15.5078	−0.534435
$843$	−39.6097	−1.36423
$844$	25.8050	0.888245
$845$	35.6174	1.22528
$846$	−88.4070	−3.03950
$847$	−4.46748	−0.153504
$848$	−12.1771	−0.418163
$849$	−105.498	−3.62069
$850$	−0.416066	−0.0142710
$851$	−3.11002	−0.106610
$852$	−9.76425	−0.334518
$853$	−23.6340	−0.809214	−0.404607	−	0.914491i	$-0.632592\pi$
$853$	−23.6340	−0.809214	−0.404607	+	0.914491i	$0.632592\pi$
$854$	−4.19899	−0.143686
$855$	41.9995	1.43635
$856$	9.08797	0.310620
$857$	−37.6634	−1.28656	−0.643278	−	0.765633i	$-0.722426\pi$
$857$	−37.6634	−1.28656	−0.643278	+	0.765633i	$0.722426\pi$
$858$	−65.5572	−2.23808
$859$	16.0217	0.546654	0.273327	−	0.961921i	$-0.411876\pi$
$859$	16.0217	0.546654	0.273327	+	0.961921i	$0.411876\pi$
$860$	−6.47200	−0.220693
$861$	17.0014	0.579406
$862$	3.27709	0.111618
$863$	53.8928	1.83453	0.917266	−	0.398276i	$-0.130391\pi$
$863$	53.8928	1.83453	0.917266	+	0.398276i	$0.130391\pi$
$864$	16.7980	0.571479
$865$	38.1620	1.29755
$866$	27.4552	0.932966
$867$	55.0838	1.87074
$868$	13.7489	0.466669
$869$	9.97075	0.338235
$870$	75.1270	2.54704
$871$	51.6231	1.74918
$872$	9.45755	0.320273
$873$	−116.194	−3.93258
$874$	−1.45965	−0.0493733
$875$	16.0534	0.542706
$876$	18.0593	0.610166
$877$	23.3215	0.787510	0.393755	−	0.919215i	$-0.371176\pi$
$877$	23.3215	0.787510	0.393755	+	0.919215i	$0.371176\pi$
$878$	−18.7295	−0.632091
$879$	−47.1471	−1.59023
$880$	−8.84405	−0.298133
$881$	−5.09298	−0.171587	−0.0857934	−	0.996313i	$-0.527342\pi$
$881$	−5.09298	−0.171587	−0.0857934	+	0.996313i	$0.527342\pi$
$882$	37.0473	1.24745
$883$	22.8824	0.770053	0.385026	−	0.922906i	$-0.374192\pi$
$883$	22.8824	0.770053	0.385026	+	0.922906i	$0.374192\pi$
$884$	3.47532	0.116888
$885$	36.5865	1.22984
$886$	2.23604	0.0751213
$887$	−45.0522	−1.51270	−0.756352	−	0.654165i	$-0.773020\pi$
$887$	−45.0522	−1.51270	−0.756352	+	0.654165i	$0.773020\pi$
$888$	−15.5655	−0.522343
$889$	9.60462	0.322129
$890$	−19.9869	−0.669962
$891$	−118.073	−3.95560
$892$	−1.74963	−0.0585819
$893$	24.1244	0.807293
$894$	67.2161	2.24804
$895$	−7.64078	−0.255403
$896$	1.54900	0.0517483
$897$	−11.6868	−0.390212
$898$	36.1923	1.20775
$899$	−84.5063	−2.81844
$900$	5.10195	0.170065
$901$	−7.99671	−0.266409
$902$	12.3015	0.409597
$903$	−14.0421	−0.467290
$904$	15.7655	0.524353
$905$	21.8840	0.727448
$906$	43.3637	1.44066
$907$	−3.69362	−0.122645	−0.0613224	−	0.998118i	$-0.519532\pi$
$907$	−3.69362	−0.122645	−0.0613224	+	0.998118i	$0.519532\pi$
$908$	−10.5074	−0.348701
$909$	40.4725	1.34239
$910$	19.4567	0.644983
$911$	−4.97526	−0.164838	−0.0824189	−	0.996598i	$-0.526265\pi$
$911$	−4.97526	−0.164838	−0.0824189	+	0.996598i	$0.526265\pi$
$912$	−7.30543	−0.241907
$913$	15.4524	0.511401
$914$	41.1830	1.36221
$915$	21.3905	0.707148
$916$	14.4544	0.477588
$917$	32.7433	1.08128
$918$	11.0313	0.364086
$919$	33.9748	1.12072	0.560362	−	0.828248i	$-0.310662\pi$
$919$	33.9748	1.12072	0.560362	+	0.828248i	$0.310662\pi$
$920$	−1.57662	−0.0519797
$921$	22.7688	0.750256
$922$	−7.36302	−0.242488
$923$	15.5429	0.511600
$924$	−19.1886	−0.631258
$925$	−2.96635	−0.0975330
$926$	−13.9817	−0.459466
$927$	81.5828	2.67953
$928$	−9.52073	−0.312533
$929$	6.98427	0.229146	0.114573	−	0.993415i	$-0.463450\pi$
$929$	6.98427	0.229146	0.114573	+	0.993415i	$0.463450\pi$
$930$	−70.0398	−2.29670
$931$	−10.1094	−0.331324
$932$	28.5701	0.935846
$933$	4.34324	0.142191
$934$	29.3165	0.959265
$935$	−5.80789	−0.189938
$936$	−42.6156	−1.39293
$937$	3.43675	0.112274	0.0561369	−	0.998423i	$-0.482122\pi$
$937$	3.43675	0.112274	0.0561369	+	0.998423i	$0.482122\pi$
$938$	15.1101	0.493362
$939$	−36.3723	−1.18697
$940$	26.0578	0.849911
$941$	−35.2737	−1.14989	−0.574944	−	0.818193i	$-0.694976\pi$
$941$	−35.2737	−1.14989	−0.574944	+	0.818193i	$0.694976\pi$
$942$	−68.5428	−2.23324
$943$	2.19299	0.0714135
$944$	−4.63655	−0.150907
$945$	61.7588	2.00901
$946$	−10.1603	−0.330339
$947$	−23.0915	−0.750373	−0.375186	−	0.926949i	$-0.622421\pi$
$947$	−23.0915	−0.750373	−0.375186	+	0.926949i	$0.622421\pi$
$948$	8.89617	0.288934
$949$	−28.7470	−0.933169
$950$	−1.39221	−0.0451694
$951$	−113.580	−3.68307
$952$	1.01723	0.0329685
$953$	7.87412	0.255068	0.127534	−	0.991834i	$-0.459294\pi$
$953$	7.87412	0.255068	0.127534	+	0.991834i	$0.459294\pi$
$954$	98.0583	3.17476
$955$	−29.1085	−0.941930
$956$	−0.659452	−0.0213282
$957$	117.941	3.81248
$958$	−27.3079	−0.882278
$959$	27.3157	0.882070
$960$	−7.89089	−0.254677
$961$	47.7840	1.54142
$962$	24.7774	0.798854
$963$	−73.1826	−2.35828
$964$	22.4029	0.721549
$965$	8.93490	0.287625
$966$	−3.42074	−0.110060
$967$	−59.5354	−1.91453	−0.957265	−	0.289212i	$-0.906607\pi$
$967$	−59.5354	−1.91453	−0.957265	+	0.289212i	$0.906607\pi$
$968$	−2.88412	−0.0926990
$969$	−4.79748	−0.154117
$970$	34.2481	1.09964
$971$	44.5994	1.43126	0.715631	−	0.698479i	$-0.246139\pi$
$971$	44.5994	1.43126	0.715631	+	0.698479i	$0.246139\pi$
$972$	−54.9540	−1.76265
$973$	−14.9924	−0.480634
$974$	13.7250	0.439776
$975$	−11.1469	−0.356988
$976$	−2.71078	−0.0867701
$977$	−17.8605	−0.571409	−0.285705	−	0.958318i	$-0.592228\pi$
$977$	−17.8605	−0.571409	−0.285705	+	0.958318i	$0.592228\pi$
$978$	20.6244	0.659495
$979$	−31.3771	−1.00282
$980$	−10.9196	−0.348815
$981$	−76.1588	−2.43156
$982$	−27.8718	−0.889424
$983$	5.23157	0.166861	0.0834307	−	0.996514i	$-0.473412\pi$
$983$	5.23157	0.166861	0.0834307	+	0.996514i	$0.473412\pi$
$984$	10.9758	0.349894
$985$	−33.9667	−1.08227
$986$	−6.25227	−0.199113
$987$	56.5366	1.79958
$988$	11.6289	0.369965
$989$	−1.81127	−0.0575949
$990$	71.2184	2.26347
$991$	−47.8812	−1.52100	−0.760499	−	0.649340i	$-0.775045\pi$
$991$	−47.8812	−1.52100	−0.760499	+	0.649340i	$0.775045\pi$
$992$	8.87603	0.281814
$993$	36.8262	1.16864
$994$	4.54941	0.144298
$995$	58.5036	1.85469
$996$	13.7871	0.436860
$997$	−32.4695	−1.02832	−0.514160	−	0.857694i	$-0.671896\pi$
$997$	−32.4695	−1.02832	−0.514160	+	0.857694i	$0.671896\pi$
$998$	28.8904	0.914509
$999$	78.6475	2.48830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.2	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.2	✓	89	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.32456 q^{3} +1.00000 q^{4} +2.37351 q^{5} +3.32456 q^{6} -1.54900 q^{7} -1.00000 q^{8} +8.05269 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.32456 q^{3} +1.00000 q^{4} +2.37351 q^{5} +3.32456 q^{6} -1.54900 q^{7} -1.00000 q^{8} +8.05269 q^{9} -2.37351 q^{10} -3.72614 q^{11} -3.32456 q^{12} +5.29209 q^{13} +1.54900 q^{14} -7.89089 q^{15} +1.00000 q^{16} +0.656701 q^{17} -8.05269 q^{18} +2.19741 q^{19} +2.37351 q^{20} +5.14973 q^{21} +3.72614 q^{22} +0.664257 q^{23} +3.32456 q^{24} +0.633570 q^{25} -5.29209 q^{26} -16.7980 q^{27} -1.54900 q^{28} +9.52073 q^{29} +7.89089 q^{30} -8.87603 q^{31} -1.00000 q^{32} +12.3878 q^{33} -0.656701 q^{34} -3.67656 q^{35} +8.05269 q^{36} -4.68196 q^{37} -2.19741 q^{38} -17.5939 q^{39} -2.37351 q^{40} +3.30142 q^{41} -5.14973 q^{42} -2.72676 q^{43} -3.72614 q^{44} +19.1132 q^{45} -0.664257 q^{46} +10.9786 q^{47} -3.32456 q^{48} -4.60061 q^{49} -0.633570 q^{50} -2.18324 q^{51} +5.29209 q^{52} -12.1771 q^{53} +16.7980 q^{54} -8.84405 q^{55} +1.54900 q^{56} -7.30543 q^{57} -9.52073 q^{58} -4.63655 q^{59} -7.89089 q^{60} -2.71078 q^{61} +8.87603 q^{62} -12.4736 q^{63} +1.00000 q^{64} +12.5608 q^{65} -12.3878 q^{66} +9.75477 q^{67} +0.656701 q^{68} -2.20836 q^{69} +3.67656 q^{70} +2.93701 q^{71} -8.05269 q^{72} -5.43208 q^{73} +4.68196 q^{74} -2.10634 q^{75} +2.19741 q^{76} +5.77177 q^{77} +17.5939 q^{78} -2.67589 q^{79} +2.37351 q^{80} +31.6878 q^{81} -3.30142 q^{82} -4.14703 q^{83} +5.14973 q^{84} +1.55869 q^{85} +2.72676 q^{86} -31.6522 q^{87} +3.72614 q^{88} +8.42081 q^{89} -19.1132 q^{90} -8.19742 q^{91} +0.664257 q^{92} +29.5089 q^{93} -10.9786 q^{94} +5.21559 q^{95} +3.32456 q^{96} -14.4293 q^{97} +4.60061 q^{98} -30.0055 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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