LMFDB - Embedded newform 8038.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.74439 q^{3} +1.00000 q^{4} +2.72118 q^{5} -2.74439 q^{6} +4.78598 q^{7} +1.00000 q^{8} +4.53169 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.74439 q^{3} +1.00000 q^{4} +2.72118 q^{5} -2.74439 q^{6} +4.78598 q^{7} +1.00000 q^{8} +4.53169 q^{9} +2.72118 q^{10} +3.46596 q^{11} -2.74439 q^{12} -0.939390 q^{13} +4.78598 q^{14} -7.46798 q^{15} +1.00000 q^{16} -2.10747 q^{17} +4.53169 q^{18} +4.03856 q^{19} +2.72118 q^{20} -13.1346 q^{21} +3.46596 q^{22} -2.35478 q^{23} -2.74439 q^{24} +2.40482 q^{25} -0.939390 q^{26} -4.20356 q^{27} +4.78598 q^{28} +7.86853 q^{29} -7.46798 q^{30} -7.50956 q^{31} +1.00000 q^{32} -9.51197 q^{33} -2.10747 q^{34} +13.0235 q^{35} +4.53169 q^{36} -3.55514 q^{37} +4.03856 q^{38} +2.57805 q^{39} +2.72118 q^{40} +1.16230 q^{41} -13.1346 q^{42} +3.57234 q^{43} +3.46596 q^{44} +12.3315 q^{45} -2.35478 q^{46} +1.07298 q^{47} -2.74439 q^{48} +15.9056 q^{49} +2.40482 q^{50} +5.78371 q^{51} -0.939390 q^{52} +12.0274 q^{53} -4.20356 q^{54} +9.43151 q^{55} +4.78598 q^{56} -11.0834 q^{57} +7.86853 q^{58} +6.98143 q^{59} -7.46798 q^{60} +10.0283 q^{61} -7.50956 q^{62} +21.6886 q^{63} +1.00000 q^{64} -2.55625 q^{65} -9.51197 q^{66} -8.41179 q^{67} -2.10747 q^{68} +6.46244 q^{69} +13.0235 q^{70} -10.5966 q^{71} +4.53169 q^{72} -12.8990 q^{73} -3.55514 q^{74} -6.59976 q^{75} +4.03856 q^{76} +16.5880 q^{77} +2.57805 q^{78} +2.07464 q^{79} +2.72118 q^{80} -2.05885 q^{81} +1.16230 q^{82} +2.96141 q^{83} -13.1346 q^{84} -5.73479 q^{85} +3.57234 q^{86} -21.5943 q^{87} +3.46596 q^{88} -2.23799 q^{89} +12.3315 q^{90} -4.49590 q^{91} -2.35478 q^{92} +20.6092 q^{93} +1.07298 q^{94} +10.9896 q^{95} -2.74439 q^{96} +12.3949 q^{97} +15.9056 q^{98} +15.7067 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) $ 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.74439	−1.58448	−0.792238	−	0.610212i	$-0.791084\pi$
$3$	−2.74439	−1.58448	−0.792238	+	0.610212i	$0.791084\pi$
$4$	1.00000	0.500000
$5$	2.72118	1.21695	0.608474	−	0.793574i	$-0.291782\pi$
$5$	2.72118	1.21695	0.608474	+	0.793574i	$0.291782\pi$
$6$	−2.74439	−1.12039
$7$	4.78598	1.80893	0.904466	−	0.426546i	$-0.140270\pi$
$7$	4.78598	1.80893	0.904466	+	0.426546i	$0.140270\pi$
$8$	1.00000	0.353553
$9$	4.53169	1.51056
$10$	2.72118	0.860512
$11$	3.46596	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$11$	3.46596	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$12$	−2.74439	−0.792238
$13$	−0.939390	−0.260540	−0.130270	−	0.991479i	$-0.541584\pi$
$13$	−0.939390	−0.260540	−0.130270	+	0.991479i	$0.541584\pi$
$14$	4.78598	1.27911
$15$	−7.46798	−1.92823
$16$	1.00000	0.250000
$17$	−2.10747	−0.511135	−0.255568	−	0.966791i	$-0.582262\pi$
$17$	−2.10747	−0.511135	−0.255568	+	0.966791i	$0.582262\pi$
$18$	4.53169	1.06813
$19$	4.03856	0.926509	0.463255	−	0.886225i	$-0.346682\pi$
$19$	4.03856	0.926509	0.463255	+	0.886225i	$0.346682\pi$
$20$	2.72118	0.608474
$21$	−13.1346	−2.86621
$22$	3.46596	0.738946
$23$	−2.35478	−0.491006	−0.245503	−	0.969396i	$-0.578953\pi$
$23$	−2.35478	−0.491006	−0.245503	+	0.969396i	$0.578953\pi$
$24$	−2.74439	−0.560197
$25$	2.40482	0.480963
$26$	−0.939390	−0.184230
$27$	−4.20356	−0.808976
$28$	4.78598	0.904466
$29$	7.86853	1.46115	0.730575	−	0.682833i	$-0.239252\pi$
$29$	7.86853	1.46115	0.730575	+	0.682833i	$0.239252\pi$
$30$	−7.46798	−1.36346
$31$	−7.50956	−1.34876	−0.674379	−	0.738386i	$-0.735588\pi$
$31$	−7.50956	−1.34876	−0.674379	+	0.738386i	$0.735588\pi$
$32$	1.00000	0.176777
$33$	−9.51197	−1.65582
$34$	−2.10747	−0.361427
$35$	13.0235	2.20138
$36$	4.53169	0.755282
$37$	−3.55514	−0.584462	−0.292231	−	0.956348i	$-0.594398\pi$
$37$	−3.55514	−0.584462	−0.292231	+	0.956348i	$0.594398\pi$
$38$	4.03856	0.655141
$39$	2.57805	0.412819
$40$	2.72118	0.430256
$41$	1.16230	0.181521	0.0907606	−	0.995873i	$-0.471070\pi$
$41$	1.16230	0.181521	0.0907606	+	0.995873i	$0.471070\pi$
$42$	−13.1346	−2.02672
$43$	3.57234	0.544776	0.272388	−	0.962187i	$-0.412187\pi$
$43$	3.57234	0.544776	0.272388	+	0.962187i	$0.412187\pi$
$44$	3.46596	0.522514
$45$	12.3315	1.83828
$46$	−2.35478	−0.347194
$47$	1.07298	0.156511	0.0782553	−	0.996933i	$-0.475065\pi$
$47$	1.07298	0.156511	0.0782553	+	0.996933i	$0.475065\pi$
$48$	−2.74439	−0.396119
$49$	15.9056	2.27223
$50$	2.40482	0.340092
$51$	5.78371	0.809882
$52$	−0.939390	−0.130270
$53$	12.0274	1.65209	0.826047	−	0.563601i	$-0.190585\pi$
$53$	12.0274	1.65209	0.826047	+	0.563601i	$0.190585\pi$
$54$	−4.20356	−0.572032
$55$	9.43151	1.27174
$56$	4.78598	0.639554
$57$	−11.0834	−1.46803
$58$	7.86853	1.03319
$59$	6.98143	0.908904	0.454452	−	0.890771i	$-0.349835\pi$
$59$	6.98143	0.908904	0.454452	+	0.890771i	$0.349835\pi$
$60$	−7.46798	−0.964113
$61$	10.0283	1.28399	0.641994	−	0.766710i	$-0.278107\pi$
$61$	10.0283	1.28399	0.641994	+	0.766710i	$0.278107\pi$
$62$	−7.50956	−0.953715
$63$	21.6886	2.73251
$64$	1.00000	0.125000
$65$	−2.55625	−0.317064
$66$	−9.51197	−1.17084
$67$	−8.41179	−1.02766	−0.513832	−	0.857891i	$-0.671775\pi$
$67$	−8.41179	−1.02766	−0.513832	+	0.857891i	$0.671775\pi$
$68$	−2.10747	−0.255568
$69$	6.46244	0.777987
$70$	13.0235	1.55661
$71$	−10.5966	−1.25758	−0.628792	−	0.777574i	$-0.716450\pi$
$71$	−10.5966	−1.25758	−0.628792	+	0.777574i	$0.716450\pi$
$72$	4.53169	0.534065
$73$	−12.8990	−1.50972	−0.754859	−	0.655887i	$-0.772295\pi$
$73$	−12.8990	−1.50972	−0.754859	+	0.655887i	$0.772295\pi$
$74$	−3.55514	−0.413277
$75$	−6.59976	−0.762075
$76$	4.03856	0.463255
$77$	16.5880	1.89038
$78$	2.57805	0.291907
$79$	2.07464	0.233416	0.116708	−	0.993166i	$-0.462766\pi$
$79$	2.07464	0.233416	0.116708	+	0.993166i	$0.462766\pi$
$80$	2.72118	0.304237
$81$	−2.05885	−0.228761
$82$	1.16230	0.128355
$83$	2.96141	0.325057	0.162529	−	0.986704i	$-0.448035\pi$
$83$	2.96141	0.325057	0.162529	+	0.986704i	$0.448035\pi$
$84$	−13.1346	−1.43310
$85$	−5.73479	−0.622025
$86$	3.57234	0.385215
$87$	−21.5943	−2.31516
$88$	3.46596	0.369473
$89$	−2.23799	−0.237226	−0.118613	−	0.992941i	$-0.537845\pi$
$89$	−2.23799	−0.237226	−0.118613	+	0.992941i	$0.537845\pi$
$90$	12.3315	1.29986
$91$	−4.49590	−0.471299
$92$	−2.35478	−0.245503
$93$	20.6092	2.13707
$94$	1.07298	0.110670
$95$	10.9896	1.12751
$96$	−2.74439	−0.280098
$97$	12.3949	1.25851	0.629254	−	0.777199i	$-0.283360\pi$
$97$	12.3949	1.25851	0.629254	+	0.777199i	$0.283360\pi$
$98$	15.9056	1.60671
$99$	15.7067	1.57858
$100$	2.40482	0.240482
$101$	−7.51477	−0.747747	−0.373874	−	0.927480i	$-0.621971\pi$
$101$	−7.51477	−0.747747	−0.373874	+	0.927480i	$0.621971\pi$
$102$	5.78371	0.572673
$103$	12.7158	1.25293	0.626464	−	0.779450i	$-0.284502\pi$
$103$	12.7158	1.25293	0.626464	+	0.779450i	$0.284502\pi$
$104$	−0.939390	−0.0921148
$105$	−35.7416	−3.48803
$106$	12.0274	1.16821
$107$	−4.50881	−0.435883	−0.217942	−	0.975962i	$-0.569934\pi$
$107$	−4.50881	−0.435883	−0.217942	+	0.975962i	$0.569934\pi$
$108$	−4.20356	−0.404488
$109$	0.124663	0.0119405	0.00597027	−	0.999982i	$-0.498100\pi$
$109$	0.124663	0.0119405	0.00597027	+	0.999982i	$0.498100\pi$
$110$	9.43151	0.899259
$111$	9.75671	0.926066
$112$	4.78598	0.452233
$113$	3.91536	0.368326	0.184163	−	0.982896i	$-0.441043\pi$
$113$	3.91536	0.368326	0.184163	+	0.982896i	$0.441043\pi$
$114$	−11.0834	−1.03806
$115$	−6.40778	−0.597529
$116$	7.86853	0.730575
$117$	−4.25703	−0.393562
$118$	6.98143	0.642692
$119$	−10.0863	−0.924609
$120$	−7.46798	−0.681731
$121$	1.01291	0.0920826
$122$	10.0283	0.907917
$123$	−3.18982	−0.287616
$124$	−7.50956	−0.674379
$125$	−7.06196	−0.631641
$126$	21.6886	1.93217
$127$	13.0904	1.16159	0.580794	−	0.814051i	$-0.302742\pi$
$127$	13.0904	1.16159	0.580794	+	0.814051i	$0.302742\pi$
$128$	1.00000	0.0883883
$129$	−9.80389	−0.863185
$130$	−2.55625	−0.224198
$131$	−14.2524	−1.24523	−0.622617	−	0.782526i	$-0.713931\pi$
$131$	−14.2524	−1.24523	−0.622617	+	0.782526i	$0.713931\pi$
$132$	−9.51197	−0.827910
$133$	19.3285	1.67599
$134$	−8.41179	−0.726668
$135$	−11.4386	−0.984482
$136$	−2.10747	−0.180714
$137$	−17.3749	−1.48444	−0.742220	−	0.670156i	$-0.766227\pi$
$137$	−17.3749	−1.48444	−0.742220	+	0.670156i	$0.766227\pi$
$138$	6.46244	0.550120
$139$	0.604558	0.0512779	0.0256390	−	0.999671i	$-0.491838\pi$
$139$	0.604558	0.0512779	0.0256390	+	0.999671i	$0.491838\pi$
$140$	13.0235	1.10069
$141$	−2.94469	−0.247987
$142$	−10.5966	−0.889246
$143$	−3.25589	−0.272271
$144$	4.53169	0.377641
$145$	21.4117	1.77814
$146$	−12.8990	−1.06753
$147$	−43.6513	−3.60030
$148$	−3.55514	−0.292231
$149$	−20.8058	−1.70448	−0.852238	−	0.523155i	$-0.824755\pi$
$149$	−20.8058	−1.70448	−0.852238	+	0.523155i	$0.824755\pi$
$150$	−6.59976	−0.538868
$151$	5.16874	0.420626	0.210313	−	0.977634i	$-0.432552\pi$
$151$	5.16874	0.420626	0.210313	+	0.977634i	$0.432552\pi$
$152$	4.03856	0.327570
$153$	−9.55038	−0.772103
$154$	16.5880	1.33670
$155$	−20.4349	−1.64137
$156$	2.57805	0.206410
$157$	4.30411	0.343505	0.171753	−	0.985140i	$-0.445057\pi$
$157$	4.30411	0.343505	0.171753	+	0.985140i	$0.445057\pi$
$158$	2.07464	0.165050
$159$	−33.0080	−2.61770
$160$	2.72118	0.215128
$161$	−11.2699	−0.888196
$162$	−2.05885	−0.161758
$163$	1.71796	0.134561	0.0672807	−	0.997734i	$-0.478568\pi$
$163$	1.71796	0.134561	0.0672807	+	0.997734i	$0.478568\pi$
$164$	1.16230	0.0907606
$165$	−25.8838	−2.01505
$166$	2.96141	0.229850
$167$	24.0879	1.86398	0.931989	−	0.362487i	$-0.118072\pi$
$167$	24.0879	1.86398	0.931989	+	0.362487i	$0.118072\pi$
$168$	−13.1346	−1.01336
$169$	−12.1175	−0.932119
$170$	−5.73479	−0.439838
$171$	18.3015	1.39955
$172$	3.57234	0.272388
$173$	19.4698	1.48026	0.740130	−	0.672463i	$-0.234764\pi$
$173$	19.4698	1.48026	0.740130	+	0.672463i	$0.234764\pi$
$174$	−21.5943	−1.63706
$175$	11.5094	0.870030
$176$	3.46596	0.261257
$177$	−19.1598	−1.44014
$178$	−2.23799	−0.167744
$179$	−15.9483	−1.19203	−0.596017	−	0.802972i	$-0.703251\pi$
$179$	−15.9483	−1.19203	−0.596017	+	0.802972i	$0.703251\pi$
$180$	12.3315	0.919139
$181$	−4.68494	−0.348229	−0.174114	−	0.984725i	$-0.555706\pi$
$181$	−4.68494	−0.348229	−0.174114	+	0.984725i	$0.555706\pi$
$182$	−4.49590	−0.333259
$183$	−27.5215	−2.03445
$184$	−2.35478	−0.173597
$185$	−9.67418	−0.711260
$186$	20.6092	1.51114
$187$	−7.30440	−0.534151
$188$	1.07298	0.0782553
$189$	−20.1182	−1.46338
$190$	10.9896	0.797273
$191$	−26.1879	−1.89489	−0.947446	−	0.319916i	$-0.896345\pi$
$191$	−26.1879	−1.89489	−0.947446	+	0.319916i	$0.896345\pi$
$192$	−2.74439	−0.198059
$193$	−3.69553	−0.266010	−0.133005	−	0.991115i	$-0.542463\pi$
$193$	−3.69553	−0.266010	−0.133005	+	0.991115i	$0.542463\pi$
$194$	12.3949	0.889900
$195$	7.01535	0.502380
$196$	15.9056	1.13612
$197$	−16.0973	−1.14688	−0.573442	−	0.819246i	$-0.694392\pi$
$197$	−16.0973	−1.14688	−0.573442	+	0.819246i	$0.694392\pi$
$198$	15.7067	1.11623
$199$	23.0058	1.63084	0.815419	−	0.578872i	$-0.196507\pi$
$199$	23.0058	1.63084	0.815419	+	0.578872i	$0.196507\pi$
$200$	2.40482	0.170046
$201$	23.0853	1.62831
$202$	−7.51477	−0.528737
$203$	37.6586	2.64312
$204$	5.78371	0.404941
$205$	3.16283	0.220902
$206$	12.7158	0.885954
$207$	−10.6711	−0.741696
$208$	−0.939390	−0.0651350
$209$	13.9975	0.968228
$210$	−35.7416	−2.46641
$211$	−18.7382	−1.28999	−0.644996	−	0.764186i	$-0.723141\pi$
$211$	−18.7382	−1.28999	−0.644996	+	0.764186i	$0.723141\pi$
$212$	12.0274	0.826047
$213$	29.0812	1.99261
$214$	−4.50881	−0.308216
$215$	9.72097	0.662964
$216$	−4.20356	−0.286016
$217$	−35.9406	−2.43981
$218$	0.124663	0.00844324
$219$	35.4000	2.39211
$220$	9.43151	0.635872
$221$	1.97973	0.133171
$222$	9.75671	0.654828
$223$	−8.97574	−0.601060	−0.300530	−	0.953772i	$-0.597164\pi$
$223$	−8.97574	−0.601060	−0.300530	+	0.953772i	$0.597164\pi$
$224$	4.78598	0.319777
$225$	10.8979	0.726526
$226$	3.91536	0.260446
$227$	12.7848	0.848556	0.424278	−	0.905532i	$-0.360528\pi$
$227$	12.7848	0.848556	0.424278	+	0.905532i	$0.360528\pi$
$228$	−11.0834	−0.734016
$229$	−8.72670	−0.576676	−0.288338	−	0.957529i	$-0.593103\pi$
$229$	−8.72670	−0.576676	−0.288338	+	0.957529i	$0.593103\pi$
$230$	−6.40778	−0.422517
$231$	−45.5241	−2.99527
$232$	7.86853	0.516594
$233$	−10.5957	−0.694149	−0.347074	−	0.937838i	$-0.612825\pi$
$233$	−10.5957	−0.694149	−0.347074	+	0.937838i	$0.612825\pi$
$234$	−4.25703	−0.278290
$235$	2.91978	0.190465
$236$	6.98143	0.454452
$237$	−5.69364	−0.369842
$238$	−10.0863	−0.653797
$239$	−9.37310	−0.606295	−0.303148	−	0.952944i	$-0.598038\pi$
$239$	−9.37310	−0.606295	−0.303148	+	0.952944i	$0.598038\pi$
$240$	−7.46798	−0.482056
$241$	−21.9420	−1.41341	−0.706706	−	0.707508i	$-0.749820\pi$
$241$	−21.9420	−1.41341	−0.706706	+	0.707508i	$0.749820\pi$
$242$	1.01291	0.0651122
$243$	18.2610	1.17144
$244$	10.0283	0.641994
$245$	43.2821	2.76519
$246$	−3.18982	−0.203375
$247$	−3.79378	−0.241393
$248$	−7.50956	−0.476858
$249$	−8.12727	−0.515045
$250$	−7.06196	−0.446638
$251$	23.0663	1.45593	0.727967	−	0.685612i	$-0.240465\pi$
$251$	23.0663	1.45593	0.727967	+	0.685612i	$0.240465\pi$
$252$	21.6886	1.36625
$253$	−8.16159	−0.513115
$254$	13.0904	0.821366
$255$	15.7385	0.985584
$256$	1.00000	0.0625000
$257$	−12.8974	−0.804516	−0.402258	−	0.915526i	$-0.631775\pi$
$257$	−12.8974	−0.804516	−0.402258	+	0.915526i	$0.631775\pi$
$258$	−9.80389	−0.610364
$259$	−17.0149	−1.05725
$260$	−2.55625	−0.158532
$261$	35.6577	2.20716
$262$	−14.2524	−0.880514
$263$	14.5624	0.897956	0.448978	−	0.893543i	$-0.351788\pi$
$263$	14.5624	0.897956	0.448978	+	0.893543i	$0.351788\pi$
$264$	−9.51197	−0.585421
$265$	32.7288	2.01051
$266$	19.3285	1.18511
$267$	6.14192	0.375879
$268$	−8.41179	−0.513832
$269$	−10.8032	−0.658682	−0.329341	−	0.944211i	$-0.606827\pi$
$269$	−10.8032	−0.658682	−0.329341	+	0.944211i	$0.606827\pi$
$270$	−11.4386	−0.696134
$271$	28.0300	1.70270	0.851352	−	0.524595i	$-0.175783\pi$
$271$	28.0300	1.70270	0.851352	+	0.524595i	$0.175783\pi$
$272$	−2.10747	−0.127784
$273$	12.3385	0.746762
$274$	−17.3749	−1.04966
$275$	8.33501	0.502620
$276$	6.46244	0.388993
$277$	17.3463	1.04224	0.521118	−	0.853485i	$-0.325515\pi$
$277$	17.3463	1.04224	0.521118	+	0.853485i	$0.325515\pi$
$278$	0.604558	0.0362590
$279$	−34.0310	−2.03738
$280$	13.0235	0.778304
$281$	8.45531	0.504401	0.252201	−	0.967675i	$-0.418846\pi$
$281$	8.45531	0.504401	0.252201	+	0.967675i	$0.418846\pi$
$282$	−2.94469	−0.175353
$283$	−21.5735	−1.28241	−0.641204	−	0.767370i	$-0.721565\pi$
$283$	−21.5735	−1.28241	−0.641204	+	0.767370i	$0.721565\pi$
$284$	−10.5966	−0.628792
$285$	−30.1599	−1.78652
$286$	−3.25589	−0.192525
$287$	5.56276	0.328359
$288$	4.53169	0.267032
$289$	−12.5586	−0.738741
$290$	21.4117	1.25734
$291$	−34.0164	−1.99408
$292$	−12.8990	−0.754859
$293$	14.9975	0.876165	0.438082	−	0.898935i	$-0.355658\pi$
$293$	14.9975	0.876165	0.438082	+	0.898935i	$0.355658\pi$
$294$	−43.6513	−2.54580
$295$	18.9977	1.10609
$296$	−3.55514	−0.206639
$297$	−14.5694	−0.845402
$298$	−20.8058	−1.20525
$299$	2.21206	0.127927
$300$	−6.59976	−0.381037
$301$	17.0971	0.985463
$302$	5.16874	0.297428
$303$	20.6235	1.18479
$304$	4.03856	0.231627
$305$	27.2887	1.56255
$306$	−9.55038	−0.545959
$307$	8.36727	0.477545	0.238773	−	0.971075i	$-0.423255\pi$
$307$	8.36727	0.477545	0.238773	+	0.971075i	$0.423255\pi$
$308$	16.5880	0.945192
$309$	−34.8972	−1.98523
$310$	−20.4349	−1.16062
$311$	19.4619	1.10359	0.551793	−	0.833981i	$-0.313944\pi$
$311$	19.4619	1.10359	0.551793	+	0.833981i	$0.313944\pi$
$312$	2.57805	0.145954
$313$	−5.94971	−0.336298	−0.168149	−	0.985762i	$-0.553779\pi$
$313$	−5.94971	−0.336298	−0.168149	+	0.985762i	$0.553779\pi$
$314$	4.30411	0.242895
$315$	59.0186	3.32532
$316$	2.07464	0.116708
$317$	7.70344	0.432668	0.216334	−	0.976319i	$-0.430590\pi$
$317$	7.70344	0.432668	0.216334	+	0.976319i	$0.430590\pi$
$318$	−33.0080	−1.85100
$319$	27.2720	1.52694
$320$	2.72118	0.152119
$321$	12.3739	0.690646
$322$	−11.2699	−0.628049
$323$	−8.51113	−0.473572
$324$	−2.05885	−0.114380
$325$	−2.25906	−0.125310
$326$	1.71796	0.0951493
$327$	−0.342124	−0.0189195
$328$	1.16230	0.0641774
$329$	5.13528	0.283117
$330$	−25.8838	−1.42485
$331$	9.82635	0.540105	0.270053	−	0.962846i	$-0.412959\pi$
$331$	9.82635	0.540105	0.270053	+	0.962846i	$0.412959\pi$
$332$	2.96141	0.162529
$333$	−16.1108	−0.882867
$334$	24.0879	1.31803
$335$	−22.8900	−1.25061
$336$	−13.1346	−0.716552
$337$	23.3447	1.27167	0.635834	−	0.771826i	$-0.280656\pi$
$337$	23.3447	1.27167	0.635834	+	0.771826i	$0.280656\pi$
$338$	−12.1175	−0.659108
$339$	−10.7453	−0.583604
$340$	−5.73479	−0.311013
$341$	−26.0279	−1.40949
$342$	18.3015	0.989632
$343$	42.6222	2.30138
$344$	3.57234	0.192607
$345$	17.5855	0.946770
$346$	19.4698	1.04670
$347$	17.1020	0.918083	0.459042	−	0.888415i	$-0.348193\pi$
$347$	17.1020	0.918083	0.459042	+	0.888415i	$0.348193\pi$
$348$	−21.5943	−1.15758
$349$	−4.82512	−0.258283	−0.129141	−	0.991626i	$-0.541222\pi$
$349$	−4.82512	−0.258283	−0.129141	+	0.991626i	$0.541222\pi$
$350$	11.5094	0.615204
$351$	3.94878	0.210771
$352$	3.46596	0.184737
$353$	3.86808	0.205877	0.102939	−	0.994688i	$-0.467175\pi$
$353$	3.86808	0.205877	0.102939	+	0.994688i	$0.467175\pi$
$354$	−19.1598	−1.01833
$355$	−28.8352	−1.53041
$356$	−2.23799	−0.118613
$357$	27.6808	1.46502
$358$	−15.9483	−0.842896
$359$	−33.4255	−1.76413	−0.882065	−	0.471128i	$-0.843847\pi$
$359$	−33.4255	−1.76413	−0.882065	+	0.471128i	$0.843847\pi$
$360$	12.3315	0.649929
$361$	−2.69003	−0.141581
$362$	−4.68494	−0.246235
$363$	−2.77982	−0.145903
$364$	−4.49590	−0.235649
$365$	−35.1006	−1.83725
$366$	−27.5215	−1.43857
$367$	−24.2281	−1.26470	−0.632349	−	0.774684i	$-0.717909\pi$
$367$	−24.2281	−1.26470	−0.632349	+	0.774684i	$0.717909\pi$
$368$	−2.35478	−0.122751
$369$	5.26720	0.274199
$370$	−9.67418	−0.502937
$371$	57.5630	2.98852
$372$	20.6092	1.06854
$373$	8.12575	0.420735	0.210368	−	0.977622i	$-0.432534\pi$
$373$	8.12575	0.420735	0.210368	+	0.977622i	$0.432534\pi$
$374$	−7.30440	−0.377702
$375$	19.3808	1.00082
$376$	1.07298	0.0553348
$377$	−7.39162	−0.380688
$378$	−20.1182	−1.03477
$379$	6.29036	0.323114	0.161557	−	0.986863i	$-0.448348\pi$
$379$	6.29036	0.323114	0.161557	+	0.986863i	$0.448348\pi$
$380$	10.9896	0.563757
$381$	−35.9253	−1.84051
$382$	−26.1879	−1.33989
$383$	24.1796	1.23552	0.617761	−	0.786366i	$-0.288040\pi$
$383$	24.1796	1.23552	0.617761	+	0.786366i	$0.288040\pi$
$384$	−2.74439	−0.140049
$385$	45.1390	2.30050
$386$	−3.69553	−0.188098
$387$	16.1887	0.822919
$388$	12.3949	0.629254
$389$	18.8416	0.955307	0.477654	−	0.878548i	$-0.341487\pi$
$389$	18.8416	0.955307	0.477654	+	0.878548i	$0.341487\pi$
$390$	7.01535	0.355236
$391$	4.96262	0.250970
$392$	15.9056	0.803356
$393$	39.1141	1.97304
$394$	−16.0973	−0.810970
$395$	5.64548	0.284055
$396$	15.7067	0.789290
$397$	−19.0715	−0.957173	−0.478587	−	0.878040i	$-0.658851\pi$
$397$	−19.0715	−0.957173	−0.478587	+	0.878040i	$0.658851\pi$
$398$	23.0058	1.15318
$399$	−53.0449	−2.65557
$400$	2.40482	0.120241
$401$	−18.1166	−0.904700	−0.452350	−	0.891841i	$-0.649414\pi$
$401$	−18.1166	−0.904700	−0.452350	+	0.891841i	$0.649414\pi$
$402$	23.0853	1.15139
$403$	7.05441	0.351405
$404$	−7.51477	−0.373874
$405$	−5.60249	−0.278390
$406$	37.6586	1.86897
$407$	−12.3220	−0.610779
$408$	5.78371	0.286336
$409$	−18.2458	−0.902199	−0.451099	−	0.892474i	$-0.648968\pi$
$409$	−18.2458	−0.902199	−0.451099	+	0.892474i	$0.648968\pi$
$410$	3.16283	0.156201
$411$	47.6836	2.35206
$412$	12.7158	0.626464
$413$	33.4130	1.64415
$414$	−10.6711	−0.524458
$415$	8.05853	0.395578
$416$	−0.939390	−0.0460574
$417$	−1.65914	−0.0812486
$418$	13.9975	0.684640
$419$	28.3817	1.38653	0.693267	−	0.720680i	$-0.256170\pi$
$419$	28.3817	1.38653	0.693267	+	0.720680i	$0.256170\pi$
$420$	−35.7416	−1.74401
$421$	−16.1836	−0.788739	−0.394370	−	0.918952i	$-0.629037\pi$
$421$	−16.1836	−0.788739	−0.394370	+	0.918952i	$0.629037\pi$
$422$	−18.7382	−0.912162
$423$	4.86243	0.236419
$424$	12.0274	0.584103
$425$	−5.06807	−0.245837
$426$	29.0812	1.40899
$427$	47.9951	2.32265
$428$	−4.50881	−0.217942
$429$	8.93545	0.431407
$430$	9.72097	0.468787
$431$	−4.25759	−0.205081	−0.102540	−	0.994729i	$-0.532697\pi$
$431$	−4.25759	−0.205081	−0.102540	+	0.994729i	$0.532697\pi$
$432$	−4.20356	−0.202244
$433$	−7.30345	−0.350981	−0.175491	−	0.984481i	$-0.556151\pi$
$433$	−7.30345	−0.350981	−0.175491	+	0.984481i	$0.556151\pi$
$434$	−35.9406	−1.72521
$435$	−58.7620	−2.81742
$436$	0.124663	0.00597027
$437$	−9.50992	−0.454921
$438$	35.4000	1.69148
$439$	25.0238	1.19432	0.597161	−	0.802122i	$-0.296296\pi$
$439$	25.0238	1.19432	0.597161	+	0.802122i	$0.296296\pi$
$440$	9.43151	0.449630
$441$	72.0794	3.43235
$442$	1.97973	0.0941662
$443$	−12.5646	−0.596962	−0.298481	−	0.954416i	$-0.596480\pi$
$443$	−12.5646	−0.596962	−0.298481	+	0.954416i	$0.596480\pi$
$444$	9.75671	0.463033
$445$	−6.08997	−0.288692
$446$	−8.97574	−0.425014
$447$	57.0992	2.70070
$448$	4.78598	0.226116
$449$	23.9089	1.12833	0.564164	−	0.825663i	$-0.309198\pi$
$449$	23.9089	1.12833	0.564164	+	0.825663i	$0.309198\pi$
$450$	10.8979	0.513731
$451$	4.02850	0.189695
$452$	3.91536	0.184163
$453$	−14.1851	−0.666472
$454$	12.7848	0.600020
$455$	−12.2342	−0.573546
$456$	−11.0834	−0.519028
$457$	−4.32224	−0.202186	−0.101093	−	0.994877i	$-0.532234\pi$
$457$	−4.32224	−0.202186	−0.101093	+	0.994877i	$0.532234\pi$
$458$	−8.72670	−0.407772
$459$	8.85886	0.413496
$460$	−6.40778	−0.298764
$461$	−14.6170	−0.680781	−0.340390	−	0.940284i	$-0.610559\pi$
$461$	−14.6170	−0.680781	−0.340390	+	0.940284i	$0.610559\pi$
$462$	−45.5241	−2.11797
$463$	24.7338	1.14948	0.574739	−	0.818337i	$-0.305103\pi$
$463$	24.7338	1.14948	0.574739	+	0.818337i	$0.305103\pi$
$464$	7.86853	0.365287
$465$	56.0813	2.60071
$466$	−10.5957	−0.490837
$467$	−32.8807	−1.52154	−0.760769	−	0.649023i	$-0.775178\pi$
$467$	−32.8807	−1.52154	−0.760769	+	0.649023i	$0.775178\pi$
$468$	−4.25703	−0.196781
$469$	−40.2587	−1.85897
$470$	2.91978	0.134679
$471$	−11.8122	−0.544276
$472$	6.98143	0.321346
$473$	12.3816	0.569306
$474$	−5.69364	−0.261518
$475$	9.71200	0.445617
$476$	−10.0863	−0.462305
$477$	54.5046	2.49559
$478$	−9.37310	−0.428716
$479$	−12.9377	−0.591140	−0.295570	−	0.955321i	$-0.595510\pi$
$479$	−12.9377	−0.591140	−0.295570	+	0.955321i	$0.595510\pi$
$480$	−7.46798	−0.340865
$481$	3.33967	0.152276
$482$	−21.9420	−0.999433
$483$	30.9291	1.40732
$484$	1.01291	0.0460413
$485$	33.7287	1.53154
$486$	18.2610	0.828335
$487$	14.8392	0.672431	0.336215	−	0.941785i	$-0.390853\pi$
$487$	14.8392	0.672431	0.336215	+	0.941785i	$0.390853\pi$
$488$	10.0283	0.453958
$489$	−4.71477	−0.213209
$490$	43.2821	1.95528
$491$	23.4329	1.05751	0.528757	−	0.848774i	$-0.322658\pi$
$491$	23.4329	1.05751	0.528757	+	0.848774i	$0.322658\pi$
$492$	−3.18982	−0.143808
$493$	−16.5827	−0.746845
$494$	−3.79378	−0.170690
$495$	42.7407	1.92105
$496$	−7.50956	−0.337189
$497$	−50.7151	−2.27488
$498$	−8.12727	−0.364192
$499$	−31.8213	−1.42452	−0.712258	−	0.701917i	$-0.752328\pi$
$499$	−31.8213	−1.42452	−0.712258	+	0.701917i	$0.752328\pi$
$500$	−7.06196	−0.315820
$501$	−66.1067	−2.95343
$502$	23.0663	1.02950
$503$	−11.5590	−0.515389	−0.257694	−	0.966226i	$-0.582963\pi$
$503$	−11.5590	−0.515389	−0.257694	+	0.966226i	$0.582963\pi$
$504$	21.6886	0.966087
$505$	−20.4490	−0.909970
$506$	−8.16159	−0.362827
$507$	33.2553	1.47692
$508$	13.0904	0.580794
$509$	−12.4076	−0.549956	−0.274978	−	0.961450i	$-0.588671\pi$
$509$	−12.4076	−0.549956	−0.274978	+	0.961450i	$0.588671\pi$
$510$	15.7385	0.696913
$511$	−61.7345	−2.73098
$512$	1.00000	0.0441942
$513$	−16.9763	−0.749524
$514$	−12.8974	−0.568879
$515$	34.6021	1.52475
$516$	−9.80389	−0.431592
$517$	3.71892	0.163558
$518$	−17.0149	−0.747590
$519$	−53.4327	−2.34544
$520$	−2.55625	−0.112099
$521$	−16.3741	−0.717363	−0.358682	−	0.933460i	$-0.616774\pi$
$521$	−16.3741	−0.717363	−0.358682	+	0.933460i	$0.616774\pi$
$522$	35.6577	1.56070
$523$	10.2260	0.447154	0.223577	−	0.974686i	$-0.428227\pi$
$523$	10.2260	0.447154	0.223577	+	0.974686i	$0.428227\pi$
$524$	−14.2524	−0.622617
$525$	−31.5863	−1.37854
$526$	14.5624	0.634951
$527$	15.8261	0.689398
$528$	−9.51197	−0.413955
$529$	−17.4550	−0.758913
$530$	32.7288	1.42165
$531$	31.6377	1.37296
$532$	19.3285	0.837996
$533$	−1.09186	−0.0472935
$534$	6.14192	0.265787
$535$	−12.2693	−0.530447
$536$	−8.41179	−0.363334
$537$	43.7685	1.88875
$538$	−10.8032	−0.465758
$539$	55.1283	2.37455
$540$	−11.4386	−0.492241
$541$	8.09160	0.347885	0.173943	−	0.984756i	$-0.444349\pi$
$541$	8.09160	0.347885	0.173943	+	0.984756i	$0.444349\pi$
$542$	28.0300	1.20399
$543$	12.8573	0.551760
$544$	−2.10747	−0.0903568
$545$	0.339230	0.0145310
$546$	12.3385	0.528040
$547$	−24.0963	−1.03028	−0.515142	−	0.857105i	$-0.672261\pi$
$547$	−24.0963	−1.03028	−0.515142	+	0.857105i	$0.672261\pi$
$548$	−17.3749	−0.742220
$549$	45.4450	1.93955
$550$	8.33501	0.355406
$551$	31.7775	1.35377
$552$	6.46244	0.275060
$553$	9.92921	0.422233
$554$	17.3463	0.736972
$555$	26.5498	1.12697
$556$	0.604558	0.0256390
$557$	40.8400	1.73045	0.865224	−	0.501385i	$-0.167176\pi$
$557$	40.8400	1.73045	0.865224	+	0.501385i	$0.167176\pi$
$558$	−34.0310	−1.44065
$559$	−3.35582	−0.141936
$560$	13.0235	0.550344
$561$	20.0461	0.846349
$562$	8.45531	0.356666
$563$	22.7116	0.957180	0.478590	−	0.878039i	$-0.341148\pi$
$563$	22.7116	0.957180	0.478590	+	0.878039i	$0.341148\pi$
$564$	−2.94469	−0.123994
$565$	10.6544	0.448234
$566$	−21.5735	−0.906800
$567$	−9.85361	−0.413813
$568$	−10.5966	−0.444623
$569$	4.35481	0.182563	0.0912815	−	0.995825i	$-0.470904\pi$
$569$	4.35481	0.182563	0.0912815	+	0.995825i	$0.470904\pi$
$570$	−30.1599	−1.26326
$571$	−7.27276	−0.304355	−0.152178	−	0.988353i	$-0.548629\pi$
$571$	−7.27276	−0.304355	−0.152178	+	0.988353i	$0.548629\pi$
$572$	−3.25589	−0.136136
$573$	71.8700	3.00241
$574$	5.56276	0.232185
$575$	−5.66282	−0.236156
$576$	4.53169	0.188820
$577$	−6.86752	−0.285899	−0.142949	−	0.989730i	$-0.545659\pi$
$577$	−6.86752	−0.285899	−0.142949	+	0.989730i	$0.545659\pi$
$578$	−12.5586	−0.522368
$579$	10.1420	0.421487
$580$	21.4117	0.889071
$581$	14.1733	0.588006
$582$	−34.0164	−1.41003
$583$	41.6866	1.72648
$584$	−12.8990	−0.533766
$585$	−11.5841	−0.478945
$586$	14.9975	0.619542
$587$	6.69901	0.276498	0.138249	−	0.990398i	$-0.455853\pi$
$587$	6.69901	0.276498	0.138249	+	0.990398i	$0.455853\pi$
$588$	−43.6513	−1.80015
$589$	−30.3278	−1.24964
$590$	18.9977	0.782123
$591$	44.1773	1.81721
$592$	−3.55514	−0.146116
$593$	1.93677	0.0795336	0.0397668	−	0.999209i	$-0.487338\pi$
$593$	1.93677	0.0795336	0.0397668	+	0.999209i	$0.487338\pi$
$594$	−14.5694	−0.597790
$595$	−27.4466	−1.12520
$596$	−20.8058	−0.852238
$597$	−63.1369	−2.58402
$598$	2.21206	0.0904578
$599$	−36.8852	−1.50709	−0.753545	−	0.657397i	$-0.771658\pi$
$599$	−36.8852	−1.50709	−0.753545	+	0.657397i	$0.771658\pi$
$600$	−6.59976	−0.269434
$601$	24.8822	1.01497	0.507483	−	0.861662i	$-0.330576\pi$
$601$	24.8822	1.01497	0.507483	+	0.861662i	$0.330576\pi$
$602$	17.0971	0.696827
$603$	−38.1196	−1.55235
$604$	5.16874	0.210313
$605$	2.75631	0.112060
$606$	20.6235	0.837771
$607$	8.17500	0.331813	0.165906	−	0.986142i	$-0.446945\pi$
$607$	8.17500	0.331813	0.165906	+	0.986142i	$0.446945\pi$
$608$	4.03856	0.163785
$609$	−103.350	−4.18796
$610$	27.2887	1.10489
$611$	−1.00795	−0.0407773
$612$	−9.55038	−0.386051
$613$	−10.1033	−0.408067	−0.204033	−	0.978964i	$-0.565405\pi$
$613$	−10.1033	−0.408067	−0.204033	+	0.978964i	$0.565405\pi$
$614$	8.36727	0.337676
$615$	−8.68006	−0.350014
$616$	16.5880	0.668351
$617$	11.3611	0.457379	0.228690	−	0.973499i	$-0.426556\pi$
$617$	11.3611	0.457379	0.228690	+	0.973499i	$0.426556\pi$
$618$	−34.8972	−1.40377
$619$	26.6170	1.06983	0.534913	−	0.844907i	$-0.320344\pi$
$619$	26.6170	1.06983	0.534913	+	0.844907i	$0.320344\pi$
$620$	−20.4349	−0.820684
$621$	9.89847	0.397212
$622$	19.4619	0.780353
$623$	−10.7110	−0.429126
$624$	2.57805	0.103205
$625$	−31.2409	−1.24964
$626$	−5.94971	−0.237798
$627$	−38.4147	−1.53413
$628$	4.30411	0.171753
$629$	7.49234	0.298739
$630$	59.0186	2.35136
$631$	13.5911	0.541053	0.270526	−	0.962713i	$-0.412802\pi$
$631$	13.5911	0.541053	0.270526	+	0.962713i	$0.412802\pi$
$632$	2.07464	0.0825249
$633$	51.4250	2.04396
$634$	7.70344	0.305943
$635$	35.6214	1.41359
$636$	−33.0080	−1.30885
$637$	−14.9416	−0.592007
$638$	27.2720	1.07971
$639$	−48.0205	−1.89966
$640$	2.72118	0.107564
$641$	16.0263	0.633000	0.316500	−	0.948592i	$-0.397492\pi$
$641$	16.0263	0.633000	0.316500	+	0.948592i	$0.397492\pi$
$642$	12.3739	0.488361
$643$	0.927611	0.0365814	0.0182907	−	0.999833i	$-0.494178\pi$
$643$	0.927611	0.0365814	0.0182907	+	0.999833i	$0.494178\pi$
$644$	−11.2699	−0.444098
$645$	−26.6781	−1.05045
$646$	−8.51113	−0.334866
$647$	45.7576	1.79892	0.899459	−	0.437004i	$-0.143961\pi$
$647$	45.7576	1.79892	0.899459	+	0.437004i	$0.143961\pi$
$648$	−2.05885	−0.0808791
$649$	24.1974	0.949830
$650$	−2.25906	−0.0886076
$651$	98.6352	3.86582
$652$	1.71796	0.0672807
$653$	−30.8571	−1.20753	−0.603767	−	0.797161i	$-0.706334\pi$
$653$	−30.8571	−1.20753	−0.603767	+	0.797161i	$0.706334\pi$
$654$	−0.342124	−0.0133781
$655$	−38.7832	−1.51539
$656$	1.16230	0.0453803
$657$	−58.4544	−2.28052
$658$	5.13528	0.200194
$659$	20.3944	0.794452	0.397226	−	0.917721i	$-0.369973\pi$
$659$	20.3944	0.794452	0.397226	+	0.917721i	$0.369973\pi$
$660$	−25.8838	−1.00752
$661$	42.5697	1.65577	0.827884	−	0.560899i	$-0.189544\pi$
$661$	42.5697	1.65577	0.827884	+	0.560899i	$0.189544\pi$
$662$	9.82635	0.381912
$663$	−5.43316	−0.211007
$664$	2.96141	0.114925
$665$	52.5963	2.03960
$666$	−16.1108	−0.624281
$667$	−18.5287	−0.717433
$668$	24.0879	0.931989
$669$	24.6330	0.952365
$670$	−22.8900	−0.884318
$671$	34.7576	1.34180
$672$	−13.1346	−0.506679
$673$	−18.1524	−0.699724	−0.349862	−	0.936801i	$-0.613772\pi$
$673$	−18.1524	−0.699724	−0.349862	+	0.936801i	$0.613772\pi$
$674$	23.3447	0.899205
$675$	−10.1088	−0.389088
$676$	−12.1175	−0.466059
$677$	1.76271	0.0677466	0.0338733	−	0.999426i	$-0.489216\pi$
$677$	1.76271	0.0677466	0.0338733	+	0.999426i	$0.489216\pi$
$678$	−10.7453	−0.412670
$679$	59.3217	2.27656
$680$	−5.73479	−0.219919
$681$	−35.0865	−1.34452
$682$	−26.0279	−0.996659
$683$	−1.25515	−0.0480270	−0.0240135	−	0.999712i	$-0.507644\pi$
$683$	−1.25515	−0.0480270	−0.0240135	+	0.999712i	$0.507644\pi$
$684$	18.3015	0.699776
$685$	−47.2803	−1.80649
$686$	42.6222	1.62732
$687$	23.9495	0.913730
$688$	3.57234	0.136194
$689$	−11.2984	−0.430436
$690$	17.5855	0.669467
$691$	−11.6752	−0.444145	−0.222072	−	0.975030i	$-0.571282\pi$
$691$	−11.6752	−0.444145	−0.222072	+	0.975030i	$0.571282\pi$
$692$	19.4698	0.740130
$693$	75.1719	2.85554
$694$	17.1020	0.649183
$695$	1.64511	0.0624026
$696$	−21.5943	−0.818531
$697$	−2.44951	−0.0927819
$698$	−4.82512	−0.182633
$699$	29.0788	1.09986
$700$	11.5094	0.435015
$701$	−40.1035	−1.51469	−0.757343	−	0.653017i	$-0.773503\pi$
$701$	−40.1035	−1.51469	−0.757343	+	0.653017i	$0.773503\pi$
$702$	3.94878	0.149037
$703$	−14.3577	−0.541509
$704$	3.46596	0.130628
$705$	−8.01302	−0.301788
$706$	3.86808	0.145577
$707$	−35.9656	−1.35262
$708$	−19.1598	−0.720068
$709$	5.53876	0.208012	0.104006	−	0.994577i	$-0.466834\pi$
$709$	5.53876	0.208012	0.104006	+	0.994577i	$0.466834\pi$
$710$	−28.8352	−1.08217
$711$	9.40165	0.352589
$712$	−2.23799	−0.0838721
$713$	17.6834	0.662248
$714$	27.6808	1.03593
$715$	−8.85987	−0.331340
$716$	−15.9483	−0.596017
$717$	25.7235	0.960660
$718$	−33.4255	−1.24743
$719$	32.9011	1.22700	0.613501	−	0.789694i	$-0.289760\pi$
$719$	32.9011	1.22700	0.613501	+	0.789694i	$0.289760\pi$
$720$	12.3315	0.459570
$721$	60.8577	2.26646
$722$	−2.69003	−0.100113
$723$	60.2176	2.23952
$724$	−4.68494	−0.174114
$725$	18.9224	0.702759
$726$	−2.77982	−0.103169
$727$	−13.1368	−0.487219	−0.243609	−	0.969873i	$-0.578331\pi$
$727$	−13.1368	−0.487219	−0.243609	+	0.969873i	$0.578331\pi$
$728$	−4.49590	−0.166629
$729$	−43.9387	−1.62736
$730$	−35.1006	−1.29913
$731$	−7.52857	−0.278454
$732$	−27.5215	−1.01722
$733$	−44.9046	−1.65859	−0.829295	−	0.558812i	$-0.811258\pi$
$733$	−44.9046	−1.65859	−0.829295	+	0.558812i	$0.811258\pi$
$734$	−24.2281	−0.894276
$735$	−118.783	−4.38138
$736$	−2.35478	−0.0867984
$737$	−29.1550	−1.07394
$738$	5.26720	0.193888
$739$	−25.2605	−0.929222	−0.464611	−	0.885515i	$-0.653806\pi$
$739$	−25.2605	−0.929222	−0.464611	+	0.885515i	$0.653806\pi$
$740$	−9.67418	−0.355630
$741$	10.4116	0.382481
$742$	57.5630	2.11321
$743$	38.4102	1.40913	0.704567	−	0.709637i	$-0.251141\pi$
$743$	38.4102	1.40913	0.704567	+	0.709637i	$0.251141\pi$
$744$	20.6092	0.755569
$745$	−56.6162	−2.07426
$746$	8.12575	0.297505
$747$	13.4202	0.491019
$748$	−7.30440	−0.267075
$749$	−21.5791	−0.788483
$750$	19.3808	0.707686
$751$	−48.8995	−1.78437	−0.892183	−	0.451674i	$-0.850827\pi$
$751$	−48.8995	−1.78437	−0.892183	+	0.451674i	$0.850827\pi$
$752$	1.07298	0.0391276
$753$	−63.3031	−2.30689
$754$	−7.39162	−0.269187
$755$	14.0651	0.511881
$756$	−20.1182	−0.731691
$757$	8.13013	0.295495	0.147747	−	0.989025i	$-0.452798\pi$
$757$	8.13013	0.295495	0.147747	+	0.989025i	$0.452798\pi$
$758$	6.29036	0.228476
$759$	22.3986	0.813018
$760$	10.9896	0.398636
$761$	−37.2241	−1.34937	−0.674686	−	0.738105i	$-0.735721\pi$
$761$	−37.2241	−1.34937	−0.674686	+	0.738105i	$0.735721\pi$
$762$	−35.9253	−1.30144
$763$	0.596634	0.0215996
$764$	−26.1879	−0.947446
$765$	−25.9883	−0.939609
$766$	24.1796	0.873645
$767$	−6.55828	−0.236806
$768$	−2.74439	−0.0990297
$769$	38.9820	1.40573	0.702864	−	0.711324i	$-0.251904\pi$
$769$	38.9820	1.40573	0.702864	+	0.711324i	$0.251904\pi$
$770$	45.1390	1.62670
$771$	35.3955	1.27474
$772$	−3.69553	−0.133005
$773$	−16.1041	−0.579224	−0.289612	−	0.957144i	$-0.593526\pi$
$773$	−16.1041	−0.579224	−0.289612	+	0.957144i	$0.593526\pi$
$774$	16.1887	0.581892
$775$	−18.0591	−0.648703
$776$	12.3949	0.444950
$777$	46.6954	1.67519
$778$	18.8416	0.675504
$779$	4.69403	0.168181
$780$	7.01535	0.251190
$781$	−36.7274	−1.31421
$782$	4.96262	0.177463
$783$	−33.0759	−1.18203
$784$	15.9056	0.568058
$785$	11.7123	0.418028
$786$	39.1141	1.39515
$787$	13.4023	0.477741	0.238871	−	0.971051i	$-0.423223\pi$
$787$	13.4023	0.477741	0.238871	+	0.971051i	$0.423223\pi$
$788$	−16.0973	−0.573442
$789$	−39.9650	−1.42279
$790$	5.64548	0.200857
$791$	18.7388	0.666277
$792$	15.7067	0.558113
$793$	−9.42045	−0.334530
$794$	−19.0715	−0.676824
$795$	−89.8206	−3.18561
$796$	23.0058	0.815419
$797$	18.8099	0.666280	0.333140	−	0.942877i	$-0.391892\pi$
$797$	18.8099	0.666280	0.333140	+	0.942877i	$0.391892\pi$
$798$	−53.0449	−1.87777
$799$	−2.26127	−0.0799981
$800$	2.40482	0.0850231
$801$	−10.1419	−0.358345
$802$	−18.1166	−0.639719
$803$	−44.7076	−1.57770
$804$	23.0853	0.814154
$805$	−30.6675	−1.08089
$806$	7.05441	0.248481
$807$	29.6482	1.04367
$808$	−7.51477	−0.264369
$809$	14.1164	0.496305	0.248153	−	0.968721i	$-0.420177\pi$
$809$	14.1164	0.496305	0.248153	+	0.968721i	$0.420177\pi$
$810$	−5.60249	−0.196851
$811$	15.4285	0.541767	0.270883	−	0.962612i	$-0.412684\pi$
$811$	15.4285	0.541767	0.270883	+	0.962612i	$0.412684\pi$
$812$	37.6586	1.32156
$813$	−76.9254	−2.69789
$814$	−12.3220	−0.431886
$815$	4.67489	0.163754
$816$	5.78371	0.202470
$817$	14.4271	0.504740
$818$	−18.2458	−0.637951
$819$	−20.3740	−0.711927
$820$	3.16283	0.110451
$821$	−45.1122	−1.57443	−0.787214	−	0.616680i	$-0.788477\pi$
$821$	−45.1122	−1.57443	−0.787214	+	0.616680i	$0.788477\pi$
$822$	47.6836	1.66316
$823$	11.8369	0.412609	0.206304	−	0.978488i	$-0.433856\pi$
$823$	11.8369	0.412609	0.206304	+	0.978488i	$0.433856\pi$
$824$	12.7158	0.442977
$825$	−22.8745	−0.796389
$826$	33.4130	1.16259
$827$	30.5543	1.06248	0.531239	−	0.847222i	$-0.321727\pi$
$827$	30.5543	1.06248	0.531239	+	0.847222i	$0.321727\pi$
$828$	−10.6711	−0.370848
$829$	−36.8174	−1.27872	−0.639360	−	0.768908i	$-0.720801\pi$
$829$	−36.8174	−1.27872	−0.639360	+	0.768908i	$0.720801\pi$
$830$	8.05853	0.279716
$831$	−47.6050	−1.65140
$832$	−0.939390	−0.0325675
$833$	−33.5206	−1.16142
$834$	−1.65914	−0.0574515
$835$	65.5475	2.26836
$836$	13.9975	0.484114
$837$	31.5669	1.09111
$838$	28.3817	0.980428
$839$	2.04083	0.0704572	0.0352286	−	0.999379i	$-0.488784\pi$
$839$	2.04083	0.0704572	0.0352286	+	0.999379i	$0.488784\pi$
$840$	−35.7416	−1.23320
$841$	32.9137	1.13496
$842$	−16.1836	−0.557723
$843$	−23.2047	−0.799212
$844$	−18.7382	−0.644996
$845$	−32.9740	−1.13434
$846$	4.86243	0.167174
$847$	4.84776	0.166571
$848$	12.0274	0.413023
$849$	59.2060	2.03195
$850$	−5.06807	−0.173833
$851$	8.37159	0.286974
$852$	29.0812	0.996306
$853$	0.420554	0.0143995	0.00719975	−	0.999974i	$-0.497708\pi$
$853$	0.420554	0.0143995	0.00719975	+	0.999974i	$0.497708\pi$
$854$	47.9951	1.64236
$855$	49.8017	1.70318
$856$	−4.50881	−0.154108
$857$	35.1652	1.20122	0.600610	−	0.799542i	$-0.294925\pi$
$857$	35.1652	1.20122	0.600610	+	0.799542i	$0.294925\pi$
$858$	8.93545	0.305051
$859$	−39.0064	−1.33088	−0.665441	−	0.746450i	$-0.731757\pi$
$859$	−39.0064	−1.33088	−0.665441	+	0.746450i	$0.731757\pi$
$860$	9.72097	0.331482
$861$	−15.2664	−0.520278
$862$	−4.25759	−0.145014
$863$	32.9427	1.12138	0.560691	−	0.828025i	$-0.310536\pi$
$863$	32.9427	1.12138	0.560691	+	0.828025i	$0.310536\pi$
$864$	−4.20356	−0.143008
$865$	52.9808	1.80140
$866$	−7.30345	−0.248181
$867$	34.4657	1.17052
$868$	−35.9406	−1.21990
$869$	7.19065	0.243926
$870$	−58.7620	−1.99222
$871$	7.90195	0.267747
$872$	0.124663	0.00422162
$873$	56.1697	1.90106
$874$	−9.50992	−0.321678
$875$	−33.7984	−1.14259
$876$	35.4000	1.19606
$877$	−20.9434	−0.707208	−0.353604	−	0.935395i	$-0.615044\pi$
$877$	−20.9434	−0.707208	−0.353604	+	0.935395i	$0.615044\pi$
$878$	25.0238	0.844513
$879$	−41.1591	−1.38826
$880$	9.43151	0.317936
$881$	−22.1479	−0.746181	−0.373091	−	0.927795i	$-0.621702\pi$
$881$	−22.1479	−0.746181	−0.373091	+	0.927795i	$0.621702\pi$
$882$	72.0794	2.42704
$883$	14.2620	0.479956	0.239978	−	0.970778i	$-0.422860\pi$
$883$	14.2620	0.479956	0.239978	+	0.970778i	$0.422860\pi$
$884$	1.97973	0.0665856
$885$	−52.1372	−1.75257
$886$	−12.5646	−0.422116
$887$	−31.6764	−1.06359	−0.531795	−	0.846873i	$-0.678482\pi$
$887$	−31.6764	−1.06359	−0.531795	+	0.846873i	$0.678482\pi$
$888$	9.75671	0.327414
$889$	62.6506	2.10123
$890$	−6.08997	−0.204136
$891$	−7.13589	−0.239061
$892$	−8.97574	−0.300530
$893$	4.33330	0.145009
$894$	57.0992	1.90968
$895$	−43.3983	−1.45064
$896$	4.78598	0.159888
$897$	−6.07075	−0.202697
$898$	23.9089	0.797849
$899$	−59.0892	−1.97074
$900$	10.8979	0.363263
$901$	−25.3474	−0.844444
$902$	4.02850	0.134134
$903$	−46.9213	−1.56144
$904$	3.91536	0.130223
$905$	−12.7485	−0.423776
$906$	−14.1851	−0.471267
$907$	−32.6973	−1.08570	−0.542848	−	0.839831i	$-0.682654\pi$
$907$	−32.6973	−1.08570	−0.542848	+	0.839831i	$0.682654\pi$
$908$	12.7848	0.424278
$909$	−34.0546	−1.12952
$910$	−12.2342	−0.405558
$911$	−39.4090	−1.30568	−0.652839	−	0.757496i	$-0.726422\pi$
$911$	−39.4090	−1.30568	−0.652839	+	0.757496i	$0.726422\pi$
$912$	−11.0834	−0.367008
$913$	10.2641	0.339694
$914$	−4.32224	−0.142967
$915$	−74.8909	−2.47582
$916$	−8.72670	−0.288338
$917$	−68.2116	−2.25254
$918$	8.85886	0.292386
$919$	48.0813	1.58606	0.793028	−	0.609185i	$-0.208503\pi$
$919$	48.0813	1.58606	0.793028	+	0.609185i	$0.208503\pi$
$920$	−6.40778	−0.211258
$921$	−22.9631	−0.756659
$922$	−14.6170	−0.481385
$923$	9.95433	0.327651
$924$	−45.5241	−1.49763
$925$	−8.54947	−0.281105
$926$	24.7338	0.812804
$927$	57.6242	1.89263
$928$	7.86853	0.258297
$929$	−7.75808	−0.254534	−0.127267	−	0.991868i	$-0.540621\pi$
$929$	−7.75808	−0.254534	−0.127267	+	0.991868i	$0.540621\pi$
$930$	56.0813	1.83898
$931$	64.2358	2.10524
$932$	−10.5957	−0.347074
$933$	−53.4112	−1.74860
$934$	−32.8807	−1.07589
$935$	−19.8766	−0.650034
$936$	−4.25703	−0.139145
$937$	35.3699	1.15548	0.577742	−	0.816220i	$-0.303934\pi$
$937$	35.3699	1.15548	0.577742	+	0.816220i	$0.303934\pi$
$938$	−40.2587	−1.31449
$939$	16.3283	0.532855
$940$	2.91978	0.0952327
$941$	−39.8313	−1.29846	−0.649232	−	0.760590i	$-0.724910\pi$
$941$	−39.8313	−1.29846	−0.649232	+	0.760590i	$0.724910\pi$
$942$	−11.8122	−0.384861
$943$	−2.73697	−0.0891280
$944$	6.98143	0.227226
$945$	−54.7452	−1.78086
$946$	12.3816	0.402560
$947$	−39.7907	−1.29303	−0.646513	−	0.762903i	$-0.723773\pi$
$947$	−39.7907	−1.29303	−0.646513	+	0.762903i	$0.723773\pi$
$948$	−5.69364	−0.184921
$949$	12.1172	0.393342
$950$	9.71200	0.315099
$951$	−21.1413	−0.685553
$952$	−10.0863	−0.326899
$953$	50.7546	1.64410	0.822051	−	0.569414i	$-0.192830\pi$
$953$	50.7546	1.64410	0.822051	+	0.569414i	$0.192830\pi$
$954$	54.5046	1.76465
$955$	−71.2620	−2.30599
$956$	−9.37310	−0.303148
$957$	−74.8452	−2.41940
$958$	−12.9377	−0.417999
$959$	−83.1561	−2.68525
$960$	−7.46798	−0.241028
$961$	25.3935	0.819146
$962$	3.33967	0.107675
$963$	−20.4325	−0.658429
$964$	−21.9420	−0.706706
$965$	−10.0562	−0.323721
$966$	30.9291	0.995129
$967$	−55.1863	−1.77467	−0.887335	−	0.461125i	$-0.847446\pi$
$967$	−55.1863	−1.77467	−0.887335	+	0.461125i	$0.847446\pi$
$968$	1.01291	0.0325561
$969$	23.3579	0.750363
$970$	33.7287	1.08296
$971$	4.41525	0.141692	0.0708461	−	0.997487i	$-0.477430\pi$
$971$	4.41525	0.141692	0.0708461	+	0.997487i	$0.477430\pi$
$972$	18.2610	0.585721
$973$	2.89340	0.0927582
$974$	14.8392	0.475480
$975$	6.19975	0.198551
$976$	10.0283	0.320997
$977$	48.2331	1.54311	0.771557	−	0.636160i	$-0.219478\pi$
$977$	48.2331	1.54311	0.771557	+	0.636160i	$0.219478\pi$
$978$	−4.71477	−0.150762
$979$	−7.75678	−0.247908
$980$	43.2821	1.38260
$981$	0.564934	0.0180369
$982$	23.4329	0.747775
$983$	−29.6419	−0.945429	−0.472714	−	0.881216i	$-0.656726\pi$
$983$	−29.6419	−0.945429	−0.472714	+	0.881216i	$0.656726\pi$
$984$	−3.18982	−0.101688
$985$	−43.8036	−1.39570
$986$	−16.5827	−0.528099
$987$	−14.0932	−0.448592
$988$	−3.79378	−0.120696
$989$	−8.41207	−0.267488
$990$	42.7407	1.35839
$991$	20.1564	0.640289	0.320145	−	0.947369i	$-0.396268\pi$
$991$	20.1564	0.640289	0.320145	+	0.947369i	$0.396268\pi$
$992$	−7.50956	−0.238429
$993$	−26.9674	−0.855783
$994$	−50.7151	−1.60859
$995$	62.6029	1.98465
$996$	−8.12727	−0.257523
$997$	−10.7082	−0.339132	−0.169566	−	0.985519i	$-0.554237\pi$
$997$	−10.7082	−0.339132	−0.169566	+	0.985519i	$0.554237\pi$
$998$	−31.8213	−1.00729
$999$	14.9443	0.472816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.d.1.5	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.d.1.5	✓	92	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.74439 q^{3} +1.00000 q^{4} +2.72118 q^{5} -2.74439 q^{6} +4.78598 q^{7} +1.00000 q^{8} +4.53169 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.74439 q^{3} +1.00000 q^{4} +2.72118 q^{5} -2.74439 q^{6} +4.78598 q^{7} +1.00000 q^{8} +4.53169 q^{9} +2.72118 q^{10} +3.46596 q^{11} -2.74439 q^{12} -0.939390 q^{13} +4.78598 q^{14} -7.46798 q^{15} +1.00000 q^{16} -2.10747 q^{17} +4.53169 q^{18} +4.03856 q^{19} +2.72118 q^{20} -13.1346 q^{21} +3.46596 q^{22} -2.35478 q^{23} -2.74439 q^{24} +2.40482 q^{25} -0.939390 q^{26} -4.20356 q^{27} +4.78598 q^{28} +7.86853 q^{29} -7.46798 q^{30} -7.50956 q^{31} +1.00000 q^{32} -9.51197 q^{33} -2.10747 q^{34} +13.0235 q^{35} +4.53169 q^{36} -3.55514 q^{37} +4.03856 q^{38} +2.57805 q^{39} +2.72118 q^{40} +1.16230 q^{41} -13.1346 q^{42} +3.57234 q^{43} +3.46596 q^{44} +12.3315 q^{45} -2.35478 q^{46} +1.07298 q^{47} -2.74439 q^{48} +15.9056 q^{49} +2.40482 q^{50} +5.78371 q^{51} -0.939390 q^{52} +12.0274 q^{53} -4.20356 q^{54} +9.43151 q^{55} +4.78598 q^{56} -11.0834 q^{57} +7.86853 q^{58} +6.98143 q^{59} -7.46798 q^{60} +10.0283 q^{61} -7.50956 q^{62} +21.6886 q^{63} +1.00000 q^{64} -2.55625 q^{65} -9.51197 q^{66} -8.41179 q^{67} -2.10747 q^{68} +6.46244 q^{69} +13.0235 q^{70} -10.5966 q^{71} +4.53169 q^{72} -12.8990 q^{73} -3.55514 q^{74} -6.59976 q^{75} +4.03856 q^{76} +16.5880 q^{77} +2.57805 q^{78} +2.07464 q^{79} +2.72118 q^{80} -2.05885 q^{81} +1.16230 q^{82} +2.96141 q^{83} -13.1346 q^{84} -5.73479 q^{85} +3.57234 q^{86} -21.5943 q^{87} +3.46596 q^{88} -2.23799 q^{89} +12.3315 q^{90} -4.49590 q^{91} -2.35478 q^{92} +20.6092 q^{93} +1.07298 q^{94} +10.9896 q^{95} -2.74439 q^{96} +12.3949 q^{97} +15.9056 q^{98} +15.7067 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

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