LMFDB - Embedded newform 8045.2.a.e.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$0$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.20465 q^{2} -2.96481 q^{3} +2.86047 q^{4} +1.00000 q^{5} +6.53637 q^{6} +0.166097 q^{7} -1.89703 q^{8} +5.79013 q^{9} +O(q^{10})$	\(q-2.20465 q^{2} -2.96481 q^{3} +2.86047 q^{4} +1.00000 q^{5} +6.53637 q^{6} +0.166097 q^{7} -1.89703 q^{8} +5.79013 q^{9} -2.20465 q^{10} -2.81530 q^{11} -8.48077 q^{12} -2.55354 q^{13} -0.366186 q^{14} -2.96481 q^{15} -1.53865 q^{16} +6.07626 q^{17} -12.7652 q^{18} +1.50804 q^{19} +2.86047 q^{20} -0.492447 q^{21} +6.20675 q^{22} +7.69146 q^{23} +5.62435 q^{24} +1.00000 q^{25} +5.62964 q^{26} -8.27221 q^{27} +0.475116 q^{28} +4.01356 q^{29} +6.53637 q^{30} -7.71270 q^{31} +7.18625 q^{32} +8.34686 q^{33} -13.3960 q^{34} +0.166097 q^{35} +16.5625 q^{36} +1.11294 q^{37} -3.32470 q^{38} +7.57076 q^{39} -1.89703 q^{40} -9.56860 q^{41} +1.08567 q^{42} +7.43429 q^{43} -8.05309 q^{44} +5.79013 q^{45} -16.9570 q^{46} +9.62358 q^{47} +4.56181 q^{48} -6.97241 q^{49} -2.20465 q^{50} -18.0150 q^{51} -7.30431 q^{52} -12.1846 q^{53} +18.2373 q^{54} -2.81530 q^{55} -0.315092 q^{56} -4.47106 q^{57} -8.84847 q^{58} +2.82209 q^{59} -8.48077 q^{60} +13.2485 q^{61} +17.0038 q^{62} +0.961723 q^{63} -12.7658 q^{64} -2.55354 q^{65} -18.4019 q^{66} +6.52724 q^{67} +17.3810 q^{68} -22.8038 q^{69} -0.366186 q^{70} +5.03499 q^{71} -10.9841 q^{72} -12.1836 q^{73} -2.45364 q^{74} -2.96481 q^{75} +4.31371 q^{76} -0.467614 q^{77} -16.6909 q^{78} +4.88368 q^{79} -1.53865 q^{80} +7.15519 q^{81} +21.0954 q^{82} +12.6293 q^{83} -1.40863 q^{84} +6.07626 q^{85} -16.3900 q^{86} -11.8994 q^{87} +5.34073 q^{88} -2.85425 q^{89} -12.7652 q^{90} -0.424135 q^{91} +22.0012 q^{92} +22.8667 q^{93} -21.2166 q^{94} +1.50804 q^{95} -21.3059 q^{96} +17.7038 q^{97} +15.3717 q^{98} -16.3010 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) $ 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9 + 21 * q^10 + 36 * q^11 + 55 * q^12 + 57 * q^13 + 2 * q^14 + 33 * q^15 + 179 * q^16 + 55 * q^17 + 65 * q^18 + 130 * q^19 + 157 * q^20 + 28 * q^21 + 30 * q^22 + 117 * q^23 + 21 * q^24 + 142 * q^25 + 21 * q^26 + 120 * q^27 + 135 * q^28 + 12 * q^29 + 15 * q^30 + 74 * q^31 + 126 * q^32 + 55 * q^33 + 35 * q^34 + 63 * q^35 + 186 * q^36 + 75 * q^37 + 65 * q^38 + 23 * q^39 + 60 * q^40 + 22 * q^41 + 10 * q^42 + 190 * q^43 + 22 * q^44 + 157 * q^45 + 56 * q^46 + 102 * q^47 + 78 * q^48 + 197 * q^49 + 21 * q^50 + 30 * q^51 + 120 * q^52 + 56 * q^53 - 6 * q^54 + 36 * q^55 + 3 * q^56 + 68 * q^57 + 31 * q^58 + 55 * q^59 + 55 * q^60 + 90 * q^61 + 68 * q^62 + 167 * q^63 + 180 * q^64 + 57 * q^65 + 17 * q^66 + 151 * q^67 + 119 * q^68 + 21 * q^69 + 2 * q^70 + 4 * q^71 + 130 * q^72 + 143 * q^73 - 46 * q^74 + 33 * q^75 + 213 * q^76 + 75 * q^77 - 24 * q^78 + 47 * q^79 + 179 * q^80 + 150 * q^81 + 69 * q^82 + 201 * q^83 - 31 * q^84 + 55 * q^85 - 4 * q^86 + 153 * q^87 + 37 * q^88 + 25 * q^89 + 65 * q^90 + 132 * q^91 + 194 * q^92 + 52 * q^93 + 18 * q^94 + 130 * q^95 + 13 * q^96 + 80 * q^97 + 58 * q^98 + 103 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.20465	−1.55892	−0.779461	−	0.626451i	$-0.784507\pi$
$2$	−2.20465	−1.55892	−0.779461	+	0.626451i	$0.784507\pi$
$3$	−2.96481	−1.71174	−0.855868	−	0.517194i	$-0.826977\pi$
$3$	−2.96481	−1.71174	−0.855868	+	0.517194i	$0.826977\pi$
$4$	2.86047	1.43024
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	6.53637	2.66846
$7$	0.166097	0.0627788	0.0313894	−	0.999507i	$-0.490007\pi$
$7$	0.166097	0.0627788	0.0313894	+	0.999507i	$0.490007\pi$
$8$	−1.89703	−0.670703
$9$	5.79013	1.93004
$10$	−2.20465	−0.697171
$11$	−2.81530	−0.848846	−0.424423	−	0.905464i	$-0.639523\pi$
$11$	−2.81530	−0.848846	−0.424423	+	0.905464i	$0.639523\pi$
$12$	−8.48077	−2.44819
$13$	−2.55354	−0.708223	−0.354112	−	0.935203i	$-0.615217\pi$
$13$	−2.55354	−0.708223	−0.354112	+	0.935203i	$0.615217\pi$
$14$	−0.366186	−0.0978672
$15$	−2.96481	−0.765512
$16$	−1.53865	−0.384662
$17$	6.07626	1.47371	0.736855	−	0.676051i	$-0.236310\pi$
$17$	6.07626	1.47371	0.736855	+	0.676051i	$0.236310\pi$
$18$	−12.7652	−3.00878
$19$	1.50804	0.345968	0.172984	−	0.984925i	$-0.444659\pi$
$19$	1.50804	0.345968	0.172984	+	0.984925i	$0.444659\pi$
$20$	2.86047	0.639621
$21$	−0.492447	−0.107461
$22$	6.20675	1.32328
$23$	7.69146	1.60378	0.801891	−	0.597471i	$-0.203828\pi$
$23$	7.69146	1.60378	0.801891	+	0.597471i	$0.203828\pi$
$24$	5.62435	1.14807
$25$	1.00000	0.200000
$26$	5.62964	1.10406
$27$	−8.27221	−1.59199
$28$	0.475116	0.0897884
$29$	4.01356	0.745298	0.372649	−	0.927972i	$-0.378449\pi$
$29$	4.01356	0.745298	0.372649	+	0.927972i	$0.378449\pi$
$30$	6.53637	1.19337
$31$	−7.71270	−1.38524	−0.692621	−	0.721302i	$-0.743544\pi$
$31$	−7.71270	−1.38524	−0.692621	+	0.721302i	$0.743544\pi$
$32$	7.18625	1.27036
$33$	8.34686	1.45300
$34$	−13.3960	−2.29740
$35$	0.166097	0.0280755
$36$	16.5625	2.76041
$37$	1.11294	0.182966	0.0914829	−	0.995807i	$-0.470839\pi$
$37$	1.11294	0.182966	0.0914829	+	0.995807i	$0.470839\pi$
$38$	−3.32470	−0.539338
$39$	7.57076	1.21229
$40$	−1.89703	−0.299947
$41$	−9.56860	−1.49436	−0.747182	−	0.664619i	$-0.768594\pi$
$41$	−9.56860	−1.49436	−0.747182	+	0.664619i	$0.768594\pi$
$42$	1.08567	0.167523
$43$	7.43429	1.13372	0.566859	−	0.823815i	$-0.308158\pi$
$43$	7.43429	1.13372	0.566859	+	0.823815i	$0.308158\pi$
$44$	−8.05309	−1.21405
$45$	5.79013	0.863141
$46$	−16.9570	−2.50017
$47$	9.62358	1.40374	0.701872	−	0.712304i	$-0.252348\pi$
$47$	9.62358	1.40374	0.701872	+	0.712304i	$0.252348\pi$
$48$	4.56181	0.658441
$49$	−6.97241	−0.996059
$50$	−2.20465	−0.311784
$51$	−18.0150	−2.52260
$52$	−7.30431	−1.01293
$53$	−12.1846	−1.67368	−0.836842	−	0.547444i	$-0.815601\pi$
$53$	−12.1846	−1.67368	−0.836842	+	0.547444i	$0.815601\pi$
$54$	18.2373	2.48178
$55$	−2.81530	−0.379616
$56$	−0.315092	−0.0421059
$57$	−4.47106	−0.592207
$58$	−8.84847	−1.16186
$59$	2.82209	0.367405	0.183702	−	0.982982i	$-0.441192\pi$
$59$	2.82209	0.367405	0.183702	+	0.982982i	$0.441192\pi$
$60$	−8.48077	−1.09486
$61$	13.2485	1.69629	0.848147	−	0.529761i	$-0.177718\pi$
$61$	13.2485	1.69629	0.848147	+	0.529761i	$0.177718\pi$
$62$	17.0038	2.15948
$63$	0.961723	0.121166
$64$	−12.7658	−1.59573
$65$	−2.55354	−0.316727
$66$	−18.4019	−2.26511
$67$	6.52724	0.797429	0.398715	−	0.917075i	$-0.369456\pi$
$67$	6.52724	0.797429	0.398715	+	0.917075i	$0.369456\pi$
$68$	17.3810	2.10775
$69$	−22.8038	−2.74525
$70$	−0.366186	−0.0437675
$71$	5.03499	0.597544	0.298772	−	0.954325i	$-0.403423\pi$
$71$	5.03499	0.597544	0.298772	+	0.954325i	$0.403423\pi$
$72$	−10.9841	−1.29448
$73$	−12.1836	−1.42598	−0.712989	−	0.701175i	$-0.752659\pi$
$73$	−12.1836	−1.42598	−0.712989	+	0.701175i	$0.752659\pi$
$74$	−2.45364	−0.285229
$75$	−2.96481	−0.342347
$76$	4.31371	0.494816
$77$	−0.467614	−0.0532895
$78$	−16.6909	−1.88987
$79$	4.88368	0.549457	0.274728	−	0.961522i	$-0.411412\pi$
$79$	4.88368	0.549457	0.274728	+	0.961522i	$0.411412\pi$
$80$	−1.53865	−0.172026
$81$	7.15519	0.795021
$82$	21.0954	2.32960
$83$	12.6293	1.38625	0.693123	−	0.720819i	$-0.256234\pi$
$83$	12.6293	1.38625	0.693123	+	0.720819i	$0.256234\pi$
$84$	−1.40863	−0.153694
$85$	6.07626	0.659063
$86$	−16.3900	−1.76738
$87$	−11.8994	−1.27575
$88$	5.34073	0.569324
$89$	−2.85425	−0.302550	−0.151275	−	0.988492i	$-0.548338\pi$
$89$	−2.85425	−0.302550	−0.151275	+	0.988492i	$0.548338\pi$
$90$	−12.7652	−1.34557
$91$	−0.424135	−0.0444614
$92$	22.0012	2.29378
$93$	22.8667	2.37117
$94$	−21.2166	−2.18832
$95$	1.50804	0.154722
$96$	−21.3059	−2.17452
$97$	17.7038	1.79755	0.898774	−	0.438412i	$-0.144459\pi$
$97$	17.7038	1.79755	0.898774	+	0.438412i	$0.144459\pi$
$98$	15.3717	1.55278
$99$	−16.3010	−1.63831
$100$	2.86047	0.286047
$101$	0.407319	0.0405298	0.0202649	−	0.999795i	$-0.493549\pi$
$101$	0.407319	0.0405298	0.0202649	+	0.999795i	$0.493549\pi$
$102$	39.7167	3.93254
$103$	−2.01561	−0.198604	−0.0993019	−	0.995057i	$-0.531661\pi$
$103$	−2.01561	−0.198604	−0.0993019	+	0.995057i	$0.531661\pi$
$104$	4.84414	0.475007
$105$	−0.492447	−0.0480579
$106$	26.8628	2.60914
$107$	16.5310	1.59811	0.799055	−	0.601257i	$-0.205333\pi$
$107$	16.5310	1.59811	0.799055	+	0.601257i	$0.205333\pi$
$108$	−23.6624	−2.27692
$109$	18.2611	1.74910	0.874548	−	0.484939i	$-0.161158\pi$
$109$	18.2611	1.74910	0.874548	+	0.484939i	$0.161158\pi$
$110$	6.20675	0.591791
$111$	−3.29965	−0.313189
$112$	−0.255565	−0.0241486
$113$	−8.23860	−0.775022	−0.387511	−	0.921865i	$-0.626665\pi$
$113$	−8.23860	−0.775022	−0.387511	+	0.921865i	$0.626665\pi$
$114$	9.85712	0.923204
$115$	7.69146	0.717233
$116$	11.4807	1.06595
$117$	−14.7853	−1.36690
$118$	−6.22172	−0.572755
$119$	1.00925	0.0925177
$120$	5.62435	0.513431
$121$	−3.07406	−0.279460
$122$	−29.2082	−2.64439
$123$	28.3691	2.55796
$124$	−22.0619	−1.98122
$125$	1.00000	0.0894427
$126$	−2.12026	−0.188888
$127$	−2.59661	−0.230412	−0.115206	−	0.993342i	$-0.536753\pi$
$127$	−2.59661	−0.230412	−0.115206	+	0.993342i	$0.536753\pi$
$128$	13.7717	1.21726
$129$	−22.0413	−1.94063
$130$	5.62964	0.493753
$131$	−8.39017	−0.733053	−0.366526	−	0.930408i	$-0.619453\pi$
$131$	−8.39017	−0.733053	−0.366526	+	0.930408i	$0.619453\pi$
$132$	23.8759	2.07813
$133$	0.250481	0.0217195
$134$	−14.3903	−1.24313
$135$	−8.27221	−0.711959
$136$	−11.5269	−0.988421
$137$	19.0137	1.62445	0.812224	−	0.583345i	$-0.198256\pi$
$137$	19.0137	1.62445	0.812224	+	0.583345i	$0.198256\pi$
$138$	50.2743	4.27963
$139$	9.25765	0.785224	0.392612	−	0.919704i	$-0.371572\pi$
$139$	9.25765	0.785224	0.392612	+	0.919704i	$0.371572\pi$
$140$	0.475116	0.0401546
$141$	−28.5321	−2.40284
$142$	−11.1004	−0.931524
$143$	7.18898	0.601173
$144$	−8.90898	−0.742415
$145$	4.01356	0.333308
$146$	26.8605	2.22299
$147$	20.6719	1.70499
$148$	3.18353	0.261684
$149$	13.6796	1.12068	0.560339	−	0.828264i	$-0.310671\pi$
$149$	13.6796	1.12068	0.560339	+	0.828264i	$0.310671\pi$
$150$	6.53637	0.533693
$151$	−10.6217	−0.864381	−0.432190	−	0.901782i	$-0.642259\pi$
$151$	−10.6217	−0.864381	−0.432190	+	0.901782i	$0.642259\pi$
$152$	−2.86081	−0.232042
$153$	35.1823	2.84432
$154$	1.03092	0.0830742
$155$	−7.71270	−0.619499
$156$	21.6559	1.73386
$157$	−6.13532	−0.489652	−0.244826	−	0.969567i	$-0.578731\pi$
$157$	−6.13532	−0.489652	−0.244826	+	0.969567i	$0.578731\pi$
$158$	−10.7668	−0.856560
$159$	36.1251	2.86491
$160$	7.18625	0.568123
$161$	1.27753	0.100683
$162$	−15.7747	−1.23938
$163$	5.13612	0.402292	0.201146	−	0.979561i	$-0.435533\pi$
$163$	5.13612	0.402292	0.201146	+	0.979561i	$0.435533\pi$
$164$	−27.3707	−2.13729
$165$	8.34686	0.649802
$166$	−27.8432	−2.16105
$167$	8.47795	0.656044	0.328022	−	0.944670i	$-0.393618\pi$
$167$	8.47795	0.656044	0.328022	+	0.944670i	$0.393618\pi$
$168$	0.934189	0.0720742
$169$	−6.47946	−0.498420
$170$	−13.3960	−1.02743
$171$	8.73175	0.667734
$172$	21.2656	1.62148
$173$	−13.5605	−1.03098	−0.515491	−	0.856895i	$-0.672390\pi$
$173$	−13.5605	−1.03098	−0.515491	+	0.856895i	$0.672390\pi$
$174$	26.2341	1.98880
$175$	0.166097	0.0125558
$176$	4.33177	0.326519
$177$	−8.36698	−0.628901
$178$	6.29261	0.471651
$179$	−9.05276	−0.676635	−0.338318	−	0.941032i	$-0.609858\pi$
$179$	−9.05276	−0.676635	−0.338318	+	0.941032i	$0.609858\pi$
$180$	16.5625	1.23449
$181$	−12.5210	−0.930676	−0.465338	−	0.885133i	$-0.654067\pi$
$181$	−12.5210	−0.930676	−0.465338	+	0.885133i	$0.654067\pi$
$182$	0.935068	0.0693118
$183$	−39.2793	−2.90361
$184$	−14.5910	−1.07566
$185$	1.11294	0.0818248
$186$	−50.4130	−3.69646
$187$	−17.1065	−1.25095
$188$	27.5280	2.00768
$189$	−1.37399	−0.0999431
$190$	−3.32470	−0.241199
$191$	−4.55269	−0.329421	−0.164710	−	0.986342i	$-0.552669\pi$
$191$	−4.55269	−0.329421	−0.164710	+	0.986342i	$0.552669\pi$
$192$	37.8484	2.73147
$193$	−24.0023	−1.72772	−0.863861	−	0.503731i	$-0.831960\pi$
$193$	−24.0023	−1.72772	−0.863861	+	0.503731i	$0.831960\pi$
$194$	−39.0306	−2.80224
$195$	7.57076	0.542153
$196$	−19.9444	−1.42460
$197$	−14.9736	−1.06683	−0.533413	−	0.845855i	$-0.679091\pi$
$197$	−14.9736	−1.06683	−0.533413	+	0.845855i	$0.679091\pi$
$198$	35.9379	2.55399
$199$	26.9154	1.90798	0.953990	−	0.299838i	$-0.0969327\pi$
$199$	26.9154	1.90798	0.953990	+	0.299838i	$0.0969327\pi$
$200$	−1.89703	−0.134141
$201$	−19.3521	−1.36499
$202$	−0.897995	−0.0631827
$203$	0.666640	0.0467889
$204$	−51.5313	−3.60792
$205$	−9.56860	−0.668300
$206$	4.44371	0.309608
$207$	44.5346	3.09537
$208$	3.92900	0.272427
$209$	−4.24560	−0.293674
$210$	1.08567	0.0749185
$211$	19.4310	1.33768	0.668842	−	0.743405i	$-0.266790\pi$
$211$	19.4310	1.33768	0.668842	+	0.743405i	$0.266790\pi$
$212$	−34.8537	−2.39376
$213$	−14.9278	−1.02284
$214$	−36.4450	−2.49133
$215$	7.43429	0.507014
$216$	15.6927	1.06775
$217$	−1.28106	−0.0869638
$218$	−40.2593	−2.72670
$219$	36.1220	2.44090
$220$	−8.05309	−0.542940
$221$	−15.5159	−1.04372
$222$	7.27457	0.488237
$223$	2.80134	0.187591	0.0937957	−	0.995591i	$-0.470100\pi$
$223$	2.80134	0.187591	0.0937957	+	0.995591i	$0.470100\pi$
$224$	1.19361	0.0797517
$225$	5.79013	0.386008
$226$	18.1632	1.20820
$227$	−25.9724	−1.72385	−0.861924	−	0.507037i	$-0.830741\pi$
$227$	−25.9724	−1.72385	−0.861924	+	0.507037i	$0.830741\pi$
$228$	−12.7893	−0.846995
$229$	22.6330	1.49563	0.747814	−	0.663908i	$-0.231104\pi$
$229$	22.6330	1.49563	0.747814	+	0.663908i	$0.231104\pi$
$230$	−16.9570	−1.11811
$231$	1.38639	0.0912177
$232$	−7.61385	−0.499874
$233$	−14.1892	−0.929567	−0.464784	−	0.885424i	$-0.653868\pi$
$233$	−14.1892	−0.929567	−0.464784	+	0.885424i	$0.653868\pi$
$234$	32.5964	2.13089
$235$	9.62358	0.627773
$236$	8.07251	0.525476
$237$	−14.4792	−0.940526
$238$	−2.22504	−0.144228
$239$	−23.0008	−1.48780	−0.743899	−	0.668292i	$-0.767026\pi$
$239$	−23.0008	−1.48780	−0.743899	+	0.668292i	$0.767026\pi$
$240$	4.56181	0.294464
$241$	1.27010	0.0818141	0.0409070	−	0.999163i	$-0.486975\pi$
$241$	1.27010	0.0818141	0.0409070	+	0.999163i	$0.486975\pi$
$242$	6.77722	0.435656
$243$	3.60282	0.231121
$244$	37.8969	2.42610
$245$	−6.97241	−0.445451
$246$	−62.5439	−3.98766
$247$	−3.85084	−0.245023
$248$	14.6312	0.929085
$249$	−37.4436	−2.37289
$250$	−2.20465	−0.139434
$251$	−21.8266	−1.37768	−0.688841	−	0.724912i	$-0.741880\pi$
$251$	−21.8266	−1.37768	−0.688841	+	0.724912i	$0.741880\pi$
$252$	2.75098	0.173296
$253$	−21.6538	−1.36136
$254$	5.72461	0.359194
$255$	−18.0150	−1.12814
$256$	−4.83003	−0.301877
$257$	7.43856	0.464004	0.232002	−	0.972715i	$-0.425472\pi$
$257$	7.43856	0.464004	0.232002	+	0.972715i	$0.425472\pi$
$258$	48.5933	3.02529
$259$	0.184856	0.0114864
$260$	−7.30431	−0.452994
$261$	23.2390	1.43846
$262$	18.4974	1.14277
$263$	15.7746	0.972706	0.486353	−	0.873762i	$-0.338327\pi$
$263$	15.7746	0.972706	0.486353	+	0.873762i	$0.338327\pi$
$264$	−15.8343	−0.974532
$265$	−12.1846	−0.748495
$266$	−0.552223	−0.0338590
$267$	8.46232	0.517886
$268$	18.6710	1.14051
$269$	−3.99278	−0.243444	−0.121722	−	0.992564i	$-0.538842\pi$
$269$	−3.99278	−0.243444	−0.121722	+	0.992564i	$0.538842\pi$
$270$	18.2373	1.10989
$271$	−3.02035	−0.183473	−0.0917366	−	0.995783i	$-0.529242\pi$
$271$	−3.02035	−0.183473	−0.0917366	+	0.995783i	$0.529242\pi$
$272$	−9.34924	−0.566881
$273$	1.25748	0.0761062
$274$	−41.9185	−2.53239
$275$	−2.81530	−0.169769
$276$	−65.2295	−3.92635
$277$	−16.7738	−1.00784	−0.503919	−	0.863751i	$-0.668109\pi$
$277$	−16.7738	−1.00784	−0.503919	+	0.863751i	$0.668109\pi$
$278$	−20.4099	−1.22410
$279$	−44.6575	−2.67357
$280$	−0.315092	−0.0188303
$281$	−2.58644	−0.154294	−0.0771469	−	0.997020i	$-0.524581\pi$
$281$	−2.58644	−0.154294	−0.0771469	+	0.997020i	$0.524581\pi$
$282$	62.9033	3.74584
$283$	23.3312	1.38690	0.693449	−	0.720506i	$-0.256090\pi$
$283$	23.3312	1.38690	0.693449	+	0.720506i	$0.256090\pi$
$284$	14.4024	0.854628
$285$	−4.47106	−0.264843
$286$	−15.8492	−0.937181
$287$	−1.58932	−0.0938144
$288$	41.6093	2.45185
$289$	19.9209	1.17182
$290$	−8.84847	−0.519600
$291$	−52.4885	−3.07693
$292$	−34.8507	−2.03948
$293$	4.22145	0.246620	0.123310	−	0.992368i	$-0.460649\pi$
$293$	4.22145	0.246620	0.123310	+	0.992368i	$0.460649\pi$
$294$	−45.5743	−2.65795
$295$	2.82209	0.164309
$296$	−2.11128	−0.122716
$297$	23.2888	1.35135
$298$	−30.1587	−1.74705
$299$	−19.6404	−1.13583
$300$	−8.48077	−0.489637
$301$	1.23481	0.0711735
$302$	23.4171	1.34750
$303$	−1.20763	−0.0693763
$304$	−2.32035	−0.133081
$305$	13.2485	0.758606
$306$	−77.5646	−4.43407
$307$	−5.97636	−0.341089	−0.170544	−	0.985350i	$-0.554553\pi$
$307$	−5.97636	−0.341089	−0.170544	+	0.985350i	$0.554553\pi$
$308$	−1.33760	−0.0762166
$309$	5.97591	0.339958
$310$	17.0038	0.965750
$311$	4.02857	0.228439	0.114220	−	0.993456i	$-0.463563\pi$
$311$	4.02857	0.228439	0.114220	+	0.993456i	$0.463563\pi$
$312$	−14.3620	−0.813087
$313$	26.7905	1.51429	0.757145	−	0.653247i	$-0.226594\pi$
$313$	26.7905	1.51429	0.757145	+	0.653247i	$0.226594\pi$
$314$	13.5262	0.763328
$315$	0.961723	0.0541870
$316$	13.9696	0.785853
$317$	−1.64588	−0.0924416	−0.0462208	−	0.998931i	$-0.514718\pi$
$317$	−1.64588	−0.0924416	−0.0462208	+	0.998931i	$0.514718\pi$
$318$	−79.6431	−4.46616
$319$	−11.2994	−0.632644
$320$	−12.7658	−0.713632
$321$	−49.0113	−2.73554
$322$	−2.81650	−0.156958
$323$	9.16325	0.509857
$324$	20.4672	1.13707
$325$	−2.55354	−0.141645
$326$	−11.3233	−0.627141
$327$	−54.1408	−2.99399
$328$	18.1520	1.00227
$329$	1.59845	0.0881253
$330$	−18.4019	−1.01299
$331$	−18.4633	−1.01484	−0.507418	−	0.861700i	$-0.669400\pi$
$331$	−18.4633	−1.01484	−0.507418	+	0.861700i	$0.669400\pi$
$332$	36.1258	1.98266
$333$	6.44405	0.353132
$334$	−18.6909	−1.02272
$335$	6.52724	0.356621
$336$	0.757704	0.0413361
$337$	−5.26586	−0.286850	−0.143425	−	0.989661i	$-0.545812\pi$
$337$	−5.26586	−0.286850	−0.143425	+	0.989661i	$0.545812\pi$
$338$	14.2849	0.776997
$339$	24.4259	1.32663
$340$	17.3810	0.942615
$341$	21.7136	1.17586
$342$	−19.2504	−1.04094
$343$	−2.32078	−0.125310
$344$	−14.1031	−0.760388
$345$	−22.8038	−1.22771
$346$	29.8960	1.60722
$347$	−12.7016	−0.681857	−0.340929	−	0.940089i	$-0.610741\pi$
$347$	−12.7016	−0.681857	−0.340929	+	0.940089i	$0.610741\pi$
$348$	−34.0380	−1.82463
$349$	−4.52725	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$349$	−4.52725	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$350$	−0.366186	−0.0195734
$351$	21.1234	1.12748
$352$	−20.2315	−1.07834
$353$	−36.2732	−1.93063	−0.965315	−	0.261088i	$-0.915919\pi$
$353$	−36.2732	−1.93063	−0.965315	+	0.261088i	$0.915919\pi$
$354$	18.4462	0.980406
$355$	5.03499	0.267230
$356$	−8.16449	−0.432717
$357$	−2.99224	−0.158366
$358$	19.9582	1.05482
$359$	−11.7924	−0.622377	−0.311189	−	0.950348i	$-0.600727\pi$
$359$	−11.7924	−0.622377	−0.311189	+	0.950348i	$0.600727\pi$
$360$	−10.9841	−0.578911
$361$	−16.7258	−0.880306
$362$	27.6043	1.45085
$363$	9.11403	0.478362
$364$	−1.21322	−0.0635903
$365$	−12.1836	−0.637717
$366$	86.5970	4.52650
$367$	−16.9865	−0.886688	−0.443344	−	0.896351i	$-0.646208\pi$
$367$	−16.9865	−0.886688	−0.443344	+	0.896351i	$0.646208\pi$
$368$	−11.8345	−0.616914
$369$	−55.4034	−2.88419
$370$	−2.45364	−0.127558
$371$	−2.02383	−0.105072
$372$	65.4096	3.39133
$373$	−29.7733	−1.54160	−0.770800	−	0.637077i	$-0.780143\pi$
$373$	−29.7733	−1.54160	−0.770800	+	0.637077i	$0.780143\pi$
$374$	37.7138	1.95014
$375$	−2.96481	−0.153102
$376$	−18.2563	−0.941494
$377$	−10.2488	−0.527838
$378$	3.02916	0.155803
$379$	1.85456	0.0952625	0.0476312	−	0.998865i	$-0.484833\pi$
$379$	1.85456	0.0952625	0.0476312	+	0.998865i	$0.484833\pi$
$380$	4.31371	0.221289
$381$	7.69846	0.394404
$382$	10.0371	0.513541
$383$	17.1117	0.874369	0.437184	−	0.899372i	$-0.355976\pi$
$383$	17.1117	0.874369	0.437184	+	0.899372i	$0.355976\pi$
$384$	−40.8305	−2.08362
$385$	−0.467614	−0.0238318
$386$	52.9166	2.69338
$387$	43.0455	2.18813
$388$	50.6412	2.57092
$389$	−39.4030	−1.99781	−0.998905	−	0.0467846i	$-0.985103\pi$
$389$	−39.4030	−1.99781	−0.998905	+	0.0467846i	$0.985103\pi$
$390$	−16.6909	−0.845174
$391$	46.7353	2.36351
$392$	13.2269	0.668059
$393$	24.8753	1.25479
$394$	33.0115	1.66310
$395$	4.88368	0.245725
$396$	−46.6284	−2.34317
$397$	28.4898	1.42986	0.714932	−	0.699194i	$-0.246458\pi$
$397$	28.4898	1.42986	0.714932	+	0.699194i	$0.246458\pi$
$398$	−59.3389	−2.97439
$399$	−0.742631	−0.0371780
$400$	−1.53865	−0.0769325
$401$	9.62550	0.480675	0.240337	−	0.970689i	$-0.422742\pi$
$401$	9.62550	0.480675	0.240337	+	0.970689i	$0.422742\pi$
$402$	42.6645	2.12791
$403$	19.6946	0.981060
$404$	1.16512	0.0579671
$405$	7.15519	0.355544
$406$	−1.46971	−0.0729403
$407$	−3.13326	−0.155310
$408$	34.1750	1.69192
$409$	−13.0701	−0.646275	−0.323138	−	0.946352i	$-0.604738\pi$
$409$	−13.0701	−0.646275	−0.323138	+	0.946352i	$0.604738\pi$
$410$	21.0954	1.04183
$411$	−56.3721	−2.78063
$412$	−5.76559	−0.284050
$413$	0.468741	0.0230652
$414$	−98.1830	−4.82543
$415$	12.6293	0.619948
$416$	−18.3503	−0.899699
$417$	−27.4472	−1.34410
$418$	9.36004	0.457815
$419$	32.0188	1.56422	0.782111	−	0.623139i	$-0.214143\pi$
$419$	32.0188	1.56422	0.782111	+	0.623139i	$0.214143\pi$
$420$	−1.40863	−0.0687341
$421$	26.2344	1.27859	0.639293	−	0.768963i	$-0.279227\pi$
$421$	26.2344	1.27859	0.639293	+	0.768963i	$0.279227\pi$
$422$	−42.8385	−2.08534
$423$	55.7217	2.70928
$424$	23.1146	1.12255
$425$	6.07626	0.294742
$426$	32.9106	1.59452
$427$	2.20053	0.106491
$428$	47.2864	2.28567
$429$	−21.3140	−1.02905
$430$	−16.3900	−0.790396
$431$	−19.6251	−0.945306	−0.472653	−	0.881249i	$-0.656704\pi$
$431$	−19.6251	−0.945306	−0.472653	+	0.881249i	$0.656704\pi$
$432$	12.7280	0.612378
$433$	29.3116	1.40863	0.704314	−	0.709889i	$-0.251255\pi$
$433$	29.3116	1.40863	0.704314	+	0.709889i	$0.251255\pi$
$434$	2.82428	0.135570
$435$	−11.8994	−0.570535
$436$	52.2353	2.50162
$437$	11.5990	0.554858
$438$	−79.6363	−3.80517
$439$	24.4385	1.16638	0.583192	−	0.812334i	$-0.301804\pi$
$439$	24.4385	1.16638	0.583192	+	0.812334i	$0.301804\pi$
$440$	5.34073	0.254609
$441$	−40.3712	−1.92244
$442$	34.2072	1.62707
$443$	30.8256	1.46457	0.732284	−	0.680999i	$-0.238454\pi$
$443$	30.8256	1.46457	0.732284	+	0.680999i	$0.238454\pi$
$444$	−9.43856	−0.447934
$445$	−2.85425	−0.135304
$446$	−6.17596	−0.292440
$447$	−40.5575	−1.91830
$448$	−2.12037	−0.100178
$449$	−33.2346	−1.56844	−0.784219	−	0.620484i	$-0.786936\pi$
$449$	−33.2346	−1.56844	−0.784219	+	0.620484i	$0.786936\pi$
$450$	−12.7652	−0.601757
$451$	26.9385	1.26849
$452$	−23.5663	−1.10846
$453$	31.4913	1.47959
$454$	57.2600	2.68734
$455$	−0.424135	−0.0198837
$456$	8.48176	0.397195
$457$	−23.2859	−1.08927	−0.544635	−	0.838673i	$-0.683332\pi$
$457$	−23.2859	−1.08927	−0.544635	+	0.838673i	$0.683332\pi$
$458$	−49.8977	−2.33157
$459$	−50.2641	−2.34613
$460$	22.0012	1.02581
$461$	30.2498	1.40888	0.704438	−	0.709766i	$-0.251199\pi$
$461$	30.2498	1.40888	0.704438	+	0.709766i	$0.251199\pi$
$462$	−3.05650	−0.142201
$463$	−12.6747	−0.589044	−0.294522	−	0.955645i	$-0.595160\pi$
$463$	−12.6747	−0.589044	−0.294522	+	0.955645i	$0.595160\pi$
$464$	−6.17545	−0.286688
$465$	22.8667	1.06042
$466$	31.2822	1.44912
$467$	4.44062	0.205488	0.102744	−	0.994708i	$-0.467238\pi$
$467$	4.44062	0.205488	0.102744	+	0.994708i	$0.467238\pi$
$468$	−42.2929	−1.95499
$469$	1.08416	0.0500616
$470$	−21.2166	−0.978649
$471$	18.1901	0.838154
$472$	−5.35360	−0.246420
$473$	−20.9298	−0.962353
$474$	31.9215	1.46621
$475$	1.50804	0.0691937
$476$	2.88693	0.132322
$477$	−70.5504	−3.23028
$478$	50.7087	2.31936
$479$	−7.84862	−0.358613	−0.179306	−	0.983793i	$-0.557385\pi$
$479$	−7.84862	−0.358613	−0.179306	+	0.983793i	$0.557385\pi$
$480$	−21.3059	−0.972477
$481$	−2.84193	−0.129581
$482$	−2.80011	−0.127542
$483$	−3.78764	−0.172344
$484$	−8.79326	−0.399694
$485$	17.7038	0.803888
$486$	−7.94294	−0.360299
$487$	23.0847	1.04607	0.523034	−	0.852312i	$-0.324800\pi$
$487$	23.0847	1.04607	0.523034	+	0.852312i	$0.324800\pi$
$488$	−25.1328	−1.13771
$489$	−15.2276	−0.688618
$490$	15.3717	0.694423
$491$	−32.0743	−1.44749	−0.723747	−	0.690066i	$-0.757582\pi$
$491$	−32.0743	−1.44749	−0.723747	+	0.690066i	$0.757582\pi$
$492$	81.1491	3.65848
$493$	24.3874	1.09835
$494$	8.48974	0.381971
$495$	−16.3010	−0.732674
$496$	11.8671	0.532850
$497$	0.836298	0.0375131
$498$	82.5498	3.69915
$499$	−0.784409	−0.0351150	−0.0175575	−	0.999846i	$-0.505589\pi$
$499$	−0.784409	−0.0351150	−0.0175575	+	0.999846i	$0.505589\pi$
$500$	2.86047	0.127924
$501$	−25.1356	−1.12297
$502$	48.1200	2.14770
$503$	−13.3851	−0.596812	−0.298406	−	0.954439i	$-0.596455\pi$
$503$	−13.3851	−0.596812	−0.298406	+	0.954439i	$0.596455\pi$
$504$	−1.82442	−0.0812662
$505$	0.407319	0.0181255
$506$	47.7390	2.12226
$507$	19.2104	0.853164
$508$	−7.42752	−0.329543
$509$	14.9701	0.663536	0.331768	−	0.943361i	$-0.392355\pi$
$509$	14.9701	0.663536	0.331768	+	0.943361i	$0.392355\pi$
$510$	39.7167	1.75868
$511$	−2.02365	−0.0895212
$512$	−16.8949	−0.746654
$513$	−12.4748	−0.550777
$514$	−16.3994	−0.723346
$515$	−2.01561	−0.0888183
$516$	−63.0485	−2.77555
$517$	−27.0933	−1.19156
$518$	−0.407542	−0.0179064
$519$	40.2042	1.76477
$520$	4.84414	0.212430
$521$	5.37276	0.235385	0.117692	−	0.993050i	$-0.462450\pi$
$521$	5.37276	0.235385	0.117692	+	0.993050i	$0.462450\pi$
$522$	−51.2338	−2.24244
$523$	18.7249	0.818781	0.409391	−	0.912359i	$-0.365741\pi$
$523$	18.7249	0.818781	0.409391	+	0.912359i	$0.365741\pi$
$524$	−23.9998	−1.04844
$525$	−0.492447	−0.0214922
$526$	−34.7775	−1.51637
$527$	−46.8643	−2.04144
$528$	−12.8429	−0.558915
$529$	36.1586	1.57211
$530$	26.8628	1.16684
$531$	16.3403	0.709107
$532$	0.716494	0.0310640
$533$	24.4338	1.05834
$534$	−18.6564	−0.807343
$535$	16.5310	0.714697
$536$	−12.3824	−0.534838
$537$	26.8398	1.15822
$538$	8.80268	0.379510
$539$	19.6295	0.845501
$540$	−23.6624	−1.01827
$541$	19.1724	0.824284	0.412142	−	0.911120i	$-0.364781\pi$
$541$	19.1724	0.824284	0.412142	+	0.911120i	$0.364781\pi$
$542$	6.65881	0.286020
$543$	37.1224	1.59307
$544$	43.6655	1.87214
$545$	18.2611	0.782220
$546$	−2.77230	−0.118644
$547$	−16.6951	−0.713831	−0.356915	−	0.934137i	$-0.616172\pi$
$547$	−16.6951	−0.713831	−0.356915	+	0.934137i	$0.616172\pi$
$548$	54.3881	2.32334
$549$	76.7104	3.27392
$550$	6.20675	0.264657
$551$	6.05261	0.257850
$552$	43.2595	1.84125
$553$	0.811165	0.0344942
$554$	36.9802	1.57114
$555$	−3.29965	−0.140063
$556$	26.4812	1.12305
$557$	17.2278	0.729965	0.364983	−	0.931014i	$-0.381075\pi$
$557$	17.2278	0.729965	0.364983	+	0.931014i	$0.381075\pi$
$558$	98.4540	4.16789
$559$	−18.9837	−0.802926
$560$	−0.255565	−0.0107996
$561$	50.7177	2.14130
$562$	5.70218	0.240532
$563$	−2.40323	−0.101284	−0.0506421	−	0.998717i	$-0.516127\pi$
$563$	−2.40323	−0.101284	−0.0506421	+	0.998717i	$0.516127\pi$
$564$	−81.6153	−3.43662
$565$	−8.23860	−0.346601
$566$	−51.4371	−2.16206
$567$	1.18846	0.0499105
$568$	−9.55155	−0.400774
$569$	−33.7562	−1.41514	−0.707568	−	0.706645i	$-0.750208\pi$
$569$	−33.7562	−1.41514	−0.707568	+	0.706645i	$0.750208\pi$
$570$	9.85712	0.412869
$571$	30.7194	1.28557	0.642784	−	0.766047i	$-0.277779\pi$
$571$	30.7194	1.28557	0.642784	+	0.766047i	$0.277779\pi$
$572$	20.5639	0.859818
$573$	13.4979	0.563882
$574$	3.50388	0.146249
$575$	7.69146	0.320756
$576$	−73.9159	−3.07983
$577$	−13.8782	−0.577756	−0.288878	−	0.957366i	$-0.593282\pi$
$577$	−13.8782	−0.577756	−0.288878	+	0.957366i	$0.593282\pi$
$578$	−43.9186	−1.82677
$579$	71.1623	2.95740
$580$	11.4807	0.476708
$581$	2.09769	0.0870269
$582$	115.719	4.79669
$583$	34.3034	1.42070
$584$	23.1126	0.956408
$585$	−14.7853	−0.611297
$586$	−9.30681	−0.384461
$587$	−14.0041	−0.578009	−0.289005	−	0.957328i	$-0.593324\pi$
$587$	−14.0041	−0.578009	−0.289005	+	0.957328i	$0.593324\pi$
$588$	59.1314	2.43854
$589$	−11.6311	−0.479250
$590$	−6.22172	−0.256144
$591$	44.3940	1.82612
$592$	−1.71242	−0.0703801
$593$	13.6063	0.558745	0.279372	−	0.960183i	$-0.409874\pi$
$593$	13.6063	0.558745	0.279372	+	0.960183i	$0.409874\pi$
$594$	−51.3436	−2.10665
$595$	1.00925	0.0413752
$596$	39.1301	1.60283
$597$	−79.7991	−3.26596
$598$	43.3002	1.77068
$599$	31.2991	1.27884	0.639422	−	0.768856i	$-0.279174\pi$
$599$	31.2991	1.27884	0.639422	+	0.768856i	$0.279174\pi$
$600$	5.62435	0.229613
$601$	4.11686	0.167930	0.0839651	−	0.996469i	$-0.473242\pi$
$601$	4.11686	0.167930	0.0839651	+	0.996469i	$0.473242\pi$
$602$	−2.72233	−0.110954
$603$	37.7935	1.53907
$604$	−30.3830	−1.23627
$605$	−3.07406	−0.124978
$606$	2.66239	0.108152
$607$	31.4014	1.27454	0.637272	−	0.770639i	$-0.280063\pi$
$607$	31.4014	1.27454	0.637272	+	0.770639i	$0.280063\pi$
$608$	10.8372	0.439505
$609$	−1.97646	−0.0800903
$610$	−29.2082	−1.18261
$611$	−24.5741	−0.994163
$612$	100.638	4.06805
$613$	−41.3440	−1.66987	−0.834934	−	0.550350i	$-0.814494\pi$
$613$	−41.3440	−1.66987	−0.834934	+	0.550350i	$0.814494\pi$
$614$	13.1758	0.531731
$615$	28.3691	1.14395
$616$	0.887079	0.0357414
$617$	−0.571392	−0.0230034	−0.0115017	−	0.999934i	$-0.503661\pi$
$617$	−0.571392	−0.0230034	−0.0115017	+	0.999934i	$0.503661\pi$
$618$	−13.1748	−0.529967
$619$	−23.6466	−0.950437	−0.475218	−	0.879868i	$-0.657631\pi$
$619$	−23.6466	−0.950437	−0.475218	+	0.879868i	$0.657631\pi$
$620$	−22.0619	−0.886029
$621$	−63.6254	−2.55320
$622$	−8.88157	−0.356119
$623$	−0.474082	−0.0189937
$624$	−11.6487	−0.466323
$625$	1.00000	0.0400000
$626$	−59.0636	−2.36066
$627$	12.5874	0.502693
$628$	−17.5499	−0.700317
$629$	6.76250	0.269638
$630$	−2.12026	−0.0844732
$631$	45.2168	1.80005	0.900026	−	0.435836i	$-0.143547\pi$
$631$	45.2168	1.80005	0.900026	+	0.435836i	$0.143547\pi$
$632$	−9.26451	−0.368522
$633$	−57.6093	−2.28976
$634$	3.62858	0.144109
$635$	−2.59661	−0.103043
$636$	103.335	4.09749
$637$	17.8043	0.705432
$638$	24.9111	0.986242
$639$	29.1532	1.15328
$640$	13.7717	0.544374
$641$	−22.4174	−0.885434	−0.442717	−	0.896661i	$-0.645985\pi$
$641$	−22.4174	−0.885434	−0.442717	+	0.896661i	$0.645985\pi$
$642$	108.053	4.26450
$643$	30.1650	1.18959	0.594796	−	0.803877i	$-0.297233\pi$
$643$	30.1650	1.18959	0.594796	+	0.803877i	$0.297233\pi$
$644$	3.65434	0.144001
$645$	−22.0413	−0.867875
$646$	−20.2017	−0.794827
$647$	3.04158	0.119577	0.0597883	−	0.998211i	$-0.480957\pi$
$647$	3.04158	0.119577	0.0597883	+	0.998211i	$0.480957\pi$
$648$	−13.5736	−0.533223
$649$	−7.94505	−0.311870
$650$	5.62964	0.220813
$651$	3.79809	0.148859
$652$	14.6917	0.575372
$653$	41.1501	1.61033	0.805163	−	0.593053i	$-0.202078\pi$
$653$	41.1501	1.61033	0.805163	+	0.593053i	$0.202078\pi$
$654$	119.361	4.66740
$655$	−8.39017	−0.327831
$656$	14.7227	0.574826
$657$	−70.5444	−2.75220
$658$	−3.52401	−0.137380
$659$	30.6141	1.19256	0.596279	−	0.802778i	$-0.296645\pi$
$659$	30.6141	1.19256	0.596279	+	0.802778i	$0.296645\pi$
$660$	23.8759	0.929369
$661$	−0.296264	−0.0115233	−0.00576167	−	0.999983i	$-0.501834\pi$
$661$	−0.296264	−0.0115233	−0.00576167	+	0.999983i	$0.501834\pi$
$662$	40.7051	1.58205
$663$	46.0019	1.78657
$664$	−23.9582	−0.929759
$665$	0.250481	0.00971325
$666$	−14.2069	−0.550505
$667$	30.8701	1.19530
$668$	24.2509	0.938297
$669$	−8.30545	−0.321107
$670$	−14.3903	−0.555944
$671$	−37.2985	−1.43989
$672$	−3.53885	−0.136514
$673$	−29.1157	−1.12233	−0.561163	−	0.827705i	$-0.689646\pi$
$673$	−29.1157	−1.12233	−0.561163	+	0.827705i	$0.689646\pi$
$674$	11.6094	0.447176
$675$	−8.27221	−0.318398
$676$	−18.5343	−0.712858
$677$	−41.6993	−1.60263	−0.801317	−	0.598240i	$-0.795867\pi$
$677$	−41.6993	−1.60263	−0.801317	+	0.598240i	$0.795867\pi$
$678$	−53.8506	−2.06812
$679$	2.94055	0.112848
$680$	−11.5269	−0.442035
$681$	77.0033	2.95077
$682$	−47.8708	−1.83307
$683$	−20.7460	−0.793823	−0.396911	−	0.917857i	$-0.629918\pi$
$683$	−20.7460	−0.793823	−0.396911	+	0.917857i	$0.629918\pi$
$684$	24.9769	0.955016
$685$	19.0137	0.726476
$686$	5.11649	0.195349
$687$	−67.1025	−2.56012
$688$	−11.4388	−0.436099
$689$	31.1138	1.18534
$690$	50.2743	1.91391
$691$	−14.5445	−0.553301	−0.276650	−	0.960971i	$-0.589224\pi$
$691$	−14.5445	−0.553301	−0.276650	+	0.960971i	$0.589224\pi$
$692$	−38.7893	−1.47455
$693$	−2.70754	−0.102851
$694$	28.0025	1.06296
$695$	9.25765	0.351163
$696$	22.5737	0.855652
$697$	−58.1413	−2.20226
$698$	9.98099	0.377786
$699$	42.0684	1.59117
$700$	0.475116	0.0179577
$701$	8.75495	0.330670	0.165335	−	0.986237i	$-0.447129\pi$
$701$	8.75495	0.330670	0.165335	+	0.986237i	$0.447129\pi$
$702$	−46.5696	−1.75766
$703$	1.67836	0.0633004
$704$	35.9397	1.35453
$705$	−28.5321	−1.07458
$706$	79.9697	3.00970
$707$	0.0676545	0.00254441
$708$	−23.9335	−0.899476
$709$	31.5256	1.18397	0.591984	−	0.805950i	$-0.298345\pi$
$709$	31.5256	1.18397	0.591984	+	0.805950i	$0.298345\pi$
$710$	−11.1004	−0.416590
$711$	28.2771	1.06048
$712$	5.41461	0.202921
$713$	−59.3219	−2.22162
$714$	6.59683	0.246880
$715$	7.18898	0.268853
$716$	−25.8952	−0.967748
$717$	68.1931	2.54672
$718$	25.9980	0.970237
$719$	−31.8379	−1.18735	−0.593677	−	0.804703i	$-0.702324\pi$
$719$	−31.8379	−1.18735	−0.593677	+	0.804703i	$0.702324\pi$
$720$	−8.90898	−0.332018
$721$	−0.334787	−0.0124681
$722$	36.8745	1.37233
$723$	−3.76560	−0.140044
$724$	−35.8159	−1.33109
$725$	4.01356	0.149060
$726$	−20.0932	−0.745729
$727$	−23.9906	−0.889763	−0.444881	−	0.895590i	$-0.646754\pi$
$727$	−23.9906	−0.889763	−0.444881	+	0.895590i	$0.646754\pi$
$728$	0.804598	0.0298204
$729$	−32.1473	−1.19064
$730$	26.8605	0.994150
$731$	45.1727	1.67077
$732$	−112.357	−4.15284
$733$	−17.0682	−0.630428	−0.315214	−	0.949021i	$-0.602076\pi$
$733$	−17.0682	−0.630428	−0.315214	+	0.949021i	$0.602076\pi$
$734$	37.4493	1.38228
$735$	20.6719	0.762495
$736$	55.2728	2.03738
$737$	−18.3762	−0.676895
$738$	122.145	4.49622
$739$	−12.2773	−0.451628	−0.225814	−	0.974170i	$-0.572504\pi$
$739$	−12.2773	−0.451628	−0.225814	+	0.974170i	$0.572504\pi$
$740$	3.18353	0.117029
$741$	11.4170	0.419415
$742$	4.46183	0.163799
$743$	25.3552	0.930193	0.465097	−	0.885260i	$-0.346020\pi$
$743$	25.3552	0.930193	0.465097	+	0.885260i	$0.346020\pi$
$744$	−43.3789	−1.59035
$745$	13.6796	0.501182
$746$	65.6395	2.40323
$747$	73.1253	2.67551
$748$	−48.9327	−1.78916
$749$	2.74575	0.100327
$750$	6.53637	0.238675
$751$	−25.7727	−0.940458	−0.470229	−	0.882545i	$-0.655829\pi$
$751$	−25.7727	−0.940458	−0.470229	+	0.882545i	$0.655829\pi$
$752$	−14.8073	−0.539967
$753$	64.7118	2.35823
$754$	22.5949	0.822857
$755$	−10.6217	−0.386563
$756$	−3.93026	−0.142942
$757$	4.42381	0.160786	0.0803931	−	0.996763i	$-0.474382\pi$
$757$	4.42381	0.160786	0.0803931	+	0.996763i	$0.474382\pi$
$758$	−4.08866	−0.148507
$759$	64.1995	2.33030
$760$	−2.86081	−0.103772
$761$	45.0126	1.63170	0.815852	−	0.578261i	$-0.196268\pi$
$761$	45.0126	1.63170	0.815852	+	0.578261i	$0.196268\pi$
$762$	−16.9724	−0.614845
$763$	3.03312	0.109806
$764$	−13.0228	−0.471149
$765$	35.1823	1.27202
$766$	−37.7253	−1.36307
$767$	−7.20631	−0.260205
$768$	14.3202	0.516734
$769$	23.4305	0.844925	0.422462	−	0.906380i	$-0.361166\pi$
$769$	23.4305	0.844925	0.422462	+	0.906380i	$0.361166\pi$
$770$	1.03092	0.0371519
$771$	−22.0539	−0.794253
$772$	−68.6578	−2.47105
$773$	9.78683	0.352008	0.176004	−	0.984389i	$-0.443683\pi$
$773$	9.78683	0.352008	0.176004	+	0.984389i	$0.443683\pi$
$774$	−94.9001	−3.41111
$775$	−7.71270	−0.277048
$776$	−33.5847	−1.20562
$777$	−0.548063	−0.0196616
$778$	86.8697	3.11443
$779$	−14.4298	−0.517003
$780$	21.6559	0.775407
$781$	−14.1750	−0.507223
$782$	−103.035	−3.68452
$783$	−33.2010	−1.18651
$784$	10.7281	0.383146
$785$	−6.13532	−0.218979
$786$	−54.8413	−1.95612
$787$	36.6545	1.30659	0.653296	−	0.757102i	$-0.273386\pi$
$787$	36.6545	1.30659	0.653296	+	0.757102i	$0.273386\pi$
$788$	−42.8316	−1.52581
$789$	−46.7689	−1.66502
$790$	−10.7668	−0.383065
$791$	−1.36841	−0.0486550
$792$	30.9235	1.09882
$793$	−33.8305	−1.20135
$794$	−62.8101	−2.22905
$795$	36.1251	1.28123
$796$	76.9906	2.72886
$797$	40.3593	1.42960	0.714799	−	0.699330i	$-0.246518\pi$
$797$	40.3593	1.42960	0.714799	+	0.699330i	$0.246518\pi$
$798$	1.63724	0.0579576
$799$	58.4754	2.06871
$800$	7.18625	0.254072
$801$	−16.5265	−0.583934
$802$	−21.2208	−0.749334
$803$	34.3004	1.21044
$804$	−55.3560	−1.95226
$805$	1.27753	0.0450270
$806$	−43.4197	−1.52939
$807$	11.8379	0.416712
$808$	−0.772699	−0.0271834
$809$	−18.6410	−0.655383	−0.327692	−	0.944785i	$-0.606271\pi$
$809$	−18.6410	−0.655383	−0.327692	+	0.944785i	$0.606271\pi$
$810$	−15.7747	−0.554266
$811$	−49.9239	−1.75307	−0.876533	−	0.481342i	$-0.840150\pi$
$811$	−49.9239	−1.75307	−0.876533	+	0.481342i	$0.840150\pi$
$812$	1.90690	0.0669192
$813$	8.95478	0.314058
$814$	6.90773	0.242116
$815$	5.13612	0.179910
$816$	27.7188	0.970350
$817$	11.2112	0.392231
$818$	28.8150	1.00749
$819$	−2.45579	−0.0858124
$820$	−27.3707	−0.955827
$821$	−17.8809	−0.624046	−0.312023	−	0.950075i	$-0.601007\pi$
$821$	−17.8809	−0.624046	−0.312023	+	0.950075i	$0.601007\pi$
$822$	124.281	4.33478
$823$	48.3279	1.68461	0.842303	−	0.539005i	$-0.181200\pi$
$823$	48.3279	1.68461	0.842303	+	0.539005i	$0.181200\pi$
$824$	3.82368	0.133204
$825$	8.34686	0.290600
$826$	−1.03341	−0.0359569
$827$	38.4860	1.33829	0.669144	−	0.743132i	$-0.266661\pi$
$827$	38.4860	1.33829	0.669144	+	0.743132i	$0.266661\pi$
$828$	127.390	4.42710
$829$	−31.2187	−1.08427	−0.542135	−	0.840291i	$-0.682384\pi$
$829$	−31.2187	−1.08427	−0.542135	+	0.840291i	$0.682384\pi$
$830$	−27.8432	−0.966450
$831$	49.7311	1.72515
$832$	32.5980	1.13013
$833$	−42.3662	−1.46790
$834$	60.5115	2.09534
$835$	8.47795	0.293392
$836$	−12.1444	−0.420023
$837$	63.8010	2.20529
$838$	−70.5902	−2.43850
$839$	−9.59471	−0.331246	−0.165623	−	0.986189i	$-0.552964\pi$
$839$	−9.59471	−0.331246	−0.165623	+	0.986189i	$0.552964\pi$
$840$	0.934189	0.0322326
$841$	−12.8914	−0.444530
$842$	−57.8376	−1.99321
$843$	7.66830	0.264110
$844$	55.5817	1.91320
$845$	−6.47946	−0.222900
$846$	−122.847	−4.22356
$847$	−0.510593	−0.0175442
$848$	18.7478	0.643804
$849$	−69.1727	−2.37400
$850$	−13.3960	−0.459479
$851$	8.56012	0.293437
$852$	−42.7006	−1.46290
$853$	0.169739	0.00581176	0.00290588	−	0.999996i	$-0.499075\pi$
$853$	0.169739	0.00581176	0.00290588	+	0.999996i	$0.499075\pi$
$854$	−4.85140	−0.166011
$855$	8.73175	0.298620
$856$	−31.3598	−1.07186
$857$	−41.2074	−1.40762	−0.703810	−	0.710389i	$-0.748519\pi$
$857$	−41.2074	−1.40762	−0.703810	+	0.710389i	$0.748519\pi$
$858$	46.9898	1.60421
$859$	−19.5959	−0.668603	−0.334301	−	0.942466i	$-0.608500\pi$
$859$	−19.5959	−0.668603	−0.334301	+	0.942466i	$0.608500\pi$
$860$	21.2656	0.725150
$861$	4.71203	0.160586
$862$	43.2663	1.47366
$863$	−20.3121	−0.691432	−0.345716	−	0.938339i	$-0.612364\pi$
$863$	−20.3121	−0.691432	−0.345716	+	0.938339i	$0.612364\pi$
$864$	−59.4462	−2.02240
$865$	−13.5605	−0.461069
$866$	−64.6218	−2.19594
$867$	−59.0619	−2.00585
$868$	−3.66442	−0.124379
$869$	−13.7490	−0.466404
$870$	26.2341	0.889419
$871$	−16.6675	−0.564758
$872$	−34.6419	−1.17312
$873$	102.507	3.46934
$874$	−25.5718	−0.864979
$875$	0.166097	0.00561511
$876$	103.326	3.49106
$877$	−18.8644	−0.637004	−0.318502	−	0.947922i	$-0.603180\pi$
$877$	−18.8644	−0.637004	−0.318502	+	0.947922i	$0.603180\pi$
$878$	−53.8782	−1.81830
$879$	−12.5158	−0.422148
$880$	4.33177	0.146024
$881$	9.50379	0.320191	0.160095	−	0.987102i	$-0.448820\pi$
$881$	9.50379	0.320191	0.160095	+	0.987102i	$0.448820\pi$
$882$	89.0042	2.99693
$883$	−37.3696	−1.25759	−0.628793	−	0.777572i	$-0.716451\pi$
$883$	−37.3696	−1.25759	−0.628793	+	0.777572i	$0.716451\pi$
$884$	−44.3829	−1.49276
$885$	−8.36698	−0.281253
$886$	−67.9596	−2.28315
$887$	−34.2878	−1.15127	−0.575636	−	0.817706i	$-0.695246\pi$
$887$	−34.2878	−1.15127	−0.575636	+	0.817706i	$0.695246\pi$
$888$	6.25956	0.210057
$889$	−0.431289	−0.0144650
$890$	6.29261	0.210929
$891$	−20.1440	−0.674851
$892$	8.01314	0.268300
$893$	14.5128	0.485651
$894$	89.4150	2.99049
$895$	−9.05276	−0.302600
$896$	2.28744	0.0764179
$897$	58.2302	1.94425
$898$	73.2706	2.44507
$899$	−30.9553	−1.03242
$900$	16.5625	0.552083
$901$	−74.0369	−2.46653
$902$	−59.3900	−1.97747
$903$	−3.66099	−0.121830
$904$	15.6289	0.519810
$905$	−12.5210	−0.416211
$906$	−69.4273	−2.30657
$907$	30.0477	0.997717	0.498859	−	0.866683i	$-0.333753\pi$
$907$	30.0477	0.997717	0.498859	+	0.866683i	$0.333753\pi$
$908$	−74.2933	−2.46551
$909$	2.35843	0.0782242
$910$	0.935068	0.0309972
$911$	−1.33585	−0.0442586	−0.0221293	−	0.999755i	$-0.507045\pi$
$911$	−1.33585	−0.0442586	−0.0221293	+	0.999755i	$0.507045\pi$
$912$	6.87940	0.227800
$913$	−35.5553	−1.17671
$914$	51.3372	1.69809
$915$	−39.2793	−1.29853
$916$	64.7409	2.13910
$917$	−1.39358	−0.0460202
$918$	110.815	3.65743
$919$	−41.8763	−1.38137	−0.690686	−	0.723155i	$-0.742691\pi$
$919$	−41.8763	−1.38137	−0.690686	+	0.723155i	$0.742691\pi$
$920$	−14.5910	−0.481050
$921$	17.7188	0.583854
$922$	−66.6902	−2.19633
$923$	−12.8570	−0.423194
$924$	3.96572	0.130463
$925$	1.11294	0.0365932
$926$	27.9433	0.918272
$927$	−11.6706	−0.383314
$928$	28.8424	0.946798
$929$	10.3710	0.340261	0.170130	−	0.985422i	$-0.445581\pi$
$929$	10.3710	0.340261	0.170130	+	0.985422i	$0.445581\pi$
$930$	−50.4130	−1.65311
$931$	−10.5147	−0.344605
$932$	−40.5879	−1.32950
$933$	−11.9440	−0.391028
$934$	−9.79001	−0.320339
$935$	−17.1065	−0.559443
$936$	28.0482	0.916784
$937$	38.9097	1.27112	0.635562	−	0.772050i	$-0.280768\pi$
$937$	38.9097	1.27112	0.635562	+	0.772050i	$0.280768\pi$
$938$	−2.39018	−0.0780422
$939$	−79.4289	−2.59206
$940$	27.5280	0.897863
$941$	58.9839	1.92282	0.961410	−	0.275118i	$-0.0887170\pi$
$941$	58.9839	1.92282	0.961410	+	0.275118i	$0.0887170\pi$
$942$	−40.1027	−1.30662
$943$	−73.5966	−2.39663
$944$	−4.34221	−0.141327
$945$	−1.37399	−0.0446959
$946$	46.1428	1.50023
$947$	34.9890	1.13699	0.568496	−	0.822686i	$-0.307526\pi$
$947$	34.9890	1.13699	0.568496	+	0.822686i	$0.307526\pi$
$948$	−41.4173	−1.34517
$949$	31.1112	1.00991
$950$	−3.32470	−0.107868
$951$	4.87972	0.158236
$952$	−1.91458	−0.0620519
$953$	40.7153	1.31890	0.659449	−	0.751749i	$-0.270790\pi$
$953$	40.7153	1.31890	0.659449	+	0.751749i	$0.270790\pi$
$954$	155.539	5.03576
$955$	−4.55269	−0.147321
$956$	−65.7931	−2.12790
$957$	33.5006	1.08292
$958$	17.3034	0.559049
$959$	3.15812	0.101981
$960$	37.8484	1.22155
$961$	28.4857	0.918893
$962$	6.26544	0.202006
$963$	95.7165	3.08442
$964$	3.63307	0.117013
$965$	−24.0023	−0.772661
$966$	8.35041	0.268670
$967$	50.7765	1.63286	0.816432	−	0.577442i	$-0.195949\pi$
$967$	50.7765	1.63286	0.816432	+	0.577442i	$0.195949\pi$
$968$	5.83160	0.187435
$969$	−27.1673	−0.872741
$970$	−39.0306	−1.25320
$971$	−26.5463	−0.851911	−0.425956	−	0.904744i	$-0.640062\pi$
$971$	−26.5463	−0.851911	−0.425956	+	0.904744i	$0.640062\pi$
$972$	10.3057	0.330557
$973$	1.53767	0.0492954
$974$	−50.8937	−1.63074
$975$	7.57076	0.242458
$976$	−20.3848	−0.652500
$977$	18.0834	0.578538	0.289269	−	0.957248i	$-0.406588\pi$
$977$	18.0834	0.578538	0.289269	+	0.957248i	$0.406588\pi$
$978$	33.5716	1.07350
$979$	8.03558	0.256818
$980$	−19.9444	−0.637100
$981$	105.734	3.37583
$982$	70.7126	2.25653
$983$	16.4845	0.525775	0.262888	−	0.964826i	$-0.415325\pi$
$983$	16.4845	0.525775	0.262888	+	0.964826i	$0.415325\pi$
$984$	−53.8172	−1.71563
$985$	−14.9736	−0.477099
$986$	−53.7656	−1.71225
$987$	−4.73910	−0.150847
$988$	−11.0152	−0.350440
$989$	57.1806	1.81824
$990$	35.9379	1.14218
$991$	−22.9756	−0.729842	−0.364921	−	0.931038i	$-0.618904\pi$
$991$	−22.9756	−0.729842	−0.364921	+	0.931038i	$0.618904\pi$
$992$	−55.4253	−1.75976
$993$	54.7403	1.73713
$994$	−1.84374	−0.0584799
$995$	26.9154	0.853275
$996$	−107.106	−3.39379
$997$	42.5764	1.34841	0.674204	−	0.738545i	$-0.264487\pi$
$997$	42.5764	1.34841	0.674204	+	0.738545i	$0.264487\pi$
$998$	1.72934	0.0547415
$999$	−9.20645	−0.291279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.e.1.14	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.e.1.14	✓	142	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.20465 q^{2} -2.96481 q^{3} +2.86047 q^{4} +1.00000 q^{5} +6.53637 q^{6} +0.166097 q^{7} -1.89703 q^{8} +5.79013 q^{9} +O(q^{10})\)	\(q-2.20465 q^{2} -2.96481 q^{3} +2.86047 q^{4} +1.00000 q^{5} +6.53637 q^{6} +0.166097 q^{7} -1.89703 q^{8} +5.79013 q^{9} -2.20465 q^{10} -2.81530 q^{11} -8.48077 q^{12} -2.55354 q^{13} -0.366186 q^{14} -2.96481 q^{15} -1.53865 q^{16} +6.07626 q^{17} -12.7652 q^{18} +1.50804 q^{19} +2.86047 q^{20} -0.492447 q^{21} +6.20675 q^{22} +7.69146 q^{23} +5.62435 q^{24} +1.00000 q^{25} +5.62964 q^{26} -8.27221 q^{27} +0.475116 q^{28} +4.01356 q^{29} +6.53637 q^{30} -7.71270 q^{31} +7.18625 q^{32} +8.34686 q^{33} -13.3960 q^{34} +0.166097 q^{35} +16.5625 q^{36} +1.11294 q^{37} -3.32470 q^{38} +7.57076 q^{39} -1.89703 q^{40} -9.56860 q^{41} +1.08567 q^{42} +7.43429 q^{43} -8.05309 q^{44} +5.79013 q^{45} -16.9570 q^{46} +9.62358 q^{47} +4.56181 q^{48} -6.97241 q^{49} -2.20465 q^{50} -18.0150 q^{51} -7.30431 q^{52} -12.1846 q^{53} +18.2373 q^{54} -2.81530 q^{55} -0.315092 q^{56} -4.47106 q^{57} -8.84847 q^{58} +2.82209 q^{59} -8.48077 q^{60} +13.2485 q^{61} +17.0038 q^{62} +0.961723 q^{63} -12.7658 q^{64} -2.55354 q^{65} -18.4019 q^{66} +6.52724 q^{67} +17.3810 q^{68} -22.8038 q^{69} -0.366186 q^{70} +5.03499 q^{71} -10.9841 q^{72} -12.1836 q^{73} -2.45364 q^{74} -2.96481 q^{75} +4.31371 q^{76} -0.467614 q^{77} -16.6909 q^{78} +4.88368 q^{79} -1.53865 q^{80} +7.15519 q^{81} +21.0954 q^{82} +12.6293 q^{83} -1.40863 q^{84} +6.07626 q^{85} -16.3900 q^{86} -11.8994 q^{87} +5.34073 q^{88} -2.85425 q^{89} -12.7652 q^{90} -0.424135 q^{91} +22.0012 q^{92} +22.8667 q^{93} -21.2166 q^{94} +1.50804 q^{95} -21.3059 q^{96} +17.7038 q^{97} +15.3717 q^{98} -16.3010 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9 + 21 * q^10 + 36 * q^11 + 55 * q^12 + 57 * q^13 + 2 * q^14 + 33 * q^15 + 179 * q^16 + 55 * q^17 + 65 * q^18 + 130 * q^19 + 157 * q^20 + 28 * q^21 + 30 * q^22 + 117 * q^23 + 21 * q^24 + 142 * q^25 + 21 * q^26 + 120 * q^27 + 135 * q^28 + 12 * q^29 + 15 * q^30 + 74 * q^31 + 126 * q^32 + 55 * q^33 + 35 * q^34 + 63 * q^35 + 186 * q^36 + 75 * q^37 + 65 * q^38 + 23 * q^39 + 60 * q^40 + 22 * q^41 + 10 * q^42 + 190 * q^43 + 22 * q^44 + 157 * q^45 + 56 * q^46 + 102 * q^47 + 78 * q^48 + 197 * q^49 + 21 * q^50 + 30 * q^51 + 120 * q^52 + 56 * q^53 - 6 * q^54 + 36 * q^55 + 3 * q^56 + 68 * q^57 + 31 * q^58 + 55 * q^59 + 55 * q^60 + 90 * q^61 + 68 * q^62 + 167 * q^63 + 180 * q^64 + 57 * q^65 + 17 * q^66 + 151 * q^67 + 119 * q^68 + 21 * q^69 + 2 * q^70 + 4 * q^71 + 130 * q^72 + 143 * q^73 - 46 * q^74 + 33 * q^75 + 213 * q^76 + 75 * q^77 - 24 * q^78 + 47 * q^79 + 179 * q^80 + 150 * q^81 + 69 * q^82 + 201 * q^83 - 31 * q^84 + 55 * q^85 - 4 * q^86 + 153 * q^87 + 37 * q^88 + 25 * q^89 + 65 * q^90 + 132 * q^91 + 194 * q^92 + 52 * q^93 + 18 * q^94 + 130 * q^95 + 13 * q^96 + 80 * q^97 + 58 * q^98 + 103 * q^99

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