LMFDB - Embedded newform 8038.2.a.b.1.20

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:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.59172 q^{3} +1.00000 q^{4} +0.670129 q^{5} +1.59172 q^{6} -4.47737 q^{7} -1.00000 q^{8} -0.466431 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.59172 q^{3} +1.00000 q^{4} +0.670129 q^{5} +1.59172 q^{6} -4.47737 q^{7} -1.00000 q^{8} -0.466431 q^{9} -0.670129 q^{10} +4.13526 q^{11} -1.59172 q^{12} +6.62792 q^{13} +4.47737 q^{14} -1.06666 q^{15} +1.00000 q^{16} +4.34613 q^{17} +0.466431 q^{18} +0.344252 q^{19} +0.670129 q^{20} +7.12671 q^{21} -4.13526 q^{22} -0.0584939 q^{23} +1.59172 q^{24} -4.55093 q^{25} -6.62792 q^{26} +5.51758 q^{27} -4.47737 q^{28} -3.02823 q^{29} +1.06666 q^{30} +3.59551 q^{31} -1.00000 q^{32} -6.58217 q^{33} -4.34613 q^{34} -3.00041 q^{35} -0.466431 q^{36} +8.23071 q^{37} -0.344252 q^{38} -10.5498 q^{39} -0.670129 q^{40} +2.81771 q^{41} -7.12671 q^{42} -0.130402 q^{43} +4.13526 q^{44} -0.312569 q^{45} +0.0584939 q^{46} -6.77902 q^{47} -1.59172 q^{48} +13.0468 q^{49} +4.55093 q^{50} -6.91781 q^{51} +6.62792 q^{52} +8.30912 q^{53} -5.51758 q^{54} +2.77115 q^{55} +4.47737 q^{56} -0.547953 q^{57} +3.02823 q^{58} +13.2730 q^{59} -1.06666 q^{60} -4.55104 q^{61} -3.59551 q^{62} +2.08838 q^{63} +1.00000 q^{64} +4.44156 q^{65} +6.58217 q^{66} +6.01455 q^{67} +4.34613 q^{68} +0.0931058 q^{69} +3.00041 q^{70} +9.18621 q^{71} +0.466431 q^{72} +6.66999 q^{73} -8.23071 q^{74} +7.24380 q^{75} +0.344252 q^{76} -18.5151 q^{77} +10.5498 q^{78} -15.1530 q^{79} +0.670129 q^{80} -7.38315 q^{81} -2.81771 q^{82} -1.46270 q^{83} +7.12671 q^{84} +2.91247 q^{85} +0.130402 q^{86} +4.82009 q^{87} -4.13526 q^{88} -3.73587 q^{89} +0.312569 q^{90} -29.6756 q^{91} -0.0584939 q^{92} -5.72305 q^{93} +6.77902 q^{94} +0.230693 q^{95} +1.59172 q^{96} -0.250519 q^{97} -13.0468 q^{98} -1.92881 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * 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
$3$	−1.59172	−0.918979	−0.459490	−	0.888183i	$-0.651968\pi$
$3$	−1.59172	−0.918979	−0.459490	+	0.888183i	$0.651968\pi$
$4$	1.00000	0.500000
$5$	0.670129	0.299691	0.149845	−	0.988709i	$-0.452122\pi$
$5$	0.670129	0.299691	0.149845	+	0.988709i	$0.452122\pi$
$6$	1.59172	0.649816
$7$	−4.47737	−1.69229	−0.846143	−	0.532956i	$-0.821081\pi$
$7$	−4.47737	−1.69229	−0.846143	+	0.532956i	$0.821081\pi$
$8$	−1.00000	−0.353553
$9$	−0.466431	−0.155477
$10$	−0.670129	−0.211913
$11$	4.13526	1.24683	0.623413	−	0.781892i	$-0.285745\pi$
$11$	4.13526	1.24683	0.623413	+	0.781892i	$0.285745\pi$
$12$	−1.59172	−0.459490
$13$	6.62792	1.83825	0.919127	−	0.393961i	$-0.128895\pi$
$13$	6.62792	1.83825	0.919127	+	0.393961i	$0.128895\pi$
$14$	4.47737	1.19663
$15$	−1.06666	−0.275410
$16$	1.00000	0.250000
$17$	4.34613	1.05409	0.527045	−	0.849837i	$-0.323300\pi$
$17$	4.34613	1.05409	0.527045	+	0.849837i	$0.323300\pi$
$18$	0.466431	0.109939
$19$	0.344252	0.0789769	0.0394884	−	0.999220i	$-0.487427\pi$
$19$	0.344252	0.0789769	0.0394884	+	0.999220i	$0.487427\pi$
$20$	0.670129	0.149845
$21$	7.12671	1.55518
$22$	−4.13526	−0.881640
$23$	−0.0584939	−0.0121968	−0.00609841	−	0.999981i	$-0.501941\pi$
$23$	−0.0584939	−0.0121968	−0.00609841	+	0.999981i	$0.501941\pi$
$24$	1.59172	0.324908
$25$	−4.55093	−0.910185
$26$	−6.62792	−1.29984
$27$	5.51758	1.06186
$28$	−4.47737	−0.846143
$29$	−3.02823	−0.562328	−0.281164	−	0.959660i	$-0.590721\pi$
$29$	−3.02823	−0.562328	−0.281164	+	0.959660i	$0.590721\pi$
$30$	1.06666	0.194744
$31$	3.59551	0.645774	0.322887	−	0.946438i	$-0.395347\pi$
$31$	3.59551	0.645774	0.322887	+	0.946438i	$0.395347\pi$
$32$	−1.00000	−0.176777
$33$	−6.58217	−1.14581
$34$	−4.34613	−0.745355
$35$	−3.00041	−0.507162
$36$	−0.466431	−0.0777385
$37$	8.23071	1.35312	0.676561	−	0.736387i	$-0.263470\pi$
$37$	8.23071	1.35312	0.676561	+	0.736387i	$0.263470\pi$
$38$	−0.344252	−0.0558451
$39$	−10.5498	−1.68932
$40$	−0.670129	−0.105957
$41$	2.81771	0.440052	0.220026	−	0.975494i	$-0.429386\pi$
$41$	2.81771	0.440052	0.220026	+	0.975494i	$0.429386\pi$
$42$	−7.12671	−1.09967
$43$	−0.130402	−0.0198861	−0.00994307	−	0.999951i	$-0.503165\pi$
$43$	−0.130402	−0.0198861	−0.00994307	+	0.999951i	$0.503165\pi$
$44$	4.13526	0.623413
$45$	−0.312569	−0.0465950
$46$	0.0584939	0.00862445
$47$	−6.77902	−0.988822	−0.494411	−	0.869228i	$-0.664616\pi$
$47$	−6.77902	−0.988822	−0.494411	+	0.869228i	$0.664616\pi$
$48$	−1.59172	−0.229745
$49$	13.0468	1.86383
$50$	4.55093	0.643598
$51$	−6.91781	−0.968688
$52$	6.62792	0.919127
$53$	8.30912	1.14135	0.570673	−	0.821177i	$-0.306682\pi$
$53$	8.30912	1.14135	0.570673	+	0.821177i	$0.306682\pi$
$54$	−5.51758	−0.750848
$55$	2.77115	0.373662
$56$	4.47737	0.598313
$57$	−0.547953	−0.0725781
$58$	3.02823	0.397626
$59$	13.2730	1.72800	0.864000	−	0.503491i	$-0.167951\pi$
$59$	13.2730	1.72800	0.864000	+	0.503491i	$0.167951\pi$
$60$	−1.06666	−0.137705
$61$	−4.55104	−0.582701	−0.291350	−	0.956616i	$-0.594105\pi$
$61$	−4.55104	−0.582701	−0.291350	+	0.956616i	$0.594105\pi$
$62$	−3.59551	−0.456631
$63$	2.08838	0.263112
$64$	1.00000	0.125000
$65$	4.44156	0.550908
$66$	6.58217	0.810209
$67$	6.01455	0.734794	0.367397	−	0.930064i	$-0.380249\pi$
$67$	6.01455	0.734794	0.367397	+	0.930064i	$0.380249\pi$
$68$	4.34613	0.527045
$69$	0.0931058	0.0112086
$70$	3.00041	0.358618
$71$	9.18621	1.09020	0.545101	−	0.838370i	$-0.316491\pi$
$71$	9.18621	1.09020	0.545101	+	0.838370i	$0.316491\pi$
$72$	0.466431	0.0549694
$73$	6.66999	0.780663	0.390332	−	0.920674i	$-0.372360\pi$
$73$	6.66999	0.780663	0.390332	+	0.920674i	$0.372360\pi$
$74$	−8.23071	−0.956801
$75$	7.24380	0.836442
$76$	0.344252	0.0394884
$77$	−18.5151	−2.10999
$78$	10.5498	1.19453
$79$	−15.1530	−1.70484	−0.852421	−	0.522856i	$-0.824866\pi$
$79$	−15.1530	−1.70484	−0.852421	+	0.522856i	$0.824866\pi$
$80$	0.670129	0.0749227
$81$	−7.38315	−0.820350
$82$	−2.81771	−0.311164
$83$	−1.46270	−0.160553	−0.0802763	−	0.996773i	$-0.525580\pi$
$83$	−1.46270	−0.160553	−0.0802763	+	0.996773i	$0.525580\pi$
$84$	7.12671	0.777588
$85$	2.91247	0.315901
$86$	0.130402	0.0140616
$87$	4.82009	0.516768
$88$	−4.13526	−0.440820
$89$	−3.73587	−0.396002	−0.198001	−	0.980202i	$-0.563445\pi$
$89$	−3.73587	−0.396002	−0.198001	+	0.980202i	$0.563445\pi$
$90$	0.312569	0.0329477
$91$	−29.6756	−3.11085
$92$	−0.0584939	−0.00609841
$93$	−5.72305	−0.593453
$94$	6.77902	0.699203
$95$	0.230693	0.0236686
$96$	1.59172	0.162454
$97$	−0.250519	−0.0254364	−0.0127182	−	0.999919i	$-0.504048\pi$
$97$	−0.250519	−0.0254364	−0.0127182	+	0.999919i	$0.504048\pi$
$98$	−13.0468	−1.31793
$99$	−1.92881	−0.193853
$100$	−4.55093	−0.455093
$101$	−4.45480	−0.443270	−0.221635	−	0.975130i	$-0.571139\pi$
$101$	−4.45480	−0.443270	−0.221635	+	0.975130i	$0.571139\pi$
$102$	6.91781	0.684966
$103$	1.04778	0.103240	0.0516202	−	0.998667i	$-0.483561\pi$
$103$	1.04778	0.103240	0.0516202	+	0.998667i	$0.483561\pi$
$104$	−6.62792	−0.649921
$105$	4.77581	0.466072
$106$	−8.30912	−0.807053
$107$	−15.5579	−1.50404	−0.752020	−	0.659140i	$-0.770921\pi$
$107$	−15.5579	−1.50404	−0.752020	+	0.659140i	$0.770921\pi$
$108$	5.51758	0.530930
$109$	5.98184	0.572957	0.286478	−	0.958087i	$-0.407515\pi$
$109$	5.98184	0.572957	0.286478	+	0.958087i	$0.407515\pi$
$110$	−2.77115	−0.264219
$111$	−13.1010	−1.24349
$112$	−4.47737	−0.423071
$113$	−4.12358	−0.387913	−0.193957	−	0.981010i	$-0.562132\pi$
$113$	−4.12358	−0.387913	−0.193957	+	0.981010i	$0.562132\pi$
$114$	0.547953	0.0513205
$115$	−0.0391984	−0.00365527
$116$	−3.02823	−0.281164
$117$	−3.09147	−0.285806
$118$	−13.2730	−1.22188
$119$	−19.4592	−1.78382
$120$	1.06666	0.0973720
$121$	6.10035	0.554578
$122$	4.55104	0.412032
$123$	−4.48500	−0.404399
$124$	3.59551	0.322887
$125$	−6.40035	−0.572465
$126$	−2.08838	−0.186048
$127$	−5.43731	−0.482484	−0.241242	−	0.970465i	$-0.577555\pi$
$127$	−5.43731	−0.482484	−0.241242	+	0.970465i	$0.577555\pi$
$128$	−1.00000	−0.0883883
$129$	0.207564	0.0182750
$130$	−4.44156	−0.389551
$131$	−8.10912	−0.708497	−0.354249	−	0.935151i	$-0.615263\pi$
$131$	−8.10912	−0.708497	−0.354249	+	0.935151i	$0.615263\pi$
$132$	−6.58217	−0.572904
$133$	−1.54134	−0.133651
$134$	−6.01455	−0.519578
$135$	3.69749	0.318229
$136$	−4.34613	−0.372677
$137$	−1.38251	−0.118116	−0.0590579	−	0.998255i	$-0.518810\pi$
$137$	−1.38251	−0.118116	−0.0590579	+	0.998255i	$0.518810\pi$
$138$	−0.0931058	−0.00792569
$139$	−2.75898	−0.234014	−0.117007	−	0.993131i	$-0.537330\pi$
$139$	−2.75898	−0.234014	−0.117007	+	0.993131i	$0.537330\pi$
$140$	−3.00041	−0.253581
$141$	10.7903	0.908707
$142$	−9.18621	−0.770890
$143$	27.4082	2.29199
$144$	−0.466431	−0.0388693
$145$	−2.02930	−0.168524
$146$	−6.66999	−0.552012
$147$	−20.7669	−1.71282
$148$	8.23071	0.676561
$149$	0.974098	0.0798012	0.0399006	−	0.999204i	$-0.487296\pi$
$149$	0.974098	0.0798012	0.0399006	+	0.999204i	$0.487296\pi$
$150$	−7.24380	−0.591454
$151$	−10.1673	−0.827402	−0.413701	−	0.910413i	$-0.635764\pi$
$151$	−10.1673	−0.827402	−0.413701	+	0.910413i	$0.635764\pi$
$152$	−0.344252	−0.0279225
$153$	−2.02717	−0.163887
$154$	18.5151	1.49199
$155$	2.40946	0.193532
$156$	−10.5498	−0.844659
$157$	11.0009	0.877967	0.438984	−	0.898495i	$-0.355339\pi$
$157$	11.0009	0.877967	0.438984	+	0.898495i	$0.355339\pi$
$158$	15.1530	1.20550
$159$	−13.2258	−1.04887
$160$	−0.670129	−0.0529783
$161$	0.261899	0.0206405
$162$	7.38315	0.580075
$163$	0.827948	0.0648499	0.0324249	−	0.999474i	$-0.489677\pi$
$163$	0.827948	0.0648499	0.0324249	+	0.999474i	$0.489677\pi$
$164$	2.81771	0.220026
$165$	−4.41090	−0.343388
$166$	1.46270	0.113528
$167$	−15.5475	−1.20310	−0.601550	−	0.798835i	$-0.705450\pi$
$167$	−15.5475	−1.20310	−0.601550	+	0.798835i	$0.705450\pi$
$168$	−7.12671	−0.549837
$169$	30.9293	2.37918
$170$	−2.91247	−0.223376
$171$	−0.160570	−0.0122791
$172$	−0.130402	−0.00994307
$173$	−5.63052	−0.428080	−0.214040	−	0.976825i	$-0.568662\pi$
$173$	−5.63052	−0.428080	−0.214040	+	0.976825i	$0.568662\pi$
$174$	−4.82009	−0.365410
$175$	20.3762	1.54029
$176$	4.13526	0.311707
$177$	−21.1269	−1.58800
$178$	3.73587	0.280016
$179$	4.31878	0.322801	0.161400	−	0.986889i	$-0.448399\pi$
$179$	4.31878	0.322801	0.161400	+	0.986889i	$0.448399\pi$
$180$	−0.312569	−0.0232975
$181$	19.1418	1.42280	0.711401	−	0.702786i	$-0.248061\pi$
$181$	19.1418	1.42280	0.711401	+	0.702786i	$0.248061\pi$
$182$	29.6756	2.19970
$183$	7.24397	0.535490
$184$	0.0584939	0.00431223
$185$	5.51564	0.405518
$186$	5.72305	0.419634
$187$	17.9724	1.31427
$188$	−6.77902	−0.494411
$189$	−24.7042	−1.79697
$190$	−0.230693	−0.0167363
$191$	−3.40779	−0.246579	−0.123289	−	0.992371i	$-0.539344\pi$
$191$	−3.40779	−0.246579	−0.123289	+	0.992371i	$0.539344\pi$
$192$	−1.59172	−0.114872
$193$	0.919547	0.0661904	0.0330952	−	0.999452i	$-0.489464\pi$
$193$	0.919547	0.0661904	0.0330952	+	0.999452i	$0.489464\pi$
$194$	0.250519	0.0179862
$195$	−7.06972	−0.506273
$196$	13.0468	0.931915
$197$	6.37966	0.454532	0.227266	−	0.973833i	$-0.427021\pi$
$197$	6.37966	0.454532	0.227266	+	0.973833i	$0.427021\pi$
$198$	1.92881	0.137075
$199$	18.0412	1.27891	0.639455	−	0.768828i	$-0.279160\pi$
$199$	18.0412	1.27891	0.639455	+	0.768828i	$0.279160\pi$
$200$	4.55093	0.321799
$201$	−9.57347	−0.675260
$202$	4.45480	0.313439
$203$	13.5585	0.951620
$204$	−6.91781	−0.484344
$205$	1.88823	0.131880
$206$	−1.04778	−0.0730020
$207$	0.0272834	0.00189633
$208$	6.62792	0.459564
$209$	1.42357	0.0984705
$210$	−4.77581	−0.329562
$211$	−19.6712	−1.35422	−0.677111	−	0.735881i	$-0.736768\pi$
$211$	−19.6712	−1.35422	−0.677111	+	0.735881i	$0.736768\pi$
$212$	8.30912	0.570673
$213$	−14.6219	−1.00187
$214$	15.5579	1.06352
$215$	−0.0873863	−0.00595969
$216$	−5.51758	−0.375424
$217$	−16.0984	−1.09283
$218$	−5.98184	−0.405142
$219$	−10.6167	−0.717413
$220$	2.77115	0.186831
$221$	28.8058	1.93769
$222$	13.1010	0.879280
$223$	12.3470	0.826814	0.413407	−	0.910546i	$-0.364339\pi$
$223$	12.3470	0.826814	0.413407	+	0.910546i	$0.364339\pi$
$224$	4.47737	0.299157
$225$	2.12269	0.141513
$226$	4.12358	0.274296
$227$	11.9523	0.793303	0.396652	−	0.917969i	$-0.370172\pi$
$227$	11.9523	0.793303	0.396652	+	0.917969i	$0.370172\pi$
$228$	−0.547953	−0.0362891
$229$	−19.1395	−1.26477	−0.632386	−	0.774653i	$-0.717924\pi$
$229$	−19.1395	−1.26477	−0.632386	+	0.774653i	$0.717924\pi$
$230$	0.0391984	0.00258467
$231$	29.4708	1.93903
$232$	3.02823	0.198813
$233$	4.92002	0.322322	0.161161	−	0.986928i	$-0.448476\pi$
$233$	4.92002	0.322322	0.161161	+	0.986928i	$0.448476\pi$
$234$	3.09147	0.202096
$235$	−4.54282	−0.296341
$236$	13.2730	0.864000
$237$	24.1193	1.56671
$238$	19.4592	1.26135
$239$	19.6864	1.27341	0.636703	−	0.771109i	$-0.280298\pi$
$239$	19.6864	1.27341	0.636703	+	0.771109i	$0.280298\pi$
$240$	−1.06666	−0.0688524
$241$	−2.63544	−0.169764	−0.0848819	−	0.996391i	$-0.527051\pi$
$241$	−2.63544	−0.169764	−0.0848819	+	0.996391i	$0.527051\pi$
$242$	−6.10035	−0.392146
$243$	−4.80085	−0.307975
$244$	−4.55104	−0.291350
$245$	8.74304	0.558572
$246$	4.48500	0.285953
$247$	2.28168	0.145180
$248$	−3.59551	−0.228315
$249$	2.32821	0.147544
$250$	6.40035	0.404794
$251$	−30.3826	−1.91773	−0.958866	−	0.283858i	$-0.908385\pi$
$251$	−30.3826	−1.91773	−0.958866	+	0.283858i	$0.908385\pi$
$252$	2.08838	0.131556
$253$	−0.241887	−0.0152073
$254$	5.43731	0.341167
$255$	−4.63583	−0.290307
$256$	1.00000	0.0625000
$257$	11.5092	0.717925	0.358962	−	0.933352i	$-0.383131\pi$
$257$	11.5092	0.717925	0.358962	+	0.933352i	$0.383131\pi$
$258$	−0.207564	−0.0129223
$259$	−36.8519	−2.28987
$260$	4.44156	0.275454
$261$	1.41246	0.0874291
$262$	8.10912	0.500983
$263$	21.4723	1.32404	0.662021	−	0.749486i	$-0.269699\pi$
$263$	21.4723	1.32404	0.662021	+	0.749486i	$0.269699\pi$
$264$	6.58217	0.405104
$265$	5.56818	0.342051
$266$	1.54134	0.0945058
$267$	5.94646	0.363917
$268$	6.01455	0.367397
$269$	−11.8556	−0.722849	−0.361424	−	0.932401i	$-0.617709\pi$
$269$	−11.8556	−0.722849	−0.361424	+	0.932401i	$0.617709\pi$
$270$	−3.69749	−0.225022
$271$	−23.0271	−1.39880	−0.699399	−	0.714731i	$-0.746549\pi$
$271$	−23.0271	−1.39880	−0.699399	+	0.714731i	$0.746549\pi$
$272$	4.34613	0.263523
$273$	47.2353	2.85881
$274$	1.38251	0.0835205
$275$	−18.8193	−1.13484
$276$	0.0931058	0.00560431
$277$	28.8060	1.73079	0.865393	−	0.501094i	$-0.167069\pi$
$277$	28.8060	1.73079	0.865393	+	0.501094i	$0.167069\pi$
$278$	2.75898	0.165473
$279$	−1.67706	−0.100403
$280$	3.00041	0.179309
$281$	−5.33537	−0.318282	−0.159141	−	0.987256i	$-0.550872\pi$
$281$	−5.33537	−0.318282	−0.159141	+	0.987256i	$0.550872\pi$
$282$	−10.7903	−0.642553
$283$	−23.0668	−1.37118	−0.685589	−	0.727989i	$-0.740455\pi$
$283$	−23.0668	−1.37118	−0.685589	+	0.727989i	$0.740455\pi$
$284$	9.18621	0.545101
$285$	−0.367199	−0.0217510
$286$	−27.4082	−1.62068
$287$	−12.6159	−0.744694
$288$	0.466431	0.0274847
$289$	1.88883	0.111108
$290$	2.02930	0.119165
$291$	0.398756	0.0233755
$292$	6.66999	0.390332
$293$	−33.9907	−1.98576	−0.992879	−	0.119129i	$-0.961990\pi$
$293$	−33.9907	−1.98576	−0.992879	+	0.119129i	$0.961990\pi$
$294$	20.7669	1.21115
$295$	8.89463	0.517866
$296$	−8.23071	−0.478401
$297$	22.8166	1.32396
$298$	−0.974098	−0.0564280
$299$	−0.387693	−0.0224209
$300$	7.24380	0.418221
$301$	0.583858	0.0336530
$302$	10.1673	0.585061
$303$	7.09079	0.407356
$304$	0.344252	0.0197442
$305$	−3.04978	−0.174630
$306$	2.02717	0.115886
$307$	−18.8122	−1.07367	−0.536834	−	0.843688i	$-0.680380\pi$
$307$	−18.8122	−1.07367	−0.536834	+	0.843688i	$0.680380\pi$
$308$	−18.5151	−1.05499
$309$	−1.66776	−0.0948758
$310$	−2.40946	−0.136848
$311$	−14.3352	−0.812874	−0.406437	−	0.913679i	$-0.633229\pi$
$311$	−14.3352	−0.812874	−0.406437	+	0.913679i	$0.633229\pi$
$312$	10.5498	0.597264
$313$	29.9580	1.69332	0.846662	−	0.532131i	$-0.178608\pi$
$313$	29.9580	1.69332	0.846662	+	0.532131i	$0.178608\pi$
$314$	−11.0009	−0.620817
$315$	1.39949	0.0788521
$316$	−15.1530	−0.852421
$317$	26.1895	1.47095	0.735474	−	0.677552i	$-0.236959\pi$
$317$	26.1895	1.47095	0.735474	+	0.677552i	$0.236959\pi$
$318$	13.2258	0.741665
$319$	−12.5225	−0.701126
$320$	0.670129	0.0374613
$321$	24.7638	1.38218
$322$	−0.261899	−0.0145950
$323$	1.49616	0.0832488
$324$	−7.38315	−0.410175
$325$	−30.1632	−1.67315
$326$	−0.827948	−0.0458558
$327$	−9.52141	−0.526535
$328$	−2.81771	−0.155582
$329$	30.3522	1.67337
$330$	4.41090	0.242812
$331$	8.91749	0.490149	0.245075	−	0.969504i	$-0.421188\pi$
$331$	8.91749	0.490149	0.245075	+	0.969504i	$0.421188\pi$
$332$	−1.46270	−0.0802763
$333$	−3.83906	−0.210379
$334$	15.5475	0.850720
$335$	4.03052	0.220211
$336$	7.12671	0.388794
$337$	8.19477	0.446398	0.223199	−	0.974773i	$-0.428350\pi$
$337$	8.19477	0.446398	0.223199	+	0.974773i	$0.428350\pi$
$338$	−30.9293	−1.68233
$339$	6.56357	0.356484
$340$	2.91247	0.157951
$341$	14.8684	0.805168
$342$	0.160570	0.00868263
$343$	−27.0738	−1.46185
$344$	0.130402	0.00703082
$345$	0.0623929	0.00335912
$346$	5.63052	0.302699
$347$	17.6186	0.945818	0.472909	−	0.881111i	$-0.343204\pi$
$347$	17.6186	0.945818	0.472909	+	0.881111i	$0.343204\pi$
$348$	4.82009	0.258384
$349$	19.6658	1.05268	0.526342	−	0.850273i	$-0.323563\pi$
$349$	19.6658	1.05268	0.526342	+	0.850273i	$0.323563\pi$
$350$	−20.3762	−1.08915
$351$	36.5701	1.95197
$352$	−4.13526	−0.220410
$353$	0.834652	0.0444241	0.0222120	−	0.999753i	$-0.492929\pi$
$353$	0.834652	0.0444241	0.0222120	+	0.999753i	$0.492929\pi$
$354$	21.1269	1.12288
$355$	6.15594	0.326724
$356$	−3.73587	−0.198001
$357$	30.9736	1.63930
$358$	−4.31878	−0.228255
$359$	28.0804	1.48203	0.741013	−	0.671490i	$-0.234346\pi$
$359$	28.0804	1.48203	0.741013	+	0.671490i	$0.234346\pi$
$360$	0.312569	0.0164738
$361$	−18.8815	−0.993763
$362$	−19.1418	−1.00607
$363$	−9.71005	−0.509645
$364$	−29.6756	−1.55543
$365$	4.46975	0.233958
$366$	−7.24397	−0.378649
$367$	14.1544	0.738852	0.369426	−	0.929260i	$-0.379554\pi$
$367$	14.1544	0.738852	0.369426	+	0.929260i	$0.379554\pi$
$368$	−0.0584939	−0.00304920
$369$	−1.31427	−0.0684180
$370$	−5.51564	−0.286744
$371$	−37.2030	−1.93148
$372$	−5.72305	−0.296726
$373$	21.9581	1.13695	0.568473	−	0.822702i	$-0.307534\pi$
$373$	21.9581	1.13695	0.568473	+	0.822702i	$0.307534\pi$
$374$	−17.9724	−0.929328
$375$	10.1876	0.526083
$376$	6.77902	0.349601
$377$	−20.0709	−1.03370
$378$	24.7042	1.27065
$379$	−6.67360	−0.342800	−0.171400	−	0.985202i	$-0.554829\pi$
$379$	−6.67360	−0.342800	−0.171400	+	0.985202i	$0.554829\pi$
$380$	0.230693	0.0118343
$381$	8.65467	0.443392
$382$	3.40779	0.174358
$383$	−4.84595	−0.247616	−0.123808	−	0.992306i	$-0.539511\pi$
$383$	−4.84595	−0.247616	−0.123808	+	0.992306i	$0.539511\pi$
$384$	1.59172	0.0812271
$385$	−12.4075	−0.632343
$386$	−0.919547	−0.0468037
$387$	0.0608236	0.00309184
$388$	−0.250519	−0.0127182
$389$	7.55301	0.382953	0.191476	−	0.981497i	$-0.438672\pi$
$389$	7.55301	0.382953	0.191476	+	0.981497i	$0.438672\pi$
$390$	7.06972	0.357989
$391$	−0.254222	−0.0128566
$392$	−13.0468	−0.658963
$393$	12.9074	0.651095
$394$	−6.37966	−0.321403
$395$	−10.1544	−0.510925
$396$	−1.92881	−0.0969265
$397$	10.8035	0.542212	0.271106	−	0.962549i	$-0.412611\pi$
$397$	10.8035	0.542212	0.271106	+	0.962549i	$0.412611\pi$
$398$	−18.0412	−0.904326
$399$	2.45339	0.122823
$400$	−4.55093	−0.227546
$401$	−13.1616	−0.657258	−0.328629	−	0.944459i	$-0.606587\pi$
$401$	−13.1616	−0.657258	−0.328629	+	0.944459i	$0.606587\pi$
$402$	9.57347	0.477481
$403$	23.8308	1.18710
$404$	−4.45480	−0.221635
$405$	−4.94766	−0.245851
$406$	−13.5585	−0.672897
$407$	34.0361	1.68711
$408$	6.91781	0.342483
$409$	23.0532	1.13991	0.569953	−	0.821677i	$-0.306961\pi$
$409$	23.0532	1.13991	0.569953	+	0.821677i	$0.306961\pi$
$410$	−1.88823	−0.0932530
$411$	2.20057	0.108546
$412$	1.04778	0.0516202
$413$	−59.4282	−2.92427
$414$	−0.0272834	−0.00134090
$415$	−0.980199	−0.0481161
$416$	−6.62792	−0.324961
$417$	4.39152	0.215054
$418$	−1.42357	−0.0696292
$419$	14.2275	0.695057	0.347528	−	0.937669i	$-0.387021\pi$
$419$	14.2275	0.695057	0.347528	+	0.937669i	$0.387021\pi$
$420$	4.77581	0.233036
$421$	17.8193	0.868462	0.434231	−	0.900802i	$-0.357020\pi$
$421$	17.8193	0.868462	0.434231	+	0.900802i	$0.357020\pi$
$422$	19.6712	0.957580
$423$	3.16195	0.153739
$424$	−8.30912	−0.403527
$425$	−19.7789	−0.959418
$426$	14.6219	0.708432
$427$	20.3767	0.986096
$428$	−15.5579	−0.752020
$429$	−43.6261	−2.10629
$430$	0.0873863	0.00421414
$431$	−16.9147	−0.814752	−0.407376	−	0.913261i	$-0.633556\pi$
$431$	−16.9147	−0.814752	−0.407376	+	0.913261i	$0.633556\pi$
$432$	5.51758	0.265465
$433$	−15.9547	−0.766734	−0.383367	−	0.923596i	$-0.625236\pi$
$433$	−15.9547	−0.766734	−0.383367	+	0.923596i	$0.625236\pi$
$434$	16.0984	0.772750
$435$	3.23008	0.154871
$436$	5.98184	0.286478
$437$	−0.0201367	−0.000963267 0
$438$	10.6167	0.507288
$439$	13.3820	0.638686	0.319343	−	0.947639i	$-0.396538\pi$
$439$	13.3820	0.638686	0.319343	+	0.947639i	$0.396538\pi$
$440$	−2.77115	−0.132110
$441$	−6.08544	−0.289783
$442$	−28.8058	−1.37015
$443$	−7.68009	−0.364892	−0.182446	−	0.983216i	$-0.558402\pi$
$443$	−7.68009	−0.364892	−0.182446	+	0.983216i	$0.558402\pi$
$444$	−13.1010	−0.621745
$445$	−2.50352	−0.118678
$446$	−12.3470	−0.584645
$447$	−1.55049	−0.0733357
$448$	−4.47737	−0.211536
$449$	25.8114	1.21812	0.609058	−	0.793126i	$-0.291548\pi$
$449$	25.8114	1.21812	0.609058	+	0.793126i	$0.291548\pi$
$450$	−2.12269	−0.100065
$451$	11.6520	0.548669
$452$	−4.12358	−0.193957
$453$	16.1835	0.760365
$454$	−11.9523	−0.560950
$455$	−19.8865	−0.932293
$456$	0.547953	0.0256602
$457$	33.0954	1.54814	0.774070	−	0.633100i	$-0.218218\pi$
$457$	33.0954	1.54814	0.774070	+	0.633100i	$0.218218\pi$
$458$	19.1395	0.894329
$459$	23.9801	1.11930
$460$	−0.0391984	−0.00182764
$461$	−9.58209	−0.446282	−0.223141	−	0.974786i	$-0.571631\pi$
$461$	−9.58209	−0.446282	−0.223141	+	0.974786i	$0.571631\pi$
$462$	−29.4708	−1.37110
$463$	14.1492	0.657568	0.328784	−	0.944405i	$-0.393361\pi$
$463$	14.1492	0.657568	0.328784	+	0.944405i	$0.393361\pi$
$464$	−3.02823	−0.140582
$465$	−3.83518	−0.177852
$466$	−4.92002	−0.227916
$467$	23.2323	1.07506	0.537531	−	0.843244i	$-0.319357\pi$
$467$	23.2323	1.07506	0.537531	+	0.843244i	$0.319357\pi$
$468$	−3.09147	−0.142903
$469$	−26.9293	−1.24348
$470$	4.54282	0.209545
$471$	−17.5103	−0.806834
$472$	−13.2730	−0.610940
$473$	−0.539247	−0.0247946
$474$	−24.1193	−1.10783
$475$	−1.56667	−0.0718836
$476$	−19.4592	−0.891911
$477$	−3.87563	−0.177453
$478$	−19.6864	−0.900435
$479$	15.5433	0.710191	0.355095	−	0.934830i	$-0.384448\pi$
$479$	15.5433	0.710191	0.355095	+	0.934830i	$0.384448\pi$
$480$	1.06666	0.0486860
$481$	54.5525	2.48738
$482$	2.63544	0.120041
$483$	−0.416869	−0.0189682
$484$	6.10035	0.277289
$485$	−0.167880	−0.00762304
$486$	4.80085	0.217771
$487$	17.9004	0.811146	0.405573	−	0.914063i	$-0.367072\pi$
$487$	17.9004	0.811146	0.405573	+	0.914063i	$0.367072\pi$
$488$	4.55104	0.206016
$489$	−1.31786	−0.0595957
$490$	−8.74304	−0.394970
$491$	−0.827037	−0.0373236	−0.0186618	−	0.999826i	$-0.505941\pi$
$491$	−0.827037	−0.0373236	−0.0186618	+	0.999826i	$0.505941\pi$
$492$	−4.48500	−0.202200
$493$	−13.1611	−0.592745
$494$	−2.28168	−0.102658
$495$	−1.29255	−0.0580959
$496$	3.59551	0.161443
$497$	−41.1300	−1.84493
$498$	−2.32821	−0.104330
$499$	13.9285	0.623526	0.311763	−	0.950160i	$-0.399080\pi$
$499$	13.9285	0.623526	0.311763	+	0.950160i	$0.399080\pi$
$500$	−6.40035	−0.286232
$501$	24.7472	1.10562
$502$	30.3826	1.35604
$503$	−4.69332	−0.209265	−0.104632	−	0.994511i	$-0.533367\pi$
$503$	−4.69332	−0.209265	−0.104632	+	0.994511i	$0.533367\pi$
$504$	−2.08838	−0.0930240
$505$	−2.98529	−0.132844
$506$	0.241887	0.0107532
$507$	−49.2308	−2.18642
$508$	−5.43731	−0.241242
$509$	27.2155	1.20631	0.603153	−	0.797626i	$-0.293911\pi$
$509$	27.2155	1.20631	0.603153	+	0.797626i	$0.293911\pi$
$510$	4.63583	0.205278
$511$	−29.8640	−1.32111
$512$	−1.00000	−0.0441942
$513$	1.89944	0.0838624
$514$	−11.5092	−0.507649
$515$	0.702145	0.0309402
$516$	0.207564	0.00913748
$517$	−28.0330	−1.23289
$518$	36.8519	1.61918
$519$	8.96220	0.393397
$520$	−4.44156	−0.194775
$521$	5.32218	0.233169	0.116585	−	0.993181i	$-0.462805\pi$
$521$	5.32218	0.233169	0.116585	+	0.993181i	$0.462805\pi$
$522$	−1.41246	−0.0618217
$523$	−1.91525	−0.0837481	−0.0418740	−	0.999123i	$-0.513333\pi$
$523$	−1.91525	−0.0837481	−0.0418740	+	0.999123i	$0.513333\pi$
$524$	−8.10912	−0.354249
$525$	−32.4331	−1.41550
$526$	−21.4723	−0.936239
$527$	15.6266	0.680704
$528$	−6.58217	−0.286452
$529$	−22.9966	−0.999851
$530$	−5.56818	−0.241866
$531$	−6.19095	−0.268664
$532$	−1.54134	−0.0668257
$533$	18.6756	0.808928
$534$	−5.94646	−0.257328
$535$	−10.4258	−0.450747
$536$	−6.01455	−0.259789
$537$	−6.87428	−0.296647
$538$	11.8556	0.511131
$539$	53.9519	2.32387
$540$	3.69749	0.159115
$541$	−14.3582	−0.617308	−0.308654	−	0.951174i	$-0.599879\pi$
$541$	−14.3582	−0.617308	−0.308654	+	0.951174i	$0.599879\pi$
$542$	23.0271	0.989100
$543$	−30.4684	−1.30753
$544$	−4.34613	−0.186339
$545$	4.00861	0.171710
$546$	−47.2353	−2.02148
$547$	2.21372	0.0946520	0.0473260	−	0.998879i	$-0.484930\pi$
$547$	2.21372	0.0946520	0.0473260	+	0.998879i	$0.484930\pi$
$548$	−1.38251	−0.0590579
$549$	2.12275	0.0905966
$550$	18.8193	0.802456
$551$	−1.04247	−0.0444109
$552$	−0.0931058	−0.00396285
$553$	67.8454	2.88508
$554$	−28.8060	−1.22385
$555$	−8.77935	−0.372662
$556$	−2.75898	−0.117007
$557$	5.66393	0.239988	0.119994	−	0.992775i	$-0.461712\pi$
$557$	5.66393	0.239988	0.119994	+	0.992775i	$0.461712\pi$
$558$	1.67706	0.0709956
$559$	−0.864296	−0.0365558
$560$	−3.00041	−0.126791
$561$	−28.6069	−1.20779
$562$	5.33537	0.225059
$563$	1.26168	0.0531734	0.0265867	−	0.999647i	$-0.491536\pi$
$563$	1.26168	0.0531734	0.0265867	+	0.999647i	$0.491536\pi$
$564$	10.7903	0.454354
$565$	−2.76333	−0.116254
$566$	23.0668	0.969569
$567$	33.0571	1.38827
$568$	−9.18621	−0.385445
$569$	9.23123	0.386993	0.193497	−	0.981101i	$-0.438017\pi$
$569$	9.23123	0.386993	0.193497	+	0.981101i	$0.438017\pi$
$570$	0.367199	0.0153803
$571$	−25.0411	−1.04794	−0.523968	−	0.851738i	$-0.675549\pi$
$571$	−25.0411	−1.04794	−0.523968	+	0.851738i	$0.675549\pi$
$572$	27.4082	1.14599
$573$	5.42424	0.226601
$574$	12.6159	0.526578
$575$	0.266201	0.0111014
$576$	−0.466431	−0.0194346
$577$	−31.7089	−1.32006	−0.660029	−	0.751240i	$-0.729456\pi$
$577$	−31.7089	−1.32006	−0.660029	+	0.751240i	$0.729456\pi$
$578$	−1.88883	−0.0785649
$579$	−1.46366	−0.0608276
$580$	−2.02930	−0.0842622
$581$	6.54906	0.271701
$582$	−0.398756	−0.0165290
$583$	34.3604	1.42306
$584$	−6.66999	−0.276006
$585$	−2.07168	−0.0856535
$586$	33.9907	1.40414
$587$	5.61845	0.231898	0.115949	−	0.993255i	$-0.463009\pi$
$587$	5.61845	0.231898	0.115949	+	0.993255i	$0.463009\pi$
$588$	−20.7669	−0.856410
$589$	1.23776	0.0510012
$590$	−8.89463	−0.366186
$591$	−10.1546	−0.417706
$592$	8.23071	0.338280
$593$	35.2410	1.44718	0.723588	−	0.690232i	$-0.242492\pi$
$593$	35.2410	1.44718	0.723588	+	0.690232i	$0.242492\pi$
$594$	−22.8166	−0.936178
$595$	−13.0402	−0.534595
$596$	0.974098	0.0399006
$597$	−28.7166	−1.17529
$598$	0.387693	0.0158539
$599$	31.0333	1.26798	0.633992	−	0.773339i	$-0.281415\pi$
$599$	31.0333	1.26798	0.633992	+	0.773339i	$0.281415\pi$
$600$	−7.24380	−0.295727
$601$	5.89256	0.240362	0.120181	−	0.992752i	$-0.461652\pi$
$601$	5.89256	0.240362	0.120181	+	0.992752i	$0.461652\pi$
$602$	−0.583858	−0.0237963
$603$	−2.80537	−0.114244
$604$	−10.1673	−0.413701
$605$	4.08802	0.166202
$606$	−7.09079	−0.288044
$607$	−46.1155	−1.87177	−0.935886	−	0.352304i	$-0.885398\pi$
$607$	−46.1155	−1.87177	−0.935886	+	0.352304i	$0.885398\pi$
$608$	−0.344252	−0.0139613
$609$	−21.5813	−0.874519
$610$	3.04978	0.123482
$611$	−44.9308	−1.81771
$612$	−2.02717	−0.0819435
$613$	29.4705	1.19030	0.595151	−	0.803614i	$-0.297092\pi$
$613$	29.4705	1.19030	0.595151	+	0.803614i	$0.297092\pi$
$614$	18.8122	0.759198
$615$	−3.00553	−0.121195
$616$	18.5151	0.745993
$617$	14.0403	0.565241	0.282620	−	0.959232i	$-0.408796\pi$
$617$	14.0403	0.565241	0.282620	+	0.959232i	$0.408796\pi$
$618$	1.66776	0.0670873
$619$	−14.2922	−0.574451	−0.287226	−	0.957863i	$-0.592733\pi$
$619$	−14.2922	−0.574451	−0.287226	+	0.957863i	$0.592733\pi$
$620$	2.40946	0.0967662
$621$	−0.322745	−0.0129513
$622$	14.3352	0.574789
$623$	16.7269	0.670148
$624$	−10.5498	−0.422330
$625$	18.4656	0.738623
$626$	−29.9580	−1.19736
$627$	−2.26593	−0.0904924
$628$	11.0009	0.438984
$629$	35.7717	1.42631
$630$	−1.39949	−0.0557568
$631$	−17.3106	−0.689124	−0.344562	−	0.938764i	$-0.611973\pi$
$631$	−17.3106	−0.689124	−0.344562	+	0.938764i	$0.611973\pi$
$632$	15.1530	0.602752
$633$	31.3110	1.24450
$634$	−26.1895	−1.04012
$635$	−3.64370	−0.144596
$636$	−13.2258	−0.524437
$637$	86.4732	3.42619
$638$	12.5225	0.495771
$639$	−4.28473	−0.169501
$640$	−0.670129	−0.0264892
$641$	5.31405	0.209892	0.104946	−	0.994478i	$-0.466533\pi$
$641$	5.31405	0.209892	0.104946	+	0.994478i	$0.466533\pi$
$642$	−24.7638	−0.977351
$643$	43.9635	1.73375	0.866875	−	0.498526i	$-0.166125\pi$
$643$	43.9635	1.73375	0.866875	+	0.498526i	$0.166125\pi$
$644$	0.261899	0.0103202
$645$	0.139094	0.00547683
$646$	−1.49616	−0.0588658
$647$	−2.91327	−0.114532	−0.0572662	−	0.998359i	$-0.518238\pi$
$647$	−2.91327	−0.114532	−0.0572662	+	0.998359i	$0.518238\pi$
$648$	7.38315	0.290037
$649$	54.8874	2.15452
$650$	30.1632	1.18310
$651$	25.6242	1.00429
$652$	0.827948	0.0324249
$653$	25.6976	1.00563	0.502813	−	0.864395i	$-0.332298\pi$
$653$	25.6976	1.00563	0.502813	+	0.864395i	$0.332298\pi$
$654$	9.52141	0.372317
$655$	−5.43416	−0.212330
$656$	2.81771	0.110013
$657$	−3.11109	−0.121375
$658$	−30.3522	−1.18325
$659$	−38.0152	−1.48086	−0.740430	−	0.672133i	$-0.765378\pi$
$659$	−38.0152	−1.48086	−0.740430	+	0.672133i	$0.765378\pi$
$660$	−4.41090	−0.171694
$661$	−13.6824	−0.532182	−0.266091	−	0.963948i	$-0.585732\pi$
$661$	−13.6824	−0.532182	−0.266091	+	0.963948i	$0.585732\pi$
$662$	−8.91749	−0.346588
$663$	−45.8507	−1.78069
$664$	1.46270	0.0567639
$665$	−1.03290	−0.0400541
$666$	3.83906	0.148761
$667$	0.177133	0.00685861
$668$	−15.5475	−0.601550
$669$	−19.6529	−0.759825
$670$	−4.03052	−0.155713
$671$	−18.8197	−0.726527
$672$	−7.12671	−0.274919
$673$	21.1548	0.815457	0.407728	−	0.913103i	$-0.366321\pi$
$673$	21.1548	0.815457	0.407728	+	0.913103i	$0.366321\pi$
$674$	−8.19477	−0.315651
$675$	−25.1101	−0.966489
$676$	30.9293	1.18959
$677$	2.47640	0.0951757	0.0475878	−	0.998867i	$-0.484847\pi$
$677$	2.47640	0.0951757	0.0475878	+	0.998867i	$0.484847\pi$
$678$	−6.56357	−0.252073
$679$	1.12167	0.0430456
$680$	−2.91247	−0.111688
$681$	−19.0247	−0.729029
$682$	−14.8684	−0.569340
$683$	−1.54066	−0.0589516	−0.0294758	−	0.999565i	$-0.509384\pi$
$683$	−1.54066	−0.0589516	−0.0294758	+	0.999565i	$0.509384\pi$
$684$	−0.160570	−0.00613955
$685$	−0.926460	−0.0353982
$686$	27.0738	1.03368
$687$	30.4647	1.16230
$688$	−0.130402	−0.00497154
$689$	55.0722	2.09808
$690$	−0.0623929	−0.00237526
$691$	−29.4543	−1.12049	−0.560247	−	0.828326i	$-0.689294\pi$
$691$	−29.4543	−1.12049	−0.560247	+	0.828326i	$0.689294\pi$
$692$	−5.63052	−0.214040
$693$	8.63600	0.328055
$694$	−17.6186	−0.668795
$695$	−1.84887	−0.0701317
$696$	−4.82009	−0.182705
$697$	12.2461	0.463855
$698$	−19.6658	−0.744360
$699$	−7.83130	−0.296207
$700$	20.3762	0.770147
$701$	−42.0930	−1.58983	−0.794915	−	0.606721i	$-0.792485\pi$
$701$	−42.0930	−1.58983	−0.794915	+	0.606721i	$0.792485\pi$
$702$	−36.5701	−1.38025
$703$	2.83344	0.106865
$704$	4.13526	0.155853
$705$	7.23089	0.272331
$706$	−0.834652	−0.0314126
$707$	19.9458	0.750139
$708$	−21.1269	−0.793998
$709$	−33.5228	−1.25897	−0.629487	−	0.777011i	$-0.716735\pi$
$709$	−33.5228	−1.25897	−0.629487	+	0.777011i	$0.716735\pi$
$710$	−6.15594	−0.231028
$711$	7.06781	0.265064
$712$	3.73587	0.140008
$713$	−0.210316	−0.00787638
$714$	−30.9736	−1.15916
$715$	18.3670	0.686887
$716$	4.31878	0.161400
$717$	−31.3352	−1.17023
$718$	−28.0804	−1.04795
$719$	2.38440	0.0889230	0.0444615	−	0.999011i	$-0.485843\pi$
$719$	2.38440	0.0889230	0.0444615	+	0.999011i	$0.485843\pi$
$720$	−0.312569	−0.0116488
$721$	−4.69128	−0.174712
$722$	18.8815	0.702696
$723$	4.19488	0.156009
$724$	19.1418	0.711401
$725$	13.7813	0.511823
$726$	9.71005	0.360374
$727$	46.5843	1.72772	0.863858	−	0.503736i	$-0.168042\pi$
$727$	46.5843	1.72772	0.863858	+	0.503736i	$0.168042\pi$
$728$	29.6756	1.09985
$729$	29.7911	1.10337
$730$	−4.46975	−0.165433
$731$	−0.566745	−0.0209618
$732$	7.24397	0.267745
$733$	9.15268	0.338062	0.169031	−	0.985611i	$-0.445936\pi$
$733$	9.15268	0.338062	0.169031	+	0.985611i	$0.445936\pi$
$734$	−14.1544	−0.522447
$735$	−13.9165	−0.513316
$736$	0.0584939	0.00215611
$737$	24.8717	0.916161
$738$	1.31427	0.0483789
$739$	−35.7671	−1.31571	−0.657857	−	0.753143i	$-0.728537\pi$
$739$	−35.7671	−1.31571	−0.657857	+	0.753143i	$0.728537\pi$
$740$	5.51564	0.202759
$741$	−3.63179	−0.133417
$742$	37.2030	1.36576
$743$	29.4495	1.08040	0.540198	−	0.841538i	$-0.318349\pi$
$743$	29.4495	1.08040	0.540198	+	0.841538i	$0.318349\pi$
$744$	5.72305	0.209817
$745$	0.652771	0.0239157
$746$	−21.9581	−0.803943
$747$	0.682250	0.0249622
$748$	17.9724	0.657134
$749$	69.6585	2.54527
$750$	−10.1876	−0.371997
$751$	−44.8472	−1.63650	−0.818248	−	0.574865i	$-0.805054\pi$
$751$	−44.8472	−1.63650	−0.818248	+	0.574865i	$0.805054\pi$
$752$	−6.77902	−0.247206
$753$	48.3605	1.76236
$754$	20.0709	0.730938
$755$	−6.81339	−0.247965
$756$	−24.7042	−0.898485
$757$	−1.40023	−0.0508922	−0.0254461	−	0.999676i	$-0.508101\pi$
$757$	−1.40023	−0.0508922	−0.0254461	+	0.999676i	$0.508101\pi$
$758$	6.67360	0.242396
$759$	0.385017	0.0139752
$760$	−0.230693	−0.00836813
$761$	22.6298	0.820330	0.410165	−	0.912011i	$-0.365471\pi$
$761$	22.6298	0.820330	0.410165	+	0.912011i	$0.365471\pi$
$762$	−8.65467	−0.313526
$763$	−26.7829	−0.969606
$764$	−3.40779	−0.123289
$765$	−1.35846	−0.0491154
$766$	4.84595	0.175091
$767$	87.9726	3.17650
$768$	−1.59172	−0.0574362
$769$	28.9181	1.04281	0.521406	−	0.853309i	$-0.325408\pi$
$769$	28.9181	1.04281	0.521406	+	0.853309i	$0.325408\pi$
$770$	12.4075	0.447134
$771$	−18.3194	−0.659758
$772$	0.919547	0.0330952
$773$	1.27840	0.0459808	0.0229904	−	0.999736i	$-0.492681\pi$
$773$	1.27840	0.0459808	0.0229904	+	0.999736i	$0.492681\pi$
$774$	−0.0608236	−0.00218626
$775$	−16.3629	−0.587774
$776$	0.250519	0.00899311
$777$	58.6579	2.10434
$778$	−7.55301	−0.270788
$779$	0.970003	0.0347540
$780$	−7.06972	−0.253136
$781$	37.9873	1.35929
$782$	0.254222	0.00909096
$783$	−16.7085	−0.597113
$784$	13.0468	0.465957
$785$	7.37202	0.263119
$786$	−12.9074	−0.460393
$787$	6.55313	0.233594	0.116797	−	0.993156i	$-0.462737\pi$
$787$	6.55313	0.233594	0.116797	+	0.993156i	$0.462737\pi$
$788$	6.37966	0.227266
$789$	−34.1779	−1.21677
$790$	10.1544	0.361279
$791$	18.4628	0.656460
$792$	1.92881	0.0685374
$793$	−30.1639	−1.07115
$794$	−10.8035	−0.383402
$795$	−8.86298	−0.314338
$796$	18.0412	0.639455
$797$	−50.5109	−1.78919	−0.894594	−	0.446880i	$-0.852535\pi$
$797$	−50.5109	−1.78919	−0.894594	+	0.446880i	$0.852535\pi$
$798$	−2.45339	−0.0868489
$799$	−29.4625	−1.04231
$800$	4.55093	0.160900
$801$	1.74253	0.0615692
$802$	13.1616	0.464752
$803$	27.5821	0.973352
$804$	−9.57347	−0.337630
$805$	0.175506	0.00618576
$806$	−23.8308	−0.839404
$807$	18.8708	0.664283
$808$	4.45480	0.156719
$809$	24.7254	0.869299	0.434649	−	0.900600i	$-0.356872\pi$
$809$	24.7254	0.869299	0.434649	+	0.900600i	$0.356872\pi$
$810$	4.94766	0.173843
$811$	43.8140	1.53852	0.769259	−	0.638937i	$-0.220626\pi$
$811$	43.8140	1.53852	0.769259	+	0.638937i	$0.220626\pi$
$812$	13.5585	0.475810
$813$	36.6527	1.28547
$814$	−34.0361	−1.19297
$815$	0.554831	0.0194349
$816$	−6.91781	−0.242172
$817$	−0.0448913	−0.00157055
$818$	−23.0532	−0.806036
$819$	13.8416	0.483666
$820$	1.88823	0.0659398
$821$	31.8754	1.11246	0.556229	−	0.831029i	$-0.312248\pi$
$821$	31.8754	1.11246	0.556229	+	0.831029i	$0.312248\pi$
$822$	−2.20057	−0.0767536
$823$	20.5120	0.715002	0.357501	−	0.933913i	$-0.383629\pi$
$823$	20.5120	0.715002	0.357501	+	0.933913i	$0.383629\pi$
$824$	−1.04778	−0.0365010
$825$	29.9550	1.04290
$826$	59.4282	2.06777
$827$	−8.22849	−0.286133	−0.143066	−	0.989713i	$-0.545696\pi$
$827$	−8.22849	−0.286133	−0.143066	+	0.989713i	$0.545696\pi$
$828$	0.0272834	0.000948163 0
$829$	−48.0071	−1.66736	−0.833678	−	0.552251i	$-0.813769\pi$
$829$	−48.0071	−1.66736	−0.833678	+	0.552251i	$0.813769\pi$
$830$	0.980199	0.0340232
$831$	−45.8511	−1.59056
$832$	6.62792	0.229782
$833$	56.7031	1.96465
$834$	−4.39152	−0.152066
$835$	−10.4188	−0.360558
$836$	1.42357	0.0492353
$837$	19.8386	0.685721
$838$	−14.2275	−0.491479
$839$	−26.1464	−0.902674	−0.451337	−	0.892354i	$-0.649053\pi$
$839$	−26.1464	−0.902674	−0.451337	+	0.892354i	$0.649053\pi$
$840$	−4.77581	−0.164781
$841$	−19.8298	−0.683787
$842$	−17.8193	−0.614095
$843$	8.49241	0.292494
$844$	−19.6712	−0.677111
$845$	20.7266	0.713018
$846$	−3.16195	−0.108710
$847$	−27.3135	−0.938503
$848$	8.30912	0.285336
$849$	36.7158	1.26008
$850$	19.7789	0.678411
$851$	−0.481446	−0.0165038
$852$	−14.6219	−0.500937
$853$	−24.0146	−0.822246	−0.411123	−	0.911580i	$-0.634863\pi$
$853$	−24.0146	−0.822246	−0.411123	+	0.911580i	$0.634863\pi$
$854$	−20.3767	−0.697275
$855$	−0.107603	−0.00367993
$856$	15.5579	0.531759
$857$	−0.383812	−0.0131108	−0.00655538	−	0.999979i	$-0.502087\pi$
$857$	−0.383812	−0.0131108	−0.00655538	+	0.999979i	$0.502087\pi$
$858$	43.6261	1.48937
$859$	23.9826	0.818277	0.409139	−	0.912472i	$-0.365829\pi$
$859$	23.9826	0.818277	0.409139	+	0.912472i	$0.365829\pi$
$860$	−0.0873863	−0.00297985
$861$	20.0810	0.684359
$862$	16.9147	0.576116
$863$	22.7289	0.773700	0.386850	−	0.922143i	$-0.373563\pi$
$863$	22.7289	0.773700	0.386850	+	0.922143i	$0.373563\pi$
$864$	−5.51758	−0.187712
$865$	−3.77317	−0.128292
$866$	15.9547	0.542163
$867$	−3.00648	−0.102106
$868$	−16.0984	−0.546417
$869$	−62.6614	−2.12564
$870$	−3.23008	−0.109510
$871$	39.8639	1.35074
$872$	−5.98184	−0.202571
$873$	0.116850	0.00395477
$874$	0.0201367	0.000681132 0
$875$	28.6567	0.968774
$876$	−10.6167	−0.358707
$877$	−57.1171	−1.92871	−0.964353	−	0.264618i	$-0.914754\pi$
$877$	−57.1171	−1.92871	−0.964353	+	0.264618i	$0.914754\pi$
$878$	−13.3820	−0.451619
$879$	54.1036	1.82487
$880$	2.77115	0.0934156
$881$	−41.5628	−1.40029	−0.700143	−	0.714003i	$-0.746880\pi$
$881$	−41.5628	−1.40029	−0.700143	+	0.714003i	$0.746880\pi$
$882$	6.08544	0.204907
$883$	−9.81440	−0.330281	−0.165140	−	0.986270i	$-0.552808\pi$
$883$	−9.81440	−0.330281	−0.165140	+	0.986270i	$0.552808\pi$
$884$	28.8058	0.968844
$885$	−14.1578	−0.475908
$886$	7.68009	0.258018
$887$	10.7057	0.359463	0.179731	−	0.983716i	$-0.442477\pi$
$887$	10.7057	0.359463	0.179731	+	0.983716i	$0.442477\pi$
$888$	13.1010	0.439640
$889$	24.3448	0.816500
$890$	2.50352	0.0839180
$891$	−30.5312	−1.02283
$892$	12.3470	0.413407
$893$	−2.33369	−0.0780941
$894$	1.55049	0.0518562
$895$	2.89414	0.0967404
$896$	4.47737	0.149578
$897$	0.617098	0.0206043
$898$	−25.8114	−0.861338
$899$	−10.8880	−0.363137
$900$	2.12269	0.0707565
$901$	36.1125	1.20308
$902$	−11.6520	−0.387968
$903$	−0.929338	−0.0309264
$904$	4.12358	0.137148
$905$	12.8275	0.426401
$906$	−16.1835	−0.537659
$907$	−35.5186	−1.17938	−0.589688	−	0.807631i	$-0.700749\pi$
$907$	−35.5186	−1.17938	−0.589688	+	0.807631i	$0.700749\pi$
$908$	11.9523	0.396652
$909$	2.07786	0.0689182
$910$	19.8865	0.659231
$911$	18.6959	0.619423	0.309712	−	0.950831i	$-0.399767\pi$
$911$	18.6959	0.619423	0.309712	+	0.950831i	$0.399767\pi$
$912$	−0.547953	−0.0181445
$913$	−6.04865	−0.200181
$914$	−33.0954	−1.09470
$915$	4.85440	0.160481
$916$	−19.1395	−0.632386
$917$	36.3075	1.19898
$918$	−23.9801	−0.791462
$919$	−45.4205	−1.49828	−0.749141	−	0.662410i	$-0.769534\pi$
$919$	−45.4205	−1.49828	−0.749141	+	0.662410i	$0.769534\pi$
$920$	0.0391984	0.00129233
$921$	29.9437	0.986679
$922$	9.58209	0.315569
$923$	60.8855	2.00407
$924$	29.4708	0.969517
$925$	−37.4574	−1.23159
$926$	−14.1492	−0.464971
$927$	−0.488715	−0.0160515
$928$	3.02823	0.0994065
$929$	22.2368	0.729564	0.364782	−	0.931093i	$-0.381143\pi$
$929$	22.2368	0.729564	0.364782	+	0.931093i	$0.381143\pi$
$930$	3.83518	0.125760
$931$	4.49139	0.147199
$932$	4.92002	0.161161
$933$	22.8176	0.747014
$934$	−23.2323	−0.760184
$935$	12.0438	0.393874
$936$	3.09147	0.101048
$937$	−30.5654	−0.998530	−0.499265	−	0.866449i	$-0.666396\pi$
$937$	−30.5654	−0.998530	−0.499265	+	0.866449i	$0.666396\pi$
$938$	26.9293	0.879274
$939$	−47.6847	−1.55613
$940$	−4.54282	−0.148170
$941$	56.9888	1.85778	0.928890	−	0.370355i	$-0.120764\pi$
$941$	56.9888	1.85778	0.928890	+	0.370355i	$0.120764\pi$
$942$	17.5103	0.570518
$943$	−0.164819	−0.00536724
$944$	13.2730	0.432000
$945$	−16.5550	−0.538535
$946$	0.539247	0.0175324
$947$	−15.9637	−0.518751	−0.259375	−	0.965777i	$-0.583517\pi$
$947$	−15.9637	−0.518751	−0.259375	+	0.965777i	$0.583517\pi$
$948$	24.1193	0.783357
$949$	44.2082	1.43506
$950$	1.56667	0.0508294
$951$	−41.6863	−1.35177
$952$	19.4592	0.630677
$953$	−19.1141	−0.619165	−0.309583	−	0.950873i	$-0.600189\pi$
$953$	−19.1141	−0.619165	−0.309583	+	0.950873i	$0.600189\pi$
$954$	3.87563	0.125478
$955$	−2.28366	−0.0738974
$956$	19.6864	0.636703
$957$	19.9323	0.644320
$958$	−15.5433	−0.502181
$959$	6.19000	0.199886
$960$	−1.06666	−0.0344262
$961$	−18.0723	−0.582977
$962$	−54.5525	−1.75884
$963$	7.25670	0.233844
$964$	−2.63544	−0.0848819
$965$	0.616215	0.0198367
$966$	0.416869	0.0134125
$967$	47.8467	1.53865	0.769323	−	0.638860i	$-0.220594\pi$
$967$	47.8467	1.53865	0.769323	+	0.638860i	$0.220594\pi$
$968$	−6.10035	−0.196073
$969$	−2.38147	−0.0765039
$970$	0.167880	0.00539030
$971$	36.9710	1.18646	0.593228	−	0.805034i	$-0.297853\pi$
$971$	36.9710	1.18646	0.593228	+	0.805034i	$0.297853\pi$
$972$	−4.80085	−0.153987
$973$	12.3530	0.396018
$974$	−17.9004	−0.573567
$975$	48.0113	1.53759
$976$	−4.55104	−0.145675
$977$	6.59263	0.210917	0.105458	−	0.994424i	$-0.466369\pi$
$977$	6.59263	0.210917	0.105458	+	0.994424i	$0.466369\pi$
$978$	1.31786	0.0421405
$979$	−15.4488	−0.493746
$980$	8.74304	0.279286
$981$	−2.79012	−0.0890816
$982$	0.827037	0.0263918
$983$	13.4182	0.427975	0.213988	−	0.976836i	$-0.431355\pi$
$983$	13.4182	0.427975	0.213988	+	0.976836i	$0.431355\pi$
$984$	4.48500	0.142977
$985$	4.27520	0.136219
$986$	13.1611	0.419134
$987$	−48.3121	−1.53779
$988$	2.28168	0.0725898
$989$	0.00762773	0.000242548 0
$990$	1.29255	0.0410800
$991$	3.50038	0.111193	0.0555965	−	0.998453i	$-0.482294\pi$
$991$	3.50038	0.111193	0.0555965	+	0.998453i	$0.482294\pi$
$992$	−3.59551	−0.114158
$993$	−14.1941	−0.450437
$994$	41.1300	1.30457
$995$	12.0900	0.383277
$996$	2.32821	0.0737722
$997$	−48.9843	−1.55135	−0.775674	−	0.631134i	$-0.782590\pi$
$997$	−48.9843	−1.55135	−0.775674	+	0.631134i	$0.782590\pi$
$998$	−13.9285	−0.440900
$999$	45.4137	1.43682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.20	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.20	✓	83	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.59172 q^{3} +1.00000 q^{4} +0.670129 q^{5} +1.59172 q^{6} -4.47737 q^{7} -1.00000 q^{8} -0.466431 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.59172 q^{3} +1.00000 q^{4} +0.670129 q^{5} +1.59172 q^{6} -4.47737 q^{7} -1.00000 q^{8} -0.466431 q^{9} -0.670129 q^{10} +4.13526 q^{11} -1.59172 q^{12} +6.62792 q^{13} +4.47737 q^{14} -1.06666 q^{15} +1.00000 q^{16} +4.34613 q^{17} +0.466431 q^{18} +0.344252 q^{19} +0.670129 q^{20} +7.12671 q^{21} -4.13526 q^{22} -0.0584939 q^{23} +1.59172 q^{24} -4.55093 q^{25} -6.62792 q^{26} +5.51758 q^{27} -4.47737 q^{28} -3.02823 q^{29} +1.06666 q^{30} +3.59551 q^{31} -1.00000 q^{32} -6.58217 q^{33} -4.34613 q^{34} -3.00041 q^{35} -0.466431 q^{36} +8.23071 q^{37} -0.344252 q^{38} -10.5498 q^{39} -0.670129 q^{40} +2.81771 q^{41} -7.12671 q^{42} -0.130402 q^{43} +4.13526 q^{44} -0.312569 q^{45} +0.0584939 q^{46} -6.77902 q^{47} -1.59172 q^{48} +13.0468 q^{49} +4.55093 q^{50} -6.91781 q^{51} +6.62792 q^{52} +8.30912 q^{53} -5.51758 q^{54} +2.77115 q^{55} +4.47737 q^{56} -0.547953 q^{57} +3.02823 q^{58} +13.2730 q^{59} -1.06666 q^{60} -4.55104 q^{61} -3.59551 q^{62} +2.08838 q^{63} +1.00000 q^{64} +4.44156 q^{65} +6.58217 q^{66} +6.01455 q^{67} +4.34613 q^{68} +0.0931058 q^{69} +3.00041 q^{70} +9.18621 q^{71} +0.466431 q^{72} +6.66999 q^{73} -8.23071 q^{74} +7.24380 q^{75} +0.344252 q^{76} -18.5151 q^{77} +10.5498 q^{78} -15.1530 q^{79} +0.670129 q^{80} -7.38315 q^{81} -2.81771 q^{82} -1.46270 q^{83} +7.12671 q^{84} +2.91247 q^{85} +0.130402 q^{86} +4.82009 q^{87} -4.13526 q^{88} -3.73587 q^{89} +0.312569 q^{90} -29.6756 q^{91} -0.0584939 q^{92} -5.72305 q^{93} +6.77902 q^{94} +0.230693 q^{95} +1.59172 q^{96} -0.250519 q^{97} -13.0468 q^{98} -1.92881 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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