LMFDB - Embedded newform 8047.2.a.c.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, 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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$151$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66878 q^{2} +1.69053 q^{3} +5.12237 q^{4} -3.40805 q^{5} -4.51165 q^{6} -4.86127 q^{7} -8.33291 q^{8} -0.142111 q^{9} +O(q^{10})$	\(q-2.66878 q^{2} +1.69053 q^{3} +5.12237 q^{4} -3.40805 q^{5} -4.51165 q^{6} -4.86127 q^{7} -8.33291 q^{8} -0.142111 q^{9} +9.09532 q^{10} -1.84059 q^{11} +8.65952 q^{12} -1.00000 q^{13} +12.9736 q^{14} -5.76140 q^{15} +11.9939 q^{16} +0.0257198 q^{17} +0.379262 q^{18} -1.04515 q^{19} -17.4573 q^{20} -8.21811 q^{21} +4.91212 q^{22} -1.37947 q^{23} -14.0870 q^{24} +6.61479 q^{25} +2.66878 q^{26} -5.31183 q^{27} -24.9012 q^{28} +1.77687 q^{29} +15.3759 q^{30} +7.99657 q^{31} -15.3433 q^{32} -3.11157 q^{33} -0.0686403 q^{34} +16.5674 q^{35} -0.727943 q^{36} -2.01207 q^{37} +2.78926 q^{38} -1.69053 q^{39} +28.3990 q^{40} +11.0724 q^{41} +21.9323 q^{42} +2.76799 q^{43} -9.42818 q^{44} +0.484320 q^{45} +3.68150 q^{46} +5.86528 q^{47} +20.2761 q^{48} +16.6319 q^{49} -17.6534 q^{50} +0.0434800 q^{51} -5.12237 q^{52} -1.85491 q^{53} +14.1761 q^{54} +6.27282 q^{55} +40.5085 q^{56} -1.76685 q^{57} -4.74207 q^{58} -9.37698 q^{59} -29.5121 q^{60} +1.74301 q^{61} -21.3411 q^{62} +0.690838 q^{63} +16.9601 q^{64} +3.40805 q^{65} +8.30409 q^{66} -1.91921 q^{67} +0.131746 q^{68} -2.33204 q^{69} -44.2148 q^{70} -3.35314 q^{71} +1.18420 q^{72} -8.66437 q^{73} +5.36977 q^{74} +11.1825 q^{75} -5.35363 q^{76} +8.94760 q^{77} +4.51165 q^{78} +16.2247 q^{79} -40.8759 q^{80} -8.55347 q^{81} -29.5498 q^{82} -3.18677 q^{83} -42.0962 q^{84} -0.0876542 q^{85} -7.38714 q^{86} +3.00385 q^{87} +15.3375 q^{88} +0.735015 q^{89} -1.29254 q^{90} +4.86127 q^{91} -7.06616 q^{92} +13.5184 q^{93} -15.6531 q^{94} +3.56191 q^{95} -25.9384 q^{96} +16.1510 q^{97} -44.3869 q^{98} +0.261567 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) $ 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * 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 $

$2$	−2.66878	−1.88711	−0.943555	−	0.331215i	$-0.892541\pi$
$2$	−2.66878	−1.88711	−0.943555	+	0.331215i	$0.892541\pi$
$3$	1.69053	0.976028	0.488014	−	0.872836i	$-0.337722\pi$
$3$	1.69053	0.976028	0.488014	+	0.872836i	$0.337722\pi$
$4$	5.12237	2.56119
$5$	−3.40805	−1.52413	−0.762063	−	0.647503i	$-0.775813\pi$
$5$	−3.40805	−1.52413	−0.762063	+	0.647503i	$0.775813\pi$
$6$	−4.51165	−1.84187
$7$	−4.86127	−1.83739	−0.918693	−	0.394972i	$-0.870754\pi$
$7$	−4.86127	−1.83739	−0.918693	+	0.394972i	$0.870754\pi$
$8$	−8.33291	−2.94613
$9$	−0.142111	−0.0473702
$10$	9.09532	2.87619
$11$	−1.84059	−0.554959	−0.277479	−	0.960732i	$-0.589499\pi$
$11$	−1.84059	−0.554959	−0.277479	+	0.960732i	$0.589499\pi$
$12$	8.65952	2.49979
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	12.9736	3.46735
$15$	−5.76140	−1.48759
$16$	11.9939	2.99849
$17$	0.0257198	0.00623796	0.00311898	−	0.999995i	$-0.499007\pi$
$17$	0.0257198	0.00623796	0.00311898	+	0.999995i	$0.499007\pi$
$18$	0.379262	0.0893928
$19$	−1.04515	−0.239773	−0.119887	−	0.992788i	$-0.538253\pi$
$19$	−1.04515	−0.239773	−0.119887	+	0.992788i	$0.538253\pi$
$20$	−17.4573	−3.90357
$21$	−8.21811	−1.79334
$22$	4.91212	1.04727
$23$	−1.37947	−0.287640	−0.143820	−	0.989604i	$-0.545939\pi$
$23$	−1.37947	−0.287640	−0.143820	+	0.989604i	$0.545939\pi$
$24$	−14.0870	−2.87550
$25$	6.61479	1.32296
$26$	2.66878	0.523390
$27$	−5.31183	−1.02226
$28$	−24.9012	−4.70589
$29$	1.77687	0.329957	0.164978	−	0.986297i	$-0.447245\pi$
$29$	1.77687	0.329957	0.164978	+	0.986297i	$0.447245\pi$
$30$	15.3759	2.80724
$31$	7.99657	1.43623	0.718113	−	0.695926i	$-0.245006\pi$
$31$	7.99657	1.43623	0.718113	+	0.695926i	$0.245006\pi$
$32$	−15.3433	−2.71234
$33$	−3.11157	−0.541655
$34$	−0.0686403	−0.0117717
$35$	16.5674	2.80041
$36$	−0.727943	−0.121324
$37$	−2.01207	−0.330783	−0.165391	−	0.986228i	$-0.552889\pi$
$37$	−2.01207	−0.330783	−0.165391	+	0.986228i	$0.552889\pi$
$38$	2.78926	0.452478
$39$	−1.69053	−0.270701
$40$	28.3990	4.49027
$41$	11.0724	1.72922	0.864610	−	0.502443i	$-0.167565\pi$
$41$	11.0724	1.72922	0.864610	+	0.502443i	$0.167565\pi$
$42$	21.9323	3.38423
$43$	2.76799	0.422114	0.211057	−	0.977474i	$-0.432309\pi$
$43$	2.76799	0.422114	0.211057	+	0.977474i	$0.432309\pi$
$44$	−9.42818	−1.42135
$45$	0.484320	0.0721981
$46$	3.68150	0.542808
$47$	5.86528	0.855538	0.427769	−	0.903888i	$-0.359300\pi$
$47$	5.86528	0.855538	0.427769	+	0.903888i	$0.359300\pi$
$48$	20.2761	2.92660
$49$	16.6319	2.37599
$50$	−17.6534	−2.49657
$51$	0.0434800	0.00608842
$52$	−5.12237	−0.710345
$53$	−1.85491	−0.254792	−0.127396	−	0.991852i	$-0.540662\pi$
$53$	−1.85491	−0.254792	−0.127396	+	0.991852i	$0.540662\pi$
$54$	14.1761	1.92912
$55$	6.27282	0.845827
$56$	40.5085	5.41318
$57$	−1.76685	−0.234025
$58$	−4.74207	−0.622665
$59$	−9.37698	−1.22078	−0.610389	−	0.792102i	$-0.708987\pi$
$59$	−9.37698	−1.22078	−0.610389	+	0.792102i	$0.708987\pi$
$60$	−29.5121	−3.80999
$61$	1.74301	0.223169	0.111585	−	0.993755i	$-0.464407\pi$
$61$	1.74301	0.223169	0.111585	+	0.993755i	$0.464407\pi$
$62$	−21.3411	−2.71032
$63$	0.690838	0.0870373
$64$	16.9601	2.12001
$65$	3.40805	0.422716
$66$	8.30409	1.02216
$67$	−1.91921	−0.234469	−0.117235	−	0.993104i	$-0.537403\pi$
$67$	−1.91921	−0.234469	−0.117235	+	0.993104i	$0.537403\pi$
$68$	0.131746	0.0159766
$69$	−2.33204	−0.280744
$70$	−44.2148	−5.28468
$71$	−3.35314	−0.397945	−0.198972	−	0.980005i	$-0.563760\pi$
$71$	−3.35314	−0.397945	−0.198972	+	0.980005i	$0.563760\pi$
$72$	1.18420	0.139559
$73$	−8.66437	−1.01409	−0.507044	−	0.861920i	$-0.669262\pi$
$73$	−8.66437	−1.01409	−0.507044	+	0.861920i	$0.669262\pi$
$74$	5.36977	0.624223
$75$	11.1825	1.29124
$76$	−5.35363	−0.614103
$77$	8.94760	1.01967
$78$	4.51165	0.510843
$79$	16.2247	1.82542	0.912710	−	0.408608i	$-0.133986\pi$
$79$	16.2247	1.82542	0.912710	+	0.408608i	$0.133986\pi$
$80$	−40.8759	−4.57007
$81$	−8.55347	−0.950386
$82$	−29.5498	−3.26323
$83$	−3.18677	−0.349794	−0.174897	−	0.984587i	$-0.555959\pi$
$83$	−3.18677	−0.349794	−0.174897	+	0.984587i	$0.555959\pi$
$84$	−42.0962	−4.59307
$85$	−0.0876542	−0.00950743
$86$	−7.38714	−0.796576
$87$	3.00385	0.322047
$88$	15.3375	1.63498
$89$	0.735015	0.0779115	0.0389557	−	0.999241i	$-0.487597\pi$
$89$	0.735015	0.0779115	0.0389557	+	0.999241i	$0.487597\pi$
$90$	−1.29254	−0.136246
$91$	4.86127	0.509599
$92$	−7.06616	−0.736699
$93$	13.5184	1.40180
$94$	−15.6531	−1.61450
$95$	3.56191	0.365444
$96$	−25.9384	−2.64732
$97$	16.1510	1.63988	0.819941	−	0.572448i	$-0.194006\pi$
$97$	16.1510	1.63988	0.819941	+	0.572448i	$0.194006\pi$
$98$	−44.3869	−4.48375
$99$	0.261567	0.0262885
$100$	33.8834	3.38834
$101$	−8.08535	−0.804522	−0.402261	−	0.915525i	$-0.631776\pi$
$101$	−8.08535	−0.804522	−0.402261	+	0.915525i	$0.631776\pi$
$102$	−0.116038	−0.0114895
$103$	2.78785	0.274695	0.137348	−	0.990523i	$-0.456142\pi$
$103$	2.78785	0.274695	0.137348	+	0.990523i	$0.456142\pi$
$104$	8.33291	0.817109
$105$	28.0077	2.73327
$106$	4.95035	0.480821
$107$	−8.60976	−0.832337	−0.416168	−	0.909288i	$-0.636627\pi$
$107$	−8.60976	−0.832337	−0.416168	+	0.909288i	$0.636627\pi$
$108$	−27.2092	−2.61820
$109$	14.2022	1.36032	0.680160	−	0.733064i	$-0.261910\pi$
$109$	14.2022	1.36032	0.680160	+	0.733064i	$0.261910\pi$
$110$	−16.7408	−1.59617
$111$	−3.40147	−0.322853
$112$	−58.3057	−5.50938
$113$	−7.28804	−0.685601	−0.342800	−	0.939408i	$-0.611375\pi$
$113$	−7.28804	−0.685601	−0.342800	+	0.939408i	$0.611375\pi$
$114$	4.71533	0.441631
$115$	4.70130	0.438399
$116$	9.10179	0.845080
$117$	0.142111	0.0131381
$118$	25.0251	2.30374
$119$	−0.125031	−0.0114615
$120$	48.0093	4.38263
$121$	−7.61223	−0.692021
$122$	−4.65170	−0.421145
$123$	18.7182	1.68777
$124$	40.9614	3.67844
$125$	−5.50328	−0.492229
$126$	−1.84369	−0.164249
$127$	11.5533	1.02519	0.512595	−	0.858630i	$-0.328684\pi$
$127$	11.5533	1.02519	0.512595	+	0.858630i	$0.328684\pi$
$128$	−14.5759	−1.28834
$129$	4.67936	0.411995
$130$	−9.09532	−0.797712
$131$	15.9570	1.39417	0.697083	−	0.716990i	$-0.254481\pi$
$131$	15.9570	1.39417	0.697083	+	0.716990i	$0.254481\pi$
$132$	−15.9386	−1.38728
$133$	5.08074	0.440556
$134$	5.12195	0.442469
$135$	18.1030	1.55806
$136$	−0.214320	−0.0183778
$137$	−16.8473	−1.43936	−0.719680	−	0.694306i	$-0.755712\pi$
$137$	−16.8473	−1.43936	−0.719680	+	0.694306i	$0.755712\pi$
$138$	6.22369	0.529795
$139$	−2.43930	−0.206899	−0.103449	−	0.994635i	$-0.532988\pi$
$139$	−2.43930	−0.206899	−0.103449	+	0.994635i	$0.532988\pi$
$140$	84.8645	7.17236
$141$	9.91542	0.835029
$142$	8.94879	0.750966
$143$	1.84059	0.153918
$144$	−1.70447	−0.142039
$145$	−6.05566	−0.502895
$146$	23.1233	1.91370
$147$	28.1167	2.31903
$148$	−10.3066	−0.847195
$149$	8.56477	0.701654	0.350827	−	0.936440i	$-0.385901\pi$
$149$	8.56477	0.701654	0.350827	+	0.936440i	$0.385901\pi$
$150$	−29.8436	−2.43672
$151$	−2.97643	−0.242218	−0.121109	−	0.992639i	$-0.538645\pi$
$151$	−2.97643	−0.242218	−0.121109	+	0.992639i	$0.538645\pi$
$152$	8.70911	0.706402
$153$	−0.00365505	−0.000295493 0
$154$	−23.8791	−1.92424
$155$	−27.2527	−2.18899
$156$	−8.65952	−0.693316
$157$	6.79319	0.542155	0.271078	−	0.962557i	$-0.412620\pi$
$157$	6.79319	0.542155	0.271078	+	0.962557i	$0.412620\pi$
$158$	−43.3001	−3.44477
$159$	−3.13579	−0.248684
$160$	52.2908	4.13395
$161$	6.70598	0.528505
$162$	22.8273	1.79348
$163$	0.140051	0.0109697	0.00548483	−	0.999985i	$-0.498254\pi$
$163$	0.140051	0.0109697	0.00548483	+	0.999985i	$0.498254\pi$
$164$	56.7170	4.42886
$165$	10.6044	0.825550
$166$	8.50479	0.660100
$167$	9.13870	0.707174	0.353587	−	0.935402i	$-0.384962\pi$
$167$	9.13870	0.707174	0.353587	+	0.935402i	$0.384962\pi$
$168$	68.4808	5.28341
$169$	1.00000	0.0769231
$170$	0.233929	0.0179416
$171$	0.148526	0.0113581
$172$	14.1787	1.08111
$173$	5.41170	0.411444	0.205722	−	0.978610i	$-0.434046\pi$
$173$	5.41170	0.411444	0.205722	+	0.978610i	$0.434046\pi$
$174$	−8.01661	−0.607738
$175$	−32.1563	−2.43078
$176$	−22.0759	−1.66404
$177$	−15.8521	−1.19151
$178$	−1.96159	−0.147028
$179$	−18.8477	−1.40874	−0.704371	−	0.709832i	$-0.748771\pi$
$179$	−18.8477	−1.40874	−0.704371	+	0.709832i	$0.748771\pi$
$180$	2.48087	0.184913
$181$	4.51757	0.335788	0.167894	−	0.985805i	$-0.446303\pi$
$181$	4.51757	0.335788	0.167894	+	0.985805i	$0.446303\pi$
$182$	−12.9736	−0.961670
$183$	2.94661	0.217819
$184$	11.4950	0.847424
$185$	6.85724	0.504154
$186$	−36.0777	−2.64534
$187$	−0.0473395	−0.00346181
$188$	30.0441	2.19119
$189$	25.8222	1.87829
$190$	−9.50594	−0.689633
$191$	−24.7272	−1.78920	−0.894598	−	0.446873i	$-0.852538\pi$
$191$	−24.7272	−1.78920	−0.894598	+	0.446873i	$0.852538\pi$
$192$	28.6715	2.06918
$193$	−3.85032	−0.277152	−0.138576	−	0.990352i	$-0.544253\pi$
$193$	−3.85032	−0.277152	−0.138576	+	0.990352i	$0.544253\pi$
$194$	−43.1033	−3.09464
$195$	5.76140	0.412583
$196$	85.1948	6.08534
$197$	9.16293	0.652832	0.326416	−	0.945226i	$-0.394159\pi$
$197$	9.16293	0.652832	0.326416	+	0.945226i	$0.394159\pi$
$198$	−0.698065	−0.0496093
$199$	22.0728	1.56470	0.782351	−	0.622838i	$-0.214020\pi$
$199$	22.0728	1.56470	0.782351	+	0.622838i	$0.214020\pi$
$200$	−55.1205	−3.89761
$201$	−3.24448	−0.228848
$202$	21.5780	1.51822
$203$	−8.63784	−0.606258
$204$	0.222721	0.0155936
$205$	−37.7353	−2.63555
$206$	−7.44016	−0.518380
$207$	0.196038	0.0136255
$208$	−11.9939	−0.831630
$209$	1.92369	0.133064
$210$	−74.7464	−5.15799
$211$	−18.6610	−1.28467	−0.642337	−	0.766422i	$-0.722035\pi$
$211$	−18.6610	−1.28467	−0.642337	+	0.766422i	$0.722035\pi$
$212$	−9.50156	−0.652570
$213$	−5.66858	−0.388405
$214$	22.9775	1.57071
$215$	−9.43343	−0.643355
$216$	44.2630	3.01172
$217$	−38.8735	−2.63890
$218$	−37.9024	−2.56707
$219$	−14.6474	−0.989777
$220$	32.1317	2.16632
$221$	−0.0257198	−0.00173010
$222$	9.07775	0.609259
$223$	3.37589	0.226066	0.113033	−	0.993591i	$-0.463943\pi$
$223$	3.37589	0.226066	0.113033	+	0.993591i	$0.463943\pi$
$224$	74.5880	4.98362
$225$	−0.940032	−0.0626688
$226$	19.4501	1.29380
$227$	8.16141	0.541692	0.270846	−	0.962623i	$-0.412697\pi$
$227$	8.16141	0.541692	0.270846	+	0.962623i	$0.412697\pi$
$228$	−9.05046	−0.599382
$229$	6.77430	0.447658	0.223829	−	0.974628i	$-0.428144\pi$
$229$	6.77430	0.447658	0.223829	+	0.974628i	$0.428144\pi$
$230$	−12.5467	−0.827307
$231$	15.1262	0.995229
$232$	−14.8065	−0.972095
$233$	−14.7407	−0.965694	−0.482847	−	0.875705i	$-0.660397\pi$
$233$	−14.7407	−0.965694	−0.482847	+	0.875705i	$0.660397\pi$
$234$	−0.379262	−0.0247931
$235$	−19.9891	−1.30395
$236$	−48.0323	−3.12664
$237$	27.4283	1.78166
$238$	0.333679	0.0216292
$239$	7.45296	0.482092	0.241046	−	0.970514i	$-0.422510\pi$
$239$	7.45296	0.482092	0.241046	+	0.970514i	$0.422510\pi$
$240$	−69.1020	−4.46051
$241$	18.7329	1.20670	0.603348	−	0.797478i	$-0.293833\pi$
$241$	18.7329	1.20670	0.603348	+	0.797478i	$0.293833\pi$
$242$	20.3153	1.30592
$243$	1.47559	0.0946594
$244$	8.92834	0.571578
$245$	−56.6823	−3.62130
$246$	−49.9548	−3.18500
$247$	1.04515	0.0665011
$248$	−66.6347	−4.23131
$249$	−5.38734	−0.341408
$250$	14.6870	0.928890
$251$	24.9107	1.57235	0.786174	−	0.618005i	$-0.212059\pi$
$251$	24.9107	1.57235	0.786174	+	0.618005i	$0.212059\pi$
$252$	3.53873	0.222919
$253$	2.53904	0.159628
$254$	−30.8332	−1.93465
$255$	−0.148182	−0.00927951
$256$	4.97980	0.311238
$257$	−19.0125	−1.18596	−0.592982	−	0.805216i	$-0.702049\pi$
$257$	−19.0125	−1.18596	−0.592982	+	0.805216i	$0.702049\pi$
$258$	−12.4882	−0.777480
$259$	9.78122	0.607775
$260$	17.4573	1.08265
$261$	−0.252512	−0.0156301
$262$	−42.5856	−2.63095
$263$	−15.4653	−0.953629	−0.476814	−	0.879004i	$-0.658209\pi$
$263$	−15.4653	−0.953629	−0.476814	+	0.879004i	$0.658209\pi$
$264$	25.9284	1.59579
$265$	6.32163	0.388335
$266$	−13.5594	−0.831377
$267$	1.24256	0.0760437
$268$	−9.83092	−0.600519
$269$	−11.1799	−0.681649	−0.340824	−	0.940127i	$-0.610706\pi$
$269$	−11.1799	−0.681649	−0.340824	+	0.940127i	$0.610706\pi$
$270$	−48.3128	−2.94022
$271$	−23.3378	−1.41767	−0.708836	−	0.705373i	$-0.750779\pi$
$271$	−23.3378	−1.41767	−0.708836	+	0.705373i	$0.750779\pi$
$272$	0.308481	0.0187044
$273$	8.21811	0.497383
$274$	44.9616	2.71623
$275$	−12.1751	−0.734187
$276$	−11.9456	−0.719038
$277$	1.51410	0.0909737	0.0454868	−	0.998965i	$-0.485516\pi$
$277$	1.51410	0.0909737	0.0454868	+	0.998965i	$0.485516\pi$
$278$	6.50995	0.390441
$279$	−1.13640	−0.0680343
$280$	−138.055	−8.25036
$281$	17.8315	1.06374	0.531870	−	0.846826i	$-0.321490\pi$
$281$	17.8315	1.06374	0.531870	+	0.846826i	$0.321490\pi$
$282$	−26.4620	−1.57579
$283$	−9.31228	−0.553557	−0.276779	−	0.960934i	$-0.589267\pi$
$283$	−9.31228	−0.553557	−0.276779	+	0.960934i	$0.589267\pi$
$284$	−17.1760	−1.01921
$285$	6.02151	0.356684
$286$	−4.91212	−0.290460
$287$	−53.8260	−3.17725
$288$	2.18045	0.128484
$289$	−16.9993	−0.999961
$290$	16.1612	0.949019
$291$	27.3037	1.60057
$292$	−44.3821	−2.59727
$293$	−28.7794	−1.68131	−0.840654	−	0.541572i	$-0.817829\pi$
$293$	−28.7794	−1.68131	−0.840654	+	0.541572i	$0.817829\pi$
$294$	−75.0373	−4.37626
$295$	31.9572	1.86062
$296$	16.7664	0.974528
$297$	9.77690	0.567313
$298$	−22.8575	−1.32410
$299$	1.37947	0.0797769
$300$	57.2809	3.30711
$301$	−13.4559	−0.775586
$302$	7.94342	0.457092
$303$	−13.6685	−0.785236
$304$	−12.5354	−0.718956
$305$	−5.94026	−0.340138
$306$	0.00975451	0.000557628 0
$307$	21.0305	1.20028	0.600138	−	0.799896i	$-0.295112\pi$
$307$	21.0305	1.20028	0.600138	+	0.799896i	$0.295112\pi$
$308$	45.8329	2.61157
$309$	4.71295	0.268110
$310$	72.7314	4.13086
$311$	−2.59309	−0.147041	−0.0735204	−	0.997294i	$-0.523423\pi$
$311$	−2.59309	−0.147041	−0.0735204	+	0.997294i	$0.523423\pi$
$312$	14.0870	0.797521
$313$	3.38605	0.191391	0.0956953	−	0.995411i	$-0.469493\pi$
$313$	3.38605	0.191391	0.0956953	+	0.995411i	$0.469493\pi$
$314$	−18.1295	−1.02311
$315$	−2.35441	−0.132656
$316$	83.1089	4.67524
$317$	−4.36670	−0.245258	−0.122629	−	0.992453i	$-0.539133\pi$
$317$	−4.36670	−0.245258	−0.122629	+	0.992453i	$0.539133\pi$
$318$	8.36871	0.469294
$319$	−3.27049	−0.183112
$320$	−57.8007	−3.23116
$321$	−14.5550	−0.812384
$322$	−17.8968	−0.997347
$323$	−0.0268809	−0.00149569
$324$	−43.8141	−2.43411
$325$	−6.61479	−0.366923
$326$	−0.373766	−0.0207010
$327$	24.0092	1.32771
$328$	−92.2655	−5.09451
$329$	−28.5127	−1.57195
$330$	−28.3007	−1.55790
$331$	−5.07978	−0.279210	−0.139605	−	0.990207i	$-0.544583\pi$
$331$	−5.07978	−0.279210	−0.139605	+	0.990207i	$0.544583\pi$
$332$	−16.3238	−0.895887
$333$	0.285937	0.0156692
$334$	−24.3892	−1.33452
$335$	6.54077	0.357360
$336$	−98.5676	−5.37730
$337$	−17.1186	−0.932510	−0.466255	−	0.884650i	$-0.654397\pi$
$337$	−17.1186	−0.932510	−0.466255	+	0.884650i	$0.654397\pi$
$338$	−2.66878	−0.145162
$339$	−12.3206	−0.669165
$340$	−0.448997	−0.0243503
$341$	−14.7184	−0.797046
$342$	−0.396384	−0.0214340
$343$	−46.8233	−2.52822
$344$	−23.0654	−1.24360
$345$	7.94769	0.427889
$346$	−14.4426	−0.776440
$347$	3.09339	0.166062	0.0830310	−	0.996547i	$-0.473540\pi$
$347$	3.09339	0.166062	0.0830310	+	0.996547i	$0.473540\pi$
$348$	15.3868	0.824821
$349$	24.5635	1.31485	0.657426	−	0.753519i	$-0.271645\pi$
$349$	24.5635	1.31485	0.657426	+	0.753519i	$0.271645\pi$
$350$	85.8179	4.58716
$351$	5.31183	0.283525
$352$	28.2408	1.50524
$353$	−14.5968	−0.776909	−0.388455	−	0.921468i	$-0.626991\pi$
$353$	−14.5968	−0.776909	−0.388455	+	0.921468i	$0.626991\pi$
$354$	42.3056	2.24852
$355$	11.4277	0.606518
$356$	3.76502	0.199546
$357$	−0.211368	−0.0111868
$358$	50.3002	2.65845
$359$	24.6301	1.29993	0.649964	−	0.759965i	$-0.274784\pi$
$359$	24.6301	1.29993	0.649964	+	0.759965i	$0.274784\pi$
$360$	−4.03579	−0.212705
$361$	−17.9077	−0.942509
$362$	−12.0564	−0.633669
$363$	−12.8687	−0.675431
$364$	24.9012	1.30518
$365$	29.5286	1.54560
$366$	−7.86384	−0.411049
$367$	19.0709	0.995491	0.497746	−	0.867323i	$-0.334161\pi$
$367$	19.0709	0.995491	0.497746	+	0.867323i	$0.334161\pi$
$368$	−16.5453	−0.862483
$369$	−1.57351	−0.0819135
$370$	−18.3004	−0.951394
$371$	9.01723	0.468151
$372$	69.2464	3.59026
$373$	−14.1965	−0.735066	−0.367533	−	0.930011i	$-0.619797\pi$
$373$	−14.1965	−0.735066	−0.367533	+	0.930011i	$0.619797\pi$
$374$	0.126339	0.00653281
$375$	−9.30346	−0.480429
$376$	−48.8748	−2.52053
$377$	−1.77687	−0.0915135
$378$	−68.9138	−3.54454
$379$	−4.25697	−0.218666	−0.109333	−	0.994005i	$-0.534871\pi$
$379$	−4.25697	−0.218666	−0.109333	+	0.994005i	$0.534871\pi$
$380$	18.2454	0.935970
$381$	19.5312	1.00061
$382$	65.9913	3.37641
$383$	−25.7618	−1.31636	−0.658182	−	0.752858i	$-0.728674\pi$
$383$	−25.7618	−1.31636	−0.658182	+	0.752858i	$0.728674\pi$
$384$	−24.6410	−1.25746
$385$	−30.4938	−1.55411
$386$	10.2756	0.523017
$387$	−0.393360	−0.0199956
$388$	82.7313	4.20004
$389$	−32.4491	−1.64524	−0.822618	−	0.568594i	$-0.807487\pi$
$389$	−32.4491	−1.64524	−0.822618	+	0.568594i	$0.807487\pi$
$390$	−15.3759	−0.778589
$391$	−0.0354797	−0.00179428
$392$	−138.592	−6.99997
$393$	26.9757	1.36075
$394$	−24.4538	−1.23197
$395$	−55.2945	−2.78217
$396$	1.33984	0.0673297
$397$	27.2163	1.36595	0.682973	−	0.730444i	$-0.260687\pi$
$397$	27.2163	1.36595	0.682973	+	0.730444i	$0.260687\pi$
$398$	−58.9075	−2.95276
$399$	8.58913	0.429994
$400$	79.3374	3.96687
$401$	−11.5571	−0.577135	−0.288567	−	0.957460i	$-0.593179\pi$
$401$	−11.5571	−0.577135	−0.288567	+	0.957460i	$0.593179\pi$
$402$	8.65881	0.431862
$403$	−7.99657	−0.398338
$404$	−41.4162	−2.06053
$405$	29.1506	1.44851
$406$	23.0525	1.14408
$407$	3.70340	0.183571
$408$	−0.362315	−0.0179373
$409$	−6.81384	−0.336923	−0.168461	−	0.985708i	$-0.553880\pi$
$409$	−6.81384	−0.336923	−0.168461	+	0.985708i	$0.553880\pi$
$410$	100.707	4.97357
$411$	−28.4808	−1.40486
$412$	14.2804	0.703546
$413$	45.5840	2.24304
$414$	−0.523180	−0.0257129
$415$	10.8607	0.533130
$416$	15.3433	0.752269
$417$	−4.12371	−0.201939
$418$	−5.13389	−0.251107
$419$	8.69779	0.424915	0.212458	−	0.977170i	$-0.431853\pi$
$419$	8.69779	0.424915	0.212458	+	0.977170i	$0.431853\pi$
$420$	143.466	7.00042
$421$	−9.50802	−0.463392	−0.231696	−	0.972788i	$-0.574428\pi$
$421$	−9.50802	−0.463392	−0.231696	+	0.972788i	$0.574428\pi$
$422$	49.8020	2.42432
$423$	−0.833518	−0.0405270
$424$	15.4568	0.750650
$425$	0.170131	0.00825256
$426$	15.1282	0.732963
$427$	−8.47323	−0.410048
$428$	−44.1024	−2.13177
$429$	3.11157	0.150228
$430$	25.1757	1.21408
$431$	−21.4137	−1.03146	−0.515731	−	0.856750i	$-0.672480\pi$
$431$	−21.4137	−1.03146	−0.515731	+	0.856750i	$0.672480\pi$
$432$	−63.7098	−3.06524
$433$	7.85813	0.377638	0.188819	−	0.982012i	$-0.439534\pi$
$433$	7.85813	0.377638	0.188819	+	0.982012i	$0.439534\pi$
$434$	103.745	4.97990
$435$	−10.2373	−0.490840
$436$	72.7487	3.48403
$437$	1.44175	0.0689682
$438$	39.0906	1.86782
$439$	30.7885	1.46945	0.734727	−	0.678363i	$-0.237310\pi$
$439$	30.7885	1.46945	0.734727	+	0.678363i	$0.237310\pi$
$440$	−52.2708	−2.49191
$441$	−2.36357	−0.112551
$442$	0.0686403	0.00326489
$443$	37.0743	1.76145	0.880725	−	0.473627i	$-0.157056\pi$
$443$	37.0743	1.76145	0.880725	+	0.473627i	$0.157056\pi$
$444$	−17.4236	−0.826886
$445$	−2.50497	−0.118747
$446$	−9.00949	−0.426612
$447$	14.4790	0.684833
$448$	−82.4473	−3.89527
$449$	15.2284	0.718671	0.359335	−	0.933209i	$-0.383003\pi$
$449$	15.2284	0.718671	0.359335	+	0.933209i	$0.383003\pi$
$450$	2.50874	0.118263
$451$	−20.3798	−0.959646
$452$	−37.3320	−1.75595
$453$	−5.03174	−0.236412
$454$	−21.7810	−1.02223
$455$	−16.5674	−0.776693
$456$	14.7230	0.689468
$457$	−1.97519	−0.0923955	−0.0461977	−	0.998932i	$-0.514710\pi$
$457$	−1.97519	−0.0923955	−0.0461977	+	0.998932i	$0.514710\pi$
$458$	−18.0791	−0.844780
$459$	−0.136619	−0.00637683
$460$	24.0818	1.12282
$461$	5.09076	0.237100	0.118550	−	0.992948i	$-0.462175\pi$
$461$	5.09076	0.237100	0.118550	+	0.992948i	$0.462175\pi$
$462$	−40.3684	−1.87811
$463$	−10.3928	−0.482992	−0.241496	−	0.970402i	$-0.577638\pi$
$463$	−10.3928	−0.482992	−0.241496	+	0.970402i	$0.577638\pi$
$464$	21.3117	0.989370
$465$	−46.0715	−2.13651
$466$	39.3396	1.82237
$467$	27.2112	1.25918	0.629592	−	0.776926i	$-0.283222\pi$
$467$	27.2112	1.25918	0.629592	+	0.776926i	$0.283222\pi$
$468$	0.727943	0.0336492
$469$	9.32980	0.430810
$470$	53.3466	2.46069
$471$	11.4841	0.529158
$472$	78.1375	3.59657
$473$	−5.09473	−0.234256
$474$	−73.2000	−3.36219
$475$	−6.91342	−0.317210
$476$	−0.640453	−0.0293551
$477$	0.263603	0.0120695
$478$	−19.8903	−0.909760
$479$	−25.6812	−1.17340	−0.586702	−	0.809803i	$-0.699574\pi$
$479$	−25.6812	−1.17340	−0.586702	+	0.809803i	$0.699574\pi$
$480$	88.3992	4.03485
$481$	2.01207	0.0917426
$482$	−49.9941	−2.27717
$483$	11.3367	0.515836
$484$	−38.9927	−1.77239
$485$	−55.0433	−2.49939
$486$	−3.93803	−0.178633
$487$	−14.5670	−0.660094	−0.330047	−	0.943964i	$-0.607065\pi$
$487$	−14.5670	−0.660094	−0.330047	+	0.943964i	$0.607065\pi$
$488$	−14.5243	−0.657486
$489$	0.236761	0.0107067
$490$	151.273	6.83380
$491$	12.2811	0.554240	0.277120	−	0.960835i	$-0.410620\pi$
$491$	12.2811	0.554240	0.277120	+	0.960835i	$0.410620\pi$
$492$	95.8818	4.32268
$493$	0.0457007	0.00205826
$494$	−2.78926	−0.125495
$495$	−0.891434	−0.0400670
$496$	95.9104	4.30650
$497$	16.3005	0.731178
$498$	14.3776	0.644275
$499$	−39.3886	−1.76328	−0.881639	−	0.471925i	$-0.843559\pi$
$499$	−39.3886	−1.76328	−0.881639	+	0.471925i	$0.843559\pi$
$500$	−28.1899	−1.26069
$501$	15.4492	0.690221
$502$	−66.4810	−2.96719
$503$	−19.3187	−0.861378	−0.430689	−	0.902500i	$-0.641729\pi$
$503$	−19.3187	−0.861378	−0.430689	+	0.902500i	$0.641729\pi$
$504$	−5.75669	−0.256423
$505$	27.5553	1.22619
$506$	−6.77613	−0.301236
$507$	1.69053	0.0750790
$508$	59.1803	2.62570
$509$	−33.5317	−1.48627	−0.743133	−	0.669143i	$-0.766661\pi$
$509$	−33.5317	−1.48627	−0.743133	+	0.669143i	$0.766661\pi$
$510$	0.395465	0.0175115
$511$	42.1198	1.86327
$512$	15.8619	0.701002
$513$	5.55164	0.245111
$514$	50.7400	2.23805
$515$	−9.50114	−0.418670
$516$	23.9694	1.05520
$517$	−10.7956	−0.474788
$518$	−26.1039	−1.14694
$519$	9.14864	0.401581
$520$	−28.3990	−1.24538
$521$	12.4033	0.543397	0.271699	−	0.962382i	$-0.412415\pi$
$521$	12.4033	0.543397	0.271699	+	0.962382i	$0.412415\pi$
$522$	0.673899	0.0294957
$523$	41.2618	1.80425	0.902126	−	0.431473i	$-0.142006\pi$
$523$	41.2618	1.80425	0.902126	+	0.431473i	$0.142006\pi$
$524$	81.7375	3.57072
$525$	−54.3611	−2.37251
$526$	41.2733	1.79960
$527$	0.205670	0.00895912
$528$	−37.3200	−1.62414
$529$	−21.0971	−0.917263
$530$	−16.8710	−0.732831
$531$	1.33257	0.0578285
$532$	26.0254	1.12834
$533$	−11.0724	−0.479600
$534$	−3.31613	−0.143503
$535$	29.3425	1.26859
$536$	15.9926	0.690776
$537$	−31.8625	−1.37497
$538$	29.8366	1.28635
$539$	−30.6125	−1.31857
$540$	92.7301	3.99047
$541$	19.2532	0.827760	0.413880	−	0.910331i	$-0.364173\pi$
$541$	19.2532	0.827760	0.413880	+	0.910331i	$0.364173\pi$
$542$	62.2835	2.67530
$543$	7.63708	0.327738
$544$	−0.394627	−0.0169195
$545$	−48.4016	−2.07330
$546$	−21.9323	−0.938616
$547$	−0.410457	−0.0175499	−0.00877494	−	0.999961i	$-0.502793\pi$
$547$	−0.410457	−0.0175499	−0.00877494	+	0.999961i	$0.502793\pi$
$548$	−86.2980	−3.68647
$549$	−0.247700	−0.0105716
$550$	32.4927	1.38549
$551$	−1.85709	−0.0791147
$552$	19.4327	0.827109
$553$	−78.8725	−3.35400
$554$	−4.04081	−0.171677
$555$	11.5924	0.492068
$556$	−12.4950	−0.529906
$557$	−8.87469	−0.376033	−0.188016	−	0.982166i	$-0.560206\pi$
$557$	−8.87469	−0.376033	−0.188016	+	0.982166i	$0.560206\pi$
$558$	3.03279	0.128388
$559$	−2.76799	−0.117073
$560$	198.709	8.39698
$561$	−0.0800288	−0.00337882
$562$	−47.5883	−2.00739
$563$	13.2870	0.559982	0.279991	−	0.960003i	$-0.409669\pi$
$563$	13.2870	0.559982	0.279991	+	0.960003i	$0.409669\pi$
$564$	50.7905	2.13866
$565$	24.8380	1.04494
$566$	24.8524	1.04462
$567$	41.5807	1.74623
$568$	27.9414	1.17240
$569$	32.4651	1.36101	0.680504	−	0.732745i	$-0.261761\pi$
$569$	32.4651	1.36101	0.680504	+	0.732745i	$0.261761\pi$
$570$	−16.0701	−0.673101
$571$	−32.2064	−1.34780	−0.673898	−	0.738824i	$-0.735381\pi$
$571$	−32.2064	−1.34780	−0.673898	+	0.738824i	$0.735381\pi$
$572$	9.42818	0.394212
$573$	−41.8020	−1.74630
$574$	143.649	5.99581
$575$	−9.12491	−0.380535
$576$	−2.41020	−0.100425
$577$	−31.5147	−1.31197	−0.655987	−	0.754772i	$-0.727748\pi$
$577$	−31.5147	−1.31197	−0.655987	+	0.754772i	$0.727748\pi$
$578$	45.3674	1.88704
$579$	−6.50908	−0.270508
$580$	−31.0193	−1.28801
$581$	15.4918	0.642706
$582$	−72.8675	−3.02045
$583$	3.41413	0.141399
$584$	72.1994	2.98763
$585$	−0.484320	−0.0200242
$586$	76.8057	3.17281
$587$	45.3777	1.87294	0.936468	−	0.350753i	$-0.114074\pi$
$587$	45.3777	1.87294	0.936468	+	0.350753i	$0.114074\pi$
$588$	144.024	5.93946
$589$	−8.35759	−0.344368
$590$	−85.2866	−3.51119
$591$	15.4902	0.637182
$592$	−24.1327	−0.991847
$593$	−35.9849	−1.47772	−0.738862	−	0.673857i	$-0.764636\pi$
$593$	−35.9849	−1.47772	−0.738862	+	0.673857i	$0.764636\pi$
$594$	−26.0924	−1.07058
$595$	0.426110	0.0174688
$596$	43.8719	1.79706
$597$	37.3148	1.52719
$598$	−3.68150	−0.150548
$599$	−8.42382	−0.344188	−0.172094	−	0.985081i	$-0.555053\pi$
$599$	−8.42382	−0.344188	−0.172094	+	0.985081i	$0.555053\pi$
$600$	−93.1828	−3.80417
$601$	−27.3258	−1.11464	−0.557322	−	0.830297i	$-0.688171\pi$
$601$	−27.3258	−1.11464	−0.557322	+	0.830297i	$0.688171\pi$
$602$	35.9108	1.46362
$603$	0.272740	0.0111068
$604$	−15.2464	−0.620366
$605$	25.9428	1.05473
$606$	36.4782	1.48183
$607$	−2.49617	−0.101316	−0.0506582	−	0.998716i	$-0.516132\pi$
$607$	−2.49617	−0.101316	−0.0506582	+	0.998716i	$0.516132\pi$
$608$	16.0360	0.650347
$609$	−14.6025	−0.591724
$610$	15.8532	0.641878
$611$	−5.86528	−0.237284
$612$	−0.0187225	−0.000756813 0
$613$	2.87964	0.116308	0.0581538	−	0.998308i	$-0.481479\pi$
$613$	2.87964	0.116308	0.0581538	+	0.998308i	$0.481479\pi$
$614$	−56.1258	−2.26505
$615$	−63.7927	−2.57237
$616$	−74.5595	−3.00409
$617$	−6.86878	−0.276527	−0.138263	−	0.990395i	$-0.544152\pi$
$617$	−6.86878	−0.276527	−0.138263	+	0.990395i	$0.544152\pi$
$618$	−12.5778	−0.505954
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−139.598	−5.60641
$621$	7.32752	0.294043
$622$	6.92039	0.277482
$623$	−3.57310	−0.143153
$624$	−20.2761	−0.811694
$625$	−14.3185	−0.572740
$626$	−9.03660	−0.361175
$627$	3.25205	0.129874
$628$	34.7972	1.38856
$629$	−0.0517500	−0.00206341
$630$	6.28339	0.250336
$631$	23.6478	0.941402	0.470701	−	0.882293i	$-0.344001\pi$
$631$	23.6478	0.941402	0.470701	+	0.882293i	$0.344001\pi$
$632$	−135.199	−5.37792
$633$	−31.5469	−1.25388
$634$	11.6537	0.462829
$635$	−39.3742	−1.56252
$636$	−16.0627	−0.636926
$637$	−16.6319	−0.658980
$638$	8.72821	0.345553
$639$	0.476517	0.0188507
$640$	49.6755	1.96360
$641$	33.9856	1.34235	0.671176	−	0.741298i	$-0.265790\pi$
$641$	33.9856	1.34235	0.671176	+	0.741298i	$0.265790\pi$
$642$	38.8442	1.53306
$643$	23.8857	0.941959	0.470980	−	0.882144i	$-0.343901\pi$
$643$	23.8857	0.941959	0.470980	+	0.882144i	$0.343901\pi$
$644$	34.3505	1.35360
$645$	−15.9475	−0.627932
$646$	0.0717392	0.00282254
$647$	−27.4466	−1.07904	−0.539519	−	0.841973i	$-0.681394\pi$
$647$	−27.4466	−1.07904	−0.539519	+	0.841973i	$0.681394\pi$
$648$	71.2753	2.79996
$649$	17.2592	0.677481
$650$	17.6534	0.692423
$651$	−65.7167	−2.57564
$652$	0.717395	0.0280953
$653$	−11.3511	−0.444204	−0.222102	−	0.975023i	$-0.571292\pi$
$653$	−11.3511	−0.444204	−0.222102	+	0.975023i	$0.571292\pi$
$654$	−64.0751	−2.50553
$655$	−54.3821	−2.12489
$656$	132.802	5.18504
$657$	1.23130	0.0480375
$658$	76.0940	2.96645
$659$	37.6404	1.46626	0.733131	−	0.680088i	$-0.238058\pi$
$659$	37.6404	1.46626	0.733131	+	0.680088i	$0.238058\pi$
$660$	54.3196	2.11439
$661$	28.0931	1.09270	0.546348	−	0.837558i	$-0.316018\pi$
$661$	28.0931	1.09270	0.546348	+	0.837558i	$0.316018\pi$
$662$	13.5568	0.526900
$663$	−0.0434800	−0.00168862
$664$	26.5551	1.03054
$665$	−17.3154	−0.671462
$666$	−0.763101	−0.0295696
$667$	−2.45114	−0.0949086
$668$	46.8118	1.81120
$669$	5.70704	0.220647
$670$	−17.4558	−0.674378
$671$	−3.20816	−0.123850
$672$	126.093	4.86415
$673$	20.8532	0.803832	0.401916	−	0.915677i	$-0.368344\pi$
$673$	20.8532	0.803832	0.401916	+	0.915677i	$0.368344\pi$
$674$	45.6858	1.75975
$675$	−35.1366	−1.35241
$676$	5.12237	0.197014
$677$	29.4801	1.13301	0.566507	−	0.824057i	$-0.308294\pi$
$677$	29.4801	1.13301	0.566507	+	0.824057i	$0.308294\pi$
$678$	32.8810	1.26279
$679$	−78.5142	−3.01310
$680$	0.730414	0.0280101
$681$	13.7971	0.528706
$682$	39.2801	1.50411
$683$	−30.2576	−1.15777	−0.578887	−	0.815408i	$-0.696513\pi$
$683$	−30.2576	−1.15777	−0.578887	+	0.815408i	$0.696513\pi$
$684$	0.760807	0.0290902
$685$	57.4164	2.19377
$686$	124.961	4.77103
$687$	11.4521	0.436927
$688$	33.1991	1.26570
$689$	1.85491	0.0706666
$690$	−21.2106	−0.807475
$691$	−39.9106	−1.51827	−0.759136	−	0.650932i	$-0.774378\pi$
$691$	−39.9106	−1.51827	−0.759136	+	0.650932i	$0.774378\pi$
$692$	27.7207	1.05378
$693$	−1.27155	−0.0483021
$694$	−8.25558	−0.313377
$695$	8.31325	0.315340
$696$	−25.0308	−0.948791
$697$	0.284780	0.0107868
$698$	−65.5544	−2.48127
$699$	−24.9196	−0.942544
$700$	−164.716	−6.22569
$701$	50.8642	1.92111	0.960556	−	0.278086i	$-0.0896999\pi$
$701$	50.8642	1.92111	0.960556	+	0.278086i	$0.0896999\pi$
$702$	−14.1761	−0.535042
$703$	2.10291	0.0793127
$704$	−31.2165	−1.17652
$705$	−33.7922	−1.27269
$706$	38.9556	1.46611
$707$	39.3050	1.47822
$708$	−81.2001	−3.05169
$709$	39.4315	1.48088	0.740441	−	0.672122i	$-0.234617\pi$
$709$	39.4315	1.48088	0.740441	+	0.672122i	$0.234617\pi$
$710$	−30.4979	−1.14457
$711$	−2.30570	−0.0864705
$712$	−6.12482	−0.229537
$713$	−11.0310	−0.413116
$714$	0.564094	0.0211107
$715$	−6.27282	−0.234590
$716$	−96.5448	−3.60805
$717$	12.5994	0.470535
$718$	−65.7323	−2.45311
$719$	18.5524	0.691887	0.345943	−	0.938255i	$-0.387559\pi$
$719$	18.5524	0.691887	0.345943	+	0.938255i	$0.387559\pi$
$720$	5.80890	0.216485
$721$	−13.5525	−0.504721
$722$	47.7916	1.77862
$723$	31.6686	1.17777
$724$	23.1406	0.860015
$725$	11.7536	0.436519
$726$	34.3437	1.27461
$727$	−39.0783	−1.44933	−0.724666	−	0.689100i	$-0.758006\pi$
$727$	−39.0783	−1.44933	−0.724666	+	0.689100i	$0.758006\pi$
$728$	−40.5085	−1.50135
$729$	28.1550	1.04278
$730$	−78.8052	−2.91671
$731$	0.0711919	0.00263313
$732$	15.0936	0.557876
$733$	15.9098	0.587642	0.293821	−	0.955861i	$-0.405073\pi$
$733$	15.9098	0.587642	0.293821	+	0.955861i	$0.405073\pi$
$734$	−50.8959	−1.87860
$735$	−95.8232	−3.53449
$736$	21.1657	0.780178
$737$	3.53248	0.130121
$738$	4.19934	0.154580
$739$	−25.5626	−0.940335	−0.470168	−	0.882577i	$-0.655807\pi$
$739$	−25.5626	−0.940335	−0.470168	+	0.882577i	$0.655807\pi$
$740$	35.1253	1.29123
$741$	1.76685	0.0649069
$742$	−24.0650	−0.883453
$743$	47.4337	1.74017	0.870087	−	0.492898i	$-0.164062\pi$
$743$	47.4337	1.74017	0.870087	+	0.492898i	$0.164062\pi$
$744$	−112.648	−4.12987
$745$	−29.1892	−1.06941
$746$	37.8872	1.38715
$747$	0.452874	0.0165698
$748$	−0.242491	−0.00886633
$749$	41.8543	1.52932
$750$	24.8289	0.906622
$751$	−11.2068	−0.408942	−0.204471	−	0.978873i	$-0.565547\pi$
$751$	−11.2068	−0.408942	−0.204471	+	0.978873i	$0.565547\pi$
$752$	70.3478	2.56532
$753$	42.1122	1.53465
$754$	4.74207	0.172696
$755$	10.1438	0.369171
$756$	132.271	4.81065
$757$	−31.6885	−1.15174	−0.575869	−	0.817542i	$-0.695336\pi$
$757$	−31.6885	−1.15174	−0.575869	+	0.817542i	$0.695336\pi$
$758$	11.3609	0.412647
$759$	4.29232	0.155801
$760$	−29.6811	−1.07665
$761$	8.37015	0.303418	0.151709	−	0.988425i	$-0.451522\pi$
$761$	8.37015	0.303418	0.151709	+	0.988425i	$0.451522\pi$
$762$	−52.1245	−1.88827
$763$	−69.0405	−2.49943
$764$	−126.662	−4.58246
$765$	0.0124566	0.000450369 0
$766$	68.7524	2.48413
$767$	9.37698	0.338583
$768$	8.41851	0.303777
$769$	−16.9755	−0.612152	−0.306076	−	0.952007i	$-0.599016\pi$
$769$	−16.9755	−0.612152	−0.306076	+	0.952007i	$0.599016\pi$
$770$	81.3813	2.93278
$771$	−32.1411	−1.15753
$772$	−19.7228	−0.709838
$773$	42.4878	1.52818	0.764089	−	0.645110i	$-0.223189\pi$
$773$	42.4878	1.52818	0.764089	+	0.645110i	$0.223189\pi$
$774$	1.04979	0.0377339
$775$	52.8956	1.90007
$776$	−134.585	−4.83131
$777$	16.5354	0.593205
$778$	86.5995	3.10474
$779$	−11.5723	−0.414621
$780$	29.5121	1.05670
$781$	6.17176	0.220843
$782$	0.0946873	0.00338601
$783$	−9.43844	−0.337302
$784$	199.482	7.12436
$785$	−23.1515	−0.826313
$786$	−71.9922	−2.56788
$787$	−2.09400	−0.0746432	−0.0373216	−	0.999303i	$-0.511883\pi$
$787$	−2.09400	−0.0746432	−0.0373216	+	0.999303i	$0.511883\pi$
$788$	46.9359	1.67202
$789$	−26.1445	−0.930768
$790$	147.569	5.25026
$791$	35.4291	1.25971
$792$	−2.17962	−0.0774493
$793$	−1.74301	−0.0618961
$794$	−72.6341	−2.57769
$795$	10.6869	0.379026
$796$	113.065	4.00749
$797$	29.2471	1.03599	0.517993	−	0.855385i	$-0.326680\pi$
$797$	29.2471	1.03599	0.517993	+	0.855385i	$0.326680\pi$
$798$	−22.9225	−0.811447
$799$	0.150853	0.00533681
$800$	−101.493	−3.58832
$801$	−0.104453	−0.00369068
$802$	30.8434	1.08912
$803$	15.9475	0.562777
$804$	−16.6195	−0.586123
$805$	−22.8543	−0.805508
$806$	21.3411	0.751707
$807$	−18.8999	−0.665308
$808$	67.3745	2.37023
$809$	−28.2173	−0.992068	−0.496034	−	0.868303i	$-0.665211\pi$
$809$	−28.2173	−0.992068	−0.496034	+	0.868303i	$0.665211\pi$
$810$	−77.7966	−2.73349
$811$	7.31996	0.257038	0.128519	−	0.991707i	$-0.458978\pi$
$811$	7.31996	0.257038	0.128519	+	0.991707i	$0.458978\pi$
$812$	−44.2462	−1.55274
$813$	−39.4533	−1.38369
$814$	−9.88354	−0.346418
$815$	−0.477302	−0.0167191
$816$	0.521497	0.0182560
$817$	−2.89295	−0.101212
$818$	18.1846	0.635810
$819$	−0.690838	−0.0241398
$820$	−193.294	−6.75013
$821$	32.1611	1.12243	0.561215	−	0.827670i	$-0.310334\pi$
$821$	32.1611	1.12243	0.561215	+	0.827670i	$0.310334\pi$
$822$	76.0090	2.65112
$823$	28.7592	1.00248	0.501242	−	0.865307i	$-0.332877\pi$
$823$	28.7592	1.00248	0.501242	+	0.865307i	$0.332877\pi$
$824$	−23.2309	−0.809288
$825$	−20.5824	−0.716587
$826$	−121.653	−4.23287
$827$	32.0192	1.11342	0.556708	−	0.830708i	$-0.312064\pi$
$827$	32.0192	1.11342	0.556708	+	0.830708i	$0.312064\pi$
$828$	1.00418	0.0348976
$829$	−20.8180	−0.723040	−0.361520	−	0.932364i	$-0.617742\pi$
$829$	−20.8180	−0.723040	−0.361520	+	0.932364i	$0.617742\pi$
$830$	−28.9847	−1.00607
$831$	2.55964	0.0887928
$832$	−16.9601	−0.587984
$833$	0.427769	0.0148213
$834$	11.0053	0.381081
$835$	−31.1451	−1.07782
$836$	9.85383	0.340802
$837$	−42.4764	−1.46820
$838$	−23.2125	−0.801862
$839$	−13.4258	−0.463510	−0.231755	−	0.972774i	$-0.574447\pi$
$839$	−13.4258	−0.463510	−0.231755	+	0.972774i	$0.574447\pi$
$840$	−233.386	−8.05258
$841$	−25.8427	−0.891129
$842$	25.3748	0.874473
$843$	30.1447	1.03824
$844$	−95.5884	−3.29029
$845$	−3.40805	−0.117240
$846$	2.22447	0.0764790
$847$	37.0051	1.27151
$848$	−22.2477	−0.763990
$849$	−15.7427	−0.540287
$850$	−0.454041	−0.0155735
$851$	2.77560	0.0951462
$852$	−29.0366	−0.994777
$853$	11.0718	0.379091	0.189546	−	0.981872i	$-0.439299\pi$
$853$	11.0718	0.379091	0.189546	+	0.981872i	$0.439299\pi$
$854$	22.6132	0.773806
$855$	−0.506185	−0.0173112
$856$	71.7443	2.45217
$857$	−54.9925	−1.87851	−0.939255	−	0.343221i	$-0.888482\pi$
$857$	−54.9925	−1.87851	−0.939255	+	0.343221i	$0.888482\pi$
$858$	−8.30409	−0.283497
$859$	−9.10042	−0.310502	−0.155251	−	0.987875i	$-0.549619\pi$
$859$	−9.10042	−0.310502	−0.155251	+	0.987875i	$0.549619\pi$
$860$	−48.3215	−1.64775
$861$	−90.9944	−3.10108
$862$	57.1484	1.94648
$863$	−17.1627	−0.584224	−0.292112	−	0.956384i	$-0.594358\pi$
$863$	−17.1627	−0.584224	−0.292112	+	0.956384i	$0.594358\pi$
$864$	81.5012	2.77273
$865$	−18.4433	−0.627092
$866$	−20.9716	−0.712644
$867$	−28.7379	−0.975990
$868$	−199.124	−6.75872
$869$	−29.8630	−1.01303
$870$	27.3210	0.926269
$871$	1.91921	0.0650300
$872$	−118.345	−4.00768
$873$	−2.29522	−0.0776816
$874$	−3.84771	−0.130151
$875$	26.7529	0.904414
$876$	−75.0293	−2.53500
$877$	−20.8154	−0.702885	−0.351443	−	0.936209i	$-0.614309\pi$
$877$	−20.8154	−0.702885	−0.351443	+	0.936209i	$0.614309\pi$
$878$	−82.1676	−2.77302
$879$	−48.6524	−1.64100
$880$	75.2358	2.53620
$881$	−26.3376	−0.887338	−0.443669	−	0.896191i	$-0.646323\pi$
$881$	−26.3376	−0.887338	−0.443669	+	0.896191i	$0.646323\pi$
$882$	6.30784	0.212396
$883$	26.8259	0.902765	0.451382	−	0.892331i	$-0.350931\pi$
$883$	26.8259	0.902765	0.451382	+	0.892331i	$0.350931\pi$
$884$	−0.131746	−0.00443110
$885$	54.0246	1.81602
$886$	−98.9429	−3.32405
$887$	38.2342	1.28378	0.641890	−	0.766797i	$-0.278151\pi$
$887$	38.2342	1.28378	0.641890	+	0.766797i	$0.278151\pi$
$888$	28.3441	0.951166
$889$	−56.1637	−1.88367
$890$	6.68520	0.224088
$891$	15.7434	0.527425
$892$	17.2925	0.578997
$893$	−6.13007	−0.205135
$894$	−38.6412	−1.29236
$895$	64.2338	2.14710
$896$	70.8575	2.36718
$897$	2.33204	0.0778644
$898$	−40.6411	−1.35621
$899$	14.2089	0.473892
$900$	−4.81519	−0.160506
$901$	−0.0477079	−0.00158938
$902$	54.3891	1.81096
$903$	−22.7476	−0.756994
$904$	60.7306	2.01987
$905$	−15.3961	−0.511783
$906$	13.4286	0.446135
$907$	−16.0249	−0.532099	−0.266049	−	0.963959i	$-0.585718\pi$
$907$	−16.0249	−0.532099	−0.266049	+	0.963959i	$0.585718\pi$
$908$	41.8058	1.38737
$909$	1.14901	0.0381104
$910$	44.2148	1.46571
$911$	−2.39834	−0.0794606	−0.0397303	−	0.999210i	$-0.512650\pi$
$911$	−2.39834	−0.0794606	−0.0397303	+	0.999210i	$0.512650\pi$
$912$	−21.1915	−0.701721
$913$	5.86554	0.194121
$914$	5.27134	0.174360
$915$	−10.0422	−0.331984
$916$	34.7005	1.14654
$917$	−77.5711	−2.56162
$918$	0.364606	0.0120338
$919$	29.6596	0.978380	0.489190	−	0.872177i	$-0.337292\pi$
$919$	29.6596	0.978380	0.489190	+	0.872177i	$0.337292\pi$
$920$	−39.1756	−1.29158
$921$	35.5528	1.17150
$922$	−13.5861	−0.447434
$923$	3.35314	0.110370
$924$	77.4819	2.54897
$925$	−13.3094	−0.437611
$926$	27.7360	0.911460
$927$	−0.396184	−0.0130124
$928$	−27.2631	−0.894956
$929$	−51.7433	−1.69764	−0.848821	−	0.528681i	$-0.822687\pi$
$929$	−51.7433	−1.69764	−0.848821	+	0.528681i	$0.822687\pi$
$930$	122.954	4.03184
$931$	−17.3828	−0.569698
$932$	−75.5072	−2.47332
$933$	−4.38370	−0.143516
$934$	−72.6206	−2.37622
$935$	0.161335	0.00527623
$936$	−1.18420	−0.0387066
$937$	32.6778	1.06754	0.533769	−	0.845630i	$-0.320775\pi$
$937$	32.6778	1.06754	0.533769	+	0.845630i	$0.320775\pi$
$938$	−24.8992	−0.812986
$939$	5.72421	0.186803
$940$	−102.392	−3.33965
$941$	−7.00942	−0.228501	−0.114250	−	0.993452i	$-0.536447\pi$
$941$	−7.00942	−0.228501	−0.114250	+	0.993452i	$0.536447\pi$
$942$	−30.6485	−0.998580
$943$	−15.2741	−0.497393
$944$	−112.467	−3.66049
$945$	−88.0034	−2.86275
$946$	13.5967	0.442067
$947$	−6.89226	−0.223968	−0.111984	−	0.993710i	$-0.535721\pi$
$947$	−6.89226	−0.223968	−0.111984	+	0.993710i	$0.535721\pi$
$948$	140.498	4.56316
$949$	8.66437	0.281257
$950$	18.4504	0.598610
$951$	−7.38203	−0.239379
$952$	1.04187	0.0337672
$953$	−8.45736	−0.273961	−0.136980	−	0.990574i	$-0.543740\pi$
$953$	−8.45736	−0.273961	−0.136980	+	0.990574i	$0.543740\pi$
$954$	−0.703497	−0.0227766
$955$	84.2714	2.72696
$956$	38.1768	1.23473
$957$	−5.52886	−0.178723
$958$	68.5374	2.21434
$959$	81.8991	2.64466
$960$	−97.7137	−3.15370
$961$	32.9451	1.06275
$962$	−5.36977	−0.173128
$963$	1.22354	0.0394280
$964$	95.9571	3.09057
$965$	13.1221	0.422415
$966$	−30.2550	−0.973439
$967$	−13.0606	−0.420002	−0.210001	−	0.977701i	$-0.567347\pi$
$967$	−13.0606	−0.420002	−0.210001	+	0.977701i	$0.567347\pi$
$968$	63.4320	2.03878
$969$	−0.0454430	−0.00145984
$970$	146.898	4.71662
$971$	14.7911	0.474669	0.237334	−	0.971428i	$-0.423726\pi$
$971$	14.7911	0.474669	0.237334	+	0.971428i	$0.423726\pi$
$972$	7.55854	0.242440
$973$	11.8581	0.380153
$974$	38.8761	1.24567
$975$	−11.1825	−0.358127
$976$	20.9055	0.669170
$977$	−34.6062	−1.10715	−0.553575	−	0.832799i	$-0.686737\pi$
$977$	−34.6062	−1.10715	−0.553575	+	0.832799i	$0.686737\pi$
$978$	−0.631862	−0.0202047
$979$	−1.35286	−0.0432376
$980$	−290.348	−9.27483
$981$	−2.01828	−0.0644386
$982$	−32.7756	−1.04591
$983$	−38.2859	−1.22113	−0.610566	−	0.791965i	$-0.709058\pi$
$983$	−38.2859	−1.22113	−0.610566	+	0.791965i	$0.709058\pi$
$984$	−155.977	−4.97238
$985$	−31.2277	−0.994997
$986$	−0.121965	−0.00388415
$987$	−48.2015	−1.53427
$988$	5.35363	0.170322
$989$	−3.81836	−0.121417
$990$	2.37904	0.0756108
$991$	−39.9217	−1.26816	−0.634078	−	0.773269i	$-0.718620\pi$
$991$	−39.9217	−1.26816	−0.634078	+	0.773269i	$0.718620\pi$
$992$	−122.694	−3.89554
$993$	−8.58752	−0.272517
$994$	−43.5024	−1.37981
$995$	−75.2253	−2.38480
$996$	−27.5959	−0.874410
$997$	−37.1830	−1.17760	−0.588798	−	0.808280i	$-0.700399\pi$
$997$	−37.1830	−1.17760	−0.588798	+	0.808280i	$0.700399\pi$
$998$	105.120	3.32750
$999$	10.6878	0.338146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.c.1.7	✓	151

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.c.1.7	✓	151	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66878 q^{2} +1.69053 q^{3} +5.12237 q^{4} -3.40805 q^{5} -4.51165 q^{6} -4.86127 q^{7} -8.33291 q^{8} -0.142111 q^{9} +O(q^{10})\)	\(q-2.66878 q^{2} +1.69053 q^{3} +5.12237 q^{4} -3.40805 q^{5} -4.51165 q^{6} -4.86127 q^{7} -8.33291 q^{8} -0.142111 q^{9} +9.09532 q^{10} -1.84059 q^{11} +8.65952 q^{12} -1.00000 q^{13} +12.9736 q^{14} -5.76140 q^{15} +11.9939 q^{16} +0.0257198 q^{17} +0.379262 q^{18} -1.04515 q^{19} -17.4573 q^{20} -8.21811 q^{21} +4.91212 q^{22} -1.37947 q^{23} -14.0870 q^{24} +6.61479 q^{25} +2.66878 q^{26} -5.31183 q^{27} -24.9012 q^{28} +1.77687 q^{29} +15.3759 q^{30} +7.99657 q^{31} -15.3433 q^{32} -3.11157 q^{33} -0.0686403 q^{34} +16.5674 q^{35} -0.727943 q^{36} -2.01207 q^{37} +2.78926 q^{38} -1.69053 q^{39} +28.3990 q^{40} +11.0724 q^{41} +21.9323 q^{42} +2.76799 q^{43} -9.42818 q^{44} +0.484320 q^{45} +3.68150 q^{46} +5.86528 q^{47} +20.2761 q^{48} +16.6319 q^{49} -17.6534 q^{50} +0.0434800 q^{51} -5.12237 q^{52} -1.85491 q^{53} +14.1761 q^{54} +6.27282 q^{55} +40.5085 q^{56} -1.76685 q^{57} -4.74207 q^{58} -9.37698 q^{59} -29.5121 q^{60} +1.74301 q^{61} -21.3411 q^{62} +0.690838 q^{63} +16.9601 q^{64} +3.40805 q^{65} +8.30409 q^{66} -1.91921 q^{67} +0.131746 q^{68} -2.33204 q^{69} -44.2148 q^{70} -3.35314 q^{71} +1.18420 q^{72} -8.66437 q^{73} +5.36977 q^{74} +11.1825 q^{75} -5.35363 q^{76} +8.94760 q^{77} +4.51165 q^{78} +16.2247 q^{79} -40.8759 q^{80} -8.55347 q^{81} -29.5498 q^{82} -3.18677 q^{83} -42.0962 q^{84} -0.0876542 q^{85} -7.38714 q^{86} +3.00385 q^{87} +15.3375 q^{88} +0.735015 q^{89} -1.29254 q^{90} +4.86127 q^{91} -7.06616 q^{92} +13.5184 q^{93} -15.6531 q^{94} +3.56191 q^{95} -25.9384 q^{96} +16.1510 q^{97} -44.3869 q^{98} +0.261567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * q^99

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