LMFDB - Embedded newform 8047.2.a.c.1.16

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			1.16
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)

$f(q)$	$=$	$q-2.36725 q^{2} +0.254210 q^{3} +3.60387 q^{4} +1.17067 q^{5} -0.601779 q^{6} -2.57756 q^{7} -3.79676 q^{8} -2.93538 q^{9} +O(q^{10})$	\(q-2.36725 q^{2} +0.254210 q^{3} +3.60387 q^{4} +1.17067 q^{5} -0.601779 q^{6} -2.57756 q^{7} -3.79676 q^{8} -2.93538 q^{9} -2.77126 q^{10} -2.66016 q^{11} +0.916140 q^{12} -1.00000 q^{13} +6.10173 q^{14} +0.297596 q^{15} +1.78014 q^{16} +3.09731 q^{17} +6.94877 q^{18} -6.16603 q^{19} +4.21894 q^{20} -0.655243 q^{21} +6.29727 q^{22} +8.37256 q^{23} -0.965175 q^{24} -3.62953 q^{25} +2.36725 q^{26} -1.50883 q^{27} -9.28920 q^{28} -2.73366 q^{29} -0.704484 q^{30} +0.535016 q^{31} +3.37949 q^{32} -0.676240 q^{33} -7.33211 q^{34} -3.01747 q^{35} -10.5787 q^{36} +10.5604 q^{37} +14.5965 q^{38} -0.254210 q^{39} -4.44475 q^{40} -4.18761 q^{41} +1.55112 q^{42} +8.17993 q^{43} -9.58688 q^{44} -3.43635 q^{45} -19.8199 q^{46} +9.39988 q^{47} +0.452529 q^{48} -0.356173 q^{49} +8.59201 q^{50} +0.787368 q^{51} -3.60387 q^{52} -0.512259 q^{53} +3.57178 q^{54} -3.11417 q^{55} +9.78638 q^{56} -1.56747 q^{57} +6.47126 q^{58} +3.78589 q^{59} +1.07250 q^{60} +2.00673 q^{61} -1.26652 q^{62} +7.56612 q^{63} -11.5604 q^{64} -1.17067 q^{65} +1.60083 q^{66} -9.55948 q^{67} +11.1623 q^{68} +2.12839 q^{69} +7.14311 q^{70} -3.63682 q^{71} +11.1449 q^{72} +10.8378 q^{73} -24.9990 q^{74} -0.922665 q^{75} -22.2216 q^{76} +6.85673 q^{77} +0.601779 q^{78} +11.7148 q^{79} +2.08395 q^{80} +8.42257 q^{81} +9.91312 q^{82} +0.465564 q^{83} -2.36141 q^{84} +3.62593 q^{85} -19.3639 q^{86} -0.694925 q^{87} +10.1000 q^{88} +0.472409 q^{89} +8.13471 q^{90} +2.57756 q^{91} +30.1736 q^{92} +0.136006 q^{93} -22.2519 q^{94} -7.21838 q^{95} +0.859101 q^{96} +1.20908 q^{97} +0.843150 q^{98} +7.80858 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.36725	−1.67390	−0.836949	−	0.547281i	$-0.815663\pi$
$2$	−2.36725	−1.67390	−0.836949	+	0.547281i	$0.815663\pi$
$3$	0.254210	0.146768	0.0733842	−	0.997304i	$-0.476620\pi$
$3$	0.254210	0.146768	0.0733842	+	0.997304i	$0.476620\pi$
$4$	3.60387	1.80193
$5$	1.17067	0.523539	0.261770	−	0.965130i	$-0.415694\pi$
$5$	1.17067	0.523539	0.261770	+	0.965130i	$0.415694\pi$
$6$	−0.601779	−0.245675
$7$	−2.57756	−0.974227	−0.487113	−	0.873339i	$-0.661950\pi$
$7$	−2.57756	−0.974227	−0.487113	+	0.873339i	$0.661950\pi$
$8$	−3.79676	−1.34236
$9$	−2.93538	−0.978459
$10$	−2.77126	−0.876351
$11$	−2.66016	−0.802069	−0.401035	−	0.916063i	$-0.631349\pi$
$11$	−2.66016	−0.802069	−0.401035	+	0.916063i	$0.631349\pi$
$12$	0.916140	0.264467
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	6.10173	1.63076
$15$	0.297596	0.0768389
$16$	1.78014	0.445034
$17$	3.09731	0.751209	0.375604	−	0.926780i	$-0.377435\pi$
$17$	3.09731	0.751209	0.375604	+	0.926780i	$0.377435\pi$
$18$	6.94877	1.63784
$19$	−6.16603	−1.41458	−0.707292	−	0.706922i	$-0.750083\pi$
$19$	−6.16603	−1.41458	−0.707292	+	0.706922i	$0.750083\pi$
$20$	4.21894	0.943383
$21$	−0.655243	−0.142986
$22$	6.29727	1.34258
$23$	8.37256	1.74580	0.872899	−	0.487900i	$-0.162237\pi$
$23$	8.37256	1.74580	0.872899	+	0.487900i	$0.162237\pi$
$24$	−0.965175	−0.197015
$25$	−3.62953	−0.725907
$26$	2.36725	0.464256
$27$	−1.50883	−0.290375
$28$	−9.28920	−1.75549
$29$	−2.73366	−0.507629	−0.253814	−	0.967253i	$-0.581685\pi$
$29$	−2.73366	−0.507629	−0.253814	+	0.967253i	$0.581685\pi$
$30$	−0.704484	−0.128621
$31$	0.535016	0.0960917	0.0480458	−	0.998845i	$-0.484701\pi$
$31$	0.535016	0.0960917	0.0480458	+	0.998845i	$0.484701\pi$
$32$	3.37949	0.597416
$33$	−0.676240	−0.117718
$34$	−7.33211	−1.25745
$35$	−3.01747	−0.510046
$36$	−10.5787	−1.76312
$37$	10.5604	1.73611	0.868056	−	0.496467i	$-0.165370\pi$
$37$	10.5604	1.73611	0.868056	+	0.496467i	$0.165370\pi$
$38$	14.5965	2.36787
$39$	−0.254210	−0.0407062
$40$	−4.44475	−0.702776
$41$	−4.18761	−0.653995	−0.326998	−	0.945025i	$-0.606037\pi$
$41$	−4.18761	−0.653995	−0.326998	+	0.945025i	$0.606037\pi$
$42$	1.55112	0.239343
$43$	8.17993	1.24743	0.623714	−	0.781653i	$-0.285623\pi$
$43$	8.17993	1.24743	0.623714	+	0.781653i	$0.285623\pi$
$44$	−9.58688	−1.44528
$45$	−3.43635	−0.512261
$46$	−19.8199	−2.92229
$47$	9.39988	1.37111	0.685557	−	0.728019i	$-0.259559\pi$
$47$	9.39988	1.37111	0.685557	+	0.728019i	$0.259559\pi$
$48$	0.452529	0.0653169
$49$	−0.356173	−0.0508818
$50$	8.59201	1.21509
$51$	0.787368	0.110254
$52$	−3.60387	−0.499767
$53$	−0.512259	−0.0703642	−0.0351821	−	0.999381i	$-0.511201\pi$
$53$	−0.512259	−0.0703642	−0.0351821	+	0.999381i	$0.511201\pi$
$54$	3.57178	0.486058
$55$	−3.11417	−0.419914
$56$	9.78638	1.30776
$57$	−1.56747	−0.207616
$58$	6.47126	0.849718
$59$	3.78589	0.492881	0.246441	−	0.969158i	$-0.420739\pi$
$59$	3.78589	0.492881	0.246441	+	0.969158i	$0.420739\pi$
$60$	1.07250	0.138459
$61$	2.00673	0.256935	0.128468	−	0.991714i	$-0.458994\pi$
$61$	2.00673	0.256935	0.128468	+	0.991714i	$0.458994\pi$
$62$	−1.26652	−0.160848
$63$	7.56612	0.953241
$64$	−11.5604	−1.44505
$65$	−1.17067	−0.145204
$66$	1.60083	0.197048
$67$	−9.55948	−1.16788	−0.583938	−	0.811798i	$-0.698489\pi$
$67$	−9.55948	−1.16788	−0.583938	+	0.811798i	$0.698489\pi$
$68$	11.1623	1.35363
$69$	2.12839	0.256228
$70$	7.14311	0.853765
$71$	−3.63682	−0.431611	−0.215806	−	0.976436i	$-0.569238\pi$
$71$	−3.63682	−0.431611	−0.215806	+	0.976436i	$0.569238\pi$
$72$	11.1449	1.31344
$73$	10.8378	1.26847	0.634237	−	0.773139i	$-0.281314\pi$
$73$	10.8378	1.26847	0.634237	+	0.773139i	$0.281314\pi$
$74$	−24.9990	−2.90607
$75$	−0.922665	−0.106540
$76$	−22.2216	−2.54899
$77$	6.85673	0.781397
$78$	0.601779	0.0681380
$79$	11.7148	1.31802	0.659012	−	0.752133i	$-0.270975\pi$
$79$	11.7148	1.31802	0.659012	+	0.752133i	$0.270975\pi$
$80$	2.08395	0.232993
$81$	8.42257	0.935841
$82$	9.91312	1.09472
$83$	0.465564	0.0511023	0.0255512	−	0.999674i	$-0.491866\pi$
$83$	0.465564	0.0511023	0.0255512	+	0.999674i	$0.491866\pi$
$84$	−2.36141	−0.257651
$85$	3.62593	0.393287
$86$	−19.3639	−2.08807
$87$	−0.694925	−0.0745038
$88$	10.1000	1.07666
$89$	0.472409	0.0500753	0.0250376	−	0.999687i	$-0.492029\pi$
$89$	0.472409	0.0500753	0.0250376	+	0.999687i	$0.492029\pi$
$90$	8.13471	0.857473
$91$	2.57756	0.270202
$92$	30.1736	3.14582
$93$	0.136006	0.0141032
$94$	−22.2519	−2.29510
$95$	−7.21838	−0.740590
$96$	0.859101	0.0876817
$97$	1.20908	0.122764	0.0613819	−	0.998114i	$-0.480449\pi$
$97$	1.20908	0.122764	0.0613819	+	0.998114i	$0.480449\pi$
$98$	0.843150	0.0851710
$99$	7.80858	0.784792
$100$	−13.0804	−1.30804
$101$	16.1150	1.60351	0.801753	−	0.597655i	$-0.203901\pi$
$101$	16.1150	1.60351	0.801753	+	0.597655i	$0.203901\pi$
$102$	−1.86390	−0.184553
$103$	−1.82301	−0.179627	−0.0898134	−	0.995959i	$-0.528627\pi$
$103$	−1.82301	−0.179627	−0.0898134	+	0.995959i	$0.528627\pi$
$104$	3.79676	0.372303
$105$	−0.767072	−0.0748586
$106$	1.21264	0.117782
$107$	1.22156	0.118093	0.0590465	−	0.998255i	$-0.481194\pi$
$107$	1.22156	0.118093	0.0590465	+	0.998255i	$0.481194\pi$
$108$	−5.43764	−0.523237
$109$	−10.7167	−1.02647	−0.513237	−	0.858247i	$-0.671554\pi$
$109$	−10.7167	−1.02647	−0.513237	+	0.858247i	$0.671554\pi$
$110$	7.37201	0.702894
$111$	2.68455	0.254806
$112$	−4.58841	−0.433564
$113$	10.8624	1.02185	0.510925	−	0.859625i	$-0.329303\pi$
$113$	10.8624	1.02185	0.510925	+	0.859625i	$0.329303\pi$
$114$	3.71059	0.347528
$115$	9.80149	0.913994
$116$	−9.85177	−0.914713
$117$	2.93538	0.271376
$118$	−8.96215	−0.825033
$119$	−7.98352	−0.731848
$120$	−1.12990	−0.103145
$121$	−3.92354	−0.356685
$122$	−4.75043	−0.430083
$123$	−1.06453	−0.0959858
$124$	1.92813	0.173151
$125$	−10.1023	−0.903580
$126$	−17.9109	−1.59563
$127$	4.25493	0.377564	0.188782	−	0.982019i	$-0.439546\pi$
$127$	4.25493	0.377564	0.188782	+	0.982019i	$0.439546\pi$
$128$	20.6073	1.82145
$129$	2.07942	0.183083
$130$	2.77126	0.243056
$131$	16.2566	1.42035	0.710174	−	0.704026i	$-0.248616\pi$
$131$	16.2566	1.42035	0.710174	+	0.704026i	$0.248616\pi$
$132$	−2.43708	−0.212121
$133$	15.8933	1.37813
$134$	22.6297	1.95491
$135$	−1.76634	−0.152023
$136$	−11.7597	−1.00839
$137$	−10.8802	−0.929561	−0.464780	−	0.885426i	$-0.653867\pi$
$137$	−10.8802	−0.929561	−0.464780	+	0.885426i	$0.653867\pi$
$138$	−5.03843	−0.428899
$139$	4.68524	0.397397	0.198699	−	0.980061i	$-0.436329\pi$
$139$	4.68524	0.397397	0.198699	+	0.980061i	$0.436329\pi$
$140$	−10.8746	−0.919069
$141$	2.38955	0.201236
$142$	8.60926	0.722473
$143$	2.66016	0.222454
$144$	−5.22537	−0.435447
$145$	−3.20021	−0.265763
$146$	−25.6559	−2.12330
$147$	−0.0905428	−0.00746784
$148$	38.0581	3.12836
$149$	3.99003	0.326876	0.163438	−	0.986554i	$-0.447742\pi$
$149$	3.99003	0.326876	0.163438	+	0.986554i	$0.447742\pi$
$150$	2.18418	0.178337
$151$	−6.15106	−0.500566	−0.250283	−	0.968173i	$-0.580524\pi$
$151$	−6.15106	−0.500566	−0.250283	+	0.968173i	$0.580524\pi$
$152$	23.4109	1.89888
$153$	−9.09178	−0.735027
$154$	−16.2316	−1.30798
$155$	0.626326	0.0503077
$156$	−0.916140	−0.0733499
$157$	−7.53876	−0.601659	−0.300829	−	0.953678i	$-0.597263\pi$
$157$	−7.53876	−0.601659	−0.300829	+	0.953678i	$0.597263\pi$
$158$	−27.7320	−2.20624
$159$	−0.130221	−0.0103272
$160$	3.95627	0.312770
$161$	−21.5808	−1.70080
$162$	−19.9383	−1.56650
$163$	−22.4555	−1.75885	−0.879425	−	0.476038i	$-0.842072\pi$
$163$	−22.4555	−1.75885	−0.879425	+	0.476038i	$0.842072\pi$
$164$	−15.0916	−1.17846
$165$	−0.791653	−0.0616301
$166$	−1.10211	−0.0855401
$167$	1.94999	0.150895	0.0754475	−	0.997150i	$-0.475961\pi$
$167$	1.94999	0.150895	0.0754475	+	0.997150i	$0.475961\pi$
$168$	2.48780	0.191938
$169$	1.00000	0.0769231
$170$	−8.58347	−0.658322
$171$	18.0996	1.38411
$172$	29.4794	2.24778
$173$	−17.5205	−1.33206	−0.666031	−	0.745924i	$-0.732008\pi$
$173$	−17.5205	−1.33206	−0.666031	+	0.745924i	$0.732008\pi$
$174$	1.64506	0.124712
$175$	9.35535	0.707198
$176$	−4.73545	−0.356948
$177$	0.962413	0.0723394
$178$	−1.11831	−0.0838209
$179$	−3.89016	−0.290764	−0.145382	−	0.989376i	$-0.546441\pi$
$179$	−3.89016	−0.290764	−0.145382	+	0.989376i	$0.546441\pi$
$180$	−12.3842	−0.923062
$181$	12.7087	0.944631	0.472316	−	0.881429i	$-0.343418\pi$
$181$	12.7087	0.944631	0.472316	+	0.881429i	$0.343418\pi$
$182$	−6.10173	−0.452291
$183$	0.510131	0.0377099
$184$	−31.7886	−2.34349
$185$	12.3627	0.908922
$186$	−0.321961	−0.0236073
$187$	−8.23935	−0.602521
$188$	33.8759	2.47066
$189$	3.88911	0.282891
$190$	17.0877	1.23967
$191$	−23.7992	−1.72205	−0.861024	−	0.508564i	$-0.830177\pi$
$191$	−23.7992	−1.72205	−0.861024	+	0.508564i	$0.830177\pi$
$192$	−2.93876	−0.212087
$193$	−23.6873	−1.70505	−0.852523	−	0.522690i	$-0.824929\pi$
$193$	−23.6873	−1.70505	−0.852523	+	0.522690i	$0.824929\pi$
$194$	−2.86220	−0.205494
$195$	−0.297596	−0.0213113
$196$	−1.28360	−0.0916858
$197$	−18.5672	−1.32286	−0.661428	−	0.750009i	$-0.730049\pi$
$197$	−18.5672	−1.32286	−0.661428	+	0.750009i	$0.730049\pi$
$198$	−18.4849	−1.31366
$199$	−28.1323	−1.99425	−0.997123	−	0.0758013i	$-0.975849\pi$
$199$	−28.1323	−1.99425	−0.997123	+	0.0758013i	$0.975849\pi$
$200$	13.7805	0.974426
$201$	−2.43012	−0.171407
$202$	−38.1483	−2.68411
$203$	7.04619	0.494545
$204$	2.83757	0.198670
$205$	−4.90231	−0.342392
$206$	4.31553	0.300677
$207$	−24.5766	−1.70819
$208$	−1.78014	−0.123430
$209$	16.4026	1.13459
$210$	1.81585	0.125306
$211$	−17.5114	−1.20554	−0.602768	−	0.797917i	$-0.705935\pi$
$211$	−17.5114	−1.20554	−0.602768	+	0.797917i	$0.705935\pi$
$212$	−1.84611	−0.126792
$213$	−0.924517	−0.0633468
$214$	−2.89174	−0.197676
$215$	9.57599	0.653077
$216$	5.72868	0.389787
$217$	−1.37904	−0.0936151
$218$	25.3691	1.71821
$219$	2.75509	0.186172
$220$	−11.2231	−0.756658
$221$	−3.09731	−0.208348
$222$	−6.35500	−0.426520
$223$	17.5161	1.17296	0.586482	−	0.809962i	$-0.300512\pi$
$223$	17.5161	1.17296	0.586482	+	0.809962i	$0.300512\pi$
$224$	−8.71085	−0.582018
$225$	10.6541	0.710270
$226$	−25.7141	−1.71047
$227$	−6.85422	−0.454931	−0.227465	−	0.973786i	$-0.573044\pi$
$227$	−6.85422	−0.454931	−0.227465	+	0.973786i	$0.573044\pi$
$228$	−5.64895	−0.374111
$229$	−24.8303	−1.64083	−0.820416	−	0.571767i	$-0.806258\pi$
$229$	−24.8303	−1.64083	−0.820416	+	0.571767i	$0.806258\pi$
$230$	−23.2026	−1.52993
$231$	1.74305	0.114684
$232$	10.3791	0.681419
$233$	14.5096	0.950559	0.475279	−	0.879835i	$-0.342347\pi$
$233$	14.5096	0.950559	0.475279	+	0.879835i	$0.342347\pi$
$234$	−6.94877	−0.454255
$235$	11.0041	0.717831
$236$	13.6439	0.888140
$237$	2.97803	0.193444
$238$	18.8990	1.22504
$239$	−18.9850	−1.22804	−0.614020	−	0.789290i	$-0.710449\pi$
$239$	−18.9850	−1.22804	−0.614020	+	0.789290i	$0.710449\pi$
$240$	0.529761	0.0341959
$241$	17.2473	1.11099	0.555497	−	0.831519i	$-0.312528\pi$
$241$	17.2473	1.11099	0.555497	+	0.831519i	$0.312528\pi$
$242$	9.28799	0.597055
$243$	6.66760	0.427727
$244$	7.23198	0.462980
$245$	−0.416961	−0.0266386
$246$	2.52002	0.160670
$247$	6.16603	0.392335
$248$	−2.03133	−0.128989
$249$	0.118351	0.00750020
$250$	23.9147	1.51250
$251$	−28.6309	−1.80716	−0.903582	−	0.428415i	$-0.859072\pi$
$251$	−28.6309	−1.80716	−0.903582	+	0.428415i	$0.859072\pi$
$252$	27.2673	1.71768
$253$	−22.2724	−1.40025
$254$	−10.0725	−0.632003
$255$	0.921748	0.0577221
$256$	−25.6619	−1.60387
$257$	−10.9530	−0.683231	−0.341615	−	0.939840i	$-0.610974\pi$
$257$	−10.9530	−0.683231	−0.341615	+	0.939840i	$0.610974\pi$
$258$	−4.92251	−0.306462
$259$	−27.2200	−1.69137
$260$	−4.21894	−0.261647
$261$	8.02433	0.496694
$262$	−38.4835	−2.37752
$263$	−13.7632	−0.848674	−0.424337	−	0.905504i	$-0.639493\pi$
$263$	−13.7632	−0.848674	−0.424337	+	0.905504i	$0.639493\pi$
$264$	2.56752	0.158020
$265$	−0.599685	−0.0368384
$266$	−37.6235	−2.30684
$267$	0.120091	0.00734947
$268$	−34.4511	−2.10444
$269$	−6.23289	−0.380026	−0.190013	−	0.981782i	$-0.560853\pi$
$269$	−6.23289	−0.380026	−0.190013	+	0.981782i	$0.560853\pi$
$270$	4.18138	0.254470
$271$	5.34314	0.324573	0.162286	−	0.986744i	$-0.448113\pi$
$271$	5.34314	0.324573	0.162286	+	0.986744i	$0.448113\pi$
$272$	5.51364	0.334313
$273$	0.655243	0.0396571
$274$	25.7562	1.55599
$275$	9.65515	0.582227
$276$	7.67044	0.461706
$277$	2.51617	0.151182	0.0755909	−	0.997139i	$-0.475916\pi$
$277$	2.51617	0.151182	0.0755909	+	0.997139i	$0.475916\pi$
$278$	−11.0911	−0.665202
$279$	−1.57047	−0.0940218
$280$	11.4566	0.684664
$281$	5.17983	0.309003	0.154501	−	0.987993i	$-0.450623\pi$
$281$	5.17983	0.309003	0.154501	+	0.987993i	$0.450623\pi$
$282$	−5.65665	−0.336849
$283$	27.2815	1.62172	0.810860	−	0.585241i	$-0.199000\pi$
$283$	27.2815	1.62172	0.810860	+	0.585241i	$0.199000\pi$
$284$	−13.1066	−0.777735
$285$	−1.83498	−0.108695
$286$	−6.29727	−0.372365
$287$	10.7938	0.637140
$288$	−9.92009	−0.584547
$289$	−7.40666	−0.435686
$290$	7.57571	0.444861
$291$	0.307361	0.0180178
$292$	39.0582	2.28571
$293$	−6.46192	−0.377509	−0.188755	−	0.982024i	$-0.560445\pi$
$293$	−6.46192	−0.377509	−0.188755	+	0.982024i	$0.560445\pi$
$294$	0.214337	0.0125004
$295$	4.43203	0.258043
$296$	−40.0951	−2.33048
$297$	4.01374	0.232901
$298$	−9.44538	−0.547156
$299$	−8.37256	−0.484197
$300$	−3.32516	−0.191978
$301$	−21.0843	−1.21528
$302$	14.5611	0.837897
$303$	4.09661	0.235344
$304$	−10.9764	−0.629538
$305$	2.34921	0.134516
$306$	21.5225	1.23036
$307$	19.5316	1.11472	0.557362	−	0.830269i	$-0.311813\pi$
$307$	19.5316	1.11472	0.557362	+	0.830269i	$0.311813\pi$
$308$	24.7108	1.40803
$309$	−0.463429	−0.0263635
$310$	−1.48267	−0.0842100
$311$	−14.8043	−0.839474	−0.419737	−	0.907646i	$-0.637878\pi$
$311$	−14.8043	−0.839474	−0.419737	+	0.907646i	$0.637878\pi$
$312$	0.965175	0.0546423
$313$	4.74272	0.268074	0.134037	−	0.990976i	$-0.457206\pi$
$313$	4.74272	0.268074	0.134037	+	0.990976i	$0.457206\pi$
$314$	17.8461	1.00712
$315$	8.85742	0.499059
$316$	42.2188	2.37499
$317$	−14.0678	−0.790124	−0.395062	−	0.918654i	$-0.629277\pi$
$317$	−14.0678	−0.790124	−0.395062	+	0.918654i	$0.629277\pi$
$318$	0.308267	0.0172867
$319$	7.27199	0.407153
$320$	−13.5334	−0.756538
$321$	0.310534	0.0173323
$322$	51.0871	2.84697
$323$	−19.0981	−1.06265
$324$	30.3538	1.68632
$325$	3.62953	0.201330
$326$	53.1577	2.94413
$327$	−2.72430	−0.150654
$328$	15.8994	0.877895
$329$	−24.2288	−1.33578
$330$	1.87404	0.103163
$331$	10.8321	0.595388	0.297694	−	0.954661i	$-0.403783\pi$
$331$	10.8321	0.595388	0.297694	+	0.954661i	$0.403783\pi$
$332$	1.67783	0.0920830
$333$	−30.9986	−1.69871
$334$	−4.61612	−0.252583
$335$	−11.1910	−0.611429
$336$	−1.16642	−0.0636335
$337$	13.9217	0.758363	0.379182	−	0.925322i	$-0.376206\pi$
$337$	13.9217	0.758363	0.379182	+	0.925322i	$0.376206\pi$
$338$	−2.36725	−0.128761
$339$	2.76134	0.149975
$340$	13.0674	0.708678
$341$	−1.42323	−0.0770722
$342$	−42.8463	−2.31686
$343$	18.9610	1.02380
$344$	−31.0572	−1.67449
$345$	2.49164	0.134145
$346$	41.4755	2.22974
$347$	12.5692	0.674748	0.337374	−	0.941371i	$-0.390461\pi$
$347$	12.5692	0.674748	0.337374	+	0.941371i	$0.390461\pi$
$348$	−2.50442	−0.134251
$349$	19.6054	1.04946	0.524728	−	0.851270i	$-0.324167\pi$
$349$	19.6054	1.04946	0.524728	+	0.851270i	$0.324167\pi$
$350$	−22.1464	−1.18378
$351$	1.50883	0.0805356
$352$	−8.99000	−0.479168
$353$	−31.3603	−1.66914	−0.834569	−	0.550903i	$-0.814283\pi$
$353$	−31.3603	−1.66914	−0.834569	+	0.550903i	$0.814283\pi$
$354$	−2.27827	−0.121089
$355$	−4.25751	−0.225965
$356$	1.70250	0.0902324
$357$	−2.02949	−0.107412
$358$	9.20898	0.486710
$359$	−12.4242	−0.655726	−0.327863	−	0.944725i	$-0.606328\pi$
$359$	−12.4242	−0.655726	−0.327863	+	0.944725i	$0.606328\pi$
$360$	13.0470	0.687638
$361$	19.0199	1.00105
$362$	−30.0847	−1.58122
$363$	−0.997403	−0.0523501
$364$	9.28920	0.486886
$365$	12.6875	0.664095
$366$	−1.20761	−0.0631226
$367$	27.9206	1.45744	0.728722	−	0.684810i	$-0.240115\pi$
$367$	27.9206	1.45744	0.728722	+	0.684810i	$0.240115\pi$
$368$	14.9043	0.776940
$369$	12.2922	0.639908
$370$	−29.2655	−1.52144
$371$	1.32038	0.0685507
$372$	0.490150	0.0254131
$373$	−15.1658	−0.785257	−0.392629	−	0.919697i	$-0.628434\pi$
$373$	−15.1658	−0.785257	−0.392629	+	0.919697i	$0.628434\pi$
$374$	19.5046	1.00856
$375$	−2.56811	−0.132617
$376$	−35.6891	−1.84052
$377$	2.73366	0.140791
$378$	−9.20650	−0.473531
$379$	−1.91443	−0.0983379	−0.0491690	−	0.998790i	$-0.515657\pi$
$379$	−1.91443	−0.0983379	−0.0491690	+	0.998790i	$0.515657\pi$
$380$	−26.0141	−1.33449
$381$	1.08165	0.0554144
$382$	56.3386	2.88253
$383$	−30.8204	−1.57485	−0.787425	−	0.616411i	$-0.788586\pi$
$383$	−30.8204	−1.57485	−0.787425	+	0.616411i	$0.788586\pi$
$384$	5.23859	0.267330
$385$	8.02696	0.409092
$386$	56.0736	2.85407
$387$	−24.0112	−1.22056
$388$	4.35738	0.221212
$389$	15.4395	0.782816	0.391408	−	0.920217i	$-0.371988\pi$
$389$	15.4395	0.782816	0.391408	+	0.920217i	$0.371988\pi$
$390$	0.704484	0.0356729
$391$	25.9324	1.31146
$392$	1.35230	0.0683016
$393$	4.13260	0.208462
$394$	43.9531	2.21433
$395$	13.7142	0.690037
$396$	28.1411	1.41414
$397$	−8.74317	−0.438807	−0.219404	−	0.975634i	$-0.570411\pi$
$397$	−8.74317	−0.438807	−0.219404	+	0.975634i	$0.570411\pi$
$398$	66.5962	3.33816
$399$	4.04024	0.202265
$400$	−6.46106	−0.323053
$401$	23.0924	1.15318	0.576590	−	0.817033i	$-0.304383\pi$
$401$	23.0924	1.15318	0.576590	+	0.817033i	$0.304383\pi$
$402$	5.75269	0.286918
$403$	−0.535016	−0.0266510
$404$	58.0765	2.88941
$405$	9.86004	0.489949
$406$	−16.6801	−0.827819
$407$	−28.0923	−1.39248
$408$	−2.98945	−0.148000
$409$	0.519002	0.0256630	0.0128315	−	0.999918i	$-0.495915\pi$
$409$	0.519002	0.0256630	0.0128315	+	0.999918i	$0.495915\pi$
$410$	11.6050	0.573129
$411$	−2.76586	−0.136430
$412$	−6.56990	−0.323676
$413$	−9.75838	−0.480178
$414$	58.1790	2.85934
$415$	0.545022	0.0267541
$416$	−3.37949	−0.165693
$417$	1.19104	0.0583253
$418$	−38.8291	−1.89919
$419$	−14.4316	−0.705032	−0.352516	−	0.935806i	$-0.614674\pi$
$419$	−14.4316	−0.705032	−0.352516	+	0.935806i	$0.614674\pi$
$420$	−2.76443	−0.134890
$421$	−4.55122	−0.221813	−0.110906	−	0.993831i	$-0.535375\pi$
$421$	−4.55122	−0.221813	−0.110906	+	0.993831i	$0.535375\pi$
$422$	41.4539	2.01794
$423$	−27.5922	−1.34158
$424$	1.94492	0.0944538
$425$	−11.2418	−0.545308
$426$	2.18856	0.106036
$427$	−5.17247	−0.250313
$428$	4.40235	0.212796
$429$	0.676240	0.0326492
$430$	−22.6687	−1.09318
$431$	10.7454	0.517589	0.258794	−	0.965932i	$-0.416675\pi$
$431$	10.7454	0.517589	0.258794	+	0.965932i	$0.416675\pi$
$432$	−2.68593	−0.129227
$433$	19.0730	0.916589	0.458294	−	0.888801i	$-0.348461\pi$
$433$	19.0730	0.916589	0.458294	+	0.888801i	$0.348461\pi$
$434$	3.26452	0.156702
$435$	−0.813527	−0.0390056
$436$	−38.6216	−1.84964
$437$	−51.6254	−2.46958
$438$	−6.52199	−0.311633
$439$	5.78468	0.276088	0.138044	−	0.990426i	$-0.455918\pi$
$439$	5.78468	0.276088	0.138044	+	0.990426i	$0.455918\pi$
$440$	11.8237	0.563675
$441$	1.04550	0.0497858
$442$	7.33211	0.348753
$443$	3.97399	0.188810	0.0944049	−	0.995534i	$-0.469905\pi$
$443$	3.97399	0.188810	0.0944049	+	0.995534i	$0.469905\pi$
$444$	9.67477	0.459144
$445$	0.553035	0.0262164
$446$	−41.4650	−1.96342
$447$	1.01431	0.0479750
$448$	29.7976	1.40780
$449$	−20.9464	−0.988523	−0.494262	−	0.869313i	$-0.664562\pi$
$449$	−20.9464	−0.988523	−0.494262	+	0.869313i	$0.664562\pi$
$450$	−25.2208	−1.18892
$451$	11.1397	0.524549
$452$	39.1467	1.84131
$453$	−1.56366	−0.0734673
$454$	16.2257	0.761508
$455$	3.01747	0.141461
$456$	5.95129	0.278695
$457$	14.7098	0.688094	0.344047	−	0.938952i	$-0.388202\pi$
$457$	14.7098	0.688094	0.344047	+	0.938952i	$0.388202\pi$
$458$	58.7795	2.74659
$459$	−4.67333	−0.218132
$460$	35.3233	1.64696
$461$	13.0128	0.606064	0.303032	−	0.952980i	$-0.402001\pi$
$461$	13.0128	0.606064	0.303032	+	0.952980i	$0.402001\pi$
$462$	−4.12624	−0.191970
$463$	−26.3695	−1.22549	−0.612747	−	0.790279i	$-0.709936\pi$
$463$	−26.3695	−1.22549	−0.612747	+	0.790279i	$0.709936\pi$
$464$	−4.86629	−0.225912
$465$	0.159219	0.00738358
$466$	−34.3480	−1.59114
$467$	10.8100	0.500226	0.250113	−	0.968217i	$-0.419532\pi$
$467$	10.8100	0.500226	0.250113	+	0.968217i	$0.419532\pi$
$468$	10.5787	0.489001
$469$	24.6402	1.13778
$470$	−26.0496	−1.20158
$471$	−1.91643	−0.0883044
$472$	−14.3741	−0.661623
$473$	−21.7599	−1.00052
$474$	−7.04975	−0.323806
$475$	22.3798	1.02686
$476$	−28.7715	−1.31874
$477$	1.50367	0.0688484
$478$	44.9423	2.05561
$479$	26.3405	1.20353	0.601765	−	0.798673i	$-0.294464\pi$
$479$	26.3405	1.20353	0.601765	+	0.798673i	$0.294464\pi$
$480$	1.00572	0.0459048
$481$	−10.5604	−0.481511
$482$	−40.8286	−1.85969
$483$	−5.48606	−0.249624
$484$	−14.1399	−0.642723
$485$	1.41544	0.0642716
$486$	−15.7839	−0.715971
$487$	−36.6265	−1.65970	−0.829852	−	0.557984i	$-0.811575\pi$
$487$	−36.6265	−1.65970	−0.829852	+	0.557984i	$0.811575\pi$
$488$	−7.61906	−0.344899
$489$	−5.70841	−0.258143
$490$	0.987049	0.0445904
$491$	19.3589	0.873653	0.436826	−	0.899546i	$-0.356102\pi$
$491$	19.3589	0.873653	0.436826	+	0.899546i	$0.356102\pi$
$492$	−3.83644	−0.172960
$493$	−8.46701	−0.381335
$494$	−14.5965	−0.656729
$495$	9.14126	0.410869
$496$	0.952401	0.0427641
$497$	9.37413	0.420487
$498$	−0.280167	−0.0125546
$499$	12.2245	0.547243	0.273622	−	0.961837i	$-0.411778\pi$
$499$	12.2245	0.547243	0.273622	+	0.961837i	$0.411778\pi$
$500$	−36.4075	−1.62819
$501$	0.495708	0.0221466
$502$	67.7764	3.02501
$503$	42.2576	1.88417	0.942086	−	0.335371i	$-0.108862\pi$
$503$	42.2576	1.88417	0.942086	+	0.335371i	$0.108862\pi$
$504$	−28.7267	−1.27959
$505$	18.8654	0.839498
$506$	52.7242	2.34388
$507$	0.254210	0.0112899
$508$	15.3342	0.680345
$509$	−37.5291	−1.66345	−0.831724	−	0.555189i	$-0.812646\pi$
$509$	−37.5291	−1.66345	−0.831724	+	0.555189i	$0.812646\pi$
$510$	−2.18201	−0.0966209
$511$	−27.9352	−1.23578
$512$	19.5334	0.863264
$513$	9.30351	0.410760
$514$	25.9285	1.14366
$515$	−2.13415	−0.0940417
$516$	7.49396	0.329903
$517$	−25.0052	−1.09973
$518$	64.4365	2.83118
$519$	−4.45390	−0.195504
$520$	4.44475	0.194915
$521$	−9.99951	−0.438087	−0.219043	−	0.975715i	$-0.570294\pi$
$521$	−9.99951	−0.438087	−0.219043	+	0.975715i	$0.570294\pi$
$522$	−18.9956	−0.831415
$523$	10.5497	0.461306	0.230653	−	0.973036i	$-0.425914\pi$
$523$	10.5497	0.461306	0.230653	+	0.973036i	$0.425914\pi$
$524$	58.5867	2.55937
$525$	2.37823	0.103794
$526$	32.5809	1.42059
$527$	1.65711	0.0721849
$528$	−1.20380	−0.0523886
$529$	47.0997	2.04781
$530$	1.41960	0.0616637
$531$	−11.1130	−0.482264
$532$	57.2774	2.48329
$533$	4.18761	0.181386
$534$	−0.284286	−0.0123023
$535$	1.43005	0.0618263
$536$	36.2950	1.56771
$537$	−0.988919	−0.0426750
$538$	14.7548	0.636124
$539$	0.947478	0.0408108
$540$	−6.36567	−0.273935
$541$	−0.372435	−0.0160122	−0.00800612	−	0.999968i	$-0.502548\pi$
$541$	−0.372435	−0.0160122	−0.00800612	+	0.999968i	$0.502548\pi$
$542$	−12.6485	−0.543301
$543$	3.23068	0.138642
$544$	10.4673	0.448784
$545$	−12.5457	−0.537399
$546$	−1.55112	−0.0663819
$547$	−9.31392	−0.398235	−0.199117	−	0.979976i	$-0.563807\pi$
$547$	−9.31392	−0.398235	−0.199117	+	0.979976i	$0.563807\pi$
$548$	−39.2109	−1.67501
$549$	−5.89050	−0.251401
$550$	−22.8561	−0.974589
$551$	16.8558	0.718083
$552$	−8.08098	−0.343949
$553$	−30.1957	−1.28405
$554$	−5.95639	−0.253063
$555$	3.14272	0.133401
$556$	16.8850	0.716083
$557$	21.8874	0.927397	0.463699	−	0.885993i	$-0.346522\pi$
$557$	21.8874	0.927397	0.463699	+	0.885993i	$0.346522\pi$
$558$	3.71770	0.157383
$559$	−8.17993	−0.345974
$560$	−5.37151	−0.226988
$561$	−2.09453	−0.0884310
$562$	−12.2619	−0.517239
$563$	21.4734	0.904997	0.452498	−	0.891765i	$-0.350533\pi$
$563$	21.4734	0.904997	0.452498	+	0.891765i	$0.350533\pi$
$564$	8.61161	0.362614
$565$	12.7163	0.534979
$566$	−64.5822	−2.71459
$567$	−21.7097	−0.911722
$568$	13.8081	0.579376
$569$	−35.8201	−1.50166	−0.750829	−	0.660496i	$-0.770346\pi$
$569$	−35.8201	−1.50166	−0.750829	+	0.660496i	$0.770346\pi$
$570$	4.34387	0.181945
$571$	4.45961	0.186629	0.0933144	−	0.995637i	$-0.470254\pi$
$571$	4.45961	0.186629	0.0933144	+	0.995637i	$0.470254\pi$
$572$	9.58688	0.400847
$573$	−6.04999	−0.252742
$574$	−25.5517	−1.06651
$575$	−30.3885	−1.26729
$576$	33.9341	1.41392
$577$	−9.95024	−0.414234	−0.207117	−	0.978316i	$-0.566408\pi$
$577$	−9.95024	−0.414234	−0.207117	+	0.978316i	$0.566408\pi$
$578$	17.5334	0.729293
$579$	−6.02154	−0.250247
$580$	−11.5332	−0.478888
$581$	−1.20002	−0.0497853
$582$	−0.727601	−0.0301600
$583$	1.36269	0.0564369
$584$	−41.1487	−1.70274
$585$	3.43635	0.142076
$586$	15.2970	0.631912
$587$	−20.0626	−0.828073	−0.414036	−	0.910260i	$-0.635881\pi$
$587$	−20.0626	−0.828073	−0.414036	+	0.910260i	$0.635881\pi$
$588$	−0.326304	−0.0134566
$589$	−3.29892	−0.135930
$590$	−10.4917	−0.431937
$591$	−4.71996	−0.194153
$592$	18.7989	0.772628
$593$	−40.0251	−1.64363	−0.821817	−	0.569751i	$-0.807039\pi$
$593$	−40.0251	−1.64363	−0.821817	+	0.569751i	$0.807039\pi$
$594$	−9.50153	−0.389852
$595$	−9.34605	−0.383151
$596$	14.3795	0.589008
$597$	−7.15152	−0.292692
$598$	19.8199	0.810497
$599$	19.8508	0.811083	0.405541	−	0.914077i	$-0.367083\pi$
$599$	19.8508	0.811083	0.405541	+	0.914077i	$0.367083\pi$
$600$	3.50313	0.143015
$601$	7.19057	0.293310	0.146655	−	0.989188i	$-0.453149\pi$
$601$	7.19057	0.293310	0.146655	+	0.989188i	$0.453149\pi$
$602$	49.9117	2.03425
$603$	28.0607	1.14272
$604$	−22.1676	−0.901988
$605$	−4.59316	−0.186739
$606$	−9.69769	−0.393942
$607$	5.62132	0.228162	0.114081	−	0.993471i	$-0.463608\pi$
$607$	5.62132	0.228162	0.114081	+	0.993471i	$0.463608\pi$
$608$	−20.8380	−0.845094
$609$	1.79121	0.0725836
$610$	−5.56117	−0.225165
$611$	−9.39988	−0.380278
$612$	−32.7656	−1.32447
$613$	−28.8128	−1.16374	−0.581870	−	0.813282i	$-0.697679\pi$
$613$	−28.8128	−1.16374	−0.581870	+	0.813282i	$0.697679\pi$
$614$	−46.2361	−1.86594
$615$	−1.24622	−0.0502523
$616$	−26.0334	−1.04891
$617$	−7.81520	−0.314628	−0.157314	−	0.987549i	$-0.550283\pi$
$617$	−7.81520	−0.314628	−0.157314	+	0.987549i	$0.550283\pi$
$618$	1.09705	0.0441299
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	2.25720	0.0906513
$621$	−12.6328	−0.506936
$622$	35.0454	1.40519
$623$	−1.21766	−0.0487847
$624$	−0.452529	−0.0181156
$625$	6.32119	0.252848
$626$	−11.2272	−0.448729
$627$	4.16972	0.166522
$628$	−27.1687	−1.08415
$629$	32.7087	1.30418
$630$	−20.9677	−0.835374
$631$	−44.9706	−1.79025	−0.895125	−	0.445815i	$-0.852914\pi$
$631$	−44.9706	−1.79025	−0.895125	+	0.445815i	$0.852914\pi$
$632$	−44.4784	−1.76926
$633$	−4.45158	−0.176934
$634$	33.3019	1.32259
$635$	4.98111	0.197669
$636$	−0.469301	−0.0186090
$637$	0.356173	0.0141121
$638$	−17.2146	−0.681533
$639$	10.6754	0.422314
$640$	24.1243	0.953598
$641$	−8.39444	−0.331560	−0.165780	−	0.986163i	$-0.553014\pi$
$641$	−8.39444	−0.331560	−0.165780	+	0.986163i	$0.553014\pi$
$642$	−0.735111	−0.0290125
$643$	−11.2465	−0.443517	−0.221759	−	0.975102i	$-0.571180\pi$
$643$	−11.2465	−0.443517	−0.221759	+	0.975102i	$0.571180\pi$
$644$	−77.7743	−3.06474
$645$	2.43431	0.0958510
$646$	45.2100	1.77876
$647$	−35.8394	−1.40899	−0.704497	−	0.709707i	$-0.748827\pi$
$647$	−35.8394	−1.40899	−0.704497	+	0.709707i	$0.748827\pi$
$648$	−31.9785	−1.25623
$649$	−10.0711	−0.395325
$650$	−8.59201	−0.337006
$651$	−0.350565	−0.0137397
$652$	−80.9266	−3.16933
$653$	−39.7379	−1.55506	−0.777531	−	0.628844i	$-0.783528\pi$
$653$	−39.7379	−1.55506	−0.777531	+	0.628844i	$0.783528\pi$
$654$	6.44909	0.252179
$655$	19.0311	0.743607
$656$	−7.45452	−0.291050
$657$	−31.8132	−1.24115
$658$	57.3556	2.23595
$659$	−5.49785	−0.214166	−0.107083	−	0.994250i	$-0.534151\pi$
$659$	−5.49785	−0.214166	−0.107083	+	0.994250i	$0.534151\pi$
$660$	−2.85302	−0.111053
$661$	33.3406	1.29680	0.648399	−	0.761301i	$-0.275439\pi$
$661$	33.3406	1.29680	0.648399	+	0.761301i	$0.275439\pi$
$662$	−25.6424	−0.996619
$663$	−0.787368	−0.0305789
$664$	−1.76764	−0.0685976
$665$	18.6058	0.721503
$666$	73.3815	2.84347
$667$	−22.8878	−0.886217
$668$	7.02752	0.271903
$669$	4.45277	0.172154
$670$	26.4919	1.02347
$671$	−5.33822	−0.206080
$672$	−2.21439	−0.0854218
$673$	−23.3947	−0.901800	−0.450900	−	0.892574i	$-0.648897\pi$
$673$	−23.3947	−0.901800	−0.450900	+	0.892574i	$0.648897\pi$
$674$	−32.9561	−1.26942
$675$	5.47636	0.210785
$676$	3.60387	0.138610
$677$	−0.287333	−0.0110431	−0.00552155	−	0.999985i	$-0.501758\pi$
$677$	−0.287333	−0.0110431	−0.00552155	+	0.999985i	$0.501758\pi$
$678$	−6.53677	−0.251043
$679$	−3.11649	−0.119600
$680$	−13.7668	−0.527932
$681$	−1.74241	−0.0667694
$682$	3.36914	0.129011
$683$	44.7800	1.71346	0.856729	−	0.515767i	$-0.172493\pi$
$683$	44.7800	1.71346	0.856729	+	0.515767i	$0.172493\pi$
$684$	65.2287	2.49408
$685$	−12.7371	−0.486661
$686$	−44.8854	−1.71373
$687$	−6.31211	−0.240822
$688$	14.5614	0.555147
$689$	0.512259	0.0195155
$690$	−5.89833	−0.224546
$691$	33.9291	1.29073	0.645363	−	0.763876i	$-0.276706\pi$
$691$	33.9291	1.29073	0.645363	+	0.763876i	$0.276706\pi$
$692$	−63.1417	−2.40029
$693$	−20.1271	−0.764565
$694$	−29.7543	−1.12946
$695$	5.48487	0.208053
$696$	2.63846	0.100011
$697$	−12.9703	−0.491287
$698$	−46.4110	−1.75668
$699$	3.68850	0.139512
$700$	33.7155	1.27432
$701$	−46.7646	−1.76627	−0.883137	−	0.469115i	$-0.844573\pi$
$701$	−46.7646	−1.76627	−0.883137	+	0.469115i	$0.844573\pi$
$702$	−3.57178	−0.134808
$703$	−65.1154	−2.45588
$704$	30.7525	1.15903
$705$	2.79737	0.105355
$706$	74.2375	2.79397
$707$	−41.5375	−1.56218
$708$	3.46841	0.130351
$709$	−6.07482	−0.228145	−0.114072	−	0.993472i	$-0.536390\pi$
$709$	−6.07482	−0.228145	−0.114072	+	0.993472i	$0.536390\pi$
$710$	10.0786	0.378243
$711$	−34.3875	−1.28963
$712$	−1.79362	−0.0672189
$713$	4.47945	0.167757
$714$	4.80431	0.179797
$715$	3.11417	0.116463
$716$	−14.0196	−0.523938
$717$	−4.82619	−0.180237
$718$	29.4112	1.09762
$719$	−23.6083	−0.880441	−0.440221	−	0.897890i	$-0.645100\pi$
$719$	−23.6083	−0.880441	−0.440221	+	0.897890i	$0.645100\pi$
$720$	−6.11718	−0.227974
$721$	4.69893	0.174997
$722$	−45.0248	−1.67565
$723$	4.38443	0.163059
$724$	45.8005	1.70216
$725$	9.92193	0.368491
$726$	2.36110	0.0876287
$727$	42.1944	1.56490	0.782452	−	0.622711i	$-0.213969\pi$
$727$	42.1944	1.56490	0.782452	+	0.622711i	$0.213969\pi$
$728$	−9.78638	−0.362707
$729$	−23.5727	−0.873064
$730$	−30.0345	−1.11163
$731$	25.3358	0.937078
$732$	1.83844	0.0679509
$733$	10.9044	0.402763	0.201381	−	0.979513i	$-0.435457\pi$
$733$	10.9044	0.402763	0.201381	+	0.979513i	$0.435457\pi$
$734$	−66.0950	−2.43961
$735$	−0.105996	−0.00390971
$736$	28.2950	1.04297
$737$	25.4298	0.936718
$738$	−29.0988	−1.07114
$739$	29.8737	1.09892	0.549460	−	0.835520i	$-0.314833\pi$
$739$	29.8737	1.09892	0.549460	+	0.835520i	$0.314833\pi$
$740$	44.5535	1.63782
$741$	1.56747	0.0575823
$742$	−3.12567	−0.114747
$743$	−36.3622	−1.33400	−0.666999	−	0.745058i	$-0.732422\pi$
$743$	−36.3622	−1.33400	−0.666999	+	0.745058i	$0.732422\pi$
$744$	−0.516384	−0.0189315
$745$	4.67100	0.171132
$746$	35.9013	1.31444
$747$	−1.36661	−0.0500015
$748$	−29.6936	−1.08570
$749$	−3.14866	−0.115049
$750$	6.07937	0.221987
$751$	17.3540	0.633257	0.316628	−	0.948550i	$-0.397449\pi$
$751$	17.3540	0.633257	0.316628	+	0.948550i	$0.397449\pi$
$752$	16.7331	0.610192
$753$	−7.27826	−0.265234
$754$	−6.47126	−0.235669
$755$	−7.20086	−0.262066
$756$	14.0159	0.509752
$757$	40.7801	1.48218	0.741090	−	0.671406i	$-0.234309\pi$
$757$	40.7801	1.48218	0.741090	+	0.671406i	$0.234309\pi$
$758$	4.53194	0.164608
$759$	−5.66186	−0.205513
$760$	27.4064	0.994136
$761$	6.63784	0.240621	0.120311	−	0.992736i	$-0.461611\pi$
$761$	6.63784	0.240621	0.120311	+	0.992736i	$0.461611\pi$
$762$	−2.56053	−0.0927581
$763$	27.6230	1.00002
$764$	−85.7691	−3.10302
$765$	−10.6435	−0.384815
$766$	72.9596	2.63614
$767$	−3.78589	−0.136701
$768$	−6.52351	−0.235397
$769$	−19.9729	−0.720242	−0.360121	−	0.932906i	$-0.617265\pi$
$769$	−19.9729	−0.720242	−0.360121	+	0.932906i	$0.617265\pi$
$770$	−19.0018	−0.684778
$771$	−2.78437	−0.100277
$772$	−85.3658	−3.07238
$773$	−14.2866	−0.513852	−0.256926	−	0.966431i	$-0.582710\pi$
$773$	−14.2866	−0.513852	−0.256926	+	0.966431i	$0.582710\pi$
$774$	56.8404	2.04309
$775$	−1.94186	−0.0697536
$776$	−4.59060	−0.164793
$777$	−6.91959	−0.248239
$778$	−36.5492	−1.31035
$779$	25.8209	0.925131
$780$	−1.07250	−0.0384015
$781$	9.67453	0.346182
$782$	−61.3885	−2.19525
$783$	4.12464	0.147403
$784$	−0.634036	−0.0226441
$785$	−8.82539	−0.314992
$786$	−9.78289	−0.348944
$787$	−8.47130	−0.301969	−0.150985	−	0.988536i	$-0.548244\pi$
$787$	−8.47130	−0.301969	−0.150985	+	0.988536i	$0.548244\pi$
$788$	−66.9136	−2.38370
$789$	−3.49874	−0.124558
$790$	−32.4649	−1.15505
$791$	−27.9986	−0.995514
$792$	−29.6473	−1.05347
$793$	−2.00673	−0.0712610
$794$	20.6973	0.734518
$795$	−0.152446	−0.00540671
$796$	−101.385	−3.59350
$797$	−16.4470	−0.582582	−0.291291	−	0.956634i	$-0.594085\pi$
$797$	−16.4470	−0.582582	−0.291291	+	0.956634i	$0.594085\pi$
$798$	−9.56426	−0.338571
$799$	29.1144	1.02999
$800$	−12.2660	−0.433668
$801$	−1.38670	−0.0489966
$802$	−54.6655	−1.93031
$803$	−28.8304	−1.01740
$804$	−8.75783	−0.308865
$805$	−25.2640	−0.890437
$806$	1.26652	0.0446111
$807$	−1.58446	−0.0557757
$808$	−61.1849	−2.15248
$809$	43.9842	1.54640	0.773202	−	0.634160i	$-0.218654\pi$
$809$	43.9842	1.54640	0.773202	+	0.634160i	$0.218654\pi$
$810$	−23.3412	−0.820125
$811$	45.2574	1.58920	0.794602	−	0.607131i	$-0.207680\pi$
$811$	45.2574	1.58920	0.794602	+	0.607131i	$0.207680\pi$
$812$	25.3935	0.891138
$813$	1.35828	0.0476370
$814$	66.5014	2.33087
$815$	−26.2879	−0.920826
$816$	1.40162	0.0490666
$817$	−50.4377	−1.76459
$818$	−1.22861	−0.0429573
$819$	−7.56612	−0.264382
$820$	−17.6673	−0.616968
$821$	−0.572243	−0.0199714	−0.00998571	−	0.999950i	$-0.503179\pi$
$821$	−0.572243	−0.0199714	−0.00998571	+	0.999950i	$0.503179\pi$
$822$	6.54749	0.228370
$823$	−35.6804	−1.24374	−0.621871	−	0.783120i	$-0.713627\pi$
$823$	−35.6804	−1.24374	−0.621871	+	0.783120i	$0.713627\pi$
$824$	6.92154	0.241123
$825$	2.45444	0.0854525
$826$	23.1005	0.803770
$827$	−11.1929	−0.389215	−0.194608	−	0.980881i	$-0.562343\pi$
$827$	−11.1929	−0.389215	−0.194608	+	0.980881i	$0.562343\pi$
$828$	−88.5709	−3.07805
$829$	26.5929	0.923611	0.461805	−	0.886981i	$-0.347202\pi$
$829$	26.5929	0.923611	0.461805	+	0.886981i	$0.347202\pi$
$830$	−1.29020	−0.0447836
$831$	0.639635	0.0221887
$832$	11.5604	0.400784
$833$	−1.10318	−0.0382229
$834$	−2.81948	−0.0976306
$835$	2.28280	0.0789994
$836$	59.1129	2.04446
$837$	−0.807250	−0.0279026
$838$	34.1633	1.18015
$839$	44.3237	1.53022	0.765112	−	0.643897i	$-0.222683\pi$
$839$	44.3237	1.53022	0.765112	+	0.643897i	$0.222683\pi$
$840$	2.91239	0.100487
$841$	−21.5271	−0.742313
$842$	10.7739	0.371292
$843$	1.31677	0.0453518
$844$	−63.1089	−2.17230
$845$	1.17067	0.0402722
$846$	65.3176	2.24567
$847$	10.1132	0.347492
$848$	−0.911890	−0.0313144
$849$	6.93524	0.238017
$850$	26.6122	0.912789
$851$	88.4172	3.03090
$852$	−3.33184	−0.114147
$853$	4.07104	0.139390	0.0696948	−	0.997568i	$-0.477797\pi$
$853$	4.07104	0.139390	0.0696948	+	0.997568i	$0.477797\pi$
$854$	12.2445	0.418999
$855$	21.1887	0.724637
$856$	−4.63798	−0.158523
$857$	5.86818	0.200453	0.100227	−	0.994965i	$-0.468043\pi$
$857$	5.86818	0.200453	0.100227	+	0.994965i	$0.468043\pi$
$858$	−1.60083	−0.0546514
$859$	11.2912	0.385251	0.192625	−	0.981272i	$-0.438300\pi$
$859$	11.2912	0.385251	0.192625	+	0.981272i	$0.438300\pi$
$860$	34.5106	1.17680
$861$	2.74390	0.0935119
$862$	−25.4371	−0.866391
$863$	56.3109	1.91685	0.958423	−	0.285353i	$-0.0921107\pi$
$863$	56.3109	1.91685	0.958423	+	0.285353i	$0.0921107\pi$
$864$	−5.09909	−0.173475
$865$	−20.5107	−0.697386
$866$	−45.1505	−1.53428
$867$	−1.88285	−0.0639448
$868$	−4.96987	−0.168688
$869$	−31.1634	−1.05715
$870$	1.92582	0.0652915
$871$	9.55948	0.323911
$872$	40.6888	1.37790
$873$	−3.54911	−0.120119
$874$	122.210	4.13382
$875$	26.0394	0.880292
$876$	9.92899	0.335469
$877$	−38.2820	−1.29269	−0.646346	−	0.763044i	$-0.723704\pi$
$877$	−38.2820	−1.29269	−0.646346	+	0.763044i	$0.723704\pi$
$878$	−13.6938	−0.462143
$879$	−1.64268	−0.0554064
$880$	−5.54364	−0.186876
$881$	−47.6529	−1.60547	−0.802734	−	0.596338i	$-0.796622\pi$
$881$	−47.6529	−1.60547	−0.802734	+	0.596338i	$0.796622\pi$
$882$	−2.47496	−0.0833364
$883$	17.6246	0.593114	0.296557	−	0.955015i	$-0.404162\pi$
$883$	17.6246	0.593114	0.296557	+	0.955015i	$0.404162\pi$
$884$	−11.1623	−0.375429
$885$	1.12667	0.0378725
$886$	−9.40742	−0.316048
$887$	12.0489	0.404564	0.202282	−	0.979327i	$-0.435164\pi$
$887$	12.0489	0.404564	0.202282	+	0.979327i	$0.435164\pi$
$888$	−10.1926	−0.342041
$889$	−10.9673	−0.367833
$890$	−1.30917	−0.0438835
$891$	−22.4054	−0.750609
$892$	63.1257	2.11360
$893$	−57.9599	−1.93955
$894$	−2.40111	−0.0803052
$895$	−4.55409	−0.152226
$896$	−53.1166	−1.77450
$897$	−2.12839	−0.0710648
$898$	49.5854	1.65469
$899$	−1.46255	−0.0487789
$900$	38.3958	1.27986
$901$	−1.58663	−0.0528582
$902$	−26.3705	−0.878042
$903$	−5.35984	−0.178364
$904$	−41.2420	−1.37169
$905$	14.8777	0.494551
$906$	3.70158	0.122977
$907$	12.7168	0.422256	0.211128	−	0.977458i	$-0.432286\pi$
$907$	12.7168	0.422256	0.211128	+	0.977458i	$0.432286\pi$
$908$	−24.7017	−0.819756
$909$	−47.3037	−1.56897
$910$	−7.14311	−0.236792
$911$	3.44702	0.114205	0.0571024	−	0.998368i	$-0.481814\pi$
$911$	3.44702	0.114205	0.0571024	+	0.998368i	$0.481814\pi$
$912$	−2.79030	−0.0923962
$913$	−1.23848	−0.0409876
$914$	−34.8217	−1.15180
$915$	0.597194	0.0197426
$916$	−89.4851	−2.95667
$917$	−41.9025	−1.38374
$918$	11.0629	0.365131
$919$	−44.1399	−1.45604	−0.728020	−	0.685556i	$-0.759559\pi$
$919$	−44.1399	−1.45604	−0.728020	+	0.685556i	$0.759559\pi$
$920$	−37.2139	−1.22691
$921$	4.96512	0.163606
$922$	−30.8044	−1.01449
$923$	3.63682	0.119707
$924$	6.28173	0.206654
$925$	−38.3292	−1.26026
$926$	62.4232	2.05135
$927$	5.35123	0.175758
$928$	−9.23839	−0.303265
$929$	−29.1404	−0.956064	−0.478032	−	0.878342i	$-0.658650\pi$
$929$	−29.1404	−0.956064	−0.478032	+	0.878342i	$0.658650\pi$
$930$	−0.376910	−0.0123594
$931$	2.19617	0.0719766
$932$	52.2909	1.71284
$933$	−3.76340	−0.123208
$934$	−25.5899	−0.837327
$935$	−9.64555	−0.315443
$936$	−11.1449	−0.364283
$937$	4.85175	0.158500	0.0792499	−	0.996855i	$-0.474748\pi$
$937$	4.85175	0.158500	0.0792499	+	0.996855i	$0.474748\pi$
$938$	−58.3294	−1.90452
$939$	1.20565	0.0393448
$940$	39.6575	1.29349
$941$	39.0103	1.27170	0.635849	−	0.771813i	$-0.280650\pi$
$941$	39.0103	1.27170	0.635849	+	0.771813i	$0.280650\pi$
$942$	4.53667	0.147813
$943$	−35.0610	−1.14174
$944$	6.73940	0.219349
$945$	4.55286	0.148105
$946$	51.5112	1.67477
$947$	57.3354	1.86315	0.931575	−	0.363550i	$-0.118435\pi$
$947$	57.3354	1.86315	0.931575	+	0.363550i	$0.118435\pi$
$948$	10.7324	0.348574
$949$	−10.8378	−0.351811
$950$	−52.9786	−1.71885
$951$	−3.57617	−0.115965
$952$	30.3115	0.982401
$953$	−56.2867	−1.82330	−0.911652	−	0.410964i	$-0.865192\pi$
$953$	−56.2867	−1.82330	−0.911652	+	0.410964i	$0.865192\pi$
$954$	−3.55957	−0.115245
$955$	−27.8610	−0.901560
$956$	−68.4196	−2.21285
$957$	1.84861	0.0597572
$958$	−62.3546	−2.01459
$959$	28.0445	0.905603
$960$	−3.44032	−0.111036
$961$	−30.7138	−0.990766
$962$	24.9990	0.806000
$963$	−3.58575	−0.115549
$964$	62.1569	2.00194
$965$	−27.7299	−0.892658
$966$	12.9869	0.417845
$967$	−60.0770	−1.93195	−0.965973	−	0.258642i	$-0.916725\pi$
$967$	−60.0770	−1.93195	−0.965973	+	0.258642i	$0.916725\pi$
$968$	14.8967	0.478799
$969$	−4.85494	−0.155963
$970$	−3.35069	−0.107584
$971$	33.6589	1.08017	0.540083	−	0.841612i	$-0.318393\pi$
$971$	33.6589	1.08017	0.540083	+	0.841612i	$0.318393\pi$
$972$	24.0292	0.770736
$973$	−12.0765	−0.387155
$974$	86.7040	2.77817
$975$	0.922665	0.0295489
$976$	3.57225	0.114345
$977$	18.8800	0.604026	0.302013	−	0.953304i	$-0.402341\pi$
$977$	18.8800	0.604026	0.302013	+	0.953304i	$0.402341\pi$
$978$	13.5132	0.432106
$979$	−1.25669	−0.0401638
$980$	−1.50267	−0.0480011
$981$	31.4576	1.00436
$982$	−45.8272	−1.46241
$983$	−5.58231	−0.178048	−0.0890240	−	0.996029i	$-0.528375\pi$
$983$	−5.58231	−0.178048	−0.0890240	+	0.996029i	$0.528375\pi$
$984$	4.04178	0.128847
$985$	−21.7360	−0.692567
$986$	20.0435	0.638316
$987$	−6.15920	−0.196050
$988$	22.2216	0.706962
$989$	68.4869	2.17776
$990$	−21.6396	−0.687753
$991$	−6.12039	−0.194421	−0.0972103	−	0.995264i	$-0.530992\pi$
$991$	−6.12039	−0.194421	−0.0972103	+	0.995264i	$0.530992\pi$
$992$	1.80808	0.0574067
$993$	2.75364	0.0873841
$994$	−22.1909	−0.703853
$995$	−32.9336	−1.04407
$996$	0.426522	0.0135149
$997$	−46.8442	−1.48357	−0.741786	−	0.670637i	$-0.766021\pi$
$997$	−46.8442	−1.48357	−0.741786	+	0.670637i	$0.766021\pi$
$998$	−28.9384	−0.916029
$999$	−15.9338	−0.504124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.c.1.16	✓	151

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.c.1.16	✓	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)

\(f(q)\)	\(=\)	\(q-2.36725 q^{2} +0.254210 q^{3} +3.60387 q^{4} +1.17067 q^{5} -0.601779 q^{6} -2.57756 q^{7} -3.79676 q^{8} -2.93538 q^{9} +O(q^{10})\)	\(q-2.36725 q^{2} +0.254210 q^{3} +3.60387 q^{4} +1.17067 q^{5} -0.601779 q^{6} -2.57756 q^{7} -3.79676 q^{8} -2.93538 q^{9} -2.77126 q^{10} -2.66016 q^{11} +0.916140 q^{12} -1.00000 q^{13} +6.10173 q^{14} +0.297596 q^{15} +1.78014 q^{16} +3.09731 q^{17} +6.94877 q^{18} -6.16603 q^{19} +4.21894 q^{20} -0.655243 q^{21} +6.29727 q^{22} +8.37256 q^{23} -0.965175 q^{24} -3.62953 q^{25} +2.36725 q^{26} -1.50883 q^{27} -9.28920 q^{28} -2.73366 q^{29} -0.704484 q^{30} +0.535016 q^{31} +3.37949 q^{32} -0.676240 q^{33} -7.33211 q^{34} -3.01747 q^{35} -10.5787 q^{36} +10.5604 q^{37} +14.5965 q^{38} -0.254210 q^{39} -4.44475 q^{40} -4.18761 q^{41} +1.55112 q^{42} +8.17993 q^{43} -9.58688 q^{44} -3.43635 q^{45} -19.8199 q^{46} +9.39988 q^{47} +0.452529 q^{48} -0.356173 q^{49} +8.59201 q^{50} +0.787368 q^{51} -3.60387 q^{52} -0.512259 q^{53} +3.57178 q^{54} -3.11417 q^{55} +9.78638 q^{56} -1.56747 q^{57} +6.47126 q^{58} +3.78589 q^{59} +1.07250 q^{60} +2.00673 q^{61} -1.26652 q^{62} +7.56612 q^{63} -11.5604 q^{64} -1.17067 q^{65} +1.60083 q^{66} -9.55948 q^{67} +11.1623 q^{68} +2.12839 q^{69} +7.14311 q^{70} -3.63682 q^{71} +11.1449 q^{72} +10.8378 q^{73} -24.9990 q^{74} -0.922665 q^{75} -22.2216 q^{76} +6.85673 q^{77} +0.601779 q^{78} +11.7148 q^{79} +2.08395 q^{80} +8.42257 q^{81} +9.91312 q^{82} +0.465564 q^{83} -2.36141 q^{84} +3.62593 q^{85} -19.3639 q^{86} -0.694925 q^{87} +10.1000 q^{88} +0.472409 q^{89} +8.13471 q^{90} +2.57756 q^{91} +30.1736 q^{92} +0.136006 q^{93} -22.2519 q^{94} -7.21838 q^{95} +0.859101 q^{96} +1.20908 q^{97} +0.843150 q^{98} +7.80858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) 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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