LMFDB - Embedded newform 8007.2.a.f.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00652 q^{2} -1.00000 q^{3} +2.02613 q^{4} -1.83243 q^{5} +2.00652 q^{6} -1.92586 q^{7} -0.0524220 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.00652 q^{2} -1.00000 q^{3} +2.02613 q^{4} -1.83243 q^{5} +2.00652 q^{6} -1.92586 q^{7} -0.0524220 q^{8} +1.00000 q^{9} +3.67682 q^{10} +3.86289 q^{11} -2.02613 q^{12} +6.14415 q^{13} +3.86428 q^{14} +1.83243 q^{15} -3.94707 q^{16} -1.00000 q^{17} -2.00652 q^{18} +0.890505 q^{19} -3.71274 q^{20} +1.92586 q^{21} -7.75098 q^{22} -5.99726 q^{23} +0.0524220 q^{24} -1.64219 q^{25} -12.3284 q^{26} -1.00000 q^{27} -3.90204 q^{28} +6.30206 q^{29} -3.67682 q^{30} -6.65982 q^{31} +8.02471 q^{32} -3.86289 q^{33} +2.00652 q^{34} +3.52902 q^{35} +2.02613 q^{36} +3.43677 q^{37} -1.78682 q^{38} -6.14415 q^{39} +0.0960598 q^{40} +10.4479 q^{41} -3.86428 q^{42} +9.99772 q^{43} +7.82671 q^{44} -1.83243 q^{45} +12.0336 q^{46} -1.34629 q^{47} +3.94707 q^{48} -3.29105 q^{49} +3.29508 q^{50} +1.00000 q^{51} +12.4488 q^{52} -11.2524 q^{53} +2.00652 q^{54} -7.07850 q^{55} +0.100958 q^{56} -0.890505 q^{57} -12.6452 q^{58} +2.14834 q^{59} +3.71274 q^{60} -14.3474 q^{61} +13.3631 q^{62} -1.92586 q^{63} -8.20762 q^{64} -11.2587 q^{65} +7.75098 q^{66} -4.33338 q^{67} -2.02613 q^{68} +5.99726 q^{69} -7.08105 q^{70} +8.04138 q^{71} -0.0524220 q^{72} -0.195770 q^{73} -6.89594 q^{74} +1.64219 q^{75} +1.80428 q^{76} -7.43940 q^{77} +12.3284 q^{78} -11.0344 q^{79} +7.23274 q^{80} +1.00000 q^{81} -20.9639 q^{82} -9.85496 q^{83} +3.90204 q^{84} +1.83243 q^{85} -20.0606 q^{86} -6.30206 q^{87} -0.202501 q^{88} -8.91778 q^{89} +3.67682 q^{90} -11.8328 q^{91} -12.1512 q^{92} +6.65982 q^{93} +2.70136 q^{94} -1.63179 q^{95} -8.02471 q^{96} -11.5640 q^{97} +6.60356 q^{98} +3.86289 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	−2.00652	−1.41882	−0.709412	−	0.704794i	$-0.751040\pi$
$2$	−2.00652	−1.41882	−0.709412	+	0.704794i	$0.751040\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.02613	1.01306
$5$	−1.83243	−0.819489	−0.409745	−	0.912200i	$-0.634382\pi$
$5$	−1.83243	−0.819489	−0.409745	+	0.912200i	$0.634382\pi$
$6$	2.00652	0.819159
$7$	−1.92586	−0.727908	−0.363954	−	0.931417i	$-0.618573\pi$
$7$	−1.92586	−0.727908	−0.363954	+	0.931417i	$0.618573\pi$
$8$	−0.0524220	−0.0185340
$9$	1.00000	0.333333
$10$	3.67682	1.16271
$11$	3.86289	1.16471	0.582353	−	0.812936i	$-0.302132\pi$
$11$	3.86289	1.16471	0.582353	+	0.812936i	$0.302132\pi$
$12$	−2.02613	−0.584892
$13$	6.14415	1.70408	0.852040	−	0.523477i	$-0.175365\pi$
$13$	6.14415	1.70408	0.852040	+	0.523477i	$0.175365\pi$
$14$	3.86428	1.03277
$15$	1.83243	0.473132
$16$	−3.94707	−0.986766
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.00652	−0.472941
$19$	0.890505	0.204296	0.102148	−	0.994769i	$-0.467428\pi$
$19$	0.890505	0.204296	0.102148	+	0.994769i	$0.467428\pi$
$20$	−3.71274	−0.830194
$21$	1.92586	0.420258
$22$	−7.75098	−1.65251
$23$	−5.99726	−1.25052	−0.625258	−	0.780418i	$-0.715006\pi$
$23$	−5.99726	−1.25052	−0.625258	+	0.780418i	$0.715006\pi$
$24$	0.0524220	0.0107006
$25$	−1.64219	−0.328437
$26$	−12.3284	−2.41779
$27$	−1.00000	−0.192450
$28$	−3.90204	−0.737416
$29$	6.30206	1.17026	0.585131	−	0.810939i	$-0.301043\pi$
$29$	6.30206	1.17026	0.585131	+	0.810939i	$0.301043\pi$
$30$	−3.67682	−0.671292
$31$	−6.65982	−1.19614	−0.598069	−	0.801444i	$-0.704065\pi$
$31$	−6.65982	−1.19614	−0.598069	+	0.801444i	$0.704065\pi$
$32$	8.02471	1.41858
$33$	−3.86289	−0.672443
$34$	2.00652	0.344115
$35$	3.52902	0.596513
$36$	2.02613	0.337688
$37$	3.43677	0.565001	0.282500	−	0.959267i	$-0.408836\pi$
$37$	3.43677	0.565001	0.282500	+	0.959267i	$0.408836\pi$
$38$	−1.78682	−0.289860
$39$	−6.14415	−0.983851
$40$	0.0960598	0.0151884
$41$	10.4479	1.63168	0.815842	−	0.578274i	$-0.196274\pi$
$41$	10.4479	1.63168	0.815842	+	0.578274i	$0.196274\pi$
$42$	−3.86428	−0.596272
$43$	9.99772	1.52464	0.762319	−	0.647202i	$-0.224061\pi$
$43$	9.99772	1.52464	0.762319	+	0.647202i	$0.224061\pi$
$44$	7.82671	1.17992
$45$	−1.83243	−0.273163
$46$	12.0336	1.77426
$47$	−1.34629	−0.196376	−0.0981882	−	0.995168i	$-0.531305\pi$
$47$	−1.34629	−0.196376	−0.0981882	+	0.995168i	$0.531305\pi$
$48$	3.94707	0.569710
$49$	−3.29105	−0.470150
$50$	3.29508	0.465995
$51$	1.00000	0.140028
$52$	12.4488	1.72634
$53$	−11.2524	−1.54563	−0.772817	−	0.634629i	$-0.781153\pi$
$53$	−11.2524	−1.54563	−0.772817	+	0.634629i	$0.781153\pi$
$54$	2.00652	0.273053
$55$	−7.07850	−0.954464
$56$	0.100958	0.0134910
$57$	−0.890505	−0.117950
$58$	−12.6452	−1.66040
$59$	2.14834	0.279690	0.139845	−	0.990173i	$-0.455340\pi$
$59$	2.14834	0.279690	0.139845	+	0.990173i	$0.455340\pi$
$60$	3.71274	0.479313
$61$	−14.3474	−1.83700	−0.918501	−	0.395420i	$-0.870599\pi$
$61$	−14.3474	−1.83700	−0.918501	+	0.395420i	$0.870599\pi$
$62$	13.3631	1.69711
$63$	−1.92586	−0.242636
$64$	−8.20762	−1.02595
$65$	−11.2587	−1.39648
$66$	7.75098	0.954079
$67$	−4.33338	−0.529406	−0.264703	−	0.964330i	$-0.585274\pi$
$67$	−4.33338	−0.529406	−0.264703	+	0.964330i	$0.585274\pi$
$68$	−2.02613	−0.245704
$69$	5.99726	0.721985
$70$	−7.08105	−0.846347
$71$	8.04138	0.954337	0.477168	−	0.878812i	$-0.341663\pi$
$71$	8.04138	0.954337	0.477168	+	0.878812i	$0.341663\pi$
$72$	−0.0524220	−0.00617799
$73$	−0.195770	−0.0229131	−0.0114565	−	0.999934i	$-0.503647\pi$
$73$	−0.195770	−0.0229131	−0.0114565	+	0.999934i	$0.503647\pi$
$74$	−6.89594	−0.801637
$75$	1.64219	0.189623
$76$	1.80428	0.206965
$77$	−7.43940	−0.847799
$78$	12.3284	1.39591
$79$	−11.0344	−1.24147	−0.620736	−	0.784020i	$-0.713166\pi$
$79$	−11.0344	−1.24147	−0.620736	+	0.784020i	$0.713166\pi$
$80$	7.23274	0.808645
$81$	1.00000	0.111111
$82$	−20.9639	−2.31507
$83$	−9.85496	−1.08172	−0.540861	−	0.841112i	$-0.681902\pi$
$83$	−9.85496	−1.08172	−0.540861	+	0.841112i	$0.681902\pi$
$84$	3.90204	0.425748
$85$	1.83243	0.198755
$86$	−20.0606	−2.16319
$87$	−6.30206	−0.675651
$88$	−0.202501	−0.0215866
$89$	−8.91778	−0.945283	−0.472641	−	0.881255i	$-0.656699\pi$
$89$	−8.91778	−0.945283	−0.472641	+	0.881255i	$0.656699\pi$
$90$	3.67682	0.387571
$91$	−11.8328	−1.24041
$92$	−12.1512	−1.26685
$93$	6.65982	0.690591
$94$	2.70136	0.278624
$95$	−1.63179	−0.167418
$96$	−8.02471	−0.819019
$97$	−11.5640	−1.17415	−0.587074	−	0.809534i	$-0.699720\pi$
$97$	−11.5640	−1.17415	−0.587074	+	0.809534i	$0.699720\pi$
$98$	6.60356	0.667061
$99$	3.86289	0.388235
$100$	−3.32728	−0.332728
$101$	−0.668211	−0.0664895	−0.0332448	−	0.999447i	$-0.510584\pi$
$101$	−0.668211	−0.0664895	−0.0332448	+	0.999447i	$0.510584\pi$
$102$	−2.00652	−0.198675
$103$	5.30849	0.523061	0.261531	−	0.965195i	$-0.415773\pi$
$103$	5.30849	0.523061	0.261531	+	0.965195i	$0.415773\pi$
$104$	−0.322088	−0.0315834
$105$	−3.52902	−0.344397
$106$	22.5781	2.19298
$107$	−5.00162	−0.483524	−0.241762	−	0.970336i	$-0.577725\pi$
$107$	−5.00162	−0.483524	−0.241762	+	0.970336i	$0.577725\pi$
$108$	−2.02613	−0.194964
$109$	−5.75725	−0.551445	−0.275722	−	0.961237i	$-0.588917\pi$
$109$	−5.75725	−0.551445	−0.275722	+	0.961237i	$0.588917\pi$
$110$	14.2032	1.35422
$111$	−3.43677	−0.326203
$112$	7.60151	0.718275
$113$	8.38572	0.788862	0.394431	−	0.918925i	$-0.370942\pi$
$113$	8.38572	0.788862	0.394431	+	0.918925i	$0.370942\pi$
$114$	1.78682	0.167351
$115$	10.9896	1.02478
$116$	12.7688	1.18555
$117$	6.14415	0.568027
$118$	−4.31069	−0.396831
$119$	1.92586	0.176544
$120$	−0.0960598	−0.00876902
$121$	3.92194	0.356540
$122$	28.7884	2.60638
$123$	−10.4479	−0.942054
$124$	−13.4936	−1.21176
$125$	12.1714	1.08864
$126$	3.86428	0.344258
$127$	4.29246	0.380894	0.190447	−	0.981697i	$-0.439006\pi$
$127$	4.29246	0.380894	0.190447	+	0.981697i	$0.439006\pi$
$128$	0.419340	0.0370648
$129$	−9.99772	−0.880250
$130$	22.5909	1.98135
$131$	8.79709	0.768605	0.384303	−	0.923207i	$-0.374442\pi$
$131$	8.79709	0.768605	0.384303	+	0.923207i	$0.374442\pi$
$132$	−7.82671	−0.681227
$133$	−1.71499	−0.148709
$134$	8.69501	0.751135
$135$	1.83243	0.157711
$136$	0.0524220	0.00449515
$137$	−7.81121	−0.667356	−0.333678	−	0.942687i	$-0.608290\pi$
$137$	−7.81121	−0.667356	−0.333678	+	0.942687i	$0.608290\pi$
$138$	−12.0336	−1.02437
$139$	−7.10591	−0.602716	−0.301358	−	0.953511i	$-0.597440\pi$
$139$	−7.10591	−0.602716	−0.301358	+	0.953511i	$0.597440\pi$
$140$	7.15023	0.604305
$141$	1.34629	0.113378
$142$	−16.1352	−1.35404
$143$	23.7342	1.98475
$144$	−3.94707	−0.328922
$145$	−11.5481	−0.959018
$146$	0.392816	0.0325096
$147$	3.29105	0.271441
$148$	6.96332	0.572381
$149$	−6.36133	−0.521140	−0.260570	−	0.965455i	$-0.583911\pi$
$149$	−6.36133	−0.521140	−0.260570	+	0.965455i	$0.583911\pi$
$150$	−3.29508	−0.269042
$151$	23.0557	1.87624	0.938122	−	0.346306i	$-0.112564\pi$
$151$	23.0557	1.87624	0.938122	+	0.346306i	$0.112564\pi$
$152$	−0.0466821	−0.00378641
$153$	−1.00000	−0.0808452
$154$	14.9273	1.20288
$155$	12.2037	0.980223
$156$	−12.4488	−0.996703
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	22.1408	1.76143
$159$	11.2524	0.892372
$160$	−14.7048	−1.16251
$161$	11.5499	0.910260
$162$	−2.00652	−0.157647
$163$	−12.4528	−0.975380	−0.487690	−	0.873017i	$-0.662160\pi$
$163$	−12.4528	−0.975380	−0.487690	+	0.873017i	$0.662160\pi$
$164$	21.1687	1.65300
$165$	7.07850	0.551060
$166$	19.7742	1.53477
$167$	16.1183	1.24727	0.623635	−	0.781716i	$-0.285655\pi$
$167$	16.1183	1.24727	0.623635	+	0.781716i	$0.285655\pi$
$168$	−0.100958	−0.00778905
$169$	24.7506	1.90389
$170$	−3.67682	−0.281999
$171$	0.890505	0.0680986
$172$	20.2566	1.54455
$173$	19.4724	1.48046	0.740230	−	0.672353i	$-0.234716\pi$
$173$	19.4724	1.48046	0.740230	+	0.672353i	$0.234716\pi$
$174$	12.6452	0.958631
$175$	3.16263	0.239072
$176$	−15.2471	−1.14929
$177$	−2.14834	−0.161479
$178$	17.8937	1.34119
$179$	15.4589	1.15545	0.577725	−	0.816231i	$-0.303940\pi$
$179$	15.4589	1.15545	0.577725	+	0.816231i	$0.303940\pi$
$180$	−3.71274	−0.276731
$181$	5.64307	0.419446	0.209723	−	0.977761i	$-0.432744\pi$
$181$	5.64307	0.419446	0.209723	+	0.977761i	$0.432744\pi$
$182$	23.7427	1.75993
$183$	14.3474	1.06059
$184$	0.314388	0.0231770
$185$	−6.29765	−0.463012
$186$	−13.3631	−0.979827
$187$	−3.86289	−0.282483
$188$	−2.72775	−0.198942
$189$	1.92586	0.140086
$190$	3.27422	0.237537
$191$	24.8861	1.80069	0.900346	−	0.435175i	$-0.143313\pi$
$191$	24.8861	1.80069	0.900346	+	0.435175i	$0.143313\pi$
$192$	8.20762	0.592334
$193$	−6.33620	−0.456089	−0.228045	−	0.973651i	$-0.573233\pi$
$193$	−6.33620	−0.456089	−0.228045	+	0.973651i	$0.573233\pi$
$194$	23.2034	1.66591
$195$	11.2587	0.806256
$196$	−6.66808	−0.476292
$197$	−25.6014	−1.82403	−0.912013	−	0.410162i	$-0.865472\pi$
$197$	−25.6014	−1.82403	−0.912013	+	0.410162i	$0.865472\pi$
$198$	−7.75098	−0.550838
$199$	5.72472	0.405814	0.202907	−	0.979198i	$-0.434961\pi$
$199$	5.72472	0.405814	0.202907	+	0.979198i	$0.434961\pi$
$200$	0.0860866	0.00608725
$201$	4.33338	0.305653
$202$	1.34078	0.0943370
$203$	−12.1369	−0.851843
$204$	2.02613	0.141857
$205$	−19.1450	−1.33715
$206$	−10.6516	−0.742132
$207$	−5.99726	−0.416838
$208$	−24.2514	−1.68153
$209$	3.43993	0.237945
$210$	7.08105	0.488639
$211$	0.0723951	0.00498388	0.00249194	−	0.999997i	$-0.499207\pi$
$211$	0.0723951	0.00498388	0.00249194	+	0.999997i	$0.499207\pi$
$212$	−22.7988	−1.56582
$213$	−8.04138	−0.550987
$214$	10.0358	0.686036
$215$	−18.3202	−1.24942
$216$	0.0524220	0.00356686
$217$	12.8259	0.870679
$218$	11.5520	0.782403
$219$	0.195770	0.0132289
$220$	−14.3419	−0.966932
$221$	−6.14415	−0.413300
$222$	6.89594	0.462825
$223$	−22.9352	−1.53585	−0.767926	−	0.640538i	$-0.778711\pi$
$223$	−22.9352	−1.53585	−0.767926	+	0.640538i	$0.778711\pi$
$224$	−15.4545	−1.03260
$225$	−1.64219	−0.109479
$226$	−16.8261	−1.11926
$227$	24.2755	1.61122	0.805609	−	0.592447i	$-0.201838\pi$
$227$	24.2755	1.61122	0.805609	+	0.592447i	$0.201838\pi$
$228$	−1.80428	−0.119491
$229$	5.32925	0.352167	0.176083	−	0.984375i	$-0.443657\pi$
$229$	5.32925	0.352167	0.176083	+	0.984375i	$0.443657\pi$
$230$	−22.0508	−1.45399
$231$	7.43940	0.489477
$232$	−0.330366	−0.0216896
$233$	−5.24595	−0.343674	−0.171837	−	0.985125i	$-0.554970\pi$
$233$	−5.24595	−0.343674	−0.171837	+	0.985125i	$0.554970\pi$
$234$	−12.3284	−0.805930
$235$	2.46699	0.160928
$236$	4.35281	0.283344
$237$	11.0344	0.716764
$238$	−3.86428	−0.250484
$239$	−2.19209	−0.141795	−0.0708974	−	0.997484i	$-0.522586\pi$
$239$	−2.19209	−0.141795	−0.0708974	+	0.997484i	$0.522586\pi$
$240$	−7.23274	−0.466871
$241$	7.77245	0.500668	0.250334	−	0.968160i	$-0.419460\pi$
$241$	7.77245	0.500668	0.250334	+	0.968160i	$0.419460\pi$
$242$	−7.86946	−0.505868
$243$	−1.00000	−0.0641500
$244$	−29.0697	−1.86100
$245$	6.03063	0.385283
$246$	20.9639	1.33661
$247$	5.47140	0.348137
$248$	0.349121	0.0221692
$249$	9.85496	0.624533
$250$	−24.4221	−1.54459
$251$	25.2611	1.59447	0.797234	−	0.603670i	$-0.206296\pi$
$251$	25.2611	1.59447	0.797234	+	0.603670i	$0.206296\pi$
$252$	−3.90204	−0.245805
$253$	−23.1668	−1.45648
$254$	−8.61291	−0.540422
$255$	−1.83243	−0.114751
$256$	15.5738	0.973365
$257$	−9.97646	−0.622314	−0.311157	−	0.950358i	$-0.600717\pi$
$257$	−9.97646	−0.622314	−0.311157	+	0.950358i	$0.600717\pi$
$258$	20.0606	1.24892
$259$	−6.61874	−0.411268
$260$	−22.8116	−1.41472
$261$	6.30206	0.390088
$262$	−17.6515	−1.09052
$263$	14.1949	0.875293	0.437646	−	0.899147i	$-0.355812\pi$
$263$	14.1949	0.875293	0.437646	+	0.899147i	$0.355812\pi$
$264$	0.202501	0.0124630
$265$	20.6193	1.26663
$266$	3.44117	0.210991
$267$	8.91778	0.545759
$268$	−8.77997	−0.536322
$269$	−8.01771	−0.488849	−0.244424	−	0.969668i	$-0.578599\pi$
$269$	−8.01771	−0.488849	−0.244424	+	0.969668i	$0.578599\pi$
$270$	−3.67682	−0.223764
$271$	6.63486	0.403039	0.201519	−	0.979485i	$-0.435412\pi$
$271$	6.63486	0.403039	0.201519	+	0.979485i	$0.435412\pi$
$272$	3.94707	0.239326
$273$	11.8328	0.716153
$274$	15.6733	0.946862
$275$	−6.34359	−0.382533
$276$	12.1512	0.731416
$277$	11.7417	0.705490	0.352745	−	0.935719i	$-0.385248\pi$
$277$	11.7417	0.705490	0.352745	+	0.935719i	$0.385248\pi$
$278$	14.2582	0.855148
$279$	−6.65982	−0.398713
$280$	−0.184998	−0.0110558
$281$	−25.2715	−1.50757	−0.753785	−	0.657121i	$-0.771774\pi$
$281$	−25.2715	−1.50757	−0.753785	+	0.657121i	$0.771774\pi$
$282$	−2.70136	−0.160863
$283$	−16.3299	−0.970709	−0.485355	−	0.874317i	$-0.661309\pi$
$283$	−16.3299	−0.970709	−0.485355	+	0.874317i	$0.661309\pi$
$284$	16.2929	0.966803
$285$	1.63179	0.0966590
$286$	−47.6231	−2.81602
$287$	−20.1212	−1.18772
$288$	8.02471	0.472861
$289$	1.00000	0.0588235
$290$	23.1715	1.36068
$291$	11.5640	0.677894
$292$	−0.396654	−0.0232124
$293$	16.9866	0.992370	0.496185	−	0.868217i	$-0.334734\pi$
$293$	16.9866	0.992370	0.496185	+	0.868217i	$0.334734\pi$
$294$	−6.60356	−0.385128
$295$	−3.93669	−0.229203
$296$	−0.180162	−0.0104717
$297$	−3.86289	−0.224148
$298$	12.7641	0.739407
$299$	−36.8481	−2.13098
$300$	3.32728	0.192100
$301$	−19.2542	−1.10980
$302$	−46.2617	−2.66206
$303$	0.668211	0.0383877
$304$	−3.51488	−0.201592
$305$	26.2907	1.50540
$306$	2.00652	0.114705
$307$	−20.3919	−1.16382	−0.581912	−	0.813251i	$-0.697695\pi$
$307$	−20.3919	−1.16382	−0.581912	+	0.813251i	$0.697695\pi$
$308$	−15.0732	−0.858873
$309$	−5.30849	−0.301990
$310$	−24.4869	−1.39076
$311$	0.253731	0.0143878	0.00719389	−	0.999974i	$-0.497710\pi$
$311$	0.253731	0.0143878	0.00719389	+	0.999974i	$0.497710\pi$
$312$	0.322088	0.0182347
$313$	−18.9071	−1.06869	−0.534345	−	0.845266i	$-0.679442\pi$
$313$	−18.9071	−1.06869	−0.534345	+	0.845266i	$0.679442\pi$
$314$	2.00652	0.113235
$315$	3.52902	0.198838
$316$	−22.3572	−1.25769
$317$	−23.7699	−1.33505	−0.667525	−	0.744588i	$-0.732646\pi$
$317$	−23.7699	−1.33505	−0.667525	+	0.744588i	$0.732646\pi$
$318$	−22.5781	−1.26612
$319$	24.3442	1.36301
$320$	15.0399	0.840758
$321$	5.00162	0.279163
$322$	−23.1751	−1.29150
$323$	−0.890505	−0.0495490
$324$	2.02613	0.112563
$325$	−10.0898	−0.559683
$326$	24.9868	1.38389
$327$	5.75725	0.318377
$328$	−0.547699	−0.0302416
$329$	2.59277	0.142944
$330$	−14.2032	−0.781858
$331$	5.64108	0.310062	0.155031	−	0.987910i	$-0.450452\pi$
$331$	5.64108	0.310062	0.155031	+	0.987910i	$0.450452\pi$
$332$	−19.9674	−1.09585
$333$	3.43677	0.188334
$334$	−32.3417	−1.76966
$335$	7.94063	0.433843
$336$	−7.60151	−0.414696
$337$	6.97492	0.379948	0.189974	−	0.981789i	$-0.439160\pi$
$337$	6.97492	0.379948	0.189974	+	0.981789i	$0.439160\pi$
$338$	−49.6625	−2.70128
$339$	−8.38572	−0.455450
$340$	3.71274	0.201352
$341$	−25.7262	−1.39315
$342$	−1.78682	−0.0966200
$343$	19.8192	1.07013
$344$	−0.524100	−0.0282576
$345$	−10.9896	−0.591659
$346$	−39.0718	−2.10051
$347$	−26.4785	−1.42144	−0.710720	−	0.703475i	$-0.751631\pi$
$347$	−26.4785	−1.42144	−0.710720	+	0.703475i	$0.751631\pi$
$348$	−12.7688	−0.684477
$349$	−36.0079	−1.92746	−0.963729	−	0.266884i	$-0.914006\pi$
$349$	−36.0079	−1.92746	−0.963729	+	0.266884i	$0.914006\pi$
$350$	−6.34587	−0.339201
$351$	−6.14415	−0.327950
$352$	30.9986	1.65223
$353$	24.4545	1.30158	0.650790	−	0.759258i	$-0.274438\pi$
$353$	24.4545	1.30158	0.650790	+	0.759258i	$0.274438\pi$
$354$	4.31069	0.229110
$355$	−14.7353	−0.782069
$356$	−18.0685	−0.957631
$357$	−1.92586	−0.101927
$358$	−31.0186	−1.63938
$359$	24.7684	1.30722	0.653612	−	0.756830i	$-0.273253\pi$
$359$	24.7684	1.30722	0.653612	+	0.756830i	$0.273253\pi$
$360$	0.0960598	0.00506280
$361$	−18.2070	−0.958263
$362$	−11.3229	−0.595120
$363$	−3.92194	−0.205849
$364$	−23.9747	−1.25662
$365$	0.358735	0.0187770
$366$	−28.7884	−1.50480
$367$	22.0639	1.15172	0.575862	−	0.817547i	$-0.304667\pi$
$367$	22.0639	1.15172	0.575862	+	0.817547i	$0.304667\pi$
$368$	23.6716	1.23397
$369$	10.4479	0.543895
$370$	12.6364	0.656933
$371$	21.6706	1.12508
$372$	13.4936	0.699612
$373$	0.584279	0.0302528	0.0151264	−	0.999886i	$-0.495185\pi$
$373$	0.584279	0.0302528	0.0151264	+	0.999886i	$0.495185\pi$
$374$	7.75098	0.400793
$375$	−12.1714	−0.628527
$376$	0.0705751	0.00363963
$377$	38.7208	1.99422
$378$	−3.86428	−0.198757
$379$	−18.6260	−0.956752	−0.478376	−	0.878155i	$-0.658774\pi$
$379$	−18.6260	−0.956752	−0.478376	+	0.878155i	$0.658774\pi$
$380$	−3.30622	−0.169605
$381$	−4.29246	−0.219909
$382$	−49.9344	−2.55487
$383$	22.8502	1.16759	0.583796	−	0.811901i	$-0.301567\pi$
$383$	22.8502	1.16759	0.583796	+	0.811901i	$0.301567\pi$
$384$	−0.419340	−0.0213994
$385$	13.6322	0.694762
$386$	12.7137	0.647111
$387$	9.99772	0.508213
$388$	−23.4301	−1.18949
$389$	2.92273	0.148188	0.0740942	−	0.997251i	$-0.476393\pi$
$389$	2.92273	0.148188	0.0740942	+	0.997251i	$0.476393\pi$
$390$	−22.5909	−1.14394
$391$	5.99726	0.303294
$392$	0.172523	0.00871375
$393$	−8.79709	−0.443754
$394$	51.3698	2.58797
$395$	20.2199	1.01737
$396$	7.82671	0.393307
$397$	−7.77571	−0.390252	−0.195126	−	0.980778i	$-0.562512\pi$
$397$	−7.77571	−0.390252	−0.195126	+	0.980778i	$0.562512\pi$
$398$	−11.4868	−0.575779
$399$	1.71499	0.0858570
$400$	6.48182	0.324091
$401$	35.4766	1.77162	0.885809	−	0.464050i	$-0.153604\pi$
$401$	35.4766	1.77162	0.885809	+	0.464050i	$0.153604\pi$
$402$	−8.69501	−0.433668
$403$	−40.9189	−2.03832
$404$	−1.35388	−0.0673581
$405$	−1.83243	−0.0910544
$406$	24.3529	1.20862
$407$	13.2759	0.658060
$408$	−0.0524220	−0.00259528
$409$	−8.22238	−0.406571	−0.203285	−	0.979120i	$-0.565162\pi$
$409$	−8.22238	−0.406571	−0.203285	+	0.979120i	$0.565162\pi$
$410$	38.4149	1.89718
$411$	7.81121	0.385298
$412$	10.7557	0.529894
$413$	−4.13741	−0.203589
$414$	12.0336	0.591420
$415$	18.0586	0.886460
$416$	49.3050	2.41738
$417$	7.10591	0.347978
$418$	−6.90228	−0.337602
$419$	−6.38908	−0.312127	−0.156064	−	0.987747i	$-0.549880\pi$
$419$	−6.38908	−0.312127	−0.156064	+	0.987747i	$0.549880\pi$
$420$	−7.15023	−0.348896
$421$	−3.46000	−0.168630	−0.0843151	−	0.996439i	$-0.526870\pi$
$421$	−3.46000	−0.168630	−0.0843151	+	0.996439i	$0.526870\pi$
$422$	−0.145262	−0.00707126
$423$	−1.34629	−0.0654588
$424$	0.589873	0.0286467
$425$	1.64219	0.0796577
$426$	16.1352	0.781753
$427$	27.6312	1.33717
$428$	−10.1339	−0.489841
$429$	−23.7342	−1.14590
$430$	36.7598	1.77271
$431$	−32.6003	−1.57030	−0.785152	−	0.619304i	$-0.787415\pi$
$431$	−32.6003	−1.57030	−0.785152	+	0.619304i	$0.787415\pi$
$432$	3.94707	0.189903
$433$	27.3070	1.31229	0.656145	−	0.754635i	$-0.272186\pi$
$433$	27.3070	1.31229	0.656145	+	0.754635i	$0.272186\pi$
$434$	−25.7354	−1.23534
$435$	11.5481	0.553689
$436$	−11.6649	−0.558648
$437$	−5.34059	−0.255475
$438$	−0.392816	−0.0187695
$439$	−11.6003	−0.553650	−0.276825	−	0.960920i	$-0.589282\pi$
$439$	−11.6003	−0.553650	−0.276825	+	0.960920i	$0.589282\pi$
$440$	0.371069	0.0176900
$441$	−3.29105	−0.156717
$442$	12.3284	0.586400
$443$	−24.5529	−1.16654	−0.583272	−	0.812277i	$-0.698228\pi$
$443$	−24.5529	−1.16654	−0.583272	+	0.812277i	$0.698228\pi$
$444$	−6.96332	−0.330465
$445$	16.3412	0.774649
$446$	46.0199	2.17910
$447$	6.36133	0.300881
$448$	15.8068	0.746799
$449$	−21.5051	−1.01489	−0.507444	−	0.861685i	$-0.669409\pi$
$449$	−21.5051	−1.01489	−0.507444	+	0.861685i	$0.669409\pi$
$450$	3.29508	0.155332
$451$	40.3590	1.90043
$452$	16.9905	0.799167
$453$	−23.0557	−1.08325
$454$	−48.7092	−2.28604
$455$	21.6828	1.01651
$456$	0.0466821	0.00218609
$457$	−32.7765	−1.53322	−0.766609	−	0.642114i	$-0.778057\pi$
$457$	−32.7765	−1.53322	−0.766609	+	0.642114i	$0.778057\pi$
$458$	−10.6933	−0.499663
$459$	1.00000	0.0466760
$460$	22.2663	1.03817
$461$	20.3967	0.949967	0.474983	−	0.879995i	$-0.342454\pi$
$461$	20.3967	0.949967	0.474983	+	0.879995i	$0.342454\pi$
$462$	−14.9273	−0.694482
$463$	−25.3223	−1.17683	−0.588414	−	0.808560i	$-0.700247\pi$
$463$	−25.3223	−1.17683	−0.588414	+	0.808560i	$0.700247\pi$
$464$	−24.8746	−1.15478
$465$	−12.2037	−0.565932
$466$	10.5261	0.487613
$467$	−25.1858	−1.16546	−0.582730	−	0.812666i	$-0.698015\pi$
$467$	−25.1858	−1.16546	−0.582730	+	0.812666i	$0.698015\pi$
$468$	12.4488	0.575447
$469$	8.34549	0.385359
$470$	−4.95006	−0.228329
$471$	1.00000	0.0460776
$472$	−0.112620	−0.00518377
$473$	38.6201	1.77576
$474$	−22.1408	−1.01696
$475$	−1.46238	−0.0670984
$476$	3.90204	0.178850
$477$	−11.2524	−0.515211
$478$	4.39848	0.201182
$479$	−2.19637	−0.100355	−0.0501775	−	0.998740i	$-0.515979\pi$
$479$	−2.19637	−0.100355	−0.0501775	+	0.998740i	$0.515979\pi$
$480$	14.7048	0.671177
$481$	21.1160	0.962807
$482$	−15.5956	−0.710359
$483$	−11.5499	−0.525539
$484$	7.94635	0.361198
$485$	21.1903	0.962201
$486$	2.00652	0.0910176
$487$	−14.5496	−0.659306	−0.329653	−	0.944102i	$-0.606932\pi$
$487$	−14.5496	−0.659306	−0.329653	+	0.944102i	$0.606932\pi$
$488$	0.752121	0.0340469
$489$	12.4528	0.563136
$490$	−12.1006	−0.546649
$491$	38.3968	1.73282	0.866412	−	0.499330i	$-0.166420\pi$
$491$	38.3968	1.73282	0.866412	+	0.499330i	$0.166420\pi$
$492$	−21.1687	−0.954360
$493$	−6.30206	−0.283830
$494$	−10.9785	−0.493945
$495$	−7.07850	−0.318155
$496$	26.2867	1.18031
$497$	−15.4866	−0.694669
$498$	−19.7742	−0.886102
$499$	1.97846	0.0885679	0.0442840	−	0.999019i	$-0.485899\pi$
$499$	1.97846	0.0885679	0.0442840	+	0.999019i	$0.485899\pi$
$500$	24.6607	1.10286
$501$	−16.1183	−0.720111
$502$	−50.6870	−2.26227
$503$	−9.12840	−0.407015	−0.203508	−	0.979073i	$-0.565234\pi$
$503$	−9.12840	−0.407015	−0.203508	+	0.979073i	$0.565234\pi$
$504$	0.100958	0.00449701
$505$	1.22445	0.0544875
$506$	46.4846	2.06649
$507$	−24.7506	−1.09921
$508$	8.69706	0.385870
$509$	−36.3313	−1.61036	−0.805179	−	0.593032i	$-0.797931\pi$
$509$	−36.3313	−1.61036	−0.805179	+	0.593032i	$0.797931\pi$
$510$	3.67682	0.162812
$511$	0.377025	0.0166786
$512$	−32.0879	−1.41810
$513$	−0.890505	−0.0393168
$514$	20.0180	0.882955
$515$	−9.72746	−0.428643
$516$	−20.2566	−0.891749
$517$	−5.20057	−0.228721
$518$	13.2806	0.583518
$519$	−19.4724	−0.854744
$520$	0.590206	0.0258822
$521$	−16.0917	−0.704991	−0.352495	−	0.935814i	$-0.614667\pi$
$521$	−16.0917	−0.704991	−0.352495	+	0.935814i	$0.614667\pi$
$522$	−12.6452	−0.553466
$523$	4.30024	0.188036	0.0940182	−	0.995570i	$-0.470029\pi$
$523$	4.30024	0.188036	0.0940182	+	0.995570i	$0.470029\pi$
$524$	17.8240	0.778645
$525$	−3.16263	−0.138028
$526$	−28.4823	−1.24189
$527$	6.65982	0.290106
$528$	15.2471	0.663545
$529$	12.9671	0.563788
$530$	−41.3730	−1.79713
$531$	2.14834	0.0932300
$532$	−3.47479	−0.150651
$533$	64.1933	2.78052
$534$	−17.8937	−0.774336
$535$	9.16513	0.396243
$536$	0.227164	0.00981200
$537$	−15.4589	−0.667100
$538$	16.0877	0.693590
$539$	−12.7130	−0.547587
$540$	3.71274	0.159771
$541$	25.1234	1.08014	0.540069	−	0.841620i	$-0.318398\pi$
$541$	25.1234	1.08014	0.540069	+	0.841620i	$0.318398\pi$
$542$	−13.3130	−0.571841
$543$	−5.64307	−0.242167
$544$	−8.02471	−0.344057
$545$	10.5498	0.451903
$546$	−23.7427	−1.01610
$547$	−37.1417	−1.58807	−0.794033	−	0.607875i	$-0.792022\pi$
$547$	−37.1417	−1.58807	−0.794033	+	0.607875i	$0.792022\pi$
$548$	−15.8265	−0.676074
$549$	−14.3474	−0.612334
$550$	12.7285	0.542747
$551$	5.61202	0.239080
$552$	−0.314388	−0.0133813
$553$	21.2508	0.903677
$554$	−23.5600	−1.00097
$555$	6.29765	0.267320
$556$	−14.3975	−0.610589
$557$	−27.2639	−1.15521	−0.577603	−	0.816317i	$-0.696012\pi$
$557$	−27.2639	−1.15521	−0.577603	+	0.816317i	$0.696012\pi$
$558$	13.3631	0.565704
$559$	61.4275	2.59811
$560$	−13.9293	−0.588619
$561$	3.86289	0.163091
$562$	50.7078	2.13898
$563$	−21.7338	−0.915971	−0.457986	−	0.888960i	$-0.651429\pi$
$563$	−21.7338	−0.915971	−0.457986	+	0.888960i	$0.651429\pi$
$564$	2.72775	0.114859
$565$	−15.3663	−0.646464
$566$	32.7662	1.37727
$567$	−1.92586	−0.0808786
$568$	−0.421545	−0.0176876
$569$	−8.37425	−0.351067	−0.175533	−	0.984473i	$-0.556165\pi$
$569$	−8.37425	−0.351067	−0.175533	+	0.984473i	$0.556165\pi$
$570$	−3.27422	−0.137142
$571$	−18.8762	−0.789944	−0.394972	−	0.918693i	$-0.629246\pi$
$571$	−18.8762	−0.789944	−0.394972	+	0.918693i	$0.629246\pi$
$572$	48.0885	2.01068
$573$	−24.8861	−1.03963
$574$	40.3736	1.68516
$575$	9.84861	0.410716
$576$	−8.20762	−0.341984
$577$	−15.8753	−0.660898	−0.330449	−	0.943824i	$-0.607200\pi$
$577$	−15.8753	−0.660898	−0.330449	+	0.943824i	$0.607200\pi$
$578$	−2.00652	−0.0834603
$579$	6.33620	0.263323
$580$	−23.3979	−0.971545
$581$	18.9793	0.787394
$582$	−23.2034	−0.961813
$583$	−43.4668	−1.80021
$584$	0.0102626	0.000424671 0
$585$	−11.2587	−0.465492
$586$	−34.0840	−1.40800
$587$	5.14095	0.212190	0.106095	−	0.994356i	$-0.466165\pi$
$587$	5.14095	0.212190	0.106095	+	0.994356i	$0.466165\pi$
$588$	6.66808	0.274987
$589$	−5.93060	−0.244366
$590$	7.89905	0.325199
$591$	25.6014	1.05310
$592$	−13.5651	−0.557524
$593$	−10.1063	−0.415017	−0.207509	−	0.978233i	$-0.566536\pi$
$593$	−10.1063	−0.415017	−0.207509	+	0.978233i	$0.566536\pi$
$594$	7.75098	0.318026
$595$	−3.52902	−0.144676
$596$	−12.8889	−0.527948
$597$	−5.72472	−0.234297
$598$	73.9364	3.02348
$599$	37.2839	1.52338	0.761688	−	0.647943i	$-0.224371\pi$
$599$	37.2839	1.52338	0.761688	+	0.647943i	$0.224371\pi$
$600$	−0.0860866	−0.00351447
$601$	−44.1884	−1.80248	−0.901241	−	0.433318i	$-0.857343\pi$
$601$	−44.1884	−1.80248	−0.901241	+	0.433318i	$0.857343\pi$
$602$	38.6340	1.57461
$603$	−4.33338	−0.176469
$604$	46.7137	1.90075
$605$	−7.18670	−0.292181
$606$	−1.34078	−0.0544655
$607$	−29.4875	−1.19686	−0.598430	−	0.801175i	$-0.704209\pi$
$607$	−29.4875	−1.19686	−0.598430	+	0.801175i	$0.704209\pi$
$608$	7.14605	0.289811
$609$	12.1369	0.491812
$610$	−52.7529	−2.13590
$611$	−8.27180	−0.334641
$612$	−2.02613	−0.0819013
$613$	−15.9150	−0.642802	−0.321401	−	0.946943i	$-0.604154\pi$
$613$	−15.9150	−0.642802	−0.321401	+	0.946943i	$0.604154\pi$
$614$	40.9167	1.65126
$615$	19.1450	0.772003
$616$	0.389988	0.0157131
$617$	−42.5319	−1.71227	−0.856134	−	0.516753i	$-0.827140\pi$
$617$	−42.5319	−1.71227	−0.856134	+	0.516753i	$0.827140\pi$
$618$	10.6516	0.428470
$619$	−2.25945	−0.0908151	−0.0454075	−	0.998969i	$-0.514459\pi$
$619$	−2.25945	−0.0908151	−0.0454075	+	0.998969i	$0.514459\pi$
$620$	24.7262	0.993027
$621$	5.99726	0.240662
$622$	−0.509117	−0.0204137
$623$	17.1744	0.688079
$624$	24.2514	0.970831
$625$	−14.0923	−0.563692
$626$	37.9374	1.51628
$627$	−3.43993	−0.137377
$628$	−2.02613	−0.0808512
$629$	−3.43677	−0.137033
$630$	−7.08105	−0.282116
$631$	12.9756	0.516551	0.258276	−	0.966071i	$-0.416846\pi$
$631$	12.9756	0.516551	0.258276	+	0.966071i	$0.416846\pi$
$632$	0.578447	0.0230094
$633$	−0.0723951	−0.00287745
$634$	47.6948	1.89420
$635$	−7.86565	−0.312139
$636$	22.7988	0.904029
$637$	−20.2207	−0.801174
$638$	−48.8471	−1.93387
$639$	8.04138	0.318112
$640$	−0.768413	−0.0303742
$641$	−26.5958	−1.05047	−0.525235	−	0.850957i	$-0.676022\pi$
$641$	−26.5958	−1.05047	−0.525235	+	0.850957i	$0.676022\pi$
$642$	−10.0358	−0.396083
$643$	−18.4224	−0.726509	−0.363254	−	0.931690i	$-0.618334\pi$
$643$	−18.4224	−0.726509	−0.363254	+	0.931690i	$0.618334\pi$
$644$	23.4016	0.922150
$645$	18.3202	0.721356
$646$	1.78682	0.0703014
$647$	29.4426	1.15751	0.578755	−	0.815502i	$-0.303539\pi$
$647$	29.4426	1.15751	0.578755	+	0.815502i	$0.303539\pi$
$648$	−0.0524220	−0.00205933
$649$	8.29881	0.325757
$650$	20.2455	0.794092
$651$	−12.8259	−0.502687
$652$	−25.2310	−0.988121
$653$	19.1072	0.747722	0.373861	−	0.927485i	$-0.378034\pi$
$653$	19.1072	0.747722	0.373861	+	0.927485i	$0.378034\pi$
$654$	−11.5520	−0.451721
$655$	−16.1201	−0.629864
$656$	−41.2385	−1.61009
$657$	−0.195770	−0.00763770
$658$	−5.20244	−0.202812
$659$	5.01331	0.195291	0.0976455	−	0.995221i	$-0.468869\pi$
$659$	5.01331	0.195291	0.0976455	+	0.995221i	$0.468869\pi$
$660$	14.3419	0.558259
$661$	17.5057	0.680893	0.340447	−	0.940264i	$-0.389422\pi$
$661$	17.5057	0.680893	0.340447	+	0.940264i	$0.389422\pi$
$662$	−11.3189	−0.439923
$663$	6.14415	0.238619
$664$	0.516616	0.0200486
$665$	3.14261	0.121865
$666$	−6.89594	−0.267212
$667$	−37.7951	−1.46343
$668$	32.6576	1.26356
$669$	22.9352	0.886725
$670$	−15.9330	−0.615547
$671$	−55.4226	−2.13957
$672$	15.4545	0.596170
$673$	−34.2041	−1.31847	−0.659235	−	0.751937i	$-0.729120\pi$
$673$	−34.2041	−1.31847	−0.659235	+	0.751937i	$0.729120\pi$
$674$	−13.9953	−0.539080
$675$	1.64219	0.0632078
$676$	50.1477	1.92876
$677$	−36.9819	−1.42133	−0.710664	−	0.703531i	$-0.751605\pi$
$677$	−36.9819	−1.42133	−0.710664	+	0.703531i	$0.751605\pi$
$678$	16.8261	0.646204
$679$	22.2707	0.854671
$680$	−0.0960598	−0.00368373
$681$	−24.2755	−0.930237
$682$	51.6201	1.97664
$683$	−15.5900	−0.596535	−0.298267	−	0.954482i	$-0.596409\pi$
$683$	−15.5900	−0.596535	−0.298267	+	0.954482i	$0.596409\pi$
$684$	1.80428	0.0689882
$685$	14.3135	0.546891
$686$	−39.7675	−1.51833
$687$	−5.32925	−0.203324
$688$	−39.4617	−1.50446
$689$	−69.1363	−2.63388
$690$	22.0508	0.839461
$691$	−12.4194	−0.472457	−0.236229	−	0.971698i	$-0.575911\pi$
$691$	−12.4194	−0.472457	−0.236229	+	0.971698i	$0.575911\pi$
$692$	39.4536	1.49980
$693$	−7.43940	−0.282600
$694$	53.1297	2.01678
$695$	13.0211	0.493919
$696$	0.330366	0.0125225
$697$	−10.4479	−0.395742
$698$	72.2505	2.73472
$699$	5.24595	0.198420
$700$	6.40788	0.242195
$701$	45.8308	1.73101	0.865504	−	0.500903i	$-0.166999\pi$
$701$	45.8308	1.73101	0.865504	+	0.500903i	$0.166999\pi$
$702$	12.3284	0.465304
$703$	3.06046	0.115427
$704$	−31.7052	−1.19493
$705$	−2.46699	−0.0929120
$706$	−49.0684	−1.84671
$707$	1.28688	0.0483982
$708$	−4.35281	−0.163588
$709$	12.1609	0.456712	0.228356	−	0.973578i	$-0.426665\pi$
$709$	12.1609	0.456712	0.228356	+	0.973578i	$0.426665\pi$
$710$	29.5667	1.10962
$711$	−11.0344	−0.413824
$712$	0.467488	0.0175198
$713$	39.9407	1.49579
$714$	3.86428	0.144617
$715$	−43.4913	−1.62648
$716$	31.3216	1.17054
$717$	2.19209	0.0818653
$718$	−49.6982	−1.85472
$719$	47.2832	1.76337	0.881683	−	0.471843i	$-0.156411\pi$
$719$	47.2832	1.76337	0.881683	+	0.471843i	$0.156411\pi$
$720$	7.23274	0.269548
$721$	−10.2234	−0.380740
$722$	36.5327	1.35961
$723$	−7.77245	−0.289061
$724$	11.4336	0.424925
$725$	−10.3491	−0.384358
$726$	7.86946	0.292063
$727$	10.2019	0.378366	0.189183	−	0.981942i	$-0.439416\pi$
$727$	10.2019	0.378366	0.189183	+	0.981942i	$0.439416\pi$
$728$	0.620298	0.0229898
$729$	1.00000	0.0370370
$730$	−0.719809	−0.0266413
$731$	−9.99772	−0.369779
$732$	29.0697	1.07445
$733$	−7.70062	−0.284429	−0.142214	−	0.989836i	$-0.545422\pi$
$733$	−7.70062	−0.284429	−0.142214	+	0.989836i	$0.545422\pi$
$734$	−44.2716	−1.63409
$735$	−6.03063	−0.222443
$736$	−48.1263	−1.77396
$737$	−16.7394	−0.616603
$738$	−20.9639	−0.771691
$739$	−16.0844	−0.591675	−0.295838	−	0.955238i	$-0.595599\pi$
$739$	−16.0844	−0.591675	−0.295838	+	0.955238i	$0.595599\pi$
$740$	−12.7598	−0.469060
$741$	−5.47140	−0.200997
$742$	−43.4824	−1.59629
$743$	1.92699	0.0706943	0.0353471	−	0.999375i	$-0.488746\pi$
$743$	1.92699	0.0706943	0.0353471	+	0.999375i	$0.488746\pi$
$744$	−0.349121	−0.0127994
$745$	11.6567	0.427069
$746$	−1.17237	−0.0429235
$747$	−9.85496	−0.360574
$748$	−7.82671	−0.286173
$749$	9.63243	0.351961
$750$	24.4221	0.891769
$751$	28.0063	1.02196	0.510982	−	0.859591i	$-0.329282\pi$
$751$	28.0063	1.02196	0.510982	+	0.859591i	$0.329282\pi$
$752$	5.31389	0.193778
$753$	−25.2611	−0.920567
$754$	−77.6940	−2.82945
$755$	−42.2480	−1.53756
$756$	3.90204	0.141916
$757$	7.75500	0.281860	0.140930	−	0.990020i	$-0.454991\pi$
$757$	7.75500	0.281860	0.140930	+	0.990020i	$0.454991\pi$
$758$	37.3734	1.35746
$759$	23.1668	0.840901
$760$	0.0855418	0.00310293
$761$	15.5413	0.563373	0.281686	−	0.959507i	$-0.409106\pi$
$761$	15.5413	0.563373	0.281686	+	0.959507i	$0.409106\pi$
$762$	8.61291	0.312013
$763$	11.0877	0.401401
$764$	50.4223	1.82421
$765$	1.83243	0.0662518
$766$	−45.8494	−1.65661
$767$	13.1997	0.476614
$768$	−15.5738	−0.561972
$769$	−13.4400	−0.484657	−0.242329	−	0.970194i	$-0.577911\pi$
$769$	−13.4400	−0.484657	−0.242329	+	0.970194i	$0.577911\pi$
$770$	−27.3533	−0.985745
$771$	9.97646	0.359293
$772$	−12.8379	−0.462047
$773$	−35.4138	−1.27374	−0.636872	−	0.770969i	$-0.719772\pi$
$773$	−35.4138	−1.27374	−0.636872	+	0.770969i	$0.719772\pi$
$774$	−20.0606	−0.721065
$775$	10.9367	0.392856
$776$	0.606208	0.0217616
$777$	6.61874	0.237446
$778$	−5.86453	−0.210253
$779$	9.30389	0.333347
$780$	22.8116	0.816788
$781$	31.0630	1.11152
$782$	−12.0336	−0.430322
$783$	−6.30206	−0.225217
$784$	12.9900	0.463928
$785$	1.83243	0.0654024
$786$	17.6515	0.629610
$787$	39.4330	1.40564	0.702818	−	0.711370i	$-0.251925\pi$
$787$	39.4330	1.40564	0.702818	+	0.711370i	$0.251925\pi$
$788$	−51.8717	−1.84785
$789$	−14.1949	−0.505351
$790$	−40.5716	−1.44347
$791$	−16.1498	−0.574219
$792$	−0.202501	−0.00719554
$793$	−88.1528	−3.13040
$794$	15.6021	0.553698
$795$	−20.6193	−0.731290
$796$	11.5990	0.411116
$797$	36.5285	1.29390	0.646952	−	0.762530i	$-0.276043\pi$
$797$	36.5285	1.29390	0.646952	+	0.762530i	$0.276043\pi$
$798$	−3.44117	−0.121816
$799$	1.34629	0.0476283
$800$	−13.1781	−0.465915
$801$	−8.91778	−0.315094
$802$	−71.1846	−2.51362
$803$	−0.756237	−0.0266870
$804$	8.77997	0.309646
$805$	−21.1644	−0.745948
$806$	82.1046	2.89201
$807$	8.01771	0.282237
$808$	0.0350290	0.00123231
$809$	−48.4476	−1.70333	−0.851663	−	0.524090i	$-0.824406\pi$
$809$	−48.4476	−1.70333	−0.851663	+	0.524090i	$0.824406\pi$
$810$	3.67682	0.129190
$811$	9.64546	0.338698	0.169349	−	0.985556i	$-0.445834\pi$
$811$	9.64546	0.338698	0.169349	+	0.985556i	$0.445834\pi$
$812$	−24.5909	−0.862971
$813$	−6.63486	−0.232695
$814$	−26.6383	−0.933671
$815$	22.8190	0.799314
$816$	−3.94707	−0.138175
$817$	8.90302	0.311477
$818$	16.4984	0.576852
$819$	−11.8328	−0.413471
$820$	−38.7903	−1.35462
$821$	44.8788	1.56628	0.783141	−	0.621845i	$-0.213616\pi$
$821$	44.8788	1.56628	0.783141	+	0.621845i	$0.213616\pi$
$822$	−15.6733	−0.546671
$823$	−21.3854	−0.745450	−0.372725	−	0.927942i	$-0.621576\pi$
$823$	−21.3854	−0.745450	−0.372725	+	0.927942i	$0.621576\pi$
$824$	−0.278282	−0.00969440
$825$	6.34359	0.220855
$826$	8.30180	0.288856
$827$	11.8470	0.411960	0.205980	−	0.978556i	$-0.433962\pi$
$827$	11.8470	0.411960	0.205980	+	0.978556i	$0.433962\pi$
$828$	−12.1512	−0.422283
$829$	2.36211	0.0820394	0.0410197	−	0.999158i	$-0.486939\pi$
$829$	2.36211	0.0820394	0.0410197	+	0.999158i	$0.486939\pi$
$830$	−36.2349	−1.25773
$831$	−11.7417	−0.407315
$832$	−50.4289	−1.74831
$833$	3.29105	0.114028
$834$	−14.2582	−0.493720
$835$	−29.5357	−1.02212
$836$	6.96972	0.241053
$837$	6.65982	0.230197
$838$	12.8198	0.442854
$839$	−12.0025	−0.414373	−0.207187	−	0.978301i	$-0.566431\pi$
$839$	−12.0025	−0.414373	−0.207187	+	0.978301i	$0.566431\pi$
$840$	0.184998	0.00638304
$841$	10.7159	0.369515
$842$	6.94257	0.239257
$843$	25.2715	0.870396
$844$	0.146682	0.00504899
$845$	−45.3538	−1.56022
$846$	2.70136	0.0928745
$847$	−7.55313	−0.259529
$848$	44.4139	1.52518
$849$	16.3299	0.560439
$850$	−3.29508	−0.113020
$851$	−20.6112	−0.706542
$852$	−16.2929	−0.558184
$853$	40.3937	1.38305	0.691527	−	0.722351i	$-0.256938\pi$
$853$	40.3937	1.38305	0.691527	+	0.722351i	$0.256938\pi$
$854$	−55.4426	−1.89721
$855$	−1.63179	−0.0558061
$856$	0.262195	0.00896163
$857$	−4.99326	−0.170566	−0.0852832	−	0.996357i	$-0.527179\pi$
$857$	−4.99326	−0.170566	−0.0852832	+	0.996357i	$0.527179\pi$
$858$	47.6231	1.62583
$859$	45.4208	1.54974	0.774868	−	0.632123i	$-0.217816\pi$
$859$	45.4208	1.54974	0.774868	+	0.632123i	$0.217816\pi$
$860$	−37.1190	−1.26575
$861$	20.1212	0.685728
$862$	65.4133	2.22798
$863$	37.4242	1.27393	0.636967	−	0.770891i	$-0.280189\pi$
$863$	37.4242	1.27393	0.636967	+	0.770891i	$0.280189\pi$
$864$	−8.02471	−0.273006
$865$	−35.6819	−1.21322
$866$	−54.7920	−1.86191
$867$	−1.00000	−0.0339618
$868$	25.9869	0.882052
$869$	−42.6249	−1.44595
$870$	−23.1715	−0.785588
$871$	−26.6249	−0.902151
$872$	0.301807	0.0102205
$873$	−11.5640	−0.391382
$874$	10.7160	0.362474
$875$	−23.4404	−0.792430
$876$	0.396654	0.0134017
$877$	−23.0996	−0.780018	−0.390009	−	0.920811i	$-0.627528\pi$
$877$	−23.0996	−0.780018	−0.390009	+	0.920811i	$0.627528\pi$
$878$	23.2762	0.785532
$879$	−16.9866	−0.572945
$880$	27.9393	0.941833
$881$	−28.8715	−0.972705	−0.486353	−	0.873763i	$-0.661673\pi$
$881$	−28.8715	−0.972705	−0.486353	+	0.873763i	$0.661673\pi$
$882$	6.60356	0.222354
$883$	−21.2169	−0.714004	−0.357002	−	0.934104i	$-0.616201\pi$
$883$	−21.2169	−0.714004	−0.357002	+	0.934104i	$0.616201\pi$
$884$	−12.4488	−0.418699
$885$	3.93669	0.132330
$886$	49.2659	1.65512
$887$	2.45408	0.0823999	0.0412000	−	0.999151i	$-0.486882\pi$
$887$	2.45408	0.0823999	0.0412000	+	0.999151i	$0.486882\pi$
$888$	0.180162	0.00604584
$889$	−8.26669	−0.277256
$890$	−32.7890	−1.09909
$891$	3.86289	0.129412
$892$	−46.4695	−1.55591
$893$	−1.19888	−0.0401189
$894$	−12.7641	−0.426897
$895$	−28.3274	−0.946880
$896$	−0.807592	−0.0269797
$897$	36.8481	1.23032
$898$	43.1504	1.43995
$899$	−41.9706	−1.39980
$900$	−3.32728	−0.110909
$901$	11.2524	0.374871
$902$	−80.9813	−2.69638
$903$	19.2542	0.640741
$904$	−0.439596	−0.0146208
$905$	−10.3406	−0.343732
$906$	46.2617	1.53694
$907$	−1.66296	−0.0552176	−0.0276088	−	0.999619i	$-0.508789\pi$
$907$	−1.66296	−0.0552176	−0.0276088	+	0.999619i	$0.508789\pi$
$908$	49.1851	1.63227
$909$	−0.668211	−0.0221632
$910$	−43.5070	−1.44224
$911$	−35.6558	−1.18133	−0.590665	−	0.806917i	$-0.701135\pi$
$911$	−35.6558	−1.18133	−0.590665	+	0.806917i	$0.701135\pi$
$912$	3.51488	0.116389
$913$	−38.0686	−1.25989
$914$	65.7666	2.17537
$915$	−26.2907	−0.869145
$916$	10.7977	0.356767
$917$	−16.9420	−0.559474
$918$	−2.00652	−0.0662251
$919$	23.0236	0.759480	0.379740	−	0.925093i	$-0.376014\pi$
$919$	23.0236	0.759480	0.379740	+	0.925093i	$0.376014\pi$
$920$	−0.576096	−0.0189933
$921$	20.3919	0.671935
$922$	−40.9263	−1.34784
$923$	49.4075	1.62627
$924$	15.0732	0.495871
$925$	−5.64381	−0.185567
$926$	50.8098	1.66971
$927$	5.30849	0.174354
$928$	50.5722	1.66011
$929$	44.0824	1.44630	0.723149	−	0.690693i	$-0.242694\pi$
$929$	44.0824	1.44630	0.723149	+	0.690693i	$0.242694\pi$
$930$	24.4869	0.802958
$931$	−2.93070	−0.0960498
$932$	−10.6290	−0.348163
$933$	−0.253731	−0.00830679
$934$	50.5358	1.65358
$935$	7.07850	0.231492
$936$	−0.322088	−0.0105278
$937$	27.4483	0.896698	0.448349	−	0.893859i	$-0.352012\pi$
$937$	27.4483	0.896698	0.448349	+	0.893859i	$0.352012\pi$
$938$	−16.7454	−0.546757
$939$	18.9071	0.617009
$940$	4.99842	0.163031
$941$	−48.8533	−1.59257	−0.796285	−	0.604921i	$-0.793205\pi$
$941$	−48.8533	−1.59257	−0.796285	+	0.604921i	$0.793205\pi$
$942$	−2.00652	−0.0653760
$943$	−62.6586	−2.04045
$944$	−8.47964	−0.275989
$945$	−3.52902	−0.114799
$946$	−77.4921	−2.51948
$947$	−20.3405	−0.660977	−0.330488	−	0.943810i	$-0.607213\pi$
$947$	−20.3405	−0.660977	−0.330488	+	0.943810i	$0.607213\pi$
$948$	22.3572	0.726127
$949$	−1.20284	−0.0390457
$950$	2.93429	0.0952008
$951$	23.7699	0.770791
$952$	−0.100958	−0.00327205
$953$	19.8704	0.643667	0.321833	−	0.946796i	$-0.395701\pi$
$953$	19.8704	0.643667	0.321833	+	0.946796i	$0.395701\pi$
$954$	22.5781	0.730995
$955$	−45.6021	−1.47565
$956$	−4.44146	−0.143647
$957$	−24.3442	−0.786935
$958$	4.40707	0.142386
$959$	15.0433	0.485774
$960$	−15.0399	−0.485412
$961$	13.3532	0.430747
$962$	−42.3697	−1.36605
$963$	−5.00162	−0.161175
$964$	15.7480	0.507208
$965$	11.6107	0.373760
$966$	23.1751	0.745647
$967$	−39.1328	−1.25843	−0.629213	−	0.777233i	$-0.716623\pi$
$967$	−39.1328	−1.25843	−0.629213	+	0.777233i	$0.716623\pi$
$968$	−0.205596	−0.00660811
$969$	0.890505	0.0286072
$970$	−42.5187	−1.36519
$971$	−39.1577	−1.25663	−0.628315	−	0.777959i	$-0.716255\pi$
$971$	−39.1577	−1.25663	−0.628315	+	0.777959i	$0.716255\pi$
$972$	−2.02613	−0.0649880
$973$	13.6850	0.438721
$974$	29.1941	0.935440
$975$	10.0898	0.323133
$976$	56.6303	1.81269
$977$	6.32794	0.202449	0.101224	−	0.994864i	$-0.467724\pi$
$977$	6.32794	0.202449	0.101224	+	0.994864i	$0.467724\pi$
$978$	−24.9868	−0.798991
$979$	−34.4484	−1.10098
$980$	12.2188	0.390316
$981$	−5.75725	−0.183815
$982$	−77.0440	−2.45857
$983$	−14.0117	−0.446903	−0.223451	−	0.974715i	$-0.571732\pi$
$983$	−14.0117	−0.446903	−0.223451	+	0.974715i	$0.571732\pi$
$984$	0.547699	0.0174600
$985$	46.9129	1.49477
$986$	12.6452	0.402706
$987$	−2.59277	−0.0825287
$988$	11.0857	0.352684
$989$	−59.9589	−1.90658
$990$	14.2032	0.451406
$991$	−61.5751	−1.95600	−0.977999	−	0.208611i	$-0.933106\pi$
$991$	−61.5751	−1.95600	−0.977999	+	0.208611i	$0.933106\pi$
$992$	−53.4431	−1.69682
$993$	−5.64108	−0.179014
$994$	31.0742	0.985614
$995$	−10.4902	−0.332561
$996$	19.9674	0.632691
$997$	17.1467	0.543042	0.271521	−	0.962433i	$-0.412473\pi$
$997$	17.1467	0.543042	0.271521	+	0.962433i	$0.412473\pi$
$998$	−3.96982	−0.125662
$999$	−3.43677	−0.108734

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.9	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.9	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00652 q^{2} -1.00000 q^{3} +2.02613 q^{4} -1.83243 q^{5} +2.00652 q^{6} -1.92586 q^{7} -0.0524220 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00652 q^{2} -1.00000 q^{3} +2.02613 q^{4} -1.83243 q^{5} +2.00652 q^{6} -1.92586 q^{7} -0.0524220 q^{8} +1.00000 q^{9} +3.67682 q^{10} +3.86289 q^{11} -2.02613 q^{12} +6.14415 q^{13} +3.86428 q^{14} +1.83243 q^{15} -3.94707 q^{16} -1.00000 q^{17} -2.00652 q^{18} +0.890505 q^{19} -3.71274 q^{20} +1.92586 q^{21} -7.75098 q^{22} -5.99726 q^{23} +0.0524220 q^{24} -1.64219 q^{25} -12.3284 q^{26} -1.00000 q^{27} -3.90204 q^{28} +6.30206 q^{29} -3.67682 q^{30} -6.65982 q^{31} +8.02471 q^{32} -3.86289 q^{33} +2.00652 q^{34} +3.52902 q^{35} +2.02613 q^{36} +3.43677 q^{37} -1.78682 q^{38} -6.14415 q^{39} +0.0960598 q^{40} +10.4479 q^{41} -3.86428 q^{42} +9.99772 q^{43} +7.82671 q^{44} -1.83243 q^{45} +12.0336 q^{46} -1.34629 q^{47} +3.94707 q^{48} -3.29105 q^{49} +3.29508 q^{50} +1.00000 q^{51} +12.4488 q^{52} -11.2524 q^{53} +2.00652 q^{54} -7.07850 q^{55} +0.100958 q^{56} -0.890505 q^{57} -12.6452 q^{58} +2.14834 q^{59} +3.71274 q^{60} -14.3474 q^{61} +13.3631 q^{62} -1.92586 q^{63} -8.20762 q^{64} -11.2587 q^{65} +7.75098 q^{66} -4.33338 q^{67} -2.02613 q^{68} +5.99726 q^{69} -7.08105 q^{70} +8.04138 q^{71} -0.0524220 q^{72} -0.195770 q^{73} -6.89594 q^{74} +1.64219 q^{75} +1.80428 q^{76} -7.43940 q^{77} +12.3284 q^{78} -11.0344 q^{79} +7.23274 q^{80} +1.00000 q^{81} -20.9639 q^{82} -9.85496 q^{83} +3.90204 q^{84} +1.83243 q^{85} -20.0606 q^{86} -6.30206 q^{87} -0.202501 q^{88} -8.91778 q^{89} +3.67682 q^{90} -11.8328 q^{91} -12.1512 q^{92} +6.65982 q^{93} +2.70136 q^{94} -1.63179 q^{95} -8.02471 q^{96} -11.5640 q^{97} +6.60356 q^{98} +3.86289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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