LMFDB - Embedded newform 8007.2.a.f.1.34

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			1.34
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)

$f(q)$	$=$	$q+1.18815 q^{2} -1.00000 q^{3} -0.588311 q^{4} -3.84337 q^{5} -1.18815 q^{6} -5.02013 q^{7} -3.07529 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.18815 q^{2} -1.00000 q^{3} -0.588311 q^{4} -3.84337 q^{5} -1.18815 q^{6} -5.02013 q^{7} -3.07529 q^{8} +1.00000 q^{9} -4.56649 q^{10} +1.17485 q^{11} +0.588311 q^{12} +4.56005 q^{13} -5.96465 q^{14} +3.84337 q^{15} -2.47727 q^{16} -1.00000 q^{17} +1.18815 q^{18} -5.09079 q^{19} +2.26110 q^{20} +5.02013 q^{21} +1.39589 q^{22} +0.118540 q^{23} +3.07529 q^{24} +9.77153 q^{25} +5.41801 q^{26} -1.00000 q^{27} +2.95340 q^{28} +0.483859 q^{29} +4.56649 q^{30} +2.83970 q^{31} +3.20722 q^{32} -1.17485 q^{33} -1.18815 q^{34} +19.2942 q^{35} -0.588311 q^{36} -0.774252 q^{37} -6.04860 q^{38} -4.56005 q^{39} +11.8195 q^{40} +1.91141 q^{41} +5.96465 q^{42} +0.0806862 q^{43} -0.691175 q^{44} -3.84337 q^{45} +0.140843 q^{46} +0.641314 q^{47} +2.47727 q^{48} +18.2017 q^{49} +11.6100 q^{50} +1.00000 q^{51} -2.68273 q^{52} +1.13291 q^{53} -1.18815 q^{54} -4.51538 q^{55} +15.4384 q^{56} +5.09079 q^{57} +0.574895 q^{58} +14.7903 q^{59} -2.26110 q^{60} +4.40403 q^{61} +3.37397 q^{62} -5.02013 q^{63} +8.76519 q^{64} -17.5260 q^{65} -1.39589 q^{66} +3.88247 q^{67} +0.588311 q^{68} -0.118540 q^{69} +22.9244 q^{70} -2.48521 q^{71} -3.07529 q^{72} -9.15915 q^{73} -0.919924 q^{74} -9.77153 q^{75} +2.99496 q^{76} -5.89789 q^{77} -5.41801 q^{78} +0.193340 q^{79} +9.52107 q^{80} +1.00000 q^{81} +2.27104 q^{82} -6.84589 q^{83} -2.95340 q^{84} +3.84337 q^{85} +0.0958669 q^{86} -0.483859 q^{87} -3.61299 q^{88} -4.55261 q^{89} -4.56649 q^{90} -22.8921 q^{91} -0.0697385 q^{92} -2.83970 q^{93} +0.761974 q^{94} +19.5658 q^{95} -3.20722 q^{96} -1.32321 q^{97} +21.6263 q^{98} +1.17485 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.18815	0.840146	0.420073	−	0.907490i	$-0.362005\pi$
$2$	1.18815	0.840146	0.420073	+	0.907490i	$0.362005\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.588311	−0.294155
$5$	−3.84337	−1.71881	−0.859405	−	0.511296i	$-0.829166\pi$
$5$	−3.84337	−1.71881	−0.859405	+	0.511296i	$0.829166\pi$
$6$	−1.18815	−0.485058
$7$	−5.02013	−1.89743	−0.948716	−	0.316130i	$-0.897616\pi$
$7$	−5.02013	−1.89743	−0.948716	+	0.316130i	$0.897616\pi$
$8$	−3.07529	−1.08728
$9$	1.00000	0.333333
$10$	−4.56649	−1.44405
$11$	1.17485	0.354230	0.177115	−	0.984190i	$-0.443324\pi$
$11$	1.17485	0.354230	0.177115	+	0.984190i	$0.443324\pi$
$12$	0.588311	0.169831
$13$	4.56005	1.26473	0.632366	−	0.774670i	$-0.282084\pi$
$13$	4.56005	1.26473	0.632366	+	0.774670i	$0.282084\pi$
$14$	−5.96465	−1.59412
$15$	3.84337	0.992355
$16$	−2.47727	−0.619317
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.18815	0.280049
$19$	−5.09079	−1.16791	−0.583953	−	0.811787i	$-0.698495\pi$
$19$	−5.09079	−1.16791	−0.583953	+	0.811787i	$0.698495\pi$
$20$	2.26110	0.505597
$21$	5.02013	1.09548
$22$	1.39589	0.297604
$23$	0.118540	0.0247174	0.0123587	−	0.999924i	$-0.496066\pi$
$23$	0.118540	0.0247174	0.0123587	+	0.999924i	$0.496066\pi$
$24$	3.07529	0.627741
$25$	9.77153	1.95431
$26$	5.41801	1.06256
$27$	−1.00000	−0.192450
$28$	2.95340	0.558140
$29$	0.483859	0.0898504	0.0449252	−	0.998990i	$-0.485695\pi$
$29$	0.483859	0.0898504	0.0449252	+	0.998990i	$0.485695\pi$
$30$	4.56649	0.833723
$31$	2.83970	0.510025	0.255012	−	0.966938i	$-0.417920\pi$
$31$	2.83970	0.510025	0.255012	+	0.966938i	$0.417920\pi$
$32$	3.20722	0.566962
$33$	−1.17485	−0.204515
$34$	−1.18815	−0.203765
$35$	19.2942	3.26132
$36$	−0.588311	−0.0980518
$37$	−0.774252	−0.127286	−0.0636432	−	0.997973i	$-0.520272\pi$
$37$	−0.774252	−0.127286	−0.0636432	+	0.997973i	$0.520272\pi$
$38$	−6.04860	−0.981212
$39$	−4.56005	−0.730193
$40$	11.8195	1.86883
$41$	1.91141	0.298513	0.149256	−	0.988799i	$-0.452312\pi$
$41$	1.91141	0.298513	0.149256	+	0.988799i	$0.452312\pi$
$42$	5.96465	0.920365
$43$	0.0806862	0.0123045	0.00615226	−	0.999981i	$-0.498042\pi$
$43$	0.0806862	0.0123045	0.00615226	+	0.999981i	$0.498042\pi$
$44$	−0.691175	−0.104199
$45$	−3.84337	−0.572936
$46$	0.140843	0.0207662
$47$	0.641314	0.0935453	0.0467726	−	0.998906i	$-0.485106\pi$
$47$	0.641314	0.0935453	0.0467726	+	0.998906i	$0.485106\pi$
$48$	2.47727	0.357563
$49$	18.2017	2.60025
$50$	11.6100	1.64190
$51$	1.00000	0.140028
$52$	−2.68273	−0.372028
$53$	1.13291	0.155617	0.0778084	−	0.996968i	$-0.475208\pi$
$53$	1.13291	0.155617	0.0778084	+	0.996968i	$0.475208\pi$
$54$	−1.18815	−0.161686
$55$	−4.51538	−0.608853
$56$	15.4384	2.06304
$57$	5.09079	0.674291
$58$	0.574895	0.0754875
$59$	14.7903	1.92553	0.962767	−	0.270333i	$-0.0871339\pi$
$59$	14.7903	1.92553	0.962767	+	0.270333i	$0.0871339\pi$
$60$	−2.26110	−0.291907
$61$	4.40403	0.563879	0.281939	−	0.959432i	$-0.409022\pi$
$61$	4.40403	0.563879	0.281939	+	0.959432i	$0.409022\pi$
$62$	3.37397	0.428495
$63$	−5.02013	−0.632477
$64$	8.76519	1.09565
$65$	−17.5260	−2.17383
$66$	−1.39589	−0.171822
$67$	3.88247	0.474320	0.237160	−	0.971471i	$-0.423784\pi$
$67$	3.88247	0.474320	0.237160	+	0.971471i	$0.423784\pi$
$68$	0.588311	0.0713431
$69$	−0.118540	−0.0142706
$70$	22.9244	2.73999
$71$	−2.48521	−0.294940	−0.147470	−	0.989067i	$-0.547113\pi$
$71$	−2.48521	−0.294940	−0.147470	+	0.989067i	$0.547113\pi$
$72$	−3.07529	−0.362426
$73$	−9.15915	−1.07200	−0.535999	−	0.844219i	$-0.680065\pi$
$73$	−9.15915	−1.07200	−0.535999	+	0.844219i	$0.680065\pi$
$74$	−0.919924	−0.106939
$75$	−9.77153	−1.12832
$76$	2.99496	0.343546
$77$	−5.89789	−0.672127
$78$	−5.41801	−0.613469
$79$	0.193340	0.0217525	0.0108762	−	0.999941i	$-0.496538\pi$
$79$	0.193340	0.0217525	0.0108762	+	0.999941i	$0.496538\pi$
$80$	9.52107	1.06449
$81$	1.00000	0.111111
$82$	2.27104	0.250794
$83$	−6.84589	−0.751434	−0.375717	−	0.926735i	$-0.622603\pi$
$83$	−6.84589	−0.751434	−0.375717	+	0.926735i	$0.622603\pi$
$84$	−2.95340	−0.322242
$85$	3.84337	0.416873
$86$	0.0958669	0.0103376
$87$	−0.483859	−0.0518752
$88$	−3.61299	−0.385146
$89$	−4.55261	−0.482576	−0.241288	−	0.970454i	$-0.577570\pi$
$89$	−4.55261	−0.482576	−0.241288	+	0.970454i	$0.577570\pi$
$90$	−4.56649	−0.481350
$91$	−22.8921	−2.39974
$92$	−0.0697385	−0.00727074
$93$	−2.83970	−0.294463
$94$	0.761974	0.0785917
$95$	19.5658	2.00741
$96$	−3.20722	−0.327336
$97$	−1.32321	−0.134351	−0.0671756	−	0.997741i	$-0.521399\pi$
$97$	−1.32321	−0.134351	−0.0671756	+	0.997741i	$0.521399\pi$
$98$	21.6263	2.18459
$99$	1.17485	0.118077
$100$	−5.74870	−0.574870
$101$	0.241677	0.0240478	0.0120239	−	0.999928i	$-0.496173\pi$
$101$	0.241677	0.0240478	0.0120239	+	0.999928i	$0.496173\pi$
$102$	1.18815	0.117644
$103$	−7.68238	−0.756967	−0.378484	−	0.925608i	$-0.623554\pi$
$103$	−7.68238	−0.756967	−0.378484	+	0.925608i	$0.623554\pi$
$104$	−14.0235	−1.37512
$105$	−19.2942	−1.88293
$106$	1.34606	0.130741
$107$	−7.24061	−0.699976	−0.349988	−	0.936754i	$-0.613814\pi$
$107$	−7.24061	−0.699976	−0.349988	+	0.936754i	$0.613814\pi$
$108$	0.588311	0.0566102
$109$	−2.12835	−0.203859	−0.101929	−	0.994792i	$-0.532502\pi$
$109$	−2.12835	−0.203859	−0.101929	+	0.994792i	$0.532502\pi$
$110$	−5.36492	−0.511525
$111$	0.774252	0.0734888
$112$	12.4362	1.17511
$113$	−4.12701	−0.388237	−0.194118	−	0.980978i	$-0.562185\pi$
$113$	−4.12701	−0.388237	−0.194118	+	0.980978i	$0.562185\pi$
$114$	6.04860	0.566503
$115$	−0.455595	−0.0424844
$116$	−0.284660	−0.0264300
$117$	4.56005	0.421577
$118$	17.5730	1.61773
$119$	5.02013	0.460195
$120$	−11.8195	−1.07897
$121$	−9.61973	−0.874521
$122$	5.23263	0.473740
$123$	−1.91141	−0.172346
$124$	−1.67062	−0.150026
$125$	−18.3388	−1.64027
$126$	−5.96465	−0.531373
$127$	0.0280070	0.00248522	0.00124261	−	0.999999i	$-0.499604\pi$
$127$	0.0280070	0.00248522	0.00124261	+	0.999999i	$0.499604\pi$
$128$	3.99987	0.353542
$129$	−0.0806862	−0.00710402
$130$	−20.8234	−1.82634
$131$	15.4637	1.35107	0.675533	−	0.737329i	$-0.263913\pi$
$131$	15.4637	1.35107	0.675533	+	0.737329i	$0.263913\pi$
$132$	0.691175	0.0601591
$133$	25.5564	2.21602
$134$	4.61294	0.398498
$135$	3.84337	0.330785
$136$	3.07529	0.263704
$137$	5.63978	0.481839	0.240920	−	0.970545i	$-0.422551\pi$
$137$	5.63978	0.481839	0.240920	+	0.970545i	$0.422551\pi$
$138$	−0.140843	−0.0119894
$139$	1.70309	0.144454	0.0722270	−	0.997388i	$-0.476989\pi$
$139$	1.70309	0.144454	0.0722270	+	0.997388i	$0.476989\pi$
$140$	−11.3510	−0.959336
$141$	−0.641314	−0.0540084
$142$	−2.95279	−0.247792
$143$	5.35737	0.448005
$144$	−2.47727	−0.206439
$145$	−1.85965	−0.154436
$146$	−10.8824	−0.900634
$147$	−18.2017	−1.50125
$148$	0.455501	0.0374419
$149$	−18.3377	−1.50229	−0.751143	−	0.660139i	$-0.770497\pi$
$149$	−18.3377	−1.50229	−0.751143	+	0.660139i	$0.770497\pi$
$150$	−11.6100	−0.947952
$151$	22.6194	1.84074	0.920371	−	0.391046i	$-0.127887\pi$
$151$	22.6194	1.84074	0.920371	+	0.391046i	$0.127887\pi$
$152$	15.6556	1.26984
$153$	−1.00000	−0.0808452
$154$	−7.00755	−0.564684
$155$	−10.9140	−0.876635
$156$	2.68273	0.214790
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	0.229716	0.0182753
$159$	−1.13291	−0.0898454
$160$	−12.3266	−0.974500
$161$	−0.595088	−0.0468995
$162$	1.18815	0.0933495
$163$	−2.05607	−0.161044	−0.0805218	−	0.996753i	$-0.525659\pi$
$163$	−2.05607	−0.161044	−0.0805218	+	0.996753i	$0.525659\pi$
$164$	−1.12451	−0.0878092
$165$	4.51538	0.351522
$166$	−8.13391	−0.631314
$167$	15.6503	1.21105	0.605527	−	0.795825i	$-0.292962\pi$
$167$	15.6503	1.21105	0.605527	+	0.795825i	$0.292962\pi$
$168$	−15.4384	−1.19110
$169$	7.79410	0.599546
$170$	4.56649	0.350234
$171$	−5.09079	−0.389302
$172$	−0.0474685	−0.00361944
$173$	−20.3537	−1.54746	−0.773731	−	0.633514i	$-0.781612\pi$
$173$	−20.3537	−1.54746	−0.773731	+	0.633514i	$0.781612\pi$
$174$	−0.574895	−0.0435827
$175$	−49.0544	−3.70816
$176$	−2.91041	−0.219381
$177$	−14.7903	−1.11171
$178$	−5.40916	−0.405434
$179$	12.9224	0.965869	0.482934	−	0.875657i	$-0.339571\pi$
$179$	12.9224	0.965869	0.482934	+	0.875657i	$0.339571\pi$
$180$	2.26110	0.168532
$181$	−17.7219	−1.31726	−0.658630	−	0.752467i	$-0.728864\pi$
$181$	−17.7219	−1.31726	−0.658630	+	0.752467i	$0.728864\pi$
$182$	−27.1991	−2.01613
$183$	−4.40403	−0.325555
$184$	−0.364546	−0.0268747
$185$	2.97574	0.218781
$186$	−3.37397	−0.247392
$187$	−1.17485	−0.0859133
$188$	−0.377292	−0.0275168
$189$	5.02013	0.365161
$190$	23.2470	1.68652
$191$	20.7123	1.49869	0.749344	−	0.662181i	$-0.230369\pi$
$191$	20.7123	1.49869	0.749344	+	0.662181i	$0.230369\pi$
$192$	−8.76519	−0.632573
$193$	−22.7026	−1.63417	−0.817083	−	0.576520i	$-0.804410\pi$
$193$	−22.7026	−1.63417	−0.817083	+	0.576520i	$0.804410\pi$
$194$	−1.57216	−0.112875
$195$	17.5260	1.25506
$196$	−10.7083	−0.764876
$197$	7.53823	0.537077	0.268538	−	0.963269i	$-0.413459\pi$
$197$	7.53823	0.537077	0.268538	+	0.963269i	$0.413459\pi$
$198$	1.39589	0.0992015
$199$	12.3704	0.876915	0.438458	−	0.898752i	$-0.355525\pi$
$199$	12.3704	0.876915	0.438458	+	0.898752i	$0.355525\pi$
$200$	−30.0503	−2.12488
$201$	−3.88247	−0.273849
$202$	0.287147	0.0202036
$203$	−2.42904	−0.170485
$204$	−0.588311	−0.0411900
$205$	−7.34628	−0.513087
$206$	−9.12778	−0.635963
$207$	0.118540	0.00823912
$208$	−11.2965	−0.783270
$209$	−5.98090	−0.413707
$210$	−22.9244	−1.58193
$211$	−22.7571	−1.56667	−0.783333	−	0.621602i	$-0.786482\pi$
$211$	−22.7571	−1.56667	−0.783333	+	0.621602i	$0.786482\pi$
$212$	−0.666501	−0.0457755
$213$	2.48521	0.170284
$214$	−8.60290	−0.588082
$215$	−0.310107	−0.0211491
$216$	3.07529	0.209247
$217$	−14.2557	−0.967737
$218$	−2.52878	−0.171271
$219$	9.15915	0.618918
$220$	2.65644	0.179097
$221$	−4.56005	−0.306742
$222$	0.919924	0.0617413
$223$	14.7887	0.990326	0.495163	−	0.868800i	$-0.335108\pi$
$223$	14.7887	0.990326	0.495163	+	0.868800i	$0.335108\pi$
$224$	−16.1007	−1.07577
$225$	9.77153	0.651435
$226$	−4.90349	−0.326175
$227$	19.3291	1.28292	0.641459	−	0.767157i	$-0.278330\pi$
$227$	19.3291	1.28292	0.641459	+	0.767157i	$0.278330\pi$
$228$	−2.99496	−0.198346
$229$	23.1244	1.52810	0.764052	−	0.645154i	$-0.223207\pi$
$229$	23.1244	1.52810	0.764052	+	0.645154i	$0.223207\pi$
$230$	−0.541313	−0.0356931
$231$	5.89789	0.388052
$232$	−1.48801	−0.0976925
$233$	−23.2370	−1.52231	−0.761155	−	0.648570i	$-0.775367\pi$
$233$	−23.2370	−1.52231	−0.761155	+	0.648570i	$0.775367\pi$
$234$	5.41801	0.354186
$235$	−2.46481	−0.160787
$236$	−8.70129	−0.566406
$237$	−0.193340	−0.0125588
$238$	5.96465	0.386631
$239$	19.2054	1.24230	0.621148	−	0.783693i	$-0.286667\pi$
$239$	19.2054	1.24230	0.621148	+	0.783693i	$0.286667\pi$
$240$	−9.52107	−0.614583
$241$	24.0043	1.54625	0.773126	−	0.634253i	$-0.218692\pi$
$241$	24.0043	1.54625	0.773126	+	0.634253i	$0.218692\pi$
$242$	−11.4296	−0.734725
$243$	−1.00000	−0.0641500
$244$	−2.59094	−0.165868
$245$	−69.9561	−4.46933
$246$	−2.27104	−0.144796
$247$	−23.2143	−1.47709
$248$	−8.73289	−0.554539
$249$	6.84589	0.433840
$250$	−21.7891	−1.37807
$251$	−20.2627	−1.27897	−0.639485	−	0.768804i	$-0.720852\pi$
$251$	−20.2627	−1.27897	−0.639485	+	0.768804i	$0.720852\pi$
$252$	2.95340	0.186047
$253$	0.139267	0.00875562
$254$	0.0332764	0.00208794
$255$	−3.84337	−0.240681
$256$	−12.7779	−0.798622
$257$	22.3132	1.39186	0.695928	−	0.718111i	$-0.254993\pi$
$257$	22.3132	1.39186	0.695928	+	0.718111i	$0.254993\pi$
$258$	−0.0958669	−0.00596841
$259$	3.88685	0.241517
$260$	10.3107	0.639445
$261$	0.483859	0.0299501
$262$	18.3731	1.13509
$263$	−19.1595	−1.18143	−0.590714	−	0.806881i	$-0.701154\pi$
$263$	−19.1595	−1.18143	−0.590714	+	0.806881i	$0.701154\pi$
$264$	3.61299	0.222364
$265$	−4.35419	−0.267476
$266$	30.3648	1.86178
$267$	4.55261	0.278615
$268$	−2.28410	−0.139524
$269$	16.5414	1.00854	0.504272	−	0.863545i	$-0.331761\pi$
$269$	16.5414	1.00854	0.504272	+	0.863545i	$0.331761\pi$
$270$	4.56649	0.277908
$271$	23.9458	1.45461	0.727303	−	0.686317i	$-0.240774\pi$
$271$	23.9458	1.45461	0.727303	+	0.686317i	$0.240774\pi$
$272$	2.47727	0.150207
$273$	22.8921	1.38549
$274$	6.70088	0.404815
$275$	11.4801	0.692273
$276$	0.0697385	0.00419777
$277$	−30.0068	−1.80293	−0.901466	−	0.432850i	$-0.857508\pi$
$277$	−30.0068	−1.80293	−0.901466	+	0.432850i	$0.857508\pi$
$278$	2.02351	0.121362
$279$	2.83970	0.170008
$280$	−59.3354	−3.54597
$281$	−25.4712	−1.51949	−0.759743	−	0.650224i	$-0.774675\pi$
$281$	−25.4712	−1.51949	−0.759743	+	0.650224i	$0.774675\pi$
$282$	−0.761974	−0.0453749
$283$	21.9793	1.30653	0.653267	−	0.757127i	$-0.273398\pi$
$283$	21.9793	1.30653	0.653267	+	0.757127i	$0.273398\pi$
$284$	1.46207	0.0867581
$285$	−19.5658	−1.15898
$286$	6.36533	0.376390
$287$	−9.59556	−0.566408
$288$	3.20722	0.188987
$289$	1.00000	0.0588235
$290$	−2.20954	−0.129749
$291$	1.32321	0.0775677
$292$	5.38843	0.315334
$293$	2.20695	0.128931	0.0644657	−	0.997920i	$-0.479466\pi$
$293$	2.20695	0.128931	0.0644657	+	0.997920i	$0.479466\pi$
$294$	−21.6263	−1.26127
$295$	−56.8447	−3.30963
$296$	2.38105	0.138396
$297$	−1.17485	−0.0681715
$298$	−21.7879	−1.26214
$299$	0.540550	0.0312608
$300$	5.74870	0.331901
$301$	−0.405055	−0.0233470
$302$	26.8752	1.54649
$303$	−0.241677	−0.0138840
$304$	12.6113	0.723305
$305$	−16.9263	−0.969200
$306$	−1.18815	−0.0679217
$307$	0.998472	0.0569858	0.0284929	−	0.999594i	$-0.490929\pi$
$307$	0.998472	0.0569858	0.0284929	+	0.999594i	$0.490929\pi$
$308$	3.46979	0.197710
$309$	7.68238	0.437035
$310$	−12.9674	−0.736501
$311$	9.90215	0.561500	0.280750	−	0.959781i	$-0.409417\pi$
$311$	9.90215	0.561500	0.280750	+	0.959781i	$0.409417\pi$
$312$	14.0235	0.793924
$313$	−8.86140	−0.500876	−0.250438	−	0.968133i	$-0.580575\pi$
$313$	−8.86140	−0.500876	−0.250438	+	0.968133i	$0.580575\pi$
$314$	−1.18815	−0.0670509
$315$	19.2942	1.08711
$316$	−0.113744	−0.00639861
$317$	6.77340	0.380432	0.190216	−	0.981742i	$-0.439081\pi$
$317$	6.77340	0.380432	0.190216	+	0.981742i	$0.439081\pi$
$318$	−1.34606	−0.0754832
$319$	0.568461	0.0318277
$320$	−33.6879	−1.88321
$321$	7.24061	0.404132
$322$	−0.707051	−0.0394024
$323$	5.09079	0.283259
$324$	−0.588311	−0.0326839
$325$	44.5587	2.47167
$326$	−2.44291	−0.135300
$327$	2.12835	0.117698
$328$	−5.87815	−0.324567
$329$	−3.21948	−0.177496
$330$	5.36492	0.295329
$331$	−8.31021	−0.456771	−0.228385	−	0.973571i	$-0.573345\pi$
$331$	−8.31021	−0.456771	−0.228385	+	0.973571i	$0.573345\pi$
$332$	4.02751	0.221038
$333$	−0.774252	−0.0424288
$334$	18.5948	1.01746
$335$	−14.9218	−0.815265
$336$	−12.4362	−0.678451
$337$	7.68995	0.418898	0.209449	−	0.977820i	$-0.432833\pi$
$337$	7.68995	0.418898	0.209449	+	0.977820i	$0.432833\pi$
$338$	9.26053	0.503706
$339$	4.12701	0.224149
$340$	−2.26110	−0.122625
$341$	3.33621	0.180666
$342$	−6.04860	−0.327071
$343$	−56.2342	−3.03636
$344$	−0.248133	−0.0133784
$345$	0.455595	0.0245284
$346$	−24.1831	−1.30009
$347$	16.5830	0.890224	0.445112	−	0.895475i	$-0.353164\pi$
$347$	16.5830	0.890224	0.445112	+	0.895475i	$0.353164\pi$
$348$	0.284660	0.0152594
$349$	0.549394	0.0294084	0.0147042	−	0.999892i	$-0.495319\pi$
$349$	0.549394	0.0294084	0.0147042	+	0.999892i	$0.495319\pi$
$350$	−58.2837	−3.11540
$351$	−4.56005	−0.243398
$352$	3.76800	0.200835
$353$	−11.7068	−0.623091	−0.311545	−	0.950231i	$-0.600847\pi$
$353$	−11.7068	−0.623091	−0.311545	+	0.950231i	$0.600847\pi$
$354$	−17.5730	−0.933996
$355$	9.55158	0.506945
$356$	2.67835	0.141952
$357$	−5.02013	−0.265694
$358$	15.3537	0.811470
$359$	0.235266	0.0124169	0.00620843	−	0.999981i	$-0.498024\pi$
$359$	0.235266	0.0124169	0.00620843	+	0.999981i	$0.498024\pi$
$360$	11.8195	0.622942
$361$	6.91612	0.364006
$362$	−21.0562	−1.10669
$363$	9.61973	0.504905
$364$	13.4677	0.705897
$365$	35.2021	1.84256
$366$	−5.23263	−0.273514
$367$	−14.7870	−0.771875	−0.385937	−	0.922525i	$-0.626122\pi$
$367$	−14.7870	−0.771875	−0.385937	+	0.922525i	$0.626122\pi$
$368$	−0.293656	−0.0153079
$369$	1.91141	0.0995043
$370$	3.53561	0.183808
$371$	−5.68734	−0.295272
$372$	1.67062	0.0866178
$373$	12.1448	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$373$	12.1448	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$374$	−1.39589	−0.0721797
$375$	18.3388	0.947010
$376$	−1.97223	−0.101710
$377$	2.20643	0.113637
$378$	5.96465	0.306788
$379$	−3.26264	−0.167591	−0.0837953	−	0.996483i	$-0.526704\pi$
$379$	−3.26264	−0.167591	−0.0837953	+	0.996483i	$0.526704\pi$
$380$	−11.5108	−0.590490
$381$	−0.0280070	−0.00143484
$382$	24.6092	1.25912
$383$	−9.91788	−0.506780	−0.253390	−	0.967364i	$-0.581546\pi$
$383$	−9.91788	−0.506780	−0.253390	+	0.967364i	$0.581546\pi$
$384$	−3.99987	−0.204117
$385$	22.6678	1.15526
$386$	−26.9740	−1.37294
$387$	0.0806862	0.00410151
$388$	0.778456	0.0395201
$389$	−9.22013	−0.467479	−0.233740	−	0.972299i	$-0.575096\pi$
$389$	−9.22013	−0.467479	−0.233740	+	0.972299i	$0.575096\pi$
$390$	20.8234	1.05444
$391$	−0.118540	−0.00599484
$392$	−55.9756	−2.82719
$393$	−15.4637	−0.780039
$394$	8.95652	0.451223
$395$	−0.743079	−0.0373884
$396$	−0.691175	−0.0347328
$397$	−7.06405	−0.354535	−0.177267	−	0.984163i	$-0.556726\pi$
$397$	−7.06405	−0.354535	−0.177267	+	0.984163i	$0.556726\pi$
$398$	14.6978	0.736737
$399$	−25.5564	−1.27942
$400$	−24.2067	−1.21034
$401$	5.98441	0.298847	0.149424	−	0.988773i	$-0.452258\pi$
$401$	5.98441	0.298847	0.149424	+	0.988773i	$0.452258\pi$
$402$	−4.61294	−0.230073
$403$	12.9492	0.645044
$404$	−0.142181	−0.00707378
$405$	−3.84337	−0.190979
$406$	−2.88605	−0.143232
$407$	−0.909628	−0.0450886
$408$	−3.07529	−0.152250
$409$	−36.0175	−1.78095	−0.890477	−	0.455029i	$-0.849629\pi$
$409$	−36.0175	−1.78095	−0.890477	+	0.455029i	$0.849629\pi$
$410$	−8.72845	−0.431068
$411$	−5.63978	−0.278190
$412$	4.51963	0.222666
$413$	−74.2493	−3.65357
$414$	0.140843	0.00692206
$415$	26.3113	1.29157
$416$	14.6251	0.717055
$417$	−1.70309	−0.0834005
$418$	−7.10617	−0.347574
$419$	−32.3489	−1.58035	−0.790175	−	0.612882i	$-0.790010\pi$
$419$	−32.3489	−1.58035	−0.790175	+	0.612882i	$0.790010\pi$
$420$	11.3510	0.553873
$421$	−7.80652	−0.380467	−0.190233	−	0.981739i	$-0.560924\pi$
$421$	−7.80652	−0.380467	−0.190233	+	0.981739i	$0.560924\pi$
$422$	−27.0388	−1.31623
$423$	0.641314	0.0311818
$424$	−3.48402	−0.169199
$425$	−9.77153	−0.473989
$426$	2.95279	0.143063
$427$	−22.1088	−1.06992
$428$	4.25973	0.205902
$429$	−5.35737	−0.258656
$430$	−0.368452	−0.0177683
$431$	7.23554	0.348524	0.174262	−	0.984699i	$-0.444246\pi$
$431$	7.23554	0.348524	0.174262	+	0.984699i	$0.444246\pi$
$432$	2.47727	0.119188
$433$	−18.1709	−0.873236	−0.436618	−	0.899647i	$-0.643824\pi$
$433$	−18.1709	−0.873236	−0.436618	+	0.899647i	$0.643824\pi$
$434$	−16.9378	−0.813040
$435$	1.85965	0.0891635
$436$	1.25213	0.0599661
$437$	−0.603463	−0.0288676
$438$	10.8824	0.519981
$439$	9.03632	0.431280	0.215640	−	0.976473i	$-0.430816\pi$
$439$	9.03632	0.431280	0.215640	+	0.976473i	$0.430816\pi$
$440$	13.8861	0.661993
$441$	18.2017	0.866749
$442$	−5.41801	−0.257708
$443$	−8.46783	−0.402319	−0.201159	−	0.979559i	$-0.564471\pi$
$443$	−8.46783	−0.402319	−0.201159	+	0.979559i	$0.564471\pi$
$444$	−0.455501	−0.0216171
$445$	17.4974	0.829456
$446$	17.5711	0.832018
$447$	18.3377	0.867345
$448$	−44.0024	−2.07892
$449$	14.4508	0.681974	0.340987	−	0.940068i	$-0.389239\pi$
$449$	14.4508	0.681974	0.340987	+	0.940068i	$0.389239\pi$
$450$	11.6100	0.547301
$451$	2.24562	0.105742
$452$	2.42797	0.114202
$453$	−22.6194	−1.06275
$454$	22.9658	1.07784
$455$	87.9828	4.12470
$456$	−15.6556	−0.733143
$457$	21.2804	0.995457	0.497729	−	0.867333i	$-0.334168\pi$
$457$	21.2804	0.995457	0.497729	+	0.867333i	$0.334168\pi$
$458$	27.4752	1.28383
$459$	1.00000	0.0466760
$460$	0.268031	0.0124970
$461$	−15.7070	−0.731546	−0.365773	−	0.930704i	$-0.619195\pi$
$461$	−15.7070	−0.731546	−0.365773	+	0.930704i	$0.619195\pi$
$462$	7.00755	0.326021
$463$	−22.5481	−1.04790	−0.523949	−	0.851749i	$-0.675542\pi$
$463$	−22.5481	−1.04790	−0.523949	+	0.851749i	$0.675542\pi$
$464$	−1.19865	−0.0556459
$465$	10.9140	0.506126
$466$	−27.6090	−1.27896
$467$	−15.8490	−0.733406	−0.366703	−	0.930338i	$-0.619513\pi$
$467$	−15.8490	−0.733406	−0.366703	+	0.930338i	$0.619513\pi$
$468$	−2.68273	−0.124009
$469$	−19.4905	−0.899989
$470$	−2.92855	−0.135084
$471$	1.00000	0.0460776
$472$	−45.4845	−2.09359
$473$	0.0947939	0.00435863
$474$	−0.229716	−0.0105512
$475$	−49.7448	−2.28245
$476$	−2.95340	−0.135369
$477$	1.13291	0.0518723
$478$	22.8188	1.04371
$479$	17.6303	0.805551	0.402775	−	0.915299i	$-0.368046\pi$
$479$	17.6303	0.805551	0.402775	+	0.915299i	$0.368046\pi$
$480$	12.3266	0.562628
$481$	−3.53063	−0.160983
$482$	28.5206	1.29908
$483$	0.595088	0.0270774
$484$	5.65939	0.257245
$485$	5.08557	0.230924
$486$	−1.18815	−0.0538954
$487$	−29.8625	−1.35320	−0.676600	−	0.736350i	$-0.736548\pi$
$487$	−29.8625	−1.35320	−0.676600	+	0.736350i	$0.736548\pi$
$488$	−13.5437	−0.613093
$489$	2.05607	0.0929786
$490$	−83.1180	−3.75489
$491$	−24.4846	−1.10497	−0.552487	−	0.833522i	$-0.686321\pi$
$491$	−24.4846	−1.10497	−0.552487	+	0.833522i	$0.686321\pi$
$492$	1.12451	0.0506966
$493$	−0.483859	−0.0217919
$494$	−27.5819	−1.24097
$495$	−4.51538	−0.202951
$496$	−7.03469	−0.315867
$497$	12.4761	0.559628
$498$	8.13391	0.364489
$499$	21.4869	0.961886	0.480943	−	0.876752i	$-0.340294\pi$
$499$	21.4869	0.961886	0.480943	+	0.876752i	$0.340294\pi$
$500$	10.7889	0.482494
$501$	−15.6503	−0.699203
$502$	−24.0750	−1.07452
$503$	37.4588	1.67021	0.835103	−	0.550094i	$-0.185408\pi$
$503$	37.4588	1.67021	0.835103	+	0.550094i	$0.185408\pi$
$504$	15.4384	0.687679
$505$	−0.928855	−0.0413335
$506$	0.165469	0.00735600
$507$	−7.79410	−0.346148
$508$	−0.0164768	−0.000731040 0
$509$	22.6036	1.00189	0.500944	−	0.865479i	$-0.332986\pi$
$509$	22.6036	1.00189	0.500944	+	0.865479i	$0.332986\pi$
$510$	−4.56649	−0.202207
$511$	45.9802	2.03404
$512$	−23.1818	−1.02450
$513$	5.09079	0.224764
$514$	26.5113	1.16936
$515$	29.5263	1.30108
$516$	0.0474685	0.00208969
$517$	0.753446	0.0331365
$518$	4.61814	0.202910
$519$	20.3537	0.893428
$520$	53.8975	2.36356
$521$	−13.2533	−0.580638	−0.290319	−	0.956930i	$-0.593761\pi$
$521$	−13.2533	−0.580638	−0.290319	+	0.956930i	$0.593761\pi$
$522$	0.574895	0.0251625
$523$	−5.14882	−0.225142	−0.112571	−	0.993644i	$-0.535909\pi$
$523$	−5.14882	−0.225142	−0.112571	+	0.993644i	$0.535909\pi$
$524$	−9.09744	−0.397424
$525$	49.0544	2.14091
$526$	−22.7643	−0.992572
$527$	−2.83970	−0.123699
$528$	2.91041	0.126659
$529$	−22.9859	−0.999389
$530$	−5.17341	−0.224718
$531$	14.7903	0.641845
$532$	−15.0351	−0.651855
$533$	8.71616	0.377539
$534$	5.40916	0.234077
$535$	27.8284	1.20313
$536$	−11.9397	−0.515718
$537$	−12.9224	−0.557645
$538$	19.6535	0.847324
$539$	21.3842	0.921085
$540$	−2.26110	−0.0973022
$541$	−27.3251	−1.17480	−0.587400	−	0.809297i	$-0.699848\pi$
$541$	−27.3251	−1.17480	−0.587400	+	0.809297i	$0.699848\pi$
$542$	28.4511	1.22208
$543$	17.7219	0.760521
$544$	−3.20722	−0.137509
$545$	8.18003	0.350394
$546$	27.1991	1.16401
$547$	28.8887	1.23519	0.617597	−	0.786495i	$-0.288106\pi$
$547$	28.8887	1.23519	0.617597	+	0.786495i	$0.288106\pi$
$548$	−3.31794	−0.141736
$549$	4.40403	0.187960
$550$	13.6400	0.581610
$551$	−2.46323	−0.104937
$552$	0.364546	0.0155161
$553$	−0.970594	−0.0412738
$554$	−35.6524	−1.51473
$555$	−2.97574	−0.126313
$556$	−1.00194	−0.0424919
$557$	32.6810	1.38474	0.692369	−	0.721544i	$-0.256567\pi$
$557$	32.6810	1.38474	0.692369	+	0.721544i	$0.256567\pi$
$558$	3.37397	0.142832
$559$	0.367933	0.0155619
$560$	−47.7971	−2.01979
$561$	1.17485	0.0496021
$562$	−30.2635	−1.27659
$563$	−6.79956	−0.286567	−0.143284	−	0.989682i	$-0.545766\pi$
$563$	−6.79956	−0.286567	−0.143284	+	0.989682i	$0.545766\pi$
$564$	0.377292	0.0158869
$565$	15.8617	0.667305
$566$	26.1146	1.09768
$567$	−5.02013	−0.210826
$568$	7.64273	0.320682
$569$	−3.86595	−0.162069	−0.0810344	−	0.996711i	$-0.525822\pi$
$569$	−3.86595	−0.162069	−0.0810344	+	0.996711i	$0.525822\pi$
$570$	−23.2470	−0.973711
$571$	−25.7320	−1.07685	−0.538424	−	0.842674i	$-0.680980\pi$
$571$	−25.7320	−1.07685	−0.538424	+	0.842674i	$0.680980\pi$
$572$	−3.15180	−0.131783
$573$	−20.7123	−0.865268
$574$	−11.4009	−0.475865
$575$	1.15832	0.0483053
$576$	8.76519	0.365216
$577$	14.6372	0.609355	0.304678	−	0.952456i	$-0.401451\pi$
$577$	14.6372	0.609355	0.304678	+	0.952456i	$0.401451\pi$
$578$	1.18815	0.0494203
$579$	22.7026	0.943487
$580$	1.09405	0.0454281
$581$	34.3673	1.42579
$582$	1.57216	0.0651681
$583$	1.33099	0.0551241
$584$	28.1670	1.16556
$585$	−17.5260	−0.724611
$586$	2.62218	0.108321
$587$	30.4250	1.25577	0.627887	−	0.778305i	$-0.283920\pi$
$587$	30.4250	1.25577	0.627887	+	0.778305i	$0.283920\pi$
$588$	10.7083	0.441602
$589$	−14.4563	−0.595661
$590$	−67.5397	−2.78057
$591$	−7.53823	−0.310081
$592$	1.91803	0.0788306
$593$	9.22355	0.378766	0.189383	−	0.981903i	$-0.439351\pi$
$593$	9.22355	0.378766	0.189383	+	0.981903i	$0.439351\pi$
$594$	−1.39589	−0.0572740
$595$	−19.2942	−0.790987
$596$	10.7883	0.441906
$597$	−12.3704	−0.506287
$598$	0.642252	0.0262636
$599$	27.2671	1.11410	0.557052	−	0.830477i	$-0.311932\pi$
$599$	27.2671	1.11410	0.557052	+	0.830477i	$0.311932\pi$
$600$	30.0503	1.22680
$601$	−17.8387	−0.727658	−0.363829	−	0.931466i	$-0.618531\pi$
$601$	−17.8387	−0.727658	−0.363829	+	0.931466i	$0.618531\pi$
$602$	−0.481264	−0.0196149
$603$	3.88247	0.158107
$604$	−13.3072	−0.541464
$605$	36.9722	1.50314
$606$	−0.287147	−0.0116646
$607$	0.574338	0.0233117	0.0116558	−	0.999932i	$-0.496290\pi$
$607$	0.574338	0.0233117	0.0116558	+	0.999932i	$0.496290\pi$
$608$	−16.3273	−0.662159
$609$	2.42904	0.0984296
$610$	−20.1110	−0.814269
$611$	2.92443	0.118310
$612$	0.588311	0.0237810
$613$	42.2247	1.70544	0.852720	−	0.522368i	$-0.174951\pi$
$613$	42.2247	1.70544	0.852720	+	0.522368i	$0.174951\pi$
$614$	1.18633	0.0478764
$615$	7.34628	0.296231
$616$	18.1377	0.730789
$617$	−8.90341	−0.358438	−0.179219	−	0.983809i	$-0.557357\pi$
$617$	−8.90341	−0.358438	−0.179219	+	0.983809i	$0.557357\pi$
$618$	9.12778	0.367173
$619$	19.5496	0.785763	0.392882	−	0.919589i	$-0.371478\pi$
$619$	19.5496	0.785763	0.392882	+	0.919589i	$0.371478\pi$
$620$	6.42083	0.257867
$621$	−0.118540	−0.00475686
$622$	11.7652	0.471741
$623$	22.8547	0.915655
$624$	11.2965	0.452221
$625$	21.6251	0.865006
$626$	−10.5286	−0.420809
$627$	5.98090	0.238854
$628$	0.588311	0.0234762
$629$	0.774252	0.0308715
$630$	22.9244	0.913329
$631$	−0.935282	−0.0372330	−0.0186165	−	0.999827i	$-0.505926\pi$
$631$	−0.935282	−0.0372330	−0.0186165	+	0.999827i	$0.505926\pi$
$632$	−0.594577	−0.0236510
$633$	22.7571	0.904515
$634$	8.04778	0.319618
$635$	−0.107641	−0.00427161
$636$	0.666501	0.0264285
$637$	83.0009	3.28861
$638$	0.675414	0.0267399
$639$	−2.48521	−0.0983133
$640$	−15.3730	−0.607671
$641$	−29.2071	−1.15361	−0.576805	−	0.816882i	$-0.695701\pi$
$641$	−29.2071	−1.15361	−0.576805	+	0.816882i	$0.695701\pi$
$642$	8.60290	0.339529
$643$	24.1361	0.951837	0.475918	−	0.879490i	$-0.342116\pi$
$643$	24.1361	0.951837	0.475918	+	0.879490i	$0.342116\pi$
$644$	0.350097	0.0137957
$645$	0.310107	0.0122105
$646$	6.04860	0.237979
$647$	22.5190	0.885312	0.442656	−	0.896691i	$-0.354036\pi$
$647$	22.5190	0.885312	0.442656	+	0.896691i	$0.354036\pi$
$648$	−3.07529	−0.120809
$649$	17.3763	0.682081
$650$	52.9422	2.07656
$651$	14.2557	0.558723
$652$	1.20961	0.0473719
$653$	10.2446	0.400901	0.200451	−	0.979704i	$-0.435759\pi$
$653$	10.2446	0.400901	0.200451	+	0.979704i	$0.435759\pi$
$654$	2.52878	0.0988833
$655$	−59.4327	−2.32223
$656$	−4.73509	−0.184874
$657$	−9.15915	−0.357333
$658$	−3.82521	−0.149122
$659$	39.1301	1.52429	0.762145	−	0.647406i	$-0.224146\pi$
$659$	39.1301	1.52429	0.762145	+	0.647406i	$0.224146\pi$
$660$	−2.65644	−0.103402
$661$	−43.7425	−1.70139	−0.850694	−	0.525661i	$-0.823818\pi$
$661$	−43.7425	−1.70139	−0.850694	+	0.525661i	$0.823818\pi$
$662$	−9.87374	−0.383754
$663$	4.56005	0.177098
$664$	21.0531	0.817018
$665$	−98.2229	−3.80892
$666$	−0.919924	−0.0356463
$667$	0.0573568	0.00222087
$668$	−9.20722	−0.356238
$669$	−14.7887	−0.571765
$670$	−17.7293	−0.684941
$671$	5.17406	0.199742
$672$	16.1007	0.621097
$673$	13.0045	0.501285	0.250642	−	0.968080i	$-0.419358\pi$
$673$	13.0045	0.501285	0.250642	+	0.968080i	$0.419358\pi$
$674$	9.13677	0.351935
$675$	−9.77153	−0.376106
$676$	−4.58535	−0.176360
$677$	12.4576	0.478785	0.239393	−	0.970923i	$-0.423052\pi$
$677$	12.4576	0.478785	0.239393	+	0.970923i	$0.423052\pi$
$678$	4.90349	0.188317
$679$	6.64267	0.254922
$680$	−11.8195	−0.453257
$681$	−19.3291	−0.740693
$682$	3.96390	0.151786
$683$	3.41846	0.130804	0.0654018	−	0.997859i	$-0.479167\pi$
$683$	3.41846	0.130804	0.0654018	+	0.997859i	$0.479167\pi$
$684$	2.99496	0.114515
$685$	−21.6758	−0.828190
$686$	−66.8143	−2.55098
$687$	−23.1244	−0.882252
$688$	−0.199881	−0.00762040
$689$	5.16612	0.196813
$690$	0.541313	0.0206074
$691$	33.9308	1.29079	0.645394	−	0.763850i	$-0.276693\pi$
$691$	33.9308	1.29079	0.645394	+	0.763850i	$0.276693\pi$
$692$	11.9743	0.455194
$693$	−5.89789	−0.224042
$694$	19.7031	0.747918
$695$	−6.54560	−0.248289
$696$	1.48801	0.0564028
$697$	−1.91141	−0.0724000
$698$	0.652760	0.0247073
$699$	23.2370	0.878906
$700$	28.8592	1.09078
$701$	−41.2533	−1.55812	−0.779059	−	0.626951i	$-0.784303\pi$
$701$	−41.2533	−1.55812	−0.779059	+	0.626951i	$0.784303\pi$
$702$	−5.41801	−0.204490
$703$	3.94155	0.148659
$704$	10.2977	0.388111
$705$	2.46481	0.0928301
$706$	−13.9094	−0.523487
$707$	−1.21325	−0.0456290
$708$	8.70129	0.327015
$709$	−6.13360	−0.230352	−0.115176	−	0.993345i	$-0.536743\pi$
$709$	−6.13360	−0.230352	−0.115176	+	0.993345i	$0.536743\pi$
$710$	11.3487	0.425908
$711$	0.193340	0.00725083
$712$	14.0006	0.524695
$713$	0.336619	0.0126065
$714$	−5.96465	−0.223221
$715$	−20.5904	−0.770036
$716$	−7.60241	−0.284115
$717$	−19.2054	−0.717240
$718$	0.279530	0.0104320
$719$	15.8157	0.589825	0.294912	−	0.955524i	$-0.404710\pi$
$719$	15.8157	0.589825	0.294912	+	0.955524i	$0.404710\pi$
$720$	9.52107	0.354829
$721$	38.5666	1.43629
$722$	8.21736	0.305818
$723$	−24.0043	−0.892728
$724$	10.4260	0.387479
$725$	4.72805	0.175595
$726$	11.4296	0.424194
$727$	−33.5663	−1.24491	−0.622453	−	0.782657i	$-0.713864\pi$
$727$	−33.5663	−1.24491	−0.622453	+	0.782657i	$0.713864\pi$
$728$	70.3998	2.60919
$729$	1.00000	0.0370370
$730$	41.8252	1.54802
$731$	−0.0806862	−0.00298429
$732$	2.59094	0.0957639
$733$	−24.4196	−0.901958	−0.450979	−	0.892535i	$-0.648925\pi$
$733$	−24.4196	−0.901958	−0.450979	+	0.892535i	$0.648925\pi$
$734$	−17.5691	−0.648487
$735$	69.9561	2.58037
$736$	0.380185	0.0140138
$737$	4.56131	0.168018
$738$	2.27104	0.0835981
$739$	30.3577	1.11673	0.558363	−	0.829597i	$-0.311430\pi$
$739$	30.3577	1.11673	0.558363	+	0.829597i	$0.311430\pi$
$740$	−1.75066	−0.0643556
$741$	23.2143	0.852798
$742$	−6.75739	−0.248072
$743$	−15.9453	−0.584977	−0.292489	−	0.956269i	$-0.594483\pi$
$743$	−15.9453	−0.584977	−0.292489	+	0.956269i	$0.594483\pi$
$744$	8.73289	0.320163
$745$	70.4788	2.58214
$746$	14.4298	0.528311
$747$	−6.84589	−0.250478
$748$	0.691175	0.0252719
$749$	36.3488	1.32816
$750$	21.7891	0.795627
$751$	0.273758	0.00998958	0.00499479	−	0.999988i	$-0.498410\pi$
$751$	0.273758	0.00998958	0.00499479	+	0.999988i	$0.498410\pi$
$752$	−1.58871	−0.0579342
$753$	20.2627	0.738413
$754$	2.62155	0.0954714
$755$	−86.9349	−3.16389
$756$	−2.95340	−0.107414
$757$	19.1900	0.697472	0.348736	−	0.937221i	$-0.386611\pi$
$757$	19.1900	0.697472	0.348736	+	0.937221i	$0.386611\pi$
$758$	−3.87649	−0.140801
$759$	−0.139267	−0.00505506
$760$	−60.1705	−2.18261
$761$	−28.1802	−1.02153	−0.510766	−	0.859720i	$-0.670638\pi$
$761$	−28.1802	−1.02153	−0.510766	+	0.859720i	$0.670638\pi$
$762$	−0.0332764	−0.00120548
$763$	10.6846	0.386808
$764$	−12.1853	−0.440847
$765$	3.84337	0.138958
$766$	−11.7839	−0.425769
$767$	67.4446	2.43528
$768$	12.7779	0.461084
$769$	2.44796	0.0882759	0.0441379	−	0.999025i	$-0.485946\pi$
$769$	2.44796	0.0882759	0.0441379	+	0.999025i	$0.485946\pi$
$770$	26.9326	0.970584
$771$	−22.3132	−0.803589
$772$	13.3562	0.480699
$773$	34.8672	1.25409	0.627043	−	0.778984i	$-0.284265\pi$
$773$	34.8672	1.25409	0.627043	+	0.778984i	$0.284265\pi$
$774$	0.0958669	0.00344586
$775$	27.7482	0.996744
$776$	4.06924	0.146077
$777$	−3.88685	−0.139440
$778$	−10.9549	−0.392751
$779$	−9.73061	−0.348635
$780$	−10.3107	−0.369183
$781$	−2.91974	−0.104476
$782$	−0.140843	−0.00503654
$783$	−0.483859	−0.0172917
$784$	−45.0906	−1.61038
$785$	3.84337	0.137176
$786$	−18.3731	−0.655346
$787$	−47.9465	−1.70911	−0.854554	−	0.519363i	$-0.826169\pi$
$787$	−47.9465	−1.70911	−0.854554	+	0.519363i	$0.826169\pi$
$788$	−4.43482	−0.157984
$789$	19.1595	0.682098
$790$	−0.882886	−0.0314117
$791$	20.7182	0.736653
$792$	−3.61299	−0.128382
$793$	20.0826	0.713155
$794$	−8.39312	−0.297861
$795$	4.35419	0.154427
$796$	−7.27765	−0.257949
$797$	−16.4717	−0.583456	−0.291728	−	0.956501i	$-0.594230\pi$
$797$	−16.4717	−0.583456	−0.291728	+	0.956501i	$0.594230\pi$
$798$	−30.3648	−1.07490
$799$	−0.641314	−0.0226881
$800$	31.3395	1.10802
$801$	−4.55261	−0.160859
$802$	7.11035	0.251075
$803$	−10.7606	−0.379733
$804$	2.28410	0.0805540
$805$	2.28715	0.0806113
$806$	15.3855	0.541931
$807$	−16.5414	−0.582283
$808$	−0.743227	−0.0261466
$809$	−35.8117	−1.25907	−0.629536	−	0.776971i	$-0.716755\pi$
$809$	−35.8117	−1.25907	−0.629536	+	0.776971i	$0.716755\pi$
$810$	−4.56649	−0.160450
$811$	−37.2845	−1.30924	−0.654618	−	0.755960i	$-0.727170\pi$
$811$	−37.2845	−1.30924	−0.654618	+	0.755960i	$0.727170\pi$
$812$	1.42903	0.0501491
$813$	−23.9458	−0.839817
$814$	−1.08077	−0.0378810
$815$	7.90224	0.276803
$816$	−2.47727	−0.0867218
$817$	−0.410756	−0.0143705
$818$	−42.7941	−1.49626
$819$	−22.8921	−0.799914
$820$	4.32190	0.150927
$821$	44.1705	1.54156	0.770780	−	0.637101i	$-0.219867\pi$
$821$	44.1705	1.54156	0.770780	+	0.637101i	$0.219867\pi$
$822$	−6.70088	−0.233720
$823$	32.8716	1.14583	0.572917	−	0.819614i	$-0.305812\pi$
$823$	32.8716	1.14583	0.572917	+	0.819614i	$0.305812\pi$
$824$	23.6255	0.823035
$825$	−11.4801	−0.399684
$826$	−88.2189	−3.06953
$827$	27.9463	0.971789	0.485894	−	0.874017i	$-0.338494\pi$
$827$	27.9463	0.971789	0.485894	+	0.874017i	$0.338494\pi$
$828$	−0.0697385	−0.00242358
$829$	16.7122	0.580438	0.290219	−	0.956960i	$-0.406272\pi$
$829$	16.7122	0.580438	0.290219	+	0.956960i	$0.406272\pi$
$830$	31.2617	1.08511
$831$	30.0068	1.04092
$832$	39.9697	1.38570
$833$	−18.2017	−0.630652
$834$	−2.02351	−0.0700686
$835$	−60.1499	−2.08157
$836$	3.51862	0.121694
$837$	−2.83970	−0.0981543
$838$	−38.4352	−1.32772
$839$	−33.7159	−1.16400	−0.582001	−	0.813188i	$-0.697730\pi$
$839$	−33.7159	−1.16400	−0.582001	+	0.813188i	$0.697730\pi$
$840$	59.3354	2.04727
$841$	−28.7659	−0.991927
$842$	−9.27528	−0.319647
$843$	25.4712	0.877275
$844$	13.3883	0.460843
$845$	−29.9557	−1.03051
$846$	0.761974	0.0261972
$847$	48.2923	1.65934
$848$	−2.80652	−0.0963762
$849$	−21.9793	−0.754328
$850$	−11.6100	−0.398220
$851$	−0.0917801	−0.00314618
$852$	−1.46207	−0.0500898
$853$	4.61320	0.157953	0.0789765	−	0.996876i	$-0.474835\pi$
$853$	4.61320	0.157953	0.0789765	+	0.996876i	$0.474835\pi$
$854$	−26.2685	−0.898889
$855$	19.5658	0.669136
$856$	22.2670	0.761070
$857$	−22.2479	−0.759972	−0.379986	−	0.924992i	$-0.624071\pi$
$857$	−22.2479	−0.759972	−0.379986	+	0.924992i	$0.624071\pi$
$858$	−6.36533	−0.217309
$859$	−47.4050	−1.61744	−0.808719	−	0.588196i	$-0.799839\pi$
$859$	−47.4050	−1.61744	−0.808719	+	0.588196i	$0.799839\pi$
$860$	0.182439	0.00622113
$861$	9.59556	0.327016
$862$	8.59687	0.292811
$863$	16.6358	0.566289	0.283144	−	0.959077i	$-0.408622\pi$
$863$	16.6358	0.566289	0.283144	+	0.959077i	$0.408622\pi$
$864$	−3.20722	−0.109112
$865$	78.2269	2.65979
$866$	−21.5896	−0.733646
$867$	−1.00000	−0.0339618
$868$	8.38675	0.284665
$869$	0.227145	0.00770537
$870$	2.20954	0.0749104
$871$	17.7043	0.599887
$872$	6.54528	0.221651
$873$	−1.32321	−0.0447837
$874$	−0.717002	−0.0242530
$875$	92.0631	3.11230
$876$	−5.38843	−0.182058
$877$	0.731181	0.0246902	0.0123451	−	0.999924i	$-0.496070\pi$
$877$	0.731181	0.0246902	0.0123451	+	0.999924i	$0.496070\pi$
$878$	10.7365	0.362338
$879$	−2.20695	−0.0744386
$880$	11.1858	0.377073
$881$	13.8625	0.467038	0.233519	−	0.972352i	$-0.424976\pi$
$881$	13.8625	0.467038	0.233519	+	0.972352i	$0.424976\pi$
$882$	21.6263	0.728195
$883$	−17.8901	−0.602048	−0.301024	−	0.953617i	$-0.597328\pi$
$883$	−17.8901	−0.602048	−0.301024	+	0.953617i	$0.597328\pi$
$884$	2.68273	0.0902299
$885$	56.8447	1.91081
$886$	−10.0610	−0.338006
$887$	17.5822	0.590353	0.295177	−	0.955443i	$-0.404621\pi$
$887$	17.5822	0.590353	0.295177	+	0.955443i	$0.404621\pi$
$888$	−2.38105	−0.0799028
$889$	−0.140599	−0.00471553
$890$	20.7894	0.696864
$891$	1.17485	0.0393588
$892$	−8.70036	−0.291310
$893$	−3.26479	−0.109252
$894$	21.7879	0.728697
$895$	−49.6658	−1.66014
$896$	−20.0799	−0.670821
$897$	−0.540550	−0.0180484
$898$	17.1696	0.572957
$899$	1.37401	0.0458259
$900$	−5.74870	−0.191623
$901$	−1.13291	−0.0377426
$902$	2.66812	0.0888388
$903$	0.405055	0.0134794
$904$	12.6918	0.422122
$905$	68.1120	2.26412
$906$	−26.8752	−0.892867
$907$	13.1856	0.437822	0.218911	−	0.975745i	$-0.429750\pi$
$907$	13.1856	0.437822	0.218911	+	0.975745i	$0.429750\pi$
$908$	−11.3715	−0.377377
$909$	0.241677	0.00801592
$910$	104.536	3.46535
$911$	−38.3718	−1.27131	−0.635657	−	0.771971i	$-0.719271\pi$
$911$	−38.3718	−1.27131	−0.635657	+	0.771971i	$0.719271\pi$
$912$	−12.6113	−0.417600
$913$	−8.04287	−0.266180
$914$	25.2843	0.836329
$915$	16.9263	0.559568
$916$	−13.6043	−0.449500
$917$	−77.6297	−2.56356
$918$	1.18815	0.0392146
$919$	52.9985	1.74826	0.874129	−	0.485693i	$-0.161433\pi$
$919$	52.9985	1.74826	0.874129	+	0.485693i	$0.161433\pi$
$920$	1.40109	0.0461924
$921$	−0.998472	−0.0329008
$922$	−18.6621	−0.614605
$923$	−11.3327	−0.373020
$924$	−3.46979	−0.114148
$925$	−7.56563	−0.248756
$926$	−26.7904	−0.880388
$927$	−7.68238	−0.252322
$928$	1.55184	0.0509418
$929$	−40.0709	−1.31468	−0.657342	−	0.753593i	$-0.728319\pi$
$929$	−40.0709	−1.31468	−0.657342	+	0.753593i	$0.728319\pi$
$930$	12.9674	0.425219
$931$	−92.6611	−3.03685
$932$	13.6706	0.447795
$933$	−9.90215	−0.324182
$934$	−18.8310	−0.616168
$935$	4.51538	0.147669
$936$	−14.0235	−0.458372
$937$	−0.530328	−0.0173251	−0.00866254	−	0.999962i	$-0.502757\pi$
$937$	−0.530328	−0.0173251	−0.00866254	+	0.999962i	$0.502757\pi$
$938$	−23.1576	−0.756122
$939$	8.86140	0.289181
$940$	1.45007	0.0472962
$941$	1.09490	0.0356928	0.0178464	−	0.999841i	$-0.494319\pi$
$941$	1.09490	0.0356928	0.0178464	+	0.999841i	$0.494319\pi$
$942$	1.18815	0.0387119
$943$	0.226580	0.00737845
$944$	−36.6396	−1.19252
$945$	−19.2942	−0.627642
$946$	0.112629	0.00366188
$947$	−33.5395	−1.08989	−0.544944	−	0.838472i	$-0.683449\pi$
$947$	−33.5395	−1.08989	−0.544944	+	0.838472i	$0.683449\pi$
$948$	0.113744	0.00369424
$949$	−41.7662	−1.35579
$950$	−59.1040	−1.91759
$951$	−6.77340	−0.219642
$952$	−15.4384	−0.500360
$953$	36.0895	1.16905	0.584527	−	0.811374i	$-0.301280\pi$
$953$	36.0895	1.16905	0.584527	+	0.811374i	$0.301280\pi$
$954$	1.34606	0.0435803
$955$	−79.6050	−2.57596
$956$	−11.2988	−0.365428
$957$	−0.568461	−0.0183757
$958$	20.9474	0.676780
$959$	−28.3124	−0.914257
$960$	33.6879	1.08727
$961$	−22.9361	−0.739875
$962$	−4.19491	−0.135249
$963$	−7.24061	−0.233325
$964$	−14.1220	−0.454838
$965$	87.2545	2.80882
$966$	0.707051	0.0227490
$967$	35.1226	1.12947	0.564733	−	0.825274i	$-0.308979\pi$
$967$	35.1226	1.12947	0.564733	+	0.825274i	$0.308979\pi$
$968$	29.5835	0.950849
$969$	−5.09079	−0.163540
$970$	6.04240	0.194010
$971$	47.7957	1.53384	0.766919	−	0.641744i	$-0.221789\pi$
$971$	47.7957	1.53384	0.766919	+	0.641744i	$0.221789\pi$
$972$	0.588311	0.0188701
$973$	−8.54972	−0.274091
$974$	−35.4810	−1.13689
$975$	−44.5587	−1.42702
$976$	−10.9100	−0.349220
$977$	32.8823	1.05200	0.525998	−	0.850486i	$-0.323692\pi$
$977$	32.8823	1.05200	0.525998	+	0.850486i	$0.323692\pi$
$978$	2.44291	0.0781156
$979$	−5.34862	−0.170943
$980$	41.1559	1.31468
$981$	−2.12835	−0.0679529
$982$	−29.0912	−0.928338
$983$	5.08816	0.162287	0.0811436	−	0.996702i	$-0.474143\pi$
$983$	5.08816	0.162287	0.0811436	+	0.996702i	$0.474143\pi$
$984$	5.87815	0.187389
$985$	−28.9723	−0.923133
$986$	−0.574895	−0.0183084
$987$	3.21948	0.102477
$988$	13.6572	0.434494
$989$	0.00956456	0.000304135 0
$990$	−5.36492	−0.170508
$991$	−4.73117	−0.150291	−0.0751453	−	0.997173i	$-0.523942\pi$
$991$	−4.73117	−0.150291	−0.0751453	+	0.997173i	$0.523942\pi$
$992$	9.10754	0.289165
$993$	8.31021	0.263717
$994$	14.8234	0.470169
$995$	−47.5441	−1.50725
$996$	−4.02751	−0.127616
$997$	−36.8310	−1.16645	−0.583224	−	0.812311i	$-0.698209\pi$
$997$	−36.8310	−1.16645	−0.583224	+	0.812311i	$0.698209\pi$
$998$	25.5296	0.808124
$999$	0.774252	0.0244963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.34	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.34	✓	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)

\(f(q)\)	\(=\)	\(q+1.18815 q^{2} -1.00000 q^{3} -0.588311 q^{4} -3.84337 q^{5} -1.18815 q^{6} -5.02013 q^{7} -3.07529 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.18815 q^{2} -1.00000 q^{3} -0.588311 q^{4} -3.84337 q^{5} -1.18815 q^{6} -5.02013 q^{7} -3.07529 q^{8} +1.00000 q^{9} -4.56649 q^{10} +1.17485 q^{11} +0.588311 q^{12} +4.56005 q^{13} -5.96465 q^{14} +3.84337 q^{15} -2.47727 q^{16} -1.00000 q^{17} +1.18815 q^{18} -5.09079 q^{19} +2.26110 q^{20} +5.02013 q^{21} +1.39589 q^{22} +0.118540 q^{23} +3.07529 q^{24} +9.77153 q^{25} +5.41801 q^{26} -1.00000 q^{27} +2.95340 q^{28} +0.483859 q^{29} +4.56649 q^{30} +2.83970 q^{31} +3.20722 q^{32} -1.17485 q^{33} -1.18815 q^{34} +19.2942 q^{35} -0.588311 q^{36} -0.774252 q^{37} -6.04860 q^{38} -4.56005 q^{39} +11.8195 q^{40} +1.91141 q^{41} +5.96465 q^{42} +0.0806862 q^{43} -0.691175 q^{44} -3.84337 q^{45} +0.140843 q^{46} +0.641314 q^{47} +2.47727 q^{48} +18.2017 q^{49} +11.6100 q^{50} +1.00000 q^{51} -2.68273 q^{52} +1.13291 q^{53} -1.18815 q^{54} -4.51538 q^{55} +15.4384 q^{56} +5.09079 q^{57} +0.574895 q^{58} +14.7903 q^{59} -2.26110 q^{60} +4.40403 q^{61} +3.37397 q^{62} -5.02013 q^{63} +8.76519 q^{64} -17.5260 q^{65} -1.39589 q^{66} +3.88247 q^{67} +0.588311 q^{68} -0.118540 q^{69} +22.9244 q^{70} -2.48521 q^{71} -3.07529 q^{72} -9.15915 q^{73} -0.919924 q^{74} -9.77153 q^{75} +2.99496 q^{76} -5.89789 q^{77} -5.41801 q^{78} +0.193340 q^{79} +9.52107 q^{80} +1.00000 q^{81} +2.27104 q^{82} -6.84589 q^{83} -2.95340 q^{84} +3.84337 q^{85} +0.0958669 q^{86} -0.483859 q^{87} -3.61299 q^{88} -4.55261 q^{89} -4.56649 q^{90} -22.8921 q^{91} -0.0697385 q^{92} -2.83970 q^{93} +0.761974 q^{94} +19.5658 q^{95} -3.20722 q^{96} -1.32321 q^{97} +21.6263 q^{98} +1.17485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 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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