LMFDB - Embedded newform 4015.2.a.f.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, 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("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.773764 q^{2} -2.02237 q^{3} -1.40129 q^{4} -1.00000 q^{5} -1.56484 q^{6} -2.27321 q^{7} -2.63180 q^{8} +1.08999 q^{9} +O(q^{10})$	\(q+0.773764 q^{2} -2.02237 q^{3} -1.40129 q^{4} -1.00000 q^{5} -1.56484 q^{6} -2.27321 q^{7} -2.63180 q^{8} +1.08999 q^{9} -0.773764 q^{10} +1.00000 q^{11} +2.83393 q^{12} +6.36030 q^{13} -1.75893 q^{14} +2.02237 q^{15} +0.766190 q^{16} -5.57219 q^{17} +0.843395 q^{18} -4.88762 q^{19} +1.40129 q^{20} +4.59727 q^{21} +0.773764 q^{22} +4.99417 q^{23} +5.32247 q^{24} +1.00000 q^{25} +4.92137 q^{26} +3.86275 q^{27} +3.18542 q^{28} +7.12359 q^{29} +1.56484 q^{30} -1.82434 q^{31} +5.85644 q^{32} -2.02237 q^{33} -4.31156 q^{34} +2.27321 q^{35} -1.52739 q^{36} -4.38574 q^{37} -3.78186 q^{38} -12.8629 q^{39} +2.63180 q^{40} +1.00535 q^{41} +3.55720 q^{42} -0.387387 q^{43} -1.40129 q^{44} -1.08999 q^{45} +3.86431 q^{46} +8.76444 q^{47} -1.54952 q^{48} -1.83253 q^{49} +0.773764 q^{50} +11.2691 q^{51} -8.91262 q^{52} +6.13459 q^{53} +2.98886 q^{54} -1.00000 q^{55} +5.98261 q^{56} +9.88458 q^{57} +5.51198 q^{58} +12.4101 q^{59} -2.83393 q^{60} -11.1114 q^{61} -1.41161 q^{62} -2.47777 q^{63} +2.99912 q^{64} -6.36030 q^{65} -1.56484 q^{66} -8.25643 q^{67} +7.80826 q^{68} -10.1001 q^{69} +1.75893 q^{70} +11.6565 q^{71} -2.86863 q^{72} +1.00000 q^{73} -3.39353 q^{74} -2.02237 q^{75} +6.84897 q^{76} -2.27321 q^{77} -9.95285 q^{78} +6.89304 q^{79} -0.766190 q^{80} -11.0819 q^{81} +0.777903 q^{82} -0.604287 q^{83} -6.44210 q^{84} +5.57219 q^{85} -0.299746 q^{86} -14.4066 q^{87} -2.63180 q^{88} -3.63458 q^{89} -0.843395 q^{90} -14.4583 q^{91} -6.99828 q^{92} +3.68950 q^{93} +6.78161 q^{94} +4.88762 q^{95} -11.8439 q^{96} +6.31756 q^{97} -1.41795 q^{98} +1.08999 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9} + 7 q^{10} + 31 q^{11} - 4 q^{12} - 24 q^{13} - 9 q^{14} + 4 q^{15} + 43 q^{16} - 49 q^{17} - 35 q^{18} - 22 q^{19} - 39 q^{20} - 8 q^{21} - 7 q^{22} - q^{23} - 13 q^{24} + 31 q^{25} - 9 q^{26} - 22 q^{27} - 34 q^{28} - 12 q^{29} + 5 q^{30} + 4 q^{31} - 45 q^{32} - 4 q^{33} + 2 q^{34} + 11 q^{35} + 34 q^{36} - 18 q^{37} - 7 q^{38} - q^{39} + 24 q^{40} - 58 q^{41} - 21 q^{42} - 41 q^{43} + 39 q^{44} - 31 q^{45} + 23 q^{46} - 31 q^{47} - 29 q^{48} + 44 q^{49} - 7 q^{50} + 8 q^{51} - 89 q^{52} - 46 q^{53} - 47 q^{54} - 31 q^{55} + 10 q^{56} - 47 q^{57} - 34 q^{58} - 9 q^{59} + 4 q^{60} - 5 q^{61} - 50 q^{62} - 61 q^{63} + 78 q^{64} + 24 q^{65} - 5 q^{66} + q^{67} - 115 q^{68} - 19 q^{69} + 9 q^{70} - 8 q^{71} - 93 q^{72} + 31 q^{73} - 19 q^{74} - 4 q^{75} - 7 q^{76} - 11 q^{77} + 57 q^{78} - 43 q^{80} + 43 q^{81} + 20 q^{82} - 29 q^{83} - 32 q^{84} + 49 q^{85} + 25 q^{86} - 62 q^{87} - 24 q^{88} - 77 q^{89} + 35 q^{90} - 11 q^{91} - 25 q^{92} - 38 q^{94} + 22 q^{95} - 23 q^{96} - 39 q^{97} - 65 q^{98} + 31 q^{99}+O(q^{100}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9 + 7 * q^10 + 31 * q^11 - 4 * q^12 - 24 * q^13 - 9 * q^14 + 4 * q^15 + 43 * q^16 - 49 * q^17 - 35 * q^18 - 22 * q^19 - 39 * q^20 - 8 * q^21 - 7 * q^22 - q^23 - 13 * q^24 + 31 * q^25 - 9 * q^26 - 22 * q^27 - 34 * q^28 - 12 * q^29 + 5 * q^30 + 4 * q^31 - 45 * q^32 - 4 * q^33 + 2 * q^34 + 11 * q^35 + 34 * q^36 - 18 * q^37 - 7 * q^38 - q^39 + 24 * q^40 - 58 * q^41 - 21 * q^42 - 41 * q^43 + 39 * q^44 - 31 * q^45 + 23 * q^46 - 31 * q^47 - 29 * q^48 + 44 * q^49 - 7 * q^50 + 8 * q^51 - 89 * q^52 - 46 * q^53 - 47 * q^54 - 31 * q^55 + 10 * q^56 - 47 * q^57 - 34 * q^58 - 9 * q^59 + 4 * q^60 - 5 * q^61 - 50 * q^62 - 61 * q^63 + 78 * q^64 + 24 * q^65 - 5 * q^66 + q^67 - 115 * q^68 - 19 * q^69 + 9 * q^70 - 8 * q^71 - 93 * q^72 + 31 * q^73 - 19 * q^74 - 4 * q^75 - 7 * q^76 - 11 * q^77 + 57 * q^78 - 43 * q^80 + 43 * q^81 + 20 * q^82 - 29 * q^83 - 32 * q^84 + 49 * q^85 + 25 * q^86 - 62 * q^87 - 24 * q^88 - 77 * q^89 + 35 * q^90 - 11 * q^91 - 25 * q^92 - 38 * q^94 + 22 * q^95 - 23 * q^96 - 39 * q^97 - 65 * q^98 + 31 * 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 $

$2$	0.773764	0.547134	0.273567	−	0.961853i	$-0.411797\pi$
$2$	0.773764	0.547134	0.273567	+	0.961853i	$0.411797\pi$
$3$	−2.02237	−1.16762	−0.583809	−	0.811891i	$-0.698438\pi$
$3$	−2.02237	−1.16762	−0.583809	+	0.811891i	$0.698438\pi$
$4$	−1.40129	−0.700645
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.56484	−0.638843
$7$	−2.27321	−0.859191	−0.429596	−	0.903021i	$-0.641344\pi$
$7$	−2.27321	−0.859191	−0.429596	+	0.903021i	$0.641344\pi$
$8$	−2.63180	−0.930480
$9$	1.08999	0.363330
$10$	−0.773764	−0.244686
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.83393	0.818085
$13$	6.36030	1.76403	0.882015	−	0.471221i	$-0.156186\pi$
$13$	6.36030	1.76403	0.882015	+	0.471221i	$0.156186\pi$
$14$	−1.75893	−0.470093
$15$	2.02237	0.522174
$16$	0.766190	0.191547
$17$	−5.57219	−1.35146	−0.675728	−	0.737151i	$-0.736171\pi$
$17$	−5.57219	−1.35146	−0.675728	+	0.737151i	$0.736171\pi$
$18$	0.843395	0.198790
$19$	−4.88762	−1.12130	−0.560648	−	0.828054i	$-0.689448\pi$
$19$	−4.88762	−1.12130	−0.560648	+	0.828054i	$0.689448\pi$
$20$	1.40129	0.313338
$21$	4.59727	1.00321
$22$	0.773764	0.164967
$23$	4.99417	1.04136	0.520679	−	0.853753i	$-0.325679\pi$
$23$	4.99417	1.04136	0.520679	+	0.853753i	$0.325679\pi$
$24$	5.32247	1.08644
$25$	1.00000	0.200000
$26$	4.92137	0.965161
$27$	3.86275	0.743387
$28$	3.18542	0.601988
$29$	7.12359	1.32282	0.661409	−	0.750025i	$-0.269959\pi$
$29$	7.12359	1.32282	0.661409	+	0.750025i	$0.269959\pi$
$30$	1.56484	0.285699
$31$	−1.82434	−0.327662	−0.163831	−	0.986488i	$-0.552385\pi$
$31$	−1.82434	−0.327662	−0.163831	+	0.986488i	$0.552385\pi$
$32$	5.85644	1.03528
$33$	−2.02237	−0.352050
$34$	−4.31156	−0.739427
$35$	2.27321	0.384242
$36$	−1.52739	−0.254565
$37$	−4.38574	−0.721011	−0.360506	−	0.932757i	$-0.617396\pi$
$37$	−4.38574	−0.721011	−0.360506	+	0.932757i	$0.617396\pi$
$38$	−3.78186	−0.613499
$39$	−12.8629	−2.05971
$40$	2.63180	0.416123
$41$	1.00535	0.157009	0.0785046	−	0.996914i	$-0.474985\pi$
$41$	1.00535	0.157009	0.0785046	+	0.996914i	$0.474985\pi$
$42$	3.55720	0.548888
$43$	−0.387387	−0.0590759	−0.0295380	−	0.999564i	$-0.509404\pi$
$43$	−0.387387	−0.0590759	−0.0295380	+	0.999564i	$0.509404\pi$
$44$	−1.40129	−0.211252
$45$	−1.08999	−0.162486
$46$	3.86431	0.569762
$47$	8.76444	1.27843	0.639213	−	0.769030i	$-0.279260\pi$
$47$	8.76444	1.27843	0.639213	+	0.769030i	$0.279260\pi$
$48$	−1.54952	−0.223654
$49$	−1.83253	−0.261790
$50$	0.773764	0.109427
$51$	11.2691	1.57798
$52$	−8.91262	−1.23596
$53$	6.13459	0.842651	0.421325	−	0.906910i	$-0.361565\pi$
$53$	6.13459	0.842651	0.421325	+	0.906910i	$0.361565\pi$
$54$	2.98886	0.406732
$55$	−1.00000	−0.134840
$56$	5.98261	0.799460
$57$	9.88458	1.30925
$58$	5.51198	0.723758
$59$	12.4101	1.61566	0.807829	−	0.589417i	$-0.200642\pi$
$59$	12.4101	1.61566	0.807829	+	0.589417i	$0.200642\pi$
$60$	−2.83393	−0.365859
$61$	−11.1114	−1.42266	−0.711332	−	0.702856i	$-0.751908\pi$
$61$	−11.1114	−1.42266	−0.711332	+	0.702856i	$0.751908\pi$
$62$	−1.41161	−0.179275
$63$	−2.47777	−0.312170
$64$	2.99912	0.374890
$65$	−6.36030	−0.788899
$66$	−1.56484	−0.192618
$67$	−8.25643	−1.00868	−0.504342	−	0.863504i	$-0.668265\pi$
$67$	−8.25643	−1.00868	−0.504342	+	0.863504i	$0.668265\pi$
$68$	7.80826	0.946890
$69$	−10.1001	−1.21591
$70$	1.75893	0.210232
$71$	11.6565	1.38338	0.691688	−	0.722196i	$-0.256867\pi$
$71$	11.6565	1.38338	0.691688	+	0.722196i	$0.256867\pi$
$72$	−2.86863	−0.338071
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	−3.39353	−0.394490
$75$	−2.02237	−0.233523
$76$	6.84897	0.785630
$77$	−2.27321	−0.259056
$78$	−9.95285	−1.12694
$79$	6.89304	0.775528	0.387764	−	0.921759i	$-0.373248\pi$
$79$	6.89304	0.775528	0.387764	+	0.921759i	$0.373248\pi$
$80$	−0.766190	−0.0856626
$81$	−11.0819	−1.23132
$82$	0.777903	0.0859050
$83$	−0.604287	−0.0663292	−0.0331646	−	0.999450i	$-0.510559\pi$
$83$	−0.604287	−0.0663292	−0.0331646	+	0.999450i	$0.510559\pi$
$84$	−6.44210	−0.702891
$85$	5.57219	0.604389
$86$	−0.299746	−0.0323224
$87$	−14.4066	−1.54454
$88$	−2.63180	−0.280550
$89$	−3.63458	−0.385265	−0.192632	−	0.981271i	$-0.561702\pi$
$89$	−3.63458	−0.385265	−0.192632	+	0.981271i	$0.561702\pi$
$90$	−0.843395	−0.0889016
$91$	−14.4583	−1.51564
$92$	−6.99828	−0.729621
$93$	3.68950	0.382583
$94$	6.78161	0.699470
$95$	4.88762	0.501459
$96$	−11.8439	−1.20881
$97$	6.31756	0.641452	0.320726	−	0.947172i	$-0.396073\pi$
$97$	6.31756	0.641452	0.320726	+	0.947172i	$0.396073\pi$
$98$	−1.41795	−0.143234
$99$	1.08999	0.109548
$100$	−1.40129	−0.140129
$101$	−11.3536	−1.12973	−0.564864	−	0.825184i	$-0.691071\pi$
$101$	−11.3536	−1.12973	−0.564864	+	0.825184i	$0.691071\pi$
$102$	8.71959	0.863368
$103$	−17.0506	−1.68005	−0.840024	−	0.542549i	$-0.817459\pi$
$103$	−17.0506	−1.68005	−0.840024	+	0.542549i	$0.817459\pi$
$104$	−16.7390	−1.64140
$105$	−4.59727	−0.448648
$106$	4.74673	0.461043
$107$	−13.8416	−1.33812	−0.669060	−	0.743209i	$-0.733303\pi$
$107$	−13.8416	−1.33812	−0.669060	+	0.743209i	$0.733303\pi$
$108$	−5.41283	−0.520850
$109$	0.736585	0.0705520	0.0352760	−	0.999378i	$-0.488769\pi$
$109$	0.736585	0.0705520	0.0352760	+	0.999378i	$0.488769\pi$
$110$	−0.773764	−0.0737755
$111$	8.86960	0.841865
$112$	−1.74171	−0.164576
$113$	−13.4300	−1.26339	−0.631696	−	0.775216i	$-0.717641\pi$
$113$	−13.4300	−1.26339	−0.631696	+	0.775216i	$0.717641\pi$
$114$	7.64834	0.716332
$115$	−4.99417	−0.465709
$116$	−9.98221	−0.926825
$117$	6.93266	0.640925
$118$	9.60250	0.883981
$119$	12.6667	1.16116
$120$	−5.32247	−0.485873
$121$	1.00000	0.0909091
$122$	−8.59757	−0.778387
$123$	−2.03319	−0.183327
$124$	2.55643	0.229574
$125$	−1.00000	−0.0894427
$126$	−1.91721	−0.170799
$127$	−3.05570	−0.271150	−0.135575	−	0.990767i	$-0.543288\pi$
$127$	−3.05570	−0.271150	−0.135575	+	0.990767i	$0.543288\pi$
$128$	−9.39227	−0.830167
$129$	0.783440	0.0689781
$130$	−4.92137	−0.431633
$131$	3.78058	0.330311	0.165155	−	0.986268i	$-0.447187\pi$
$131$	3.78058	0.330311	0.165155	+	0.986268i	$0.447187\pi$
$132$	2.83393	0.246662
$133$	11.1106	0.963408
$134$	−6.38853	−0.551885
$135$	−3.86275	−0.332453
$136$	14.6649	1.25750
$137$	−0.0896275	−0.00765740	−0.00382870	−	0.999993i	$-0.501219\pi$
$137$	−0.0896275	−0.00765740	−0.00382870	+	0.999993i	$0.501219\pi$
$138$	−7.81508	−0.665264
$139$	9.62781	0.816620	0.408310	−	0.912843i	$-0.366118\pi$
$139$	9.62781	0.816620	0.408310	+	0.912843i	$0.366118\pi$
$140$	−3.18542	−0.269217
$141$	−17.7250	−1.49271
$142$	9.01941	0.756892
$143$	6.36030	0.531875
$144$	0.835139	0.0695949
$145$	−7.12359	−0.591582
$146$	0.773764	0.0640372
$147$	3.70606	0.305671
$148$	6.14569	0.505173
$149$	−15.3621	−1.25852	−0.629258	−	0.777196i	$-0.716641\pi$
$149$	−15.3621	−1.25852	−0.629258	+	0.777196i	$0.716641\pi$
$150$	−1.56484	−0.127769
$151$	−0.452322	−0.0368095	−0.0184047	−	0.999831i	$-0.505859\pi$
$151$	−0.452322	−0.0368095	−0.0184047	+	0.999831i	$0.505859\pi$
$152$	12.8632	1.04334
$153$	−6.07363	−0.491024
$154$	−1.75893	−0.141738
$155$	1.82434	0.146535
$156$	18.0246	1.44313
$157$	−8.81215	−0.703286	−0.351643	−	0.936134i	$-0.614377\pi$
$157$	−8.81215	−0.703286	−0.351643	+	0.936134i	$0.614377\pi$
$158$	5.33359	0.424317
$159$	−12.4064	−0.983894
$160$	−5.85644	−0.462992
$161$	−11.3528	−0.894725
$162$	−8.57477	−0.673697
$163$	11.9354	0.934851	0.467425	−	0.884032i	$-0.345182\pi$
$163$	11.9354	0.934851	0.467425	+	0.884032i	$0.345182\pi$
$164$	−1.40878	−0.110008
$165$	2.02237	0.157441
$166$	−0.467576	−0.0362909
$167$	−20.8167	−1.61084	−0.805422	−	0.592702i	$-0.798061\pi$
$167$	−20.8167	−1.61084	−0.805422	+	0.592702i	$0.798061\pi$
$168$	−12.0991	−0.933464
$169$	27.4535	2.11180
$170$	4.31156	0.330682
$171$	−5.32745	−0.407400
$172$	0.542841	0.0413912
$173$	−18.3809	−1.39747	−0.698737	−	0.715379i	$-0.746254\pi$
$173$	−18.3809	−1.39747	−0.698737	+	0.715379i	$0.746254\pi$
$174$	−11.1473	−0.845073
$175$	−2.27321	−0.171838
$176$	0.766190	0.0577537
$177$	−25.0979	−1.88647
$178$	−2.81231	−0.210791
$179$	12.6214	0.943367	0.471683	−	0.881768i	$-0.343646\pi$
$179$	12.6214	0.943367	0.471683	+	0.881768i	$0.343646\pi$
$180$	1.52739	0.113845
$181$	10.7632	0.800022	0.400011	−	0.916510i	$-0.369006\pi$
$181$	10.7632	0.800022	0.400011	+	0.916510i	$0.369006\pi$
$182$	−11.1873	−0.829258
$183$	22.4713	1.66113
$184$	−13.1436	−0.968962
$185$	4.38574	0.322446
$186$	2.85480	0.209324
$187$	−5.57219	−0.407479
$188$	−12.2815	−0.895722
$189$	−8.78083	−0.638712
$190$	3.78186	0.274365
$191$	18.9922	1.37423	0.687114	−	0.726550i	$-0.258877\pi$
$191$	18.9922	1.37423	0.687114	+	0.726550i	$0.258877\pi$
$192$	−6.06534	−0.437728
$193$	16.5465	1.19104	0.595522	−	0.803339i	$-0.296945\pi$
$193$	16.5465	1.19104	0.595522	+	0.803339i	$0.296945\pi$
$194$	4.88830	0.350960
$195$	12.8629	0.921132
$196$	2.56791	0.183422
$197$	−16.4240	−1.17016	−0.585082	−	0.810974i	$-0.698938\pi$
$197$	−16.4240	−1.17016	−0.585082	+	0.810974i	$0.698938\pi$
$198$	0.843395	0.0599374
$199$	−27.0937	−1.92062	−0.960311	−	0.278933i	$-0.910019\pi$
$199$	−27.0937	−1.92062	−0.960311	+	0.278933i	$0.910019\pi$
$200$	−2.63180	−0.186096
$201$	16.6976	1.17776
$202$	−8.78503	−0.618112
$203$	−16.1934	−1.13655
$204$	−15.7912	−1.10561
$205$	−1.00535	−0.0702166
$206$	−13.1932	−0.919211
$207$	5.44360	0.378356
$208$	4.87320	0.337896
$209$	−4.88762	−0.338084
$210$	−3.55720	−0.245470
$211$	−26.2156	−1.80476	−0.902379	−	0.430943i	$-0.858181\pi$
$211$	−26.2156	−1.80476	−0.902379	+	0.430943i	$0.858181\pi$
$212$	−8.59634	−0.590399
$213$	−23.5739	−1.61525
$214$	−10.7101	−0.732130
$215$	0.387387	0.0264196
$216$	−10.1660	−0.691707
$217$	4.14711	0.281524
$218$	0.569943	0.0386014
$219$	−2.02237	−0.136659
$220$	1.40129	0.0944749
$221$	−35.4408	−2.38401
$222$	6.86298	0.460613
$223$	3.13115	0.209678	0.104839	−	0.994489i	$-0.466567\pi$
$223$	3.13115	0.209678	0.104839	+	0.994489i	$0.466567\pi$
$224$	−13.3129	−0.889505
$225$	1.08999	0.0726660
$226$	−10.3917	−0.691244
$227$	−13.6290	−0.904585	−0.452293	−	0.891870i	$-0.649394\pi$
$227$	−13.6290	−0.904585	−0.452293	+	0.891870i	$0.649394\pi$
$228$	−13.8512	−0.917316
$229$	9.05458	0.598343	0.299172	−	0.954199i	$-0.403290\pi$
$229$	9.05458	0.598343	0.299172	+	0.954199i	$0.403290\pi$
$230$	−3.86431	−0.254805
$231$	4.59727	0.302478
$232$	−18.7478	−1.23086
$233$	1.57225	0.103002	0.0515009	−	0.998673i	$-0.483600\pi$
$233$	1.57225	0.103002	0.0515009	+	0.998673i	$0.483600\pi$
$234$	5.36425	0.350672
$235$	−8.76444	−0.571729
$236$	−17.3902	−1.13200
$237$	−13.9403	−0.905520
$238$	9.80107	0.635309
$239$	27.0581	1.75024	0.875120	−	0.483906i	$-0.160782\pi$
$239$	27.0581	1.75024	0.875120	+	0.483906i	$0.160782\pi$
$240$	1.54952	0.100021
$241$	−19.5098	−1.25674	−0.628369	−	0.777915i	$-0.716277\pi$
$241$	−19.5098	−1.25674	−0.628369	+	0.777915i	$0.716277\pi$
$242$	0.773764	0.0497394
$243$	10.8235	0.694325
$244$	15.5702	0.996782
$245$	1.83253	0.117076
$246$	−1.57321	−0.100304
$247$	−31.0867	−1.97800
$248$	4.80130	0.304883
$249$	1.22209	0.0774471
$250$	−0.773764	−0.0489371
$251$	5.15312	0.325262	0.162631	−	0.986687i	$-0.448002\pi$
$251$	5.15312	0.325262	0.162631	+	0.986687i	$0.448002\pi$
$252$	3.47207	0.218720
$253$	4.99417	0.313981
$254$	−2.36439	−0.148355
$255$	−11.2691	−0.705695
$256$	−13.2656	−0.829103
$257$	−14.1830	−0.884709	−0.442354	−	0.896840i	$-0.645857\pi$
$257$	−14.1830	−0.884709	−0.442354	+	0.896840i	$0.645857\pi$
$258$	0.606198	0.0377402
$259$	9.96969	0.619487
$260$	8.91262	0.552738
$261$	7.76464	0.480619
$262$	2.92528	0.180724
$263$	23.0370	1.42052	0.710262	−	0.703938i	$-0.248577\pi$
$263$	23.0370	1.42052	0.710262	+	0.703938i	$0.248577\pi$
$264$	5.32247	0.327575
$265$	−6.13459	−0.376845
$266$	8.59696	0.527113
$267$	7.35047	0.449842
$268$	11.5697	0.706729
$269$	−3.01572	−0.183872	−0.0919358	−	0.995765i	$-0.529305\pi$
$269$	−3.01572	−0.183872	−0.0919358	+	0.995765i	$0.529305\pi$
$270$	−2.98886	−0.181896
$271$	17.2046	1.04510	0.522551	−	0.852608i	$-0.324980\pi$
$271$	17.2046	1.04510	0.522551	+	0.852608i	$0.324980\pi$
$272$	−4.26936	−0.258868
$273$	29.2400	1.76969
$274$	−0.0693505	−0.00418962
$275$	1.00000	0.0603023
$276$	14.1531	0.851918
$277$	−9.95229	−0.597975	−0.298988	−	0.954257i	$-0.596649\pi$
$277$	−9.95229	−0.597975	−0.298988	+	0.954257i	$0.596649\pi$
$278$	7.44965	0.446801
$279$	−1.98851	−0.119049
$280$	−5.98261	−0.357530
$281$	−9.04386	−0.539512	−0.269756	−	0.962929i	$-0.586943\pi$
$281$	−9.04386	−0.539512	−0.269756	+	0.962929i	$0.586943\pi$
$282$	−13.7149	−0.816713
$283$	11.3497	0.674667	0.337334	−	0.941385i	$-0.390475\pi$
$283$	11.3497	0.674667	0.337334	+	0.941385i	$0.390475\pi$
$284$	−16.3342	−0.969256
$285$	−9.88458	−0.585512
$286$	4.92137	0.291007
$287$	−2.28537	−0.134901
$288$	6.38346	0.376149
$289$	14.0493	0.826432
$290$	−5.51198	−0.323675
$291$	−12.7765	−0.748970
$292$	−1.40129	−0.0820042
$293$	−24.6240	−1.43855	−0.719275	−	0.694725i	$-0.755526\pi$
$293$	−24.6240	−1.43855	−0.719275	+	0.694725i	$0.755526\pi$
$294$	2.86762	0.167243
$295$	−12.4101	−0.722544
$296$	11.5424	0.670887
$297$	3.86275	0.224140
$298$	−11.8867	−0.688577
$299$	31.7645	1.83699
$300$	2.83393	0.163617
$301$	0.880610	0.0507575
$302$	−0.349991	−0.0201397
$303$	22.9613	1.31909
$304$	−3.74484	−0.214782
$305$	11.1114	0.636235
$306$	−4.69956	−0.268656
$307$	24.5444	1.40082	0.700410	−	0.713741i	$-0.253000\pi$
$307$	24.5444	1.40082	0.700410	+	0.713741i	$0.253000\pi$
$308$	3.18542	0.181506
$309$	34.4827	1.96165
$310$	1.41161	0.0801741
$311$	33.0013	1.87133	0.935665	−	0.352888i	$-0.114800\pi$
$311$	33.0013	1.87133	0.935665	+	0.352888i	$0.114800\pi$
$312$	33.8525	1.91652
$313$	14.0535	0.794352	0.397176	−	0.917743i	$-0.369990\pi$
$313$	14.0535	0.794352	0.397176	+	0.917743i	$0.369990\pi$
$314$	−6.81853	−0.384792
$315$	2.47777	0.139607
$316$	−9.65915	−0.543369
$317$	19.5753	1.09946	0.549729	−	0.835343i	$-0.314731\pi$
$317$	19.5753	1.09946	0.549729	+	0.835343i	$0.314731\pi$
$318$	−9.59965	−0.538321
$319$	7.12359	0.398845
$320$	−2.99912	−0.167656
$321$	27.9929	1.56241
$322$	−8.78438	−0.489534
$323$	27.2348	1.51538
$324$	15.5289	0.862719
$325$	6.36030	0.352806
$326$	9.23517	0.511489
$327$	−1.48965	−0.0823778
$328$	−2.64587	−0.146094
$329$	−19.9234	−1.09841
$330$	1.56484	0.0861415
$331$	8.67038	0.476567	0.238283	−	0.971196i	$-0.423415\pi$
$331$	8.67038	0.476567	0.238283	+	0.971196i	$0.423415\pi$
$332$	0.846781	0.0464732
$333$	−4.78041	−0.261965
$334$	−16.1072	−0.881347
$335$	8.25643	0.451097
$336$	3.52238	0.192162
$337$	−19.5157	−1.06309	−0.531544	−	0.847031i	$-0.678388\pi$
$337$	−19.5157	−1.06309	−0.531544	+	0.847031i	$0.678388\pi$
$338$	21.2425	1.15544
$339$	27.1605	1.47516
$340$	−7.80826	−0.423462
$341$	−1.82434	−0.0987937
$342$	−4.12219	−0.222903
$343$	20.0782	1.08412
$344$	1.01952	0.0549690
$345$	10.1001	0.543770
$346$	−14.2225	−0.764605
$347$	−35.5633	−1.90914	−0.954569	−	0.297991i	$-0.903684\pi$
$347$	−35.5633	−1.90914	−0.954569	+	0.297991i	$0.903684\pi$
$348$	20.1878	1.08218
$349$	−0.0464784	−0.00248793	−0.00124397	−	0.999999i	$-0.500396\pi$
$349$	−0.0464784	−0.00248793	−0.00124397	+	0.999999i	$0.500396\pi$
$350$	−1.75893	−0.0940185
$351$	24.5683	1.31136
$352$	5.85644	0.312149
$353$	−12.7064	−0.676293	−0.338146	−	0.941094i	$-0.609800\pi$
$353$	−12.7064	−0.676293	−0.338146	+	0.941094i	$0.609800\pi$
$354$	−19.4198	−1.03215
$355$	−11.6565	−0.618665
$356$	5.09310	0.269934
$357$	−25.6169	−1.35579
$358$	9.76598	0.516148
$359$	−15.0254	−0.793010	−0.396505	−	0.918033i	$-0.629777\pi$
$359$	−15.0254	−0.793010	−0.396505	+	0.918033i	$0.629777\pi$
$360$	2.86863	0.151190
$361$	4.88881	0.257306
$362$	8.32818	0.437719
$363$	−2.02237	−0.106147
$364$	20.2602	1.06192
$365$	−1.00000	−0.0523424
$366$	17.3875	0.908859
$367$	−33.5922	−1.75350	−0.876750	−	0.480946i	$-0.840293\pi$
$367$	−33.5922	−1.75350	−0.876750	+	0.480946i	$0.840293\pi$
$368$	3.82649	0.199469
$369$	1.09582	0.0570461
$370$	3.39353	0.176421
$371$	−13.9452	−0.723998
$372$	−5.17006	−0.268055
$373$	0.912942	0.0472703	0.0236352	−	0.999721i	$-0.492476\pi$
$373$	0.912942	0.0472703	0.0236352	+	0.999721i	$0.492476\pi$
$374$	−4.31156	−0.222946
$375$	2.02237	0.104435
$376$	−23.0662	−1.18955
$377$	45.3082	2.33349
$378$	−6.79429	−0.349461
$379$	28.8619	1.48254	0.741268	−	0.671209i	$-0.234225\pi$
$379$	28.8619	1.48254	0.741268	+	0.671209i	$0.234225\pi$
$380$	−6.84897	−0.351345
$381$	6.17977	0.316599
$382$	14.6955	0.751886
$383$	−6.24508	−0.319108	−0.159554	−	0.987189i	$-0.551006\pi$
$383$	−6.24508	−0.319108	−0.159554	+	0.987189i	$0.551006\pi$
$384$	18.9947	0.969317
$385$	2.27321	0.115853
$386$	12.8031	0.651660
$387$	−0.422247	−0.0214640
$388$	−8.85274	−0.449430
$389$	−13.0129	−0.659779	−0.329890	−	0.944020i	$-0.607012\pi$
$389$	−13.0129	−0.659779	−0.329890	+	0.944020i	$0.607012\pi$
$390$	9.95285	0.503982
$391$	−27.8285	−1.40735
$392$	4.82285	0.243591
$393$	−7.64574	−0.385676
$394$	−12.7083	−0.640236
$395$	−6.89304	−0.346827
$396$	−1.52739	−0.0767543
$397$	−10.2963	−0.516756	−0.258378	−	0.966044i	$-0.583188\pi$
$397$	−10.2963	−0.516756	−0.258378	+	0.966044i	$0.583188\pi$
$398$	−20.9641	−1.05084
$399$	−22.4697	−1.12489
$400$	0.766190	0.0383095
$401$	−3.37327	−0.168453	−0.0842265	−	0.996447i	$-0.526842\pi$
$401$	−3.37327	−0.168453	−0.0842265	+	0.996447i	$0.526842\pi$
$402$	12.9200	0.644391
$403$	−11.6034	−0.578005
$404$	15.9097	0.791538
$405$	11.0819	0.550664
$406$	−12.5299	−0.621847
$407$	−4.38574	−0.217393
$408$	−29.6578	−1.46828
$409$	−0.0478601	−0.00236653	−0.00118326	−	0.999999i	$-0.500377\pi$
$409$	−0.0478601	−0.00236653	−0.00118326	+	0.999999i	$0.500377\pi$
$410$	−0.777903	−0.0384179
$411$	0.181260	0.00894091
$412$	23.8929	1.17712
$413$	−28.2107	−1.38816
$414$	4.21206	0.207011
$415$	0.604287	0.0296633
$416$	37.2487	1.82627
$417$	−19.4710	−0.953500
$418$	−3.78186	−0.184977
$419$	8.46772	0.413675	0.206838	−	0.978375i	$-0.433683\pi$
$419$	8.46772	0.413675	0.206838	+	0.978375i	$0.433683\pi$
$420$	6.44210	0.314342
$421$	−20.1006	−0.979644	−0.489822	−	0.871822i	$-0.662938\pi$
$421$	−20.1006	−0.979644	−0.489822	+	0.871822i	$0.662938\pi$
$422$	−20.2847	−0.987444
$423$	9.55315	0.464490
$424$	−16.1450	−0.784070
$425$	−5.57219	−0.270291
$426$	−18.2406	−0.883760
$427$	25.2584	1.22234
$428$	19.3961	0.937546
$429$	−12.8629	−0.621027
$430$	0.299746	0.0144550
$431$	17.2832	0.832503	0.416251	−	0.909250i	$-0.363344\pi$
$431$	17.2832	0.832503	0.416251	+	0.909250i	$0.363344\pi$
$432$	2.95960	0.142394
$433$	24.5886	1.18165	0.590827	−	0.806798i	$-0.298802\pi$
$433$	24.5886	1.18165	0.590827	+	0.806798i	$0.298802\pi$
$434$	3.20888	0.154031
$435$	14.4066	0.690741
$436$	−1.03217	−0.0494319
$437$	−24.4096	−1.16767
$438$	−1.56484	−0.0747709
$439$	−16.8448	−0.803960	−0.401980	−	0.915648i	$-0.631678\pi$
$439$	−16.8448	−0.803960	−0.401980	+	0.915648i	$0.631678\pi$
$440$	2.63180	0.125466
$441$	−1.99744	−0.0951163
$442$	−27.4229	−1.30437
$443$	−32.0894	−1.52461	−0.762307	−	0.647215i	$-0.775933\pi$
$443$	−32.0894	−1.52461	−0.762307	+	0.647215i	$0.775933\pi$
$444$	−12.4289	−0.589848
$445$	3.63458	0.172296
$446$	2.42278	0.114722
$447$	31.0680	1.46946
$448$	−6.81763	−0.322103
$449$	0.662758	0.0312775	0.0156387	−	0.999878i	$-0.495022\pi$
$449$	0.662758	0.0312775	0.0156387	+	0.999878i	$0.495022\pi$
$450$	0.843395	0.0397580
$451$	1.00535	0.0473400
$452$	18.8194	0.885189
$453$	0.914764	0.0429794
$454$	−10.5456	−0.494929
$455$	14.4583	0.677815
$456$	−26.0142	−1.21823
$457$	−11.1288	−0.520584	−0.260292	−	0.965530i	$-0.583819\pi$
$457$	−11.1288	−0.520584	−0.260292	+	0.965530i	$0.583819\pi$
$458$	7.00611	0.327374
$459$	−21.5240	−1.00465
$460$	6.99828	0.326297
$461$	37.1269	1.72917	0.864585	−	0.502486i	$-0.167581\pi$
$461$	37.1269	1.72917	0.864585	+	0.502486i	$0.167581\pi$
$462$	3.55720	0.165496
$463$	−4.20128	−0.195250	−0.0976251	−	0.995223i	$-0.531125\pi$
$463$	−4.20128	−0.195250	−0.0976251	+	0.995223i	$0.531125\pi$
$464$	5.45802	0.253382
$465$	−3.68950	−0.171096
$466$	1.21655	0.0563558
$467$	−31.1172	−1.43993	−0.719966	−	0.694010i	$-0.755842\pi$
$467$	−31.1172	−1.43993	−0.719966	+	0.694010i	$0.755842\pi$
$468$	−9.71467	−0.449061
$469$	18.7686	0.866652
$470$	−6.78161	−0.312812
$471$	17.8215	0.821169
$472$	−32.6609	−1.50334
$473$	−0.387387	−0.0178121
$474$	−10.7865	−0.495440
$475$	−4.88762	−0.224259
$476$	−17.7498	−0.813560
$477$	6.68664	0.306160
$478$	20.9366	0.957616
$479$	15.8107	0.722410	0.361205	−	0.932486i	$-0.382365\pi$
$479$	15.8107	0.722410	0.361205	+	0.932486i	$0.382365\pi$
$480$	11.8439	0.540598
$481$	−27.8946	−1.27189
$482$	−15.0960	−0.687604
$483$	22.9596	1.04470
$484$	−1.40129	−0.0636950
$485$	−6.31756	−0.286866
$486$	8.37480	0.379889
$487$	5.30086	0.240205	0.120102	−	0.992762i	$-0.461678\pi$
$487$	5.30086	0.240205	0.120102	+	0.992762i	$0.461678\pi$
$488$	29.2428	1.32376
$489$	−24.1378	−1.09155
$490$	1.41795	0.0640564
$491$	−30.0105	−1.35435	−0.677177	−	0.735821i	$-0.736797\pi$
$491$	−30.0105	−1.35435	−0.677177	+	0.735821i	$0.736797\pi$
$492$	2.84909	0.128447
$493$	−39.6940	−1.78773
$494$	−24.0538	−1.08223
$495$	−1.08999	−0.0489914
$496$	−1.39779	−0.0627628
$497$	−26.4977	−1.18859
$498$	0.945613	0.0423739
$499$	22.9716	1.02835	0.514175	−	0.857685i	$-0.328098\pi$
$499$	22.9716	1.02835	0.514175	+	0.857685i	$0.328098\pi$
$500$	1.40129	0.0626676
$501$	42.0991	1.88085
$502$	3.98730	0.177962
$503$	−22.0731	−0.984190	−0.492095	−	0.870541i	$-0.663769\pi$
$503$	−22.0731	−0.984190	−0.492095	+	0.870541i	$0.663769\pi$
$504$	6.52099	0.290468
$505$	11.3536	0.505230
$506$	3.86431	0.171790
$507$	−55.5211	−2.46578
$508$	4.28192	0.189980
$509$	−8.88905	−0.394000	−0.197000	−	0.980403i	$-0.563120\pi$
$509$	−8.88905	−0.394000	−0.197000	+	0.980403i	$0.563120\pi$
$510$	−8.71959	−0.386110
$511$	−2.27321	−0.100561
$512$	8.52006	0.376537
$513$	−18.8797	−0.833557
$514$	−10.9743	−0.484054
$515$	17.0506	0.751341
$516$	−1.09783	−0.0483291
$517$	8.76444	0.385460
$518$	7.71419	0.338942
$519$	37.1730	1.63171
$520$	16.7390	0.734054
$521$	36.7430	1.60974	0.804871	−	0.593450i	$-0.202235\pi$
$521$	36.7430	1.60974	0.804871	+	0.593450i	$0.202235\pi$
$522$	6.00800	0.262963
$523$	39.9025	1.74481	0.872406	−	0.488782i	$-0.162559\pi$
$523$	39.9025	1.74481	0.872406	+	0.488782i	$0.162559\pi$
$524$	−5.29768	−0.231430
$525$	4.59727	0.200641
$526$	17.8252	0.777216
$527$	10.1656	0.442820
$528$	−1.54952	−0.0674343
$529$	1.94178	0.0844250
$530$	−4.74673	−0.206185
$531$	13.5269	0.587017
$532$	−15.5691	−0.675007
$533$	6.39433	0.276969
$534$	5.68753	0.246124
$535$	13.8416	0.598425
$536$	21.7292	0.938560
$537$	−25.5251	−1.10149
$538$	−2.33345	−0.100602
$539$	−1.83253	−0.0789328
$540$	5.41283	0.232931
$541$	24.7059	1.06219	0.531095	−	0.847312i	$-0.321781\pi$
$541$	24.7059	1.06219	0.531095	+	0.847312i	$0.321781\pi$
$542$	13.3123	0.571811
$543$	−21.7672	−0.934120
$544$	−32.6332	−1.39914
$545$	−0.736585	−0.0315518
$546$	22.6249	0.968256
$547$	3.59213	0.153588	0.0767941	−	0.997047i	$-0.475532\pi$
$547$	3.59213	0.153588	0.0767941	+	0.997047i	$0.475532\pi$
$548$	0.125594	0.00536511
$549$	−12.1113	−0.516896
$550$	0.773764	0.0329934
$551$	−34.8174	−1.48327
$552$	26.5813	1.13138
$553$	−15.6693	−0.666327
$554$	−7.70072	−0.327172
$555$	−8.86960	−0.376494
$556$	−13.4913	−0.572161
$557$	−0.569929	−0.0241487	−0.0120743	−	0.999927i	$-0.503843\pi$
$557$	−0.569929	−0.0241487	−0.0120743	+	0.999927i	$0.503843\pi$
$558$	−1.53864	−0.0651359
$559$	−2.46390	−0.104212
$560$	1.74171	0.0736006
$561$	11.2691	0.475780
$562$	−6.99782	−0.295185
$563$	5.99352	0.252597	0.126298	−	0.991992i	$-0.459690\pi$
$563$	5.99352	0.252597	0.126298	+	0.991992i	$0.459690\pi$
$564$	24.8378	1.04586
$565$	13.4300	0.565006
$566$	8.78196	0.369133
$567$	25.1914	1.05794
$568$	−30.6776	−1.28720
$569$	−1.85488	−0.0777607	−0.0388804	−	0.999244i	$-0.512379\pi$
$569$	−1.85488	−0.0777607	−0.0388804	+	0.999244i	$0.512379\pi$
$570$	−7.64834	−0.320354
$571$	−4.81394	−0.201457	−0.100728	−	0.994914i	$-0.532117\pi$
$571$	−4.81394	−0.201457	−0.100728	+	0.994914i	$0.532117\pi$
$572$	−8.91262	−0.372656
$573$	−38.4093	−1.60457
$574$	−1.76833	−0.0738088
$575$	4.99417	0.208271
$576$	3.26901	0.136209
$577$	17.8676	0.743836	0.371918	−	0.928266i	$-0.378700\pi$
$577$	17.8676	0.743836	0.371918	+	0.928266i	$0.378700\pi$
$578$	10.8709	0.452169
$579$	−33.4632	−1.39068
$580$	9.98221	0.414489
$581$	1.37367	0.0569894
$582$	−9.88597	−0.409787
$583$	6.13459	0.254069
$584$	−2.63180	−0.108904
$585$	−6.93266	−0.286630
$586$	−19.0532	−0.787079
$587$	−13.0211	−0.537440	−0.268720	−	0.963218i	$-0.586601\pi$
$587$	−13.0211	−0.537440	−0.268720	+	0.963218i	$0.586601\pi$
$588$	−5.19327	−0.214167
$589$	8.91669	0.367406
$590$	−9.60250	−0.395328
$591$	33.2155	1.36630
$592$	−3.36031	−0.138108
$593$	−16.2575	−0.667618	−0.333809	−	0.942641i	$-0.608334\pi$
$593$	−16.2575	−0.667618	−0.333809	+	0.942641i	$0.608334\pi$
$594$	2.98886	0.122634
$595$	−12.6667	−0.519286
$596$	21.5268	0.881773
$597$	54.7935	2.24255
$598$	24.5782	1.00508
$599$	−28.6623	−1.17111	−0.585555	−	0.810632i	$-0.699124\pi$
$599$	−28.6623	−1.17111	−0.585555	+	0.810632i	$0.699124\pi$
$600$	5.32247	0.217289
$601$	−23.2983	−0.950356	−0.475178	−	0.879890i	$-0.657616\pi$
$601$	−23.2983	−0.950356	−0.475178	+	0.879890i	$0.657616\pi$
$602$	0.681384	0.0277711
$603$	−8.99943	−0.366485
$604$	0.633835	0.0257904
$605$	−1.00000	−0.0406558
$606$	17.7666	0.721719
$607$	−21.4328	−0.869932	−0.434966	−	0.900447i	$-0.643240\pi$
$607$	−21.4328	−0.869932	−0.434966	+	0.900447i	$0.643240\pi$
$608$	−28.6240	−1.16086
$609$	32.7491	1.32706
$610$	8.59757	0.348105
$611$	55.7445	2.25518
$612$	8.51092	0.344033
$613$	−2.71165	−0.109523	−0.0547613	−	0.998499i	$-0.517440\pi$
$613$	−2.71165	−0.109523	−0.0547613	+	0.998499i	$0.517440\pi$
$614$	18.9915	0.766436
$615$	2.03319	0.0819861
$616$	5.98261	0.241046
$617$	33.2040	1.33674	0.668371	−	0.743828i	$-0.266992\pi$
$617$	33.2040	1.33674	0.668371	+	0.743828i	$0.266992\pi$
$618$	26.6815	1.07329
$619$	−41.4482	−1.66594	−0.832972	−	0.553315i	$-0.813363\pi$
$619$	−41.4482	−1.66594	−0.832972	+	0.553315i	$0.813363\pi$
$620$	−2.55643	−0.102669
$621$	19.2913	0.774132
$622$	25.5352	1.02387
$623$	8.26215	0.331016
$624$	−9.85543	−0.394533
$625$	1.00000	0.0400000
$626$	10.8741	0.434617
$627$	9.88458	0.394752
$628$	12.3484	0.492754
$629$	24.4382	0.974415
$630$	1.91721	0.0763835
$631$	−36.5206	−1.45386	−0.726931	−	0.686710i	$-0.759054\pi$
$631$	−36.5206	−1.45386	−0.726931	+	0.686710i	$0.759054\pi$
$632$	−18.1411	−0.721613
$633$	53.0178	2.10727
$634$	15.1467	0.601551
$635$	3.05570	0.121262
$636$	17.3850	0.689360
$637$	−11.6555	−0.461806
$638$	5.51198	0.218221
$639$	12.7055	0.502622
$640$	9.39227	0.371262
$641$	−0.196876	−0.00777612	−0.00388806	−	0.999992i	$-0.501238\pi$
$641$	−0.196876	−0.00777612	−0.00388806	+	0.999992i	$0.501238\pi$
$642$	21.6599	0.854848
$643$	31.3514	1.23638	0.618189	−	0.786029i	$-0.287867\pi$
$643$	31.3514	1.23638	0.618189	+	0.786029i	$0.287867\pi$
$644$	15.9085	0.626884
$645$	−0.783440	−0.0308479
$646$	21.0733	0.829117
$647$	−41.0739	−1.61478	−0.807391	−	0.590017i	$-0.799121\pi$
$647$	−41.0739	−1.61478	−0.807391	+	0.590017i	$0.799121\pi$
$648$	29.1653	1.14572
$649$	12.4101	0.487139
$650$	4.92137	0.193032
$651$	−8.38699	−0.328712
$652$	−16.7249	−0.654998
$653$	−17.7115	−0.693105	−0.346552	−	0.938031i	$-0.612648\pi$
$653$	−17.7115	−0.693105	−0.346552	+	0.938031i	$0.612648\pi$
$654$	−1.15264	−0.0450717
$655$	−3.78058	−0.147719
$656$	0.770288	0.0300747
$657$	1.08999	0.0425245
$658$	−15.4160	−0.600978
$659$	−37.7448	−1.47033	−0.735164	−	0.677889i	$-0.762895\pi$
$659$	−37.7448	−1.47033	−0.735164	+	0.677889i	$0.762895\pi$
$660$	−2.83393	−0.110311
$661$	−39.6171	−1.54093	−0.770464	−	0.637483i	$-0.779975\pi$
$661$	−39.6171	−1.54093	−0.770464	+	0.637483i	$0.779975\pi$
$662$	6.70883	0.260746
$663$	71.6746	2.78361
$664$	1.59036	0.0617180
$665$	−11.1106	−0.430849
$666$	−3.69891	−0.143330
$667$	35.5765	1.37753
$668$	29.1702	1.12863
$669$	−6.33236	−0.244823
$670$	6.38853	0.246811
$671$	−11.1114	−0.428949
$672$	26.9236	1.03860
$673$	−11.4966	−0.443163	−0.221582	−	0.975142i	$-0.571122\pi$
$673$	−11.4966	−0.443163	−0.221582	+	0.975142i	$0.571122\pi$
$674$	−15.1005	−0.581651
$675$	3.86275	0.148677
$676$	−38.4702	−1.47962
$677$	−13.6580	−0.524920	−0.262460	−	0.964943i	$-0.584534\pi$
$677$	−13.6580	−0.524920	−0.262460	+	0.964943i	$0.584534\pi$
$678$	21.0158	0.807109
$679$	−14.3611	−0.551130
$680$	−14.6649	−0.562372
$681$	27.5628	1.05621
$682$	−1.41161	−0.0540534
$683$	30.1560	1.15389	0.576944	−	0.816783i	$-0.304245\pi$
$683$	30.1560	1.15389	0.576944	+	0.816783i	$0.304245\pi$
$684$	7.46530	0.285443
$685$	0.0896275	0.00342449
$686$	15.5358	0.593158
$687$	−18.3117	−0.698636
$688$	−0.296812	−0.0113158
$689$	39.0179	1.48646
$690$	7.81508	0.297515
$691$	−4.91310	−0.186903	−0.0934516	−	0.995624i	$-0.529790\pi$
$691$	−4.91310	−0.186903	−0.0934516	+	0.995624i	$0.529790\pi$
$692$	25.7570	0.979133
$693$	−2.47777	−0.0941227
$694$	−27.5176	−1.04455
$695$	−9.62781	−0.365204
$696$	37.9151	1.43717
$697$	−5.60200	−0.212191
$698$	−0.0359633	−0.00136123
$699$	−3.17968	−0.120267
$700$	3.18542	0.120398
$701$	−21.6716	−0.818526	−0.409263	−	0.912416i	$-0.634214\pi$
$701$	−21.6716	−0.818526	−0.409263	+	0.912416i	$0.634214\pi$
$702$	19.0101	0.717488
$703$	21.4358	0.808467
$704$	2.99912	0.113034
$705$	17.7250	0.667561
$706$	−9.83174	−0.370023
$707$	25.8091	0.970652
$708$	35.1694	1.32175
$709$	13.3481	0.501298	0.250649	−	0.968078i	$-0.419356\pi$
$709$	13.3481	0.501298	0.250649	+	0.968078i	$0.419356\pi$
$710$	−9.01941	−0.338493
$711$	7.51334	0.281772
$712$	9.56547	0.358481
$713$	−9.11109	−0.341213
$714$	−19.8214	−0.741798
$715$	−6.36030	−0.237862
$716$	−17.6862	−0.660965
$717$	−54.7215	−2.04361
$718$	−11.6261	−0.433883
$719$	−42.1064	−1.57030	−0.785152	−	0.619303i	$-0.787415\pi$
$719$	−42.1064	−1.57030	−0.785152	+	0.619303i	$0.787415\pi$
$720$	−0.835139	−0.0311238
$721$	38.7596	1.44348
$722$	3.78279	0.140781
$723$	39.4561	1.46739
$724$	−15.0824	−0.560531
$725$	7.12359	0.264564
$726$	−1.56484	−0.0580766
$727$	2.47814	0.0919093	0.0459546	−	0.998944i	$-0.485367\pi$
$727$	2.47814	0.0919093	0.0459546	+	0.998944i	$0.485367\pi$
$728$	38.0512	1.41027
$729$	11.3566	0.420616
$730$	−0.773764	−0.0286383
$731$	2.15859	0.0798385
$732$	−31.4888	−1.16386
$733$	−12.7763	−0.471905	−0.235952	−	0.971765i	$-0.575821\pi$
$733$	−12.7763	−0.471905	−0.235952	+	0.971765i	$0.575821\pi$
$734$	−25.9925	−0.959399
$735$	−3.70606	−0.136700
$736$	29.2481	1.07810
$737$	−8.25643	−0.304130
$738$	0.847906	0.0312119
$739$	−5.77067	−0.212278	−0.106139	−	0.994351i	$-0.533849\pi$
$739$	−5.77067	−0.212278	−0.106139	+	0.994351i	$0.533849\pi$
$740$	−6.14569	−0.225920
$741$	62.8690	2.30955
$742$	−10.7903	−0.396124
$743$	−47.5642	−1.74496	−0.872480	−	0.488649i	$-0.837490\pi$
$743$	−47.5642	−1.74496	−0.872480	+	0.488649i	$0.837490\pi$
$744$	−9.71001	−0.355986
$745$	15.3621	0.562826
$746$	0.706401	0.0258632
$747$	−0.658667	−0.0240994
$748$	7.80826	0.285498
$749$	31.4648	1.14970
$750$	1.56484	0.0571398
$751$	−6.01791	−0.219597	−0.109798	−	0.993954i	$-0.535021\pi$
$751$	−6.01791	−0.219597	−0.109798	+	0.993954i	$0.535021\pi$
$752$	6.71523	0.244879
$753$	−10.4215	−0.379782
$754$	35.0579	1.27673
$755$	0.452322	0.0164617
$756$	12.3045	0.447510
$757$	−15.6469	−0.568695	−0.284347	−	0.958721i	$-0.591777\pi$
$757$	−15.6469	−0.568695	−0.284347	+	0.958721i	$0.591777\pi$
$758$	22.3323	0.811145
$759$	−10.1001	−0.366610
$760$	−12.8632	−0.466598
$761$	−22.2642	−0.807078	−0.403539	−	0.914962i	$-0.632220\pi$
$761$	−22.2642	−0.807078	−0.403539	+	0.914962i	$0.632220\pi$
$762$	4.78168	0.173222
$763$	−1.67441	−0.0606177
$764$	−26.6136	−0.962845
$765$	6.07363	0.219593
$766$	−4.83222	−0.174595
$767$	78.9321	2.85007
$768$	26.8281	0.968075
$769$	4.25508	0.153442	0.0767211	−	0.997053i	$-0.475555\pi$
$769$	4.25508	0.153442	0.0767211	+	0.997053i	$0.475555\pi$
$770$	1.75893	0.0633873
$771$	28.6832	1.03300
$772$	−23.1864	−0.834498
$773$	−26.4826	−0.952511	−0.476256	−	0.879307i	$-0.658006\pi$
$773$	−26.4826	−0.952511	−0.476256	+	0.879307i	$0.658006\pi$
$774$	−0.326720	−0.0117437
$775$	−1.82434	−0.0655323
$776$	−16.6265	−0.596858
$777$	−20.1624	−0.723323
$778$	−10.0689	−0.360988
$779$	−4.91376	−0.176054
$780$	−18.0246	−0.645386
$781$	11.6565	0.417104
$782$	−21.5327	−0.770008
$783$	27.5167	0.983366
$784$	−1.40407	−0.0501453
$785$	8.81215	0.314519
$786$	−5.91600	−0.211017
$787$	−18.0190	−0.642307	−0.321154	−	0.947027i	$-0.604071\pi$
$787$	−18.0190	−0.642307	−0.321154	+	0.947027i	$0.604071\pi$
$788$	23.0148	0.819869
$789$	−46.5894	−1.65863
$790$	−5.33359	−0.189761
$791$	30.5292	1.08549
$792$	−2.86863	−0.101932
$793$	−70.6716	−2.50962
$794$	−7.96689	−0.282734
$795$	12.4064	0.440011
$796$	37.9661	1.34567
$797$	30.1850	1.06921	0.534604	−	0.845103i	$-0.320461\pi$
$797$	30.1850	1.06921	0.534604	+	0.845103i	$0.320461\pi$
$798$	−17.3862	−0.615466
$799$	−48.8372	−1.72774
$800$	5.85644	0.207056
$801$	−3.96165	−0.139978
$802$	−2.61011	−0.0921663
$803$	1.00000	0.0352892
$804$	−23.3981	−0.825189
$805$	11.3528	0.400133
$806$	−8.97827	−0.316246
$807$	6.09890	0.214692
$808$	29.8804	1.05119
$809$	2.00076	0.0703430	0.0351715	−	0.999381i	$-0.488802\pi$
$809$	2.00076	0.0703430	0.0351715	+	0.999381i	$0.488802\pi$
$810$	8.57477	0.301287
$811$	44.5566	1.56459	0.782297	−	0.622905i	$-0.214048\pi$
$811$	44.5566	1.56459	0.782297	+	0.622905i	$0.214048\pi$
$812$	22.6916	0.796320
$813$	−34.7940	−1.22028
$814$	−3.39353	−0.118943
$815$	−11.9354	−0.418078
$816$	8.63423	0.302259
$817$	1.89340	0.0662416
$818$	−0.0370324	−0.00129481
$819$	−15.7594	−0.550677
$820$	1.40878	0.0491969
$821$	26.9951	0.942136	0.471068	−	0.882097i	$-0.343869\pi$
$821$	26.9951	0.942136	0.471068	+	0.882097i	$0.343869\pi$
$822$	0.140253	0.00489187
$823$	8.40357	0.292930	0.146465	−	0.989216i	$-0.453210\pi$
$823$	8.40357	0.292930	0.146465	+	0.989216i	$0.453210\pi$
$824$	44.8738	1.56325
$825$	−2.02237	−0.0704100
$826$	−21.8285	−0.759509
$827$	54.1040	1.88138	0.940691	−	0.339265i	$-0.110178\pi$
$827$	54.1040	1.88138	0.940691	+	0.339265i	$0.110178\pi$
$828$	−7.62805	−0.265093
$829$	−31.1085	−1.08044	−0.540221	−	0.841523i	$-0.681659\pi$
$829$	−31.1085	−1.08044	−0.540221	+	0.841523i	$0.681659\pi$
$830$	0.467576	0.0162298
$831$	20.1272	0.698206
$832$	19.0753	0.661318
$833$	10.2112	0.353798
$834$	−15.0660	−0.521692
$835$	20.8167	0.720392
$836$	6.84897	0.236876
$837$	−7.04698	−0.243579
$838$	6.55202	0.226336
$839$	47.8501	1.65197	0.825985	−	0.563692i	$-0.190620\pi$
$839$	47.8501	1.65197	0.825985	+	0.563692i	$0.190620\pi$
$840$	12.0991	0.417458
$841$	21.7456	0.749847
$842$	−15.5531	−0.535997
$843$	18.2901	0.629943
$844$	36.7357	1.26449
$845$	−27.4535	−0.944428
$846$	7.39189	0.254138
$847$	−2.27321	−0.0781083
$848$	4.70026	0.161408
$849$	−22.9532	−0.787753
$850$	−4.31156	−0.147885
$851$	−21.9032	−0.750830
$852$	33.0338	1.13172
$853$	13.4339	0.459966	0.229983	−	0.973195i	$-0.426133\pi$
$853$	13.4339	0.459966	0.229983	+	0.973195i	$0.426133\pi$
$854$	19.5441	0.668784
$855$	5.32745	0.182195
$856$	36.4283	1.24509
$857$	−38.7029	−1.32207	−0.661033	−	0.750357i	$-0.729882\pi$
$857$	−38.7029	−1.32207	−0.661033	+	0.750357i	$0.729882\pi$
$858$	−9.95285	−0.339785
$859$	10.9548	0.373772	0.186886	−	0.982382i	$-0.440161\pi$
$859$	10.9548	0.373772	0.186886	+	0.982382i	$0.440161\pi$
$860$	−0.542841	−0.0185107
$861$	4.62186	0.157513
$862$	13.3731	0.455490
$863$	−26.8155	−0.912810	−0.456405	−	0.889772i	$-0.650863\pi$
$863$	−26.8155	−0.912810	−0.456405	+	0.889772i	$0.650863\pi$
$864$	22.6220	0.769615
$865$	18.3809	0.624969
$866$	19.0258	0.646523
$867$	−28.4130	−0.964956
$868$	−5.81130	−0.197248
$869$	6.89304	0.233830
$870$	11.1473	0.377928
$871$	−52.5134	−1.77935
$872$	−1.93854	−0.0656473
$873$	6.88608	0.233058
$874$	−18.8873	−0.638872
$875$	2.27321	0.0768484
$876$	2.83393	0.0957496
$877$	−26.9325	−0.909447	−0.454723	−	0.890633i	$-0.650262\pi$
$877$	−26.9325	−0.909447	−0.454723	+	0.890633i	$0.650262\pi$
$878$	−13.0339	−0.439874
$879$	49.7989	1.67968
$880$	−0.766190	−0.0258283
$881$	−27.0076	−0.909910	−0.454955	−	0.890514i	$-0.650345\pi$
$881$	−27.0076	−0.909910	−0.454955	+	0.890514i	$0.650345\pi$
$882$	−1.54555	−0.0520413
$883$	−45.7694	−1.54026	−0.770131	−	0.637886i	$-0.779809\pi$
$883$	−45.7694	−1.54026	−0.770131	+	0.637886i	$0.779809\pi$
$884$	49.6629	1.67034
$885$	25.0979	0.843655
$886$	−24.8296	−0.834168
$887$	−25.0094	−0.839734	−0.419867	−	0.907586i	$-0.637923\pi$
$887$	−25.0094	−0.839734	−0.419867	+	0.907586i	$0.637923\pi$
$888$	−23.3430	−0.783339
$889$	6.94624	0.232969
$890$	2.81231	0.0942687
$891$	−11.0819	−0.371257
$892$	−4.38765	−0.146910
$893$	−42.8373	−1.43349
$894$	24.0393	0.803994
$895$	−12.6214	−0.421887
$896$	21.3506	0.713272
$897$	−64.2396	−2.14490
$898$	0.512818	0.0171130
$899$	−12.9959	−0.433437
$900$	−1.52739	−0.0509130
$901$	−34.1831	−1.13881
$902$	0.777903	0.0259013
$903$	−1.78092	−0.0592653
$904$	35.3451	1.17556
$905$	−10.7632	−0.357781
$906$	0.707812	0.0235155
$907$	16.0633	0.533374	0.266687	−	0.963783i	$-0.414071\pi$
$907$	16.0633	0.533374	0.266687	+	0.963783i	$0.414071\pi$
$908$	19.0981	0.633793
$909$	−12.3753	−0.410464
$910$	11.1873	0.370855
$911$	−29.1891	−0.967078	−0.483539	−	0.875323i	$-0.660649\pi$
$911$	−29.1891	−0.967078	−0.483539	+	0.875323i	$0.660649\pi$
$912$	7.57347	0.250783
$913$	−0.604287	−0.0199990
$914$	−8.61108	−0.284829
$915$	−22.4713	−0.742878
$916$	−12.6881	−0.419226
$917$	−8.59403	−0.283800
$918$	−16.6545	−0.549680
$919$	26.7713	0.883105	0.441552	−	0.897236i	$-0.354428\pi$
$919$	26.7713	0.883105	0.441552	+	0.897236i	$0.354428\pi$
$920$	13.1436	0.433333
$921$	−49.6378	−1.63562
$922$	28.7274	0.946088
$923$	74.1391	2.44032
$924$	−6.44210	−0.211930
$925$	−4.38574	−0.144202
$926$	−3.25080	−0.106828
$927$	−18.5850	−0.610412
$928$	41.7189	1.36949
$929$	−12.1833	−0.399723	−0.199861	−	0.979824i	$-0.564049\pi$
$929$	−12.1833	−0.399723	−0.199861	+	0.979824i	$0.564049\pi$
$930$	−2.85480	−0.0936127
$931$	8.95672	0.293545
$932$	−2.20318	−0.0721677
$933$	−66.7409	−2.18500
$934$	−24.0774	−0.787835
$935$	5.57219	0.182230
$936$	−18.2454	−0.596368
$937$	48.0527	1.56981	0.784907	−	0.619614i	$-0.212711\pi$
$937$	48.0527	1.56981	0.784907	+	0.619614i	$0.212711\pi$
$938$	14.5225	0.474175
$939$	−28.4214	−0.927499
$940$	12.2815	0.400579
$941$	−11.2125	−0.365518	−0.182759	−	0.983158i	$-0.558503\pi$
$941$	−11.2125	−0.365518	−0.182759	+	0.983158i	$0.558503\pi$
$942$	13.7896	0.449289
$943$	5.02089	0.163503
$944$	9.50850	0.309475
$945$	8.78083	0.285641
$946$	−0.299746	−0.00974558
$947$	−45.8248	−1.48911	−0.744554	−	0.667563i	$-0.767338\pi$
$947$	−45.8248	−1.48911	−0.744554	+	0.667563i	$0.767338\pi$
$948$	19.5344	0.634447
$949$	6.36030	0.206464
$950$	−3.78186	−0.122700
$951$	−39.5886	−1.28375
$952$	−33.3363	−1.08044
$953$	19.4683	0.630639	0.315320	−	0.948986i	$-0.397888\pi$
$953$	19.4683	0.630639	0.315320	+	0.948986i	$0.397888\pi$
$954$	5.17388	0.167511
$955$	−18.9922	−0.614573
$956$	−37.9162	−1.22630
$957$	−14.4066	−0.465698
$958$	12.2338	0.395255
$959$	0.203742	0.00657917
$960$	6.06534	0.195758
$961$	−27.6718	−0.892638
$962$	−21.5839	−0.695892
$963$	−15.0872	−0.486179
$964$	27.3389	0.880527
$965$	−16.5465	−0.532651
$966$	17.7653	0.571589
$967$	24.7706	0.796568	0.398284	−	0.917262i	$-0.369606\pi$
$967$	24.7706	0.796568	0.398284	+	0.917262i	$0.369606\pi$
$968$	−2.63180	−0.0845891
$969$	−55.0788	−1.76939
$970$	−4.88830	−0.156954
$971$	−59.6534	−1.91437	−0.957185	−	0.289478i	$-0.906518\pi$
$971$	−59.6534	−1.91437	−0.957185	+	0.289478i	$0.906518\pi$
$972$	−15.1668	−0.486475
$973$	−21.8860	−0.701633
$974$	4.10161	0.131424
$975$	−12.8629	−0.411943
$976$	−8.51341	−0.272508
$977$	38.6343	1.23602	0.618010	−	0.786170i	$-0.287939\pi$
$977$	38.6343	1.23602	0.618010	+	0.786170i	$0.287939\pi$
$978$	−18.6769	−0.597223
$979$	−3.63458	−0.116162
$980$	−2.56791	−0.0820288
$981$	0.802870	0.0256337
$982$	−23.2210	−0.741012
$983$	26.0902	0.832147	0.416073	−	0.909331i	$-0.363406\pi$
$983$	26.0902	0.832147	0.416073	+	0.909331i	$0.363406\pi$
$984$	5.35094	0.170582
$985$	16.4240	0.523313
$986$	−30.7138	−0.978127
$987$	40.2925	1.28252
$988$	43.5615	1.38588
$989$	−1.93468	−0.0615191
$990$	−0.843395	−0.0268048
$991$	3.83669	0.121876	0.0609382	−	0.998142i	$-0.480591\pi$
$991$	3.83669	0.121876	0.0609382	+	0.998142i	$0.480591\pi$
$992$	−10.6842	−0.339222
$993$	−17.5347	−0.556448
$994$	−20.5030	−0.650315
$995$	27.0937	0.858928
$996$	−1.71251	−0.0542629
$997$	−37.4543	−1.18619	−0.593094	−	0.805133i	$-0.702094\pi$
$997$	−37.4543	−1.18619	−0.593094	+	0.805133i	$0.702094\pi$
$998$	17.7746	0.562645
$999$	−16.9410	−0.535990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.f.1.20	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.f.1.20	✓	31	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:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q+0.773764 q^{2} -2.02237 q^{3} -1.40129 q^{4} -1.00000 q^{5} -1.56484 q^{6} -2.27321 q^{7} -2.63180 q^{8} +1.08999 q^{9} +O(q^{10})\)	\(q+0.773764 q^{2} -2.02237 q^{3} -1.40129 q^{4} -1.00000 q^{5} -1.56484 q^{6} -2.27321 q^{7} -2.63180 q^{8} +1.08999 q^{9} -0.773764 q^{10} +1.00000 q^{11} +2.83393 q^{12} +6.36030 q^{13} -1.75893 q^{14} +2.02237 q^{15} +0.766190 q^{16} -5.57219 q^{17} +0.843395 q^{18} -4.88762 q^{19} +1.40129 q^{20} +4.59727 q^{21} +0.773764 q^{22} +4.99417 q^{23} +5.32247 q^{24} +1.00000 q^{25} +4.92137 q^{26} +3.86275 q^{27} +3.18542 q^{28} +7.12359 q^{29} +1.56484 q^{30} -1.82434 q^{31} +5.85644 q^{32} -2.02237 q^{33} -4.31156 q^{34} +2.27321 q^{35} -1.52739 q^{36} -4.38574 q^{37} -3.78186 q^{38} -12.8629 q^{39} +2.63180 q^{40} +1.00535 q^{41} +3.55720 q^{42} -0.387387 q^{43} -1.40129 q^{44} -1.08999 q^{45} +3.86431 q^{46} +8.76444 q^{47} -1.54952 q^{48} -1.83253 q^{49} +0.773764 q^{50} +11.2691 q^{51} -8.91262 q^{52} +6.13459 q^{53} +2.98886 q^{54} -1.00000 q^{55} +5.98261 q^{56} +9.88458 q^{57} +5.51198 q^{58} +12.4101 q^{59} -2.83393 q^{60} -11.1114 q^{61} -1.41161 q^{62} -2.47777 q^{63} +2.99912 q^{64} -6.36030 q^{65} -1.56484 q^{66} -8.25643 q^{67} +7.80826 q^{68} -10.1001 q^{69} +1.75893 q^{70} +11.6565 q^{71} -2.86863 q^{72} +1.00000 q^{73} -3.39353 q^{74} -2.02237 q^{75} +6.84897 q^{76} -2.27321 q^{77} -9.95285 q^{78} +6.89304 q^{79} -0.766190 q^{80} -11.0819 q^{81} +0.777903 q^{82} -0.604287 q^{83} -6.44210 q^{84} +5.57219 q^{85} -0.299746 q^{86} -14.4066 q^{87} -2.63180 q^{88} -3.63458 q^{89} -0.843395 q^{90} -14.4583 q^{91} -6.99828 q^{92} +3.68950 q^{93} +6.78161 q^{94} +4.88762 q^{95} -11.8439 q^{96} +6.31756 q^{97} -1.41795 q^{98} +1.08999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9} + 7 q^{10} + 31 q^{11} - 4 q^{12} - 24 q^{13} - 9 q^{14} + 4 q^{15} + 43 q^{16} - 49 q^{17} - 35 q^{18} - 22 q^{19} - 39 q^{20} - 8 q^{21} - 7 q^{22} - q^{23} - 13 q^{24} + 31 q^{25} - 9 q^{26} - 22 q^{27} - 34 q^{28} - 12 q^{29} + 5 q^{30} + 4 q^{31} - 45 q^{32} - 4 q^{33} + 2 q^{34} + 11 q^{35} + 34 q^{36} - 18 q^{37} - 7 q^{38} - q^{39} + 24 q^{40} - 58 q^{41} - 21 q^{42} - 41 q^{43} + 39 q^{44} - 31 q^{45} + 23 q^{46} - 31 q^{47} - 29 q^{48} + 44 q^{49} - 7 q^{50} + 8 q^{51} - 89 q^{52} - 46 q^{53} - 47 q^{54} - 31 q^{55} + 10 q^{56} - 47 q^{57} - 34 q^{58} - 9 q^{59} + 4 q^{60} - 5 q^{61} - 50 q^{62} - 61 q^{63} + 78 q^{64} + 24 q^{65} - 5 q^{66} + q^{67} - 115 q^{68} - 19 q^{69} + 9 q^{70} - 8 q^{71} - 93 q^{72} + 31 q^{73} - 19 q^{74} - 4 q^{75} - 7 q^{76} - 11 q^{77} + 57 q^{78} - 43 q^{80} + 43 q^{81} + 20 q^{82} - 29 q^{83} - 32 q^{84} + 49 q^{85} + 25 q^{86} - 62 q^{87} - 24 q^{88} - 77 q^{89} + 35 q^{90} - 11 q^{91} - 25 q^{92} - 38 q^{94} + 22 q^{95} - 23 q^{96} - 39 q^{97} - 65 q^{98} + 31 q^{99}+O(q^{100}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9 + 7 * q^10 + 31 * q^11 - 4 * q^12 - 24 * q^13 - 9 * q^14 + 4 * q^15 + 43 * q^16 - 49 * q^17 - 35 * q^18 - 22 * q^19 - 39 * q^20 - 8 * q^21 - 7 * q^22 - q^23 - 13 * q^24 + 31 * q^25 - 9 * q^26 - 22 * q^27 - 34 * q^28 - 12 * q^29 + 5 * q^30 + 4 * q^31 - 45 * q^32 - 4 * q^33 + 2 * q^34 + 11 * q^35 + 34 * q^36 - 18 * q^37 - 7 * q^38 - q^39 + 24 * q^40 - 58 * q^41 - 21 * q^42 - 41 * q^43 + 39 * q^44 - 31 * q^45 + 23 * q^46 - 31 * q^47 - 29 * q^48 + 44 * q^49 - 7 * q^50 + 8 * q^51 - 89 * q^52 - 46 * q^53 - 47 * q^54 - 31 * q^55 + 10 * q^56 - 47 * q^57 - 34 * q^58 - 9 * q^59 + 4 * q^60 - 5 * q^61 - 50 * q^62 - 61 * q^63 + 78 * q^64 + 24 * q^65 - 5 * q^66 + q^67 - 115 * q^68 - 19 * q^69 + 9 * q^70 - 8 * q^71 - 93 * q^72 + 31 * q^73 - 19 * q^74 - 4 * q^75 - 7 * q^76 - 11 * q^77 + 57 * q^78 - 43 * q^80 + 43 * q^81 + 20 * q^82 - 29 * q^83 - 32 * q^84 + 49 * q^85 + 25 * q^86 - 62 * q^87 - 24 * q^88 - 77 * q^89 + 35 * q^90 - 11 * q^91 - 25 * q^92 - 38 * q^94 + 22 * q^95 - 23 * q^96 - 39 * q^97 - 65 * q^98 + 31 * q^99

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