LMFDB - Embedded newform 4015.2.a.g.1.19

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:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+0.467946 q^{2} +3.03596 q^{3} -1.78103 q^{4} -1.00000 q^{5} +1.42067 q^{6} -2.71430 q^{7} -1.76932 q^{8} +6.21706 q^{9} +O(q^{10})$	\(q+0.467946 q^{2} +3.03596 q^{3} -1.78103 q^{4} -1.00000 q^{5} +1.42067 q^{6} -2.71430 q^{7} -1.76932 q^{8} +6.21706 q^{9} -0.467946 q^{10} -1.00000 q^{11} -5.40713 q^{12} +1.34721 q^{13} -1.27015 q^{14} -3.03596 q^{15} +2.73411 q^{16} +1.16571 q^{17} +2.90925 q^{18} -0.0442350 q^{19} +1.78103 q^{20} -8.24052 q^{21} -0.467946 q^{22} -8.58427 q^{23} -5.37158 q^{24} +1.00000 q^{25} +0.630422 q^{26} +9.76688 q^{27} +4.83424 q^{28} -2.26840 q^{29} -1.42067 q^{30} +7.20850 q^{31} +4.81805 q^{32} -3.03596 q^{33} +0.545490 q^{34} +2.71430 q^{35} -11.0728 q^{36} +1.51664 q^{37} -0.0206996 q^{38} +4.09008 q^{39} +1.76932 q^{40} -12.1019 q^{41} -3.85612 q^{42} -5.91089 q^{43} +1.78103 q^{44} -6.21706 q^{45} -4.01698 q^{46} +3.28302 q^{47} +8.30064 q^{48} +0.367437 q^{49} +0.467946 q^{50} +3.53905 q^{51} -2.39941 q^{52} -13.3259 q^{53} +4.57038 q^{54} +1.00000 q^{55} +4.80246 q^{56} -0.134296 q^{57} -1.06149 q^{58} -11.6189 q^{59} +5.40713 q^{60} +6.52023 q^{61} +3.37319 q^{62} -16.8750 q^{63} -3.21362 q^{64} -1.34721 q^{65} -1.42067 q^{66} -14.4784 q^{67} -2.07616 q^{68} -26.0615 q^{69} +1.27015 q^{70} -7.29807 q^{71} -11.0000 q^{72} -1.00000 q^{73} +0.709706 q^{74} +3.03596 q^{75} +0.0787836 q^{76} +2.71430 q^{77} +1.91394 q^{78} +11.0304 q^{79} -2.73411 q^{80} +11.0007 q^{81} -5.66303 q^{82} -1.57287 q^{83} +14.6766 q^{84} -1.16571 q^{85} -2.76598 q^{86} -6.88678 q^{87} +1.76932 q^{88} -18.7534 q^{89} -2.90925 q^{90} -3.65673 q^{91} +15.2888 q^{92} +21.8847 q^{93} +1.53628 q^{94} +0.0442350 q^{95} +14.6274 q^{96} +5.70827 q^{97} +0.171941 q^{98} -6.21706 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9}+O(q^{10}) $ 32 * q - 5 * q^2 - 7 * q^3 + 37 * q^4 - 32 * q^5 - 3 * q^6 - 18 * q^8 + 29 * q^9	\( 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9} + 5 q^{10} - 32 q^{11} - 24 q^{12} - q^{13} - 5 q^{14} + 7 q^{15} + 47 q^{16} - 30 q^{17} - 11 q^{18} + 16 q^{19} - 37 q^{20} + q^{21} + 5 q^{22} - 26 q^{23} - 21 q^{24} + 32 q^{25} - q^{26} - 31 q^{27} - 24 q^{28} - 10 q^{29} + 3 q^{30} - 2 q^{31} - 31 q^{32} + 7 q^{33} - 14 q^{34} + 38 q^{36} - 28 q^{37} - 63 q^{38} - 2 q^{39} + 18 q^{40} - 62 q^{41} - 9 q^{42} + 8 q^{43} - 37 q^{44} - 29 q^{45} + 19 q^{46} - 21 q^{47} - 79 q^{48} + 34 q^{49} - 5 q^{50} + 17 q^{51} + 15 q^{52} - 32 q^{53} + 5 q^{54} + 32 q^{55} - 52 q^{56} - 57 q^{57} + 4 q^{58} - 37 q^{59} + 24 q^{60} + 15 q^{61} - 22 q^{62} + 5 q^{63} + 70 q^{64} + q^{65} + 3 q^{66} - 42 q^{67} - 81 q^{68} - 8 q^{69} + 5 q^{70} - 40 q^{71} - 27 q^{72} - 32 q^{73} - 17 q^{74} - 7 q^{75} + 21 q^{76} - 105 q^{78} + 18 q^{79} - 47 q^{80} + 12 q^{81} - 70 q^{82} - 26 q^{83} + 22 q^{84} + 30 q^{85} - 45 q^{86} - 18 q^{87} + 18 q^{88} - 83 q^{89} + 11 q^{90} - 18 q^{91} - 73 q^{92} - 68 q^{93} + 56 q^{94} - 16 q^{95} - 35 q^{96} - 99 q^{97} - 61 q^{98} - 29 q^{99}+O(q^{100}) \) 32 * q - 5 * q^2 - 7 * q^3 + 37 * q^4 - 32 * q^5 - 3 * q^6 - 18 * q^8 + 29 * q^9 + 5 * q^10 - 32 * q^11 - 24 * q^12 - q^13 - 5 * q^14 + 7 * q^15 + 47 * q^16 - 30 * q^17 - 11 * q^18 + 16 * q^19 - 37 * q^20 + q^21 + 5 * q^22 - 26 * q^23 - 21 * q^24 + 32 * q^25 - q^26 - 31 * q^27 - 24 * q^28 - 10 * q^29 + 3 * q^30 - 2 * q^31 - 31 * q^32 + 7 * q^33 - 14 * q^34 + 38 * q^36 - 28 * q^37 - 63 * q^38 - 2 * q^39 + 18 * q^40 - 62 * q^41 - 9 * q^42 + 8 * q^43 - 37 * q^44 - 29 * q^45 + 19 * q^46 - 21 * q^47 - 79 * q^48 + 34 * q^49 - 5 * q^50 + 17 * q^51 + 15 * q^52 - 32 * q^53 + 5 * q^54 + 32 * q^55 - 52 * q^56 - 57 * q^57 + 4 * q^58 - 37 * q^59 + 24 * q^60 + 15 * q^61 - 22 * q^62 + 5 * q^63 + 70 * q^64 + q^65 + 3 * q^66 - 42 * q^67 - 81 * q^68 - 8 * q^69 + 5 * q^70 - 40 * q^71 - 27 * q^72 - 32 * q^73 - 17 * q^74 - 7 * q^75 + 21 * q^76 - 105 * q^78 + 18 * q^79 - 47 * q^80 + 12 * q^81 - 70 * q^82 - 26 * q^83 + 22 * q^84 + 30 * q^85 - 45 * q^86 - 18 * q^87 + 18 * q^88 - 83 * q^89 + 11 * q^90 - 18 * q^91 - 73 * q^92 - 68 * q^93 + 56 * q^94 - 16 * q^95 - 35 * q^96 - 99 * q^97 - 61 * q^98 - 29 * 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 $

$2$	0.467946	0.330888	0.165444	−	0.986219i	$-0.447094\pi$
$2$	0.467946	0.330888	0.165444	+	0.986219i	$0.447094\pi$
$3$	3.03596	1.75281	0.876407	−	0.481572i	$-0.159934\pi$
$3$	3.03596	1.75281	0.876407	+	0.481572i	$0.159934\pi$
$4$	−1.78103	−0.890513
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.42067	0.579985
$7$	−2.71430	−1.02591	−0.512955	−	0.858416i	$-0.671449\pi$
$7$	−2.71430	−1.02591	−0.512955	+	0.858416i	$0.671449\pi$
$8$	−1.76932	−0.625548
$9$	6.21706	2.07235
$10$	−0.467946	−0.147978
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−5.40713	−1.56090
$13$	1.34721	0.373649	0.186824	−	0.982393i	$-0.440180\pi$
$13$	1.34721	0.373649	0.186824	+	0.982393i	$0.440180\pi$
$14$	−1.27015	−0.339461
$15$	−3.03596	−0.783882
$16$	2.73411	0.683527
$17$	1.16571	0.282726	0.141363	−	0.989958i	$-0.454851\pi$
$17$	1.16571	0.282726	0.141363	+	0.989958i	$0.454851\pi$
$18$	2.90925	0.685718
$19$	−0.0442350	−0.0101482	−0.00507410	−	0.999987i	$-0.501615\pi$
$19$	−0.0442350	−0.0101482	−0.00507410	+	0.999987i	$0.501615\pi$
$20$	1.78103	0.398250
$21$	−8.24052	−1.79823
$22$	−0.467946	−0.0997665
$23$	−8.58427	−1.78994	−0.894972	−	0.446122i	$-0.852805\pi$
$23$	−8.58427	−1.78994	−0.894972	+	0.446122i	$0.852805\pi$
$24$	−5.37158	−1.09647
$25$	1.00000	0.200000
$26$	0.630422	0.123636
$27$	9.76688	1.87964
$28$	4.83424	0.913586
$29$	−2.26840	−0.421232	−0.210616	−	0.977569i	$-0.567547\pi$
$29$	−2.26840	−0.421232	−0.210616	+	0.977569i	$0.567547\pi$
$30$	−1.42067	−0.259377
$31$	7.20850	1.29468	0.647342	−	0.762199i	$-0.275880\pi$
$31$	7.20850	1.29468	0.647342	+	0.762199i	$0.275880\pi$
$32$	4.81805	0.851719
$33$	−3.03596	−0.528493
$34$	0.545490	0.0935508
$35$	2.71430	0.458801
$36$	−11.0728	−1.84546
$37$	1.51664	0.249334	0.124667	−	0.992199i	$-0.460214\pi$
$37$	1.51664	0.249334	0.124667	+	0.992199i	$0.460214\pi$
$38$	−0.0206996	−0.00335792
$39$	4.09008	0.654936
$40$	1.76932	0.279754
$41$	−12.1019	−1.88999	−0.944997	−	0.327079i	$-0.893936\pi$
$41$	−12.1019	−1.88999	−0.944997	+	0.327079i	$0.893936\pi$
$42$	−3.85612	−0.595012
$43$	−5.91089	−0.901402	−0.450701	−	0.892675i	$-0.648826\pi$
$43$	−5.91089	−0.901402	−0.450701	+	0.892675i	$0.648826\pi$
$44$	1.78103	0.268500
$45$	−6.21706	−0.926785
$46$	−4.01698	−0.592271
$47$	3.28302	0.478877	0.239439	−	0.970912i	$-0.423037\pi$
$47$	3.28302	0.478877	0.239439	+	0.970912i	$0.423037\pi$
$48$	8.30064	1.19809
$49$	0.367437	0.0524910
$50$	0.467946	0.0661776
$51$	3.53905	0.495567
$52$	−2.39941	−0.332739
$53$	−13.3259	−1.83045	−0.915226	−	0.402942i	$-0.867988\pi$
$53$	−13.3259	−1.83045	−0.915226	+	0.402942i	$0.867988\pi$
$54$	4.57038	0.621950
$55$	1.00000	0.134840
$56$	4.80246	0.641756
$57$	−0.134296	−0.0177879
$58$	−1.06149	−0.139381
$59$	−11.6189	−1.51265	−0.756327	−	0.654193i	$-0.773008\pi$
$59$	−11.6189	−1.51265	−0.756327	+	0.654193i	$0.773008\pi$
$60$	5.40713	0.698057
$61$	6.52023	0.834830	0.417415	−	0.908716i	$-0.362936\pi$
$61$	6.52023	0.834830	0.417415	+	0.908716i	$0.362936\pi$
$62$	3.37319	0.428396
$63$	−16.8750	−2.12605
$64$	−3.21362	−0.401703
$65$	−1.34721	−0.167101
$66$	−1.42067	−0.174872
$67$	−14.4784	−1.76881	−0.884407	−	0.466716i	$-0.845437\pi$
$67$	−14.4784	−1.76881	−0.884407	+	0.466716i	$0.845437\pi$
$68$	−2.07616	−0.251772
$69$	−26.0615	−3.13744
$70$	1.27015	0.151812
$71$	−7.29807	−0.866122	−0.433061	−	0.901365i	$-0.642566\pi$
$71$	−7.29807	−0.866122	−0.433061	+	0.901365i	$0.642566\pi$
$72$	−11.0000	−1.29636
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	0.709706	0.0825017
$75$	3.03596	0.350563
$76$	0.0787836	0.00903710
$77$	2.71430	0.309323
$78$	1.91394	0.216711
$79$	11.0304	1.24101	0.620506	−	0.784201i	$-0.286927\pi$
$79$	11.0304	1.24101	0.620506	+	0.784201i	$0.286927\pi$
$80$	−2.73411	−0.305682
$81$	11.0007	1.22230
$82$	−5.66303	−0.625377
$83$	−1.57287	−0.172644	−0.0863222	−	0.996267i	$-0.527511\pi$
$83$	−1.57287	−0.172644	−0.0863222	+	0.996267i	$0.527511\pi$
$84$	14.6766	1.60135
$85$	−1.16571	−0.126439
$86$	−2.76598	−0.298263
$87$	−6.88678	−0.738340
$88$	1.76932	0.188610
$89$	−18.7534	−1.98786	−0.993931	−	0.110008i	$-0.964912\pi$
$89$	−18.7534	−1.98786	−0.993931	+	0.110008i	$0.964912\pi$
$90$	−2.90925	−0.306662
$91$	−3.65673	−0.383330
$92$	15.2888	1.59397
$93$	21.8847	2.26934
$94$	1.53628	0.158455
$95$	0.0442350	0.00453841
$96$	14.6274	1.49290
$97$	5.70827	0.579587	0.289794	−	0.957089i	$-0.406413\pi$
$97$	5.70827	0.579587	0.289794	+	0.957089i	$0.406413\pi$
$98$	0.171941	0.0173686
$99$	−6.21706	−0.624838
$100$	−1.78103	−0.178103
$101$	3.75133	0.373272	0.186636	−	0.982429i	$-0.440242\pi$
$101$	3.75133	0.373272	0.186636	+	0.982429i	$0.440242\pi$
$102$	1.65609	0.163977
$103$	−16.1925	−1.59549	−0.797745	−	0.602995i	$-0.793974\pi$
$103$	−16.1925	−1.59549	−0.797745	+	0.602995i	$0.793974\pi$
$104$	−2.38364	−0.233735
$105$	8.24052	0.804192
$106$	−6.23580	−0.605675
$107$	−9.49626	−0.918038	−0.459019	−	0.888427i	$-0.651799\pi$
$107$	−9.49626	−0.918038	−0.459019	+	0.888427i	$0.651799\pi$
$108$	−17.3951	−1.67384
$109$	18.6147	1.78297	0.891483	−	0.453054i	$-0.149666\pi$
$109$	18.6147	1.78297	0.891483	+	0.453054i	$0.149666\pi$
$110$	0.467946	0.0446169
$111$	4.60446	0.437036
$112$	−7.42119	−0.701237
$113$	−2.99888	−0.282111	−0.141055	−	0.990002i	$-0.545050\pi$
$113$	−2.99888	−0.282111	−0.141055	+	0.990002i	$0.545050\pi$
$114$	−0.0628432	−0.00588580
$115$	8.58427	0.800487
$116$	4.04008	0.375112
$117$	8.37568	0.774332
$118$	−5.43703	−0.500519
$119$	−3.16409	−0.290052
$120$	5.37158	0.490356
$121$	1.00000	0.0909091
$122$	3.05112	0.276235
$123$	−36.7408	−3.31281
$124$	−12.8385	−1.15293
$125$	−1.00000	−0.0894427
$126$	−7.89659	−0.703484
$127$	−12.4601	−1.10566	−0.552829	−	0.833294i	$-0.686452\pi$
$127$	−12.4601	−1.10566	−0.552829	+	0.833294i	$0.686452\pi$
$128$	−11.1399	−0.984638
$129$	−17.9452	−1.57999
$130$	−0.630422	−0.0552916
$131$	12.6194	1.10257	0.551283	−	0.834318i	$-0.314138\pi$
$131$	12.6194	1.10257	0.551283	+	0.834318i	$0.314138\pi$
$132$	5.40713	0.470630
$133$	0.120067	0.0104111
$134$	−6.77510	−0.585280
$135$	−9.76688	−0.840599
$136$	−2.06251	−0.176859
$137$	16.4566	1.40598	0.702992	−	0.711198i	$-0.251847\pi$
$137$	16.4566	1.40598	0.702992	+	0.711198i	$0.251847\pi$
$138$	−12.1954	−1.03814
$139$	9.31487	0.790077	0.395039	−	0.918665i	$-0.370731\pi$
$139$	9.31487	0.790077	0.395039	+	0.918665i	$0.370731\pi$
$140$	−4.83424	−0.408568
$141$	9.96711	0.839382
$142$	−3.41511	−0.286589
$143$	−1.34721	−0.112659
$144$	16.9981	1.41651
$145$	2.26840	0.188381
$146$	−0.467946	−0.0387275
$147$	1.11552	0.0920069
$148$	−2.70117	−0.222035
$149$	−15.0286	−1.23119	−0.615594	−	0.788064i	$-0.711084\pi$
$149$	−15.0286	−1.23119	−0.615594	+	0.788064i	$0.711084\pi$
$150$	1.42067	0.115997
$151$	21.8148	1.77526	0.887630	−	0.460557i	$-0.152350\pi$
$151$	21.8148	1.77526	0.887630	+	0.460557i	$0.152350\pi$
$152$	0.0782657	0.00634819
$153$	7.24730	0.585909
$154$	1.27015	0.102351
$155$	−7.20850	−0.579001
$156$	−7.28453	−0.583229
$157$	8.07469	0.644431	0.322215	−	0.946666i	$-0.395572\pi$
$157$	8.07469	0.644431	0.322215	+	0.946666i	$0.395572\pi$
$158$	5.16162	0.410636
$159$	−40.4569	−3.20844
$160$	−4.81805	−0.380900
$161$	23.3003	1.83632
$162$	5.14774	0.404444
$163$	−8.62025	−0.675190	−0.337595	−	0.941291i	$-0.609613\pi$
$163$	−8.62025	−0.675190	−0.337595	+	0.941291i	$0.609613\pi$
$164$	21.5537	1.68306
$165$	3.03596	0.236349
$166$	−0.736017	−0.0571260
$167$	23.2949	1.80261	0.901307	−	0.433181i	$-0.142609\pi$
$167$	23.2949	1.80261	0.901307	+	0.433181i	$0.142609\pi$
$168$	14.5801	1.12488
$169$	−11.1850	−0.860387
$170$	−0.545490	−0.0418372
$171$	−0.275012	−0.0210307
$172$	10.5274	0.802710
$173$	−14.3961	−1.09452	−0.547258	−	0.836964i	$-0.684328\pi$
$173$	−14.3961	−1.09452	−0.547258	+	0.836964i	$0.684328\pi$
$174$	−3.22265	−0.244308
$175$	−2.71430	−0.205182
$176$	−2.73411	−0.206091
$177$	−35.2746	−2.65140
$178$	−8.77561	−0.657760
$179$	−5.34442	−0.399461	−0.199730	−	0.979851i	$-0.564007\pi$
$179$	−5.34442	−0.399461	−0.199730	+	0.979851i	$0.564007\pi$
$180$	11.0728	0.825314
$181$	4.91040	0.364987	0.182493	−	0.983207i	$-0.441583\pi$
$181$	4.91040	0.364987	0.182493	+	0.983207i	$0.441583\pi$
$182$	−1.71116	−0.126839
$183$	19.7952	1.46330
$184$	15.1883	1.11970
$185$	−1.51664	−0.111506
$186$	10.2409	0.750898
$187$	−1.16571	−0.0852452
$188$	−5.84714	−0.426446
$189$	−26.5103	−1.92834
$190$	0.0206996	0.00150171
$191$	15.6038	1.12905	0.564526	−	0.825415i	$-0.309059\pi$
$191$	15.6038	1.12905	0.564526	+	0.825415i	$0.309059\pi$
$192$	−9.75643	−0.704110
$193$	−16.5649	−1.19237	−0.596184	−	0.802848i	$-0.703317\pi$
$193$	−16.5649	−1.19237	−0.596184	+	0.802848i	$0.703317\pi$
$194$	2.67117	0.191779
$195$	−4.09008	−0.292896
$196$	−0.654415	−0.0467439
$197$	−9.30599	−0.663024	−0.331512	−	0.943451i	$-0.607559\pi$
$197$	−9.30599	−0.663024	−0.331512	+	0.943451i	$0.607559\pi$
$198$	−2.90925	−0.206752
$199$	11.7744	0.834667	0.417333	−	0.908753i	$-0.362965\pi$
$199$	11.7744	0.834667	0.417333	+	0.908753i	$0.362965\pi$
$200$	−1.76932	−0.125110
$201$	−43.9558	−3.10040
$202$	1.75542	0.123511
$203$	6.15713	0.432146
$204$	−6.30315	−0.441309
$205$	12.1019	0.845231
$206$	−7.57720	−0.527929
$207$	−53.3690	−3.70940
$208$	3.68341	0.255399
$209$	0.0442350	0.00305980
$210$	3.85612	0.266098
$211$	16.0748	1.10664	0.553319	−	0.832970i	$-0.313361\pi$
$211$	16.0748	1.10664	0.553319	+	0.832970i	$0.313361\pi$
$212$	23.7338	1.63004
$213$	−22.1567	−1.51815
$214$	−4.44374	−0.303768
$215$	5.91089	0.403119
$216$	−17.2807	−1.17580
$217$	−19.5660	−1.32823
$218$	8.71069	0.589962
$219$	−3.03596	−0.205151
$220$	−1.78103	−0.120077
$221$	1.57046	0.105640
$222$	2.15464	0.144610
$223$	15.2906	1.02394	0.511968	−	0.859004i	$-0.328916\pi$
$223$	15.2906	1.02394	0.511968	+	0.859004i	$0.328916\pi$
$224$	−13.0776	−0.873787
$225$	6.21706	0.414471
$226$	−1.40331	−0.0933471
$227$	−6.70595	−0.445090	−0.222545	−	0.974922i	$-0.571436\pi$
$227$	−6.70595	−0.445090	−0.222545	+	0.974922i	$0.571436\pi$
$228$	0.239184	0.0158404
$229$	14.5426	0.961004	0.480502	−	0.876994i	$-0.340454\pi$
$229$	14.5426	0.961004	0.480502	+	0.876994i	$0.340454\pi$
$230$	4.01698	0.264872
$231$	8.24052	0.542186
$232$	4.01352	0.263501
$233$	−11.4121	−0.747632	−0.373816	−	0.927503i	$-0.621951\pi$
$233$	−11.4121	−0.747632	−0.373816	+	0.927503i	$0.621951\pi$
$234$	3.91937	0.256217
$235$	−3.28302	−0.214160
$236$	20.6936	1.34704
$237$	33.4878	2.17526
$238$	−1.48063	−0.0959747
$239$	−1.62094	−0.104850	−0.0524249	−	0.998625i	$-0.516695\pi$
$239$	−1.62094	−0.104850	−0.0524249	+	0.998625i	$0.516695\pi$
$240$	−8.30064	−0.535804
$241$	−22.8086	−1.46923	−0.734614	−	0.678485i	$-0.762637\pi$
$241$	−22.8086	−1.46923	−0.734614	+	0.678485i	$0.762637\pi$
$242$	0.467946	0.0300807
$243$	4.09703	0.262825
$244$	−11.6127	−0.743427
$245$	−0.367437	−0.0234747
$246$	−17.1927	−1.09617
$247$	−0.0595937	−0.00379186
$248$	−12.7541	−0.809888
$249$	−4.77516	−0.302613
$250$	−0.467946	−0.0295955
$251$	−15.6156	−0.985647	−0.492823	−	0.870129i	$-0.664035\pi$
$251$	−15.6156	−0.985647	−0.492823	+	0.870129i	$0.664035\pi$
$252$	30.0548	1.89327
$253$	8.58427	0.539688
$254$	−5.83068	−0.365849
$255$	−3.53905	−0.221624
$256$	1.21437	0.0758978
$257$	0.471522	0.0294127	0.0147064	−	0.999892i	$-0.495319\pi$
$257$	0.471522	0.0294127	0.0147064	+	0.999892i	$0.495319\pi$
$258$	−8.39741	−0.522800
$259$	−4.11662	−0.255794
$260$	2.39941	0.148805
$261$	−14.1028	−0.872941
$262$	5.90522	0.364826
$263$	27.6848	1.70712	0.853560	−	0.520995i	$-0.174439\pi$
$263$	27.6848	1.70712	0.853560	+	0.520995i	$0.174439\pi$
$264$	5.37158	0.330598
$265$	13.3259	0.818603
$266$	0.0561850	0.00344492
$267$	−56.9347	−3.48435
$268$	25.7864	1.57515
$269$	24.2676	1.47962	0.739812	−	0.672814i	$-0.234915\pi$
$269$	24.2676	1.47962	0.739812	+	0.672814i	$0.234915\pi$
$270$	−4.57038	−0.278144
$271$	−28.5674	−1.73534	−0.867672	−	0.497137i	$-0.834384\pi$
$271$	−28.5674	−1.73534	−0.867672	+	0.497137i	$0.834384\pi$
$272$	3.18718	0.193251
$273$	−11.1017	−0.671905
$274$	7.70081	0.465223
$275$	−1.00000	−0.0603023
$276$	46.4162	2.79393
$277$	−1.62635	−0.0977177	−0.0488589	−	0.998806i	$-0.515558\pi$
$277$	−1.62635	−0.0977177	−0.0488589	+	0.998806i	$0.515558\pi$
$278$	4.35886	0.261427
$279$	44.8157	2.68305
$280$	−4.80246	−0.287002
$281$	21.1863	1.26387	0.631935	−	0.775022i	$-0.282261\pi$
$281$	21.1863	1.26387	0.631935	+	0.775022i	$0.282261\pi$
$282$	4.66407	0.277742
$283$	21.5637	1.28183	0.640915	−	0.767612i	$-0.278555\pi$
$283$	21.5637	1.28183	0.640915	+	0.767612i	$0.278555\pi$
$284$	12.9981	0.771293
$285$	0.134296	0.00795499
$286$	−0.630422	−0.0372776
$287$	32.8481	1.93896
$288$	29.9541	1.76506
$289$	−15.6411	−0.920066
$290$	1.06149	0.0623329
$291$	17.3301	1.01591
$292$	1.78103	0.104227
$293$	3.48128	0.203379	0.101689	−	0.994816i	$-0.467575\pi$
$293$	3.48128	0.203379	0.101689	+	0.994816i	$0.467575\pi$
$294$	0.522006	0.0304440
$295$	11.6189	0.676480
$296$	−2.68342	−0.155970
$297$	−9.76688	−0.566732
$298$	−7.03256	−0.407385
$299$	−11.5648	−0.668810
$300$	−5.40713	−0.312181
$301$	16.0439	0.924757
$302$	10.2081	0.587413
$303$	11.3889	0.654275
$304$	−0.120943	−0.00693656
$305$	−6.52023	−0.373347
$306$	3.39135	0.193870
$307$	33.4817	1.91090	0.955450	−	0.295152i	$-0.0953704\pi$
$307$	33.4817	1.91090	0.955450	+	0.295152i	$0.0953704\pi$
$308$	−4.83424	−0.275457
$309$	−49.1597	−2.79660
$310$	−3.37319	−0.191584
$311$	8.81887	0.500072	0.250036	−	0.968236i	$-0.419557\pi$
$311$	8.81887	0.500072	0.250036	+	0.968236i	$0.419557\pi$
$312$	−7.23664	−0.409694
$313$	−19.2764	−1.08956	−0.544782	−	0.838578i	$-0.683388\pi$
$313$	−19.2764	−1.08956	−0.544782	+	0.838578i	$0.683388\pi$
$314$	3.77852	0.213234
$315$	16.8750	0.950798
$316$	−19.6454	−1.10514
$317$	−18.4858	−1.03826	−0.519132	−	0.854694i	$-0.673745\pi$
$317$	−18.4858	−1.03826	−0.519132	+	0.854694i	$0.673745\pi$
$318$	−18.9317	−1.06163
$319$	2.26840	0.127006
$320$	3.21362	0.179647
$321$	−28.8303	−1.60915
$322$	10.9033	0.607617
$323$	−0.0515652	−0.00286916
$324$	−19.5925	−1.08847
$325$	1.34721	0.0747297
$326$	−4.03382	−0.223412
$327$	56.5136	3.12521
$328$	21.4120	1.18228
$329$	−8.91110	−0.491285
$330$	1.42067	0.0782052
$331$	−23.1623	−1.27311	−0.636557	−	0.771230i	$-0.719642\pi$
$331$	−23.1623	−1.27311	−0.636557	+	0.771230i	$0.719642\pi$
$332$	2.80132	0.153742
$333$	9.42904	0.516708
$334$	10.9008	0.596463
$335$	14.4784	0.791038
$336$	−22.5305	−1.22914
$337$	4.09417	0.223024	0.111512	−	0.993763i	$-0.464431\pi$
$337$	4.09417	0.223024	0.111512	+	0.993763i	$0.464431\pi$
$338$	−5.23399	−0.284692
$339$	−9.10448	−0.494488
$340$	2.07616	0.112596
$341$	−7.20850	−0.390362
$342$	−0.128691	−0.00695880
$343$	18.0028	0.972059
$344$	10.4582	0.563871
$345$	26.0615	1.40310
$346$	−6.73661	−0.362163
$347$	1.88989	0.101455	0.0507273	−	0.998713i	$-0.483846\pi$
$347$	1.88989	0.101455	0.0507273	+	0.998713i	$0.483846\pi$
$348$	12.2655	0.657502
$349$	5.71523	0.305929	0.152965	−	0.988232i	$-0.451118\pi$
$349$	5.71523	0.305929	0.152965	+	0.988232i	$0.451118\pi$
$350$	−1.27015	−0.0678923
$351$	13.1580	0.702324
$352$	−4.81805	−0.256803
$353$	−4.29471	−0.228584	−0.114292	−	0.993447i	$-0.536460\pi$
$353$	−4.29471	−0.228584	−0.114292	+	0.993447i	$0.536460\pi$
$354$	−16.5066	−0.877317
$355$	7.29807	0.387341
$356$	33.4004	1.77022
$357$	−9.60606	−0.508407
$358$	−2.50090	−0.132177
$359$	−15.6492	−0.825935	−0.412968	−	0.910746i	$-0.635508\pi$
$359$	−15.6492	−0.825935	−0.412968	+	0.910746i	$0.635508\pi$
$360$	11.0000	0.579749
$361$	−18.9980	−0.999897
$362$	2.29780	0.120770
$363$	3.03596	0.159347
$364$	6.51274	0.341360
$365$	1.00000	0.0523424
$366$	9.26308	0.484189
$367$	−10.7729	−0.562342	−0.281171	−	0.959658i	$-0.590723\pi$
$367$	−10.7729	−0.562342	−0.281171	+	0.959658i	$0.590723\pi$
$368$	−23.4703	−1.22347
$369$	−75.2381	−3.91674
$370$	−0.709706	−0.0368959
$371$	36.1705	1.87788
$372$	−38.9773	−2.02088
$373$	−3.00464	−0.155575	−0.0777873	−	0.996970i	$-0.524785\pi$
$373$	−3.00464	−0.155575	−0.0777873	+	0.996970i	$0.524785\pi$
$374$	−0.545490	−0.0282066
$375$	−3.03596	−0.156776
$376$	−5.80870	−0.299561
$377$	−3.05601	−0.157393
$378$	−12.4054	−0.638064
$379$	−21.6738	−1.11331	−0.556654	−	0.830745i	$-0.687915\pi$
$379$	−21.6738	−1.11331	−0.556654	+	0.830745i	$0.687915\pi$
$380$	−0.0787836	−0.00404151
$381$	−37.8285	−1.93801
$382$	7.30175	0.373590
$383$	25.5644	1.30628	0.653140	−	0.757238i	$-0.273451\pi$
$383$	25.5644	1.30628	0.653140	+	0.757238i	$0.273451\pi$
$384$	−33.8203	−1.72589
$385$	−2.71430	−0.138334
$386$	−7.75149	−0.394541
$387$	−36.7484	−1.86803
$388$	−10.1666	−0.516130
$389$	−24.8088	−1.25786	−0.628928	−	0.777463i	$-0.716506\pi$
$389$	−24.8088	−1.25786	−0.628928	+	0.777463i	$0.716506\pi$
$390$	−1.91394	−0.0969159
$391$	−10.0068	−0.506064
$392$	−0.650113	−0.0328357
$393$	38.3121	1.93259
$394$	−4.35471	−0.219387
$395$	−11.0304	−0.554998
$396$	11.0728	0.556427
$397$	27.6508	1.38775	0.693877	−	0.720093i	$-0.255901\pi$
$397$	27.6508	1.38775	0.693877	+	0.720093i	$0.255901\pi$
$398$	5.50980	0.276181
$399$	0.364519	0.0182488
$400$	2.73411	0.136705
$401$	9.45204	0.472013	0.236006	−	0.971752i	$-0.424161\pi$
$401$	9.45204	0.472013	0.236006	+	0.971752i	$0.424161\pi$
$402$	−20.5689	−1.02589
$403$	9.71136	0.483757
$404$	−6.68122	−0.332403
$405$	−11.0007	−0.546629
$406$	2.88121	0.142992
$407$	−1.51664	−0.0751770
$408$	−6.26171	−0.310001
$409$	−23.4600	−1.16002	−0.580011	−	0.814609i	$-0.696952\pi$
$409$	−23.4600	−1.16002	−0.580011	+	0.814609i	$0.696952\pi$
$410$	5.66303	0.279677
$411$	49.9617	2.46443
$412$	28.8392	1.42080
$413$	31.5373	1.55185
$414$	−24.9738	−1.22740
$415$	1.57287	0.0772089
$416$	6.49092	0.318244
$417$	28.2796	1.38486
$418$	0.0206996	0.00101245
$419$	27.6640	1.35148	0.675738	−	0.737142i	$-0.263825\pi$
$419$	27.6640	1.35148	0.675738	+	0.737142i	$0.263825\pi$
$420$	−14.6766	−0.716144
$421$	7.03424	0.342828	0.171414	−	0.985199i	$-0.445166\pi$
$421$	7.03424	0.342828	0.171414	+	0.985199i	$0.445166\pi$
$422$	7.52216	0.366173
$423$	20.4107	0.992403
$424$	23.5777	1.14504
$425$	1.16571	0.0565453
$426$	−10.3681	−0.502338
$427$	−17.6979	−0.856460
$428$	16.9131	0.817525
$429$	−4.09008	−0.197471
$430$	2.76598	0.133387
$431$	1.77077	0.0852952	0.0426476	−	0.999090i	$-0.486421\pi$
$431$	1.77077	0.0852952	0.0426476	+	0.999090i	$0.486421\pi$
$432$	26.7037	1.28478
$433$	15.9056	0.764374	0.382187	−	0.924085i	$-0.375171\pi$
$433$	15.9056	0.764374	0.382187	+	0.924085i	$0.375171\pi$
$434$	−9.15586	−0.439496
$435$	6.88678	0.330196
$436$	−33.1533	−1.58775
$437$	0.379725	0.0181647
$438$	−1.42067	−0.0678821
$439$	26.7087	1.27474	0.637369	−	0.770558i	$-0.280023\pi$
$439$	26.7087	1.27474	0.637369	+	0.770558i	$0.280023\pi$
$440$	−1.76932	−0.0843489
$441$	2.28438	0.108780
$442$	0.734889	0.0349551
$443$	25.8742	1.22932	0.614660	−	0.788792i	$-0.289293\pi$
$443$	25.8742	1.22932	0.614660	+	0.788792i	$0.289293\pi$
$444$	−8.20066	−0.389186
$445$	18.7534	0.888999
$446$	7.15520	0.338809
$447$	−45.6261	−2.15804
$448$	8.72274	0.412111
$449$	7.54582	0.356109	0.178055	−	0.984021i	$-0.443020\pi$
$449$	7.54582	0.356109	0.178055	+	0.984021i	$0.443020\pi$
$450$	2.90925	0.137144
$451$	12.1019	0.569855
$452$	5.34108	0.251223
$453$	66.2288	3.11170
$454$	−3.13803	−0.147275
$455$	3.65673	0.171430
$456$	0.237612	0.0111272
$457$	−18.2666	−0.854475	−0.427237	−	0.904139i	$-0.640513\pi$
$457$	−18.2666	−0.854475	−0.427237	+	0.904139i	$0.640513\pi$
$458$	6.80518	0.317985
$459$	11.3854	0.531423
$460$	−15.2888	−0.712844
$461$	14.2237	0.662465	0.331233	−	0.943549i	$-0.392536\pi$
$461$	14.2237	0.662465	0.331233	+	0.943549i	$0.392536\pi$
$462$	3.85612	0.179403
$463$	−21.7442	−1.01054	−0.505270	−	0.862961i	$-0.668607\pi$
$463$	−21.7442	−1.01054	−0.505270	+	0.862961i	$0.668607\pi$
$464$	−6.20205	−0.287923
$465$	−21.8847	−1.01488
$466$	−5.34026	−0.247383
$467$	29.2781	1.35483	0.677415	−	0.735601i	$-0.263100\pi$
$467$	29.2781	1.35483	0.677415	+	0.735601i	$0.263100\pi$
$468$	−14.9173	−0.689553
$469$	39.2987	1.81464
$470$	−1.53628	−0.0708631
$471$	24.5145	1.12957
$472$	20.5576	0.946238
$473$	5.91089	0.271783
$474$	15.6705	0.719769
$475$	−0.0442350	−0.00202964
$476$	5.63533	0.258295
$477$	−82.8479	−3.79334
$478$	−0.758513	−0.0346936
$479$	−15.3374	−0.700784	−0.350392	−	0.936603i	$-0.613952\pi$
$479$	−15.3374	−0.700784	−0.350392	+	0.936603i	$0.613952\pi$
$480$	−14.6274	−0.667647
$481$	2.04323	0.0931633
$482$	−10.6732	−0.486150
$483$	70.7388	3.21873
$484$	−1.78103	−0.0809557
$485$	−5.70827	−0.259199
$486$	1.91719	0.0869656
$487$	−7.61790	−0.345200	−0.172600	−	0.984992i	$-0.555217\pi$
$487$	−7.61790	−0.345200	−0.172600	+	0.984992i	$0.555217\pi$
$488$	−11.5364	−0.522226
$489$	−26.1708	−1.18348
$490$	−0.171941	−0.00776749
$491$	−12.5391	−0.565880	−0.282940	−	0.959138i	$-0.591310\pi$
$491$	−12.5391	−0.565880	−0.282940	+	0.959138i	$0.591310\pi$
$492$	65.4363	2.95010
$493$	−2.64430	−0.119093
$494$	−0.0278867	−0.00125468
$495$	6.21706	0.279436
$496$	19.7088	0.884951
$497$	19.8092	0.888563
$498$	−2.23452	−0.100131
$499$	−18.8274	−0.842830	−0.421415	−	0.906868i	$-0.638466\pi$
$499$	−18.8274	−0.842830	−0.421415	+	0.906868i	$0.638466\pi$
$500$	1.78103	0.0796499
$501$	70.7224	3.15965
$502$	−7.30726	−0.326139
$503$	−33.9291	−1.51282	−0.756412	−	0.654095i	$-0.773050\pi$
$503$	−33.9291	−1.51282	−0.756412	+	0.654095i	$0.773050\pi$
$504$	29.8572	1.32995
$505$	−3.75133	−0.166932
$506$	4.01698	0.178576
$507$	−33.9573	−1.50810
$508$	22.1918	0.984604
$509$	−6.40494	−0.283894	−0.141947	−	0.989874i	$-0.545336\pi$
$509$	−6.40494	−0.283894	−0.141947	+	0.989874i	$0.545336\pi$
$510$	−1.65609	−0.0733328
$511$	2.71430	0.120074
$512$	22.8481	1.00975
$513$	−0.432038	−0.0190749
$514$	0.220647	0.00973233
$515$	16.1925	0.713525
$516$	31.9609	1.40700
$517$	−3.28302	−0.144387
$518$	−1.92636	−0.0846393
$519$	−43.7061	−1.91848
$520$	2.38364	0.104530
$521$	8.18951	0.358789	0.179394	−	0.983777i	$-0.442586\pi$
$521$	8.18951	0.358789	0.179394	+	0.983777i	$0.442586\pi$
$522$	−6.59936	−0.288846
$523$	10.2044	0.446206	0.223103	−	0.974795i	$-0.428381\pi$
$523$	10.2044	0.446206	0.223103	+	0.974795i	$0.428381\pi$
$524$	−22.4756	−0.981849
$525$	−8.24052	−0.359646
$526$	12.9550	0.564866
$527$	8.40303	0.366042
$528$	−8.30064	−0.361239
$529$	50.6897	2.20390
$530$	6.23580	0.270866
$531$	−72.2356	−3.13476
$532$	−0.213843	−0.00927125
$533$	−16.3037	−0.706194
$534$	−26.6424	−1.15293
$535$	9.49626	0.410559
$536$	25.6168	1.10648
$537$	−16.2254	−0.700180
$538$	11.3559	0.489590
$539$	−0.367437	−0.0158266
$540$	17.3951	0.748565
$541$	35.7467	1.53687	0.768436	−	0.639927i	$-0.221035\pi$
$541$	35.7467	1.53687	0.768436	+	0.639927i	$0.221035\pi$
$542$	−13.3680	−0.574205
$543$	14.9078	0.639754
$544$	5.61645	0.240803
$545$	−18.6147	−0.797367
$546$	−5.19500	−0.222326
$547$	−4.50350	−0.192556	−0.0962779	−	0.995354i	$-0.530694\pi$
$547$	−4.50350	−0.192556	−0.0962779	+	0.995354i	$0.530694\pi$
$548$	−29.3097	−1.25205
$549$	40.5367	1.73006
$550$	−0.467946	−0.0199533
$551$	0.100343	0.00427474
$552$	46.1111	1.96262
$553$	−29.9397	−1.27317
$554$	−0.761043	−0.0323336
$555$	−4.60446	−0.195448
$556$	−16.5900	−0.703574
$557$	−38.7105	−1.64022	−0.820108	−	0.572209i	$-0.806087\pi$
$557$	−38.7105	−1.64022	−0.820108	+	0.572209i	$0.806087\pi$
$558$	20.9714	0.887788
$559$	−7.96320	−0.336808
$560$	7.42119	0.313603
$561$	−3.53905	−0.149419
$562$	9.91406	0.418199
$563$	31.4769	1.32659	0.663297	−	0.748356i	$-0.269157\pi$
$563$	31.4769	1.32659	0.663297	+	0.748356i	$0.269157\pi$
$564$	−17.7517	−0.747481
$565$	2.99888	0.126164
$566$	10.0907	0.424142
$567$	−29.8592	−1.25397
$568$	12.9126	0.541801
$569$	36.0559	1.51154	0.755771	−	0.654836i	$-0.227262\pi$
$569$	36.0559	1.51154	0.755771	+	0.654836i	$0.227262\pi$
$570$	0.0628432	0.00263221
$571$	43.6947	1.82857	0.914283	−	0.405077i	$-0.132755\pi$
$571$	43.6947	1.82857	0.914283	+	0.405077i	$0.132755\pi$
$572$	2.39941	0.100325
$573$	47.3726	1.97902
$574$	15.3712	0.641580
$575$	−8.58427	−0.357989
$576$	−19.9793	−0.832471
$577$	−11.3827	−0.473869	−0.236935	−	0.971526i	$-0.576143\pi$
$577$	−11.3827	−0.473869	−0.236935	+	0.971526i	$0.576143\pi$
$578$	−7.31921	−0.304439
$579$	−50.2904	−2.09000
$580$	−4.04008	−0.167755
$581$	4.26923	0.177118
$582$	8.10956	0.336152
$583$	13.3259	0.551902
$584$	1.76932	0.0732149
$585$	−8.37568	−0.346292
$586$	1.62905	0.0672956
$587$	10.5015	0.433445	0.216722	−	0.976233i	$-0.430463\pi$
$587$	10.5015	0.433445	0.216722	+	0.976233i	$0.430463\pi$
$588$	−1.98678	−0.0819334
$589$	−0.318868	−0.0131387
$590$	5.43703	0.223839
$591$	−28.2526	−1.16216
$592$	4.14665	0.170426
$593$	−16.0327	−0.658383	−0.329192	−	0.944263i	$-0.606776\pi$
$593$	−16.0327	−0.658383	−0.329192	+	0.944263i	$0.606776\pi$
$594$	−4.57038	−0.187525
$595$	3.16409	0.129715
$596$	26.7663	1.09639
$597$	35.7467	1.46302
$598$	−5.41171	−0.221301
$599$	−13.5508	−0.553672	−0.276836	−	0.960917i	$-0.589286\pi$
$599$	−13.5508	−0.553672	−0.276836	+	0.960917i	$0.589286\pi$
$600$	−5.37158	−0.219294
$601$	−25.4287	−1.03726	−0.518628	−	0.855000i	$-0.673557\pi$
$601$	−25.4287	−1.03726	−0.518628	+	0.855000i	$0.673557\pi$
$602$	7.50771	0.305991
$603$	−90.0129	−3.66561
$604$	−38.8527	−1.58089
$605$	−1.00000	−0.0406558
$606$	5.32940	0.216492
$607$	−7.75852	−0.314909	−0.157454	−	0.987526i	$-0.550329\pi$
$607$	−7.75852	−0.314909	−0.157454	+	0.987526i	$0.550329\pi$
$608$	−0.213126	−0.00864341
$609$	18.6928	0.757471
$610$	−3.05112	−0.123536
$611$	4.42291	0.178932
$612$	−12.9076	−0.521760
$613$	−8.67477	−0.350371	−0.175185	−	0.984535i	$-0.556052\pi$
$613$	−8.67477	−0.350371	−0.175185	+	0.984535i	$0.556052\pi$
$614$	15.6676	0.632294
$615$	36.7408	1.48153
$616$	−4.80246	−0.193497
$617$	31.5321	1.26944	0.634718	−	0.772744i	$-0.281116\pi$
$617$	31.5321	1.26944	0.634718	+	0.772744i	$0.281116\pi$
$618$	−23.0041	−0.925361
$619$	−23.7558	−0.954828	−0.477414	−	0.878678i	$-0.658426\pi$
$619$	−23.7558	−0.954828	−0.477414	+	0.878678i	$0.658426\pi$
$620$	12.8385	0.515608
$621$	−83.8416	−3.36445
$622$	4.12676	0.165468
$623$	50.9025	2.03937
$624$	11.1827	0.447666
$625$	1.00000	0.0400000
$626$	−9.02031	−0.360524
$627$	0.134296	0.00536325
$628$	−14.3812	−0.573874
$629$	1.76796	0.0704933
$630$	7.89659	0.314608
$631$	−17.9929	−0.716288	−0.358144	−	0.933666i	$-0.616590\pi$
$631$	−17.9929	−0.716288	−0.358144	+	0.933666i	$0.616590\pi$
$632$	−19.5162	−0.776314
$633$	48.8026	1.93973
$634$	−8.65034	−0.343549
$635$	12.4601	0.494466
$636$	72.0548	2.85716
$637$	0.495014	0.0196132
$638$	1.06149	0.0420248
$639$	−45.3726	−1.79491
$640$	11.1399	0.440343
$641$	−45.7239	−1.80598	−0.902992	−	0.429658i	$-0.858634\pi$
$641$	−45.7239	−1.80598	−0.902992	+	0.429658i	$0.858634\pi$
$642$	−13.4910	−0.532448
$643$	−3.68400	−0.145283	−0.0726413	−	0.997358i	$-0.523143\pi$
$643$	−3.68400	−0.145283	−0.0726413	+	0.997358i	$0.523143\pi$
$644$	−41.4984	−1.63527
$645$	17.9452	0.706593
$646$	−0.0241297	−0.000949372 0
$647$	9.97867	0.392302	0.196151	−	0.980574i	$-0.437156\pi$
$647$	9.97867	0.392302	0.196151	+	0.980574i	$0.437156\pi$
$648$	−19.4637	−0.764607
$649$	11.6189	0.456082
$650$	0.630422	0.0247272
$651$	−59.4018	−2.32814
$652$	15.3529	0.601266
$653$	−7.31221	−0.286149	−0.143074	−	0.989712i	$-0.545699\pi$
$653$	−7.31221	−0.286149	−0.143074	+	0.989712i	$0.545699\pi$
$654$	26.4453	1.03409
$655$	−12.6194	−0.493082
$656$	−33.0878	−1.29186
$657$	−6.21706	−0.242551
$658$	−4.16992	−0.162560
$659$	−33.9729	−1.32339	−0.661697	−	0.749771i	$-0.730164\pi$
$659$	−33.9729	−1.32339	−0.661697	+	0.749771i	$0.730164\pi$
$660$	−5.40713	−0.210472
$661$	−25.4590	−0.990241	−0.495121	−	0.868824i	$-0.664876\pi$
$661$	−25.4590	−0.990241	−0.495121	+	0.868824i	$0.664876\pi$
$662$	−10.8387	−0.421258
$663$	4.76784	0.185168
$664$	2.78290	0.107997
$665$	−0.120067	−0.00465600
$666$	4.41229	0.170973
$667$	19.4726	0.753981
$668$	−41.4888	−1.60525
$669$	46.4218	1.79477
$670$	6.77510	0.261745
$671$	−6.52023	−0.251711
$672$	−39.7032	−1.53159
$673$	−19.9963	−0.770803	−0.385401	−	0.922749i	$-0.625937\pi$
$673$	−19.9963	−0.770803	−0.385401	+	0.922749i	$0.625937\pi$
$674$	1.91585	0.0737959
$675$	9.76688	0.375928
$676$	19.9208	0.766186
$677$	−33.6433	−1.29302	−0.646508	−	0.762907i	$-0.723771\pi$
$677$	−33.6433	−1.29302	−0.646508	+	0.762907i	$0.723771\pi$
$678$	−4.26041	−0.163620
$679$	−15.4940	−0.594604
$680$	2.06251	0.0790938
$681$	−20.3590	−0.780159
$682$	−3.37319	−0.129166
$683$	−24.7873	−0.948460	−0.474230	−	0.880401i	$-0.657273\pi$
$683$	−24.7873	−0.948460	−0.474230	+	0.880401i	$0.657273\pi$
$684$	0.489803	0.0187281
$685$	−16.4566	−0.628775
$686$	8.42434	0.321643
$687$	44.1509	1.68446
$688$	−16.1610	−0.616132
$689$	−17.9528	−0.683946
$690$	12.1954	0.464271
$691$	−39.8619	−1.51642	−0.758208	−	0.652012i	$-0.773925\pi$
$691$	−39.8619	−1.51642	−0.758208	+	0.652012i	$0.773925\pi$
$692$	25.6399	0.974681
$693$	16.8750	0.641028
$694$	0.884367	0.0335701
$695$	−9.31487	−0.353333
$696$	12.1849	0.461868
$697$	−14.1073	−0.534351
$698$	2.67442	0.101228
$699$	−34.6467	−1.31046
$700$	4.83424	0.182717
$701$	−46.9970	−1.77505	−0.887525	−	0.460759i	$-0.847577\pi$
$701$	−46.9970	−1.77505	−0.887525	+	0.460759i	$0.847577\pi$
$702$	6.15726	0.232391
$703$	−0.0670885	−0.00253029
$704$	3.21362	0.121118
$705$	−9.96711	−0.375383
$706$	−2.00969	−0.0756358
$707$	−10.1823	−0.382943
$708$	62.8250	2.36111
$709$	−32.9410	−1.23713	−0.618563	−	0.785735i	$-0.712285\pi$
$709$	−32.9410	−1.23713	−0.618563	+	0.785735i	$0.712285\pi$
$710$	3.41511	0.128167
$711$	68.5765	2.57182
$712$	33.1808	1.24350
$713$	−61.8797	−2.31741
$714$	−4.49512	−0.168226
$715$	1.34721	0.0503828
$716$	9.51855	0.355725
$717$	−4.92111	−0.183782
$718$	−7.32301	−0.273292
$719$	33.6062	1.25330	0.626650	−	0.779301i	$-0.284426\pi$
$719$	33.6062	1.25330	0.626650	+	0.779301i	$0.284426\pi$
$720$	−16.9981	−0.633482
$721$	43.9512	1.63683
$722$	−8.89007	−0.330854
$723$	−69.2459	−2.57528
$724$	−8.74555	−0.325026
$725$	−2.26840	−0.0842463
$726$	1.42067	0.0527259
$727$	−21.4020	−0.793758	−0.396879	−	0.917871i	$-0.629907\pi$
$727$	−21.4020	−0.793758	−0.396879	+	0.917871i	$0.629907\pi$
$728$	6.46992	0.239791
$729$	−20.5636	−0.761616
$730$	0.467946	0.0173195
$731$	−6.89039	−0.254850
$732$	−35.2557	−1.30309
$733$	−20.7807	−0.767552	−0.383776	−	0.923426i	$-0.625376\pi$
$733$	−20.7807	−0.767552	−0.383776	+	0.923426i	$0.625376\pi$
$734$	−5.04115	−0.186072
$735$	−1.11552	−0.0411467
$736$	−41.3594	−1.52453
$737$	14.4784	0.533318
$738$	−35.2074	−1.29600
$739$	31.1723	1.14669	0.573346	−	0.819314i	$-0.305645\pi$
$739$	31.1723	1.14669	0.573346	+	0.819314i	$0.305645\pi$
$740$	2.70117	0.0992971
$741$	−0.180924	−0.00664642
$742$	16.9259	0.621368
$743$	−1.23517	−0.0453141	−0.0226570	−	0.999743i	$-0.507213\pi$
$743$	−1.23517	−0.0453141	−0.0226570	+	0.999743i	$0.507213\pi$
$744$	−38.7210	−1.41958
$745$	15.0286	0.550604
$746$	−1.40601	−0.0514778
$747$	−9.77861	−0.357781
$748$	2.07616	0.0759120
$749$	25.7757	0.941824
$750$	−1.42067	−0.0518754
$751$	18.0574	0.658922	0.329461	−	0.944169i	$-0.393133\pi$
$751$	18.0574	0.658922	0.329461	+	0.944169i	$0.393133\pi$
$752$	8.97611	0.327325
$753$	−47.4083	−1.72766
$754$	−1.43005	−0.0520793
$755$	−21.8148	−0.793920
$756$	47.2155	1.71721
$757$	20.3124	0.738266	0.369133	−	0.929377i	$-0.379655\pi$
$757$	20.3124	0.738266	0.369133	+	0.929377i	$0.379655\pi$
$758$	−10.1422	−0.368380
$759$	26.0615	0.945973
$760$	−0.0782657	−0.00283900
$761$	10.1543	0.368094	0.184047	−	0.982917i	$-0.441080\pi$
$761$	10.1543	0.368094	0.184047	+	0.982917i	$0.441080\pi$
$762$	−17.7017	−0.641266
$763$	−50.5260	−1.82916
$764$	−27.7908	−1.00544
$765$	−7.24730	−0.262027
$766$	11.9628	0.432232
$767$	−15.6531	−0.565201
$768$	3.68677	0.133035
$769$	9.81336	0.353879	0.176939	−	0.984222i	$-0.443380\pi$
$769$	9.81336	0.353879	0.176939	+	0.984222i	$0.443380\pi$
$770$	−1.27015	−0.0457730
$771$	1.43152	0.0515550
$772$	29.5025	1.06182
$773$	−24.3516	−0.875867	−0.437934	−	0.899007i	$-0.644290\pi$
$773$	−24.3516	−0.875867	−0.437934	+	0.899007i	$0.644290\pi$
$774$	−17.1963	−0.618107
$775$	7.20850	0.258937
$776$	−10.0997	−0.362560
$777$	−12.4979	−0.448359
$778$	−11.6092	−0.416210
$779$	0.535326	0.0191800
$780$	7.28453	0.260828
$781$	7.29807	0.261146
$782$	−4.68264	−0.167451
$783$	−22.1552	−0.791763
$784$	1.00461	0.0358790
$785$	−8.07469	−0.288198
$786$	17.9280	0.639472
$787$	−1.71879	−0.0612681	−0.0306340	−	0.999531i	$-0.509753\pi$
$787$	−1.71879	−0.0612681	−0.0306340	+	0.999531i	$0.509753\pi$
$788$	16.5742	0.590432
$789$	84.0501	2.99226
$790$	−5.16162	−0.183642
$791$	8.13986	0.289420
$792$	11.0000	0.390867
$793$	8.78411	0.311933
$794$	12.9391	0.459191
$795$	40.4569	1.43486
$796$	−20.9706	−0.743282
$797$	−15.1378	−0.536208	−0.268104	−	0.963390i	$-0.586397\pi$
$797$	−15.1378	−0.536208	−0.268104	+	0.963390i	$0.586397\pi$
$798$	0.170575	0.00603830
$799$	3.82705	0.135391
$800$	4.81805	0.170344
$801$	−116.591	−4.11955
$802$	4.42305	0.156183
$803$	1.00000	0.0352892
$804$	78.2864	2.76095
$805$	−23.3003	−0.821228
$806$	4.54440	0.160069
$807$	73.6756	2.59350
$808$	−6.63730	−0.233499
$809$	46.2433	1.62583	0.812915	−	0.582383i	$-0.197880\pi$
$809$	46.2433	1.62583	0.812915	+	0.582383i	$0.197880\pi$
$810$	−5.14774	−0.180873
$811$	43.9761	1.54421	0.772105	−	0.635495i	$-0.219204\pi$
$811$	43.9761	1.54421	0.772105	+	0.635495i	$0.219204\pi$
$812$	−10.9660	−0.384831
$813$	−86.7294	−3.04173
$814$	−0.709706	−0.0248752
$815$	8.62025	0.301954
$816$	9.67615	0.338733
$817$	0.261468	0.00914761
$818$	−10.9780	−0.383838
$819$	−22.7341	−0.794395
$820$	−21.5537	−0.752689
$821$	−4.87335	−0.170081	−0.0850406	−	0.996377i	$-0.527102\pi$
$821$	−4.87335	−0.170081	−0.0850406	+	0.996377i	$0.527102\pi$
$822$	23.3794	0.815449
$823$	26.0553	0.908231	0.454116	−	0.890943i	$-0.349955\pi$
$823$	26.0553	0.908231	0.454116	+	0.890943i	$0.349955\pi$
$824$	28.6496	0.998056
$825$	−3.03596	−0.105699
$826$	14.7577	0.513488
$827$	−32.2625	−1.12188	−0.560938	−	0.827858i	$-0.689559\pi$
$827$	−32.2625	−1.12188	−0.560938	+	0.827858i	$0.689559\pi$
$828$	95.0515	3.30327
$829$	−2.85812	−0.0992668	−0.0496334	−	0.998768i	$-0.515805\pi$
$829$	−2.85812	−0.0992668	−0.0496334	+	0.998768i	$0.515805\pi$
$830$	0.736017	0.0255475
$831$	−4.93753	−0.171281
$832$	−4.32942	−0.150096
$833$	0.428325	0.0148406
$834$	13.2333	0.458233
$835$	−23.2949	−0.806153
$836$	−0.0787836	−0.00272479
$837$	70.4046	2.43354
$838$	12.9453	0.447187
$839$	−39.2700	−1.35575	−0.677875	−	0.735177i	$-0.737099\pi$
$839$	−39.2700	−1.35575	−0.677875	+	0.735177i	$0.737099\pi$
$840$	−14.5801	−0.503061
$841$	−23.8544	−0.822564
$842$	3.29165	0.113438
$843$	64.3208	2.21533
$844$	−28.6297	−0.985475
$845$	11.1850	0.384777
$846$	9.55112	0.328374
$847$	−2.71430	−0.0932645
$848$	−36.4344	−1.25116
$849$	65.4666	2.24681
$850$	0.545490	0.0187102
$851$	−13.0192	−0.446294
$852$	39.4616	1.35193
$853$	45.0248	1.54162	0.770811	−	0.637064i	$-0.219851\pi$
$853$	45.0248	1.54162	0.770811	+	0.637064i	$0.219851\pi$
$854$	−8.28165	−0.283392
$855$	0.275012	0.00940520
$856$	16.8019	0.574277
$857$	0.996084	0.0340256	0.0170128	−	0.999855i	$-0.494584\pi$
$857$	0.996084	0.0340256	0.0170128	+	0.999855i	$0.494584\pi$
$858$	−1.91394	−0.0653407
$859$	28.6706	0.978228	0.489114	−	0.872220i	$-0.337320\pi$
$859$	28.6706	0.978228	0.489114	+	0.872220i	$0.337320\pi$
$860$	−10.5274	−0.358983
$861$	99.7257	3.39864
$862$	0.828628	0.0282232
$863$	5.43227	0.184917	0.0924583	−	0.995717i	$-0.470528\pi$
$863$	5.43227	0.184917	0.0924583	+	0.995717i	$0.470528\pi$
$864$	47.0573	1.60092
$865$	14.3961	0.489483
$866$	7.44297	0.252922
$867$	−47.4858	−1.61270
$868$	34.8476	1.18281
$869$	−11.0304	−0.374179
$870$	3.22265	0.109258
$871$	−19.5054	−0.660915
$872$	−32.9353	−1.11533
$873$	35.4887	1.20111
$874$	0.177691	0.00601048
$875$	2.71430	0.0917602
$876$	5.40713	0.182690
$877$	−35.9738	−1.21475	−0.607374	−	0.794416i	$-0.707777\pi$
$877$	−35.9738	−1.21475	−0.607374	+	0.794416i	$0.707777\pi$
$878$	12.4983	0.421796
$879$	10.5690	0.356485
$880$	2.73411	0.0921667
$881$	−7.29110	−0.245643	−0.122822	−	0.992429i	$-0.539194\pi$
$881$	−7.29110	−0.245643	−0.122822	+	0.992429i	$0.539194\pi$
$882$	1.06897	0.0359940
$883$	−15.9444	−0.536573	−0.268286	−	0.963339i	$-0.586457\pi$
$883$	−15.9444	−0.536573	−0.268286	+	0.963339i	$0.586457\pi$
$884$	−2.79702	−0.0940741
$885$	35.2746	1.18574
$886$	12.1077	0.406767
$887$	49.9617	1.67755	0.838775	−	0.544478i	$-0.183272\pi$
$887$	49.9617	1.67755	0.838775	+	0.544478i	$0.183272\pi$
$888$	−8.14675	−0.273387
$889$	33.8206	1.13431
$890$	8.77561	0.294159
$891$	−11.0007	−0.368537
$892$	−27.2330	−0.911829
$893$	−0.145224	−0.00485974
$894$	−21.3506	−0.714071
$895$	5.34442	0.178644
$896$	30.2371	1.01015
$897$	−35.1103	−1.17230
$898$	3.53104	0.117832
$899$	−16.3518	−0.545362
$900$	−11.0728	−0.369092
$901$	−15.5341	−0.517517
$902$	5.66303	0.188558
$903$	48.7088	1.62093
$904$	5.30597	0.176474
$905$	−4.91040	−0.163227
$906$	30.9915	1.02962
$907$	−57.3002	−1.90262	−0.951311	−	0.308233i	$-0.900262\pi$
$907$	−57.3002	−1.90262	−0.951311	+	0.308233i	$0.900262\pi$
$908$	11.9435	0.396358
$909$	23.3223	0.773551
$910$	1.71116	0.0567242
$911$	40.9247	1.35590	0.677948	−	0.735110i	$-0.262869\pi$
$911$	40.9247	1.35590	0.677948	+	0.735110i	$0.262869\pi$
$912$	−0.367179	−0.0121585
$913$	1.57287	0.0520543
$914$	−8.54778	−0.282736
$915$	−19.7952	−0.654408
$916$	−25.9008	−0.855787
$917$	−34.2530	−1.13113
$918$	5.32774	0.175842
$919$	−49.8030	−1.64285	−0.821424	−	0.570318i	$-0.806820\pi$
$919$	−49.8030	−1.64285	−0.821424	+	0.570318i	$0.806820\pi$
$920$	−15.1883	−0.500743
$921$	101.649	3.34945
$922$	6.65595	0.219202
$923$	−9.83203	−0.323625
$924$	−14.6766	−0.482824
$925$	1.51664	0.0498668
$926$	−10.1751	−0.334376
$927$	−100.670	−3.30642
$928$	−10.9293	−0.358771
$929$	31.6616	1.03878	0.519392	−	0.854536i	$-0.326158\pi$
$929$	31.6616	1.03878	0.519392	+	0.854536i	$0.326158\pi$
$930$	−10.2409	−0.335812
$931$	−0.0162536	−0.000532689 0
$932$	20.3253	0.665776
$933$	26.7738	0.876534
$934$	13.7006	0.448297
$935$	1.16571	0.0381228
$936$	−14.8192	−0.484382
$937$	−20.9242	−0.683564	−0.341782	−	0.939779i	$-0.611031\pi$
$937$	−20.9242	−0.683564	−0.341782	+	0.939779i	$0.611031\pi$
$938$	18.3897	0.600444
$939$	−58.5223	−1.90980
$940$	5.84714	0.190713
$941$	−36.1102	−1.17716	−0.588579	−	0.808440i	$-0.700312\pi$
$941$	−36.1102	−1.17716	−0.588579	+	0.808440i	$0.700312\pi$
$942$	11.4715	0.373760
$943$	103.886	3.38298
$944$	−31.7674	−1.03394
$945$	26.5103	0.862379
$946$	2.76598	0.0899298
$947$	−26.0630	−0.846935	−0.423467	−	0.905911i	$-0.639187\pi$
$947$	−26.0630	−0.846935	−0.423467	+	0.905911i	$0.639187\pi$
$948$	−59.6426	−1.93710
$949$	−1.34721	−0.0437323
$950$	−0.0206996	−0.000671584 0
$951$	−56.1220	−1.81988
$952$	5.59828	0.181441
$953$	−25.0079	−0.810084	−0.405042	−	0.914298i	$-0.632743\pi$
$953$	−25.0079	−0.810084	−0.405042	+	0.914298i	$0.632743\pi$
$954$	−38.7684	−1.25517
$955$	−15.6038	−0.504928
$956$	2.88694	0.0933702
$957$	6.88678	0.222618
$958$	−7.17708	−0.231881
$959$	−44.6682	−1.44241
$960$	9.75643	0.314888
$961$	20.9625	0.676209
$962$	0.956122	0.0308266
$963$	−59.0388	−1.90250
$964$	40.6226	1.30837
$965$	16.5649	0.533243
$966$	33.1020	1.06504
$967$	24.4540	0.786389	0.393195	−	0.919455i	$-0.371370\pi$
$967$	24.4540	0.786389	0.393195	+	0.919455i	$0.371370\pi$
$968$	−1.76932	−0.0568680
$969$	−0.156550	−0.00502911
$970$	−2.67117	−0.0857660
$971$	1.73551	0.0556951	0.0278475	−	0.999612i	$-0.491135\pi$
$971$	1.73551	0.0556951	0.0278475	+	0.999612i	$0.491135\pi$
$972$	−7.29692	−0.234049
$973$	−25.2834	−0.810548
$974$	−3.56477	−0.114223
$975$	4.09008	0.130987
$976$	17.8270	0.570628
$977$	2.23235	0.0714191	0.0357096	−	0.999362i	$-0.488631\pi$
$977$	2.23235	0.0714191	0.0357096	+	0.999362i	$0.488631\pi$
$978$	−12.2465	−0.391600
$979$	18.7534	0.599363
$980$	0.654415	0.0209045
$981$	115.729	3.69494
$982$	−5.86761	−0.187243
$983$	4.81493	0.153573	0.0767863	−	0.997048i	$-0.475534\pi$
$983$	4.81493	0.153573	0.0767863	+	0.997048i	$0.475534\pi$
$984$	65.0062	2.07232
$985$	9.30599	0.296514
$986$	−1.23739	−0.0394066
$987$	−27.0537	−0.861130
$988$	0.106138	0.00337670
$989$	50.7407	1.61346
$990$	2.90925	0.0924621
$991$	8.57526	0.272402	0.136201	−	0.990681i	$-0.456511\pi$
$991$	8.57526	0.272402	0.136201	+	0.990681i	$0.456511\pi$
$992$	34.7309	1.10271
$993$	−70.3198	−2.23153
$994$	9.26963	0.294015
$995$	−11.7744	−0.373274
$996$	8.50469	0.269481
$997$	12.5728	0.398186	0.199093	−	0.979981i	$-0.436200\pi$
$997$	12.5728	0.398186	0.199093	+	0.979981i	$0.436200\pi$
$998$	−8.81022	−0.278883
$999$	14.8128	0.468658

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.g.1.19	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.g.1.19	✓	32	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)

\(f(q)\)	\(=\)	\(q+0.467946 q^{2} +3.03596 q^{3} -1.78103 q^{4} -1.00000 q^{5} +1.42067 q^{6} -2.71430 q^{7} -1.76932 q^{8} +6.21706 q^{9} +O(q^{10})\)	\(q+0.467946 q^{2} +3.03596 q^{3} -1.78103 q^{4} -1.00000 q^{5} +1.42067 q^{6} -2.71430 q^{7} -1.76932 q^{8} +6.21706 q^{9} -0.467946 q^{10} -1.00000 q^{11} -5.40713 q^{12} +1.34721 q^{13} -1.27015 q^{14} -3.03596 q^{15} +2.73411 q^{16} +1.16571 q^{17} +2.90925 q^{18} -0.0442350 q^{19} +1.78103 q^{20} -8.24052 q^{21} -0.467946 q^{22} -8.58427 q^{23} -5.37158 q^{24} +1.00000 q^{25} +0.630422 q^{26} +9.76688 q^{27} +4.83424 q^{28} -2.26840 q^{29} -1.42067 q^{30} +7.20850 q^{31} +4.81805 q^{32} -3.03596 q^{33} +0.545490 q^{34} +2.71430 q^{35} -11.0728 q^{36} +1.51664 q^{37} -0.0206996 q^{38} +4.09008 q^{39} +1.76932 q^{40} -12.1019 q^{41} -3.85612 q^{42} -5.91089 q^{43} +1.78103 q^{44} -6.21706 q^{45} -4.01698 q^{46} +3.28302 q^{47} +8.30064 q^{48} +0.367437 q^{49} +0.467946 q^{50} +3.53905 q^{51} -2.39941 q^{52} -13.3259 q^{53} +4.57038 q^{54} +1.00000 q^{55} +4.80246 q^{56} -0.134296 q^{57} -1.06149 q^{58} -11.6189 q^{59} +5.40713 q^{60} +6.52023 q^{61} +3.37319 q^{62} -16.8750 q^{63} -3.21362 q^{64} -1.34721 q^{65} -1.42067 q^{66} -14.4784 q^{67} -2.07616 q^{68} -26.0615 q^{69} +1.27015 q^{70} -7.29807 q^{71} -11.0000 q^{72} -1.00000 q^{73} +0.709706 q^{74} +3.03596 q^{75} +0.0787836 q^{76} +2.71430 q^{77} +1.91394 q^{78} +11.0304 q^{79} -2.73411 q^{80} +11.0007 q^{81} -5.66303 q^{82} -1.57287 q^{83} +14.6766 q^{84} -1.16571 q^{85} -2.76598 q^{86} -6.88678 q^{87} +1.76932 q^{88} -18.7534 q^{89} -2.90925 q^{90} -3.65673 q^{91} +15.2888 q^{92} +21.8847 q^{93} +1.53628 q^{94} +0.0442350 q^{95} +14.6274 q^{96} +5.70827 q^{97} +0.171941 q^{98} -6.21706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9}+O(q^{10}) \) 32 * q - 5 * q^2 - 7 * q^3 + 37 * q^4 - 32 * q^5 - 3 * q^6 - 18 * q^8 + 29 * q^9	\( 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9} + 5 q^{10} - 32 q^{11} - 24 q^{12} - q^{13} - 5 q^{14} + 7 q^{15} + 47 q^{16} - 30 q^{17} - 11 q^{18} + 16 q^{19} - 37 q^{20} + q^{21} + 5 q^{22} - 26 q^{23} - 21 q^{24} + 32 q^{25} - q^{26} - 31 q^{27} - 24 q^{28} - 10 q^{29} + 3 q^{30} - 2 q^{31} - 31 q^{32} + 7 q^{33} - 14 q^{34} + 38 q^{36} - 28 q^{37} - 63 q^{38} - 2 q^{39} + 18 q^{40} - 62 q^{41} - 9 q^{42} + 8 q^{43} - 37 q^{44} - 29 q^{45} + 19 q^{46} - 21 q^{47} - 79 q^{48} + 34 q^{49} - 5 q^{50} + 17 q^{51} + 15 q^{52} - 32 q^{53} + 5 q^{54} + 32 q^{55} - 52 q^{56} - 57 q^{57} + 4 q^{58} - 37 q^{59} + 24 q^{60} + 15 q^{61} - 22 q^{62} + 5 q^{63} + 70 q^{64} + q^{65} + 3 q^{66} - 42 q^{67} - 81 q^{68} - 8 q^{69} + 5 q^{70} - 40 q^{71} - 27 q^{72} - 32 q^{73} - 17 q^{74} - 7 q^{75} + 21 q^{76} - 105 q^{78} + 18 q^{79} - 47 q^{80} + 12 q^{81} - 70 q^{82} - 26 q^{83} + 22 q^{84} + 30 q^{85} - 45 q^{86} - 18 q^{87} + 18 q^{88} - 83 q^{89} + 11 q^{90} - 18 q^{91} - 73 q^{92} - 68 q^{93} + 56 q^{94} - 16 q^{95} - 35 q^{96} - 99 q^{97} - 61 q^{98} - 29 q^{99}+O(q^{100}) \) 32 * q - 5 * q^2 - 7 * q^3 + 37 * q^4 - 32 * q^5 - 3 * q^6 - 18 * q^8 + 29 * q^9 + 5 * q^10 - 32 * q^11 - 24 * q^12 - q^13 - 5 * q^14 + 7 * q^15 + 47 * q^16 - 30 * q^17 - 11 * q^18 + 16 * q^19 - 37 * q^20 + q^21 + 5 * q^22 - 26 * q^23 - 21 * q^24 + 32 * q^25 - q^26 - 31 * q^27 - 24 * q^28 - 10 * q^29 + 3 * q^30 - 2 * q^31 - 31 * q^32 + 7 * q^33 - 14 * q^34 + 38 * q^36 - 28 * q^37 - 63 * q^38 - 2 * q^39 + 18 * q^40 - 62 * q^41 - 9 * q^42 + 8 * q^43 - 37 * q^44 - 29 * q^45 + 19 * q^46 - 21 * q^47 - 79 * q^48 + 34 * q^49 - 5 * q^50 + 17 * q^51 + 15 * q^52 - 32 * q^53 + 5 * q^54 + 32 * q^55 - 52 * q^56 - 57 * q^57 + 4 * q^58 - 37 * q^59 + 24 * q^60 + 15 * q^61 - 22 * q^62 + 5 * q^63 + 70 * q^64 + q^65 + 3 * q^66 - 42 * q^67 - 81 * q^68 - 8 * q^69 + 5 * q^70 - 40 * q^71 - 27 * q^72 - 32 * q^73 - 17 * q^74 - 7 * q^75 + 21 * q^76 - 105 * q^78 + 18 * q^79 - 47 * q^80 + 12 * q^81 - 70 * q^82 - 26 * q^83 + 22 * q^84 + 30 * q^85 - 45 * q^86 - 18 * q^87 + 18 * q^88 - 83 * q^89 + 11 * q^90 - 18 * q^91 - 73 * q^92 - 68 * q^93 + 56 * q^94 - 16 * q^95 - 35 * q^96 - 99 * q^97 - 61 * q^98 - 29 * q^99

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