LMFDB - Embedded newform 4009.2.a.c.1.46

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.46
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.839330 q^{2} +1.22529 q^{3} -1.29552 q^{4} +2.59839 q^{5} +1.02842 q^{6} +2.35945 q^{7} -2.76603 q^{8} -1.49867 q^{9} +O(q^{10})$	\(q+0.839330 q^{2} +1.22529 q^{3} -1.29552 q^{4} +2.59839 q^{5} +1.02842 q^{6} +2.35945 q^{7} -2.76603 q^{8} -1.49867 q^{9} +2.18091 q^{10} -2.43979 q^{11} -1.58739 q^{12} -6.49746 q^{13} +1.98035 q^{14} +3.18377 q^{15} +0.269435 q^{16} -3.77784 q^{17} -1.25788 q^{18} +1.00000 q^{19} -3.36628 q^{20} +2.89100 q^{21} -2.04779 q^{22} +0.232392 q^{23} -3.38918 q^{24} +1.75162 q^{25} -5.45352 q^{26} -5.51216 q^{27} -3.05672 q^{28} +4.72772 q^{29} +2.67223 q^{30} +2.74538 q^{31} +5.75821 q^{32} -2.98944 q^{33} -3.17085 q^{34} +6.13075 q^{35} +1.94157 q^{36} -6.70024 q^{37} +0.839330 q^{38} -7.96126 q^{39} -7.18723 q^{40} -0.748434 q^{41} +2.42650 q^{42} -6.31190 q^{43} +3.16081 q^{44} -3.89413 q^{45} +0.195054 q^{46} +0.547458 q^{47} +0.330135 q^{48} -1.43302 q^{49} +1.47019 q^{50} -4.62894 q^{51} +8.41763 q^{52} -8.40082 q^{53} -4.62653 q^{54} -6.33952 q^{55} -6.52631 q^{56} +1.22529 q^{57} +3.96812 q^{58} +1.15255 q^{59} -4.12465 q^{60} -13.6742 q^{61} +2.30428 q^{62} -3.53604 q^{63} +4.29417 q^{64} -16.8829 q^{65} -2.50913 q^{66} -5.34592 q^{67} +4.89428 q^{68} +0.284747 q^{69} +5.14573 q^{70} +10.5796 q^{71} +4.14538 q^{72} -8.51824 q^{73} -5.62371 q^{74} +2.14624 q^{75} -1.29552 q^{76} -5.75655 q^{77} -6.68212 q^{78} +9.20120 q^{79} +0.700096 q^{80} -2.25797 q^{81} -0.628184 q^{82} -0.396884 q^{83} -3.74536 q^{84} -9.81629 q^{85} -5.29777 q^{86} +5.79281 q^{87} +6.74853 q^{88} -11.9203 q^{89} -3.26846 q^{90} -15.3304 q^{91} -0.301070 q^{92} +3.36388 q^{93} +0.459498 q^{94} +2.59839 q^{95} +7.05546 q^{96} -1.12187 q^{97} -1.20277 q^{98} +3.65644 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	0.839330	0.593496	0.296748	−	0.954956i	$-0.404098\pi$
$2$	0.839330	0.593496	0.296748	+	0.954956i	$0.404098\pi$
$3$	1.22529	0.707420	0.353710	−	0.935355i	$-0.384920\pi$
$3$	1.22529	0.707420	0.353710	+	0.935355i	$0.384920\pi$
$4$	−1.29552	−0.647762
$5$	2.59839	1.16203	0.581017	−	0.813891i	$-0.302655\pi$
$5$	2.59839	1.16203	0.581017	+	0.813891i	$0.302655\pi$
$6$	1.02842	0.419851
$7$	2.35945	0.891787	0.445893	−	0.895086i	$-0.352886\pi$
$7$	2.35945	0.891787	0.445893	+	0.895086i	$0.352886\pi$
$8$	−2.76603	−0.977940
$9$	−1.49867	−0.499557
$10$	2.18091	0.689663
$11$	−2.43979	−0.735624	−0.367812	−	0.929900i	$-0.619893\pi$
$11$	−2.43979	−0.735624	−0.367812	+	0.929900i	$0.619893\pi$
$12$	−1.58739	−0.458240
$13$	−6.49746	−1.80207	−0.901036	−	0.433744i	$-0.857192\pi$
$13$	−6.49746	−1.80207	−0.901036	+	0.433744i	$0.857192\pi$
$14$	1.98035	0.529272
$15$	3.18377	0.822046
$16$	0.269435	0.0673587
$17$	−3.77784	−0.916261	−0.458130	−	0.888885i	$-0.651481\pi$
$17$	−3.77784	−0.916261	−0.458130	+	0.888885i	$0.651481\pi$
$18$	−1.25788	−0.296485
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.36628	−0.752722
$21$	2.89100	0.630867
$22$	−2.04779	−0.436590
$23$	0.232392	0.0484572	0.0242286	−	0.999706i	$-0.492287\pi$
$23$	0.232392	0.0484572	0.0242286	+	0.999706i	$0.492287\pi$
$24$	−3.38918	−0.691814
$25$	1.75162	0.350324
$26$	−5.45352	−1.06952
$27$	−5.51216	−1.06082
$28$	−3.05672	−0.577666
$29$	4.72772	0.877916	0.438958	−	0.898508i	$-0.355348\pi$
$29$	4.72772	0.877916	0.438958	+	0.898508i	$0.355348\pi$
$30$	2.67223	0.487881
$31$	2.74538	0.493085	0.246543	−	0.969132i	$-0.420705\pi$
$31$	2.74538	0.493085	0.246543	+	0.969132i	$0.420705\pi$
$32$	5.75821	1.01792
$33$	−2.98944	−0.520395
$34$	−3.17085	−0.543797
$35$	6.13075	1.03629
$36$	1.94157	0.323595
$37$	−6.70024	−1.10151	−0.550756	−	0.834666i	$-0.685661\pi$
$37$	−6.70024	−1.10151	−0.550756	+	0.834666i	$0.685661\pi$
$38$	0.839330	0.136157
$39$	−7.96126	−1.27482
$40$	−7.18723	−1.13640
$41$	−0.748434	−0.116886	−0.0584429	−	0.998291i	$-0.518614\pi$
$41$	−0.748434	−0.116886	−0.0584429	+	0.998291i	$0.518614\pi$
$42$	2.42650	0.374417
$43$	−6.31190	−0.962556	−0.481278	−	0.876568i	$-0.659827\pi$
$43$	−6.31190	−0.962556	−0.481278	+	0.876568i	$0.659827\pi$
$44$	3.16081	0.476509
$45$	−3.89413	−0.580503
$46$	0.195054	0.0287591
$47$	0.547458	0.0798550	0.0399275	−	0.999203i	$-0.487287\pi$
$47$	0.547458	0.0798550	0.0399275	+	0.999203i	$0.487287\pi$
$48$	0.330135	0.0476508
$49$	−1.43302	−0.204717
$50$	1.47019	0.207916
$51$	−4.62894	−0.648181
$52$	8.41763	1.16731
$53$	−8.40082	−1.15394	−0.576971	−	0.816765i	$-0.695765\pi$
$53$	−8.40082	−1.15394	−0.576971	+	0.816765i	$0.695765\pi$
$54$	−4.62653	−0.629590
$55$	−6.33952	−0.854820
$56$	−6.52631	−0.872114
$57$	1.22529	0.162293
$58$	3.96812	0.521040
$59$	1.15255	0.150049	0.0750246	−	0.997182i	$-0.476096\pi$
$59$	1.15255	0.150049	0.0750246	+	0.997182i	$0.476096\pi$
$60$	−4.12465	−0.532491
$61$	−13.6742	−1.75080	−0.875399	−	0.483401i	$-0.839401\pi$
$61$	−13.6742	−1.75080	−0.875399	+	0.483401i	$0.839401\pi$
$62$	2.30428	0.292644
$63$	−3.53604	−0.445499
$64$	4.29417	0.536771
$65$	−16.8829	−2.09407
$66$	−2.50913	−0.308852
$67$	−5.34592	−0.653108	−0.326554	−	0.945179i	$-0.605887\pi$
$67$	−5.34592	−0.653108	−0.326554	+	0.945179i	$0.605887\pi$
$68$	4.89428	0.593519
$69$	0.284747	0.0342796
$70$	5.14573	0.615032
$71$	10.5796	1.25557	0.627785	−	0.778386i	$-0.283962\pi$
$71$	10.5796	1.25557	0.627785	+	0.778386i	$0.283962\pi$
$72$	4.14538	0.488537
$73$	−8.51824	−0.996985	−0.498492	−	0.866894i	$-0.666113\pi$
$73$	−8.51824	−0.996985	−0.498492	+	0.866894i	$0.666113\pi$
$74$	−5.62371	−0.653743
$75$	2.14624	0.247826
$76$	−1.29552	−0.148607
$77$	−5.75655	−0.656019
$78$	−6.68212	−0.756602
$79$	9.20120	1.03522	0.517608	−	0.855618i	$-0.326823\pi$
$79$	9.20120	1.03522	0.517608	+	0.855618i	$0.326823\pi$
$80$	0.700096	0.0782731
$81$	−2.25797	−0.250885
$82$	−0.628184	−0.0693713
$83$	−0.396884	−0.0435637	−0.0217818	−	0.999763i	$-0.506934\pi$
$83$	−0.396884	−0.0435637	−0.0217818	+	0.999763i	$0.506934\pi$
$84$	−3.74536	−0.408652
$85$	−9.81629	−1.06473
$86$	−5.29777	−0.571273
$87$	5.79281	0.621055
$88$	6.74853	0.719396
$89$	−11.9203	−1.26355	−0.631774	−	0.775153i	$-0.717673\pi$
$89$	−11.9203	−1.26355	−0.631774	+	0.775153i	$0.717673\pi$
$90$	−3.26846	−0.344526
$91$	−15.3304	−1.60706
$92$	−0.301070	−0.0313887
$93$	3.36388	0.348818
$94$	0.459498	0.0473936
$95$	2.59839	0.266589
$96$	7.05546	0.720095
$97$	−1.12187	−0.113909	−0.0569545	−	0.998377i	$-0.518139\pi$
$97$	−1.12187	−0.113909	−0.0569545	+	0.998377i	$0.518139\pi$
$98$	−1.20277	−0.121499
$99$	3.65644	0.367486
$100$	−2.26927	−0.226927
$101$	6.95247	0.691797	0.345898	−	0.938272i	$-0.387574\pi$
$101$	6.95247	0.691797	0.345898	+	0.938272i	$0.387574\pi$
$102$	−3.88521	−0.384693
$103$	4.89589	0.482407	0.241203	−	0.970475i	$-0.422458\pi$
$103$	4.89589	0.482407	0.241203	+	0.970475i	$0.422458\pi$
$104$	17.9722	1.76232
$105$	7.51193	0.733090
$106$	−7.05106	−0.684860
$107$	−2.73564	−0.264464	−0.132232	−	0.991219i	$-0.542214\pi$
$107$	−2.73564	−0.264464	−0.132232	+	0.991219i	$0.542214\pi$
$108$	7.14115	0.687157
$109$	14.0103	1.34194	0.670970	−	0.741485i	$-0.265878\pi$
$109$	14.0103	1.34194	0.670970	+	0.741485i	$0.265878\pi$
$110$	−5.32095	−0.507332
$111$	−8.20972	−0.779232
$112$	0.635716	0.0600696
$113$	2.07692	0.195380	0.0976899	−	0.995217i	$-0.468855\pi$
$113$	2.07692	0.195380	0.0976899	+	0.995217i	$0.468855\pi$
$114$	1.02842	0.0963204
$115$	0.603846	0.0563089
$116$	−6.12488	−0.568681
$117$	9.73757	0.900239
$118$	0.967369	0.0890536
$119$	−8.91361	−0.817109
$120$	−8.80642	−0.803912
$121$	−5.04744	−0.458858
$122$	−11.4771	−1.03909
$123$	−0.917047	−0.0826873
$124$	−3.55671	−0.319402
$125$	−8.44055	−0.754946
$126$	−2.96790	−0.264402
$127$	19.3658	1.71843	0.859217	−	0.511612i	$-0.170951\pi$
$127$	19.3658	1.71843	0.859217	+	0.511612i	$0.170951\pi$
$128$	−7.91220	−0.699346
$129$	−7.73389	−0.680931
$130$	−14.1704	−1.24282
$131$	16.6314	1.45309	0.726544	−	0.687120i	$-0.241125\pi$
$131$	16.6314	1.45309	0.726544	+	0.687120i	$0.241125\pi$
$132$	3.87289	0.337092
$133$	2.35945	0.204590
$134$	−4.48699	−0.387617
$135$	−14.3227	−1.23271
$136$	10.4496	0.896048
$137$	−10.9865	−0.938641	−0.469321	−	0.883028i	$-0.655501\pi$
$137$	−10.9865	−0.938641	−0.469321	+	0.883028i	$0.655501\pi$
$138$	0.238997	0.0203448
$139$	−6.40496	−0.543262	−0.271631	−	0.962402i	$-0.587563\pi$
$139$	−6.40496	−0.543262	−0.271631	+	0.962402i	$0.587563\pi$
$140$	−7.94255	−0.671268
$141$	0.670793	0.0564910
$142$	8.87980	0.745176
$143$	15.8524	1.32565
$144$	−0.403794	−0.0336495
$145$	12.2845	1.02017
$146$	−7.14962	−0.591707
$147$	−1.75586	−0.144821
$148$	8.68033	0.713519
$149$	23.1246	1.89444	0.947220	−	0.320583i	$-0.103879\pi$
$149$	23.1246	1.89444	0.947220	+	0.320583i	$0.103879\pi$
$150$	1.80140	0.147084
$151$	−18.4125	−1.49839	−0.749194	−	0.662351i	$-0.769559\pi$
$151$	−18.4125	−1.49839	−0.749194	+	0.662351i	$0.769559\pi$
$152$	−2.76603	−0.224355
$153$	5.66174	0.457725
$154$	−4.83164	−0.389345
$155$	7.13357	0.572982
$156$	10.3140	0.825782
$157$	−3.04619	−0.243112	−0.121556	−	0.992585i	$-0.538788\pi$
$157$	−3.04619	−0.243112	−0.121556	+	0.992585i	$0.538788\pi$
$158$	7.72284	0.614396
$159$	−10.2934	−0.816321
$160$	14.9621	1.18286
$161$	0.548317	0.0432135
$162$	−1.89518	−0.148899
$163$	2.71109	0.212349	0.106175	−	0.994348i	$-0.466140\pi$
$163$	2.71109	0.212349	0.106175	+	0.994348i	$0.466140\pi$
$164$	0.969615	0.0757142
$165$	−7.76772	−0.604717
$166$	−0.333117	−0.0258549
$167$	3.29107	0.254671	0.127335	−	0.991860i	$-0.459358\pi$
$167$	3.29107	0.254671	0.127335	+	0.991860i	$0.459358\pi$
$168$	−7.99660	−0.616951
$169$	29.2170	2.24747
$170$	−8.23911	−0.631911
$171$	−1.49867	−0.114606
$172$	8.17722	0.623507
$173$	14.2999	1.08720	0.543602	−	0.839343i	$-0.317060\pi$
$173$	14.2999	1.08720	0.543602	+	0.839343i	$0.317060\pi$
$174$	4.86208	0.368594
$175$	4.13285	0.312414
$176$	−0.657363	−0.0495506
$177$	1.41220	0.106148
$178$	−10.0051	−0.749911
$179$	−14.6846	−1.09758	−0.548789	−	0.835961i	$-0.684911\pi$
$179$	−14.6846	−1.09758	−0.548789	+	0.835961i	$0.684911\pi$
$180$	5.04494	0.376028
$181$	−23.2505	−1.72820	−0.864098	−	0.503323i	$-0.832111\pi$
$181$	−23.2505	−1.72820	−0.864098	+	0.503323i	$0.832111\pi$
$182$	−12.8673	−0.953786
$183$	−16.7548	−1.23855
$184$	−0.642805	−0.0473882
$185$	−17.4098	−1.28000
$186$	2.82341	0.207022
$187$	9.21713	0.674023
$188$	−0.709246	−0.0517271
$189$	−13.0057	−0.946022
$190$	2.18091	0.158220
$191$	−17.8805	−1.29379	−0.646894	−	0.762580i	$-0.723932\pi$
$191$	−17.8805	−1.29379	−0.646894	+	0.762580i	$0.723932\pi$
$192$	5.26159	0.379723
$193$	−9.17057	−0.660112	−0.330056	−	0.943961i	$-0.607068\pi$
$193$	−9.17057	−0.660112	−0.330056	+	0.943961i	$0.607068\pi$
$194$	−0.941622	−0.0676045
$195$	−20.6864	−1.48139
$196$	1.85651	0.132608
$197$	−9.17425	−0.653638	−0.326819	−	0.945087i	$-0.605977\pi$
$197$	−9.17425	−0.653638	−0.326819	+	0.945087i	$0.605977\pi$
$198$	3.06896	0.218102
$199$	10.6905	0.757828	0.378914	−	0.925432i	$-0.376298\pi$
$199$	10.6905	0.757828	0.378914	+	0.925432i	$0.376298\pi$
$200$	−4.84504	−0.342596
$201$	−6.55028	−0.462021
$202$	5.83542	0.410578
$203$	11.1548	0.782914
$204$	5.99690	0.419867
$205$	−1.94472	−0.135825
$206$	4.10927	0.286306
$207$	−0.348280	−0.0242071
$208$	−1.75064	−0.121385
$209$	−2.43979	−0.168764
$210$	6.30499	0.435086
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	10.8835	0.747480
$213$	12.9631	0.888216
$214$	−2.29611	−0.156959
$215$	−16.4008	−1.11852
$216$	15.2468	1.03742
$217$	6.47758	0.439727
$218$	11.7592	0.796436
$219$	−10.4373	−0.705287
$220$	8.21300	0.553720
$221$	24.5464	1.65117
$222$	−6.89066	−0.462471
$223$	25.0279	1.67599	0.837996	−	0.545677i	$-0.183727\pi$
$223$	25.0279	1.67599	0.837996	+	0.545677i	$0.183727\pi$
$224$	13.5862	0.907765
$225$	−2.62510	−0.175007
$226$	1.74322	0.115957
$227$	12.4136	0.823917	0.411958	−	0.911203i	$-0.364845\pi$
$227$	12.4136	0.823917	0.411958	+	0.911203i	$0.364845\pi$
$228$	−1.58739	−0.105127
$229$	7.64770	0.505374	0.252687	−	0.967548i	$-0.418686\pi$
$229$	7.64770	0.505374	0.252687	+	0.967548i	$0.418686\pi$
$230$	0.506826	0.0334191
$231$	−7.05342	−0.464081
$232$	−13.0770	−0.858549
$233$	−2.58749	−0.169512	−0.0847560	−	0.996402i	$-0.527011\pi$
$233$	−2.58749	−0.169512	−0.0847560	+	0.996402i	$0.527011\pi$
$234$	8.17304	0.534288
$235$	1.42251	0.0927942
$236$	−1.49316	−0.0971962
$237$	11.2741	0.732332
$238$	−7.48146	−0.484951
$239$	−5.57185	−0.360413	−0.180206	−	0.983629i	$-0.557677\pi$
$239$	−5.57185	−0.360413	−0.180206	+	0.983629i	$0.557677\pi$
$240$	0.857818	0.0553719
$241$	−23.9561	−1.54315	−0.771574	−	0.636140i	$-0.780530\pi$
$241$	−23.9561	−1.54315	−0.771574	+	0.636140i	$0.780530\pi$
$242$	−4.23646	−0.272330
$243$	13.7698	0.883335
$244$	17.7152	1.13410
$245$	−3.72353	−0.237888
$246$	−0.769705	−0.0490746
$247$	−6.49746	−0.413424
$248$	−7.59382	−0.482208
$249$	−0.486297	−0.0308178
$250$	−7.08441	−0.448057
$251$	−9.57005	−0.604056	−0.302028	−	0.953299i	$-0.597664\pi$
$251$	−9.57005	−0.604056	−0.302028	+	0.953299i	$0.597664\pi$
$252$	4.58102	0.288577
$253$	−0.566988	−0.0356462
$254$	16.2543	1.01988
$255$	−12.0278	−0.753208
$256$	−15.2293	−0.951830
$257$	8.79485	0.548608	0.274304	−	0.961643i	$-0.411553\pi$
$257$	8.79485	0.548608	0.274304	+	0.961643i	$0.411553\pi$
$258$	−6.49129	−0.404130
$259$	−15.8089	−0.982314
$260$	21.8723	1.35646
$261$	−7.08530	−0.438569
$262$	13.9592	0.862402
$263$	−20.7003	−1.27644	−0.638218	−	0.769856i	$-0.720328\pi$
$263$	−20.7003	−1.27644	−0.638218	+	0.769856i	$0.720328\pi$
$264$	8.26889	0.508915
$265$	−21.8286	−1.34092
$266$	1.98035	0.121423
$267$	−14.6058	−0.893859
$268$	6.92577	0.423059
$269$	16.8431	1.02694	0.513472	−	0.858106i	$-0.328359\pi$
$269$	16.8431	1.02694	0.513472	+	0.858106i	$0.328359\pi$
$270$	−12.0215	−0.731606
$271$	9.60067	0.583199	0.291600	−	0.956540i	$-0.405813\pi$
$271$	9.60067	0.583199	0.291600	+	0.956540i	$0.405813\pi$
$272$	−1.01788	−0.0617181
$273$	−18.7842	−1.13687
$274$	−9.22131	−0.557080
$275$	−4.27358	−0.257707
$276$	−0.368897	−0.0222050
$277$	13.5708	0.815389	0.407695	−	0.913118i	$-0.366333\pi$
$277$	13.5708	0.815389	0.407695	+	0.913118i	$0.366333\pi$
$278$	−5.37588	−0.322424
$279$	−4.11443	−0.246324
$280$	−16.9579	−1.01343
$281$	−11.5780	−0.690688	−0.345344	−	0.938476i	$-0.612238\pi$
$281$	−11.5780	−0.690688	−0.345344	+	0.938476i	$0.612238\pi$
$282$	0.563017	0.0335272
$283$	16.2405	0.965400	0.482700	−	0.875786i	$-0.339656\pi$
$283$	16.2405	0.965400	0.482700	+	0.875786i	$0.339656\pi$
$284$	−13.7062	−0.813312
$285$	3.18377	0.188590
$286$	13.3054	0.786766
$287$	−1.76589	−0.104237
$288$	−8.62967	−0.508508
$289$	−2.72793	−0.160467
$290$	10.3107	0.605466
$291$	−1.37462	−0.0805815
$292$	11.0356	0.645809
$293$	−1.72993	−0.101064	−0.0505318	−	0.998722i	$-0.516092\pi$
$293$	−1.72993	−0.101064	−0.0505318	+	0.998722i	$0.516092\pi$
$294$	−1.47374	−0.0859504
$295$	2.99477	0.174362
$296$	18.5331	1.07721
$297$	13.4485	0.780362
$298$	19.4092	1.12434
$299$	−1.50996	−0.0873233
$300$	−2.78050	−0.160532
$301$	−14.8926	−0.858394
$302$	−15.4542	−0.889287
$303$	8.51877	0.489390
$304$	0.269435	0.0154531
$305$	−35.5308	−2.03449
$306$	4.75207	0.271658
$307$	20.9738	1.19704	0.598518	−	0.801109i	$-0.295757\pi$
$307$	20.9738	1.19704	0.598518	+	0.801109i	$0.295757\pi$
$308$	7.45775	0.424945
$309$	5.99887	0.341264
$310$	5.98742	0.340062
$311$	−11.9727	−0.678909	−0.339454	−	0.940623i	$-0.610242\pi$
$311$	−11.9727	−0.678909	−0.339454	+	0.940623i	$0.610242\pi$
$312$	22.0211	1.24670
$313$	13.2893	0.751158	0.375579	−	0.926790i	$-0.377444\pi$
$313$	13.2893	0.751158	0.375579	+	0.926790i	$0.377444\pi$
$314$	−2.55676	−0.144286
$315$	−9.18799	−0.517685
$316$	−11.9204	−0.670574
$317$	−30.8154	−1.73077	−0.865383	−	0.501111i	$-0.832925\pi$
$317$	−30.8154	−1.73077	−0.865383	+	0.501111i	$0.832925\pi$
$318$	−8.63957	−0.484483
$319$	−11.5346	−0.645816
$320$	11.1579	0.623747
$321$	−3.35194	−0.187087
$322$	0.460219	0.0256470
$323$	−3.77784	−0.210205
$324$	2.92525	0.162514
$325$	−11.3811	−0.631309
$326$	2.27550	0.126028
$327$	17.1666	0.949314
$328$	2.07019	0.114307
$329$	1.29170	0.0712136
$330$	−6.51969	−0.358897
$331$	−6.31204	−0.346941	−0.173470	−	0.984839i	$-0.555498\pi$
$331$	−6.31204	−0.346941	−0.173470	+	0.984839i	$0.555498\pi$
$332$	0.514173	0.0282189
$333$	10.0415	0.550269
$334$	2.76229	0.151146
$335$	−13.8908	−0.758934
$336$	0.778935	0.0424944
$337$	−28.5569	−1.55560	−0.777798	−	0.628515i	$-0.783663\pi$
$337$	−28.5569	−1.55560	−0.777798	+	0.628515i	$0.783663\pi$
$338$	24.5227	1.33386
$339$	2.54482	0.138216
$340$	12.7173	0.689690
$341$	−6.69815	−0.362725
$342$	−1.25788	−0.0680184
$343$	−19.8972	−1.07435
$344$	17.4589	0.941322
$345$	0.739884	0.0398340
$346$	12.0024	0.645251
$347$	12.8664	0.690707	0.345353	−	0.938473i	$-0.387759\pi$
$347$	12.8664	0.690707	0.345353	+	0.938473i	$0.387759\pi$
$348$	−7.50474	−0.402296
$349$	−25.5401	−1.36713	−0.683564	−	0.729891i	$-0.739571\pi$
$349$	−25.5401	−1.36713	−0.683564	+	0.729891i	$0.739571\pi$
$350$	3.46883	0.185417
$351$	35.8151	1.91167
$352$	−14.0488	−0.748804
$353$	14.7409	0.784579	0.392290	−	0.919842i	$-0.371683\pi$
$353$	14.7409	0.784579	0.392290	+	0.919842i	$0.371683\pi$
$354$	1.18530	0.0629982
$355$	27.4900	1.45902
$356$	15.4430	0.818479
$357$	−10.9217	−0.578039
$358$	−12.3252	−0.651408
$359$	−22.4193	−1.18324	−0.591622	−	0.806215i	$-0.701512\pi$
$359$	−22.4193	−1.18324	−0.591622	+	0.806215i	$0.701512\pi$
$360$	10.7713	0.567697
$361$	1.00000	0.0526316
$362$	−19.5149	−1.02568
$363$	−6.18456	−0.324605
$364$	19.8609	1.04100
$365$	−22.1337	−1.15853
$366$	−14.0628	−0.735074
$367$	−6.35728	−0.331847	−0.165924	−	0.986139i	$-0.553061\pi$
$367$	−6.35728	−0.331847	−0.165924	+	0.986139i	$0.553061\pi$
$368$	0.0626146	0.00326401
$369$	1.12166	0.0583912
$370$	−14.6126	−0.759672
$371$	−19.8213	−1.02907
$372$	−4.35799	−0.225951
$373$	37.3094	1.93181	0.965904	−	0.258900i	$-0.0833600\pi$
$373$	37.3094	1.93181	0.965904	+	0.258900i	$0.0833600\pi$
$374$	7.73621	0.400030
$375$	−10.3421	−0.534064
$376$	−1.51429	−0.0780934
$377$	−30.7182	−1.58207
$378$	−10.9160	−0.561460
$379$	−26.5062	−1.36153	−0.680766	−	0.732501i	$-0.738353\pi$
$379$	−26.5062	−1.36153	−0.680766	+	0.732501i	$0.738353\pi$
$380$	−3.36628	−0.172686
$381$	23.7286	1.21565
$382$	−15.0076	−0.767857
$383$	−4.48903	−0.229379	−0.114689	−	0.993401i	$-0.536587\pi$
$383$	−4.48903	−0.229379	−0.114689	+	0.993401i	$0.536587\pi$
$384$	−9.69471	−0.494731
$385$	−14.9577	−0.762317
$386$	−7.69714	−0.391774
$387$	9.45947	0.480852
$388$	1.45342	0.0737860
$389$	7.91387	0.401249	0.200625	−	0.979668i	$-0.435703\pi$
$389$	7.91387	0.401249	0.200625	+	0.979668i	$0.435703\pi$
$390$	−17.3628	−0.879197
$391$	−0.877941	−0.0443994
$392$	3.96377	0.200201
$393$	20.3782	1.02794
$394$	−7.70023	−0.387932
$395$	23.9083	1.20296
$396$	−4.73701	−0.238044
$397$	−10.2793	−0.515905	−0.257952	−	0.966158i	$-0.583048\pi$
$397$	−10.2793	−0.515905	−0.257952	+	0.966158i	$0.583048\pi$
$398$	8.97284	0.449768
$399$	2.89100	0.144731
$400$	0.471947	0.0235974
$401$	−5.01472	−0.250423	−0.125212	−	0.992130i	$-0.539961\pi$
$401$	−5.01472	−0.250423	−0.125212	+	0.992130i	$0.539961\pi$
$402$	−5.49785	−0.274208
$403$	−17.8380	−0.888575
$404$	−9.00710	−0.448120
$405$	−5.86707	−0.291537
$406$	9.36256	0.464656
$407$	16.3472	0.810299
$408$	12.8038	0.633882
$409$	−31.0284	−1.53426	−0.767129	−	0.641493i	$-0.778315\pi$
$409$	−31.0284	−1.53426	−0.767129	+	0.641493i	$0.778315\pi$
$410$	−1.63226	−0.0806118
$411$	−13.4616	−0.664013
$412$	−6.34275	−0.312485
$413$	2.71938	0.133812
$414$	−0.292322	−0.0143668
$415$	−1.03126	−0.0506225
$416$	−37.4138	−1.83436
$417$	−7.84791	−0.384314
$418$	−2.04779	−0.100161
$419$	−12.1190	−0.592054	−0.296027	−	0.955180i	$-0.595662\pi$
$419$	−12.1190	−0.592054	−0.296027	+	0.955180i	$0.595662\pi$
$420$	−9.73190	−0.474868
$421$	4.51497	0.220046	0.110023	−	0.993929i	$-0.464907\pi$
$421$	4.51497	0.220046	0.110023	+	0.993929i	$0.464907\pi$
$422$	0.839330	0.0408580
$423$	−0.820460	−0.0398922
$424$	23.2370	1.12849
$425$	−6.61734	−0.320988
$426$	10.8803	0.527152
$427$	−32.2635	−1.56134
$428$	3.54409	0.171310
$429$	19.4238	0.937789
$430$	−13.7657	−0.663839
$431$	32.3348	1.55751	0.778756	−	0.627326i	$-0.215851\pi$
$431$	32.3348	1.55751	0.778756	+	0.627326i	$0.215851\pi$
$432$	−1.48517	−0.0714552
$433$	−27.7317	−1.33270	−0.666351	−	0.745639i	$-0.732145\pi$
$433$	−27.7317	−1.33270	−0.666351	+	0.745639i	$0.732145\pi$
$434$	5.43683	0.260976
$435$	15.0520	0.721687
$436$	−18.1506	−0.869258
$437$	0.232392	0.0111168
$438$	−8.76033	−0.418585
$439$	−8.00868	−0.382233	−0.191117	−	0.981567i	$-0.561211\pi$
$439$	−8.00868	−0.382233	−0.191117	+	0.981567i	$0.561211\pi$
$440$	17.5353	0.835963
$441$	2.14762	0.102268
$442$	20.6025	0.979962
$443$	37.9010	1.80073	0.900365	−	0.435136i	$-0.143300\pi$
$443$	37.9010	1.80073	0.900365	+	0.435136i	$0.143300\pi$
$444$	10.6359	0.504757
$445$	−30.9735	−1.46829
$446$	21.0067	0.994694
$447$	28.3343	1.34016
$448$	10.1319	0.478686
$449$	29.5000	1.39219	0.696095	−	0.717950i	$-0.254919\pi$
$449$	29.5000	1.39219	0.696095	+	0.717950i	$0.254919\pi$
$450$	−2.20333	−0.103866
$451$	1.82602	0.0859840
$452$	−2.69070	−0.126560
$453$	−22.5606	−1.05999
$454$	10.4191	0.488991
$455$	−39.8344	−1.86746
$456$	−3.38918	−0.158713
$457$	22.8679	1.06972	0.534858	−	0.844942i	$-0.320365\pi$
$457$	22.8679	1.06972	0.534858	+	0.844942i	$0.320365\pi$
$458$	6.41895	0.299938
$459$	20.8241	0.971984
$460$	−0.782297	−0.0364748
$461$	−3.14734	−0.146586	−0.0732930	−	0.997310i	$-0.523351\pi$
$461$	−3.14734	−0.146586	−0.0732930	+	0.997310i	$0.523351\pi$
$462$	−5.92015	−0.275430
$463$	40.1992	1.86822	0.934108	−	0.356990i	$-0.116197\pi$
$463$	40.1992	1.86822	0.934108	+	0.356990i	$0.116197\pi$
$464$	1.27381	0.0591352
$465$	8.74067	0.405339
$466$	−2.17176	−0.100605
$467$	32.7221	1.51420	0.757099	−	0.653300i	$-0.226616\pi$
$467$	32.7221	1.51420	0.757099	+	0.653300i	$0.226616\pi$
$468$	−12.6153	−0.583141
$469$	−12.6134	−0.582433
$470$	1.19395	0.0550730
$471$	−3.73245	−0.171982
$472$	−3.18799	−0.146739
$473$	15.3997	0.708079
$474$	9.46270	0.434636
$475$	1.75162	0.0803698
$476$	11.5478	0.529292
$477$	12.5901	0.576460
$478$	−4.67662	−0.213904
$479$	27.2658	1.24581	0.622903	−	0.782299i	$-0.285953\pi$
$479$	27.2658	1.24581	0.622903	+	0.782299i	$0.285953\pi$
$480$	18.3328	0.836775
$481$	43.5346	1.98501
$482$	−20.1071	−0.915852
$483$	0.671846	0.0305700
$484$	6.53908	0.297231
$485$	−2.91506	−0.132366
$486$	11.5574	0.524256
$487$	32.5256	1.47388	0.736938	−	0.675961i	$-0.236271\pi$
$487$	32.5256	1.47388	0.736938	+	0.675961i	$0.236271\pi$
$488$	37.8232	1.71218
$489$	3.32187	0.150220
$490$	−3.12527	−0.141185
$491$	−34.7941	−1.57023	−0.785117	−	0.619347i	$-0.787397\pi$
$491$	−34.7941	−1.57023	−0.785117	+	0.619347i	$0.787397\pi$
$492$	1.18806	0.0535617
$493$	−17.8606	−0.804400
$494$	−5.45352	−0.245365
$495$	9.50086	0.427032
$496$	0.739701	0.0332136
$497$	24.9621	1.11970
$498$	−0.408163	−0.0182902
$499$	−26.8788	−1.20326	−0.601630	−	0.798775i	$-0.705482\pi$
$499$	−26.8788	−1.20326	−0.601630	+	0.798775i	$0.705482\pi$
$500$	10.9349	0.489026
$501$	4.03251	0.180159
$502$	−8.03243	−0.358505
$503$	38.5693	1.71972	0.859859	−	0.510531i	$-0.170551\pi$
$503$	38.5693	1.71972	0.859859	+	0.510531i	$0.170551\pi$
$504$	9.78079	0.435671
$505$	18.0652	0.803891
$506$	−0.475890	−0.0211559
$507$	35.7993	1.58990
$508$	−25.0888	−1.11314
$509$	−9.74643	−0.432003	−0.216002	−	0.976393i	$-0.569302\pi$
$509$	−9.74643	−0.432003	−0.216002	+	0.976393i	$0.569302\pi$
$510$	−10.0953	−0.447026
$511$	−20.0983	−0.889098
$512$	3.04199	0.134438
$513$	−5.51216	−0.243368
$514$	7.38178	0.325596
$515$	12.7214	0.560573
$516$	10.0194	0.441081
$517$	−1.33568	−0.0587432
$518$	−13.2688	−0.583000
$519$	17.5215	0.769110
$520$	46.6988	2.04788
$521$	−39.6139	−1.73552	−0.867758	−	0.496987i	$-0.834440\pi$
$521$	−39.6139	−1.73552	−0.867758	+	0.496987i	$0.834440\pi$
$522$	−5.94691	−0.260289
$523$	−11.7762	−0.514936	−0.257468	−	0.966287i	$-0.582888\pi$
$523$	−11.7762	−0.514936	−0.257468	+	0.966287i	$0.582888\pi$
$524$	−21.5463	−0.941256
$525$	5.06393	0.221008
$526$	−17.3744	−0.757559
$527$	−10.3716	−0.451794
$528$	−0.805459	−0.0350531
$529$	−22.9460	−0.997652
$530$	−18.3214	−0.795831
$531$	−1.72729	−0.0749581
$532$	−3.05672	−0.132526
$533$	4.86293	0.210637
$534$	−12.2591	−0.530502
$535$	−7.10826	−0.307317
$536$	14.7870	0.638701
$537$	−17.9928	−0.776448
$538$	14.1369	0.609487
$539$	3.49626	0.150594
$540$	18.5555	0.798500
$541$	17.6084	0.757043	0.378522	−	0.925592i	$-0.376433\pi$
$541$	17.6084	0.757043	0.378522	+	0.925592i	$0.376433\pi$
$542$	8.05813	0.346126
$543$	−28.4885	−1.22256
$544$	−21.7536	−0.932678
$545$	36.4041	1.55938
$546$	−15.7661	−0.674727
$547$	−10.5603	−0.451524	−0.225762	−	0.974183i	$-0.572487\pi$
$547$	−10.5603	−0.451524	−0.225762	+	0.974183i	$0.572487\pi$
$548$	14.2333	0.608016
$549$	20.4931	0.874624
$550$	−3.58695	−0.152948
$551$	4.72772	0.201408
$552$	−0.787621	−0.0335234
$553$	21.7097	0.923191
$554$	11.3904	0.483930
$555$	−21.3320	−0.905494
$556$	8.29778	0.351904
$557$	24.9068	1.05533	0.527667	−	0.849451i	$-0.323067\pi$
$557$	24.9068	1.05533	0.527667	+	0.849451i	$0.323067\pi$
$558$	−3.45336	−0.146193
$559$	41.0113	1.73460
$560$	1.65184	0.0698029
$561$	11.2936	0.476817
$562$	−9.71780	−0.409921
$563$	10.2380	0.431479	0.215740	−	0.976451i	$-0.430784\pi$
$563$	10.2380	0.431479	0.215740	+	0.976451i	$0.430784\pi$
$564$	−0.869029	−0.0365927
$565$	5.39664	0.227038
$566$	13.6312	0.572961
$567$	−5.32755	−0.223736
$568$	−29.2636	−1.22787
$569$	−18.7019	−0.784024	−0.392012	−	0.919960i	$-0.628221\pi$
$569$	−18.7019	−0.784024	−0.392012	+	0.919960i	$0.628221\pi$
$570$	2.67223	0.111928
$571$	29.1505	1.21991	0.609956	−	0.792435i	$-0.291187\pi$
$571$	29.1505	1.21991	0.609956	+	0.792435i	$0.291187\pi$
$572$	−20.5372	−0.858705
$573$	−21.9087	−0.915250
$574$	−1.48217	−0.0618644
$575$	0.407063	0.0169757
$576$	−6.43555	−0.268148
$577$	16.9969	0.707588	0.353794	−	0.935323i	$-0.384891\pi$
$577$	16.9969	0.707588	0.353794	+	0.935323i	$0.384891\pi$
$578$	−2.28963	−0.0952363
$579$	−11.2366	−0.466976
$580$	−15.9148	−0.660827
$581$	−0.936426	−0.0388495
$582$	−1.15376	−0.0478248
$583$	20.4962	0.848867
$584$	23.5617	0.974992
$585$	25.3020	1.04611
$586$	−1.45198	−0.0599809
$587$	15.7906	0.651749	0.325874	−	0.945413i	$-0.394341\pi$
$587$	15.7906	0.651749	0.325874	+	0.945413i	$0.394341\pi$
$588$	2.27476	0.0938093
$589$	2.74538	0.113121
$590$	2.51360	0.103483
$591$	−11.2411	−0.462397
$592$	−1.80528	−0.0741964
$593$	−17.0004	−0.698121	−0.349060	−	0.937100i	$-0.613499\pi$
$593$	−17.0004	−0.698121	−0.349060	+	0.937100i	$0.613499\pi$
$594$	11.2877	0.463142
$595$	−23.1610	−0.949509
$596$	−29.9585	−1.22715
$597$	13.0989	0.536103
$598$	−1.26736	−0.0518261
$599$	−11.6127	−0.474483	−0.237241	−	0.971451i	$-0.576243\pi$
$599$	−11.6127	−0.474483	−0.237241	+	0.971451i	$0.576243\pi$
$600$	−5.93656	−0.242359
$601$	−2.43112	−0.0991676	−0.0495838	−	0.998770i	$-0.515789\pi$
$601$	−2.43112	−0.0991676	−0.0495838	+	0.998770i	$0.515789\pi$
$602$	−12.4998	−0.509454
$603$	8.01178	0.326265
$604$	23.8539	0.970600
$605$	−13.1152	−0.533208
$606$	7.15006	0.290451
$607$	−12.9386	−0.525162	−0.262581	−	0.964910i	$-0.584574\pi$
$607$	−12.9386	−0.525162	−0.262581	+	0.964910i	$0.584574\pi$
$608$	5.75821	0.233526
$609$	13.6678	0.553848
$610$	−29.8221	−1.20746
$611$	−3.55709	−0.143904
$612$	−7.33493	−0.296497
$613$	−25.7474	−1.03993	−0.519963	−	0.854189i	$-0.674054\pi$
$613$	−25.7474	−1.03993	−0.519963	+	0.854189i	$0.674054\pi$
$614$	17.6039	0.710436
$615$	−2.38284	−0.0960855
$616$	15.9228	0.641548
$617$	3.76843	0.151711	0.0758557	−	0.997119i	$-0.475831\pi$
$617$	3.76843	0.151711	0.0758557	+	0.997119i	$0.475831\pi$
$618$	5.03504	0.202539
$619$	44.1164	1.77319	0.886594	−	0.462548i	$-0.153065\pi$
$619$	44.1164	1.77319	0.886594	+	0.462548i	$0.153065\pi$
$620$	−9.24171	−0.371156
$621$	−1.28099	−0.0514042
$622$	−10.0490	−0.402930
$623$	−28.1253	−1.12682
$624$	−2.14504	−0.0858703
$625$	−30.6899	−1.22760
$626$	11.1542	0.445809
$627$	−2.98944	−0.119387
$628$	3.94641	0.157479
$629$	25.3124	1.00927
$630$	−7.71176	−0.307244
$631$	−25.8970	−1.03094	−0.515471	−	0.856907i	$-0.672383\pi$
$631$	−25.8970	−1.03094	−0.515471	+	0.856907i	$0.672383\pi$
$632$	−25.4508	−1.01238
$633$	1.22529	0.0487008
$634$	−25.8643	−1.02720
$635$	50.3198	1.99688
$636$	13.3354	0.528782
$637$	9.31097	0.368914
$638$	−9.68137	−0.383289
$639$	−15.8554	−0.627230
$640$	−20.5590	−0.812664
$641$	−1.18358	−0.0467486	−0.0233743	−	0.999727i	$-0.507441\pi$
$641$	−1.18358	−0.0467486	−0.0233743	+	0.999727i	$0.507441\pi$
$642$	−2.81339	−0.111036
$643$	−2.82042	−0.111227	−0.0556133	−	0.998452i	$-0.517711\pi$
$643$	−2.82042	−0.111227	−0.0556133	+	0.998452i	$0.517711\pi$
$644$	−0.710359	−0.0279921
$645$	−20.0956	−0.791265
$646$	−3.17085	−0.124756
$647$	7.64689	0.300630	0.150315	−	0.988638i	$-0.451971\pi$
$647$	7.64689	0.300630	0.150315	+	0.988638i	$0.451971\pi$
$648$	6.24561	0.245351
$649$	−2.81198	−0.110380
$650$	−9.55249	−0.374680
$651$	7.93689	0.311071
$652$	−3.51229	−0.137552
$653$	48.0823	1.88161	0.940804	−	0.338952i	$-0.110073\pi$
$653$	48.0823	1.88161	0.940804	+	0.338952i	$0.110073\pi$
$654$	14.4084	0.563414
$655$	43.2147	1.68854
$656$	−0.201654	−0.00787327
$657$	12.7661	0.498051
$658$	1.08416	0.0422650
$659$	−8.85320	−0.344872	−0.172436	−	0.985021i	$-0.555164\pi$
$659$	−8.85320	−0.344872	−0.172436	+	0.985021i	$0.555164\pi$
$660$	10.0633	0.391713
$661$	32.4214	1.26104	0.630522	−	0.776171i	$-0.282841\pi$
$661$	32.4214	1.26104	0.630522	+	0.776171i	$0.282841\pi$
$662$	−5.29788	−0.205908
$663$	30.0764	1.16807
$664$	1.09779	0.0426027
$665$	6.13075	0.237740
$666$	8.42810	0.326582
$667$	1.09869	0.0425413
$668$	−4.26366	−0.164966
$669$	30.6663	1.18563
$670$	−11.6589	−0.450424
$671$	33.3621	1.28793
$672$	16.6470	0.642171
$673$	−36.0735	−1.39053	−0.695266	−	0.718752i	$-0.744714\pi$
$673$	−36.0735	−1.39053	−0.695266	+	0.718752i	$0.744714\pi$
$674$	−23.9687	−0.923240
$675$	−9.65522	−0.371629
$676$	−37.8514	−1.45582
$677$	−22.1935	−0.852966	−0.426483	−	0.904495i	$-0.640248\pi$
$677$	−22.1935	−0.852966	−0.426483	+	0.904495i	$0.640248\pi$
$678$	2.13594	0.0820304
$679$	−2.64700	−0.101583
$680$	27.1522	1.04124
$681$	15.2102	0.582855
$682$	−5.62196	−0.215276
$683$	−44.7194	−1.71114	−0.855571	−	0.517686i	$-0.826793\pi$
$683$	−44.7194	−1.71114	−0.855571	+	0.517686i	$0.826793\pi$
$684$	1.94157	0.0742377
$685$	−28.5472	−1.09073
$686$	−16.7004	−0.637623
$687$	9.37063	0.357512
$688$	−1.70064	−0.0648365
$689$	54.5840	2.07949
$690$	0.621007	0.0236413
$691$	−28.2485	−1.07462	−0.537312	−	0.843383i	$-0.680560\pi$
$691$	−28.2485	−1.07462	−0.537312	+	0.843383i	$0.680560\pi$
$692$	−18.5259	−0.704250
$693$	8.62718	0.327719
$694$	10.7992	0.409932
$695$	−16.6426	−0.631289
$696$	−16.0231	−0.607355
$697$	2.82747	0.107098
$698$	−21.4365	−0.811385
$699$	−3.17042	−0.119916
$700$	−5.35421	−0.202370
$701$	−7.81040	−0.294995	−0.147497	−	0.989062i	$-0.547122\pi$
$701$	−7.81040	−0.294995	−0.147497	+	0.989062i	$0.547122\pi$
$702$	30.0607	1.13457
$703$	−6.70024	−0.252704
$704$	−10.4769	−0.394862
$705$	1.74298	0.0656445
$706$	12.3725	0.465645
$707$	16.4040	0.616935
$708$	−1.82954	−0.0687585
$709$	10.3171	0.387466	0.193733	−	0.981054i	$-0.437940\pi$
$709$	10.3171	0.387466	0.193733	+	0.981054i	$0.437940\pi$
$710$	23.0732	0.865921
$711$	−13.7896	−0.517150
$712$	32.9719	1.23567
$713$	0.638006	0.0238935
$714$	−9.16693	−0.343064
$715$	41.1908	1.54045
$716$	19.0243	0.710970
$717$	−6.82711	−0.254963
$718$	−18.8172	−0.702251
$719$	−25.8620	−0.964490	−0.482245	−	0.876036i	$-0.660179\pi$
$719$	−25.8620	−0.964490	−0.482245	+	0.876036i	$0.660179\pi$
$720$	−1.04921	−0.0391019
$721$	11.5516	0.430204
$722$	0.839330	0.0312366
$723$	−29.3531	−1.09165
$724$	30.1216	1.11946
$725$	8.28117	0.307555
$726$	−5.19088	−0.192652
$727$	20.9068	0.775392	0.387696	−	0.921787i	$-0.373271\pi$
$727$	20.9068	0.775392	0.387696	+	0.921787i	$0.373271\pi$
$728$	42.4044	1.57161
$729$	23.6459	0.875774
$730$	−18.5775	−0.687583
$731$	23.8453	0.881952
$732$	21.7062	0.802286
$733$	22.1689	0.818826	0.409413	−	0.912349i	$-0.365734\pi$
$733$	22.1689	0.818826	0.409413	+	0.912349i	$0.365734\pi$
$734$	−5.33586	−0.196950
$735$	−4.56240	−0.168287
$736$	1.33816	0.0493254
$737$	13.0429	0.480442
$738$	0.941441	0.0346549
$739$	31.6864	1.16560	0.582802	−	0.812614i	$-0.301956\pi$
$739$	31.6864	1.16560	0.582802	+	0.812614i	$0.301956\pi$
$740$	22.5549	0.829133
$741$	−7.96126	−0.292464
$742$	−16.6366	−0.610749
$743$	−17.4426	−0.639908	−0.319954	−	0.947433i	$-0.603668\pi$
$743$	−17.4426	−0.639908	−0.319954	+	0.947433i	$0.603668\pi$
$744$	−9.30460	−0.341123
$745$	60.0867	2.20141
$746$	31.3149	1.14652
$747$	0.594799	0.0217626
$748$	−11.9410	−0.436607
$749$	−6.45459	−0.235846
$750$	−8.68043	−0.316965
$751$	−35.7196	−1.30343	−0.651713	−	0.758466i	$-0.725949\pi$
$751$	−35.7196	−1.30343	−0.651713	+	0.758466i	$0.725949\pi$
$752$	0.147504	0.00537893
$753$	−11.7261	−0.427321
$754$	−25.7827	−0.938951
$755$	−47.8428	−1.74118
$756$	16.8491	0.612797
$757$	10.6591	0.387411	0.193705	−	0.981060i	$-0.437949\pi$
$757$	10.6591	0.387411	0.193705	+	0.981060i	$0.437949\pi$
$758$	−22.2475	−0.808064
$759$	−0.694723	−0.0252169
$760$	−7.18723	−0.260708
$761$	−28.3257	−1.02681	−0.513404	−	0.858147i	$-0.671616\pi$
$761$	−28.3257	−1.02681	−0.513404	+	0.858147i	$0.671616\pi$
$762$	19.9161	0.721486
$763$	33.0564	1.19672
$764$	23.1646	0.838067
$765$	14.7114	0.531892
$766$	−3.76778	−0.136136
$767$	−7.48865	−0.270399
$768$	−18.6602	−0.673344
$769$	−47.2862	−1.70518	−0.852592	−	0.522578i	$-0.824971\pi$
$769$	−47.2862	−1.70518	−0.852592	+	0.522578i	$0.824971\pi$
$770$	−12.5545	−0.452432
$771$	10.7762	0.388096
$772$	11.8807	0.427596
$773$	35.4459	1.27490	0.637450	−	0.770492i	$-0.279989\pi$
$773$	35.4459	1.27490	0.637450	+	0.770492i	$0.279989\pi$
$774$	7.93962	0.285384
$775$	4.80887	0.172740
$776$	3.10314	0.111396
$777$	−19.3704	−0.694908
$778$	6.64235	0.238140
$779$	−0.748434	−0.0268154
$780$	26.7998	0.959587
$781$	−25.8121	−0.923628
$782$	−0.736882	−0.0263509
$783$	−26.0600	−0.931307
$784$	−0.386104	−0.0137894
$785$	−7.91517	−0.282505
$786$	17.1040	0.610080
$787$	−24.1191	−0.859752	−0.429876	−	0.902888i	$-0.641443\pi$
$787$	−24.1191	−0.859752	−0.429876	+	0.902888i	$0.641443\pi$
$788$	11.8855	0.423402
$789$	−25.3638	−0.902975
$790$	20.0669	0.713950
$791$	4.90037	0.174237
$792$	−10.1138	−0.359380
$793$	88.8474	3.15507
$794$	−8.62775	−0.306187
$795$	−26.7463	−0.948593
$796$	−13.8498	−0.490893
$797$	35.9808	1.27451	0.637253	−	0.770655i	$-0.280071\pi$
$797$	35.9808	1.27451	0.637253	+	0.770655i	$0.280071\pi$
$798$	2.42650	0.0858972
$799$	−2.06821	−0.0731680
$800$	10.0862	0.356601
$801$	17.8646	0.631215
$802$	−4.20901	−0.148625
$803$	20.7827	0.733406
$804$	8.48606	0.299280
$805$	1.42474	0.0502155
$806$	−14.9720	−0.527366
$807$	20.6377	0.726480
$808$	−19.2308	−0.676536
$809$	24.9976	0.878869	0.439435	−	0.898275i	$-0.355179\pi$
$809$	24.9976	0.878869	0.439435	+	0.898275i	$0.355179\pi$
$810$	−4.92441	−0.173026
$811$	−45.5950	−1.60106	−0.800529	−	0.599295i	$-0.795448\pi$
$811$	−45.5950	−1.60106	−0.800529	+	0.599295i	$0.795448\pi$
$812$	−14.4513	−0.507142
$813$	11.7636	0.412567
$814$	13.7207	0.480909
$815$	7.04447	0.246757
$816$	−1.24720	−0.0436606
$817$	−6.31190	−0.220825
$818$	−26.0431	−0.910576
$819$	22.9753	0.802821
$820$	2.51944	0.0879826
$821$	0.943805	0.0329390	0.0164695	−	0.999864i	$-0.494757\pi$
$821$	0.943805	0.0329390	0.0164695	+	0.999864i	$0.494757\pi$
$822$	−11.2988	−0.394089
$823$	8.85870	0.308795	0.154397	−	0.988009i	$-0.450656\pi$
$823$	8.85870	0.308795	0.154397	+	0.988009i	$0.450656\pi$
$824$	−13.5422	−0.471765
$825$	−5.23636	−0.182307
$826$	2.28245	0.0794168
$827$	−17.9070	−0.622688	−0.311344	−	0.950297i	$-0.600779\pi$
$827$	−17.9070	−0.622688	−0.311344	+	0.950297i	$0.600779\pi$
$828$	0.451205	0.0156805
$829$	36.8342	1.27930	0.639652	−	0.768665i	$-0.279078\pi$
$829$	36.8342	1.27930	0.639652	+	0.768665i	$0.279078\pi$
$830$	−0.865566	−0.0300442
$831$	16.6281	0.576822
$832$	−27.9012	−0.967301
$833$	5.41371	0.187574
$834$	−6.58699	−0.228089
$835$	8.55148	0.295936
$836$	3.16081	0.109319
$837$	−15.1330	−0.523073
$838$	−10.1719	−0.351382
$839$	−15.4204	−0.532371	−0.266186	−	0.963922i	$-0.585763\pi$
$839$	−15.4204	−0.532371	−0.266186	+	0.963922i	$0.585763\pi$
$840$	−20.7783	−0.716918
$841$	−6.64865	−0.229264
$842$	3.78955	0.130597
$843$	−14.1864	−0.488606
$844$	−1.29552	−0.0445938
$845$	75.9172	2.61163
$846$	−0.688637	−0.0236758
$847$	−11.9091	−0.409203
$848$	−2.26347	−0.0777280
$849$	19.8993	0.682943
$850$	−5.55413	−0.190505
$851$	−1.55709	−0.0533762
$852$	−16.7940	−0.575353
$853$	24.0916	0.824881	0.412440	−	0.910985i	$-0.364677\pi$
$853$	24.0916	0.824881	0.412440	+	0.910985i	$0.364677\pi$
$854$	−27.0797	−0.926648
$855$	−3.89413	−0.133176
$856$	7.56687	0.258630
$857$	−53.5018	−1.82759	−0.913793	−	0.406181i	$-0.866860\pi$
$857$	−53.5018	−1.82759	−0.913793	+	0.406181i	$0.866860\pi$
$858$	16.3030	0.556574
$859$	6.47416	0.220895	0.110448	−	0.993882i	$-0.464772\pi$
$859$	6.47416	0.220895	0.110448	+	0.993882i	$0.464772\pi$
$860$	21.2476	0.724537
$861$	−2.16372	−0.0737395
$862$	27.1396	0.924378
$863$	−7.59495	−0.258535	−0.129267	−	0.991610i	$-0.541263\pi$
$863$	−7.59495	−0.258535	−0.129267	+	0.991610i	$0.541263\pi$
$864$	−31.7402	−1.07982
$865$	37.1568	1.26337
$866$	−23.2761	−0.790953
$867$	−3.34250	−0.113517
$868$	−8.39186	−0.284838
$869$	−22.4490	−0.761529
$870$	12.6336	0.428318
$871$	34.7349	1.17695
$872$	−38.7528	−1.31234
$873$	1.68132	0.0569041
$874$	0.195054	0.00659780
$875$	−19.9150	−0.673251
$876$	13.5218	0.456858
$877$	−30.3749	−1.02569	−0.512844	−	0.858482i	$-0.671408\pi$
$877$	−30.3749	−1.02569	−0.512844	+	0.858482i	$0.671408\pi$
$878$	−6.72193	−0.226854
$879$	−2.11966	−0.0714944
$880$	−1.70809	−0.0575795
$881$	−2.84597	−0.0958830	−0.0479415	−	0.998850i	$-0.515266\pi$
$881$	−2.84597	−0.0958830	−0.0479415	+	0.998850i	$0.515266\pi$
$882$	1.80256	0.0606955
$883$	5.07217	0.170692	0.0853460	−	0.996351i	$-0.472800\pi$
$883$	5.07217	0.170692	0.0853460	+	0.996351i	$0.472800\pi$
$884$	−31.8004	−1.06956
$885$	3.66945	0.123347
$886$	31.8114	1.06873
$887$	−18.4626	−0.619914	−0.309957	−	0.950751i	$-0.600315\pi$
$887$	−18.4626	−0.619914	−0.309957	+	0.950751i	$0.600315\pi$
$888$	22.7083	0.762042
$889$	45.6925	1.53248
$890$	−25.9970	−0.871422
$891$	5.50896	0.184557
$892$	−32.4243	−1.08564
$893$	0.547458	0.0183200
$894$	23.7818	0.795382
$895$	−38.1563	−1.27542
$896$	−18.6684	−0.623667
$897$	−1.85014	−0.0617742
$898$	24.7602	0.826259
$899$	12.9794	0.432887
$900$	3.40089	0.113363
$901$	31.7369	1.05731
$902$	1.53263	0.0510312
$903$	−18.2477	−0.607245
$904$	−5.74482	−0.191070
$905$	−60.4138	−2.00822
$906$	−18.9358	−0.629099
$907$	−11.8762	−0.394342	−0.197171	−	0.980369i	$-0.563175\pi$
$907$	−11.8762	−0.394342	−0.197171	+	0.980369i	$0.563175\pi$
$908$	−16.0821	−0.533702
$909$	−10.4195	−0.345592
$910$	−33.4342	−1.10833
$911$	−32.7728	−1.08581	−0.542906	−	0.839794i	$-0.682676\pi$
$911$	−32.7728	−1.08581	−0.542906	+	0.839794i	$0.682676\pi$
$912$	0.330135	0.0109319
$913$	0.968313	0.0320465
$914$	19.1937	0.634872
$915$	−43.5354	−1.43924
$916$	−9.90779	−0.327363
$917$	39.2408	1.29584
$918$	17.4783	0.576869
$919$	21.7590	0.717764	0.358882	−	0.933383i	$-0.383158\pi$
$919$	21.7590	0.717764	0.358882	+	0.933383i	$0.383158\pi$
$920$	−1.67026	−0.0550668
$921$	25.6989	0.846807
$922$	−2.64165	−0.0869983
$923$	−68.7408	−2.26263
$924$	9.13788	0.300614
$925$	−11.7363	−0.385886
$926$	33.7404	1.10878
$927$	−7.33734	−0.240990
$928$	27.2232	0.893646
$929$	−37.7354	−1.23806	−0.619029	−	0.785368i	$-0.712474\pi$
$929$	−37.7354	−1.23806	−0.619029	+	0.785368i	$0.712474\pi$
$930$	7.33630	0.240567
$931$	−1.43302	−0.0469652
$932$	3.35216	0.109804
$933$	−14.6700	−0.480273
$934$	27.4646	0.898670
$935$	23.9497	0.783238
$936$	−26.9344	−0.880380
$937$	40.1178	1.31059	0.655296	−	0.755372i	$-0.272544\pi$
$937$	40.1178	1.31059	0.655296	+	0.755372i	$0.272544\pi$
$938$	−10.5868	−0.345672
$939$	16.2833	0.531384
$940$	−1.84290	−0.0601086
$941$	−4.53780	−0.147928	−0.0739641	−	0.997261i	$-0.523565\pi$
$941$	−4.53780	−0.147928	−0.0739641	+	0.997261i	$0.523565\pi$
$942$	−3.13276	−0.102071
$943$	−0.173930	−0.00566396
$944$	0.310537	0.0101071
$945$	−33.7937	−1.09931
$946$	12.9254	0.420242
$947$	0.885506	0.0287751	0.0143875	−	0.999896i	$-0.495420\pi$
$947$	0.885506	0.0287751	0.0143875	+	0.999896i	$0.495420\pi$
$948$	−14.6059	−0.474377
$949$	55.3470	1.79664
$950$	1.47019	0.0476992
$951$	−37.7577	−1.22438
$952$	24.6553	0.799084
$953$	−47.1689	−1.52795	−0.763975	−	0.645245i	$-0.776755\pi$
$953$	−47.1689	−1.52795	−0.763975	+	0.645245i	$0.776755\pi$
$954$	10.5672	0.342127
$955$	−46.4605	−1.50343
$956$	7.21847	0.233462
$957$	−14.1332	−0.456863
$958$	22.8850	0.739380
$959$	−25.9221	−0.837068
$960$	13.6717	0.441251
$961$	−23.4629	−0.756867
$962$	36.5399	1.17809
$963$	4.09983	0.132115
$964$	31.0357	0.999593
$965$	−23.8287	−0.767073
$966$	0.563901	0.0181432
$967$	−51.9337	−1.67008	−0.835038	−	0.550192i	$-0.814555\pi$
$967$	−51.9337	−1.67008	−0.835038	+	0.550192i	$0.814555\pi$
$968$	13.9614	0.448736
$969$	−4.62894	−0.148703
$970$	−2.44670	−0.0785588
$971$	49.2427	1.58027	0.790137	−	0.612930i	$-0.210009\pi$
$971$	49.2427	1.58027	0.790137	+	0.612930i	$0.210009\pi$
$972$	−17.8392	−0.572192
$973$	−15.1122	−0.484473
$974$	27.2997	0.874739
$975$	−13.9451	−0.446601
$976$	−3.68430	−0.117931
$977$	36.3458	1.16280	0.581402	−	0.813616i	$-0.302504\pi$
$977$	36.3458	1.16280	0.581402	+	0.813616i	$0.302504\pi$
$978$	2.78814	0.0891550
$979$	29.0830	0.929496
$980$	4.82393	0.154095
$981$	−20.9968	−0.670376
$982$	−29.2037	−0.931928
$983$	−3.01699	−0.0962270	−0.0481135	−	0.998842i	$-0.515321\pi$
$983$	−3.01699	−0.0962270	−0.0481135	+	0.998842i	$0.515321\pi$
$984$	2.53658	0.0808633
$985$	−23.8383	−0.759550
$986$	−14.9909	−0.477408
$987$	1.58270	0.0503779
$988$	8.41763	0.267800
$989$	−1.46684	−0.0466427
$990$	7.97435	0.253442
$991$	10.4593	0.332251	0.166125	−	0.986105i	$-0.446874\pi$
$991$	10.4593	0.332251	0.166125	+	0.986105i	$0.446874\pi$
$992$	15.8085	0.501920
$993$	−7.73406	−0.245433
$994$	20.9514	0.664538
$995$	27.7780	0.880622
$996$	0.630009	0.0199626
$997$	−32.9373	−1.04314	−0.521568	−	0.853210i	$-0.674653\pi$
$997$	−32.9373	−1.04314	−0.521568	+	0.853210i	$0.674653\pi$
$998$	−22.5602	−0.714130
$999$	36.9328	1.16850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.46	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.46	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q+0.839330 q^{2} +1.22529 q^{3} -1.29552 q^{4} +2.59839 q^{5} +1.02842 q^{6} +2.35945 q^{7} -2.76603 q^{8} -1.49867 q^{9} +O(q^{10})\)	\(q+0.839330 q^{2} +1.22529 q^{3} -1.29552 q^{4} +2.59839 q^{5} +1.02842 q^{6} +2.35945 q^{7} -2.76603 q^{8} -1.49867 q^{9} +2.18091 q^{10} -2.43979 q^{11} -1.58739 q^{12} -6.49746 q^{13} +1.98035 q^{14} +3.18377 q^{15} +0.269435 q^{16} -3.77784 q^{17} -1.25788 q^{18} +1.00000 q^{19} -3.36628 q^{20} +2.89100 q^{21} -2.04779 q^{22} +0.232392 q^{23} -3.38918 q^{24} +1.75162 q^{25} -5.45352 q^{26} -5.51216 q^{27} -3.05672 q^{28} +4.72772 q^{29} +2.67223 q^{30} +2.74538 q^{31} +5.75821 q^{32} -2.98944 q^{33} -3.17085 q^{34} +6.13075 q^{35} +1.94157 q^{36} -6.70024 q^{37} +0.839330 q^{38} -7.96126 q^{39} -7.18723 q^{40} -0.748434 q^{41} +2.42650 q^{42} -6.31190 q^{43} +3.16081 q^{44} -3.89413 q^{45} +0.195054 q^{46} +0.547458 q^{47} +0.330135 q^{48} -1.43302 q^{49} +1.47019 q^{50} -4.62894 q^{51} +8.41763 q^{52} -8.40082 q^{53} -4.62653 q^{54} -6.33952 q^{55} -6.52631 q^{56} +1.22529 q^{57} +3.96812 q^{58} +1.15255 q^{59} -4.12465 q^{60} -13.6742 q^{61} +2.30428 q^{62} -3.53604 q^{63} +4.29417 q^{64} -16.8829 q^{65} -2.50913 q^{66} -5.34592 q^{67} +4.89428 q^{68} +0.284747 q^{69} +5.14573 q^{70} +10.5796 q^{71} +4.14538 q^{72} -8.51824 q^{73} -5.62371 q^{74} +2.14624 q^{75} -1.29552 q^{76} -5.75655 q^{77} -6.68212 q^{78} +9.20120 q^{79} +0.700096 q^{80} -2.25797 q^{81} -0.628184 q^{82} -0.396884 q^{83} -3.74536 q^{84} -9.81629 q^{85} -5.29777 q^{86} +5.79281 q^{87} +6.74853 q^{88} -11.9203 q^{89} -3.26846 q^{90} -15.3304 q^{91} -0.301070 q^{92} +3.36388 q^{93} +0.459498 q^{94} +2.59839 q^{95} +7.05546 q^{96} -1.12187 q^{97} -1.20277 q^{98} +3.65644 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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