LMFDB - Embedded newform 4009.2.a.d.1.4

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:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.62786 q^{2} +2.23579 q^{3} +4.90563 q^{4} +3.65029 q^{5} -5.87534 q^{6} -3.99765 q^{7} -7.63559 q^{8} +1.99877 q^{9} +O(q^{10})$	\(q-2.62786 q^{2} +2.23579 q^{3} +4.90563 q^{4} +3.65029 q^{5} -5.87534 q^{6} -3.99765 q^{7} -7.63559 q^{8} +1.99877 q^{9} -9.59245 q^{10} -2.03134 q^{11} +10.9680 q^{12} +3.70185 q^{13} +10.5053 q^{14} +8.16130 q^{15} +10.2540 q^{16} -4.66978 q^{17} -5.25248 q^{18} -1.00000 q^{19} +17.9070 q^{20} -8.93792 q^{21} +5.33806 q^{22} -3.05084 q^{23} -17.0716 q^{24} +8.32463 q^{25} -9.72794 q^{26} -2.23854 q^{27} -19.6110 q^{28} -3.40761 q^{29} -21.4467 q^{30} -1.60717 q^{31} -11.6748 q^{32} -4.54165 q^{33} +12.2715 q^{34} -14.5926 q^{35} +9.80523 q^{36} -7.58254 q^{37} +2.62786 q^{38} +8.27658 q^{39} -27.8721 q^{40} -9.09659 q^{41} +23.4876 q^{42} -1.04627 q^{43} -9.96499 q^{44} +7.29610 q^{45} +8.01718 q^{46} +8.52041 q^{47} +22.9257 q^{48} +8.98120 q^{49} -21.8759 q^{50} -10.4407 q^{51} +18.1599 q^{52} +11.2237 q^{53} +5.88257 q^{54} -7.41497 q^{55} +30.5244 q^{56} -2.23579 q^{57} +8.95471 q^{58} -6.45628 q^{59} +40.0363 q^{60} +0.0337109 q^{61} +4.22341 q^{62} -7.99038 q^{63} +10.1717 q^{64} +13.5128 q^{65} +11.9348 q^{66} -6.13755 q^{67} -22.9082 q^{68} -6.82106 q^{69} +38.3472 q^{70} -12.3952 q^{71} -15.2618 q^{72} -4.44423 q^{73} +19.9258 q^{74} +18.6122 q^{75} -4.90563 q^{76} +8.12058 q^{77} -21.7497 q^{78} +10.6122 q^{79} +37.4300 q^{80} -11.0012 q^{81} +23.9045 q^{82} +6.58699 q^{83} -43.8461 q^{84} -17.0461 q^{85} +2.74946 q^{86} -7.61871 q^{87} +15.5105 q^{88} +4.42602 q^{89} -19.1731 q^{90} -14.7987 q^{91} -14.9663 q^{92} -3.59329 q^{93} -22.3904 q^{94} -3.65029 q^{95} -26.1024 q^{96} -11.8247 q^{97} -23.6013 q^{98} -4.06018 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.62786	−1.85818	−0.929088	−	0.369859i	$-0.879406\pi$
$2$	−2.62786	−1.85818	−0.929088	+	0.369859i	$0.879406\pi$
$3$	2.23579	1.29084	0.645418	−	0.763830i	$-0.276683\pi$
$3$	2.23579	1.29084	0.645418	+	0.763830i	$0.276683\pi$
$4$	4.90563	2.45282
$5$	3.65029	1.63246	0.816230	−	0.577727i	$-0.196060\pi$
$5$	3.65029	1.63246	0.816230	+	0.577727i	$0.196060\pi$
$6$	−5.87534	−2.39860
$7$	−3.99765	−1.51097	−0.755485	−	0.655166i	$-0.772599\pi$
$7$	−3.99765	−1.51097	−0.755485	+	0.655166i	$0.772599\pi$
$8$	−7.63559	−2.69959
$9$	1.99877	0.666257
$10$	−9.59245	−3.03340
$11$	−2.03134	−0.612471	−0.306236	−	0.951956i	$-0.599070\pi$
$11$	−2.03134	−0.612471	−0.306236	+	0.951956i	$0.599070\pi$
$12$	10.9680	3.16618
$13$	3.70185	1.02671	0.513355	−	0.858177i	$-0.328403\pi$
$13$	3.70185	1.02671	0.513355	+	0.858177i	$0.328403\pi$
$14$	10.5053	2.80765
$15$	8.16130	2.10724
$16$	10.2540	2.56349
$17$	−4.66978	−1.13259	−0.566294	−	0.824203i	$-0.691623\pi$
$17$	−4.66978	−1.13259	−0.566294	+	0.824203i	$0.691623\pi$
$18$	−5.25248	−1.23802
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	17.9070	4.00412
$21$	−8.93792	−1.95041
$22$	5.33806	1.13808
$23$	−3.05084	−0.636145	−0.318072	−	0.948066i	$-0.603036\pi$
$23$	−3.05084	−0.636145	−0.318072	+	0.948066i	$0.603036\pi$
$24$	−17.0716	−3.48472
$25$	8.32463	1.66493
$26$	−9.72794	−1.90781
$27$	−2.23854	−0.430808
$28$	−19.6110	−3.70613
$29$	−3.40761	−0.632777	−0.316389	−	0.948630i	$-0.602470\pi$
$29$	−3.40761	−0.632777	−0.316389	+	0.948630i	$0.602470\pi$
$30$	−21.4467	−3.91562
$31$	−1.60717	−0.288656	−0.144328	−	0.989530i	$-0.546102\pi$
$31$	−1.60717	−0.288656	−0.144328	+	0.989530i	$0.546102\pi$
$32$	−11.6748	−2.06383
$33$	−4.54165	−0.790600
$34$	12.2715	2.10455
$35$	−14.5926	−2.46660
$36$	9.80523	1.63421
$37$	−7.58254	−1.24656	−0.623281	−	0.781998i	$-0.714201\pi$
$37$	−7.58254	−1.24656	−0.623281	+	0.781998i	$0.714201\pi$
$38$	2.62786	0.426295
$39$	8.27658	1.32531
$40$	−27.8721	−4.40697
$41$	−9.09659	−1.42065	−0.710324	−	0.703875i	$-0.751452\pi$
$41$	−9.09659	−1.42065	−0.710324	+	0.703875i	$0.751452\pi$
$42$	23.4876	3.62421
$43$	−1.04627	−0.159555	−0.0797777	−	0.996813i	$-0.525421\pi$
$43$	−1.04627	−0.159555	−0.0797777	+	0.996813i	$0.525421\pi$
$44$	−9.96499	−1.50228
$45$	7.29610	1.08764
$46$	8.01718	1.18207
$47$	8.52041	1.24283	0.621415	−	0.783482i	$-0.286558\pi$
$47$	8.52041	1.24283	0.621415	+	0.783482i	$0.286558\pi$
$48$	22.9257	3.30905
$49$	8.98120	1.28303
$50$	−21.8759	−3.09373
$51$	−10.4407	−1.46199
$52$	18.1599	2.51833
$53$	11.2237	1.54169	0.770845	−	0.637023i	$-0.219834\pi$
$53$	11.2237	1.54169	0.770845	+	0.637023i	$0.219834\pi$
$54$	5.88257	0.800516
$55$	−7.41497	−0.999835
$56$	30.5244	4.07899
$57$	−2.23579	−0.296138
$58$	8.95471	1.17581
$59$	−6.45628	−0.840536	−0.420268	−	0.907400i	$-0.638064\pi$
$59$	−6.45628	−0.840536	−0.420268	+	0.907400i	$0.638064\pi$
$60$	40.0363	5.16867
$61$	0.0337109	0.00431624	0.00215812	−	0.999998i	$-0.499313\pi$
$61$	0.0337109	0.00431624	0.00215812	+	0.999998i	$0.499313\pi$
$62$	4.22341	0.536373
$63$	−7.99038	−1.00669
$64$	10.1717	1.27147
$65$	13.5128	1.67606
$66$	11.9348	1.46907
$67$	−6.13755	−0.749821	−0.374911	−	0.927061i	$-0.622327\pi$
$67$	−6.13755	−0.749821	−0.374911	+	0.927061i	$0.622327\pi$
$68$	−22.9082	−2.77803
$69$	−6.82106	−0.821159
$70$	38.3472	4.58337
$71$	−12.3952	−1.47104	−0.735521	−	0.677501i	$-0.763063\pi$
$71$	−12.3952	−1.47104	−0.735521	+	0.677501i	$0.763063\pi$
$72$	−15.2618	−1.79862
$73$	−4.44423	−0.520158	−0.260079	−	0.965587i	$-0.583749\pi$
$73$	−4.44423	−0.520158	−0.260079	+	0.965587i	$0.583749\pi$
$74$	19.9258	2.31633
$75$	18.6122	2.14915
$76$	−4.90563	−0.562715
$77$	8.12058	0.925426
$78$	−21.7497	−2.46266
$79$	10.6122	1.19396	0.596981	−	0.802255i	$-0.296367\pi$
$79$	10.6122	1.19396	0.596981	+	0.802255i	$0.296367\pi$
$80$	37.4300	4.18480
$81$	−11.0012	−1.22236
$82$	23.9045	2.63981
$83$	6.58699	0.723016	0.361508	−	0.932369i	$-0.382262\pi$
$83$	6.58699	0.723016	0.361508	+	0.932369i	$0.382262\pi$
$84$	−43.8461	−4.78401
$85$	−17.0461	−1.84891
$86$	2.74946	0.296482
$87$	−7.61871	−0.816812
$88$	15.5105	1.65342
$89$	4.42602	0.469157	0.234579	−	0.972097i	$-0.424629\pi$
$89$	4.42602	0.469157	0.234579	+	0.972097i	$0.424629\pi$
$90$	−19.1731	−2.02102
$91$	−14.7987	−1.55133
$92$	−14.9663	−1.56035
$93$	−3.59329	−0.372607
$94$	−22.3904	−2.30940
$95$	−3.65029	−0.374512
$96$	−26.1024	−2.66406
$97$	−11.8247	−1.20062	−0.600309	−	0.799768i	$-0.704956\pi$
$97$	−11.8247	−1.20062	−0.600309	+	0.799768i	$0.704956\pi$
$98$	−23.6013	−2.38409
$99$	−4.06018	−0.408063
$100$	40.8376	4.08376
$101$	−6.96337	−0.692881	−0.346441	−	0.938072i	$-0.612610\pi$
$101$	−6.96337	−0.692881	−0.346441	+	0.938072i	$0.612610\pi$
$102$	27.4366	2.71662
$103$	−2.13583	−0.210449	−0.105225	−	0.994448i	$-0.533556\pi$
$103$	−2.13583	−0.210449	−0.105225	+	0.994448i	$0.533556\pi$
$104$	−28.2658	−2.77169
$105$	−32.6260	−3.18397
$106$	−29.4942	−2.86473
$107$	−13.2667	−1.28254	−0.641270	−	0.767315i	$-0.721592\pi$
$107$	−13.2667	−1.28254	−0.641270	+	0.767315i	$0.721592\pi$
$108$	−10.9815	−1.05669
$109$	8.78254	0.841215	0.420608	−	0.907243i	$-0.361817\pi$
$109$	8.78254	0.841215	0.420608	+	0.907243i	$0.361817\pi$
$110$	19.4855	1.85787
$111$	−16.9530	−1.60911
$112$	−40.9918	−3.87336
$113$	18.1811	1.71033	0.855167	−	0.518353i	$-0.173455\pi$
$113$	18.1811	1.71033	0.855167	+	0.518353i	$0.173455\pi$
$114$	5.87534	0.550276
$115$	−11.1365	−1.03848
$116$	−16.7165	−1.55209
$117$	7.39915	0.684052
$118$	16.9662	1.56186
$119$	18.6681	1.71131
$120$	−62.3163	−5.68867
$121$	−6.87367	−0.624879
$122$	−0.0885875	−0.00802033
$123$	−20.3381	−1.83382
$124$	−7.88417	−0.708020
$125$	12.1359	1.08547
$126$	20.9976	1.87061
$127$	9.48055	0.841263	0.420631	−	0.907232i	$-0.361809\pi$
$127$	9.48055	0.841263	0.420631	+	0.907232i	$0.361809\pi$
$128$	−3.38027	−0.298777
$129$	−2.33925	−0.205960
$130$	−35.5098	−3.11442
$131$	−4.68477	−0.409310	−0.204655	−	0.978834i	$-0.565607\pi$
$131$	−4.68477	−0.409310	−0.204655	+	0.978834i	$0.565607\pi$
$132$	−22.2797	−1.93920
$133$	3.99765	0.346640
$134$	16.1286	1.39330
$135$	−8.17133	−0.703276
$136$	35.6565	3.05752
$137$	−16.2135	−1.38521	−0.692607	−	0.721315i	$-0.743538\pi$
$137$	−16.2135	−1.38521	−0.692607	+	0.721315i	$0.743538\pi$
$138$	17.9248	1.52586
$139$	−19.2035	−1.62882	−0.814411	−	0.580289i	$-0.802940\pi$
$139$	−19.2035	−1.62882	−0.814411	+	0.580289i	$0.802940\pi$
$140$	−71.5859	−6.05011
$141$	19.0499	1.60429
$142$	32.5729	2.73346
$143$	−7.51971	−0.628830
$144$	20.4953	1.70794
$145$	−12.4388	−1.03298
$146$	11.6788	0.966545
$147$	20.0801	1.65618
$148$	−37.1972	−3.05759
$149$	−18.4668	−1.51286	−0.756430	−	0.654075i	$-0.773058\pi$
$149$	−18.4668	−1.51286	−0.756430	+	0.654075i	$0.773058\pi$
$150$	−48.9101	−3.99349
$151$	−7.81395	−0.635890	−0.317945	−	0.948109i	$-0.602993\pi$
$151$	−7.81395	−0.635890	−0.317945	+	0.948109i	$0.602993\pi$
$152$	7.63559	0.619328
$153$	−9.33382	−0.754594
$154$	−21.3397	−1.71960
$155$	−5.86663	−0.471219
$156$	40.6018	3.25075
$157$	9.16296	0.731284	0.365642	−	0.930756i	$-0.380849\pi$
$157$	9.16296	0.731284	0.365642	+	0.930756i	$0.380849\pi$
$158$	−27.8873	−2.21859
$159$	25.0938	1.99007
$160$	−42.6164	−3.36912
$161$	12.1962	0.961196
$162$	28.9097	2.27136
$163$	7.37545	0.577690	0.288845	−	0.957376i	$-0.406729\pi$
$163$	7.37545	0.577690	0.288845	+	0.957376i	$0.406729\pi$
$164$	−44.6245	−3.48459
$165$	−16.5783	−1.29062
$166$	−17.3097	−1.34349
$167$	−5.81601	−0.450056	−0.225028	−	0.974352i	$-0.572247\pi$
$167$	−5.81601	−0.450056	−0.225028	+	0.974352i	$0.572247\pi$
$168$	68.2462	5.26531
$169$	0.703718	0.0541321
$170$	44.7946	3.43559
$171$	−1.99877	−0.152850
$172$	−5.13264	−0.391360
$173$	−9.77772	−0.743387	−0.371693	−	0.928356i	$-0.621223\pi$
$173$	−9.77772	−0.743387	−0.371693	+	0.928356i	$0.621223\pi$
$174$	20.0209	1.51778
$175$	−33.2790	−2.51565
$176$	−20.8293	−1.57006
$177$	−14.4349	−1.08499
$178$	−11.6310	−0.871777
$179$	14.5633	1.08851	0.544255	−	0.838920i	$-0.316813\pi$
$179$	14.5633	1.08851	0.544255	+	0.838920i	$0.316813\pi$
$180$	35.7920	2.66778
$181$	7.77597	0.577984	0.288992	−	0.957332i	$-0.406680\pi$
$181$	7.77597	0.577984	0.288992	+	0.957332i	$0.406680\pi$
$182$	38.8889	2.88264
$183$	0.0753706	0.00557156
$184$	23.2950	1.71733
$185$	−27.6785	−2.03496
$186$	9.44266	0.692370
$187$	9.48590	0.693678
$188$	41.7980	3.04843
$189$	8.94891	0.650937
$190$	9.59245	0.695909
$191$	−23.1501	−1.67508	−0.837540	−	0.546376i	$-0.816007\pi$
$191$	−23.1501	−1.67508	−0.837540	+	0.546376i	$0.816007\pi$
$192$	22.7419	1.64125
$193$	15.1857	1.09309	0.546546	−	0.837429i	$-0.315942\pi$
$193$	15.1857	1.09309	0.546546	+	0.837429i	$0.315942\pi$
$194$	31.0737	2.23096
$195$	30.2119	2.16352
$196$	44.0585	3.14703
$197$	19.9193	1.41919	0.709596	−	0.704609i	$-0.248878\pi$
$197$	19.9193	1.41919	0.709596	+	0.704609i	$0.248878\pi$
$198$	10.6696	0.758253
$199$	8.90208	0.631052	0.315526	−	0.948917i	$-0.397819\pi$
$199$	8.90208	0.631052	0.315526	+	0.948917i	$0.397819\pi$
$200$	−63.5634	−4.49461
$201$	−13.7223	−0.967896
$202$	18.2987	1.28750
$203$	13.6224	0.956108
$204$	−51.2181	−3.58598
$205$	−33.2052	−2.31915
$206$	5.61265	0.391052
$207$	−6.09794	−0.423836
$208$	37.9587	2.63196
$209$	2.03134	0.140511
$210$	85.7365	5.91638
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	55.0592	3.78148
$213$	−27.7132	−1.89887
$214$	34.8630	2.38319
$215$	−3.81921	−0.260468
$216$	17.0926	1.16300
$217$	6.42489	0.436150
$218$	−23.0793	−1.56313
$219$	−9.93639	−0.671439
$220$	−36.3751	−2.45241
$221$	−17.2868	−1.16284
$222$	44.5500	2.99000
$223$	13.2471	0.887095	0.443547	−	0.896251i	$-0.353720\pi$
$223$	13.2471	0.887095	0.443547	+	0.896251i	$0.353720\pi$
$224$	46.6717	3.11838
$225$	16.6390	1.10927
$226$	−47.7773	−3.17810
$227$	−29.8899	−1.98386	−0.991930	−	0.126788i	$-0.959533\pi$
$227$	−29.8899	−1.98386	−0.991930	+	0.126788i	$0.959533\pi$
$228$	−10.9680	−0.726372
$229$	10.0808	0.666155	0.333077	−	0.942899i	$-0.391913\pi$
$229$	10.0808	0.666155	0.333077	+	0.942899i	$0.391913\pi$
$230$	29.2651	1.92968
$231$	18.1559	1.19457
$232$	26.0191	1.70824
$233$	−5.42515	−0.355413	−0.177707	−	0.984084i	$-0.556868\pi$
$233$	−5.42515	−0.355413	−0.177707	+	0.984084i	$0.556868\pi$
$234$	−19.4439	−1.27109
$235$	31.1020	2.02887
$236$	−31.6721	−2.06168
$237$	23.7266	1.54121
$238$	−49.0572	−3.17991
$239$	−13.2763	−0.858776	−0.429388	−	0.903120i	$-0.641271\pi$
$239$	−13.2763	−0.858776	−0.429388	+	0.903120i	$0.641271\pi$
$240$	83.6857	5.40189
$241$	3.84587	0.247734	0.123867	−	0.992299i	$-0.460470\pi$
$241$	3.84587	0.247734	0.123867	+	0.992299i	$0.460470\pi$
$242$	18.0630	1.16113
$243$	−17.8808	−1.14706
$244$	0.165373	0.0105869
$245$	32.7840	2.09449
$246$	53.4456	3.40757
$247$	−3.70185	−0.235543
$248$	12.2717	0.779251
$249$	14.7271	0.933295
$250$	−31.8913	−2.01699
$251$	13.1690	0.831223	0.415611	−	0.909542i	$-0.363568\pi$
$251$	13.1690	0.831223	0.415611	+	0.909542i	$0.363568\pi$
$252$	−39.1979	−2.46923
$253$	6.19729	0.389621
$254$	−24.9135	−1.56321
$255$	−38.1115	−2.38663
$256$	−11.4606	−0.716286
$257$	20.6537	1.28834	0.644172	−	0.764880i	$-0.277202\pi$
$257$	20.6537	1.28834	0.644172	+	0.764880i	$0.277202\pi$
$258$	6.14722	0.382709
$259$	30.3123	1.88352
$260$	66.2891	4.11107
$261$	−6.81103	−0.421592
$262$	12.3109	0.760571
$263$	18.2811	1.12726	0.563630	−	0.826027i	$-0.309404\pi$
$263$	18.2811	1.12726	0.563630	+	0.826027i	$0.309404\pi$
$264$	34.6782	2.13429
$265$	40.9697	2.51675
$266$	−10.5053	−0.644118
$267$	9.89567	0.605605
$268$	−30.1086	−1.83917
$269$	−9.71052	−0.592061	−0.296030	−	0.955179i	$-0.595663\pi$
$269$	−9.71052	−0.592061	−0.296030	+	0.955179i	$0.595663\pi$
$270$	21.4731	1.30681
$271$	−6.47350	−0.393237	−0.196619	−	0.980480i	$-0.562996\pi$
$271$	−6.47350	−0.393237	−0.196619	+	0.980480i	$0.562996\pi$
$272$	−47.8838	−2.90338
$273$	−33.0869	−2.00251
$274$	42.6068	2.57397
$275$	−16.9101	−1.01972
$276$	−33.4616	−2.01415
$277$	24.0761	1.44659	0.723296	−	0.690538i	$-0.242626\pi$
$277$	24.0761	1.44659	0.723296	+	0.690538i	$0.242626\pi$
$278$	50.4641	3.02664
$279$	−3.21236	−0.192319
$280$	111.423	6.65880
$281$	−31.5672	−1.88314	−0.941571	−	0.336814i	$-0.890651\pi$
$281$	−31.5672	−1.88314	−0.941571	+	0.336814i	$0.890651\pi$
$282$	−50.0603	−2.98105
$283$	−1.80518	−0.107307	−0.0536534	−	0.998560i	$-0.517087\pi$
$283$	−1.80518	−0.107307	−0.0536534	+	0.998560i	$0.517087\pi$
$284$	−60.8064	−3.60820
$285$	−8.16130	−0.483434
$286$	19.7607	1.16848
$287$	36.3650	2.14656
$288$	−23.3352	−1.37504
$289$	4.80685	0.282756
$290$	32.6873	1.91947
$291$	−26.4376	−1.54980
$292$	−21.8018	−1.27585
$293$	−0.296765	−0.0173372	−0.00866861	−	0.999962i	$-0.502759\pi$
$293$	−0.296765	−0.0173372	−0.00866861	+	0.999962i	$0.502759\pi$
$294$	−52.7677	−3.07747
$295$	−23.5673	−1.37214
$296$	57.8972	3.36520
$297$	4.54723	0.263857
$298$	48.5281	2.81116
$299$	−11.2938	−0.653136
$300$	91.3044	5.27146
$301$	4.18264	0.241083
$302$	20.5339	1.18160
$303$	−15.5687	−0.894396
$304$	−10.2540	−0.588105
$305$	0.123055	0.00704609
$306$	24.5279	1.40217
$307$	−8.03837	−0.458774	−0.229387	−	0.973335i	$-0.573672\pi$
$307$	−8.03837	−0.458774	−0.229387	+	0.973335i	$0.573672\pi$
$308$	39.8366	2.26990
$309$	−4.77527	−0.271656
$310$	15.4167	0.875608
$311$	−27.8479	−1.57911	−0.789554	−	0.613682i	$-0.789688\pi$
$311$	−27.8479	−1.57911	−0.789554	+	0.613682i	$0.789688\pi$
$312$	−63.1965	−3.57780
$313$	−32.7330	−1.85018	−0.925091	−	0.379747i	$-0.876011\pi$
$313$	−32.7330	−1.85018	−0.925091	+	0.379747i	$0.876011\pi$
$314$	−24.0790	−1.35885
$315$	−29.1672	−1.64339
$316$	52.0594	2.92857
$317$	−12.7329	−0.715150	−0.357575	−	0.933884i	$-0.616396\pi$
$317$	−12.7329	−0.715150	−0.357575	+	0.933884i	$0.616396\pi$
$318$	−65.9429	−3.69790
$319$	6.92201	0.387558
$320$	37.1298	2.07562
$321$	−29.6616	−1.65555
$322$	−32.0499	−1.78607
$323$	4.66978	0.259834
$324$	−53.9680	−2.99822
$325$	30.8166	1.70940
$326$	−19.3816	−1.07345
$327$	19.6359	1.08587
$328$	69.4578	3.83516
$329$	−34.0616	−1.87788
$330$	43.5655	2.39820
$331$	32.7965	1.80266	0.901329	−	0.433135i	$-0.142593\pi$
$331$	32.7965	1.80266	0.901329	+	0.433135i	$0.142593\pi$
$332$	32.3134	1.77343
$333$	−15.1558	−0.830531
$334$	15.2836	0.836283
$335$	−22.4039	−1.22405
$336$	−91.6491	−4.99987
$337$	6.79831	0.370328	0.185164	−	0.982708i	$-0.440718\pi$
$337$	6.79831	0.370328	0.185164	+	0.982708i	$0.440718\pi$
$338$	−1.84927	−0.100587
$339$	40.6492	2.20776
$340$	−83.6217	−4.53502
$341$	3.26470	0.176793
$342$	5.25248	0.284022
$343$	−7.92016	−0.427649
$344$	7.98892	0.430733
$345$	−24.8988	−1.34051
$346$	25.6945	1.38134
$347$	−14.5574	−0.781485	−0.390742	−	0.920500i	$-0.627782\pi$
$347$	−14.5574	−0.781485	−0.390742	+	0.920500i	$0.627782\pi$
$348$	−37.3746	−2.00349
$349$	30.8973	1.65389	0.826946	−	0.562281i	$-0.190076\pi$
$349$	30.8973	1.65389	0.826946	+	0.562281i	$0.190076\pi$
$350$	87.4524	4.67453
$351$	−8.28676	−0.442314
$352$	23.7154	1.26404
$353$	10.8108	0.575402	0.287701	−	0.957720i	$-0.407109\pi$
$353$	10.8108	0.575402	0.287701	+	0.957720i	$0.407109\pi$
$354$	37.9328	2.01611
$355$	−45.2462	−2.40142
$356$	21.7124	1.15076
$357$	41.7381	2.20902
$358$	−38.2702	−2.02264
$359$	−5.12561	−0.270519	−0.135260	−	0.990810i	$-0.543187\pi$
$359$	−5.12561	−0.270519	−0.135260	+	0.990810i	$0.543187\pi$
$360$	−55.7100	−2.93617
$361$	1.00000	0.0526316
$362$	−20.4341	−1.07399
$363$	−15.3681	−0.806616
$364$	−72.5970	−3.80512
$365$	−16.2228	−0.849138
$366$	−0.198063	−0.0103529
$367$	−25.9757	−1.35592	−0.677961	−	0.735098i	$-0.737136\pi$
$367$	−25.9757	−1.35592	−0.677961	+	0.735098i	$0.737136\pi$
$368$	−31.2832	−1.63075
$369$	−18.1820	−0.946516
$370$	72.7351	3.78132
$371$	−44.8683	−2.32945
$372$	−17.6274	−0.913937
$373$	10.2159	0.528961	0.264481	−	0.964391i	$-0.414799\pi$
$373$	10.2159	0.528961	0.264481	+	0.964391i	$0.414799\pi$
$374$	−24.9276	−1.28897
$375$	27.1333	1.40116
$376$	−65.0583	−3.35513
$377$	−12.6145	−0.649679
$378$	−23.5165	−1.20956
$379$	11.9859	0.615675	0.307837	−	0.951439i	$-0.400395\pi$
$379$	11.9859	0.615675	0.307837	+	0.951439i	$0.400395\pi$
$380$	−17.9070	−0.918609
$381$	21.1966	1.08593
$382$	60.8351	3.11259
$383$	8.38578	0.428493	0.214247	−	0.976780i	$-0.431270\pi$
$383$	8.38578	0.428493	0.214247	+	0.976780i	$0.431270\pi$
$384$	−7.55759	−0.385671
$385$	29.6425	1.51072
$386$	−39.9059	−2.03116
$387$	−2.09126	−0.106305
$388$	−58.0078	−2.94490
$389$	22.0985	1.12044	0.560219	−	0.828345i	$-0.310717\pi$
$389$	22.0985	1.12044	0.560219	+	0.828345i	$0.310717\pi$
$390$	−79.3926	−4.02020
$391$	14.2468	0.720490
$392$	−68.5768	−3.46365
$393$	−10.4742	−0.528352
$394$	−52.3451	−2.63711
$395$	38.7375	1.94910
$396$	−19.9177	−1.00090
$397$	7.64498	0.383690	0.191845	−	0.981425i	$-0.438553\pi$
$397$	7.64498	0.383690	0.191845	+	0.981425i	$0.438553\pi$
$398$	−23.3934	−1.17260
$399$	8.93792	0.447456
$400$	85.3605	4.26802
$401$	3.32713	0.166149	0.0830745	−	0.996543i	$-0.473526\pi$
$401$	3.32713	0.166149	0.0830745	+	0.996543i	$0.473526\pi$
$402$	36.0602	1.79852
$403$	−5.94950	−0.296366
$404$	−34.1597	−1.69951
$405$	−40.1577	−1.99545
$406$	−35.7978	−1.77662
$407$	15.4027	0.763484
$408$	79.7206	3.94676
$409$	26.2632	1.29863	0.649316	−	0.760519i	$-0.275055\pi$
$409$	26.2632	1.29863	0.649316	+	0.760519i	$0.275055\pi$
$410$	87.2585	4.30939
$411$	−36.2501	−1.78808
$412$	−10.4776	−0.516194
$413$	25.8099	1.27002
$414$	16.0245	0.787561
$415$	24.0444	1.18030
$416$	−43.2183	−2.11895
$417$	−42.9351	−2.10254
$418$	−5.33806	−0.261093
$419$	5.14419	0.251310	0.125655	−	0.992074i	$-0.459897\pi$
$419$	5.14419	0.251310	0.125655	+	0.992074i	$0.459897\pi$
$420$	−160.051	−7.80970
$421$	34.1155	1.66269	0.831344	−	0.555759i	$-0.187572\pi$
$421$	34.1155	1.66269	0.831344	+	0.555759i	$0.187572\pi$
$422$	2.62786	0.127922
$423$	17.0303	0.828044
$424$	−85.6993	−4.16193
$425$	−38.8742	−1.88568
$426$	72.8262	3.52844
$427$	−0.134764	−0.00652171
$428$	−65.0816	−3.14584
$429$	−16.8125	−0.811716
$430$	10.0363	0.483995
$431$	−26.8350	−1.29260	−0.646298	−	0.763085i	$-0.723684\pi$
$431$	−26.8350	−1.29260	−0.646298	+	0.763085i	$0.723684\pi$
$432$	−22.9539	−1.10437
$433$	31.3484	1.50651	0.753254	−	0.657729i	$-0.228483\pi$
$433$	31.3484	1.50651	0.753254	+	0.657729i	$0.228483\pi$
$434$	−16.8837	−0.810443
$435$	−27.8105	−1.33341
$436$	43.0839	2.06335
$437$	3.05084	0.145942
$438$	26.1114	1.24765
$439$	−16.3588	−0.780761	−0.390380	−	0.920654i	$-0.627656\pi$
$439$	−16.3588	−0.780761	−0.390380	+	0.920654i	$0.627656\pi$
$440$	56.6177	2.69914
$441$	17.9514	0.854827
$442$	45.4273	2.16076
$443$	21.7997	1.03574	0.517868	−	0.855461i	$-0.326726\pi$
$443$	21.7997	1.03574	0.517868	+	0.855461i	$0.326726\pi$
$444$	−83.1652	−3.94684
$445$	16.1563	0.765881
$446$	−34.8116	−1.64838
$447$	−41.2879	−1.95285
$448$	−40.6630	−1.92115
$449$	0.810412	0.0382457	0.0191229	−	0.999817i	$-0.493913\pi$
$449$	0.810412	0.0382457	0.0191229	+	0.999817i	$0.493913\pi$
$450$	−43.7250	−2.06122
$451$	18.4782	0.870106
$452$	89.1898	4.19513
$453$	−17.4704	−0.820830
$454$	78.5463	3.68636
$455$	−54.0196	−2.53248
$456$	17.0716	0.799450
$457$	1.90419	0.0890742	0.0445371	−	0.999008i	$-0.485819\pi$
$457$	1.90419	0.0890742	0.0445371	+	0.999008i	$0.485819\pi$
$458$	−26.4908	−1.23783
$459$	10.4535	0.487928
$460$	−54.6314	−2.54720
$461$	−29.9279	−1.39388	−0.696940	−	0.717130i	$-0.745456\pi$
$461$	−29.9279	−1.39388	−0.696940	+	0.717130i	$0.745456\pi$
$462$	−47.7112	−2.21973
$463$	9.19342	0.427254	0.213627	−	0.976915i	$-0.431472\pi$
$463$	9.19342	0.427254	0.213627	+	0.976915i	$0.431472\pi$
$464$	−34.9415	−1.62212
$465$	−13.1166	−0.608266
$466$	14.2565	0.660420
$467$	−29.1635	−1.34953	−0.674763	−	0.738034i	$-0.735754\pi$
$467$	−29.1635	−1.34953	−0.674763	+	0.738034i	$0.735754\pi$
$468$	36.2975	1.67785
$469$	24.5358	1.13296
$470$	−81.7316	−3.77000
$471$	20.4865	0.943967
$472$	49.2975	2.26910
$473$	2.12534	0.0977230
$474$	−62.3502	−2.86384
$475$	−8.32463	−0.381960
$476$	91.5791	4.19752
$477$	22.4335	1.02716
$478$	34.8883	1.59576
$479$	33.9078	1.54929	0.774644	−	0.632398i	$-0.217929\pi$
$479$	33.9078	1.54929	0.774644	+	0.632398i	$0.217929\pi$
$480$	−95.2813	−4.34898
$481$	−28.0695	−1.27986
$482$	−10.1064	−0.460333
$483$	27.2682	1.24075
$484$	−33.7197	−1.53271
$485$	−43.1637	−1.95996
$486$	46.9883	2.13143
$487$	−14.1619	−0.641738	−0.320869	−	0.947124i	$-0.603975\pi$
$487$	−14.1619	−0.641738	−0.320869	+	0.947124i	$0.603975\pi$
$488$	−0.257403	−0.0116521
$489$	16.4900	0.745703
$490$	−86.1517	−3.89194
$491$	−21.5009	−0.970321	−0.485161	−	0.874425i	$-0.661239\pi$
$491$	−21.5009	−0.970321	−0.485161	+	0.874425i	$0.661239\pi$
$492$	−99.7712	−4.49803
$493$	15.9128	0.716676
$494$	9.72794	0.437681
$495$	−14.8208	−0.666147
$496$	−16.4798	−0.739966
$497$	49.5518	2.22270
$498$	−38.7008	−1.73423
$499$	−37.7234	−1.68873	−0.844367	−	0.535766i	$-0.820023\pi$
$499$	−37.7234	−1.68873	−0.844367	+	0.535766i	$0.820023\pi$
$500$	59.5341	2.66245
$501$	−13.0034	−0.580949
$502$	−34.6064	−1.54456
$503$	−9.49884	−0.423533	−0.211766	−	0.977320i	$-0.567922\pi$
$503$	−9.49884	−0.423533	−0.211766	+	0.977320i	$0.567922\pi$
$504$	61.0113	2.71766
$505$	−25.4183	−1.13110
$506$	−16.2856	−0.723983
$507$	1.57337	0.0698757
$508$	46.5081	2.06346
$509$	−20.1462	−0.892965	−0.446483	−	0.894792i	$-0.647324\pi$
$509$	−20.1462	−0.892965	−0.446483	+	0.894792i	$0.647324\pi$
$510$	100.151	4.43478
$511$	17.7665	0.785943
$512$	36.8773	1.62976
$513$	2.23854	0.0988341
$514$	−54.2751	−2.39397
$515$	−7.79640	−0.343550
$516$	−11.4755	−0.505181
$517$	−17.3078	−0.761197
$518$	−79.6565	−3.49991
$519$	−21.8610	−0.959590
$520$	−103.178	−4.52468
$521$	27.9467	1.22437	0.612184	−	0.790715i	$-0.290291\pi$
$521$	27.9467	1.22437	0.612184	+	0.790715i	$0.290291\pi$
$522$	17.8984	0.783392
$523$	−15.7643	−0.689323	−0.344661	−	0.938727i	$-0.612006\pi$
$523$	−15.7643	−0.689323	−0.344661	+	0.938727i	$0.612006\pi$
$524$	−22.9818	−1.00396
$525$	−74.4049	−3.24729
$526$	−48.0401	−2.09465
$527$	7.50512	0.326928
$528$	−46.5699	−2.02670
$529$	−13.6924	−0.595320
$530$	−107.662	−4.67656
$531$	−12.9046	−0.560013
$532$	19.6110	0.850245
$533$	−33.6742	−1.45859
$534$	−26.0044	−1.12532
$535$	−48.4273	−2.09370
$536$	46.8638	2.02421
$537$	32.5604	1.40509
$538$	25.5179	1.10015
$539$	−18.2439	−0.785819
$540$	−40.0856	−1.72501
$541$	20.9670	0.901440	0.450720	−	0.892665i	$-0.351167\pi$
$541$	20.9670	0.901440	0.450720	+	0.892665i	$0.351167\pi$
$542$	17.0114	0.730704
$543$	17.3855	0.746082
$544$	54.5187	2.33747
$545$	32.0588	1.37325
$546$	86.9475	3.72101
$547$	36.2304	1.54910	0.774549	−	0.632514i	$-0.217977\pi$
$547$	36.2304	1.54910	0.774549	+	0.632514i	$0.217977\pi$
$548$	−79.5375	−3.39767
$549$	0.0673804	0.00287572
$550$	44.4374	1.89482
$551$	3.40761	0.145169
$552$	52.0828	2.21679
$553$	−42.4238	−1.80404
$554$	−63.2685	−2.68802
$555$	−61.8834	−2.62680
$556$	−94.2054	−3.99520
$557$	6.14000	0.260160	0.130080	−	0.991503i	$-0.458477\pi$
$557$	6.14000	0.260160	0.130080	+	0.991503i	$0.458477\pi$
$558$	8.44162	0.357362
$559$	−3.87315	−0.163817
$560$	−149.632	−6.32310
$561$	21.2085	0.895424
$562$	82.9541	3.49921
$563$	21.7611	0.917119	0.458560	−	0.888664i	$-0.348366\pi$
$563$	21.7611	0.917119	0.458560	+	0.888664i	$0.348366\pi$
$564$	93.4517	3.93503
$565$	66.3663	2.79205
$566$	4.74376	0.199395
$567$	43.9791	1.84695
$568$	94.6448	3.97121
$569$	−0.00944600	−0.000395997 0	−0.000197999	−	1.00000i	$-0.500063\pi$
$569$	−0.00944600	−0.000395997 0	−0.000197999		1.00000i	$0.500063\pi$
$570$	21.4467	0.898304
$571$	−9.16977	−0.383743	−0.191871	−	0.981420i	$-0.561456\pi$
$571$	−9.16977	−0.383743	−0.191871	+	0.981420i	$0.561456\pi$
$572$	−36.8889	−1.54240
$573$	−51.7588	−2.16225
$574$	−95.5619	−3.98868
$575$	−25.3972	−1.05913
$576$	20.3309	0.847122
$577$	−11.0953	−0.461903	−0.230952	−	0.972965i	$-0.574184\pi$
$577$	−11.0953	−0.461903	−0.230952	+	0.972965i	$0.574184\pi$
$578$	−12.6317	−0.525410
$579$	33.9521	1.41100
$580$	−61.0201	−2.53372
$581$	−26.3325	−1.09246
$582$	69.4743	2.87980
$583$	−22.7991	−0.944241
$584$	33.9343	1.40421
$585$	27.0091	1.11669
$586$	0.779857	0.0322156
$587$	5.37931	0.222028	0.111014	−	0.993819i	$-0.464590\pi$
$587$	5.37931	0.222028	0.111014	+	0.993819i	$0.464590\pi$
$588$	98.5057	4.06231
$589$	1.60717	0.0662222
$590$	61.9315	2.54968
$591$	44.5354	1.83194
$592$	−77.7511	−3.19555
$593$	27.5496	1.13133	0.565664	−	0.824636i	$-0.308620\pi$
$593$	27.5496	1.13133	0.565664	+	0.824636i	$0.308620\pi$
$594$	−11.9495	−0.490293
$595$	68.1442	2.79364
$596$	−90.5913	−3.71077
$597$	19.9032	0.814584
$598$	29.6784	1.21364
$599$	−36.9344	−1.50910	−0.754549	−	0.656244i	$-0.772144\pi$
$599$	−36.9344	−1.50910	−0.754549	+	0.656244i	$0.772144\pi$
$600$	−142.115	−5.80181
$601$	28.5751	1.16560	0.582802	−	0.812614i	$-0.301956\pi$
$601$	28.5751	1.16560	0.582802	+	0.812614i	$0.301956\pi$
$602$	−10.9914	−0.447975
$603$	−12.2676	−0.499573
$604$	−38.3324	−1.55972
$605$	−25.0909	−1.02009
$606$	40.9122	1.66194
$607$	33.8028	1.37201	0.686006	−	0.727596i	$-0.259362\pi$
$607$	33.8028	1.37201	0.686006	+	0.727596i	$0.259362\pi$
$608$	11.6748	0.473475
$609$	30.4569	1.23418
$610$	−0.323370	−0.0130929
$611$	31.5413	1.27602
$612$	−45.7883	−1.85088
$613$	−26.0627	−1.05266	−0.526331	−	0.850280i	$-0.676433\pi$
$613$	−26.0627	−1.05266	−0.526331	+	0.850280i	$0.676433\pi$
$614$	21.1237	0.852483
$615$	−74.2399	−2.99364
$616$	−62.0054	−2.49827
$617$	−28.8351	−1.16086	−0.580429	−	0.814311i	$-0.697115\pi$
$617$	−28.8351	−1.16086	−0.580429	+	0.814311i	$0.697115\pi$
$618$	12.5487	0.504784
$619$	28.7496	1.15554	0.577771	−	0.816199i	$-0.303923\pi$
$619$	28.7496	1.15554	0.577771	+	0.816199i	$0.303923\pi$
$620$	−28.7795	−1.15581
$621$	6.82944	0.274056
$622$	73.1802	2.93426
$623$	−17.6937	−0.708883
$624$	84.8677	3.39743
$625$	2.67633	0.107053
$626$	86.0178	3.43796
$627$	4.54165	0.181376
$628$	44.9501	1.79371
$629$	35.4088	1.41184
$630$	76.6473	3.05370
$631$	−29.1247	−1.15944	−0.579718	−	0.814817i	$-0.696837\pi$
$631$	−29.1247	−1.15944	−0.579718	+	0.814817i	$0.696837\pi$
$632$	−81.0302	−3.22321
$633$	−2.23579	−0.0888648
$634$	33.4602	1.32887
$635$	34.6068	1.37333
$636$	123.101	4.88127
$637$	33.2471	1.31730
$638$	−18.1900	−0.720151
$639$	−24.7752	−0.980092
$640$	−12.3390	−0.487741
$641$	47.1090	1.86069	0.930347	−	0.366680i	$-0.119506\pi$
$641$	47.1090	1.86069	0.930347	+	0.366680i	$0.119506\pi$
$642$	77.9464	3.07630
$643$	−16.3181	−0.643521	−0.321761	−	0.946821i	$-0.604275\pi$
$643$	−16.3181	−0.643521	−0.321761	+	0.946821i	$0.604275\pi$
$644$	59.8301	2.35764
$645$	−8.53895	−0.336221
$646$	−12.2715	−0.482816
$647$	−28.9402	−1.13776	−0.568878	−	0.822422i	$-0.692622\pi$
$647$	−28.9402	−1.13776	−0.568878	+	0.822422i	$0.692622\pi$
$648$	84.0008	3.29986
$649$	13.1149	0.514804
$650$	−80.9815	−3.17636
$651$	14.3647	0.562998
$652$	36.1813	1.41697
$653$	37.7318	1.47656	0.738280	−	0.674494i	$-0.235638\pi$
$653$	37.7318	1.47656	0.738280	+	0.674494i	$0.235638\pi$
$654$	−51.6005	−2.01774
$655$	−17.1008	−0.668183
$656$	−93.2761	−3.64182
$657$	−8.88300	−0.346559
$658$	89.5091	3.48943
$659$	−16.3902	−0.638473	−0.319237	−	0.947675i	$-0.603426\pi$
$659$	−16.3902	−0.638473	−0.319237	+	0.947675i	$0.603426\pi$
$660$	−81.3273	−3.16566
$661$	16.6032	0.645789	0.322895	−	0.946435i	$-0.395344\pi$
$661$	16.6032	0.645789	0.322895	+	0.946435i	$0.395344\pi$
$662$	−86.1845	−3.34966
$663$	−38.6498	−1.50103
$664$	−50.2955	−1.95185
$665$	14.5926	0.565876
$666$	39.8272	1.54327
$667$	10.3961	0.402538
$668$	−28.5312	−1.10391
$669$	29.6179	1.14509
$670$	58.8741	2.27451
$671$	−0.0684782	−0.00264357
$672$	104.348	4.02532
$673$	22.7041	0.875180	0.437590	−	0.899175i	$-0.355832\pi$
$673$	22.7041	0.875180	0.437590	+	0.899175i	$0.355832\pi$
$674$	−17.8650	−0.688134
$675$	−18.6350	−0.717263
$676$	3.45218	0.132776
$677$	28.9732	1.11353	0.556764	−	0.830670i	$-0.312043\pi$
$677$	28.9732	1.11353	0.556764	+	0.830670i	$0.312043\pi$
$678$	−106.820	−4.10240
$679$	47.2711	1.81410
$680$	130.157	4.99128
$681$	−66.8275	−2.56084
$682$	−8.57916	−0.328513
$683$	−20.5064	−0.784655	−0.392328	−	0.919826i	$-0.628330\pi$
$683$	−20.5064	−0.784655	−0.392328	+	0.919826i	$0.628330\pi$
$684$	−9.80523	−0.374912
$685$	−59.1841	−2.26131
$686$	20.8131	0.794646
$687$	22.5385	0.859896
$688$	−10.7285	−0.409019
$689$	41.5484	1.58287
$690$	65.4306	2.49090
$691$	6.42718	0.244501	0.122251	−	0.992499i	$-0.460989\pi$
$691$	6.42718	0.244501	0.122251	+	0.992499i	$0.460989\pi$
$692$	−47.9659	−1.82339
$693$	16.2312	0.616571
$694$	38.2549	1.45214
$695$	−70.0984	−2.65899
$696$	58.1733	2.20505
$697$	42.4791	1.60901
$698$	−81.1936	−3.07322
$699$	−12.1295	−0.458780
$700$	−163.254	−6.17043
$701$	4.62692	0.174756	0.0873781	−	0.996175i	$-0.472151\pi$
$701$	4.62692	0.174756	0.0873781	+	0.996175i	$0.472151\pi$
$702$	21.7764	0.821898
$703$	7.58254	0.285981
$704$	−20.6622	−0.778736
$705$	69.5376	2.61894
$706$	−28.4093	−1.06920
$707$	27.8371	1.04692
$708$	−70.8123	−2.66129
$709$	12.7887	0.480288	0.240144	−	0.970737i	$-0.422805\pi$
$709$	12.7887	0.480288	0.240144	+	0.970737i	$0.422805\pi$
$710$	118.901	4.46226
$711$	21.2113	0.795486
$712$	−33.7953	−1.26653
$713$	4.90322	0.183627
$714$	−109.682	−4.10474
$715$	−27.4491	−1.02654
$716$	71.4420	2.66991
$717$	−29.6832	−1.10854
$718$	13.4694	0.502672
$719$	−17.6369	−0.657745	−0.328873	−	0.944374i	$-0.606669\pi$
$719$	−17.6369	−0.657745	−0.328873	+	0.944374i	$0.606669\pi$
$720$	74.8139	2.78815
$721$	8.53829	0.317983
$722$	−2.62786	−0.0977987
$723$	8.59856	0.319784
$724$	38.1461	1.41769
$725$	−28.3671	−1.05353
$726$	40.3852	1.49883
$727$	−13.9208	−0.516295	−0.258147	−	0.966106i	$-0.583112\pi$
$727$	−13.9208	−0.516295	−0.258147	+	0.966106i	$0.583112\pi$
$728$	112.997	4.18794
$729$	−6.97418	−0.258303
$730$	42.6311	1.57785
$731$	4.88587	0.180710
$732$	0.369741	0.0136660
$733$	−7.18038	−0.265213	−0.132607	−	0.991169i	$-0.542335\pi$
$733$	−7.18038	−0.265213	−0.132607	+	0.991169i	$0.542335\pi$
$734$	68.2605	2.51954
$735$	73.2983	2.70365
$736$	35.6179	1.31289
$737$	12.4674	0.459244
$738$	47.7797	1.75879
$739$	−15.6990	−0.577498	−0.288749	−	0.957405i	$-0.593239\pi$
$739$	−15.6990	−0.577498	−0.288749	+	0.957405i	$0.593239\pi$
$740$	−135.781	−4.99139
$741$	−8.27658	−0.304048
$742$	117.907	4.32852
$743$	−12.6673	−0.464718	−0.232359	−	0.972630i	$-0.574644\pi$
$743$	−12.6673	−0.464718	−0.232359	+	0.972630i	$0.574644\pi$
$744$	27.4369	1.00589
$745$	−67.4092	−2.46968
$746$	−26.8460	−0.982903
$747$	13.1659	0.481714
$748$	46.5343	1.70146
$749$	53.0356	1.93788
$750$	−71.3025	−2.60360
$751$	−39.1222	−1.42759	−0.713795	−	0.700355i	$-0.753025\pi$
$751$	−39.1222	−1.42759	−0.713795	+	0.700355i	$0.753025\pi$
$752$	87.3680	3.18598
$753$	29.4433	1.07297
$754$	33.1490	1.20722
$755$	−28.5232	−1.03807
$756$	43.9001	1.59663
$757$	34.8371	1.26618	0.633088	−	0.774080i	$-0.281787\pi$
$757$	34.8371	1.26618	0.633088	+	0.774080i	$0.281787\pi$
$758$	−31.4972	−1.14403
$759$	13.8559	0.502936
$760$	27.8721	1.01103
$761$	−4.35906	−0.158016	−0.0790079	−	0.996874i	$-0.525175\pi$
$761$	−4.35906	−0.158016	−0.0790079	+	0.996874i	$0.525175\pi$
$762$	−55.7015	−2.01785
$763$	−35.1095	−1.27105
$764$	−113.566	−4.10866
$765$	−34.0712	−1.23185
$766$	−22.0366	−0.796216
$767$	−23.9002	−0.862986
$768$	−25.6235	−0.924607
$769$	−14.2099	−0.512421	−0.256211	−	0.966621i	$-0.582474\pi$
$769$	−14.2099	−0.512421	−0.256211	+	0.966621i	$0.582474\pi$
$770$	−77.8962	−2.80718
$771$	46.1775	1.66304
$772$	74.4956	2.68115
$773$	−6.20665	−0.223238	−0.111619	−	0.993751i	$-0.535604\pi$
$773$	−6.20665	−0.223238	−0.111619	+	0.993751i	$0.535604\pi$
$774$	5.49554	0.197533
$775$	−13.3791	−0.480591
$776$	90.2887	3.24118
$777$	67.7721	2.43131
$778$	−58.0717	−2.08197
$779$	9.09659	0.325919
$780$	148.209	5.30672
$781$	25.1789	0.900972
$782$	−37.4385	−1.33880
$783$	7.62808	0.272605
$784$	92.0930	3.28903
$785$	33.4475	1.19379
$786$	27.5246	0.981772
$787$	−44.0895	−1.57162	−0.785811	−	0.618467i	$-0.787754\pi$
$787$	−44.0895	−1.57162	−0.785811	+	0.618467i	$0.787754\pi$
$788$	97.7167	3.48101
$789$	40.8727	1.45511
$790$	−101.797	−3.62176
$791$	−72.6816	−2.58426
$792$	31.0018	1.10160
$793$	0.124793	0.00443152
$794$	−20.0899	−0.712964
$795$	91.5997	3.24871
$796$	43.6703	1.54785
$797$	52.6891	1.86634	0.933171	−	0.359432i	$-0.117030\pi$
$797$	52.6891	1.86634	0.933171	+	0.359432i	$0.117030\pi$
$798$	−23.4876	−0.831451
$799$	−39.7884	−1.40761
$800$	−97.1882	−3.43612
$801$	8.84660	0.312579
$802$	−8.74323	−0.308734
$803$	9.02774	0.318582
$804$	−67.3165	−2.37407
$805$	44.5197	1.56911
$806$	15.6344	0.550699
$807$	−21.7107	−0.764253
$808$	53.1694	1.87049
$809$	−20.4676	−0.719603	−0.359801	−	0.933029i	$-0.617156\pi$
$809$	−20.4676	−0.719603	−0.359801	+	0.933029i	$0.617156\pi$
$810$	105.529	3.70790
$811$	41.2599	1.44883	0.724416	−	0.689363i	$-0.242109\pi$
$811$	41.2599	1.44883	0.724416	+	0.689363i	$0.242109\pi$
$812$	66.8267	2.34516
$813$	−14.4734	−0.507605
$814$	−40.4761	−1.41869
$815$	26.9226	0.943056
$816$	−107.058	−3.74779
$817$	1.04627	0.0366045
$818$	−69.0160	−2.41309
$819$	−29.5792	−1.03358
$820$	−162.892	−5.68845
$821$	36.5490	1.27557	0.637785	−	0.770215i	$-0.279851\pi$
$821$	36.5490	1.27557	0.637785	+	0.770215i	$0.279851\pi$
$822$	95.2600	3.32257
$823$	31.8636	1.11070	0.555348	−	0.831618i	$-0.312585\pi$
$823$	31.8636	1.11070	0.555348	+	0.831618i	$0.312585\pi$
$824$	16.3083	0.568127
$825$	−37.8076	−1.31629
$826$	−67.8248	−2.35993
$827$	−25.5670	−0.889051	−0.444525	−	0.895766i	$-0.646628\pi$
$827$	−25.5670	−0.889051	−0.444525	+	0.895766i	$0.646628\pi$
$828$	−29.9142	−1.03959
$829$	−47.5454	−1.65132	−0.825659	−	0.564169i	$-0.809197\pi$
$829$	−47.5454	−1.65132	−0.825659	+	0.564169i	$0.809197\pi$
$830$	−63.1853	−2.19320
$831$	53.8291	1.86731
$832$	37.6542	1.30543
$833$	−41.9403	−1.45314
$834$	112.827	3.90689
$835$	−21.2301	−0.734699
$836$	9.96499	0.344647
$837$	3.59771	0.124355
$838$	−13.5182	−0.466978
$839$	−21.3124	−0.735787	−0.367893	−	0.929868i	$-0.619921\pi$
$839$	−21.3124	−0.735787	−0.367893	+	0.929868i	$0.619921\pi$
$840$	249.119	8.59541
$841$	−17.3882	−0.599593
$842$	−89.6506	−3.08956
$843$	−70.5778	−2.43083
$844$	−4.90563	−0.168859
$845$	2.56877	0.0883685
$846$	−44.7533	−1.53865
$847$	27.4785	0.944173
$848$	115.087	3.95211
$849$	−4.03601	−0.138516
$850$	102.156	3.50392
$851$	23.1332	0.792994
$852$	−135.951	−4.65759
$853$	17.3514	0.594100	0.297050	−	0.954862i	$-0.403997\pi$
$853$	17.3514	0.594100	0.297050	+	0.954862i	$0.403997\pi$
$854$	0.354142	0.0121185
$855$	−7.29610	−0.249521
$856$	101.299	3.46233
$857$	19.8774	0.679000	0.339500	−	0.940606i	$-0.389742\pi$
$857$	19.8774	0.679000	0.339500	+	0.940606i	$0.389742\pi$
$858$	44.1809	1.50831
$859$	−51.5750	−1.75972	−0.879858	−	0.475237i	$-0.842362\pi$
$859$	−51.5750	−1.75972	−0.879858	+	0.475237i	$0.842362\pi$
$860$	−18.7356	−0.638879
$861$	81.3045	2.77085
$862$	70.5185	2.40187
$863$	−53.2743	−1.81348	−0.906739	−	0.421692i	$-0.861436\pi$
$863$	−53.2743	−1.81348	−0.906739	+	0.421692i	$0.861436\pi$
$864$	26.1345	0.889113
$865$	−35.6916	−1.21355
$866$	−82.3791	−2.79936
$867$	10.7471	0.364991
$868$	31.5182	1.06980
$869$	−21.5569	−0.731268
$870$	73.0821	2.47771
$871$	−22.7203	−0.769848
$872$	−67.0599	−2.27093
$873$	−23.6349	−0.799920
$874$	−8.01718	−0.271185
$875$	−48.5150	−1.64011
$876$	−48.7443	−1.64692
$877$	−34.8448	−1.17662	−0.588312	−	0.808634i	$-0.700207\pi$
$877$	−34.8448	−1.17662	−0.588312	+	0.808634i	$0.700207\pi$
$878$	42.9885	1.45079
$879$	−0.663506	−0.0223795
$880$	−76.0329	−2.56307
$881$	−12.5090	−0.421440	−0.210720	−	0.977547i	$-0.567581\pi$
$881$	−12.5090	−0.421440	−0.210720	+	0.977547i	$0.567581\pi$
$882$	−47.1736	−1.58842
$883$	−2.03852	−0.0686018	−0.0343009	−	0.999412i	$-0.510920\pi$
$883$	−2.03852	−0.0686018	−0.0343009	+	0.999412i	$0.510920\pi$
$884$	−84.8029	−2.85223
$885$	−52.6916	−1.77121
$886$	−57.2865	−1.92458
$887$	13.1234	0.440640	0.220320	−	0.975428i	$-0.429290\pi$
$887$	13.1234	0.440640	0.220320	+	0.975428i	$0.429290\pi$
$888$	129.446	4.34393
$889$	−37.8999	−1.27112
$890$	−42.4564	−1.42314
$891$	22.3472	0.748660
$892$	64.9856	2.17588
$893$	−8.52041	−0.285125
$894$	108.499	3.62874
$895$	53.1602	1.77695
$896$	13.5131	0.451442
$897$	−25.2505	−0.843091
$898$	−2.12965	−0.0710673
$899$	5.47660	0.182655
$900$	81.6249	2.72083
$901$	−52.4121	−1.74610
$902$	−48.5582	−1.61681
$903$	9.35151	0.311199
$904$	−138.823	−4.61719
$905$	28.3846	0.943535
$906$	45.9096	1.52525
$907$	−20.7804	−0.690003	−0.345002	−	0.938602i	$-0.612122\pi$
$907$	−20.7804	−0.690003	−0.345002	+	0.938602i	$0.612122\pi$
$908$	−146.629	−4.86604
$909$	−13.9182	−0.461637
$910$	141.956	4.70579
$911$	−44.5652	−1.47651	−0.738255	−	0.674521i	$-0.764350\pi$
$911$	−44.5652	−1.47651	−0.738255	+	0.674521i	$0.764350\pi$
$912$	−22.9257	−0.759147
$913$	−13.3804	−0.442827
$914$	−5.00394	−0.165516
$915$	0.275125	0.00909534
$916$	49.4525	1.63396
$917$	18.7281	0.618456
$918$	−27.4703	−0.906655
$919$	47.5418	1.56826	0.784130	−	0.620597i	$-0.213110\pi$
$919$	47.5418	1.56826	0.784130	+	0.620597i	$0.213110\pi$
$920$	85.0335	2.80347
$921$	−17.9721	−0.592202
$922$	78.6462	2.59007
$923$	−45.8853	−1.51033
$924$	89.0663	2.93007
$925$	−63.1219	−2.07543
$926$	−24.1590	−0.793913
$927$	−4.26903	−0.140213
$928$	39.7831	1.30594
$929$	−49.4519	−1.62247	−0.811233	−	0.584723i	$-0.801203\pi$
$929$	−49.4519	−1.62247	−0.811233	+	0.584723i	$0.801203\pi$
$930$	34.4685	1.13027
$931$	−8.98120	−0.294347
$932$	−26.6138	−0.871763
$933$	−62.2621	−2.03837
$934$	76.6376	2.50766
$935$	34.6263	1.13240
$936$	−56.4969	−1.84666
$937$	−28.1404	−0.919308	−0.459654	−	0.888098i	$-0.652027\pi$
$937$	−28.1404	−0.919308	−0.459654	+	0.888098i	$0.652027\pi$
$938$	−64.4765	−2.10523
$939$	−73.1843	−2.38828
$940$	152.575	4.97644
$941$	−39.5189	−1.28828	−0.644140	−	0.764908i	$-0.722785\pi$
$941$	−39.5189	−1.28828	−0.644140	+	0.764908i	$0.722785\pi$
$942$	−53.8356	−1.75406
$943$	27.7523	0.903738
$944$	−66.2024	−2.15471
$945$	32.6661	1.06263
$946$	−5.58508	−0.181587
$947$	−3.07663	−0.0999771	−0.0499885	−	0.998750i	$-0.515918\pi$
$947$	−3.07663	−0.0999771	−0.0499885	+	0.998750i	$0.515918\pi$
$948$	116.394	3.78030
$949$	−16.4519	−0.534051
$950$	21.8759	0.709749
$951$	−28.4681	−0.923141
$952$	−142.542	−4.61982
$953$	1.15689	0.0374754	0.0187377	−	0.999824i	$-0.494035\pi$
$953$	1.15689	0.0374754	0.0187377	+	0.999824i	$0.494035\pi$
$954$	−58.9521	−1.90865
$955$	−84.5045	−2.73450
$956$	−65.1289	−2.10642
$957$	15.4762	0.500274
$958$	−89.1049	−2.87885
$959$	64.8159	2.09302
$960$	83.0144	2.67928
$961$	−28.4170	−0.916678
$962$	73.7625	2.37820
$963$	−26.5171	−0.854501
$964$	18.8664	0.607646
$965$	55.4323	1.78443
$966$	−71.6569	−2.30552
$967$	−46.2474	−1.48722	−0.743609	−	0.668615i	$-0.766887\pi$
$967$	−46.2474	−1.48722	−0.743609	+	0.668615i	$0.766887\pi$
$968$	52.4845	1.68692
$969$	10.4407	0.335402
$970$	113.428	3.64195
$971$	51.8675	1.66451	0.832254	−	0.554395i	$-0.187050\pi$
$971$	51.8675	1.66451	0.832254	+	0.554395i	$0.187050\pi$
$972$	−87.7168	−2.81352
$973$	76.7689	2.46110
$974$	37.2155	1.19246
$975$	68.8995	2.20655
$976$	0.345671	0.0110646
$977$	−18.8704	−0.603716	−0.301858	−	0.953353i	$-0.597607\pi$
$977$	−18.8704	−0.603716	−0.301858	+	0.953353i	$0.597607\pi$
$978$	−43.3333	−1.38565
$979$	−8.99074	−0.287345
$980$	160.826	5.13741
$981$	17.5543	0.560465
$982$	56.5013	1.80303
$983$	23.6305	0.753696	0.376848	−	0.926275i	$-0.377008\pi$
$983$	23.6305	0.753696	0.376848	+	0.926275i	$0.377008\pi$
$984$	155.293	4.95057
$985$	72.7112	2.31677
$986$	−41.8166	−1.33171
$987$	−76.1547	−2.42403
$988$	−18.1599	−0.577744
$989$	3.19202	0.101500
$990$	38.9470	1.23782
$991$	44.6344	1.41786	0.708930	−	0.705279i	$-0.249178\pi$
$991$	44.6344	1.41786	0.708930	+	0.705279i	$0.249178\pi$
$992$	18.7633	0.595736
$993$	73.3262	2.32694
$994$	−130.215	−4.13017
$995$	32.4952	1.03017
$996$	72.2460	2.28920
$997$	−30.1821	−0.955877	−0.477939	−	0.878393i	$-0.658616\pi$
$997$	−30.1821	−0.955877	−0.477939	+	0.878393i	$0.658616\pi$
$998$	99.1318	3.13796
$999$	16.9738	0.537029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.4	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.4	✓	75	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)

\(f(q)\)	\(=\)	\(q-2.62786 q^{2} +2.23579 q^{3} +4.90563 q^{4} +3.65029 q^{5} -5.87534 q^{6} -3.99765 q^{7} -7.63559 q^{8} +1.99877 q^{9} +O(q^{10})\)	\(q-2.62786 q^{2} +2.23579 q^{3} +4.90563 q^{4} +3.65029 q^{5} -5.87534 q^{6} -3.99765 q^{7} -7.63559 q^{8} +1.99877 q^{9} -9.59245 q^{10} -2.03134 q^{11} +10.9680 q^{12} +3.70185 q^{13} +10.5053 q^{14} +8.16130 q^{15} +10.2540 q^{16} -4.66978 q^{17} -5.25248 q^{18} -1.00000 q^{19} +17.9070 q^{20} -8.93792 q^{21} +5.33806 q^{22} -3.05084 q^{23} -17.0716 q^{24} +8.32463 q^{25} -9.72794 q^{26} -2.23854 q^{27} -19.6110 q^{28} -3.40761 q^{29} -21.4467 q^{30} -1.60717 q^{31} -11.6748 q^{32} -4.54165 q^{33} +12.2715 q^{34} -14.5926 q^{35} +9.80523 q^{36} -7.58254 q^{37} +2.62786 q^{38} +8.27658 q^{39} -27.8721 q^{40} -9.09659 q^{41} +23.4876 q^{42} -1.04627 q^{43} -9.96499 q^{44} +7.29610 q^{45} +8.01718 q^{46} +8.52041 q^{47} +22.9257 q^{48} +8.98120 q^{49} -21.8759 q^{50} -10.4407 q^{51} +18.1599 q^{52} +11.2237 q^{53} +5.88257 q^{54} -7.41497 q^{55} +30.5244 q^{56} -2.23579 q^{57} +8.95471 q^{58} -6.45628 q^{59} +40.0363 q^{60} +0.0337109 q^{61} +4.22341 q^{62} -7.99038 q^{63} +10.1717 q^{64} +13.5128 q^{65} +11.9348 q^{66} -6.13755 q^{67} -22.9082 q^{68} -6.82106 q^{69} +38.3472 q^{70} -12.3952 q^{71} -15.2618 q^{72} -4.44423 q^{73} +19.9258 q^{74} +18.6122 q^{75} -4.90563 q^{76} +8.12058 q^{77} -21.7497 q^{78} +10.6122 q^{79} +37.4300 q^{80} -11.0012 q^{81} +23.9045 q^{82} +6.58699 q^{83} -43.8461 q^{84} -17.0461 q^{85} +2.74946 q^{86} -7.61871 q^{87} +15.5105 q^{88} +4.42602 q^{89} -19.1731 q^{90} -14.7987 q^{91} -14.9663 q^{92} -3.59329 q^{93} -22.3904 q^{94} -3.65029 q^{95} -26.1024 q^{96} -11.8247 q^{97} -23.6013 q^{98} -4.06018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

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