LMFDB - Embedded newform 4009.2.a.c.1.23

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			1.23
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)

$f(q)$	$=$	$q-1.34769 q^{2} +2.03763 q^{3} -0.183722 q^{4} +1.29260 q^{5} -2.74611 q^{6} +2.31006 q^{7} +2.94299 q^{8} +1.15196 q^{9} +O(q^{10})$	\(q-1.34769 q^{2} +2.03763 q^{3} -0.183722 q^{4} +1.29260 q^{5} -2.74611 q^{6} +2.31006 q^{7} +2.94299 q^{8} +1.15196 q^{9} -1.74203 q^{10} -6.55654 q^{11} -0.374358 q^{12} -3.41994 q^{13} -3.11325 q^{14} +2.63384 q^{15} -3.59880 q^{16} +1.81461 q^{17} -1.55248 q^{18} +1.00000 q^{19} -0.237478 q^{20} +4.70705 q^{21} +8.83621 q^{22} +4.29320 q^{23} +5.99673 q^{24} -3.32919 q^{25} +4.60903 q^{26} -3.76564 q^{27} -0.424407 q^{28} -7.12441 q^{29} -3.54961 q^{30} +4.81238 q^{31} -1.03589 q^{32} -13.3598 q^{33} -2.44554 q^{34} +2.98597 q^{35} -0.211639 q^{36} -6.94875 q^{37} -1.34769 q^{38} -6.96858 q^{39} +3.80410 q^{40} +0.533072 q^{41} -6.34366 q^{42} +9.45380 q^{43} +1.20458 q^{44} +1.48902 q^{45} -5.78591 q^{46} +1.10974 q^{47} -7.33305 q^{48} -1.66364 q^{49} +4.48673 q^{50} +3.69751 q^{51} +0.628317 q^{52} -0.446950 q^{53} +5.07493 q^{54} -8.47497 q^{55} +6.79846 q^{56} +2.03763 q^{57} +9.60152 q^{58} -6.24897 q^{59} -0.483894 q^{60} -8.86497 q^{61} -6.48562 q^{62} +2.66108 q^{63} +8.59367 q^{64} -4.42061 q^{65} +18.0050 q^{66} +16.0729 q^{67} -0.333383 q^{68} +8.74797 q^{69} -4.02418 q^{70} -13.3388 q^{71} +3.39019 q^{72} -13.1311 q^{73} +9.36478 q^{74} -6.78367 q^{75} -0.183722 q^{76} -15.1460 q^{77} +9.39152 q^{78} -5.65970 q^{79} -4.65181 q^{80} -11.1289 q^{81} -0.718418 q^{82} +5.19356 q^{83} -0.864787 q^{84} +2.34556 q^{85} -12.7408 q^{86} -14.5169 q^{87} -19.2958 q^{88} -0.604811 q^{89} -2.00674 q^{90} -7.90024 q^{91} -0.788753 q^{92} +9.80588 q^{93} -1.49559 q^{94} +1.29260 q^{95} -2.11077 q^{96} -17.4046 q^{97} +2.24208 q^{98} -7.55284 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.34769	−0.952963	−0.476482	−	0.879184i	$-0.658088\pi$
$2$	−1.34769	−0.952963	−0.476482	+	0.879184i	$0.658088\pi$
$3$	2.03763	1.17643	0.588215	−	0.808705i	$-0.299831\pi$
$3$	2.03763	1.17643	0.588215	+	0.808705i	$0.299831\pi$
$4$	−0.183722	−0.0918608
$5$	1.29260	0.578068	0.289034	−	0.957319i	$-0.406666\pi$
$5$	1.29260	0.578068	0.289034	+	0.957319i	$0.406666\pi$
$6$	−2.74611	−1.12109
$7$	2.31006	0.873119	0.436559	−	0.899675i	$-0.356197\pi$
$7$	2.31006	0.873119	0.436559	+	0.899675i	$0.356197\pi$
$8$	2.94299	1.04050
$9$	1.15196	0.383985
$10$	−1.74203	−0.550877
$11$	−6.55654	−1.97687	−0.988436	−	0.151640i	$-0.951544\pi$
$11$	−6.55654	−1.97687	−0.988436	+	0.151640i	$0.951544\pi$
$12$	−0.374358	−0.108068
$13$	−3.41994	−0.948520	−0.474260	−	0.880385i	$-0.657284\pi$
$13$	−3.41994	−0.948520	−0.474260	+	0.880385i	$0.657284\pi$
$14$	−3.11325	−0.832050
$15$	2.63384	0.680055
$16$	−3.59880	−0.899701
$17$	1.81461	0.440108	0.220054	−	0.975488i	$-0.429377\pi$
$17$	1.81461	0.440108	0.220054	+	0.975488i	$0.429377\pi$
$18$	−1.55248	−0.365924
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.237478	−0.0531018
$21$	4.70705	1.02716
$22$	8.83621	1.88389
$23$	4.29320	0.895193	0.447597	−	0.894236i	$-0.352280\pi$
$23$	4.29320	0.895193	0.447597	+	0.894236i	$0.352280\pi$
$24$	5.99673	1.22408
$25$	−3.32919	−0.665838
$26$	4.60903	0.903905
$27$	−3.76564	−0.724698
$28$	−0.424407	−0.0802054
$29$	−7.12441	−1.32297	−0.661485	−	0.749959i	$-0.730073\pi$
$29$	−7.12441	−1.32297	−0.661485	+	0.749959i	$0.730073\pi$
$30$	−3.54961	−0.648068
$31$	4.81238	0.864329	0.432165	−	0.901795i	$-0.357750\pi$
$31$	4.81238	0.864329	0.432165	+	0.901795i	$0.357750\pi$
$32$	−1.03589	−0.183122
$33$	−13.3598	−2.32565
$34$	−2.44554	−0.419406
$35$	2.98597	0.504722
$36$	−0.211639	−0.0352732
$37$	−6.94875	−1.14237	−0.571184	−	0.820822i	$-0.693516\pi$
$37$	−6.94875	−1.14237	−0.571184	+	0.820822i	$0.693516\pi$
$38$	−1.34769	−0.218625
$39$	−6.96858	−1.11587
$40$	3.80410	0.601481
$41$	0.533072	0.0832518	0.0416259	−	0.999133i	$-0.486746\pi$
$41$	0.533072	0.0832518	0.0416259	+	0.999133i	$0.486746\pi$
$42$	−6.34366	−0.978848
$43$	9.45380	1.44169	0.720846	−	0.693095i	$-0.243754\pi$
$43$	9.45380	1.44169	0.720846	+	0.693095i	$0.243754\pi$
$44$	1.20458	0.181597
$45$	1.48902	0.221969
$46$	−5.78591	−0.853087
$47$	1.10974	0.161872	0.0809361	−	0.996719i	$-0.474209\pi$
$47$	1.10974	0.161872	0.0809361	+	0.996719i	$0.474209\pi$
$48$	−7.33305	−1.05843
$49$	−1.66364	−0.237664
$50$	4.48673	0.634519
$51$	3.69751	0.517755
$52$	0.628317	0.0871318
$53$	−0.446950	−0.0613933	−0.0306966	−	0.999529i	$-0.509773\pi$
$53$	−0.446950	−0.0613933	−0.0306966	+	0.999529i	$0.509773\pi$
$54$	5.07493	0.690610
$55$	−8.47497	−1.14277
$56$	6.79846	0.908483
$57$	2.03763	0.269891
$58$	9.60152	1.26074
$59$	−6.24897	−0.813546	−0.406773	−	0.913529i	$-0.633346\pi$
$59$	−6.24897	−0.813546	−0.406773	+	0.913529i	$0.633346\pi$
$60$	−0.483894	−0.0624705
$61$	−8.86497	−1.13504	−0.567521	−	0.823359i	$-0.692098\pi$
$61$	−8.86497	−1.13504	−0.567521	+	0.823359i	$0.692098\pi$
$62$	−6.48562	−0.823674
$63$	2.66108	0.335265
$64$	8.59367	1.07421
$65$	−4.42061	−0.548309
$66$	18.0050	2.21626
$67$	16.0729	1.96362	0.981808	−	0.189877i	$-0.0608088\pi$
$67$	16.0729	1.96362	0.981808	+	0.189877i	$0.0608088\pi$
$68$	−0.333383	−0.0404287
$69$	8.74797	1.05313
$70$	−4.02418	−0.480981
$71$	−13.3388	−1.58302	−0.791510	−	0.611157i	$-0.790705\pi$
$71$	−13.3388	−1.58302	−0.791510	+	0.611157i	$0.790705\pi$
$72$	3.39019	0.399538
$73$	−13.1311	−1.53688	−0.768440	−	0.639922i	$-0.778967\pi$
$73$	−13.1311	−1.53688	−0.768440	+	0.639922i	$0.778967\pi$
$74$	9.36478	1.08863
$75$	−6.78367	−0.783311
$76$	−0.183722	−0.0210743
$77$	−15.1460	−1.72604
$78$	9.39152	1.06338
$79$	−5.65970	−0.636766	−0.318383	−	0.947962i	$-0.603140\pi$
$79$	−5.65970	−0.636766	−0.318383	+	0.947962i	$0.603140\pi$
$80$	−4.65181	−0.520088
$81$	−11.1289	−1.23654
$82$	−0.718418	−0.0793360
$83$	5.19356	0.570068	0.285034	−	0.958517i	$-0.407995\pi$
$83$	5.19356	0.570068	0.285034	+	0.958517i	$0.407995\pi$
$84$	−0.864787	−0.0943560
$85$	2.34556	0.254412
$86$	−12.7408	−1.37388
$87$	−14.5169	−1.55638
$88$	−19.2958	−2.05694
$89$	−0.604811	−0.0641098	−0.0320549	−	0.999486i	$-0.510205\pi$
$89$	−0.604811	−0.0641098	−0.0320549	+	0.999486i	$0.510205\pi$
$90$	−2.00674	−0.211529
$91$	−7.90024	−0.828171
$92$	−0.788753	−0.0822332
$93$	9.80588	1.01682
$94$	−1.49559	−0.154258
$95$	1.29260	0.132618
$96$	−2.11077	−0.215429
$97$	−17.4046	−1.76717	−0.883585	−	0.468272i	$-0.844877\pi$
$97$	−17.4046	−1.76717	−0.883585	+	0.468272i	$0.844877\pi$
$98$	2.24208	0.226485
$99$	−7.55284	−0.759089
$100$	0.611644	0.0611644
$101$	1.73126	0.172267	0.0861334	−	0.996284i	$-0.472549\pi$
$101$	1.73126	0.172267	0.0861334	+	0.996284i	$0.472549\pi$
$102$	−4.98312	−0.493402
$103$	−5.22342	−0.514679	−0.257339	−	0.966321i	$-0.582846\pi$
$103$	−5.22342	−0.514679	−0.257339	+	0.966321i	$0.582846\pi$
$104$	−10.0648	−0.986938
$105$	6.08432	0.593769
$106$	0.602351	0.0585055
$107$	2.05079	0.198257	0.0991286	−	0.995075i	$-0.468394\pi$
$107$	2.05079	0.198257	0.0991286	+	0.995075i	$0.468394\pi$
$108$	0.691830	0.0665713
$109$	−15.3636	−1.47157	−0.735784	−	0.677216i	$-0.763186\pi$
$109$	−15.3636	−1.47157	−0.735784	+	0.677216i	$0.763186\pi$
$110$	11.4217	1.08901
$111$	−14.1590	−1.34391
$112$	−8.31343	−0.785546
$113$	−8.44215	−0.794171	−0.397086	−	0.917782i	$-0.629978\pi$
$113$	−8.44215	−0.794171	−0.397086	+	0.917782i	$0.629978\pi$
$114$	−2.74611	−0.257197
$115$	5.54938	0.517482
$116$	1.30891	0.121529
$117$	−3.93962	−0.364218
$118$	8.42169	0.775280
$119$	4.19185	0.384266
$120$	7.75137	0.707600
$121$	31.9882	2.90802
$122$	11.9473	1.08165
$123$	1.08621	0.0979399
$124$	−0.884139	−0.0793980
$125$	−10.7663	−0.962967
$126$	−3.58632	−0.319495
$127$	9.78465	0.868247	0.434124	−	0.900853i	$-0.357058\pi$
$127$	9.78465	0.868247	0.434124	+	0.900853i	$0.357058\pi$
$128$	−9.50985	−0.840560
$129$	19.2634	1.69605
$130$	5.95762	0.522518
$131$	−3.40397	−0.297406	−0.148703	−	0.988882i	$-0.547510\pi$
$131$	−3.40397	−0.297406	−0.148703	+	0.988882i	$0.547510\pi$
$132$	2.45449	0.213636
$133$	2.31006	0.200307
$134$	−21.6613	−1.87125
$135$	−4.86746	−0.418924
$136$	5.34038	0.457933
$137$	−2.96680	−0.253471	−0.126735	−	0.991937i	$-0.540450\pi$
$137$	−2.96680	−0.253471	−0.126735	+	0.991937i	$0.540450\pi$
$138$	−11.7896	−1.00360
$139$	7.04057	0.597173	0.298587	−	0.954383i	$-0.403485\pi$
$139$	7.04057	0.597173	0.298587	+	0.954383i	$0.403485\pi$
$140$	−0.548588	−0.0463642
$141$	2.26125	0.190431
$142$	17.9766	1.50856
$143$	22.4230	1.87510
$144$	−4.14566	−0.345472
$145$	−9.20900	−0.764766
$146$	17.6967	1.46459
$147$	−3.38990	−0.279594
$148$	1.27664	0.104939
$149$	−12.3700	−1.01339	−0.506696	−	0.862125i	$-0.669133\pi$
$149$	−12.3700	−1.01339	−0.506696	+	0.862125i	$0.669133\pi$
$150$	9.14231	0.746467
$151$	13.0667	1.06335	0.531675	−	0.846949i	$-0.321563\pi$
$151$	13.0667	1.06335	0.531675	+	0.846949i	$0.321563\pi$
$152$	2.94299	0.238708
$153$	2.09035	0.168995
$154$	20.4121	1.64486
$155$	6.22048	0.499641
$156$	1.28028	0.102504
$157$	1.45407	0.116047	0.0580237	−	0.998315i	$-0.481520\pi$
$157$	1.45407	0.116047	0.0580237	+	0.998315i	$0.481520\pi$
$158$	7.62754	0.606815
$159$	−0.910720	−0.0722248
$160$	−1.33899	−0.105857
$161$	9.91752	0.781610
$162$	14.9983	1.17838
$163$	−5.69988	−0.446449	−0.223224	−	0.974767i	$-0.571658\pi$
$163$	−5.69988	−0.446449	−0.223224	+	0.974767i	$0.571658\pi$
$164$	−0.0979368	−0.00764758
$165$	−17.2689	−1.34438
$166$	−6.99933	−0.543253
$167$	20.1326	1.55791	0.778954	−	0.627081i	$-0.215750\pi$
$167$	20.1326	1.55791	0.778954	+	0.627081i	$0.215750\pi$
$168$	13.8528	1.06877
$169$	−1.30403	−0.100310
$170$	−3.16110	−0.242445
$171$	1.15196	0.0880923
$172$	−1.73687	−0.132435
$173$	−9.34276	−0.710317	−0.355158	−	0.934806i	$-0.615573\pi$
$173$	−9.34276	−0.710317	−0.355158	+	0.934806i	$0.615573\pi$
$174$	19.5644	1.48317
$175$	−7.69061	−0.581356
$176$	23.5957	1.77859
$177$	−12.7331	−0.957079
$178$	0.815100	0.0610943
$179$	−4.04160	−0.302084	−0.151042	−	0.988527i	$-0.548263\pi$
$179$	−4.04160	−0.302084	−0.151042	+	0.988527i	$0.548263\pi$
$180$	−0.273565	−0.0203903
$181$	17.4673	1.29833	0.649167	−	0.760646i	$-0.275117\pi$
$181$	17.4673	1.29833	0.649167	+	0.760646i	$0.275117\pi$
$182$	10.6471	0.789216
$183$	−18.0636	−1.33530
$184$	12.6348	0.931452
$185$	−8.98194	−0.660365
$186$	−13.2153	−0.968994
$187$	−11.8976	−0.870036
$188$	−0.203883	−0.0148697
$189$	−8.69884	−0.632747
$190$	−1.74203	−0.126380
$191$	−18.0683	−1.30738	−0.653689	−	0.756764i	$-0.726779\pi$
$191$	−18.0683	−1.30738	−0.653689	+	0.756764i	$0.726779\pi$
$192$	17.5108	1.26373
$193$	−20.8233	−1.49890	−0.749448	−	0.662063i	$-0.769681\pi$
$193$	−20.8233	−1.49890	−0.749448	+	0.662063i	$0.769681\pi$
$194$	23.4561	1.68405
$195$	−9.00758	−0.645046
$196$	0.305648	0.0218320
$197$	5.26849	0.375364	0.187682	−	0.982230i	$-0.439903\pi$
$197$	5.26849	0.375364	0.187682	+	0.982230i	$0.439903\pi$
$198$	10.1789	0.723384
$199$	19.0126	1.34776	0.673882	−	0.738839i	$-0.264626\pi$
$199$	19.0126	1.34776	0.673882	+	0.738839i	$0.264626\pi$
$200$	−9.79776	−0.692807
$201$	32.7507	2.31005
$202$	−2.33321	−0.164164
$203$	−16.4578	−1.15511
$204$	−0.679313	−0.0475614
$205$	0.689048	0.0481252
$206$	7.03957	0.490470
$207$	4.94557	0.343741
$208$	12.3077	0.853384
$209$	−6.55654	−0.453525
$210$	−8.19981	−0.565840
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	0.0821143	0.00563964
$213$	−27.1795	−1.86231
$214$	−2.76383	−0.188932
$215$	12.2200	0.833395
$216$	−11.0822	−0.754050
$217$	11.1169	0.754662
$218$	20.7055	1.40235
$219$	−26.7564	−1.80803
$220$	1.55704	0.104975
$221$	−6.20585	−0.417451
$222$	19.0820	1.28070
$223$	4.55356	0.304929	0.152465	−	0.988309i	$-0.451279\pi$
$223$	4.55356	0.304929	0.152465	+	0.988309i	$0.451279\pi$
$224$	−2.39297	−0.159887
$225$	−3.83508	−0.255672
$226$	11.3774	0.756816
$227$	6.63418	0.440326	0.220163	−	0.975463i	$-0.429341\pi$
$227$	6.63418	0.440326	0.220163	+	0.975463i	$0.429341\pi$
$228$	−0.374358	−0.0247924
$229$	27.9461	1.84673	0.923367	−	0.383920i	$-0.125426\pi$
$229$	27.9461	1.84673	0.923367	+	0.383920i	$0.125426\pi$
$230$	−7.47886	−0.493142
$231$	−30.8620	−2.03057
$232$	−20.9670	−1.37655
$233$	12.6978	0.831862	0.415931	−	0.909396i	$-0.363456\pi$
$233$	12.6978	0.831862	0.415931	+	0.909396i	$0.363456\pi$
$234$	5.30940	0.347086
$235$	1.43445	0.0935731
$236$	1.14807	0.0747330
$237$	−11.5324	−0.749110
$238$	−5.64933	−0.366192
$239$	11.3627	0.734991	0.367495	−	0.930025i	$-0.380215\pi$
$239$	11.3627	0.734991	0.367495	+	0.930025i	$0.380215\pi$
$240$	−9.47868	−0.611846
$241$	−6.30795	−0.406331	−0.203165	−	0.979144i	$-0.565123\pi$
$241$	−6.30795	−0.406331	−0.203165	+	0.979144i	$0.565123\pi$
$242$	−43.1103	−2.77124
$243$	−11.3796	−0.730005
$244$	1.62869	0.104266
$245$	−2.15042	−0.137386
$246$	−1.46387	−0.0933331
$247$	−3.41994	−0.217605
$248$	14.1628	0.899337
$249$	10.5826	0.670644
$250$	14.5097	0.917672
$251$	6.09663	0.384816	0.192408	−	0.981315i	$-0.438370\pi$
$251$	6.09663	0.384816	0.192408	+	0.981315i	$0.438370\pi$
$252$	−0.488898	−0.0307977
$253$	−28.1485	−1.76968
$254$	−13.1867	−0.827408
$255$	4.77940	0.299298
$256$	−4.37097	−0.273186
$257$	−17.5168	−1.09267	−0.546334	−	0.837567i	$-0.683977\pi$
$257$	−17.5168	−1.09267	−0.546334	+	0.837567i	$0.683977\pi$
$258$	−25.9612	−1.61627
$259$	−16.0520	−0.997422
$260$	0.812161	0.0503681
$261$	−8.20700	−0.508001
$262$	4.58751	0.283417
$263$	−15.8802	−0.979216	−0.489608	−	0.871943i	$-0.662860\pi$
$263$	−15.8802	−0.979216	−0.489608	+	0.871943i	$0.662860\pi$
$264$	−39.3178	−2.41985
$265$	−0.577726	−0.0354895
$266$	−3.11325	−0.190885
$267$	−1.23238	−0.0754206
$268$	−2.95294	−0.180379
$269$	23.0497	1.40537	0.702683	−	0.711504i	$-0.251985\pi$
$269$	23.0497	1.40537	0.702683	+	0.711504i	$0.251985\pi$
$270$	6.55984	0.399219
$271$	−16.6787	−1.01316	−0.506581	−	0.862193i	$-0.669091\pi$
$271$	−16.6787	−1.01316	−0.506581	+	0.862193i	$0.669091\pi$
$272$	−6.53042	−0.395965
$273$	−16.0978	−0.974284
$274$	3.99834	0.241548
$275$	21.8280	1.31628
$276$	−1.60719	−0.0967415
$277$	4.87912	0.293158	0.146579	−	0.989199i	$-0.453174\pi$
$277$	4.87912	0.293158	0.146579	+	0.989199i	$0.453174\pi$
$278$	−9.48853	−0.569084
$279$	5.54365	0.331890
$280$	8.78768	0.525165
$281$	−11.9055	−0.710225	−0.355112	−	0.934824i	$-0.615557\pi$
$281$	−11.9055	−0.710225	−0.355112	+	0.934824i	$0.615557\pi$
$282$	−3.04747	−0.181474
$283$	−2.42416	−0.144102	−0.0720508	−	0.997401i	$-0.522954\pi$
$283$	−2.42416	−0.144102	−0.0720508	+	0.997401i	$0.522954\pi$
$284$	2.45062	0.145417
$285$	2.63384	0.156015
$286$	−30.2193	−1.78690
$287$	1.23143	0.0726887
$288$	−1.19330	−0.0703160
$289$	−13.7072	−0.806305
$290$	12.4109	0.728794
$291$	−35.4642	−2.07895
$292$	2.41247	0.141179
$293$	−14.9049	−0.870751	−0.435375	−	0.900249i	$-0.643384\pi$
$293$	−14.9049	−0.870751	−0.435375	+	0.900249i	$0.643384\pi$
$294$	4.56855	0.266443
$295$	−8.07740	−0.470285
$296$	−20.4501	−1.18864
$297$	24.6896	1.43263
$298$	16.6710	0.965724
$299$	−14.6825	−0.849109
$300$	1.24631	0.0719556
$301$	21.8388	1.25877
$302$	−17.6098	−1.01333
$303$	3.52768	0.202660
$304$	−3.59880	−0.206406
$305$	−11.4588	−0.656131
$306$	−2.81715	−0.161046
$307$	−4.87861	−0.278437	−0.139218	−	0.990262i	$-0.544459\pi$
$307$	−4.87861	−0.278437	−0.139218	+	0.990262i	$0.544459\pi$
$308$	2.78264	0.158556
$309$	−10.6434	−0.605483
$310$	−8.38330	−0.476139
$311$	−24.6606	−1.39837	−0.699187	−	0.714939i	$-0.746455\pi$
$311$	−24.6606	−1.39837	−0.699187	+	0.714939i	$0.746455\pi$
$312$	−20.5085	−1.16106
$313$	14.3667	0.812053	0.406026	−	0.913861i	$-0.366914\pi$
$313$	14.3667	0.812053	0.406026	+	0.913861i	$0.366914\pi$
$314$	−1.95964	−0.110589
$315$	3.43971	0.193806
$316$	1.03981	0.0584938
$317$	16.7603	0.941352	0.470676	−	0.882306i	$-0.344010\pi$
$317$	16.7603	0.941352	0.470676	+	0.882306i	$0.344010\pi$
$318$	1.22737	0.0688276
$319$	46.7115	2.61534
$320$	11.1082	0.620965
$321$	4.17876	0.233235
$322$	−13.3658	−0.744846
$323$	1.81461	0.100968
$324$	2.04461	0.113590
$325$	11.3856	0.631561
$326$	7.68169	0.425449
$327$	−31.3055	−1.73120
$328$	1.56882	0.0866238
$329$	2.56356	0.141334
$330$	23.2732	1.28115
$331$	−16.9060	−0.929240	−0.464620	−	0.885510i	$-0.653809\pi$
$331$	−16.9060	−0.929240	−0.464620	+	0.885510i	$0.653809\pi$
$332$	−0.954170	−0.0523669
$333$	−8.00465	−0.438652
$334$	−27.1326	−1.48463
$335$	20.7758	1.13510
$336$	−16.9397	−0.924139
$337$	7.50119	0.408616	0.204308	−	0.978907i	$-0.434506\pi$
$337$	7.50119	0.408616	0.204308	+	0.978907i	$0.434506\pi$
$338$	1.75743	0.0955915
$339$	−17.2020	−0.934286
$340$	−0.430931	−0.0233705
$341$	−31.5526	−1.70867
$342$	−1.55248	−0.0839487
$343$	−20.0135	−1.08063
$344$	27.8224	1.50008
$345$	11.3076	0.608781
$346$	12.5912	0.676906
$347$	−34.0863	−1.82985	−0.914924	−	0.403626i	$-0.867750\pi$
$347$	−34.0863	−1.82985	−0.914924	+	0.403626i	$0.867750\pi$
$348$	2.66708	0.142970
$349$	−0.195633	−0.0104720	−0.00523600	−	0.999986i	$-0.501667\pi$
$349$	−0.195633	−0.0104720	−0.00523600	+	0.999986i	$0.501667\pi$
$350$	10.3646	0.554011
$351$	12.8783	0.687390
$352$	6.79187	0.362008
$353$	4.13428	0.220045	0.110023	−	0.993929i	$-0.464908\pi$
$353$	4.13428	0.220045	0.110023	+	0.993929i	$0.464908\pi$
$354$	17.1603	0.912061
$355$	−17.2417	−0.915092
$356$	0.111117	0.00588918
$357$	8.54146	0.452062
$358$	5.44684	0.287875
$359$	−5.81832	−0.307079	−0.153540	−	0.988142i	$-0.549067\pi$
$359$	−5.81832	−0.307079	−0.153540	+	0.988142i	$0.549067\pi$
$360$	4.38216	0.230960
$361$	1.00000	0.0526316
$362$	−23.5406	−1.23727
$363$	65.1803	3.42108
$364$	1.45145	0.0760764
$365$	−16.9733	−0.888421
$366$	24.3442	1.27249
$367$	−5.41878	−0.282858	−0.141429	−	0.989948i	$-0.545170\pi$
$367$	−5.41878	−0.282858	−0.141429	+	0.989948i	$0.545170\pi$
$368$	−15.4504	−0.805406
$369$	0.614075	0.0319675
$370$	12.1049	0.629304
$371$	−1.03248	−0.0536036
$372$	−1.80155	−0.0934061
$373$	10.3755	0.537223	0.268612	−	0.963249i	$-0.413435\pi$
$373$	10.3755	0.537223	0.268612	+	0.963249i	$0.413435\pi$
$374$	16.0343	0.829113
$375$	−21.9378	−1.13286
$376$	3.26595	0.168429
$377$	24.3650	1.25486
$378$	11.7234	0.602985
$379$	32.6736	1.67833	0.839166	−	0.543875i	$-0.183043\pi$
$379$	32.6736	1.67833	0.839166	+	0.543875i	$0.183043\pi$
$380$	−0.237478	−0.0121824
$381$	19.9375	1.02143
$382$	24.3506	1.24588
$383$	−3.68080	−0.188080	−0.0940400	−	0.995568i	$-0.529978\pi$
$383$	−3.68080	−0.188080	−0.0940400	+	0.995568i	$0.529978\pi$
$384$	−19.3776	−0.988859
$385$	−19.5777	−0.997770
$386$	28.0635	1.42839
$387$	10.8904	0.553588
$388$	3.19760	0.162334
$389$	−10.9892	−0.557172	−0.278586	−	0.960411i	$-0.589866\pi$
$389$	−10.9892	−0.557172	−0.278586	+	0.960411i	$0.589866\pi$
$390$	12.1395	0.614705
$391$	7.79048	0.393981
$392$	−4.89609	−0.247290
$393$	−6.93605	−0.349877
$394$	−7.10031	−0.357708
$395$	−7.31572	−0.368094
$396$	1.38762	0.0697306
$397$	−2.78775	−0.139913	−0.0699565	−	0.997550i	$-0.522286\pi$
$397$	−2.78775	−0.139913	−0.0699565	+	0.997550i	$0.522286\pi$
$398$	−25.6231	−1.28437
$399$	4.70705	0.235647
$400$	11.9811	0.599055
$401$	36.1922	1.80735	0.903676	−	0.428217i	$-0.140858\pi$
$401$	36.1922	1.80735	0.903676	+	0.428217i	$0.140858\pi$
$402$	−44.1379	−2.20140
$403$	−16.4580	−0.819834
$404$	−0.318070	−0.0158246
$405$	−14.3852	−0.714804
$406$	22.1800	1.10078
$407$	45.5597	2.25831
$408$	10.8817	0.538726
$409$	24.1313	1.19322	0.596608	−	0.802533i	$-0.296515\pi$
$409$	24.1313	1.19322	0.596608	+	0.802533i	$0.296515\pi$
$410$	−0.928625	−0.0458615
$411$	−6.04525	−0.298190
$412$	0.959655	0.0472788
$413$	−14.4355	−0.710322
$414$	−6.66512	−0.327573
$415$	6.71319	0.329538
$416$	3.54268	0.173694
$417$	14.3461	0.702532
$418$	8.83621	0.432193
$419$	−16.8304	−0.822221	−0.411110	−	0.911586i	$-0.634859\pi$
$419$	−16.8304	−0.822221	−0.411110	+	0.911586i	$0.634859\pi$
$420$	−1.11782	−0.0545441
$421$	30.2153	1.47260	0.736302	−	0.676653i	$-0.236570\pi$
$421$	30.2153	1.47260	0.736302	+	0.676653i	$0.236570\pi$
$422$	−1.34769	−0.0656047
$423$	1.27837	0.0621566
$424$	−1.31537	−0.0638799
$425$	−6.04118	−0.293040
$426$	36.6297	1.77471
$427$	−20.4786	−0.991027
$428$	−0.376774	−0.0182121
$429$	45.6898	2.20592
$430$	−16.4688	−0.794195
$431$	−24.8253	−1.19579	−0.597895	−	0.801574i	$-0.703996\pi$
$431$	−24.8253	−1.19579	−0.597895	+	0.801574i	$0.703996\pi$
$432$	13.5518	0.652011
$433$	−37.5107	−1.80265	−0.901323	−	0.433147i	$-0.857403\pi$
$433$	−37.5107	−1.80265	−0.901323	+	0.433147i	$0.857403\pi$
$434$	−14.9821	−0.719165
$435$	−18.7646	−0.899693
$436$	2.82263	0.135179
$437$	4.29320	0.205371
$438$	36.0594	1.72299
$439$	−28.4622	−1.35843	−0.679214	−	0.733941i	$-0.737679\pi$
$439$	−28.4622	−1.35843	−0.679214	+	0.733941i	$0.737679\pi$
$440$	−24.9417	−1.18905
$441$	−1.91645	−0.0912593
$442$	8.36359	0.397815
$443$	−15.8532	−0.753206	−0.376603	−	0.926375i	$-0.622908\pi$
$443$	−15.8532	−0.753206	−0.376603	+	0.926375i	$0.622908\pi$
$444$	2.60132	0.123453
$445$	−0.781777	−0.0370598
$446$	−6.13681	−0.290586
$447$	−25.2056	−1.19218
$448$	19.8519	0.937912
$449$	−25.6840	−1.21210	−0.606052	−	0.795425i	$-0.707248\pi$
$449$	−25.6840	−1.21210	−0.606052	+	0.795425i	$0.707248\pi$
$450$	5.16851	0.243646
$451$	−3.49511	−0.164578
$452$	1.55101	0.0729532
$453$	26.6251	1.25095
$454$	−8.94084	−0.419614
$455$	−10.2118	−0.478739
$456$	5.99673	0.280823
$457$	−3.23542	−0.151347	−0.0756733	−	0.997133i	$-0.524111\pi$
$457$	−3.23542	−0.151347	−0.0756733	+	0.997133i	$0.524111\pi$
$458$	−37.6628	−1.75987
$459$	−6.83317	−0.318945
$460$	−1.01954	−0.0475364
$461$	20.4283	0.951439	0.475719	−	0.879597i	$-0.342188\pi$
$461$	20.4283	0.951439	0.475719	+	0.879597i	$0.342188\pi$
$462$	41.5925	1.93506
$463$	−19.3851	−0.900901	−0.450451	−	0.892801i	$-0.648737\pi$
$463$	−19.3851	−0.900901	−0.450451	+	0.892801i	$0.648737\pi$
$464$	25.6393	1.19028
$465$	12.6751	0.587792
$466$	−17.1128	−0.792734
$467$	−20.0869	−0.929511	−0.464756	−	0.885439i	$-0.653858\pi$
$467$	−20.0869	−0.929511	−0.464756	+	0.885439i	$0.653858\pi$
$468$	0.723793	0.0334573
$469$	37.1293	1.71447
$470$	−1.93320	−0.0891717
$471$	2.96286	0.136521
$472$	−18.3906	−0.846497
$473$	−61.9843	−2.85004
$474$	15.5421	0.713874
$475$	−3.32919	−0.152754
$476$	−0.770134	−0.0352990
$477$	−0.514866	−0.0235741
$478$	−15.3134	−0.700419
$479$	21.4885	0.981833	0.490917	−	0.871207i	$-0.336662\pi$
$479$	21.4885	0.981833	0.490917	+	0.871207i	$0.336662\pi$
$480$	−2.72838	−0.124533
$481$	23.7643	1.08356
$482$	8.50119	0.387219
$483$	20.2083	0.919509
$484$	−5.87693	−0.267133
$485$	−22.4972	−1.02154
$486$	15.3363	0.695668
$487$	−21.1784	−0.959685	−0.479842	−	0.877355i	$-0.659306\pi$
$487$	−21.1784	−0.959685	−0.479842	+	0.877355i	$0.659306\pi$
$488$	−26.0895	−1.18102
$489$	−11.6143	−0.525215
$490$	2.89811	0.130923
$491$	32.0482	1.44632	0.723159	−	0.690682i	$-0.242690\pi$
$491$	32.0482	1.44632	0.723159	+	0.690682i	$0.242690\pi$
$492$	−0.199560	−0.00899684
$493$	−12.9280	−0.582249
$494$	4.60903	0.207370
$495$	−9.76279	−0.438805
$496$	−17.3188	−0.777638
$497$	−30.8133	−1.38216
$498$	−14.2621	−0.639099
$499$	36.6253	1.63957	0.819787	−	0.572668i	$-0.194092\pi$
$499$	36.6253	1.63957	0.819787	+	0.572668i	$0.194092\pi$
$500$	1.97800	0.0884589
$501$	41.0229	1.83277
$502$	−8.21639	−0.366715
$503$	7.36139	0.328228	0.164114	−	0.986441i	$-0.447523\pi$
$503$	7.36139	0.328228	0.164114	+	0.986441i	$0.447523\pi$
$504$	7.83153	0.348844
$505$	2.23782	0.0995819
$506$	37.9356	1.68644
$507$	−2.65713	−0.118007
$508$	−1.79765	−0.0797579
$509$	−10.9821	−0.486773	−0.243386	−	0.969929i	$-0.578258\pi$
$509$	−10.9821	−0.486773	−0.243386	+	0.969929i	$0.578258\pi$
$510$	−6.44117	−0.285220
$511$	−30.3336	−1.34188
$512$	24.9104	1.10090
$513$	−3.76564	−0.166257
$514$	23.6073	1.04127
$515$	−6.75178	−0.297519
$516$	−3.53910	−0.155800
$517$	−7.27606	−0.320001
$518$	21.6332	0.950507
$519$	−19.0371	−0.835637
$520$	−13.0098	−0.570517
$521$	1.71211	0.0750089	0.0375044	−	0.999296i	$-0.488059\pi$
$521$	1.71211	0.0750089	0.0375044	+	0.999296i	$0.488059\pi$
$522$	11.0605	0.484106
$523$	−26.2298	−1.14695	−0.573475	−	0.819223i	$-0.694405\pi$
$523$	−26.2298	−1.14695	−0.573475	+	0.819223i	$0.694405\pi$
$524$	0.625383	0.0273200
$525$	−15.6707	−0.683924
$526$	21.4017	0.933157
$527$	8.73260	0.380398
$528$	48.0794	2.09239
$529$	−4.56846	−0.198629
$530$	0.778598	0.0338202
$531$	−7.19853	−0.312390
$532$	−0.424407	−0.0184004
$533$	−1.82307	−0.0789660
$534$	1.66088	0.0718731
$535$	2.65085	0.114606
$536$	47.3023	2.04315
$537$	−8.23531	−0.355380
$538$	−31.0639	−1.33926
$539$	10.9078	0.469830
$540$	0.894258	0.0384827
$541$	39.0813	1.68023	0.840117	−	0.542405i	$-0.182486\pi$
$541$	39.0813	1.68023	0.840117	+	0.542405i	$0.182486\pi$
$542$	22.4778	0.965506
$543$	35.5920	1.52740
$544$	−1.87974	−0.0805932
$545$	−19.8590	−0.850666
$546$	21.6949	0.928457
$547$	32.7848	1.40178	0.700889	−	0.713271i	$-0.252787\pi$
$547$	32.7848	1.40178	0.700889	+	0.713271i	$0.252787\pi$
$548$	0.545065	0.0232840
$549$	−10.2120	−0.435840
$550$	−29.4174	−1.25436
$551$	−7.12441	−0.303510
$552$	25.7452	1.09579
$553$	−13.0742	−0.555972
$554$	−6.57556	−0.279369
$555$	−18.3019	−0.776873
$556$	−1.29351	−0.0548568
$557$	−21.6825	−0.918716	−0.459358	−	0.888251i	$-0.651920\pi$
$557$	−21.6825	−0.918716	−0.459358	+	0.888251i	$0.651920\pi$
$558$	−7.47114	−0.316279
$559$	−32.3314	−1.36747
$560$	−10.7459	−0.454098
$561$	−24.2429	−1.02354
$562$	16.0450	0.676818
$563$	19.2047	0.809383	0.404691	−	0.914453i	$-0.367379\pi$
$563$	19.2047	0.809383	0.404691	+	0.914453i	$0.367379\pi$
$564$	−0.415440	−0.0174932
$565$	−10.9123	−0.459085
$566$	3.26703	0.137324
$567$	−25.7083	−1.07965
$568$	−39.2558	−1.64714
$569$	−37.5407	−1.57379	−0.786894	−	0.617088i	$-0.788312\pi$
$569$	−37.5407	−1.57379	−0.786894	+	0.617088i	$0.788312\pi$
$570$	−3.54961	−0.148677
$571$	−1.67881	−0.0702561	−0.0351280	−	0.999383i	$-0.511184\pi$
$571$	−1.67881	−0.0702561	−0.0351280	+	0.999383i	$0.511184\pi$
$572$	−4.11958	−0.172248
$573$	−36.8166	−1.53804
$574$	−1.65958	−0.0692697
$575$	−14.2929	−0.596054
$576$	9.89953	0.412480
$577$	−47.4207	−1.97415	−0.987075	−	0.160260i	$-0.948767\pi$
$577$	−47.4207	−1.97415	−0.987075	+	0.160260i	$0.948767\pi$
$578$	18.4731	0.768379
$579$	−42.4303	−1.76334
$580$	1.69189	0.0702520
$581$	11.9974	0.497737
$582$	47.7949	1.98116
$583$	2.93044	0.121367
$584$	−38.6447	−1.59913
$585$	−5.09234	−0.210542
$586$	20.0872	0.829794
$587$	−3.00658	−0.124095	−0.0620474	−	0.998073i	$-0.519763\pi$
$587$	−3.00658	−0.124095	−0.0620474	+	0.998073i	$0.519763\pi$
$588$	0.622798	0.0256838
$589$	4.81238	0.198291
$590$	10.8859	0.448164
$591$	10.7353	0.441589
$592$	25.0072	1.02779
$593$	29.9699	1.23071	0.615357	−	0.788248i	$-0.289012\pi$
$593$	29.9699	1.23071	0.615357	+	0.788248i	$0.289012\pi$
$594$	−33.2740	−1.36525
$595$	5.41838	0.222132
$596$	2.27264	0.0930909
$597$	38.7406	1.58555
$598$	19.7875	0.809170
$599$	7.06890	0.288827	0.144414	−	0.989517i	$-0.453870\pi$
$599$	7.06890	0.288827	0.144414	+	0.989517i	$0.453870\pi$
$600$	−19.9643	−0.815038
$601$	12.5708	0.512772	0.256386	−	0.966574i	$-0.417468\pi$
$601$	12.5708	0.512772	0.256386	+	0.966574i	$0.417468\pi$
$602$	−29.4320	−1.19956
$603$	18.5153	0.754000
$604$	−2.40063	−0.0976801
$605$	41.3479	1.68103
$606$	−4.75423	−0.193127
$607$	5.21078	0.211499	0.105749	−	0.994393i	$-0.466276\pi$
$607$	5.21078	0.211499	0.105749	+	0.994393i	$0.466276\pi$
$608$	−1.03589	−0.0420110
$609$	−33.5349	−1.35890
$610$	15.4430	0.625269
$611$	−3.79524	−0.153539
$612$	−0.384043	−0.0155240
$613$	−18.2877	−0.738632	−0.369316	−	0.929304i	$-0.620408\pi$
$613$	−18.2877	−0.738632	−0.369316	+	0.929304i	$0.620408\pi$
$614$	6.57487	0.265340
$615$	1.40403	0.0566159
$616$	−44.5744	−1.79595
$617$	−34.1921	−1.37652	−0.688261	−	0.725463i	$-0.741626\pi$
$617$	−34.1921	−1.37652	−0.688261	+	0.725463i	$0.741626\pi$
$618$	14.3441	0.577003
$619$	42.4186	1.70495	0.852474	−	0.522770i	$-0.175101\pi$
$619$	42.4186	1.70495	0.852474	+	0.522770i	$0.175101\pi$
$620$	−1.14284	−0.0458974
$621$	−16.1666	−0.648745
$622$	33.2349	1.33260
$623$	−1.39715	−0.0559755
$624$	25.0786	1.00395
$625$	2.72945	0.109178
$626$	−19.3619	−0.773857
$627$	−13.3598	−0.533540
$628$	−0.267144	−0.0106602
$629$	−12.6093	−0.502764
$630$	−4.63567	−0.184690
$631$	39.8439	1.58616	0.793080	−	0.609118i	$-0.208476\pi$
$631$	39.8439	1.58616	0.793080	+	0.609118i	$0.208476\pi$
$632$	−16.6564	−0.662557
$633$	2.03763	0.0809887
$634$	−22.5878	−0.897074
$635$	12.6476	0.501905
$636$	0.167319	0.00663463
$637$	5.68956	0.225429
$638$	−62.9528	−2.49232
$639$	−15.3657	−0.607856
$640$	−12.2924	−0.485901
$641$	39.4806	1.55939	0.779696	−	0.626159i	$-0.215374\pi$
$641$	39.4806	1.55939	0.779696	+	0.626159i	$0.215374\pi$
$642$	−5.63168	−0.222265
$643$	−31.6227	−1.24708	−0.623540	−	0.781792i	$-0.714306\pi$
$643$	−31.6227	−1.24708	−0.623540	+	0.781792i	$0.714306\pi$
$644$	−1.82206	−0.0717994
$645$	24.8998	0.980430
$646$	−2.44554	−0.0962184
$647$	4.50126	0.176963	0.0884813	−	0.996078i	$-0.471799\pi$
$647$	4.50126	0.176963	0.0884813	+	0.996078i	$0.471799\pi$
$648$	−32.7521	−1.28662
$649$	40.9716	1.60828
$650$	−15.3443	−0.601854
$651$	22.6521	0.887806
$652$	1.04719	0.0410112
$653$	11.1263	0.435406	0.217703	−	0.976015i	$-0.430144\pi$
$653$	11.1263	0.435406	0.217703	+	0.976015i	$0.430144\pi$
$654$	42.1902	1.64977
$655$	−4.39997	−0.171921
$656$	−1.91842	−0.0749017
$657$	−15.1265	−0.590139
$658$	−3.45490	−0.134686
$659$	−15.8557	−0.617651	−0.308826	−	0.951119i	$-0.599936\pi$
$659$	−15.8557	−0.617651	−0.308826	+	0.951119i	$0.599936\pi$
$660$	3.17267	0.123496
$661$	−16.5510	−0.643757	−0.321879	−	0.946781i	$-0.604314\pi$
$661$	−16.5510	−0.643757	−0.321879	+	0.946781i	$0.604314\pi$
$662$	22.7842	0.885532
$663$	−12.6453	−0.491101
$664$	15.2846	0.593157
$665$	2.98597	0.115791
$666$	10.7878	0.418019
$667$	−30.5865	−1.18431
$668$	−3.69880	−0.143111
$669$	9.27850	0.358728
$670$	−27.9994	−1.08171
$671$	58.1235	2.24383
$672$	−4.87599	−0.188096
$673$	−13.4817	−0.519680	−0.259840	−	0.965652i	$-0.583670\pi$
$673$	−13.4817	−0.519680	−0.259840	+	0.965652i	$0.583670\pi$
$674$	−10.1093	−0.389396
$675$	12.5365	0.482531
$676$	0.239578	0.00921454
$677$	34.4389	1.32359	0.661796	−	0.749684i	$-0.269794\pi$
$677$	34.4389	1.32359	0.661796	+	0.749684i	$0.269794\pi$
$678$	23.1831	0.890340
$679$	−40.2056	−1.54295
$680$	6.90296	0.264716
$681$	13.5180	0.518012
$682$	42.5232	1.62830
$683$	−19.6063	−0.750216	−0.375108	−	0.926981i	$-0.622394\pi$
$683$	−19.6063	−0.750216	−0.375108	+	0.926981i	$0.622394\pi$
$684$	−0.211639	−0.00809223
$685$	−3.83488	−0.146523
$686$	26.9721	1.02980
$687$	56.9440	2.17255
$688$	−34.0224	−1.29709
$689$	1.52854	0.0582327
$690$	−15.2392	−0.580146
$691$	26.3602	1.00279	0.501394	−	0.865219i	$-0.332821\pi$
$691$	26.3602	1.00279	0.501394	+	0.865219i	$0.332821\pi$
$692$	1.71647	0.0652503
$693$	−17.4475	−0.662775
$694$	45.9379	1.74378
$695$	9.10063	0.345207
$696$	−42.7232	−1.61942
$697$	0.967318	0.0366398
$698$	0.263653	0.00997942
$699$	25.8735	0.978627
$700$	1.41293	0.0534038
$701$	0.423485	0.0159948	0.00799740	−	0.999968i	$-0.497454\pi$
$701$	0.423485	0.0159948	0.00799740	+	0.999968i	$0.497454\pi$
$702$	−17.3559	−0.655058
$703$	−6.94875	−0.262077
$704$	−56.3448	−2.12357
$705$	2.92288	0.110082
$706$	−5.57174	−0.209695
$707$	3.99931	0.150409
$708$	2.33935	0.0879181
$709$	−16.9423	−0.636281	−0.318140	−	0.948044i	$-0.603058\pi$
$709$	−16.9423	−0.636281	−0.318140	+	0.948044i	$0.603058\pi$
$710$	23.2365	0.872049
$711$	−6.51972	−0.244509
$712$	−1.77995	−0.0667065
$713$	20.6605	0.773742
$714$	−11.5113	−0.430798
$715$	28.9839	1.08394
$716$	0.742530	0.0277497
$717$	23.1530	0.864664
$718$	7.84131	0.292635
$719$	−18.3278	−0.683513	−0.341756	−	0.939789i	$-0.611022\pi$
$719$	−18.3278	−0.683513	−0.341756	+	0.939789i	$0.611022\pi$
$720$	−5.35868	−0.199706
$721$	−12.0664	−0.449376
$722$	−1.34769	−0.0501560
$723$	−12.8533	−0.478020
$724$	−3.20912	−0.119266
$725$	23.7185	0.880883
$726$	−87.8431	−3.26016
$727$	23.4512	0.869755	0.434878	−	0.900490i	$-0.356792\pi$
$727$	23.4512	0.869755	0.434878	+	0.900490i	$0.356792\pi$
$728$	−23.2503	−0.861714
$729$	10.1990	0.377742
$730$	22.8747	0.846632
$731$	17.1550	0.634499
$732$	3.31867	0.122662
$733$	21.2660	0.785477	0.392738	−	0.919650i	$-0.371528\pi$
$733$	21.2660	0.785477	0.392738	+	0.919650i	$0.371528\pi$
$734$	7.30285	0.269553
$735$	−4.38178	−0.161624
$736$	−4.44729	−0.163929
$737$	−105.383	−3.88182
$738$	−0.827585	−0.0304638
$739$	−1.54623	−0.0568792	−0.0284396	−	0.999596i	$-0.509054\pi$
$739$	−1.54623	−0.0568792	−0.0284396	+	0.999596i	$0.509054\pi$
$740$	1.65018	0.0606617
$741$	−6.96858	−0.255997
$742$	1.39146	0.0510823
$743$	−21.7111	−0.796502	−0.398251	−	0.917277i	$-0.630383\pi$
$743$	−21.7111	−0.796502	−0.398251	+	0.917277i	$0.630383\pi$
$744$	28.8586	1.05801
$745$	−15.9895	−0.585808
$746$	−13.9830	−0.511954
$747$	5.98275	0.218898
$748$	2.18584	0.0799222
$749$	4.73743	0.173102
$750$	29.5654	1.07958
$751$	16.4493	0.600243	0.300122	−	0.953901i	$-0.402973\pi$
$751$	16.4493	0.600243	0.300122	+	0.953901i	$0.402973\pi$
$752$	−3.99374	−0.145637
$753$	12.4227	0.452708
$754$	−32.8366	−1.19584
$755$	16.8899	0.614688
$756$	1.59816	0.0581247
$757$	48.7996	1.77365	0.886827	−	0.462102i	$-0.152905\pi$
$757$	48.7996	1.77365	0.886827	+	0.462102i	$0.152905\pi$
$758$	−44.0341	−1.59939
$759$	−57.3564	−2.08191
$760$	3.80410	0.137989
$761$	−26.0934	−0.945885	−0.472943	−	0.881093i	$-0.656808\pi$
$761$	−26.0934	−0.945885	−0.472943	+	0.881093i	$0.656808\pi$
$762$	−26.8697	−0.973386
$763$	−35.4908	−1.28485
$764$	3.31954	0.120097
$765$	2.70198	0.0976904
$766$	4.96059	0.179233
$767$	21.3711	0.771665
$768$	−8.90645	−0.321384
$769$	20.4733	0.738285	0.369142	−	0.929373i	$-0.379652\pi$
$769$	20.4733	0.738285	0.369142	+	0.929373i	$0.379652\pi$
$770$	26.3847	0.950838
$771$	−35.6929	−1.28545
$772$	3.82570	0.137690
$773$	−13.2231	−0.475601	−0.237800	−	0.971314i	$-0.576426\pi$
$773$	−13.2231	−0.475601	−0.237800	+	0.971314i	$0.576426\pi$
$774$	−14.6769	−0.527549
$775$	−16.0213	−0.575503
$776$	−51.2215	−1.83875
$777$	−32.7081	−1.17340
$778$	14.8100	0.530965
$779$	0.533072	0.0190993
$780$	1.65489	0.0592545
$781$	87.4561	3.12943
$782$	−10.4992	−0.375450
$783$	26.8280	0.958753
$784$	5.98713	0.213826
$785$	1.87953	0.0670832
$786$	9.34767	0.333420
$787$	37.5911	1.33998	0.669989	−	0.742371i	$-0.266299\pi$
$787$	37.5911	1.33998	0.669989	+	0.742371i	$0.266299\pi$
$788$	−0.967935	−0.0344813
$789$	−32.3581	−1.15198
$790$	9.85935	0.350780
$791$	−19.5018	−0.693406
$792$	−22.2279	−0.789835
$793$	30.3176	1.07661
$794$	3.75703	0.133332
$795$	−1.17720	−0.0417508
$796$	−3.49302	−0.123807
$797$	15.1208	0.535606	0.267803	−	0.963474i	$-0.413702\pi$
$797$	15.1208	0.535606	0.267803	+	0.963474i	$0.413702\pi$
$798$	−6.34366	−0.224563
$799$	2.01375	0.0712412
$800$	3.44868	0.121929
$801$	−0.696715	−0.0246172
$802$	−48.7760	−1.72234
$803$	86.0947	3.03822
$804$	−6.01701	−0.212204
$805$	12.8194	0.451824
$806$	22.1804	0.781271
$807$	46.9669	1.65331
$808$	5.09508	0.179244
$809$	−24.7843	−0.871369	−0.435685	−	0.900099i	$-0.643494\pi$
$809$	−24.7843	−0.871369	−0.435685	+	0.900099i	$0.643494\pi$
$810$	19.3868	0.681182
$811$	49.9012	1.75227	0.876134	−	0.482067i	$-0.160114\pi$
$811$	49.9012	1.75227	0.876134	+	0.482067i	$0.160114\pi$
$812$	3.02365	0.106109
$813$	−33.9852	−1.19191
$814$	−61.4006	−2.15209
$815$	−7.36765	−0.258078
$816$	−13.3066	−0.465825
$817$	9.45380	0.330747
$818$	−32.5216	−1.13709
$819$	−9.10073	−0.318005
$820$	−0.126593	−0.00442082
$821$	−7.74058	−0.270148	−0.135074	−	0.990836i	$-0.543127\pi$
$821$	−7.74058	−0.270148	−0.135074	+	0.990836i	$0.543127\pi$
$822$	8.14715	0.284164
$823$	28.5513	0.995235	0.497617	−	0.867397i	$-0.334208\pi$
$823$	28.5513	0.995235	0.497617	+	0.867397i	$0.334208\pi$
$824$	−15.3725	−0.535525
$825$	44.4774	1.54851
$826$	19.4546	0.676911
$827$	−30.5991	−1.06403	−0.532017	−	0.846734i	$-0.678566\pi$
$827$	−30.5991	−1.06403	−0.532017	+	0.846734i	$0.678566\pi$
$828$	−0.908609	−0.0315763
$829$	22.1221	0.768332	0.384166	−	0.923264i	$-0.374489\pi$
$829$	22.1221	0.768332	0.384166	+	0.923264i	$0.374489\pi$
$830$	−9.04732	−0.314037
$831$	9.94187	0.344880
$832$	−29.3898	−1.01891
$833$	−3.01887	−0.104598
$834$	−19.3342	−0.669487
$835$	26.0234	0.900577
$836$	1.20458	0.0416612
$837$	−18.1217	−0.626377
$838$	22.6823	0.783546
$839$	−52.5170	−1.81309	−0.906544	−	0.422111i	$-0.861289\pi$
$839$	−52.5170	−1.81309	−0.906544	+	0.422111i	$0.861289\pi$
$840$	17.9061	0.617819
$841$	21.7572	0.750248
$842$	−40.7210	−1.40334
$843$	−24.2591	−0.835529
$844$	−0.183722	−0.00632396
$845$	−1.68558	−0.0579858
$846$	−1.72285	−0.0592329
$847$	73.8946	2.53905
$848$	1.60848	0.0552356
$849$	−4.93956	−0.169525
$850$	8.14166	0.279257
$851$	−29.8323	−1.02264
$852$	4.99347	0.171073
$853$	12.1350	0.415494	0.207747	−	0.978183i	$-0.433387\pi$
$853$	12.1350	0.415494	0.207747	+	0.978183i	$0.433387\pi$
$854$	27.5988	0.944413
$855$	1.48902	0.0509233
$856$	6.03544	0.206287
$857$	−52.2901	−1.78620	−0.893098	−	0.449862i	$-0.851473\pi$
$857$	−52.2901	−1.78620	−0.893098	+	0.449862i	$0.851473\pi$
$858$	−61.5759	−2.10217
$859$	17.3849	0.593165	0.296583	−	0.955007i	$-0.404153\pi$
$859$	17.3849	0.593165	0.296583	+	0.955007i	$0.404153\pi$
$860$	−2.24507	−0.0765564
$861$	2.50920	0.0855132
$862$	33.4568	1.13954
$863$	−8.77317	−0.298642	−0.149321	−	0.988789i	$-0.547709\pi$
$863$	−8.77317	−0.298642	−0.149321	+	0.988789i	$0.547709\pi$
$864$	3.90079	0.132708
$865$	−12.0764	−0.410611
$866$	50.5529	1.71786
$867$	−27.9302	−0.948561
$868$	−2.04241	−0.0693239
$869$	37.1081	1.25880
$870$	25.2889	0.857374
$871$	−54.9683	−1.86253
$872$	−45.2150	−1.53117
$873$	−20.0493	−0.678567
$874$	−5.78591	−0.195711
$875$	−24.8707	−0.840784
$876$	4.91573	0.166087
$877$	−32.6999	−1.10420	−0.552099	−	0.833778i	$-0.686173\pi$
$877$	−32.6999	−1.10420	−0.552099	+	0.833778i	$0.686173\pi$
$878$	38.3583	1.29453
$879$	−30.3706	−1.02438
$880$	30.4998	1.02815
$881$	−3.20479	−0.107972	−0.0539860	−	0.998542i	$-0.517193\pi$
$881$	−3.20479	−0.107972	−0.0539860	+	0.998542i	$0.517193\pi$
$882$	2.58278	0.0869668
$883$	19.9346	0.670853	0.335426	−	0.942066i	$-0.391120\pi$
$883$	19.9346	0.670853	0.335426	+	0.942066i	$0.391120\pi$
$884$	1.14015	0.0383474
$885$	−16.4588	−0.553256
$886$	21.3652	0.717778
$887$	−2.07109	−0.0695403	−0.0347701	−	0.999395i	$-0.511070\pi$
$887$	−2.07109	−0.0695403	−0.0347701	+	0.999395i	$0.511070\pi$
$888$	−41.6698	−1.39835
$889$	22.6031	0.758083
$890$	1.05360	0.0353166
$891$	72.9669	2.44448
$892$	−0.836588	−0.0280111
$893$	1.10974	0.0371360
$894$	33.9694	1.13611
$895$	−5.22417	−0.174625
$896$	−21.9683	−0.733909
$897$	−29.9175	−0.998916
$898$	34.6142	1.15509
$899$	−34.2854	−1.14348
$900$	0.704587	0.0234862
$901$	−0.811040	−0.0270196
$902$	4.71033	0.156837
$903$	44.4995	1.48085
$904$	−24.8452	−0.826338
$905$	22.5782	0.750525
$906$	−35.8824	−1.19211
$907$	14.9304	0.495757	0.247878	−	0.968791i	$-0.420267\pi$
$907$	14.9304	0.495757	0.247878	+	0.968791i	$0.420267\pi$
$908$	−1.21884	−0.0404487
$909$	1.99434	0.0661479
$910$	13.7624	0.456220
$911$	20.6759	0.685024	0.342512	−	0.939513i	$-0.388722\pi$
$911$	20.6759	0.685024	0.342512	+	0.939513i	$0.388722\pi$
$912$	−7.33305	−0.242821
$913$	−34.0518	−1.12695
$914$	4.36035	0.144228
$915$	−23.3489	−0.771892
$916$	−5.13431	−0.169642
$917$	−7.86336	−0.259671
$918$	9.20902	0.303943
$919$	−42.6318	−1.40629	−0.703146	−	0.711046i	$-0.748222\pi$
$919$	−42.6318	−1.40629	−0.703146	+	0.711046i	$0.748222\pi$
$920$	16.3318	0.538442
$921$	−9.94082	−0.327561
$922$	−27.5310	−0.906686
$923$	45.6177	1.50153
$924$	5.67001	0.186530
$925$	23.1337	0.760631
$926$	26.1252	0.858526
$927$	−6.01715	−0.197629
$928$	7.38012	0.242264
$929$	6.26526	0.205556	0.102778	−	0.994704i	$-0.467227\pi$
$929$	6.26526	0.205556	0.102778	+	0.994704i	$0.467227\pi$
$930$	−17.0821	−0.560144
$931$	−1.66364	−0.0545238
$932$	−2.33287	−0.0764156
$933$	−50.2493	−1.64509
$934$	27.0710	0.885790
$935$	−15.3788	−0.502940
$936$	−11.5942	−0.378970
$937$	10.8452	0.354296	0.177148	−	0.984184i	$-0.443313\pi$
$937$	10.8452	0.354296	0.177148	+	0.984184i	$0.443313\pi$
$938$	−50.0389	−1.63383
$939$	29.2741	0.955323
$940$	−0.263539	−0.00859570
$941$	27.6817	0.902397	0.451199	−	0.892424i	$-0.350997\pi$
$941$	27.6817	0.902397	0.451199	+	0.892424i	$0.350997\pi$
$942$	−3.99303	−0.130100
$943$	2.28858	0.0745265
$944$	22.4888	0.731948
$945$	−11.2441	−0.365771
$946$	83.5358	2.71598
$947$	41.4462	1.34682	0.673410	−	0.739269i	$-0.264829\pi$
$947$	41.4462	1.34682	0.673410	+	0.739269i	$0.264829\pi$
$948$	2.11875	0.0688139
$949$	44.9076	1.45776
$950$	4.48673	0.145569
$951$	34.1514	1.10743
$952$	12.3366	0.399830
$953$	−13.6570	−0.442394	−0.221197	−	0.975229i	$-0.570996\pi$
$953$	−13.6570	−0.442394	−0.221197	+	0.975229i	$0.570996\pi$
$954$	0.693882	0.0224653
$955$	−23.3551	−0.755752
$956$	−2.08757	−0.0675168
$957$	95.1809	3.07676
$958$	−28.9599	−0.935651
$959$	−6.85347	−0.221310
$960$	22.6344	0.730522
$961$	−7.84099	−0.252935
$962$	−32.0270	−1.03259
$963$	2.36242	0.0761278
$964$	1.15891	0.0373259
$965$	−26.9162	−0.866463
$966$	−27.2346	−0.876258
$967$	−17.5676	−0.564937	−0.282468	−	0.959277i	$-0.591153\pi$
$967$	−17.5676	−0.564937	−0.282468	+	0.959277i	$0.591153\pi$
$968$	94.1410	3.02581
$969$	3.69751	0.118781
$970$	30.3193	0.973493
$971$	19.3594	0.621273	0.310637	−	0.950529i	$-0.399458\pi$
$971$	19.3594	0.621273	0.310637	+	0.950529i	$0.399458\pi$
$972$	2.09069	0.0670588
$973$	16.2641	0.521403
$974$	28.5420	0.914544
$975$	23.1997	0.742986
$976$	31.9033	1.02120
$977$	31.4809	1.00716	0.503581	−	0.863948i	$-0.332015\pi$
$977$	31.4809	1.00716	0.503581	+	0.863948i	$0.332015\pi$
$978$	15.6525	0.500511
$979$	3.96547	0.126737
$980$	0.395080	0.0126204
$981$	−17.6982	−0.565060
$982$	−43.1912	−1.37829
$983$	42.2203	1.34662	0.673310	−	0.739361i	$-0.264872\pi$
$983$	42.2203	1.34662	0.673310	+	0.739361i	$0.264872\pi$
$984$	3.19669	0.101907
$985$	6.81004	0.216986
$986$	17.4230	0.554862
$987$	5.22360	0.166269
$988$	0.628317	0.0199894
$989$	40.5870	1.29059
$990$	13.1573	0.418165
$991$	−5.22733	−0.166052	−0.0830258	−	0.996547i	$-0.526458\pi$
$991$	−5.22733	−0.166052	−0.0830258	+	0.996547i	$0.526458\pi$
$992$	−4.98511	−0.158277
$993$	−34.4483	−1.09319
$994$	41.5268	1.31715
$995$	24.5756	0.779099
$996$	−1.94425	−0.0616059
$997$	46.8427	1.48352	0.741761	−	0.670664i	$-0.233991\pi$
$997$	46.8427	1.48352	0.741761	+	0.670664i	$0.233991\pi$
$998$	−49.3597	−1.56245
$999$	26.1665	0.827871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.23	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.23	✓	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)

\(f(q)\)	\(=\)	\(q-1.34769 q^{2} +2.03763 q^{3} -0.183722 q^{4} +1.29260 q^{5} -2.74611 q^{6} +2.31006 q^{7} +2.94299 q^{8} +1.15196 q^{9} +O(q^{10})\)	\(q-1.34769 q^{2} +2.03763 q^{3} -0.183722 q^{4} +1.29260 q^{5} -2.74611 q^{6} +2.31006 q^{7} +2.94299 q^{8} +1.15196 q^{9} -1.74203 q^{10} -6.55654 q^{11} -0.374358 q^{12} -3.41994 q^{13} -3.11325 q^{14} +2.63384 q^{15} -3.59880 q^{16} +1.81461 q^{17} -1.55248 q^{18} +1.00000 q^{19} -0.237478 q^{20} +4.70705 q^{21} +8.83621 q^{22} +4.29320 q^{23} +5.99673 q^{24} -3.32919 q^{25} +4.60903 q^{26} -3.76564 q^{27} -0.424407 q^{28} -7.12441 q^{29} -3.54961 q^{30} +4.81238 q^{31} -1.03589 q^{32} -13.3598 q^{33} -2.44554 q^{34} +2.98597 q^{35} -0.211639 q^{36} -6.94875 q^{37} -1.34769 q^{38} -6.96858 q^{39} +3.80410 q^{40} +0.533072 q^{41} -6.34366 q^{42} +9.45380 q^{43} +1.20458 q^{44} +1.48902 q^{45} -5.78591 q^{46} +1.10974 q^{47} -7.33305 q^{48} -1.66364 q^{49} +4.48673 q^{50} +3.69751 q^{51} +0.628317 q^{52} -0.446950 q^{53} +5.07493 q^{54} -8.47497 q^{55} +6.79846 q^{56} +2.03763 q^{57} +9.60152 q^{58} -6.24897 q^{59} -0.483894 q^{60} -8.86497 q^{61} -6.48562 q^{62} +2.66108 q^{63} +8.59367 q^{64} -4.42061 q^{65} +18.0050 q^{66} +16.0729 q^{67} -0.333383 q^{68} +8.74797 q^{69} -4.02418 q^{70} -13.3388 q^{71} +3.39019 q^{72} -13.1311 q^{73} +9.36478 q^{74} -6.78367 q^{75} -0.183722 q^{76} -15.1460 q^{77} +9.39152 q^{78} -5.65970 q^{79} -4.65181 q^{80} -11.1289 q^{81} -0.718418 q^{82} +5.19356 q^{83} -0.864787 q^{84} +2.34556 q^{85} -12.7408 q^{86} -14.5169 q^{87} -19.2958 q^{88} -0.604811 q^{89} -2.00674 q^{90} -7.90024 q^{91} -0.788753 q^{92} +9.80588 q^{93} -1.49559 q^{94} +1.29260 q^{95} -2.11077 q^{96} -17.4046 q^{97} +2.24208 q^{98} -7.55284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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