LMFDB - Embedded newform 4009.2.a.e.1.20

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:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-1.51298 q^{2} +0.189655 q^{3} +0.289107 q^{4} -0.503823 q^{5} -0.286944 q^{6} -1.74123 q^{7} +2.58855 q^{8} -2.96403 q^{9} +O(q^{10})$	\(q-1.51298 q^{2} +0.189655 q^{3} +0.289107 q^{4} -0.503823 q^{5} -0.286944 q^{6} -1.74123 q^{7} +2.58855 q^{8} -2.96403 q^{9} +0.762274 q^{10} -2.87254 q^{11} +0.0548306 q^{12} -4.23631 q^{13} +2.63444 q^{14} -0.0955526 q^{15} -4.49463 q^{16} -5.18447 q^{17} +4.48452 q^{18} +1.00000 q^{19} -0.145659 q^{20} -0.330233 q^{21} +4.34609 q^{22} -1.46216 q^{23} +0.490931 q^{24} -4.74616 q^{25} +6.40945 q^{26} -1.13111 q^{27} -0.503401 q^{28} +1.40446 q^{29} +0.144569 q^{30} -1.00015 q^{31} +1.62319 q^{32} -0.544792 q^{33} +7.84399 q^{34} +0.877271 q^{35} -0.856922 q^{36} -7.73876 q^{37} -1.51298 q^{38} -0.803438 q^{39} -1.30417 q^{40} -5.17206 q^{41} +0.499636 q^{42} +0.0105236 q^{43} -0.830471 q^{44} +1.49335 q^{45} +2.21222 q^{46} -7.70806 q^{47} -0.852430 q^{48} -3.96812 q^{49} +7.18085 q^{50} -0.983261 q^{51} -1.22475 q^{52} -0.534407 q^{53} +1.71135 q^{54} +1.44725 q^{55} -4.50725 q^{56} +0.189655 q^{57} -2.12493 q^{58} -0.874825 q^{59} -0.0276249 q^{60} +8.73830 q^{61} +1.51321 q^{62} +5.16106 q^{63} +6.53341 q^{64} +2.13435 q^{65} +0.824259 q^{66} +6.40498 q^{67} -1.49887 q^{68} -0.277307 q^{69} -1.32729 q^{70} -2.56425 q^{71} -7.67253 q^{72} +6.75029 q^{73} +11.7086 q^{74} -0.900134 q^{75} +0.289107 q^{76} +5.00175 q^{77} +1.21559 q^{78} -7.16825 q^{79} +2.26450 q^{80} +8.67757 q^{81} +7.82521 q^{82} -7.76565 q^{83} -0.0954727 q^{84} +2.61205 q^{85} -0.0159221 q^{86} +0.266364 q^{87} -7.43570 q^{88} +4.42254 q^{89} -2.25940 q^{90} +7.37639 q^{91} -0.422722 q^{92} -0.189684 q^{93} +11.6621 q^{94} -0.503823 q^{95} +0.307847 q^{96} -7.34010 q^{97} +6.00369 q^{98} +8.51429 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * 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 $

$2$	−1.51298	−1.06984	−0.534919	−	0.844903i	$-0.679658\pi$
$2$	−1.51298	−1.06984	−0.534919	+	0.844903i	$0.679658\pi$
$3$	0.189655	0.109497	0.0547487	−	0.998500i	$-0.482564\pi$
$3$	0.189655	0.109497	0.0547487	+	0.998500i	$0.482564\pi$
$4$	0.289107	0.144553
$5$	−0.503823	−0.225316	−0.112658	−	0.993634i	$-0.535937\pi$
$5$	−0.503823	−0.225316	−0.112658	+	0.993634i	$0.535937\pi$
$6$	−0.286944	−0.117145
$7$	−1.74123	−0.658123	−0.329061	−	0.944309i	$-0.606732\pi$
$7$	−1.74123	−0.658123	−0.329061	+	0.944309i	$0.606732\pi$
$8$	2.58855	0.915189
$9$	−2.96403	−0.988010
$10$	0.762274	0.241052
$11$	−2.87254	−0.866103	−0.433052	−	0.901369i	$-0.642563\pi$
$11$	−2.87254	−0.866103	−0.433052	+	0.901369i	$0.642563\pi$
$12$	0.0548306	0.0158282
$13$	−4.23631	−1.17494	−0.587471	−	0.809245i	$-0.699876\pi$
$13$	−4.23631	−1.17494	−0.587471	+	0.809245i	$0.699876\pi$
$14$	2.63444	0.704084
$15$	−0.0955526	−0.0246716
$16$	−4.49463	−1.12366
$17$	−5.18447	−1.25742	−0.628709	−	0.777641i	$-0.716416\pi$
$17$	−5.18447	−1.25742	−0.628709	+	0.777641i	$0.716416\pi$
$18$	4.48452	1.05701
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.145659	−0.0325703
$21$	−0.330233	−0.0720628
$22$	4.34609	0.926590
$23$	−1.46216	−0.304882	−0.152441	−	0.988313i	$-0.548713\pi$
$23$	−1.46216	−0.304882	−0.152441	+	0.988313i	$0.548713\pi$
$24$	0.490931	0.100211
$25$	−4.74616	−0.949233
$26$	6.40945	1.25700
$27$	−1.13111	−0.217682
$28$	−0.503401	−0.0951339
$29$	1.40446	0.260803	0.130401	−	0.991461i	$-0.458373\pi$
$29$	1.40446	0.260803	0.130401	+	0.991461i	$0.458373\pi$
$30$	0.144569	0.0263946
$31$	−1.00015	−0.179633	−0.0898163	−	0.995958i	$-0.528628\pi$
$31$	−1.00015	−0.179633	−0.0898163	+	0.995958i	$0.528628\pi$
$32$	1.62319	0.286943
$33$	−0.544792	−0.0948361
$34$	7.84399	1.34523
$35$	0.877271	0.148286
$36$	−0.856922	−0.142820
$37$	−7.73876	−1.27224	−0.636122	−	0.771588i	$-0.719463\pi$
$37$	−7.73876	−1.27224	−0.636122	+	0.771588i	$0.719463\pi$
$38$	−1.51298	−0.245438
$39$	−0.803438	−0.128653
$40$	−1.30417	−0.206207
$41$	−5.17206	−0.807739	−0.403870	−	0.914817i	$-0.632335\pi$
$41$	−5.17206	−0.807739	−0.403870	+	0.914817i	$0.632335\pi$
$42$	0.499636	0.0770955
$43$	0.0105236	0.00160484	0.000802420	−	1.00000i	$-0.499745\pi$
$43$	0.0105236	0.00160484	0.000802420		1.00000i	$0.499745\pi$
$44$	−0.830471	−0.125198
$45$	1.49335	0.222615
$46$	2.21222	0.326175
$47$	−7.70806	−1.12434	−0.562168	−	0.827023i	$-0.690033\pi$
$47$	−7.70806	−1.12434	−0.562168	+	0.827023i	$0.690033\pi$
$48$	−0.852430	−0.123038
$49$	−3.96812	−0.566875
$50$	7.18085	1.01553
$51$	−0.983261	−0.137684
$52$	−1.22475	−0.169842
$53$	−0.534407	−0.0734064	−0.0367032	−	0.999326i	$-0.511686\pi$
$53$	−0.534407	−0.0734064	−0.0367032	+	0.999326i	$0.511686\pi$
$54$	1.71135	0.232885
$55$	1.44725	0.195147
$56$	−4.50725	−0.602307
$57$	0.189655	0.0251204
$58$	−2.12493	−0.279016
$59$	−0.874825	−0.113893	−0.0569463	−	0.998377i	$-0.518136\pi$
$59$	−0.874825	−0.113893	−0.0569463	+	0.998377i	$0.518136\pi$
$60$	−0.0276249	−0.00356636
$61$	8.73830	1.11882	0.559412	−	0.828890i	$-0.311027\pi$
$61$	8.73830	1.11882	0.559412	+	0.828890i	$0.311027\pi$
$62$	1.51321	0.192178
$63$	5.16106	0.650232
$64$	6.53341	0.816676
$65$	2.13435	0.264734
$66$	0.824259	0.101459
$67$	6.40498	0.782493	0.391247	−	0.920286i	$-0.372044\pi$
$67$	6.40498	0.782493	0.391247	+	0.920286i	$0.372044\pi$
$68$	−1.49887	−0.181764
$69$	−0.277307	−0.0333838
$70$	−1.32729	−0.158642
$71$	−2.56425	−0.304321	−0.152160	−	0.988356i	$-0.548623\pi$
$71$	−2.56425	−0.304321	−0.152160	+	0.988356i	$0.548623\pi$
$72$	−7.67253	−0.904216
$73$	6.75029	0.790062	0.395031	−	0.918668i	$-0.370734\pi$
$73$	6.75029	0.790062	0.395031	+	0.918668i	$0.370734\pi$
$74$	11.7086	1.36110
$75$	−0.900134	−0.103939
$76$	0.289107	0.0331628
$77$	5.00175	0.570002
$78$	1.21559	0.137638
$79$	−7.16825	−0.806492	−0.403246	−	0.915092i	$-0.632118\pi$
$79$	−7.16825	−0.806492	−0.403246	+	0.915092i	$0.632118\pi$
$80$	2.26450	0.253179
$81$	8.67757	0.964175
$82$	7.82521	0.864150
$83$	−7.76565	−0.852390	−0.426195	−	0.904631i	$-0.640146\pi$
$83$	−7.76565	−0.852390	−0.426195	+	0.904631i	$0.640146\pi$
$84$	−0.0954727	−0.0104169
$85$	2.61205	0.283317
$86$	−0.0159221	−0.00171692
$87$	0.266364	0.0285572
$88$	−7.43570	−0.792648
$89$	4.42254	0.468788	0.234394	−	0.972142i	$-0.424689\pi$
$89$	4.42254	0.468788	0.234394	+	0.972142i	$0.424689\pi$
$90$	−2.25940	−0.238162
$91$	7.37639	0.773255
$92$	−0.422722	−0.0440718
$93$	−0.189684	−0.0196693
$94$	11.6621	1.20286
$95$	−0.503823	−0.0516911
$96$	0.307847	0.0314195
$97$	−7.34010	−0.745274	−0.372637	−	0.927977i	$-0.621546\pi$
$97$	−7.34010	−0.745274	−0.372637	+	0.927977i	$0.621546\pi$
$98$	6.00369	0.606464
$99$	8.51429	0.855719
$100$	−1.37215	−0.137215
$101$	5.13030	0.510484	0.255242	−	0.966877i	$-0.417845\pi$
$101$	5.13030	0.510484	0.255242	+	0.966877i	$0.417845\pi$
$102$	1.48765	0.147300
$103$	−7.90037	−0.778446	−0.389223	−	0.921144i	$-0.627256\pi$
$103$	−7.90037	−0.778446	−0.389223	+	0.921144i	$0.627256\pi$
$104$	−10.9659	−1.07529
$105$	0.166379	0.0162369
$106$	0.808547	0.0785330
$107$	−17.6663	−1.70786	−0.853931	−	0.520386i	$-0.825788\pi$
$107$	−17.6663	−1.70786	−0.853931	+	0.520386i	$0.825788\pi$
$108$	−0.327012	−0.0314667
$109$	19.3850	1.85674	0.928371	−	0.371654i	$-0.121209\pi$
$109$	19.3850	1.85674	0.928371	+	0.371654i	$0.121209\pi$
$110$	−2.18966	−0.208776
$111$	−1.46770	−0.139308
$112$	7.82618	0.739504
$113$	−2.30085	−0.216446	−0.108223	−	0.994127i	$-0.534516\pi$
$113$	−2.30085	−0.216446	−0.108223	+	0.994127i	$0.534516\pi$
$114$	−0.286944	−0.0268748
$115$	0.736671	0.0686950
$116$	0.406040	0.0376999
$117$	12.5566	1.16085
$118$	1.32359	0.121847
$119$	9.02734	0.827535
$120$	−0.247342	−0.0225792
$121$	−2.74852	−0.249865
$122$	−13.2209	−1.19696
$123$	−0.980907	−0.0884454
$124$	−0.289151	−0.0259665
$125$	4.91034	0.439194
$126$	−7.80857	−0.695643
$127$	−6.68761	−0.593429	−0.296715	−	0.954966i	$-0.595891\pi$
$127$	−6.68761	−0.593429	−0.296715	+	0.954966i	$0.595891\pi$
$128$	−13.1313	−1.16065
$129$	0.00199586	0.000175726 0
$130$	−3.22923	−0.283222
$131$	−3.58712	−0.313408	−0.156704	−	0.987646i	$-0.550087\pi$
$131$	−3.58712	−0.313408	−0.156704	+	0.987646i	$0.550087\pi$
$132$	−0.157503	−0.0137089
$133$	−1.74123	−0.150984
$134$	−9.69061	−0.837141
$135$	0.569879	0.0490474
$136$	−13.4202	−1.15078
$137$	12.9737	1.10841	0.554207	−	0.832379i	$-0.313021\pi$
$137$	12.9737	1.10841	0.554207	+	0.832379i	$0.313021\pi$
$138$	0.419560	0.0357153
$139$	−6.47087	−0.548852	−0.274426	−	0.961608i	$-0.588488\pi$
$139$	−6.47087	−0.548852	−0.274426	+	0.961608i	$0.588488\pi$
$140$	0.253625	0.0214352
$141$	−1.46187	−0.123112
$142$	3.87966	0.325574
$143$	12.1690	1.01762
$144$	13.3222	1.11019
$145$	−0.707601	−0.0587631
$146$	−10.2131	−0.845239
$147$	−0.752575	−0.0620714
$148$	−2.23733	−0.183907
$149$	−7.06193	−0.578536	−0.289268	−	0.957248i	$-0.593412\pi$
$149$	−7.06193	−0.578536	−0.289268	+	0.957248i	$0.593412\pi$
$150$	1.36188	0.111197
$151$	−20.1931	−1.64329	−0.821647	−	0.569997i	$-0.806944\pi$
$151$	−20.1931	−1.64329	−0.821647	+	0.569997i	$0.806944\pi$
$152$	2.58855	0.209959
$153$	15.3669	1.24234
$154$	−7.56754	−0.609810
$155$	0.503900	0.0404742
$156$	−0.232280	−0.0185973
$157$	−3.05531	−0.243840	−0.121920	−	0.992540i	$-0.538905\pi$
$157$	−3.05531	−0.243840	−0.121920	+	0.992540i	$0.538905\pi$
$158$	10.8454	0.862815
$159$	−0.101353	−0.00803782
$160$	−0.817801	−0.0646529
$161$	2.54596	0.200650
$162$	−13.1290	−1.03151
$163$	4.40294	0.344865	0.172433	−	0.985021i	$-0.444837\pi$
$163$	4.40294	0.344865	0.172433	+	0.985021i	$0.444837\pi$
$164$	−1.49528	−0.116762
$165$	0.274479	0.0213681
$166$	11.7493	0.911920
$167$	−6.68542	−0.517334	−0.258667	−	0.965967i	$-0.583283\pi$
$167$	−6.68542	−0.517334	−0.258667	+	0.965967i	$0.583283\pi$
$168$	−0.854823	−0.0659511
$169$	4.94633	0.380487
$170$	−3.95198	−0.303103
$171$	−2.96403	−0.226665
$172$	0.00304246	0.000231985 0
$173$	9.25237	0.703444	0.351722	−	0.936104i	$-0.385596\pi$
$173$	9.25237	0.703444	0.351722	+	0.936104i	$0.385596\pi$
$174$	−0.403003	−0.0305516
$175$	8.26415	0.624711
$176$	12.9110	0.973204
$177$	−0.165915	−0.0124709
$178$	−6.69121	−0.501527
$179$	5.56553	0.415987	0.207994	−	0.978130i	$-0.433307\pi$
$179$	5.56553	0.415987	0.207994	+	0.978130i	$0.433307\pi$
$180$	0.431737	0.0321798
$181$	−17.1102	−1.27179	−0.635895	−	0.771776i	$-0.719369\pi$
$181$	−17.1102	−1.27179	−0.635895	+	0.771776i	$0.719369\pi$
$182$	−11.1603	−0.827258
$183$	1.65726	0.122508
$184$	−3.78488	−0.279025
$185$	3.89896	0.286658
$186$	0.286988	0.0210430
$187$	14.8926	1.08905
$188$	−2.22845	−0.162527
$189$	1.96952	0.143261
$190$	0.762274	0.0553011
$191$	13.2123	0.956008	0.478004	−	0.878358i	$-0.341360\pi$
$191$	13.2123	0.956008	0.478004	+	0.878358i	$0.341360\pi$
$192$	1.23909	0.0894239
$193$	0.987167	0.0710578	0.0355289	−	0.999369i	$-0.488688\pi$
$193$	0.987167	0.0710578	0.0355289	+	0.999369i	$0.488688\pi$
$194$	11.1054	0.797323
$195$	0.404791	0.0289877
$196$	−1.14721	−0.0819437
$197$	11.4595	0.816452	0.408226	−	0.912881i	$-0.366147\pi$
$197$	11.4595	0.816452	0.408226	+	0.912881i	$0.366147\pi$
$198$	−12.8820	−0.915481
$199$	1.54881	0.109792	0.0548960	−	0.998492i	$-0.482517\pi$
$199$	1.54881	0.109792	0.0548960	+	0.998492i	$0.482517\pi$
$200$	−12.2857	−0.868727
$201$	1.21474	0.0856810
$202$	−7.76203	−0.546135
$203$	−2.44549	−0.171640
$204$	−0.284268	−0.0199027
$205$	2.60580	0.181997
$206$	11.9531	0.832811
$207$	4.33390	0.301227
$208$	19.0407	1.32023
$209$	−2.87254	−0.198698
$210$	−0.251728	−0.0173709
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−0.154501	−0.0106112
$213$	−0.486324	−0.0333224
$214$	26.7287	1.82714
$215$	−0.00530205	−0.000361597 0
$216$	−2.92793	−0.199220
$217$	1.74149	0.118220
$218$	−29.3290	−1.98641
$219$	1.28023	0.0865098
$220$	0.418410	0.0282092
$221$	21.9630	1.47739
$222$	2.22059	0.149037
$223$	−2.49565	−0.167121	−0.0835606	−	0.996503i	$-0.526629\pi$
$223$	−2.49565	−0.167121	−0.0835606	+	0.996503i	$0.526629\pi$
$224$	−2.82635	−0.188843
$225$	14.0678	0.937852
$226$	3.48114	0.231562
$227$	6.58955	0.437363	0.218682	−	0.975796i	$-0.429824\pi$
$227$	6.58955	0.437363	0.218682	+	0.975796i	$0.429824\pi$
$228$	0.0548306	0.00363125
$229$	−11.3473	−0.749847	−0.374924	−	0.927056i	$-0.622331\pi$
$229$	−11.3473	−0.749847	−0.374924	+	0.927056i	$0.622331\pi$
$230$	−1.11457	−0.0734925
$231$	0.948607	0.0624138
$232$	3.63552	0.238684
$233$	5.99664	0.392853	0.196427	−	0.980519i	$-0.437066\pi$
$233$	5.99664	0.392853	0.196427	+	0.980519i	$0.437066\pi$
$234$	−18.9978	−1.24193
$235$	3.88350	0.253331
$236$	−0.252918	−0.0164636
$237$	−1.35950	−0.0883088
$238$	−13.6582	−0.885329
$239$	−30.1085	−1.94756	−0.973778	−	0.227501i	$-0.926944\pi$
$239$	−30.1085	−1.94756	−0.973778	+	0.227501i	$0.926944\pi$
$240$	0.429474	0.0277224
$241$	−10.5579	−0.680093	−0.340047	−	0.940409i	$-0.610443\pi$
$241$	−10.5579	−0.680093	−0.340047	+	0.940409i	$0.610443\pi$
$242$	4.15845	0.267315
$243$	5.03907	0.323257
$244$	2.52630	0.161730
$245$	1.99923	0.127726
$246$	1.48409	0.0946223
$247$	−4.23631	−0.269550
$248$	−2.58894	−0.164398
$249$	−1.47279	−0.0933346
$250$	−7.42924	−0.469867
$251$	14.2059	0.896671	0.448336	−	0.893865i	$-0.352017\pi$
$251$	14.2059	0.896671	0.448336	+	0.893865i	$0.352017\pi$
$252$	1.49210	0.0939933
$253$	4.20012	0.264059
$254$	10.1182	0.634873
$255$	0.495389	0.0310225
$256$	6.80057	0.425035
$257$	21.8837	1.36507	0.682535	−	0.730853i	$-0.260878\pi$
$257$	21.8837	1.36507	0.682535	+	0.730853i	$0.260878\pi$
$258$	−0.00301970	−0.000187998 0
$259$	13.4750	0.837293
$260$	0.617055	0.0382682
$261$	−4.16288	−0.257676
$262$	5.42724	0.335296
$263$	10.3139	0.635984	0.317992	−	0.948093i	$-0.396991\pi$
$263$	10.3139	0.635984	0.317992	+	0.948093i	$0.396991\pi$
$264$	−1.41022	−0.0867930
$265$	0.269246	0.0165397
$266$	2.63444	0.161528
$267$	0.838757	0.0513311
$268$	1.85172	0.113112
$269$	18.3718	1.12015	0.560075	−	0.828442i	$-0.310772\pi$
$269$	18.3718	1.12015	0.560075	+	0.828442i	$0.310772\pi$
$270$	−0.862215	−0.0524727
$271$	−19.6750	−1.19517	−0.597584	−	0.801806i	$-0.703873\pi$
$271$	−19.6750	−1.19517	−0.597584	+	0.801806i	$0.703873\pi$
$272$	23.3023	1.41291
$273$	1.39897	0.0846695
$274$	−19.6289	−1.18582
$275$	13.6335	0.822133
$276$	−0.0801714	−0.00482575
$277$	−27.3334	−1.64231	−0.821154	−	0.570707i	$-0.806669\pi$
$277$	−27.3334	−1.64231	−0.821154	+	0.570707i	$0.806669\pi$
$278$	9.79029	0.587183
$279$	2.96448	0.177479
$280$	2.27086	0.135710
$281$	−5.81817	−0.347083	−0.173541	−	0.984827i	$-0.555521\pi$
$281$	−5.81817	−0.347083	−0.173541	+	0.984827i	$0.555521\pi$
$282$	2.21179	0.131710
$283$	27.9077	1.65894	0.829471	−	0.558550i	$-0.188642\pi$
$283$	27.9077	1.65894	0.829471	+	0.558550i	$0.188642\pi$
$284$	−0.741343	−0.0439906
$285$	−0.0955526	−0.00566005
$286$	−18.4114	−1.08869
$287$	9.00573	0.531592
$288$	−4.81119	−0.283502
$289$	9.87870	0.581100
$290$	1.07059	0.0628670
$291$	−1.39209	−0.0816057
$292$	1.95156	0.114206
$293$	−25.7117	−1.50209	−0.751046	−	0.660249i	$-0.770451\pi$
$293$	−25.7117	−1.50209	−0.751046	+	0.660249i	$0.770451\pi$
$294$	1.13863	0.0664063
$295$	0.440757	0.0256619
$296$	−20.0321	−1.16434
$297$	3.24916	0.188535
$298$	10.6846	0.618939
$299$	6.19418	0.358219
$300$	−0.260235	−0.0150247
$301$	−0.0183241	−0.00105618
$302$	30.5518	1.75806
$303$	0.972987	0.0558967
$304$	−4.49463	−0.257785
$305$	−4.40255	−0.252090
$306$	−23.2498	−1.32910
$307$	−4.04044	−0.230600	−0.115300	−	0.993331i	$-0.536783\pi$
$307$	−4.04044	−0.230600	−0.115300	+	0.993331i	$0.536783\pi$
$308$	1.44604	0.0823958
$309$	−1.49835	−0.0852379
$310$	−0.762390	−0.0433008
$311$	−0.709669	−0.0402416	−0.0201208	−	0.999798i	$-0.506405\pi$
$311$	−0.709669	−0.0402416	−0.0201208	+	0.999798i	$0.506405\pi$
$312$	−2.07974	−0.117742
$313$	9.51663	0.537912	0.268956	−	0.963152i	$-0.413321\pi$
$313$	9.51663	0.537912	0.268956	+	0.963152i	$0.413321\pi$
$314$	4.62263	0.260870
$315$	−2.60026	−0.146508
$316$	−2.07239	−0.116581
$317$	14.1779	0.796308	0.398154	−	0.917319i	$-0.369651\pi$
$317$	14.1779	0.796308	0.398154	+	0.917319i	$0.369651\pi$
$318$	0.153345	0.00859917
$319$	−4.03438	−0.225882
$320$	−3.29168	−0.184010
$321$	−3.35050	−0.187007
$322$	−3.85199	−0.214663
$323$	−5.18447	−0.288471
$324$	2.50875	0.139375
$325$	20.1062	1.11529
$326$	−6.66156	−0.368950
$327$	3.67646	0.203309
$328$	−13.3881	−0.739234
$329$	13.4215	0.739951
$330$	−0.415281	−0.0228604
$331$	30.0122	1.64962	0.824809	−	0.565412i	$-0.191283\pi$
$331$	30.0122	1.64962	0.824809	+	0.565412i	$0.191283\pi$
$332$	−2.24510	−0.123216
$333$	22.9379	1.25699
$334$	10.1149	0.553463
$335$	−3.22698	−0.176309
$336$	1.48428	0.0809739
$337$	−14.9158	−0.812515	−0.406258	−	0.913759i	$-0.633166\pi$
$337$	−14.9158	−0.812515	−0.406258	+	0.913759i	$0.633166\pi$
$338$	−7.48370	−0.407060
$339$	−0.436369	−0.0237003
$340$	0.755162	0.0409544
$341$	2.87298	0.155580
$342$	4.48452	0.242495
$343$	19.0980	1.03120
$344$	0.0272409	0.00146873
$345$	0.139714	0.00752193
$346$	−13.9986	−0.752572
$347$	−5.67698	−0.304756	−0.152378	−	0.988322i	$-0.548693\pi$
$347$	−5.67698	−0.304756	−0.152378	+	0.988322i	$0.548693\pi$
$348$	0.0770077	0.00412804
$349$	29.8435	1.59749	0.798744	−	0.601672i	$-0.205498\pi$
$349$	29.8435	1.59749	0.798744	+	0.601672i	$0.205498\pi$
$350$	−12.5035	−0.668340
$351$	4.79173	0.255764
$352$	−4.66268	−0.248522
$353$	−33.1170	−1.76264	−0.881320	−	0.472520i	$-0.843345\pi$
$353$	−33.1170	−1.76264	−0.881320	+	0.472520i	$0.843345\pi$
$354$	0.251026	0.0133419
$355$	1.29193	0.0685685
$356$	1.27859	0.0677649
$357$	1.71208	0.0906130
$358$	−8.42053	−0.445039
$359$	21.4228	1.13065	0.565327	−	0.824867i	$-0.308750\pi$
$359$	21.4228	1.13065	0.565327	+	0.824867i	$0.308750\pi$
$360$	3.86560	0.203735
$361$	1.00000	0.0526316
$362$	25.8874	1.36061
$363$	−0.521271	−0.0273596
$364$	2.13256	0.111777
$365$	−3.40095	−0.178014
$366$	−2.50741	−0.131064
$367$	4.41654	0.230542	0.115271	−	0.993334i	$-0.463226\pi$
$367$	4.41654	0.230542	0.115271	+	0.993334i	$0.463226\pi$
$368$	6.57189	0.342583
$369$	15.3301	0.798055
$370$	−5.89905	−0.306677
$371$	0.930525	0.0483104
$372$	−0.0548390	−0.00284327
$373$	−11.3595	−0.588171	−0.294085	−	0.955779i	$-0.595015\pi$
$373$	−11.3595	−0.588171	−0.294085	+	0.955779i	$0.595015\pi$
$374$	−22.5322	−1.16511
$375$	0.931271	0.0480906
$376$	−19.9527	−1.02898
$377$	−5.94975	−0.306428
$378$	−2.97984	−0.153267
$379$	22.3586	1.14848	0.574242	−	0.818686i	$-0.305297\pi$
$379$	22.3586	1.14848	0.574242	+	0.818686i	$0.305297\pi$
$380$	−0.145659	−0.00747213
$381$	−1.26834	−0.0649790
$382$	−19.9899	−1.02277
$383$	24.0336	1.22806	0.614030	−	0.789283i	$-0.289548\pi$
$383$	24.0336	1.22806	0.614030	+	0.789283i	$0.289548\pi$
$384$	−2.49042	−0.127089
$385$	−2.51999	−0.128431
$386$	−1.49356	−0.0760204
$387$	−0.0311924	−0.00158560
$388$	−2.12207	−0.107732
$389$	38.1073	1.93212	0.966060	−	0.258319i	$-0.0831687\pi$
$389$	38.1073	1.93212	0.966060	+	0.258319i	$0.0831687\pi$
$390$	−0.612440	−0.0310121
$391$	7.58054	0.383364
$392$	−10.2717	−0.518798
$393$	−0.680317	−0.0343174
$394$	−17.3379	−0.873472
$395$	3.61153	0.181716
$396$	2.46154	0.123697
$397$	−27.7379	−1.39213	−0.696063	−	0.717981i	$-0.745067\pi$
$397$	−27.7379	−1.39213	−0.696063	+	0.717981i	$0.745067\pi$
$398$	−2.34331	−0.117460
$399$	−0.330233	−0.0165323
$400$	21.3322	1.06661
$401$	−24.4227	−1.21961	−0.609806	−	0.792551i	$-0.708753\pi$
$401$	−24.4227	−1.21961	−0.609806	+	0.792551i	$0.708753\pi$
$402$	−1.83787	−0.0916648
$403$	4.23696	0.211058
$404$	1.48320	0.0737922
$405$	−4.37196	−0.217244
$406$	3.69998	0.183627
$407$	22.2299	1.10189
$408$	−2.54522	−0.126007
$409$	−6.49052	−0.320935	−0.160468	−	0.987041i	$-0.551300\pi$
$409$	−6.49052	−0.320935	−0.160468	+	0.987041i	$0.551300\pi$
$410$	−3.94252	−0.194707
$411$	2.46052	0.121369
$412$	−2.28405	−0.112527
$413$	1.52327	0.0749553
$414$	−6.55710	−0.322264
$415$	3.91251	0.192058
$416$	−6.87635	−0.337141
$417$	−1.22723	−0.0600979
$418$	4.34609	0.212574
$419$	4.99180	0.243865	0.121933	−	0.992538i	$-0.461091\pi$
$419$	4.99180	0.243865	0.121933	+	0.992538i	$0.461091\pi$
$420$	0.0481013	0.00234710
$421$	−3.20597	−0.156250	−0.0781248	−	0.996944i	$-0.524893\pi$
$421$	−3.20597	−0.156250	−0.0781248	+	0.996944i	$0.524893\pi$
$422$	1.51298	0.0736507
$423$	22.8469	1.11086
$424$	−1.38334	−0.0671808
$425$	24.6063	1.19358
$426$	0.735798	0.0356495
$427$	−15.2154	−0.736324
$428$	−5.10744	−0.246877
$429$	2.30791	0.111427
$430$	0.00802189	0.000386850 0
$431$	6.03157	0.290531	0.145265	−	0.989393i	$-0.453596\pi$
$431$	6.03157	0.290531	0.145265	+	0.989393i	$0.453596\pi$
$432$	5.08392	0.244600
$433$	0.0673072	0.00323458	0.00161729	−	0.999999i	$-0.499485\pi$
$433$	0.0673072	0.00323458	0.00161729	+	0.999999i	$0.499485\pi$
$434$	−2.63484	−0.126477
$435$	−0.134200	−0.00643441
$436$	5.60433	0.268399
$437$	−1.46216	−0.0699448
$438$	−1.93696	−0.0925515
$439$	1.70449	0.0813507	0.0406754	−	0.999172i	$-0.487049\pi$
$439$	1.70449	0.0813507	0.0406754	+	0.999172i	$0.487049\pi$
$440$	3.74628	0.178597
$441$	11.7616	0.560078
$442$	−33.2296	−1.58057
$443$	−9.50602	−0.451645	−0.225822	−	0.974169i	$-0.572507\pi$
$443$	−9.50602	−0.451645	−0.225822	+	0.974169i	$0.572507\pi$
$444$	−0.424321	−0.0201374
$445$	−2.22818	−0.105626
$446$	3.77587	0.178793
$447$	−1.33933	−0.0633482
$448$	−11.3762	−0.537473
$449$	−2.44932	−0.115591	−0.0577953	−	0.998328i	$-0.518407\pi$
$449$	−2.44932	−0.115591	−0.0577953	+	0.998328i	$0.518407\pi$
$450$	−21.2843	−1.00335
$451$	14.8569	0.699586
$452$	−0.665192	−0.0312880
$453$	−3.82973	−0.179936
$454$	−9.96985	−0.467908
$455$	−3.71639	−0.174227
$456$	0.490931	0.0229900
$457$	18.2434	0.853392	0.426696	−	0.904395i	$-0.359677\pi$
$457$	18.2434	0.853392	0.426696	+	0.904395i	$0.359677\pi$
$458$	17.1682	0.802215
$459$	5.86420	0.273717
$460$	0.212977	0.00993009
$461$	−12.7199	−0.592427	−0.296213	−	0.955122i	$-0.595724\pi$
$461$	−12.7199	−0.592427	−0.296213	+	0.955122i	$0.595724\pi$
$462$	−1.43522	−0.0667726
$463$	−14.2827	−0.663772	−0.331886	−	0.943320i	$-0.607685\pi$
$463$	−14.2827	−0.663772	−0.331886	+	0.943320i	$0.607685\pi$
$464$	−6.31255	−0.293053
$465$	0.0955672	0.00443182
$466$	−9.07280	−0.420289
$467$	−14.0363	−0.649522	−0.324761	−	0.945796i	$-0.605284\pi$
$467$	−14.0363	−0.649522	−0.324761	+	0.945796i	$0.605284\pi$
$468$	3.63019	0.167805
$469$	−11.1525	−0.514976
$470$	−5.87565	−0.271024
$471$	−0.579456	−0.0266999
$472$	−2.26453	−0.104233
$473$	−0.0302296	−0.00138996
$474$	2.05689	0.0944761
$475$	−4.74616	−0.217769
$476$	2.60987	0.119623
$477$	1.58400	0.0725263
$478$	45.5535	2.08357
$479$	−7.09470	−0.324165	−0.162083	−	0.986777i	$-0.551821\pi$
$479$	−7.09470	−0.324165	−0.162083	+	0.986777i	$0.551821\pi$
$480$	−0.155100	−0.00707933
$481$	32.7838	1.49481
$482$	15.9739	0.727590
$483$	0.482855	0.0219707
$484$	−0.794616	−0.0361189
$485$	3.69811	0.167923
$486$	−7.62402	−0.345832
$487$	−18.7531	−0.849783	−0.424891	−	0.905244i	$-0.639688\pi$
$487$	−18.7531	−0.849783	−0.424891	+	0.905244i	$0.639688\pi$
$488$	22.6195	1.02394
$489$	0.835041	0.0377619
$490$	−3.02480	−0.136646
$491$	3.16939	0.143032	0.0715162	−	0.997439i	$-0.477216\pi$
$491$	3.16939	0.143032	0.0715162	+	0.997439i	$0.477216\pi$
$492$	−0.283587	−0.0127851
$493$	−7.28140	−0.327938
$494$	6.40945	0.288375
$495$	−4.28970	−0.192807
$496$	4.49532	0.201846
$497$	4.46495	0.200280
$498$	2.22831	0.0998529
$499$	17.5407	0.785228	0.392614	−	0.919703i	$-0.371571\pi$
$499$	17.5407	0.785228	0.392614	+	0.919703i	$0.371571\pi$
$500$	1.41961	0.0634870
$501$	−1.26793	−0.0566467
$502$	−21.4933	−0.959293
$503$	10.4070	0.464026	0.232013	−	0.972713i	$-0.425469\pi$
$503$	10.4070	0.464026	0.232013	+	0.972713i	$0.425469\pi$
$504$	13.3596	0.595085
$505$	−2.58476	−0.115020
$506$	−6.35470	−0.282501
$507$	0.938098	0.0416624
$508$	−1.93343	−0.0857822
$509$	33.2401	1.47334	0.736671	−	0.676252i	$-0.236397\pi$
$509$	33.2401	1.47334	0.736671	+	0.676252i	$0.236397\pi$
$510$	−0.749514	−0.0331890
$511$	−11.7538	−0.519958
$512$	15.9735	0.705934
$513$	−1.13111	−0.0499397
$514$	−33.1096	−1.46040
$515$	3.98038	0.175397
$516$	0.000577018 0	2.54018e−5 0
$517$	22.1417	0.973791
$518$	−20.3873	−0.895768
$519$	1.75476	0.0770254
$520$	5.52486	0.242281
$521$	−28.9232	−1.26715	−0.633574	−	0.773682i	$-0.718413\pi$
$521$	−28.9232	−1.26715	−0.633574	+	0.773682i	$0.718413\pi$
$522$	6.29835	0.275671
$523$	−27.5799	−1.20598	−0.602991	−	0.797748i	$-0.706025\pi$
$523$	−27.5799	−1.20598	−0.602991	+	0.797748i	$0.706025\pi$
$524$	−1.03706	−0.0453043
$525$	1.56734	0.0684043
$526$	−15.6048	−0.680400
$527$	5.18526	0.225873
$528$	2.44864	0.106563
$529$	−20.8621	−0.907047
$530$	−0.407364	−0.0176948
$531$	2.59301	0.112527
$532$	−0.503401	−0.0218252
$533$	21.9104	0.949047
$534$	−1.26902	−0.0549160
$535$	8.90067	0.384809
$536$	16.5796	0.716129
$537$	1.05553	0.0455495
$538$	−27.7962	−1.19838
$539$	11.3986	0.490972
$540$	0.164756	0.00708996
$541$	−33.8734	−1.45633	−0.728165	−	0.685402i	$-0.759626\pi$
$541$	−33.8734	−1.45633	−0.728165	+	0.685402i	$0.759626\pi$
$542$	29.7678	1.27864
$543$	−3.24503	−0.139258
$544$	−8.41539	−0.360807
$545$	−9.76658	−0.418355
$546$	−2.11661	−0.0905827
$547$	−28.5345	−1.22005	−0.610024	−	0.792383i	$-0.708840\pi$
$547$	−28.5345	−1.22005	−0.610024	+	0.792383i	$0.708840\pi$
$548$	3.75078	0.160225
$549$	−25.9006	−1.10541
$550$	−20.6273	−0.879549
$551$	1.40446	0.0598322
$552$	−0.717822	−0.0305525
$553$	12.4816	0.530770
$554$	41.3549	1.75700
$555$	0.739459	0.0313883
$556$	−1.87077	−0.0793385
$557$	20.4106	0.864826	0.432413	−	0.901676i	$-0.357662\pi$
$557$	20.4106	0.864826	0.432413	+	0.901676i	$0.357662\pi$
$558$	−4.48520	−0.189874
$559$	−0.0445814	−0.00188559
$560$	−3.94301	−0.166622
$561$	2.82446	0.119249
$562$	8.80277	0.371322
$563$	35.9091	1.51339	0.756693	−	0.653770i	$-0.226814\pi$
$563$	35.9091	1.51339	0.756693	+	0.653770i	$0.226814\pi$
$564$	−0.422638	−0.0177963
$565$	1.15922	0.0487688
$566$	−42.2238	−1.77480
$567$	−15.1096	−0.634545
$568$	−6.63768	−0.278511
$569$	−42.7840	−1.79360	−0.896799	−	0.442439i	$-0.854113\pi$
$569$	−42.7840	−1.79360	−0.896799	+	0.442439i	$0.854113\pi$
$570$	0.144569	0.00605533
$571$	34.4904	1.44338	0.721689	−	0.692218i	$-0.243366\pi$
$571$	34.4904	1.44338	0.721689	+	0.692218i	$0.243366\pi$
$572$	3.51813	0.147101
$573$	2.50578	0.104681
$574$	−13.6255	−0.568717
$575$	6.93967	0.289404
$576$	−19.3652	−0.806884
$577$	−5.92656	−0.246726	−0.123363	−	0.992362i	$-0.539368\pi$
$577$	−5.92656	−0.246726	−0.123363	+	0.992362i	$0.539368\pi$
$578$	−14.9463	−0.621683
$579$	0.187221	0.00778065
$580$	−0.204572	−0.00849441
$581$	13.5218	0.560977
$582$	2.10620	0.0873048
$583$	1.53510	0.0635775
$584$	17.4734	0.723056
$585$	−6.32628	−0.261559
$586$	38.9013	1.60700
$587$	15.4812	0.638977	0.319489	−	0.947590i	$-0.396489\pi$
$587$	15.4812	0.638977	0.319489	+	0.947590i	$0.396489\pi$
$588$	−0.217575	−0.00897263
$589$	−1.00015	−0.0412106
$590$	−0.666856	−0.0274540
$591$	2.17334	0.0893995
$592$	34.7829	1.42957
$593$	2.25748	0.0927034	0.0463517	−	0.998925i	$-0.485240\pi$
$593$	2.25748	0.0927034	0.0463517	+	0.998925i	$0.485240\pi$
$594$	−4.91591	−0.201702
$595$	−4.54818	−0.186457
$596$	−2.04165	−0.0836293
$597$	0.293739	0.0120219
$598$	−9.37167	−0.383236
$599$	−7.14134	−0.291787	−0.145894	−	0.989300i	$-0.546606\pi$
$599$	−7.14134	−0.291787	−0.145894	+	0.989300i	$0.546606\pi$
$600$	−2.33004	−0.0951235
$601$	5.40390	0.220430	0.110215	−	0.993908i	$-0.464846\pi$
$601$	5.40390	0.220430	0.110215	+	0.993908i	$0.464846\pi$
$602$	0.0277239	0.00112994
$603$	−18.9846	−0.773111
$604$	−5.83797	−0.237544
$605$	1.38477	0.0562988
$606$	−1.47211	−0.0598004
$607$	27.5261	1.11725	0.558626	−	0.829420i	$-0.311329\pi$
$607$	27.5261	1.11725	0.558626	+	0.829420i	$0.311329\pi$
$608$	1.62319	0.0658291
$609$	−0.463801	−0.0187941
$610$	6.66097	0.269695
$611$	32.6537	1.32103
$612$	4.44268	0.179585
$613$	17.6384	0.712406	0.356203	−	0.934409i	$-0.384071\pi$
$613$	17.6384	0.712406	0.356203	+	0.934409i	$0.384071\pi$
$614$	6.11310	0.246705
$615$	0.494203	0.0199282
$616$	12.9473	0.521660
$617$	−13.4987	−0.543439	−0.271720	−	0.962376i	$-0.587592\pi$
$617$	−13.4987	−0.543439	−0.271720	+	0.962376i	$0.587592\pi$
$618$	2.26697	0.0911907
$619$	−29.8385	−1.19931	−0.599655	−	0.800259i	$-0.704695\pi$
$619$	−29.8385	−1.19931	−0.599655	+	0.800259i	$0.704695\pi$
$620$	0.145681	0.00585068
$621$	1.65387	0.0663674
$622$	1.07371	0.0430520
$623$	−7.70065	−0.308520
$624$	3.61116	0.144562
$625$	21.2569	0.850275
$626$	−14.3985	−0.575479
$627$	−0.544792	−0.0217569
$628$	−0.883312	−0.0352480
$629$	40.1214	1.59974
$630$	3.93414	0.156740
$631$	46.4378	1.84866	0.924331	−	0.381592i	$-0.124624\pi$
$631$	46.4378	1.84866	0.924331	+	0.381592i	$0.124624\pi$
$632$	−18.5554	−0.738092
$633$	−0.189655	−0.00753812
$634$	−21.4508	−0.851921
$635$	3.36937	0.133709
$636$	−0.0293019	−0.00116189
$637$	16.8102	0.666045
$638$	6.10393	0.241657
$639$	7.60052	0.300672
$640$	6.61584	0.261514
$641$	−0.496242	−0.0196004	−0.00980019	−	0.999952i	$-0.503120\pi$
$641$	−0.496242	−0.0196004	−0.00980019	+	0.999952i	$0.503120\pi$
$642$	5.06924	0.200067
$643$	−44.6920	−1.76248	−0.881241	−	0.472668i	$-0.843291\pi$
$643$	−44.6920	−1.76248	−0.881241	+	0.472668i	$0.843291\pi$
$644$	0.736055	0.0290046
$645$	−0.00100556	−3.95939e−5 0
$646$	7.84399	0.308618
$647$	−6.57681	−0.258561	−0.129280	−	0.991608i	$-0.541267\pi$
$647$	−6.57681	−0.258561	−0.129280	+	0.991608i	$0.541267\pi$
$648$	22.4623	0.882402
$649$	2.51297	0.0986427
$650$	−30.4203	−1.19318
$651$	0.330283	0.0129448
$652$	1.27292	0.0498515
$653$	−22.6869	−0.887805	−0.443903	−	0.896075i	$-0.646406\pi$
$653$	−22.6869	−0.887805	−0.443903	+	0.896075i	$0.646406\pi$
$654$	−5.56241	−0.217507
$655$	1.80727	0.0706161
$656$	23.2465	0.907623
$657$	−20.0081	−0.780590
$658$	−20.3064	−0.791628
$659$	46.0560	1.79409	0.897043	−	0.441942i	$-0.145710\pi$
$659$	46.0560	1.79409	0.897043	+	0.441942i	$0.145710\pi$
$660$	0.0793537	0.00308884
$661$	−5.20104	−0.202297	−0.101149	−	0.994871i	$-0.532252\pi$
$661$	−5.20104	−0.202297	−0.101149	+	0.994871i	$0.532252\pi$
$662$	−45.4078	−1.76482
$663$	4.16540	0.161771
$664$	−20.1017	−0.780098
$665$	0.877271	0.0340191
$666$	−34.7046	−1.34478
$667$	−2.05356	−0.0795141
$668$	−1.93280	−0.0747823
$669$	−0.473313	−0.0182994
$670$	4.88235	0.188622
$671$	−25.1011	−0.969018
$672$	−0.536032	−0.0206779
$673$	1.75836	0.0677800	0.0338900	−	0.999426i	$-0.489210\pi$
$673$	1.75836	0.0677800	0.0338900	+	0.999426i	$0.489210\pi$
$674$	22.5673	0.869260
$675$	5.36843	0.206631
$676$	1.43002	0.0550008
$677$	7.29534	0.280383	0.140191	−	0.990124i	$-0.455228\pi$
$677$	7.29534	0.280383	0.140191	+	0.990124i	$0.455228\pi$
$678$	0.660217	0.0253555
$679$	12.7808	0.490482
$680$	6.76142	0.259289
$681$	1.24974	0.0478902
$682$	−4.34675	−0.166446
$683$	−17.8920	−0.684620	−0.342310	−	0.939587i	$-0.611209\pi$
$683$	−17.8920	−0.684620	−0.342310	+	0.939587i	$0.611209\pi$
$684$	−0.856922	−0.0327652
$685$	−6.53643	−0.249744
$686$	−28.8949	−1.10321
$687$	−2.15206	−0.0821064
$688$	−0.0472999	−0.00180329
$689$	2.26391	0.0862483
$690$	−0.211384	−0.00804724
$691$	16.0732	0.611453	0.305726	−	0.952119i	$-0.401101\pi$
$691$	16.0732	0.611453	0.305726	+	0.952119i	$0.401101\pi$
$692$	2.67492	0.101685
$693$	−14.8253	−0.563168
$694$	8.58916	0.326040
$695$	3.26017	0.123665
$696$	0.689496	0.0261353
$697$	26.8144	1.01567
$698$	−45.1526	−1.70905
$699$	1.13729	0.0430164
$700$	2.38922	0.0903042
$701$	−11.8442	−0.447351	−0.223676	−	0.974664i	$-0.571806\pi$
$701$	−11.8442	−0.447351	−0.223676	+	0.974664i	$0.571806\pi$
$702$	−7.24979	−0.273626
$703$	−7.73876	−0.291873
$704$	−18.7675	−0.707325
$705$	0.736525	0.0277392
$706$	50.1053	1.88574
$707$	−8.93302	−0.335961
$708$	−0.0479672	−0.00180272
$709$	−36.7013	−1.37835	−0.689173	−	0.724597i	$-0.742026\pi$
$709$	−36.7013	−1.37835	−0.689173	+	0.724597i	$0.742026\pi$
$710$	−1.95466	−0.0733571
$711$	21.2469	0.796822
$712$	11.4479	0.429030
$713$	1.46239	0.0547668
$714$	−2.59035	−0.0969412
$715$	−6.13100	−0.229287
$716$	1.60903	0.0601324
$717$	−5.71023	−0.213252
$718$	−32.4123	−1.20962
$719$	6.94504	0.259006	0.129503	−	0.991579i	$-0.458662\pi$
$719$	6.94504	0.259006	0.129503	+	0.991579i	$0.458662\pi$
$720$	−6.71204	−0.250143
$721$	13.7563	0.512313
$722$	−1.51298	−0.0563073
$723$	−2.00236	−0.0744685
$724$	−4.94667	−0.183842
$725$	−6.66582	−0.247562
$726$	0.788672	0.0292704
$727$	5.22414	0.193753	0.0968763	−	0.995296i	$-0.469115\pi$
$727$	5.22414	0.193753	0.0968763	+	0.995296i	$0.469115\pi$
$728$	19.0941	0.707675
$729$	−25.0770	−0.928779
$730$	5.14557	0.190446
$731$	−0.0545595	−0.00201795
$732$	0.479126	0.0177090
$733$	−16.8730	−0.623219	−0.311609	−	0.950210i	$-0.600868\pi$
$733$	−16.8730	−0.623219	−0.311609	+	0.950210i	$0.600868\pi$
$734$	−6.68214	−0.246642
$735$	0.379165	0.0139857
$736$	−2.37337	−0.0874837
$737$	−18.3986	−0.677720
$738$	−23.1942	−0.853789
$739$	−19.8629	−0.730667	−0.365333	−	0.930877i	$-0.619045\pi$
$739$	−19.8629	−0.730667	−0.365333	+	0.930877i	$0.619045\pi$
$740$	1.12722	0.0414373
$741$	−0.803438	−0.0295150
$742$	−1.40786	−0.0516843
$743$	−21.8405	−0.801249	−0.400625	−	0.916242i	$-0.631207\pi$
$743$	−21.8405	−0.801249	−0.400625	+	0.916242i	$0.631207\pi$
$744$	−0.491006	−0.0180012
$745$	3.55796	0.130354
$746$	17.1866	0.629248
$747$	23.0176	0.842170
$748$	4.30555	0.157426
$749$	30.7610	1.12398
$750$	−1.40899	−0.0514492
$751$	−10.4412	−0.381006	−0.190503	−	0.981687i	$-0.561012\pi$
$751$	−10.4412	−0.381006	−0.190503	+	0.981687i	$0.561012\pi$
$752$	34.6449	1.26337
$753$	2.69423	0.0981833
$754$	9.00185	0.327828
$755$	10.1738	0.370261
$756$	0.569402	0.0207089
$757$	−42.5772	−1.54749	−0.773747	−	0.633494i	$-0.781620\pi$
$757$	−42.5772	−1.54749	−0.773747	+	0.633494i	$0.781620\pi$
$758$	−33.8281	−1.22869
$759$	0.796575	0.0289138
$760$	−1.30417	−0.0473072
$761$	−16.1516	−0.585496	−0.292748	−	0.956190i	$-0.594570\pi$
$761$	−16.1516	−0.585496	−0.292748	+	0.956190i	$0.594570\pi$
$762$	1.91897	0.0695170
$763$	−33.7536	−1.22196
$764$	3.81977	0.138194
$765$	−7.74220	−0.279920
$766$	−36.3623	−1.31382
$767$	3.70603	0.133817
$768$	1.28976	0.0465403
$769$	−20.1773	−0.727611	−0.363806	−	0.931475i	$-0.618523\pi$
$769$	−20.1773	−0.727611	−0.363806	+	0.931475i	$0.618523\pi$
$770$	3.81270	0.137400
$771$	4.15036	0.149472
$772$	0.285397	0.0102717
$773$	42.5111	1.52902	0.764508	−	0.644614i	$-0.222982\pi$
$773$	42.5111	1.52902	0.764508	+	0.644614i	$0.222982\pi$
$774$	0.0471934	0.00169633
$775$	4.74689	0.170513
$776$	−19.0002	−0.682067
$777$	2.55559	0.0916814
$778$	−57.6556	−2.06705
$779$	−5.17206	−0.185308
$780$	0.117028	0.00419027
$781$	7.36591	0.263573
$782$	−11.4692	−0.410138
$783$	−1.58860	−0.0567720
$784$	17.8353	0.636973
$785$	1.53934	0.0549413
$786$	1.02931	0.0367141
$787$	8.43174	0.300559	0.150279	−	0.988644i	$-0.451983\pi$
$787$	8.43174	0.300559	0.150279	+	0.988644i	$0.451983\pi$
$788$	3.31301	0.118021
$789$	1.95609	0.0696387
$790$	−5.46417	−0.194406
$791$	4.00631	0.142448
$792$	22.0396	0.783145
$793$	−37.0182	−1.31455
$794$	41.9669	1.48935
$795$	0.0510640	0.00181105
$796$	0.447770	0.0158708
$797$	−30.4362	−1.07811	−0.539053	−	0.842272i	$-0.681218\pi$
$797$	−30.4362	−1.07811	−0.539053	+	0.842272i	$0.681218\pi$
$798$	0.499636	0.0176869
$799$	39.9622	1.41376
$800$	−7.70393	−0.272375
$801$	−13.1085	−0.463167
$802$	36.9510	1.30479
$803$	−19.3905	−0.684275
$804$	0.351189	0.0123855
$805$	−1.28271	−0.0452097
$806$	−6.41043	−0.225798
$807$	3.48431	0.122654
$808$	13.2800	0.467189
$809$	17.5005	0.615285	0.307643	−	0.951502i	$-0.400460\pi$
$809$	17.5005	0.615285	0.307643	+	0.951502i	$0.400460\pi$
$810$	6.61468	0.232416
$811$	−16.8595	−0.592017	−0.296009	−	0.955185i	$-0.595656\pi$
$811$	−16.8595	−0.592017	−0.296009	+	0.955185i	$0.595656\pi$
$812$	−0.707009	−0.0248112
$813$	−3.73146	−0.130868
$814$	−33.6334	−1.17885
$815$	−2.21830	−0.0777038
$816$	4.41940	0.154710
$817$	0.0105236	0.000368176 0
$818$	9.82002	0.343349
$819$	−21.8638	−0.763984
$820$	0.753355	0.0263083
$821$	23.4064	0.816890	0.408445	−	0.912783i	$-0.366071\pi$
$821$	23.4064	0.816890	0.408445	+	0.912783i	$0.366071\pi$
$822$	−3.72272	−0.129845
$823$	−12.7045	−0.442853	−0.221426	−	0.975177i	$-0.571071\pi$
$823$	−12.7045	−0.442853	−0.221426	+	0.975177i	$0.571071\pi$
$824$	−20.4505	−0.712426
$825$	2.58567	0.0900215
$826$	−2.30468	−0.0801900
$827$	−1.53438	−0.0533555	−0.0266778	−	0.999644i	$-0.508493\pi$
$827$	−1.53438	−0.0533555	−0.0266778	+	0.999644i	$0.508493\pi$
$828$	1.25296	0.0435434
$829$	−42.5136	−1.47656	−0.738280	−	0.674495i	$-0.764362\pi$
$829$	−42.5136	−1.47656	−0.738280	+	0.674495i	$0.764362\pi$
$830$	−5.91955	−0.205470
$831$	−5.18393	−0.179828
$832$	−27.6775	−0.959546
$833$	20.5726	0.712799
$834$	1.85678	0.0642950
$835$	3.36827	0.116564
$836$	−0.830471	−0.0287224
$837$	1.13128	0.0391028
$838$	−7.55249	−0.260896
$839$	35.5903	1.22871	0.614357	−	0.789028i	$-0.289415\pi$
$839$	35.5903	1.22871	0.614357	+	0.789028i	$0.289415\pi$
$840$	0.430680	0.0148599
$841$	−27.0275	−0.931982
$842$	4.85057	0.167162
$843$	−1.10345	−0.0380047
$844$	−0.289107	−0.00995147
$845$	−2.49208	−0.0857300
$846$	−34.5669	−1.18844
$847$	4.78580	0.164442
$848$	2.40196	0.0824837
$849$	5.29284	0.181650
$850$	−37.2289	−1.27694
$851$	11.3153	0.387885
$852$	−0.140600	−0.00481686
$853$	−0.670534	−0.0229587	−0.0114793	−	0.999934i	$-0.503654\pi$
$853$	−0.670534	−0.0229587	−0.0114793	+	0.999934i	$0.503654\pi$
$854$	23.0206	0.787747
$855$	1.49335	0.0510714
$856$	−45.7300	−1.56302
$857$	−2.12815	−0.0726962	−0.0363481	−	0.999339i	$-0.511573\pi$
$857$	−2.12815	−0.0726962	−0.0363481	+	0.999339i	$0.511573\pi$
$858$	−3.49182	−0.119209
$859$	−33.3262	−1.13707	−0.568537	−	0.822657i	$-0.692491\pi$
$859$	−33.3262	−1.13707	−0.568537	+	0.822657i	$0.692491\pi$
$860$	−0.00153286	−5.22701e−5 0
$861$	1.70798	0.0582079
$862$	−9.12565	−0.310821
$863$	−11.9066	−0.405304	−0.202652	−	0.979251i	$-0.564956\pi$
$863$	−11.9066	−0.405304	−0.202652	+	0.979251i	$0.564956\pi$
$864$	−1.83601	−0.0624623
$865$	−4.66155	−0.158498
$866$	−0.101834	−0.00346048
$867$	1.87355	0.0636290
$868$	0.503478	0.0170892
$869$	20.5911	0.698505
$870$	0.203042	0.00688378
$871$	−27.1335	−0.919384
$872$	50.1789	1.69927
$873$	21.7563	0.736339
$874$	2.21222	0.0748296
$875$	−8.55002	−0.289043
$876$	0.370123	0.0125053
$877$	−54.7348	−1.84826	−0.924131	−	0.382075i	$-0.875209\pi$
$877$	−54.7348	−1.84826	−0.924131	+	0.382075i	$0.875209\pi$
$878$	−2.57885	−0.0870321
$879$	−4.87635	−0.164475
$880$	−6.50486	−0.219279
$881$	18.9363	0.637979	0.318990	−	0.947758i	$-0.396657\pi$
$881$	18.9363	0.637979	0.318990	+	0.947758i	$0.396657\pi$
$882$	−17.7951	−0.599193
$883$	−5.53248	−0.186183	−0.0930914	−	0.995658i	$-0.529675\pi$
$883$	−5.53248	−0.186183	−0.0930914	+	0.995658i	$0.529675\pi$
$884$	6.34966	0.213562
$885$	0.0835918	0.00280991
$886$	14.3824	0.483187
$887$	12.8467	0.431349	0.215674	−	0.976465i	$-0.430805\pi$
$887$	12.8467	0.431349	0.215674	+	0.976465i	$0.430805\pi$
$888$	−3.79920	−0.127493
$889$	11.6446	0.390549
$890$	3.37118	0.113002
$891$	−24.9267	−0.835075
$892$	−0.721510	−0.0241579
$893$	−7.70806	−0.257940
$894$	2.02638	0.0677723
$895$	−2.80404	−0.0937287
$896$	22.8646	0.763852
$897$	1.17476	0.0392240
$898$	3.70578	0.123663
$899$	−1.40468	−0.0468487
$900$	4.06709	0.135570
$901$	2.77062	0.0923026
$902$	−22.4782	−0.748443
$903$	−0.00347525	−0.000115649 0
$904$	−5.95586	−0.198089
$905$	8.62050	0.286555
$906$	5.79431	0.192503
$907$	33.3832	1.10847	0.554235	−	0.832360i	$-0.313011\pi$
$907$	33.3832	1.10847	0.554235	+	0.832360i	$0.313011\pi$
$908$	1.90508	0.0632224
$909$	−15.2064	−0.504363
$910$	5.62282	0.186395
$911$	−40.7444	−1.34992	−0.674961	−	0.737853i	$-0.735840\pi$
$911$	−40.7444	−1.34992	−0.674961	+	0.737853i	$0.735840\pi$
$912$	−0.852430	−0.0282268
$913$	22.3071	0.738258
$914$	−27.6020	−0.912992
$915$	−0.834967	−0.0276032
$916$	−3.28057	−0.108393
$917$	6.24600	0.206261
$918$	−8.87241	−0.292833
$919$	33.4716	1.10413	0.552063	−	0.833803i	$-0.313841\pi$
$919$	33.4716	1.10413	0.552063	+	0.833803i	$0.313841\pi$
$920$	1.90691	0.0628689
$921$	−0.766290	−0.0252501
$922$	19.2450	0.633800
$923$	10.8630	0.357559
$924$	0.274249	0.00902213
$925$	36.7294	1.20766
$926$	21.6094	0.710128
$927$	23.4169	0.769113
$928$	2.27972	0.0748354
$929$	−11.8943	−0.390241	−0.195120	−	0.980779i	$-0.562510\pi$
$929$	−11.8943	−0.390241	−0.195120	+	0.980779i	$0.562510\pi$
$930$	−0.144591	−0.00474133
$931$	−3.96812	−0.130050
$932$	1.73367	0.0567883
$933$	−0.134592	−0.00440636
$934$	21.2366	0.694884
$935$	−7.50322	−0.245382
$936$	32.5032	1.06240
$937$	1.58899	0.0519101	0.0259550	−	0.999663i	$-0.491737\pi$
$937$	1.58899	0.0519101	0.0259550	+	0.999663i	$0.491737\pi$
$938$	16.8736	0.550941
$939$	1.80488	0.0589000
$940$	1.12275	0.0366199
$941$	−19.2450	−0.627370	−0.313685	−	0.949527i	$-0.601564\pi$
$941$	−19.2450	−0.627370	−0.313685	+	0.949527i	$0.601564\pi$
$942$	0.876705	0.0285646
$943$	7.56239	0.246265
$944$	3.93202	0.127976
$945$	−0.992289	−0.0322792
$946$	0.0457367	0.00148703
$947$	28.1607	0.915099	0.457550	−	0.889184i	$-0.348727\pi$
$947$	28.1607	0.915099	0.457550	+	0.889184i	$0.348727\pi$
$948$	−0.393040	−0.0127653
$949$	−28.5963	−0.928277
$950$	7.18085	0.232977
$951$	2.68891	0.0871938
$952$	23.3677	0.757351
$953$	37.1441	1.20322	0.601608	−	0.798792i	$-0.294527\pi$
$953$	37.1441	1.20322	0.601608	+	0.798792i	$0.294527\pi$
$954$	−2.39656	−0.0775914
$955$	−6.65666	−0.215404
$956$	−8.70457	−0.281526
$957$	−0.765141	−0.0247335
$958$	10.7341	0.346804
$959$	−22.5901	−0.729473
$960$	−0.624284	−0.0201487
$961$	−29.9997	−0.967732
$962$	−49.6012	−1.59921
$963$	52.3634	1.68739
$964$	−3.05236	−0.0983098
$965$	−0.497357	−0.0160105
$966$	−0.730549	−0.0235050
$967$	−27.2569	−0.876523	−0.438262	−	0.898847i	$-0.644406\pi$
$967$	−27.2569	−0.876523	−0.438262	+	0.898847i	$0.644406\pi$
$968$	−7.11467	−0.228674
$969$	−0.983261	−0.0315869
$970$	−5.59516	−0.179650
$971$	−26.8333	−0.861121	−0.430560	−	0.902562i	$-0.641684\pi$
$971$	−26.8333	−0.861121	−0.430560	+	0.902562i	$0.641684\pi$
$972$	1.45683	0.0467279
$973$	11.2673	0.361212
$974$	28.3730	0.909130
$975$	3.81325	0.122122
$976$	−39.2754	−1.25718
$977$	−22.2068	−0.710458	−0.355229	−	0.934779i	$-0.615597\pi$
$977$	−22.2068	−0.710458	−0.355229	+	0.934779i	$0.615597\pi$
$978$	−1.26340	−0.0403991
$979$	−12.7039	−0.406019
$980$	0.577992	0.0184633
$981$	−57.4576	−1.83448
$982$	−4.79522	−0.153021
$983$	3.64180	0.116155	0.0580777	−	0.998312i	$-0.481503\pi$
$983$	3.64180	0.116155	0.0580777	+	0.998312i	$0.481503\pi$
$984$	−2.53912	−0.0809443
$985$	−5.77353	−0.183960
$986$	11.0166	0.350840
$987$	2.54546	0.0810228
$988$	−1.22475	−0.0389644
$989$	−0.0153873	−0.000489287 0
$990$	6.49022	0.206273
$991$	−5.61218	−0.178277	−0.0891383	−	0.996019i	$-0.528411\pi$
$991$	−5.61218	−0.178277	−0.0891383	+	0.996019i	$0.528411\pi$
$992$	−1.62344	−0.0515443
$993$	5.69196	0.180629
$994$	−6.75538	−0.214268
$995$	−0.780324	−0.0247379
$996$	−0.425795	−0.0134918
$997$	−55.5716	−1.75997	−0.879985	−	0.475002i	$-0.842447\pi$
$997$	−55.5716	−1.75997	−0.879985	+	0.475002i	$0.842447\pi$
$998$	−26.5386	−0.840066
$999$	8.75339	0.276945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.20	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.20	✓	82	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)

\(f(q)\)	\(=\)	\(q-1.51298 q^{2} +0.189655 q^{3} +0.289107 q^{4} -0.503823 q^{5} -0.286944 q^{6} -1.74123 q^{7} +2.58855 q^{8} -2.96403 q^{9} +O(q^{10})\)	\(q-1.51298 q^{2} +0.189655 q^{3} +0.289107 q^{4} -0.503823 q^{5} -0.286944 q^{6} -1.74123 q^{7} +2.58855 q^{8} -2.96403 q^{9} +0.762274 q^{10} -2.87254 q^{11} +0.0548306 q^{12} -4.23631 q^{13} +2.63444 q^{14} -0.0955526 q^{15} -4.49463 q^{16} -5.18447 q^{17} +4.48452 q^{18} +1.00000 q^{19} -0.145659 q^{20} -0.330233 q^{21} +4.34609 q^{22} -1.46216 q^{23} +0.490931 q^{24} -4.74616 q^{25} +6.40945 q^{26} -1.13111 q^{27} -0.503401 q^{28} +1.40446 q^{29} +0.144569 q^{30} -1.00015 q^{31} +1.62319 q^{32} -0.544792 q^{33} +7.84399 q^{34} +0.877271 q^{35} -0.856922 q^{36} -7.73876 q^{37} -1.51298 q^{38} -0.803438 q^{39} -1.30417 q^{40} -5.17206 q^{41} +0.499636 q^{42} +0.0105236 q^{43} -0.830471 q^{44} +1.49335 q^{45} +2.21222 q^{46} -7.70806 q^{47} -0.852430 q^{48} -3.96812 q^{49} +7.18085 q^{50} -0.983261 q^{51} -1.22475 q^{52} -0.534407 q^{53} +1.71135 q^{54} +1.44725 q^{55} -4.50725 q^{56} +0.189655 q^{57} -2.12493 q^{58} -0.874825 q^{59} -0.0276249 q^{60} +8.73830 q^{61} +1.51321 q^{62} +5.16106 q^{63} +6.53341 q^{64} +2.13435 q^{65} +0.824259 q^{66} +6.40498 q^{67} -1.49887 q^{68} -0.277307 q^{69} -1.32729 q^{70} -2.56425 q^{71} -7.67253 q^{72} +6.75029 q^{73} +11.7086 q^{74} -0.900134 q^{75} +0.289107 q^{76} +5.00175 q^{77} +1.21559 q^{78} -7.16825 q^{79} +2.26450 q^{80} +8.67757 q^{81} +7.82521 q^{82} -7.76565 q^{83} -0.0954727 q^{84} +2.61205 q^{85} -0.0159221 q^{86} +0.266364 q^{87} -7.43570 q^{88} +4.42254 q^{89} -2.25940 q^{90} +7.37639 q^{91} -0.422722 q^{92} -0.189684 q^{93} +11.6621 q^{94} -0.503823 q^{95} +0.307847 q^{96} -7.34010 q^{97} +6.00369 q^{98} +8.51429 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

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