LMFDB - Embedded newform 4009.2.a.e.1.7

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			1.7
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39206 q^{2} -0.112128 q^{3} +3.72195 q^{4} +0.840987 q^{5} +0.268217 q^{6} +2.66760 q^{7} -4.11902 q^{8} -2.98743 q^{9} +O(q^{10})$	\(q-2.39206 q^{2} -0.112128 q^{3} +3.72195 q^{4} +0.840987 q^{5} +0.268217 q^{6} +2.66760 q^{7} -4.11902 q^{8} -2.98743 q^{9} -2.01169 q^{10} +0.850704 q^{11} -0.417335 q^{12} +2.82726 q^{13} -6.38107 q^{14} -0.0942980 q^{15} +2.40904 q^{16} +2.10684 q^{17} +7.14611 q^{18} +1.00000 q^{19} +3.13011 q^{20} -0.299113 q^{21} -2.03494 q^{22} +5.59891 q^{23} +0.461857 q^{24} -4.29274 q^{25} -6.76297 q^{26} +0.671357 q^{27} +9.92870 q^{28} -4.17538 q^{29} +0.225567 q^{30} -1.42022 q^{31} +2.47548 q^{32} -0.0953877 q^{33} -5.03969 q^{34} +2.24342 q^{35} -11.1191 q^{36} +3.76440 q^{37} -2.39206 q^{38} -0.317014 q^{39} -3.46404 q^{40} +5.63563 q^{41} +0.715496 q^{42} -2.40605 q^{43} +3.16628 q^{44} -2.51239 q^{45} -13.3929 q^{46} +1.86537 q^{47} -0.270120 q^{48} +0.116115 q^{49} +10.2685 q^{50} -0.236236 q^{51} +10.5229 q^{52} +13.1509 q^{53} -1.60593 q^{54} +0.715431 q^{55} -10.9879 q^{56} -0.112128 q^{57} +9.98777 q^{58} +0.155588 q^{59} -0.350973 q^{60} -8.27792 q^{61} +3.39725 q^{62} -7.96928 q^{63} -10.7396 q^{64} +2.37769 q^{65} +0.228173 q^{66} -2.91489 q^{67} +7.84157 q^{68} -0.627794 q^{69} -5.36640 q^{70} -5.31927 q^{71} +12.3053 q^{72} -6.70490 q^{73} -9.00468 q^{74} +0.481336 q^{75} +3.72195 q^{76} +2.26934 q^{77} +0.758318 q^{78} +8.94377 q^{79} +2.02597 q^{80} +8.88700 q^{81} -13.4808 q^{82} +7.77171 q^{83} -1.11328 q^{84} +1.77182 q^{85} +5.75541 q^{86} +0.468177 q^{87} -3.50407 q^{88} +12.1312 q^{89} +6.00978 q^{90} +7.54200 q^{91} +20.8389 q^{92} +0.159246 q^{93} -4.46207 q^{94} +0.840987 q^{95} -0.277570 q^{96} +11.2791 q^{97} -0.277753 q^{98} -2.54142 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.39206	−1.69144	−0.845721	−	0.533625i	$-0.820829\pi$
$2$	−2.39206	−1.69144	−0.845721	+	0.533625i	$0.820829\pi$
$3$	−0.112128	−0.0647371	−0.0323685	−	0.999476i	$-0.510305\pi$
$3$	−0.112128	−0.0647371	−0.0323685	+	0.999476i	$0.510305\pi$
$4$	3.72195	1.86098
$5$	0.840987	0.376101	0.188050	−	0.982159i	$-0.439783\pi$
$5$	0.840987	0.376101	0.188050	+	0.982159i	$0.439783\pi$
$6$	0.268217	0.109499
$7$	2.66760	1.00826	0.504130	−	0.863628i	$-0.331813\pi$
$7$	2.66760	1.00826	0.504130	+	0.863628i	$0.331813\pi$
$8$	−4.11902	−1.45629
$9$	−2.98743	−0.995809
$10$	−2.01169	−0.636153
$11$	0.850704	0.256497	0.128249	−	0.991742i	$-0.459064\pi$
$11$	0.850704	0.256497	0.128249	+	0.991742i	$0.459064\pi$
$12$	−0.417335	−0.120474
$13$	2.82726	0.784140	0.392070	−	0.919935i	$-0.371759\pi$
$13$	2.82726	0.784140	0.392070	+	0.919935i	$0.371759\pi$
$14$	−6.38107	−1.70541
$15$	−0.0942980	−0.0243477
$16$	2.40904	0.602260
$17$	2.10684	0.510984	0.255492	−	0.966811i	$-0.417763\pi$
$17$	2.10684	0.510984	0.255492	+	0.966811i	$0.417763\pi$
$18$	7.14611	1.68435
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.13011	0.699915
$21$	−0.299113	−0.0652718
$22$	−2.03494	−0.433850
$23$	5.59891	1.16745	0.583727	−	0.811950i	$-0.301594\pi$
$23$	5.59891	1.16745	0.583727	+	0.811950i	$0.301594\pi$
$24$	0.461857	0.0942762
$25$	−4.29274	−0.858548
$26$	−6.76297	−1.32633
$27$	0.671357	0.129203
$28$	9.92870	1.87635
$29$	−4.17538	−0.775349	−0.387675	−	0.921796i	$-0.626722\pi$
$29$	−4.17538	−0.775349	−0.387675	+	0.921796i	$0.626722\pi$
$30$	0.225567	0.0411826
$31$	−1.42022	−0.255079	−0.127540	−	0.991833i	$-0.540708\pi$
$31$	−1.42022	−0.255079	−0.127540	+	0.991833i	$0.540708\pi$
$32$	2.47548	0.437606
$33$	−0.0953877	−0.0166049
$34$	−5.03969	−0.864300
$35$	2.24342	0.379207
$36$	−11.1191	−1.85318
$37$	3.76440	0.618864	0.309432	−	0.950922i	$-0.399861\pi$
$37$	3.76440	0.618864	0.309432	+	0.950922i	$0.399861\pi$
$38$	−2.39206	−0.388044
$39$	−0.317014	−0.0507629
$40$	−3.46404	−0.547713
$41$	5.63563	0.880137	0.440069	−	0.897964i	$-0.354954\pi$
$41$	5.63563	0.880137	0.440069	+	0.897964i	$0.354954\pi$
$42$	0.715496	0.110403
$43$	−2.40605	−0.366919	−0.183459	−	0.983027i	$-0.558730\pi$
$43$	−2.40605	−0.366919	−0.183459	+	0.983027i	$0.558730\pi$
$44$	3.16628	0.477335
$45$	−2.51239	−0.374524
$46$	−13.3929	−1.97468
$47$	1.86537	0.272092	0.136046	−	0.990703i	$-0.456561\pi$
$47$	1.86537	0.272092	0.136046	+	0.990703i	$0.456561\pi$
$48$	−0.270120	−0.0389885
$49$	0.116115	0.0165878
$50$	10.2685	1.45218
$51$	−0.236236	−0.0330796
$52$	10.5229	1.45927
$53$	13.1509	1.80642	0.903211	−	0.429197i	$-0.141203\pi$
$53$	13.1509	1.80642	0.903211	+	0.429197i	$0.141203\pi$
$54$	−1.60593	−0.218539
$55$	0.715431	0.0964687
$56$	−10.9879	−1.46832
$57$	−0.112128	−0.0148517
$58$	9.98777	1.31146
$59$	0.155588	0.0202558	0.0101279	−	0.999949i	$-0.496776\pi$
$59$	0.155588	0.0202558	0.0101279	+	0.999949i	$0.496776\pi$
$60$	−0.350973	−0.0453104
$61$	−8.27792	−1.05988	−0.529940	−	0.848035i	$-0.677785\pi$
$61$	−8.27792	−1.05988	−0.529940	+	0.848035i	$0.677785\pi$
$62$	3.39725	0.431451
$63$	−7.96928	−1.00403
$64$	−10.7396	−1.34245
$65$	2.37769	0.294916
$66$	0.228173	0.0280862
$67$	−2.91489	−0.356110	−0.178055	−	0.984021i	$-0.556981\pi$
$67$	−2.91489	−0.356110	−0.178055	+	0.984021i	$0.556981\pi$
$68$	7.84157	0.950930
$69$	−0.627794	−0.0755775
$70$	−5.36640	−0.641407
$71$	−5.31927	−0.631281	−0.315640	−	0.948879i	$-0.602219\pi$
$71$	−5.31927	−0.631281	−0.315640	+	0.948879i	$0.602219\pi$
$72$	12.3053	1.45019
$73$	−6.70490	−0.784749	−0.392375	−	0.919806i	$-0.628346\pi$
$73$	−6.70490	−0.784749	−0.392375	+	0.919806i	$0.628346\pi$
$74$	−9.00468	−1.04677
$75$	0.481336	0.0555799
$76$	3.72195	0.426938
$77$	2.26934	0.258616
$78$	0.758318	0.0858626
$79$	8.94377	1.00625	0.503126	−	0.864213i	$-0.332183\pi$
$79$	8.94377	1.00625	0.503126	+	0.864213i	$0.332183\pi$
$80$	2.02597	0.226510
$81$	8.88700	0.987445
$82$	−13.4808	−1.48870
$83$	7.77171	0.853056	0.426528	−	0.904474i	$-0.359736\pi$
$83$	7.77171	0.853056	0.426528	+	0.904474i	$0.359736\pi$
$84$	−1.11328	−0.121469
$85$	1.77182	0.192181
$86$	5.75541	0.620622
$87$	0.468177	0.0501938
$88$	−3.50407	−0.373535
$89$	12.1312	1.28590	0.642950	−	0.765908i	$-0.277710\pi$
$89$	12.1312	1.28590	0.642950	+	0.765908i	$0.277710\pi$
$90$	6.00978	0.633487
$91$	7.54200	0.790617
$92$	20.8389	2.17261
$93$	0.159246	0.0165131
$94$	−4.46207	−0.460228
$95$	0.840987	0.0862834
$96$	−0.277570	−0.0283294
$97$	11.2791	1.14522	0.572608	−	0.819829i	$-0.305932\pi$
$97$	11.2791	1.14522	0.572608	+	0.819829i	$0.305932\pi$
$98$	−0.277753	−0.0280573
$99$	−2.54142	−0.255422
$100$	−15.9774	−1.59774
$101$	17.3017	1.72158	0.860792	−	0.508956i	$-0.169969\pi$
$101$	17.3017	1.72158	0.860792	+	0.508956i	$0.169969\pi$
$102$	0.565090	0.0559522
$103$	−4.97753	−0.490450	−0.245225	−	0.969466i	$-0.578862\pi$
$103$	−4.97753	−0.490450	−0.245225	+	0.969466i	$0.578862\pi$
$104$	−11.6455	−1.14194
$105$	−0.251550	−0.0245488
$106$	−31.4579	−3.05546
$107$	2.83917	0.274473	0.137237	−	0.990538i	$-0.456178\pi$
$107$	2.83917	0.274473	0.137237	+	0.990538i	$0.456178\pi$
$108$	2.49876	0.240444
$109$	1.93762	0.185590	0.0927949	−	0.995685i	$-0.470420\pi$
$109$	1.93762	0.185590	0.0927949	+	0.995685i	$0.470420\pi$
$110$	−1.71135	−0.163171
$111$	−0.422095	−0.0400634
$112$	6.42636	0.607234
$113$	12.5023	1.17612	0.588058	−	0.808818i	$-0.299892\pi$
$113$	12.5023	1.17612	0.588058	+	0.808818i	$0.299892\pi$
$114$	0.268217	0.0251208
$115$	4.70861	0.439080
$116$	−15.5406	−1.44291
$117$	−8.44623	−0.780854
$118$	−0.372175	−0.0342615
$119$	5.62022	0.515205
$120$	0.388416	0.0354573
$121$	−10.2763	−0.934209
$122$	19.8013	1.79273
$123$	−0.631911	−0.0569775
$124$	−5.28599	−0.474696
$125$	−7.81507	−0.699001
$126$	19.0630	1.69827
$127$	13.9531	1.23814	0.619069	−	0.785336i	$-0.287510\pi$
$127$	13.9531	1.23814	0.619069	+	0.785336i	$0.287510\pi$
$128$	20.7387	1.83306
$129$	0.269785	0.0237533
$130$	−5.68757	−0.498833
$131$	−19.2728	−1.68387	−0.841937	−	0.539576i	$-0.818585\pi$
$131$	−19.2728	−1.68387	−0.841937	+	0.539576i	$0.818585\pi$
$132$	−0.355029	−0.0309013
$133$	2.66760	0.231311
$134$	6.97259	0.602340
$135$	0.564603	0.0485933
$136$	−8.67812	−0.744143
$137$	−5.22218	−0.446161	−0.223080	−	0.974800i	$-0.571611\pi$
$137$	−5.22218	−0.446161	−0.223080	+	0.974800i	$0.571611\pi$
$138$	1.50172	0.127835
$139$	14.6770	1.24489	0.622445	−	0.782664i	$-0.286139\pi$
$139$	14.6770	1.24489	0.622445	+	0.782664i	$0.286139\pi$
$140$	8.34991	0.705696
$141$	−0.209160	−0.0176144
$142$	12.7240	1.06778
$143$	2.40516	0.201130
$144$	−7.19683	−0.599736
$145$	−3.51144	−0.291609
$146$	16.0385	1.32736
$147$	−0.0130197	−0.00107385
$148$	14.0109	1.15169
$149$	−2.06799	−0.169416	−0.0847081	−	0.996406i	$-0.526996\pi$
$149$	−2.06799	−0.169416	−0.0847081	+	0.996406i	$0.526996\pi$
$150$	−1.15138	−0.0940102
$151$	−6.82413	−0.555340	−0.277670	−	0.960676i	$-0.589562\pi$
$151$	−6.82413	−0.555340	−0.277670	+	0.960676i	$0.589562\pi$
$152$	−4.11902	−0.334097
$153$	−6.29403	−0.508842
$154$	−5.42841	−0.437433
$155$	−1.19439	−0.0959354
$156$	−1.17991	−0.0944686
$157$	−22.7174	−1.81305	−0.906524	−	0.422153i	$-0.861274\pi$
$157$	−22.7174	−1.81305	−0.906524	+	0.422153i	$0.861274\pi$
$158$	−21.3940	−1.70202
$159$	−1.47459	−0.116942
$160$	2.08184	0.164584
$161$	14.9357	1.17710
$162$	−21.2583	−1.67021
$163$	−11.4637	−0.897908	−0.448954	−	0.893555i	$-0.648203\pi$
$163$	−11.4637	−0.897908	−0.448954	+	0.893555i	$0.648203\pi$
$164$	20.9756	1.63792
$165$	−0.0802198	−0.00624510
$166$	−18.5904	−1.44290
$167$	−10.9824	−0.849841	−0.424921	−	0.905231i	$-0.639698\pi$
$167$	−10.9824	−0.849841	−0.424921	+	0.905231i	$0.639698\pi$
$168$	1.23205	0.0950549
$169$	−5.00662	−0.385124
$170$	−4.23831	−0.325064
$171$	−2.98743	−0.228454
$172$	−8.95520	−0.682828
$173$	−1.55575	−0.118282	−0.0591409	−	0.998250i	$-0.518836\pi$
$173$	−1.55575	−0.118282	−0.0591409	+	0.998250i	$0.518836\pi$
$174$	−1.11991	−0.0849000
$175$	−11.4513	−0.865640
$176$	2.04938	0.154478
$177$	−0.0174457	−0.00131130
$178$	−29.0185	−2.17503
$179$	−19.8942	−1.48696	−0.743480	−	0.668758i	$-0.766826\pi$
$179$	−19.8942	−1.48696	−0.743480	+	0.668758i	$0.766826\pi$
$180$	−9.35099	−0.696982
$181$	−3.41327	−0.253706	−0.126853	−	0.991921i	$-0.540488\pi$
$181$	−3.41327	−0.253706	−0.126853	+	0.991921i	$0.540488\pi$
$182$	−18.0409	−1.33728
$183$	0.928186	0.0686135
$184$	−23.0620	−1.70016
$185$	3.16581	0.232755
$186$	−0.380927	−0.0279309
$187$	1.79230	0.131066
$188$	6.94281	0.506357
$189$	1.79092	0.130270
$190$	−2.01169	−0.145943
$191$	8.65740	0.626428	0.313214	−	0.949683i	$-0.398594\pi$
$191$	8.65740	0.626428	0.313214	+	0.949683i	$0.398594\pi$
$192$	1.20420	0.0869060
$193$	3.17561	0.228585	0.114293	−	0.993447i	$-0.463540\pi$
$193$	3.17561	0.228585	0.114293	+	0.993447i	$0.463540\pi$
$194$	−26.9802	−1.93707
$195$	−0.266605	−0.0190920
$196$	0.432174	0.0308696
$197$	7.32007	0.521533	0.260767	−	0.965402i	$-0.416025\pi$
$197$	7.32007	0.521533	0.260767	+	0.965402i	$0.416025\pi$
$198$	6.07922	0.432032
$199$	19.2694	1.36597	0.682987	−	0.730431i	$-0.260681\pi$
$199$	19.2694	1.36597	0.682987	+	0.730431i	$0.260681\pi$
$200$	17.6819	1.25030
$201$	0.326840	0.0230535
$202$	−41.3867	−2.91196
$203$	−11.1383	−0.781753
$204$	−0.879258	−0.0615604
$205$	4.73949	0.331020
$206$	11.9065	0.829568
$207$	−16.7263	−1.16256
$208$	6.81097	0.472256
$209$	0.850704	0.0588444
$210$	0.601723	0.0415228
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	48.9472	3.36171
$213$	0.596438	0.0408673
$214$	−6.79148	−0.464256
$215$	−2.02345	−0.137998
$216$	−2.76534	−0.188157
$217$	−3.78858	−0.257186
$218$	−4.63489	−0.313915
$219$	0.751806	0.0508023
$220$	2.66280	0.179526
$221$	5.95658	0.400683
$222$	1.00968	0.0677650
$223$	20.8725	1.39773	0.698864	−	0.715255i	$-0.253689\pi$
$223$	20.8725	1.39773	0.698864	+	0.715255i	$0.253689\pi$
$224$	6.60359	0.441221
$225$	12.8243	0.854950
$226$	−29.9062	−1.98933
$227$	−4.49792	−0.298538	−0.149269	−	0.988797i	$-0.547692\pi$
$227$	−4.49792	−0.298538	−0.149269	+	0.988797i	$0.547692\pi$
$228$	−0.417335	−0.0276387
$229$	18.2510	1.20606	0.603029	−	0.797720i	$-0.293960\pi$
$229$	18.2510	1.20606	0.603029	+	0.797720i	$0.293960\pi$
$230$	−11.2633	−0.742679
$231$	−0.254457	−0.0167420
$232$	17.1985	1.12914
$233$	7.95854	0.521381	0.260691	−	0.965422i	$-0.416050\pi$
$233$	7.95854	0.521381	0.260691	+	0.965422i	$0.416050\pi$
$234$	20.2039	1.32077
$235$	1.56875	0.102334
$236$	0.579091	0.0376956
$237$	−1.00285	−0.0651418
$238$	−13.4439	−0.871439
$239$	−7.42264	−0.480131	−0.240065	−	0.970757i	$-0.577169\pi$
$239$	−7.42264	−0.480131	−0.240065	+	0.970757i	$0.577169\pi$
$240$	−0.227168	−0.0146636
$241$	−26.4340	−1.70276	−0.851381	−	0.524548i	$-0.824234\pi$
$241$	−26.4340	−1.70276	−0.851381	+	0.524548i	$0.824234\pi$
$242$	24.5815	1.58016
$243$	−3.01055	−0.193127
$244$	−30.8101	−1.97241
$245$	0.0976509	0.00623869
$246$	1.51157	0.0963742
$247$	2.82726	0.179894
$248$	5.84991	0.371470
$249$	−0.871426	−0.0552244
$250$	18.6941	1.18232
$251$	0.792619	0.0500297	0.0250148	−	0.999687i	$-0.492037\pi$
$251$	0.792619	0.0500297	0.0250148	+	0.999687i	$0.492037\pi$
$252$	−29.6613	−1.86849
$253$	4.76302	0.299448
$254$	−33.3767	−2.09424
$255$	−0.198671	−0.0124413
$256$	−28.1292	−1.75807
$257$	26.6617	1.66311	0.831557	−	0.555440i	$-0.187450\pi$
$257$	26.6617	1.66311	0.831557	+	0.555440i	$0.187450\pi$
$258$	−0.645342	−0.0401773
$259$	10.0419	0.623976
$260$	8.84964	0.548831
$261$	12.4737	0.772100
$262$	46.1018	2.84818
$263$	−17.3176	−1.06785	−0.533924	−	0.845532i	$-0.679283\pi$
$263$	−17.3176	−1.06785	−0.533924	+	0.845532i	$0.679283\pi$
$264$	0.392904	0.0241816
$265$	11.0598	0.679396
$266$	−6.38107	−0.391249
$267$	−1.36024	−0.0832454
$268$	−10.8491	−0.662713
$269$	14.0069	0.854013	0.427006	−	0.904249i	$-0.359568\pi$
$269$	14.0069	0.854013	0.427006	+	0.904249i	$0.359568\pi$
$270$	−1.35056	−0.0821927
$271$	−7.57013	−0.459853	−0.229926	−	0.973208i	$-0.573849\pi$
$271$	−7.57013	−0.459853	−0.229926	+	0.973208i	$0.573849\pi$
$272$	5.07546	0.307745
$273$	−0.845669	−0.0511822
$274$	12.4918	0.754655
$275$	−3.65185	−0.220215
$276$	−2.33662	−0.140648
$277$	29.6749	1.78299	0.891495	−	0.453031i	$-0.149657\pi$
$277$	29.6749	1.78299	0.891495	+	0.453031i	$0.149657\pi$
$278$	−35.1084	−2.10566
$279$	4.24280	0.254010
$280$	−9.24069	−0.552237
$281$	17.4389	1.04032	0.520158	−	0.854070i	$-0.325873\pi$
$281$	17.4389	1.04032	0.520158	+	0.854070i	$0.325873\pi$
$282$	0.500323	0.0297938
$283$	−6.30789	−0.374965	−0.187482	−	0.982268i	$-0.560033\pi$
$283$	−6.30789	−0.374965	−0.187482	+	0.982268i	$0.560033\pi$
$284$	−19.7981	−1.17480
$285$	−0.0942980	−0.00558573
$286$	−5.75329	−0.340199
$287$	15.0336	0.887407
$288$	−7.39530	−0.435772
$289$	−12.5612	−0.738895
$290$	8.39958	0.493240
$291$	−1.26470	−0.0741379
$292$	−24.9553	−1.46040
$293$	−21.4237	−1.25159	−0.625793	−	0.779989i	$-0.715224\pi$
$293$	−21.4237	−1.25159	−0.625793	+	0.779989i	$0.715224\pi$
$294$	0.0311439	0.00181635
$295$	0.130847	0.00761822
$296$	−15.5057	−0.901248
$297$	0.571127	0.0331401
$298$	4.94675	0.286558
$299$	15.8296	0.915447
$300$	1.79151	0.103433
$301$	−6.41839	−0.369950
$302$	16.3237	0.939326
$303$	−1.94000	−0.111450
$304$	2.40904	0.138168
$305$	−6.96162	−0.398621
$306$	15.0557	0.860678
$307$	4.55977	0.260240	0.130120	−	0.991498i	$-0.458464\pi$
$307$	4.55977	0.260240	0.130120	+	0.991498i	$0.458464\pi$
$308$	8.44639	0.481278
$309$	0.558119	0.0317503
$310$	2.85704	0.162269
$311$	−5.55373	−0.314923	−0.157462	−	0.987525i	$-0.550331\pi$
$311$	−5.55373	−0.314923	−0.157462	+	0.987525i	$0.550331\pi$
$312$	1.30579	0.0739257
$313$	7.52860	0.425542	0.212771	−	0.977102i	$-0.431751\pi$
$313$	7.52860	0.425542	0.212771	+	0.977102i	$0.431751\pi$
$314$	54.3415	3.06667
$315$	−6.70205	−0.377618
$316$	33.2883	1.87261
$317$	9.58530	0.538364	0.269182	−	0.963089i	$-0.413247\pi$
$317$	9.58530	0.538364	0.269182	+	0.963089i	$0.413247\pi$
$318$	3.52730	0.197801
$319$	−3.55202	−0.198875
$320$	−9.03183	−0.504895
$321$	−0.318350	−0.0177686
$322$	−35.7271	−1.99099
$323$	2.10684	0.117228
$324$	33.0770	1.83761
$325$	−12.1367	−0.673222
$326$	27.4219	1.51876
$327$	−0.217261	−0.0120145
$328$	−23.2133	−1.28174
$329$	4.97606	0.274339
$330$	0.191891	0.0105632
$331$	17.6859	0.972107	0.486054	−	0.873929i	$-0.338436\pi$
$331$	17.6859	0.972107	0.486054	+	0.873929i	$0.338436\pi$
$332$	28.9260	1.58752
$333$	−11.2459	−0.616270
$334$	26.2705	1.43746
$335$	−2.45138	−0.133933
$336$	−0.720574	−0.0393105
$337$	−27.1868	−1.48096	−0.740481	−	0.672077i	$-0.765402\pi$
$337$	−27.1868	−1.48096	−0.740481	+	0.672077i	$0.765402\pi$
$338$	11.9761	0.651416
$339$	−1.40186	−0.0761383
$340$	6.59465	0.357645
$341$	−1.20819	−0.0654270
$342$	7.14611	0.386417
$343$	−18.3635	−0.991535
$344$	9.91056	0.534342
$345$	−0.527967	−0.0284248
$346$	3.72146	0.200067
$347$	16.0414	0.861147	0.430573	−	0.902556i	$-0.358311\pi$
$347$	16.0414	0.861147	0.430573	+	0.902556i	$0.358311\pi$
$348$	1.74253	0.0934096
$349$	15.7846	0.844928	0.422464	−	0.906380i	$-0.361165\pi$
$349$	15.7846	0.844928	0.422464	+	0.906380i	$0.361165\pi$
$350$	27.3923	1.46418
$351$	1.89810	0.101313
$352$	2.10590	0.112245
$353$	2.14250	0.114034	0.0570170	−	0.998373i	$-0.481841\pi$
$353$	2.14250	0.114034	0.0570170	+	0.998373i	$0.481841\pi$
$354$	0.0417312	0.00221799
$355$	−4.47343	−0.237425
$356$	45.1516	2.39303
$357$	−0.630183	−0.0333528
$358$	47.5881	2.51511
$359$	−27.2266	−1.43696	−0.718481	−	0.695546i	$-0.755162\pi$
$359$	−27.2266	−1.43696	−0.718481	+	0.695546i	$0.755162\pi$
$360$	10.3486	0.545418
$361$	1.00000	0.0526316
$362$	8.16475	0.429130
$363$	1.15226	0.0604780
$364$	28.0710	1.47132
$365$	−5.63873	−0.295145
$366$	−2.22028	−0.116056
$367$	11.2931	0.589498	0.294749	−	0.955575i	$-0.404764\pi$
$367$	11.2931	0.589498	0.294749	+	0.955575i	$0.404764\pi$
$368$	13.4880	0.703110
$369$	−16.8360	−0.876449
$370$	−7.57282	−0.393692
$371$	35.0815	1.82134
$372$	0.592707	0.0307304
$373$	−18.6661	−0.966494	−0.483247	−	0.875484i	$-0.660543\pi$
$373$	−18.6661	−0.966494	−0.483247	+	0.875484i	$0.660543\pi$
$374$	−4.28729	−0.221690
$375$	0.876287	0.0452513
$376$	−7.68349	−0.396246
$377$	−11.8049	−0.607982
$378$	−4.28398	−0.220344
$379$	19.0045	0.976193	0.488097	−	0.872790i	$-0.337691\pi$
$379$	19.0045	0.976193	0.488097	+	0.872790i	$0.337691\pi$
$380$	3.13011	0.160571
$381$	−1.56453	−0.0801535
$382$	−20.7090	−1.05957
$383$	−1.58550	−0.0810153	−0.0405077	−	0.999179i	$-0.512898\pi$
$383$	−1.58550	−0.0810153	−0.0405077	+	0.999179i	$0.512898\pi$
$384$	−2.32539	−0.118667
$385$	1.90849	0.0972655
$386$	−7.59625	−0.386639
$387$	7.18789	0.365381
$388$	41.9802	2.13122
$389$	36.5981	1.85560	0.927798	−	0.373084i	$-0.121700\pi$
$389$	36.5981	1.85560	0.927798	+	0.373084i	$0.121700\pi$
$390$	0.637735	0.0322930
$391$	11.7960	0.596550
$392$	−0.478279	−0.0241567
$393$	2.16102	0.109009
$394$	−17.5100	−0.882143
$395$	7.52159	0.378452
$396$	−9.45904	−0.475335
$397$	−12.2686	−0.615745	−0.307872	−	0.951428i	$-0.599617\pi$
$397$	−12.2686	−0.615745	−0.307872	+	0.951428i	$0.599617\pi$
$398$	−46.0936	−2.31047
$399$	−0.299113	−0.0149744
$400$	−10.3414	−0.517069
$401$	−26.1817	−1.30745	−0.653727	−	0.756731i	$-0.726795\pi$
$401$	−26.1817	−1.30745	−0.653727	+	0.756731i	$0.726795\pi$
$402$	−0.781822	−0.0389937
$403$	−4.01533	−0.200018
$404$	64.3962	3.20383
$405$	7.47385	0.371379
$406$	26.6434	1.32229
$407$	3.20239	0.158737
$408$	0.973059	0.0481736
$409$	20.4820	1.01277	0.506385	−	0.862307i	$-0.330981\pi$
$409$	20.4820	1.01277	0.506385	+	0.862307i	$0.330981\pi$
$410$	−11.3371	−0.559902
$411$	0.585552	0.0288831
$412$	−18.5261	−0.912717
$413$	0.415047	0.0204231
$414$	40.0104	1.96641
$415$	6.53591	0.320835
$416$	6.99881	0.343145
$417$	−1.64570	−0.0805905
$418$	−2.03494	−0.0995320
$419$	37.7103	1.84227	0.921134	−	0.389245i	$-0.127264\pi$
$419$	37.7103	1.84227	0.921134	+	0.389245i	$0.127264\pi$
$420$	−0.936257	−0.0456847
$421$	33.5443	1.63485	0.817425	−	0.576035i	$-0.195401\pi$
$421$	33.5443	1.63485	0.817425	+	0.576035i	$0.195401\pi$
$422$	2.39206	0.116444
$423$	−5.57265	−0.270951
$424$	−54.1690	−2.63068
$425$	−9.04412	−0.438704
$426$	−1.42672	−0.0691246
$427$	−22.0822	−1.06863
$428$	10.5673	0.510789
$429$	−0.269685	−0.0130205
$430$	4.84023	0.233416
$431$	15.9014	0.765943	0.382972	−	0.923760i	$-0.374901\pi$
$431$	15.9014	0.765943	0.382972	+	0.923760i	$0.374901\pi$
$432$	1.61733	0.0778136
$433$	6.71705	0.322801	0.161400	−	0.986889i	$-0.448399\pi$
$433$	6.71705	0.322801	0.161400	+	0.986889i	$0.448399\pi$
$434$	9.06253	0.435015
$435$	0.393730	0.0188779
$436$	7.21172	0.345379
$437$	5.59891	0.267832
$438$	−1.79837	−0.0859292
$439$	10.4578	0.499122	0.249561	−	0.968359i	$-0.419714\pi$
$439$	10.4578	0.499122	0.249561	+	0.968359i	$0.419714\pi$
$440$	−2.94687	−0.140487
$441$	−0.346884	−0.0165183
$442$	−14.2485	−0.677732
$443$	39.9449	1.89784	0.948920	−	0.315515i	$-0.102177\pi$
$443$	39.9449	1.89784	0.948920	+	0.315515i	$0.102177\pi$
$444$	−1.57102	−0.0745572
$445$	10.2021	0.483628
$446$	−49.9284	−2.36418
$447$	0.231879	0.0109675
$448$	−28.6489	−1.35353
$449$	13.4576	0.635104	0.317552	−	0.948241i	$-0.397139\pi$
$449$	13.4576	0.635104	0.317552	+	0.948241i	$0.397139\pi$
$450$	−30.6764	−1.44610
$451$	4.79425	0.225753
$452$	46.5330	2.18873
$453$	0.765176	0.0359511
$454$	10.7593	0.504959
$455$	6.34272	0.297352
$456$	0.461857	0.0216284
$457$	6.07403	0.284131	0.142066	−	0.989857i	$-0.454626\pi$
$457$	6.07403	0.284131	0.142066	+	0.989857i	$0.454626\pi$
$458$	−43.6574	−2.03998
$459$	1.41444	0.0660206
$460$	17.5252	0.817118
$461$	13.5418	0.630704	0.315352	−	0.948975i	$-0.397877\pi$
$461$	13.5418	0.630704	0.315352	+	0.948975i	$0.397877\pi$
$462$	0.608676	0.0283182
$463$	2.38817	0.110988	0.0554939	−	0.998459i	$-0.482327\pi$
$463$	2.38817	0.110988	0.0554939	+	0.998459i	$0.482327\pi$
$464$	−10.0587	−0.466961
$465$	0.133924	0.00621057
$466$	−19.0373	−0.881887
$467$	−16.4063	−0.759192	−0.379596	−	0.925152i	$-0.623937\pi$
$467$	−16.4063	−0.759192	−0.379596	+	0.925152i	$0.623937\pi$
$468$	−31.4365	−1.45315
$469$	−7.77577	−0.359052
$470$	−3.75254	−0.173092
$471$	2.54726	0.117371
$472$	−0.640869	−0.0294984
$473$	−2.04684	−0.0941136
$474$	2.39887	0.110184
$475$	−4.29274	−0.196964
$476$	20.9182	0.958784
$477$	−39.2875	−1.79885
$478$	17.7554	0.812114
$479$	13.9893	0.639186	0.319593	−	0.947555i	$-0.396454\pi$
$479$	13.9893	0.639186	0.319593	+	0.947555i	$0.396454\pi$
$480$	−0.233433	−0.0106547
$481$	10.6429	0.485276
$482$	63.2316	2.88012
$483$	−1.67471	−0.0762018
$484$	−38.2479	−1.73854
$485$	9.48555	0.430716
$486$	7.20143	0.326663
$487$	11.7624	0.533007	0.266503	−	0.963834i	$-0.414132\pi$
$487$	11.7624	0.533007	0.266503	+	0.963834i	$0.414132\pi$
$488$	34.0969	1.54350
$489$	1.28540	0.0581279
$490$	−0.233587	−0.0105524
$491$	34.3109	1.54843	0.774214	−	0.632924i	$-0.218145\pi$
$491$	34.3109	1.54843	0.774214	+	0.632924i	$0.218145\pi$
$492$	−2.35194	−0.106034
$493$	−8.79687	−0.396191
$494$	−6.76297	−0.304280
$495$	−2.13730	−0.0960644
$496$	−3.42136	−0.153624
$497$	−14.1897	−0.636495
$498$	2.08450	0.0934088
$499$	18.0849	0.809592	0.404796	−	0.914407i	$-0.367343\pi$
$499$	18.0849	0.809592	0.404796	+	0.914407i	$0.367343\pi$
$500$	−29.0873	−1.30083
$501$	1.23143	0.0550162
$502$	−1.89599	−0.0846223
$503$	1.63886	0.0730733	0.0365366	−	0.999332i	$-0.488367\pi$
$503$	1.63886	0.0730733	0.0365366	+	0.999332i	$0.488367\pi$
$504$	32.8256	1.46217
$505$	14.5505	0.647489
$506$	−11.3934	−0.506500
$507$	0.561381	0.0249318
$508$	51.9329	2.30415
$509$	−12.0499	−0.534100	−0.267050	−	0.963683i	$-0.586049\pi$
$509$	−12.0499	−0.534100	−0.267050	+	0.963683i	$0.586049\pi$
$510$	0.475233	0.0210437
$511$	−17.8860	−0.791231
$512$	25.8093	1.14062
$513$	0.671357	0.0296412
$514$	−63.7765	−2.81306
$515$	−4.18603	−0.184459
$516$	1.00413	0.0442043
$517$	1.58688	0.0697907
$518$	−24.0209	−1.05542
$519$	0.174443	0.00765722
$520$	−9.79374	−0.429484
$521$	39.9477	1.75014	0.875070	−	0.483997i	$-0.160815\pi$
$521$	39.9477	1.75014	0.875070	+	0.483997i	$0.160815\pi$
$522$	−29.8377	−1.30596
$523$	−21.6782	−0.947921	−0.473961	−	0.880546i	$-0.657176\pi$
$523$	−21.6782	−0.947921	−0.473961	+	0.880546i	$0.657176\pi$
$524$	−71.7326	−3.13365
$525$	1.28401	0.0560390
$526$	41.4247	1.80620
$527$	−2.99218	−0.130341
$528$	−0.229793	−0.0100004
$529$	8.34783	0.362949
$530$	−26.4556	−1.14916
$531$	−0.464807	−0.0201709
$532$	9.92870	0.430464
$533$	15.9334	0.690151
$534$	3.25378	0.140805
$535$	2.38771	0.103230
$536$	12.0065	0.518601
$537$	2.23069	0.0962614
$538$	−33.5052	−1.44451
$539$	0.0987793	0.00425473
$540$	2.10143	0.0904310
$541$	−37.4020	−1.60804	−0.804019	−	0.594604i	$-0.797309\pi$
$541$	−37.4020	−1.60804	−0.804019	+	0.594604i	$0.797309\pi$
$542$	18.1082	0.777814
$543$	0.382723	0.0164242
$544$	5.21543	0.223610
$545$	1.62951	0.0698005
$546$	2.02289	0.0865718
$547$	38.4055	1.64210	0.821051	−	0.570855i	$-0.193388\pi$
$547$	38.4055	1.64210	0.821051	+	0.570855i	$0.193388\pi$
$548$	−19.4367	−0.830295
$549$	24.7297	1.05544
$550$	8.73546	0.372481
$551$	−4.17538	−0.177877
$552$	2.58590	0.110063
$553$	23.8584	1.01456
$554$	−70.9841	−3.01582
$555$	−0.354976	−0.0150679
$556$	54.6273	2.31671
$557$	−37.7220	−1.59833	−0.799166	−	0.601111i	$-0.794725\pi$
$557$	−37.7220	−1.59833	−0.799166	+	0.601111i	$0.794725\pi$
$558$	−10.1490	−0.429643
$559$	−6.80252	−0.287716
$560$	5.40448	0.228381
$561$	−0.200967	−0.00848482
$562$	−41.7149	−1.75963
$563$	15.7089	0.662051	0.331026	−	0.943622i	$-0.392605\pi$
$563$	15.7089	0.662051	0.331026	+	0.943622i	$0.392605\pi$
$564$	−0.778483	−0.0327800
$565$	10.5143	0.442338
$566$	15.0888	0.634232
$567$	23.7070	0.995601
$568$	21.9102	0.919330
$569$	1.51723	0.0636055	0.0318027	−	0.999494i	$-0.489875\pi$
$569$	1.51723	0.0636055	0.0318027	+	0.999494i	$0.489875\pi$
$570$	0.225567	0.00944795
$571$	40.8596	1.70992	0.854961	−	0.518692i	$-0.173581\pi$
$571$	40.8596	1.70992	0.854961	+	0.518692i	$0.173581\pi$
$572$	8.95190	0.374298
$573$	−0.970736	−0.0405531
$574$	−35.9614	−1.50100
$575$	−24.0347	−1.00232
$576$	32.0837	1.33682
$577$	12.7866	0.532313	0.266157	−	0.963930i	$-0.414246\pi$
$577$	12.7866	0.532313	0.266157	+	0.963930i	$0.414246\pi$
$578$	30.0472	1.24980
$579$	−0.356074	−0.0147980
$580$	−13.0694	−0.542678
$581$	20.7319	0.860102
$582$	3.02523	0.125400
$583$	11.1876	0.463342
$584$	27.6176	1.14283
$585$	−7.10316	−0.293680
$586$	51.2468	2.11699
$587$	−27.2544	−1.12491	−0.562455	−	0.826828i	$-0.690143\pi$
$587$	−27.2544	−1.12491	−0.562455	+	0.826828i	$0.690143\pi$
$588$	−0.0484587	−0.00199840
$589$	−1.42022	−0.0585191
$590$	−0.312995	−0.0128858
$591$	−0.820784	−0.0337625
$592$	9.06859	0.372717
$593$	26.7585	1.09884	0.549420	−	0.835546i	$-0.314849\pi$
$593$	26.7585	1.09884	0.549420	+	0.835546i	$0.314849\pi$
$594$	−1.36617	−0.0560546
$595$	4.72653	0.193769
$596$	−7.69695	−0.315280
$597$	−2.16064	−0.0884291
$598$	−37.8653	−1.54843
$599$	−26.3264	−1.07567	−0.537835	−	0.843050i	$-0.680757\pi$
$599$	−26.3264	−1.07567	−0.537835	+	0.843050i	$0.680757\pi$
$600$	−1.98263	−0.0809406
$601$	11.4460	0.466892	0.233446	−	0.972370i	$-0.425000\pi$
$601$	11.4460	0.466892	0.233446	+	0.972370i	$0.425000\pi$
$602$	15.3532	0.625748
$603$	8.70802	0.354618
$604$	−25.3991	−1.03348
$605$	−8.64223	−0.351357
$606$	4.64061	0.188512
$607$	33.2704	1.35040	0.675202	−	0.737633i	$-0.264056\pi$
$607$	33.2704	1.35040	0.675202	+	0.737633i	$0.264056\pi$
$608$	2.47548	0.100394
$609$	1.24891	0.0506084
$610$	16.6526	0.674245
$611$	5.27387	0.213358
$612$	−23.4261	−0.946944
$613$	8.86434	0.358027	0.179014	−	0.983847i	$-0.442709\pi$
$613$	8.86434	0.358027	0.179014	+	0.983847i	$0.442709\pi$
$614$	−10.9073	−0.440181
$615$	−0.531429	−0.0214293
$616$	−9.34747	−0.376620
$617$	−9.00447	−0.362506	−0.181253	−	0.983436i	$-0.558015\pi$
$617$	−9.00447	−0.362506	−0.181253	+	0.983436i	$0.558015\pi$
$618$	−1.33506	−0.0537038
$619$	−8.30411	−0.333770	−0.166885	−	0.985976i	$-0.553371\pi$
$619$	−8.30411	−0.333770	−0.166885	+	0.985976i	$0.553371\pi$
$620$	−4.44545	−0.178534
$621$	3.75887	0.150838
$622$	13.2849	0.532674
$623$	32.3611	1.29652
$624$	−0.763700	−0.0305725
$625$	14.8913	0.595653
$626$	−18.0089	−0.719779
$627$	−0.0953877	−0.00380942
$628$	−84.5533	−3.37404
$629$	7.93100	0.316230
$630$	16.0317	0.638719
$631$	−13.6483	−0.543331	−0.271665	−	0.962392i	$-0.587574\pi$
$631$	−13.6483	−0.543331	−0.271665	+	0.962392i	$0.587574\pi$
$632$	−36.8396	−1.46540
$633$	0.112128	0.00445668
$634$	−22.9286	−0.910612
$635$	11.7344	0.465665
$636$	−5.48835	−0.217627
$637$	0.328286	0.0130072
$638$	8.49664	0.336385
$639$	15.8909	0.628635
$640$	17.4410	0.689416
$641$	−37.6091	−1.48547	−0.742734	−	0.669586i	$-0.766471\pi$
$641$	−37.6091	−1.48547	−0.742734	+	0.669586i	$0.766471\pi$
$642$	0.761514	0.0300545
$643$	−42.8317	−1.68912	−0.844558	−	0.535464i	$-0.820137\pi$
$643$	−42.8317	−1.68912	−0.844558	+	0.535464i	$0.820137\pi$
$644$	55.5899	2.19055
$645$	0.226886	0.00893361
$646$	−5.03969	−0.198284
$647$	44.4971	1.74936	0.874681	−	0.484698i	$-0.161071\pi$
$647$	44.4971	1.74936	0.874681	+	0.484698i	$0.161071\pi$
$648$	−36.6058	−1.43801
$649$	0.132359	0.00519555
$650$	29.0317	1.13872
$651$	0.424806	0.0166495
$652$	−42.6674	−1.67099
$653$	−16.1389	−0.631564	−0.315782	−	0.948832i	$-0.602267\pi$
$653$	−16.1389	−0.631564	−0.315782	+	0.948832i	$0.602267\pi$
$654$	0.519701	0.0203219
$655$	−16.2082	−0.633306
$656$	13.5764	0.530071
$657$	20.0304	0.781460
$658$	−11.9030	−0.464029
$659$	36.3450	1.41580	0.707900	−	0.706312i	$-0.249643\pi$
$659$	36.3450	1.41580	0.707900	+	0.706312i	$0.249643\pi$
$660$	−0.298574	−0.0116220
$661$	−17.9190	−0.696967	−0.348484	−	0.937315i	$-0.613303\pi$
$661$	−17.9190	−0.696967	−0.348484	+	0.937315i	$0.613303\pi$
$662$	−42.3058	−1.64426
$663$	−0.667899	−0.0259390
$664$	−32.0118	−1.24230
$665$	2.24342	0.0869961
$666$	26.9008	1.04239
$667$	−23.3776	−0.905185
$668$	−40.8759	−1.58154
$669$	−2.34039	−0.0904848
$670$	5.86386	0.226541
$671$	−7.04207	−0.271856
$672$	−0.740447	−0.0285633
$673$	8.92344	0.343973	0.171987	−	0.985099i	$-0.444981\pi$
$673$	8.92344	0.343973	0.171987	+	0.985099i	$0.444981\pi$
$674$	65.0326	2.50496
$675$	−2.88196	−0.110927
$676$	−18.6344	−0.716708
$677$	−30.6229	−1.17693	−0.588467	−	0.808521i	$-0.700268\pi$
$677$	−30.6229	−1.17693	−0.588467	+	0.808521i	$0.700268\pi$
$678$	3.35332	0.128784
$679$	30.0881	1.15468
$680$	−7.29818	−0.279873
$681$	0.504342	0.0193264
$682$	2.89006	0.110666
$683$	25.2268	0.965276	0.482638	−	0.875820i	$-0.339679\pi$
$683$	25.2268	0.965276	0.482638	+	0.875820i	$0.339679\pi$
$684$	−11.1191	−0.425148
$685$	−4.39178	−0.167801
$686$	43.9266	1.67712
$687$	−2.04644	−0.0780766
$688$	−5.79626	−0.220980
$689$	37.1811	1.41649
$690$	1.26293	0.0480788
$691$	−40.2491	−1.53115	−0.765575	−	0.643347i	$-0.777545\pi$
$691$	−40.2491	−1.53115	−0.765575	+	0.643347i	$0.777545\pi$
$692$	−5.79045	−0.220120
$693$	−6.77950	−0.257532
$694$	−38.3720	−1.45658
$695$	12.3432	0.468204
$696$	−1.92843	−0.0730970
$697$	11.8734	0.449736
$698$	−37.7576	−1.42915
$699$	−0.892374	−0.0337527
$700$	−42.6214	−1.61094
$701$	−41.7195	−1.57572	−0.787862	−	0.615852i	$-0.788812\pi$
$701$	−41.7195	−1.57572	−0.787862	+	0.615852i	$0.788812\pi$
$702$	−4.54037	−0.171365
$703$	3.76440	0.141977
$704$	−9.13619	−0.344333
$705$	−0.175900	−0.00662480
$706$	−5.12500	−0.192882
$707$	46.1541	1.73580
$708$	−0.0649322	−0.00244030
$709$	−18.7833	−0.705421	−0.352711	−	0.935732i	$-0.614740\pi$
$709$	−18.7833	−0.705421	−0.352711	+	0.935732i	$0.614740\pi$
$710$	10.7007	0.401591
$711$	−26.7188	−1.00204
$712$	−49.9685	−1.87265
$713$	−7.95169	−0.297793
$714$	1.50744	0.0564144
$715$	2.02271	0.0756450
$716$	−74.0452	−2.76720
$717$	0.832285	0.0310823
$718$	65.1276	2.43054
$719$	−5.34902	−0.199485	−0.0997424	−	0.995013i	$-0.531802\pi$
$719$	−5.34902	−0.199485	−0.0997424	+	0.995013i	$0.531802\pi$
$720$	−6.05244	−0.225561
$721$	−13.2781	−0.494501
$722$	−2.39206	−0.0890233
$723$	2.96398	0.110232
$724$	−12.7040	−0.472142
$725$	17.9238	0.665675
$726$	−2.75628	−0.102295
$727$	−50.8973	−1.88768	−0.943838	−	0.330409i	$-0.892813\pi$
$727$	−50.8973	−1.88768	−0.943838	+	0.330409i	$0.892813\pi$
$728$	−31.0657	−1.15137
$729$	−26.3234	−0.974942
$730$	13.4882	0.499220
$731$	−5.06916	−0.187490
$732$	3.45467	0.127688
$733$	15.2312	0.562578	0.281289	−	0.959623i	$-0.409238\pi$
$733$	15.2312	0.562578	0.281289	+	0.959623i	$0.409238\pi$
$734$	−27.0139	−0.997101
$735$	−0.0109494	−0.000403874 0
$736$	13.8600	0.510885
$737$	−2.47971	−0.0913412
$738$	40.2728	1.48246
$739$	−16.3253	−0.600534	−0.300267	−	0.953855i	$-0.597076\pi$
$739$	−16.3253	−0.600534	−0.300267	+	0.953855i	$0.597076\pi$
$740$	11.7830	0.433152
$741$	−0.317014	−0.0116458
$742$	−83.9172	−3.08070
$743$	−20.1840	−0.740478	−0.370239	−	0.928937i	$-0.620724\pi$
$743$	−20.1840	−0.740478	−0.370239	+	0.928937i	$0.620724\pi$
$744$	−0.655938	−0.0240479
$745$	−1.73915	−0.0637175
$746$	44.6505	1.63477
$747$	−23.2174	−0.849481
$748$	6.67085	0.243911
$749$	7.57379	0.276740
$750$	−2.09613	−0.0765399
$751$	37.7255	1.37662	0.688312	−	0.725415i	$-0.258352\pi$
$751$	37.7255	1.37662	0.688312	+	0.725415i	$0.258352\pi$
$752$	4.49374	0.163870
$753$	−0.0888747	−0.00323877
$754$	28.2380	1.02837
$755$	−5.73901	−0.208864
$756$	6.66571	0.242430
$757$	0.198137	0.00720143	0.00360071	−	0.999994i	$-0.498854\pi$
$757$	0.198137	0.00720143	0.00360071	+	0.999994i	$0.498854\pi$
$758$	−45.4598	−1.65117
$759$	−0.534067	−0.0193854
$760$	−3.46404	−0.125654
$761$	52.2781	1.89508	0.947539	−	0.319640i	$-0.103562\pi$
$761$	52.2781	1.89508	0.947539	+	0.319640i	$0.103562\pi$
$762$	3.74246	0.135575
$763$	5.16879	0.187123
$764$	32.2225	1.16577
$765$	−5.29320	−0.191376
$766$	3.79262	0.137033
$767$	0.439887	0.0158834
$768$	3.15407	0.113813
$769$	38.8995	1.40275	0.701376	−	0.712791i	$-0.252569\pi$
$769$	38.8995	1.40275	0.701376	+	0.712791i	$0.252569\pi$
$770$	−4.56522	−0.164519
$771$	−2.98952	−0.107665
$772$	11.8195	0.425392
$773$	34.0319	1.22404	0.612022	−	0.790841i	$-0.290356\pi$
$773$	34.0319	1.22404	0.612022	+	0.790841i	$0.290356\pi$
$774$	−17.1939	−0.618021
$775$	6.09664	0.218998
$776$	−46.4587	−1.66777
$777$	−1.12598	−0.0403944
$778$	−87.5448	−3.13863
$779$	5.63563	0.201917
$780$	−0.992291	−0.0355297
$781$	−4.52512	−0.161922
$782$	−28.2168	−1.00903
$783$	−2.80317	−0.100177
$784$	0.279725	0.00999017
$785$	−19.1051	−0.681889
$786$	−5.16929	−0.184383
$787$	34.4264	1.22717	0.613584	−	0.789629i	$-0.289727\pi$
$787$	34.4264	1.22717	0.613584	+	0.789629i	$0.289727\pi$
$788$	27.2450	0.970562
$789$	1.94179	0.0691294
$790$	−17.9921	−0.640130
$791$	33.3512	1.18583
$792$	10.4682	0.371970
$793$	−23.4038	−0.831094
$794$	29.3473	1.04150
$795$	−1.24011	−0.0439821
$796$	71.7199	2.54205
$797$	−42.2901	−1.49799	−0.748996	−	0.662575i	$-0.769464\pi$
$797$	−42.2901	−1.49799	−0.748996	+	0.662575i	$0.769464\pi$
$798$	0.715496	0.0253283
$799$	3.93003	0.139035
$800$	−10.6266	−0.375706
$801$	−36.2409	−1.28051
$802$	62.6283	2.21148
$803$	−5.70389	−0.201286
$804$	1.21648	0.0429021
$805$	12.5607	0.442707
$806$	9.60491	0.338318
$807$	−1.57056	−0.0552863
$808$	−71.2661	−2.50713
$809$	−30.9500	−1.08814	−0.544072	−	0.839038i	$-0.683118\pi$
$809$	−30.9500	−1.08814	−0.544072	+	0.839038i	$0.683118\pi$
$810$	−17.8779	−0.628166
$811$	−4.25878	−0.149546	−0.0747731	−	0.997201i	$-0.523823\pi$
$811$	−4.25878	−0.149546	−0.0747731	+	0.997201i	$0.523823\pi$
$812$	−41.4561	−1.45483
$813$	0.848823	0.0297695
$814$	−7.66032	−0.268494
$815$	−9.64083	−0.337704
$816$	−0.569101	−0.0199225
$817$	−2.40605	−0.0841770
$818$	−48.9942	−1.71304
$819$	−22.5312	−0.787303
$820$	17.6402	0.616021
$821$	23.9817	0.836967	0.418484	−	0.908224i	$-0.362562\pi$
$821$	23.9817	0.836967	0.418484	+	0.908224i	$0.362562\pi$
$822$	−1.40068	−0.0488542
$823$	−26.9179	−0.938300	−0.469150	−	0.883118i	$-0.655440\pi$
$823$	−26.9179	−0.938300	−0.469150	+	0.883118i	$0.655440\pi$
$824$	20.5025	0.714240
$825$	0.409475	0.0142561
$826$	−0.992817	−0.0345445
$827$	−0.546858	−0.0190161	−0.00950806	−	0.999955i	$-0.503027\pi$
$827$	−0.546858	−0.0190161	−0.00950806	+	0.999955i	$0.503027\pi$
$828$	−62.2547	−2.16350
$829$	−29.7604	−1.03362	−0.516810	−	0.856100i	$-0.672881\pi$
$829$	−29.7604	−1.03362	−0.516810	+	0.856100i	$0.672881\pi$
$830$	−15.6343	−0.542674
$831$	−3.32738	−0.115426
$832$	−30.3635	−1.05267
$833$	0.244635	0.00847611
$834$	3.93663	0.136314
$835$	−9.23603	−0.319626
$836$	3.16628	0.109508
$837$	−0.953475	−0.0329569
$838$	−90.2053	−3.11609
$839$	30.3968	1.04941	0.524707	−	0.851283i	$-0.324175\pi$
$839$	30.3968	1.04941	0.524707	+	0.851283i	$0.324175\pi$
$840$	1.03614	0.0357502
$841$	−11.5662	−0.398834
$842$	−80.2400	−2.76525
$843$	−1.95538	−0.0673470
$844$	−3.72195	−0.128115
$845$	−4.21050	−0.144846
$846$	13.3301	0.458299
$847$	−27.4131	−0.941926
$848$	31.6811	1.08793
$849$	0.707290	0.0242741
$850$	21.6341	0.742043
$851$	21.0766	0.722495
$852$	2.21992	0.0760531
$853$	−41.3695	−1.41646	−0.708232	−	0.705980i	$-0.750507\pi$
$853$	−41.3695	−1.41646	−0.708232	+	0.705980i	$0.750507\pi$
$854$	52.8220	1.80753
$855$	−2.51239	−0.0859218
$856$	−11.6946	−0.399714
$857$	−4.22076	−0.144178	−0.0720892	−	0.997398i	$-0.522967\pi$
$857$	−4.22076	−0.144178	−0.0720892	+	0.997398i	$0.522967\pi$
$858$	0.645104	0.0220235
$859$	24.5511	0.837672	0.418836	−	0.908062i	$-0.362438\pi$
$859$	24.5511	0.837672	0.418836	+	0.908062i	$0.362438\pi$
$860$	−7.53121	−0.256812
$861$	−1.68569	−0.0574481
$862$	−38.0371	−1.29555
$863$	−22.5604	−0.767965	−0.383983	−	0.923340i	$-0.625448\pi$
$863$	−22.5604	−0.767965	−0.383983	+	0.923340i	$0.625448\pi$
$864$	1.66193	0.0565400
$865$	−1.30837	−0.0444859
$866$	−16.0676	−0.545999
$867$	1.40846	0.0478339
$868$	−14.1009	−0.478617
$869$	7.60850	0.258101
$870$	−0.941827	−0.0319309
$871$	−8.24114	−0.279240
$872$	−7.98108	−0.270273
$873$	−33.6954	−1.14042
$874$	−13.3929	−0.453023
$875$	−20.8475	−0.704775
$876$	2.79819	0.0945420
$877$	−19.0711	−0.643986	−0.321993	−	0.946742i	$-0.604353\pi$
$877$	−19.0711	−0.643986	−0.321993	+	0.946742i	$0.604353\pi$
$878$	−25.0156	−0.844236
$879$	2.40219	0.0810240
$880$	1.72350	0.0580992
$881$	−27.7779	−0.935861	−0.467930	−	0.883765i	$-0.655000\pi$
$881$	−27.7779	−0.935861	−0.467930	+	0.883765i	$0.655000\pi$
$882$	0.829768	0.0279398
$883$	−3.16501	−0.106511	−0.0532555	−	0.998581i	$-0.516960\pi$
$883$	−3.16501	−0.106511	−0.0532555	+	0.998581i	$0.516960\pi$
$884$	22.1701	0.745662
$885$	−0.0146716	−0.000493181 0
$886$	−95.5507	−3.21009
$887$	−17.9878	−0.603972	−0.301986	−	0.953312i	$-0.597650\pi$
$887$	−17.9878	−0.603972	−0.301986	+	0.953312i	$0.597650\pi$
$888$	1.73862	0.0583441
$889$	37.2214	1.24837
$890$	−24.4041	−0.818028
$891$	7.56021	0.253277
$892$	77.6866	2.60114
$893$	1.86537	0.0624221
$894$	−0.554669	−0.0185509
$895$	−16.7307	−0.559247
$896$	55.3228	1.84820
$897$	−1.77494	−0.0592634
$898$	−32.1914	−1.07424
$899$	5.92996	0.197775
$900$	47.7313	1.59104
$901$	27.7070	0.923052
$902$	−11.4681	−0.381848
$903$	0.719680	0.0239494
$904$	−51.4972	−1.71277
$905$	−2.87052	−0.0954192
$906$	−1.83035	−0.0608092
$907$	−20.3393	−0.675354	−0.337677	−	0.941262i	$-0.609641\pi$
$907$	−20.3393	−0.675354	−0.337677	+	0.941262i	$0.609641\pi$
$908$	−16.7411	−0.555572
$909$	−51.6876	−1.71437
$910$	−15.1722	−0.502953
$911$	−20.0863	−0.665487	−0.332744	−	0.943017i	$-0.607974\pi$
$911$	−20.0863	−0.665487	−0.332744	+	0.943017i	$0.607974\pi$
$912$	−0.270120	−0.00894458
$913$	6.61143	0.218806
$914$	−14.5295	−0.480592
$915$	0.780592	0.0258056
$916$	67.9292	2.24445
$917$	−51.4123	−1.69778
$918$	−3.38343	−0.111670
$919$	−30.3216	−1.00022	−0.500109	−	0.865963i	$-0.666707\pi$
$919$	−30.3216	−1.00022	−0.500109	+	0.865963i	$0.666707\pi$
$920$	−19.3949	−0.639430
$921$	−0.511278	−0.0168472
$922$	−32.3928	−1.06680
$923$	−15.0389	−0.495013
$924$	−0.947076	−0.0311565
$925$	−16.1596	−0.531325
$926$	−5.71266	−0.187730
$927$	14.8700	0.488395
$928$	−10.3361	−0.339298
$929$	30.0446	0.985733	0.492866	−	0.870105i	$-0.335949\pi$
$929$	30.0446	0.985733	0.492866	+	0.870105i	$0.335949\pi$
$930$	−0.320354	−0.0105048
$931$	0.116115	0.00380551
$932$	29.6213	0.970279
$933$	0.622728	0.0203872
$934$	39.2448	1.28413
$935$	1.50730	0.0492940
$936$	34.7902	1.13715
$937$	23.6042	0.771114	0.385557	−	0.922684i	$-0.374009\pi$
$937$	23.6042	0.771114	0.385557	+	0.922684i	$0.374009\pi$
$938$	18.6001	0.607315
$939$	−0.844166	−0.0275483
$940$	5.83881	0.190441
$941$	2.23627	0.0729003	0.0364502	−	0.999335i	$-0.488395\pi$
$941$	2.23627	0.0729003	0.0364502	+	0.999335i	$0.488395\pi$
$942$	−6.09320	−0.198527
$943$	31.5534	1.02752
$944$	0.374817	0.0121993
$945$	1.50614	0.0489946
$946$	4.89616	0.159188
$947$	24.1854	0.785920	0.392960	−	0.919556i	$-0.371451\pi$
$947$	24.1854	0.785920	0.392960	+	0.919556i	$0.371451\pi$
$948$	−3.73255	−0.121227
$949$	−18.9565	−0.615353
$950$	10.2685	0.333154
$951$	−1.07478	−0.0348521
$952$	−23.1498	−0.750289
$953$	12.1018	0.392015	0.196008	−	0.980602i	$-0.437202\pi$
$953$	12.1018	0.392015	0.196008	+	0.980602i	$0.437202\pi$
$954$	93.9781	3.04265
$955$	7.28076	0.235600
$956$	−27.6267	−0.893513
$957$	0.398280	0.0128746
$958$	−33.4632	−1.08115
$959$	−13.9307	−0.449846
$960$	1.01272	0.0326854
$961$	−28.9830	−0.934935
$962$	−25.4585	−0.820817
$963$	−8.48182	−0.273323
$964$	−98.3860	−3.16880
$965$	2.67065	0.0859711
$966$	4.00600	0.128891
$967$	20.7249	0.666469	0.333235	−	0.942844i	$-0.391860\pi$
$967$	20.7249	0.666469	0.333235	+	0.942844i	$0.391860\pi$
$968$	42.3283	1.36048
$969$	−0.236236	−0.00758898
$970$	−22.6900	−0.728532
$971$	−10.2305	−0.328313	−0.164157	−	0.986434i	$-0.552490\pi$
$971$	−10.2305	−0.328313	−0.164157	+	0.986434i	$0.552490\pi$
$972$	−11.2051	−0.359405
$973$	39.1525	1.25517
$974$	−28.1365	−0.901550
$975$	1.36086	0.0435824
$976$	−19.9418	−0.638323
$977$	28.8951	0.924438	0.462219	−	0.886766i	$-0.347053\pi$
$977$	28.8951	0.924438	0.462219	+	0.886766i	$0.347053\pi$
$978$	−3.07476	−0.0983200
$979$	10.3200	0.329829
$980$	0.363452	0.0116101
$981$	−5.78848	−0.184812
$982$	−82.0737	−2.61908
$983$	−5.70887	−0.182085	−0.0910423	−	0.995847i	$-0.529020\pi$
$983$	−5.70887	−0.182085	−0.0910423	+	0.995847i	$0.529020\pi$
$984$	2.60285	0.0829760
$985$	6.15608	0.196149
$986$	21.0426	0.670134
$987$	−0.557955	−0.0177599
$988$	10.5229	0.334779
$989$	−13.4713	−0.428361
$990$	5.11255	0.162487
$991$	40.7712	1.29514	0.647569	−	0.762007i	$-0.275786\pi$
$991$	40.7712	1.29514	0.647569	+	0.762007i	$0.275786\pi$
$992$	−3.51572	−0.111624
$993$	−1.98309	−0.0629314
$994$	33.9426	1.07659
$995$	16.2053	0.513744
$996$	−3.24341	−0.102771
$997$	3.33224	0.105533	0.0527665	−	0.998607i	$-0.483196\pi$
$997$	3.33224	0.105533	0.0527665	+	0.998607i	$0.483196\pi$
$998$	−43.2602	−1.36938
$999$	2.52726	0.0799590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.7	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-2.39206 q^{2} -0.112128 q^{3} +3.72195 q^{4} +0.840987 q^{5} +0.268217 q^{6} +2.66760 q^{7} -4.11902 q^{8} -2.98743 q^{9} +O(q^{10})\)	\(q-2.39206 q^{2} -0.112128 q^{3} +3.72195 q^{4} +0.840987 q^{5} +0.268217 q^{6} +2.66760 q^{7} -4.11902 q^{8} -2.98743 q^{9} -2.01169 q^{10} +0.850704 q^{11} -0.417335 q^{12} +2.82726 q^{13} -6.38107 q^{14} -0.0942980 q^{15} +2.40904 q^{16} +2.10684 q^{17} +7.14611 q^{18} +1.00000 q^{19} +3.13011 q^{20} -0.299113 q^{21} -2.03494 q^{22} +5.59891 q^{23} +0.461857 q^{24} -4.29274 q^{25} -6.76297 q^{26} +0.671357 q^{27} +9.92870 q^{28} -4.17538 q^{29} +0.225567 q^{30} -1.42022 q^{31} +2.47548 q^{32} -0.0953877 q^{33} -5.03969 q^{34} +2.24342 q^{35} -11.1191 q^{36} +3.76440 q^{37} -2.39206 q^{38} -0.317014 q^{39} -3.46404 q^{40} +5.63563 q^{41} +0.715496 q^{42} -2.40605 q^{43} +3.16628 q^{44} -2.51239 q^{45} -13.3929 q^{46} +1.86537 q^{47} -0.270120 q^{48} +0.116115 q^{49} +10.2685 q^{50} -0.236236 q^{51} +10.5229 q^{52} +13.1509 q^{53} -1.60593 q^{54} +0.715431 q^{55} -10.9879 q^{56} -0.112128 q^{57} +9.98777 q^{58} +0.155588 q^{59} -0.350973 q^{60} -8.27792 q^{61} +3.39725 q^{62} -7.96928 q^{63} -10.7396 q^{64} +2.37769 q^{65} +0.228173 q^{66} -2.91489 q^{67} +7.84157 q^{68} -0.627794 q^{69} -5.36640 q^{70} -5.31927 q^{71} +12.3053 q^{72} -6.70490 q^{73} -9.00468 q^{74} +0.481336 q^{75} +3.72195 q^{76} +2.26934 q^{77} +0.758318 q^{78} +8.94377 q^{79} +2.02597 q^{80} +8.88700 q^{81} -13.4808 q^{82} +7.77171 q^{83} -1.11328 q^{84} +1.77182 q^{85} +5.75541 q^{86} +0.468177 q^{87} -3.50407 q^{88} +12.1312 q^{89} +6.00978 q^{90} +7.54200 q^{91} +20.8389 q^{92} +0.159246 q^{93} -4.46207 q^{94} +0.840987 q^{95} -0.277570 q^{96} +11.2791 q^{97} -0.277753 q^{98} -2.54142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 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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