LMFDB - Embedded newform 1045.4.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.55697$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.55697 q^{2} +3.83974 q^{3} +4.65205 q^{4} +5.00000 q^{5} -13.6579 q^{6} -6.66228 q^{7} +11.9086 q^{8} -12.2564 q^{9} +O(q^{10})$	\(q-3.55697 q^{2} +3.83974 q^{3} +4.65205 q^{4} +5.00000 q^{5} -13.6579 q^{6} -6.66228 q^{7} +11.9086 q^{8} -12.2564 q^{9} -17.7849 q^{10} +11.0000 q^{11} +17.8627 q^{12} -52.2535 q^{13} +23.6976 q^{14} +19.1987 q^{15} -79.5748 q^{16} +85.6957 q^{17} +43.5956 q^{18} +19.0000 q^{19} +23.2603 q^{20} -25.5815 q^{21} -39.1267 q^{22} +64.2035 q^{23} +45.7258 q^{24} +25.0000 q^{25} +185.864 q^{26} -150.734 q^{27} -30.9933 q^{28} -18.6081 q^{29} -68.2893 q^{30} +73.3559 q^{31} +187.777 q^{32} +42.2372 q^{33} -304.817 q^{34} -33.3114 q^{35} -57.0173 q^{36} -75.0767 q^{37} -67.5825 q^{38} -200.640 q^{39} +59.5428 q^{40} -149.195 q^{41} +90.9925 q^{42} -98.8672 q^{43} +51.1726 q^{44} -61.2818 q^{45} -228.370 q^{46} -125.350 q^{47} -305.547 q^{48} -298.614 q^{49} -88.9243 q^{50} +329.049 q^{51} -243.086 q^{52} +748.312 q^{53} +536.158 q^{54} +55.0000 q^{55} -79.3381 q^{56} +72.9551 q^{57} +66.1886 q^{58} +490.679 q^{59} +89.3135 q^{60} -585.978 q^{61} -260.925 q^{62} +81.6554 q^{63} -31.3193 q^{64} -261.268 q^{65} -150.236 q^{66} -395.417 q^{67} +398.661 q^{68} +246.525 q^{69} +118.488 q^{70} -535.682 q^{71} -145.956 q^{72} -894.238 q^{73} +267.046 q^{74} +95.9936 q^{75} +88.3890 q^{76} -73.2851 q^{77} +713.671 q^{78} -720.669 q^{79} -397.874 q^{80} -247.859 q^{81} +530.682 q^{82} +356.541 q^{83} -119.006 q^{84} +428.478 q^{85} +351.668 q^{86} -71.4505 q^{87} +130.994 q^{88} -10.2924 q^{89} +217.978 q^{90} +348.128 q^{91} +298.678 q^{92} +281.668 q^{93} +445.866 q^{94} +95.0000 q^{95} +721.016 q^{96} -1538.87 q^{97} +1062.16 q^{98} -134.820 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * 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 $

$2$	−3.55697	−1.25758	−0.628790	−	0.777575i	$-0.716449\pi$
$2$	−3.55697	−1.25758	−0.628790	+	0.777575i	$0.716449\pi$
$3$	3.83974	0.738959	0.369480	−	0.929239i	$-0.379536\pi$
$3$	3.83974	0.738959	0.369480	+	0.929239i	$0.379536\pi$
$4$	4.65205	0.581507
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−13.6579	−0.929300
$7$	−6.66228	−0.359729	−0.179865	−	0.983691i	$-0.557566\pi$
$7$	−6.66228	−0.359729	−0.179865	+	0.983691i	$0.557566\pi$
$8$	11.9086	0.526289
$9$	−12.2564	−0.453940
$10$	−17.7849	−0.562407
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	17.8627	0.429710
$13$	−52.2535	−1.11481	−0.557405	−	0.830241i	$-0.688203\pi$
$13$	−52.2535	−1.11481	−0.557405	+	0.830241i	$0.688203\pi$
$14$	23.6976	0.452388
$15$	19.1987	0.330473
$16$	−79.5748	−1.24336
$17$	85.6957	1.22260	0.611301	−	0.791398i	$-0.290646\pi$
$17$	85.6957	1.22260	0.611301	+	0.791398i	$0.290646\pi$
$18$	43.5956	0.570865
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	23.2603	0.260058
$21$	−25.5815	−0.265825
$22$	−39.1267	−0.379175
$23$	64.2035	0.582058	0.291029	−	0.956714i	$-0.406002\pi$
$23$	64.2035	0.582058	0.291029	+	0.956714i	$0.406002\pi$
$24$	45.7258	0.388906
$25$	25.0000	0.200000
$26$	185.864	1.40196
$27$	−150.734	−1.07440
$28$	−30.9933	−0.209185
$29$	−18.6081	−0.119153	−0.0595766	−	0.998224i	$-0.518975\pi$
$29$	−18.6081	−0.119153	−0.0595766	+	0.998224i	$0.518975\pi$
$30$	−68.2893	−0.415596
$31$	73.3559	0.425003	0.212502	−	0.977161i	$-0.431839\pi$
$31$	73.3559	0.425003	0.212502	+	0.977161i	$0.431839\pi$
$32$	187.777	1.03733
$33$	42.2372	0.222805
$34$	−304.817	−1.53752
$35$	−33.3114	−0.160876
$36$	−57.0173	−0.263969
$37$	−75.0767	−0.333582	−0.166791	−	0.985992i	$-0.553340\pi$
$37$	−75.0767	−0.333582	−0.166791	+	0.985992i	$0.553340\pi$
$38$	−67.5825	−0.288509
$39$	−200.640	−0.823798
$40$	59.5428	0.235363
$41$	−149.195	−0.568301	−0.284150	−	0.958780i	$-0.591711\pi$
$41$	−149.195	−0.568301	−0.284150	+	0.958780i	$0.591711\pi$
$42$	90.9925	0.334297
$43$	−98.8672	−0.350630	−0.175315	−	0.984512i	$-0.556094\pi$
$43$	−98.8672	−0.350630	−0.175315	+	0.984512i	$0.556094\pi$
$44$	51.1726	0.175331
$45$	−61.2818	−0.203008
$46$	−228.370	−0.731985
$47$	−125.350	−0.389024	−0.194512	−	0.980900i	$-0.562312\pi$
$47$	−125.350	−0.389024	−0.194512	+	0.980900i	$0.562312\pi$
$48$	−305.547	−0.918790
$49$	−298.614	−0.870595
$50$	−88.9243	−0.251516
$51$	329.049	0.903453
$52$	−243.086	−0.648269
$53$	748.312	1.93941	0.969703	−	0.244288i	$-0.0785541\pi$
$53$	748.312	1.93941	0.969703	+	0.244288i	$0.0785541\pi$
$54$	536.158	1.35115
$55$	55.0000	0.134840
$56$	−79.3381	−0.189321
$57$	72.9551	0.169529
$58$	66.1886	0.149845
$59$	490.679	1.08273	0.541364	−	0.840788i	$-0.317908\pi$
$59$	490.679	1.08273	0.541364	+	0.840788i	$0.317908\pi$
$60$	89.3135	0.192172
$61$	−585.978	−1.22995	−0.614974	−	0.788547i	$-0.710834\pi$
$61$	−585.978	−1.22995	−0.614974	+	0.788547i	$0.710834\pi$
$62$	−260.925	−0.534476
$63$	81.6554	0.163295
$64$	−31.3193	−0.0611705
$65$	−261.268	−0.498558
$66$	−150.236	−0.280194
$67$	−395.417	−0.721013	−0.360506	−	0.932757i	$-0.617396\pi$
$67$	−395.417	−0.721013	−0.360506	+	0.932757i	$0.617396\pi$
$68$	398.661	0.710952
$69$	246.525	0.430117
$70$	118.488	0.202314
$71$	−535.682	−0.895406	−0.447703	−	0.894182i	$-0.647758\pi$
$71$	−535.682	−0.895406	−0.447703	+	0.894182i	$0.647758\pi$
$72$	−145.956	−0.238903
$73$	−894.238	−1.43374	−0.716868	−	0.697209i	$-0.754425\pi$
$73$	−894.238	−1.43374	−0.716868	+	0.697209i	$0.754425\pi$
$74$	267.046	0.419506
$75$	95.9936	0.147792
$76$	88.3890	0.133407
$77$	−73.2851	−0.108463
$78$	713.671	1.03599
$79$	−720.669	−1.02635	−0.513175	−	0.858284i	$-0.671531\pi$
$79$	−720.669	−1.02635	−0.513175	+	0.858284i	$0.671531\pi$
$80$	−397.874	−0.556046
$81$	−247.859	−0.339999
$82$	530.682	0.714683
$83$	356.541	0.471512	0.235756	−	0.971812i	$-0.424243\pi$
$83$	356.541	0.471512	0.235756	+	0.971812i	$0.424243\pi$
$84$	−119.006	−0.154579
$85$	428.478	0.546765
$86$	351.668	0.440946
$87$	−71.4505	−0.0880493
$88$	130.994	0.158682
$89$	−10.2924	−0.0122583	−0.00612915	−	0.999981i	$-0.501951\pi$
$89$	−10.2924	−0.0122583	−0.00612915	+	0.999981i	$0.501951\pi$
$90$	217.978	0.255299
$91$	348.128	0.401030
$92$	298.678	0.338471
$93$	281.668	0.314060
$94$	445.866	0.489229
$95$	95.0000	0.102598
$96$	721.016	0.766546
$97$	−1538.87	−1.61081	−0.805407	−	0.592722i	$-0.798053\pi$
$97$	−1538.87	−1.61081	−0.805407	+	0.592722i	$0.798053\pi$
$98$	1062.16	1.09484
$99$	−134.820	−0.136868
$100$	116.301	0.116301
$101$	610.350	0.601307	0.300654	−	0.953733i	$-0.402795\pi$
$101$	610.350	0.601307	0.300654	+	0.953733i	$0.402795\pi$
$102$	−1170.42	−1.13616
$103$	229.017	0.219085	0.109542	−	0.993982i	$-0.465061\pi$
$103$	229.017	0.219085	0.109542	+	0.993982i	$0.465061\pi$
$104$	−622.264	−0.586711
$105$	−127.907	−0.118881
$106$	−2661.72	−2.43896
$107$	1107.05	1.00021	0.500106	−	0.865964i	$-0.333294\pi$
$107$	1107.05	1.00021	0.500106	+	0.865964i	$0.333294\pi$
$108$	−701.225	−0.624772
$109$	2071.91	1.82067	0.910334	−	0.413874i	$-0.135824\pi$
$109$	2071.91	1.82067	0.910334	+	0.413874i	$0.135824\pi$
$110$	−195.633	−0.169572
$111$	−288.275	−0.246503
$112$	530.150	0.447272
$113$	25.1665	0.0209510	0.0104755	−	0.999945i	$-0.496665\pi$
$113$	25.1665	0.0209510	0.0104755	+	0.999945i	$0.496665\pi$
$114$	−259.499	−0.213196
$115$	321.017	0.260304
$116$	−86.5660	−0.0692884
$117$	640.438	0.506056
$118$	−1745.33	−1.36162
$119$	−570.929	−0.439806
$120$	228.629	0.173924
$121$	121.000	0.0909091
$122$	2084.31	1.54676
$123$	−572.870	−0.419951
$124$	341.256	0.247142
$125$	125.000	0.0894427
$126$	−290.446	−0.205357
$127$	86.8610	0.0606903	0.0303452	−	0.999539i	$-0.490339\pi$
$127$	86.8610	0.0606903	0.0303452	+	0.999539i	$0.490339\pi$
$128$	−1390.81	−0.960405
$129$	−379.625	−0.259102
$130$	929.322	0.626976
$131$	−552.530	−0.368510	−0.184255	−	0.982879i	$-0.558987\pi$
$131$	−552.530	−0.368510	−0.184255	+	0.982879i	$0.558987\pi$
$132$	196.490	0.129562
$133$	−126.583	−0.0825276
$134$	1406.49	0.906731
$135$	−753.672	−0.480487
$136$	1020.51	0.643442
$137$	1872.76	1.16789	0.583943	−	0.811795i	$-0.301509\pi$
$137$	1872.76	1.16789	0.583943	+	0.811795i	$0.301509\pi$
$138$	−876.882	−0.540907
$139$	−993.198	−0.606057	−0.303029	−	0.952981i	$-0.597998\pi$
$139$	−993.198	−0.606057	−0.303029	+	0.952981i	$0.597998\pi$
$140$	−154.967	−0.0935504
$141$	−481.311	−0.287473
$142$	1905.41	1.12604
$143$	−574.789	−0.336128
$144$	975.298	0.564409
$145$	−93.0407	−0.0532869
$146$	3180.78	1.80304
$147$	−1146.60	−0.643334
$148$	−349.261	−0.193980
$149$	−2519.53	−1.38529	−0.692644	−	0.721280i	$-0.743554\pi$
$149$	−2519.53	−1.38529	−0.692644	+	0.721280i	$0.743554\pi$
$150$	−341.447	−0.185860
$151$	−2221.75	−1.19737	−0.598687	−	0.800983i	$-0.704311\pi$
$151$	−2221.75	−1.19737	−0.598687	+	0.800983i	$0.704311\pi$
$152$	226.262	0.120739
$153$	−1050.32	−0.554988
$154$	260.673	0.136400
$155$	366.779	0.190067
$156$	−933.389	−0.479044
$157$	−3074.00	−1.56262	−0.781311	−	0.624142i	$-0.785449\pi$
$157$	−3074.00	−1.56262	−0.781311	+	0.624142i	$0.785449\pi$
$158$	2563.40	1.29072
$159$	2873.32	1.43314
$160$	938.885	0.463909
$161$	−427.742	−0.209384
$162$	881.629	0.427576
$163$	1786.81	0.858612	0.429306	−	0.903159i	$-0.358758\pi$
$163$	1786.81	0.858612	0.429306	+	0.903159i	$0.358758\pi$
$164$	−694.063	−0.330471
$165$	211.186	0.0996412
$166$	−1268.21	−0.592964
$167$	−3285.65	−1.52246	−0.761232	−	0.648480i	$-0.775405\pi$
$167$	−3285.65	−1.52246	−0.761232	+	0.648480i	$0.775405\pi$
$168$	−304.638	−0.139901
$169$	533.431	0.242800
$170$	−1524.09	−0.687600
$171$	−232.871	−0.104141
$172$	−459.936	−0.203894
$173$	−4038.35	−1.77474	−0.887371	−	0.461057i	$-0.847470\pi$
$173$	−4038.35	−1.77474	−0.887371	+	0.461057i	$0.847470\pi$
$174$	254.147	0.110729
$175$	−166.557	−0.0719459
$176$	−875.323	−0.374886
$177$	1884.08	0.800092
$178$	36.6097	0.0154158
$179$	4563.08	1.90537	0.952684	−	0.303963i	$-0.0983100\pi$
$179$	4563.08	1.90537	0.952684	+	0.303963i	$0.0983100\pi$
$180$	−285.086	−0.118051
$181$	−1061.61	−0.435960	−0.217980	−	0.975953i	$-0.569947\pi$
$181$	−1061.61	−0.435960	−0.217980	+	0.975953i	$0.569947\pi$
$182$	−1238.28	−0.504327
$183$	−2250.01	−0.908881
$184$	764.570	0.306331
$185$	−375.383	−0.149182
$186$	−1001.88	−0.394956
$187$	942.652	0.368629
$188$	−583.134	−0.226220
$189$	1004.24	0.386494
$190$	−337.912	−0.129025
$191$	−3661.40	−1.38707	−0.693533	−	0.720425i	$-0.743947\pi$
$191$	−3661.40	−1.38707	−0.693533	+	0.720425i	$0.743947\pi$
$192$	−120.258	−0.0452025
$193$	−3413.97	−1.27328	−0.636640	−	0.771161i	$-0.719676\pi$
$193$	−3413.97	−1.27328	−0.636640	+	0.771161i	$0.719676\pi$
$194$	5473.73	2.02573
$195$	−1003.20	−0.368414
$196$	−1389.17	−0.506257
$197$	−2034.54	−0.735811	−0.367906	−	0.929863i	$-0.619925\pi$
$197$	−2034.54	−0.735811	−0.367906	+	0.929863i	$0.619925\pi$
$198$	479.551	0.172122
$199$	3750.06	1.33585	0.667927	−	0.744227i	$-0.267182\pi$
$199$	3750.06	1.33585	0.667927	+	0.744227i	$0.267182\pi$
$200$	297.714	0.105258
$201$	−1518.30	−0.532799
$202$	−2171.00	−0.756192
$203$	123.973	0.0428629
$204$	1530.76	0.525364
$205$	−745.974	−0.254152
$206$	−814.608	−0.275517
$207$	−786.901	−0.264219
$208$	4158.06	1.38611
$209$	209.000	0.0691714
$210$	454.963	0.149502
$211$	−1327.48	−0.433115	−0.216557	−	0.976270i	$-0.569483\pi$
$211$	−1327.48	−0.433115	−0.216557	+	0.976270i	$0.569483\pi$
$212$	3481.19	1.12778
$213$	−2056.88	−0.661668
$214$	−3937.75	−1.25785
$215$	−494.336	−0.156807
$216$	−1795.03	−0.565445
$217$	−488.718	−0.152886
$218$	−7369.72	−2.28964
$219$	−3433.65	−1.05947
$220$	255.863	0.0784104
$221$	−4477.90	−1.36297
$222$	1025.39	0.309998
$223$	−3419.31	−1.02679	−0.513395	−	0.858153i	$-0.671612\pi$
$223$	−3419.31	−1.02679	−0.513395	+	0.858153i	$0.671612\pi$
$224$	−1251.02	−0.373159
$225$	−306.409	−0.0907879
$226$	−89.5164	−0.0263475
$227$	−2624.41	−0.767349	−0.383675	−	0.923468i	$-0.625342\pi$
$227$	−2624.41	−0.767349	−0.383675	+	0.923468i	$0.625342\pi$
$228$	339.391	0.0985822
$229$	5505.28	1.58864	0.794322	−	0.607497i	$-0.207826\pi$
$229$	5505.28	1.58864	0.794322	+	0.607497i	$0.207826\pi$
$230$	−1141.85	−0.327354
$231$	−281.396	−0.0801493
$232$	−221.596	−0.0627090
$233$	−6118.35	−1.72028	−0.860142	−	0.510054i	$-0.829625\pi$
$233$	−6118.35	−1.72028	−0.860142	+	0.510054i	$0.829625\pi$
$234$	−2278.02	−0.636406
$235$	−626.749	−0.173977
$236$	2282.67	0.629614
$237$	−2767.18	−0.758430
$238$	2030.78	0.553091
$239$	−1200.62	−0.324944	−0.162472	−	0.986713i	$-0.551947\pi$
$239$	−1200.62	−0.324944	−0.162472	+	0.986713i	$0.551947\pi$
$240$	−1527.73	−0.410895
$241$	−3024.11	−0.808299	−0.404149	−	0.914693i	$-0.632432\pi$
$241$	−3024.11	−0.808299	−0.404149	+	0.914693i	$0.632432\pi$
$242$	−430.394	−0.114325
$243$	3118.11	0.823156
$244$	−2726.00	−0.715223
$245$	−1493.07	−0.389342
$246$	2037.68	0.528122
$247$	−992.817	−0.255755
$248$	873.562	0.223674
$249$	1369.03	0.348428
$250$	−444.622	−0.112481
$251$	−4071.02	−1.02375	−0.511873	−	0.859061i	$-0.671048\pi$
$251$	−4071.02	−1.02375	−0.511873	+	0.859061i	$0.671048\pi$
$252$	379.865	0.0949574
$253$	706.238	0.175497
$254$	−308.962	−0.0763229
$255$	1645.25	0.404037
$256$	5197.64	1.26896
$257$	−7294.51	−1.77050	−0.885252	−	0.465113i	$-0.846014\pi$
$257$	−7294.51	−1.77050	−0.885252	+	0.465113i	$0.846014\pi$
$258$	1350.32	0.325841
$259$	500.182	0.119999
$260$	−1215.43	−0.289915
$261$	228.068	0.0540884
$262$	1965.33	0.463430
$263$	5840.43	1.36934	0.684670	−	0.728854i	$-0.259947\pi$
$263$	5840.43	1.36934	0.684670	+	0.728854i	$0.259947\pi$
$264$	502.984	0.117259
$265$	3741.56	0.867329
$266$	450.254	0.103785
$267$	−39.5201	−0.00905839
$268$	−1839.50	−0.419274
$269$	−3308.22	−0.749835	−0.374918	−	0.927058i	$-0.622329\pi$
$269$	−3308.22	−0.749835	−0.374918	+	0.927058i	$0.622329\pi$
$270$	2680.79	0.604251
$271$	−3363.20	−0.753874	−0.376937	−	0.926239i	$-0.623023\pi$
$271$	−3363.20	−0.753874	−0.376937	+	0.926239i	$0.623023\pi$
$272$	−6819.22	−1.52013
$273$	1336.72	0.296345
$274$	−6661.34	−1.46871
$275$	275.000	0.0603023
$276$	1146.85	0.250116
$277$	3334.39	0.723264	0.361632	−	0.932321i	$-0.382220\pi$
$277$	3334.39	0.723264	0.361632	+	0.932321i	$0.382220\pi$
$278$	3532.78	0.762165
$279$	−899.077	−0.192926
$280$	−396.691	−0.0846671
$281$	−2711.69	−0.575679	−0.287839	−	0.957679i	$-0.592937\pi$
$281$	−2711.69	−0.575679	−0.287839	+	0.957679i	$0.592937\pi$
$282$	1712.01	0.361520
$283$	−7468.32	−1.56871	−0.784356	−	0.620311i	$-0.787006\pi$
$283$	−7468.32	−1.56871	−0.784356	+	0.620311i	$0.787006\pi$
$284$	−2492.02	−0.520684
$285$	364.776	0.0758156
$286$	2044.51	0.422707
$287$	993.978	0.204434
$288$	−2301.46	−0.470886
$289$	2430.74	0.494758
$290$	330.943	0.0670126
$291$	−5908.88	−1.19033
$292$	−4160.05	−0.833727
$293$	−2856.36	−0.569524	−0.284762	−	0.958598i	$-0.591915\pi$
$293$	−2856.36	−0.569524	−0.284762	+	0.958598i	$0.591915\pi$
$294$	4078.43	0.809044
$295$	2453.39	0.484211
$296$	−894.054	−0.175560
$297$	−1658.08	−0.323944
$298$	8961.90	1.74211
$299$	−3354.86	−0.648884
$300$	446.567	0.0859419
$301$	658.681	0.126132
$302$	7902.70	1.50579
$303$	2343.59	0.444342
$304$	−1511.92	−0.285246
$305$	−2929.89	−0.550049
$306$	3735.95	0.697941
$307$	−5278.55	−0.981311	−0.490656	−	0.871354i	$-0.663243\pi$
$307$	−5278.55	−0.981311	−0.490656	+	0.871354i	$0.663243\pi$
$308$	−340.926	−0.0630717
$309$	879.367	0.161895
$310$	−1304.62	−0.239025
$311$	−5888.35	−1.07363	−0.536813	−	0.843701i	$-0.680372\pi$
$311$	−5888.35	−1.07363	−0.536813	+	0.843701i	$0.680372\pi$
$312$	−2389.33	−0.433556
$313$	−1412.25	−0.255032	−0.127516	−	0.991836i	$-0.540700\pi$
$313$	−1412.25	−0.255032	−0.127516	+	0.991836i	$0.540700\pi$
$314$	10934.1	1.96512
$315$	408.277	0.0730279
$316$	−3352.59	−0.596829
$317$	−3076.05	−0.545010	−0.272505	−	0.962154i	$-0.587852\pi$
$317$	−3076.05	−0.545010	−0.272505	+	0.962154i	$0.587852\pi$
$318$	−10220.3	−1.80229
$319$	−204.689	−0.0359260
$320$	−156.597	−0.0273563
$321$	4250.79	0.739116
$322$	1521.46	0.263317
$323$	1628.22	0.280484
$324$	−1153.06	−0.197712
$325$	−1306.34	−0.222962
$326$	−6355.63	−1.07977
$327$	7955.60	1.34540
$328$	−1776.69	−0.299090
$329$	835.115	0.139943
$330$	−751.182	−0.125307
$331$	9637.91	1.60045	0.800223	−	0.599703i	$-0.204715\pi$
$331$	9637.91	1.60045	0.800223	+	0.599703i	$0.204715\pi$
$332$	1658.65	0.274187
$333$	920.167	0.151426
$334$	11687.0	1.91462
$335$	−1977.08	−0.322447
$336$	2035.64	0.330516
$337$	5094.71	0.823521	0.411760	−	0.911292i	$-0.364914\pi$
$337$	5094.71	0.823521	0.411760	+	0.911292i	$0.364914\pi$
$338$	−1897.40	−0.305340
$339$	96.6327	0.0154819
$340$	1993.30	0.317947
$341$	806.915	0.128143
$342$	828.316	0.130965
$343$	4274.61	0.672908
$344$	−1177.37	−0.184533
$345$	1232.62	0.192354
$346$	14364.3	2.23188
$347$	−6158.68	−0.952782	−0.476391	−	0.879233i	$-0.658055\pi$
$347$	−6158.68	−0.952782	−0.476391	+	0.879233i	$0.658055\pi$
$348$	−332.391	−0.0512013
$349$	2307.74	0.353956	0.176978	−	0.984215i	$-0.443368\pi$
$349$	2307.74	0.353956	0.176978	+	0.984215i	$0.443368\pi$
$350$	592.439	0.0904777
$351$	7876.40	1.19775
$352$	2065.55	0.312767
$353$	4445.44	0.670274	0.335137	−	0.942169i	$-0.391217\pi$
$353$	4445.44	0.670274	0.335137	+	0.942169i	$0.391217\pi$
$354$	−6701.63	−1.00618
$355$	−2678.41	−0.400438
$356$	−47.8807	−0.00712829
$357$	−2192.22	−0.324999
$358$	−16230.8	−2.39615
$359$	9168.47	1.34789	0.673946	−	0.738781i	$-0.264598\pi$
$359$	9168.47	1.34789	0.673946	+	0.738781i	$0.264598\pi$
$360$	−729.778	−0.106841
$361$	361.000	0.0526316
$362$	3776.11	0.548255
$363$	464.609	0.0671781
$364$	1619.51	0.233201
$365$	−4471.19	−0.641186
$366$	8003.21	1.14299
$367$	1992.24	0.283363	0.141681	−	0.989912i	$-0.454749\pi$
$367$	1992.24	0.283363	0.141681	+	0.989912i	$0.454749\pi$
$368$	−5108.98	−0.723706
$369$	1828.59	0.257974
$370$	1335.23	0.187609
$371$	−4985.46	−0.697661
$372$	1310.33	0.182628
$373$	−9139.34	−1.26868	−0.634339	−	0.773055i	$-0.718728\pi$
$373$	−9139.34	−1.26868	−0.634339	+	0.773055i	$0.718728\pi$
$374$	−3352.99	−0.463580
$375$	479.968	0.0660945
$376$	−1492.73	−0.204739
$377$	972.340	0.132833
$378$	−3572.04	−0.486047
$379$	−12387.3	−1.67887	−0.839436	−	0.543458i	$-0.817115\pi$
$379$	−12387.3	−1.67887	−0.839436	+	0.543458i	$0.817115\pi$
$380$	441.945	0.0596613
$381$	333.524	0.0448476
$382$	13023.5	1.74435
$383$	11246.7	1.50047	0.750235	−	0.661171i	$-0.229940\pi$
$383$	11246.7	1.50047	0.750235	+	0.661171i	$0.229940\pi$
$384$	−5340.37	−0.709700
$385$	−366.426	−0.0485059
$386$	12143.4	1.60125
$387$	1211.75	0.159165
$388$	−7158.93	−0.936700
$389$	13061.9	1.70248	0.851241	−	0.524775i	$-0.175850\pi$
$389$	13061.9	1.70248	0.851241	+	0.524775i	$0.175850\pi$
$390$	3568.36	0.463310
$391$	5501.96	0.711626
$392$	−3556.06	−0.458184
$393$	−2121.57	−0.272313
$394$	7236.80	0.925341
$395$	−3603.35	−0.458997
$396$	−627.190	−0.0795896
$397$	1075.56	0.135972	0.0679860	−	0.997686i	$-0.478343\pi$
$397$	1075.56	0.135972	0.0679860	+	0.997686i	$0.478343\pi$
$398$	−13338.9	−1.67994
$399$	−486.048	−0.0609845
$400$	−1989.37	−0.248671
$401$	−4614.48	−0.574654	−0.287327	−	0.957833i	$-0.592767\pi$
$401$	−4614.48	−0.574654	−0.287327	+	0.957833i	$0.592767\pi$
$402$	5400.55	0.670037
$403$	−3833.10	−0.473798
$404$	2839.38	0.349664
$405$	−1239.30	−0.152052
$406$	−440.967	−0.0539035
$407$	−825.843	−0.100579
$408$	3918.50	0.475477
$409$	16173.3	1.95531	0.977653	−	0.210225i	$-0.0674198\pi$
$409$	16173.3	1.95531	0.977653	+	0.210225i	$0.0674198\pi$
$410$	2653.41	0.319616
$411$	7190.90	0.863019
$412$	1065.40	0.127399
$413$	−3269.04	−0.389489
$414$	2798.99	0.332277
$415$	1782.71	0.210867
$416$	−9812.01	−1.15643
$417$	−3813.62	−0.447851
$418$	−743.407	−0.0869886
$419$	4558.51	0.531498	0.265749	−	0.964042i	$-0.414381\pi$
$419$	4558.51	0.531498	0.265749	+	0.964042i	$0.414381\pi$
$420$	−595.032	−0.0691299
$421$	12225.6	1.41529	0.707645	−	0.706568i	$-0.249757\pi$
$421$	12225.6	1.41529	0.707645	+	0.706568i	$0.249757\pi$
$422$	4721.80	0.544677
$423$	1536.33	0.176593
$424$	8911.31	1.02069
$425$	2142.39	0.244521
$426$	7316.28	0.832100
$427$	3903.95	0.442448
$428$	5150.06	0.581630
$429$	−2207.04	−0.248385
$430$	1758.34	0.197197
$431$	6600.18	0.737632	0.368816	−	0.929502i	$-0.379763\pi$
$431$	6600.18	0.737632	0.368816	+	0.929502i	$0.379763\pi$
$432$	11994.7	1.33586
$433$	−7005.03	−0.777461	−0.388730	−	0.921352i	$-0.627086\pi$
$433$	−7005.03	−0.777461	−0.388730	+	0.921352i	$0.627086\pi$
$434$	1738.36	0.192267
$435$	−357.252	−0.0393769
$436$	9638.63	1.05873
$437$	1219.87	0.133533
$438$	12213.4	1.33237
$439$	18341.5	1.99406	0.997030	−	0.0770186i	$-0.0245401\pi$
$439$	18341.5	1.99406	0.997030	+	0.0770186i	$0.0245401\pi$
$440$	654.970	0.0709647
$441$	3659.92	0.395197
$442$	15927.8	1.71404
$443$	−9193.66	−0.986014	−0.493007	−	0.870025i	$-0.664102\pi$
$443$	−9193.66	−0.986014	−0.493007	+	0.870025i	$0.664102\pi$
$444$	−1341.07	−0.143343
$445$	−51.4619	−0.00548208
$446$	12162.4	1.29127
$447$	−9674.35	−1.02367
$448$	208.658	0.0220048
$449$	12291.2	1.29189	0.645943	−	0.763386i	$-0.276464\pi$
$449$	12291.2	1.29189	0.645943	+	0.763386i	$0.276464\pi$
$450$	1089.89	0.114173
$451$	−1641.14	−0.171349
$452$	117.076	0.0121831
$453$	−8530.95	−0.884810
$454$	9334.96	0.965003
$455$	1740.64	0.179346
$456$	868.790	0.0892211
$457$	−7656.44	−0.783705	−0.391852	−	0.920028i	$-0.628166\pi$
$457$	−7656.44	−0.783705	−0.391852	+	0.920028i	$0.628166\pi$
$458$	−19582.1	−1.99785
$459$	−12917.3	−1.31357
$460$	1493.39	0.151369
$461$	−2594.98	−0.262170	−0.131085	−	0.991371i	$-0.541846\pi$
$461$	−2594.98	−0.262170	−0.131085	+	0.991371i	$0.541846\pi$
$462$	1000.92	0.100794
$463$	−8735.81	−0.876863	−0.438432	−	0.898765i	$-0.644466\pi$
$463$	−8735.81	−0.876863	−0.438432	+	0.898765i	$0.644466\pi$
$464$	1480.74	0.148150
$465$	1408.34	0.140452
$466$	21762.8	2.16340
$467$	−3120.38	−0.309195	−0.154597	−	0.987978i	$-0.549408\pi$
$467$	−3120.38	−0.309195	−0.154597	+	0.987978i	$0.549408\pi$
$468$	2979.35	0.294275
$469$	2634.38	0.259370
$470$	2229.33	0.218790
$471$	−11803.4	−1.15471
$472$	5843.28	0.569827
$473$	−1087.54	−0.105719
$474$	9842.80	0.953786
$475$	475.000	0.0458831
$476$	−2655.99	−0.255750
$477$	−9171.58	−0.880373
$478$	4270.57	0.408643
$479$	−14135.1	−1.34833	−0.674166	−	0.738579i	$-0.735497\pi$
$479$	−14135.1	−1.34833	−0.674166	+	0.738579i	$0.735497\pi$
$480$	3605.08	0.342810
$481$	3923.02	0.371880
$482$	10756.7	1.01650
$483$	−1642.42	−0.154726
$484$	562.899	0.0528643
$485$	−7694.37	−0.720378
$486$	−11091.0	−1.03518
$487$	10060.7	0.936129	0.468065	−	0.883694i	$-0.344951\pi$
$487$	10060.7	0.936129	0.468065	+	0.883694i	$0.344951\pi$
$488$	−6978.15	−0.647308
$489$	6860.89	0.634479
$490$	5310.81	0.489628
$491$	4902.73	0.450625	0.225313	−	0.974287i	$-0.427660\pi$
$491$	4902.73	0.450625	0.225313	+	0.974287i	$0.427660\pi$
$492$	−2665.02	−0.244204
$493$	−1594.64	−0.145677
$494$	3531.42	0.321632
$495$	−674.100	−0.0612092
$496$	−5837.28	−0.528431
$497$	3568.87	0.322104
$498$	−4869.59	−0.438176
$499$	−11993.0	−1.07591	−0.537957	−	0.842972i	$-0.680804\pi$
$499$	−11993.0	−1.07591	−0.537957	+	0.842972i	$0.680804\pi$
$500$	581.507	0.0520116
$501$	−12616.1	−1.12504
$502$	14480.5	1.28744
$503$	−10154.7	−0.900146	−0.450073	−	0.892992i	$-0.648602\pi$
$503$	−10154.7	−0.900146	−0.450073	+	0.892992i	$0.648602\pi$
$504$	972.397	0.0859405
$505$	3051.75	0.268913
$506$	−2512.07	−0.220702
$507$	2048.24	0.179419
$508$	404.082	0.0352918
$509$	17330.6	1.50916	0.754582	−	0.656206i	$-0.227840\pi$
$509$	17330.6	1.50916	0.754582	+	0.656206i	$0.227840\pi$
$510$	−5852.10	−0.508108
$511$	5957.67	0.515757
$512$	−7361.36	−0.635409
$513$	−2863.95	−0.246485
$514$	25946.4	2.22655
$515$	1145.09	0.0979777
$516$	−1766.04	−0.150669
$517$	−1378.85	−0.117295
$518$	−1779.13	−0.150909
$519$	−15506.2	−1.31146
$520$	−3111.32	−0.262385
$521$	8032.95	0.675489	0.337745	−	0.941238i	$-0.390336\pi$
$521$	8032.95	0.675489	0.337745	+	0.941238i	$0.390336\pi$
$522$	−811.232	−0.0680204
$523$	−16656.0	−1.39257	−0.696287	−	0.717764i	$-0.745166\pi$
$523$	−16656.0	−1.39257	−0.696287	+	0.717764i	$0.745166\pi$
$524$	−2570.40	−0.214291
$525$	−639.536	−0.0531651
$526$	−20774.2	−1.72205
$527$	6286.28	0.519610
$528$	−3361.02	−0.277025
$529$	−8044.92	−0.661208
$530$	−13308.6	−1.09073
$531$	−6013.94	−0.491493
$532$	−588.873	−0.0479904
$533$	7795.96	0.633547
$534$	140.572	0.0113916
$535$	5535.26	0.447309
$536$	−4708.84	−0.379461
$537$	17521.1	1.40799
$538$	11767.2	0.942977
$539$	−3284.75	−0.262494
$540$	−3506.12	−0.279407
$541$	−2209.49	−0.175588	−0.0877941	−	0.996139i	$-0.527982\pi$
$541$	−2209.49	−0.175588	−0.0877941	+	0.996139i	$0.527982\pi$
$542$	11962.8	0.948057
$543$	−4076.31	−0.322157
$544$	16091.7	1.26824
$545$	10359.5	0.814228
$546$	−4754.68	−0.372677
$547$	−4225.60	−0.330299	−0.165150	−	0.986269i	$-0.552811\pi$
$547$	−4225.60	−0.330299	−0.165150	+	0.986269i	$0.552811\pi$
$548$	8712.16	0.679133
$549$	7181.97	0.558322
$550$	−978.167	−0.0758349
$551$	−353.554	−0.0273356
$552$	2935.75	0.226366
$553$	4801.30	0.369208
$554$	−11860.3	−0.909562
$555$	−1441.38	−0.110240
$556$	−4620.41	−0.352426
$557$	−12319.5	−0.937154	−0.468577	−	0.883423i	$-0.655233\pi$
$557$	−12319.5	−0.937154	−0.468577	+	0.883423i	$0.655233\pi$
$558$	3197.99	0.242620
$559$	5166.16	0.390886
$560$	2650.75	0.200026
$561$	3619.54	0.272401
$562$	9645.40	0.723962
$563$	24530.1	1.83627	0.918134	−	0.396269i	$-0.129695\pi$
$563$	24530.1	1.83627	0.918134	+	0.396269i	$0.129695\pi$
$564$	−2239.08	−0.167167
$565$	125.832	0.00936956
$566$	26564.6	1.97278
$567$	1651.31	0.122308
$568$	−6379.20	−0.471242
$569$	−20243.0	−1.49144	−0.745720	−	0.666259i	$-0.767894\pi$
$569$	−20243.0	−1.49144	−0.745720	+	0.666259i	$0.767894\pi$
$570$	−1297.50	−0.0953442
$571$	26613.4	1.95050	0.975249	−	0.221110i	$-0.0709679\pi$
$571$	26613.4	1.95050	0.975249	+	0.221110i	$0.0709679\pi$
$572$	−2673.95	−0.195461
$573$	−14058.8	−1.02498
$574$	−3535.55	−0.257093
$575$	1605.09	0.116412
$576$	383.861	0.0277677
$577$	−3876.25	−0.279671	−0.139836	−	0.990175i	$-0.544657\pi$
$577$	−3876.25	−0.279671	−0.139836	+	0.990175i	$0.544657\pi$
$578$	−8646.09	−0.622197
$579$	−13108.8	−0.940902
$580$	−432.830	−0.0309867
$581$	−2375.38	−0.169617
$582$	21017.7	1.49693
$583$	8231.43	0.584753
$584$	−10649.1	−0.754559
$585$	3202.19	0.226315
$586$	10160.0	0.716221
$587$	14994.3	1.05431	0.527156	−	0.849768i	$-0.323258\pi$
$587$	14994.3	1.05431	0.527156	+	0.849768i	$0.323258\pi$
$588$	−5334.05	−0.374103
$589$	1393.76	0.0975025
$590$	−8726.66	−0.608934
$591$	−7812.10	−0.543734
$592$	5974.21	0.414761
$593$	−5825.54	−0.403417	−0.201709	−	0.979446i	$-0.564649\pi$
$593$	−5825.54	−0.403417	−0.201709	+	0.979446i	$0.564649\pi$
$594$	5897.74	0.407386
$595$	−2854.64	−0.196687
$596$	−11721.0	−0.805554
$597$	14399.3	0.987141
$598$	11933.1	0.816024
$599$	11495.2	0.784112	0.392056	−	0.919941i	$-0.371764\pi$
$599$	11495.2	0.784112	0.392056	+	0.919941i	$0.371764\pi$
$600$	1143.14	0.0777811
$601$	−20814.0	−1.41268	−0.706339	−	0.707874i	$-0.749654\pi$
$601$	−20814.0	−1.41268	−0.706339	+	0.707874i	$0.749654\pi$
$602$	−2342.91	−0.158621
$603$	4846.38	0.327296
$604$	−10335.7	−0.696281
$605$	605.000	0.0406558
$606$	−8336.07	−0.558795
$607$	−15582.6	−1.04198	−0.520988	−	0.853564i	$-0.674436\pi$
$607$	−15582.6	−1.04198	−0.520988	+	0.853564i	$0.674436\pi$
$608$	3567.76	0.237980
$609$	476.023	0.0316739
$610$	10421.5	0.691731
$611$	6549.96	0.433688
$612$	−4886.13	−0.322729
$613$	5816.82	0.383261	0.191631	−	0.981467i	$-0.438622\pi$
$613$	5816.82	0.383261	0.191631	+	0.981467i	$0.438622\pi$
$614$	18775.6	1.23408
$615$	−2864.35	−0.187808
$616$	−872.719	−0.0570826
$617$	23096.7	1.50703	0.753515	−	0.657430i	$-0.228357\pi$
$617$	23096.7	1.50703	0.753515	+	0.657430i	$0.228357\pi$
$618$	−3127.88	−0.203595
$619$	1576.26	0.102351	0.0511756	−	0.998690i	$-0.483703\pi$
$619$	1576.26	0.102351	0.0511756	+	0.998690i	$0.483703\pi$
$620$	1706.28	0.110525
$621$	−9677.67	−0.625365
$622$	20944.7	1.35017
$623$	68.5707	0.00440967
$624$	15965.9	1.02428
$625$	625.000	0.0400000
$626$	5023.33	0.320723
$627$	802.506	0.0511149
$628$	−14300.4	−0.908676
$629$	−6433.74	−0.407838
$630$	−1452.23	−0.0918385
$631$	−6994.32	−0.441267	−0.220633	−	0.975357i	$-0.570812\pi$
$631$	−6994.32	−0.441267	−0.220633	+	0.975357i	$0.570812\pi$
$632$	−8582.12	−0.540156
$633$	−5097.17	−0.320054
$634$	10941.4	0.685393
$635$	434.305	0.0271415
$636$	13366.9	0.833381
$637$	15603.6	0.970547
$638$	728.075	0.0451799
$639$	6565.52	0.406460
$640$	−6954.07	−0.429506
$641$	−1699.95	−0.104749	−0.0523745	−	0.998628i	$-0.516679\pi$
$641$	−1699.95	−0.104749	−0.0523745	+	0.998628i	$0.516679\pi$
$642$	−15120.0	−0.929497
$643$	−12960.7	−0.794901	−0.397450	−	0.917624i	$-0.630105\pi$
$643$	−12960.7	−0.794901	−0.397450	+	0.917624i	$0.630105\pi$
$644$	−1989.88	−0.121758
$645$	−1898.12	−0.115874
$646$	−5791.52	−0.352731
$647$	−12452.8	−0.756677	−0.378339	−	0.925667i	$-0.623505\pi$
$647$	−12452.8	−0.756677	−0.378339	+	0.925667i	$0.623505\pi$
$648$	−2951.65	−0.178938
$649$	5397.47	0.326455
$650$	4646.61	0.280392
$651$	−1876.55	−0.112977
$652$	8312.33	0.499288
$653$	−14013.3	−0.839787	−0.419894	−	0.907573i	$-0.637933\pi$
$653$	−14013.3	−0.839787	−0.419894	+	0.907573i	$0.637933\pi$
$654$	−28297.8	−1.69195
$655$	−2762.65	−0.164803
$656$	11872.2	0.706600
$657$	10960.1	0.650829
$658$	−2970.48	−0.175990
$659$	24920.3	1.47307	0.736537	−	0.676397i	$-0.236460\pi$
$659$	24920.3	1.47307	0.736537	+	0.676397i	$0.236460\pi$
$660$	982.448	0.0579420
$661$	14820.7	0.872099	0.436049	−	0.899923i	$-0.356377\pi$
$661$	14820.7	0.872099	0.436049	+	0.899923i	$0.356377\pi$
$662$	−34281.8	−2.01269
$663$	−17194.0	−1.00718
$664$	4245.89	0.248151
$665$	−632.917	−0.0369075
$666$	−3273.01	−0.190430
$667$	−1194.71	−0.0693541
$668$	−15285.0	−0.885323
$669$	−13129.3	−0.758756
$670$	7032.44	0.405502
$671$	−6445.76	−0.370843
$672$	−4803.61	−0.275749
$673$	28003.2	1.60393	0.801964	−	0.597373i	$-0.203789\pi$
$673$	28003.2	1.60393	0.801964	+	0.597373i	$0.203789\pi$
$674$	−18121.7	−1.03564
$675$	−3768.36	−0.214880
$676$	2481.55	0.141190
$677$	4673.35	0.265305	0.132653	−	0.991163i	$-0.457651\pi$
$677$	4673.35	0.265305	0.132653	+	0.991163i	$0.457651\pi$
$678$	−343.720	−0.0194697
$679$	10252.4	0.579457
$680$	5102.56	0.287756
$681$	−10077.1	−0.567040
$682$	−2870.17	−0.161151
$683$	−312.173	−0.0174890	−0.00874449	−	0.999962i	$-0.502783\pi$
$683$	−312.173	−0.0174890	−0.00874449	+	0.999962i	$0.502783\pi$
$684$	−1083.33	−0.0605586
$685$	9363.78	0.522294
$686$	−15204.7	−0.846235
$687$	21138.9	1.17394
$688$	7867.34	0.435959
$689$	−39101.9	−2.16207
$690$	−4384.41	−0.241901
$691$	15522.8	0.854579	0.427290	−	0.904115i	$-0.359468\pi$
$691$	15522.8	0.854579	0.427290	+	0.904115i	$0.359468\pi$
$692$	−18786.6	−1.03202
$693$	898.209	0.0492354
$694$	21906.3	1.19820
$695$	−4965.99	−0.271037
$696$	−850.871	−0.0463394
$697$	−12785.4	−0.694806
$698$	−8208.57	−0.445128
$699$	−23492.9	−1.27122
$700$	−774.833	−0.0418370
$701$	34791.9	1.87457	0.937285	−	0.348565i	$-0.113331\pi$
$701$	34791.9	1.87457	0.937285	+	0.348565i	$0.113331\pi$
$702$	−28016.2	−1.50627
$703$	−1426.46	−0.0765289
$704$	−344.512	−0.0184436
$705$	−2406.55	−0.128562
$706$	−15812.3	−0.842924
$707$	−4066.32	−0.216308
$708$	8764.85	0.465259
$709$	−13563.7	−0.718471	−0.359235	−	0.933247i	$-0.616962\pi$
$709$	−13563.7	−0.718471	−0.359235	+	0.933247i	$0.616962\pi$
$710$	9527.04	0.503582
$711$	8832.79	0.465901
$712$	−122.567	−0.00645141
$713$	4709.70	0.247377
$714$	7797.67	0.408712
$715$	−2873.94	−0.150321
$716$	21227.7	1.10798
$717$	−4610.07	−0.240120
$718$	−32612.0	−1.69508
$719$	−15675.2	−0.813053	−0.406526	−	0.913639i	$-0.633260\pi$
$719$	−15675.2	−0.813053	−0.406526	+	0.913639i	$0.633260\pi$
$720$	4876.49	0.252411
$721$	−1525.78	−0.0788112
$722$	−1284.07	−0.0661884
$723$	−11611.8	−0.597300
$724$	−4938.66	−0.253514
$725$	−465.203	−0.0238306
$726$	−1652.60	−0.0844818
$727$	−22093.5	−1.12710	−0.563550	−	0.826082i	$-0.690565\pi$
$727$	−22093.5	−1.12710	−0.563550	+	0.826082i	$0.690565\pi$
$728$	4145.70	0.211057
$729$	18665.0	0.948278
$730$	15903.9	0.806343
$731$	−8472.49	−0.428682
$732$	−10467.2	−0.528521
$733$	−2353.87	−0.118611	−0.0593057	−	0.998240i	$-0.518889\pi$
$733$	−2353.87	−0.118611	−0.0593057	+	0.998240i	$0.518889\pi$
$734$	−7086.35	−0.356352
$735$	−5733.01	−0.287708
$736$	12055.9	0.603788
$737$	−4349.59	−0.217394
$738$	−6504.24	−0.324423
$739$	−6033.38	−0.300327	−0.150163	−	0.988661i	$-0.547980\pi$
$739$	−6033.38	−0.300327	−0.150163	+	0.988661i	$0.547980\pi$
$740$	−1746.30	−0.0867506
$741$	−3812.16	−0.188992
$742$	17733.2	0.877365
$743$	19105.4	0.943351	0.471675	−	0.881772i	$-0.343649\pi$
$743$	19105.4	0.943351	0.471675	+	0.881772i	$0.343649\pi$
$744$	3354.26	0.165286
$745$	−12597.6	−0.619520
$746$	32508.4	1.59546
$747$	−4369.90	−0.214038
$748$	4385.27	0.214360
$749$	−7375.49	−0.359806
$750$	−1707.23	−0.0831191
$751$	33895.9	1.64698	0.823489	−	0.567333i	$-0.192025\pi$
$751$	33895.9	1.64698	0.823489	+	0.567333i	$0.192025\pi$
$752$	9974.68	0.483696
$753$	−15631.7	−0.756506
$754$	−3458.59	−0.167048
$755$	−11108.7	−0.535482
$756$	4671.76	0.224749
$757$	12975.4	0.622984	0.311492	−	0.950249i	$-0.399171\pi$
$757$	12975.4	0.622984	0.311492	+	0.950249i	$0.399171\pi$
$758$	44061.3	2.11132
$759$	2711.77	0.129685
$760$	1131.31	0.0539961
$761$	−23371.1	−1.11327	−0.556637	−	0.830756i	$-0.687909\pi$
$761$	−23371.1	−1.11327	−0.556637	+	0.830756i	$0.687909\pi$
$762$	−1186.34	−0.0563995
$763$	−13803.6	−0.654948
$764$	−17033.0	−0.806588
$765$	−5251.59	−0.248198
$766$	−40004.2	−1.88696
$767$	−25639.7	−1.20704
$768$	19957.6	0.937707
$769$	−10106.3	−0.473917	−0.236959	−	0.971520i	$-0.576151\pi$
$769$	−10106.3	−0.473917	−0.236959	+	0.971520i	$0.576151\pi$
$770$	1303.37	0.0610000
$771$	−28009.1	−1.30833
$772$	−15882.0	−0.740421
$773$	−379.460	−0.0176562	−0.00882808	−	0.999961i	$-0.502810\pi$
$773$	−379.460	−0.0176562	−0.00882808	+	0.999961i	$0.502810\pi$
$774$	−4310.17	−0.200163
$775$	1833.90	0.0850007
$776$	−18325.8	−0.847753
$777$	1920.57	0.0886745
$778$	−46460.9	−2.14101
$779$	−2834.70	−0.130377
$780$	−4666.94	−0.214235
$781$	−5892.51	−0.269975
$782$	−19570.3	−0.894927
$783$	2804.89	0.128018
$784$	23762.2	1.08246
$785$	−15370.0	−0.698826
$786$	7546.38	0.342456
$787$	−40747.8	−1.84562	−0.922810	−	0.385254i	$-0.874114\pi$
$787$	−40747.8	−1.84562	−0.922810	+	0.385254i	$0.874114\pi$
$788$	−9464.78	−0.427879
$789$	22425.7	1.01189
$790$	12817.0	0.577226
$791$	−167.666	−0.00753669
$792$	−1605.51	−0.0720320
$793$	30619.4	1.37116
$794$	−3825.74	−0.170996
$795$	14366.6	0.640920
$796$	17445.5	0.776808
$797$	−6151.33	−0.273389	−0.136695	−	0.990613i	$-0.543648\pi$
$797$	−6151.33	−0.273389	−0.136695	+	0.990613i	$0.543648\pi$
$798$	1728.86	0.0766929
$799$	−10741.9	−0.475622
$800$	4694.43	0.207466
$801$	126.147	0.00556453
$802$	16413.6	0.722673
$803$	−9836.62	−0.432288
$804$	−7063.21	−0.309826
$805$	−2138.71	−0.0936392
$806$	13634.2	0.595838
$807$	−12702.7	−0.554097
$808$	7268.38	0.316461
$809$	1448.37	0.0629444	0.0314722	−	0.999505i	$-0.489980\pi$
$809$	1448.37	0.0629444	0.0314722	+	0.999505i	$0.489980\pi$
$810$	4408.15	0.191218
$811$	29234.7	1.26581	0.632903	−	0.774231i	$-0.281863\pi$
$811$	29234.7	1.26581	0.632903	+	0.774231i	$0.281863\pi$
$812$	576.727	0.0249251
$813$	−12913.8	−0.557082
$814$	2937.50	0.126486
$815$	8934.04	0.383983
$816$	−26184.0	−1.12331
$817$	−1878.48	−0.0804401
$818$	−57528.1	−2.45895
$819$	−4266.78	−0.182043
$820$	−3470.31	−0.147791
$821$	−36843.6	−1.56620	−0.783100	−	0.621896i	$-0.786363\pi$
$821$	−36843.6	−1.56620	−0.783100	+	0.621896i	$0.786363\pi$
$822$	−25577.8	−1.08532
$823$	−9141.97	−0.387204	−0.193602	−	0.981080i	$-0.562017\pi$
$823$	−9141.97	−0.387204	−0.193602	+	0.981080i	$0.562017\pi$
$824$	2727.26	0.115302
$825$	1055.93	0.0445609
$826$	11627.9	0.489814
$827$	−27310.5	−1.14834	−0.574171	−	0.818735i	$-0.694676\pi$
$827$	−27310.5	−1.14834	−0.574171	+	0.818735i	$0.694676\pi$
$828$	−3660.71	−0.153645
$829$	35184.1	1.47406	0.737029	−	0.675861i	$-0.236228\pi$
$829$	35184.1	1.47406	0.737029	+	0.675861i	$0.236228\pi$
$830$	−6341.04	−0.265181
$831$	12803.2	0.534463
$832$	1636.54	0.0681935
$833$	−25589.9	−1.06439
$834$	13565.0	0.563209
$835$	−16428.3	−0.680867
$836$	972.279	0.0402237
$837$	−11057.3	−0.456624
$838$	−16214.5	−0.668401
$839$	−2897.10	−0.119212	−0.0596061	−	0.998222i	$-0.518984\pi$
$839$	−2897.10	−0.119212	−0.0596061	+	0.998222i	$0.518984\pi$
$840$	−1523.19	−0.0625655
$841$	−24042.7	−0.985803
$842$	−43486.0	−1.77984
$843$	−10412.2	−0.425403
$844$	−6175.49	−0.251859
$845$	2667.15	0.108583
$846$	−5464.69	−0.222080
$847$	−806.136	−0.0327027
$848$	−59546.8	−2.41137
$849$	−28676.4	−1.15921
$850$	−7620.43	−0.307504
$851$	−4820.18	−0.194164
$852$	−9568.73	−0.384764
$853$	5048.43	0.202644	0.101322	−	0.994854i	$-0.467693\pi$
$853$	5048.43	0.202644	0.101322	+	0.994854i	$0.467693\pi$
$854$	−13886.3	−0.556414
$855$	−1164.36	−0.0465732
$856$	13183.4	0.526400
$857$	42800.5	1.70599	0.852997	−	0.521916i	$-0.174783\pi$
$857$	42800.5	1.70599	0.852997	+	0.521916i	$0.174783\pi$
$858$	7850.39	0.312363
$859$	−37196.1	−1.47743	−0.738717	−	0.674016i	$-0.764568\pi$
$859$	−37196.1	−1.47743	−0.738717	+	0.674016i	$0.764568\pi$
$860$	−2299.68	−0.0911842
$861$	3816.62	0.151069
$862$	−23476.7	−0.927631
$863$	23350.3	0.921036	0.460518	−	0.887650i	$-0.347664\pi$
$863$	23350.3	0.921036	0.460518	+	0.887650i	$0.347664\pi$
$864$	−28304.5	−1.11451
$865$	−20191.8	−0.793688
$866$	24916.7	0.977719
$867$	9333.44	0.365606
$868$	−2273.54	−0.0889044
$869$	−7927.36	−0.309456
$870$	1270.74	0.0495195
$871$	20661.9	0.803792
$872$	24673.4	0.958197
$873$	18861.0	0.731212
$874$	−4339.03	−0.167929
$875$	−832.785	−0.0321752
$876$	−15973.5	−0.616090
$877$	49765.8	1.91616	0.958080	−	0.286501i	$-0.0924921\pi$
$877$	49765.8	1.91616	0.958080	+	0.286501i	$0.0924921\pi$
$878$	−65240.2	−2.50769
$879$	−10967.7	−0.420855
$880$	−4376.62	−0.167654
$881$	−12687.5	−0.485189	−0.242594	−	0.970128i	$-0.577998\pi$
$881$	−12687.5	−0.485189	−0.242594	+	0.970128i	$0.577998\pi$
$882$	−13018.2	−0.496992
$883$	35433.5	1.35043	0.675216	−	0.737620i	$-0.264050\pi$
$883$	35433.5	1.35043	0.675216	+	0.737620i	$0.264050\pi$
$884$	−20831.4	−0.792576
$885$	9420.41	0.357812
$886$	32701.6	1.23999
$887$	−2534.56	−0.0959440	−0.0479720	−	0.998849i	$-0.515276\pi$
$887$	−2534.56	−0.0959440	−0.0479720	+	0.998849i	$0.515276\pi$
$888$	−3432.94	−0.129732
$889$	−578.693	−0.0218321
$890$	183.048	0.00689416
$891$	−2726.45	−0.102514
$892$	−15906.8	−0.597085
$893$	−2381.64	−0.0892483
$894$	34411.4	1.28735
$895$	22815.4	0.852106
$896$	9266.00	0.345486
$897$	−12881.8	−0.479499
$898$	−43719.4	−1.62465
$899$	−1365.02	−0.0506405
$900$	−1425.43	−0.0527938
$901$	64127.0	2.37112
$902$	5837.50	0.215485
$903$	2529.17	0.0932065
$904$	299.696	0.0110263
$905$	−5308.05	−0.194967
$906$	30344.3	1.11272
$907$	12557.4	0.459717	0.229858	−	0.973224i	$-0.426174\pi$
$907$	12557.4	0.459717	0.229858	+	0.973224i	$0.426174\pi$
$908$	−12208.9	−0.446219
$909$	−7480.67	−0.272957
$910$	−6191.40	−0.225542
$911$	3858.25	0.140318	0.0701590	−	0.997536i	$-0.477649\pi$
$911$	3858.25	0.140318	0.0701590	+	0.997536i	$0.477649\pi$
$912$	−5805.39	−0.210785
$913$	3921.95	0.142166
$914$	27233.7	0.985571
$915$	−11250.0	−0.406464
$916$	25610.9	0.923807
$917$	3681.11	0.132564
$918$	45946.4	1.65191
$919$	−21109.3	−0.757707	−0.378853	−	0.925457i	$-0.623682\pi$
$919$	−21109.3	−0.757707	−0.378853	+	0.925457i	$0.623682\pi$
$920$	3822.85	0.136995
$921$	−20268.3	−0.725149
$922$	9230.28	0.329699
$923$	27991.3	0.998206
$924$	−1309.07	−0.0466074
$925$	−1876.92	−0.0667164
$926$	31073.0	1.10273
$927$	−2806.92	−0.0994512
$928$	−3494.18	−0.123601
$929$	−1914.39	−0.0676093	−0.0338046	−	0.999428i	$-0.510762\pi$
$929$	−1914.39	−0.0676093	−0.0338046	+	0.999428i	$0.510762\pi$
$930$	−5009.42	−0.176630
$931$	−5673.67	−0.199728
$932$	−28462.9	−1.00036
$933$	−22609.8	−0.793366
$934$	11099.1	0.388837
$935$	4713.26	0.164856
$936$	7626.69	0.266331
$937$	−23508.9	−0.819638	−0.409819	−	0.912167i	$-0.634408\pi$
$937$	−23508.9	−0.819638	−0.409819	+	0.912167i	$0.634408\pi$
$938$	−9370.41	−0.326178
$939$	−5422.68	−0.188458
$940$	−2915.67	−0.101169
$941$	−1549.71	−0.0536865	−0.0268433	−	0.999640i	$-0.508546\pi$
$941$	−1549.71	−0.0536865	−0.0268433	+	0.999640i	$0.508546\pi$
$942$	41984.2	1.45214
$943$	−9578.83	−0.330784
$944$	−39045.7	−1.34622
$945$	5021.18	0.172845
$946$	3868.35	0.132950
$947$	10570.6	0.362724	0.181362	−	0.983416i	$-0.441949\pi$
$947$	10570.6	0.362724	0.181362	+	0.983416i	$0.441949\pi$
$948$	−12873.1	−0.441032
$949$	46727.1	1.59834
$950$	−1689.56	−0.0577017
$951$	−11811.2	−0.402740
$952$	−6798.93	−0.231465
$953$	24021.7	0.816515	0.408257	−	0.912867i	$-0.366137\pi$
$953$	24021.7	0.816515	0.408257	+	0.912867i	$0.366137\pi$
$954$	32623.1	1.10714
$955$	−18307.0	−0.620315
$956$	−5585.35	−0.188957
$957$	−785.955	−0.0265479
$958$	50278.3	1.69564
$959$	−12476.8	−0.420123
$960$	−601.291	−0.0202152
$961$	−24409.9	−0.819372
$962$	−13954.1	−0.467669
$963$	−13568.4	−0.454036
$964$	−14068.3	−0.470031
$965$	−17069.9	−0.569428
$966$	5842.04	0.194580
$967$	10157.2	0.337780	0.168890	−	0.985635i	$-0.445982\pi$
$967$	10157.2	0.337780	0.168890	+	0.985635i	$0.445982\pi$
$968$	1440.93	0.0478444
$969$	6251.94	0.207266
$970$	27368.7	0.905933
$971$	−10439.2	−0.345015	−0.172508	−	0.985008i	$-0.555187\pi$
$971$	−10439.2	−0.345015	−0.172508	+	0.985008i	$0.555187\pi$
$972$	14505.6	0.478671
$973$	6616.96	0.218017
$974$	−35785.7	−1.17726
$975$	−5016.00	−0.164760
$976$	46629.1	1.52926
$977$	3242.88	0.106191	0.0530956	−	0.998589i	$-0.483091\pi$
$977$	3242.88	0.106191	0.0530956	+	0.998589i	$0.483091\pi$
$978$	−24404.0	−0.797908
$979$	−113.216	−0.00369602
$980$	−6945.84	−0.226405
$981$	−25394.1	−0.826474
$982$	−17438.9	−0.566697
$983$	25053.1	0.812888	0.406444	−	0.913676i	$-0.366769\pi$
$983$	25053.1	0.812888	0.406444	+	0.913676i	$0.366769\pi$
$984$	−6822.05	−0.221015
$985$	−10172.7	−0.329065
$986$	5672.08	0.183201
$987$	3206.63	0.103412
$988$	−4618.64	−0.148723
$989$	−6347.62	−0.204087
$990$	2397.76	0.0769754
$991$	18014.7	0.577452	0.288726	−	0.957412i	$-0.406768\pi$
$991$	18014.7	0.577452	0.288726	+	0.957412i	$0.406768\pi$
$992$	13774.6	0.440870
$993$	37007.1	1.18266
$994$	−12694.4	−0.405071
$995$	18750.3	0.597412
$996$	6368.79	0.202613
$997$	−1579.87	−0.0501855	−0.0250928	−	0.999685i	$-0.507988\pi$
$997$	−1579.87	−0.0501855	−0.0250928	+	0.999685i	$0.507988\pi$
$998$	42658.8	1.35305
$999$	11316.6	0.358401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.5	✓	20	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-3.55697 q^{2} +3.83974 q^{3} +4.65205 q^{4} +5.00000 q^{5} -13.6579 q^{6} -6.66228 q^{7} +11.9086 q^{8} -12.2564 q^{9} +O(q^{10})\)	\(q-3.55697 q^{2} +3.83974 q^{3} +4.65205 q^{4} +5.00000 q^{5} -13.6579 q^{6} -6.66228 q^{7} +11.9086 q^{8} -12.2564 q^{9} -17.7849 q^{10} +11.0000 q^{11} +17.8627 q^{12} -52.2535 q^{13} +23.6976 q^{14} +19.1987 q^{15} -79.5748 q^{16} +85.6957 q^{17} +43.5956 q^{18} +19.0000 q^{19} +23.2603 q^{20} -25.5815 q^{21} -39.1267 q^{22} +64.2035 q^{23} +45.7258 q^{24} +25.0000 q^{25} +185.864 q^{26} -150.734 q^{27} -30.9933 q^{28} -18.6081 q^{29} -68.2893 q^{30} +73.3559 q^{31} +187.777 q^{32} +42.2372 q^{33} -304.817 q^{34} -33.3114 q^{35} -57.0173 q^{36} -75.0767 q^{37} -67.5825 q^{38} -200.640 q^{39} +59.5428 q^{40} -149.195 q^{41} +90.9925 q^{42} -98.8672 q^{43} +51.1726 q^{44} -61.2818 q^{45} -228.370 q^{46} -125.350 q^{47} -305.547 q^{48} -298.614 q^{49} -88.9243 q^{50} +329.049 q^{51} -243.086 q^{52} +748.312 q^{53} +536.158 q^{54} +55.0000 q^{55} -79.3381 q^{56} +72.9551 q^{57} +66.1886 q^{58} +490.679 q^{59} +89.3135 q^{60} -585.978 q^{61} -260.925 q^{62} +81.6554 q^{63} -31.3193 q^{64} -261.268 q^{65} -150.236 q^{66} -395.417 q^{67} +398.661 q^{68} +246.525 q^{69} +118.488 q^{70} -535.682 q^{71} -145.956 q^{72} -894.238 q^{73} +267.046 q^{74} +95.9936 q^{75} +88.3890 q^{76} -73.2851 q^{77} +713.671 q^{78} -720.669 q^{79} -397.874 q^{80} -247.859 q^{81} +530.682 q^{82} +356.541 q^{83} -119.006 q^{84} +428.478 q^{85} +351.668 q^{86} -71.4505 q^{87} +130.994 q^{88} -10.2924 q^{89} +217.978 q^{90} +348.128 q^{91} +298.678 q^{92} +281.668 q^{93} +445.866 q^{94} +95.0000 q^{95} +721.016 q^{96} -1538.87 q^{97} +1062.16 q^{98} -134.820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

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