LMFDB - Embedded newform 177.4.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.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:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 $ x^8 - 2x^7 - 45x^6 + 47x^5 + 654x^4 - 157x^3 - 2898x^2 + 96*x + 2432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.04902$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04902 q^{2} +3.00000 q^{3} -3.80150 q^{4} +16.1855 q^{5} +6.14707 q^{6} -1.13960 q^{7} -24.1816 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.04902 q^{2} +3.00000 q^{3} -3.80150 q^{4} +16.1855 q^{5} +6.14707 q^{6} -1.13960 q^{7} -24.1816 q^{8} +9.00000 q^{9} +33.1645 q^{10} +36.3258 q^{11} -11.4045 q^{12} +78.1440 q^{13} -2.33506 q^{14} +48.5565 q^{15} -19.1366 q^{16} -43.6811 q^{17} +18.4412 q^{18} -18.4272 q^{19} -61.5292 q^{20} -3.41879 q^{21} +74.4325 q^{22} -4.45159 q^{23} -72.5447 q^{24} +136.971 q^{25} +160.119 q^{26} +27.0000 q^{27} +4.33218 q^{28} -161.921 q^{29} +99.4936 q^{30} +245.150 q^{31} +154.241 q^{32} +108.977 q^{33} -89.5037 q^{34} -18.4450 q^{35} -34.2135 q^{36} -173.192 q^{37} -37.7579 q^{38} +234.432 q^{39} -391.391 q^{40} +128.039 q^{41} -7.00519 q^{42} -229.176 q^{43} -138.092 q^{44} +145.670 q^{45} -9.12142 q^{46} -138.962 q^{47} -57.4099 q^{48} -341.701 q^{49} +280.657 q^{50} -131.043 q^{51} -297.064 q^{52} -168.587 q^{53} +55.3237 q^{54} +587.952 q^{55} +27.5572 q^{56} -55.2817 q^{57} -331.780 q^{58} +59.0000 q^{59} -184.588 q^{60} -878.522 q^{61} +502.318 q^{62} -10.2564 q^{63} +469.137 q^{64} +1264.80 q^{65} +223.297 q^{66} +361.058 q^{67} +166.054 q^{68} -13.3548 q^{69} -37.7942 q^{70} -761.904 q^{71} -217.634 q^{72} -404.740 q^{73} -354.874 q^{74} +410.913 q^{75} +70.0511 q^{76} -41.3968 q^{77} +480.357 q^{78} -493.825 q^{79} -309.736 q^{80} +81.0000 q^{81} +262.356 q^{82} +398.428 q^{83} +12.9965 q^{84} -707.002 q^{85} -469.588 q^{86} -485.762 q^{87} -878.415 q^{88} -1327.94 q^{89} +298.481 q^{90} -89.0527 q^{91} +16.9227 q^{92} +735.449 q^{93} -284.736 q^{94} -298.254 q^{95} +462.723 q^{96} -1462.81 q^{97} -700.154 q^{98} +326.932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 6 * q^2 + 24 * q^3 + 34 * q^4 + 42 * q^5 + 18 * q^6 + 53 * q^7 + 51 * q^8 + 72 * q^9	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9} + 21 q^{10} + 67 q^{11} + 102 q^{12} + 33 q^{13} + 79 q^{14} + 126 q^{15} - 30 q^{16} + 139 q^{17} + 54 q^{18} + 64 q^{19} + 117 q^{20} + 159 q^{21} - 84 q^{22} + 226 q^{23} + 153 q^{24} + 96 q^{25} + 24 q^{26} + 216 q^{27} + 34 q^{28} + 456 q^{29} + 63 q^{30} + 124 q^{31} + 174 q^{32} + 201 q^{33} - 114 q^{34} + 556 q^{35} + 306 q^{36} + 127 q^{37} + 237 q^{38} + 99 q^{39} - 188 q^{40} + 425 q^{41} + 237 q^{42} - 115 q^{43} + 510 q^{44} + 378 q^{45} - 711 q^{46} + 420 q^{47} - 90 q^{48} + 171 q^{49} - 137 q^{50} + 417 q^{51} - 922 q^{52} + 98 q^{53} + 162 q^{54} - 616 q^{55} - 412 q^{56} + 192 q^{57} - 1548 q^{58} + 472 q^{59} + 351 q^{60} - 1254 q^{61} - 766 q^{62} + 477 q^{63} - 2019 q^{64} - 734 q^{65} - 252 q^{66} - 1010 q^{67} - 503 q^{68} + 678 q^{69} - 2956 q^{70} - 17 q^{71} + 459 q^{72} - 1180 q^{73} - 1228 q^{74} + 288 q^{75} - 2008 q^{76} + 441 q^{77} + 72 q^{78} - 873 q^{79} - 865 q^{80} + 648 q^{81} - 3645 q^{82} + 759 q^{83} + 102 q^{84} - 850 q^{85} - 1226 q^{86} + 1368 q^{87} - 3047 q^{88} + 988 q^{89} + 189 q^{90} - 2111 q^{91} - 1062 q^{92} + 372 q^{93} - 2240 q^{94} + 1822 q^{95} + 522 q^{96} - 668 q^{97} - 1368 q^{98} + 603 q^{99}+O(q^{100}) \) 8 * q + 6 * q^2 + 24 * q^3 + 34 * q^4 + 42 * q^5 + 18 * q^6 + 53 * q^7 + 51 * q^8 + 72 * q^9 + 21 * q^10 + 67 * q^11 + 102 * q^12 + 33 * q^13 + 79 * q^14 + 126 * q^15 - 30 * q^16 + 139 * q^17 + 54 * q^18 + 64 * q^19 + 117 * q^20 + 159 * q^21 - 84 * q^22 + 226 * q^23 + 153 * q^24 + 96 * q^25 + 24 * q^26 + 216 * q^27 + 34 * q^28 + 456 * q^29 + 63 * q^30 + 124 * q^31 + 174 * q^32 + 201 * q^33 - 114 * q^34 + 556 * q^35 + 306 * q^36 + 127 * q^37 + 237 * q^38 + 99 * q^39 - 188 * q^40 + 425 * q^41 + 237 * q^42 - 115 * q^43 + 510 * q^44 + 378 * q^45 - 711 * q^46 + 420 * q^47 - 90 * q^48 + 171 * q^49 - 137 * q^50 + 417 * q^51 - 922 * q^52 + 98 * q^53 + 162 * q^54 - 616 * q^55 - 412 * q^56 + 192 * q^57 - 1548 * q^58 + 472 * q^59 + 351 * q^60 - 1254 * q^61 - 766 * q^62 + 477 * q^63 - 2019 * q^64 - 734 * q^65 - 252 * q^66 - 1010 * q^67 - 503 * q^68 + 678 * q^69 - 2956 * q^70 - 17 * q^71 + 459 * q^72 - 1180 * q^73 - 1228 * q^74 + 288 * q^75 - 2008 * q^76 + 441 * q^77 + 72 * q^78 - 873 * q^79 - 865 * q^80 + 648 * q^81 - 3645 * q^82 + 759 * q^83 + 102 * q^84 - 850 * q^85 - 1226 * q^86 + 1368 * q^87 - 3047 * q^88 + 988 * q^89 + 189 * q^90 - 2111 * q^91 - 1062 * q^92 + 372 * q^93 - 2240 * q^94 + 1822 * q^95 + 522 * q^96 - 668 * q^97 - 1368 * q^98 + 603 * 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 $

$2$	2.04902	0.724440	0.362220	−	0.932093i	$-0.382019\pi$
$2$	2.04902	0.724440	0.362220	+	0.932093i	$0.382019\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−3.80150	−0.475187
$5$	16.1855	1.44768	0.723838	−	0.689970i	$-0.242376\pi$
$5$	16.1855	1.44768	0.723838	+	0.689970i	$0.242376\pi$
$6$	6.14707	0.418255
$7$	−1.13960	−0.0615325	−0.0307662	−	0.999527i	$-0.509795\pi$
$7$	−1.13960	−0.0615325	−0.0307662	+	0.999527i	$0.509795\pi$
$8$	−24.1816	−1.06868
$9$	9.00000	0.333333
$10$	33.1645	1.04875
$11$	36.3258	0.995695	0.497847	−	0.867265i	$-0.334124\pi$
$11$	36.3258	0.995695	0.497847	+	0.867265i	$0.334124\pi$
$12$	−11.4045	−0.274349
$13$	78.1440	1.66717	0.833586	−	0.552389i	$-0.186284\pi$
$13$	78.1440	1.66717	0.833586	+	0.552389i	$0.186284\pi$
$14$	−2.33506	−0.0445766
$15$	48.5565	0.835816
$16$	−19.1366	−0.299010
$17$	−43.6811	−0.623190	−0.311595	−	0.950215i	$-0.600863\pi$
$17$	−43.6811	−0.623190	−0.311595	+	0.950215i	$0.600863\pi$
$18$	18.4412	0.241480
$19$	−18.4272	−0.222500	−0.111250	−	0.993792i	$-0.535485\pi$
$19$	−18.4272	−0.222500	−0.111250	+	0.993792i	$0.535485\pi$
$20$	−61.5292	−0.687917
$21$	−3.41879	−0.0355258
$22$	74.4325	0.721321
$23$	−4.45159	−0.0403574	−0.0201787	−	0.999796i	$-0.506424\pi$
$23$	−4.45159	−0.0403574	−0.0201787	+	0.999796i	$0.506424\pi$
$24$	−72.5447	−0.617005
$25$	136.971	1.09577
$26$	160.119	1.20777
$27$	27.0000	0.192450
$28$	4.33218	0.0292394
$29$	−161.921	−1.03682	−0.518412	−	0.855131i	$-0.673477\pi$
$29$	−161.921	−1.03682	−0.518412	+	0.855131i	$0.673477\pi$
$30$	99.4936	0.605499
$31$	245.150	1.42033	0.710164	−	0.704036i	$-0.248621\pi$
$31$	245.150	1.42033	0.710164	+	0.704036i	$0.248621\pi$
$32$	154.241	0.852069
$33$	108.977	0.574865
$34$	−89.5037	−0.451464
$35$	−18.4450	−0.0890791
$36$	−34.2135	−0.158396
$37$	−173.192	−0.769528	−0.384764	−	0.923015i	$-0.625717\pi$
$37$	−173.192	−0.769528	−0.384764	+	0.923015i	$0.625717\pi$
$38$	−37.7579	−0.161188
$39$	234.432	0.962542
$40$	−391.391	−1.54711
$41$	128.039	0.487716	0.243858	−	0.969811i	$-0.421587\pi$
$41$	128.039	0.487716	0.243858	+	0.969811i	$0.421587\pi$
$42$	−7.00519	−0.0257363
$43$	−229.176	−0.812769	−0.406384	−	0.913702i	$-0.633211\pi$
$43$	−229.176	−0.812769	−0.406384	+	0.913702i	$0.633211\pi$
$44$	−138.092	−0.473141
$45$	145.670	0.482559
$46$	−9.12142	−0.0292365
$47$	−138.962	−0.431269	−0.215635	−	0.976474i	$-0.569182\pi$
$47$	−138.962	−0.431269	−0.215635	+	0.976474i	$0.569182\pi$
$48$	−57.4099	−0.172634
$49$	−341.701	−0.996214
$50$	280.657	0.793817
$51$	−131.043	−0.359799
$52$	−297.064	−0.792219
$53$	−168.587	−0.436929	−0.218464	−	0.975845i	$-0.570105\pi$
$53$	−168.587	−0.436929	−0.218464	+	0.975845i	$0.570105\pi$
$54$	55.3237	0.139418
$55$	587.952	1.44144
$56$	27.5572	0.0657588
$57$	−55.2817	−0.128460
$58$	−331.780	−0.751117
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	−184.588	−0.397169
$61$	−878.522	−1.84399	−0.921994	−	0.387205i	$-0.873441\pi$
$61$	−878.522	−1.84399	−0.921994	+	0.387205i	$0.873441\pi$
$62$	502.318	1.02894
$63$	−10.2564	−0.0205108
$64$	469.137	0.916283
$65$	1264.80	2.41353
$66$	223.297	0.416455
$67$	361.058	0.658362	0.329181	−	0.944267i	$-0.393227\pi$
$67$	361.058	0.658362	0.329181	+	0.944267i	$0.393227\pi$
$68$	166.054	0.296132
$69$	−13.3548	−0.0233004
$70$	−37.7942	−0.0645324
$71$	−761.904	−1.27354	−0.636771	−	0.771053i	$-0.719730\pi$
$71$	−761.904	−1.27354	−0.636771	+	0.771053i	$0.719730\pi$
$72$	−217.634	−0.356228
$73$	−404.740	−0.648921	−0.324460	−	0.945899i	$-0.605183\pi$
$73$	−404.740	−0.648921	−0.324460	+	0.945899i	$0.605183\pi$
$74$	−354.874	−0.557477
$75$	410.913	0.632641
$76$	70.0511	0.105729
$77$	−41.3968	−0.0612676
$78$	480.357	0.697304
$79$	−493.825	−0.703286	−0.351643	−	0.936134i	$-0.614377\pi$
$79$	−493.825	−0.703286	−0.351643	+	0.936134i	$0.614377\pi$
$80$	−309.736	−0.432870
$81$	81.0000	0.111111
$82$	262.356	0.353321
$83$	398.428	0.526906	0.263453	−	0.964672i	$-0.415139\pi$
$83$	398.428	0.526906	0.263453	+	0.964672i	$0.415139\pi$
$84$	12.9965	0.0168814
$85$	−707.002	−0.902177
$86$	−469.588	−0.588802
$87$	−485.762	−0.598611
$88$	−878.415	−1.06408
$89$	−1327.94	−1.58159	−0.790796	−	0.612080i	$-0.790333\pi$
$89$	−1327.94	−1.58159	−0.790796	+	0.612080i	$0.790333\pi$
$90$	298.481	0.349585
$91$	−89.0527	−0.102585
$92$	16.9227	0.0191773
$93$	735.449	0.820027
$94$	−284.736	−0.312429
$95$	−298.254	−0.322108
$96$	462.723	0.491942
$97$	−1462.81	−1.53120	−0.765598	−	0.643320i	$-0.777557\pi$
$97$	−1462.81	−1.53120	−0.765598	+	0.643320i	$0.777557\pi$
$98$	−700.154	−0.721697
$99$	326.932	0.331898
$100$	−520.694	−0.520694
$101$	228.118	0.224738	0.112369	−	0.993667i	$-0.464156\pi$
$101$	228.118	0.224738	0.112369	+	0.993667i	$0.464156\pi$
$102$	−268.511	−0.260653
$103$	764.688	0.731524	0.365762	−	0.930708i	$-0.380808\pi$
$103$	764.688	0.731524	0.365762	+	0.930708i	$0.380808\pi$
$104$	−1889.64	−1.78168
$105$	−55.3349	−0.0514299
$106$	−345.439	−0.316529
$107$	1557.21	1.40692	0.703462	−	0.710733i	$-0.251637\pi$
$107$	1557.21	1.40692	0.703462	+	0.710733i	$0.251637\pi$
$108$	−102.640	−0.0914498
$109$	1589.34	1.39662	0.698310	−	0.715796i	$-0.253936\pi$
$109$	1589.34	1.39662	0.698310	+	0.715796i	$0.253936\pi$
$110$	1204.73	1.04424
$111$	−519.575	−0.444287
$112$	21.8081	0.0183988
$113$	1409.48	1.17339	0.586694	−	0.809809i	$-0.300429\pi$
$113$	1409.48	1.17339	0.586694	+	0.809809i	$0.300429\pi$
$114$	−113.274	−0.0930618
$115$	−72.0513	−0.0584245
$116$	615.541	0.492686
$117$	703.296	0.555724
$118$	120.892	0.0943140
$119$	49.7789	0.0383464
$120$	−1174.17	−0.893224
$121$	−11.4358	−0.00859188
$122$	−1800.11	−1.33586
$123$	384.118	0.281583
$124$	−931.935	−0.674921
$125$	193.755	0.138640
$126$	−21.0156	−0.0148589
$127$	−1270.35	−0.887603	−0.443802	−	0.896125i	$-0.646370\pi$
$127$	−1270.35	−0.887603	−0.443802	+	0.896125i	$0.646370\pi$
$128$	−272.655	−0.188278
$129$	−687.529	−0.469252
$130$	2591.61	1.74845
$131$	2502.45	1.66901	0.834504	−	0.551002i	$-0.185754\pi$
$131$	2502.45	1.66901	0.834504	+	0.551002i	$0.185754\pi$
$132$	−414.277	−0.273168
$133$	20.9996	0.0136910
$134$	739.817	0.476944
$135$	437.009	0.278605
$136$	1056.28	0.665993
$137$	458.017	0.285628	0.142814	−	0.989750i	$-0.454385\pi$
$137$	458.017	0.285628	0.142814	+	0.989750i	$0.454385\pi$
$138$	−27.3643	−0.0168797
$139$	286.953	0.175101	0.0875505	−	0.996160i	$-0.472096\pi$
$139$	286.953	0.175101	0.0875505	+	0.996160i	$0.472096\pi$
$140$	70.1185	0.0423293
$141$	−416.885	−0.248993
$142$	−1561.16	−0.922604
$143$	2838.64	1.65999
$144$	−172.230	−0.0996700
$145$	−2620.77	−1.50099
$146$	−829.322	−0.470104
$147$	−1025.10	−0.575164
$148$	658.388	0.365670
$149$	2607.82	1.43383	0.716915	−	0.697160i	$-0.245553\pi$
$149$	2607.82	1.43383	0.716915	+	0.697160i	$0.245553\pi$
$150$	841.970	0.458311
$151$	−20.5789	−0.0110907	−0.00554533	−	0.999985i	$-0.501765\pi$
$151$	−20.5789	−0.0110907	−0.00554533	+	0.999985i	$0.501765\pi$
$152$	445.600	0.237782
$153$	−393.130	−0.207730
$154$	−84.8231	−0.0443847
$155$	3967.87	2.05617
$156$	−891.192	−0.457388
$157$	−1903.56	−0.967649	−0.483825	−	0.875165i	$-0.660753\pi$
$157$	−1903.56	−0.967649	−0.483825	+	0.875165i	$0.660753\pi$
$158$	−1011.86	−0.509488
$159$	−505.762	−0.252261
$160$	2496.47	1.23352
$161$	5.07302	0.00248329
$162$	165.971	0.0804933
$163$	−195.798	−0.0940862	−0.0470431	−	0.998893i	$-0.514980\pi$
$163$	−195.798	−0.0940862	−0.0470431	+	0.998893i	$0.514980\pi$
$164$	−486.741	−0.231757
$165$	1763.86	0.832218
$166$	816.390	0.381712
$167$	−2098.12	−0.972198	−0.486099	−	0.873904i	$-0.661581\pi$
$167$	−2098.12	−0.972198	−0.486099	+	0.873904i	$0.661581\pi$
$168$	82.6717	0.0379659
$169$	3909.48	1.77946
$170$	−1448.66	−0.653573
$171$	−165.845	−0.0741667
$172$	871.213	0.386217
$173$	1996.80	0.877537	0.438769	−	0.898600i	$-0.355415\pi$
$173$	1996.80	0.877537	0.438769	+	0.898600i	$0.355415\pi$
$174$	−995.339	−0.433658
$175$	−156.092	−0.0674253
$176$	−695.154	−0.297723
$177$	177.000	0.0751646
$178$	−2720.99	−1.14577
$179$	−2474.76	−1.03336	−0.516682	−	0.856177i	$-0.672833\pi$
$179$	−2474.76	−1.03336	−0.516682	+	0.856177i	$0.672833\pi$
$180$	−553.763	−0.229306
$181$	618.103	0.253830	0.126915	−	0.991914i	$-0.459492\pi$
$181$	618.103	0.253830	0.126915	+	0.991914i	$0.459492\pi$
$182$	−182.471	−0.0743168
$183$	−2635.57	−1.06463
$184$	107.646	0.0431294
$185$	−2803.20	−1.11403
$186$	1506.95	0.594060
$187$	−1586.75	−0.620507
$188$	528.263	0.204934
$189$	−30.7691	−0.0118419
$190$	−611.131	−0.233348
$191$	552.196	0.209191	0.104596	−	0.994515i	$-0.466645\pi$
$191$	552.196	0.209191	0.104596	+	0.994515i	$0.466645\pi$
$192$	1407.41	0.529016
$193$	4002.75	1.49287	0.746436	−	0.665457i	$-0.231763\pi$
$193$	4002.75	1.49287	0.746436	+	0.665457i	$0.231763\pi$
$194$	−2997.34	−1.10926
$195$	3794.40	1.39345
$196$	1298.98	0.473388
$197$	1380.23	0.499173	0.249586	−	0.968353i	$-0.419705\pi$
$197$	1380.23	0.499173	0.249586	+	0.968353i	$0.419705\pi$
$198$	669.892	0.240440
$199$	2132.94	0.759800	0.379900	−	0.925028i	$-0.375958\pi$
$199$	2132.94	0.759800	0.379900	+	0.925028i	$0.375958\pi$
$200$	−3312.17	−1.17103
$201$	1083.17	0.380106
$202$	467.419	0.162809
$203$	184.524	0.0637984
$204$	498.161	0.170972
$205$	2072.38	0.706055
$206$	1566.86	0.529945
$207$	−40.0643	−0.0134525
$208$	−1495.41	−0.498501
$209$	−669.385	−0.221542
$210$	−113.383	−0.0372578
$211$	6000.66	1.95783	0.978916	−	0.204265i	$-0.0654806\pi$
$211$	6000.66	1.95783	0.978916	+	0.204265i	$0.0654806\pi$
$212$	640.884	0.207623
$213$	−2285.71	−0.735279
$214$	3190.76	1.01923
$215$	−3709.34	−1.17663
$216$	−652.902	−0.205668
$217$	−279.372	−0.0873963
$218$	3256.60	1.01177
$219$	−1214.22	−0.374655
$220$	−2235.10	−0.684956
$221$	−3413.42	−1.03897
$222$	−1064.62	−0.321859
$223$	−4919.66	−1.47733	−0.738666	−	0.674071i	$-0.764544\pi$
$223$	−4919.66	−1.47733	−0.738666	+	0.674071i	$0.764544\pi$
$224$	−175.773	−0.0524299
$225$	1232.74	0.365256
$226$	2888.06	0.850048
$227$	1021.59	0.298701	0.149350	−	0.988784i	$-0.452282\pi$
$227$	1021.59	0.298701	0.149350	+	0.988784i	$0.452282\pi$
$228$	210.153	0.0610427
$229$	1745.58	0.503716	0.251858	−	0.967764i	$-0.418958\pi$
$229$	1745.58	0.503716	0.251858	+	0.967764i	$0.418958\pi$
$230$	−147.635	−0.0423250
$231$	−124.190	−0.0353728
$232$	3915.50	1.10804
$233$	−1961.28	−0.551450	−0.275725	−	0.961237i	$-0.588918\pi$
$233$	−1961.28	−0.551450	−0.275725	+	0.961237i	$0.588918\pi$
$234$	1441.07	0.402589
$235$	−2249.17	−0.624339
$236$	−224.288	−0.0618641
$237$	−1481.47	−0.406042
$238$	101.998	0.0277797
$239$	1822.98	0.493385	0.246692	−	0.969094i	$-0.420656\pi$
$239$	1822.98	0.493385	0.246692	+	0.969094i	$0.420656\pi$
$240$	−929.209	−0.249917
$241$	−6154.85	−1.64510	−0.822549	−	0.568695i	$-0.807449\pi$
$241$	−6154.85	−1.64510	−0.822549	+	0.568695i	$0.807449\pi$
$242$	−23.4322	−0.00622430
$243$	243.000	0.0641500
$244$	3339.70	0.876239
$245$	−5530.61	−1.44220
$246$	787.067	0.203990
$247$	−1439.98	−0.370946
$248$	−5928.10	−1.51788
$249$	1195.29	0.304209
$250$	397.009	0.100436
$251$	4896.77	1.23140	0.615700	−	0.787981i	$-0.288873\pi$
$251$	4896.77	1.23140	0.615700	+	0.787981i	$0.288873\pi$
$252$	38.9896	0.00974648
$253$	−161.708	−0.0401837
$254$	−2602.98	−0.643015
$255$	−2121.00	−0.520872
$256$	−4311.77	−1.05268
$257$	−6238.24	−1.51413	−0.757063	−	0.653341i	$-0.773367\pi$
$257$	−6238.24	−1.51413	−0.757063	+	0.653341i	$0.773367\pi$
$258$	−1408.76	−0.339945
$259$	197.369	0.0473510
$260$	−4808.14	−1.14688
$261$	−1457.29	−0.345608
$262$	5127.58	1.20910
$263$	−5359.90	−1.25667	−0.628337	−	0.777941i	$-0.716264\pi$
$263$	−5359.90	−1.25667	−0.628337	+	0.777941i	$0.716264\pi$
$264$	−2635.24	−0.614349
$265$	−2728.67	−0.632532
$266$	43.0288	0.00991829
$267$	−3983.83	−0.913132
$268$	−1372.56	−0.312845
$269$	−7201.10	−1.63219	−0.816094	−	0.577919i	$-0.803865\pi$
$269$	−7201.10	−1.63219	−0.816094	+	0.577919i	$0.803865\pi$
$270$	895.442	0.201833
$271$	−1135.65	−0.254560	−0.127280	−	0.991867i	$-0.540625\pi$
$271$	−1135.65	−0.254560	−0.127280	+	0.991867i	$0.540625\pi$
$272$	835.910	0.186340
$273$	−267.158	−0.0592276
$274$	938.489	0.206920
$275$	4975.58	1.09105
$276$	50.7682	0.0110720
$277$	1044.58	0.226580	0.113290	−	0.993562i	$-0.463861\pi$
$277$	1044.58	0.226580	0.113290	+	0.993562i	$0.463861\pi$
$278$	587.974	0.126850
$279$	2206.35	0.473443
$280$	446.028	0.0951974
$281$	6506.31	1.38126	0.690630	−	0.723209i	$-0.257333\pi$
$281$	6506.31	1.38126	0.690630	+	0.723209i	$0.257333\pi$
$282$	−854.208	−0.180381
$283$	−8390.09	−1.76233	−0.881164	−	0.472811i	$-0.843239\pi$
$283$	−8390.09	−1.76233	−0.881164	+	0.472811i	$0.843239\pi$
$284$	2896.38	0.605170
$285$	−894.763	−0.185969
$286$	5816.45	1.20257
$287$	−145.913	−0.0300104
$288$	1388.17	0.284023
$289$	−3004.96	−0.611634
$290$	−5370.02	−1.08737
$291$	−4388.43	−0.884036
$292$	1538.62	0.308359
$293$	3643.33	0.726436	0.363218	−	0.931704i	$-0.381678\pi$
$293$	3643.33	0.726436	0.363218	+	0.931704i	$0.381678\pi$
$294$	−2100.46	−0.416672
$295$	954.945	0.188471
$296$	4188.05	0.822383
$297$	980.797	0.191622
$298$	5343.48	1.03872
$299$	−347.865	−0.0672828
$300$	−1562.08	−0.300623
$301$	261.169	0.0500117
$302$	−42.1667	−0.00803451
$303$	684.354	0.129753
$304$	352.636	0.0665297
$305$	−14219.3	−2.66950
$306$	−805.533	−0.150488
$307$	−521.575	−0.0969637	−0.0484819	−	0.998824i	$-0.515438\pi$
$307$	−521.575	−0.0969637	−0.0484819	+	0.998824i	$0.515438\pi$
$308$	157.370	0.0291136
$309$	2294.06	0.422345
$310$	8130.27	1.48957
$311$	5344.76	0.974513	0.487256	−	0.873259i	$-0.337998\pi$
$311$	5344.76	0.974513	0.487256	+	0.873259i	$0.337998\pi$
$312$	−5668.93	−1.02865
$313$	4368.71	0.788927	0.394463	−	0.918912i	$-0.370931\pi$
$313$	4368.71	0.788927	0.394463	+	0.918912i	$0.370931\pi$
$314$	−3900.45	−0.701004
$315$	−166.005	−0.0296930
$316$	1877.27	0.334193
$317$	5679.69	1.00632	0.503160	−	0.864193i	$-0.332171\pi$
$317$	5679.69	1.00632	0.503160	+	0.864193i	$0.332171\pi$
$318$	−1036.32	−0.182748
$319$	−5881.90	−1.03236
$320$	7593.22	1.32648
$321$	4671.62	0.812288
$322$	10.3948	0.00179900
$323$	804.923	0.138660
$324$	−307.921	−0.0527986
$325$	10703.5	1.82683
$326$	−401.194	−0.0681598
$327$	4768.03	0.806339
$328$	−3096.19	−0.521215
$329$	158.361	0.0265371
$330$	3614.18	0.602892
$331$	3620.41	0.601196	0.300598	−	0.953751i	$-0.402814\pi$
$331$	3620.41	0.601196	0.300598	+	0.953751i	$0.402814\pi$
$332$	−1514.62	−0.250379
$333$	−1558.73	−0.256509
$334$	−4299.09	−0.704299
$335$	5843.91	0.953096
$336$	65.4242	0.0106226
$337$	−4951.57	−0.800383	−0.400191	−	0.916432i	$-0.631056\pi$
$337$	−4951.57	−0.800383	−0.400191	+	0.916432i	$0.631056\pi$
$338$	8010.63	1.28911
$339$	4228.44	0.677455
$340$	2687.66	0.428703
$341$	8905.26	1.41421
$342$	−339.821	−0.0537293
$343$	780.284	0.122832
$344$	5541.84	0.868593
$345$	−216.154	−0.0337314
$346$	4091.49	0.635723
$347$	−8903.74	−1.37746	−0.688729	−	0.725019i	$-0.741831\pi$
$347$	−8903.74	−1.37746	−0.688729	+	0.725019i	$0.741831\pi$
$348$	1846.62	0.284452
$349$	−10124.5	−1.55287	−0.776433	−	0.630200i	$-0.782973\pi$
$349$	−10124.5	−1.55287	−0.776433	+	0.630200i	$0.782973\pi$
$350$	−319.836	−0.0488455
$351$	2109.89	0.320847
$352$	5602.93	0.848401
$353$	10278.4	1.54975	0.774877	−	0.632112i	$-0.217812\pi$
$353$	10278.4	1.54975	0.774877	+	0.632112i	$0.217812\pi$
$354$	362.677	0.0544522
$355$	−12331.8	−1.84368
$356$	5048.17	0.751552
$357$	149.337	0.0221393
$358$	−5070.84	−0.748610
$359$	11976.1	1.76065	0.880324	−	0.474374i	$-0.157325\pi$
$359$	11976.1	1.76065	0.880324	+	0.474374i	$0.157325\pi$
$360$	−3522.52	−0.515703
$361$	−6519.44	−0.950494
$362$	1266.51	0.183885
$363$	−34.3074	−0.00496052
$364$	338.534	0.0487472
$365$	−6550.92	−0.939427
$366$	−5400.34	−0.771258
$367$	11740.3	1.66987	0.834933	−	0.550352i	$-0.185507\pi$
$367$	11740.3	1.66987	0.834933	+	0.550352i	$0.185507\pi$
$368$	85.1885	0.0120673
$369$	1152.35	0.162572
$370$	−5743.82	−0.807046
$371$	192.122	0.0268853
$372$	−2795.81	−0.389666
$373$	−4055.41	−0.562952	−0.281476	−	0.959568i	$-0.590824\pi$
$373$	−4055.41	−0.562952	−0.281476	+	0.959568i	$0.590824\pi$
$374$	−3251.29	−0.449520
$375$	581.265	0.0800437
$376$	3360.31	0.460891
$377$	−12653.1	−1.72857
$378$	−63.0467	−0.00857876
$379$	−1798.03	−0.243691	−0.121845	−	0.992549i	$-0.538881\pi$
$379$	−1798.03	−0.243691	−0.121845	+	0.992549i	$0.538881\pi$
$380$	1133.81	0.153062
$381$	−3811.06	−0.512458
$382$	1131.46	0.151546
$383$	7620.06	1.01662	0.508312	−	0.861173i	$-0.330270\pi$
$383$	7620.06	1.01662	0.508312	+	0.861173i	$0.330270\pi$
$384$	−817.966	−0.108702
$385$	−670.028	−0.0886956
$386$	8201.73	1.08150
$387$	−2062.59	−0.270923
$388$	5560.87	0.727604
$389$	−978.747	−0.127569	−0.0637846	−	0.997964i	$-0.520317\pi$
$389$	−978.747	−0.127569	−0.0637846	+	0.997964i	$0.520317\pi$
$390$	7774.82	1.00947
$391$	194.451	0.0251504
$392$	8262.87	1.06464
$393$	7507.35	0.963602
$394$	2828.12	0.361621
$395$	−7992.80	−1.01813
$396$	−1242.83	−0.157714
$397$	−2554.43	−0.322930	−0.161465	−	0.986878i	$-0.551622\pi$
$397$	−2554.43	−0.322930	−0.161465	+	0.986878i	$0.551622\pi$
$398$	4370.45	0.550429
$399$	62.9989	0.00790449
$400$	−2621.16	−0.327645
$401$	−12779.6	−1.59148	−0.795740	−	0.605639i	$-0.792918\pi$
$401$	−12779.6	−1.59148	−0.795740	+	0.605639i	$0.792918\pi$
$402$	2219.45	0.275364
$403$	19157.0	2.36793
$404$	−867.190	−0.106793
$405$	1311.03	0.160853
$406$	378.095	0.0462181
$407$	−6291.33	−0.766215
$408$	3168.83	0.384511
$409$	−14112.8	−1.70620	−0.853099	−	0.521749i	$-0.825280\pi$
$409$	−14112.8	−1.70620	−0.853099	+	0.521749i	$0.825280\pi$
$410$	4246.36	0.511495
$411$	1374.05	0.164908
$412$	−2906.96	−0.347611
$413$	−67.2362	−0.00801085
$414$	−82.0928	−0.00974551
$415$	6448.77	0.762789
$416$	12053.0	1.42055
$417$	860.859	0.101095
$418$	−1371.59	−0.160494
$419$	3577.00	0.417060	0.208530	−	0.978016i	$-0.433132\pi$
$419$	3577.00	0.417060	0.208530	+	0.978016i	$0.433132\pi$
$420$	210.356	0.0244388
$421$	−10454.2	−1.21023	−0.605116	−	0.796137i	$-0.706873\pi$
$421$	−10454.2	−1.21023	−0.605116	+	0.796137i	$0.706873\pi$
$422$	12295.5	1.41833
$423$	−1250.66	−0.143756
$424$	4076.70	0.466939
$425$	−5983.04	−0.682871
$426$	−4683.48	−0.532666
$427$	1001.16	0.113465
$428$	−5919.72	−0.668552
$429$	8515.93	0.958398
$430$	−7600.52	−0.852395
$431$	−12521.0	−1.39934	−0.699670	−	0.714466i	$-0.746670\pi$
$431$	−12521.0	−1.39934	−0.699670	+	0.714466i	$0.746670\pi$
$432$	−516.689	−0.0575445
$433$	13582.0	1.50741	0.753707	−	0.657210i	$-0.228264\pi$
$433$	13582.0	1.50741	0.753707	+	0.657210i	$0.228264\pi$
$434$	−572.440	−0.0633133
$435$	−7862.31	−0.866595
$436$	−6041.89	−0.663656
$437$	82.0306	0.00897953
$438$	−2487.97	−0.271415
$439$	1232.27	0.133970	0.0669851	−	0.997754i	$-0.478662\pi$
$439$	1232.27	0.133970	0.0669851	+	0.997754i	$0.478662\pi$
$440$	−14217.6	−1.54045
$441$	−3075.31	−0.332071
$442$	−6994.18	−0.752668
$443$	12543.1	1.34524	0.672622	−	0.739986i	$-0.265168\pi$
$443$	12543.1	1.34524	0.672622	+	0.739986i	$0.265168\pi$
$444$	1975.16	0.211120
$445$	−21493.4	−2.28963
$446$	−10080.5	−1.07024
$447$	7823.45	0.827822
$448$	−534.627	−0.0563812
$449$	−1370.68	−0.144068	−0.0720340	−	0.997402i	$-0.522949\pi$
$449$	−1370.68	−0.144068	−0.0720340	+	0.997402i	$0.522949\pi$
$450$	2525.91	0.264606
$451$	4651.13	0.485617
$452$	−5358.14	−0.557579
$453$	−61.7368	−0.00640319
$454$	2093.25	0.216391
$455$	−1441.36	−0.148510
$456$	1336.80	0.137284
$457$	−7052.45	−0.721881	−0.360941	−	0.932589i	$-0.617544\pi$
$457$	−7052.45	−0.721881	−0.360941	+	0.932589i	$0.617544\pi$
$458$	3576.73	0.364912
$459$	−1179.39	−0.119933
$460$	273.903	0.0277626
$461$	5082.44	0.513476	0.256738	−	0.966481i	$-0.417352\pi$
$461$	5082.44	0.513476	0.256738	+	0.966481i	$0.417352\pi$
$462$	−254.469	−0.0256255
$463$	−5699.92	−0.572133	−0.286067	−	0.958210i	$-0.592348\pi$
$463$	−5699.92	−0.572133	−0.286067	+	0.958210i	$0.592348\pi$
$464$	3098.62	0.310021
$465$	11903.6	1.18713
$466$	−4018.71	−0.399492
$467$	16957.5	1.68030	0.840149	−	0.542355i	$-0.182467\pi$
$467$	16957.5	1.68030	0.840149	+	0.542355i	$0.182467\pi$
$468$	−2673.58	−0.264073
$469$	−411.461	−0.0405107
$470$	−4608.60	−0.452296
$471$	−5710.69	−0.558673
$472$	−1426.71	−0.139131
$473$	−8325.02	−0.809270
$474$	−3035.58	−0.294153
$475$	−2524.00	−0.243808
$476$	−189.234	−0.0182217
$477$	−1517.28	−0.145643
$478$	3735.34	0.357427
$479$	15755.7	1.50291	0.751456	−	0.659783i	$-0.229352\pi$
$479$	15755.7	1.50291	0.751456	+	0.659783i	$0.229352\pi$
$480$	7489.41	0.712174
$481$	−13533.9	−1.28294
$482$	−12611.4	−1.19177
$483$	15.2191	0.00143373
$484$	43.4731	0.00408275
$485$	−23676.4	−2.21668
$486$	497.913	0.0464728
$487$	−6390.04	−0.594580	−0.297290	−	0.954787i	$-0.596083\pi$
$487$	−6390.04	−0.594580	−0.297290	+	0.954787i	$0.596083\pi$
$488$	21244.0	1.97064
$489$	−587.393	−0.0543207
$490$	−11332.4	−1.04478
$491$	15901.1	1.46152	0.730760	−	0.682635i	$-0.239166\pi$
$491$	15901.1	1.46152	0.730760	+	0.682635i	$0.239166\pi$
$492$	−1460.22	−0.133805
$493$	7072.88	0.646139
$494$	−2950.55	−0.268728
$495$	5291.57	0.480481
$496$	−4691.34	−0.424692
$497$	868.264	0.0783641
$498$	2449.17	0.220381
$499$	3902.49	0.350100	0.175050	−	0.984560i	$-0.443991\pi$
$499$	3902.49	0.350100	0.175050	+	0.984560i	$0.443991\pi$
$500$	−736.559	−0.0658799
$501$	−6294.35	−0.561299
$502$	10033.6	0.892075
$503$	−3985.69	−0.353306	−0.176653	−	0.984273i	$-0.556527\pi$
$503$	−3985.69	−0.353306	−0.176653	+	0.984273i	$0.556527\pi$
$504$	248.015	0.0219196
$505$	3692.21	0.325349
$506$	−331.343	−0.0291107
$507$	11728.4	1.02737
$508$	4829.24	0.421778
$509$	−4706.40	−0.409838	−0.204919	−	0.978779i	$-0.565693\pi$
$509$	−4706.40	−0.409838	−0.204919	+	0.978779i	$0.565693\pi$
$510$	−4345.99	−0.377341
$511$	461.240	0.0399297
$512$	−6653.69	−0.574325
$513$	−497.536	−0.0428201
$514$	−12782.3	−1.09689
$515$	12376.9	1.05901
$516$	2613.64	0.222983
$517$	−5047.90	−0.429413
$518$	404.414	0.0343029
$519$	5990.40	0.506646
$520$	−30584.9	−2.57930
$521$	3269.28	0.274913	0.137456	−	0.990508i	$-0.456107\pi$
$521$	3269.28	0.274913	0.137456	+	0.990508i	$0.456107\pi$
$522$	−2986.02	−0.250372
$523$	6507.07	0.544043	0.272021	−	0.962291i	$-0.412308\pi$
$523$	6507.07	0.544043	0.272021	+	0.962291i	$0.412308\pi$
$524$	−9513.06	−0.793091
$525$	−468.275	−0.0389280
$526$	−10982.6	−0.910385
$527$	−10708.4	−0.885134
$528$	−2085.46	−0.171890
$529$	−12147.2	−0.998371
$530$	−5591.11	−0.458231
$531$	531.000	0.0433963
$532$	−79.8301	−0.00650578
$533$	10005.5	0.813107
$534$	−8162.96	−0.661509
$535$	25204.2	2.03677
$536$	−8730.95	−0.703581
$537$	−7424.28	−0.596613
$538$	−14755.2	−1.18242
$539$	−12412.6	−0.991925
$540$	−1661.29	−0.132390
$541$	17118.2	1.36039	0.680193	−	0.733033i	$-0.261896\pi$
$541$	17118.2	1.36039	0.680193	+	0.733033i	$0.261896\pi$
$542$	−2326.97	−0.184413
$543$	1854.31	0.146549
$544$	−6737.42	−0.531001
$545$	25724.3	2.02185
$546$	−547.413	−0.0429068
$547$	18217.3	1.42398	0.711988	−	0.702192i	$-0.247795\pi$
$547$	18217.3	1.42398	0.711988	+	0.702192i	$0.247795\pi$
$548$	−1741.15	−0.135727
$549$	−7906.70	−0.614662
$550$	10195.1	0.790400
$551$	2983.75	0.230694
$552$	322.939	0.0249007
$553$	562.761	0.0432749
$554$	2140.37	0.164144
$555$	−8409.59	−0.643184
$556$	−1090.85	−0.0832057
$557$	11664.8	0.887347	0.443673	−	0.896189i	$-0.353675\pi$
$557$	11664.8	0.887347	0.443673	+	0.896189i	$0.353675\pi$
$558$	4520.86	0.342981
$559$	−17908.8	−1.35503
$560$	352.975	0.0266355
$561$	−4760.26	−0.358250
$562$	13331.6	1.00064
$563$	11698.6	0.875733	0.437867	−	0.899040i	$-0.355734\pi$
$563$	11698.6	0.875733	0.437867	+	0.899040i	$0.355734\pi$
$564$	1584.79	0.118319
$565$	22813.2	1.69869
$566$	−17191.5	−1.27670
$567$	−92.3074	−0.00683694
$568$	18424.0	1.36101
$569$	3094.45	0.227990	0.113995	−	0.993481i	$-0.463635\pi$
$569$	3094.45	0.227990	0.113995	+	0.993481i	$0.463635\pi$
$570$	−1833.39	−0.134723
$571$	20401.7	1.49525	0.747623	−	0.664123i	$-0.231195\pi$
$571$	20401.7	1.49525	0.747623	+	0.664123i	$0.231195\pi$
$572$	−10791.1	−0.788808
$573$	1656.59	0.120777
$574$	−298.980	−0.0217407
$575$	−609.739	−0.0442224
$576$	4222.23	0.305428
$577$	−20917.5	−1.50920	−0.754599	−	0.656187i	$-0.772168\pi$
$577$	−20917.5	−1.50920	−0.754599	+	0.656187i	$0.772168\pi$
$578$	−6157.24	−0.443092
$579$	12008.3	0.861910
$580$	9962.85	0.713250
$581$	−454.048	−0.0324218
$582$	−8992.01	−0.640431
$583$	−6124.07	−0.435048
$584$	9787.24	0.693491
$585$	11383.2	0.804509
$586$	7465.28	0.526259
$587$	−18046.9	−1.26895	−0.634475	−	0.772943i	$-0.718784\pi$
$587$	−18046.9	−1.26895	−0.634475	+	0.772943i	$0.718784\pi$
$588$	3896.93	0.273311
$589$	−4517.43	−0.316023
$590$	1956.71	0.136536
$591$	4140.68	0.288198
$592$	3314.31	0.230097
$593$	5854.76	0.405441	0.202720	−	0.979237i	$-0.435022\pi$
$593$	5854.76	0.405441	0.202720	+	0.979237i	$0.435022\pi$
$594$	2009.68	0.138818
$595$	805.697	0.0555132
$596$	−9913.61	−0.681338
$597$	6398.82	0.438671
$598$	−712.784	−0.0487423
$599$	−5850.74	−0.399090	−0.199545	−	0.979889i	$-0.563946\pi$
$599$	−5850.74	−0.399090	−0.199545	+	0.979889i	$0.563946\pi$
$600$	−9936.51	−0.676094
$601$	−5057.35	−0.343251	−0.171625	−	0.985162i	$-0.554902\pi$
$601$	−5057.35	−0.343251	−0.171625	+	0.985162i	$0.554902\pi$
$602$	535.141	0.0362305
$603$	3249.52	0.219454
$604$	78.2307	0.00527014
$605$	−185.094	−0.0124383
$606$	1402.26	0.0939981
$607$	7535.66	0.503893	0.251946	−	0.967741i	$-0.418929\pi$
$607$	7535.66	0.503893	0.251946	+	0.967741i	$0.418929\pi$
$608$	−2842.24	−0.189585
$609$	553.573	0.0368340
$610$	−29135.8	−1.93389
$611$	−10859.0	−0.719000
$612$	1494.48	0.0987106
$613$	−9172.53	−0.604364	−0.302182	−	0.953250i	$-0.597715\pi$
$613$	−9172.53	−0.604364	−0.302182	+	0.953250i	$0.597715\pi$
$614$	−1068.72	−0.0702444
$615$	6217.14	0.407641
$616$	1001.04	0.0654757
$617$	6795.93	0.443426	0.221713	−	0.975112i	$-0.428835\pi$
$617$	6795.93	0.443426	0.221713	+	0.975112i	$0.428835\pi$
$618$	4700.59	0.305964
$619$	−18841.8	−1.22345	−0.611726	−	0.791070i	$-0.709524\pi$
$619$	−18841.8	−1.22345	−0.611726	+	0.791070i	$0.709524\pi$
$620$	−15083.9	−0.977068
$621$	−120.193	−0.00776679
$622$	10951.5	0.705976
$623$	1513.32	0.0973192
$624$	−4486.24	−0.287810
$625$	−13985.3	−0.895062
$626$	8951.59	0.571530
$627$	−2008.15	−0.127907
$628$	7236.39	0.459815
$629$	7565.21	0.479562
$630$	−340.148	−0.0215108
$631$	4816.45	0.303867	0.151933	−	0.988391i	$-0.451450\pi$
$631$	4816.45	0.303867	0.151933	+	0.988391i	$0.451450\pi$
$632$	11941.4	0.751591
$633$	18002.0	1.13035
$634$	11637.8	0.729018
$635$	−20561.3	−1.28496
$636$	1922.65	0.119871
$637$	−26701.9	−1.66086
$638$	−12052.2	−0.747883
$639$	−6857.14	−0.424514
$640$	−4413.07	−0.272565
$641$	−4109.33	−0.253212	−0.126606	−	0.991953i	$-0.540408\pi$
$641$	−4109.33	−0.253212	−0.126606	+	0.991953i	$0.540408\pi$
$642$	9572.27	0.588454
$643$	−20097.1	−1.23259	−0.616293	−	0.787517i	$-0.711366\pi$
$643$	−20097.1	−1.23259	−0.616293	+	0.787517i	$0.711366\pi$
$644$	−19.2851	−0.00118003
$645$	−11128.0	−0.679326
$646$	1649.31	0.100451
$647$	−11607.1	−0.705290	−0.352645	−	0.935757i	$-0.614718\pi$
$647$	−11607.1	−0.705290	−0.352645	+	0.935757i	$0.614718\pi$
$648$	−1958.71	−0.118743
$649$	2143.22	0.129628
$650$	21931.6	1.32343
$651$	−838.115	−0.0504583
$652$	744.324	0.0447085
$653$	28649.4	1.71691	0.858453	−	0.512893i	$-0.171426\pi$
$653$	28649.4	1.71691	0.858453	+	0.512893i	$0.171426\pi$
$654$	9769.81	0.584144
$655$	40503.4	2.41618
$656$	−2450.24	−0.145832
$657$	−3642.66	−0.216307
$658$	324.485	0.0192245
$659$	−2146.45	−0.126880	−0.0634398	−	0.997986i	$-0.520207\pi$
$659$	−2146.45	−0.126880	−0.0634398	+	0.997986i	$0.520207\pi$
$660$	−6705.29	−0.395459
$661$	7235.64	0.425769	0.212885	−	0.977077i	$-0.431714\pi$
$661$	7235.64	0.425769	0.212885	+	0.977077i	$0.431714\pi$
$662$	7418.32	0.435530
$663$	−10240.3	−0.599847
$664$	−9634.62	−0.563096
$665$	339.890	0.0198201
$666$	−3193.87	−0.185826
$667$	720.805	0.0418436
$668$	7975.98	0.461976
$669$	−14759.0	−0.852938
$670$	11974.3	0.690460
$671$	−31913.0	−1.83605
$672$	−527.318	−0.0302704
$673$	16869.8	0.966247	0.483124	−	0.875552i	$-0.339502\pi$
$673$	16869.8	0.966247	0.483124	+	0.875552i	$0.339502\pi$
$674$	−10145.9	−0.579829
$675$	3698.21	0.210880
$676$	−14861.9	−0.845578
$677$	269.724	0.0153121	0.00765607	−	0.999971i	$-0.497563\pi$
$677$	269.724	0.0153121	0.00765607	+	0.999971i	$0.497563\pi$
$678$	8664.18	0.490776
$679$	1667.02	0.0942183
$680$	17096.4	0.964143
$681$	3064.76	0.172455
$682$	18247.1	1.02451
$683$	−2809.90	−0.157420	−0.0787099	−	0.996898i	$-0.525080\pi$
$683$	−2809.90	−0.157420	−0.0787099	+	0.996898i	$0.525080\pi$
$684$	630.460	0.0352430
$685$	7413.25	0.413497
$686$	1598.82	0.0889844
$687$	5236.73	0.290820
$688$	4385.67	0.243026
$689$	−13174.1	−0.728436
$690$	−442.905	−0.0244364
$691$	24053.0	1.32420	0.662098	−	0.749418i	$-0.269666\pi$
$691$	24053.0	1.32420	0.662098	+	0.749418i	$0.269666\pi$
$692$	−7590.83	−0.416994
$693$	−372.571	−0.0204225
$694$	−18244.0	−0.997885
$695$	4644.48	0.253490
$696$	11746.5	0.639726
$697$	−5592.90	−0.303940
$698$	−20745.3	−1.12496
$699$	−5883.84	−0.318380
$700$	593.382	0.0320396
$701$	32736.0	1.76379	0.881897	−	0.471441i	$-0.156266\pi$
$701$	32736.0	1.76379	0.881897	+	0.471441i	$0.156266\pi$
$702$	4323.21	0.232435
$703$	3191.45	0.171220
$704$	17041.8	0.912338
$705$	−6747.50	−0.360462
$706$	21060.7	1.12270
$707$	−259.963	−0.0138287
$708$	−672.865	−0.0357173
$709$	−20629.3	−1.09273	−0.546367	−	0.837546i	$-0.683989\pi$
$709$	−20629.3	−1.09273	−0.546367	+	0.837546i	$0.683989\pi$
$710$	−25268.2	−1.33563
$711$	−4444.42	−0.234429
$712$	32111.7	1.69022
$713$	−1091.31	−0.0573208
$714$	305.995	0.0160386
$715$	45944.9	2.40314
$716$	9407.79	0.491041
$717$	5468.95	0.284856
$718$	24539.2	1.27548
$719$	6316.05	0.327606	0.163803	−	0.986493i	$-0.447624\pi$
$719$	6316.05	0.327606	0.163803	+	0.986493i	$0.447624\pi$
$720$	−2787.63	−0.144290
$721$	−871.437	−0.0450125
$722$	−13358.5	−0.688575
$723$	−18464.5	−0.949798
$724$	−2349.72	−0.120617
$725$	−22178.4	−1.13612
$726$	−70.2966	−0.00359360
$727$	36426.5	1.85830	0.929150	−	0.369703i	$-0.120541\pi$
$727$	36426.5	1.85830	0.929150	+	0.369703i	$0.120541\pi$
$728$	2153.43	0.109631
$729$	729.000	0.0370370
$730$	−13423.0	−0.680558
$731$	10010.7	0.506509
$732$	10019.1	0.505897
$733$	−14013.9	−0.706161	−0.353081	−	0.935593i	$-0.614866\pi$
$733$	−14013.9	−0.706161	−0.353081	+	0.935593i	$0.614866\pi$
$734$	24056.2	1.20972
$735$	−16591.8	−0.832652
$736$	−686.618	−0.0343873
$737$	13115.7	0.655528
$738$	2361.20	0.117774
$739$	−23407.6	−1.16517	−0.582587	−	0.812768i	$-0.697960\pi$
$739$	−23407.6	−1.16517	−0.582587	+	0.812768i	$0.697960\pi$
$740$	10656.3	0.529372
$741$	−4319.94	−0.214166
$742$	393.662	0.0194768
$743$	9526.11	0.470362	0.235181	−	0.971952i	$-0.424432\pi$
$743$	9526.11	0.470362	0.235181	+	0.971952i	$0.424432\pi$
$744$	−17784.3	−0.876349
$745$	42208.9	2.07572
$746$	−8309.64	−0.407825
$747$	3585.86	0.175635
$748$	6032.03	0.294857
$749$	−1774.59	−0.0865715
$750$	1191.03	0.0579868
$751$	−17166.3	−0.834098	−0.417049	−	0.908884i	$-0.636936\pi$
$751$	−17166.3	−0.834098	−0.417049	+	0.908884i	$0.636936\pi$
$752$	2659.26	0.128954
$753$	14690.3	0.710949
$754$	−25926.6	−1.25224
$755$	−333.080	−0.0160557
$756$	116.969	0.00562713
$757$	9502.71	0.456251	0.228125	−	0.973632i	$-0.426740\pi$
$757$	9502.71	0.456251	0.228125	+	0.973632i	$0.426740\pi$
$758$	−3684.21	−0.176539
$759$	−485.123	−0.0232001
$760$	7212.26	0.344232
$761$	23314.4	1.11057	0.555287	−	0.831659i	$-0.312608\pi$
$761$	23314.4	1.11057	0.555287	+	0.831659i	$0.312608\pi$
$762$	−7808.95	−0.371245
$763$	−1811.21	−0.0859375
$764$	−2099.17	−0.0994049
$765$	−6363.01	−0.300726
$766$	15613.7	0.736483
$767$	4610.50	0.217047
$768$	−12935.3	−0.607764
$769$	36770.4	1.72428	0.862142	−	0.506667i	$-0.169123\pi$
$769$	36770.4	1.72428	0.862142	+	0.506667i	$0.169123\pi$
$770$	−1372.90	−0.0642546
$771$	−18714.7	−0.874182
$772$	−15216.4	−0.709393
$773$	30333.2	1.41140	0.705698	−	0.708513i	$-0.250634\pi$
$773$	30333.2	1.41140	0.705698	+	0.708513i	$0.250634\pi$
$774$	−4226.29	−0.196267
$775$	33578.4	1.55635
$776$	35373.1	1.63636
$777$	592.107	0.0273381
$778$	−2005.48	−0.0924162
$779$	−2359.41	−0.108517
$780$	−14424.4	−0.662150
$781$	−27676.8	−1.26806
$782$	398.434	0.0182199
$783$	−4371.86	−0.199537
$784$	6539.02	0.297878
$785$	−30810.2	−1.40084
$786$	15382.7	0.698072
$787$	−10130.6	−0.458851	−0.229426	−	0.973326i	$-0.573685\pi$
$787$	−10130.6	−0.458851	−0.229426	+	0.973326i	$0.573685\pi$
$788$	−5246.93	−0.237201
$789$	−16079.7	−0.725541
$790$	−16377.5	−0.737574
$791$	−1606.24	−0.0722014
$792$	−7905.73	−0.354694
$793$	−68651.2	−3.07424
$794$	−5234.09	−0.233943
$795$	−8186.01	−0.365192
$796$	−8108.37	−0.361047
$797$	−24869.8	−1.10531	−0.552655	−	0.833410i	$-0.686385\pi$
$797$	−24869.8	−1.10531	−0.552655	+	0.833410i	$0.686385\pi$
$798$	129.086	0.00572633
$799$	6070.01	0.268763
$800$	21126.5	0.933670
$801$	−11951.5	−0.527197
$802$	−26185.7	−1.15293
$803$	−14702.5	−0.646127
$804$	−4117.68	−0.180621
$805$	82.1095	0.00359501
$806$	39253.1	1.71542
$807$	−21603.3	−0.942344
$808$	−5516.25	−0.240174
$809$	−28593.0	−1.24262	−0.621308	−	0.783567i	$-0.713398\pi$
$809$	−28593.0	−1.24262	−0.621308	+	0.783567i	$0.713398\pi$
$810$	2686.33	0.116528
$811$	−28737.4	−1.24428	−0.622138	−	0.782908i	$-0.713736\pi$
$811$	−28737.4	−1.24428	−0.622138	+	0.782908i	$0.713736\pi$
$812$	−701.469	−0.0303162
$813$	−3406.94	−0.146970
$814$	−12891.1	−0.555077
$815$	−3169.08	−0.136206
$816$	2507.73	0.107583
$817$	4223.09	0.180841
$818$	−28917.6	−1.23604
$819$	−801.474	−0.0341951
$820$	−7878.15	−0.335508
$821$	28082.4	1.19377	0.596884	−	0.802328i	$-0.296405\pi$
$821$	28082.4	1.19377	0.596884	+	0.802328i	$0.296405\pi$
$822$	2815.47	0.119466
$823$	16565.4	0.701621	0.350810	−	0.936447i	$-0.385906\pi$
$823$	16565.4	0.701621	0.350810	+	0.936447i	$0.385906\pi$
$824$	−18491.3	−0.781768
$825$	14926.7	0.629918
$826$	−137.769	−0.00580338
$827$	−25792.9	−1.08453	−0.542265	−	0.840207i	$-0.682433\pi$
$827$	−25792.9	−1.08453	−0.542265	+	0.840207i	$0.682433\pi$
$828$	152.304	0.00639245
$829$	34787.4	1.45744	0.728720	−	0.684812i	$-0.240116\pi$
$829$	34787.4	1.45744	0.728720	+	0.684812i	$0.240116\pi$
$830$	13213.7	0.552595
$831$	3133.74	0.130816
$832$	36660.2	1.52760
$833$	14925.9	0.620830
$834$	1763.92	0.0732369
$835$	−33959.1	−1.40743
$836$	2544.66	0.105274
$837$	6619.04	0.273342
$838$	7329.37	0.302135
$839$	−40608.0	−1.67097	−0.835485	−	0.549513i	$-0.814813\pi$
$839$	−40608.0	−1.67097	−0.835485	+	0.549513i	$0.814813\pi$
$840$	1338.08	0.0549623
$841$	1829.31	0.0750056
$842$	−21421.0	−0.876740
$843$	19518.9	0.797470
$844$	−22811.5	−0.930336
$845$	63277.0	2.57609
$846$	−2562.63	−0.104143
$847$	13.0322	0.000528680 0
$848$	3226.19	0.130646
$849$	−25170.3	−1.01748
$850$	−12259.4	−0.494699
$851$	770.979	0.0310562
$852$	8689.13	0.349395
$853$	18149.1	0.728505	0.364252	−	0.931300i	$-0.381325\pi$
$853$	18149.1	0.728505	0.364252	+	0.931300i	$0.381325\pi$
$854$	2051.40	0.0821986
$855$	−2684.29	−0.107369
$856$	−37655.7	−1.50356
$857$	−4806.87	−0.191598	−0.0957990	−	0.995401i	$-0.530541\pi$
$857$	−4806.87	−0.191598	−0.0957990	+	0.995401i	$0.530541\pi$
$858$	17449.4	0.694302
$859$	15908.4	0.631885	0.315942	−	0.948778i	$-0.397679\pi$
$859$	15908.4	0.631885	0.315942	+	0.948778i	$0.397679\pi$
$860$	14101.0	0.559118
$861$	−437.740	−0.0173265
$862$	−25655.8	−1.01374
$863$	8799.82	0.347102	0.173551	−	0.984825i	$-0.444476\pi$
$863$	8799.82	0.347102	0.173551	+	0.984825i	$0.444476\pi$
$864$	4164.51	0.163981
$865$	32319.2	1.27039
$866$	27829.9	1.09203
$867$	−9014.88	−0.353127
$868$	1062.03	0.0415296
$869$	−17938.6	−0.700258
$870$	−16110.1	−0.627796
$871$	28214.5	1.09760
$872$	−38432.8	−1.49255
$873$	−13165.3	−0.510399
$874$	168.083	0.00650513
$875$	−220.803	−0.00853085
$876$	4615.85	0.178031
$877$	8805.63	0.339048	0.169524	−	0.985526i	$-0.445777\pi$
$877$	8805.63	0.339048	0.169524	+	0.985526i	$0.445777\pi$
$878$	2524.95	0.0970533
$879$	10930.0	0.419408
$880$	−11251.4	−0.431006
$881$	−37620.6	−1.43867	−0.719335	−	0.694663i	$-0.755553\pi$
$881$	−37620.6	−1.43867	−0.719335	+	0.694663i	$0.755553\pi$
$882$	−6301.39	−0.240566
$883$	15013.0	0.572173	0.286086	−	0.958204i	$-0.407646\pi$
$883$	15013.0	0.572173	0.286086	+	0.958204i	$0.407646\pi$
$884$	12976.1	0.493703
$885$	2864.84	0.108814
$886$	25701.2	0.974548
$887$	32943.4	1.24705	0.623525	−	0.781804i	$-0.285700\pi$
$887$	32943.4	1.24705	0.623525	+	0.781804i	$0.285700\pi$
$888$	12564.1	0.474803
$889$	1447.69	0.0546164
$890$	−44040.6	−1.65870
$891$	2942.39	0.110633
$892$	18702.1	0.702009
$893$	2560.68	0.0959574
$894$	16030.4	0.599707
$895$	−40055.3	−1.49598
$896$	310.717	0.0115852
$897$	−1043.60	−0.0388458
$898$	−2808.56	−0.104369
$899$	−39694.8	−1.47263
$900$	−4686.25	−0.173565
$901$	7364.08	0.272290
$902$	9530.28	0.351800
$903$	783.506	0.0288743
$904$	−34083.4	−1.25398
$905$	10004.3	0.367464
$906$	−126.500	−0.00463873
$907$	46052.7	1.68595	0.842973	−	0.537955i	$-0.180803\pi$
$907$	46052.7	1.68595	0.842973	+	0.537955i	$0.180803\pi$
$908$	−3883.55	−0.141939
$909$	2053.06	0.0749128
$910$	−2953.39	−0.107587
$911$	−20657.9	−0.751290	−0.375645	−	0.926764i	$-0.622579\pi$
$911$	−20657.9	−0.751290	−0.375645	+	0.926764i	$0.622579\pi$
$912$	1057.91	0.0384110
$913$	14473.2	0.524638
$914$	−14450.7	−0.522960
$915$	−42658.0	−1.54123
$916$	−6635.80	−0.239359
$917$	−2851.79	−0.102698
$918$	−2416.60	−0.0868842
$919$	−47631.7	−1.70971	−0.854855	−	0.518867i	$-0.826354\pi$
$919$	−47631.7	−1.70971	−0.854855	+	0.518867i	$0.826354\pi$
$920$	1742.31	0.0624374
$921$	−1564.73	−0.0559820
$922$	10414.0	0.371983
$923$	−59538.3	−2.12321
$924$	472.109	0.0168087
$925$	−23722.2	−0.843224
$926$	−11679.3	−0.414476
$927$	6882.19	0.243841
$928$	−24974.8	−0.883447
$929$	−27546.0	−0.972827	−0.486413	−	0.873729i	$-0.661695\pi$
$929$	−27546.0	−0.972827	−0.486413	+	0.873729i	$0.661695\pi$
$930$	24390.8	0.860006
$931$	6296.61	0.221658
$932$	7455.81	0.262042
$933$	16034.3	0.562635
$934$	34746.3	1.21727
$935$	−25682.4	−0.898293
$936$	−17006.8	−0.593894
$937$	−8405.09	−0.293044	−0.146522	−	0.989207i	$-0.546808\pi$
$937$	−8405.09	−0.293044	−0.146522	+	0.989207i	$0.546808\pi$
$938$	−843.094	−0.0293475
$939$	13106.1	0.455487
$940$	8550.21	0.296678
$941$	−12903.8	−0.447027	−0.223514	−	0.974701i	$-0.571753\pi$
$941$	−12903.8	−0.447027	−0.223514	+	0.974701i	$0.571753\pi$
$942$	−11701.3	−0.404725
$943$	−569.978	−0.0196830
$944$	−1129.06	−0.0389278
$945$	−498.014	−0.0171433
$946$	−17058.2	−0.586267
$947$	−26496.9	−0.909223	−0.454612	−	0.890690i	$-0.650222\pi$
$947$	−26496.9	−0.909223	−0.454612	+	0.890690i	$0.650222\pi$
$948$	5631.82	0.192946
$949$	−31628.0	−1.08186
$950$	−5171.73	−0.176624
$951$	17039.1	0.580999
$952$	−1203.73	−0.0409802
$953$	3951.58	0.134317	0.0671585	−	0.997742i	$-0.478607\pi$
$953$	3951.58	0.134317	0.0671585	+	0.997742i	$0.478607\pi$
$954$	−3108.95	−0.105510
$955$	8937.57	0.302841
$956$	−6930.06	−0.234450
$957$	−17645.7	−0.596034
$958$	32283.8	1.08877
$959$	−521.955	−0.0175754
$960$	22779.7	0.765844
$961$	30307.3	1.01733
$962$	−27731.3	−0.929410
$963$	14014.9	0.468975
$964$	23397.6	0.781729
$965$	64786.6	2.16120
$966$	31.1843	0.00103865
$967$	46296.1	1.53959	0.769793	−	0.638293i	$-0.220359\pi$
$967$	46296.1	1.53959	0.769793	+	0.638293i	$0.220359\pi$
$968$	276.535	0.00918200
$969$	2414.77	0.0800553
$970$	−48513.4	−1.60585
$971$	−49931.5	−1.65023	−0.825117	−	0.564961i	$-0.808891\pi$
$971$	−49931.5	−1.65023	−0.825117	+	0.564961i	$0.808891\pi$
$972$	−923.764	−0.0304833
$973$	−327.011	−0.0107744
$974$	−13093.4	−0.430737
$975$	32110.4	1.05472
$976$	16812.0	0.551371
$977$	−13238.2	−0.433497	−0.216748	−	0.976227i	$-0.569545\pi$
$977$	−13238.2	−0.433497	−0.216748	+	0.976227i	$0.569545\pi$
$978$	−1203.58	−0.0393521
$979$	−48238.6	−1.57478
$980$	21024.6	0.685313
$981$	14304.1	0.465540
$982$	32581.7	1.05878
$983$	30187.5	0.979483	0.489742	−	0.871868i	$-0.337091\pi$
$983$	30187.5	0.979483	0.489742	+	0.871868i	$0.337091\pi$
$984$	−9288.56	−0.300923
$985$	22339.7	0.722641
$986$	14492.5	0.468089
$987$	475.082	0.0153212
$988$	5474.07	0.176269
$989$	1020.20	0.0328013
$990$	10842.6	0.348080
$991$	−57834.2	−1.85385	−0.926924	−	0.375250i	$-0.877557\pi$
$991$	−57834.2	−1.85385	−0.926924	+	0.375250i	$0.877557\pi$
$992$	37812.1	1.21022
$993$	10861.2	0.347101
$994$	1779.10	0.0567701
$995$	34522.7	1.09994
$996$	−4543.87	−0.144556
$997$	22267.8	0.707351	0.353675	−	0.935368i	$-0.384932\pi$
$997$	22267.8	0.707351	0.353675	+	0.935368i	$0.384932\pi$
$998$	7996.31	0.253626
$999$	−4676.18	−0.148096

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.d.1.5	✓	8
3.2	odd	2		531.4.a.e.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.d.1.5	✓	8	1.1	even	1	trivial
531.4.a.e.1.4		8	3.2	odd	2

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+2.04902 q^{2} +3.00000 q^{3} -3.80150 q^{4} +16.1855 q^{5} +6.14707 q^{6} -1.13960 q^{7} -24.1816 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.04902 q^{2} +3.00000 q^{3} -3.80150 q^{4} +16.1855 q^{5} +6.14707 q^{6} -1.13960 q^{7} -24.1816 q^{8} +9.00000 q^{9} +33.1645 q^{10} +36.3258 q^{11} -11.4045 q^{12} +78.1440 q^{13} -2.33506 q^{14} +48.5565 q^{15} -19.1366 q^{16} -43.6811 q^{17} +18.4412 q^{18} -18.4272 q^{19} -61.5292 q^{20} -3.41879 q^{21} +74.4325 q^{22} -4.45159 q^{23} -72.5447 q^{24} +136.971 q^{25} +160.119 q^{26} +27.0000 q^{27} +4.33218 q^{28} -161.921 q^{29} +99.4936 q^{30} +245.150 q^{31} +154.241 q^{32} +108.977 q^{33} -89.5037 q^{34} -18.4450 q^{35} -34.2135 q^{36} -173.192 q^{37} -37.7579 q^{38} +234.432 q^{39} -391.391 q^{40} +128.039 q^{41} -7.00519 q^{42} -229.176 q^{43} -138.092 q^{44} +145.670 q^{45} -9.12142 q^{46} -138.962 q^{47} -57.4099 q^{48} -341.701 q^{49} +280.657 q^{50} -131.043 q^{51} -297.064 q^{52} -168.587 q^{53} +55.3237 q^{54} +587.952 q^{55} +27.5572 q^{56} -55.2817 q^{57} -331.780 q^{58} +59.0000 q^{59} -184.588 q^{60} -878.522 q^{61} +502.318 q^{62} -10.2564 q^{63} +469.137 q^{64} +1264.80 q^{65} +223.297 q^{66} +361.058 q^{67} +166.054 q^{68} -13.3548 q^{69} -37.7942 q^{70} -761.904 q^{71} -217.634 q^{72} -404.740 q^{73} -354.874 q^{74} +410.913 q^{75} +70.0511 q^{76} -41.3968 q^{77} +480.357 q^{78} -493.825 q^{79} -309.736 q^{80} +81.0000 q^{81} +262.356 q^{82} +398.428 q^{83} +12.9965 q^{84} -707.002 q^{85} -469.588 q^{86} -485.762 q^{87} -878.415 q^{88} -1327.94 q^{89} +298.481 q^{90} -89.0527 q^{91} +16.9227 q^{92} +735.449 q^{93} -284.736 q^{94} -298.254 q^{95} +462.723 q^{96} -1462.81 q^{97} -700.154 q^{98} +326.932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 6 * q^2 + 24 * q^3 + 34 * q^4 + 42 * q^5 + 18 * q^6 + 53 * q^7 + 51 * q^8 + 72 * q^9	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9} + 21 q^{10} + 67 q^{11} + 102 q^{12} + 33 q^{13} + 79 q^{14} + 126 q^{15} - 30 q^{16} + 139 q^{17} + 54 q^{18} + 64 q^{19} + 117 q^{20} + 159 q^{21} - 84 q^{22} + 226 q^{23} + 153 q^{24} + 96 q^{25} + 24 q^{26} + 216 q^{27} + 34 q^{28} + 456 q^{29} + 63 q^{30} + 124 q^{31} + 174 q^{32} + 201 q^{33} - 114 q^{34} + 556 q^{35} + 306 q^{36} + 127 q^{37} + 237 q^{38} + 99 q^{39} - 188 q^{40} + 425 q^{41} + 237 q^{42} - 115 q^{43} + 510 q^{44} + 378 q^{45} - 711 q^{46} + 420 q^{47} - 90 q^{48} + 171 q^{49} - 137 q^{50} + 417 q^{51} - 922 q^{52} + 98 q^{53} + 162 q^{54} - 616 q^{55} - 412 q^{56} + 192 q^{57} - 1548 q^{58} + 472 q^{59} + 351 q^{60} - 1254 q^{61} - 766 q^{62} + 477 q^{63} - 2019 q^{64} - 734 q^{65} - 252 q^{66} - 1010 q^{67} - 503 q^{68} + 678 q^{69} - 2956 q^{70} - 17 q^{71} + 459 q^{72} - 1180 q^{73} - 1228 q^{74} + 288 q^{75} - 2008 q^{76} + 441 q^{77} + 72 q^{78} - 873 q^{79} - 865 q^{80} + 648 q^{81} - 3645 q^{82} + 759 q^{83} + 102 q^{84} - 850 q^{85} - 1226 q^{86} + 1368 q^{87} - 3047 q^{88} + 988 q^{89} + 189 q^{90} - 2111 q^{91} - 1062 q^{92} + 372 q^{93} - 2240 q^{94} + 1822 q^{95} + 522 q^{96} - 668 q^{97} - 1368 q^{98} + 603 q^{99}+O(q^{100}) \) 8 * q + 6 * q^2 + 24 * q^3 + 34 * q^4 + 42 * q^5 + 18 * q^6 + 53 * q^7 + 51 * q^8 + 72 * q^9 + 21 * q^10 + 67 * q^11 + 102 * q^12 + 33 * q^13 + 79 * q^14 + 126 * q^15 - 30 * q^16 + 139 * q^17 + 54 * q^18 + 64 * q^19 + 117 * q^20 + 159 * q^21 - 84 * q^22 + 226 * q^23 + 153 * q^24 + 96 * q^25 + 24 * q^26 + 216 * q^27 + 34 * q^28 + 456 * q^29 + 63 * q^30 + 124 * q^31 + 174 * q^32 + 201 * q^33 - 114 * q^34 + 556 * q^35 + 306 * q^36 + 127 * q^37 + 237 * q^38 + 99 * q^39 - 188 * q^40 + 425 * q^41 + 237 * q^42 - 115 * q^43 + 510 * q^44 + 378 * q^45 - 711 * q^46 + 420 * q^47 - 90 * q^48 + 171 * q^49 - 137 * q^50 + 417 * q^51 - 922 * q^52 + 98 * q^53 + 162 * q^54 - 616 * q^55 - 412 * q^56 + 192 * q^57 - 1548 * q^58 + 472 * q^59 + 351 * q^60 - 1254 * q^61 - 766 * q^62 + 477 * q^63 - 2019 * q^64 - 734 * q^65 - 252 * q^66 - 1010 * q^67 - 503 * q^68 + 678 * q^69 - 2956 * q^70 - 17 * q^71 + 459 * q^72 - 1180 * q^73 - 1228 * q^74 + 288 * q^75 - 2008 * q^76 + 441 * q^77 + 72 * q^78 - 873 * q^79 - 865 * q^80 + 648 * q^81 - 3645 * q^82 + 759 * q^83 + 102 * q^84 - 850 * q^85 - 1226 * q^86 + 1368 * q^87 - 3047 * q^88 + 988 * q^89 + 189 * q^90 - 2111 * q^91 - 1062 * q^92 + 372 * q^93 - 2240 * q^94 + 1822 * q^95 + 522 * q^96 - 668 * q^97 - 1368 * q^98 + 603 * q^99

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