LMFDB - Embedded newform 177.4.a.d.1.6

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			1.6
Root			$-3.06139$ 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)

$f(q)$	$=$	$q+4.06139 q^{2} +3.00000 q^{3} +8.49485 q^{4} +16.2722 q^{5} +12.1842 q^{6} +6.77038 q^{7} +2.00979 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.06139 q^{2} +3.00000 q^{3} +8.49485 q^{4} +16.2722 q^{5} +12.1842 q^{6} +6.77038 q^{7} +2.00979 q^{8} +9.00000 q^{9} +66.0875 q^{10} -16.5845 q^{11} +25.4846 q^{12} -82.3410 q^{13} +27.4971 q^{14} +48.8165 q^{15} -59.7963 q^{16} +6.48327 q^{17} +36.5525 q^{18} +151.026 q^{19} +138.230 q^{20} +20.3111 q^{21} -67.3561 q^{22} -154.283 q^{23} +6.02938 q^{24} +139.783 q^{25} -334.419 q^{26} +27.0000 q^{27} +57.5134 q^{28} +220.887 q^{29} +198.263 q^{30} -79.4162 q^{31} -258.934 q^{32} -49.7535 q^{33} +26.3311 q^{34} +110.169 q^{35} +76.4537 q^{36} +414.092 q^{37} +613.375 q^{38} -247.023 q^{39} +32.7037 q^{40} -521.903 q^{41} +82.4914 q^{42} -265.009 q^{43} -140.883 q^{44} +146.449 q^{45} -626.603 q^{46} +31.6419 q^{47} -179.389 q^{48} -297.162 q^{49} +567.714 q^{50} +19.4498 q^{51} -699.475 q^{52} -256.788 q^{53} +109.657 q^{54} -269.866 q^{55} +13.6071 q^{56} +453.078 q^{57} +897.107 q^{58} +59.0000 q^{59} +414.689 q^{60} -68.7236 q^{61} -322.540 q^{62} +60.9334 q^{63} -573.261 q^{64} -1339.87 q^{65} -202.068 q^{66} +198.971 q^{67} +55.0745 q^{68} -462.849 q^{69} +447.438 q^{70} +725.821 q^{71} +18.0882 q^{72} +571.563 q^{73} +1681.79 q^{74} +419.350 q^{75} +1282.94 q^{76} -112.283 q^{77} -1003.26 q^{78} -304.888 q^{79} -973.015 q^{80} +81.0000 q^{81} -2119.65 q^{82} +731.213 q^{83} +172.540 q^{84} +105.497 q^{85} -1076.30 q^{86} +662.661 q^{87} -33.3315 q^{88} -694.016 q^{89} +594.788 q^{90} -557.480 q^{91} -1310.61 q^{92} -238.249 q^{93} +128.510 q^{94} +2457.52 q^{95} -776.802 q^{96} -597.160 q^{97} -1206.89 q^{98} -149.261 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.06139	1.43592	0.717958	−	0.696086i	$-0.245077\pi$
$2$	4.06139	1.43592	0.717958	+	0.696086i	$0.245077\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	8.49485	1.06186
$5$	16.2722	1.45543	0.727713	−	0.685882i	$-0.240583\pi$
$5$	16.2722	1.45543	0.727713	+	0.685882i	$0.240583\pi$
$6$	12.1842	0.829027
$7$	6.77038	0.365566	0.182783	−	0.983153i	$-0.441489\pi$
$7$	6.77038	0.365566	0.182783	+	0.983153i	$0.441489\pi$
$8$	2.00979	0.0888212
$9$	9.00000	0.333333
$10$	66.0875	2.08987
$11$	−16.5845	−0.454583	−0.227292	−	0.973827i	$-0.572987\pi$
$11$	−16.5845	−0.454583	−0.227292	+	0.973827i	$0.572987\pi$
$12$	25.4846	0.613063
$13$	−82.3410	−1.75672	−0.878358	−	0.478004i	$-0.841360\pi$
$13$	−82.3410	−1.75672	−0.878358	+	0.478004i	$0.841360\pi$
$14$	27.4971	0.524923
$15$	48.8165	0.840291
$16$	−59.7963	−0.934317
$17$	6.48327	0.0924956	0.0462478	−	0.998930i	$-0.485274\pi$
$17$	6.48327	0.0924956	0.0462478	+	0.998930i	$0.485274\pi$
$18$	36.5525	0.478639
$19$	151.026	1.82356	0.911782	−	0.410674i	$-0.134706\pi$
$19$	151.026	1.82356	0.911782	+	0.410674i	$0.134706\pi$
$20$	138.230	1.54545
$21$	20.3111	0.211060
$22$	−67.3561	−0.652744
$23$	−154.283	−1.39871	−0.699353	−	0.714777i	$-0.746528\pi$
$23$	−154.283	−1.39871	−0.699353	+	0.714777i	$0.746528\pi$
$24$	6.02938	0.0512809
$25$	139.783	1.11827
$26$	−334.419	−2.52250
$27$	27.0000	0.192450
$28$	57.5134	0.388179
$29$	220.887	1.41440	0.707201	−	0.707012i	$-0.249958\pi$
$29$	220.887	1.41440	0.707201	+	0.707012i	$0.249958\pi$
$30$	198.263	1.20659
$31$	−79.4162	−0.460115	−0.230058	−	0.973177i	$-0.573891\pi$
$31$	−79.4162	−0.460115	−0.230058	+	0.973177i	$0.573891\pi$
$32$	−258.934	−1.43042
$33$	−49.7535	−0.262454
$34$	26.3311	0.132816
$35$	110.169	0.532055
$36$	76.4537	0.353952
$37$	414.092	1.83990	0.919949	−	0.392038i	$-0.128230\pi$
$37$	414.092	1.83990	0.919949	+	0.392038i	$0.128230\pi$
$38$	613.375	2.61849
$39$	−247.023	−1.01424
$40$	32.7037	0.129273
$41$	−521.903	−1.98799	−0.993995	−	0.109423i	$-0.965100\pi$
$41$	−521.903	−1.98799	−0.993995	+	0.109423i	$0.965100\pi$
$42$	82.4914	0.303064
$43$	−265.009	−0.939848	−0.469924	−	0.882707i	$-0.655719\pi$
$43$	−265.009	−0.939848	−0.469924	+	0.882707i	$0.655719\pi$
$44$	−140.883	−0.482702
$45$	146.449	0.485142
$46$	−626.603	−2.00843
$47$	31.6419	0.0982011	0.0491005	−	0.998794i	$-0.484365\pi$
$47$	31.6419	0.0982011	0.0491005	+	0.998794i	$0.484365\pi$
$48$	−179.389	−0.539428
$49$	−297.162	−0.866361
$50$	567.714	1.60574
$51$	19.4498	0.0534023
$52$	−699.475	−1.86538
$53$	−256.788	−0.665519	−0.332760	−	0.943012i	$-0.607980\pi$
$53$	−256.788	−0.665519	−0.332760	+	0.943012i	$0.607980\pi$
$54$	109.657	0.276342
$55$	−269.866	−0.661613
$56$	13.6071	0.0324700
$57$	453.078	1.05284
$58$	897.107	2.03096
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	414.689	0.892268
$61$	−68.7236	−0.144248	−0.0721242	−	0.997396i	$-0.522978\pi$
$61$	−68.7236	−0.144248	−0.0721242	+	0.997396i	$0.522978\pi$
$62$	−322.540	−0.660687
$63$	60.9334	0.121855
$64$	−573.261	−1.11965
$65$	−1339.87	−2.55677
$66$	−202.068	−0.376862
$67$	198.971	0.362808	0.181404	−	0.983409i	$-0.441936\pi$
$67$	198.971	0.362808	0.181404	+	0.983409i	$0.441936\pi$
$68$	55.0745	0.0982170
$69$	−462.849	−0.807543
$70$	447.438	0.763986
$71$	725.821	1.21323	0.606613	−	0.794997i	$-0.292528\pi$
$71$	725.821	1.21323	0.606613	+	0.794997i	$0.292528\pi$
$72$	18.0882	0.0296071
$73$	571.563	0.916388	0.458194	−	0.888852i	$-0.348496\pi$
$73$	571.563	0.916388	0.458194	+	0.888852i	$0.348496\pi$
$74$	1681.79	2.64194
$75$	419.350	0.645631
$76$	1282.94	1.93636
$77$	−112.283	−0.166180
$78$	−1003.26	−1.45636
$79$	−304.888	−0.434209	−0.217105	−	0.976148i	$-0.569661\pi$
$79$	−304.888	−0.434209	−0.217105	+	0.976148i	$0.569661\pi$
$80$	−973.015	−1.35983
$81$	81.0000	0.111111
$82$	−2119.65	−2.85459
$83$	731.213	0.967001	0.483500	−	0.875344i	$-0.339365\pi$
$83$	731.213	0.967001	0.483500	+	0.875344i	$0.339365\pi$
$84$	172.540	0.224115
$85$	105.497	0.134620
$86$	−1076.30	−1.34954
$87$	662.661	0.816606
$88$	−33.3315	−0.0403766
$89$	−694.016	−0.826580	−0.413290	−	0.910600i	$-0.635620\pi$
$89$	−694.016	−0.826580	−0.413290	+	0.910600i	$0.635620\pi$
$90$	594.788	0.696624
$91$	−557.480	−0.642196
$92$	−1310.61	−1.48523
$93$	−238.249	−0.265648
$94$	128.510	0.141009
$95$	2457.52	2.65406
$96$	−776.802	−0.825855
$97$	−597.160	−0.625076	−0.312538	−	0.949905i	$-0.601179\pi$
$97$	−597.160	−0.625076	−0.312538	+	0.949905i	$0.601179\pi$
$98$	−1206.89	−1.24402
$99$	−149.261	−0.151528
$100$	1187.44	1.18744
$101$	1186.90	1.16932	0.584659	−	0.811279i	$-0.301228\pi$
$101$	1186.90	1.16932	0.584659	+	0.811279i	$0.301228\pi$
$102$	78.9932	0.0766813
$103$	−1037.17	−0.992190	−0.496095	−	0.868268i	$-0.665233\pi$
$103$	−1037.17	−0.992190	−0.496095	+	0.868268i	$0.665233\pi$
$104$	−165.489	−0.156034
$105$	330.506	0.307182
$106$	−1042.91	−0.955630
$107$	1786.99	1.61453	0.807266	−	0.590188i	$-0.200946\pi$
$107$	1786.99	1.61453	0.807266	+	0.590188i	$0.200946\pi$
$108$	229.361	0.204354
$109$	−1548.23	−1.36050	−0.680248	−	0.732982i	$-0.738128\pi$
$109$	−1548.23	−1.36050	−0.680248	+	0.732982i	$0.738128\pi$
$110$	−1096.03	−0.950021
$111$	1242.27	1.06227
$112$	−404.844	−0.341555
$113$	1414.84	1.17785	0.588924	−	0.808189i	$-0.299552\pi$
$113$	1414.84	1.17785	0.588924	+	0.808189i	$0.299552\pi$
$114$	1840.12	1.51178
$115$	−2510.52	−2.03571
$116$	1876.40	1.50189
$117$	−741.069	−0.585572
$118$	239.622	0.186940
$119$	43.8942	0.0338133
$120$	98.1111	0.0746356
$121$	−1055.95	−0.793354
$122$	−279.113	−0.207129
$123$	−1565.71	−1.14777
$124$	−674.629	−0.488576
$125$	240.555	0.172127
$126$	247.474	0.174974
$127$	1389.63	0.970942	0.485471	−	0.874253i	$-0.338648\pi$
$127$	1389.63	0.970942	0.485471	+	0.874253i	$0.338648\pi$
$128$	−256.762	−0.177303
$129$	−795.026	−0.542621
$130$	−5441.72	−3.67131
$131$	−398.137	−0.265537	−0.132769	−	0.991147i	$-0.542387\pi$
$131$	−398.137	−0.265537	−0.132769	+	0.991147i	$0.542387\pi$
$132$	−422.649	−0.278688
$133$	1022.50	0.666634
$134$	808.096	0.520962
$135$	439.348	0.280097
$136$	13.0300	0.00821557
$137$	−42.8475	−0.0267205	−0.0133603	−	0.999911i	$-0.504253\pi$
$137$	−42.8475	−0.0267205	−0.0133603	+	0.999911i	$0.504253\pi$
$138$	−1879.81	−1.15956
$139$	−245.792	−0.149984	−0.0749922	−	0.997184i	$-0.523893\pi$
$139$	−245.792	−0.149984	−0.0749922	+	0.997184i	$0.523893\pi$
$140$	935.868	0.564966
$141$	94.9258	0.0566964
$142$	2947.84	1.74209
$143$	1365.59	0.798574
$144$	−538.167	−0.311439
$145$	3594.31	2.05856
$146$	2321.34	1.31586
$147$	−891.486	−0.500194
$148$	3517.65	1.95371
$149$	1492.30	0.820495	0.410247	−	0.911974i	$-0.365442\pi$
$149$	1492.30	0.820495	0.410247	+	0.911974i	$0.365442\pi$
$150$	1703.14	0.927072
$151$	−917.443	−0.494440	−0.247220	−	0.968959i	$-0.579517\pi$
$151$	−917.443	−0.494440	−0.247220	+	0.968959i	$0.579517\pi$
$152$	303.531	0.161971
$153$	58.3495	0.0308319
$154$	−456.027	−0.238621
$155$	−1292.27	−0.669664
$156$	−2098.43	−1.07698
$157$	2338.93	1.18896	0.594479	−	0.804111i	$-0.297358\pi$
$157$	2338.93	1.18896	0.594479	+	0.804111i	$0.297358\pi$
$158$	−1238.27	−0.623489
$159$	−770.364	−0.384238
$160$	−4213.42	−2.08187
$161$	−1044.56	−0.511320
$162$	328.972	0.159546
$163$	1425.51	0.684995	0.342498	−	0.939519i	$-0.388727\pi$
$163$	1425.51	0.684995	0.342498	+	0.939519i	$0.388727\pi$
$164$	−4433.49	−2.11096
$165$	−809.597	−0.381982
$166$	2969.74	1.38853
$167$	1937.97	0.897993	0.448996	−	0.893534i	$-0.351782\pi$
$167$	1937.97	0.897993	0.448996	+	0.893534i	$0.351782\pi$
$168$	40.8212	0.0187466
$169$	4583.05	2.08605
$170$	428.463	0.193304
$171$	1359.23	0.607855
$172$	−2251.21	−0.997984
$173$	925.049	0.406533	0.203266	−	0.979123i	$-0.434844\pi$
$173$	925.049	0.406533	0.203266	+	0.979123i	$0.434844\pi$
$174$	2691.32	1.17258
$175$	946.386	0.408800
$176$	991.692	0.424725
$177$	177.000	0.0751646
$178$	−2818.67	−1.18690
$179$	2245.15	0.937490	0.468745	−	0.883334i	$-0.344706\pi$
$179$	2245.15	0.937490	0.468745	+	0.883334i	$0.344706\pi$
$180$	1244.07	0.515151
$181$	−1772.98	−0.728090	−0.364045	−	0.931381i	$-0.618605\pi$
$181$	−1772.98	−0.728090	−0.364045	+	0.931381i	$0.618605\pi$
$182$	−2264.14	−0.922140
$183$	−206.171	−0.0832819
$184$	−310.077	−0.124235
$185$	6738.17	2.67784
$186$	−967.619	−0.381448
$187$	−107.522	−0.0420469
$188$	268.794	0.104276
$189$	182.800	0.0703533
$190$	9980.93	3.81102
$191$	−620.457	−0.235051	−0.117525	−	0.993070i	$-0.537496\pi$
$191$	−620.457	−0.235051	−0.117525	+	0.993070i	$0.537496\pi$
$192$	−1719.78	−0.646431
$193$	−2381.92	−0.888366	−0.444183	−	0.895936i	$-0.646506\pi$
$193$	−2381.92	−0.888366	−0.444183	+	0.895936i	$0.646506\pi$
$194$	−2425.30	−0.897557
$195$	−4019.60	−1.47615
$196$	−2524.35	−0.919952
$197$	2044.21	0.739311	0.369655	−	0.929169i	$-0.379476\pi$
$197$	2044.21	0.739311	0.369655	+	0.929169i	$0.379476\pi$
$198$	−606.205	−0.217581
$199$	986.827	0.351529	0.175765	−	0.984432i	$-0.443760\pi$
$199$	986.827	0.351529	0.175765	+	0.984432i	$0.443760\pi$
$200$	280.936	0.0993257
$201$	596.912	0.209467
$202$	4820.46	1.67904
$203$	1495.49	0.517058
$204$	165.223	0.0567056
$205$	−8492.50	−2.89337
$206$	−4212.35	−1.42470
$207$	−1388.55	−0.466235
$208$	4923.69	1.64133
$209$	−2504.69	−0.828962
$210$	1342.31	0.441088
$211$	2539.95	0.828706	0.414353	−	0.910116i	$-0.364008\pi$
$211$	2539.95	0.828706	0.414353	+	0.910116i	$0.364008\pi$
$212$	−2181.38	−0.706686
$213$	2177.46	0.700457
$214$	7257.66	2.31833
$215$	−4312.27	−1.36788
$216$	54.2645	0.0170936
$217$	−537.678	−0.168203
$218$	−6287.98	−1.95356
$219$	1714.69	0.529077
$220$	−2292.47	−0.702538
$221$	−533.839	−0.162488
$222$	5045.36	1.52533
$223$	5064.96	1.52096	0.760482	−	0.649359i	$-0.224963\pi$
$223$	5064.96	1.52096	0.760482	+	0.649359i	$0.224963\pi$
$224$	−1753.08	−0.522914
$225$	1258.05	0.372755
$226$	5746.20	1.69129
$227$	−875.416	−0.255962	−0.127981	−	0.991777i	$-0.540850\pi$
$227$	−875.416	−0.255962	−0.127981	+	0.991777i	$0.540850\pi$
$228$	3848.83	1.11796
$229$	−4519.88	−1.30429	−0.652145	−	0.758095i	$-0.726131\pi$
$229$	−4519.88	−1.30429	−0.652145	+	0.758095i	$0.726131\pi$
$230$	−10196.2	−2.92311
$231$	−336.850	−0.0959443
$232$	443.937	0.125629
$233$	−3300.46	−0.927983	−0.463992	−	0.885840i	$-0.653583\pi$
$233$	−3300.46	−0.927983	−0.463992	+	0.885840i	$0.653583\pi$
$234$	−3009.77	−0.840832
$235$	514.883	0.142924
$236$	501.196	0.138242
$237$	−914.663	−0.250691
$238$	178.271	0.0485530
$239$	2036.15	0.551079	0.275539	−	0.961290i	$-0.411144\pi$
$239$	2036.15	0.551079	0.275539	+	0.961290i	$0.411144\pi$
$240$	−2919.04	−0.785098
$241$	1135.42	0.303480	0.151740	−	0.988420i	$-0.451512\pi$
$241$	1135.42	0.303480	0.151740	+	0.988420i	$0.451512\pi$
$242$	−4288.64	−1.13919
$243$	243.000	0.0641500
$244$	−583.797	−0.153171
$245$	−4835.47	−1.26092
$246$	−6358.95	−1.64810
$247$	−12435.6	−3.20348
$248$	−159.610	−0.0408680
$249$	2193.64	0.558298
$250$	976.987	0.247160
$251$	6147.42	1.54590	0.772952	−	0.634465i	$-0.218780\pi$
$251$	6147.42	1.54590	0.772952	+	0.634465i	$0.218780\pi$
$252$	517.621	0.129393
$253$	2558.71	0.635828
$254$	5643.82	1.39419
$255$	316.491	0.0777232
$256$	3543.28	0.865059
$257$	866.093	0.210216	0.105108	−	0.994461i	$-0.466481\pi$
$257$	866.093	0.210216	0.105108	+	0.994461i	$0.466481\pi$
$258$	−3228.91	−0.779159
$259$	2803.56	0.672605
$260$	−11382.0	−2.71492
$261$	1987.98	0.471467
$262$	−1616.99	−0.381289
$263$	−4743.67	−1.11220	−0.556098	−	0.831117i	$-0.687702\pi$
$263$	−4743.67	−1.11220	−0.556098	+	0.831117i	$0.687702\pi$
$264$	−99.9944	−0.0233115
$265$	−4178.49	−0.968615
$266$	4152.78	0.957231
$267$	−2082.05	−0.477226
$268$	1690.23	0.385250
$269$	−5845.97	−1.32504	−0.662519	−	0.749045i	$-0.730513\pi$
$269$	−5845.97	−1.32504	−0.662519	+	0.749045i	$0.730513\pi$
$270$	1784.36	0.402196
$271$	−4680.05	−1.04905	−0.524525	−	0.851395i	$-0.675757\pi$
$271$	−4680.05	−1.04905	−0.524525	+	0.851395i	$0.675757\pi$
$272$	−387.676	−0.0864202
$273$	−1672.44	−0.370772
$274$	−174.020	−0.0383684
$275$	−2318.24	−0.508345
$276$	−3931.84	−0.857495
$277$	−3885.21	−0.842742	−0.421371	−	0.906888i	$-0.638451\pi$
$277$	−3885.21	−0.842742	−0.421371	+	0.906888i	$0.638451\pi$
$278$	−998.257	−0.215365
$279$	−714.746	−0.153372
$280$	221.417	0.0472578
$281$	−7580.94	−1.60940	−0.804700	−	0.593682i	$-0.797674\pi$
$281$	−7580.94	−1.60940	−0.804700	+	0.593682i	$0.797674\pi$
$282$	385.530	0.0814113
$283$	947.742	0.199072	0.0995361	−	0.995034i	$-0.468264\pi$
$283$	947.742	0.199072	0.0995361	+	0.995034i	$0.468264\pi$
$284$	6165.74	1.28827
$285$	7372.56	1.53232
$286$	5546.17	1.14669
$287$	−3533.49	−0.726742
$288$	−2330.41	−0.476808
$289$	−4870.97	−0.991445
$290$	14597.9	2.95592
$291$	−1791.48	−0.360888
$292$	4855.34	0.973073
$293$	−2612.95	−0.520990	−0.260495	−	0.965475i	$-0.583886\pi$
$293$	−2612.95	−0.520990	−0.260495	+	0.965475i	$0.583886\pi$
$294$	−3620.67	−0.718237
$295$	960.058	0.189480
$296$	832.239	0.163422
$297$	−447.782	−0.0874846
$298$	6060.79	1.17816
$299$	12703.8	2.45713
$300$	3562.31	0.685568
$301$	−1794.21	−0.343577
$302$	−3726.09	−0.709975
$303$	3560.70	0.675106
$304$	−9030.79	−1.70379
$305$	−1118.28	−0.209943
$306$	236.980	0.0442720
$307$	6751.14	1.25507	0.627537	−	0.778587i	$-0.284063\pi$
$307$	6751.14	1.25507	0.627537	+	0.778587i	$0.284063\pi$
$308$	−953.832	−0.176460
$309$	−3111.51	−0.572841
$310$	−5248.42	−0.961581
$311$	1176.93	0.214591	0.107295	−	0.994227i	$-0.465781\pi$
$311$	1176.93	0.214591	0.107295	+	0.994227i	$0.465781\pi$
$312$	−496.466	−0.0900860
$313$	2059.81	0.371972	0.185986	−	0.982552i	$-0.440452\pi$
$313$	2059.81	0.371972	0.185986	+	0.982552i	$0.440452\pi$
$314$	9499.28	1.70725
$315$	991.519	0.177352
$316$	−2589.98	−0.461068
$317$	5322.47	0.943027	0.471513	−	0.881859i	$-0.343708\pi$
$317$	5322.47	0.943027	0.471513	+	0.881859i	$0.343708\pi$
$318$	−3128.74	−0.551734
$319$	−3663.30	−0.642964
$320$	−9328.20	−1.62957
$321$	5360.97	0.932151
$322$	−4242.34	−0.734213
$323$	979.142	0.168672
$324$	688.083	0.117984
$325$	−11509.9	−1.96447
$326$	5789.53	0.983596
$327$	−4644.70	−0.785482
$328$	−1048.92	−0.176576
$329$	214.228	0.0358990
$330$	−3288.09	−0.548495
$331$	−10946.3	−1.81772	−0.908860	−	0.417101i	$-0.863046\pi$
$331$	−10946.3	−1.81772	−0.908860	+	0.417101i	$0.863046\pi$
$332$	6211.55	1.02682
$333$	3726.82	0.613299
$334$	7870.85	1.28944
$335$	3237.68	0.528040
$336$	−1214.53	−0.197197
$337$	333.243	0.0538661	0.0269331	−	0.999637i	$-0.491426\pi$
$337$	333.243	0.0538661	0.0269331	+	0.999637i	$0.491426\pi$
$338$	18613.5	2.99539
$339$	4244.51	0.680030
$340$	896.180	0.142948
$341$	1317.08	0.209161
$342$	5520.37	0.872829
$343$	−4334.14	−0.682279
$344$	−532.613	−0.0834784
$345$	−7531.55	−1.17532
$346$	3756.98	0.583747
$347$	−2054.05	−0.317773	−0.158886	−	0.987297i	$-0.550790\pi$
$347$	−2054.05	−0.317773	−0.158886	+	0.987297i	$0.550790\pi$
$348$	5629.21	0.867118
$349$	6861.41	1.05239	0.526193	−	0.850365i	$-0.323619\pi$
$349$	6861.41	1.05239	0.526193	+	0.850365i	$0.323619\pi$
$350$	3843.64	0.587003
$351$	−2223.21	−0.338080
$352$	4294.30	0.650246
$353$	−3523.31	−0.531237	−0.265618	−	0.964078i	$-0.585576\pi$
$353$	−3523.31	−0.531237	−0.265618	+	0.964078i	$0.585576\pi$
$354$	718.865	0.107930
$355$	11810.7	1.76576
$356$	−5895.57	−0.877709
$357$	131.683	0.0195221
$358$	9118.44	1.34616
$359$	1692.91	0.248882	0.124441	−	0.992227i	$-0.460286\pi$
$359$	1692.91	0.248882	0.124441	+	0.992227i	$0.460286\pi$
$360$	294.333	0.0430909
$361$	15949.8	2.32539
$362$	−7200.74	−1.04548
$363$	−3167.86	−0.458043
$364$	−4735.72	−0.681920
$365$	9300.56	1.33374
$366$	−837.339	−0.119586
$367$	−3865.33	−0.549778	−0.274889	−	0.961476i	$-0.588641\pi$
$367$	−3865.33	−0.549778	−0.274889	+	0.961476i	$0.588641\pi$
$368$	9225.55	1.30683
$369$	−4697.13	−0.662664
$370$	27366.3	3.84515
$371$	−1738.55	−0.243292
$372$	−2023.89	−0.282080
$373$	−6121.11	−0.849703	−0.424851	−	0.905263i	$-0.639674\pi$
$373$	−6121.11	−0.849703	−0.424851	+	0.905263i	$0.639674\pi$
$374$	−436.688	−0.0603759
$375$	721.665	0.0993777
$376$	63.5938	0.00872234
$377$	−18188.1	−2.48470
$378$	742.423	0.101021
$379$	−2939.20	−0.398356	−0.199178	−	0.979963i	$-0.563827\pi$
$379$	−2939.20	−0.398356	−0.199178	+	0.979963i	$0.563827\pi$
$380$	20876.3	2.81824
$381$	4168.89	0.560573
$382$	−2519.92	−0.337513
$383$	8175.22	1.09069	0.545345	−	0.838212i	$-0.316399\pi$
$383$	8175.22	1.09069	0.545345	+	0.838212i	$0.316399\pi$
$384$	−770.285	−0.102366
$385$	−1827.09	−0.241863
$386$	−9673.91	−1.27562
$387$	−2385.08	−0.313283
$388$	−5072.78	−0.663741
$389$	7355.93	0.958768	0.479384	−	0.877605i	$-0.340860\pi$
$389$	7355.93	0.958768	0.479384	+	0.877605i	$0.340860\pi$
$390$	−16325.1	−2.11963
$391$	−1000.26	−0.129374
$392$	−597.234	−0.0769513
$393$	−1194.41	−0.153308
$394$	8302.34	1.06159
$395$	−4961.18	−0.631960
$396$	−1267.95	−0.160901
$397$	−7487.16	−0.946523	−0.473261	−	0.880922i	$-0.656923\pi$
$397$	−7487.16	−0.946523	−0.473261	+	0.880922i	$0.656923\pi$
$398$	4007.89	0.504767
$399$	3067.51	0.384881
$400$	−8358.52	−1.04481
$401$	−9700.71	−1.20806	−0.604028	−	0.796963i	$-0.706438\pi$
$401$	−9700.71	−1.20806	−0.604028	+	0.796963i	$0.706438\pi$
$402$	2424.29	0.300777
$403$	6539.21	0.808291
$404$	10082.6	1.24165
$405$	1318.05	0.161714
$406$	6073.76	0.742452
$407$	−6867.51	−0.836387
$408$	39.0901	0.00474326
$409$	−1378.30	−0.166632	−0.0833160	−	0.996523i	$-0.526551\pi$
$409$	−1378.30	−0.166632	−0.0833160	+	0.996523i	$0.526551\pi$
$410$	−34491.3	−4.15464
$411$	−128.543	−0.0154271
$412$	−8810.62	−1.05356
$413$	399.453	0.0475927
$414$	−5639.43	−0.669475
$415$	11898.4	1.40740
$416$	21320.9	2.51285
$417$	−737.377	−0.0865935
$418$	−10172.5	−1.19032
$419$	−6367.90	−0.742463	−0.371232	−	0.928540i	$-0.621064\pi$
$419$	−6367.90	−0.742463	−0.371232	+	0.928540i	$0.621064\pi$
$420$	2807.60	0.326183
$421$	4730.02	0.547570	0.273785	−	0.961791i	$-0.411724\pi$
$421$	4730.02	0.547570	0.273785	+	0.961791i	$0.411724\pi$
$422$	10315.7	1.18995
$423$	284.777	0.0327337
$424$	−516.091	−0.0591122
$425$	906.253	0.103435
$426$	8843.52	1.00580
$427$	−465.285	−0.0527324
$428$	15180.2	1.71440
$429$	4096.76	0.461057
$430$	−17513.8	−1.96416
$431$	12035.5	1.34509	0.672543	−	0.740058i	$-0.265202\pi$
$431$	12035.5	1.34509	0.672543	+	0.740058i	$0.265202\pi$
$432$	−1614.50	−0.179809
$433$	−16014.7	−1.77741	−0.888705	−	0.458479i	$-0.848395\pi$
$433$	−16014.7	−1.77741	−0.888705	+	0.458479i	$0.848395\pi$
$434$	−2183.72	−0.241525
$435$	10782.9	1.18851
$436$	−13152.0	−1.44465
$437$	−23300.7	−2.55063
$438$	6964.01	0.759711
$439$	3032.82	0.329724	0.164862	−	0.986317i	$-0.447282\pi$
$439$	3032.82	0.329724	0.164862	+	0.986317i	$0.447282\pi$
$440$	−542.375	−0.0587652
$441$	−2674.46	−0.288787
$442$	−2168.13	−0.233320
$443$	−1755.79	−0.188308	−0.0941538	−	0.995558i	$-0.530015\pi$
$443$	−1755.79	−0.188308	−0.0941538	+	0.995558i	$0.530015\pi$
$444$	10552.9	1.12797
$445$	−11293.1	−1.20303
$446$	20570.8	2.18398
$447$	4476.89	0.473713
$448$	−3881.20	−0.409307
$449$	−11467.7	−1.20533	−0.602666	−	0.797994i	$-0.705895\pi$
$449$	−11467.7	−1.20533	−0.602666	+	0.797994i	$0.705895\pi$
$450$	5109.42	0.535246
$451$	8655.51	0.903708
$452$	12018.8	1.25071
$453$	−2752.33	−0.285465
$454$	−3555.40	−0.367540
$455$	−9071.41	−0.934669
$456$	910.593	0.0935141
$457$	920.816	0.0942537	0.0471268	−	0.998889i	$-0.484993\pi$
$457$	920.816	0.0942537	0.0471268	+	0.998889i	$0.484993\pi$
$458$	−18357.0	−1.87285
$459$	175.048	0.0178008
$460$	−21326.5	−2.16164
$461$	18808.7	1.90023	0.950117	−	0.311895i	$-0.100964\pi$
$461$	18808.7	1.90023	0.950117	+	0.311895i	$0.100964\pi$
$462$	−1368.08	−0.137768
$463$	−4494.12	−0.451101	−0.225550	−	0.974232i	$-0.572418\pi$
$463$	−4494.12	−0.451101	−0.225550	+	0.974232i	$0.572418\pi$
$464$	−13208.2	−1.32150
$465$	−3876.82	−0.386630
$466$	−13404.4	−1.33251
$467$	608.207	0.0602665	0.0301332	−	0.999546i	$-0.490407\pi$
$467$	608.207	0.0602665	0.0301332	+	0.999546i	$0.490407\pi$
$468$	−6295.28	−0.621793
$469$	1347.11	0.132630
$470$	2091.14	0.205228
$471$	7016.78	0.686446
$472$	118.578	0.0115635
$473$	4395.04	0.427239
$474$	−3714.80	−0.359971
$475$	21110.9	2.03923
$476$	372.875	0.0359048
$477$	−2311.09	−0.221840
$478$	8269.60	0.791303
$479$	−4182.01	−0.398916	−0.199458	−	0.979906i	$-0.563918\pi$
$479$	−4182.01	−0.398916	−0.199458	+	0.979906i	$0.563918\pi$
$480$	−12640.3	−1.20197
$481$	−34096.7	−3.23218
$482$	4611.37	0.435772
$483$	−3133.67	−0.295211
$484$	−8970.18	−0.842428
$485$	−9717.08	−0.909752
$486$	986.917	0.0921141
$487$	−18578.4	−1.72868	−0.864340	−	0.502908i	$-0.832264\pi$
$487$	−18578.4	−1.72868	−0.864340	+	0.502908i	$0.832264\pi$
$488$	−138.120	−0.0128123
$489$	4276.52	0.395482
$490$	−19638.7	−1.81058
$491$	14967.0	1.37567	0.687833	−	0.725869i	$-0.258562\pi$
$491$	14967.0	1.37567	0.687833	+	0.725869i	$0.258562\pi$
$492$	−13300.5	−1.21876
$493$	1432.07	0.130826
$494$	−50505.9	−4.59994
$495$	−2428.79	−0.220538
$496$	4748.79	0.429893
$497$	4914.09	0.443515
$498$	8909.22	0.801670
$499$	−14608.3	−1.31053	−0.655266	−	0.755398i	$-0.727444\pi$
$499$	−14608.3	−1.31053	−0.655266	+	0.755398i	$0.727444\pi$
$500$	2043.48	0.182774
$501$	5813.92	0.518456
$502$	24967.0	2.21979
$503$	11306.5	1.00225	0.501127	−	0.865374i	$-0.332919\pi$
$503$	11306.5	1.00225	0.501127	+	0.865374i	$0.332919\pi$
$504$	122.464	0.0108233
$505$	19313.4	1.70186
$506$	10391.9	0.912997
$507$	13749.1	1.20438
$508$	11804.7	1.03100
$509$	6037.40	0.525743	0.262871	−	0.964831i	$-0.415330\pi$
$509$	6037.40	0.525743	0.262871	+	0.964831i	$0.415330\pi$
$510$	1285.39	0.111604
$511$	3869.70	0.335001
$512$	16444.7	1.41946
$513$	4077.70	0.350945
$514$	3517.54	0.301852
$515$	−16877.0	−1.44406
$516$	−6753.63	−0.576186
$517$	−524.766	−0.0446406
$518$	11386.3	0.965805
$519$	2775.15	0.234712
$520$	−2692.86	−0.227095
$521$	12456.9	1.04750	0.523750	−	0.851872i	$-0.324532\pi$
$521$	12456.9	1.04750	0.523750	+	0.851872i	$0.324532\pi$
$522$	8073.96	0.676988
$523$	17565.0	1.46858	0.734288	−	0.678838i	$-0.237516\pi$
$523$	17565.0	1.46858	0.734288	+	0.678838i	$0.237516\pi$
$524$	−3382.11	−0.281962
$525$	2839.16	0.236021
$526$	−19265.9	−1.59702
$527$	−514.877	−0.0425586
$528$	2975.08	0.245215
$529$	11636.3	0.956378
$530$	−16970.5	−1.39085
$531$	531.000	0.0433963
$532$	8686.02	0.707870
$533$	42974.1	3.49233
$534$	−8456.00	−0.685257
$535$	29078.2	2.34983
$536$	399.890	0.0322250
$537$	6735.46	0.541260
$538$	−23742.8	−1.90264
$539$	4928.28	0.393833
$540$	3732.20	0.297423
$541$	−12600.3	−1.00135	−0.500674	−	0.865636i	$-0.666914\pi$
$541$	−12600.3	−1.00135	−0.500674	+	0.865636i	$0.666914\pi$
$542$	−19007.5	−1.50635
$543$	−5318.93	−0.420363
$544$	−1678.74	−0.132308
$545$	−25193.1	−1.98010
$546$	−6792.43	−0.532398
$547$	18529.7	1.44840	0.724198	−	0.689592i	$-0.242210\pi$
$547$	18529.7	1.44840	0.724198	+	0.689592i	$0.242210\pi$
$548$	−363.983	−0.0283734
$549$	−618.512	−0.0480828
$550$	−9415.25	−0.729941
$551$	33359.7	2.57925
$552$	−930.232	−0.0717270
$553$	−2064.21	−0.158732
$554$	−15779.3	−1.21011
$555$	20214.5	1.54605
$556$	−2087.97	−0.159262
$557$	−17686.0	−1.34538	−0.672691	−	0.739923i	$-0.734862\pi$
$557$	−17686.0	−1.34538	−0.672691	+	0.739923i	$0.734862\pi$
$558$	−2902.86	−0.220229
$559$	21821.1	1.65105
$560$	−6587.68	−0.497108
$561$	−322.566	−0.0242758
$562$	−30789.1	−2.31096
$563$	1719.56	0.128723	0.0643614	−	0.997927i	$-0.479499\pi$
$563$	1719.56	0.128723	0.0643614	+	0.997927i	$0.479499\pi$
$564$	806.381	0.0602035
$565$	23022.5	1.71427
$566$	3849.15	0.285851
$567$	548.401	0.0406185
$568$	1458.75	0.107760
$569$	4196.95	0.309218	0.154609	−	0.987976i	$-0.450588\pi$
$569$	4196.95	0.309218	0.154609	+	0.987976i	$0.450588\pi$
$570$	29942.8	2.20029
$571$	−16717.6	−1.22523	−0.612616	−	0.790380i	$-0.709883\pi$
$571$	−16717.6	−1.22523	−0.612616	+	0.790380i	$0.709883\pi$
$572$	11600.5	0.847971
$573$	−1861.37	−0.135707
$574$	−14350.8	−1.04354
$575$	−21566.2	−1.56412
$576$	−5159.35	−0.373217
$577$	4687.41	0.338196	0.169098	−	0.985599i	$-0.445915\pi$
$577$	4687.41	0.338196	0.169098	+	0.985599i	$0.445915\pi$
$578$	−19782.9	−1.42363
$579$	−7145.77	−0.512898
$580$	30533.1	2.18589
$581$	4950.59	0.353503
$582$	−7275.89	−0.518205
$583$	4258.70	0.302534
$584$	1148.72	0.0813947
$585$	−12058.8	−0.852257
$586$	−10612.2	−0.748098
$587$	15750.6	1.10749	0.553745	−	0.832686i	$-0.313198\pi$
$587$	15750.6	1.10749	0.553745	+	0.832686i	$0.313198\pi$
$588$	−7573.04	−0.531134
$589$	−11993.9	−0.839050
$590$	3899.16	0.272078
$591$	6132.64	0.426841
$592$	−24761.1	−1.71905
$593$	−3706.55	−0.256677	−0.128339	−	0.991730i	$-0.540964\pi$
$593$	−3706.55	−0.256677	−0.128339	+	0.991730i	$0.540964\pi$
$594$	−1818.61	−0.125621
$595$	714.254	0.0492127
$596$	12676.8	0.871248
$597$	2960.48	0.202956
$598$	51595.1	3.52823
$599$	263.566	0.0179783	0.00898917	−	0.999960i	$-0.497139\pi$
$599$	263.566	0.0179783	0.00898917	+	0.999960i	$0.497139\pi$
$600$	842.807	0.0573457
$601$	−11714.8	−0.795101	−0.397550	−	0.917580i	$-0.630140\pi$
$601$	−11714.8	−0.795101	−0.397550	+	0.917580i	$0.630140\pi$
$602$	−7286.98	−0.493348
$603$	1790.74	0.120936
$604$	−7793.55	−0.525025
$605$	−17182.7	−1.15467
$606$	14461.4	0.969396
$607$	7579.45	0.506821	0.253410	−	0.967359i	$-0.418448\pi$
$607$	7579.45	0.506821	0.253410	+	0.967359i	$0.418448\pi$
$608$	−39105.8	−2.60847
$609$	4486.47	0.298523
$610$	−4541.77	−0.301461
$611$	−2605.43	−0.172511
$612$	495.670	0.0327390
$613$	−15090.5	−0.994291	−0.497146	−	0.867667i	$-0.665619\pi$
$613$	−15090.5	−0.994291	−0.497146	+	0.867667i	$0.665619\pi$
$614$	27419.0	1.80218
$615$	−25477.5	−1.67049
$616$	−225.667	−0.0147603
$617$	−19397.1	−1.26564	−0.632819	−	0.774300i	$-0.718102\pi$
$617$	−19397.1	−1.26564	−0.632819	+	0.774300i	$0.718102\pi$
$618$	−12637.1	−0.822552
$619$	−14955.2	−0.971082	−0.485541	−	0.874214i	$-0.661377\pi$
$619$	−14955.2	−0.971082	−0.485541	+	0.874214i	$0.661377\pi$
$620$	−10977.7	−0.711087
$621$	−4165.64	−0.269181
$622$	4779.97	0.308134
$623$	−4698.76	−0.302170
$624$	14771.1	0.947622
$625$	−13558.6	−0.867747
$626$	8365.68	0.534121
$627$	−7514.07	−0.478602
$628$	19868.8	1.26250
$629$	2684.67	0.170182
$630$	4026.94	0.254662
$631$	−2751.59	−0.173596	−0.0867979	−	0.996226i	$-0.527663\pi$
$631$	−2751.59	−0.173596	−0.0867979	+	0.996226i	$0.527663\pi$
$632$	−612.762	−0.0385670
$633$	7619.84	0.478454
$634$	21616.6	1.35411
$635$	22612.3	1.41313
$636$	−6544.13	−0.408006
$637$	24468.6	1.52195
$638$	−14878.1	−0.923242
$639$	6532.39	0.404409
$640$	−4178.07	−0.258051
$641$	−16315.9	−1.00536	−0.502682	−	0.864471i	$-0.667653\pi$
$641$	−16315.9	−1.00536	−0.502682	+	0.864471i	$0.667653\pi$
$642$	21773.0	1.33849
$643$	24947.6	1.53007	0.765037	−	0.643987i	$-0.222721\pi$
$643$	24947.6	1.53007	0.765037	+	0.643987i	$0.222721\pi$
$644$	−8873.34	−0.542948
$645$	−12936.8	−0.789745
$646$	3976.68	0.242198
$647$	−9286.05	−0.564254	−0.282127	−	0.959377i	$-0.591040\pi$
$647$	−9286.05	−0.564254	−0.282127	+	0.959377i	$0.591040\pi$
$648$	162.793	0.00986902
$649$	−978.486	−0.0591817
$650$	−46746.1	−2.82082
$651$	−1613.03	−0.0971118
$652$	12109.5	0.727367
$653$	−7806.02	−0.467800	−0.233900	−	0.972261i	$-0.575149\pi$
$653$	−7806.02	−0.467800	−0.233900	+	0.972261i	$0.575149\pi$
$654$	−18863.9	−1.12789
$655$	−6478.54	−0.386470
$656$	31207.9	1.85741
$657$	5144.06	0.305463
$658$	870.063	0.0515480
$659$	−30652.4	−1.81191	−0.905955	−	0.423374i	$-0.860846\pi$
$659$	−30652.4	−1.81191	−0.905955	+	0.423374i	$0.860846\pi$
$660$	−6877.41	−0.405610
$661$	−1119.99	−0.0659041	−0.0329520	−	0.999457i	$-0.510491\pi$
$661$	−1119.99	−0.0659041	−0.0329520	+	0.999457i	$0.510491\pi$
$662$	−44457.3	−2.61010
$663$	−1601.52	−0.0938127
$664$	1469.59	0.0858902
$665$	16638.3	0.970237
$666$	15136.1	0.880647
$667$	−34079.1	−1.97833
$668$	16462.8	0.953540
$669$	15194.9	0.878129
$670$	13149.5	0.758222
$671$	1139.75	0.0655730
$672$	−5259.25	−0.301905
$673$	−20636.6	−1.18199	−0.590996	−	0.806675i	$-0.701265\pi$
$673$	−20636.6	−1.18199	−0.590996	+	0.806675i	$0.701265\pi$
$674$	1353.43	0.0773473
$675$	3774.15	0.215210
$676$	38932.3	2.21508
$677$	−24871.6	−1.41195	−0.705976	−	0.708236i	$-0.749491\pi$
$677$	−24871.6	−1.41195	−0.705976	+	0.708236i	$0.749491\pi$
$678$	17238.6	0.976467
$679$	−4043.00	−0.228507
$680$	212.027	0.0119572
$681$	−2626.25	−0.147780
$682$	5349.16	0.300337
$683$	17124.9	0.959392	0.479696	−	0.877435i	$-0.340747\pi$
$683$	17124.9	0.959392	0.479696	+	0.877435i	$0.340747\pi$
$684$	11546.5	0.645455
$685$	−697.222	−0.0388897
$686$	−17602.6	−0.979696
$687$	−13559.6	−0.753032
$688$	15846.5	0.878116
$689$	21144.2	1.16913
$690$	−30588.5	−1.68766
$691$	−15624.6	−0.860183	−0.430092	−	0.902785i	$-0.641519\pi$
$691$	−15624.6	−0.860183	−0.430092	+	0.902785i	$0.641519\pi$
$692$	7858.16	0.431680
$693$	−1010.55	−0.0553935
$694$	−8342.29	−0.456295
$695$	−3999.57	−0.218291
$696$	1331.81	0.0725319
$697$	−3383.64	−0.183880
$698$	27866.8	1.51114
$699$	−9901.37	−0.535771
$700$	8039.41	0.434087
$701$	21519.2	1.15944	0.579722	−	0.814814i	$-0.303161\pi$
$701$	21519.2	1.15944	0.579722	+	0.814814i	$0.303161\pi$
$702$	−9029.31	−0.485455
$703$	62538.6	3.35517
$704$	9507.25	0.508975
$705$	1544.65	0.0825175
$706$	−14309.5	−0.762812
$707$	8035.77	0.427463
$708$	1503.59	0.0798140
$709$	−28820.4	−1.52662	−0.763309	−	0.646034i	$-0.776426\pi$
$709$	−28820.4	−1.52662	−0.763309	+	0.646034i	$0.776426\pi$
$710$	47967.7	2.53549
$711$	−2743.99	−0.144736
$712$	−1394.83	−0.0734178
$713$	12252.6	0.643566
$714$	534.814	0.0280321
$715$	22221.0	1.16227
$716$	19072.3	0.995480
$717$	6108.46	0.318165
$718$	6875.57	0.357373
$719$	−4082.36	−0.211747	−0.105874	−	0.994380i	$-0.533764\pi$
$719$	−4082.36	−0.211747	−0.105874	+	0.994380i	$0.533764\pi$
$720$	−8757.13	−0.453276
$721$	−7022.05	−0.362711
$722$	64778.5	3.33906
$723$	3406.26	0.175214
$724$	−15061.2	−0.773127
$725$	30876.3	1.58168
$726$	−12865.9	−0.657712
$727$	34069.5	1.73806	0.869028	−	0.494763i	$-0.164745\pi$
$727$	34069.5	1.73806	0.869028	+	0.494763i	$0.164745\pi$
$728$	−1120.42	−0.0570406
$729$	729.000	0.0370370
$730$	37773.2	1.91513
$731$	−1718.12	−0.0869318
$732$	−1751.39	−0.0884334
$733$	23108.8	1.16445	0.582226	−	0.813027i	$-0.302182\pi$
$733$	23108.8	1.16445	0.582226	+	0.813027i	$0.302182\pi$
$734$	−15698.6	−0.789435
$735$	−14506.4	−0.727995
$736$	39949.1	2.00074
$737$	−3299.83	−0.164926
$738$	−19076.9	−0.951530
$739$	−22291.4	−1.10961	−0.554805	−	0.831981i	$-0.687207\pi$
$739$	−22291.4	−1.10961	−0.554805	+	0.831981i	$0.687207\pi$
$740$	57239.7	2.84348
$741$	−37306.9	−1.84953
$742$	−7060.93	−0.349346
$743$	2619.28	0.129330	0.0646648	−	0.997907i	$-0.479402\pi$
$743$	2619.28	0.129330	0.0646648	+	0.997907i	$0.479402\pi$
$744$	−478.831	−0.0235951
$745$	24282.9	1.19417
$746$	−24860.2	−1.22010
$747$	6580.92	0.322334
$748$	−913.383	−0.0446478
$749$	12098.6	0.590219
$750$	2930.96	0.142698
$751$	−16907.8	−0.821538	−0.410769	−	0.911739i	$-0.634740\pi$
$751$	−16907.8	−0.821538	−0.410769	+	0.911739i	$0.634740\pi$
$752$	−1892.07	−0.0917509
$753$	18442.3	0.892528
$754$	−73868.7	−3.56783
$755$	−14928.8	−0.719621
$756$	1552.86	0.0747051
$757$	3540.46	0.169987	0.0849936	−	0.996382i	$-0.472913\pi$
$757$	3540.46	0.169987	0.0849936	+	0.996382i	$0.472913\pi$
$758$	−11937.2	−0.572005
$759$	7676.12	0.367096
$760$	4939.11	0.235737
$761$	−23640.9	−1.12613	−0.563063	−	0.826414i	$-0.690377\pi$
$761$	−23640.9	−1.12613	−0.563063	+	0.826414i	$0.690377\pi$
$762$	16931.5	0.804937
$763$	−10482.1	−0.497351
$764$	−5270.69	−0.249590
$765$	949.472	0.0448735
$766$	33202.7	1.56614
$767$	−4858.12	−0.228705
$768$	10629.8	0.499442
$769$	7631.43	0.357863	0.178931	−	0.983862i	$-0.442736\pi$
$769$	7631.43	0.357863	0.178931	+	0.983862i	$0.442736\pi$
$770$	−7420.54	−0.347296
$771$	2598.28	0.121368
$772$	−20234.1	−0.943318
$773$	−18739.3	−0.871936	−0.435968	−	0.899962i	$-0.643594\pi$
$773$	−18739.3	−0.871936	−0.435968	+	0.899962i	$0.643594\pi$
$774$	−9686.73	−0.449848
$775$	−11101.0	−0.514531
$776$	−1200.17	−0.0555200
$777$	8410.68	0.388329
$778$	29875.3	1.37671
$779$	−78821.0	−3.62523
$780$	−34145.9	−1.56746
$781$	−12037.4	−0.551513
$782$	−4062.44	−0.185770
$783$	5963.95	0.272202
$784$	17769.2	0.809456
$785$	38059.4	1.73044
$786$	−4850.96	−0.220137
$787$	11942.8	0.540933	0.270467	−	0.962729i	$-0.412822\pi$
$787$	11942.8	0.540933	0.270467	+	0.962729i	$0.412822\pi$
$788$	17365.3	0.785042
$789$	−14231.0	−0.642126
$790$	−20149.3	−0.907442
$791$	9578.99	0.430581
$792$	−299.983	−0.0134589
$793$	5658.77	0.253403
$794$	−30408.2	−1.35913
$795$	−12535.5	−0.559230
$796$	8382.96	0.373274
$797$	11411.4	0.507167	0.253583	−	0.967314i	$-0.418391\pi$
$797$	11411.4	0.507167	0.253583	+	0.967314i	$0.418391\pi$
$798$	12458.3	0.552657
$799$	205.143	0.00908317
$800$	−36194.6	−1.59959
$801$	−6246.15	−0.275527
$802$	−39398.3	−1.73467
$803$	−9479.09	−0.416575
$804$	5070.68	0.222424
$805$	−16997.2	−0.744188
$806$	26558.3	1.16064
$807$	−17537.9	−0.765011
$808$	2385.43	0.103860
$809$	30559.4	1.32808	0.664038	−	0.747699i	$-0.268841\pi$
$809$	30559.4	1.32808	0.664038	+	0.747699i	$0.268841\pi$
$810$	5353.09	0.232208
$811$	−38485.1	−1.66633	−0.833165	−	0.553025i	$-0.813473\pi$
$811$	−38485.1	−1.66633	−0.833165	+	0.553025i	$0.813473\pi$
$812$	12704.0	0.549041
$813$	−14040.1	−0.605669
$814$	−27891.6	−1.20098
$815$	23196.1	0.996960
$816$	−1163.03	−0.0498947
$817$	−40023.2	−1.71387
$818$	−5597.80	−0.239270
$819$	−5017.32	−0.214065
$820$	−72142.5	−3.07235
$821$	34212.3	1.45434	0.727172	−	0.686455i	$-0.240834\pi$
$821$	34212.3	1.45434	0.727172	+	0.686455i	$0.240834\pi$
$822$	−522.061	−0.0221520
$823$	24648.8	1.04399	0.521995	−	0.852949i	$-0.325188\pi$
$823$	24648.8	1.04399	0.521995	+	0.852949i	$0.325188\pi$
$824$	−2084.50	−0.0881275
$825$	−6954.71	−0.293493
$826$	1622.33	0.0683391
$827$	−20204.0	−0.849532	−0.424766	−	0.905303i	$-0.639644\pi$
$827$	−20204.0	−0.849532	−0.424766	+	0.905303i	$0.639644\pi$
$828$	−11795.5	−0.495075
$829$	−12267.7	−0.513964	−0.256982	−	0.966416i	$-0.582728\pi$
$829$	−12267.7	−0.513964	−0.256982	+	0.966416i	$0.582728\pi$
$830$	48324.1	2.02091
$831$	−11655.6	−0.486557
$832$	47202.9	1.96691
$833$	−1926.58	−0.0801346
$834$	−2994.77	−0.124341
$835$	31535.0	1.30696
$836$	−21277.0	−0.880239
$837$	−2144.24	−0.0885492
$838$	−25862.5	−1.06612
$839$	−8916.91	−0.366920	−0.183460	−	0.983027i	$-0.558730\pi$
$839$	−8916.91	−0.366920	−0.183460	+	0.983027i	$0.558730\pi$
$840$	664.250	0.0272843
$841$	24402.0	1.00053
$842$	19210.4	0.786265
$843$	−22742.8	−0.929187
$844$	21576.5	0.879968
$845$	74576.1	3.03609
$846$	1156.59	0.0470029
$847$	−7149.21	−0.290023
$848$	15355.0	0.621806
$849$	2843.23	0.114934
$850$	3680.64	0.148524
$851$	−63887.3	−2.57348
$852$	18497.2	0.743785
$853$	5454.65	0.218949	0.109475	−	0.993990i	$-0.465083\pi$
$853$	5454.65	0.218949	0.109475	+	0.993990i	$0.465083\pi$
$854$	−1889.70	−0.0757193
$855$	22117.7	0.884688
$856$	3591.48	0.143405
$857$	30739.7	1.22526	0.612630	−	0.790370i	$-0.290112\pi$
$857$	30739.7	1.22526	0.612630	+	0.790370i	$0.290112\pi$
$858$	16638.5	0.662039
$859$	−9450.34	−0.375368	−0.187684	−	0.982229i	$-0.560098\pi$
$859$	−9450.34	−0.375368	−0.187684	+	0.982229i	$0.560098\pi$
$860$	−36632.1	−1.45249
$861$	−10600.5	−0.419585
$862$	48881.0	1.93143
$863$	20494.1	0.808373	0.404186	−	0.914677i	$-0.367555\pi$
$863$	20494.1	0.808373	0.404186	+	0.914677i	$0.367555\pi$
$864$	−6991.22	−0.275285
$865$	15052.5	0.591678
$866$	−65042.0	−2.55221
$867$	−14612.9	−0.572411
$868$	−4567.50	−0.178607
$869$	5056.41	0.197384
$870$	43793.6	1.70660
$871$	−16383.4	−0.637350
$872$	−3111.63	−0.120841
$873$	−5374.44	−0.208359
$874$	−94633.3	−3.66249
$875$	1628.65	0.0629239
$876$	14566.0	0.561804
$877$	9745.52	0.375237	0.187618	−	0.982242i	$-0.439923\pi$
$877$	9745.52	0.375237	0.187618	+	0.982242i	$0.439923\pi$
$878$	12317.5	0.473456
$879$	−7838.84	−0.300794
$880$	16137.0	0.618156
$881$	−15386.5	−0.588405	−0.294203	−	0.955743i	$-0.595054\pi$
$881$	−15386.5	−0.588405	−0.294203	+	0.955743i	$0.595054\pi$
$882$	−10862.0	−0.414674
$883$	−38659.6	−1.47339	−0.736693	−	0.676227i	$-0.763614\pi$
$883$	−38659.6	−1.47339	−0.736693	+	0.676227i	$0.763614\pi$
$884$	−4534.89	−0.172539
$885$	2880.17	0.109397
$886$	−7130.96	−0.270394
$887$	−2987.34	−0.113083	−0.0565417	−	0.998400i	$-0.518007\pi$
$887$	−2987.34	−0.113083	−0.0565417	+	0.998400i	$0.518007\pi$
$888$	2496.72	0.0943517
$889$	9408.32	0.354944
$890$	−45865.8	−1.72744
$891$	−1343.35	−0.0505093
$892$	43026.1	1.61505
$893$	4778.75	0.179076
$894$	18182.4	0.680212
$895$	36533.5	1.36445
$896$	−1738.37	−0.0648158
$897$	38111.5	1.41862
$898$	−46574.7	−1.73076
$899$	−17542.0	−0.650788
$900$	10686.9	0.395813
$901$	−1664.83	−0.0615576
$902$	35153.4	1.29765
$903$	−5382.63	−0.198364
$904$	2843.53	0.104618
$905$	−28850.2	−1.05968
$906$	−11178.3	−0.409904
$907$	−8601.67	−0.314899	−0.157450	−	0.987527i	$-0.550327\pi$
$907$	−8601.67	−0.314899	−0.157450	+	0.987527i	$0.550327\pi$
$908$	−7436.53	−0.271795
$909$	10682.1	0.389772
$910$	−36842.5	−1.34211
$911$	−6239.74	−0.226928	−0.113464	−	0.993542i	$-0.536195\pi$
$911$	−6239.74	−0.226928	−0.113464	+	0.993542i	$0.536195\pi$
$912$	−27092.4	−0.983682
$913$	−12126.8	−0.439583
$914$	3739.79	0.135340
$915$	−3354.84	−0.121211
$916$	−38395.7	−1.38497
$917$	−2695.54	−0.0970714
$918$	710.939	0.0255604
$919$	44905.0	1.61184	0.805918	−	0.592027i	$-0.201672\pi$
$919$	44905.0	1.61184	0.805918	+	0.592027i	$0.201672\pi$
$920$	−5045.63	−0.180815
$921$	20253.4	0.724617
$922$	76389.3	2.72858
$923$	−59764.9	−2.13129
$924$	−2861.50	−0.101879
$925$	57883.1	2.05750
$926$	−18252.4	−0.647743
$927$	−9334.54	−0.330730
$928$	−57195.2	−2.02319
$929$	−20966.7	−0.740470	−0.370235	−	0.928938i	$-0.620723\pi$
$929$	−20966.7	−0.740470	−0.370235	+	0.928938i	$0.620723\pi$
$930$	−15745.3	−0.555169
$931$	−44879.2	−1.57987
$932$	−28036.9	−0.985385
$933$	3530.80	0.123894
$934$	2470.16	0.0865377
$935$	−1749.61	−0.0611962
$936$	−1489.40	−0.0520112
$937$	42889.9	1.49536	0.747679	−	0.664060i	$-0.231168\pi$
$937$	42889.9	1.49536	0.747679	+	0.664060i	$0.231168\pi$
$938$	5471.12	0.190446
$939$	6179.43	0.214758
$940$	4373.85	0.151765
$941$	17830.3	0.617697	0.308848	−	0.951111i	$-0.400056\pi$
$941$	17830.3	0.617697	0.308848	+	0.951111i	$0.400056\pi$
$942$	28497.8	0.985679
$943$	80520.8	2.78061
$944$	−3527.98	−0.121638
$945$	2974.56	0.102394
$946$	17850.0	0.613480
$947$	−20658.3	−0.708874	−0.354437	−	0.935080i	$-0.615328\pi$
$947$	−20658.3	−0.708874	−0.354437	+	0.935080i	$0.615328\pi$
$948$	−7769.93	−0.266198
$949$	−47063.1	−1.60983
$950$	85739.5	2.92816
$951$	15967.4	0.544457
$952$	88.2184	0.00300333
$953$	−44095.2	−1.49883	−0.749415	−	0.662101i	$-0.769665\pi$
$953$	−44095.2	−1.49883	−0.749415	+	0.662101i	$0.769665\pi$
$954$	−9386.23	−0.318543
$955$	−10096.2	−0.342099
$956$	17296.8	0.585167
$957$	−10989.9	−0.371215
$958$	−16984.8	−0.572810
$959$	−290.094	−0.00976812
$960$	−27984.6	−0.940832
$961$	−23484.1	−0.788294
$962$	−138480.	−4.64114
$963$	16082.9	0.538177
$964$	9645.22	0.322253
$965$	−38759.1	−1.29295
$966$	−12727.0	−0.423898
$967$	45160.1	1.50181	0.750906	−	0.660409i	$-0.229617\pi$
$967$	45160.1	1.50181	0.750906	+	0.660409i	$0.229617\pi$
$968$	−2122.25	−0.0704667
$969$	2937.43	0.0973826
$970$	−39464.8	−1.30633
$971$	49299.5	1.62935	0.814673	−	0.579920i	$-0.196916\pi$
$971$	49299.5	1.62935	0.814673	+	0.579920i	$0.196916\pi$
$972$	2064.25	0.0681181
$973$	−1664.11	−0.0548292
$974$	−75454.0	−2.48224
$975$	−34529.7	−1.13419
$976$	4109.42	0.134774
$977$	27045.5	0.885632	0.442816	−	0.896613i	$-0.353980\pi$
$977$	27045.5	0.885632	0.442816	+	0.896613i	$0.353980\pi$
$978$	17368.6	0.567880
$979$	11509.9	0.375749
$980$	−41076.6	−1.33892
$981$	−13934.1	−0.453498
$982$	60786.8	1.97534
$983$	−23169.0	−0.751757	−0.375878	−	0.926669i	$-0.622659\pi$
$983$	−23169.0	−0.751757	−0.375878	+	0.926669i	$0.622659\pi$
$984$	−3146.76	−0.101946
$985$	33263.8	1.07601
$986$	5816.19	0.187855
$987$	642.684	0.0207263
$988$	−105639.	−3.40164
$989$	40886.4	1.31457
$990$	−9864.26	−0.316674
$991$	−58402.5	−1.87206	−0.936032	−	0.351915i	$-0.885531\pi$
$991$	−58402.5	−1.87206	−0.936032	+	0.351915i	$0.885531\pi$
$992$	20563.6	0.658159
$993$	−32839.0	−1.04946
$994$	19958.0	0.636851
$995$	16057.8	0.511625
$996$	18634.6	0.592833
$997$	−3853.02	−0.122394	−0.0611968	−	0.998126i	$-0.519492\pi$
$997$	−3853.02	−0.122394	−0.0611968	+	0.998126i	$0.519492\pi$
$998$	−59329.8	−1.88182
$999$	11180.5	0.354089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.d.1.6	✓	8
3.2	odd	2		531.4.a.e.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.d.1.6	✓	8	1.1	even	1	trivial
531.4.a.e.1.3		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)

\(f(q)\)	\(=\)	\(q+4.06139 q^{2} +3.00000 q^{3} +8.49485 q^{4} +16.2722 q^{5} +12.1842 q^{6} +6.77038 q^{7} +2.00979 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.06139 q^{2} +3.00000 q^{3} +8.49485 q^{4} +16.2722 q^{5} +12.1842 q^{6} +6.77038 q^{7} +2.00979 q^{8} +9.00000 q^{9} +66.0875 q^{10} -16.5845 q^{11} +25.4846 q^{12} -82.3410 q^{13} +27.4971 q^{14} +48.8165 q^{15} -59.7963 q^{16} +6.48327 q^{17} +36.5525 q^{18} +151.026 q^{19} +138.230 q^{20} +20.3111 q^{21} -67.3561 q^{22} -154.283 q^{23} +6.02938 q^{24} +139.783 q^{25} -334.419 q^{26} +27.0000 q^{27} +57.5134 q^{28} +220.887 q^{29} +198.263 q^{30} -79.4162 q^{31} -258.934 q^{32} -49.7535 q^{33} +26.3311 q^{34} +110.169 q^{35} +76.4537 q^{36} +414.092 q^{37} +613.375 q^{38} -247.023 q^{39} +32.7037 q^{40} -521.903 q^{41} +82.4914 q^{42} -265.009 q^{43} -140.883 q^{44} +146.449 q^{45} -626.603 q^{46} +31.6419 q^{47} -179.389 q^{48} -297.162 q^{49} +567.714 q^{50} +19.4498 q^{51} -699.475 q^{52} -256.788 q^{53} +109.657 q^{54} -269.866 q^{55} +13.6071 q^{56} +453.078 q^{57} +897.107 q^{58} +59.0000 q^{59} +414.689 q^{60} -68.7236 q^{61} -322.540 q^{62} +60.9334 q^{63} -573.261 q^{64} -1339.87 q^{65} -202.068 q^{66} +198.971 q^{67} +55.0745 q^{68} -462.849 q^{69} +447.438 q^{70} +725.821 q^{71} +18.0882 q^{72} +571.563 q^{73} +1681.79 q^{74} +419.350 q^{75} +1282.94 q^{76} -112.283 q^{77} -1003.26 q^{78} -304.888 q^{79} -973.015 q^{80} +81.0000 q^{81} -2119.65 q^{82} +731.213 q^{83} +172.540 q^{84} +105.497 q^{85} -1076.30 q^{86} +662.661 q^{87} -33.3315 q^{88} -694.016 q^{89} +594.788 q^{90} -557.480 q^{91} -1310.61 q^{92} -238.249 q^{93} +128.510 q^{94} +2457.52 q^{95} -776.802 q^{96} -597.160 q^{97} -1206.89 q^{98} -149.261 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9}+O(q^{10}) \) 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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