LMFDB - Embedded newform 471.4.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(1,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, 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("471.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.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:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.38179 q^{2} -3.00000 q^{3} -6.09066 q^{4} +20.1959 q^{5} +4.14537 q^{6} -9.60234 q^{7} +19.4703 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.38179 q^{2} -3.00000 q^{3} -6.09066 q^{4} +20.1959 q^{5} +4.14537 q^{6} -9.60234 q^{7} +19.4703 q^{8} +9.00000 q^{9} -27.9065 q^{10} -17.8124 q^{11} +18.2720 q^{12} +69.1039 q^{13} +13.2684 q^{14} -60.5877 q^{15} +21.8214 q^{16} +0.226776 q^{17} -12.4361 q^{18} -28.7555 q^{19} -123.006 q^{20} +28.8070 q^{21} +24.6130 q^{22} -45.3136 q^{23} -58.4110 q^{24} +282.875 q^{25} -95.4871 q^{26} -27.0000 q^{27} +58.4846 q^{28} -93.6710 q^{29} +83.7195 q^{30} -129.819 q^{31} -185.915 q^{32} +53.4373 q^{33} -0.313357 q^{34} -193.928 q^{35} -54.8159 q^{36} -45.9526 q^{37} +39.7340 q^{38} -207.312 q^{39} +393.221 q^{40} +298.241 q^{41} -39.8052 q^{42} +305.909 q^{43} +108.489 q^{44} +181.763 q^{45} +62.6138 q^{46} -78.7945 q^{47} -65.4641 q^{48} -250.795 q^{49} -390.873 q^{50} -0.680329 q^{51} -420.888 q^{52} +32.8451 q^{53} +37.3083 q^{54} -359.738 q^{55} -186.961 q^{56} +86.2664 q^{57} +129.434 q^{58} +54.9369 q^{59} +369.019 q^{60} +785.293 q^{61} +179.383 q^{62} -86.4210 q^{63} +82.3247 q^{64} +1395.62 q^{65} -73.8391 q^{66} +715.434 q^{67} -1.38122 q^{68} +135.941 q^{69} +267.968 q^{70} +885.258 q^{71} +175.233 q^{72} +951.654 q^{73} +63.4968 q^{74} -848.624 q^{75} +175.140 q^{76} +171.041 q^{77} +286.461 q^{78} -689.352 q^{79} +440.702 q^{80} +81.0000 q^{81} -412.106 q^{82} +896.713 q^{83} -175.454 q^{84} +4.57995 q^{85} -422.702 q^{86} +281.013 q^{87} -346.814 q^{88} +1065.29 q^{89} -251.158 q^{90} -663.559 q^{91} +275.989 q^{92} +389.458 q^{93} +108.877 q^{94} -580.743 q^{95} +557.745 q^{96} +1664.04 q^{97} +346.546 q^{98} -160.312 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) $ 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.38179	−0.488536	−0.244268	−	0.969708i	$-0.578548\pi$
$2$	−1.38179	−0.488536	−0.244268	+	0.969708i	$0.578548\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−6.09066	−0.761332
$5$	20.1959	1.80638	0.903188	−	0.429244i	$-0.141220\pi$
$5$	20.1959	1.80638	0.903188	+	0.429244i	$0.141220\pi$
$6$	4.14537	0.282057
$7$	−9.60234	−0.518478	−0.259239	−	0.965813i	$-0.583472\pi$
$7$	−9.60234	−0.518478	−0.259239	+	0.965813i	$0.583472\pi$
$8$	19.4703	0.860475
$9$	9.00000	0.333333
$10$	−27.9065	−0.882481
$11$	−17.8124	−0.488241	−0.244121	−	0.969745i	$-0.578499\pi$
$11$	−17.8124	−0.488241	−0.244121	+	0.969745i	$0.578499\pi$
$12$	18.2720	0.439555
$13$	69.1039	1.47431	0.737153	−	0.675726i	$-0.236170\pi$
$13$	69.1039	1.47431	0.737153	+	0.675726i	$0.236170\pi$
$14$	13.2684	0.253295
$15$	−60.5877	−1.04291
$16$	21.8214	0.340959
$17$	0.226776	0.00323537	0.00161769	−	0.999999i	$-0.499485\pi$
$17$	0.226776	0.00323537	0.00161769	+	0.999999i	$0.499485\pi$
$18$	−12.4361	−0.162845
$19$	−28.7555	−0.347208	−0.173604	−	0.984816i	$-0.555541\pi$
$19$	−28.7555	−0.347208	−0.173604	+	0.984816i	$0.555541\pi$
$20$	−123.006	−1.37525
$21$	28.8070	0.299343
$22$	24.6130	0.238524
$23$	−45.3136	−0.410806	−0.205403	−	0.978678i	$-0.565850\pi$
$23$	−45.3136	−0.410806	−0.205403	+	0.978678i	$0.565850\pi$
$24$	−58.4110	−0.496795
$25$	282.875	2.26300
$26$	−95.4871	−0.720252
$27$	−27.0000	−0.192450
$28$	58.4846	0.394734
$29$	−93.6710	−0.599802	−0.299901	−	0.953970i	$-0.596954\pi$
$29$	−93.6710	−0.599802	−0.299901	+	0.953970i	$0.596954\pi$
$30$	83.7195	0.509501
$31$	−129.819	−0.752136	−0.376068	−	0.926592i	$-0.622724\pi$
$31$	−129.819	−0.752136	−0.376068	+	0.926592i	$0.622724\pi$
$32$	−185.915	−1.02705
$33$	53.4373	0.281886
$34$	−0.313357	−0.00158060
$35$	−193.928	−0.936566
$36$	−54.8159	−0.253777
$37$	−45.9526	−0.204177	−0.102089	−	0.994775i	$-0.532553\pi$
$37$	−45.9526	−0.204177	−0.102089	+	0.994775i	$0.532553\pi$
$38$	39.7340	0.169624
$39$	−207.312	−0.851191
$40$	393.221	1.55434
$41$	298.241	1.13603	0.568017	−	0.823017i	$-0.307711\pi$
$41$	298.241	1.13603	0.568017	+	0.823017i	$0.307711\pi$
$42$	−39.8052	−0.146240
$43$	305.909	1.08490	0.542450	−	0.840088i	$-0.317497\pi$
$43$	305.909	1.08490	0.542450	+	0.840088i	$0.317497\pi$
$44$	108.489	0.371714
$45$	181.763	0.602126
$46$	62.6138	0.200693
$47$	−78.7945	−0.244540	−0.122270	−	0.992497i	$-0.539017\pi$
$47$	−78.7945	−0.244540	−0.122270	+	0.992497i	$0.539017\pi$
$48$	−65.4641	−0.196853
$49$	−250.795	−0.731181
$50$	−390.873	−1.10556
$51$	−0.680329	−0.00186794
$52$	−420.888	−1.12244
$53$	32.8451	0.0851250	0.0425625	−	0.999094i	$-0.486448\pi$
$53$	32.8451	0.0851250	0.0425625	+	0.999094i	$0.486448\pi$
$54$	37.3083	0.0940189
$55$	−359.738	−0.881948
$56$	−186.961	−0.446137
$57$	86.2664	0.200461
$58$	129.434	0.293025
$59$	54.9369	0.121223	0.0606116	−	0.998161i	$-0.480695\pi$
$59$	54.9369	0.121223	0.0606116	+	0.998161i	$0.480695\pi$
$60$	369.019	0.794003
$61$	785.293	1.64830	0.824151	−	0.566370i	$-0.191653\pi$
$61$	785.293	1.64830	0.824151	+	0.566370i	$0.191653\pi$
$62$	179.383	0.367446
$63$	−86.4210	−0.172826
$64$	82.3247	0.160790
$65$	1395.62	2.66315
$66$	−73.8391	−0.137712
$67$	715.434	1.30454	0.652270	−	0.757987i	$-0.273817\pi$
$67$	715.434	1.30454	0.652270	+	0.757987i	$0.273817\pi$
$68$	−1.38122	−0.00246319
$69$	135.941	0.237179
$70$	267.968	0.457547
$71$	885.258	1.47973	0.739865	−	0.672755i	$-0.234889\pi$
$71$	885.258	1.47973	0.739865	+	0.672755i	$0.234889\pi$
$72$	175.233	0.286825
$73$	951.654	1.52579	0.762895	−	0.646522i	$-0.223777\pi$
$73$	951.654	1.52579	0.762895	+	0.646522i	$0.223777\pi$
$74$	63.4968	0.0997480
$75$	−848.624	−1.30654
$76$	175.140	0.264341
$77$	171.041	0.253142
$78$	286.461	0.415838
$79$	−689.352	−0.981749	−0.490874	−	0.871230i	$-0.663323\pi$
$79$	−689.352	−0.981749	−0.490874	+	0.871230i	$0.663323\pi$
$80$	440.702	0.615900
$81$	81.0000	0.111111
$82$	−412.106	−0.554994
$83$	896.713	1.18587	0.592934	−	0.805251i	$-0.297969\pi$
$83$	896.713	1.18587	0.592934	+	0.805251i	$0.297969\pi$
$84$	−175.454	−0.227900
$85$	4.57995	0.00584430
$86$	−422.702	−0.530014
$87$	281.013	0.346296
$88$	−346.814	−0.420119
$89$	1065.29	1.26876	0.634382	−	0.773020i	$-0.281255\pi$
$89$	1065.29	1.26876	0.634382	+	0.773020i	$0.281255\pi$
$90$	−251.158	−0.294160
$91$	−663.559	−0.764395
$92$	275.989	0.312760
$93$	389.458	0.434246
$94$	108.877	0.119467
$95$	−580.743	−0.627189
$96$	557.745	0.592965
$97$	1664.04	1.74183	0.870913	−	0.491437i	$-0.163528\pi$
$97$	1664.04	1.74183	0.870913	+	0.491437i	$0.163528\pi$
$98$	346.546	0.357209
$99$	−160.312	−0.162747
$100$	−1722.89	−1.72289
$101$	−839.309	−0.826875	−0.413438	−	0.910532i	$-0.635672\pi$
$101$	−839.309	−0.826875	−0.413438	+	0.910532i	$0.635672\pi$
$102$	0.940071	0.000912558 0
$103$	−1017.85	−0.973709	−0.486855	−	0.873483i	$-0.661856\pi$
$103$	−1017.85	−0.973709	−0.486855	+	0.873483i	$0.661856\pi$
$104$	1345.48	1.26860
$105$	581.784	0.540727
$106$	−45.3851	−0.0415867
$107$	746.850	0.674773	0.337386	−	0.941366i	$-0.390457\pi$
$107$	746.850	0.674773	0.337386	+	0.941366i	$0.390457\pi$
$108$	164.448	0.146518
$109$	−1727.40	−1.51794	−0.758968	−	0.651128i	$-0.774296\pi$
$109$	−1727.40	−1.51794	−0.758968	+	0.651128i	$0.774296\pi$
$110$	497.083	0.430864
$111$	137.858	0.117882
$112$	−209.536	−0.176780
$113$	−417.799	−0.347816	−0.173908	−	0.984762i	$-0.555640\pi$
$113$	−417.799	−0.347816	−0.173908	+	0.984762i	$0.555640\pi$
$114$	−119.202	−0.0979324
$115$	−915.148	−0.742070
$116$	570.518	0.456649
$117$	621.935	0.491435
$118$	−75.9112	−0.0592220
$119$	−2.17758	−0.00167747
$120$	−1179.66	−0.897400
$121$	−1013.72	−0.761621
$122$	−1085.11	−0.805256
$123$	−894.723	−0.655890
$124$	790.684	0.572625
$125$	3188.42	2.28145
$126$	119.416	0.0844317
$127$	619.720	0.433002	0.216501	−	0.976282i	$-0.430536\pi$
$127$	619.720	0.433002	0.216501	+	0.976282i	$0.430536\pi$
$128$	1373.57	0.948494
$129$	−917.728	−0.626368
$130$	−1928.45	−1.30105
$131$	1074.95	0.716940	0.358470	−	0.933541i	$-0.383299\pi$
$131$	1074.95	0.716940	0.358470	+	0.933541i	$0.383299\pi$
$132$	−325.468	−0.214609
$133$	276.120	0.180020
$134$	−988.580	−0.637315
$135$	−545.290	−0.347637
$136$	4.41541	0.00278396
$137$	744.128	0.464052	0.232026	−	0.972710i	$-0.425465\pi$
$137$	744.128	0.464052	0.232026	+	0.972710i	$0.425465\pi$
$138$	−187.841	−0.115870
$139$	−680.271	−0.415107	−0.207553	−	0.978224i	$-0.566550\pi$
$139$	−680.271	−0.415107	−0.207553	+	0.978224i	$0.566550\pi$
$140$	1181.15	0.713038
$141$	236.384	0.141185
$142$	−1223.24	−0.722902
$143$	−1230.91	−0.719817
$144$	196.392	0.113653
$145$	−1891.77	−1.08347
$146$	−1314.99	−0.745404
$147$	752.385	0.422148
$148$	279.881	0.155447
$149$	−804.697	−0.442439	−0.221219	−	0.975224i	$-0.571004\pi$
$149$	−804.697	−0.442439	−0.221219	+	0.975224i	$0.571004\pi$
$150$	1172.62	0.638294
$151$	445.673	0.240188	0.120094	−	0.992763i	$-0.461680\pi$
$151$	445.673	0.240188	0.120094	+	0.992763i	$0.461680\pi$
$152$	−559.878	−0.298764
$153$	2.04099	0.00107846
$154$	−236.343	−0.123669
$155$	−2621.82	−1.35864
$156$	1262.66	0.648039
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	952.539	0.479620
$159$	−98.5354	−0.0491470
$160$	−3754.73	−1.85523
$161$	435.116	0.212993
$162$	−111.925	−0.0542818
$163$	3666.81	1.76200	0.881002	−	0.473112i	$-0.156869\pi$
$163$	3666.81	1.76200	0.881002	+	0.473112i	$0.156869\pi$
$164$	−1816.48	−0.864900
$165$	1079.22	0.509193
$166$	−1239.07	−0.579340
$167$	−2327.70	−1.07858	−0.539289	−	0.842120i	$-0.681307\pi$
$167$	−2327.70	−1.07858	−0.539289	+	0.842120i	$0.681307\pi$
$168$	560.882	0.257577
$169$	2578.35	1.17358
$170$	−6.32853	−0.00285515
$171$	−258.799	−0.115736
$172$	−1863.19	−0.825970
$173$	1408.64	0.619057	0.309529	−	0.950890i	$-0.399829\pi$
$173$	1408.64	0.619057	0.309529	+	0.950890i	$0.399829\pi$
$174$	−388.301	−0.169178
$175$	−2716.26	−1.17331
$176$	−388.692	−0.166470
$177$	−164.811	−0.0699883
$178$	−1472.00	−0.619838
$179$	4397.20	1.83610	0.918050	−	0.396465i	$-0.129763\pi$
$179$	4397.20	1.83610	0.918050	+	0.396465i	$0.129763\pi$
$180$	−1107.06	−0.458418
$181$	2235.60	0.918072	0.459036	−	0.888418i	$-0.348195\pi$
$181$	2235.60	0.918072	0.459036	+	0.888418i	$0.348195\pi$
$182$	916.899	0.373435
$183$	−2355.88	−0.951648
$184$	−882.270	−0.353488
$185$	−928.054	−0.368821
$186$	−538.148	−0.212145
$187$	−4.03944	−0.00157964
$188$	479.910	0.186176
$189$	259.263	0.0997811
$190$	802.465	0.306405
$191$	1751.31	0.663459	0.331730	−	0.943375i	$-0.392368\pi$
$191$	1751.31	0.663459	0.331730	+	0.943375i	$0.392368\pi$
$192$	−246.974	−0.0928324
$193$	−1553.00	−0.579208	−0.289604	−	0.957147i	$-0.593524\pi$
$193$	−1553.00	−0.579208	−0.289604	+	0.957147i	$0.593524\pi$
$194$	−2299.35	−0.850946
$195$	−4186.85	−1.53757
$196$	1527.51	0.556672
$197$	−3538.64	−1.27979	−0.639893	−	0.768464i	$-0.721021\pi$
$197$	−3538.64	−1.27979	−0.639893	+	0.768464i	$0.721021\pi$
$198$	221.517	0.0795079
$199$	−1594.01	−0.567820	−0.283910	−	0.958851i	$-0.591632\pi$
$199$	−1594.01	−0.567820	−0.283910	+	0.958851i	$0.591632\pi$
$200$	5507.66	1.94725
$201$	−2146.30	−0.753177
$202$	1159.75	0.403959
$203$	899.461	0.310984
$204$	4.14365	0.00142213
$205$	6023.25	2.05211
$206$	1406.46	0.475693
$207$	−407.822	−0.136935
$208$	1507.94	0.502678
$209$	512.205	0.169521
$210$	−803.903	−0.264165
$211$	−4418.51	−1.44162	−0.720812	−	0.693130i	$-0.756231\pi$
$211$	−4418.51	−1.44162	−0.720812	+	0.693130i	$0.756231\pi$
$212$	−200.048	−0.0648084
$213$	−2655.78	−0.854323
$214$	−1031.99	−0.329651
$215$	6178.12	1.95974
$216$	−525.699	−0.165598
$217$	1246.57	0.389966
$218$	2386.90	0.741567
$219$	−2854.96	−0.880915
$220$	2191.04	0.671455
$221$	15.6711	0.00476993
$222$	−190.490	−0.0575896
$223$	−3771.36	−1.13251	−0.566253	−	0.824232i	$-0.691607\pi$
$223$	−3771.36	−1.13251	−0.566253	+	0.824232i	$0.691607\pi$
$224$	1785.22	0.532500
$225$	2545.87	0.754333
$226$	577.310	0.169921
$227$	−1347.13	−0.393886	−0.196943	−	0.980415i	$-0.563101\pi$
$227$	−1347.13	−0.393886	−0.196943	+	0.980415i	$0.563101\pi$
$228$	−525.419	−0.152617
$229$	2577.10	0.743666	0.371833	−	0.928300i	$-0.378729\pi$
$229$	2577.10	0.743666	0.371833	+	0.928300i	$0.378729\pi$
$230$	1264.54	0.362528
$231$	−513.123	−0.146152
$232$	−1823.80	−0.516115
$233$	1125.78	0.316533	0.158266	−	0.987396i	$-0.449410\pi$
$233$	1125.78	0.316533	0.158266	+	0.987396i	$0.449410\pi$
$234$	−859.384	−0.240084
$235$	−1591.33	−0.441731
$236$	−334.602	−0.0922912
$237$	2068.06	0.566813
$238$	3.00896	0.000819504 0
$239$	1739.65	0.470831	0.235415	−	0.971895i	$-0.424355\pi$
$239$	1739.65	0.470831	0.235415	+	0.971895i	$0.424355\pi$
$240$	−1322.11	−0.355590
$241$	3543.58	0.947145	0.473572	−	0.880755i	$-0.342964\pi$
$241$	3543.58	0.947145	0.473572	+	0.880755i	$0.342964\pi$
$242$	1400.74	0.372079
$243$	−243.000	−0.0641500
$244$	−4782.95	−1.25491
$245$	−5065.03	−1.32079
$246$	1236.32	0.320426
$247$	−1987.12	−0.511891
$248$	−2527.62	−0.647194
$249$	−2690.14	−0.684661
$250$	−4405.73	−1.11457
$251$	511.886	0.128725	0.0643624	−	0.997927i	$-0.479499\pi$
$251$	511.886	0.128725	0.0643624	+	0.997927i	$0.479499\pi$
$252$	526.361	0.131578
$253$	807.145	0.200572
$254$	−856.323	−0.211537
$255$	−13.7399	−0.00337421
$256$	−2556.58	−0.624164
$257$	−3912.15	−0.949546	−0.474773	−	0.880108i	$-0.657470\pi$
$257$	−3912.15	−0.949546	−0.474773	+	0.880108i	$0.657470\pi$
$258$	1268.11	0.306003
$259$	441.252	0.105861
$260$	−8500.22	−2.02754
$261$	−843.039	−0.199934
$262$	−1485.36	−0.350251
$263$	−3352.52	−0.786028	−0.393014	−	0.919532i	$-0.628568\pi$
$263$	−3352.52	−0.786028	−0.393014	+	0.919532i	$0.628568\pi$
$264$	1040.44	0.242556
$265$	663.337	0.153768
$266$	−381.540	−0.0879462
$267$	−3195.86	−0.732521
$268$	−4357.46	−0.993189
$269$	−4002.43	−0.907184	−0.453592	−	0.891209i	$-0.649858\pi$
$269$	−4002.43	−0.907184	−0.453592	+	0.891209i	$0.649858\pi$
$270$	753.475	0.169834
$271$	2892.17	0.648292	0.324146	−	0.946007i	$-0.394923\pi$
$271$	2892.17	0.648292	0.324146	+	0.946007i	$0.394923\pi$
$272$	4.94857	0.00110313
$273$	1990.68	0.441323
$274$	−1028.23	−0.226706
$275$	−5038.69	−1.10489
$276$	−827.968	−0.180572
$277$	241.733	0.0524344	0.0262172	−	0.999656i	$-0.491654\pi$
$277$	241.733	0.0524344	0.0262172	+	0.999656i	$0.491654\pi$
$278$	939.991	0.202795
$279$	−1168.37	−0.250712
$280$	−3775.84	−0.805891
$281$	5132.75	1.08966	0.544830	−	0.838547i	$-0.316594\pi$
$281$	5132.75	1.08966	0.544830	+	0.838547i	$0.316594\pi$
$282$	−326.632	−0.0689740
$283$	4962.45	1.04236	0.521179	−	0.853447i	$-0.325492\pi$
$283$	4962.45	1.04236	0.521179	+	0.853447i	$0.325492\pi$
$284$	−5391.81	−1.12657
$285$	1742.23	0.362108
$286$	1700.86	0.351657
$287$	−2863.81	−0.589009
$288$	−1673.24	−0.342349
$289$	−4912.95	−0.999990
$290$	2614.03	0.529314
$291$	−4992.11	−1.00564
$292$	−5796.20	−1.16163
$293$	−3464.15	−0.690710	−0.345355	−	0.938472i	$-0.612241\pi$
$293$	−3464.15	−0.690710	−0.345355	+	0.938472i	$0.612241\pi$
$294$	−1039.64	−0.206234
$295$	1109.50	0.218975
$296$	−894.712	−0.175689
$297$	480.936	0.0939621
$298$	1111.92	0.216147
$299$	−3131.34	−0.605653
$300$	5168.68	0.994713
$301$	−2937.45	−0.562497
$302$	−615.826	−0.117340
$303$	2517.93	0.477397
$304$	−627.484	−0.118384
$305$	15859.7	2.97746
$306$	−2.82021	−0.000526866 0
$307$	4406.46	0.819185	0.409593	−	0.912269i	$-0.365671\pi$
$307$	4406.46	0.819185	0.409593	+	0.912269i	$0.365671\pi$
$308$	−1041.75	−0.192725
$309$	3053.56	0.562171
$310$	3622.80	0.663746
$311$	−2925.82	−0.533466	−0.266733	−	0.963771i	$-0.585944\pi$
$311$	−2925.82	−0.533466	−0.266733	+	0.963771i	$0.585944\pi$
$312$	−4036.43	−0.732428
$313$	2526.35	0.456223	0.228111	−	0.973635i	$-0.426745\pi$
$313$	2526.35	0.456223	0.228111	+	0.973635i	$0.426745\pi$
$314$	216.941	0.0389895
$315$	−1745.35	−0.312189
$316$	4198.61	0.747437
$317$	6045.57	1.07115	0.535573	−	0.844489i	$-0.320096\pi$
$317$	6045.57	1.07115	0.535573	+	0.844489i	$0.320096\pi$
$318$	136.155	0.0240101
$319$	1668.51	0.292848
$320$	1662.62	0.290448
$321$	−2240.55	−0.389580
$322$	−601.239	−0.104055
$323$	−6.52106	−0.00112335
$324$	−493.343	−0.0845925
$325$	19547.7	3.33635
$326$	−5066.76	−0.860803
$327$	5182.20	0.876380
$328$	5806.85	0.977529
$329$	756.612	0.126788
$330$	−1491.25	−0.248759
$331$	7663.67	1.27261	0.636305	−	0.771438i	$-0.280462\pi$
$331$	7663.67	1.27261	0.636305	+	0.771438i	$0.280462\pi$
$332$	−5461.57	−0.902840
$333$	−413.573	−0.0680591
$334$	3216.39	0.526925
$335$	14448.8	2.35649
$336$	628.608	0.102064
$337$	−6364.16	−1.02872	−0.514359	−	0.857575i	$-0.671970\pi$
$337$	−6364.16	−1.02872	−0.514359	+	0.857575i	$0.671970\pi$
$338$	−3562.74	−0.573336
$339$	1253.40	0.200812
$340$	−27.8949	−0.00444946
$341$	2312.40	0.367224
$342$	357.606	0.0565413
$343$	5701.82	0.897579
$344$	5956.15	0.933530
$345$	2745.44	0.428434
$346$	−1946.44	−0.302432
$347$	4807.58	0.743760	0.371880	−	0.928281i	$-0.378713\pi$
$347$	4807.58	0.743760	0.371880	+	0.928281i	$0.378713\pi$
$348$	−1711.55	−0.263646
$349$	−9288.67	−1.42467	−0.712337	−	0.701838i	$-0.752363\pi$
$349$	−9288.67	−1.42467	−0.712337	+	0.701838i	$0.752363\pi$
$350$	3753.30	0.573206
$351$	−1865.81	−0.283730
$352$	3311.60	0.501446
$353$	1512.47	0.228047	0.114024	−	0.993478i	$-0.463626\pi$
$353$	1512.47	0.228047	0.114024	+	0.993478i	$0.463626\pi$
$354$	227.734	0.0341918
$355$	17878.6	2.67295
$356$	−6488.29	−0.965951
$357$	6.53275	0.000968487 0
$358$	−6076.00	−0.897002
$359$	4535.67	0.666807	0.333403	−	0.942784i	$-0.391803\pi$
$359$	4535.67	0.666807	0.333403	+	0.942784i	$0.391803\pi$
$360$	3538.99	0.518114
$361$	−6032.12	−0.879446
$362$	−3089.13	−0.448512
$363$	3041.15	0.439722
$364$	4041.51	0.581958
$365$	19219.5	2.75615
$366$	3255.33	0.464915
$367$	−13384.4	−1.90370	−0.951850	−	0.306563i	$-0.900821\pi$
$367$	−13384.4	−1.90370	−0.951850	+	0.306563i	$0.900821\pi$
$368$	−988.804	−0.140068
$369$	2684.17	0.378678
$370$	1282.38	0.180183
$371$	−315.390	−0.0441354
$372$	−2372.05	−0.330605
$373$	12019.5	1.66849	0.834247	−	0.551392i	$-0.185903\pi$
$373$	12019.5	1.66849	0.834247	+	0.551392i	$0.185903\pi$
$374$	5.58166	0.000771713 0
$375$	−9565.27	−1.31720
$376$	−1534.15	−0.210420
$377$	−6473.03	−0.884292
$378$	−358.247	−0.0487467
$379$	−3405.73	−0.461584	−0.230792	−	0.973003i	$-0.574132\pi$
$379$	−3405.73	−0.461584	−0.230792	+	0.973003i	$0.574132\pi$
$380$	3537.11	0.477499
$381$	−1859.16	−0.249994
$382$	−2419.95	−0.324124
$383$	−2240.14	−0.298866	−0.149433	−	0.988772i	$-0.547745\pi$
$383$	−2240.14	−0.298866	−0.149433	+	0.988772i	$0.547745\pi$
$384$	−4120.70	−0.547613
$385$	3454.33	0.457270
$386$	2145.92	0.282964
$387$	2753.18	0.361634
$388$	−10135.1	−1.32611
$389$	7320.42	0.954140	0.477070	−	0.878865i	$-0.341699\pi$
$389$	7320.42	0.954140	0.477070	+	0.878865i	$0.341699\pi$
$390$	5785.34	0.751160
$391$	−10.2760	−0.00132911
$392$	−4883.06	−0.629163
$393$	−3224.86	−0.413926
$394$	4889.66	0.625222
$395$	−13922.1	−1.77341
$396$	976.405	0.123905
$397$	−7397.32	−0.935166	−0.467583	−	0.883949i	$-0.654875\pi$
$397$	−7397.32	−0.935166	−0.467583	+	0.883949i	$0.654875\pi$
$398$	2202.58	0.277401
$399$	−828.359	−0.103934
$400$	6172.71	0.771589
$401$	6993.89	0.870968	0.435484	−	0.900196i	$-0.356577\pi$
$401$	6993.89	0.870968	0.435484	+	0.900196i	$0.356577\pi$
$402$	2965.74	0.367954
$403$	−8971.01	−1.10888
$404$	5111.95	0.629527
$405$	1635.87	0.200709
$406$	−1242.87	−0.151927
$407$	818.528	0.0996878
$408$	−13.2462	−0.00160732
$409$	−122.130	−0.0147651	−0.00738256	−	0.999973i	$-0.502350\pi$
$409$	−122.130	−0.0147651	−0.00738256	+	0.999973i	$0.502350\pi$
$410$	−8322.86	−1.00253
$411$	−2232.38	−0.267921
$412$	6199.40	0.741316
$413$	−527.522	−0.0628515
$414$	563.524	0.0668978
$415$	18109.9	2.14213
$416$	−12847.5	−1.51418
$417$	2040.81	0.239662
$418$	−707.760	−0.0828174
$419$	11440.8	1.33394	0.666968	−	0.745086i	$-0.267592\pi$
$419$	11440.8	1.33394	0.666968	+	0.745086i	$0.267592\pi$
$420$	−3543.45	−0.411673
$421$	−7242.12	−0.838383	−0.419192	−	0.907898i	$-0.637686\pi$
$421$	−7242.12	−0.838383	−0.419192	+	0.907898i	$0.637686\pi$
$422$	6105.45	0.704286
$423$	−709.151	−0.0815132
$424$	639.505	0.0732479
$425$	64.1493	0.00732164
$426$	3669.72	0.417368
$427$	−7540.65	−0.854608
$428$	−4548.80	−0.513726
$429$	3692.73	0.415586
$430$	−8536.86	−0.957404
$431$	−3339.08	−0.373174	−0.186587	−	0.982438i	$-0.559743\pi$
$431$	−3339.08	−0.373174	−0.186587	+	0.982438i	$0.559743\pi$
$432$	−589.177	−0.0656176
$433$	−13217.6	−1.46696	−0.733482	−	0.679708i	$-0.762106\pi$
$433$	−13217.6	−1.46696	−0.733482	+	0.679708i	$0.762106\pi$
$434$	−1722.49	−0.190512
$435$	5675.31	0.625541
$436$	10521.0	1.15565
$437$	1303.01	0.142635
$438$	3944.96	0.430359
$439$	−3549.58	−0.385905	−0.192952	−	0.981208i	$-0.561806\pi$
$439$	−3549.58	−0.385905	−0.192952	+	0.981208i	$0.561806\pi$
$440$	−7004.22	−0.758894
$441$	−2257.16	−0.243727
$442$	−21.6542	−0.00233028
$443$	16129.5	1.72988	0.864938	−	0.501879i	$-0.167358\pi$
$443$	16129.5	1.72988	0.864938	+	0.501879i	$0.167358\pi$
$444$	−839.644	−0.0897472
$445$	21514.4	2.29187
$446$	5211.22	0.553270
$447$	2414.09	0.255442
$448$	−790.510	−0.0833662
$449$	15024.2	1.57915	0.789573	−	0.613657i	$-0.210302\pi$
$449$	15024.2	1.57915	0.789573	+	0.613657i	$0.210302\pi$
$450$	−3517.86	−0.368519
$451$	−5312.40	−0.554659
$452$	2544.67	0.264803
$453$	−1337.02	−0.138672
$454$	1861.45	0.192428
$455$	−13401.2	−1.38078
$456$	1679.64	0.172492
$457$	−15766.4	−1.61383	−0.806917	−	0.590665i	$-0.798866\pi$
$457$	−15766.4	−1.61383	−0.806917	+	0.590665i	$0.798866\pi$
$458$	−3561.01	−0.363308
$459$	−6.12296	−0.000622648 0
$460$	5573.85	0.564962
$461$	−6965.14	−0.703685	−0.351843	−	0.936059i	$-0.614445\pi$
$461$	−6965.14	−0.703685	−0.351843	+	0.936059i	$0.614445\pi$
$462$	709.029	0.0714004
$463$	−9569.60	−0.960555	−0.480277	−	0.877117i	$-0.659464\pi$
$463$	−9569.60	−0.960555	−0.480277	+	0.877117i	$0.659464\pi$
$464$	−2044.03	−0.204508
$465$	7865.45	0.784412
$466$	−1555.59	−0.154638
$467$	3475.68	0.344401	0.172201	−	0.985062i	$-0.444912\pi$
$467$	3475.68	0.344401	0.172201	+	0.985062i	$0.444912\pi$
$468$	−3787.99	−0.374145
$469$	−6869.84	−0.676375
$470$	2198.88	0.215802
$471$	471.000	0.0460776
$472$	1069.64	0.104310
$473$	−5448.99	−0.529693
$474$	−2857.62	−0.276909
$475$	−8134.20	−0.785732
$476$	13.2629	0.00127711
$477$	295.606	0.0283750
$478$	−2403.83	−0.230018
$479$	2559.24	0.244122	0.122061	−	0.992523i	$-0.461050\pi$
$479$	2559.24	0.244122	0.122061	+	0.992523i	$0.461050\pi$
$480$	11264.2	1.07112
$481$	−3175.50	−0.301020
$482$	−4896.48	−0.462715
$483$	−1305.35	−0.122972
$484$	6174.20	0.579846
$485$	33606.7	3.14640
$486$	335.775	0.0313396
$487$	−1170.23	−0.108888	−0.0544439	−	0.998517i	$-0.517339\pi$
$487$	−1170.23	−0.108888	−0.0544439	+	0.998517i	$0.517339\pi$
$488$	15289.9	1.41832
$489$	−11000.4	−1.01729
$490$	6998.81	0.645253
$491$	−20375.9	−1.87282	−0.936408	−	0.350912i	$-0.885872\pi$
$491$	−20375.9	−1.87282	−0.936408	+	0.350912i	$0.885872\pi$
$492$	5449.45	0.499350
$493$	−21.2424	−0.00194058
$494$	2745.78	0.250078
$495$	−3237.65	−0.293983
$496$	−2832.83	−0.256447
$497$	−8500.55	−0.767207
$498$	3717.21	0.334482
$499$	8777.77	0.787469	0.393734	−	0.919224i	$-0.371183\pi$
$499$	8777.77	0.787469	0.393734	+	0.919224i	$0.371183\pi$
$500$	−19419.6	−1.73694
$501$	6983.09	0.622718
$502$	−707.318	−0.0628868
$503$	1947.65	0.172647	0.0863237	−	0.996267i	$-0.472488\pi$
$503$	1947.65	0.172647	0.0863237	+	0.996267i	$0.472488\pi$
$504$	−1682.65	−0.148712
$505$	−16950.6	−1.49365
$506$	−1115.30	−0.0979868
$507$	−7735.05	−0.677565
$508$	−3774.50	−0.329658
$509$	14699.2	1.28002	0.640011	−	0.768366i	$-0.278930\pi$
$509$	14699.2	1.28002	0.640011	+	0.768366i	$0.278930\pi$
$510$	18.9856	0.00164842
$511$	−9138.11	−0.791088
$512$	−7455.87	−0.643567
$513$	776.398	0.0668203
$514$	5405.77	0.463888
$515$	−20556.5	−1.75889
$516$	5589.57	0.476874
$517$	1403.52	0.119394
$518$	−609.718	−0.0517171
$519$	−4225.92	−0.357413
$520$	27173.1	2.29158
$521$	−4536.50	−0.381474	−0.190737	−	0.981641i	$-0.561088\pi$
$521$	−4536.50	−0.381474	−0.190737	+	0.981641i	$0.561088\pi$
$522$	1164.90	0.0976751
$523$	−12400.8	−1.03680	−0.518401	−	0.855137i	$-0.673473\pi$
$523$	−12400.8	−1.03680	−0.518401	+	0.855137i	$0.673473\pi$
$524$	−6547.17	−0.545829
$525$	8148.78	0.677413
$526$	4632.48	0.384003
$527$	−29.4399	−0.00243344
$528$	1166.08	0.0961116
$529$	−10113.7	−0.831239
$530$	−916.593	−0.0751212
$531$	494.432	0.0404077
$532$	−1681.75	−0.137055
$533$	20609.6	1.67486
$534$	4416.00	0.357863
$535$	15083.3	1.21889
$536$	13929.7	1.12252
$537$	−13191.6	−1.06007
$538$	5530.52	0.443192
$539$	4467.27	0.356993
$540$	3321.17	0.264668
$541$	10418.5	0.827961	0.413980	−	0.910286i	$-0.364138\pi$
$541$	10418.5	0.827961	0.413980	+	0.910286i	$0.364138\pi$
$542$	−3996.38	−0.316714
$543$	−6706.81	−0.530049
$544$	−42.1611	−0.00332288
$545$	−34886.4	−2.74196
$546$	−2750.70	−0.215603
$547$	2625.47	0.205223	0.102611	−	0.994722i	$-0.467280\pi$
$547$	2625.47	0.205223	0.102611	+	0.994722i	$0.467280\pi$
$548$	−4532.23	−0.353298
$549$	7067.64	0.549434
$550$	6962.41	0.539778
$551$	2693.55	0.208256
$552$	2646.81	0.204086
$553$	6619.39	0.509015
$554$	−334.024	−0.0256161
$555$	2784.16	0.212939
$556$	4143.30	0.316034
$557$	−18347.8	−1.39573	−0.697865	−	0.716229i	$-0.745866\pi$
$557$	−18347.8	−1.39573	−0.697865	+	0.716229i	$0.745866\pi$
$558$	1614.45	0.122482
$559$	21139.5	1.59948
$560$	−4231.77	−0.319330
$561$	12.1183	0.000912007 0
$562$	−7092.38	−0.532338
$563$	6391.71	0.478470	0.239235	−	0.970962i	$-0.423103\pi$
$563$	6391.71	0.478470	0.239235	+	0.970962i	$0.423103\pi$
$564$	−1439.73	−0.107489
$565$	−8437.82	−0.628287
$566$	−6857.07	−0.509230
$567$	−777.789	−0.0576086
$568$	17236.3	1.27327
$569$	21942.8	1.61668	0.808338	−	0.588718i	$-0.200367\pi$
$569$	21942.8	1.61668	0.808338	+	0.588718i	$0.200367\pi$
$570$	−2407.39	−0.176903
$571$	−1552.25	−0.113765	−0.0568823	−	0.998381i	$-0.518116\pi$
$571$	−1552.25	−0.113765	−0.0568823	+	0.998381i	$0.518116\pi$
$572$	7497.05	0.548020
$573$	−5253.94	−0.383048
$574$	3957.18	0.287752
$575$	−12818.1	−0.929652
$576$	740.922	0.0535968
$577$	−5244.85	−0.378416	−0.189208	−	0.981937i	$-0.560592\pi$
$577$	−5244.85	−0.378416	−0.189208	+	0.981937i	$0.560592\pi$
$578$	6788.66	0.488531
$579$	4658.99	0.334406
$580$	11522.1	0.824880
$581$	−8610.54	−0.614846
$582$	6898.04	0.491294
$583$	−585.052	−0.0415615
$584$	18529.0	1.31290
$585$	12560.5	0.887717
$586$	4786.73	0.337437
$587$	−890.650	−0.0626253	−0.0313126	−	0.999510i	$-0.509969\pi$
$587$	−890.650	−0.0626253	−0.0313126	+	0.999510i	$0.509969\pi$
$588$	−4582.52	−0.321395
$589$	3733.01	0.261148
$590$	−1533.10	−0.106977
$591$	10615.9	0.738884
$592$	−1002.75	−0.0696161
$593$	−15706.9	−1.08770	−0.543849	−	0.839183i	$-0.683034\pi$
$593$	−15706.9	−1.08770	−0.543849	+	0.839183i	$0.683034\pi$
$594$	−664.552	−0.0459039
$595$	−43.9783	−0.00303014
$596$	4901.14	0.336843
$597$	4782.03	0.327831
$598$	4326.86	0.295884
$599$	−24396.2	−1.66411	−0.832053	−	0.554696i	$-0.812835\pi$
$599$	−24396.2	−1.66411	−0.832053	+	0.554696i	$0.812835\pi$
$600$	−16523.0	−1.12425
$601$	−22030.0	−1.49521	−0.747607	−	0.664142i	$-0.768797\pi$
$601$	−22030.0	−1.49521	−0.747607	+	0.664142i	$0.768797\pi$
$602$	4058.93	0.274800
$603$	6438.91	0.434847
$604$	−2714.44	−0.182863
$605$	−20472.9	−1.37577
$606$	−3479.25	−0.233226
$607$	−23738.5	−1.58734	−0.793670	−	0.608349i	$-0.791832\pi$
$607$	−23738.5	−1.58734	−0.793670	+	0.608349i	$0.791832\pi$
$608$	5346.08	0.356599
$609$	−2698.38	−0.179547
$610$	−21914.8	−1.45460
$611$	−5445.01	−0.360526
$612$	−12.4310	−0.000821064 0
$613$	−11135.9	−0.733726	−0.366863	−	0.930275i	$-0.619568\pi$
$613$	−11135.9	−0.733726	−0.366863	+	0.930275i	$0.619568\pi$
$614$	−6088.80	−0.400202
$615$	−18069.7	−1.18478
$616$	3330.23	0.217822
$617$	−15236.1	−0.994137	−0.497069	−	0.867711i	$-0.665590\pi$
$617$	−15236.1	−0.994137	−0.497069	+	0.867711i	$0.665590\pi$
$618$	−4219.38	−0.274641
$619$	−12939.8	−0.840216	−0.420108	−	0.907474i	$-0.638008\pi$
$619$	−12939.8	−0.840216	−0.420108	+	0.907474i	$0.638008\pi$
$620$	15968.6	1.03438
$621$	1223.47	0.0790596
$622$	4042.86	0.260617
$623$	−10229.2	−0.657826
$624$	−4523.83	−0.290221
$625$	29033.8	1.85816
$626$	−3490.88	−0.222881
$627$	−1536.62	−0.0978732
$628$	956.233	0.0607609
$629$	−10.4210	−0.000660590 0
$630$	2411.71	0.152516
$631$	21002.5	1.32504	0.662518	−	0.749046i	$-0.269488\pi$
$631$	21002.5	1.32504	0.662518	+	0.749046i	$0.269488\pi$
$632$	−13421.9	−0.844770
$633$	13255.5	0.832322
$634$	−8353.71	−0.523293
$635$	12515.8	0.782165
$636$	600.145	0.0374172
$637$	−17330.9	−1.07798
$638$	−2305.53	−0.143067
$639$	7967.33	0.493243
$640$	27740.4	1.71334
$641$	−14748.8	−0.908804	−0.454402	−	0.890797i	$-0.650147\pi$
$641$	−14748.8	−0.908804	−0.454402	+	0.890797i	$0.650147\pi$
$642$	3095.97	0.190324
$643$	17334.6	1.06315	0.531577	−	0.847010i	$-0.321599\pi$
$643$	17334.6	1.06315	0.531577	+	0.847010i	$0.321599\pi$
$644$	−2650.14	−0.162159
$645$	−18534.4	−1.13146
$646$	9.01073	0.000548797 0
$647$	−1719.31	−0.104472	−0.0522358	−	0.998635i	$-0.516635\pi$
$647$	−1719.31	−0.104472	−0.0522358	+	0.998635i	$0.516635\pi$
$648$	1577.10	0.0956083
$649$	−978.560	−0.0591862
$650$	−27010.9	−1.62993
$651$	−3739.70	−0.225147
$652$	−22333.3	−1.34147
$653$	−9927.27	−0.594922	−0.297461	−	0.954734i	$-0.596140\pi$
$653$	−9927.27	−0.594922	−0.297461	+	0.954734i	$0.596140\pi$
$654$	−7160.71	−0.428144
$655$	21709.7	1.29506
$656$	6508.03	0.387341
$657$	8564.89	0.508597
$658$	−1045.48	−0.0619407
$659$	17453.8	1.03172	0.515860	−	0.856673i	$-0.327472\pi$
$659$	17453.8	1.03172	0.515860	+	0.856673i	$0.327472\pi$
$660$	−6573.13	−0.387665
$661$	−6208.62	−0.365336	−0.182668	−	0.983175i	$-0.558473\pi$
$661$	−6208.62	−0.365336	−0.182668	+	0.983175i	$0.558473\pi$
$662$	−10589.6	−0.621716
$663$	−47.0134	−0.00275392
$664$	17459.3	1.02041
$665$	5576.49	0.325183
$666$	571.471	0.0332493
$667$	4244.57	0.246402
$668$	14177.2	0.821157
$669$	11314.1	0.653852
$670$	−19965.3	−1.15123
$671$	−13988.0	−0.804769
$672$	−5355.66	−0.307439
$673$	−10827.4	−0.620158	−0.310079	−	0.950711i	$-0.600356\pi$
$673$	−10827.4	−0.620158	−0.310079	+	0.950711i	$0.600356\pi$
$674$	8793.93	0.502566
$675$	−7637.62	−0.435514
$676$	−15703.8	−0.893483
$677$	24325.6	1.38096	0.690479	−	0.723352i	$-0.257400\pi$
$677$	24325.6	1.38096	0.690479	+	0.723352i	$0.257400\pi$
$678$	−1731.93	−0.0981038
$679$	−15978.6	−0.903098
$680$	89.1732	0.00502888
$681$	4041.39	0.227410
$682$	−3195.25	−0.179402
$683$	−12419.8	−0.695796	−0.347898	−	0.937532i	$-0.613104\pi$
$683$	−12419.8	−0.695796	−0.347898	+	0.937532i	$0.613104\pi$
$684$	1576.26	0.0881136
$685$	15028.3	0.838253
$686$	−7878.72	−0.438500
$687$	−7731.30	−0.429356
$688$	6675.36	0.369907
$689$	2269.73	0.125500
$690$	−3793.63	−0.209306
$691$	−28763.8	−1.58354	−0.791771	−	0.610818i	$-0.790841\pi$
$691$	−28763.8	−1.58354	−0.791771	+	0.610818i	$0.790841\pi$
$692$	−8579.54	−0.471308
$693$	1539.37	0.0843807
$694$	−6643.07	−0.363354
$695$	−13738.7	−0.749839
$696$	5471.41	0.297979
$697$	67.6340	0.00367550
$698$	12835.0	0.696005
$699$	−3377.33	−0.182750
$700$	16543.8	0.893281
$701$	24235.4	1.30579	0.652896	−	0.757448i	$-0.273554\pi$
$701$	24235.4	1.30579	0.652896	+	0.757448i	$0.273554\pi$
$702$	2578.15	0.138613
$703$	1321.39	0.0708921
$704$	−1466.40	−0.0785045
$705$	4773.98	0.255033
$706$	−2089.92	−0.111409
$707$	8059.33	0.428716
$708$	1003.80	0.0532843
$709$	23686.1	1.25466	0.627328	−	0.778755i	$-0.284149\pi$
$709$	23686.1	1.25466	0.627328	+	0.778755i	$0.284149\pi$
$710$	−24704.5	−1.30583
$711$	−6204.17	−0.327250
$712$	20741.5	1.09174
$713$	5882.57	0.308982
$714$	−9.02688	−0.000473141 0
$715$	−24859.3	−1.30026
$716$	−26781.8	−1.39788
$717$	−5218.94	−0.271834
$718$	−6267.34	−0.325759
$719$	−1561.46	−0.0809914	−0.0404957	−	0.999180i	$-0.512894\pi$
$719$	−1561.46	−0.0809914	−0.0404957	+	0.999180i	$0.512894\pi$
$720$	3966.32	0.205300
$721$	9773.77	0.504846
$722$	8335.12	0.429642
$723$	−10630.7	−0.546834
$724$	−13616.3	−0.698958
$725$	−26497.2	−1.35735
$726$	−4202.23	−0.214820
$727$	−4396.36	−0.224281	−0.112140	−	0.993692i	$-0.535771\pi$
$727$	−4396.36	−0.224281	−0.112140	+	0.993692i	$0.535771\pi$
$728$	−12919.7	−0.657742
$729$	729.000	0.0370370
$730$	−26557.3	−1.34648
$731$	69.3730	0.00351006
$732$	14348.9	0.724520
$733$	−10668.5	−0.537585	−0.268793	−	0.963198i	$-0.586625\pi$
$733$	−10668.5	−0.537585	−0.268793	+	0.963198i	$0.586625\pi$
$734$	18494.4	0.930027
$735$	15195.1	0.762558
$736$	8424.47	0.421916
$737$	−12743.6	−0.636930
$738$	−3708.96	−0.184998
$739$	14143.2	0.704012	0.352006	−	0.935998i	$-0.385500\pi$
$739$	14143.2	0.704012	0.352006	+	0.935998i	$0.385500\pi$
$740$	5652.46	0.280795
$741$	5961.35	0.295541
$742$	435.803	0.0215618
$743$	−36274.2	−1.79108	−0.895539	−	0.444984i	$-0.853209\pi$
$743$	−36274.2	−1.79108	−0.895539	+	0.444984i	$0.853209\pi$
$744$	7582.87	0.373658
$745$	−16251.6	−0.799211
$746$	−16608.5	−0.815120
$747$	8070.42	0.395289
$748$	24.6028	0.00120263
$749$	−7171.50	−0.349854
$750$	13217.2	0.643498
$751$	30337.3	1.47406	0.737032	−	0.675857i	$-0.236226\pi$
$751$	30337.3	1.47406	0.737032	+	0.675857i	$0.236226\pi$
$752$	−1719.40	−0.0833779
$753$	−1535.66	−0.0743193
$754$	8944.37	0.432009
$755$	9000.76	0.433869
$756$	−1579.08	−0.0759665
$757$	−22991.6	−1.10389	−0.551945	−	0.833881i	$-0.686114\pi$
$757$	−22991.6	−1.10389	−0.551945	+	0.833881i	$0.686114\pi$
$758$	4706.00	0.225501
$759$	−2421.44	−0.115800
$760$	−11307.3	−0.539681
$761$	−12454.2	−0.593252	−0.296626	−	0.954994i	$-0.595861\pi$
$761$	−12454.2	−0.593252	−0.296626	+	0.954994i	$0.595861\pi$
$762$	2568.97	0.122131
$763$	16587.1	0.787015
$764$	−10666.7	−0.505113
$765$	41.2196	0.00194810
$766$	3095.40	0.146007
$767$	3796.35	0.178720
$768$	7669.73	0.360361
$769$	21395.3	1.00329	0.501647	−	0.865073i	$-0.332728\pi$
$769$	21395.3	1.00329	0.501647	+	0.865073i	$0.332728\pi$
$770$	−4773.16	−0.223393
$771$	11736.5	0.548221
$772$	9458.77	0.440970
$773$	−23327.1	−1.08540	−0.542702	−	0.839926i	$-0.682599\pi$
$773$	−23327.1	−1.08540	−0.542702	+	0.839926i	$0.682599\pi$
$774$	−3804.32	−0.176671
$775$	−36722.6	−1.70208
$776$	32399.3	1.49880
$777$	−1323.76	−0.0611191
$778$	−10115.3	−0.466132
$779$	−8576.06	−0.394441
$780$	25500.7	1.17060
$781$	−15768.6	−0.722465
$782$	14.1993	0.000649318 0
$783$	2529.12	0.115432
$784$	−5472.69	−0.249303
$785$	−3170.76	−0.144165
$786$	4456.08	0.202218
$787$	22265.5	1.00849	0.504244	−	0.863561i	$-0.331771\pi$
$787$	22265.5	1.00849	0.504244	+	0.863561i	$0.331771\pi$
$788$	21552.6	0.974342
$789$	10057.6	0.453813
$790$	19237.4	0.866375
$791$	4011.84	0.180335
$792$	−3121.33	−0.140040
$793$	54266.8	2.43010
$794$	10221.5	0.456863
$795$	−1990.01	−0.0887779
$796$	9708.56	0.432300
$797$	−7210.72	−0.320473	−0.160236	−	0.987079i	$-0.551226\pi$
$797$	−7210.72	−0.320473	−0.160236	+	0.987079i	$0.551226\pi$
$798$	1144.62	0.0507758
$799$	−17.8687	−0.000791177 0
$800$	−52590.7	−2.32420
$801$	9587.57	0.422921
$802$	−9664.09	−0.425500
$803$	−16951.3	−0.744954
$804$	13072.4	0.573418
$805$	8787.56	0.384747
$806$	12396.1	0.541728
$807$	12007.3	0.523763
$808$	−16341.6	−0.711506
$809$	−28066.7	−1.21974	−0.609872	−	0.792500i	$-0.708779\pi$
$809$	−28066.7	−1.21974	−0.609872	+	0.792500i	$0.708779\pi$
$810$	−2260.43	−0.0980534
$811$	−206.581	−0.00894457	−0.00447228	−	0.999990i	$-0.501424\pi$
$811$	−206.581	−0.00894457	−0.00447228	+	0.999990i	$0.501424\pi$
$812$	−5478.31	−0.236762
$813$	−8676.52	−0.374291
$814$	−1131.03	−0.0487011
$815$	74054.6	3.18284
$816$	−14.8457	−0.000636892 0
$817$	−8796.57	−0.376687
$818$	168.758	0.00721330
$819$	−5972.03	−0.254798
$820$	−36685.5	−1.56234
$821$	23532.1	1.00033	0.500167	−	0.865929i	$-0.333272\pi$
$821$	23532.1	1.00033	0.500167	+	0.865929i	$0.333272\pi$
$822$	3084.69	0.130889
$823$	5664.80	0.239930	0.119965	−	0.992778i	$-0.461722\pi$
$823$	5664.80	0.239930	0.119965	+	0.992778i	$0.461722\pi$
$824$	−19817.9	−0.837853
$825$	15116.1	0.637908
$826$	728.925	0.0307053
$827$	43549.9	1.83117	0.915585	−	0.402125i	$-0.131728\pi$
$827$	43549.9	1.83117	0.915585	+	0.402125i	$0.131728\pi$
$828$	2483.90	0.104253
$829$	−34359.2	−1.43950	−0.719749	−	0.694234i	$-0.755743\pi$
$829$	−34359.2	−1.43950	−0.719749	+	0.694234i	$0.755743\pi$
$830$	−25024.1	−1.04651
$831$	−725.199	−0.0302730
$832$	5688.96	0.237054
$833$	−56.8744	−0.00236564
$834$	−2819.97	−0.117084
$835$	−47010.0	−1.94832
$836$	−3119.67	−0.129062
$837$	3505.12	0.144749
$838$	−15808.8	−0.651676
$839$	−4973.55	−0.204655	−0.102328	−	0.994751i	$-0.532629\pi$
$839$	−4973.55	−0.204655	−0.102328	+	0.994751i	$0.532629\pi$
$840$	11327.5	0.465282
$841$	−15614.7	−0.640237
$842$	10007.1	0.409581
$843$	−15398.3	−0.629115
$844$	26911.6	1.09756
$845$	52072.1	2.11992
$846$	979.897	0.0398222
$847$	9734.05	0.394883
$848$	716.726	0.0290241
$849$	−14887.4	−0.601806
$850$	−88.6408	−0.00357689
$851$	2082.28	0.0838772
$852$	16175.4	0.650423
$853$	8934.35	0.358624	0.179312	−	0.983792i	$-0.442613\pi$
$853$	8934.35	0.358624	0.179312	+	0.983792i	$0.442613\pi$
$854$	10419.6	0.417507
$855$	−5226.69	−0.209063
$856$	14541.4	0.580625
$857$	−20975.5	−0.836065	−0.418033	−	0.908432i	$-0.637280\pi$
$857$	−20975.5	−0.836065	−0.418033	+	0.908432i	$0.637280\pi$
$858$	−5102.57	−0.203029
$859$	−44365.1	−1.76219	−0.881093	−	0.472943i	$-0.843191\pi$
$859$	−44365.1	−1.76219	−0.881093	+	0.472943i	$0.843191\pi$
$860$	−37628.8	−1.49201
$861$	8591.43	0.340064
$862$	4613.91	0.182309
$863$	23990.7	0.946297	0.473148	−	0.880983i	$-0.343117\pi$
$863$	23990.7	0.946297	0.473148	+	0.880983i	$0.343117\pi$
$864$	5019.71	0.197655
$865$	28448.8	1.11825
$866$	18263.9	0.716666
$867$	14738.8	0.577344
$868$	−7592.42	−0.296893
$869$	12279.0	0.479330
$870$	−7842.09	−0.305600
$871$	49439.3	1.92329
$872$	−33633.0	−1.30615
$873$	14976.3	0.580609
$874$	−1800.49	−0.0696825
$875$	−30616.3	−1.18288
$876$	17388.6	0.670669
$877$	−23062.5	−0.887988	−0.443994	−	0.896030i	$-0.646439\pi$
$877$	−23062.5	−0.887988	−0.443994	+	0.896030i	$0.646439\pi$
$878$	4904.77	0.188529
$879$	10392.5	0.398781
$880$	−7849.98	−0.300708
$881$	34205.0	1.30805	0.654026	−	0.756472i	$-0.273079\pi$
$881$	34205.0	1.30805	0.654026	+	0.756472i	$0.273079\pi$
$882$	3118.91	0.119070
$883$	31210.6	1.18949	0.594744	−	0.803915i	$-0.297253\pi$
$883$	31210.6	1.18949	0.594744	+	0.803915i	$0.297253\pi$
$884$	−95.4475	−0.00363150
$885$	−3328.50	−0.126425
$886$	−22287.6	−0.845107
$887$	28974.0	1.09679	0.548395	−	0.836219i	$-0.315239\pi$
$887$	28974.0	1.09679	0.548395	+	0.836219i	$0.315239\pi$
$888$	2684.14	0.101434
$889$	−5950.76	−0.224502
$890$	−29728.4	−1.11966
$891$	−1442.81	−0.0542490
$892$	22970.0	0.862213
$893$	2265.77	0.0849062
$894$	−3335.77	−0.124793
$895$	88805.4	3.31669
$896$	−13189.4	−0.491773
$897$	9394.03	0.349674
$898$	−20760.3	−0.771470
$899$	12160.3	0.451133
$900$	−15506.0	−0.574298
$901$	7.44850	0.000275411 0
$902$	7340.62	0.270971
$903$	8812.34	0.324758
$904$	−8134.68	−0.299287
$905$	45150.0	1.65838
$906$	1847.48	0.0677465
$907$	−39687.0	−1.45290	−0.726452	−	0.687217i	$-0.758832\pi$
$907$	−39687.0	−1.45290	−0.726452	+	0.687217i	$0.758832\pi$
$908$	8204.90	0.299878
$909$	−7553.79	−0.275625
$910$	18517.6	0.674564
$911$	−816.045	−0.0296781	−0.0148391	−	0.999890i	$-0.504724\pi$
$911$	−816.045	−0.0296781	−0.0148391	+	0.999890i	$0.504724\pi$
$912$	1882.45	0.0683489
$913$	−15972.7	−0.578990
$914$	21785.9	0.788417
$915$	−47579.1	−1.71903
$916$	−15696.2	−0.566177
$917$	−10322.1	−0.371717
$918$	8.46064	0.000304186 0
$919$	3894.79	0.139801	0.0699006	−	0.997554i	$-0.477732\pi$
$919$	3894.79	0.139801	0.0699006	+	0.997554i	$0.477732\pi$
$920$	−17818.2	−0.638532
$921$	−13219.4	−0.472957
$922$	9624.36	0.343776
$923$	61174.8	2.18157
$924$	3125.26	0.111270
$925$	−12998.8	−0.462053
$926$	13223.2	0.469266
$927$	−9160.68	−0.324570
$928$	17414.9	0.616024
$929$	−33645.0	−1.18822	−0.594109	−	0.804384i	$-0.702495\pi$
$929$	−33645.0	−1.18822	−0.594109	+	0.804384i	$0.702495\pi$
$930$	−10868.4	−0.383214
$931$	7211.73	0.253872
$932$	−6856.72	−0.240986
$933$	8777.45	0.307996
$934$	−4802.66	−0.168253
$935$	−81.5802	−0.00285343
$936$	12109.3	0.422868
$937$	−24756.4	−0.863135	−0.431568	−	0.902081i	$-0.642039\pi$
$937$	−24756.4	−0.863135	−0.431568	+	0.902081i	$0.642039\pi$
$938$	9492.68	0.330434
$939$	−7579.05	−0.263400
$940$	9692.23	0.336304
$941$	21232.1	0.735543	0.367772	−	0.929916i	$-0.380121\pi$
$941$	21232.1	0.735543	0.367772	+	0.929916i	$0.380121\pi$
$942$	−650.823	−0.0225106
$943$	−13514.4	−0.466689
$944$	1198.80	0.0413321
$945$	5236.05	0.180242
$946$	7529.36	0.258774
$947$	−798.191	−0.0273894	−0.0136947	−	0.999906i	$-0.504359\pi$
$947$	−798.191	−0.0273894	−0.0136947	+	0.999906i	$0.504359\pi$
$948$	−12595.8	−0.431533
$949$	65763.0	2.24948
$950$	11239.7	0.383859
$951$	−18136.7	−0.618426
$952$	−42.3982	−0.00144342
$953$	23694.2	0.805383	0.402691	−	0.915336i	$-0.368075\pi$
$953$	23694.2	0.805383	0.402691	+	0.915336i	$0.368075\pi$
$954$	−408.466	−0.0138622
$955$	35369.4	1.19846
$956$	−10595.6	−0.358458
$957$	−5005.53	−0.169076
$958$	−3536.33	−0.119263
$959$	−7145.37	−0.240601
$960$	−4987.87	−0.167690
$961$	−12938.0	−0.434291
$962$	4387.88	0.147059
$963$	6721.65	0.224924
$964$	−21582.7	−0.721092
$965$	−31364.2	−1.04627
$966$	1803.72	0.0600762
$967$	−34814.5	−1.15776	−0.578882	−	0.815411i	$-0.696511\pi$
$967$	−34814.5	−1.15776	−0.578882	+	0.815411i	$0.696511\pi$
$968$	−19737.4	−0.655355
$969$	19.5632	0.000648565 0
$970$	−46437.4	−1.53713
$971$	−59722.5	−1.97383	−0.986913	−	0.161252i	$-0.948447\pi$
$971$	−59722.5	−1.97383	−0.986913	+	0.161252i	$0.948447\pi$
$972$	1480.03	0.0488395
$973$	6532.19	0.215223
$974$	1617.02	0.0531957
$975$	−58643.2	−1.92624
$976$	17136.2	0.562003
$977$	−16387.4	−0.536621	−0.268311	−	0.963332i	$-0.586465\pi$
$977$	−16387.4	−0.536621	−0.268311	+	0.963332i	$0.586465\pi$
$978$	15200.3	0.496985
$979$	−18975.3	−0.619463
$980$	30849.4	1.00556
$981$	−15546.6	−0.505978
$982$	28155.3	0.914939
$983$	−55379.8	−1.79689	−0.898444	−	0.439089i	$-0.855301\pi$
$983$	−55379.8	−1.79689	−0.898444	+	0.439089i	$0.855301\pi$
$984$	−17420.5	−0.564377
$985$	−71466.1	−2.31177
$986$	29.3525	0.000948046 0
$987$	−2269.84	−0.0732013
$988$	12102.8	0.389719
$989$	−13861.8	−0.445683
$990$	4473.75	0.143621
$991$	−47182.8	−1.51242	−0.756212	−	0.654327i	$-0.772952\pi$
$991$	−47182.8	−1.51242	−0.756212	+	0.654327i	$0.772952\pi$
$992$	24135.4	0.772478
$993$	−22991.0	−0.734741
$994$	11746.0	0.374809
$995$	−32192.4	−1.02570
$996$	16384.7	0.521255
$997$	23675.5	0.752067	0.376033	−	0.926606i	$-0.377288\pi$
$997$	23675.5	0.752067	0.376033	+	0.926606i	$0.377288\pi$
$998$	−12129.0	−0.384707
$999$	1240.72	0.0392939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.9	✓	22
3.2	odd	2		1413.4.a.e.1.14		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.9	✓	22	1.1	even	1	trivial
1413.4.a.e.1.14		22	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:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q-1.38179 q^{2} -3.00000 q^{3} -6.09066 q^{4} +20.1959 q^{5} +4.14537 q^{6} -9.60234 q^{7} +19.4703 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.38179 q^{2} -3.00000 q^{3} -6.09066 q^{4} +20.1959 q^{5} +4.14537 q^{6} -9.60234 q^{7} +19.4703 q^{8} +9.00000 q^{9} -27.9065 q^{10} -17.8124 q^{11} +18.2720 q^{12} +69.1039 q^{13} +13.2684 q^{14} -60.5877 q^{15} +21.8214 q^{16} +0.226776 q^{17} -12.4361 q^{18} -28.7555 q^{19} -123.006 q^{20} +28.8070 q^{21} +24.6130 q^{22} -45.3136 q^{23} -58.4110 q^{24} +282.875 q^{25} -95.4871 q^{26} -27.0000 q^{27} +58.4846 q^{28} -93.6710 q^{29} +83.7195 q^{30} -129.819 q^{31} -185.915 q^{32} +53.4373 q^{33} -0.313357 q^{34} -193.928 q^{35} -54.8159 q^{36} -45.9526 q^{37} +39.7340 q^{38} -207.312 q^{39} +393.221 q^{40} +298.241 q^{41} -39.8052 q^{42} +305.909 q^{43} +108.489 q^{44} +181.763 q^{45} +62.6138 q^{46} -78.7945 q^{47} -65.4641 q^{48} -250.795 q^{49} -390.873 q^{50} -0.680329 q^{51} -420.888 q^{52} +32.8451 q^{53} +37.3083 q^{54} -359.738 q^{55} -186.961 q^{56} +86.2664 q^{57} +129.434 q^{58} +54.9369 q^{59} +369.019 q^{60} +785.293 q^{61} +179.383 q^{62} -86.4210 q^{63} +82.3247 q^{64} +1395.62 q^{65} -73.8391 q^{66} +715.434 q^{67} -1.38122 q^{68} +135.941 q^{69} +267.968 q^{70} +885.258 q^{71} +175.233 q^{72} +951.654 q^{73} +63.4968 q^{74} -848.624 q^{75} +175.140 q^{76} +171.041 q^{77} +286.461 q^{78} -689.352 q^{79} +440.702 q^{80} +81.0000 q^{81} -412.106 q^{82} +896.713 q^{83} -175.454 q^{84} +4.57995 q^{85} -422.702 q^{86} +281.013 q^{87} -346.814 q^{88} +1065.29 q^{89} -251.158 q^{90} -663.559 q^{91} +275.989 q^{92} +389.458 q^{93} +108.877 q^{94} -580.743 q^{95} +557.745 q^{96} +1664.04 q^{97} +346.546 q^{98} -160.312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

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