LMFDB - Embedded newform 471.4.a.c.1.2

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			1.2
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)

$f(q)$	$=$	$q-5.06198 q^{2} -3.00000 q^{3} +17.6236 q^{4} +12.0952 q^{5} +15.1859 q^{6} -27.2034 q^{7} -48.7145 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.06198 q^{2} -3.00000 q^{3} +17.6236 q^{4} +12.0952 q^{5} +15.1859 q^{6} -27.2034 q^{7} -48.7145 q^{8} +9.00000 q^{9} -61.2257 q^{10} -47.0823 q^{11} -52.8708 q^{12} +43.5246 q^{13} +137.703 q^{14} -36.2856 q^{15} +105.603 q^{16} +7.57285 q^{17} -45.5578 q^{18} -35.0284 q^{19} +213.161 q^{20} +81.6101 q^{21} +238.330 q^{22} +20.2961 q^{23} +146.144 q^{24} +21.2941 q^{25} -220.320 q^{26} -27.0000 q^{27} -479.422 q^{28} +267.845 q^{29} +183.677 q^{30} -67.7471 q^{31} -144.843 q^{32} +141.247 q^{33} -38.3336 q^{34} -329.031 q^{35} +158.613 q^{36} -274.521 q^{37} +177.313 q^{38} -130.574 q^{39} -589.212 q^{40} -94.4205 q^{41} -413.109 q^{42} -504.748 q^{43} -829.761 q^{44} +108.857 q^{45} -102.738 q^{46} +70.0139 q^{47} -316.809 q^{48} +397.024 q^{49} -107.790 q^{50} -22.7185 q^{51} +767.060 q^{52} -82.4197 q^{53} +136.673 q^{54} -569.471 q^{55} +1325.20 q^{56} +105.085 q^{57} -1355.82 q^{58} +625.922 q^{59} -639.484 q^{60} -4.48285 q^{61} +342.934 q^{62} -244.830 q^{63} -111.630 q^{64} +526.439 q^{65} -714.989 q^{66} -193.692 q^{67} +133.461 q^{68} -60.8884 q^{69} +1665.55 q^{70} +304.565 q^{71} -438.431 q^{72} +689.733 q^{73} +1389.62 q^{74} -63.8824 q^{75} -617.327 q^{76} +1280.80 q^{77} +660.961 q^{78} +1175.62 q^{79} +1277.29 q^{80} +81.0000 q^{81} +477.954 q^{82} -379.294 q^{83} +1438.27 q^{84} +91.5952 q^{85} +2555.02 q^{86} -803.534 q^{87} +2293.59 q^{88} +241.104 q^{89} -551.031 q^{90} -1184.01 q^{91} +357.691 q^{92} +203.241 q^{93} -354.409 q^{94} -423.676 q^{95} +434.529 q^{96} -1074.24 q^{97} -2009.73 q^{98} -423.741 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.06198	−1.78968	−0.894840	−	0.446388i	$-0.852710\pi$
$2$	−5.06198	−1.78968	−0.894840	+	0.446388i	$0.852710\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	17.6236	2.20295
$5$	12.0952	1.08183	0.540914	−	0.841078i	$-0.318078\pi$
$5$	12.0952	1.08183	0.540914	+	0.841078i	$0.318078\pi$
$6$	15.1859	1.03327
$7$	−27.2034	−1.46884	−0.734422	−	0.678693i	$-0.762547\pi$
$7$	−27.2034	−1.46884	−0.734422	+	0.678693i	$0.762547\pi$
$8$	−48.7145	−2.15290
$9$	9.00000	0.333333
$10$	−61.2257	−1.93613
$11$	−47.0823	−1.29053	−0.645266	−	0.763958i	$-0.723254\pi$
$11$	−47.0823	−1.29053	−0.645266	+	0.763958i	$0.723254\pi$
$12$	−52.8708	−1.27187
$13$	43.5246	0.928580	0.464290	−	0.885683i	$-0.346310\pi$
$13$	43.5246	0.928580	0.464290	+	0.885683i	$0.346310\pi$
$14$	137.703	2.62876
$15$	−36.2856	−0.624594
$16$	105.603	1.65004
$17$	7.57285	0.108040	0.0540202	−	0.998540i	$-0.482796\pi$
$17$	7.57285	0.108040	0.0540202	+	0.998540i	$0.482796\pi$
$18$	−45.5578	−0.596560
$19$	−35.0284	−0.422951	−0.211475	−	0.977383i	$-0.567827\pi$
$19$	−35.0284	−0.422951	−0.211475	+	0.977383i	$0.567827\pi$
$20$	213.161	2.38322
$21$	81.6101	0.848038
$22$	238.330	2.30964
$23$	20.2961	0.184001	0.0920007	−	0.995759i	$-0.470674\pi$
$23$	20.2961	0.184001	0.0920007	+	0.995759i	$0.470674\pi$
$24$	146.144	1.24298
$25$	21.2941	0.170353
$26$	−220.320	−1.66186
$27$	−27.0000	−0.192450
$28$	−479.422	−3.23579
$29$	267.845	1.71509	0.857543	−	0.514412i	$-0.171990\pi$
$29$	267.845	1.71509	0.857543	+	0.514412i	$0.171990\pi$
$30$	183.677	1.11782
$31$	−67.7471	−0.392508	−0.196254	−	0.980553i	$-0.562878\pi$
$31$	−67.7471	−0.392508	−0.196254	+	0.980553i	$0.562878\pi$
$32$	−144.843	−0.800152
$33$	141.247	0.745090
$34$	−38.3336	−0.193358
$35$	−329.031	−1.58904
$36$	158.613	0.734317
$37$	−274.521	−1.21976	−0.609878	−	0.792495i	$-0.708782\pi$
$37$	−274.521	−1.21976	−0.609878	+	0.792495i	$0.708782\pi$
$38$	177.313	0.756946
$39$	−130.574	−0.536116
$40$	−589.212	−2.32907
$41$	−94.4205	−0.359659	−0.179829	−	0.983698i	$-0.557555\pi$
$41$	−94.4205	−0.359659	−0.179829	+	0.983698i	$0.557555\pi$
$42$	−413.109	−1.51772
$43$	−504.748	−1.79008	−0.895038	−	0.445989i	$-0.852852\pi$
$43$	−504.748	−1.79008	−0.895038	+	0.445989i	$0.852852\pi$
$44$	−829.761	−2.84298
$45$	108.857	0.360610
$46$	−102.738	−0.329304
$47$	70.0139	0.217289	0.108644	−	0.994081i	$-0.465349\pi$
$47$	70.0139	0.217289	0.108644	+	0.994081i	$0.465349\pi$
$48$	−316.809	−0.952654
$49$	397.024	1.15750
$50$	−107.790	−0.304878
$51$	−22.7185	−0.0623771
$52$	767.060	2.04562
$53$	−82.4197	−0.213608	−0.106804	−	0.994280i	$-0.534062\pi$
$53$	−82.4197	−0.213608	−0.106804	+	0.994280i	$0.534062\pi$
$54$	136.673	0.344424
$55$	−569.471	−1.39614
$56$	1325.20	3.16227
$57$	105.085	0.244191
$58$	−1355.82	−3.06945
$59$	625.922	1.38116	0.690578	−	0.723258i	$-0.257356\pi$
$59$	625.922	1.38116	0.690578	+	0.723258i	$0.257356\pi$
$60$	−639.484	−1.37595
$61$	−4.48285	−0.00940935	−0.00470468	−	0.999989i	$-0.501498\pi$
$61$	−4.48285	−0.00940935	−0.00470468	+	0.999989i	$0.501498\pi$
$62$	342.934	0.702463
$63$	−244.830	−0.489615
$64$	−111.630	−0.218028
$65$	526.439	1.00456
$66$	−714.989	−1.33347
$67$	−193.692	−0.353182	−0.176591	−	0.984284i	$-0.556507\pi$
$67$	−193.692	−0.353182	−0.176591	+	0.984284i	$0.556507\pi$
$68$	133.461	0.238008
$69$	−60.8884	−0.106233
$70$	1665.55	2.84387
$71$	304.565	0.509087	0.254543	−	0.967061i	$-0.418075\pi$
$71$	304.565	0.509087	0.254543	+	0.967061i	$0.418075\pi$
$72$	−438.431	−0.717633
$73$	689.733	1.10585	0.552926	−	0.833231i	$-0.313511\pi$
$73$	689.733	1.10585	0.552926	+	0.833231i	$0.313511\pi$
$74$	1389.62	2.18297
$75$	−63.8824	−0.0983535
$76$	−617.327	−0.931740
$77$	1280.80	1.89559
$78$	660.961	0.959475
$79$	1175.62	1.67428	0.837138	−	0.546991i	$-0.184227\pi$
$79$	1175.62	1.67428	0.837138	+	0.546991i	$0.184227\pi$
$80$	1277.29	1.78507
$81$	81.0000	0.111111
$82$	477.954	0.643674
$83$	−379.294	−0.501601	−0.250801	−	0.968039i	$-0.580694\pi$
$83$	−379.294	−0.501601	−0.250801	+	0.968039i	$0.580694\pi$
$84$	1438.27	1.86819
$85$	91.5952	0.116881
$86$	2555.02	3.20366
$87$	−803.534	−0.990205
$88$	2293.59	2.77839
$89$	241.104	0.287157	0.143578	−	0.989639i	$-0.454139\pi$
$89$	241.104	0.287157	0.143578	+	0.989639i	$0.454139\pi$
$90$	−551.031	−0.645375
$91$	−1184.01	−1.36394
$92$	357.691	0.405346
$93$	203.241	0.226614
$94$	−354.409	−0.388878
$95$	−423.676	−0.457560
$96$	434.529	0.461968
$97$	−1074.24	−1.12445	−0.562227	−	0.826983i	$-0.690055\pi$
$97$	−1074.24	−1.12445	−0.562227	+	0.826983i	$0.690055\pi$
$98$	−2009.73	−2.07156
$99$	−423.741	−0.430178
$100$	375.280	0.375280
$101$	1483.69	1.46171	0.730857	−	0.682531i	$-0.239121\pi$
$101$	1483.69	1.46171	0.730857	+	0.682531i	$0.239121\pi$
$102$	115.001	0.111635
$103$	−155.903	−0.149141	−0.0745707	−	0.997216i	$-0.523759\pi$
$103$	−155.903	−0.149141	−0.0745707	+	0.997216i	$0.523759\pi$
$104$	−2120.28	−1.99914
$105$	987.092	0.917432
$106$	417.207	0.382290
$107$	1085.29	0.980549	0.490274	−	0.871568i	$-0.336897\pi$
$107$	1085.29	0.980549	0.490274	+	0.871568i	$0.336897\pi$
$108$	−475.838	−0.423958
$109$	638.756	0.561300	0.280650	−	0.959810i	$-0.409450\pi$
$109$	638.756	0.561300	0.280650	+	0.959810i	$0.409450\pi$
$110$	2882.65	2.49863
$111$	823.563	0.704227
$112$	−2872.75	−2.42366
$113$	−987.074	−0.821735	−0.410868	−	0.911695i	$-0.634774\pi$
$113$	−987.074	−0.821735	−0.410868	+	0.911695i	$0.634774\pi$
$114$	−531.939	−0.437023
$115$	245.486	0.199058
$116$	4720.39	3.77825
$117$	391.721	0.309527
$118$	−3168.41	−2.47182
$119$	−206.007	−0.158694
$120$	1767.64	1.34469
$121$	885.747	0.665475
$122$	22.6921	0.0168397
$123$	283.262	0.207649
$124$	−1193.95	−0.864676
$125$	−1254.34	−0.897536
$126$	1239.33	0.876253
$127$	417.817	0.291932	0.145966	−	0.989290i	$-0.453371\pi$
$127$	417.817	0.291932	0.145966	+	0.989290i	$0.453371\pi$
$128$	1723.82	1.19035
$129$	1514.24	1.03350
$130$	−2664.82	−1.79785
$131$	−28.2747	−0.0188578	−0.00942891	−	0.999956i	$-0.503001\pi$
$131$	−28.2747	−0.0188578	−0.00942891	+	0.999956i	$0.503001\pi$
$132$	2489.28	1.64140
$133$	952.891	0.621249
$134$	980.463	0.632083
$135$	−326.571	−0.208198
$136$	−368.908	−0.232600
$137$	2023.32	1.26178	0.630889	−	0.775873i	$-0.282690\pi$
$137$	2023.32	1.26178	0.630889	+	0.775873i	$0.282690\pi$
$138$	308.215	0.190124
$139$	826.766	0.504499	0.252250	−	0.967662i	$-0.418830\pi$
$139$	826.766	0.504499	0.252250	+	0.967662i	$0.418830\pi$
$140$	−5798.71	−3.50057
$141$	−210.042	−0.125452
$142$	−1541.70	−0.911102
$143$	−2049.24	−1.19836
$144$	950.426	0.550015
$145$	3239.64	1.85543
$146$	−3491.41	−1.97912
$147$	−1191.07	−0.668285
$148$	−4838.05	−2.68706
$149$	798.889	0.439245	0.219623	−	0.975585i	$-0.429517\pi$
$149$	798.889	0.439245	0.219623	+	0.975585i	$0.429517\pi$
$150$	323.371	0.176021
$151$	2960.88	1.59571	0.797857	−	0.602847i	$-0.205967\pi$
$151$	2960.88	1.59571	0.797857	+	0.602847i	$0.205967\pi$
$152$	1706.39	0.910570
$153$	68.1556	0.0360134
$154$	−6483.38	−3.39250
$155$	−819.416	−0.424626
$156$	−2301.18	−1.18104
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	−5950.97	−2.99642
$159$	247.259	0.123327
$160$	−1751.91	−0.865628
$161$	−552.123	−0.270270
$162$	−410.020	−0.198853
$163$	−2436.81	−1.17095	−0.585477	−	0.810689i	$-0.699093\pi$
$163$	−2436.81	−1.17095	−0.585477	+	0.810689i	$0.699093\pi$
$164$	−1664.03	−0.792311
$165$	1708.41	0.806059
$166$	1919.98	0.897705
$167$	1674.20	0.775770	0.387885	−	0.921708i	$-0.373206\pi$
$167$	1674.20	0.775770	0.387885	+	0.921708i	$0.373206\pi$
$168$	−3975.60	−1.82574
$169$	−302.613	−0.137739
$170$	−463.653	−0.209180
$171$	−315.256	−0.140984
$172$	−8895.48	−3.94345
$173$	3209.04	1.41028	0.705141	−	0.709067i	$-0.250884\pi$
$173$	3209.04	1.41028	0.705141	+	0.709067i	$0.250884\pi$
$174$	4067.47	1.77215
$175$	−579.273	−0.250222
$176$	−4972.03	−2.12944
$177$	−1877.77	−0.797410
$178$	−1220.46	−0.513918
$179$	2844.52	1.18776	0.593880	−	0.804553i	$-0.297595\pi$
$179$	2844.52	1.18776	0.593880	+	0.804553i	$0.297595\pi$
$180$	1918.45	0.794405
$181$	4412.43	1.81201	0.906004	−	0.423270i	$-0.139118\pi$
$181$	4412.43	1.81201	0.906004	+	0.423270i	$0.139118\pi$
$182$	5993.46	2.44101
$183$	13.4486	0.00543249
$184$	−988.715	−0.396136
$185$	−3320.39	−1.31957
$186$	−1028.80	−0.405567
$187$	−356.547	−0.139430
$188$	1233.90	0.478677
$189$	734.491	0.282679
$190$	2144.64	0.818886
$191$	−4618.38	−1.74960	−0.874802	−	0.484481i	$-0.839008\pi$
$191$	−4618.38	−1.74960	−0.874802	+	0.484481i	$0.839008\pi$
$192$	334.891	0.125879
$193$	3545.77	1.32244	0.661218	−	0.750194i	$-0.270040\pi$
$193$	3545.77	1.32244	0.661218	+	0.750194i	$0.270040\pi$
$194$	5437.75	2.01241
$195$	−1579.32	−0.579985
$196$	6997.00	2.54993
$197$	637.122	0.230422	0.115211	−	0.993341i	$-0.463246\pi$
$197$	637.122	0.230422	0.115211	+	0.993341i	$0.463246\pi$
$198$	2144.97	0.769880
$199$	4057.10	1.44523	0.722614	−	0.691252i	$-0.242941\pi$
$199$	4057.10	1.44523	0.722614	+	0.691252i	$0.242941\pi$
$200$	−1037.33	−0.366753
$201$	581.075	0.203910
$202$	−7510.43	−2.61600
$203$	−7286.28	−2.51919
$204$	−400.383	−0.137414
$205$	−1142.04	−0.389089
$206$	789.177	0.266915
$207$	182.665	0.0613338
$208$	4596.32	1.53220
$209$	1649.22	0.545832
$210$	−4996.64	−1.64191
$211$	5208.47	1.69936	0.849682	−	0.527296i	$-0.176794\pi$
$211$	5208.47	1.69936	0.849682	+	0.527296i	$0.176794\pi$
$212$	−1452.53	−0.470568
$213$	−913.694	−0.293921
$214$	−5493.70	−1.75487
$215$	−6105.03	−1.93656
$216$	1315.29	0.414325
$217$	1842.95	0.576533
$218$	−3233.37	−1.00455
$219$	−2069.20	−0.638464
$220$	−10036.1	−3.07562
$221$	329.605	0.100324
$222$	−4168.86	−1.26034
$223$	5531.75	1.66114	0.830568	−	0.556917i	$-0.188016\pi$
$223$	5531.75	1.66114	0.830568	+	0.556917i	$0.188016\pi$
$224$	3940.22	1.17530
$225$	191.647	0.0567844
$226$	4996.55	1.47064
$227$	−1656.83	−0.484439	−0.242219	−	0.970221i	$-0.577875\pi$
$227$	−1656.83	−0.484439	−0.242219	+	0.970221i	$0.577875\pi$
$228$	1851.98	0.537940
$229$	4872.11	1.40593	0.702965	−	0.711225i	$-0.251859\pi$
$229$	4872.11	1.40593	0.702965	+	0.711225i	$0.251859\pi$
$230$	−1242.64	−0.356250
$231$	−3842.40	−1.09442
$232$	−13047.9	−3.69240
$233$	−3038.18	−0.854238	−0.427119	−	0.904195i	$-0.640472\pi$
$233$	−3038.18	−0.854238	−0.427119	+	0.904195i	$0.640472\pi$
$234$	−1982.88	−0.553953
$235$	846.833	0.235069
$236$	11031.0	3.04262
$237$	−3526.87	−0.966644
$238$	1042.80	0.284012
$239$	780.833	0.211330	0.105665	−	0.994402i	$-0.466303\pi$
$239$	780.833	0.211330	0.105665	+	0.994402i	$0.466303\pi$
$240$	−3831.87	−1.03061
$241$	−4739.00	−1.26666	−0.633332	−	0.773880i	$-0.718313\pi$
$241$	−4739.00	−1.26666	−0.633332	+	0.773880i	$0.718313\pi$
$242$	−4483.63	−1.19099
$243$	−243.000	−0.0641500
$244$	−79.0041	−0.0207283
$245$	4802.09	1.25222
$246$	−1433.86	−0.371625
$247$	−1524.60	−0.392744
$248$	3300.27	0.845029
$249$	1137.88	0.289600
$250$	6349.46	1.60630
$251$	4861.38	1.22250	0.611251	−	0.791437i	$-0.290667\pi$
$251$	4861.38	1.22250	0.611251	+	0.791437i	$0.290667\pi$
$252$	−4314.80	−1.07860
$253$	−955.589	−0.237460
$254$	−2114.98	−0.522464
$255$	−274.786	−0.0674813
$256$	−7832.87	−1.91232
$257$	1402.04	0.340300	0.170150	−	0.985418i	$-0.445575\pi$
$257$	1402.04	0.340300	0.170150	+	0.985418i	$0.445575\pi$
$258$	−7665.06	−1.84964
$259$	7467.90	1.79163
$260$	9277.75	2.21301
$261$	2410.60	0.571695
$262$	143.126	0.0337494
$263$	4777.52	1.12013	0.560065	−	0.828449i	$-0.310776\pi$
$263$	4777.52	1.12013	0.560065	+	0.828449i	$0.310776\pi$
$264$	−6880.78	−1.60410
$265$	−996.884	−0.231087
$266$	−4823.51	−1.11184
$267$	−723.311	−0.165790
$268$	−3413.55	−0.778043
$269$	−6020.74	−1.36465	−0.682325	−	0.731049i	$-0.739031\pi$
$269$	−6020.74	−1.36465	−0.682325	+	0.731049i	$0.739031\pi$
$270$	1653.09	0.372608
$271$	−1706.34	−0.382483	−0.191241	−	0.981543i	$-0.561251\pi$
$271$	−1706.34	−0.382483	−0.191241	+	0.981543i	$0.561251\pi$
$272$	799.714	0.178271
$273$	3552.04	0.787471
$274$	−10242.0	−2.25818
$275$	−1002.58	−0.219846
$276$	−1073.07	−0.234027
$277$	7181.71	1.55779	0.778894	−	0.627156i	$-0.215781\pi$
$277$	7181.71	1.55779	0.778894	+	0.627156i	$0.215781\pi$
$278$	−4185.07	−0.902891
$279$	−609.724	−0.130836
$280$	16028.6	3.42104
$281$	−2445.20	−0.519105	−0.259553	−	0.965729i	$-0.583575\pi$
$281$	−2445.20	−0.519105	−0.259553	+	0.965729i	$0.583575\pi$
$282$	1063.23	0.224519
$283$	−8740.97	−1.83603	−0.918015	−	0.396545i	$-0.870209\pi$
$283$	−8740.97	−1.83603	−0.918015	+	0.396545i	$0.870209\pi$
$284$	5367.53	1.12149
$285$	1271.03	0.264173
$286$	10373.2	2.14469
$287$	2568.56	0.528283
$288$	−1303.59	−0.266717
$289$	−4855.65	−0.988327
$290$	−16399.0	−3.32062
$291$	3222.71	0.649204
$292$	12155.6	2.43614
$293$	−9100.49	−1.81453	−0.907263	−	0.420563i	$-0.861832\pi$
$293$	−9100.49	−1.81453	−0.907263	+	0.420563i	$0.861832\pi$
$294$	6029.18	1.19602
$295$	7570.66	1.49417
$296$	13373.2	2.62601
$297$	1271.22	0.248363
$298$	−4043.96	−0.786108
$299$	883.379	0.170860
$300$	−1125.84	−0.216668
$301$	13730.8	2.62934
$302$	−14987.9	−2.85582
$303$	−4451.08	−0.843921
$304$	−3699.10	−0.697888
$305$	−54.2210	−0.0101793
$306$	−345.002	−0.0644525
$307$	−1870.69	−0.347772	−0.173886	−	0.984766i	$-0.555632\pi$
$307$	−1870.69	−0.347772	−0.173886	+	0.984766i	$0.555632\pi$
$308$	22572.3	4.17590
$309$	467.709	0.0861068
$310$	4147.86	0.759945
$311$	2353.14	0.429050	0.214525	−	0.976719i	$-0.431180\pi$
$311$	2353.14	0.429050	0.214525	+	0.976719i	$0.431180\pi$
$312$	6360.83	1.15420
$313$	618.044	0.111610	0.0558049	−	0.998442i	$-0.482228\pi$
$313$	618.044	0.111610	0.0558049	+	0.998442i	$0.482228\pi$
$314$	794.730	0.142832
$315$	−2961.28	−0.529679
$316$	20718.7	3.68835
$317$	6660.54	1.18010	0.590052	−	0.807365i	$-0.299107\pi$
$317$	6660.54	1.18010	0.590052	+	0.807365i	$0.299107\pi$
$318$	−1251.62	−0.220715
$319$	−12610.8	−2.21337
$320$	−1350.19	−0.235869
$321$	−3255.86	−0.566120
$322$	2794.83	0.483696
$323$	−265.265	−0.0456957
$324$	1427.51	0.244772
$325$	926.818	0.158187
$326$	12335.1	2.09563
$327$	−1916.27	−0.324067
$328$	4599.65	0.774308
$329$	−1904.62	−0.319164
$330$	−8647.95	−1.44259
$331$	−10157.4	−1.68671	−0.843357	−	0.537354i	$-0.819424\pi$
$331$	−10157.4	−1.68671	−0.843357	+	0.537354i	$0.819424\pi$
$332$	−6684.53	−1.10500
$333$	−2470.69	−0.406585
$334$	−8474.77	−1.38838
$335$	−2342.74	−0.382083
$336$	8618.26	1.39930
$337$	−11697.0	−1.89073	−0.945365	−	0.326014i	$-0.894294\pi$
$337$	−11697.0	−1.89073	−0.945365	+	0.326014i	$0.894294\pi$
$338$	1531.82	0.246509
$339$	2961.22	0.474429
$340$	1614.24	0.257483
$341$	3189.69	0.506544
$342$	1595.82	0.252315
$343$	−1469.63	−0.231349
$344$	24588.5	3.85385
$345$	−736.458	−0.114926
$346$	−16244.1	−2.52395
$347$	9673.03	1.49647	0.748235	−	0.663433i	$-0.230901\pi$
$347$	9673.03	1.49647	0.748235	+	0.663433i	$0.230901\pi$
$348$	−14161.2	−2.18137
$349$	769.768	0.118065	0.0590326	−	0.998256i	$-0.481198\pi$
$349$	769.768	0.118065	0.0590326	+	0.998256i	$0.481198\pi$
$350$	2932.27	0.447818
$351$	−1175.16	−0.178705
$352$	6819.55	1.03262
$353$	−5333.51	−0.804176	−0.402088	−	0.915601i	$-0.631715\pi$
$353$	−5333.51	−0.804176	−0.402088	+	0.915601i	$0.631715\pi$
$354$	9505.22	1.42711
$355$	3683.77	0.550745
$356$	4249.12	0.632592
$357$	618.021	0.0916223
$358$	−14398.9	−2.12571
$359$	8332.11	1.22494	0.612468	−	0.790496i	$-0.290177\pi$
$359$	8332.11	1.22494	0.612468	+	0.790496i	$0.290177\pi$
$360$	−5302.91	−0.776355
$361$	−5632.01	−0.821113
$362$	−22335.6	−3.24291
$363$	−2657.24	−0.384212
$364$	−20866.6	−3.00469
$365$	8342.47	1.19634
$366$	−68.0763	−0.00972242
$367$	7617.78	1.08350	0.541750	−	0.840539i	$-0.317762\pi$
$367$	7617.78	1.08350	0.541750	+	0.840539i	$0.317762\pi$
$368$	2143.33	0.303611
$369$	−849.785	−0.119886
$370$	16807.7	2.36160
$371$	2242.09	0.313757
$372$	3581.85	0.499221
$373$	4582.21	0.636079	0.318040	−	0.948077i	$-0.396975\pi$
$373$	4582.21	0.636079	0.318040	+	0.948077i	$0.396975\pi$
$374$	1804.84	0.249534
$375$	3763.03	0.518192
$376$	−3410.69	−0.467801
$377$	11657.8	1.59259
$378$	−3717.98	−0.505905
$379$	232.292	0.0314830	0.0157415	−	0.999876i	$-0.494989\pi$
$379$	232.292	0.0314830	0.0157415	+	0.999876i	$0.494989\pi$
$380$	−7466.70	−1.00798
$381$	−1253.45	−0.168547
$382$	23378.1	3.13123
$383$	−1507.22	−0.201085	−0.100542	−	0.994933i	$-0.532058\pi$
$383$	−1507.22	−0.201085	−0.100542	+	0.994933i	$0.532058\pi$
$384$	−5171.45	−0.687251
$385$	15491.5	2.05071
$386$	−17948.6	−2.36674
$387$	−4542.73	−0.596692
$388$	−18931.9	−2.47712
$389$	7089.30	0.924015	0.462007	−	0.886876i	$-0.347129\pi$
$389$	7089.30	0.924015	0.462007	+	0.886876i	$0.347129\pi$
$390$	7994.46	1.03799
$391$	153.699	0.0198796
$392$	−19340.8	−2.49199
$393$	84.8241	0.0108876
$394$	−3225.10	−0.412381
$395$	14219.4	1.81128
$396$	−7467.85	−0.947661
$397$	−8808.46	−1.11356	−0.556781	−	0.830659i	$-0.687964\pi$
$397$	−8808.46	−1.11356	−0.556781	+	0.830659i	$0.687964\pi$
$398$	−20537.0	−2.58649
$399$	−2858.67	−0.358678
$400$	2248.72	0.281090
$401$	2631.07	0.327654	0.163827	−	0.986489i	$-0.447616\pi$
$401$	2631.07	0.327654	0.163827	+	0.986489i	$0.447616\pi$
$402$	−2941.39	−0.364933
$403$	−2948.66	−0.364475
$404$	26148.1	3.22008
$405$	979.712	0.120203
$406$	36883.0	4.50855
$407$	12925.1	1.57414
$408$	1106.72	0.134292
$409$	−4697.18	−0.567874	−0.283937	−	0.958843i	$-0.591641\pi$
$409$	−4697.18	−0.567874	−0.283937	+	0.958843i	$0.591641\pi$
$410$	5780.96	0.696345
$411$	−6069.95	−0.728488
$412$	−2747.57	−0.328551
$413$	−17027.2	−2.02870
$414$	−924.646	−0.109768
$415$	−4587.64	−0.542647
$416$	−6304.23	−0.743006
$417$	−2480.30	−0.291273
$418$	−8348.31	−0.976864
$419$	14138.7	1.64849	0.824247	−	0.566231i	$-0.191599\pi$
$419$	14138.7	1.64849	0.824247	+	0.566231i	$0.191599\pi$
$420$	17396.1	2.02106
$421$	−2715.13	−0.314317	−0.157158	−	0.987573i	$-0.550233\pi$
$421$	−2715.13	−0.314317	−0.157158	+	0.987573i	$0.550233\pi$
$422$	−26365.1	−3.04132
$423$	630.125	0.0724297
$424$	4015.03	0.459876
$425$	161.257	0.0184050
$426$	4625.10	0.526025
$427$	121.949	0.0138209
$428$	19126.7	2.16010
$429$	6147.71	0.691875
$430$	30903.5	3.46581
$431$	−16365.0	−1.82894	−0.914469	−	0.404655i	$-0.867392\pi$
$431$	−16365.0	−1.82894	−0.914469	+	0.404655i	$0.867392\pi$
$432$	−2851.28	−0.317551
$433$	2897.62	0.321595	0.160797	−	0.986987i	$-0.448593\pi$
$433$	2897.62	0.321595	0.160797	+	0.986987i	$0.448593\pi$
$434$	−9328.97	−1.03181
$435$	−9718.91	−1.07123
$436$	11257.2	1.23652
$437$	−710.940	−0.0778236
$438$	10474.2	1.14265
$439$	8247.32	0.896636	0.448318	−	0.893874i	$-0.352023\pi$
$439$	8247.32	0.896636	0.448318	+	0.893874i	$0.352023\pi$
$440$	27741.5	3.00574
$441$	3573.21	0.385835
$442$	−1668.45	−0.179548
$443$	11850.4	1.27095	0.635473	−	0.772123i	$-0.280805\pi$
$443$	11850.4	1.27095	0.635473	+	0.772123i	$0.280805\pi$
$444$	14514.2	1.55138
$445$	2916.20	0.310654
$446$	−28001.6	−2.97290
$447$	−2396.67	−0.253598
$448$	3036.72	0.320249
$449$	−4340.76	−0.456244	−0.228122	−	0.973633i	$-0.573258\pi$
$449$	−4340.76	−0.456244	−0.228122	+	0.973633i	$0.573258\pi$
$450$	−970.114	−0.101626
$451$	4445.54	0.464151
$452$	−17395.8	−1.81024
$453$	−8882.63	−0.921286
$454$	8386.83	0.866990
$455$	−14320.9	−1.47555
$456$	−5119.17	−0.525718
$457$	15809.5	1.61824	0.809120	−	0.587644i	$-0.199944\pi$
$457$	15809.5	1.61824	0.809120	+	0.587644i	$0.199944\pi$
$458$	−24662.5	−2.51616
$459$	−204.467	−0.0207924
$460$	4326.35	0.438515
$461$	1422.45	0.143710	0.0718548	−	0.997415i	$-0.477108\pi$
$461$	1422.45	0.143710	0.0718548	+	0.997415i	$0.477108\pi$
$462$	19450.1	1.95866
$463$	−2130.56	−0.213856	−0.106928	−	0.994267i	$-0.534101\pi$
$463$	−2130.56	−0.213856	−0.106928	+	0.994267i	$0.534101\pi$
$464$	28285.1	2.82997
$465$	2458.25	0.245158
$466$	15379.2	1.52881
$467$	−5193.72	−0.514639	−0.257320	−	0.966326i	$-0.582839\pi$
$467$	−5193.72	−0.514639	−0.257320	+	0.966326i	$0.582839\pi$
$468$	6903.54	0.681872
$469$	5269.07	0.518770
$470$	−4286.65	−0.420699
$471$	471.000	0.0460776
$472$	−30491.5	−2.97349
$473$	23764.7	2.31015
$474$	17852.9	1.72998
$475$	−745.900	−0.0720510
$476$	−3630.59	−0.349596
$477$	−741.777	−0.0712026
$478$	−3952.56	−0.378213
$479$	−1655.49	−0.157915	−0.0789574	−	0.996878i	$-0.525159\pi$
$479$	−1655.49	−0.157915	−0.0789574	+	0.996878i	$0.525159\pi$
$480$	5255.72	0.499770
$481$	−11948.4	−1.13264
$482$	23988.7	2.26692
$483$	1656.37	0.156040
$484$	15610.1	1.46601
$485$	−12993.1	−1.21647
$486$	1230.06	0.114808
$487$	2042.84	0.190082	0.0950409	−	0.995473i	$-0.469702\pi$
$487$	2042.84	0.190082	0.0950409	+	0.995473i	$0.469702\pi$
$488$	218.380	0.0202574
$489$	7310.42	0.676050
$490$	−24308.1	−2.24107
$491$	2515.50	0.231208	0.115604	−	0.993295i	$-0.463120\pi$
$491$	2515.50	0.231208	0.115604	+	0.993295i	$0.463120\pi$
$492$	4992.09	0.457441
$493$	2028.35	0.185298
$494$	7717.47	0.702885
$495$	−5125.24	−0.465378
$496$	−7154.29	−0.647655
$497$	−8285.18	−0.747769
$498$	−5759.93	−0.518290
$499$	−15509.8	−1.39141	−0.695706	−	0.718326i	$-0.744909\pi$
$499$	−15509.8	−1.39141	−0.695706	+	0.718326i	$0.744909\pi$
$500$	−22106.1	−1.97723
$501$	−5022.60	−0.447891
$502$	−24608.2	−2.18788
$503$	−18014.3	−1.59686	−0.798429	−	0.602090i	$-0.794335\pi$
$503$	−18014.3	−1.59686	−0.798429	+	0.602090i	$0.794335\pi$
$504$	11926.8	1.05409
$505$	17945.6	1.58132
$506$	4837.17	0.424977
$507$	907.840	0.0795238
$508$	7363.45	0.643111
$509$	−19226.2	−1.67424	−0.837119	−	0.547021i	$-0.815762\pi$
$509$	−19226.2	−1.67424	−0.837119	+	0.547021i	$0.815762\pi$
$510$	1390.96	0.120770
$511$	−18763.1	−1.62432
$512$	25859.3	2.23209
$513$	945.767	0.0813969
$514$	−7097.11	−0.609027
$515$	−1885.68	−0.161345
$516$	26686.4	2.27675
$517$	−3296.42	−0.280419
$518$	−37802.3	−3.20645
$519$	−9627.13	−0.814227
$520$	−25645.2	−2.16272
$521$	−21331.6	−1.79377	−0.896886	−	0.442262i	$-0.854176\pi$
$521$	−21331.6	−1.79377	−0.896886	+	0.442262i	$0.854176\pi$
$522$	−12202.4	−1.02315
$523$	−3293.46	−0.275359	−0.137680	−	0.990477i	$-0.543964\pi$
$523$	−3293.46	−0.275359	−0.137680	+	0.990477i	$0.543964\pi$
$524$	−498.303	−0.0415428
$525$	1737.82	0.144466
$526$	−24183.7	−2.00467
$527$	−513.039	−0.0424067
$528$	14916.1	1.22943
$529$	−11755.1	−0.966143
$530$	5046.20	0.413572
$531$	5633.30	0.460385
$532$	16793.4	1.36858
$533$	−4109.61	−0.333972
$534$	3661.38	0.296711
$535$	13126.8	1.06079
$536$	9435.59	0.760365
$537$	−8533.55	−0.685754
$538$	30476.8	2.44229
$539$	−18692.8	−1.49380
$540$	−5755.36	−0.458650
$541$	−16756.4	−1.33163	−0.665816	−	0.746116i	$-0.731916\pi$
$541$	−16756.4	−1.33163	−0.665816	+	0.746116i	$0.731916\pi$
$542$	8637.46	0.684521
$543$	−13237.3	−1.04616
$544$	−1096.87	−0.0864487
$545$	7725.89	0.607231
$546$	−17980.4	−1.40932
$547$	9236.33	0.721969	0.360984	−	0.932572i	$-0.382441\pi$
$547$	9236.33	0.721969	0.360984	+	0.932572i	$0.382441\pi$
$548$	35658.2	2.77964
$549$	−40.3457	−0.00313645
$550$	5075.03	0.393455
$551$	−9382.17	−0.725397
$552$	2966.15	0.228709
$553$	−31980.9	−2.45925
$554$	−36353.7	−2.78794
$555$	9961.17	0.761853
$556$	14570.6	1.11139
$557$	12213.2	0.929070	0.464535	−	0.885555i	$-0.346221\pi$
$557$	12213.2	0.929070	0.464535	+	0.885555i	$0.346221\pi$
$558$	3086.41	0.234154
$559$	−21968.9	−1.66223
$560$	−34746.6	−2.62198
$561$	1069.64	0.0804997
$562$	12377.6	0.929032
$563$	−9077.70	−0.679537	−0.339768	−	0.940509i	$-0.610349\pi$
$563$	−9077.70	−0.679537	−0.339768	+	0.940509i	$0.610349\pi$
$564$	−3701.70	−0.276364
$565$	−11938.9	−0.888977
$566$	44246.6	3.28591
$567$	−2203.47	−0.163205
$568$	−14836.7	−1.09601
$569$	2559.57	0.188581	0.0942905	−	0.995545i	$-0.469942\pi$
$569$	2559.57	0.188581	0.0942905	+	0.995545i	$0.469942\pi$
$570$	−6433.91	−0.472784
$571$	20896.0	1.53147	0.765736	−	0.643156i	$-0.222375\pi$
$571$	20896.0	1.53147	0.765736	+	0.643156i	$0.222375\pi$
$572$	−36115.0	−2.63994
$573$	13855.1	1.01013
$574$	−13002.0	−0.945456
$575$	432.189	0.0313452
$576$	−1004.67	−0.0726760
$577$	2157.15	0.155639	0.0778193	−	0.996967i	$-0.475204\pi$
$577$	2157.15	0.155639	0.0778193	+	0.996967i	$0.475204\pi$
$578$	24579.2	1.76879
$579$	−10637.3	−0.763509
$580$	57094.1	4.08742
$581$	10318.1	0.736774
$582$	−16313.3	−1.16187
$583$	3880.51	0.275668
$584$	−33600.0	−2.38079
$585$	4737.95	0.334855
$586$	46066.5	3.24742
$587$	−27569.8	−1.93855	−0.969273	−	0.245988i	$-0.920888\pi$
$587$	−27569.8	−1.93855	−0.969273	+	0.245988i	$0.920888\pi$
$588$	−20991.0	−1.47220
$589$	2373.07	0.166011
$590$	−38322.5	−2.67409
$591$	−1911.37	−0.133034
$592$	−28990.2	−2.01265
$593$	16862.6	1.16773	0.583864	−	0.811852i	$-0.301540\pi$
$593$	16862.6	1.16773	0.583864	+	0.811852i	$0.301540\pi$
$594$	−6434.90	−0.444490
$595$	−2491.70	−0.171680
$596$	14079.3	0.967636
$597$	−12171.3	−0.834403
$598$	−4471.65	−0.305785
$599$	−898.713	−0.0613028	−0.0306514	−	0.999530i	$-0.509758\pi$
$599$	−898.713	−0.0613028	−0.0306514	+	0.999530i	$0.509758\pi$
$600$	3112.00	0.211745
$601$	−9194.25	−0.624029	−0.312015	−	0.950077i	$-0.601004\pi$
$601$	−9194.25	−0.624029	−0.312015	+	0.950077i	$0.601004\pi$
$602$	−69505.2	−4.70568
$603$	−1743.22	−0.117727
$604$	52181.4	3.51528
$605$	10713.3	0.719930
$606$	22531.3	1.51035
$607$	28384.5	1.89801	0.949003	−	0.315267i	$-0.102094\pi$
$607$	28384.5	1.89801	0.949003	+	0.315267i	$0.102094\pi$
$608$	5073.62	0.338425
$609$	21858.8	1.45446
$610$	274.466	0.0182177
$611$	3047.33	0.201770
$612$	1201.15	0.0793359
$613$	−9944.70	−0.655241	−0.327620	−	0.944809i	$-0.606247\pi$
$613$	−9944.70	−0.655241	−0.327620	+	0.944809i	$0.606247\pi$
$614$	9469.39	0.622400
$615$	3426.11	0.224641
$616$	−62393.5	−4.08102
$617$	−4173.07	−0.272288	−0.136144	−	0.990689i	$-0.543471\pi$
$617$	−4173.07	−0.272288	−0.136144	+	0.990689i	$0.543471\pi$
$618$	−2367.53	−0.154104
$619$	3.96796	0.000257651 0	0.000128825	−	1.00000i	$-0.499959\pi$
$619$	3.96796	0.000257651 0	0.000128825		1.00000i	$0.499959\pi$
$620$	−14441.1	−0.935431
$621$	−547.995	−0.0354111
$622$	−11911.6	−0.767862
$623$	−6558.84	−0.421788
$624$	−13788.9	−0.884615
$625$	−17833.3	−1.14133
$626$	−3128.52	−0.199746
$627$	−4947.66	−0.315136
$628$	−2766.91	−0.175815
$629$	−2078.91	−0.131783
$630$	14989.9	0.947956
$631$	−9219.48	−0.581651	−0.290825	−	0.956776i	$-0.593930\pi$
$631$	−9219.48	−0.581651	−0.290825	+	0.956776i	$0.593930\pi$
$632$	−57269.9	−3.60455
$633$	−15625.4	−0.981128
$634$	−33715.5	−2.11201
$635$	5053.59	0.315820
$636$	4357.60	0.271682
$637$	17280.3	1.07483
$638$	63835.3	3.96123
$639$	2741.08	0.169696
$640$	20849.9	1.28776
$641$	15784.1	0.972599	0.486300	−	0.873792i	$-0.338346\pi$
$641$	15784.1	0.972599	0.486300	+	0.873792i	$0.338346\pi$
$642$	16481.1	1.01317
$643$	7415.17	0.454784	0.227392	−	0.973803i	$-0.426980\pi$
$643$	7415.17	0.454784	0.227392	+	0.973803i	$0.426980\pi$
$644$	−9730.40	−0.595391
$645$	18315.1	1.11807
$646$	1342.76	0.0817807
$647$	14220.2	0.864069	0.432035	−	0.901857i	$-0.357796\pi$
$647$	14220.2	0.864069	0.432035	+	0.901857i	$0.357796\pi$
$648$	−3945.88	−0.239211
$649$	−29469.9	−1.78243
$650$	−4691.53	−0.283103
$651$	−5528.85	−0.332861
$652$	−42945.3	−2.57955
$653$	9663.06	0.579088	0.289544	−	0.957165i	$-0.406496\pi$
$653$	9663.06	0.579088	0.289544	+	0.957165i	$0.406496\pi$
$654$	9700.11	0.579976
$655$	−341.989	−0.0204009
$656$	−9971.07	−0.593453
$657$	6207.60	0.368617
$658$	9641.12	0.571201
$659$	−20631.9	−1.21958	−0.609790	−	0.792563i	$-0.708746\pi$
$659$	−20631.9	−1.21958	−0.609790	+	0.792563i	$0.708746\pi$
$660$	30108.4	1.77571
$661$	27062.0	1.59242	0.796210	−	0.605021i	$-0.206835\pi$
$661$	27062.0	1.59242	0.796210	+	0.605021i	$0.206835\pi$
$662$	51416.6	3.01868
$663$	−988.815	−0.0579221
$664$	18477.1	1.07990
$665$	11525.4	0.672085
$666$	12506.6	0.727658
$667$	5436.21	0.315578
$668$	29505.5	1.70898
$669$	−16595.3	−0.959058
$670$	11858.9	0.683805
$671$	211.063	0.0121431
$672$	−11820.7	−0.678559
$673$	318.047	0.0182167	0.00910833	−	0.999959i	$-0.497101\pi$
$673$	318.047	0.0182167	0.00910833	+	0.999959i	$0.497101\pi$
$674$	59209.9	3.38380
$675$	−574.942	−0.0327845
$676$	−5333.14	−0.303433
$677$	−8209.97	−0.466078	−0.233039	−	0.972467i	$-0.574867\pi$
$677$	−8209.97	−0.466078	−0.233039	+	0.972467i	$0.574867\pi$
$678$	−14989.6	−0.849076
$679$	29222.8	1.65165
$680$	−4462.02	−0.251633
$681$	4970.49	0.279691
$682$	−16146.2	−0.906551
$683$	29755.6	1.66701	0.833505	−	0.552512i	$-0.186331\pi$
$683$	29755.6	1.66701	0.833505	+	0.552512i	$0.186331\pi$
$684$	−5555.94	−0.310580
$685$	24472.5	1.36503
$686$	7439.24	0.414040
$687$	−14616.3	−0.811714
$688$	−53302.8	−2.95371
$689$	−3587.28	−0.198352
$690$	3727.93	0.205681
$691$	−5991.12	−0.329831	−0.164915	−	0.986308i	$-0.552735\pi$
$691$	−5991.12	−0.329831	−0.164915	+	0.986308i	$0.552735\pi$
$692$	56554.9	3.10678
$693$	11527.2	0.631864
$694$	−48964.7	−2.67820
$695$	9999.91	0.545782
$696$	39143.8	2.13181
$697$	−715.032	−0.0388576
$698$	−3896.55	−0.211299
$699$	9114.53	0.493195
$700$	−10208.9	−0.551228
$701$	16492.2	0.888591	0.444295	−	0.895880i	$-0.353454\pi$
$701$	16492.2	0.888591	0.444295	+	0.895880i	$0.353454\pi$
$702$	5948.65	0.319825
$703$	9616.03	0.515897
$704$	5255.82	0.281372
$705$	−2540.50	−0.135717
$706$	26998.1	1.43922
$707$	−40361.5	−2.14703
$708$	−33093.0	−1.75666
$709$	−18930.5	−1.00275	−0.501376	−	0.865229i	$-0.667173\pi$
$709$	−18930.5	−1.00275	−0.501376	+	0.865229i	$0.667173\pi$
$710$	−18647.2	−0.985656
$711$	10580.6	0.558092
$712$	−11745.2	−0.618219
$713$	−1375.00	−0.0722220
$714$	−3128.41	−0.163974
$715$	−24786.0	−1.29642
$716$	50130.7	2.61658
$717$	−2342.50	−0.122012
$718$	−42176.9	−2.19224
$719$	20252.0	1.05045	0.525225	−	0.850964i	$-0.323981\pi$
$719$	20252.0	1.05045	0.525225	+	0.850964i	$0.323981\pi$
$720$	11495.6	0.595022
$721$	4241.08	0.219066
$722$	28509.1	1.46953
$723$	14217.0	0.731308
$724$	77763.0	3.99176
$725$	5703.52	0.292170
$726$	13450.9	0.687617
$727$	−38281.5	−1.95293	−0.976467	−	0.215666i	$-0.930808\pi$
$727$	−38281.5	−1.95293	−0.976467	+	0.215666i	$0.930808\pi$
$728$	57678.7	2.93642
$729$	729.000	0.0370370
$730$	−42229.4	−2.14107
$731$	−3822.38	−0.193400
$732$	237.012	0.0119675
$733$	23137.7	1.16591	0.582954	−	0.812505i	$-0.301897\pi$
$733$	23137.7	1.16591	0.582954	+	0.812505i	$0.301897\pi$
$734$	−38561.0	−1.93912
$735$	−14406.3	−0.722970
$736$	−2939.75	−0.147229
$737$	9119.46	0.455793
$738$	4301.59	0.214558
$739$	16277.9	0.810272	0.405136	−	0.914256i	$-0.367224\pi$
$739$	16277.9	0.810272	0.405136	+	0.914256i	$0.367224\pi$
$740$	−58517.3	−2.90694
$741$	4573.79	0.226751
$742$	−11349.4	−0.561524
$743$	31459.9	1.55336	0.776682	−	0.629892i	$-0.216901\pi$
$743$	31459.9	1.55336	0.776682	+	0.629892i	$0.216901\pi$
$744$	−9900.80	−0.487878
$745$	9662.74	0.475188
$746$	−23195.0	−1.13838
$747$	−3413.64	−0.167200
$748$	−6283.65	−0.307157
$749$	−29523.5	−1.44027
$750$	−19048.4	−0.927398
$751$	15534.2	0.754795	0.377397	−	0.926051i	$-0.376819\pi$
$751$	15534.2	0.754795	0.377397	+	0.926051i	$0.376819\pi$
$752$	7393.67	0.358536
$753$	−14584.1	−0.705811
$754$	−59011.6	−2.85023
$755$	35812.4	1.72629
$756$	12944.4	0.622729
$757$	−2197.01	−0.105484	−0.0527422	−	0.998608i	$-0.516796\pi$
$757$	−2197.01	−0.105484	−0.0527422	+	0.998608i	$0.516796\pi$
$758$	−1175.86	−0.0563444
$759$	2866.77	0.137098
$760$	20639.2	0.985080
$761$	−3124.79	−0.148848	−0.0744241	−	0.997227i	$-0.523712\pi$
$761$	−3124.79	−0.148848	−0.0744241	+	0.997227i	$0.523712\pi$
$762$	6344.95	0.301645
$763$	−17376.3	−0.824463
$764$	−81392.5	−3.85429
$765$	824.357	0.0389604
$766$	7629.52	0.359877
$767$	27243.0	1.28251
$768$	23498.6	1.10408
$769$	−7835.53	−0.367434	−0.183717	−	0.982979i	$-0.558813\pi$
$769$	−7835.53	−0.367434	−0.183717	+	0.982979i	$0.558813\pi$
$770$	−78417.8	−3.67011
$771$	−4206.13	−0.196472
$772$	62489.3	2.91326
$773$	−8272.60	−0.384922	−0.192461	−	0.981305i	$-0.561647\pi$
$773$	−8272.60	−0.384922	−0.192461	+	0.981305i	$0.561647\pi$
$774$	22995.2	1.06789
$775$	−1442.62	−0.0668649
$776$	52330.8	2.42083
$777$	−22403.7	−1.03440
$778$	−35885.9	−1.65369
$779$	3307.40	0.152118
$780$	−27833.3	−1.27768
$781$	−14339.6	−0.656993
$782$	−778.023	−0.0355781
$783$	−7231.80	−0.330068
$784$	41926.8	1.90993
$785$	−1898.95	−0.0863393
$786$	−429.378	−0.0194852
$787$	19086.9	0.864517	0.432258	−	0.901750i	$-0.357717\pi$
$787$	19086.9	0.864517	0.432258	+	0.901750i	$0.357717\pi$
$788$	11228.4	0.507608
$789$	−14332.5	−0.646708
$790$	−71978.3	−3.24161
$791$	26851.7	1.20700
$792$	20642.3	0.926128
$793$	−195.114	−0.00873733
$794$	44588.2	1.99292
$795$	2990.65	0.133418
$796$	71500.8	3.18377
$797$	17852.0	0.793415	0.396708	−	0.917945i	$-0.370153\pi$
$797$	17852.0	0.793415	0.396708	+	0.917945i	$0.370153\pi$
$798$	14470.5	0.641919
$799$	530.205	0.0234760
$800$	−3084.31	−0.136309
$801$	2169.93	0.0957189
$802$	−13318.4	−0.586396
$803$	−32474.3	−1.42714
$804$	10240.6	0.449203
$805$	−6678.04	−0.292385
$806$	14926.1	0.652293
$807$	18062.2	0.787881
$808$	−72277.4	−3.14692
$809$	15254.1	0.662925	0.331462	−	0.943468i	$-0.392458\pi$
$809$	15254.1	0.662925	0.331462	+	0.943468i	$0.392458\pi$
$810$	−4959.28	−0.215125
$811$	−20058.3	−0.868485	−0.434242	−	0.900796i	$-0.642984\pi$
$811$	−20058.3	−0.868485	−0.434242	+	0.900796i	$0.642984\pi$
$812$	−128411.	−5.54966
$813$	5119.02	0.220826
$814$	−65426.5	−2.81720
$815$	−29473.7	−1.26677
$816$	−2399.14	−0.102925
$817$	17680.5	0.757114
$818$	23777.0	1.01631
$819$	−10656.1	−0.454646
$820$	−20126.8	−0.857144
$821$	25929.2	1.10224	0.551118	−	0.834427i	$-0.314201\pi$
$821$	25929.2	1.10224	0.551118	+	0.834427i	$0.314201\pi$
$822$	30726.0	1.30376
$823$	2589.38	0.109672	0.0548360	−	0.998495i	$-0.482536\pi$
$823$	2589.38	0.109672	0.0548360	+	0.998495i	$0.482536\pi$
$824$	7594.73	0.321086
$825$	3007.74	0.126928
$826$	86191.3	3.63073
$827$	3128.50	0.131546	0.0657731	−	0.997835i	$-0.479049\pi$
$827$	3128.50	0.131546	0.0657731	+	0.997835i	$0.479049\pi$
$828$	3219.22	0.135115
$829$	27411.7	1.14843	0.574214	−	0.818705i	$-0.305308\pi$
$829$	27411.7	1.14843	0.574214	+	0.818705i	$0.305308\pi$
$830$	23222.5	0.971163
$831$	−21545.1	−0.899389
$832$	−4858.66	−0.202457
$833$	3006.60	0.125057
$834$	12555.2	0.521285
$835$	20249.8	0.839250
$836$	29065.2	1.20244
$837$	1829.17	0.0755381
$838$	−71569.6	−2.95027
$839$	−1888.56	−0.0777118	−0.0388559	−	0.999245i	$-0.512371\pi$
$839$	−1888.56	−0.0777118	−0.0388559	+	0.999245i	$0.512371\pi$
$840$	−48085.7	−1.97514
$841$	47351.7	1.94152
$842$	13743.9	0.562526
$843$	7335.61	0.299706
$844$	91792.0	3.74362
$845$	−3660.17	−0.149010
$846$	−3189.68	−0.129626
$847$	−24095.3	−0.977479
$848$	−8703.75	−0.352462
$849$	26222.9	1.06003
$850$	−816.281	−0.0329391
$851$	−5571.71	−0.224437
$852$	−16102.6	−0.647495
$853$	11581.8	0.464891	0.232446	−	0.972609i	$-0.425327\pi$
$853$	11581.8	0.464891	0.232446	+	0.972609i	$0.425327\pi$
$854$	−617.302	−0.0247349
$855$	−3813.08	−0.152520
$856$	−52869.2	−2.11102
$857$	−40152.9	−1.60046	−0.800231	−	0.599692i	$-0.795290\pi$
$857$	−40152.9	−1.60046	−0.800231	+	0.599692i	$0.795290\pi$
$858$	−31119.6	−1.23823
$859$	−20485.1	−0.813671	−0.406835	−	0.913502i	$-0.633368\pi$
$859$	−20485.1	−0.813671	−0.406835	+	0.913502i	$0.633368\pi$
$860$	−107593.	−4.26614
$861$	−7705.67	−0.305004
$862$	82839.1	3.27321
$863$	31348.5	1.23652	0.618260	−	0.785974i	$-0.287838\pi$
$863$	31348.5	1.23652	0.618260	+	0.785974i	$0.287838\pi$
$864$	3910.76	0.153989
$865$	38814.0	1.52568
$866$	−14667.7	−0.575552
$867$	14567.0	0.570611
$868$	32479.4	1.27007
$869$	−55351.1	−2.16071
$870$	49196.9	1.91716
$871$	−8430.34	−0.327958
$872$	−31116.7	−1.20842
$873$	−9668.12	−0.374818
$874$	3598.76	0.139279
$875$	34122.4	1.31834
$876$	−36466.8	−1.40650
$877$	3203.78	0.123357	0.0616783	−	0.998096i	$-0.480355\pi$
$877$	3203.78	0.123357	0.0616783	+	0.998096i	$0.480355\pi$
$878$	−41747.8	−1.60469
$879$	27301.5	1.04762
$880$	−60137.7	−2.30369
$881$	22391.5	0.856289	0.428144	−	0.903710i	$-0.359168\pi$
$881$	22391.5	0.856289	0.428144	+	0.903710i	$0.359168\pi$
$882$	−18087.5	−0.690520
$883$	27919.7	1.06407	0.532034	−	0.846723i	$-0.321428\pi$
$883$	27919.7	1.06407	0.532034	+	0.846723i	$0.321428\pi$
$884$	5808.83	0.221009
$885$	−22712.0	−0.862661
$886$	−59986.4	−2.27458
$887$	−12452.9	−0.471393	−0.235697	−	0.971827i	$-0.575737\pi$
$887$	−12452.9	−0.471393	−0.235697	+	0.971827i	$0.575737\pi$
$888$	−40119.5	−1.51613
$889$	−11366.0	−0.428802
$890$	−14761.7	−0.555972
$891$	−3813.67	−0.143393
$892$	97489.4	3.65940
$893$	−2452.48	−0.0919025
$894$	12131.9	0.453860
$895$	34405.0	1.28495
$896$	−46893.6	−1.74844
$897$	−2650.14	−0.0986461
$898$	21972.9	0.816530
$899$	−18145.7	−0.673184
$900$	3377.52	0.125093
$901$	−624.152	−0.0230783
$902$	−22503.2	−0.830682
$903$	−41192.5	−1.51805
$904$	48084.8	1.76911
$905$	53369.3	1.96028
$906$	44963.7	1.64881
$907$	−37831.9	−1.38499	−0.692496	−	0.721422i	$-0.743489\pi$
$907$	−37831.9	−1.38499	−0.692496	+	0.721422i	$0.743489\pi$
$908$	−29199.3	−1.06720
$909$	13353.2	0.487238
$910$	72492.1	2.64076
$911$	−12524.4	−0.455492	−0.227746	−	0.973721i	$-0.573136\pi$
$911$	−12524.4	−0.455492	−0.227746	+	0.973721i	$0.573136\pi$
$912$	11097.3	0.402926
$913$	17858.0	0.647333
$914$	−80027.1	−2.89613
$915$	162.663	0.00587702
$916$	85864.1	3.09719
$917$	769.168	0.0276992
$918$	1035.01	0.0372117
$919$	−21958.3	−0.788180	−0.394090	−	0.919072i	$-0.628940\pi$
$919$	−21958.3	−0.788180	−0.394090	+	0.919072i	$0.628940\pi$
$920$	−11958.7	−0.428552
$921$	5612.07	0.200786
$922$	−7200.41	−0.257194
$923$	13256.0	0.472728
$924$	−67716.9	−2.41096
$925$	−5845.69	−0.207789
$926$	10784.8	0.382734
$927$	−1403.13	−0.0497138
$928$	−38795.4	−1.37233
$929$	−43053.5	−1.52049	−0.760247	−	0.649634i	$-0.774922\pi$
$929$	−43053.5	−1.52049	−0.760247	+	0.649634i	$0.774922\pi$
$930$	−12443.6	−0.438754
$931$	−13907.1	−0.489567
$932$	−53543.6	−1.88185
$933$	−7059.43	−0.247712
$934$	26290.5	0.921039
$935$	−4312.52	−0.150839
$936$	−19082.5	−0.666379
$937$	−41662.1	−1.45255	−0.726276	−	0.687403i	$-0.758751\pi$
$937$	−41662.1	−1.45255	−0.726276	+	0.687403i	$0.758751\pi$
$938$	−26671.9	−0.928431
$939$	−1854.13	−0.0644380
$940$	14924.3	0.517847
$941$	−22379.4	−0.775288	−0.387644	−	0.921809i	$-0.626711\pi$
$941$	−22379.4	−0.775288	−0.387644	+	0.921809i	$0.626711\pi$
$942$	−2384.19	−0.0824641
$943$	−1916.37	−0.0661777
$944$	66099.2	2.27897
$945$	8883.83	0.305811
$946$	−120296.	−4.13443
$947$	−26912.9	−0.923498	−0.461749	−	0.887011i	$-0.652778\pi$
$947$	−26912.9	−0.923498	−0.461749	+	0.887011i	$0.652778\pi$
$948$	−62156.1	−2.12947
$949$	30020.3	1.02687
$950$	3775.73	0.128948
$951$	−19981.6	−0.681334
$952$	10035.5	0.341653
$953$	21705.4	0.737783	0.368892	−	0.929472i	$-0.379737\pi$
$953$	21705.4	0.737783	0.368892	+	0.929472i	$0.379737\pi$
$954$	3754.86	0.127430
$955$	−55860.3	−1.89277
$956$	13761.1	0.465550
$957$	37832.3	1.27789
$958$	8380.04	0.282617
$959$	−55041.1	−1.85336
$960$	4050.58	0.136179
$961$	−25201.3	−0.845938
$962$	60482.6	2.02706
$963$	9767.59	0.326850
$964$	−83518.3	−2.79040
$965$	42886.9	1.43065
$966$	−8384.50	−0.279262
$967$	4242.45	0.141084	0.0705418	−	0.997509i	$-0.477527\pi$
$967$	4242.45	0.141084	0.0705418	+	0.997509i	$0.477527\pi$
$968$	−43148.7	−1.43270
$969$	795.794	0.0263824
$970$	65770.8	2.17709
$971$	−3906.00	−0.129093	−0.0645466	−	0.997915i	$-0.520560\pi$
$971$	−3906.00	−0.129093	−0.0645466	+	0.997915i	$0.520560\pi$
$972$	−4282.54	−0.141319
$973$	−22490.8	−0.741031
$974$	−10340.8	−0.340185
$975$	−2780.45	−0.0913290
$976$	−473.402	−0.0155258
$977$	14894.2	0.487725	0.243863	−	0.969810i	$-0.421585\pi$
$977$	14894.2	0.487725	0.243863	+	0.969810i	$0.421585\pi$
$978$	−37005.2	−1.20991
$979$	−11351.7	−0.370585
$980$	84630.1	2.75858
$981$	5748.80	0.187100
$982$	−12733.4	−0.413788
$983$	−8781.88	−0.284943	−0.142471	−	0.989799i	$-0.545505\pi$
$983$	−8781.88	−0.284943	−0.142471	+	0.989799i	$0.545505\pi$
$984$	−13798.9	−0.447047
$985$	7706.12	0.249277
$986$	−10267.4	−0.331625
$987$	5713.85	0.184269
$988$	−26868.9	−0.865195
$989$	−10244.4	−0.329377
$990$	25943.8	0.832878
$991$	51856.7	1.66224	0.831121	−	0.556091i	$-0.187699\pi$
$991$	51856.7	1.66224	0.831121	+	0.556091i	$0.187699\pi$
$992$	9812.70	0.314066
$993$	30472.3	0.973825
$994$	41939.4	1.33827
$995$	49071.5	1.56349
$996$	20053.6	0.637974
$997$	11925.1	0.378807	0.189404	−	0.981899i	$-0.439345\pi$
$997$	11925.1	0.378807	0.189404	+	0.981899i	$0.439345\pi$
$998$	78510.4	2.49018
$999$	7412.07	0.234742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.2	✓	22
3.2	odd	2		1413.4.a.e.1.21		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.2	✓	22	1.1	even	1	trivial
1413.4.a.e.1.21		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)

\(f(q)\)	\(=\)	\(q-5.06198 q^{2} -3.00000 q^{3} +17.6236 q^{4} +12.0952 q^{5} +15.1859 q^{6} -27.2034 q^{7} -48.7145 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.06198 q^{2} -3.00000 q^{3} +17.6236 q^{4} +12.0952 q^{5} +15.1859 q^{6} -27.2034 q^{7} -48.7145 q^{8} +9.00000 q^{9} -61.2257 q^{10} -47.0823 q^{11} -52.8708 q^{12} +43.5246 q^{13} +137.703 q^{14} -36.2856 q^{15} +105.603 q^{16} +7.57285 q^{17} -45.5578 q^{18} -35.0284 q^{19} +213.161 q^{20} +81.6101 q^{21} +238.330 q^{22} +20.2961 q^{23} +146.144 q^{24} +21.2941 q^{25} -220.320 q^{26} -27.0000 q^{27} -479.422 q^{28} +267.845 q^{29} +183.677 q^{30} -67.7471 q^{31} -144.843 q^{32} +141.247 q^{33} -38.3336 q^{34} -329.031 q^{35} +158.613 q^{36} -274.521 q^{37} +177.313 q^{38} -130.574 q^{39} -589.212 q^{40} -94.4205 q^{41} -413.109 q^{42} -504.748 q^{43} -829.761 q^{44} +108.857 q^{45} -102.738 q^{46} +70.0139 q^{47} -316.809 q^{48} +397.024 q^{49} -107.790 q^{50} -22.7185 q^{51} +767.060 q^{52} -82.4197 q^{53} +136.673 q^{54} -569.471 q^{55} +1325.20 q^{56} +105.085 q^{57} -1355.82 q^{58} +625.922 q^{59} -639.484 q^{60} -4.48285 q^{61} +342.934 q^{62} -244.830 q^{63} -111.630 q^{64} +526.439 q^{65} -714.989 q^{66} -193.692 q^{67} +133.461 q^{68} -60.8884 q^{69} +1665.55 q^{70} +304.565 q^{71} -438.431 q^{72} +689.733 q^{73} +1389.62 q^{74} -63.8824 q^{75} -617.327 q^{76} +1280.80 q^{77} +660.961 q^{78} +1175.62 q^{79} +1277.29 q^{80} +81.0000 q^{81} +477.954 q^{82} -379.294 q^{83} +1438.27 q^{84} +91.5952 q^{85} +2555.02 q^{86} -803.534 q^{87} +2293.59 q^{88} +241.104 q^{89} -551.031 q^{90} -1184.01 q^{91} +357.691 q^{92} +203.241 q^{93} -354.409 q^{94} -423.676 q^{95} +434.529 q^{96} -1074.24 q^{97} -2009.73 q^{98} -423.741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 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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