LMFDB - Embedded newform 471.4.a.c.1.17

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			1.17
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)

$f(q)$	$=$	$q+2.99489 q^{2} -3.00000 q^{3} +0.969360 q^{4} +5.29125 q^{5} -8.98467 q^{6} +8.08775 q^{7} -21.0560 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.99489 q^{2} -3.00000 q^{3} +0.969360 q^{4} +5.29125 q^{5} -8.98467 q^{6} +8.08775 q^{7} -21.0560 q^{8} +9.00000 q^{9} +15.8467 q^{10} -0.787225 q^{11} -2.90808 q^{12} +38.7726 q^{13} +24.2219 q^{14} -15.8737 q^{15} -70.8152 q^{16} +17.4063 q^{17} +26.9540 q^{18} +24.9490 q^{19} +5.12912 q^{20} -24.2633 q^{21} -2.35765 q^{22} +105.029 q^{23} +63.1680 q^{24} -97.0027 q^{25} +116.120 q^{26} -27.0000 q^{27} +7.83994 q^{28} +142.805 q^{29} -47.5401 q^{30} +137.294 q^{31} -43.6358 q^{32} +2.36168 q^{33} +52.1301 q^{34} +42.7943 q^{35} +8.72424 q^{36} +196.604 q^{37} +74.7194 q^{38} -116.318 q^{39} -111.412 q^{40} +403.589 q^{41} -72.6657 q^{42} +432.729 q^{43} -0.763104 q^{44} +47.6212 q^{45} +314.550 q^{46} +57.3946 q^{47} +212.446 q^{48} -277.588 q^{49} -290.512 q^{50} -52.2190 q^{51} +37.5846 q^{52} +67.8239 q^{53} -80.8620 q^{54} -4.16541 q^{55} -170.296 q^{56} -74.8469 q^{57} +427.685 q^{58} -498.243 q^{59} -15.3874 q^{60} +174.652 q^{61} +411.179 q^{62} +72.7898 q^{63} +435.837 q^{64} +205.156 q^{65} +7.07296 q^{66} +591.751 q^{67} +16.8730 q^{68} -315.086 q^{69} +128.164 q^{70} -198.903 q^{71} -189.504 q^{72} -89.2801 q^{73} +588.807 q^{74} +291.008 q^{75} +24.1845 q^{76} -6.36688 q^{77} -348.359 q^{78} -155.043 q^{79} -374.701 q^{80} +81.0000 q^{81} +1208.70 q^{82} -448.958 q^{83} -23.5198 q^{84} +92.1013 q^{85} +1295.97 q^{86} -428.415 q^{87} +16.5758 q^{88} +746.372 q^{89} +142.620 q^{90} +313.583 q^{91} +101.811 q^{92} -411.881 q^{93} +171.891 q^{94} +132.011 q^{95} +130.907 q^{96} -39.8738 q^{97} -831.346 q^{98} -7.08503 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.99489	1.05885	0.529427	−	0.848356i	$-0.322407\pi$
$2$	2.99489	1.05885	0.529427	+	0.848356i	$0.322407\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0.969360	0.121170
$5$	5.29125	0.473264	0.236632	−	0.971599i	$-0.423956\pi$
$5$	5.29125	0.473264	0.236632	+	0.971599i	$0.423956\pi$
$6$	−8.98467	−0.611329
$7$	8.08775	0.436697	0.218349	−	0.975871i	$-0.429933\pi$
$7$	8.08775	0.436697	0.218349	+	0.975871i	$0.429933\pi$
$8$	−21.0560	−0.930552
$9$	9.00000	0.333333
$10$	15.8467	0.501117
$11$	−0.787225	−0.0215779	−0.0107890	−	0.999942i	$-0.503434\pi$
$11$	−0.787225	−0.0215779	−0.0107890	+	0.999942i	$0.503434\pi$
$12$	−2.90808	−0.0699575
$13$	38.7726	0.827199	0.413599	−	0.910459i	$-0.364271\pi$
$13$	38.7726	0.827199	0.413599	+	0.910459i	$0.364271\pi$
$14$	24.2219	0.462399
$15$	−15.8737	−0.273239
$16$	−70.8152	−1.10649
$17$	17.4063	0.248333	0.124166	−	0.992261i	$-0.460374\pi$
$17$	17.4063	0.248333	0.124166	+	0.992261i	$0.460374\pi$
$18$	26.9540	0.352951
$19$	24.9490	0.301247	0.150623	−	0.988591i	$-0.451872\pi$
$19$	24.9490	0.301247	0.150623	+	0.988591i	$0.451872\pi$
$20$	5.12912	0.0573453
$21$	−24.2633	−0.252127
$22$	−2.35765	−0.0228479
$23$	105.029	0.952175	0.476087	−	0.879398i	$-0.342055\pi$
$23$	105.029	0.952175	0.476087	+	0.879398i	$0.342055\pi$
$24$	63.1680	0.537254
$25$	−97.0027	−0.776021
$26$	116.120	0.875882
$27$	−27.0000	−0.192450
$28$	7.83994	0.0529146
$29$	142.805	0.914422	0.457211	−	0.889358i	$-0.348848\pi$
$29$	142.805	0.914422	0.457211	+	0.889358i	$0.348848\pi$
$30$	−47.5401	−0.289320
$31$	137.294	0.795440	0.397720	−	0.917507i	$-0.369802\pi$
$31$	137.294	0.795440	0.397720	+	0.917507i	$0.369802\pi$
$32$	−43.6358	−0.241056
$33$	2.36168	0.0124580
$34$	52.1301	0.262948
$35$	42.7943	0.206673
$36$	8.72424	0.0403900
$37$	196.604	0.873555	0.436777	−	0.899570i	$-0.356120\pi$
$37$	196.604	0.873555	0.436777	+	0.899570i	$0.356120\pi$
$38$	74.7194	0.318976
$39$	−116.318	−0.477584
$40$	−111.412	−0.440397
$41$	403.589	1.53732	0.768658	−	0.639660i	$-0.220925\pi$
$41$	403.589	1.53732	0.768658	+	0.639660i	$0.220925\pi$
$42$	−72.6657	−0.266966
$43$	432.729	1.53466	0.767331	−	0.641251i	$-0.221584\pi$
$43$	432.729	1.53466	0.767331	+	0.641251i	$0.221584\pi$
$44$	−0.763104	−0.00261460
$45$	47.6212	0.157755
$46$	314.550	1.00821
$47$	57.3946	0.178125	0.0890624	−	0.996026i	$-0.471613\pi$
$47$	57.3946	0.178125	0.0890624	+	0.996026i	$0.471613\pi$
$48$	212.446	0.638831
$49$	−277.588	−0.809295
$50$	−290.512	−0.821693
$51$	−52.2190	−0.143375
$52$	37.5846	0.100232
$53$	67.8239	0.175780	0.0878898	−	0.996130i	$-0.471988\pi$
$53$	67.8239	0.175780	0.0878898	+	0.996130i	$0.471988\pi$
$54$	−80.8620	−0.203776
$55$	−4.16541	−0.0102121
$56$	−170.296	−0.406370
$57$	−74.8469	−0.173925
$58$	427.685	0.968239
$59$	−498.243	−1.09942	−0.549710	−	0.835356i	$-0.685262\pi$
$59$	−498.243	−1.09942	−0.549710	+	0.835356i	$0.685262\pi$
$60$	−15.3874	−0.0331084
$61$	174.652	0.366589	0.183295	−	0.983058i	$-0.441324\pi$
$61$	174.652	0.366589	0.183295	+	0.983058i	$0.441324\pi$
$62$	411.179	0.842254
$63$	72.7898	0.145566
$64$	435.837	0.851245
$65$	205.156	0.391483
$66$	7.07296	0.0131912
$67$	591.751	1.07901	0.539507	−	0.841981i	$-0.318611\pi$
$67$	591.751	1.07901	0.539507	+	0.841981i	$0.318611\pi$
$68$	16.8730	0.0300905
$69$	−315.086	−0.549738
$70$	128.164	0.218836
$71$	−198.903	−0.332472	−0.166236	−	0.986086i	$-0.553161\pi$
$71$	−198.903	−0.332472	−0.166236	+	0.986086i	$0.553161\pi$
$72$	−189.504	−0.310184
$73$	−89.2801	−0.143143	−0.0715716	−	0.997435i	$-0.522801\pi$
$73$	−89.2801	−0.143143	−0.0715716	+	0.997435i	$0.522801\pi$
$74$	588.807	0.924966
$75$	291.008	0.448036
$76$	24.1845	0.0365021
$77$	−6.36688	−0.00942303
$78$	−348.359	−0.505691
$79$	−155.043	−0.220806	−0.110403	−	0.993887i	$-0.535214\pi$
$79$	−155.043	−0.220806	−0.110403	+	0.993887i	$0.535214\pi$
$80$	−374.701	−0.523661
$81$	81.0000	0.111111
$82$	1208.70	1.62779
$83$	−448.958	−0.593730	−0.296865	−	0.954920i	$-0.595941\pi$
$83$	−448.958	−0.593730	−0.296865	+	0.954920i	$0.595941\pi$
$84$	−23.5198	−0.0305503
$85$	92.1013	0.117527
$86$	1295.97	1.62498
$87$	−428.415	−0.527942
$88$	16.5758	0.0200794
$89$	746.372	0.888936	0.444468	−	0.895795i	$-0.353393\pi$
$89$	746.372	0.888936	0.444468	+	0.895795i	$0.353393\pi$
$90$	142.620	0.167039
$91$	313.583	0.361236
$92$	101.811	0.115375
$93$	−411.881	−0.459248
$94$	171.891	0.188608
$95$	132.011	0.142569
$96$	130.907	0.139174
$97$	−39.8738	−0.0417378	−0.0208689	−	0.999782i	$-0.506643\pi$
$97$	−39.8738	−0.0417378	−0.0208689	+	0.999782i	$0.506643\pi$
$98$	−831.346	−0.856925
$99$	−7.08503	−0.00719265
$100$	−94.0305	−0.0940305
$101$	920.192	0.906560	0.453280	−	0.891368i	$-0.350254\pi$
$101$	920.192	0.906560	0.453280	+	0.891368i	$0.350254\pi$
$102$	−156.390	−0.151813
$103$	−1100.70	−1.05297	−0.526483	−	0.850186i	$-0.676490\pi$
$103$	−1100.70	−1.05297	−0.526483	+	0.850186i	$0.676490\pi$
$104$	−816.396	−0.769752
$105$	−128.383	−0.119323
$106$	203.125	0.186125
$107$	−10.1137	−0.00913766	−0.00456883	−	0.999990i	$-0.501454\pi$
$107$	−10.1137	−0.00913766	−0.00456883	+	0.999990i	$0.501454\pi$
$108$	−26.1727	−0.0233192
$109$	870.732	0.765147	0.382573	−	0.923925i	$-0.375038\pi$
$109$	870.732	0.765147	0.382573	+	0.923925i	$0.375038\pi$
$110$	−12.4749	−0.0108131
$111$	−589.812	−0.504347
$112$	−572.736	−0.483200
$113$	−1086.22	−0.904277	−0.452138	−	0.891948i	$-0.649339\pi$
$113$	−1086.22	−0.904277	−0.452138	+	0.891948i	$0.649339\pi$
$114$	−224.158	−0.184161
$115$	555.734	0.450630
$116$	138.430	0.110800
$117$	348.953	0.275733
$118$	−1492.18	−1.16412
$119$	140.778	0.108446
$120$	334.237	0.254263
$121$	−1330.38	−0.999534
$122$	523.064	0.388164
$123$	−1210.77	−0.887570
$124$	133.087	0.0963835
$125$	−1174.67	−0.840527
$126$	217.997	0.154133
$127$	1516.67	1.05971	0.529854	−	0.848089i	$-0.322247\pi$
$127$	1516.67	1.05971	0.529854	+	0.848089i	$0.322247\pi$
$128$	1654.37	1.14240
$129$	−1298.19	−0.886038
$130$	614.418	0.414523
$131$	−2027.63	−1.35232	−0.676162	−	0.736753i	$-0.736358\pi$
$131$	−2027.63	−1.35232	−0.676162	+	0.736753i	$0.736358\pi$
$132$	2.28931	0.00150954
$133$	201.781	0.131554
$134$	1772.23	1.14252
$135$	−142.864	−0.0910797
$136$	−366.508	−0.231087
$137$	682.445	0.425585	0.212793	−	0.977097i	$-0.431744\pi$
$137$	682.445	0.425585	0.212793	+	0.977097i	$0.431744\pi$
$138$	−943.649	−0.582092
$139$	−2857.26	−1.74352	−0.871762	−	0.489929i	$-0.837023\pi$
$139$	−2857.26	−1.74352	−0.871762	+	0.489929i	$0.837023\pi$
$140$	41.4831	0.0250426
$141$	−172.184	−0.102840
$142$	−595.694	−0.352039
$143$	−30.5228	−0.0178492
$144$	−637.337	−0.368829
$145$	755.618	0.432763
$146$	−267.384	−0.151568
$147$	832.765	0.467247
$148$	190.580	0.105849
$149$	778.660	0.428123	0.214062	−	0.976820i	$-0.431331\pi$
$149$	778.660	0.428123	0.214062	+	0.976820i	$0.431331\pi$
$150$	871.537	0.474405
$151$	−2473.50	−1.33305	−0.666526	−	0.745482i	$-0.732219\pi$
$151$	−2473.50	−1.33305	−0.666526	+	0.745482i	$0.732219\pi$
$152$	−525.325	−0.280326
$153$	156.657	0.0827776
$154$	−19.0681	−0.00997761
$155$	726.454	0.376453
$156$	−112.754	−0.0578688
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	−464.336	−0.233801
$159$	−203.472	−0.101486
$160$	−230.888	−0.114083
$161$	849.447	0.415812
$162$	242.586	0.117650
$163$	−1983.26	−0.953010	−0.476505	−	0.879172i	$-0.658097\pi$
$163$	−1983.26	−0.953010	−0.476505	+	0.879172i	$0.658097\pi$
$164$	391.223	0.186277
$165$	12.4962	0.00589593
$166$	−1344.58	−0.628672
$167$	1268.83	0.587936	0.293968	−	0.955815i	$-0.405024\pi$
$167$	1268.83	0.587936	0.293968	+	0.955815i	$0.405024\pi$
$168$	510.887	0.234618
$169$	−693.685	−0.315742
$170$	275.833	0.124444
$171$	224.541	0.100416
$172$	419.470	0.185955
$173$	−1172.52	−0.515288	−0.257644	−	0.966240i	$-0.582946\pi$
$173$	−1172.52	−0.515288	−0.257644	+	0.966240i	$0.582946\pi$
$174$	−1283.06	−0.559013
$175$	−784.533	−0.338887
$176$	55.7475	0.0238757
$177$	1494.73	0.634750
$178$	2235.30	0.941253
$179$	−1803.71	−0.753161	−0.376580	−	0.926384i	$-0.622900\pi$
$179$	−1803.71	−0.753161	−0.376580	+	0.926384i	$0.622900\pi$
$180$	46.1621	0.0191151
$181$	1535.61	0.630614	0.315307	−	0.948990i	$-0.397893\pi$
$181$	1535.61	0.630614	0.315307	+	0.948990i	$0.397893\pi$
$182$	939.147	0.382496
$183$	−523.957	−0.211650
$184$	−2211.49	−0.886048
$185$	1040.28	0.413422
$186$	−1233.54	−0.486276
$187$	−13.7027	−0.00535851
$188$	55.6360	0.0215834
$189$	−218.369	−0.0840425
$190$	395.359	0.150960
$191$	−139.703	−0.0529244	−0.0264622	−	0.999650i	$-0.508424\pi$
$191$	−139.703	−0.0529244	−0.0264622	+	0.999650i	$0.508424\pi$
$192$	−1307.51	−0.491466
$193$	−258.945	−0.0965764	−0.0482882	−	0.998833i	$-0.515377\pi$
$193$	−258.945	−0.0965764	−0.0482882	+	0.998833i	$0.515377\pi$
$194$	−119.418	−0.0441942
$195$	−615.467	−0.226023
$196$	−269.083	−0.0980623
$197$	2026.20	0.732796	0.366398	−	0.930458i	$-0.380591\pi$
$197$	2026.20	0.732796	0.366398	+	0.930458i	$0.380591\pi$
$198$	−21.2189	−0.00761596
$199$	−51.8720	−0.0184779	−0.00923896	−	0.999957i	$-0.502941\pi$
$199$	−51.8720	−0.0184779	−0.00923896	+	0.999957i	$0.502941\pi$
$200$	2042.49	0.722128
$201$	−1775.25	−0.622969
$202$	2755.87	0.959914
$203$	1154.97	0.399326
$204$	−50.6190	−0.0173728
$205$	2135.49	0.727556
$206$	−3296.48	−1.11494
$207$	945.259	0.317392
$208$	−2745.69	−0.915286
$209$	−19.6405	−0.00650028
$210$	−384.493	−0.126345
$211$	2283.86	0.745152	0.372576	−	0.928002i	$-0.378475\pi$
$211$	2283.86	0.745152	0.372576	+	0.928002i	$0.378475\pi$
$212$	65.7457	0.0212992
$213$	596.710	0.191953
$214$	−30.2895	−0.00967544
$215$	2289.68	0.726300
$216$	568.512	0.179085
$217$	1110.40	0.347367
$218$	2607.75	0.810178
$219$	267.840	0.0826437
$220$	−4.03778	−0.00123739
$221$	674.890	0.205421
$222$	−1766.42	−0.534029
$223$	2600.13	0.780796	0.390398	−	0.920646i	$-0.372337\pi$
$223$	2600.13	0.780796	0.390398	+	0.920646i	$0.372337\pi$
$224$	−352.916	−0.105269
$225$	−873.024	−0.258674
$226$	−3253.12	−0.957496
$227$	−2593.84	−0.758409	−0.379205	−	0.925313i	$-0.623802\pi$
$227$	−2593.84	−0.758409	−0.379205	+	0.925313i	$0.623802\pi$
$228$	−72.5536	−0.0210745
$229$	4310.88	1.24398	0.621989	−	0.783026i	$-0.286325\pi$
$229$	4310.88	1.24398	0.621989	+	0.783026i	$0.286325\pi$
$230$	1664.36	0.477151
$231$	19.1006	0.00544039
$232$	−3006.90	−0.850917
$233$	227.209	0.0638839	0.0319419	−	0.999490i	$-0.489831\pi$
$233$	227.209	0.0638839	0.0319419	+	0.999490i	$0.489831\pi$
$234$	1045.08	0.291961
$235$	303.689	0.0843000
$236$	−482.977	−0.133217
$237$	465.129	0.127483
$238$	421.615	0.114829
$239$	2893.96	0.783242	0.391621	−	0.920127i	$-0.371915\pi$
$239$	2893.96	0.783242	0.391621	+	0.920127i	$0.371915\pi$
$240$	1124.10	0.302336
$241$	−5385.63	−1.43950	−0.719749	−	0.694235i	$-0.755743\pi$
$241$	−5385.63	−1.43950	−0.719749	+	0.694235i	$0.755743\pi$
$242$	−3984.34	−1.05836
$243$	−243.000	−0.0641500
$244$	169.301	0.0444196
$245$	−1468.79	−0.383010
$246$	−3626.11	−0.939806
$247$	967.337	0.249191
$248$	−2890.85	−0.740198
$249$	1346.87	0.342790
$250$	−3518.01	−0.889994
$251$	3507.06	0.881927	0.440963	−	0.897525i	$-0.354637\pi$
$251$	3507.06	0.881927	0.440963	+	0.897525i	$0.354637\pi$
$252$	70.5595	0.0176382
$253$	−82.6813	−0.0205460
$254$	4542.27	1.12208
$255$	−276.304	−0.0678542
$256$	1467.96	0.358388
$257$	1110.24	0.269474	0.134737	−	0.990881i	$-0.456981\pi$
$257$	1110.24	0.269474	0.134737	+	0.990881i	$0.456981\pi$
$258$	−3887.92	−0.938184
$259$	1590.09	0.381479
$260$	198.870	0.0474360
$261$	1285.25	0.304807
$262$	−6072.51	−1.43191
$263$	6367.65	1.49295	0.746476	−	0.665413i	$-0.231744\pi$
$263$	6367.65	1.49295	0.746476	+	0.665413i	$0.231744\pi$
$264$	−49.7274	−0.0115928
$265$	358.873	0.0831902
$266$	604.312	0.139296
$267$	−2239.12	−0.513227
$268$	573.619	0.130744
$269$	2670.60	0.605314	0.302657	−	0.953100i	$-0.402126\pi$
$269$	2670.60	0.605314	0.302657	+	0.953100i	$0.402126\pi$
$270$	−427.861	−0.0964400
$271$	−7450.67	−1.67010	−0.835048	−	0.550177i	$-0.814560\pi$
$271$	−7450.67	−1.67010	−0.835048	+	0.550177i	$0.814560\pi$
$272$	−1232.63	−0.274777
$273$	−940.750	−0.208560
$274$	2043.85	0.450632
$275$	76.3630	0.0167449
$276$	−305.432	−0.0666118
$277$	−5358.17	−1.16224	−0.581122	−	0.813817i	$-0.697386\pi$
$277$	−5358.17	−1.16224	−0.581122	+	0.813817i	$0.697386\pi$
$278$	−8557.19	−1.84614
$279$	1235.64	0.265147
$280$	−901.076	−0.192320
$281$	7013.95	1.48903	0.744515	−	0.667606i	$-0.232681\pi$
$281$	7013.95	1.48903	0.744515	+	0.667606i	$0.232681\pi$
$282$	−515.672	−0.108893
$283$	−5029.91	−1.05653	−0.528264	−	0.849080i	$-0.677157\pi$
$283$	−5029.91	−1.05653	−0.528264	+	0.849080i	$0.677157\pi$
$284$	−192.809	−0.0402856
$285$	−396.034	−0.0823123
$286$	−91.4123	−0.0188997
$287$	3264.12	0.671342
$288$	−392.722	−0.0803520
$289$	−4610.02	−0.938331
$290$	2262.99	0.458232
$291$	119.621	0.0240973
$292$	−86.5445	−0.0173446
$293$	−1313.44	−0.261885	−0.130942	−	0.991390i	$-0.541800\pi$
$293$	−1313.44	−0.261885	−0.130942	+	0.991390i	$0.541800\pi$
$294$	2494.04	0.494746
$295$	−2636.33	−0.520316
$296$	−4139.69	−0.812888
$297$	21.2551	0.00415268
$298$	2332.00	0.453319
$299$	4072.24	0.787638
$300$	282.091	0.0542885
$301$	3499.80	0.670183
$302$	−7407.86	−1.41151
$303$	−2760.58	−0.523403
$304$	−1766.77	−0.333326
$305$	924.129	0.173493
$306$	469.171	0.0876494
$307$	8805.26	1.63695	0.818473	−	0.574545i	$-0.194821\pi$
$307$	8805.26	1.63695	0.818473	+	0.574545i	$0.194821\pi$
$308$	−6.17180	−0.00114179
$309$	3302.11	0.607930
$310$	2175.65	0.398608
$311$	609.810	0.111187	0.0555934	−	0.998453i	$-0.482295\pi$
$311$	609.810	0.111187	0.0555934	+	0.998453i	$0.482295\pi$
$312$	2449.19	0.444416
$313$	2378.22	0.429473	0.214736	−	0.976672i	$-0.431111\pi$
$313$	2378.22	0.429473	0.214736	+	0.976672i	$0.431111\pi$
$314$	−470.198	−0.0845057
$315$	385.149	0.0688910
$316$	−150.292	−0.0267551
$317$	2027.63	0.359252	0.179626	−	0.983735i	$-0.442511\pi$
$317$	2027.63	0.359252	0.179626	+	0.983735i	$0.442511\pi$
$318$	−609.375	−0.107459
$319$	−112.420	−0.0197313
$320$	2306.12	0.402863
$321$	30.3411	0.00527563
$322$	2544.00	0.440284
$323$	434.271	0.0748095
$324$	78.5181	0.0134633
$325$	−3761.05	−0.641924
$326$	−5939.63	−1.00910
$327$	−2612.20	−0.441758
$328$	−8497.96	−1.43055
$329$	464.193	0.0777867
$330$	37.4248	0.00624293
$331$	−9497.56	−1.57714	−0.788570	−	0.614945i	$-0.789178\pi$
$331$	−9497.56	−1.57714	−0.788570	+	0.614945i	$0.789178\pi$
$332$	−435.202	−0.0719422
$333$	1769.44	0.291185
$334$	3800.02	0.622538
$335$	3131.10	0.510658
$336$	1718.21	0.278976
$337$	−255.444	−0.0412905	−0.0206453	−	0.999787i	$-0.506572\pi$
$337$	−255.444	−0.0412905	−0.0206453	+	0.999787i	$0.506572\pi$
$338$	−2077.51	−0.334324
$339$	3258.67	0.522084
$340$	89.2793	0.0142407
$341$	−108.081	−0.0171640
$342$	672.475	0.106325
$343$	−5019.16	−0.790115
$344$	−9111.53	−1.42808
$345$	−1667.20	−0.260171
$346$	−3511.56	−0.545614
$347$	−4596.52	−0.711108	−0.355554	−	0.934656i	$-0.615708\pi$
$347$	−4596.52	−0.711108	−0.355554	+	0.934656i	$0.615708\pi$
$348$	−415.289	−0.0639707
$349$	2712.90	0.416098	0.208049	−	0.978118i	$-0.433289\pi$
$349$	2712.90	0.416098	0.208049	+	0.978118i	$0.433289\pi$
$350$	−2349.59	−0.358831
$351$	−1046.86	−0.159195
$352$	34.3512	0.00520149
$353$	2520.07	0.379971	0.189986	−	0.981787i	$-0.439156\pi$
$353$	2520.07	0.379971	0.189986	+	0.981787i	$0.439156\pi$
$354$	4476.55	0.672107
$355$	−1052.45	−0.157347
$356$	723.503	0.107712
$357$	−422.335	−0.0626115
$358$	−5401.92	−0.797487
$359$	−2112.82	−0.310614	−0.155307	−	0.987866i	$-0.549637\pi$
$359$	−2112.82	−0.310614	−0.155307	+	0.987866i	$0.549637\pi$
$360$	−1002.71	−0.146799
$361$	−6236.55	−0.909250
$362$	4598.99	0.667727
$363$	3991.14	0.577081
$364$	303.975	0.0437709
$365$	−472.403	−0.0677444
$366$	−1569.19	−0.224107
$367$	1506.75	0.214310	0.107155	−	0.994242i	$-0.465826\pi$
$367$	1506.75	0.214310	0.107155	+	0.994242i	$0.465826\pi$
$368$	−7437.64	−1.05357
$369$	3632.30	0.512439
$370$	3115.53	0.437753
$371$	548.542	0.0767626
$372$	−399.260	−0.0556470
$373$	−9431.65	−1.30926	−0.654628	−	0.755951i	$-0.727175\pi$
$373$	−9431.65	−1.30926	−0.654628	+	0.755951i	$0.727175\pi$
$374$	−41.0381	−0.00567388
$375$	3524.01	0.485278
$376$	−1208.50	−0.165754
$377$	5536.93	0.756409
$378$	−653.992	−0.0889886
$379$	−4828.57	−0.654425	−0.327213	−	0.944951i	$-0.606109\pi$
$379$	−4828.57	−0.654425	−0.327213	+	0.944951i	$0.606109\pi$
$380$	127.966	0.0172751
$381$	−4550.02	−0.611823
$382$	−418.395	−0.0560391
$383$	11471.2	1.53042	0.765210	−	0.643780i	$-0.222635\pi$
$383$	11471.2	1.53042	0.765210	+	0.643780i	$0.222635\pi$
$384$	−4963.11	−0.659565
$385$	−33.6888	−0.00445958
$386$	−775.511	−0.102260
$387$	3894.56	0.511554
$388$	−38.6520	−0.00505737
$389$	14525.7	1.89327	0.946634	−	0.322311i	$-0.104460\pi$
$389$	14525.7	1.89327	0.946634	+	0.322311i	$0.104460\pi$
$390$	−1843.25	−0.239325
$391$	1828.17	0.236456
$392$	5844.90	0.753091
$393$	6082.88	0.780765
$394$	6068.24	0.775923
$395$	−820.371	−0.104500
$396$	−6.86794	−0.000871533 0
$397$	−1431.48	−0.180966	−0.0904832	−	0.995898i	$-0.528841\pi$
$397$	−1431.48	−0.180966	−0.0904832	+	0.995898i	$0.528841\pi$
$398$	−155.351	−0.0195654
$399$	−605.343	−0.0759526
$400$	6869.27	0.858658
$401$	13528.7	1.68477	0.842383	−	0.538880i	$-0.181152\pi$
$401$	13528.7	1.68477	0.842383	+	0.538880i	$0.181152\pi$
$402$	−5316.68	−0.659632
$403$	5323.23	0.657987
$404$	891.997	0.109848
$405$	428.591	0.0525849
$406$	3459.01	0.422827
$407$	−154.772	−0.0188495
$408$	1099.52	0.133418
$409$	7804.89	0.943587	0.471793	−	0.881709i	$-0.343607\pi$
$409$	7804.89	0.943587	0.471793	+	0.881709i	$0.343607\pi$
$410$	6395.55	0.770375
$411$	−2047.33	−0.245712
$412$	−1066.98	−0.127588
$413$	−4029.67	−0.480114
$414$	2830.95	0.336071
$415$	−2375.55	−0.280991
$416$	−1691.87	−0.199401
$417$	8571.79	1.00662
$418$	−58.8210	−0.00688285
$419$	11321.6	1.32004	0.660018	−	0.751249i	$-0.270548\pi$
$419$	11321.6	1.32004	0.660018	+	0.751249i	$0.270548\pi$
$420$	−124.449	−0.0144583
$421$	6908.17	0.799724	0.399862	−	0.916575i	$-0.369058\pi$
$421$	6908.17	0.799724	0.399862	+	0.916575i	$0.369058\pi$
$422$	6839.89	0.789007
$423$	516.552	0.0593749
$424$	−1428.10	−0.163572
$425$	−1688.46	−0.192712
$426$	1787.08	0.203250
$427$	1412.54	0.160089
$428$	−9.80383	−0.00110721
$429$	91.5683	0.0103053
$430$	6857.32	0.769045
$431$	−5028.65	−0.561999	−0.281000	−	0.959708i	$-0.590666\pi$
$431$	−5028.65	−0.561999	−0.281000	+	0.959708i	$0.590666\pi$
$432$	1912.01	0.212944
$433$	4255.80	0.472334	0.236167	−	0.971712i	$-0.424109\pi$
$433$	4255.80	0.472334	0.236167	+	0.971712i	$0.424109\pi$
$434$	3325.51	0.367810
$435$	−2266.85	−0.249856
$436$	844.052	0.0927128
$437$	2620.36	0.286840
$438$	802.152	0.0875076
$439$	−5557.49	−0.604202	−0.302101	−	0.953276i	$-0.597688\pi$
$439$	−5557.49	−0.604202	−0.302101	+	0.953276i	$0.597688\pi$
$440$	87.7067	0.00950285
$441$	−2498.29	−0.269765
$442$	2021.22	0.217510
$443$	−13192.7	−1.41490	−0.707452	−	0.706761i	$-0.750155\pi$
$443$	−13192.7	−1.41490	−0.707452	+	0.706761i	$0.750155\pi$
$444$	−571.740	−0.0611117
$445$	3949.24	0.420701
$446$	7787.09	0.826748
$447$	−2335.98	−0.247177
$448$	3524.94	0.371736
$449$	9718.21	1.02145	0.510725	−	0.859744i	$-0.329377\pi$
$449$	9718.21	1.02145	0.510725	+	0.859744i	$0.329377\pi$
$450$	−2614.61	−0.273898
$451$	−317.715	−0.0331721
$452$	−1052.94	−0.109571
$453$	7420.51	0.769638
$454$	−7768.25	−0.803044
$455$	1659.25	0.170960
$456$	1575.98	0.161846
$457$	−1344.65	−0.137637	−0.0688187	−	0.997629i	$-0.521923\pi$
$457$	−1344.65	−0.137637	−0.0688187	+	0.997629i	$0.521923\pi$
$458$	12910.6	1.31719
$459$	−469.971	−0.0477917
$460$	538.706	0.0546028
$461$	4148.69	0.419141	0.209570	−	0.977794i	$-0.432793\pi$
$461$	4148.69	0.419141	0.209570	+	0.977794i	$0.432793\pi$
$462$	57.2043	0.00576057
$463$	−6945.28	−0.697137	−0.348569	−	0.937283i	$-0.613332\pi$
$463$	−6945.28	−0.697137	−0.348569	+	0.937283i	$0.613332\pi$
$464$	−10112.8	−1.01180
$465$	−2179.36	−0.217345
$466$	680.465	0.0676436
$467$	−13796.1	−1.36704	−0.683522	−	0.729930i	$-0.739553\pi$
$467$	−13796.1	−1.36704	−0.683522	+	0.729930i	$0.739553\pi$
$468$	338.261	0.0334106
$469$	4785.93	0.471202
$470$	909.516	0.0892613
$471$	471.000	0.0460776
$472$	10491.0	1.02307
$473$	−340.655	−0.0331149
$474$	1393.01	0.134985
$475$	−2420.12	−0.233774
$476$	136.465	0.0131404
$477$	610.415	0.0585932
$478$	8667.09	0.829338
$479$	−3792.19	−0.361732	−0.180866	−	0.983508i	$-0.557890\pi$
$479$	−3792.19	−0.361732	−0.180866	+	0.983508i	$0.557890\pi$
$480$	692.664	0.0658659
$481$	7622.85	0.722603
$482$	−16129.4	−1.52422
$483$	−2548.34	−0.240069
$484$	−1289.62	−0.121114
$485$	−210.982	−0.0197530
$486$	−727.758	−0.0679255
$487$	11324.4	1.05371	0.526854	−	0.849956i	$-0.323371\pi$
$487$	11324.4	1.05371	0.526854	+	0.849956i	$0.323371\pi$
$488$	−3677.48	−0.341130
$489$	5949.77	0.550220
$490$	−4398.86	−0.405551
$491$	−6792.01	−0.624276	−0.312138	−	0.950037i	$-0.601045\pi$
$491$	−6792.01	−0.624276	−0.312138	+	0.950037i	$0.601045\pi$
$492$	−1173.67	−0.107547
$493$	2485.72	0.227081
$494$	2897.07	0.263857
$495$	−37.4886	−0.00340402
$496$	−9722.47	−0.880145
$497$	−1608.68	−0.145190
$498$	4033.74	0.362964
$499$	−6283.59	−0.563712	−0.281856	−	0.959457i	$-0.590950\pi$
$499$	−6283.59	−0.563712	−0.281856	+	0.959457i	$0.590950\pi$
$500$	−1138.68	−0.101847
$501$	−3806.50	−0.339445
$502$	10503.3	0.933831
$503$	6780.08	0.601011	0.300506	−	0.953780i	$-0.402845\pi$
$503$	6780.08	0.601011	0.300506	+	0.953780i	$0.402845\pi$
$504$	−1532.66	−0.135457
$505$	4868.97	0.429042
$506$	−247.621	−0.0217552
$507$	2081.05	0.182294
$508$	1470.20	0.128405
$509$	13757.5	1.19801	0.599007	−	0.800744i	$-0.295562\pi$
$509$	13757.5	1.19801	0.599007	+	0.800744i	$0.295562\pi$
$510$	−827.500	−0.0718477
$511$	−722.075	−0.0625102
$512$	−8838.59	−0.762919
$513$	−673.622	−0.0579750
$514$	3325.05	0.285334
$515$	−5824.10	−0.498331
$516$	−1258.41	−0.107361
$517$	−45.1825	−0.00384357
$518$	4762.13	0.403930
$519$	3517.55	0.297502
$520$	−4319.75	−0.364296
$521$	−19546.4	−1.64365	−0.821825	−	0.569739i	$-0.807044\pi$
$521$	−19546.4	−1.64365	−0.821825	+	0.569739i	$0.807044\pi$
$522$	3849.17	0.322746
$523$	−457.234	−0.0382284	−0.0191142	−	0.999817i	$-0.506085\pi$
$523$	−457.234	−0.0382284	−0.0191142	+	0.999817i	$0.506085\pi$
$524$	−1965.50	−0.163861
$525$	2353.60	0.195656
$526$	19070.4	1.58082
$527$	2389.78	0.197534
$528$	−167.243	−0.0137847
$529$	−1135.95	−0.0933633
$530$	1074.78	0.0880862
$531$	−4484.19	−0.366473
$532$	195.598	0.0159404
$533$	15648.2	1.27167
$534$	−6705.91	−0.543432
$535$	−53.5142	−0.00432452
$536$	−12459.9	−1.00408
$537$	5411.14	0.434838
$538$	7998.15	0.640938
$539$	218.524	0.0174629
$540$	−138.486	−0.0110361
$541$	15563.5	1.23684	0.618418	−	0.785849i	$-0.287774\pi$
$541$	15563.5	1.23684	0.618418	+	0.785849i	$0.287774\pi$
$542$	−22313.9	−1.76839
$543$	−4606.83	−0.364085
$544$	−759.540	−0.0598622
$545$	4607.26	0.362116
$546$	−2817.44	−0.220834
$547$	5139.43	0.401730	0.200865	−	0.979619i	$-0.435625\pi$
$547$	5139.43	0.401730	0.200865	+	0.979619i	$0.435625\pi$
$548$	661.534	0.0515682
$549$	1571.87	0.122196
$550$	228.699	0.0177304
$551$	3562.84	0.275467
$552$	6634.46	0.511560
$553$	−1253.95	−0.0964255
$554$	−16047.1	−1.23064
$555$	−3120.84	−0.238689
$556$	−2769.72	−0.211263
$557$	332.601	0.0253012	0.0126506	−	0.999920i	$-0.495973\pi$
$557$	332.601	0.0253012	0.0126506	+	0.999920i	$0.495973\pi$
$558$	3700.61	0.280751
$559$	16778.0	1.26947
$560$	−3030.49	−0.228681
$561$	41.1081	0.00309374
$562$	21006.0	1.57666
$563$	−2054.13	−0.153768	−0.0768838	−	0.997040i	$-0.524497\pi$
$563$	−2054.13	−0.153768	−0.0768838	+	0.997040i	$0.524497\pi$
$564$	−166.908	−0.0124612
$565$	−5747.48	−0.427961
$566$	−15064.0	−1.11871
$567$	655.108	0.0485219
$568$	4188.11	0.309382
$569$	−10617.3	−0.782254	−0.391127	−	0.920337i	$-0.627915\pi$
$569$	−10617.3	−0.782254	−0.391127	+	0.920337i	$0.627915\pi$
$570$	−1186.08	−0.0871567
$571$	12224.3	0.895919	0.447959	−	0.894054i	$-0.352151\pi$
$571$	12224.3	0.895919	0.447959	+	0.894054i	$0.352151\pi$
$572$	−29.5875	−0.00216279
$573$	419.109	0.0305559
$574$	9775.69	0.710853
$575$	−10188.1	−0.738908
$576$	3922.54	0.283748
$577$	6850.09	0.494234	0.247117	−	0.968986i	$-0.420517\pi$
$577$	6850.09	0.494234	0.247117	+	0.968986i	$0.420517\pi$
$578$	−13806.5	−0.993554
$579$	776.834	0.0557584
$580$	732.465	0.0524379
$581$	−3631.06	−0.259280
$582$	358.253	0.0255156
$583$	−53.3927	−0.00379296
$584$	1879.88	0.133202
$585$	1846.40	0.130494
$586$	−3933.62	−0.277298
$587$	14690.2	1.03293	0.516466	−	0.856307i	$-0.327247\pi$
$587$	14690.2	1.03293	0.516466	+	0.856307i	$0.327247\pi$
$588$	807.249	0.0566163
$589$	3425.33	0.239624
$590$	−7895.52	−0.550938
$591$	−6078.60	−0.423080
$592$	−13922.6	−0.966577
$593$	−7691.87	−0.532660	−0.266330	−	0.963882i	$-0.585811\pi$
$593$	−7691.87	−0.532660	−0.266330	+	0.963882i	$0.585811\pi$
$594$	63.6566	0.00439707
$595$	744.893	0.0513237
$596$	754.802	0.0518756
$597$	155.616	0.0106682
$598$	12195.9	0.833993
$599$	−28309.5	−1.93104	−0.965521	−	0.260323i	$-0.916171\pi$
$599$	−28309.5	−1.93104	−0.965521	+	0.260323i	$0.916171\pi$
$600$	−6127.46	−0.416921
$601$	−17308.3	−1.17474	−0.587372	−	0.809317i	$-0.699837\pi$
$601$	−17308.3	−1.17474	−0.587372	+	0.809317i	$0.699837\pi$
$602$	10481.5	0.709626
$603$	5325.76	0.359671
$604$	−2397.71	−0.161526
$605$	−7039.37	−0.473043
$606$	−8267.62	−0.554206
$607$	−16527.2	−1.10514	−0.552569	−	0.833467i	$-0.686353\pi$
$607$	−16527.2	−1.10514	−0.552569	+	0.833467i	$0.686353\pi$
$608$	−1088.67	−0.0726174
$609$	−3464.92	−0.230551
$610$	2767.66	0.183704
$611$	2225.34	0.147345
$612$	151.857	0.0100302
$613$	−13015.7	−0.857585	−0.428792	−	0.903403i	$-0.641061\pi$
$613$	−13015.7	−0.857585	−0.428792	+	0.903403i	$0.641061\pi$
$614$	26370.8	1.73329
$615$	−6406.47	−0.420055
$616$	134.061	0.00876862
$617$	2503.80	0.163370	0.0816850	−	0.996658i	$-0.473970\pi$
$617$	2503.80	0.163370	0.0816850	+	0.996658i	$0.473970\pi$
$618$	9889.45	0.643709
$619$	−11571.0	−0.751337	−0.375669	−	0.926754i	$-0.622587\pi$
$619$	−11571.0	−0.751337	−0.375669	+	0.926754i	$0.622587\pi$
$620$	704.196	0.0456148
$621$	−2835.78	−0.183246
$622$	1826.31	0.117731
$623$	6036.47	0.388196
$624$	8237.07	0.528440
$625$	5909.85	0.378231
$626$	7122.50	0.454748
$627$	58.9214	0.00375294
$628$	−152.189	−0.00967042
$629$	3422.16	0.216932
$630$	1153.48	0.0729455
$631$	−5357.04	−0.337972	−0.168986	−	0.985618i	$-0.554049\pi$
$631$	−5357.04	−0.337972	−0.168986	+	0.985618i	$0.554049\pi$
$632$	3264.58	0.205472
$633$	−6851.57	−0.430214
$634$	6072.52	0.380395
$635$	8025.09	0.501521
$636$	−197.237	−0.0122971
$637$	−10762.8	−0.669448
$638$	−336.685	−0.0208926
$639$	−1790.13	−0.110824
$640$	8753.69	0.540656
$641$	−8440.69	−0.520105	−0.260052	−	0.965595i	$-0.583740\pi$
$641$	−8440.69	−0.520105	−0.260052	+	0.965595i	$0.583740\pi$
$642$	90.8684	0.00558612
$643$	−8628.91	−0.529224	−0.264612	−	0.964355i	$-0.585244\pi$
$643$	−8628.91	−0.529224	−0.264612	+	0.964355i	$0.585244\pi$
$644$	823.419	0.0503840
$645$	−6869.03	−0.419330
$646$	1300.59	0.0792123
$647$	−18057.0	−1.09721	−0.548604	−	0.836082i	$-0.684841\pi$
$647$	−18057.0	−1.09721	−0.548604	+	0.836082i	$0.684841\pi$
$648$	−1705.53	−0.103395
$649$	392.230	0.0237232
$650$	−11263.9	−0.679703
$651$	−3331.19	−0.200552
$652$	−1922.49	−0.115476
$653$	−15430.8	−0.924741	−0.462370	−	0.886687i	$-0.653001\pi$
$653$	−15430.8	−0.924741	−0.462370	+	0.886687i	$0.653001\pi$
$654$	−7823.24	−0.467756
$655$	−10728.7	−0.640006
$656$	−28580.2	−1.70102
$657$	−803.521	−0.0477144
$658$	1390.21	0.0823646
$659$	−23613.8	−1.39584	−0.697922	−	0.716174i	$-0.745892\pi$
$659$	−23613.8	−1.39584	−0.697922	+	0.716174i	$0.745892\pi$
$660$	12.1133	0.000714410 0
$661$	−13119.4	−0.771990	−0.385995	−	0.922501i	$-0.626142\pi$
$661$	−13119.4	−0.771990	−0.385995	+	0.922501i	$0.626142\pi$
$662$	−28444.1	−1.66996
$663$	−2024.67	−0.118600
$664$	9453.26	0.552496
$665$	1067.67	0.0622596
$666$	5299.27	0.308322
$667$	14998.6	0.870690
$668$	1229.96	0.0712402
$669$	−7800.38	−0.450792
$670$	9377.30	0.540712
$671$	−137.491	−0.00791024
$672$	1058.75	0.0607769
$673$	−11453.7	−0.656029	−0.328014	−	0.944673i	$-0.606379\pi$
$673$	−11453.7	−0.656029	−0.328014	+	0.944673i	$0.606379\pi$
$674$	−765.026	−0.0437206
$675$	2619.07	0.149345
$676$	−672.430	−0.0382584
$677$	9903.94	0.562244	0.281122	−	0.959672i	$-0.409293\pi$
$677$	9903.94	0.562244	0.281122	+	0.959672i	$0.409293\pi$
$678$	9759.35	0.552811
$679$	−322.489	−0.0182268
$680$	−1939.28	−0.109365
$681$	7781.51	0.437868
$682$	−323.690	−0.0181741
$683$	−34514.5	−1.93362	−0.966809	−	0.255500i	$-0.917760\pi$
$683$	−34514.5	−1.93362	−0.966809	+	0.255500i	$0.917760\pi$
$684$	217.661	0.0121674
$685$	3610.99	0.201414
$686$	−15031.8	−0.836615
$687$	−12932.6	−0.718211
$688$	−30643.8	−1.69809
$689$	2629.71	0.145405
$690$	−4993.08	−0.275483
$691$	−5287.57	−0.291098	−0.145549	−	0.989351i	$-0.546495\pi$
$691$	−5287.57	−0.291098	−0.145549	+	0.989351i	$0.546495\pi$
$692$	−1136.59	−0.0624374
$693$	−57.3019	−0.00314101
$694$	−13766.1	−0.752959
$695$	−15118.5	−0.825147
$696$	9020.71	0.491277
$697$	7025.01	0.381766
$698$	8124.83	0.440586
$699$	−681.626	−0.0368834
$700$	−760.495	−0.0410629
$701$	−16692.7	−0.899395	−0.449697	−	0.893181i	$-0.648468\pi$
$701$	−16692.7	−0.899395	−0.449697	+	0.893181i	$0.648468\pi$
$702$	−3135.23	−0.168564
$703$	4905.07	0.263155
$704$	−343.102	−0.0183681
$705$	−911.068	−0.0486706
$706$	7547.33	0.402334
$707$	7442.29	0.395892
$708$	1448.93	0.0769127
$709$	−9959.81	−0.527572	−0.263786	−	0.964581i	$-0.584971\pi$
$709$	−9959.81	−0.527572	−0.263786	+	0.964581i	$0.584971\pi$
$710$	−3151.96	−0.166607
$711$	−1395.39	−0.0736021
$712$	−15715.6	−0.827201
$713$	14419.8	0.757398
$714$	−1264.85	−0.0662964
$715$	−161.504	−0.00844740
$716$	−1748.45	−0.0912604
$717$	−8681.88	−0.452205
$718$	−6327.67	−0.328895
$719$	20241.9	1.04992	0.524962	−	0.851125i	$-0.324079\pi$
$719$	20241.9	1.04992	0.524962	+	0.851125i	$0.324079\pi$
$720$	−3372.31	−0.174554
$721$	−8902.21	−0.459828
$722$	−18677.8	−0.962763
$723$	16156.9	0.831094
$724$	1488.56	0.0764114
$725$	−13852.5	−0.709611
$726$	11953.0	0.611045
$727$	35500.6	1.81107	0.905533	−	0.424276i	$-0.139471\pi$
$727$	35500.6	1.81107	0.905533	+	0.424276i	$0.139471\pi$
$728$	−6602.80	−0.336149
$729$	729.000	0.0370370
$730$	−1414.80	−0.0717314
$731$	7532.23	0.381107
$732$	−507.903	−0.0256457
$733$	−20945.8	−1.05546	−0.527728	−	0.849413i	$-0.676956\pi$
$733$	−20945.8	−1.05546	−0.527728	+	0.849413i	$0.676956\pi$
$734$	4512.55	0.226923
$735$	4406.37	0.221131
$736$	−4583.02	−0.229528
$737$	−465.841	−0.0232829
$738$	10878.3	0.542597
$739$	−20150.1	−1.00302	−0.501511	−	0.865151i	$-0.667222\pi$
$739$	−20150.1	−1.00302	−0.501511	+	0.865151i	$0.667222\pi$
$740$	1008.41	0.0500943
$741$	−2902.01	−0.143870
$742$	1642.82	0.0812803
$743$	27924.1	1.37878	0.689392	−	0.724388i	$-0.257878\pi$
$743$	27924.1	1.37878	0.689392	+	0.724388i	$0.257878\pi$
$744$	8672.55	0.427354
$745$	4120.09	0.202615
$746$	−28246.7	−1.38631
$747$	−4040.62	−0.197910
$748$	−13.2829	−0.000649291 0
$749$	−81.7972	−0.00399039
$750$	10554.0	0.513838
$751$	−33954.6	−1.64983	−0.824914	−	0.565258i	$-0.808777\pi$
$751$	−33954.6	−1.64983	−0.824914	+	0.565258i	$0.808777\pi$
$752$	−4064.41	−0.197093
$753$	−10521.2	−0.509181
$754$	16582.5	0.800926
$755$	−13087.9	−0.630885
$756$	−211.678	−0.0101834
$757$	21410.8	1.02799	0.513995	−	0.857793i	$-0.328165\pi$
$757$	21410.8	1.02799	0.513995	+	0.857793i	$0.328165\pi$
$758$	−14461.0	−0.692940
$759$	248.044	0.0118622
$760$	−2779.63	−0.132668
$761$	4308.61	0.205239	0.102620	−	0.994721i	$-0.467278\pi$
$761$	4308.61	0.205239	0.102620	+	0.994721i	$0.467278\pi$
$762$	−13626.8	−0.647831
$763$	7042.26	0.334138
$764$	−135.422	−0.00641284
$765$	828.912	0.0391757
$766$	34355.0	1.62049
$767$	−19318.2	−0.909439
$768$	−4403.88	−0.206916
$769$	−18586.9	−0.871601	−0.435801	−	0.900043i	$-0.643535\pi$
$769$	−18586.9	−0.871601	−0.435801	+	0.900043i	$0.643535\pi$
$770$	−100.894	−0.00472204
$771$	−3330.72	−0.155581
$772$	−251.011	−0.0117022
$773$	11278.1	0.524769	0.262385	−	0.964963i	$-0.415491\pi$
$773$	11278.1	0.524769	0.262385	+	0.964963i	$0.415491\pi$
$774$	11663.8	0.541661
$775$	−13317.8	−0.617279
$776$	839.582	0.0388392
$777$	−4770.26	−0.220247
$778$	43502.8	2.00469
$779$	10069.1	0.463111
$780$	−596.609	−0.0273872
$781$	156.582	0.00717405
$782$	5475.16	0.250373
$783$	−3855.74	−0.175981
$784$	19657.5	0.895475
$785$	−830.726	−0.0377706
$786$	18217.5	0.826715
$787$	−24621.6	−1.11520	−0.557602	−	0.830108i	$-0.688278\pi$
$787$	−24621.6	−1.11520	−0.557602	+	0.830108i	$0.688278\pi$
$788$	1964.12	0.0887928
$789$	−19103.0	−0.861956
$790$	−2456.92	−0.110650
$791$	−8785.10	−0.394895
$792$	149.182	0.00669313
$793$	6771.73	0.303242
$794$	−4287.11	−0.191617
$795$	−1076.62	−0.0480299
$796$	−50.2826	−0.00223897
$797$	30567.6	1.35855	0.679273	−	0.733886i	$-0.262295\pi$
$797$	30567.6	1.35855	0.679273	+	0.733886i	$0.262295\pi$
$798$	−1812.94	−0.0804226
$799$	999.031	0.0442343
$800$	4232.79	0.187065
$801$	6717.35	0.296312
$802$	40516.9	1.78392
$803$	70.2836	0.00308873
$804$	−1720.86	−0.0754851
$805$	4494.63	0.196789
$806$	15942.5	0.696712
$807$	−8011.80	−0.349478
$808$	−19375.6	−0.843601
$809$	20459.3	0.889136	0.444568	−	0.895745i	$-0.353357\pi$
$809$	20459.3	0.889136	0.444568	+	0.895745i	$0.353357\pi$
$810$	1283.58	0.0556796
$811$	−7537.24	−0.326348	−0.163174	−	0.986597i	$-0.552173\pi$
$811$	−7537.24	−0.326348	−0.163174	+	0.986597i	$0.552173\pi$
$812$	1119.58	0.0483863
$813$	22352.0	0.964230
$814$	−463.524	−0.0199589
$815$	−10493.9	−0.451025
$816$	3697.90	0.158643
$817$	10796.1	0.462312
$818$	23374.8	0.999120
$819$	2822.25	0.120412
$820$	2070.06	0.0881579
$821$	−8022.36	−0.341026	−0.170513	−	0.985355i	$-0.554542\pi$
$821$	−8022.36	−0.341026	−0.170513	+	0.985355i	$0.554542\pi$
$822$	−6131.54	−0.260173
$823$	1080.00	0.0457428	0.0228714	−	0.999738i	$-0.492719\pi$
$823$	1080.00	0.0457428	0.0228714	+	0.999738i	$0.492719\pi$
$824$	23176.4	0.979840
$825$	−229.089	−0.00966770
$826$	−12068.4	−0.508370
$827$	−26194.6	−1.10142	−0.550710	−	0.834696i	$-0.685643\pi$
$827$	−26194.6	−1.10142	−0.550710	+	0.834696i	$0.685643\pi$
$828$	916.296	0.0384583
$829$	27601.3	1.15637	0.578186	−	0.815905i	$-0.303761\pi$
$829$	27601.3	1.15637	0.578186	+	0.815905i	$0.303761\pi$
$830$	−7114.51	−0.297528
$831$	16074.5	0.671021
$832$	16898.5	0.704149
$833$	−4831.80	−0.200975
$834$	25671.6	1.06587
$835$	6713.72	0.278249
$836$	−19.0387	−0.000787639 0
$837$	−3706.93	−0.153083
$838$	33906.9	1.39773
$839$	19772.5	0.813616	0.406808	−	0.913514i	$-0.366642\pi$
$839$	19772.5	0.813616	0.406808	+	0.913514i	$0.366642\pi$
$840$	2703.23	0.111036
$841$	−3995.70	−0.163832
$842$	20689.2	0.846790
$843$	−21041.9	−0.859692
$844$	2213.88	0.0902900
$845$	−3670.46	−0.149429
$846$	1547.01	0.0628693
$847$	−10759.8	−0.436494
$848$	−4802.96	−0.194498
$849$	15089.7	0.609986
$850$	−5056.76	−0.204053
$851$	20649.1	0.831777
$852$	578.427	0.0232589
$853$	−15079.2	−0.605277	−0.302638	−	0.953105i	$-0.597867\pi$
$853$	−15079.2	−0.605277	−0.302638	+	0.953105i	$0.597867\pi$
$854$	4230.41	0.169510
$855$	1188.10	0.0475231
$856$	212.954	0.00850307
$857$	−14113.0	−0.562533	−0.281267	−	0.959630i	$-0.590755\pi$
$857$	−14113.0	−0.562533	−0.281267	+	0.959630i	$0.590755\pi$
$858$	274.237	0.0109118
$859$	−25317.4	−1.00561	−0.502805	−	0.864400i	$-0.667699\pi$
$859$	−25317.4	−1.00561	−0.502805	+	0.864400i	$0.667699\pi$
$860$	2219.52	0.0880058
$861$	−9792.37	−0.387600
$862$	−15060.3	−0.595075
$863$	961.237	0.0379153	0.0189576	−	0.999820i	$-0.493965\pi$
$863$	961.237	0.0379153	0.0189576	+	0.999820i	$0.493965\pi$
$864$	1178.17	0.0463913
$865$	−6204.08	−0.243867
$866$	12745.6	0.500133
$867$	13830.1	0.541746
$868$	1076.37	0.0420904
$869$	122.054	0.00476454
$870$	−6788.97	−0.264561
$871$	22943.7	0.892559
$872$	−18334.1	−0.712009
$873$	−358.864	−0.0139126
$874$	7847.69	0.303721
$875$	−9500.45	−0.367056
$876$	259.634	0.0100139
$877$	5515.78	0.212377	0.106188	−	0.994346i	$-0.466135\pi$
$877$	5515.78	0.212377	0.106188	+	0.994346i	$0.466135\pi$
$878$	−16644.1	−0.639761
$879$	3940.33	0.151199
$880$	294.974	0.0112995
$881$	24095.4	0.921446	0.460723	−	0.887544i	$-0.347590\pi$
$881$	24095.4	0.921446	0.460723	+	0.887544i	$0.347590\pi$
$882$	−7482.11	−0.285642
$883$	−6695.69	−0.255185	−0.127592	−	0.991827i	$-0.540725\pi$
$883$	−6695.69	−0.255185	−0.127592	+	0.991827i	$0.540725\pi$
$884$	654.211	0.0248908
$885$	7908.99	0.300404
$886$	−39510.6	−1.49818
$887$	30715.8	1.16272	0.581361	−	0.813646i	$-0.302520\pi$
$887$	30715.8	1.16272	0.581361	+	0.813646i	$0.302520\pi$
$888$	12419.1	0.469321
$889$	12266.5	0.462772
$890$	11827.5	0.445461
$891$	−63.7652	−0.00239755
$892$	2520.46	0.0946090
$893$	1431.94	0.0536595
$894$	−6996.00	−0.261724
$895$	−9543.89	−0.356444
$896$	13380.1	0.498883
$897$	−12216.7	−0.454743
$898$	29105.0	1.08156
$899$	19606.2	0.727368
$900$	−846.274	−0.0313435
$901$	1180.57	0.0436519
$902$	−951.522	−0.0351244
$903$	−10499.4	−0.386931
$904$	22871.5	0.841476
$905$	8125.30	0.298447
$906$	22223.6	0.814933
$907$	−20022.5	−0.733005	−0.366503	−	0.930417i	$-0.619445\pi$
$907$	−20022.5	−0.733005	−0.366503	+	0.930417i	$0.619445\pi$
$908$	−2514.36	−0.0918964
$909$	8281.73	0.302187
$910$	4969.26	0.181021
$911$	24621.0	0.895422	0.447711	−	0.894178i	$-0.352239\pi$
$911$	24621.0	0.895422	0.447711	+	0.894178i	$0.352239\pi$
$912$	5300.30	0.192446
$913$	353.431	0.0128115
$914$	−4027.09	−0.145738
$915$	−2772.39	−0.100166
$916$	4178.79	0.150733
$917$	−16398.9	−0.590556
$918$	−1407.51	−0.0506044
$919$	19718.5	0.707784	0.353892	−	0.935286i	$-0.384858\pi$
$919$	19718.5	0.707784	0.353892	+	0.935286i	$0.384858\pi$
$920$	−11701.5	−0.419334
$921$	−26415.8	−0.945091
$922$	12424.9	0.443809
$923$	−7712.00	−0.275020
$924$	18.5154	0.000659212 0
$925$	−19071.1	−0.677897
$926$	−20800.3	−0.738166
$927$	−9906.33	−0.350989
$928$	−6231.42	−0.220427
$929$	−9979.16	−0.352428	−0.176214	−	0.984352i	$-0.556385\pi$
$929$	−9979.16	−0.352428	−0.176214	+	0.984352i	$0.556385\pi$
$930$	−6526.95	−0.230137
$931$	−6925.54	−0.243798
$932$	220.247	0.00774081
$933$	−1829.43	−0.0641938
$934$	−41317.9	−1.44750
$935$	−72.5045	−0.00253599
$936$	−7347.56	−0.256584
$937$	−31648.8	−1.10344	−0.551718	−	0.834030i	$-0.686028\pi$
$937$	−31648.8	−1.10344	−0.551718	+	0.834030i	$0.686028\pi$
$938$	14333.3	0.498934
$939$	−7134.66	−0.247956
$940$	294.384	0.0102146
$941$	18594.3	0.644163	0.322082	−	0.946712i	$-0.395617\pi$
$941$	18594.3	0.644163	0.322082	+	0.946712i	$0.395617\pi$
$942$	1410.59	0.0487894
$943$	42388.4	1.46379
$944$	35283.2	1.21649
$945$	−1155.45	−0.0397743
$946$	−1020.22	−0.0350638
$947$	−11902.1	−0.408412	−0.204206	−	0.978928i	$-0.565461\pi$
$947$	−11902.1	−0.408412	−0.204206	+	0.978928i	$0.565461\pi$
$948$	450.877	0.0154470
$949$	−3461.62	−0.118408
$950$	−7247.98	−0.247532
$951$	−6082.88	−0.207414
$952$	−2964.22	−0.100915
$953$	−47309.1	−1.60807	−0.804035	−	0.594582i	$-0.797318\pi$
$953$	−47309.1	−1.60807	−0.804035	+	0.594582i	$0.797318\pi$
$954$	1828.12	0.0620416
$955$	−739.204	−0.0250472
$956$	2805.29	0.0949053
$957$	337.259	0.0113919
$958$	−11357.2	−0.383021
$959$	5519.44	0.185852
$960$	−6918.37	−0.232593
$961$	−10941.5	−0.367275
$962$	22829.6	0.765131
$963$	−91.0234	−0.00304589
$964$	−5220.61	−0.174424
$965$	−1370.14	−0.0457061
$966$	−7632.00	−0.254198
$967$	−15579.6	−0.518103	−0.259051	−	0.965864i	$-0.583410\pi$
$967$	−15579.6	−0.518103	−0.259051	+	0.965864i	$0.583410\pi$
$968$	28012.5	0.930119
$969$	−1302.81	−0.0431913
$970$	−631.868	−0.0209155
$971$	−2880.82	−0.0952111	−0.0476055	−	0.998866i	$-0.515159\pi$
$971$	−2880.82	−0.0952111	−0.0476055	+	0.998866i	$0.515159\pi$
$972$	−235.554	−0.00777306
$973$	−23108.8	−0.761393
$974$	33915.2	1.11572
$975$	11283.1	0.370615
$976$	−12368.0	−0.405626
$977$	45305.3	1.48357	0.741783	−	0.670639i	$-0.233980\pi$
$977$	45305.3	1.48357	0.741783	+	0.670639i	$0.233980\pi$
$978$	17818.9	0.582603
$979$	−587.563	−0.0191814
$980$	−1423.78	−0.0464093
$981$	7836.59	0.255049
$982$	−20341.3	−0.661016
$983$	18209.1	0.590825	0.295412	−	0.955370i	$-0.404543\pi$
$983$	18209.1	0.590825	0.295412	+	0.955370i	$0.404543\pi$
$984$	25493.9	0.825930
$985$	10721.1	0.346806
$986$	7444.44	0.240446
$987$	−1392.58	−0.0449101
$988$	937.697	0.0301945
$989$	45449.0	1.46127
$990$	−112.274	−0.00360436
$991$	28241.3	0.905263	0.452632	−	0.891698i	$-0.350485\pi$
$991$	28241.3	0.905263	0.452632	+	0.891698i	$0.350485\pi$
$992$	−5990.92	−0.191746
$993$	28492.7	0.910562
$994$	−4817.82	−0.153734
$995$	−274.468	−0.00874493
$996$	1305.61	0.0415358
$997$	−22333.1	−0.709425	−0.354713	−	0.934975i	$-0.615421\pi$
$997$	−22333.1	−0.709425	−0.354713	+	0.934975i	$0.615421\pi$
$998$	−18818.7	−0.596888
$999$	−5308.31	−0.168116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.17	✓	22
3.2	odd	2		1413.4.a.e.1.6		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.17	✓	22	1.1	even	1	trivial
1413.4.a.e.1.6		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)

\(f(q)\)	\(=\)	\(q+2.99489 q^{2} -3.00000 q^{3} +0.969360 q^{4} +5.29125 q^{5} -8.98467 q^{6} +8.08775 q^{7} -21.0560 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.99489 q^{2} -3.00000 q^{3} +0.969360 q^{4} +5.29125 q^{5} -8.98467 q^{6} +8.08775 q^{7} -21.0560 q^{8} +9.00000 q^{9} +15.8467 q^{10} -0.787225 q^{11} -2.90808 q^{12} +38.7726 q^{13} +24.2219 q^{14} -15.8737 q^{15} -70.8152 q^{16} +17.4063 q^{17} +26.9540 q^{18} +24.9490 q^{19} +5.12912 q^{20} -24.2633 q^{21} -2.35765 q^{22} +105.029 q^{23} +63.1680 q^{24} -97.0027 q^{25} +116.120 q^{26} -27.0000 q^{27} +7.83994 q^{28} +142.805 q^{29} -47.5401 q^{30} +137.294 q^{31} -43.6358 q^{32} +2.36168 q^{33} +52.1301 q^{34} +42.7943 q^{35} +8.72424 q^{36} +196.604 q^{37} +74.7194 q^{38} -116.318 q^{39} -111.412 q^{40} +403.589 q^{41} -72.6657 q^{42} +432.729 q^{43} -0.763104 q^{44} +47.6212 q^{45} +314.550 q^{46} +57.3946 q^{47} +212.446 q^{48} -277.588 q^{49} -290.512 q^{50} -52.2190 q^{51} +37.5846 q^{52} +67.8239 q^{53} -80.8620 q^{54} -4.16541 q^{55} -170.296 q^{56} -74.8469 q^{57} +427.685 q^{58} -498.243 q^{59} -15.3874 q^{60} +174.652 q^{61} +411.179 q^{62} +72.7898 q^{63} +435.837 q^{64} +205.156 q^{65} +7.07296 q^{66} +591.751 q^{67} +16.8730 q^{68} -315.086 q^{69} +128.164 q^{70} -198.903 q^{71} -189.504 q^{72} -89.2801 q^{73} +588.807 q^{74} +291.008 q^{75} +24.1845 q^{76} -6.36688 q^{77} -348.359 q^{78} -155.043 q^{79} -374.701 q^{80} +81.0000 q^{81} +1208.70 q^{82} -448.958 q^{83} -23.5198 q^{84} +92.1013 q^{85} +1295.97 q^{86} -428.415 q^{87} +16.5758 q^{88} +746.372 q^{89} +142.620 q^{90} +313.583 q^{91} +101.811 q^{92} -411.881 q^{93} +171.891 q^{94} +132.011 q^{95} +130.907 q^{96} -39.8738 q^{97} -831.346 q^{98} -7.08503 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 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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