LMFDB - Embedded newform 471.4.a.c.1.19

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			1.19
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)

$f(q)$	$=$	$q+4.50022 q^{2} -3.00000 q^{3} +12.2520 q^{4} +17.0216 q^{5} -13.5007 q^{6} +15.6710 q^{7} +19.1349 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.50022 q^{2} -3.00000 q^{3} +12.2520 q^{4} +17.0216 q^{5} -13.5007 q^{6} +15.6710 q^{7} +19.1349 q^{8} +9.00000 q^{9} +76.6007 q^{10} +68.3756 q^{11} -36.7560 q^{12} -72.1818 q^{13} +70.5231 q^{14} -51.0647 q^{15} -11.9047 q^{16} -87.4682 q^{17} +40.5020 q^{18} +105.238 q^{19} +208.548 q^{20} -47.0131 q^{21} +307.705 q^{22} +74.4920 q^{23} -57.4046 q^{24} +164.733 q^{25} -324.834 q^{26} -27.0000 q^{27} +192.001 q^{28} -209.119 q^{29} -229.802 q^{30} -85.9638 q^{31} -206.653 q^{32} -205.127 q^{33} -393.626 q^{34} +266.745 q^{35} +110.268 q^{36} +361.620 q^{37} +473.593 q^{38} +216.545 q^{39} +325.705 q^{40} +301.706 q^{41} -211.569 q^{42} -295.258 q^{43} +837.736 q^{44} +153.194 q^{45} +335.230 q^{46} +369.530 q^{47} +35.7142 q^{48} -97.4191 q^{49} +741.336 q^{50} +262.405 q^{51} -884.370 q^{52} -474.006 q^{53} -121.506 q^{54} +1163.86 q^{55} +299.863 q^{56} -315.713 q^{57} -941.083 q^{58} +22.1917 q^{59} -625.643 q^{60} +765.268 q^{61} -386.856 q^{62} +141.039 q^{63} -834.746 q^{64} -1228.65 q^{65} -923.115 q^{66} -18.4848 q^{67} -1071.66 q^{68} -223.476 q^{69} +1200.41 q^{70} -314.721 q^{71} +172.214 q^{72} +935.937 q^{73} +1627.37 q^{74} -494.200 q^{75} +1289.37 q^{76} +1071.52 q^{77} +974.502 q^{78} -903.178 q^{79} -202.637 q^{80} +81.0000 q^{81} +1357.74 q^{82} -270.922 q^{83} -576.003 q^{84} -1488.84 q^{85} -1328.73 q^{86} +627.358 q^{87} +1308.36 q^{88} -1288.35 q^{89} +689.407 q^{90} -1131.16 q^{91} +912.675 q^{92} +257.891 q^{93} +1662.96 q^{94} +1791.31 q^{95} +619.959 q^{96} -195.525 q^{97} -438.407 q^{98} +615.380 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.50022	1.59107	0.795534	−	0.605909i	$-0.207190\pi$
$2$	4.50022	1.59107	0.795534	+	0.605909i	$0.207190\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	12.2520	1.53150
$5$	17.0216	1.52245	0.761227	−	0.648486i	$-0.224597\pi$
$5$	17.0216	1.52245	0.761227	+	0.648486i	$0.224597\pi$
$6$	−13.5007	−0.918604
$7$	15.6710	0.846156	0.423078	−	0.906093i	$-0.360950\pi$
$7$	15.6710	0.846156	0.423078	+	0.906093i	$0.360950\pi$
$8$	19.1349	0.845650
$9$	9.00000	0.333333
$10$	76.6007	2.42233
$11$	68.3756	1.87418	0.937091	−	0.349084i	$-0.113507\pi$
$11$	68.3756	1.87418	0.937091	+	0.349084i	$0.113507\pi$
$12$	−36.7560	−0.884211
$13$	−72.1818	−1.53997	−0.769986	−	0.638061i	$-0.779737\pi$
$13$	−72.1818	−1.53997	−0.769986	+	0.638061i	$0.779737\pi$
$14$	70.5231	1.34629
$15$	−51.0647	−0.878989
$16$	−11.9047	−0.186011
$17$	−87.4682	−1.24789	−0.623946	−	0.781468i	$-0.714471\pi$
$17$	−87.4682	−1.24789	−0.623946	+	0.781468i	$0.714471\pi$
$18$	40.5020	0.530356
$19$	105.238	1.27069	0.635347	−	0.772227i	$-0.280857\pi$
$19$	105.238	1.27069	0.635347	+	0.772227i	$0.280857\pi$
$20$	208.548	2.33164
$21$	−47.0131	−0.488528
$22$	307.705	2.98195
$23$	74.4920	0.675333	0.337666	−	0.941266i	$-0.390362\pi$
$23$	74.4920	0.675333	0.337666	+	0.941266i	$0.390362\pi$
$24$	−57.4046	−0.488236
$25$	164.733	1.31787
$26$	−324.834	−2.45020
$27$	−27.0000	−0.192450
$28$	192.001	1.29589
$29$	−209.119	−1.33905	−0.669526	−	0.742789i	$-0.733503\pi$
$29$	−209.119	−1.33905	−0.669526	+	0.742789i	$0.733503\pi$
$30$	−229.802	−1.39853
$31$	−85.9638	−0.498050	−0.249025	−	0.968497i	$-0.580110\pi$
$31$	−85.9638	−0.498050	−0.249025	+	0.968497i	$0.580110\pi$
$32$	−206.653	−1.14161
$33$	−205.127	−1.08206
$34$	−393.626	−1.98548
$35$	266.745	1.28823
$36$	110.268	0.510499
$37$	361.620	1.60676	0.803378	−	0.595469i	$-0.203034\pi$
$37$	361.620	1.60676	0.803378	+	0.595469i	$0.203034\pi$
$38$	473.593	2.02176
$39$	216.545	0.889103
$40$	325.705	1.28746
$41$	301.706	1.14923	0.574617	−	0.818422i	$-0.305151\pi$
$41$	301.706	1.14923	0.574617	+	0.818422i	$0.305151\pi$
$42$	−211.569	−0.777282
$43$	−295.258	−1.04713	−0.523563	−	0.851987i	$-0.675397\pi$
$43$	−295.258	−1.04713	−0.523563	+	0.851987i	$0.675397\pi$
$44$	837.736	2.87031
$45$	153.194	0.507485
$46$	335.230	1.07450
$47$	369.530	1.14684	0.573419	−	0.819262i	$-0.305617\pi$
$47$	369.530	1.14684	0.573419	+	0.819262i	$0.305617\pi$
$48$	35.7142	0.107394
$49$	−97.4191	−0.284021
$50$	741.336	2.09681
$51$	262.405	0.720471
$52$	−884.370	−2.35846
$53$	−474.006	−1.22849	−0.614243	−	0.789117i	$-0.710539\pi$
$53$	−474.006	−1.22849	−0.614243	+	0.789117i	$0.710539\pi$
$54$	−121.506	−0.306201
$55$	1163.86	2.85336
$56$	299.863	0.715552
$57$	−315.713	−0.733636
$58$	−941.083	−2.13052
$59$	22.1917	0.0489680	0.0244840	−	0.999700i	$-0.492206\pi$
$59$	22.1917	0.0489680	0.0244840	+	0.999700i	$0.492206\pi$
$60$	−625.643	−1.34617
$61$	765.268	1.60627	0.803135	−	0.595797i	$-0.203164\pi$
$61$	765.268	1.60627	0.803135	+	0.595797i	$0.203164\pi$
$62$	−386.856	−0.792432
$63$	141.039	0.282052
$64$	−834.746	−1.63036
$65$	−1228.65	−2.34453
$66$	−923.115	−1.72163
$67$	−18.4848	−0.0337056	−0.0168528	−	0.999858i	$-0.505365\pi$
$67$	−18.4848	−0.0337056	−0.0168528	+	0.999858i	$0.505365\pi$
$68$	−1071.66	−1.91114
$69$	−223.476	−0.389903
$70$	1200.41	2.04967
$71$	−314.721	−0.526063	−0.263031	−	0.964787i	$-0.584722\pi$
$71$	−314.721	−0.526063	−0.263031	+	0.964787i	$0.584722\pi$
$72$	172.214	0.281883
$73$	935.937	1.50059	0.750295	−	0.661103i	$-0.229911\pi$
$73$	935.937	1.50059	0.750295	+	0.661103i	$0.229911\pi$
$74$	1627.37	2.55646
$75$	−494.200	−0.760870
$76$	1289.37	1.94607
$77$	1071.52	1.58585
$78$	974.502	1.41462
$79$	−903.178	−1.28627	−0.643136	−	0.765752i	$-0.722367\pi$
$79$	−903.178	−1.28627	−0.643136	+	0.765752i	$0.722367\pi$
$80$	−202.637	−0.283194
$81$	81.0000	0.111111
$82$	1357.74	1.82851
$83$	−270.922	−0.358284	−0.179142	−	0.983823i	$-0.557332\pi$
$83$	−270.922	−0.358284	−0.179142	+	0.983823i	$0.557332\pi$
$84$	−576.003	−0.748180
$85$	−1488.84	−1.89986
$86$	−1328.73	−1.66605
$87$	627.358	0.773102
$88$	1308.36	1.58490
$89$	−1288.35	−1.53444	−0.767218	−	0.641387i	$-0.778359\pi$
$89$	−1288.35	−1.53444	−0.767218	+	0.641387i	$0.778359\pi$
$90$	689.407	0.807443
$91$	−1131.16	−1.30306
$92$	912.675	1.03427
$93$	257.891	0.287549
$94$	1662.96	1.82470
$95$	1791.31	1.93457
$96$	619.959	0.659107
$97$	−195.525	−0.204666	−0.102333	−	0.994750i	$-0.532631\pi$
$97$	−195.525	−0.204666	−0.102333	+	0.994750i	$0.532631\pi$
$98$	−438.407	−0.451896
$99$	615.380	0.624728
$100$	2018.31	2.01831
$101$	−184.531	−0.181797	−0.0908986	−	0.995860i	$-0.528974\pi$
$101$	−184.531	−0.181797	−0.0908986	+	0.995860i	$0.528974\pi$
$102$	1180.88	1.14632
$103$	288.625	0.276108	0.138054	−	0.990425i	$-0.455915\pi$
$103$	288.625	0.276108	0.138054	+	0.990425i	$0.455915\pi$
$104$	−1381.19	−1.30228
$105$	−800.235	−0.743762
$106$	−2133.13	−1.95461
$107$	82.1856	0.0742541	0.0371270	−	0.999311i	$-0.488179\pi$
$107$	82.1856	0.0742541	0.0371270	+	0.999311i	$0.488179\pi$
$108$	−330.804	−0.294737
$109$	−1439.05	−1.26455	−0.632277	−	0.774742i	$-0.717880\pi$
$109$	−1439.05	−1.26455	−0.632277	+	0.774742i	$0.717880\pi$
$110$	5237.62	4.53988
$111$	−1084.86	−0.927661
$112$	−186.559	−0.157395
$113$	−1930.17	−1.60686	−0.803429	−	0.595400i	$-0.796994\pi$
$113$	−1930.17	−1.60686	−0.803429	+	0.595400i	$0.796994\pi$
$114$	−1420.78	−1.16726
$115$	1267.97	1.02816
$116$	−2562.13	−2.05076
$117$	−649.636	−0.513324
$118$	99.8675	0.0779115
$119$	−1370.72	−1.05591
$120$	−977.116	−0.743317
$121$	3344.22	2.51256
$122$	3443.87	2.55569
$123$	−905.119	−0.663511
$124$	−1053.23	−0.762763
$125$	676.320	0.483936
$126$	634.708	0.448764
$127$	−537.498	−0.375553	−0.187776	−	0.982212i	$-0.560128\pi$
$127$	−537.498	−0.375553	−0.187776	+	0.982212i	$0.560128\pi$
$128$	−2103.32	−1.45241
$129$	885.774	0.604558
$130$	−5529.18	−3.73031
$131$	1305.60	0.870767	0.435384	−	0.900245i	$-0.356613\pi$
$131$	1305.60	0.870767	0.435384	+	0.900245i	$0.356613\pi$
$132$	−2513.21	−1.65717
$133$	1649.18	1.07521
$134$	−83.1855	−0.0536279
$135$	−459.582	−0.292996
$136$	−1673.69	−1.05528
$137$	−964.034	−0.601189	−0.300595	−	0.953752i	$-0.597185\pi$
$137$	−964.034	−0.601189	−0.300595	+	0.953752i	$0.597185\pi$
$138$	−1005.69	−0.620363
$139$	−417.817	−0.254955	−0.127478	−	0.991841i	$-0.540688\pi$
$139$	−417.817	−0.254955	−0.127478	+	0.991841i	$0.540688\pi$
$140$	3268.16	1.97293
$141$	−1108.59	−0.662128
$142$	−1416.31	−0.837002
$143$	−4935.47	−2.88619
$144$	−107.143	−0.0620038
$145$	−3559.54	−2.03864
$146$	4211.92	2.38754
$147$	292.257	0.163979
$148$	4430.56	2.46074
$149$	−2722.61	−1.49694	−0.748472	−	0.663166i	$-0.769212\pi$
$149$	−2722.61	−1.49694	−0.748472	+	0.663166i	$0.769212\pi$
$150$	−2224.01	−1.21060
$151$	565.212	0.304611	0.152306	−	0.988333i	$-0.451330\pi$
$151$	565.212	0.304611	0.152306	+	0.988333i	$0.451330\pi$
$152$	2013.71	1.07456
$153$	−787.214	−0.415964
$154$	4822.05	2.52320
$155$	−1463.24	−0.758258
$156$	2653.11	1.36166
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	−4064.50	−2.04655
$159$	1422.02	0.709267
$160$	−3517.55	−1.73804
$161$	1167.37	0.571437
$162$	364.518	0.176785
$163$	−2312.11	−1.11104	−0.555518	−	0.831505i	$-0.687480\pi$
$163$	−2312.11	−1.11104	−0.555518	+	0.831505i	$0.687480\pi$
$164$	3696.50	1.76005
$165$	−3491.57	−1.64739
$166$	−1219.21	−0.570054
$167$	−2485.26	−1.15159	−0.575795	−	0.817594i	$-0.695307\pi$
$167$	−2485.26	−1.15159	−0.575795	+	0.817594i	$0.695307\pi$
$168$	−899.589	−0.413124
$169$	3013.21	1.37151
$170$	−6700.13	−3.02280
$171$	947.140	0.423565
$172$	−3617.50	−1.60367
$173$	−596.008	−0.261929	−0.130964	−	0.991387i	$-0.541807\pi$
$173$	−596.008	−0.261929	−0.130964	+	0.991387i	$0.541807\pi$
$174$	2823.25	1.23006
$175$	2581.54	1.11512
$176$	−813.992	−0.348619
$177$	−66.5751	−0.0282717
$178$	−5797.85	−2.44139
$179$	4291.94	1.79215	0.896074	−	0.443906i	$-0.146407\pi$
$179$	4291.94	1.79215	0.896074	+	0.443906i	$0.146407\pi$
$180$	1876.93	0.777212
$181$	−110.790	−0.0454968	−0.0227484	−	0.999741i	$-0.507242\pi$
$181$	−110.790	−0.0454968	−0.0227484	+	0.999741i	$0.507242\pi$
$182$	−5090.48	−2.07325
$183$	−2295.80	−0.927381
$184$	1425.39	0.571095
$185$	6155.33	2.44621
$186$	1160.57	0.457511
$187$	−5980.69	−2.33878
$188$	4527.47	1.75638
$189$	−423.118	−0.162843
$190$	8061.29	3.07804
$191$	2257.24	0.855121	0.427561	−	0.903987i	$-0.359373\pi$
$191$	2257.24	0.855121	0.427561	+	0.903987i	$0.359373\pi$
$192$	2504.24	0.941290
$193$	2979.42	1.11121	0.555604	−	0.831447i	$-0.312487\pi$
$193$	2979.42	1.11121	0.555604	+	0.831447i	$0.312487\pi$
$194$	−879.907	−0.325637
$195$	3685.94	1.35362
$196$	−1193.58	−0.434977
$197$	1427.07	0.516115	0.258057	−	0.966130i	$-0.416918\pi$
$197$	1427.07	0.516115	0.258057	+	0.966130i	$0.416918\pi$
$198$	2769.35	0.993984
$199$	−4411.07	−1.57132	−0.785660	−	0.618659i	$-0.787676\pi$
$199$	−4411.07	−1.57132	−0.785660	+	0.618659i	$0.787676\pi$
$200$	3152.15	1.11445
$201$	55.4543	0.0194599
$202$	−830.430	−0.289252
$203$	−3277.11	−1.13305
$204$	3214.98	1.10340
$205$	5135.51	1.74966
$206$	1298.88	0.439306
$207$	670.428	0.225111
$208$	859.304	0.286452
$209$	7195.69	2.38151
$210$	−3601.24	−1.18338
$211$	87.1928	0.0284483	0.0142242	−	0.999899i	$-0.495472\pi$
$211$	87.1928	0.0284483	0.0142242	+	0.999899i	$0.495472\pi$
$212$	−5807.52	−1.88142
$213$	944.162	0.303722
$214$	369.854	0.118143
$215$	−5025.75	−1.59420
$216$	−516.642	−0.162745
$217$	−1347.14	−0.421428
$218$	−6476.06	−2.01199
$219$	−2807.81	−0.866367
$220$	14259.6	4.36991
$221$	6313.61	1.92172
$222$	−4882.11	−1.47597
$223$	−3424.65	−1.02839	−0.514197	−	0.857672i	$-0.671910\pi$
$223$	−3424.65	−1.02839	−0.514197	+	0.857672i	$0.671910\pi$
$224$	−3238.46	−0.965977
$225$	1482.60	0.439288
$226$	−8686.19	−2.55662
$227$	1296.32	0.379031	0.189515	−	0.981878i	$-0.439308\pi$
$227$	1296.32	0.379031	0.189515	+	0.981878i	$0.439308\pi$
$228$	−3868.11	−1.12356
$229$	−1691.79	−0.488194	−0.244097	−	0.969751i	$-0.578491\pi$
$229$	−1691.79	−0.488194	−0.244097	+	0.969751i	$0.578491\pi$
$230$	5706.14	1.63588
$231$	−3214.55	−0.915591
$232$	−4001.47	−1.13237
$233$	589.233	0.165674	0.0828368	−	0.996563i	$-0.473602\pi$
$233$	589.233	0.165674	0.0828368	+	0.996563i	$0.473602\pi$
$234$	−2923.51	−0.816733
$235$	6289.97	1.74601
$236$	271.892	0.0749944
$237$	2709.53	0.742629
$238$	−6168.53	−1.68003
$239$	−6422.00	−1.73810	−0.869048	−	0.494729i	$-0.835267\pi$
$239$	−6422.00	−1.73810	−0.869048	+	0.494729i	$0.835267\pi$
$240$	607.911	0.163502
$241$	5121.79	1.36898	0.684489	−	0.729024i	$-0.260026\pi$
$241$	5121.79	1.36898	0.684489	+	0.729024i	$0.260026\pi$
$242$	15049.7	3.99766
$243$	−243.000	−0.0641500
$244$	9376.05	2.46000
$245$	−1658.22	−0.432408
$246$	−4073.23	−1.05569
$247$	−7596.25	−1.95683
$248$	−1644.91	−0.421176
$249$	812.766	0.206855
$250$	3043.59	0.769974
$251$	5410.22	1.36052	0.680260	−	0.732971i	$-0.261867\pi$
$251$	5410.22	1.36052	0.680260	+	0.732971i	$0.261867\pi$
$252$	1728.01	0.431962
$253$	5093.43	1.26570
$254$	−2418.86	−0.597530
$255$	4466.53	1.09688
$256$	−2787.43	−0.680524
$257$	−3766.00	−0.914072	−0.457036	−	0.889448i	$-0.651089\pi$
$257$	−3766.00	−0.914072	−0.457036	+	0.889448i	$0.651089\pi$
$258$	3986.18	0.961894
$259$	5666.96	1.35957
$260$	−15053.4	−3.59065
$261$	−1882.07	−0.446351
$262$	5875.47	1.38545
$263$	−776.989	−0.182172	−0.0910859	−	0.995843i	$-0.529034\pi$
$263$	−776.989	−0.182172	−0.0910859	+	0.995843i	$0.529034\pi$
$264$	−3925.07	−0.915044
$265$	−8068.32	−1.87031
$266$	7421.69	1.71072
$267$	3865.05	0.885906
$268$	−226.475	−0.0516200
$269$	5698.99	1.29172	0.645862	−	0.763454i	$-0.276498\pi$
$269$	5698.99	1.29172	0.645862	+	0.763454i	$0.276498\pi$
$270$	−2068.22	−0.466177
$271$	4794.46	1.07470	0.537348	−	0.843361i	$-0.319426\pi$
$271$	4794.46	1.07470	0.537348	+	0.843361i	$0.319426\pi$
$272$	1041.29	0.232122
$273$	3393.49	0.752319
$274$	−4338.36	−0.956533
$275$	11263.7	2.46992
$276$	−2738.02	−0.597136
$277$	−4917.14	−1.06658	−0.533289	−	0.845933i	$-0.679044\pi$
$277$	−4917.14	−1.06658	−0.533289	+	0.845933i	$0.679044\pi$
$278$	−1880.27	−0.405652
$279$	−773.674	−0.166017
$280$	5104.13	1.08939
$281$	4806.62	1.02042	0.510211	−	0.860049i	$-0.329567\pi$
$281$	4806.62	1.02042	0.510211	+	0.860049i	$0.329567\pi$
$282$	−4988.89	−1.05349
$283$	−2146.44	−0.450858	−0.225429	−	0.974260i	$-0.572378\pi$
$283$	−2146.44	−0.450858	−0.225429	+	0.974260i	$0.572378\pi$
$284$	−3855.95	−0.805664
$285$	−5373.93	−1.11693
$286$	−22210.7	−4.59212
$287$	4728.04	0.972431
$288$	−1859.88	−0.380536
$289$	2737.69	0.557234
$290$	−16018.7	−3.24362
$291$	586.576	0.118164
$292$	11467.1	2.29815
$293$	6378.34	1.27176	0.635881	−	0.771787i	$-0.280637\pi$
$293$	6378.34	1.27176	0.635881	+	0.771787i	$0.280637\pi$
$294$	1315.22	0.260902
$295$	377.737	0.0745515
$296$	6919.56	1.35875
$297$	−1846.14	−0.360687
$298$	−12252.3	−2.38174
$299$	−5376.96	−1.03999
$300$	−6054.93	−1.16527
$301$	−4626.99	−0.886031
$302$	2543.58	0.484658
$303$	553.593	0.104961
$304$	−1252.83	−0.236364
$305$	13026.0	2.44547
$306$	−3542.64	−0.661827
$307$	−1774.09	−0.329813	−0.164906	−	0.986309i	$-0.552732\pi$
$307$	−1774.09	−0.329813	−0.164906	+	0.986309i	$0.552732\pi$
$308$	13128.2	2.42873
$309$	−865.876	−0.159411
$310$	−6584.89	−1.20644
$311$	4926.67	0.898282	0.449141	−	0.893461i	$-0.351730\pi$
$311$	4926.67	0.898282	0.449141	+	0.893461i	$0.351730\pi$
$312$	4143.57	0.751870
$313$	−5228.23	−0.944143	−0.472072	−	0.881560i	$-0.656494\pi$
$313$	−5228.23	−0.944143	−0.472072	+	0.881560i	$0.656494\pi$
$314$	−706.535	−0.126981
$315$	2400.71	0.429411
$316$	−11065.7	−1.96992
$317$	8060.73	1.42819	0.714094	−	0.700049i	$-0.246839\pi$
$317$	8060.73	1.42819	0.714094	+	0.700049i	$0.246839\pi$
$318$	6399.40	1.12849
$319$	−14298.7	−2.50963
$320$	−14208.7	−2.48215
$321$	−246.557	−0.0428706
$322$	5253.40	0.909195
$323$	−9204.96	−1.58569
$324$	992.411	0.170166
$325$	−11890.7	−2.02947
$326$	−10405.0	−1.76773
$327$	4317.16	0.730091
$328$	5773.11	0.971850
$329$	5790.91	0.970404
$330$	−15712.9	−2.62110
$331$	−4306.59	−0.715141	−0.357570	−	0.933886i	$-0.616395\pi$
$331$	−4306.59	−0.715141	−0.357570	+	0.933886i	$0.616395\pi$
$332$	−3319.33	−0.548711
$333$	3254.58	0.535585
$334$	−11184.2	−1.83226
$335$	−314.639	−0.0513152
$336$	559.678	0.0908718
$337$	−8071.47	−1.30469	−0.652346	−	0.757921i	$-0.726215\pi$
$337$	−8071.47	−1.30469	−0.652346	+	0.757921i	$0.726215\pi$
$338$	13560.1	2.18217
$339$	5790.51	0.927720
$340$	−18241.3	−2.90963
$341$	−5877.82	−0.933437
$342$	4262.34	0.673920
$343$	−6901.82	−1.08648
$344$	−5649.72	−0.885502
$345$	−3803.91	−0.593610
$346$	−2682.17	−0.416746
$347$	591.719	0.0915422	0.0457711	−	0.998952i	$-0.485426\pi$
$347$	591.719	0.0915422	0.0457711	+	0.998952i	$0.485426\pi$
$348$	7686.38	1.18400
$349$	6842.08	1.04942	0.524711	−	0.851280i	$-0.324173\pi$
$349$	6842.08	1.04942	0.524711	+	0.851280i	$0.324173\pi$
$350$	11617.5	1.77423
$351$	1948.91	0.296368
$352$	−14130.0	−2.13958
$353$	2622.70	0.395445	0.197722	−	0.980258i	$-0.436646\pi$
$353$	2622.70	0.395445	0.197722	+	0.980258i	$0.436646\pi$
$354$	−299.603	−0.0449822
$355$	−5357.03	−0.800906
$356$	−15784.8	−2.34998
$357$	4112.15	0.609630
$358$	19314.7	2.85143
$359$	2705.85	0.397797	0.198899	−	0.980020i	$-0.436264\pi$
$359$	2705.85	0.397797	0.198899	+	0.980020i	$0.436264\pi$
$360$	2931.35	0.429154
$361$	4215.98	0.614664
$362$	−498.578	−0.0723886
$363$	−10032.7	−1.45063
$364$	−13859.0	−1.99563
$365$	15931.1	2.28458
$366$	−10331.6	−1.47553
$367$	7386.61	1.05062	0.525310	−	0.850911i	$-0.323949\pi$
$367$	7386.61	1.05062	0.525310	+	0.850911i	$0.323949\pi$
$368$	−886.806	−0.125620
$369$	2715.36	0.383078
$370$	27700.4	3.89209
$371$	−7428.16	−1.03949
$372$	3159.68	0.440381
$373$	3121.88	0.433365	0.216682	−	0.976242i	$-0.430476\pi$
$373$	3121.88	0.433365	0.216682	+	0.976242i	$0.430476\pi$
$374$	−26914.4	−3.72115
$375$	−2028.96	−0.279400
$376$	7070.90	0.969824
$377$	15094.6	2.06210
$378$	−1904.12	−0.259094
$379$	4441.22	0.601926	0.300963	−	0.953636i	$-0.402692\pi$
$379$	4441.22	0.601926	0.300963	+	0.953636i	$0.402692\pi$
$380$	21947.1	2.96280
$381$	1612.49	0.216825
$382$	10158.1	1.36056
$383$	−7723.68	−1.03045	−0.515224	−	0.857056i	$-0.672291\pi$
$383$	−7723.68	−1.03045	−0.515224	+	0.857056i	$0.672291\pi$
$384$	6309.95	0.838550
$385$	18238.8	2.41438
$386$	13408.0	1.76801
$387$	−2657.32	−0.349042
$388$	−2395.57	−0.313445
$389$	9924.15	1.29351	0.646753	−	0.762699i	$-0.276126\pi$
$389$	9924.15	1.29351	0.646753	+	0.762699i	$0.276126\pi$
$390$	16587.5	2.15370
$391$	−6515.68	−0.842742
$392$	−1864.10	−0.240182
$393$	−3916.79	−0.502738
$394$	6422.14	0.821174
$395$	−15373.5	−1.95829
$396$	7539.63	0.956769
$397$	−8641.50	−1.09245	−0.546227	−	0.837637i	$-0.683936\pi$
$397$	−8641.50	−1.09245	−0.546227	+	0.837637i	$0.683936\pi$
$398$	−19850.8	−2.50008
$399$	−4947.55	−0.620770
$400$	−1961.10	−0.245138
$401$	24.0317	0.00299273	0.00149636	−	0.999999i	$-0.499524\pi$
$401$	24.0317	0.00299273	0.00149636	+	0.999999i	$0.499524\pi$
$402$	249.557	0.0309621
$403$	6205.02	0.766983
$404$	−2260.87	−0.278422
$405$	1378.75	0.169162
$406$	−14747.7	−1.80275
$407$	24726.0	3.01135
$408$	5021.08	0.609266
$409$	11649.7	1.40841	0.704204	−	0.709997i	$-0.251304\pi$
$409$	11649.7	1.40841	0.704204	+	0.709997i	$0.251304\pi$
$410$	23110.9	2.78382
$411$	2892.10	0.347097
$412$	3536.23	0.422859
$413$	347.767	0.0414346
$414$	3017.07	0.358167
$415$	−4611.52	−0.545471
$416$	14916.6	1.75804
$417$	1253.45	0.147199
$418$	32382.2	3.78915
$419$	4563.32	0.532059	0.266030	−	0.963965i	$-0.414288\pi$
$419$	4563.32	0.532059	0.266030	+	0.963965i	$0.414288\pi$
$420$	−9804.47	−1.13907
$421$	9222.05	1.06759	0.533795	−	0.845614i	$-0.320766\pi$
$421$	9222.05	1.06759	0.533795	+	0.845614i	$0.320766\pi$
$422$	392.387	0.0452632
$423$	3325.77	0.382280
$424$	−9070.05	−1.03887
$425$	−14408.9	−1.64455
$426$	4248.94	0.483243
$427$	11992.5	1.35915
$428$	1006.94	0.113720
$429$	14806.4	1.66634
$430$	−22617.0	−2.53648
$431$	−14282.8	−1.59624	−0.798119	−	0.602500i	$-0.794171\pi$
$431$	−14282.8	−1.59624	−0.798119	+	0.602500i	$0.794171\pi$
$432$	321.428	0.0357979
$433$	−8233.26	−0.913777	−0.456888	−	0.889524i	$-0.651036\pi$
$433$	−8233.26	−0.913777	−0.456888	+	0.889524i	$0.651036\pi$
$434$	−6062.43	−0.670520
$435$	10678.6	1.17701
$436$	−17631.3	−1.93666
$437$	7839.37	0.858141
$438$	−12635.8	−1.37845
$439$	−5530.94	−0.601315	−0.300658	−	0.953732i	$-0.597206\pi$
$439$	−5530.94	−0.601315	−0.300658	+	0.953732i	$0.597206\pi$
$440$	22270.3	2.41294
$441$	−876.771	−0.0946735
$442$	28412.6	3.05758
$443$	−4148.54	−0.444928	−0.222464	−	0.974941i	$-0.571410\pi$
$443$	−4148.54	−0.444928	−0.222464	+	0.974941i	$0.571410\pi$
$444$	−13291.7	−1.42071
$445$	−21929.7	−2.33611
$446$	−15411.7	−1.63624
$447$	8167.82	0.864261
$448$	−13081.3	−1.37954
$449$	−7651.15	−0.804187	−0.402093	−	0.915599i	$-0.631717\pi$
$449$	−7651.15	−0.804187	−0.402093	+	0.915599i	$0.631717\pi$
$450$	6672.02	0.698938
$451$	20629.3	2.15387
$452$	−23648.4	−2.46090
$453$	−1695.64	−0.175867
$454$	5833.74	0.603064
$455$	−19254.1	−1.98384
$456$	−6041.13	−0.620399
$457$	10686.4	1.09385	0.546925	−	0.837181i	$-0.315798\pi$
$457$	10686.4	1.09385	0.546925	+	0.837181i	$0.315798\pi$
$458$	−7613.41	−0.776749
$459$	2361.64	0.240157
$460$	15535.1	1.57463
$461$	12.1860	0.00123115	0.000615573	−	1.00000i	$-0.499804\pi$
$461$	12.1860	0.00123115	0.000615573		1.00000i	$0.499804\pi$
$462$	−14466.2	−1.45677
$463$	1732.43	0.173894	0.0869471	−	0.996213i	$-0.472289\pi$
$463$	1732.43	0.173894	0.0869471	+	0.996213i	$0.472289\pi$
$464$	2489.51	0.249079
$465$	4389.71	0.437781
$466$	2651.68	0.263598
$467$	3635.88	0.360276	0.180138	−	0.983641i	$-0.442346\pi$
$467$	3635.88	0.360276	0.180138	+	0.983641i	$0.442346\pi$
$468$	−7959.33	−0.786154
$469$	−289.675	−0.0285202
$470$	28306.2	2.77802
$471$	471.000	0.0460776
$472$	424.635	0.0414098
$473$	−20188.4	−1.96250
$474$	12193.5	1.18157
$475$	17336.1	1.67460
$476$	−16794.0	−1.61713
$477$	−4266.06	−0.409495
$478$	−28900.4	−2.76543
$479$	11652.6	1.11153	0.555763	−	0.831341i	$-0.312426\pi$
$479$	11652.6	1.11153	0.555763	+	0.831341i	$0.312426\pi$
$480$	10552.7	1.00346
$481$	−26102.4	−2.47436
$482$	23049.2	2.17814
$483$	−3502.10	−0.329919
$484$	40973.3	3.84798
$485$	−3328.14	−0.311594
$486$	−1093.55	−0.102067
$487$	−10838.8	−1.00853	−0.504263	−	0.863550i	$-0.668236\pi$
$487$	−10838.8	−1.00853	−0.504263	+	0.863550i	$0.668236\pi$
$488$	14643.3	1.35834
$489$	6936.34	0.641456
$490$	−7462.37	−0.687991
$491$	−364.931	−0.0335420	−0.0167710	−	0.999859i	$-0.505339\pi$
$491$	−364.931	−0.0335420	−0.0167710	+	0.999859i	$0.505339\pi$
$492$	−11089.5	−1.01617
$493$	18291.3	1.67099
$494$	−34184.8	−3.11345
$495$	10474.7	0.951119
$496$	1023.38	0.0926429
$497$	−4931.99	−0.445131
$498$	3657.63	0.329121
$499$	−20464.5	−1.83590	−0.917951	−	0.396694i	$-0.870157\pi$
$499$	−20464.5	−1.83590	−0.917951	+	0.396694i	$0.870157\pi$
$500$	8286.27	0.741146
$501$	7455.79	0.664870
$502$	24347.2	2.16468
$503$	−107.236	−0.00950582	−0.00475291	−	0.999989i	$-0.501513\pi$
$503$	−107.236	−0.00950582	−0.00475291	+	0.999989i	$0.501513\pi$
$504$	2698.77	0.238517
$505$	−3141.00	−0.276778
$506$	22921.6	2.01381
$507$	−9039.63	−0.791842
$508$	−6585.41	−0.575158
$509$	−3085.67	−0.268703	−0.134352	−	0.990934i	$-0.542895\pi$
$509$	−3085.67	−0.268703	−0.134352	+	0.990934i	$0.542895\pi$
$510$	20100.4	1.74522
$511$	14667.1	1.26973
$512$	4282.51	0.369652
$513$	−2841.42	−0.244545
$514$	−16947.8	−1.45435
$515$	4912.85	0.420361
$516$	10852.5	0.925880
$517$	25266.8	2.14939
$518$	25502.6	2.16316
$519$	1788.02	0.151225
$520$	−23510.0	−1.98266
$521$	15449.7	1.29917	0.649583	−	0.760291i	$-0.274944\pi$
$521$	15449.7	1.29917	0.649583	+	0.760291i	$0.274944\pi$
$522$	−8469.75	−0.710174
$523$	289.507	0.0242051	0.0121025	−	0.999927i	$-0.496148\pi$
$523$	289.507	0.0242051	0.0121025	+	0.999927i	$0.496148\pi$
$524$	15996.2	1.33358
$525$	−7744.61	−0.643814
$526$	−3496.62	−0.289848
$527$	7519.10	0.621513
$528$	2441.98	0.201275
$529$	−6617.95	−0.543926
$530$	−36309.2	−2.97580
$531$	199.725	0.0163227
$532$	20205.8	1.64667
$533$	−21777.7	−1.76979
$534$	17393.6	1.40954
$535$	1398.93	0.113048
$536$	−353.704	−0.0285031
$537$	−12875.8	−1.03470
$538$	25646.7	2.05522
$539$	−6661.08	−0.532306
$540$	−5630.79	−0.448723
$541$	4045.92	0.321530	0.160765	−	0.986993i	$-0.448604\pi$
$541$	4045.92	0.321530	0.160765	+	0.986993i	$0.448604\pi$
$542$	21576.1	1.70991
$543$	332.369	0.0262676
$544$	18075.6	1.42460
$545$	−24494.9	−1.92523
$546$	15271.4	1.19699
$547$	5019.87	0.392384	0.196192	−	0.980565i	$-0.437142\pi$
$547$	5019.87	0.392384	0.196192	+	0.980565i	$0.437142\pi$
$548$	−11811.3	−0.920720
$549$	6887.41	0.535423
$550$	50689.2	3.92981
$551$	−22007.3	−1.70153
$552$	−4276.18	−0.329722
$553$	−14153.7	−1.08839
$554$	−22128.2	−1.69700
$555$	−18466.0	−1.41232
$556$	−5119.09	−0.390464
$557$	11293.5	0.859101	0.429551	−	0.903043i	$-0.358672\pi$
$557$	11293.5	0.859101	0.429551	+	0.903043i	$0.358672\pi$
$558$	−3481.70	−0.264144
$559$	21312.2	1.61254
$560$	−3175.53	−0.239626
$561$	17942.1	1.35029
$562$	21630.8	1.62356
$563$	14455.5	1.08211	0.541053	−	0.840989i	$-0.318026\pi$
$563$	14455.5	1.08211	0.541053	+	0.840989i	$0.318026\pi$
$564$	−13582.4	−1.01405
$565$	−32854.5	−2.44637
$566$	−9659.47	−0.717346
$567$	1269.35	0.0940173
$568$	−6022.14	−0.444865
$569$	10536.8	0.776316	0.388158	−	0.921593i	$-0.373111\pi$
$569$	10536.8	0.776316	0.388158	+	0.921593i	$0.373111\pi$
$570$	−24183.9	−1.77711
$571$	−2894.30	−0.212124	−0.106062	−	0.994360i	$-0.533824\pi$
$571$	−2894.30	−0.212124	−0.106062	+	0.994360i	$0.533824\pi$
$572$	−60469.3	−4.42019
$573$	−6771.72	−0.493704
$574$	21277.2	1.54720
$575$	12271.3	0.889998
$576$	−7512.71	−0.543454
$577$	213.981	0.0154387	0.00771935	−	0.999970i	$-0.497543\pi$
$577$	213.981	0.0154387	0.00771935	+	0.999970i	$0.497543\pi$
$578$	12320.2	0.886597
$579$	−8938.25	−0.641556
$580$	−43611.4	−3.12218
$581$	−4245.63	−0.303164
$582$	2639.72	0.188007
$583$	−32410.4	−2.30241
$584$	17909.0	1.26897
$585$	−11057.8	−0.781512
$586$	28703.9	2.02346
$587$	18297.6	1.28658	0.643289	−	0.765623i	$-0.277569\pi$
$587$	18297.6	1.28658	0.643289	+	0.765623i	$0.277569\pi$
$588$	3580.73	0.251134
$589$	−9046.63	−0.632869
$590$	1699.90	0.118617
$591$	−4281.21	−0.297979
$592$	−4304.99	−0.298875
$593$	16316.0	1.12988	0.564939	−	0.825133i	$-0.308900\pi$
$593$	16316.0	1.12988	0.564939	+	0.825133i	$0.308900\pi$
$594$	−8308.04	−0.573877
$595$	−23331.7	−1.60758
$596$	−33357.3	−2.29257
$597$	13233.2	0.907202
$598$	−24197.5	−1.65470
$599$	12990.1	0.886081	0.443040	−	0.896502i	$-0.353900\pi$
$599$	12990.1	0.886081	0.443040	+	0.896502i	$0.353900\pi$
$600$	−9456.45	−0.643430
$601$	24784.6	1.68217	0.841084	−	0.540904i	$-0.181918\pi$
$601$	24784.6	1.68217	0.841084	+	0.540904i	$0.181918\pi$
$602$	−20822.5	−1.40974
$603$	−166.363	−0.0112352
$604$	6924.97	0.466512
$605$	56923.8	3.82526
$606$	2491.29	0.166999
$607$	25183.7	1.68398	0.841990	−	0.539493i	$-0.181384\pi$
$607$	25183.7	1.68398	0.841990	+	0.539493i	$0.181384\pi$
$608$	−21747.7	−1.45063
$609$	9831.34	0.654164
$610$	58620.1	3.89091
$611$	−26673.3	−1.76610
$612$	−9644.94	−0.637048
$613$	−17007.1	−1.12057	−0.560286	−	0.828299i	$-0.689309\pi$
$613$	−17007.1	−1.12057	−0.560286	+	0.828299i	$0.689309\pi$
$614$	−7983.78	−0.524755
$615$	−15406.5	−1.01016
$616$	20503.3	1.34107
$617$	−8961.04	−0.584696	−0.292348	−	0.956312i	$-0.594437\pi$
$617$	−8961.04	−0.584696	−0.292348	+	0.956312i	$0.594437\pi$
$618$	−3896.63	−0.253634
$619$	22710.3	1.47464	0.737322	−	0.675541i	$-0.236090\pi$
$619$	22710.3	1.47464	0.737322	+	0.675541i	$0.236090\pi$
$620$	−17927.6	−1.16127
$621$	−2011.28	−0.129968
$622$	22171.1	1.42923
$623$	−20189.7	−1.29837
$624$	−2577.91	−0.165383
$625$	−9079.63	−0.581096
$626$	−23528.2	−1.50220
$627$	−21587.1	−1.37497
$628$	−1923.56	−0.122227
$629$	−31630.3	−2.00506
$630$	10803.7	0.683222
$631$	−4067.19	−0.256596	−0.128298	−	0.991736i	$-0.540951\pi$
$631$	−4067.19	−0.256596	−0.128298	+	0.991736i	$0.540951\pi$
$632$	−17282.2	−1.08774
$633$	−261.578	−0.0164247
$634$	36275.1	2.27235
$635$	−9149.04	−0.571762
$636$	17422.6	1.08624
$637$	7031.88	0.437383
$638$	−64347.1	−3.99299
$639$	−2832.48	−0.175354
$640$	−35801.7	−2.21123
$641$	−24069.3	−1.48312	−0.741559	−	0.670887i	$-0.765913\pi$
$641$	−24069.3	−1.48312	−0.741559	+	0.670887i	$0.765913\pi$
$642$	−1109.56	−0.0682101
$643$	−19868.3	−1.21855	−0.609276	−	0.792959i	$-0.708540\pi$
$643$	−19868.3	−1.21855	−0.609276	+	0.792959i	$0.708540\pi$
$644$	14302.5	0.875154
$645$	15077.2	0.920412
$646$	−41424.3	−2.52294
$647$	21723.7	1.32001	0.660005	−	0.751262i	$-0.270554\pi$
$647$	21723.7	1.32001	0.660005	+	0.751262i	$0.270554\pi$
$648$	1549.92	0.0939611
$649$	1517.37	0.0917750
$650$	−53510.9	−3.22903
$651$	4041.42	0.243311
$652$	−28328.0	−1.70155
$653$	2059.75	0.123437	0.0617185	−	0.998094i	$-0.480342\pi$
$653$	2059.75	0.123437	0.0617185	+	0.998094i	$0.480342\pi$
$654$	19428.2	1.16162
$655$	22223.3	1.32570
$656$	−3591.73	−0.213771
$657$	8423.43	0.500197
$658$	26060.4	1.54398
$659$	12769.0	0.754792	0.377396	−	0.926052i	$-0.376820\pi$
$659$	12769.0	0.754792	0.377396	+	0.926052i	$0.376820\pi$
$660$	−42778.7	−2.52297
$661$	−24282.3	−1.42885	−0.714427	−	0.699710i	$-0.753312\pi$
$661$	−24282.3	−1.42885	−0.714427	+	0.699710i	$0.753312\pi$
$662$	−19380.6	−1.13784
$663$	−18940.8	−1.10950
$664$	−5184.06	−0.302983
$665$	28071.7	1.63695
$666$	14646.3	0.852153
$667$	−15577.7	−0.904305
$668$	−30449.4	−1.76366
$669$	10274.0	0.593743
$670$	−1415.95	−0.0816460
$671$	52325.6	3.01044
$672$	9715.39	0.557707
$673$	20729.6	1.18732	0.593661	−	0.804715i	$-0.297682\pi$
$673$	20729.6	1.18732	0.593661	+	0.804715i	$0.297682\pi$
$674$	−36323.4	−2.07585
$675$	−4447.80	−0.253623
$676$	36917.8	2.10047
$677$	6690.76	0.379833	0.189916	−	0.981800i	$-0.439178\pi$
$677$	6690.76	0.379833	0.189916	+	0.981800i	$0.439178\pi$
$678$	26058.6	1.47607
$679$	−3064.08	−0.173179
$680$	−28488.9	−1.60661
$681$	−3888.97	−0.218833
$682$	−26451.5	−1.48516
$683$	−2606.58	−0.146029	−0.0730145	−	0.997331i	$-0.523262\pi$
$683$	−2606.58	−0.146029	−0.0730145	+	0.997331i	$0.523262\pi$
$684$	11604.3	0.648689
$685$	−16409.3	−0.915283
$686$	−31059.7	−1.72867
$687$	5075.36	0.281859
$688$	3514.96	0.194777
$689$	34214.6	1.89183
$690$	−17118.4	−0.944474
$691$	−7243.88	−0.398799	−0.199400	−	0.979918i	$-0.563899\pi$
$691$	−7243.88	−0.398799	−0.199400	+	0.979918i	$0.563899\pi$
$692$	−7302.28	−0.401143
$693$	9643.64	0.528617
$694$	2662.87	0.145650
$695$	−7111.90	−0.388158
$696$	12004.4	0.653774
$697$	−26389.7	−1.43412
$698$	30790.9	1.66970
$699$	−1767.70	−0.0956517
$700$	31629.0	1.70780
$701$	−14559.9	−0.784480	−0.392240	−	0.919863i	$-0.628300\pi$
$701$	−14559.9	−0.784480	−0.392240	+	0.919863i	$0.628300\pi$
$702$	8770.52	0.471541
$703$	38056.1	2.04170
$704$	−57076.2	−3.05560
$705$	−18869.9	−1.00806
$706$	11802.7	0.629180
$707$	−2891.79	−0.153829
$708$	−815.677	−0.0432981
$709$	16345.1	0.865804	0.432902	−	0.901441i	$-0.357490\pi$
$709$	16345.1	0.865804	0.432902	+	0.901441i	$0.357490\pi$
$710$	−24107.8	−1.27430
$711$	−8128.60	−0.428757
$712$	−24652.4	−1.29760
$713$	−6403.61	−0.336349
$714$	18505.6	0.969964
$715$	−84009.4	−4.39409
$716$	52584.7	2.74467
$717$	19266.0	1.00349
$718$	12176.9	0.632923
$719$	−30412.3	−1.57745	−0.788726	−	0.614745i	$-0.789259\pi$
$719$	−30412.3	−1.57745	−0.788726	+	0.614745i	$0.789259\pi$
$720$	−1823.73	−0.0943979
$721$	4523.06	0.233630
$722$	18972.8	0.977972
$723$	−15365.4	−0.790379
$724$	−1357.39	−0.0696783
$725$	−34448.9	−1.76469
$726$	−45149.2	−2.30805
$727$	−25938.7	−1.32327	−0.661633	−	0.749828i	$-0.730136\pi$
$727$	−25938.7	−1.32327	−0.661633	+	0.749828i	$0.730136\pi$
$728$	−21644.7	−1.10193
$729$	729.000	0.0370370
$730$	71693.5	3.63492
$731$	25825.7	1.30670
$732$	−28128.1	−1.42028
$733$	25421.1	1.28097	0.640485	−	0.767971i	$-0.278733\pi$
$733$	25421.1	1.28097	0.640485	+	0.767971i	$0.278733\pi$
$734$	33241.4	1.67161
$735$	4974.67	0.249651
$736$	−15394.0	−0.770964
$737$	−1263.91	−0.0631704
$738$	12219.7	0.609503
$739$	−5871.09	−0.292248	−0.146124	−	0.989266i	$-0.546680\pi$
$739$	−5871.09	−0.292248	−0.146124	+	0.989266i	$0.546680\pi$
$740$	75415.1	3.74637
$741$	22788.7	1.12978
$742$	−33428.4	−1.65390
$743$	−12646.7	−0.624446	−0.312223	−	0.950009i	$-0.601074\pi$
$743$	−12646.7	−0.624446	−0.312223	+	0.950009i	$0.601074\pi$
$744$	4934.72	0.243166
$745$	−46343.0	−2.27903
$746$	14049.2	0.689513
$747$	−2438.30	−0.119428
$748$	−73275.3	−3.58183
$749$	1287.93	0.0628305
$750$	−9130.77	−0.444545
$751$	31508.1	1.53095	0.765477	−	0.643463i	$-0.222503\pi$
$751$	31508.1	1.53095	0.765477	+	0.643463i	$0.222503\pi$
$752$	−4399.15	−0.213325
$753$	−16230.7	−0.785496
$754$	67929.1	3.28094
$755$	9620.79	0.463757
$756$	−5184.03	−0.249393
$757$	−9258.64	−0.444533	−0.222266	−	0.974986i	$-0.571345\pi$
$757$	−9258.64	−0.444533	−0.222266	+	0.974986i	$0.571345\pi$
$758$	19986.5	0.957705
$759$	−15280.3	−0.730750
$760$	34276.5	1.63597
$761$	−16603.9	−0.790920	−0.395460	−	0.918483i	$-0.629415\pi$
$761$	−16603.9	−0.790920	−0.395460	+	0.918483i	$0.629415\pi$
$762$	7256.57	0.344984
$763$	−22551.5	−1.07001
$764$	27655.7	1.30962
$765$	−13399.6	−0.633286
$766$	−34758.3	−1.63951
$767$	−1601.84	−0.0754093
$768$	8362.28	0.392901
$769$	15422.7	0.723222	0.361611	−	0.932329i	$-0.382227\pi$
$769$	15422.7	0.723222	0.361611	+	0.932329i	$0.382227\pi$
$770$	82078.8	3.84145
$771$	11298.0	0.527740
$772$	36503.8	1.70181
$773$	7100.50	0.330384	0.165192	−	0.986261i	$-0.447176\pi$
$773$	7100.50	0.330384	0.165192	+	0.986261i	$0.447176\pi$
$774$	−11958.5	−0.555350
$775$	−14161.1	−0.656363
$776$	−3741.35	−0.173076
$777$	−17000.9	−0.784946
$778$	44660.8	2.05806
$779$	31750.9	1.46033
$780$	45160.1	2.07306
$781$	−21519.2	−0.985938
$782$	−29322.0	−1.34086
$783$	5646.22	0.257701
$784$	1159.75	0.0528310
$785$	−2672.38	−0.121505
$786$	−17626.4	−0.799890
$787$	−19647.3	−0.889897	−0.444949	−	0.895556i	$-0.646778\pi$
$787$	−19647.3	−0.889897	−0.444949	+	0.895556i	$0.646778\pi$
$788$	17484.5	0.790429
$789$	2330.97	0.105177
$790$	−69184.1	−3.11577
$791$	−30247.7	−1.35965
$792$	11775.2	0.528301
$793$	−55238.4	−2.47361
$794$	−38888.6	−1.73817
$795$	24205.0	1.07983
$796$	−54044.4	−2.40647
$797$	−27665.2	−1.22955	−0.614774	−	0.788703i	$-0.710753\pi$
$797$	−27665.2	−1.22955	−0.614774	+	0.788703i	$0.710753\pi$
$798$	−22265.1	−0.987687
$799$	−32322.1	−1.43113
$800$	−34042.6	−1.50448
$801$	−11595.1	−0.511478
$802$	108.148	0.00476163
$803$	63995.2	2.81238
$804$	679.425	0.0298028
$805$	19870.4	0.869986
$806$	27924.0	1.22032
$807$	−17097.0	−0.745777
$808$	−3530.98	−0.153737
$809$	24507.8	1.06508	0.532538	−	0.846406i	$-0.321238\pi$
$809$	24507.8	1.06508	0.532538	+	0.846406i	$0.321238\pi$
$810$	6204.66	0.269148
$811$	36148.5	1.56516	0.782581	−	0.622549i	$-0.213903\pi$
$811$	36148.5	1.56516	0.782581	+	0.622549i	$0.213903\pi$
$812$	−40151.2	−1.73526
$813$	−14383.4	−0.620476
$814$	111272.	4.79127
$815$	−39355.8	−1.69150
$816$	−3123.86	−0.134016
$817$	−31072.3	−1.33058
$818$	52426.1	2.24087
$819$	−10180.5	−0.434352
$820$	62920.2	2.67959
$821$	9573.91	0.406981	0.203491	−	0.979077i	$-0.434771\pi$
$821$	9573.91	0.406981	0.203491	+	0.979077i	$0.434771\pi$
$822$	13015.1	0.552255
$823$	28083.9	1.18948	0.594742	−	0.803917i	$-0.297254\pi$
$823$	28083.9	1.18948	0.594742	+	0.803917i	$0.297254\pi$
$824$	5522.81	0.233491
$825$	−33791.2	−1.42601
$826$	1565.03	0.0659252
$827$	−26045.7	−1.09516	−0.547580	−	0.836754i	$-0.684451\pi$
$827$	−26045.7	−1.09516	−0.547580	+	0.836754i	$0.684451\pi$
$828$	8214.07	0.344757
$829$	1620.78	0.0679034	0.0339517	−	0.999423i	$-0.489191\pi$
$829$	1620.78	0.0679034	0.0339517	+	0.999423i	$0.489191\pi$
$830$	−20752.8	−0.867881
$831$	14751.4	0.615789
$832$	60253.4	2.51071
$833$	8521.07	0.354427
$834$	5640.81	0.234203
$835$	−42303.0	−1.75324
$836$	88161.5	3.64728
$837$	2321.02	0.0958498
$838$	20535.9	0.846542
$839$	6304.01	0.259403	0.129701	−	0.991553i	$-0.458598\pi$
$839$	6304.01	0.259403	0.129701	+	0.991553i	$0.458598\pi$
$840$	−15312.4	−0.628962
$841$	19341.9	0.793059
$842$	41501.2	1.69861
$843$	−14419.9	−0.589141
$844$	1068.28	0.0435686
$845$	51289.5	2.08806
$846$	14966.7	0.608233
$847$	52407.3	2.12602
$848$	5642.91	0.228512
$849$	6439.33	0.260303
$850$	−64843.3	−2.61660
$851$	26937.8	1.08509
$852$	11567.9	0.465150
$853$	−12914.1	−0.518370	−0.259185	−	0.965828i	$-0.583454\pi$
$853$	−12914.1	−0.518370	−0.259185	+	0.965828i	$0.583454\pi$
$854$	53969.0	2.16251
$855$	16121.8	0.644858
$856$	1572.61	0.0627930
$857$	8951.16	0.356786	0.178393	−	0.983959i	$-0.442910\pi$
$857$	8951.16	0.356786	0.178393	+	0.983959i	$0.442910\pi$
$858$	66632.1	2.65126
$859$	9766.63	0.387932	0.193966	−	0.981008i	$-0.437865\pi$
$859$	9766.63	0.387932	0.193966	+	0.981008i	$0.437865\pi$
$860$	−61575.4	−2.44152
$861$	−14184.1	−0.561433
$862$	−64275.7	−2.53972
$863$	8942.70	0.352738	0.176369	−	0.984324i	$-0.443565\pi$
$863$	8942.70	0.352738	0.176369	+	0.984324i	$0.443565\pi$
$864$	5579.63	0.219702
$865$	−10145.0	−0.398774
$866$	−37051.5	−1.45388
$867$	−8213.07	−0.321719
$868$	−16505.1	−0.645416
$869$	−61755.3	−2.41071
$870$	48056.1	1.87271
$871$	1334.26	0.0519056
$872$	−27536.1	−1.06937
$873$	−1759.73	−0.0682219
$874$	35278.9	1.36536
$875$	10598.6	0.409485
$876$	−34401.3	−1.32684
$877$	−18660.2	−0.718483	−0.359241	−	0.933245i	$-0.616965\pi$
$877$	−18660.2	−0.718483	−0.359241	+	0.933245i	$0.616965\pi$
$878$	−24890.5	−0.956734
$879$	−19135.0	−0.734252
$880$	−13855.4	−0.530757
$881$	22072.0	0.844067	0.422034	−	0.906580i	$-0.361316\pi$
$881$	22072.0	0.844067	0.422034	+	0.906580i	$0.361316\pi$
$882$	−3945.67	−0.150632
$883$	−32443.2	−1.23647	−0.618234	−	0.785994i	$-0.712151\pi$
$883$	−32443.2	−1.23647	−0.618234	+	0.785994i	$0.712151\pi$
$884$	77354.3	2.94311
$885$	−1133.21	−0.0430424
$886$	−18669.3	−0.707910
$887$	−20927.9	−0.792211	−0.396105	−	0.918205i	$-0.629638\pi$
$887$	−20927.9	−0.792211	−0.396105	+	0.918205i	$0.629638\pi$
$888$	−20758.7	−0.784477
$889$	−8423.14	−0.317776
$890$	−98688.5	−3.71690
$891$	5538.42	0.208243
$892$	−41958.8	−1.57498
$893$	38888.5	1.45728
$894$	36757.0	1.37510
$895$	73055.4	2.72846
$896$	−32961.1	−1.22897
$897$	16130.9	0.600440
$898$	−34431.8	−1.27952
$899$	17976.7	0.666915
$900$	18164.8	0.672770
$901$	41460.5	1.53302
$902$	92836.6	3.42696
$903$	13881.0	0.511550
$904$	−36933.5	−1.35884
$905$	−1885.81	−0.0692668
$906$	−7630.74	−0.279817
$907$	−30389.2	−1.11252	−0.556261	−	0.831008i	$-0.687765\pi$
$907$	−30389.2	−1.11252	−0.556261	+	0.831008i	$0.687765\pi$
$908$	15882.5	0.580485
$909$	−1660.78	−0.0605990
$910$	−86647.9	−3.15643
$911$	−21258.3	−0.773128	−0.386564	−	0.922263i	$-0.626338\pi$
$911$	−21258.3	−0.773128	−0.386564	+	0.922263i	$0.626338\pi$
$912$	3758.48	0.136465
$913$	−18524.5	−0.671490
$914$	48091.3	1.74039
$915$	−39078.1	−1.41189
$916$	−20727.7	−0.747668
$917$	20460.0	0.736805
$918$	10627.9	0.382106
$919$	39280.3	1.40994	0.704971	−	0.709236i	$-0.250960\pi$
$919$	39280.3	1.40994	0.704971	+	0.709236i	$0.250960\pi$
$920$	24262.4	0.869466
$921$	5322.26	0.190417
$922$	54.8397	0.00195884
$923$	22717.1	0.810121
$924$	−39384.6	−1.40223
$925$	59570.8	2.11749
$926$	7796.33	0.276677
$927$	2597.63	0.0920360
$928$	43215.1	1.52867
$929$	27061.8	0.955727	0.477863	−	0.878434i	$-0.341411\pi$
$929$	27061.8	0.955727	0.477863	+	0.878434i	$0.341411\pi$
$930$	19754.7	0.696539
$931$	−10252.2	−0.360903
$932$	7219.28	0.253729
$933$	−14780.0	−0.518623
$934$	16362.3	0.573223
$935$	−101801.	−3.56068
$936$	−12430.7	−0.434092
$937$	10486.2	0.365602	0.182801	−	0.983150i	$-0.441484\pi$
$937$	10486.2	0.365602	0.182801	+	0.983150i	$0.441484\pi$
$938$	−1303.60	−0.0453775
$939$	15684.7	0.545101
$940$	77064.6	2.67401
$941$	−24657.9	−0.854224	−0.427112	−	0.904199i	$-0.640469\pi$
$941$	−24657.9	−0.854224	−0.427112	+	0.904199i	$0.640469\pi$
$942$	2119.60	0.0733126
$943$	22474.7	0.776115
$944$	−264.186	−0.00910861
$945$	−7202.12	−0.247921
$946$	−90852.4	−3.12248
$947$	14732.1	0.505521	0.252760	−	0.967529i	$-0.418662\pi$
$947$	14732.1	0.505521	0.252760	+	0.967529i	$0.418662\pi$
$948$	33197.2	1.13733
$949$	−67557.6	−2.31087
$950$	78016.5	2.66441
$951$	−24182.2	−0.824565
$952$	−26228.5	−0.892931
$953$	−5393.14	−0.183317	−0.0916584	−	0.995791i	$-0.529217\pi$
$953$	−5393.14	−0.183317	−0.0916584	+	0.995791i	$0.529217\pi$
$954$	−19198.2	−0.651535
$955$	38421.7	1.30188
$956$	−78682.3	−2.66189
$957$	42896.0	1.44893
$958$	52439.2	1.76851
$959$	−15107.4	−0.508700
$960$	42626.0	1.43307
$961$	−22401.2	−0.751946
$962$	−117466.	−3.93687
$963$	739.671	0.0247514
$964$	62752.1	2.09659
$965$	50714.3	1.69176
$966$	−15760.2	−0.524924
$967$	6440.11	0.214167	0.107084	−	0.994250i	$-0.465849\pi$
$967$	6440.11	0.214167	0.107084	+	0.994250i	$0.465849\pi$
$968$	63991.2	2.12475
$969$	27614.9	0.915498
$970$	−14977.4	−0.495768
$971$	−9459.15	−0.312625	−0.156312	−	0.987708i	$-0.549961\pi$
$971$	−9459.15	−0.312625	−0.156312	+	0.987708i	$0.549961\pi$
$972$	−2977.23	−0.0982457
$973$	−6547.63	−0.215732
$974$	−48777.0	−1.60463
$975$	35672.2	1.17172
$976$	−9110.30	−0.298784
$977$	25030.7	0.819656	0.409828	−	0.912163i	$-0.365589\pi$
$977$	25030.7	0.819656	0.409828	+	0.912163i	$0.365589\pi$
$978$	31215.1	1.02060
$979$	−88091.6	−2.87581
$980$	−20316.5	−0.662232
$981$	−12951.5	−0.421518
$982$	−1642.27	−0.0533676
$983$	6045.31	0.196150	0.0980750	−	0.995179i	$-0.468731\pi$
$983$	6045.31	0.196150	0.0980750	+	0.995179i	$0.468731\pi$
$984$	−17319.3	−0.561098
$985$	24291.0	0.785761
$986$	82314.9	2.65866
$987$	−17372.7	−0.560263
$988$	−93069.1	−2.99689
$989$	−21994.3	−0.707158
$990$	47138.6	1.51329
$991$	46980.8	1.50595	0.752973	−	0.658051i	$-0.228619\pi$
$991$	46980.8	1.50595	0.752973	+	0.658051i	$0.228619\pi$
$992$	17764.7	0.568577
$993$	12919.8	0.412887
$994$	−22195.1	−0.708234
$995$	−75083.3	−2.39226
$996$	9958.00	0.316799
$997$	−38932.3	−1.23671	−0.618354	−	0.785899i	$-0.712200\pi$
$997$	−38932.3	−1.23671	−0.618354	+	0.785899i	$0.712200\pi$
$998$	−92094.6	−2.92105
$999$	−9763.74	−0.309220

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.19	✓	22
3.2	odd	2		1413.4.a.e.1.4		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.19	✓	22	1.1	even	1	trivial
1413.4.a.e.1.4		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)

\(f(q)\)	\(=\)	\(q+4.50022 q^{2} -3.00000 q^{3} +12.2520 q^{4} +17.0216 q^{5} -13.5007 q^{6} +15.6710 q^{7} +19.1349 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.50022 q^{2} -3.00000 q^{3} +12.2520 q^{4} +17.0216 q^{5} -13.5007 q^{6} +15.6710 q^{7} +19.1349 q^{8} +9.00000 q^{9} +76.6007 q^{10} +68.3756 q^{11} -36.7560 q^{12} -72.1818 q^{13} +70.5231 q^{14} -51.0647 q^{15} -11.9047 q^{16} -87.4682 q^{17} +40.5020 q^{18} +105.238 q^{19} +208.548 q^{20} -47.0131 q^{21} +307.705 q^{22} +74.4920 q^{23} -57.4046 q^{24} +164.733 q^{25} -324.834 q^{26} -27.0000 q^{27} +192.001 q^{28} -209.119 q^{29} -229.802 q^{30} -85.9638 q^{31} -206.653 q^{32} -205.127 q^{33} -393.626 q^{34} +266.745 q^{35} +110.268 q^{36} +361.620 q^{37} +473.593 q^{38} +216.545 q^{39} +325.705 q^{40} +301.706 q^{41} -211.569 q^{42} -295.258 q^{43} +837.736 q^{44} +153.194 q^{45} +335.230 q^{46} +369.530 q^{47} +35.7142 q^{48} -97.4191 q^{49} +741.336 q^{50} +262.405 q^{51} -884.370 q^{52} -474.006 q^{53} -121.506 q^{54} +1163.86 q^{55} +299.863 q^{56} -315.713 q^{57} -941.083 q^{58} +22.1917 q^{59} -625.643 q^{60} +765.268 q^{61} -386.856 q^{62} +141.039 q^{63} -834.746 q^{64} -1228.65 q^{65} -923.115 q^{66} -18.4848 q^{67} -1071.66 q^{68} -223.476 q^{69} +1200.41 q^{70} -314.721 q^{71} +172.214 q^{72} +935.937 q^{73} +1627.37 q^{74} -494.200 q^{75} +1289.37 q^{76} +1071.52 q^{77} +974.502 q^{78} -903.178 q^{79} -202.637 q^{80} +81.0000 q^{81} +1357.74 q^{82} -270.922 q^{83} -576.003 q^{84} -1488.84 q^{85} -1328.73 q^{86} +627.358 q^{87} +1308.36 q^{88} -1288.35 q^{89} +689.407 q^{90} -1131.16 q^{91} +912.675 q^{92} +257.891 q^{93} +1662.96 q^{94} +1791.31 q^{95} +619.959 q^{96} -195.525 q^{97} -438.407 q^{98} +615.380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 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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