LMFDB - Embedded newform 1045.4.a.f.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.45595 q^{2} +5.90454 q^{3} +11.8555 q^{4} -5.00000 q^{5} +26.3103 q^{6} -31.8736 q^{7} +17.1798 q^{8} +7.86359 q^{9} +O(q^{10})$	\(q+4.45595 q^{2} +5.90454 q^{3} +11.8555 q^{4} -5.00000 q^{5} +26.3103 q^{6} -31.8736 q^{7} +17.1798 q^{8} +7.86359 q^{9} -22.2797 q^{10} +11.0000 q^{11} +70.0011 q^{12} -20.2994 q^{13} -142.027 q^{14} -29.5227 q^{15} -18.2915 q^{16} -53.3119 q^{17} +35.0397 q^{18} -19.0000 q^{19} -59.2774 q^{20} -188.199 q^{21} +49.0154 q^{22} +94.3006 q^{23} +101.439 q^{24} +25.0000 q^{25} -90.4529 q^{26} -112.992 q^{27} -377.876 q^{28} -140.392 q^{29} -131.552 q^{30} +36.2541 q^{31} -218.944 q^{32} +64.9499 q^{33} -237.555 q^{34} +159.368 q^{35} +93.2265 q^{36} +115.863 q^{37} -84.6630 q^{38} -119.858 q^{39} -85.8990 q^{40} -87.4492 q^{41} -838.604 q^{42} -233.831 q^{43} +130.410 q^{44} -39.3179 q^{45} +420.199 q^{46} -598.927 q^{47} -108.003 q^{48} +672.925 q^{49} +111.399 q^{50} -314.782 q^{51} -240.658 q^{52} +347.531 q^{53} -503.485 q^{54} -55.0000 q^{55} -547.582 q^{56} -112.186 q^{57} -625.580 q^{58} +314.814 q^{59} -350.006 q^{60} -444.371 q^{61} +161.546 q^{62} -250.641 q^{63} -829.273 q^{64} +101.497 q^{65} +289.414 q^{66} +206.250 q^{67} -632.038 q^{68} +556.802 q^{69} +710.135 q^{70} +362.745 q^{71} +135.095 q^{72} +648.154 q^{73} +516.280 q^{74} +147.613 q^{75} -225.254 q^{76} -350.609 q^{77} -534.082 q^{78} +619.914 q^{79} +91.4576 q^{80} -879.481 q^{81} -389.669 q^{82} -1018.71 q^{83} -2231.19 q^{84} +266.559 q^{85} -1041.94 q^{86} -828.951 q^{87} +188.978 q^{88} -631.513 q^{89} -175.199 q^{90} +647.013 q^{91} +1117.98 q^{92} +214.064 q^{93} -2668.79 q^{94} +95.0000 q^{95} -1292.77 q^{96} -999.295 q^{97} +2998.52 q^{98} +86.4994 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * 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 $

$2$	4.45595	1.57542	0.787708	−	0.616049i	$-0.211268\pi$
$2$	4.45595	1.57542	0.787708	+	0.616049i	$0.211268\pi$
$3$	5.90454	1.13633	0.568165	−	0.822915i	$-0.307654\pi$
$3$	5.90454	1.13633	0.568165	+	0.822915i	$0.307654\pi$
$4$	11.8555	1.48193
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	26.3103	1.79019
$7$	−31.8736	−1.72101	−0.860506	−	0.509441i	$-0.829852\pi$
$7$	−31.8736	−1.72101	−0.860506	+	0.509441i	$0.829852\pi$
$8$	17.1798	0.759247
$9$	7.86359	0.291244
$10$	−22.2797	−0.704547
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	70.0011	1.68397
$13$	−20.2994	−0.433079	−0.216539	−	0.976274i	$-0.569477\pi$
$13$	−20.2994	−0.433079	−0.216539	+	0.976274i	$0.569477\pi$
$14$	−142.027	−2.71131
$15$	−29.5227	−0.508182
$16$	−18.2915	−0.285805
$17$	−53.3119	−0.760590	−0.380295	−	0.924865i	$-0.624178\pi$
$17$	−53.3119	−0.760590	−0.380295	+	0.924865i	$0.624178\pi$
$18$	35.0397	0.458830
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−59.2774	−0.662741
$21$	−188.199	−1.95564
$22$	49.0154	0.475006
$23$	94.3006	0.854915	0.427457	−	0.904036i	$-0.359409\pi$
$23$	94.3006	0.854915	0.427457	+	0.904036i	$0.359409\pi$
$24$	101.439	0.862754
$25$	25.0000	0.200000
$26$	−90.4529	−0.682279
$27$	−112.992	−0.805380
$28$	−377.876	−2.55043
$29$	−140.392	−0.898972	−0.449486	−	0.893287i	$-0.648393\pi$
$29$	−140.392	−0.898972	−0.449486	+	0.893287i	$0.648393\pi$
$30$	−131.552	−0.800598
$31$	36.2541	0.210046	0.105023	−	0.994470i	$-0.466508\pi$
$31$	36.2541	0.210046	0.105023	+	0.994470i	$0.466508\pi$
$32$	−218.944	−1.20951
$33$	64.9499	0.342616
$34$	−237.555	−1.19825
$35$	159.368	0.769660
$36$	93.2265	0.431604
$37$	115.863	0.514805	0.257403	−	0.966304i	$-0.417133\pi$
$37$	115.863	0.514805	0.257403	+	0.966304i	$0.417133\pi$
$38$	−84.6630	−0.361425
$39$	−119.858	−0.492120
$40$	−85.8990	−0.339545
$41$	−87.4492	−0.333104	−0.166552	−	0.986033i	$-0.553263\pi$
$41$	−87.4492	−0.333104	−0.166552	+	0.986033i	$0.553263\pi$
$42$	−838.604	−3.08094
$43$	−233.831	−0.829275	−0.414638	−	0.909987i	$-0.636092\pi$
$43$	−233.831	−0.829275	−0.414638	+	0.909987i	$0.636092\pi$
$44$	130.410	0.446820
$45$	−39.3179	−0.130248
$46$	420.199	1.34685
$47$	−598.927	−1.85878	−0.929388	−	0.369103i	$-0.879665\pi$
$47$	−598.927	−1.85878	−0.929388	+	0.369103i	$0.879665\pi$
$48$	−108.003	−0.324769
$49$	672.925	1.96188
$50$	111.399	0.315083
$51$	−314.782	−0.864281
$52$	−240.658	−0.641795
$53$	347.531	0.900699	0.450350	−	0.892852i	$-0.351299\pi$
$53$	347.531	0.900699	0.450350	+	0.892852i	$0.351299\pi$
$54$	−503.485	−1.26881
$55$	−55.0000	−0.134840
$56$	−547.582	−1.30667
$57$	−112.186	−0.260692
$58$	−625.580	−1.41625
$59$	314.814	0.694665	0.347333	−	0.937742i	$-0.387088\pi$
$59$	314.814	0.694665	0.347333	+	0.937742i	$0.387088\pi$
$60$	−350.006	−0.753092
$61$	−444.371	−0.932719	−0.466359	−	0.884595i	$-0.654435\pi$
$61$	−444.371	−0.932719	−0.466359	+	0.884595i	$0.654435\pi$
$62$	161.546	0.330909
$63$	−250.641	−0.501234
$64$	−829.273	−1.61967
$65$	101.497	0.193679
$66$	289.414	0.539763
$67$	206.250	0.376082	0.188041	−	0.982161i	$-0.439786\pi$
$67$	206.250	0.376082	0.188041	+	0.982161i	$0.439786\pi$
$68$	−632.038	−1.12714
$69$	556.802	0.971464
$70$	710.135	1.21253
$71$	362.745	0.606337	0.303169	−	0.952937i	$-0.401955\pi$
$71$	362.745	0.606337	0.303169	+	0.952937i	$0.401955\pi$
$72$	135.095	0.221126
$73$	648.154	1.03919	0.519594	−	0.854413i	$-0.326083\pi$
$73$	648.154	1.03919	0.519594	+	0.854413i	$0.326083\pi$
$74$	516.280	0.811032
$75$	147.613	0.227266
$76$	−225.254	−0.339979
$77$	−350.609	−0.518905
$78$	−534.082	−0.775294
$79$	619.914	0.882858	0.441429	−	0.897296i	$-0.354472\pi$
$79$	619.914	0.882858	0.441429	+	0.897296i	$0.354472\pi$
$80$	91.4576	0.127816
$81$	−879.481	−1.20642
$82$	−389.669	−0.524777
$83$	−1018.71	−1.34721	−0.673604	−	0.739092i	$-0.735255\pi$
$83$	−1018.71	−1.34721	−0.673604	+	0.739092i	$0.735255\pi$
$84$	−2231.19	−2.89812
$85$	266.559	0.340146
$86$	−1041.94	−1.30645
$87$	−828.951	−1.02153
$88$	188.978	0.228922
$89$	−631.513	−0.752138	−0.376069	−	0.926592i	$-0.622724\pi$
$89$	−631.513	−0.752138	−0.376069	+	0.926592i	$0.622724\pi$
$90$	−175.199	−0.205195
$91$	647.013	0.745334
$92$	1117.98	1.26693
$93$	214.064	0.238681
$94$	−2668.79	−2.92835
$95$	95.0000	0.102598
$96$	−1292.77	−1.37440
$97$	−999.295	−1.04601	−0.523006	−	0.852329i	$-0.675189\pi$
$97$	−999.295	−1.04601	−0.523006	+	0.852329i	$0.675189\pi$
$98$	2998.52	3.09078
$99$	86.4994	0.0878133
$100$	296.387	0.296387
$101$	−481.816	−0.474678	−0.237339	−	0.971427i	$-0.576275\pi$
$101$	−481.816	−0.474678	−0.237339	+	0.971427i	$0.576275\pi$
$102$	−1402.65	−1.36160
$103$	1671.75	1.59925	0.799623	−	0.600503i	$-0.205033\pi$
$103$	1671.75	1.59925	0.799623	+	0.600503i	$0.205033\pi$
$104$	−348.739	−0.328814
$105$	940.994	0.874587
$106$	1548.58	1.41898
$107$	−829.058	−0.749047	−0.374524	−	0.927217i	$-0.622194\pi$
$107$	−829.058	−0.749047	−0.374524	+	0.927217i	$0.622194\pi$
$108$	−1339.57	−1.19352
$109$	−2078.71	−1.82664	−0.913322	−	0.407239i	$-0.866492\pi$
$109$	−2078.71	−1.82664	−0.913322	+	0.407239i	$0.866492\pi$
$110$	−245.077	−0.212429
$111$	684.119	0.584988
$112$	583.016	0.491874
$113$	1938.18	1.61353	0.806765	−	0.590873i	$-0.201216\pi$
$113$	1938.18	1.61353	0.806765	+	0.590873i	$0.201216\pi$
$114$	−499.896	−0.410698
$115$	−471.503	−0.382329
$116$	−1664.42	−1.33222
$117$	−159.626	−0.126132
$118$	1402.79	1.09439
$119$	1699.24	1.30898
$120$	−507.194	−0.385835
$121$	121.000	0.0909091
$122$	−1980.09	−1.46942
$123$	−516.347	−0.378516
$124$	429.809	0.311274
$125$	−125.000	−0.0894427
$126$	−1116.84	−0.789652
$127$	674.444	0.471238	0.235619	−	0.971846i	$-0.424288\pi$
$127$	674.444	0.471238	0.235619	+	0.971846i	$0.424288\pi$
$128$	−1943.64	−1.34215
$129$	−1380.66	−0.942330
$130$	452.264	0.305125
$131$	2203.92	1.46991	0.734953	−	0.678118i	$-0.237204\pi$
$131$	2203.92	1.46991	0.734953	+	0.678118i	$0.237204\pi$
$132$	770.012	0.507735
$133$	605.598	0.394827
$134$	919.040	0.592485
$135$	564.959	0.360177
$136$	−915.887	−0.577476
$137$	605.801	0.377789	0.188894	−	0.981997i	$-0.439510\pi$
$137$	605.801	0.377789	0.188894	+	0.981997i	$0.439510\pi$
$138$	2481.08	1.53046
$139$	−2090.54	−1.27566	−0.637831	−	0.770176i	$-0.720168\pi$
$139$	−2090.54	−1.27566	−0.637831	+	0.770176i	$0.720168\pi$
$140$	1889.38	1.14059
$141$	−3536.39	−2.11218
$142$	1616.37	0.955233
$143$	−223.293	−0.130578
$144$	−143.837	−0.0832390
$145$	701.961	0.402032
$146$	2888.14	1.63715
$147$	3973.31	2.22934
$148$	1373.61	0.762907
$149$	2520.14	1.38562	0.692811	−	0.721119i	$-0.256372\pi$
$149$	2520.14	1.38562	0.692811	+	0.721119i	$0.256372\pi$
$150$	657.758	0.358038
$151$	2539.34	1.36853	0.684267	−	0.729231i	$-0.260122\pi$
$151$	2539.34	1.36853	0.684267	+	0.729231i	$0.260122\pi$
$152$	−326.416	−0.174183
$153$	−419.223	−0.221517
$154$	−1562.30	−0.817490
$155$	−181.270	−0.0939353
$156$	−1420.98	−0.729290
$157$	2516.46	1.27921	0.639603	−	0.768706i	$-0.279099\pi$
$157$	2516.46	1.27921	0.639603	+	0.768706i	$0.279099\pi$
$158$	2762.31	1.39087
$159$	2052.01	1.02349
$160$	1094.72	0.540909
$161$	−3005.70	−1.47132
$162$	−3918.92	−1.90061
$163$	1259.71	0.605325	0.302663	−	0.953098i	$-0.402124\pi$
$163$	1259.71	0.605325	0.302663	+	0.953098i	$0.402124\pi$
$164$	−1036.75	−0.493638
$165$	−324.750	−0.153223
$166$	−4539.33	−2.12241
$167$	3555.50	1.64750	0.823752	−	0.566951i	$-0.191877\pi$
$167$	3555.50	1.64750	0.823752	+	0.566951i	$0.191877\pi$
$168$	−3233.22	−1.48481
$169$	−1784.94	−0.812443
$170$	1187.78	0.535872
$171$	−149.408	−0.0668159
$172$	−2772.17	−1.22893
$173$	−1876.78	−0.824792	−0.412396	−	0.911005i	$-0.635308\pi$
$173$	−1876.78	−0.824792	−0.412396	+	0.911005i	$0.635308\pi$
$174$	−3693.76	−1.60933
$175$	−796.840	−0.344202
$176$	−201.207	−0.0861735
$177$	1858.83	0.789368
$178$	−2813.99	−1.18493
$179$	−3641.24	−1.52044	−0.760220	−	0.649665i	$-0.774909\pi$
$179$	−3641.24	−1.52044	−0.760220	+	0.649665i	$0.774909\pi$
$180$	−466.133	−0.193019
$181$	−73.9750	−0.0303786	−0.0151893	−	0.999885i	$-0.504835\pi$
$181$	−73.9750	−0.0303786	−0.0151893	+	0.999885i	$0.504835\pi$
$182$	2883.06	1.17421
$183$	−2623.80	−1.05988
$184$	1620.06	0.649091
$185$	−579.316	−0.230228
$186$	953.856	0.376022
$187$	−586.431	−0.229327
$188$	−7100.57	−2.75459
$189$	3601.45	1.38607
$190$	423.315	0.161634
$191$	−1390.53	−0.526780	−0.263390	−	0.964690i	$-0.584841\pi$
$191$	−1390.53	−0.526780	−0.263390	+	0.964690i	$0.584841\pi$
$192$	−4896.47	−1.84048
$193$	1877.98	0.700414	0.350207	−	0.936672i	$-0.386111\pi$
$193$	1877.98	0.700414	0.350207	+	0.936672i	$0.386111\pi$
$194$	−4452.81	−1.64790
$195$	599.292	0.220083
$196$	7977.85	2.90738
$197$	1122.26	0.405877	0.202938	−	0.979191i	$-0.434951\pi$
$197$	1122.26	0.405877	0.202938	+	0.979191i	$0.434951\pi$
$198$	385.437	0.138343
$199$	−801.628	−0.285557	−0.142779	−	0.989755i	$-0.545604\pi$
$199$	−801.628	−0.285557	−0.142779	+	0.989755i	$0.545604\pi$
$200$	429.495	0.151849
$201$	1217.81	0.427353
$202$	−2146.95	−0.747815
$203$	4474.80	1.54714
$204$	−3731.89	−1.28081
$205$	437.246	0.148969
$206$	7449.23	2.51948
$207$	741.541	0.248989
$208$	371.306	0.123776
$209$	−209.000	−0.0691714
$210$	4193.02	1.37784
$211$	−4994.31	−1.62949	−0.814745	−	0.579819i	$-0.803123\pi$
$211$	−4994.31	−1.62949	−0.814745	+	0.579819i	$0.803123\pi$
$212$	4120.15	1.33478
$213$	2141.84	0.688999
$214$	−3694.24	−1.18006
$215$	1169.15	0.370863
$216$	−1941.17	−0.611482
$217$	−1155.55	−0.361491
$218$	−9262.62	−2.87772
$219$	3827.05	1.18086
$220$	−652.051	−0.199824
$221$	1082.20	0.329396
$222$	3048.40	0.921599
$223$	−3676.80	−1.10411	−0.552056	−	0.833807i	$-0.686157\pi$
$223$	−3676.80	−1.10411	−0.552056	+	0.833807i	$0.686157\pi$
$224$	6978.54	2.08158
$225$	196.590	0.0582488
$226$	8636.44	2.54198
$227$	−2863.53	−0.837265	−0.418633	−	0.908156i	$-0.637491\pi$
$227$	−2863.53	−0.837265	−0.418633	+	0.908156i	$0.637491\pi$
$228$	−1330.02	−0.386328
$229$	−3572.28	−1.03084	−0.515422	−	0.856937i	$-0.672365\pi$
$229$	−3572.28	−1.03084	−0.515422	+	0.856937i	$0.672365\pi$
$230$	−2100.99	−0.602328
$231$	−2070.19	−0.589646
$232$	−2411.91	−0.682541
$233$	936.048	0.263187	0.131593	−	0.991304i	$-0.457991\pi$
$233$	936.048	0.263187	0.131593	+	0.991304i	$0.457991\pi$
$234$	−711.284	−0.198710
$235$	2994.64	0.831270
$236$	3732.26	1.02945
$237$	3660.31	1.00322
$238$	7571.73	2.06219
$239$	3104.75	0.840291	0.420146	−	0.907457i	$-0.361979\pi$
$239$	3104.75	0.840291	0.420146	+	0.907457i	$0.361979\pi$
$240$	540.015	0.145241
$241$	−6953.75	−1.85863	−0.929317	−	0.369284i	$-0.879603\pi$
$241$	−6953.75	−1.85863	−0.929317	+	0.369284i	$0.879603\pi$
$242$	539.170	0.143220
$243$	−2142.15	−0.565511
$244$	−5268.22	−1.38223
$245$	−3364.63	−0.877380
$246$	−2300.82	−0.596320
$247$	385.688	0.0993551
$248$	622.837	0.159477
$249$	−6015.03	−1.53087
$250$	−556.994	−0.140909
$251$	−1008.37	−0.253577	−0.126789	−	0.991930i	$-0.540467\pi$
$251$	−1008.37	−0.253577	−0.126789	+	0.991930i	$0.540467\pi$
$252$	−2971.46	−0.742796
$253$	1037.31	0.257766
$254$	3005.29	0.742396
$255$	1573.91	0.386518
$256$	−2026.58	−0.494771
$257$	3012.16	0.731102	0.365551	−	0.930791i	$-0.380881\pi$
$257$	3012.16	0.731102	0.365551	+	0.930791i	$0.380881\pi$
$258$	−6152.16	−1.48456
$259$	−3692.97	−0.885985
$260$	1203.29	0.287019
$261$	−1103.99	−0.261820
$262$	9820.57	2.31571
$263$	−2288.58	−0.536578	−0.268289	−	0.963338i	$-0.586458\pi$
$263$	−2288.58	−0.536578	−0.268289	+	0.963338i	$0.586458\pi$
$264$	1115.83	0.260130
$265$	−1737.66	−0.402805
$266$	2698.51	0.622017
$267$	−3728.80	−0.854676
$268$	2445.19	0.557328
$269$	1764.62	0.399965	0.199983	−	0.979799i	$-0.435911\pi$
$269$	1764.62	0.399965	0.199983	+	0.979799i	$0.435911\pi$
$270$	2517.43	0.567428
$271$	−132.717	−0.0297489	−0.0148745	−	0.999889i	$-0.504735\pi$
$271$	−132.717	−0.0297489	−0.0148745	+	0.999889i	$0.504735\pi$
$272$	975.156	0.217381
$273$	3820.31	0.846945
$274$	2699.42	0.595174
$275$	275.000	0.0603023
$276$	6601.15	1.43965
$277$	−6374.75	−1.38275	−0.691375	−	0.722496i	$-0.742995\pi$
$277$	−6374.75	−1.38275	−0.691375	+	0.722496i	$0.742995\pi$
$278$	−9315.33	−2.00970
$279$	285.087	0.0611746
$280$	2737.91	0.584362
$281$	−2801.59	−0.594765	−0.297382	−	0.954758i	$-0.596114\pi$
$281$	−2801.59	−0.594765	−0.297382	+	0.954758i	$0.596114\pi$
$282$	−15758.0	−3.32757
$283$	−4708.47	−0.989008	−0.494504	−	0.869175i	$-0.664650\pi$
$283$	−4708.47	−0.989008	−0.494504	+	0.869175i	$0.664650\pi$
$284$	4300.52	0.898552
$285$	560.931	0.116585
$286$	−994.981	−0.205715
$287$	2787.32	0.573276
$288$	−1721.69	−0.352262
$289$	−2070.84	−0.421502
$290$	3127.90	0.633368
$291$	−5900.38	−1.18861
$292$	7684.17	1.54001
$293$	4094.09	0.816311	0.408156	−	0.912912i	$-0.366172\pi$
$293$	4094.09	0.816311	0.408156	+	0.912912i	$0.366172\pi$
$294$	17704.9	3.51214
$295$	−1574.07	−0.310664
$296$	1990.51	0.390864
$297$	−1242.91	−0.242831
$298$	11229.6	2.18293
$299$	−1914.24	−0.370246
$300$	1750.03	0.336793
$301$	7453.02	1.42719
$302$	11315.2	2.15601
$303$	−2844.90	−0.539390
$304$	347.539	0.0655682
$305$	2221.85	0.417124
$306$	−1868.03	−0.348982
$307$	−6685.43	−1.24286	−0.621430	−	0.783470i	$-0.713448\pi$
$307$	−6685.43	−1.24286	−0.621430	+	0.783470i	$0.713448\pi$
$308$	−4156.64	−0.768982
$309$	9870.91	1.81727
$310$	−807.731	−0.147987
$311$	−9452.96	−1.72356	−0.861781	−	0.507280i	$-0.830651\pi$
$311$	−9452.96	−1.72356	−0.861781	+	0.507280i	$0.830651\pi$
$312$	−2059.14	−0.373641
$313$	4197.19	0.757953	0.378976	−	0.925406i	$-0.376276\pi$
$313$	4197.19	0.757953	0.378976	+	0.925406i	$0.376276\pi$
$314$	11213.2	2.01528
$315$	1253.20	0.224159
$316$	7349.38	1.30834
$317$	−2694.70	−0.477443	−0.238722	−	0.971088i	$-0.576728\pi$
$317$	−2694.70	−0.477443	−0.238722	+	0.971088i	$0.576728\pi$
$318$	9143.65	1.61242
$319$	−1544.31	−0.271050
$320$	4146.36	0.724340
$321$	−4895.21	−0.851164
$322$	−13393.2	−2.31794
$323$	1012.93	0.174491
$324$	−10426.7	−1.78784
$325$	−507.484	−0.0866158
$326$	5613.20	0.953638
$327$	−12273.8	−2.07567
$328$	−1502.36	−0.252908
$329$	19090.0	3.19898
$330$	−1447.07	−0.241389
$331$	−4891.01	−0.812187	−0.406094	−	0.913831i	$-0.633109\pi$
$331$	−4891.01	−0.812187	−0.406094	+	0.913831i	$0.633109\pi$
$332$	−12077.3	−1.99647
$333$	911.100	0.149934
$334$	15843.1	2.59550
$335$	−1031.25	−0.168189
$336$	3442.44	0.558931
$337$	−2738.19	−0.442607	−0.221303	−	0.975205i	$-0.571031\pi$
$337$	−2738.19	−0.442607	−0.221303	+	0.975205i	$0.571031\pi$
$338$	−7953.58	−1.27993
$339$	11444.1	1.83350
$340$	3160.19	0.504074
$341$	398.795	0.0633312
$342$	−665.755	−0.105263
$343$	−10515.9	−1.65541
$344$	−4017.16	−0.629625
$345$	−2784.01	−0.434452
$346$	−8362.84	−1.29939
$347$	−5791.36	−0.895955	−0.447977	−	0.894045i	$-0.647855\pi$
$347$	−5791.36	−0.895955	−0.447977	+	0.894045i	$0.647855\pi$
$348$	−9827.61	−1.51384
$349$	−280.785	−0.0430661	−0.0215331	−	0.999768i	$-0.506855\pi$
$349$	−280.785	−0.0430661	−0.0215331	+	0.999768i	$0.506855\pi$
$350$	−3550.68	−0.542262
$351$	2293.66	0.348793
$352$	−2408.39	−0.364681
$353$	−9193.91	−1.38624	−0.693120	−	0.720823i	$-0.743764\pi$
$353$	−9193.91	−1.38624	−0.693120	+	0.720823i	$0.743764\pi$
$354$	8282.85	1.24358
$355$	−1813.73	−0.271162
$356$	−7486.89	−1.11462
$357$	10033.2	1.48744
$358$	−16225.2	−2.39533
$359$	8913.02	1.31034	0.655168	−	0.755483i	$-0.272598\pi$
$359$	8913.02	1.31034	0.655168	+	0.755483i	$0.272598\pi$
$360$	−675.474	−0.0988906
$361$	361.000	0.0526316
$362$	−329.629	−0.0478589
$363$	714.449	0.103303
$364$	7670.65	1.10454
$365$	−3240.77	−0.464739
$366$	−11691.5	−1.66974
$367$	10906.3	1.55124	0.775621	−	0.631199i	$-0.217437\pi$
$367$	10906.3	1.55124	0.775621	+	0.631199i	$0.217437\pi$
$368$	−1724.90	−0.244339
$369$	−687.664	−0.0970146
$370$	−2581.40	−0.362705
$371$	−11077.1	−1.55011
$372$	2537.82	0.353710
$373$	−8952.26	−1.24271	−0.621355	−	0.783529i	$-0.713417\pi$
$373$	−8952.26	−1.24271	−0.621355	+	0.783529i	$0.713417\pi$
$374$	−2613.11	−0.361285
$375$	−738.067	−0.101636
$376$	−10289.4	−1.41127
$377$	2849.87	0.389326
$378$	16047.9	2.18363
$379$	−2927.34	−0.396747	−0.198374	−	0.980126i	$-0.563566\pi$
$379$	−2927.34	−0.396747	−0.198374	+	0.980126i	$0.563566\pi$
$380$	1126.27	0.152043
$381$	3982.28	0.535482
$382$	−6196.11	−0.829897
$383$	9979.90	1.33146	0.665730	−	0.746193i	$-0.268120\pi$
$383$	9979.90	1.33146	0.665730	+	0.746193i	$0.268120\pi$
$384$	−11476.3	−1.52512
$385$	1753.05	0.232061
$386$	8368.17	1.10344
$387$	−1838.75	−0.241521
$388$	−11847.1	−1.55012
$389$	5131.87	0.668885	0.334443	−	0.942416i	$-0.391452\pi$
$389$	5131.87	0.668885	0.334443	+	0.942416i	$0.391452\pi$
$390$	2670.41	0.346722
$391$	−5027.34	−0.650240
$392$	11560.7	1.48955
$393$	13013.2	1.67030
$394$	5000.74	0.639425
$395$	−3099.57	−0.394826
$396$	1025.49	0.130134
$397$	−13768.7	−1.74063	−0.870316	−	0.492494i	$-0.836086\pi$
$397$	−13768.7	−1.74063	−0.870316	+	0.492494i	$0.836086\pi$
$398$	−3572.01	−0.449871
$399$	3575.78	0.448654
$400$	−457.288	−0.0571610
$401$	−9853.97	−1.22714	−0.613571	−	0.789640i	$-0.710267\pi$
$401$	−9853.97	−1.22714	−0.613571	+	0.789640i	$0.710267\pi$
$402$	5426.51	0.673258
$403$	−735.934	−0.0909664
$404$	−5712.15	−0.703441
$405$	4397.40	0.539528
$406$	19939.5	2.43739
$407$	1274.49	0.155220
$408$	−5407.89	−0.656202
$409$	9691.26	1.17164	0.585821	−	0.810440i	$-0.300772\pi$
$409$	9691.26	1.17164	0.585821	+	0.810440i	$0.300772\pi$
$410$	1948.35	0.234688
$411$	3576.97	0.429292
$412$	19819.4	2.36998
$413$	−10034.2	−1.19553
$414$	3304.27	0.392261
$415$	5093.57	0.602490
$416$	4444.43	0.523813
$417$	−12343.7	−1.44957
$418$	−931.293	−0.108974
$419$	−8602.04	−1.00295	−0.501476	−	0.865171i	$-0.667210\pi$
$419$	−8602.04	−1.00295	−0.501476	+	0.865171i	$0.667210\pi$
$420$	11155.9	1.29608
$421$	1545.60	0.178926	0.0894632	−	0.995990i	$-0.471485\pi$
$421$	1545.60	0.178926	0.0894632	+	0.995990i	$0.471485\pi$
$422$	−22254.4	−2.56712
$423$	−4709.72	−0.541357
$424$	5970.51	0.683853
$425$	−1332.80	−0.152118
$426$	9543.94	1.08546
$427$	14163.7	1.60522
$428$	−9828.88	−1.11004
$429$	−1318.44	−0.148380
$430$	5209.69	0.584264
$431$	912.028	0.101928	0.0509638	−	0.998700i	$-0.483771\pi$
$431$	912.028	0.101928	0.0509638	+	0.998700i	$0.483771\pi$
$432$	2066.79	0.230182
$433$	4383.48	0.486505	0.243253	−	0.969963i	$-0.421786\pi$
$433$	4383.48	0.486505	0.243253	+	0.969963i	$0.421786\pi$
$434$	−5149.06	−0.569499
$435$	4144.76	0.456841
$436$	−24644.1	−2.70697
$437$	−1791.71	−0.196131
$438$	17053.1	1.86034
$439$	11560.9	1.25688	0.628440	−	0.777858i	$-0.283694\pi$
$439$	11560.9	1.25688	0.628440	+	0.777858i	$0.283694\pi$
$440$	−944.889	−0.102377
$441$	5291.60	0.571386
$442$	4822.21	0.518935
$443$	−14199.7	−1.52291	−0.761453	−	0.648220i	$-0.775514\pi$
$443$	−14199.7	−1.52291	−0.761453	+	0.648220i	$0.775514\pi$
$444$	8110.55	0.866914
$445$	3157.57	0.336366
$446$	−16383.7	−1.73944
$447$	14880.3	1.57452
$448$	26431.9	2.78748
$449$	−3967.23	−0.416982	−0.208491	−	0.978024i	$-0.566855\pi$
$449$	−3967.23	−0.416982	−0.208491	+	0.978024i	$0.566855\pi$
$450$	875.993	0.0917660
$451$	−961.941	−0.100435
$452$	22978.1	2.39115
$453$	14993.6	1.55511
$454$	−12759.7	−1.31904
$455$	−3235.06	−0.333323
$456$	−1927.34	−0.197929
$457$	9168.57	0.938485	0.469242	−	0.883069i	$-0.344527\pi$
$457$	9168.57	0.938485	0.469242	+	0.883069i	$0.344527\pi$
$458$	−15917.9	−1.62401
$459$	6023.80	0.612564
$460$	−5589.89	−0.566587
$461$	−12130.9	−1.22558	−0.612788	−	0.790248i	$-0.709952\pi$
$461$	−12130.9	−1.22558	−0.612788	+	0.790248i	$0.709952\pi$
$462$	−9224.65	−0.928938
$463$	8453.96	0.848572	0.424286	−	0.905528i	$-0.360525\pi$
$463$	8453.96	0.848572	0.424286	+	0.905528i	$0.360525\pi$
$464$	2567.99	0.256931
$465$	−1070.32	−0.106741
$466$	4170.98	0.414629
$467$	5920.80	0.586685	0.293342	−	0.956007i	$-0.405232\pi$
$467$	5920.80	0.586685	0.293342	+	0.956007i	$0.405232\pi$
$468$	−1892.44	−0.186919
$469$	−6573.93	−0.647241
$470$	13343.9	1.30960
$471$	14858.5	1.45360
$472$	5408.43	0.527422
$473$	−2572.14	−0.250036
$474$	16310.1	1.58048
$475$	−475.000	−0.0458831
$476$	20145.3	1.93983
$477$	2732.84	0.262323
$478$	13834.6	1.32381
$479$	−301.208	−0.0287319	−0.0143659	−	0.999897i	$-0.504573\pi$
$479$	−301.208	−0.0287319	−0.0143659	+	0.999897i	$0.504573\pi$
$480$	6463.83	0.614650
$481$	−2351.95	−0.222951
$482$	−30985.6	−2.92812
$483$	−17747.3	−1.67190
$484$	1434.51	0.134721
$485$	4996.48	0.467790
$486$	−9545.32	−0.890915
$487$	8409.69	0.782504	0.391252	−	0.920284i	$-0.372042\pi$
$487$	8409.69	0.782504	0.391252	+	0.920284i	$0.372042\pi$
$488$	−7634.20	−0.708164
$489$	7438.00	0.687848
$490$	−14992.6	−1.38224
$491$	7143.44	0.656576	0.328288	−	0.944578i	$-0.393528\pi$
$491$	7143.44	0.656576	0.328288	+	0.944578i	$0.393528\pi$
$492$	−6121.54	−0.560936
$493$	7484.57	0.683749
$494$	1718.60	0.156526
$495$	−432.497	−0.0392713
$496$	−663.142	−0.0600322
$497$	−11562.0	−1.04351
$498$	−26802.7	−2.41176
$499$	9010.61	0.808357	0.404178	−	0.914680i	$-0.367557\pi$
$499$	9010.61	0.808357	0.404178	+	0.914680i	$0.367557\pi$
$500$	−1481.93	−0.132548
$501$	20993.6	1.87211
$502$	−4493.26	−0.399490
$503$	−17542.0	−1.55499	−0.777496	−	0.628888i	$-0.783510\pi$
$503$	−17542.0	−1.55499	−0.777496	+	0.628888i	$0.783510\pi$
$504$	−4305.95	−0.380560
$505$	2409.08	0.212282
$506$	4622.18	0.406089
$507$	−10539.2	−0.923202
$508$	7995.85	0.698344
$509$	−17649.0	−1.53689	−0.768445	−	0.639916i	$-0.778969\pi$
$509$	−17649.0	−1.53689	−0.768445	+	0.639916i	$0.778969\pi$
$510$	7013.27	0.608927
$511$	−20659.0	−1.78845
$512$	6518.78	0.562680
$513$	2146.84	0.184767
$514$	13422.0	1.15179
$515$	−8358.74	−0.715204
$516$	−16368.4	−1.39647
$517$	−6588.20	−0.560442
$518$	−16455.7	−1.39580
$519$	−11081.5	−0.937236
$520$	1743.69	0.147050
$521$	17191.5	1.44563	0.722816	−	0.691041i	$-0.242847\pi$
$521$	17191.5	1.44563	0.722816	+	0.691041i	$0.242847\pi$
$522$	−4919.30	−0.412475
$523$	275.180	0.0230072	0.0115036	−	0.999934i	$-0.496338\pi$
$523$	275.180	0.0230072	0.0115036	+	0.999934i	$0.496338\pi$
$524$	26128.6	2.17831
$525$	−4704.97	−0.391127
$526$	−10197.8	−0.845334
$527$	−1932.77	−0.159759
$528$	−1188.03	−0.0979214
$529$	−3274.39	−0.269121
$530$	−7742.90	−0.634585
$531$	2475.56	0.202317
$532$	7179.65	0.585108
$533$	1775.16	0.144260
$534$	−16615.3	−1.34647
$535$	4145.29	0.334984
$536$	3543.34	0.285539
$537$	−21499.8	−1.72772
$538$	7863.05	0.630112
$539$	7402.18	0.591529
$540$	6697.85	0.533759
$541$	−7223.39	−0.574044	−0.287022	−	0.957924i	$-0.592665\pi$
$541$	−7223.39	−0.574044	−0.287022	+	0.957924i	$0.592665\pi$
$542$	−591.379	−0.0468670
$543$	−436.788	−0.0345200
$544$	11672.3	0.919940
$545$	10393.5	0.816900
$546$	17023.1	1.33429
$547$	−1113.90	−0.0870691	−0.0435345	−	0.999052i	$-0.513862\pi$
$547$	−1113.90	−0.0870691	−0.0435345	+	0.999052i	$0.513862\pi$
$548$	7182.05	0.559858
$549$	−3494.35	−0.271649
$550$	1225.39	0.0950011
$551$	2667.45	0.206238
$552$	9565.74	0.737581
$553$	−19758.9	−1.51941
$554$	−28405.6	−2.17841
$555$	−3420.59	−0.261615
$556$	−24784.3	−1.89045
$557$	−17104.0	−1.30112	−0.650558	−	0.759457i	$-0.725465\pi$
$557$	−17104.0	−1.30112	−0.650558	+	0.759457i	$0.725465\pi$
$558$	1270.33	0.0963754
$559$	4746.61	0.359142
$560$	−2915.08	−0.219973
$561$	−3462.60	−0.260590
$562$	−12483.7	−0.937002
$563$	13517.5	1.01189	0.505946	−	0.862565i	$-0.331143\pi$
$563$	13517.5	1.01189	0.505946	+	0.862565i	$0.331143\pi$
$564$	−41925.6	−3.13012
$565$	−9690.91	−0.721592
$566$	−20980.7	−1.55810
$567$	28032.2	2.07626
$568$	6231.89	0.460360
$569$	7450.79	0.548952	0.274476	−	0.961594i	$-0.411496\pi$
$569$	7450.79	0.548952	0.274476	+	0.961594i	$0.411496\pi$
$570$	2499.48	0.183670
$571$	5920.95	0.433948	0.216974	−	0.976177i	$-0.430381\pi$
$571$	5920.95	0.433948	0.216974	+	0.976177i	$0.430381\pi$
$572$	−2647.24	−0.193508
$573$	−8210.41	−0.598595
$574$	12420.1	0.903148
$575$	2357.52	0.170983
$576$	−6521.06	−0.471720
$577$	9692.43	0.699309	0.349655	−	0.936879i	$-0.386299\pi$
$577$	9692.43	0.699309	0.349655	+	0.936879i	$0.386299\pi$
$578$	−9227.56	−0.664041
$579$	11088.6	0.795900
$580$	8322.08	0.595785
$581$	32470.0	2.31856
$582$	−26291.8	−1.87256
$583$	3822.84	0.271571
$584$	11135.2	0.789000
$585$	798.128	0.0564078
$586$	18243.0	1.28603
$587$	−9862.40	−0.693466	−0.346733	−	0.937964i	$-0.612709\pi$
$587$	−9862.40	−0.693466	−0.346733	+	0.937964i	$0.612709\pi$
$588$	47105.5	3.30374
$589$	−688.827	−0.0481878
$590$	−7013.96	−0.489424
$591$	6626.43	0.461210
$592$	−2119.31	−0.147134
$593$	9855.87	0.682516	0.341258	−	0.939970i	$-0.389147\pi$
$593$	9855.87	0.682516	0.341258	+	0.939970i	$0.389147\pi$
$594$	−5538.34	−0.382560
$595$	−8496.21	−0.585396
$596$	29877.4	2.05340
$597$	−4733.24	−0.324487
$598$	−8529.76	−0.583291
$599$	14146.2	0.964936	0.482468	−	0.875914i	$-0.339741\pi$
$599$	14146.2	0.964936	0.482468	+	0.875914i	$0.339741\pi$
$600$	2535.97	0.172551
$601$	−27762.5	−1.88429	−0.942143	−	0.335212i	$-0.891192\pi$
$601$	−27762.5	−1.88429	−0.942143	+	0.335212i	$0.891192\pi$
$602$	33210.3	2.24842
$603$	1621.87	0.109531
$604$	30105.1	2.02808
$605$	−605.000	−0.0406558
$606$	−12676.7	−0.849764
$607$	16770.3	1.12139	0.560696	−	0.828022i	$-0.310534\pi$
$607$	16770.3	1.12139	0.560696	+	0.828022i	$0.310534\pi$
$608$	4159.94	0.277480
$609$	26421.6	1.75806
$610$	9900.46	0.657144
$611$	12157.8	0.804997
$612$	−4970.08	−0.328274
$613$	−3756.04	−0.247479	−0.123740	−	0.992315i	$-0.539489\pi$
$613$	−3756.04	−0.247479	−0.123740	+	0.992315i	$0.539489\pi$
$614$	−29789.9	−1.95802
$615$	2581.74	0.169277
$616$	−6023.40	−0.393977
$617$	28309.3	1.84715	0.923573	−	0.383422i	$-0.125254\pi$
$617$	28309.3	1.84715	0.923573	+	0.383422i	$0.125254\pi$
$618$	43984.3	2.86295
$619$	4606.62	0.299121	0.149560	−	0.988753i	$-0.452214\pi$
$619$	4606.62	0.299121	0.149560	+	0.988753i	$0.452214\pi$
$620$	−2149.05	−0.139206
$621$	−10655.2	−0.688531
$622$	−42121.9	−2.71533
$623$	20128.6	1.29444
$624$	2192.39	0.140650
$625$	625.000	0.0400000
$626$	18702.5	1.19409
$627$	−1234.05	−0.0786015
$628$	29833.8	1.89570
$629$	−6176.89	−0.391556
$630$	5584.21	0.353143
$631$	1121.08	0.0707279	0.0353639	−	0.999375i	$-0.488741\pi$
$631$	1121.08	0.0707279	0.0353639	+	0.999375i	$0.488741\pi$
$632$	10650.0	0.670307
$633$	−29489.1	−1.85164
$634$	−12007.4	−0.752171
$635$	−3372.22	−0.210744
$636$	24327.6	1.51675
$637$	−13659.9	−0.849649
$638$	−6881.38	−0.427017
$639$	2852.48	0.176592
$640$	9718.21	0.600228
$641$	1293.85	0.0797254	0.0398627	−	0.999205i	$-0.487308\pi$
$641$	1293.85	0.0797254	0.0398627	+	0.999205i	$0.487308\pi$
$642$	−21812.8	−1.34094
$643$	−9377.41	−0.575131	−0.287565	−	0.957761i	$-0.592846\pi$
$643$	−9377.41	−0.575131	−0.287565	+	0.957761i	$0.592846\pi$
$644$	−35634.0	−2.18040
$645$	6903.31	0.421423
$646$	4513.55	0.274896
$647$	−27881.0	−1.69415	−0.847076	−	0.531472i	$-0.821639\pi$
$647$	−27881.0	−1.69415	−0.847076	+	0.531472i	$0.821639\pi$
$648$	−15109.3	−0.915971
$649$	3462.95	0.209449
$650$	−2261.32	−0.136456
$651$	−6822.97	−0.410773
$652$	14934.4	0.897052
$653$	31158.5	1.86727	0.933636	−	0.358224i	$-0.116618\pi$
$653$	31158.5	1.86727	0.933636	+	0.358224i	$0.116618\pi$
$654$	−54691.5	−3.27004
$655$	−11019.6	−0.657362
$656$	1599.58	0.0952029
$657$	5096.81	0.302657
$658$	85063.9	5.03972
$659$	14916.4	0.881728	0.440864	−	0.897574i	$-0.354672\pi$
$659$	14916.4	0.881728	0.440864	+	0.897574i	$0.354672\pi$
$660$	−3850.06	−0.227066
$661$	−1150.18	−0.0676807	−0.0338404	−	0.999427i	$-0.510774\pi$
$661$	−1150.18	−0.0676807	−0.0338404	+	0.999427i	$0.510774\pi$
$662$	−21794.1	−1.27953
$663$	6389.87	0.374302
$664$	−17501.3	−1.02286
$665$	−3027.99	−0.176572
$666$	4059.81	0.236208
$667$	−13239.1	−0.768544
$668$	42152.2	2.44149
$669$	−21709.8	−1.25463
$670$	−4595.20	−0.264967
$671$	−4888.08	−0.281225
$672$	41205.1	2.36536
$673$	−8169.77	−0.467937	−0.233968	−	0.972244i	$-0.575171\pi$
$673$	−8169.77	−0.467937	−0.233968	+	0.972244i	$0.575171\pi$
$674$	−12201.2	−0.697289
$675$	−2824.79	−0.161076
$676$	−21161.3	−1.20399
$677$	10249.9	0.581883	0.290942	−	0.956741i	$-0.406031\pi$
$677$	10249.9	0.581883	0.290942	+	0.956741i	$0.406031\pi$
$678$	50994.2	2.88853
$679$	31851.1	1.80020
$680$	4579.44	0.258255
$681$	−16907.8	−0.951409
$682$	1777.01	0.0997730
$683$	−25862.0	−1.44888	−0.724438	−	0.689340i	$-0.757901\pi$
$683$	−25862.0	−1.44888	−0.724438	+	0.689340i	$0.757901\pi$
$684$	−1771.30	−0.0990168
$685$	−3029.00	−0.168952
$686$	−46858.3	−2.60796
$687$	−21092.7	−1.17138
$688$	4277.12	0.237011
$689$	−7054.65	−0.390074
$690$	−12405.4	−0.684443
$691$	7203.90	0.396598	0.198299	−	0.980142i	$-0.436458\pi$
$691$	7203.90	0.396598	0.198299	+	0.980142i	$0.436458\pi$
$692$	−22250.1	−1.22229
$693$	−2757.05	−0.151128
$694$	−25806.0	−1.41150
$695$	10452.7	0.570494
$696$	−14241.2	−0.775592
$697$	4662.08	0.253356
$698$	−1251.16	−0.0678471
$699$	5526.93	0.299067
$700$	−9446.91	−0.510085
$701$	−28240.6	−1.52159	−0.760795	−	0.648992i	$-0.775191\pi$
$701$	−28240.6	−1.52159	−0.760795	+	0.648992i	$0.775191\pi$
$702$	10220.4	0.549494
$703$	−2201.40	−0.118104
$704$	−9122.00	−0.488350
$705$	17681.9	0.944597
$706$	−40967.6	−2.18390
$707$	15357.2	0.816926
$708$	22037.3	1.16979
$709$	−8600.58	−0.455573	−0.227787	−	0.973711i	$-0.573149\pi$
$709$	−8600.58	−0.455573	−0.227787	+	0.973711i	$0.573149\pi$
$710$	−8081.87	−0.427193
$711$	4874.75	0.257127
$712$	−10849.3	−0.571058
$713$	3418.78	0.179571
$714$	44707.6	2.34333
$715$	1116.46	0.0583964
$716$	−43168.6	−2.25319
$717$	18332.1	0.954847
$718$	39715.9	2.06433
$719$	−11597.3	−0.601538	−0.300769	−	0.953697i	$-0.597243\pi$
$719$	−11597.3	−0.601538	−0.300769	+	0.953697i	$0.597243\pi$
$720$	719.185	0.0372256
$721$	−53284.6	−2.75232
$722$	1608.60	0.0829166
$723$	−41058.7	−2.11202
$724$	−877.009	−0.0450190
$725$	−3509.80	−0.179794
$726$	3183.55	0.162745
$727$	−1735.75	−0.0885494	−0.0442747	−	0.999019i	$-0.514098\pi$
$727$	−1735.75	−0.0885494	−0.0442747	+	0.999019i	$0.514098\pi$
$728$	11115.5	0.565892
$729$	11097.6	0.563814
$730$	−14440.7	−0.732157
$731$	12466.0	0.630739
$732$	−31106.4	−1.57067
$733$	17513.6	0.882511	0.441255	−	0.897382i	$-0.354533\pi$
$733$	17513.6	0.882511	0.441255	+	0.897382i	$0.354533\pi$
$734$	48598.0	2.44385
$735$	−19866.6	−0.996992
$736$	−20646.6	−1.03403
$737$	2268.75	0.113393
$738$	−3064.20	−0.152838
$739$	18413.4	0.916575	0.458288	−	0.888804i	$-0.348463\pi$
$739$	18413.4	0.916575	0.458288	+	0.888804i	$0.348463\pi$
$740$	−6868.06	−0.341183
$741$	2277.31	0.112900
$742$	−49358.8	−2.44207
$743$	277.332	0.0136936	0.00684679	−	0.999977i	$-0.497821\pi$
$743$	277.332	0.0136936	0.00684679	+	0.999977i	$0.497821\pi$
$744$	3677.57	0.181218
$745$	−12600.7	−0.619669
$746$	−39890.8	−1.95778
$747$	−8010.74	−0.392366
$748$	−6952.42	−0.339847
$749$	26425.1	1.28912
$750$	−3288.79	−0.160120
$751$	−18491.4	−0.898485	−0.449243	−	0.893410i	$-0.648306\pi$
$751$	−18491.4	−0.898485	−0.449243	+	0.893410i	$0.648306\pi$
$752$	10955.3	0.531248
$753$	−5953.98	−0.288147
$754$	12698.9	0.613350
$755$	−12696.7	−0.612027
$756$	42696.9	2.05406
$757$	8912.05	0.427891	0.213946	−	0.976846i	$-0.431368\pi$
$757$	8912.05	0.427891	0.213946	+	0.976846i	$0.431368\pi$
$758$	−13044.1	−0.625042
$759$	6124.82	0.292908
$760$	1632.08	0.0778971
$761$	−1566.04	−0.0745978	−0.0372989	−	0.999304i	$-0.511875\pi$
$761$	−1566.04	−0.0745978	−0.0372989	+	0.999304i	$0.511875\pi$
$762$	17744.8	0.843606
$763$	66255.9	3.14367
$764$	−16485.3	−0.780653
$765$	2096.11	0.0990655
$766$	44469.9	2.09760
$767$	−6390.51	−0.300845
$768$	−11966.0	−0.562223
$769$	−20654.7	−0.968567	−0.484284	−	0.874911i	$-0.660920\pi$
$769$	−20654.7	−0.968567	−0.484284	+	0.874911i	$0.660920\pi$
$770$	7811.49	0.365593
$771$	17785.4	0.830772
$772$	22264.3	1.03797
$773$	38084.3	1.77205	0.886027	−	0.463635i	$-0.153455\pi$
$773$	38084.3	1.77205	0.886027	+	0.463635i	$0.153455\pi$
$774$	−8193.36	−0.380497
$775$	906.351	0.0420092
$776$	−17167.7	−0.794180
$777$	−21805.3	−1.00677
$778$	22867.4	1.05377
$779$	1661.53	0.0764193
$780$	7104.89	0.326148
$781$	3990.20	0.182818
$782$	−22401.6	−1.02440
$783$	15863.2	0.724014
$784$	−12308.8	−0.560716
$785$	−12582.3	−0.572078
$786$	57986.0	2.63141
$787$	17276.5	0.782516	0.391258	−	0.920281i	$-0.372040\pi$
$787$	17276.5	0.782516	0.391258	+	0.920281i	$0.372040\pi$
$788$	13304.9	0.601483
$789$	−13513.0	−0.609730
$790$	−13811.5	−0.622015
$791$	−61776.8	−2.77690
$792$	1486.04	0.0666720
$793$	9020.44	0.403941
$794$	−61352.6	−2.74222
$795$	−10260.1	−0.457719
$796$	−9503.68	−0.423177
$797$	−1215.30	−0.0540128	−0.0270064	−	0.999635i	$-0.508597\pi$
$797$	−1215.30	−0.0540128	−0.0270064	+	0.999635i	$0.508597\pi$
$798$	15933.5	0.706816
$799$	31929.9	1.41377
$800$	−5473.61	−0.241902
$801$	−4965.96	−0.219056
$802$	−43908.8	−1.93326
$803$	7129.69	0.313327
$804$	14437.7	0.633308
$805$	15028.5	0.657993
$806$	−3279.28	−0.143310
$807$	10419.3	0.454492
$808$	−8277.49	−0.360398
$809$	12902.2	0.560712	0.280356	−	0.959896i	$-0.409548\pi$
$809$	12902.2	0.560712	0.280356	+	0.959896i	$0.409548\pi$
$810$	19594.6	0.849981
$811$	−26805.3	−1.16062	−0.580309	−	0.814396i	$-0.697068\pi$
$811$	−26805.3	−1.16062	−0.580309	+	0.814396i	$0.697068\pi$
$812$	53050.9	2.29276
$813$	−783.631	−0.0338046
$814$	5679.08	0.244535
$815$	−6298.54	−0.270710
$816$	5757.85	0.247016
$817$	4442.78	0.190249
$818$	43183.7	1.84582
$819$	5087.84	0.217074
$820$	5183.76	0.220762
$821$	29715.4	1.26319	0.631593	−	0.775300i	$-0.282401\pi$
$821$	29715.4	1.26319	0.631593	+	0.775300i	$0.282401\pi$
$822$	15938.8	0.676314
$823$	−32908.9	−1.39384	−0.696921	−	0.717148i	$-0.745447\pi$
$823$	−32908.9	−1.39384	−0.696921	+	0.717148i	$0.745447\pi$
$824$	28720.3	1.21422
$825$	1623.75	0.0685232
$826$	−44712.0	−1.88345
$827$	−8521.96	−0.358329	−0.179164	−	0.983819i	$-0.557339\pi$
$827$	−8521.96	−0.358329	−0.179164	+	0.983819i	$0.557339\pi$
$828$	8791.32	0.368985
$829$	38044.5	1.59390	0.796949	−	0.604046i	$-0.206446\pi$
$829$	38044.5	1.59390	0.796949	+	0.604046i	$0.206446\pi$
$830$	22696.7	0.949172
$831$	−37640.0	−1.57126
$832$	16833.7	0.701447
$833$	−35874.9	−1.49219
$834$	−55002.7	−2.28368
$835$	−17777.5	−0.736786
$836$	−2477.79	−0.102508
$837$	−4096.41	−0.169167
$838$	−38330.2	−1.58007
$839$	24606.7	1.01253	0.506267	−	0.862377i	$-0.331025\pi$
$839$	24606.7	1.01253	0.506267	+	0.862377i	$0.331025\pi$
$840$	16166.1	0.664027
$841$	−4679.03	−0.191850
$842$	6887.12	0.281883
$843$	−16542.1	−0.675849
$844$	−59209.9	−2.41480
$845$	8924.68	0.363335
$846$	−20986.2	−0.852863
$847$	−3856.70	−0.156456
$848$	−6356.87	−0.257424
$849$	−27801.3	−1.12384
$850$	−5938.88	−0.239649
$851$	10926.0	0.440114
$852$	25392.6	1.02105
$853$	−36928.3	−1.48230	−0.741149	−	0.671341i	$-0.765719\pi$
$853$	−36928.3	−1.48230	−0.741149	+	0.671341i	$0.765719\pi$
$854$	63112.6	2.52889
$855$	747.041	0.0298810
$856$	−14243.0	−0.568712
$857$	−2773.03	−0.110531	−0.0552653	−	0.998472i	$-0.517600\pi$
$857$	−2773.03	−0.110531	−0.0552653	+	0.998472i	$0.517600\pi$
$858$	−5874.91	−0.233760
$859$	9087.42	0.360953	0.180477	−	0.983579i	$-0.442236\pi$
$859$	9087.42	0.360953	0.180477	+	0.983579i	$0.442236\pi$
$860$	13860.9	0.549595
$861$	16457.8	0.651430
$862$	4063.95	0.160578
$863$	3706.49	0.146200	0.0730998	−	0.997325i	$-0.476711\pi$
$863$	3706.49	0.146200	0.0730998	+	0.997325i	$0.476711\pi$
$864$	24738.9	0.974114
$865$	9383.91	0.368858
$866$	19532.6	0.766448
$867$	−12227.4	−0.478965
$868$	−13699.6	−0.535706
$869$	6819.06	0.266192
$870$	18468.8	0.719715
$871$	−4186.75	−0.162873
$872$	−35711.8	−1.38687
$873$	−7858.04	−0.304644
$874$	−7983.77	−0.308988
$875$	3984.20	0.153932
$876$	45371.5	1.74996
$877$	−3306.44	−0.127310	−0.0636548	−	0.997972i	$-0.520276\pi$
$877$	−3306.44	−0.127310	−0.0636548	+	0.997972i	$0.520276\pi$
$878$	51514.6	1.98011
$879$	24173.7	0.927598
$880$	1006.03	0.0385380
$881$	21654.6	0.828105	0.414053	−	0.910253i	$-0.364113\pi$
$881$	21654.6	0.828105	0.414053	+	0.910253i	$0.364113\pi$
$882$	23579.1	0.900170
$883$	30304.1	1.15494	0.577472	−	0.816410i	$-0.304039\pi$
$883$	30304.1	1.15494	0.577472	+	0.816410i	$0.304039\pi$
$884$	12830.0	0.488143
$885$	−9294.15	−0.353016
$886$	−63273.1	−2.39921
$887$	30582.0	1.15766	0.578830	−	0.815448i	$-0.303510\pi$
$887$	30582.0	1.15766	0.578830	+	0.815448i	$0.303510\pi$
$888$	11753.0	0.444150
$889$	−21496.9	−0.811006
$890$	14070.0	0.529917
$891$	−9674.29	−0.363750
$892$	−43590.3	−1.63622
$893$	11379.6	0.426433
$894$	66305.6	2.48053
$895$	18206.2	0.679962
$896$	61950.8	2.30986
$897$	−11302.7	−0.420721
$898$	−17677.8	−0.656921
$899$	−5089.79	−0.188825
$900$	2330.66	0.0863209
$901$	−18527.5	−0.685063
$902$	−4286.36	−0.158226
$903$	44006.7	1.62176
$904$	33297.6	1.22507
$905$	369.875	0.0135857
$906$	66810.9	2.44994
$907$	41733.2	1.52781	0.763907	−	0.645326i	$-0.223278\pi$
$907$	41733.2	1.52781	0.763907	+	0.645326i	$0.223278\pi$
$908$	−33948.5	−1.24077
$909$	−3788.80	−0.138247
$910$	−14415.3	−0.525123
$911$	2913.82	0.105971	0.0529853	−	0.998595i	$-0.483126\pi$
$911$	2913.82	0.105971	0.0529853	+	0.998595i	$0.483126\pi$
$912$	2052.06	0.0745070
$913$	−11205.8	−0.406199
$914$	40854.7	1.47850
$915$	13119.0	0.473991
$916$	−42351.1	−1.52764
$917$	−70247.0	−2.52973
$918$	26841.8	0.965043
$919$	−27470.2	−0.986026	−0.493013	−	0.870022i	$-0.664104\pi$
$919$	−27470.2	−0.986026	−0.493013	+	0.870022i	$0.664104\pi$
$920$	−8100.32	−0.290282
$921$	−39474.4	−1.41230
$922$	−54054.5	−1.93079
$923$	−7363.49	−0.262592
$924$	−24543.0	−0.873817
$925$	2896.58	0.102961
$926$	37670.4	1.33685
$927$	13145.9	0.465771
$928$	30738.1	1.08731
$929$	21194.3	0.748506	0.374253	−	0.927327i	$-0.377899\pi$
$929$	21194.3	0.748506	0.374253	+	0.927327i	$0.377899\pi$
$930$	−4769.28	−0.168162
$931$	−12785.6	−0.450086
$932$	11097.3	0.390026
$933$	−55815.4	−1.95853
$934$	26382.8	0.924273
$935$	2932.15	0.102558
$936$	−2742.34	−0.0957650
$937$	−5612.02	−0.195664	−0.0978318	−	0.995203i	$-0.531191\pi$
$937$	−5612.02	−0.195664	−0.0978318	+	0.995203i	$0.531191\pi$
$938$	−29293.1	−1.01967
$939$	24782.5	0.861284
$940$	35502.8	1.23189
$941$	1727.44	0.0598436	0.0299218	−	0.999552i	$-0.490474\pi$
$941$	1727.44	0.0598436	0.0299218	+	0.999552i	$0.490474\pi$
$942$	66208.8	2.29002
$943$	−8246.51	−0.284776
$944$	−5758.42	−0.198539
$945$	−18007.3	−0.619869
$946$	−11461.3	−0.393911
$947$	−55954.8	−1.92005	−0.960024	−	0.279917i	$-0.909693\pi$
$947$	−55954.8	−1.92005	−0.960024	+	0.279917i	$0.909693\pi$
$948$	43394.7	1.48670
$949$	−13157.1	−0.450050
$950$	−2116.58	−0.0722850
$951$	−15911.0	−0.542533
$952$	29192.6	0.993842
$953$	1695.99	0.0576481	0.0288240	−	0.999585i	$-0.490824\pi$
$953$	1695.99	0.0576481	0.0288240	+	0.999585i	$0.490824\pi$
$954$	12177.4	0.413268
$955$	6952.63	0.235583
$956$	36808.3	1.24526
$957$	−9118.46	−0.308002
$958$	−1342.17	−0.0452646
$959$	−19309.0	−0.650179
$960$	24482.4	0.823089
$961$	−28476.6	−0.955881
$962$	−10480.2	−0.351241
$963$	−6519.37	−0.218156
$964$	−82440.0	−2.75437
$965$	−9389.89	−0.313235
$966$	−79080.9	−2.63394
$967$	−10102.4	−0.335959	−0.167980	−	0.985790i	$-0.553724\pi$
$967$	−10102.4	−0.335959	−0.167980	+	0.985790i	$0.553724\pi$
$968$	2078.75	0.0690224
$969$	5980.86	0.198280
$970$	22264.0	0.736964
$971$	−51391.4	−1.69848	−0.849242	−	0.528005i	$-0.822940\pi$
$971$	−51391.4	−1.69848	−0.849242	+	0.528005i	$0.822940\pi$
$972$	−25396.2	−0.838050
$973$	66632.9	2.19543
$974$	37473.1	1.23277
$975$	−2996.46	−0.0984241
$976$	8128.22	0.266576
$977$	−6122.67	−0.200493	−0.100247	−	0.994963i	$-0.531963\pi$
$977$	−6122.67	−0.200493	−0.100247	+	0.994963i	$0.531963\pi$
$978$	33143.3	1.08365
$979$	−6946.65	−0.226778
$980$	−39889.2	−1.30022
$981$	−16346.1	−0.531999
$982$	31830.8	1.03438
$983$	−34706.1	−1.12609	−0.563047	−	0.826425i	$-0.690371\pi$
$983$	−34706.1	−1.12609	−0.563047	+	0.826425i	$0.690371\pi$
$984$	−8870.74	−0.287387
$985$	−5611.30	−0.181514
$986$	33350.9	1.07719
$987$	112717.	3.63509
$988$	4572.51	0.147238
$989$	−22050.4	−0.708960
$990$	−1927.19	−0.0618687
$991$	−6536.73	−0.209532	−0.104766	−	0.994497i	$-0.533409\pi$
$991$	−6536.73	−0.209532	−0.104766	+	0.994497i	$0.533409\pi$
$992$	−7937.62	−0.254052
$993$	−28879.1	−0.922912
$994$	−51519.6	−1.64397
$995$	4008.14	0.127705
$996$	−71311.0	−2.26865
$997$	−11198.0	−0.355711	−0.177856	−	0.984057i	$-0.556916\pi$
$997$	−11198.0	−0.355711	−0.177856	+	0.984057i	$0.556916\pi$
$998$	40150.8	1.27350
$999$	−13091.6	−0.414614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.f.1.20	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.f.1.20	✓	23	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+4.45595 q^{2} +5.90454 q^{3} +11.8555 q^{4} -5.00000 q^{5} +26.3103 q^{6} -31.8736 q^{7} +17.1798 q^{8} +7.86359 q^{9} +O(q^{10})\)	\(q+4.45595 q^{2} +5.90454 q^{3} +11.8555 q^{4} -5.00000 q^{5} +26.3103 q^{6} -31.8736 q^{7} +17.1798 q^{8} +7.86359 q^{9} -22.2797 q^{10} +11.0000 q^{11} +70.0011 q^{12} -20.2994 q^{13} -142.027 q^{14} -29.5227 q^{15} -18.2915 q^{16} -53.3119 q^{17} +35.0397 q^{18} -19.0000 q^{19} -59.2774 q^{20} -188.199 q^{21} +49.0154 q^{22} +94.3006 q^{23} +101.439 q^{24} +25.0000 q^{25} -90.4529 q^{26} -112.992 q^{27} -377.876 q^{28} -140.392 q^{29} -131.552 q^{30} +36.2541 q^{31} -218.944 q^{32} +64.9499 q^{33} -237.555 q^{34} +159.368 q^{35} +93.2265 q^{36} +115.863 q^{37} -84.6630 q^{38} -119.858 q^{39} -85.8990 q^{40} -87.4492 q^{41} -838.604 q^{42} -233.831 q^{43} +130.410 q^{44} -39.3179 q^{45} +420.199 q^{46} -598.927 q^{47} -108.003 q^{48} +672.925 q^{49} +111.399 q^{50} -314.782 q^{51} -240.658 q^{52} +347.531 q^{53} -503.485 q^{54} -55.0000 q^{55} -547.582 q^{56} -112.186 q^{57} -625.580 q^{58} +314.814 q^{59} -350.006 q^{60} -444.371 q^{61} +161.546 q^{62} -250.641 q^{63} -829.273 q^{64} +101.497 q^{65} +289.414 q^{66} +206.250 q^{67} -632.038 q^{68} +556.802 q^{69} +710.135 q^{70} +362.745 q^{71} +135.095 q^{72} +648.154 q^{73} +516.280 q^{74} +147.613 q^{75} -225.254 q^{76} -350.609 q^{77} -534.082 q^{78} +619.914 q^{79} +91.4576 q^{80} -879.481 q^{81} -389.669 q^{82} -1018.71 q^{83} -2231.19 q^{84} +266.559 q^{85} -1041.94 q^{86} -828.951 q^{87} +188.978 q^{88} -631.513 q^{89} -175.199 q^{90} +647.013 q^{91} +1117.98 q^{92} +214.064 q^{93} -2668.79 q^{94} +95.0000 q^{95} -1292.77 q^{96} -999.295 q^{97} +2998.52 q^{98} +86.4994 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * q^99

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