LMFDB - Embedded newform 1045.4.a.c.1.2

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + \cdots + 150528 $ x^20 - x^19 - 105x^18 + 103x^17 + 4500x^16 - 4345x^15 - 101844x^14 + 95592x^13 + 1317797x^12 - 1160501x^11 - 9914845x^10 + 7570653x^9 + 42786958x^8 - 23777633x^7 - 102801526x^6 + 28436356x^5 + 122325928x^4 + 411232x^3 - 47350496x^2 - 4782848x + 150528
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.63319$ of defining polynomial
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)

$f(q)$	$=$	$q-4.63319 q^{2} -7.91113 q^{3} +13.4664 q^{4} -5.00000 q^{5} +36.6537 q^{6} +11.8333 q^{7} -25.3269 q^{8} +35.5860 q^{9} +O(q^{10})$	\(q-4.63319 q^{2} -7.91113 q^{3} +13.4664 q^{4} -5.00000 q^{5} +36.6537 q^{6} +11.8333 q^{7} -25.3269 q^{8} +35.5860 q^{9} +23.1659 q^{10} -11.0000 q^{11} -106.535 q^{12} +57.0427 q^{13} -54.8258 q^{14} +39.5556 q^{15} +9.61312 q^{16} -10.8307 q^{17} -164.876 q^{18} +19.0000 q^{19} -67.3321 q^{20} -93.6146 q^{21} +50.9651 q^{22} -61.0049 q^{23} +200.365 q^{24} +25.0000 q^{25} -264.289 q^{26} -67.9246 q^{27} +159.352 q^{28} -248.803 q^{29} -183.269 q^{30} -143.802 q^{31} +158.076 q^{32} +87.0224 q^{33} +50.1805 q^{34} -59.1664 q^{35} +479.215 q^{36} +111.873 q^{37} -88.0306 q^{38} -451.272 q^{39} +126.635 q^{40} -150.164 q^{41} +433.734 q^{42} -9.53771 q^{43} -148.131 q^{44} -177.930 q^{45} +282.647 q^{46} +98.6850 q^{47} -76.0506 q^{48} -202.974 q^{49} -115.830 q^{50} +85.6828 q^{51} +768.161 q^{52} +164.534 q^{53} +314.707 q^{54} +55.0000 q^{55} -299.701 q^{56} -150.311 q^{57} +1152.75 q^{58} -83.2619 q^{59} +532.673 q^{60} +61.0571 q^{61} +666.260 q^{62} +421.099 q^{63} -809.302 q^{64} -285.213 q^{65} -403.191 q^{66} +714.520 q^{67} -145.850 q^{68} +482.618 q^{69} +274.129 q^{70} +384.213 q^{71} -901.284 q^{72} +337.256 q^{73} -518.327 q^{74} -197.778 q^{75} +255.862 q^{76} -130.166 q^{77} +2090.83 q^{78} +773.474 q^{79} -48.0656 q^{80} -423.461 q^{81} +695.738 q^{82} -461.328 q^{83} -1260.65 q^{84} +54.1533 q^{85} +44.1900 q^{86} +1968.31 q^{87} +278.596 q^{88} +1355.95 q^{89} +824.382 q^{90} +675.002 q^{91} -821.518 q^{92} +1137.63 q^{93} -457.226 q^{94} -95.0000 q^{95} -1250.56 q^{96} +954.120 q^{97} +940.414 q^{98} -391.445 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) $ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * 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 $

$2$	−4.63319	−1.63808	−0.819039	−	0.573737i	$-0.805493\pi$
$2$	−4.63319	−1.63808	−0.819039	+	0.573737i	$0.805493\pi$
$3$	−7.91113	−1.52250	−0.761249	−	0.648460i	$-0.775413\pi$
$3$	−7.91113	−1.52250	−0.761249	+	0.648460i	$0.775413\pi$
$4$	13.4664	1.68330
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	36.6537	2.49397
$7$	11.8333	0.638937	0.319468	−	0.947597i	$-0.396496\pi$
$7$	11.8333	0.638937	0.319468	+	0.947597i	$0.396496\pi$
$8$	−25.3269	−1.11930
$9$	35.5860	1.31800
$10$	23.1659	0.732571
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−106.535	−2.56282
$13$	57.0427	1.21698	0.608492	−	0.793560i	$-0.291775\pi$
$13$	57.0427	1.21698	0.608492	+	0.793560i	$0.291775\pi$
$14$	−54.8258	−1.04663
$15$	39.5556	0.680882
$16$	9.61312	0.150205
$17$	−10.8307	−0.154519	−0.0772594	−	0.997011i	$-0.524617\pi$
$17$	−10.8307	−0.154519	−0.0772594	+	0.997011i	$0.524617\pi$
$18$	−164.876	−2.15899
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−67.3321	−0.752796
$21$	−93.6146	−0.972780
$22$	50.9651	0.493899
$23$	−61.0049	−0.553061	−0.276531	−	0.961005i	$-0.589185\pi$
$23$	−61.0049	−0.553061	−0.276531	+	0.961005i	$0.589185\pi$
$24$	200.365	1.70414
$25$	25.0000	0.200000
$26$	−264.289	−1.99352
$27$	−67.9246	−0.484152
$28$	159.352	1.07552
$29$	−248.803	−1.59316	−0.796578	−	0.604536i	$-0.793359\pi$
$29$	−248.803	−1.59316	−0.796578	+	0.604536i	$0.793359\pi$
$30$	−183.269	−1.11534
$31$	−143.802	−0.833146	−0.416573	−	0.909102i	$-0.636769\pi$
$31$	−143.802	−0.833146	−0.416573	+	0.909102i	$0.636769\pi$
$32$	158.076	0.873256
$33$	87.0224	0.459050
$34$	50.1805	0.253114
$35$	−59.1664	−0.285741
$36$	479.215	2.21859
$37$	111.873	0.497074	0.248537	−	0.968622i	$-0.420050\pi$
$37$	111.873	0.497074	0.248537	+	0.968622i	$0.420050\pi$
$38$	−88.0306	−0.375801
$39$	−451.272	−1.85285
$40$	126.635	0.500568
$41$	−150.164	−0.571992	−0.285996	−	0.958231i	$-0.592324\pi$
$41$	−150.164	−0.571992	−0.285996	+	0.958231i	$0.592324\pi$
$42$	433.734	1.59349
$43$	−9.53771	−0.0338253	−0.0169126	−	0.999857i	$-0.505384\pi$
$43$	−9.53771	−0.0338253	−0.0169126	+	0.999857i	$0.505384\pi$
$44$	−148.131	−0.507535
$45$	−177.930	−0.589427
$46$	282.647	0.905958
$47$	98.6850	0.306270	0.153135	−	0.988205i	$-0.451063\pi$
$47$	98.6850	0.306270	0.153135	+	0.988205i	$0.451063\pi$
$48$	−76.0506	−0.228687
$49$	−202.974	−0.591760
$50$	−115.830	−0.327616
$51$	85.6828	0.235255
$52$	768.161	2.04855
$53$	164.534	0.426423	0.213212	−	0.977006i	$-0.431608\pi$
$53$	164.534	0.426423	0.213212	+	0.977006i	$0.431608\pi$
$54$	314.707	0.793078
$55$	55.0000	0.134840
$56$	−299.701	−0.715164
$57$	−150.311	−0.349285
$58$	1152.75	2.60971
$59$	−83.2619	−0.183725	−0.0918625	−	0.995772i	$-0.529282\pi$
$59$	−83.2619	−0.183725	−0.0918625	+	0.995772i	$0.529282\pi$
$60$	532.673	1.14613
$61$	61.0571	0.128157	0.0640784	−	0.997945i	$-0.479589\pi$
$61$	61.0571	0.128157	0.0640784	+	0.997945i	$0.479589\pi$
$62$	666.260	1.36476
$63$	421.099	0.842118
$64$	−809.302	−1.58067
$65$	−285.213	−0.544252
$66$	−403.191	−0.751961
$67$	714.520	1.30287	0.651436	−	0.758703i	$-0.274167\pi$
$67$	714.520	1.30287	0.651436	+	0.758703i	$0.274167\pi$
$68$	−145.850	−0.260102
$69$	482.618	0.842034
$70$	274.129	0.468067
$71$	384.213	0.642220	0.321110	−	0.947042i	$-0.395944\pi$
$71$	384.213	0.642220	0.321110	+	0.947042i	$0.395944\pi$
$72$	−901.284	−1.47524
$73$	337.256	0.540724	0.270362	−	0.962759i	$-0.412857\pi$
$73$	337.256	0.540724	0.270362	+	0.962759i	$0.412857\pi$
$74$	−518.327	−0.814246
$75$	−197.778	−0.304499
$76$	255.862	0.386176
$77$	−130.166	−0.192647
$78$	2090.83	3.03512
$79$	773.474	1.10155	0.550776	−	0.834653i	$-0.314332\pi$
$79$	773.474	1.10155	0.550776	+	0.834653i	$0.314332\pi$
$80$	−48.0656	−0.0671737
$81$	−423.461	−0.580879
$82$	695.738	0.936968
$83$	−461.328	−0.610089	−0.305044	−	0.952338i	$-0.598671\pi$
$83$	−461.328	−0.610089	−0.305044	+	0.952338i	$0.598671\pi$
$84$	−1260.65	−1.63748
$85$	54.1533	0.0691029
$86$	44.1900	0.0554085
$87$	1968.31	2.42558
$88$	278.596	0.337483
$89$	1355.95	1.61495	0.807475	−	0.589901i	$-0.200833\pi$
$89$	1355.95	1.61495	0.807475	+	0.589901i	$0.200833\pi$
$90$	824.382	0.965528
$91$	675.002	0.777576
$92$	−821.518	−0.930969
$93$	1137.63	1.26846
$94$	−457.226	−0.501694
$95$	−95.0000	−0.102598
$96$	−1250.56	−1.32953
$97$	954.120	0.998724	0.499362	−	0.866394i	$-0.333568\pi$
$97$	954.120	0.998724	0.499362	+	0.866394i	$0.333568\pi$
$98$	940.414	0.969349
$99$	−391.445	−0.397391
$100$	336.661	0.336661
$101$	−1249.95	−1.23143	−0.615717	−	0.787967i	$-0.711134\pi$
$101$	−1249.95	−1.23143	−0.615717	+	0.787967i	$0.711134\pi$
$102$	−396.984	−0.385366
$103$	−907.428	−0.868073	−0.434037	−	0.900895i	$-0.642911\pi$
$103$	−907.428	−0.868073	−0.434037	+	0.900895i	$0.642911\pi$
$104$	−1444.72	−1.36217
$105$	468.073	0.435040
$106$	−762.315	−0.698515
$107$	−1860.68	−1.68111	−0.840554	−	0.541728i	$-0.817770\pi$
$107$	−1860.68	−1.68111	−0.840554	+	0.541728i	$0.817770\pi$
$108$	−914.701	−0.814974
$109$	1996.44	1.75435	0.877174	−	0.480172i	$-0.159426\pi$
$109$	1996.44	1.75435	0.877174	+	0.480172i	$0.159426\pi$
$110$	−254.825	−0.220879
$111$	−885.038	−0.756794
$112$	113.755	0.0959715
$113$	1500.20	1.24891	0.624456	−	0.781060i	$-0.285321\pi$
$113$	1500.20	1.24891	0.624456	+	0.781060i	$0.285321\pi$
$114$	696.421	0.572156
$115$	305.025	0.247336
$116$	−3350.48	−2.68176
$117$	2029.92	1.60398
$118$	385.768	0.300956
$119$	−128.162	−0.0987278
$120$	−1001.82	−0.762113
$121$	121.000	0.0909091
$122$	−282.889	−0.209931
$123$	1187.97	0.870857
$124$	−1936.49	−1.40244
$125$	−125.000	−0.0894427
$126$	−1951.03	−1.37946
$127$	−1192.55	−0.833242	−0.416621	−	0.909080i	$-0.636786\pi$
$127$	−1192.55	−0.833242	−0.416621	+	0.909080i	$0.636786\pi$
$128$	2485.04	1.71600
$129$	75.4541	0.0514989
$130$	1321.45	0.891527
$131$	1325.07	0.883757	0.441878	−	0.897075i	$-0.354312\pi$
$131$	1325.07	0.883757	0.441878	+	0.897075i	$0.354312\pi$
$132$	1171.88	0.772720
$133$	224.832	0.146582
$134$	−3310.50	−2.13421
$135$	339.623	0.216519
$136$	274.308	0.172954
$137$	−231.027	−0.144073	−0.0720363	−	0.997402i	$-0.522950\pi$
$137$	−231.027	−0.144073	−0.0720363	+	0.997402i	$0.522950\pi$
$138$	−2236.06	−1.37932
$139$	−185.950	−0.113468	−0.0567341	−	0.998389i	$-0.518069\pi$
$139$	−185.950	−0.113468	−0.0567341	+	0.998389i	$0.518069\pi$
$140$	−796.760	−0.480989
$141$	−780.710	−0.466295
$142$	−1780.13	−1.05201
$143$	−627.469	−0.366934
$144$	342.092	0.197970
$145$	1244.01	0.712481
$146$	−1562.57	−0.885749
$147$	1605.75	0.900952
$148$	1506.52	0.836726
$149$	−91.3282	−0.0502141	−0.0251070	−	0.999685i	$-0.507993\pi$
$149$	−91.3282	−0.0502141	−0.0251070	+	0.999685i	$0.507993\pi$
$150$	916.343	0.498794
$151$	−1132.46	−0.610319	−0.305159	−	0.952301i	$-0.598710\pi$
$151$	−1132.46	−0.610319	−0.305159	+	0.952301i	$0.598710\pi$
$152$	−481.212	−0.256786
$153$	−385.419	−0.203656
$154$	603.084	0.315571
$155$	719.008	0.372594
$156$	−6077.02	−3.11892
$157$	−1962.04	−0.997377	−0.498689	−	0.866781i	$-0.666185\pi$
$157$	−1962.04	−0.997377	−0.498689	+	0.866781i	$0.666185\pi$
$158$	−3583.65	−1.80443
$159$	−1301.65	−0.649228
$160$	−790.381	−0.390532
$161$	−721.888	−0.353371
$162$	1961.97	0.951525
$163$	−1684.47	−0.809433	−0.404717	−	0.914442i	$-0.632630\pi$
$163$	−1684.47	−0.809433	−0.404717	+	0.914442i	$0.632630\pi$
$164$	−2022.17	−0.962836
$165$	−435.112	−0.205294
$166$	2137.42	0.999374
$167$	309.343	0.143339	0.0716696	−	0.997428i	$-0.477167\pi$
$167$	309.343	0.143339	0.0716696	+	0.997428i	$0.477167\pi$
$168$	2370.97	1.08884
$169$	1056.87	0.481050
$170$	−250.902	−0.113196
$171$	676.133	0.302370
$172$	−128.439	−0.0569382
$173$	−1025.56	−0.450704	−0.225352	−	0.974277i	$-0.572353\pi$
$173$	−1025.56	−0.450704	−0.225352	+	0.974277i	$0.572353\pi$
$174$	−9119.55	−3.97328
$175$	295.832	0.127787
$176$	−105.744	−0.0452885
$177$	658.696	0.279721
$178$	−6282.38	−2.64542
$179$	−914.294	−0.381774	−0.190887	−	0.981612i	$-0.561136\pi$
$179$	−914.294	−0.381774	−0.190887	+	0.981612i	$0.561136\pi$
$180$	−2396.08	−0.992184
$181$	2403.31	0.986944	0.493472	−	0.869762i	$-0.335728\pi$
$181$	2403.31	0.986944	0.493472	+	0.869762i	$0.335728\pi$
$182$	−3127.41	−1.27373
$183$	−483.031	−0.195118
$184$	1545.07	0.619043
$185$	−559.363	−0.222298
$186$	−5270.87	−2.07784
$187$	119.137	0.0465892
$188$	1328.93	0.515545
$189$	−803.771	−0.309342
$190$	440.153	0.168063
$191$	−37.9340	−0.0143707	−0.00718537	−	0.999974i	$-0.502287\pi$
$191$	−37.9340	−0.0143707	−0.00718537	+	0.999974i	$0.502287\pi$
$192$	6402.49	2.40656
$193$	−303.517	−0.113200	−0.0566001	−	0.998397i	$-0.518026\pi$
$193$	−303.517	−0.113200	−0.0566001	+	0.998397i	$0.518026\pi$
$194$	−4420.62	−1.63599
$195$	2256.36	0.828622
$196$	−2733.33	−0.996110
$197$	2388.64	0.863877	0.431938	−	0.901903i	$-0.357830\pi$
$197$	2388.64	0.863877	0.431938	+	0.901903i	$0.357830\pi$
$198$	1813.64	0.650959
$199$	1136.44	0.404825	0.202413	−	0.979300i	$-0.435122\pi$
$199$	1136.44	0.404825	0.202413	+	0.979300i	$0.435122\pi$
$200$	−633.174	−0.223861
$201$	−5652.66	−1.98362
$202$	5791.26	2.01719
$203$	−2944.15	−1.01793
$204$	1153.84	0.396005
$205$	750.820	0.255803
$206$	4204.28	1.42197
$207$	−2170.92	−0.728934
$208$	548.358	0.182797
$209$	−209.000	−0.0691714
$210$	−2168.67	−0.712631
$211$	−4748.75	−1.54937	−0.774686	−	0.632347i	$-0.782092\pi$
$211$	−4748.75	−1.54937	−0.774686	+	0.632347i	$0.782092\pi$
$212$	2215.68	0.717800
$213$	−3039.56	−0.977779
$214$	8620.87	2.75379
$215$	47.6886	0.0151271
$216$	1720.32	0.541913
$217$	−1701.64	−0.532328
$218$	−9249.87	−2.87376
$219$	−2668.08	−0.823251
$220$	740.653	0.226976
$221$	−617.810	−0.188047
$222$	4100.55	1.23969
$223$	6438.18	1.93333	0.966664	−	0.256048i	$-0.0824205\pi$
$223$	6438.18	1.93333	0.966664	+	0.256048i	$0.0824205\pi$
$224$	1870.56	0.557956
$225$	889.649	0.263600
$226$	−6950.71	−2.04582
$227$	171.157	0.0500444	0.0250222	−	0.999687i	$-0.492034\pi$
$227$	171.157	0.0500444	0.0250222	+	0.999687i	$0.492034\pi$
$228$	−2024.16	−0.587952
$229$	−3694.28	−1.06605	−0.533024	−	0.846100i	$-0.678945\pi$
$229$	−3694.28	−1.06605	−0.533024	+	0.846100i	$0.678945\pi$
$230$	−1413.24	−0.405157
$231$	1029.76	0.293304
$232$	6301.41	1.78322
$233$	5429.65	1.52664	0.763322	−	0.646018i	$-0.223567\pi$
$233$	5429.65	1.52664	0.763322	+	0.646018i	$0.223567\pi$
$234$	−9404.99	−2.62745
$235$	−493.425	−0.136968
$236$	−1121.24	−0.309265
$237$	−6119.05	−1.67711
$238$	593.800	0.161724
$239$	−3074.70	−0.832158	−0.416079	−	0.909329i	$-0.636596\pi$
$239$	−3074.70	−0.832158	−0.416079	+	0.909329i	$0.636596\pi$
$240$	380.253	0.102272
$241$	−4028.11	−1.07665	−0.538326	−	0.842736i	$-0.680943\pi$
$241$	−4028.11	−1.07665	−0.538326	+	0.842736i	$0.680943\pi$
$242$	−560.616	−0.148916
$243$	5184.02	1.36854
$244$	822.221	0.215727
$245$	1014.87	0.264643
$246$	−5504.07	−1.42653
$247$	1083.81	0.279195
$248$	3642.06	0.932544
$249$	3649.63	0.928859
$250$	579.148	0.146514
$251$	747.126	0.187881	0.0939406	−	0.995578i	$-0.470054\pi$
$251$	747.126	0.187881	0.0939406	+	0.995578i	$0.470054\pi$
$252$	5670.69	1.41754
$253$	671.054	0.166754
$254$	5525.31	1.36492
$255$	−428.414	−0.105209
$256$	−5039.22	−1.23028
$257$	−4881.64	−1.18486	−0.592428	−	0.805623i	$-0.701831\pi$
$257$	−4881.64	−1.18486	−0.592428	+	0.805623i	$0.701831\pi$
$258$	−349.593	−0.0843593
$259$	1323.82	0.317599
$260$	−3840.80	−0.916140
$261$	−8853.88	−2.09978
$262$	−6139.31	−1.44766
$263$	−6460.78	−1.51479	−0.757393	−	0.652959i	$-0.773527\pi$
$263$	−6460.78	−1.51479	−0.757393	+	0.652959i	$0.773527\pi$
$264$	−2204.01	−0.513817
$265$	−822.668	−0.190702
$266$	−1041.69	−0.240113
$267$	−10727.1	−2.45876
$268$	9622.02	2.19313
$269$	−6141.73	−1.39207	−0.696036	−	0.718006i	$-0.745055\pi$
$269$	−6141.73	−1.39207	−0.696036	+	0.718006i	$0.745055\pi$
$270$	−1573.54	−0.354675
$271$	3136.14	0.702977	0.351488	−	0.936192i	$-0.385676\pi$
$271$	3136.14	0.702977	0.351488	+	0.936192i	$0.385676\pi$
$272$	−104.116	−0.0232095
$273$	−5340.03	−1.18386
$274$	1070.39	0.236002
$275$	−275.000	−0.0603023
$276$	6499.13	1.41740
$277$	3785.81	0.821181	0.410590	−	0.911820i	$-0.365323\pi$
$277$	3785.81	0.821181	0.410590	+	0.911820i	$0.365323\pi$
$278$	861.541	0.185870
$279$	−5117.32	−1.09809
$280$	1498.50	0.319831
$281$	105.308	0.0223563	0.0111781	−	0.999938i	$-0.496442\pi$
$281$	105.308	0.0223563	0.0111781	+	0.999938i	$0.496442\pi$
$282$	3617.17	0.763828
$283$	−1513.77	−0.317966	−0.158983	−	0.987281i	$-0.550822\pi$
$283$	−1513.77	−0.317966	−0.158983	+	0.987281i	$0.550822\pi$
$284$	5173.97	1.08105
$285$	751.557	0.156205
$286$	2907.18	0.601068
$287$	−1776.93	−0.365467
$288$	5625.29	1.15095
$289$	−4795.70	−0.976124
$290$	−5763.75	−1.16710
$291$	−7548.17	−1.52055
$292$	4541.64	0.910203
$293$	9263.21	1.84697	0.923486	−	0.383633i	$-0.125327\pi$
$293$	9263.21	1.84697	0.923486	+	0.383633i	$0.125327\pi$
$294$	−7439.74	−1.47583
$295$	416.310	0.0821643
$296$	−2833.39	−0.556377
$297$	747.170	0.145977
$298$	423.141	0.0822546
$299$	−3479.88	−0.673066
$300$	−2663.36	−0.512565
$301$	−112.862	−0.0216122
$302$	5246.89	0.999750
$303$	9888.53	1.87486
$304$	182.649	0.0344594
$305$	−305.286	−0.0573135
$306$	1785.72	0.333604
$307$	3912.92	0.727434	0.363717	−	0.931509i	$-0.381508\pi$
$307$	3912.92	0.727434	0.363717	+	0.931509i	$0.381508\pi$
$308$	−1752.87	−0.324283
$309$	7178.78	1.32164
$310$	−3331.30	−0.610339
$311$	−3759.56	−0.685482	−0.342741	−	0.939430i	$-0.611355\pi$
$311$	−3759.56	−0.685482	−0.342741	+	0.939430i	$0.611355\pi$
$312$	11429.3	2.07391
$313$	7783.10	1.40552	0.702759	−	0.711428i	$-0.251951\pi$
$313$	7783.10	1.40552	0.702759	+	0.711428i	$0.251951\pi$
$314$	9090.52	1.63378
$315$	−2105.49	−0.376607
$316$	10415.9	1.85425
$317$	858.765	0.152155	0.0760774	−	0.997102i	$-0.475760\pi$
$317$	858.765	0.152155	0.0760774	+	0.997102i	$0.475760\pi$
$318$	6030.77	1.06349
$319$	2736.83	0.480354
$320$	4046.51	0.706896
$321$	14720.1	2.55948
$322$	3344.64	0.578850
$323$	−205.783	−0.0354491
$324$	−5702.50	−0.977795
$325$	1426.07	0.243397
$326$	7804.45	1.32592
$327$	−15794.1	−2.67099
$328$	3803.20	0.640233
$329$	1167.77	0.195687
$330$	2015.96	0.336287
$331$	8852.17	1.46997	0.734984	−	0.678084i	$-0.237190\pi$
$331$	8852.17	1.46997	0.734984	+	0.678084i	$0.237190\pi$
$332$	−6212.44	−1.02696
$333$	3981.09	0.655143
$334$	−1433.24	−0.234801
$335$	−3572.60	−0.582662
$336$	−899.928	−0.146116
$337$	989.952	0.160018	0.0800091	−	0.996794i	$-0.474505\pi$
$337$	989.952	0.160018	0.0800091	+	0.996794i	$0.474505\pi$
$338$	−4896.66	−0.787998
$339$	−11868.3	−1.90147
$340$	729.251	0.116321
$341$	1581.82	0.251203
$342$	−3132.65	−0.495305
$343$	−6460.66	−1.01703
$344$	241.561	0.0378608
$345$	−2413.09	−0.376569
$346$	4751.61	0.738289
$347$	−5745.34	−0.888836	−0.444418	−	0.895820i	$-0.646589\pi$
$347$	−5745.34	−0.888836	−0.444418	+	0.895820i	$0.646589\pi$
$348$	26506.1	4.08298
$349$	6276.87	0.962731	0.481365	−	0.876520i	$-0.340141\pi$
$349$	6276.87	0.962731	0.481365	+	0.876520i	$0.340141\pi$
$350$	−1370.64	−0.209326
$351$	−3874.60	−0.589205
$352$	−1738.84	−0.263297
$353$	−5043.26	−0.760413	−0.380207	−	0.924902i	$-0.624147\pi$
$353$	−5043.26	−0.760413	−0.380207	+	0.924902i	$0.624147\pi$
$354$	−3051.86	−0.458205
$355$	−1921.06	−0.287210
$356$	18259.8	2.71845
$357$	1013.91	0.150313
$358$	4236.09	0.625376
$359$	3119.38	0.458592	0.229296	−	0.973357i	$-0.426358\pi$
$359$	3119.38	0.458592	0.229296	+	0.973357i	$0.426358\pi$
$360$	4506.42	0.659747
$361$	361.000	0.0526316
$362$	−11135.0	−1.61669
$363$	−957.247	−0.138409
$364$	9089.86	1.30890
$365$	−1686.28	−0.241819
$366$	2237.97	0.319619
$367$	−3037.33	−0.432009	−0.216004	−	0.976392i	$-0.569303\pi$
$367$	−3037.33	−0.432009	−0.216004	+	0.976392i	$0.569303\pi$
$368$	−586.447	−0.0830725
$369$	−5343.73	−0.753885
$370$	2591.63	0.364142
$371$	1946.97	0.272458
$372$	15319.8	2.13521
$373$	−2603.55	−0.361412	−0.180706	−	0.983537i	$-0.557838\pi$
$373$	−2603.55	−0.361412	−0.180706	+	0.983537i	$0.557838\pi$
$374$	−551.985	−0.0763168
$375$	988.891	0.136176
$376$	−2499.39	−0.342809
$377$	−14192.4	−1.93884
$378$	3724.02	0.506727
$379$	−1318.14	−0.178651	−0.0893253	−	0.996003i	$-0.528471\pi$
$379$	−1318.14	−0.178651	−0.0893253	+	0.996003i	$0.528471\pi$
$380$	−1279.31	−0.172703
$381$	9434.42	1.26861
$382$	175.755	0.0235404
$383$	3028.03	0.403982	0.201991	−	0.979387i	$-0.435259\pi$
$383$	3028.03	0.403982	0.201991	+	0.979387i	$0.435259\pi$
$384$	−19659.4	−2.61261
$385$	650.830	0.0861542
$386$	1406.25	0.185431
$387$	−339.409	−0.0445817
$388$	12848.6	1.68115
$389$	−8969.92	−1.16913	−0.584567	−	0.811346i	$-0.698736\pi$
$389$	−8969.92	−1.16913	−0.584567	+	0.811346i	$0.698736\pi$
$390$	−10454.1	−1.35735
$391$	660.724	0.0854584
$392$	5140.70	0.662359
$393$	−10482.8	−1.34552
$394$	−11067.0	−1.41510
$395$	−3867.37	−0.492629
$396$	−5271.37	−0.668930
$397$	5189.24	0.656021	0.328010	−	0.944674i	$-0.393622\pi$
$397$	5189.24	0.656021	0.328010	+	0.944674i	$0.393622\pi$
$398$	−5265.35	−0.663136
$399$	−1778.68	−0.223171
$400$	240.328	0.0300410
$401$	6348.02	0.790536	0.395268	−	0.918566i	$-0.370652\pi$
$401$	6348.02	0.790536	0.395268	+	0.918566i	$0.370652\pi$
$402$	26189.8	3.24933
$403$	−8202.83	−1.01393
$404$	−16832.4	−2.07288
$405$	2117.30	0.259777
$406$	13640.8	1.66744
$407$	−1230.60	−0.149873
$408$	−2170.08	−0.263321
$409$	−3647.50	−0.440972	−0.220486	−	0.975390i	$-0.570764\pi$
$409$	−3647.50	−0.440972	−0.220486	+	0.975390i	$0.570764\pi$
$410$	−3478.69	−0.419025
$411$	1827.68	0.219350
$412$	−12219.8	−1.46123
$413$	−985.261	−0.117389
$414$	10058.3	1.19405
$415$	2306.64	0.272840
$416$	9017.09	1.06274
$417$	1471.07	0.172755
$418$	968.336	0.113308
$419$	−2570.13	−0.299663	−0.149832	−	0.988712i	$-0.547873\pi$
$419$	−2570.13	−0.299663	−0.149832	+	0.988712i	$0.547873\pi$
$420$	6303.27	0.732305
$421$	1632.30	0.188963	0.0944814	−	0.995527i	$-0.469881\pi$
$421$	1632.30	0.188963	0.0944814	+	0.995527i	$0.469881\pi$
$422$	22001.8	2.53799
$423$	3511.80	0.403663
$424$	−4167.14	−0.477297
$425$	−270.767	−0.0309038
$426$	14082.8	1.60168
$427$	722.506	0.0818841
$428$	−25056.7	−2.82981
$429$	4963.99	0.558657
$430$	−220.950	−0.0247794
$431$	3367.63	0.376364	0.188182	−	0.982134i	$-0.439741\pi$
$431$	3367.63	0.376364	0.188182	+	0.982134i	$0.439741\pi$
$432$	−652.967	−0.0727220
$433$	10012.6	1.11126	0.555632	−	0.831429i	$-0.312476\pi$
$433$	10012.6	1.11126	0.555632	+	0.831429i	$0.312476\pi$
$434$	7884.04	0.871995
$435$	−9841.55	−1.08475
$436$	26884.9	2.95310
$437$	−1159.09	−0.126881
$438$	12361.7	1.34855
$439$	752.313	0.0817903	0.0408952	−	0.999163i	$-0.486979\pi$
$439$	752.313	0.0817903	0.0408952	+	0.999163i	$0.486979\pi$
$440$	−1392.98	−0.150927
$441$	−7223.01	−0.779938
$442$	2862.43	0.308036
$443$	9882.66	1.05991	0.529954	−	0.848026i	$-0.322209\pi$
$443$	9882.66	1.05991	0.529954	+	0.848026i	$0.322209\pi$
$444$	−11918.3	−1.27391
$445$	−6779.76	−0.722228
$446$	−29829.3	−3.16694
$447$	722.509	0.0764508
$448$	−9576.69	−1.00995
$449$	−11049.7	−1.16140	−0.580700	−	0.814117i	$-0.697221\pi$
$449$	−11049.7	−1.16140	−0.580700	+	0.814117i	$0.697221\pi$
$450$	−4121.91	−0.431797
$451$	1651.80	0.172462
$452$	20202.3	2.10230
$453$	8959.02	0.929209
$454$	−793.001	−0.0819766
$455$	−3375.01	−0.347743
$456$	3806.93	0.390956
$457$	−12436.7	−1.27300	−0.636501	−	0.771276i	$-0.719619\pi$
$457$	−12436.7	−1.27300	−0.636501	+	0.771276i	$0.719619\pi$
$458$	17116.3	1.74627
$459$	735.668	0.0748106
$460$	4107.59	0.416342
$461$	11700.6	1.18211	0.591055	−	0.806632i	$-0.298712\pi$
$461$	11700.6	1.18211	0.591055	+	0.806632i	$0.298712\pi$
$462$	−4771.07	−0.480455
$463$	2309.09	0.231777	0.115888	−	0.993262i	$-0.463028\pi$
$463$	2309.09	0.231777	0.115888	+	0.993262i	$0.463028\pi$
$464$	−2391.77	−0.239300
$465$	−5688.17	−0.567274
$466$	−25156.6	−2.50076
$467$	−17746.8	−1.75851	−0.879253	−	0.476355i	$-0.841958\pi$
$467$	−17746.8	−1.75851	−0.879253	+	0.476355i	$0.841958\pi$
$468$	27335.7	2.69999
$469$	8455.11	0.832454
$470$	2286.13	0.224365
$471$	15522.0	1.51850
$472$	2108.77	0.205644
$473$	104.915	0.0101987
$474$	28350.7	2.74724
$475$	475.000	0.0458831
$476$	−1725.89	−0.166189
$477$	5855.09	0.562025
$478$	14245.7	1.36314
$479$	−3865.91	−0.368764	−0.184382	−	0.982855i	$-0.559028\pi$
$479$	−3865.91	−0.368764	−0.184382	+	0.982855i	$0.559028\pi$
$480$	6252.81	0.594584
$481$	6381.51	0.604931
$482$	18663.0	1.76364
$483$	5710.95	0.538007
$484$	1629.44	0.153028
$485$	−4770.60	−0.446643
$486$	−24018.5	−2.24177
$487$	−2030.06	−0.188893	−0.0944463	−	0.995530i	$-0.530108\pi$
$487$	−2030.06	−0.188893	−0.0944463	+	0.995530i	$0.530108\pi$
$488$	−1546.39	−0.143446
$489$	13326.0	1.23236
$490$	−4702.07	−0.433506
$491$	−620.486	−0.0570308	−0.0285154	−	0.999593i	$-0.509078\pi$
$491$	−620.486	−0.0570308	−0.0285154	+	0.999593i	$0.509078\pi$
$492$	15997.7	1.46592
$493$	2694.70	0.246173
$494$	−5021.50	−0.457344
$495$	1957.23	0.177719
$496$	−1382.38	−0.125143
$497$	4546.50	0.410338
$498$	−16909.4	−1.52154
$499$	−7167.38	−0.642998	−0.321499	−	0.946910i	$-0.604187\pi$
$499$	−7167.38	−0.642998	−0.321499	+	0.946910i	$0.604187\pi$
$500$	−1683.30	−0.150559
$501$	−2447.25	−0.218234
$502$	−3461.57	−0.307764
$503$	10420.1	0.923680	0.461840	−	0.886963i	$-0.347189\pi$
$503$	10420.1	0.923680	0.461840	+	0.886963i	$0.347189\pi$
$504$	−10665.1	−0.942586
$505$	6249.76	0.550714
$506$	−3109.12	−0.273157
$507$	−8361.01	−0.732397
$508$	−16059.4	−1.40260
$509$	−20090.4	−1.74949	−0.874747	−	0.484579i	$-0.838973\pi$
$509$	−20090.4	−1.74949	−0.874747	+	0.484579i	$0.838973\pi$
$510$	1984.92	0.172341
$511$	3990.85	0.345489
$512$	3467.37	0.299292
$513$	−1290.57	−0.111072
$514$	22617.5	1.94089
$515$	4537.14	0.388214
$516$	1016.10	0.0866882
$517$	−1085.54	−0.0923439
$518$	−6133.50	−0.520252
$519$	8113.33	0.686196
$520$	7223.58	0.609183
$521$	7043.97	0.592326	0.296163	−	0.955137i	$-0.404293\pi$
$521$	7043.97	0.592326	0.296163	+	0.955137i	$0.404293\pi$
$522$	41021.7	3.43960
$523$	−9195.27	−0.768797	−0.384399	−	0.923167i	$-0.625591\pi$
$523$	−9195.27	−0.768797	−0.384399	+	0.923167i	$0.625591\pi$
$524$	17844.0	1.48763
$525$	−2340.36	−0.194556
$526$	29934.0	2.48134
$527$	1557.47	0.128737
$528$	836.557	0.0689516
$529$	−8445.40	−0.694123
$530$	3811.58	0.312385
$531$	−2962.95	−0.242149
$532$	3027.69	0.246742
$533$	−8565.76	−0.696105
$534$	49700.7	4.02764
$535$	9303.39	0.751814
$536$	−18096.6	−1.45831
$537$	7233.10	0.581250
$538$	28455.8	2.28033
$539$	2232.71	0.178422
$540$	4573.51	0.364467
$541$	−18409.1	−1.46297	−0.731486	−	0.681856i	$-0.761173\pi$
$541$	−18409.1	−1.46297	−0.731486	+	0.681856i	$0.761173\pi$
$542$	−14530.3	−1.15153
$543$	−19012.9	−1.50262
$544$	−1712.07	−0.134935
$545$	−9982.19	−0.784569
$546$	24741.3	1.93925
$547$	−11698.1	−0.914395	−0.457198	−	0.889365i	$-0.651147\pi$
$547$	−11698.1	−0.914395	−0.457198	+	0.889365i	$0.651147\pi$
$548$	−3111.10	−0.242518
$549$	2172.78	0.168910
$550$	1274.13	0.0987799
$551$	−4727.25	−0.365495
$552$	−12223.2	−0.942492
$553$	9152.74	0.703823
$554$	−17540.3	−1.34516
$555$	4425.19	0.338448
$556$	−2504.08	−0.191001
$557$	9764.99	0.742829	0.371415	−	0.928467i	$-0.378873\pi$
$557$	9764.99	0.742829	0.371415	+	0.928467i	$0.378873\pi$
$558$	23709.5	1.79875
$559$	−544.057	−0.0411648
$560$	−568.774	−0.0429198
$561$	−942.510	−0.0709319
$562$	−487.909	−0.0366214
$563$	−13882.4	−1.03921	−0.519604	−	0.854407i	$-0.673920\pi$
$563$	−13882.4	−1.03921	−0.519604	+	0.854407i	$0.673920\pi$
$564$	−10513.4	−0.784916
$565$	−7501.01	−0.558531
$566$	7013.59	0.520854
$567$	−5010.93	−0.371145
$568$	−9730.93	−0.718840
$569$	−3669.31	−0.270343	−0.135172	−	0.990822i	$-0.543159\pi$
$569$	−3669.31	−0.270343	−0.135172	+	0.990822i	$0.543159\pi$
$570$	−3482.10	−0.255876
$571$	−23285.9	−1.70663	−0.853313	−	0.521398i	$-0.825411\pi$
$571$	−23285.9	−1.70663	−0.853313	+	0.521398i	$0.825411\pi$
$572$	−8449.77	−0.617662
$573$	300.101	0.0218794
$574$	8232.86	0.598664
$575$	−1525.12	−0.110612
$576$	−28799.8	−2.08332
$577$	−10069.7	−0.726531	−0.363265	−	0.931686i	$-0.618338\pi$
$577$	−10069.7	−0.726531	−0.363265	+	0.931686i	$0.618338\pi$
$578$	22219.4	1.59897
$579$	2401.16	0.172347
$580$	16752.4	1.19932
$581$	−5459.03	−0.389808
$582$	34972.1	2.49079
$583$	−1809.87	−0.128571
$584$	−8541.67	−0.605235
$585$	−10149.6	−0.717323
$586$	−42918.2	−3.02548
$587$	−13544.2	−0.952349	−0.476174	−	0.879351i	$-0.657977\pi$
$587$	−13544.2	−0.952349	−0.476174	+	0.879351i	$0.657977\pi$
$588$	21623.7	1.51658
$589$	−2732.23	−0.191137
$590$	−1928.84	−0.134592
$591$	−18896.9	−1.31525
$592$	1075.44	0.0746630
$593$	9765.47	0.676256	0.338128	−	0.941100i	$-0.390206\pi$
$593$	9765.47	0.676256	0.338128	+	0.941100i	$0.390206\pi$
$594$	−3461.78	−0.239122
$595$	640.811	0.0441524
$596$	−1229.86	−0.0845255
$597$	−8990.54	−0.616346
$598$	16123.0	1.10254
$599$	−8430.02	−0.575027	−0.287514	−	0.957777i	$-0.592829\pi$
$599$	−8430.02	−0.575027	−0.287514	+	0.957777i	$0.592829\pi$
$600$	5009.12	0.340827
$601$	9141.88	0.620474	0.310237	−	0.950659i	$-0.399592\pi$
$601$	9141.88	0.620474	0.310237	+	0.950659i	$0.399592\pi$
$602$	522.913	0.0354025
$603$	25426.9	1.71718
$604$	−15250.2	−1.02735
$605$	−605.000	−0.0406558
$606$	−45815.4	−3.07116
$607$	−4043.36	−0.270370	−0.135185	−	0.990820i	$-0.543163\pi$
$607$	−4043.36	−0.270370	−0.135185	+	0.990820i	$0.543163\pi$
$608$	3003.45	0.200339
$609$	23291.6	1.54979
$610$	1414.45	0.0938840
$611$	5629.26	0.372726
$612$	−5190.22	−0.342814
$613$	−11177.5	−0.736467	−0.368234	−	0.929733i	$-0.620037\pi$
$613$	−11177.5	−0.736467	−0.368234	+	0.929733i	$0.620037\pi$
$614$	−18129.3	−1.19159
$615$	−5939.84	−0.389459
$616$	3296.71	0.215630
$617$	−12089.6	−0.788830	−0.394415	−	0.918932i	$-0.629053\pi$
$617$	−12089.6	−0.788830	−0.394415	+	0.918932i	$0.629053\pi$
$618$	−33260.6	−2.16495
$619$	−20091.8	−1.30461	−0.652307	−	0.757955i	$-0.726199\pi$
$619$	−20091.8	−1.30461	−0.652307	+	0.757955i	$0.726199\pi$
$620$	9682.47	0.627189
$621$	4143.73	0.267765
$622$	17418.7	1.12287
$623$	16045.4	1.03185
$624$	−4338.13	−0.278308
$625$	625.000	0.0400000
$626$	−36060.6	−2.30235
$627$	1653.43	0.105313
$628$	−26421.7	−1.67889
$629$	−1211.65	−0.0768073
$630$	9755.14	0.616911
$631$	18403.9	1.16109	0.580545	−	0.814228i	$-0.302840\pi$
$631$	18403.9	1.16109	0.580545	+	0.814228i	$0.302840\pi$
$632$	−19589.7	−1.23297
$633$	37568.0	2.35891
$634$	−3978.82	−0.249241
$635$	5962.75	0.372637
$636$	−17528.5	−1.09285
$637$	−11578.2	−0.720162
$638$	−12680.2	−0.786859
$639$	13672.6	0.846445
$640$	−12425.2	−0.767419
$641$	−3227.79	−0.198892	−0.0994462	−	0.995043i	$-0.531707\pi$
$641$	−3227.79	−0.198892	−0.0994462	+	0.995043i	$0.531707\pi$
$642$	−68200.8	−4.19263
$643$	−24397.0	−1.49631	−0.748153	−	0.663526i	$-0.769059\pi$
$643$	−24397.0	−1.49631	−0.748153	+	0.663526i	$0.769059\pi$
$644$	−9721.25	−0.594831
$645$	−377.270	−0.0230310
$646$	953.429	0.0580684
$647$	−23252.9	−1.41293	−0.706464	−	0.707749i	$-0.749711\pi$
$647$	−23252.9	−1.41293	−0.706464	+	0.707749i	$0.749711\pi$
$648$	10725.0	0.650180
$649$	915.881	0.0553952
$650$	−6607.23	−0.398703
$651$	13461.9	0.810468
$652$	−22683.7	−1.36252
$653$	18482.4	1.10762	0.553809	−	0.832644i	$-0.313174\pi$
$653$	18482.4	1.10762	0.553809	+	0.832644i	$0.313174\pi$
$654$	73176.9	4.37529
$655$	−6625.36	−0.395228
$656$	−1443.54	−0.0859161
$657$	12001.6	0.712674
$658$	−5410.48	−0.320551
$659$	23561.5	1.39276	0.696379	−	0.717674i	$-0.254793\pi$
$659$	23561.5	1.39276	0.696379	+	0.717674i	$0.254793\pi$
$660$	−5859.40	−0.345571
$661$	−18590.5	−1.09393	−0.546963	−	0.837157i	$-0.684216\pi$
$661$	−18590.5	−1.09393	−0.546963	+	0.837157i	$0.684216\pi$
$662$	−41013.8	−2.40792
$663$	4887.57	0.286301
$664$	11684.0	0.682875
$665$	−1124.16	−0.0655536
$666$	−18445.1	−1.07318
$667$	15178.2	0.881112
$668$	4165.74	0.241283
$669$	−50933.2	−2.94349
$670$	16552.5	0.954447
$671$	−671.628	−0.0386407
$672$	−14798.2	−0.849486
$673$	4929.12	0.282323	0.141162	−	0.989987i	$-0.454916\pi$
$673$	4929.12	0.282323	0.141162	+	0.989987i	$0.454916\pi$
$674$	−4586.63	−0.262122
$675$	−1698.11	−0.0968303
$676$	14232.2	0.809753
$677$	−11379.7	−0.646023	−0.323011	−	0.946395i	$-0.604695\pi$
$677$	−11379.7	−0.646023	−0.323011	+	0.946395i	$0.604695\pi$
$678$	54988.0	3.11475
$679$	11290.4	0.638122
$680$	−1371.54	−0.0773472
$681$	−1354.04	−0.0761924
$682$	−7328.86	−0.411490
$683$	−31910.4	−1.78773	−0.893864	−	0.448338i	$-0.852016\pi$
$683$	−31910.4	−1.78773	−0.893864	+	0.448338i	$0.852016\pi$
$684$	9105.09	0.508979
$685$	1155.13	0.0644312
$686$	29933.4	1.66598
$687$	29225.9	1.62306
$688$	−91.6872	−0.00508073
$689$	9385.44	0.518950
$690$	11180.3	0.616850
$691$	1114.73	0.0613695	0.0306847	−	0.999529i	$-0.490231\pi$
$691$	1114.73	0.0613695	0.0306847	+	0.999529i	$0.490231\pi$
$692$	−13810.6	−0.758671
$693$	−4632.08	−0.253908
$694$	26619.2	1.45598
$695$	929.750	0.0507445
$696$	−49851.3	−2.71495
$697$	1626.38	0.0883836
$698$	−29081.9	−1.57703
$699$	−42954.6	−2.32431
$700$	3983.80	0.215105
$701$	−33860.4	−1.82438	−0.912189	−	0.409770i	$-0.865609\pi$
$701$	−33860.4	−1.82438	−0.912189	+	0.409770i	$0.865609\pi$
$702$	17951.7	0.965164
$703$	2125.58	0.114037
$704$	8902.32	0.476589
$705$	3903.55	0.208534
$706$	23366.4	1.24562
$707$	−14791.0	−0.786809
$708$	8870.27	0.470855
$709$	12113.0	0.641624	0.320812	−	0.947143i	$-0.396044\pi$
$709$	12113.0	0.641624	0.320812	+	0.947143i	$0.396044\pi$
$710$	8900.64	0.470472
$711$	27524.8	1.45184
$712$	−34342.1	−1.80762
$713$	8772.61	0.460781
$714$	−4697.62	−0.246224
$715$	3137.35	0.164098
$716$	−12312.3	−0.642641
$717$	24324.3	1.26696
$718$	−14452.7	−0.751209
$719$	−23108.4	−1.19861	−0.599303	−	0.800522i	$-0.704556\pi$
$719$	−23108.4	−1.19861	−0.599303	+	0.800522i	$0.704556\pi$
$720$	−1710.46	−0.0885348
$721$	−10737.8	−0.554644
$722$	−1672.58	−0.0862147
$723$	31866.9	1.63920
$724$	32364.0	1.66133
$725$	−6220.07	−0.318631
$726$	4435.10	0.226725
$727$	−16656.4	−0.849729	−0.424865	−	0.905257i	$-0.639678\pi$
$727$	−16656.4	−0.849729	−0.424865	+	0.905257i	$0.639678\pi$
$728$	−17095.7	−0.870344
$729$	−29578.0	−1.50272
$730$	7812.86	0.396119
$731$	103.300	0.00522665
$732$	−6504.70	−0.328443
$733$	−1647.60	−0.0830225	−0.0415112	−	0.999138i	$-0.513217\pi$
$733$	−1647.60	−0.0830225	−0.0415112	+	0.999138i	$0.513217\pi$
$734$	14072.5	0.707664
$735$	−8028.75	−0.402918
$736$	−9643.43	−0.482964
$737$	−7859.72	−0.392831
$738$	24758.5	1.23492
$739$	34217.6	1.70327	0.851634	−	0.524137i	$-0.175612\pi$
$739$	34217.6	1.70327	0.851634	+	0.524137i	$0.175612\pi$
$740$	−7532.62	−0.374195
$741$	−8574.17	−0.425074
$742$	−9020.69	−0.446307
$743$	−29259.4	−1.44471	−0.722357	−	0.691520i	$-0.756941\pi$
$743$	−29259.4	−1.44471	−0.722357	+	0.691520i	$0.756941\pi$
$744$	−28812.8	−1.41980
$745$	456.641	0.0224564
$746$	12062.7	0.592021
$747$	−16416.8	−0.804096
$748$	1604.35	0.0784237
$749$	−22017.9	−1.07412
$750$	−4581.72	−0.223068
$751$	23142.5	1.12448	0.562238	−	0.826976i	$-0.309941\pi$
$751$	23142.5	1.12448	0.562238	+	0.826976i	$0.309941\pi$
$752$	948.671	0.0460033
$753$	−5910.61	−0.286048
$754$	65755.9	3.17598
$755$	5662.29	0.272943
$756$	−10823.9	−0.520717
$757$	−39736.9	−1.90788	−0.953938	−	0.300003i	$-0.903012\pi$
$757$	−39736.9	−1.90788	−0.953938	+	0.300003i	$0.903012\pi$
$758$	6107.21	0.292644
$759$	−5308.79	−0.253883
$760$	2406.06	0.114838
$761$	12321.9	0.586950	0.293475	−	0.955967i	$-0.405188\pi$
$761$	12321.9	0.586950	0.293475	+	0.955967i	$0.405188\pi$
$762$	−43711.4	−2.07808
$763$	23624.4	1.12092
$764$	−510.836	−0.0241903
$765$	1927.10	0.0910776
$766$	−14029.4	−0.661754
$767$	−4749.48	−0.223590
$768$	39865.9	1.87310
$769$	32082.2	1.50444	0.752221	−	0.658911i	$-0.228983\pi$
$769$	32082.2	1.50444	0.752221	+	0.658911i	$0.228983\pi$
$770$	−3015.42	−0.141127
$771$	38619.3	1.80394
$772$	−4087.29	−0.190550
$773$	31395.6	1.46083	0.730415	−	0.683004i	$-0.239327\pi$
$773$	31395.6	1.46083	0.730415	+	0.683004i	$0.239327\pi$
$774$	1572.54	0.0730283
$775$	−3595.04	−0.166629
$776$	−24164.9	−1.11788
$777$	−10472.9	−0.483544
$778$	41559.3	1.91513
$779$	−2853.12	−0.131224
$780$	30385.1	1.39482
$781$	−4226.34	−0.193637
$782$	−3061.26	−0.139988
$783$	16899.8	0.771329
$784$	−1951.21	−0.0888852
$785$	9810.22	0.446041
$786$	48568.9	2.20406
$787$	−746.125	−0.0337948	−0.0168974	−	0.999857i	$-0.505379\pi$
$787$	−746.125	−0.0337948	−0.0168974	+	0.999857i	$0.505379\pi$
$788$	32166.5	1.45417
$789$	51112.1	2.30626
$790$	17918.3	0.806966
$791$	17752.3	0.797976
$792$	9914.12	0.444802
$793$	3482.86	0.155965
$794$	−24042.7	−1.07461
$795$	6508.23	0.290344
$796$	15303.8	0.681444
$797$	21361.0	0.949369	0.474684	−	0.880156i	$-0.342562\pi$
$797$	21361.0	0.949369	0.474684	+	0.880156i	$0.342562\pi$
$798$	8240.94	0.365572
$799$	−1068.82	−0.0473245
$800$	3951.91	0.174651
$801$	48252.8	2.12850
$802$	−29411.6	−1.29496
$803$	−3709.82	−0.163034
$804$	−76121.1	−3.33903
$805$	3609.44	0.158032
$806$	38005.2	1.66089
$807$	48588.0	2.11943
$808$	31657.5	1.37835
$809$	17340.8	0.753607	0.376804	−	0.926293i	$-0.377023\pi$
$809$	17340.8	0.753607	0.376804	+	0.926293i	$0.377023\pi$
$810$	−9809.86	−0.425535
$811$	18858.1	0.816522	0.408261	−	0.912865i	$-0.366135\pi$
$811$	18858.1	0.816522	0.408261	+	0.912865i	$0.366135\pi$
$812$	−39647.2	−1.71348
$813$	−24810.4	−1.07028
$814$	5701.59	0.245505
$815$	8422.33	0.361990
$816$	823.678	0.0353364
$817$	−181.217	−0.00776005
$818$	16899.6	0.722347
$819$	24020.6	1.02484
$820$	10110.9	0.430593
$821$	−5997.79	−0.254963	−0.127481	−	0.991841i	$-0.540689\pi$
$821$	−5997.79	−0.254963	−0.127481	+	0.991841i	$0.540689\pi$
$822$	−8468.00	−0.359313
$823$	22605.4	0.957443	0.478722	−	0.877967i	$-0.341100\pi$
$823$	22605.4	0.957443	0.478722	+	0.877967i	$0.341100\pi$
$824$	22982.4	0.971637
$825$	2175.56	0.0918100
$826$	4564.90	0.192292
$827$	−36143.9	−1.51977	−0.759883	−	0.650060i	$-0.774744\pi$
$827$	−36143.9	−1.51977	−0.759883	+	0.650060i	$0.774744\pi$
$828$	−29234.5	−1.22702
$829$	−10613.1	−0.444642	−0.222321	−	0.974974i	$-0.571363\pi$
$829$	−10613.1	−0.444642	−0.222321	+	0.974974i	$0.571363\pi$
$830$	−10687.1	−0.446933
$831$	−29950.0	−1.25025
$832$	−46164.7	−1.92365
$833$	2198.34	0.0914380
$834$	−6815.76	−0.282986
$835$	−1546.71	−0.0641032
$836$	−2814.48	−0.116436
$837$	9767.67	0.403369
$838$	11907.9	0.490872
$839$	−20614.4	−0.848257	−0.424129	−	0.905602i	$-0.639420\pi$
$839$	−20614.4	−0.848257	−0.424129	+	0.905602i	$0.639420\pi$
$840$	−11854.9	−0.486942
$841$	37513.8	1.53814
$842$	−7562.74	−0.309536
$843$	−833.101	−0.0340374
$844$	−63948.6	−2.60806
$845$	−5284.33	−0.215132
$846$	−16270.8	−0.661232
$847$	1431.83	0.0580852
$848$	1581.68	0.0640509
$849$	11975.6	0.484103
$850$	1254.51	0.0506228
$851$	−6824.78	−0.274912
$852$	−40931.9	−1.64590
$853$	−27055.9	−1.08602	−0.543012	−	0.839725i	$-0.682716\pi$
$853$	−27055.9	−1.08602	−0.543012	+	0.839725i	$0.682716\pi$
$854$	−3347.51	−0.134133
$855$	−3380.67	−0.135224
$856$	47125.3	1.88167
$857$	29093.6	1.15965	0.579825	−	0.814741i	$-0.303121\pi$
$857$	29093.6	1.15965	0.579825	+	0.814741i	$0.303121\pi$
$858$	−22999.1	−0.915124
$859$	−46726.5	−1.85598	−0.927990	−	0.372606i	$-0.878464\pi$
$859$	−46726.5	−1.85598	−0.927990	+	0.372606i	$0.878464\pi$
$860$	642.194	0.0254635
$861$	14057.5	0.556423
$862$	−15602.8	−0.616514
$863$	−14709.1	−0.580192	−0.290096	−	0.956998i	$-0.593687\pi$
$863$	−14709.1	−0.580192	−0.290096	+	0.956998i	$0.593687\pi$
$864$	−10737.3	−0.422788
$865$	5127.79	0.201561
$866$	−46390.5	−1.82034
$867$	37939.4	1.48615
$868$	−22915.1	−0.896069
$869$	−8508.22	−0.332131
$870$	45597.8	1.77691
$871$	40758.1	1.58558
$872$	−50563.7	−1.96365
$873$	33953.3	1.31632
$874$	5370.30	0.207841
$875$	−1479.16	−0.0571483
$876$	−35929.5	−1.38578
$877$	−16886.4	−0.650186	−0.325093	−	0.945682i	$-0.605396\pi$
$877$	−16886.4	−0.650186	−0.325093	+	0.945682i	$0.605396\pi$
$878$	−3485.61	−0.133979
$879$	−73282.5	−2.81201
$880$	528.722	0.0202536
$881$	−6325.39	−0.241893	−0.120947	−	0.992659i	$-0.538593\pi$
$881$	−6325.39	−0.241893	−0.120947	+	0.992659i	$0.538593\pi$
$882$	33465.5	1.27760
$883$	18598.3	0.708815	0.354407	−	0.935091i	$-0.384683\pi$
$883$	18598.3	0.708815	0.354407	+	0.935091i	$0.384683\pi$
$884$	−8319.69	−0.316540
$885$	−3293.48	−0.125095
$886$	−45788.2	−1.73621
$887$	−20369.2	−0.771062	−0.385531	−	0.922695i	$-0.625982\pi$
$887$	−20369.2	−0.771062	−0.385531	+	0.922695i	$0.625982\pi$
$888$	22415.3	0.847082
$889$	−14111.8	−0.532389
$890$	31411.9	1.18307
$891$	4658.07	0.175142
$892$	86699.2	3.25438
$893$	1875.02	0.0702632
$894$	−3347.52	−0.125232
$895$	4571.47	0.170735
$896$	29406.1	1.09642
$897$	27529.8	1.02474
$898$	51195.5	1.90247
$899$	35778.2	1.32733
$900$	11980.4	0.443718
$901$	−1782.01	−0.0658905
$902$	−7653.12	−0.282507
$903$	892.869	0.0329046
$904$	−37995.5	−1.39791
$905$	−12016.6	−0.441375
$906$	−41508.8	−1.52212
$907$	−30182.2	−1.10494	−0.552472	−	0.833531i	$-0.686315\pi$
$907$	−30182.2	−1.10494	−0.552472	+	0.833531i	$0.686315\pi$
$908$	2304.87	0.0842398
$909$	−44480.7	−1.62303
$910$	15637.0	0.569630
$911$	−15427.5	−0.561072	−0.280536	−	0.959844i	$-0.590512\pi$
$911$	−15427.5	−0.561072	−0.280536	+	0.959844i	$0.590512\pi$
$912$	−1444.96	−0.0524643
$913$	5074.61	0.183949
$914$	57621.3	2.08528
$915$	2415.15	0.0872596
$916$	−49748.8	−1.79448
$917$	15680.0	0.564665
$918$	−3408.49	−0.122546
$919$	−30990.3	−1.11238	−0.556189	−	0.831056i	$-0.687737\pi$
$919$	−30990.3	−1.11238	−0.556189	+	0.831056i	$0.687737\pi$
$920$	−7725.34	−0.276845
$921$	−30955.6	−1.10752
$922$	−54211.2	−1.93639
$923$	21916.5	0.781572
$924$	13867.2	0.493720
$925$	2796.81	0.0994148
$926$	−10698.5	−0.379669
$927$	−32291.7	−1.14412
$928$	−39329.8	−1.39123
$929$	17984.7	0.635156	0.317578	−	0.948232i	$-0.397130\pi$
$929$	17984.7	0.635156	0.317578	+	0.948232i	$0.397130\pi$
$930$	26354.3	0.929239
$931$	−3856.50	−0.135759
$932$	73117.9	2.56980
$933$	29742.3	1.04364
$934$	82224.1	2.88057
$935$	−595.686	−0.0208353
$936$	−51411.6	−1.79534
$937$	−50526.0	−1.76159	−0.880796	−	0.473497i	$-0.842992\pi$
$937$	−50526.0	−1.76159	−0.880796	+	0.473497i	$0.842992\pi$
$938$	−39174.1	−1.36362
$939$	−61573.1	−2.13990
$940$	−6644.67	−0.230559
$941$	5352.29	0.185419	0.0927097	−	0.995693i	$-0.470447\pi$
$941$	5352.29	0.185419	0.0927097	+	0.995693i	$0.470447\pi$
$942$	−71916.3	−2.48743
$943$	9160.74	0.316347
$944$	−800.407	−0.0275964
$945$	4018.85	0.138342
$946$	−486.090	−0.0167063
$947$	−54077.8	−1.85564	−0.927820	−	0.373028i	$-0.878319\pi$
$947$	−54077.8	−1.85564	−0.927820	+	0.373028i	$0.878319\pi$
$948$	−82401.8	−2.82309
$949$	19238.0	0.658053
$950$	−2200.76	−0.0751602
$951$	−6793.80	−0.231655
$952$	3245.96	0.110506
$953$	4436.21	0.150790	0.0753950	−	0.997154i	$-0.475978\pi$
$953$	4436.21	0.150790	0.0753950	+	0.997154i	$0.475978\pi$
$954$	−27127.7	−0.920642
$955$	189.670	0.00642679
$956$	−41405.2	−1.40077
$957$	−21651.4	−0.731338
$958$	17911.5	0.604064
$959$	−2733.80	−0.0920533
$960$	−32012.4	−1.07625
$961$	−9112.09	−0.305867
$962$	−29566.7	−0.990925
$963$	−66214.0	−2.21570
$964$	−54244.2	−1.81233
$965$	1517.58	0.0506246
$966$	−26459.9	−0.881297
$967$	11850.2	0.394080	0.197040	−	0.980395i	$-0.436867\pi$
$967$	11850.2	0.394080	0.197040	+	0.980395i	$0.436867\pi$
$968$	−3064.56	−0.101755
$969$	1627.97	0.0539711
$970$	22103.1	0.731636
$971$	−22890.8	−0.756542	−0.378271	−	0.925695i	$-0.623481\pi$
$971$	−22890.8	−0.756542	−0.378271	+	0.925695i	$0.623481\pi$
$972$	69810.1	2.30366
$973$	−2200.40	−0.0724990
$974$	9405.64	0.309421
$975$	−11281.8	−0.370571
$976$	586.949	0.0192498
$977$	11774.2	0.385559	0.192779	−	0.981242i	$-0.438250\pi$
$977$	11774.2	0.385559	0.192779	+	0.981242i	$0.438250\pi$
$978$	−61742.0	−2.01870
$979$	−14915.5	−0.486926
$980$	13666.6	0.445474
$981$	71045.1	2.31223
$982$	2874.83	0.0934210
$983$	−12799.5	−0.415300	−0.207650	−	0.978203i	$-0.566582\pi$
$983$	−12799.5	−0.415300	−0.207650	+	0.978203i	$0.566582\pi$
$984$	−30087.6	−0.974753
$985$	−11943.2	−0.386337
$986$	−12485.0	−0.403250
$987$	−9238.36	−0.297933
$988$	14595.1	0.469970
$989$	581.847	0.0187074
$990$	−9068.20	−0.291118
$991$	−8107.55	−0.259884	−0.129942	−	0.991522i	$-0.541479\pi$
$991$	−8107.55	−0.259884	−0.129942	+	0.991522i	$0.541479\pi$
$992$	−22731.6	−0.727550
$993$	−70030.7	−2.23802
$994$	−21064.8	−0.672167
$995$	−5682.21	−0.181043
$996$	49147.4	1.56355
$997$	231.258	0.00734604	0.00367302	−	0.999993i	$-0.498831\pi$
$997$	231.258	0.00734604	0.00367302	+	0.999993i	$0.498831\pi$
$998$	33207.8	1.05328
$999$	−7598.90	−0.240659

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.2	✓	20	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)

\(f(q)\)	\(=\)	\(q-4.63319 q^{2} -7.91113 q^{3} +13.4664 q^{4} -5.00000 q^{5} +36.6537 q^{6} +11.8333 q^{7} -25.3269 q^{8} +35.5860 q^{9} +O(q^{10})\)	\(q-4.63319 q^{2} -7.91113 q^{3} +13.4664 q^{4} -5.00000 q^{5} +36.6537 q^{6} +11.8333 q^{7} -25.3269 q^{8} +35.5860 q^{9} +23.1659 q^{10} -11.0000 q^{11} -106.535 q^{12} +57.0427 q^{13} -54.8258 q^{14} +39.5556 q^{15} +9.61312 q^{16} -10.8307 q^{17} -164.876 q^{18} +19.0000 q^{19} -67.3321 q^{20} -93.6146 q^{21} +50.9651 q^{22} -61.0049 q^{23} +200.365 q^{24} +25.0000 q^{25} -264.289 q^{26} -67.9246 q^{27} +159.352 q^{28} -248.803 q^{29} -183.269 q^{30} -143.802 q^{31} +158.076 q^{32} +87.0224 q^{33} +50.1805 q^{34} -59.1664 q^{35} +479.215 q^{36} +111.873 q^{37} -88.0306 q^{38} -451.272 q^{39} +126.635 q^{40} -150.164 q^{41} +433.734 q^{42} -9.53771 q^{43} -148.131 q^{44} -177.930 q^{45} +282.647 q^{46} +98.6850 q^{47} -76.0506 q^{48} -202.974 q^{49} -115.830 q^{50} +85.6828 q^{51} +768.161 q^{52} +164.534 q^{53} +314.707 q^{54} +55.0000 q^{55} -299.701 q^{56} -150.311 q^{57} +1152.75 q^{58} -83.2619 q^{59} +532.673 q^{60} +61.0571 q^{61} +666.260 q^{62} +421.099 q^{63} -809.302 q^{64} -285.213 q^{65} -403.191 q^{66} +714.520 q^{67} -145.850 q^{68} +482.618 q^{69} +274.129 q^{70} +384.213 q^{71} -901.284 q^{72} +337.256 q^{73} -518.327 q^{74} -197.778 q^{75} +255.862 q^{76} -130.166 q^{77} +2090.83 q^{78} +773.474 q^{79} -48.0656 q^{80} -423.461 q^{81} +695.738 q^{82} -461.328 q^{83} -1260.65 q^{84} +54.1533 q^{85} +44.1900 q^{86} +1968.31 q^{87} +278.596 q^{88} +1355.95 q^{89} +824.382 q^{90} +675.002 q^{91} -821.518 q^{92} +1137.63 q^{93} -457.226 q^{94} -95.0000 q^{95} -1250.56 q^{96} +954.120 q^{97} +940.414 q^{98} -391.445 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

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