LMFDB - Embedded newform 47.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [47,4,Mod(1,47)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(47, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("47.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 47 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	47.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:	$2.77308977027$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.68740$ of defining polynomial
Character	$\chi$	$=$	47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.90952 q^{2} -0.777884 q^{3} +16.1033 q^{4} +9.19383 q^{5} +3.81903 q^{6} -30.3255 q^{7} -39.7835 q^{8} -26.3949 q^{9} +O(q^{10})$	\(q-4.90952 q^{2} -0.777884 q^{3} +16.1033 q^{4} +9.19383 q^{5} +3.81903 q^{6} -30.3255 q^{7} -39.7835 q^{8} -26.3949 q^{9} -45.1373 q^{10} +22.4298 q^{11} -12.5265 q^{12} -62.0257 q^{13} +148.883 q^{14} -7.15173 q^{15} +66.4910 q^{16} -72.1639 q^{17} +129.586 q^{18} -25.0550 q^{19} +148.051 q^{20} +23.5897 q^{21} -110.119 q^{22} +103.176 q^{23} +30.9469 q^{24} -40.4735 q^{25} +304.516 q^{26} +41.5350 q^{27} -488.341 q^{28} -234.381 q^{29} +35.1115 q^{30} +198.714 q^{31} -8.17058 q^{32} -17.4477 q^{33} +354.290 q^{34} -278.807 q^{35} -425.046 q^{36} -203.083 q^{37} +123.008 q^{38} +48.2488 q^{39} -365.763 q^{40} +210.889 q^{41} -115.814 q^{42} +111.430 q^{43} +361.194 q^{44} -242.670 q^{45} -506.542 q^{46} +47.0000 q^{47} -51.7223 q^{48} +576.634 q^{49} +198.705 q^{50} +56.1351 q^{51} -998.822 q^{52} +499.576 q^{53} -203.917 q^{54} +206.215 q^{55} +1206.45 q^{56} +19.4898 q^{57} +1150.70 q^{58} -562.752 q^{59} -115.167 q^{60} +548.091 q^{61} -975.591 q^{62} +800.437 q^{63} -491.814 q^{64} -570.254 q^{65} +85.6600 q^{66} -760.831 q^{67} -1162.08 q^{68} -80.2586 q^{69} +1368.81 q^{70} -668.059 q^{71} +1050.08 q^{72} -1145.92 q^{73} +997.040 q^{74} +31.4836 q^{75} -403.469 q^{76} -680.193 q^{77} -236.878 q^{78} +975.010 q^{79} +611.307 q^{80} +680.353 q^{81} -1035.37 q^{82} +698.827 q^{83} +379.873 q^{84} -663.463 q^{85} -547.067 q^{86} +182.321 q^{87} -892.335 q^{88} +451.477 q^{89} +1191.39 q^{90} +1880.96 q^{91} +1661.47 q^{92} -154.577 q^{93} -230.747 q^{94} -230.351 q^{95} +6.35576 q^{96} -390.906 q^{97} -2830.99 q^{98} -592.031 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9}+O(q^{10}) $ 3 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 - 6 * q^5 - 8 * q^6 - 45 * q^7 - 39 * q^8 - 42 * q^9	\( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9} - 40 q^{10} + 2 q^{11} + 12 q^{12} - 80 q^{13} + 162 q^{14} + 14 q^{15} + 89 q^{16} - 39 q^{17} + 181 q^{18} - 24 q^{19} + 232 q^{20} + 24 q^{21} - 14 q^{22} + 120 q^{23} + 192 q^{24} - 171 q^{25} + 316 q^{26} + 64 q^{27} - 408 q^{28} - 184 q^{29} + 116 q^{30} - 4 q^{31} - 7 q^{32} - 208 q^{33} + 218 q^{34} - 156 q^{35} - 343 q^{36} - 589 q^{37} + 42 q^{38} + 60 q^{39} - 432 q^{40} - 92 q^{41} - 54 q^{42} - 250 q^{43} + 466 q^{44} - 78 q^{45} - 816 q^{46} + 141 q^{47} - 120 q^{48} + 30 q^{49} + 137 q^{50} + 317 q^{51} - 900 q^{52} + 459 q^{53} + 106 q^{54} + 448 q^{55} + 1032 q^{56} + 216 q^{57} + 684 q^{58} + 579 q^{59} - 240 q^{60} + 267 q^{61} - 244 q^{62} + 1044 q^{63} - 87 q^{64} - 424 q^{65} + 16 q^{66} - 540 q^{67} - 1334 q^{68} + 642 q^{69} + 1236 q^{70} + 749 q^{71} + 357 q^{72} - 1924 q^{73} + 950 q^{74} + 473 q^{75} - 402 q^{76} - 288 q^{77} - 152 q^{78} + 805 q^{79} + 448 q^{80} + 291 q^{81} - 938 q^{82} + 712 q^{83} + 372 q^{84} - 1038 q^{85} - 1294 q^{86} + 1216 q^{87} - 2190 q^{88} + 835 q^{89} + 764 q^{90} + 2040 q^{91} + 1596 q^{92} - 1500 q^{93} - 235 q^{94} - 312 q^{95} - 1432 q^{96} - 2243 q^{97} - 2989 q^{98} + 554 q^{99}+O(q^{100}) \) 3 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 - 6 * q^5 - 8 * q^6 - 45 * q^7 - 39 * q^8 - 42 * q^9 - 40 * q^10 + 2 * q^11 + 12 * q^12 - 80 * q^13 + 162 * q^14 + 14 * q^15 + 89 * q^16 - 39 * q^17 + 181 * q^18 - 24 * q^19 + 232 * q^20 + 24 * q^21 - 14 * q^22 + 120 * q^23 + 192 * q^24 - 171 * q^25 + 316 * q^26 + 64 * q^27 - 408 * q^28 - 184 * q^29 + 116 * q^30 - 4 * q^31 - 7 * q^32 - 208 * q^33 + 218 * q^34 - 156 * q^35 - 343 * q^36 - 589 * q^37 + 42 * q^38 + 60 * q^39 - 432 * q^40 - 92 * q^41 - 54 * q^42 - 250 * q^43 + 466 * q^44 - 78 * q^45 - 816 * q^46 + 141 * q^47 - 120 * q^48 + 30 * q^49 + 137 * q^50 + 317 * q^51 - 900 * q^52 + 459 * q^53 + 106 * q^54 + 448 * q^55 + 1032 * q^56 + 216 * q^57 + 684 * q^58 + 579 * q^59 - 240 * q^60 + 267 * q^61 - 244 * q^62 + 1044 * q^63 - 87 * q^64 - 424 * q^65 + 16 * q^66 - 540 * q^67 - 1334 * q^68 + 642 * q^69 + 1236 * q^70 + 749 * q^71 + 357 * q^72 - 1924 * q^73 + 950 * q^74 + 473 * q^75 - 402 * q^76 - 288 * q^77 - 152 * q^78 + 805 * q^79 + 448 * q^80 + 291 * q^81 - 938 * q^82 + 712 * q^83 + 372 * q^84 - 1038 * q^85 - 1294 * q^86 + 1216 * q^87 - 2190 * q^88 + 835 * q^89 + 764 * q^90 + 2040 * q^91 + 1596 * q^92 - 1500 * q^93 - 235 * q^94 - 312 * q^95 - 1432 * q^96 - 2243 * q^97 - 2989 * q^98 + 554 * 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 $

$2$	−4.90952	−1.73578	−0.867888	−	0.496760i	$-0.834523\pi$
$2$	−4.90952	−1.73578	−0.867888	+	0.496760i	$0.834523\pi$
$3$	−0.777884	−0.149704	−0.0748519	−	0.997195i	$-0.523848\pi$
$3$	−0.777884	−0.149704	−0.0748519	+	0.997195i	$0.523848\pi$
$4$	16.1033	2.01292
$5$	9.19383	0.822321	0.411161	−	0.911563i	$-0.365124\pi$
$5$	9.19383	0.822321	0.411161	+	0.911563i	$0.365124\pi$
$6$	3.81903	0.259852
$7$	−30.3255	−1.63742	−0.818711	−	0.574206i	$-0.805311\pi$
$7$	−30.3255	−1.63742	−0.818711	+	0.574206i	$0.805311\pi$
$8$	−39.7835	−1.75820
$9$	−26.3949	−0.977589
$10$	−45.1373	−1.42737
$11$	22.4298	0.614803	0.307401	−	0.951580i	$-0.400541\pi$
$11$	22.4298	0.614803	0.307401	+	0.951580i	$0.400541\pi$
$12$	−12.5265	−0.301341
$13$	−62.0257	−1.32330	−0.661648	−	0.749815i	$-0.730143\pi$
$13$	−62.0257	−1.32330	−0.661648	+	0.749815i	$0.730143\pi$
$14$	148.883	2.84220
$15$	−7.15173	−0.123105
$16$	66.4910	1.03892
$17$	−72.1639	−1.02955	−0.514774	−	0.857326i	$-0.672124\pi$
$17$	−72.1639	−1.02955	−0.514774	+	0.857326i	$0.672124\pi$
$18$	129.586	1.69688
$19$	−25.0550	−0.302526	−0.151263	−	0.988494i	$-0.548334\pi$
$19$	−25.0550	−0.302526	−0.151263	+	0.988494i	$0.548334\pi$
$20$	148.051	1.65527
$21$	23.5897	0.245128
$22$	−110.119	−1.06716
$23$	103.176	0.935373	0.467687	−	0.883894i	$-0.345088\pi$
$23$	103.176	0.935373	0.467687	+	0.883894i	$0.345088\pi$
$24$	30.9469	0.263209
$25$	−40.4735	−0.323788
$26$	304.516	2.29694
$27$	41.5350	0.296052
$28$	−488.341	−3.29600
$29$	−234.381	−1.50081	−0.750405	−	0.660978i	$-0.770142\pi$
$29$	−234.381	−1.50081	−0.750405	+	0.660978i	$0.770142\pi$
$30$	35.1115	0.213682
$31$	198.714	1.15129	0.575647	−	0.817698i	$-0.304750\pi$
$31$	198.714	1.15129	0.575647	+	0.817698i	$0.304750\pi$
$32$	−8.17058	−0.0451365
$33$	−17.4477	−0.0920383
$34$	354.290	1.78707
$35$	−278.807	−1.34649
$36$	−425.046	−1.96781
$37$	−203.083	−0.902343	−0.451171	−	0.892437i	$-0.648994\pi$
$37$	−203.083	−0.902343	−0.451171	+	0.892437i	$0.648994\pi$
$38$	123.008	0.525118
$39$	48.2488	0.198102
$40$	−365.763	−1.44580
$41$	210.889	0.803303	0.401651	−	0.915793i	$-0.368436\pi$
$41$	210.889	0.803303	0.401651	+	0.915793i	$0.368436\pi$
$42$	−115.814	−0.425487
$43$	111.430	0.395184	0.197592	−	0.980284i	$-0.436688\pi$
$43$	111.430	0.395184	0.197592	+	0.980284i	$0.436688\pi$
$44$	361.194	1.23755
$45$	−242.670	−0.803892
$46$	−506.542	−1.62360
$47$	47.0000	0.145865
$47$	47.0000	0.145865
$48$	−51.7223	−0.155531
$49$	576.634	1.68115
$50$	198.705	0.562023
$51$	56.1351	0.154127
$52$	−998.822	−2.66369
$53$	499.576	1.29476	0.647378	−	0.762169i	$-0.275866\pi$
$53$	499.576	1.29476	0.647378	+	0.762169i	$0.275866\pi$
$54$	−203.917	−0.513881
$55$	206.215	0.505565
$56$	1206.45	2.87891
$57$	19.4898	0.0452894
$58$	1150.70	2.60507
$59$	−562.752	−1.24176	−0.620882	−	0.783904i	$-0.713225\pi$
$59$	−562.752	−1.24176	−0.620882	+	0.783904i	$0.713225\pi$
$60$	−115.167	−0.247799
$61$	548.091	1.15042	0.575212	−	0.818004i	$-0.304919\pi$
$61$	548.091	1.15042	0.575212	+	0.818004i	$0.304919\pi$
$62$	−975.591	−1.99839
$63$	800.437	1.60072
$64$	−491.814	−0.960575
$65$	−570.254	−1.08817
$66$	85.6600	0.159758
$67$	−760.831	−1.38732	−0.693659	−	0.720304i	$-0.744002\pi$
$67$	−760.831	−1.38732	−0.693659	+	0.720304i	$0.744002\pi$
$68$	−1162.08	−2.07240
$69$	−80.2586	−0.140029
$70$	1368.81	2.33720
$71$	−668.059	−1.11668	−0.558338	−	0.829613i	$-0.688561\pi$
$71$	−668.059	−1.11668	−0.558338	+	0.829613i	$0.688561\pi$
$72$	1050.08	1.71880
$73$	−1145.92	−1.83726	−0.918629	−	0.395121i	$-0.870703\pi$
$73$	−1145.92	−1.83726	−0.918629	+	0.395121i	$0.870703\pi$
$74$	997.040	1.56626
$75$	31.4836	0.0484722
$76$	−403.469	−0.608961
$77$	−680.193	−1.00669
$78$	−236.878	−0.343861
$79$	975.010	1.38857	0.694286	−	0.719699i	$-0.255720\pi$
$79$	975.010	1.38857	0.694286	+	0.719699i	$0.255720\pi$
$80$	611.307	0.854328
$81$	680.353	0.933269
$82$	−1035.37	−1.39435
$83$	698.827	0.924171	0.462086	−	0.886835i	$-0.347101\pi$
$83$	698.827	0.924171	0.462086	+	0.886835i	$0.347101\pi$
$84$	379.873	0.493423
$85$	−663.463	−0.846620
$86$	−547.067	−0.685950
$87$	182.321	0.224677
$88$	−892.335	−1.08095
$89$	451.477	0.537713	0.268857	−	0.963180i	$-0.413354\pi$
$89$	451.477	0.537713	0.268857	+	0.963180i	$0.413354\pi$
$90$	1191.39	1.39538
$91$	1880.96	2.16679
$92$	1661.47	1.88283
$93$	−154.577	−0.172353
$94$	−230.747	−0.253189
$95$	−230.351	−0.248774
$96$	6.35576	0.00675710
$97$	−390.906	−0.409180	−0.204590	−	0.978848i	$-0.565586\pi$
$97$	−390.906	−0.409180	−0.204590	+	0.978848i	$0.565586\pi$
$98$	−2830.99	−2.91810
$99$	−592.031	−0.601024
$100$	−651.758	−0.651758
$101$	−1112.66	−1.09618	−0.548089	−	0.836420i	$-0.684644\pi$
$101$	−1112.66	−1.09618	−0.548089	+	0.836420i	$0.684644\pi$
$102$	−275.596	−0.267530
$103$	−882.574	−0.844298	−0.422149	−	0.906527i	$-0.638724\pi$
$103$	−882.574	−0.844298	−0.422149	+	0.906527i	$0.638724\pi$
$104$	2467.60	2.32662
$105$	216.880	0.201574
$106$	−2452.68	−2.24741
$107$	840.742	0.759604	0.379802	−	0.925068i	$-0.375992\pi$
$107$	840.742	0.759604	0.379802	+	0.925068i	$0.375992\pi$
$108$	668.853	0.595930
$109$	−1321.06	−1.16086	−0.580432	−	0.814309i	$-0.697116\pi$
$109$	−1321.06	−1.16086	−0.580432	+	0.814309i	$0.697116\pi$
$110$	−1012.42	−0.877548
$111$	157.975	0.135084
$112$	−2016.37	−1.70115
$113$	701.293	0.583824	0.291912	−	0.956445i	$-0.405709\pi$
$113$	701.293	0.583824	0.291912	+	0.956445i	$0.405709\pi$
$114$	−95.6857	−0.0786122
$115$	948.579	0.769177
$116$	−3774.32	−3.02101
$117$	1637.16	1.29364
$118$	2762.84	2.15542
$119$	2188.40	1.68580
$120$	284.521	0.216442
$121$	−827.906	−0.622018
$122$	−2690.86	−1.99688
$123$	−164.047	−0.120257
$124$	3199.96	2.31746
$125$	−1521.34	−1.08858
$126$	−3929.76	−2.77850
$127$	1142.65	0.798374	0.399187	−	0.916869i	$-0.369292\pi$
$127$	1142.65	0.798374	0.399187	+	0.916869i	$0.369292\pi$
$128$	2479.94	1.71248
$129$	−86.6795	−0.0591605
$130$	2799.67	1.88883
$131$	104.451	0.0696639	0.0348319	−	0.999393i	$-0.488910\pi$
$131$	104.451	0.0696639	0.0348319	+	0.999393i	$0.488910\pi$
$132$	−280.967	−0.185265
$133$	759.803	0.495363
$134$	3735.31	2.40807
$135$	381.866	0.243450
$136$	2870.93	1.81015
$137$	−543.914	−0.339195	−0.169598	−	0.985513i	$-0.554247\pi$
$137$	−543.914	−0.339195	−0.169598	+	0.985513i	$0.554247\pi$
$138$	394.031	0.243059
$139$	−41.8374	−0.0255295	−0.0127647	−	0.999919i	$-0.504063\pi$
$139$	−41.8374	−0.0255295	−0.0127647	+	0.999919i	$0.504063\pi$
$140$	−4489.73	−2.71037
$141$	−36.5605	−0.0218365
$142$	3279.85	1.93830
$143$	−1391.22	−0.813565
$144$	−1755.02	−1.01564
$145$	−2154.86	−1.23415
$146$	5625.91	3.18907
$147$	−448.554	−0.251674
$148$	−3270.32	−1.81634
$149$	790.139	0.434435	0.217217	−	0.976123i	$-0.430302\pi$
$149$	790.139	0.434435	0.217217	+	0.976123i	$0.430302\pi$
$150$	−154.569	−0.0841370
$151$	−2073.79	−1.11763	−0.558816	−	0.829292i	$-0.688744\pi$
$151$	−2073.79	−1.11763	−0.558816	+	0.829292i	$0.688744\pi$
$152$	996.774	0.531902
$153$	1904.76	1.00648
$154$	3339.42	1.74739
$155$	1826.95	0.946734
$156$	776.967	0.398764
$157$	2501.12	1.27141	0.635704	−	0.771933i	$-0.280710\pi$
$157$	2501.12	1.27141	0.635704	+	0.771933i	$0.280710\pi$
$158$	−4786.83	−2.41025
$159$	−388.612	−0.193830
$160$	−75.1189	−0.0371167
$161$	−3128.85	−1.53160
$162$	−3340.20	−1.61995
$163$	−97.0569	−0.0466386	−0.0233193	−	0.999728i	$-0.507423\pi$
$163$	−97.0569	−0.0466386	−0.0233193	+	0.999728i	$0.507423\pi$
$164$	3396.03	1.61698
$165$	−160.412	−0.0756850
$166$	−3430.90	−1.60415
$167$	−1826.71	−0.846437	−0.423218	−	0.906028i	$-0.639100\pi$
$167$	−1826.71	−0.846437	−0.423218	+	0.906028i	$0.639100\pi$
$168$	−938.480	−0.430984
$169$	1650.19	0.751111
$170$	3257.28	1.46954
$171$	661.323	0.295746
$172$	1794.39	0.795473
$173$	−1843.63	−0.810223	−0.405111	−	0.914267i	$-0.632767\pi$
$173$	−1843.63	−0.810223	−0.405111	+	0.914267i	$0.632767\pi$
$174$	−895.109	−0.389989
$175$	1227.38	0.530177
$176$	1491.38	0.638732
$177$	437.756	0.185897
$178$	−2216.53	−0.933350
$179$	−370.515	−0.154713	−0.0773565	−	0.997003i	$-0.524648\pi$
$179$	−370.515	−0.154713	−0.0773565	+	0.997003i	$0.524648\pi$
$180$	−3907.80	−1.61817
$181$	−866.586	−0.355872	−0.177936	−	0.984042i	$-0.556942\pi$
$181$	−866.586	−0.355872	−0.177936	+	0.984042i	$0.556942\pi$
$182$	−9234.60	−3.76107
$183$	−426.351	−0.172223
$184$	−4104.69	−1.64457
$185$	−1867.11	−0.742016
$186$	758.896	0.299166
$187$	−1618.62	−0.632969
$188$	756.857	0.293614
$189$	−1259.57	−0.484763
$190$	1130.91	0.431816
$191$	−3667.49	−1.38937	−0.694687	−	0.719312i	$-0.744457\pi$
$191$	−3667.49	−1.38937	−0.694687	+	0.719312i	$0.744457\pi$
$192$	382.574	0.143802
$193$	−4676.91	−1.74431	−0.872154	−	0.489231i	$-0.837277\pi$
$193$	−4676.91	−1.74431	−0.872154	+	0.489231i	$0.837277\pi$
$194$	1919.16	0.710245
$195$	443.591	0.162904
$196$	9285.73	3.38401
$197$	3700.99	1.33850	0.669251	−	0.743037i	$-0.266615\pi$
$197$	3700.99	1.33850	0.669251	+	0.743037i	$0.266615\pi$
$198$	2906.59	1.04324
$199$	−1361.63	−0.485042	−0.242521	−	0.970146i	$-0.577974\pi$
$199$	−1361.63	−0.485042	−0.242521	+	0.970146i	$0.577974\pi$
$200$	1610.18	0.569283
$201$	591.838	0.207687
$202$	5462.63	1.90272
$203$	7107.72	2.45746
$204$	903.964	0.310246
$205$	1938.88	0.660573
$206$	4333.01	1.46551
$207$	−2723.31	−0.914411
$208$	−4124.15	−1.37480
$209$	−561.977	−0.185994
$210$	−1064.77	−0.349887
$211$	−2873.21	−0.937442	−0.468721	−	0.883346i	$-0.655285\pi$
$211$	−2873.21	−0.937442	−0.468721	+	0.883346i	$0.655285\pi$
$212$	8044.85	2.60624
$213$	519.672	0.167171
$214$	−4127.64	−1.31850
$215$	1024.47	0.324968
$216$	−1652.41	−0.520519
$217$	−6026.10	−1.88515
$218$	6485.74	2.01500
$219$	891.393	0.275044
$220$	3320.76	1.01766
$221$	4476.02	1.36240
$222$	−775.581	−0.234476
$223$	3356.14	1.00782	0.503909	−	0.863757i	$-0.331895\pi$
$223$	3356.14	1.00782	0.503909	+	0.863757i	$0.331895\pi$
$224$	247.777	0.0739074
$225$	1068.29	0.316531
$226$	−3443.01	−1.01339
$227$	−3991.01	−1.16693	−0.583464	−	0.812139i	$-0.698303\pi$
$227$	−3991.01	−1.16693	−0.583464	+	0.812139i	$0.698303\pi$
$228$	313.852	0.0911638
$229$	−968.028	−0.279341	−0.139670	−	0.990198i	$-0.544604\pi$
$229$	−968.028	−0.279341	−0.139670	+	0.990198i	$0.544604\pi$
$230$	−4657.06	−1.33512
$231$	529.111	0.150705
$232$	9324.51	2.63872
$233$	−1227.70	−0.345189	−0.172595	−	0.984993i	$-0.555215\pi$
$233$	−1227.70	−0.345189	−0.172595	+	0.984993i	$0.555215\pi$
$234$	−8037.68	−2.24547
$235$	432.110	0.119948
$236$	−9062.19	−2.49957
$237$	−758.444	−0.207875
$238$	−10744.0	−2.92618
$239$	647.534	0.175253	0.0876265	−	0.996153i	$-0.472072\pi$
$239$	647.534	0.175253	0.0876265	+	0.996153i	$0.472072\pi$
$240$	−475.526	−0.127896
$241$	−4174.42	−1.11576	−0.557880	−	0.829922i	$-0.688385\pi$
$241$	−4174.42	−1.11576	−0.557880	+	0.829922i	$0.688385\pi$
$242$	4064.62	1.07968
$243$	−1650.68	−0.435766
$244$	8826.10	2.31571
$245$	5301.47	1.38244
$246$	805.394	0.208740
$247$	1554.05	0.400332
$248$	−7905.55	−2.02421
$249$	−543.606	−0.138352
$250$	7469.02	1.88953
$251$	−1368.39	−0.344112	−0.172056	−	0.985087i	$-0.555041\pi$
$251$	−1368.39	−0.344112	−0.172056	+	0.985087i	$0.555041\pi$
$252$	12889.7	3.22213
$253$	2314.20	0.575070
$254$	−5609.84	−1.38580
$255$	516.097	0.126742
$256$	−8240.77	−2.01191
$257$	1709.94	0.415031	0.207515	−	0.978232i	$-0.433462\pi$
$257$	1709.94	0.415031	0.207515	+	0.978232i	$0.433462\pi$
$258$	425.554	0.102689
$259$	6158.59	1.47752
$260$	−9183.00	−2.19041
$261$	6186.47	1.46718
$262$	−512.806	−0.120921
$263$	5368.95	1.25880	0.629399	−	0.777082i	$-0.283301\pi$
$263$	5368.95	1.25880	0.629399	+	0.777082i	$0.283301\pi$
$264$	694.133	0.161822
$265$	4593.02	1.06471
$266$	−3730.27	−0.859840
$267$	−351.197	−0.0804977
$268$	−12251.9	−2.79256
$269$	−4213.70	−0.955070	−0.477535	−	0.878613i	$-0.658470\pi$
$269$	−4213.70	−0.955070	−0.477535	+	0.878613i	$0.658470\pi$
$270$	−1874.78	−0.422575
$271$	4769.83	1.06918	0.534588	−	0.845113i	$-0.320467\pi$
$271$	4769.83	1.06918	0.534588	+	0.845113i	$0.320467\pi$
$272$	−4798.25	−1.06962
$273$	−1463.17	−0.324377
$274$	2670.36	0.588767
$275$	−907.810	−0.199066
$276$	−1292.43	−0.281867
$277$	5651.38	1.22584	0.612922	−	0.790144i	$-0.289994\pi$
$277$	5651.38	1.22584	0.612922	+	0.790144i	$0.289994\pi$
$278$	205.401	0.0443135
$279$	−5245.04	−1.12549
$280$	11091.9	2.36739
$281$	3264.83	0.693109	0.346554	−	0.938030i	$-0.387352\pi$
$281$	3264.83	0.693109	0.346554	+	0.938030i	$0.387352\pi$
$282$	179.495	0.0379033
$283$	2019.38	0.424169	0.212084	−	0.977251i	$-0.431975\pi$
$283$	2019.38	0.424169	0.212084	+	0.977251i	$0.431975\pi$
$284$	−10758.0	−2.24778
$285$	179.186	0.0372424
$286$	6830.23	1.41217
$287$	−6395.32	−1.31534
$288$	215.662	0.0441249
$289$	294.634	0.0599703
$290$	10579.3	2.14221
$291$	304.079	0.0612558
$292$	−18453.2	−3.69825
$293$	3432.28	0.684354	0.342177	−	0.939636i	$-0.388836\pi$
$293$	3432.28	0.684354	0.342177	+	0.939636i	$0.388836\pi$
$294$	2202.18	0.436850
$295$	−5173.85	−1.02113
$296$	8079.36	1.58650
$297$	931.621	0.182014
$298$	−3879.20	−0.754081
$299$	−6399.54	−1.23778
$300$	506.992	0.0975707
$301$	−3379.16	−0.647082
$302$	10181.3	1.93996
$303$	865.521	0.164102
$304$	−1665.93	−0.314301
$305$	5039.06	0.946019
$306$	−9351.45	−1.74702
$307$	−6327.92	−1.17640	−0.588198	−	0.808717i	$-0.700162\pi$
$307$	−6327.92	−1.17640	−0.588198	+	0.808717i	$0.700162\pi$
$308$	−10953.4	−2.02639
$309$	686.540	0.126395
$310$	−8969.42	−1.64332
$311$	9118.69	1.66262	0.831308	−	0.555812i	$-0.187592\pi$
$311$	9118.69	1.66262	0.831308	+	0.555812i	$0.187592\pi$
$312$	−1919.51	−0.348303
$313$	4611.70	0.832808	0.416404	−	0.909180i	$-0.363290\pi$
$313$	4611.70	0.832808	0.416404	+	0.909180i	$0.363290\pi$
$314$	−12279.3	−2.20688
$315$	7359.09	1.31631
$316$	15700.9	2.79508
$317$	−4243.78	−0.751907	−0.375954	−	0.926638i	$-0.622685\pi$
$317$	−4243.78	−0.751907	−0.375954	+	0.926638i	$0.622685\pi$
$318$	1907.90	0.336445
$319$	−5257.12	−0.922702
$320$	−4521.66	−0.789901
$321$	−654.000	−0.113716
$322$	15361.1	2.65851
$323$	1808.06	0.311466
$324$	10956.0	1.87859
$325$	2510.40	0.428467
$326$	476.502	0.0809541
$327$	1027.63	0.173786
$328$	−8389.92	−1.41237
$329$	−1425.30	−0.238842
$330$	787.543	0.131372
$331$	−3803.59	−0.631613	−0.315807	−	0.948824i	$-0.602275\pi$
$331$	−3803.59	−0.631613	−0.315807	+	0.948824i	$0.602275\pi$
$332$	11253.5	1.86028
$333$	5360.36	0.882120
$334$	8968.25	1.46922
$335$	−6994.95	−1.14082
$336$	1568.50	0.254669
$337$	−6419.12	−1.03760	−0.518801	−	0.854895i	$-0.673621\pi$
$337$	−6419.12	−1.03760	−0.518801	+	0.854895i	$0.673621\pi$
$338$	−8101.64	−1.30376
$339$	−545.524	−0.0874006
$340$	−10684.0	−1.70418
$341$	4457.11	0.707819
$342$	−3246.78	−0.513350
$343$	−7085.05	−1.11533
$344$	−4433.07	−0.694812
$345$	−737.884	−0.115149
$346$	9051.33	1.40637
$347$	−3495.57	−0.540783	−0.270392	−	0.962750i	$-0.587153\pi$
$347$	−3495.57	−0.540783	−0.270392	+	0.962750i	$0.587153\pi$
$348$	2935.98	0.452256
$349$	−435.838	−0.0668477	−0.0334239	−	0.999441i	$-0.510641\pi$
$349$	−435.838	−0.0668477	−0.0334239	+	0.999441i	$0.510641\pi$
$350$	−6025.83	−0.920268
$351$	−2576.24	−0.391765
$352$	−183.264	−0.0277500
$353$	1941.73	0.292771	0.146385	−	0.989228i	$-0.453236\pi$
$353$	1941.73	0.292771	0.146385	+	0.989228i	$0.453236\pi$
$354$	−2149.17	−0.322675
$355$	−6142.03	−0.918267
$356$	7270.29	1.08237
$357$	−1702.32	−0.252371
$358$	1819.05	0.268547
$359$	7188.88	1.05687	0.528433	−	0.848975i	$-0.322780\pi$
$359$	7188.88	1.05687	0.528433	+	0.848975i	$0.322780\pi$
$360$	9654.27	1.41340
$361$	−6231.25	−0.908478
$362$	4254.52	0.617714
$363$	644.014	0.0931184
$364$	30289.7	4.36158
$365$	−10535.4	−1.51082
$366$	2093.18	0.298940
$367$	1674.78	0.238209	0.119105	−	0.992882i	$-0.461998\pi$
$367$	1674.78	0.238209	0.119105	+	0.992882i	$0.461998\pi$
$368$	6860.25	0.971780
$369$	−5566.41	−0.785300
$370$	9166.62	1.28797
$371$	−15149.9	−2.12006
$372$	−2489.20	−0.346933
$373$	6028.31	0.836820	0.418410	−	0.908258i	$-0.362588\pi$
$373$	6028.31	0.836820	0.418410	+	0.908258i	$0.362588\pi$
$374$	7946.64	1.09869
$375$	1183.42	0.162964
$376$	−1869.83	−0.256460
$377$	14537.7	1.98602
$378$	6183.87	0.841439
$379$	−2065.63	−0.279959	−0.139979	−	0.990154i	$-0.544704\pi$
$379$	−2065.63	−0.279959	−0.139979	+	0.990154i	$0.544704\pi$
$380$	−3709.42	−0.500762
$381$	−888.846	−0.119520
$382$	18005.6	2.41164
$383$	7634.28	1.01852	0.509260	−	0.860613i	$-0.329919\pi$
$383$	7634.28	1.01852	0.509260	+	0.860613i	$0.329919\pi$
$384$	−1929.10	−0.256365
$385$	−6253.58	−0.827823
$386$	22961.4	3.02773
$387$	−2941.18	−0.386327
$388$	−6294.89	−0.823646
$389$	−2814.13	−0.366792	−0.183396	−	0.983039i	$-0.558709\pi$
$389$	−2814.13	−0.366792	−0.183396	+	0.983039i	$0.558709\pi$
$390$	−2177.82	−0.282764
$391$	−7445.55	−0.963012
$392$	−22940.5	−2.95579
$393$	−81.2511	−0.0104289
$394$	−18170.1	−2.32334
$395$	8964.08	1.14185
$396$	−9533.69	−1.20981
$397$	8990.52	1.13658	0.568288	−	0.822829i	$-0.307606\pi$
$397$	8990.52	1.13658	0.568288	+	0.822829i	$0.307606\pi$
$398$	6684.94	0.841924
$399$	−591.039	−0.0741577
$400$	−2691.12	−0.336390
$401$	−4829.93	−0.601484	−0.300742	−	0.953706i	$-0.597234\pi$
$401$	−4829.93	−0.601484	−0.300742	+	0.953706i	$0.597234\pi$
$402$	−2905.64	−0.360497
$403$	−12325.4	−1.52350
$404$	−17917.6	−2.20652
$405$	6255.05	0.767447
$406$	−34895.5	−4.26560
$407$	−4555.11	−0.554763
$408$	−2233.25	−0.270987
$409$	−6551.78	−0.792089	−0.396045	−	0.918231i	$-0.629617\pi$
$409$	−6551.78	−0.792089	−0.396045	+	0.918231i	$0.629617\pi$
$410$	−9518.97	−1.14661
$411$	423.102	0.0507788
$412$	−14212.4	−1.69950
$413$	17065.7	2.03329
$414$	13370.1	1.58721
$415$	6424.90	0.759966
$416$	506.786	0.0597289
$417$	32.5446	0.00382186
$418$	2759.03	0.322844
$419$	10660.5	1.24296	0.621481	−	0.783429i	$-0.286531\pi$
$419$	10660.5	1.24296	0.621481	+	0.783429i	$0.286531\pi$
$420$	3492.49	0.405752
$421$	−8654.47	−1.00188	−0.500942	−	0.865481i	$-0.667013\pi$
$421$	−8654.47	−1.00188	−0.500942	+	0.865481i	$0.667013\pi$
$422$	14106.1	1.62719
$423$	−1240.56	−0.142596
$424$	−19874.9	−2.27644
$425$	2920.73	0.333355
$426$	−2551.34	−0.290171
$427$	−16621.1	−1.88373
$428$	13538.8	1.52902
$429$	1082.21	0.121794
$430$	−5029.64	−0.564072
$431$	4990.22	0.557703	0.278852	−	0.960334i	$-0.410046\pi$
$431$	4990.22	0.557703	0.278852	+	0.960334i	$0.410046\pi$
$432$	2761.71	0.307575
$433$	−7921.49	−0.879174	−0.439587	−	0.898200i	$-0.644875\pi$
$433$	−7921.49	−0.879174	−0.439587	+	0.898200i	$0.644875\pi$
$434$	29585.2	3.27220
$435$	1676.23	0.184757
$436$	−21273.4	−2.33673
$437$	−2585.06	−0.282975
$438$	−4376.31	−0.477416
$439$	11034.1	1.19961	0.599807	−	0.800145i	$-0.295244\pi$
$439$	11034.1	1.19961	0.599807	+	0.800145i	$0.295244\pi$
$440$	−8203.97	−0.888884
$441$	−15220.2	−1.64347
$442$	−21975.1	−2.36482
$443$	9160.66	0.982474	0.491237	−	0.871026i	$-0.336545\pi$
$443$	9160.66	0.982474	0.491237	+	0.871026i	$0.336545\pi$
$444$	2543.93	0.271913
$445$	4150.80	0.442173
$446$	−16477.0	−1.74935
$447$	−614.636	−0.0650365
$448$	14914.5	1.57287
$449$	−10071.1	−1.05854	−0.529271	−	0.848453i	$-0.677535\pi$
$449$	−10071.1	−1.05854	−0.529271	+	0.848453i	$0.677535\pi$
$450$	−5244.80	−0.549427
$451$	4730.20	0.493872
$452$	11293.2	1.17519
$453$	1613.16	0.167314
$454$	19593.9	2.02553
$455$	17293.2	1.78180
$456$	−775.374	−0.0796277
$457$	−2988.54	−0.305904	−0.152952	−	0.988234i	$-0.548878\pi$
$457$	−2988.54	−0.305904	−0.152952	+	0.988234i	$0.548878\pi$
$458$	4752.55	0.484873
$459$	−2997.33	−0.304800
$460$	15275.3	1.54829
$461$	956.638	0.0966487	0.0483244	−	0.998832i	$-0.484612\pi$
$461$	956.638	0.0966487	0.0483244	+	0.998832i	$0.484612\pi$
$462$	−2597.68	−0.261591
$463$	−4352.75	−0.436910	−0.218455	−	0.975847i	$-0.570102\pi$
$463$	−4352.75	−0.436910	−0.218455	+	0.975847i	$0.570102\pi$
$464$	−15584.2	−1.55923
$465$	−1421.15	−0.141730
$466$	6027.40	0.599171
$467$	8131.35	0.805725	0.402863	−	0.915260i	$-0.368015\pi$
$467$	8131.35	0.805725	0.402863	+	0.915260i	$0.368015\pi$
$468$	26363.8	2.60399
$469$	23072.5	2.27162
$470$	−2121.45	−0.208203
$471$	−1945.58	−0.190335
$472$	22388.2	2.18327
$473$	2499.35	0.242960
$474$	3723.60	0.360824
$475$	1014.06	0.0979544
$476$	35240.6	3.39339
$477$	−13186.3	−1.26574
$478$	−3179.08	−0.304200
$479$	−12912.4	−1.23170	−0.615849	−	0.787864i	$-0.711187\pi$
$479$	−12912.4	−1.23170	−0.615849	+	0.787864i	$0.711187\pi$
$480$	58.4338	0.00555651
$481$	12596.4	1.19407
$482$	20494.4	1.93671
$483$	2433.88	0.229286
$484$	−13332.1	−1.25207
$485$	−3593.92	−0.336477
$486$	8104.04	0.756393
$487$	−11010.0	−1.02446	−0.512228	−	0.858850i	$-0.671180\pi$
$487$	−11010.0	−1.02446	−0.512228	+	0.858850i	$0.671180\pi$
$488$	−21805.0	−2.02268
$489$	75.4990	0.00698197
$490$	−26027.7	−2.39961
$491$	−14636.7	−1.34531	−0.672655	−	0.739956i	$-0.734846\pi$
$491$	−14636.7	−1.34531	−0.672655	+	0.739956i	$0.734846\pi$
$492$	−2641.71	−0.242068
$493$	16913.9	1.54516
$494$	−7629.64	−0.694887
$495$	−5443.04	−0.494235
$496$	13212.7	1.19611
$497$	20259.2	1.82847
$498$	2668.84	0.240148
$499$	−7549.78	−0.677304	−0.338652	−	0.940912i	$-0.609971\pi$
$499$	−7549.78	−0.677304	−0.338652	+	0.940912i	$0.609971\pi$
$500$	−24498.6	−2.19122
$501$	1420.97	0.126715
$502$	6718.14	0.597301
$503$	−16150.9	−1.43167	−0.715837	−	0.698267i	$-0.753955\pi$
$503$	−16150.9	−1.43167	−0.715837	+	0.698267i	$0.753955\pi$
$504$	−31844.2	−2.81439
$505$	−10229.6	−0.901410
$506$	−11361.6	−0.998193
$507$	−1283.66	−0.112444
$508$	18400.4	1.60706
$509$	19183.7	1.67053	0.835266	−	0.549846i	$-0.185314\pi$
$509$	19183.7	1.67053	0.835266	+	0.549846i	$0.185314\pi$
$510$	−2533.79	−0.219996
$511$	34750.6	3.00836
$512$	20618.7	1.77974
$513$	−1040.66	−0.0895637
$514$	−8394.96	−0.720400
$515$	−8114.24	−0.694284
$516$	−1395.83	−0.119085
$517$	1054.20	0.0896782
$518$	−30235.7	−2.56464
$519$	1434.13	0.121293
$520$	22686.7	1.91323
$521$	6867.69	0.577503	0.288751	−	0.957404i	$-0.406760\pi$
$521$	6867.69	0.577503	0.288751	+	0.957404i	$0.406760\pi$
$522$	−30372.6	−2.54669
$523$	11205.2	0.936841	0.468420	−	0.883506i	$-0.344823\pi$
$523$	11205.2	0.936841	0.468420	+	0.883506i	$0.344823\pi$
$524$	1682.02	0.140228
$525$	−954.756	−0.0793695
$526$	−26359.0	−2.18499
$527$	−14340.0	−1.18531
$528$	−1160.12	−0.0956206
$529$	−1521.81	−0.125076
$530$	−22549.5	−1.84809
$531$	14853.8	1.21393
$532$	12235.4	0.997126
$533$	−13080.6	−1.06301
$534$	1724.21	0.139726
$535$	7729.64	0.624639
$536$	30268.5	2.43918
$537$	288.218	0.0231611
$538$	20687.2	1.65779
$539$	12933.8	1.03357
$540$	6149.32	0.490045
$541$	14471.4	1.15004	0.575021	−	0.818139i	$-0.304994\pi$
$541$	14471.4	1.15004	0.575021	+	0.818139i	$0.304994\pi$
$542$	−23417.6	−1.85585
$543$	674.103	0.0532754
$544$	589.621	0.0464702
$545$	−12145.6	−0.954603
$546$	7183.44	0.563046
$547$	−13482.7	−1.05389	−0.526946	−	0.849899i	$-0.676663\pi$
$547$	−13482.7	−1.05389	−0.526946	+	0.849899i	$0.676663\pi$
$548$	−8758.84	−0.682772
$549$	−14466.8	−1.12464
$550$	4456.91	0.345533
$551$	5872.41	0.454035
$552$	3192.97	0.246199
$553$	−29567.6	−2.27368
$554$	−27745.6	−2.12779
$555$	1452.40	0.111083
$556$	−673.722	−0.0513888
$557$	10194.0	0.775466	0.387733	−	0.921772i	$-0.373258\pi$
$557$	10194.0	0.775466	0.387733	+	0.921772i	$0.373258\pi$
$558$	25750.6	1.95360
$559$	−6911.52	−0.522945
$560$	−18538.2	−1.39889
$561$	1259.10	0.0947579
$562$	−16028.7	−1.20308
$563$	3725.21	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$563$	3725.21	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$564$	−588.747	−0.0439552
$565$	6447.57	0.480091
$566$	−9914.18	−0.736262
$567$	−20632.0	−1.52815
$568$	26577.7	1.96334
$569$	−6274.57	−0.462291	−0.231145	−	0.972919i	$-0.574247\pi$
$569$	−6274.57	−0.462291	−0.231145	+	0.972919i	$0.574247\pi$
$570$	−879.718	−0.0646445
$571$	20390.6	1.49443	0.747214	−	0.664584i	$-0.231391\pi$
$571$	20390.6	1.49443	0.747214	+	0.664584i	$0.231391\pi$
$572$	−22403.3	−1.63764
$573$	2852.88	0.207995
$574$	31397.9	2.28314
$575$	−4175.87	−0.302862
$576$	12981.4	0.939048
$577$	675.385	0.0487290	0.0243645	−	0.999703i	$-0.492244\pi$
$577$	675.385	0.0487290	0.0243645	+	0.999703i	$0.492244\pi$
$578$	−1446.51	−0.104095
$579$	3638.09	0.261130
$580$	−34700.5	−2.48424
$581$	−21192.3	−1.51326
$582$	−1492.88	−0.106326
$583$	11205.4	0.796019
$584$	45588.7	3.23027
$585$	15051.8	1.06379
$586$	−16850.8	−1.18788
$587$	−5949.63	−0.418343	−0.209172	−	0.977879i	$-0.567077\pi$
$587$	−5949.63	−0.418343	−0.209172	+	0.977879i	$0.567077\pi$
$588$	−7223.22	−0.506600
$589$	−4978.78	−0.348297
$590$	25401.1	1.77245
$591$	−2878.94	−0.200379
$592$	−13503.2	−0.937464
$593$	−22071.9	−1.52847	−0.764235	−	0.644937i	$-0.776883\pi$
$593$	−22071.9	−1.52847	−0.764235	+	0.644937i	$0.776883\pi$
$594$	−4573.81	−0.315935
$595$	20119.8	1.38627
$596$	12723.9	0.874481
$597$	1059.19	0.0726126
$598$	31418.6	2.14850
$599$	−9547.79	−0.651272	−0.325636	−	0.945495i	$-0.605578\pi$
$599$	−9547.79	−0.651272	−0.325636	+	0.945495i	$0.605578\pi$
$600$	−1252.53	−0.0852239
$601$	−12303.7	−0.835076	−0.417538	−	0.908660i	$-0.637107\pi$
$601$	−12303.7	−0.835076	−0.417538	+	0.908660i	$0.637107\pi$
$602$	16590.1	1.12319
$603$	20082.0	1.35623
$604$	−33394.9	−2.24970
$605$	−7611.63	−0.511498
$606$	−4249.29	−0.284844
$607$	18252.2	1.22049	0.610243	−	0.792214i	$-0.291072\pi$
$607$	18252.2	1.22049	0.610243	+	0.792214i	$0.291072\pi$
$608$	204.714	0.0136550
$609$	−5528.98	−0.367891
$610$	−24739.3	−1.64208
$611$	−2915.21	−0.193022
$612$	30673.0	2.02595
$613$	−13990.5	−0.921809	−0.460905	−	0.887450i	$-0.652475\pi$
$613$	−13990.5	−0.921809	−0.460905	+	0.887450i	$0.652475\pi$
$614$	31067.0	2.04196
$615$	−1508.22	−0.0988902
$616$	27060.5	1.76996
$617$	−5747.16	−0.374995	−0.187498	−	0.982265i	$-0.560038\pi$
$617$	−5747.16	−0.374995	−0.187498	+	0.982265i	$0.560038\pi$
$618$	−3370.58	−0.219393
$619$	−3913.70	−0.254127	−0.127064	−	0.991895i	$-0.540555\pi$
$619$	−3913.70	−0.254127	−0.127064	+	0.991895i	$0.540555\pi$
$620$	29419.9	1.90570
$621$	4285.40	0.276920
$622$	−44768.4	−2.88593
$623$	−13691.3	−0.880463
$624$	3208.11	0.205813
$625$	−8927.71	−0.571374
$626$	−22641.2	−1.44557
$627$	437.153	0.0278440
$628$	40276.4	2.55924
$629$	14655.3	0.929006
$630$	−36129.6	−2.28482
$631$	26311.5	1.65998	0.829988	−	0.557781i	$-0.188347\pi$
$631$	26311.5	1.65998	0.829988	+	0.557781i	$0.188347\pi$
$632$	−38789.3	−2.44139
$633$	2235.03	0.140339
$634$	20834.9	1.30514
$635$	10505.3	0.656520
$636$	−6257.96	−0.390164
$637$	−35766.1	−2.22466
$638$	25809.9	1.60160
$639$	17633.4	1.09165
$640$	22800.1	1.40821
$641$	15538.7	0.957476	0.478738	−	0.877958i	$-0.341094\pi$
$641$	15538.7	0.957476	0.478738	+	0.877958i	$0.341094\pi$
$642$	3210.82	0.197385
$643$	6125.10	0.375661	0.187831	−	0.982201i	$-0.439854\pi$
$643$	6125.10	0.375661	0.187831	+	0.982201i	$0.439854\pi$
$644$	−50384.9	−3.08299
$645$	−796.917	−0.0486489
$646$	−8876.72	−0.540635
$647$	4635.86	0.281692	0.140846	−	0.990032i	$-0.455018\pi$
$647$	4635.86	0.281692	0.140846	+	0.990032i	$0.455018\pi$
$648$	−27066.8	−1.64087
$649$	−12622.4	−0.763440
$650$	−12324.8	−0.743723
$651$	4687.61	0.282215
$652$	−1562.94	−0.0938796
$653$	−11926.6	−0.714740	−0.357370	−	0.933963i	$-0.616326\pi$
$653$	−11926.6	−0.714740	−0.357370	+	0.933963i	$0.616326\pi$
$654$	−5045.15	−0.301653
$655$	960.309	0.0572861
$656$	14022.3	0.834569
$657$	30246.4	1.79608
$658$	6997.52	0.414577
$659$	881.384	0.0520999	0.0260500	−	0.999661i	$-0.491707\pi$
$659$	881.384	0.0520999	0.0260500	+	0.999661i	$0.491707\pi$
$660$	−2583.16	−0.152348
$661$	−19161.6	−1.12753	−0.563766	−	0.825935i	$-0.690648\pi$
$661$	−19161.6	−1.12753	−0.563766	+	0.825935i	$0.690648\pi$
$662$	18673.8	1.09634
$663$	−3481.82	−0.203956
$664$	−27801.8	−1.62488
$665$	6985.50	0.407348
$666$	−26316.8	−1.53116
$667$	−24182.4	−1.40382
$668$	−29416.1	−1.70381
$669$	−2610.68	−0.150874
$670$	34341.8	1.98021
$671$	12293.6	0.707284
$672$	−192.741	−0.0110642
$673$	20156.0	1.15447	0.577235	−	0.816578i	$-0.304132\pi$
$673$	20156.0	1.15447	0.577235	+	0.816578i	$0.304132\pi$
$674$	31514.8	1.80104
$675$	−1681.07	−0.0958582
$676$	26573.6	1.51192
$677$	5567.34	0.316056	0.158028	−	0.987435i	$-0.449486\pi$
$677$	5567.34	0.316056	0.158028	+	0.987435i	$0.449486\pi$
$678$	2678.26	0.151708
$679$	11854.4	0.670000
$680$	26394.9	1.48853
$681$	3104.54	0.174694
$682$	−21882.3	−1.22861
$683$	−27835.8	−1.55946	−0.779728	−	0.626118i	$-0.784643\pi$
$683$	−27835.8	−1.55946	−0.779728	+	0.626118i	$0.784643\pi$
$684$	10649.5	0.595314
$685$	−5000.66	−0.278927
$686$	34784.2	1.93596
$687$	753.013	0.0418184
$688$	7409.09	0.410565
$689$	−30986.6	−1.71334
$690$	3622.65	0.199872
$691$	17125.7	0.942824	0.471412	−	0.881913i	$-0.343745\pi$
$691$	17125.7	0.942824	0.471412	+	0.881913i	$0.343745\pi$
$692$	−29688.6	−1.63091
$693$	17953.6	0.984129
$694$	17161.5	0.938679
$695$	−384.646	−0.0209934
$696$	−7253.38	−0.395027
$697$	−15218.6	−0.827039
$698$	2139.75	0.116033
$699$	955.005	0.0516761
$700$	19764.9	1.06720
$701$	5107.63	0.275196	0.137598	−	0.990488i	$-0.456062\pi$
$701$	5107.63	0.275196	0.137598	+	0.990488i	$0.456062\pi$
$702$	12648.1	0.680016
$703$	5088.24	0.272983
$704$	−11031.3	−0.590564
$705$	−336.131	−0.0179566
$706$	−9532.97	−0.508184
$707$	33742.0	1.79490
$708$	7049.33	0.374195
$709$	11258.0	0.596335	0.298167	−	0.954514i	$-0.403625\pi$
$709$	11258.0	0.596335	0.298167	+	0.954514i	$0.403625\pi$
$710$	30154.4	1.59391
$711$	−25735.3	−1.35745
$712$	−17961.3	−0.945407
$713$	20502.5	1.07689
$714$	8357.59	0.438060
$715$	−12790.7	−0.669012
$716$	−5966.54	−0.311425
$717$	−503.706	−0.0262360
$718$	−35293.9	−1.83448
$719$	−13735.6	−0.712448	−0.356224	−	0.934401i	$-0.615936\pi$
$719$	−13735.6	−0.712448	−0.356224	+	0.934401i	$0.615936\pi$
$720$	−16135.4	−0.835181
$721$	26764.5	1.38247
$722$	30592.4	1.57691
$723$	3247.21	0.167033
$724$	−13954.9	−0.716341
$725$	9486.22	0.485944
$726$	−3161.80	−0.161633
$727$	14515.4	0.740502	0.370251	−	0.928932i	$-0.379272\pi$
$727$	14515.4	0.740502	0.370251	+	0.928932i	$0.379272\pi$
$728$	−74831.1	−3.80965
$729$	−17085.5	−0.868033
$730$	51723.7	2.62244
$731$	−8041.22	−0.406861
$732$	−6865.68	−0.346671
$733$	14218.5	0.716471	0.358236	−	0.933631i	$-0.383378\pi$
$733$	14218.5	0.716471	0.358236	+	0.933631i	$0.383378\pi$
$734$	−8222.36	−0.413478
$735$	−4123.93	−0.206957
$736$	−843.004	−0.0422195
$737$	−17065.3	−0.852926
$738$	27328.4	1.36310
$739$	8651.36	0.430643	0.215322	−	0.976543i	$-0.430920\pi$
$739$	8651.36	0.430643	0.215322	+	0.976543i	$0.430920\pi$
$740$	−30066.8	−1.49362
$741$	−1208.87	−0.0599312
$742$	74378.6	3.67995
$743$	−37263.7	−1.83993	−0.919967	−	0.391996i	$-0.871785\pi$
$743$	−37263.7	−1.83993	−0.919967	+	0.391996i	$0.871785\pi$
$744$	6149.60	0.303031
$745$	7264.41	0.357245
$746$	−29596.1	−1.45253
$747$	−18445.5	−0.903460
$748$	−26065.2	−1.27412
$749$	−25495.9	−1.24379
$750$	−5810.03	−0.282870
$751$	−33980.7	−1.65110	−0.825548	−	0.564331i	$-0.809134\pi$
$751$	−33980.7	−1.65110	−0.825548	+	0.564331i	$0.809134\pi$
$752$	3125.08	0.151542
$753$	1064.45	0.0515148
$754$	−71372.9	−3.44728
$755$	−19066.0	−0.919052
$756$	−20283.3	−0.975788
$757$	−33247.9	−1.59632	−0.798161	−	0.602445i	$-0.794193\pi$
$757$	−33247.9	−1.59632	−0.798161	+	0.602445i	$0.794193\pi$
$758$	10141.2	0.485945
$759$	−1800.18	−0.0860901
$760$	9164.18	0.437394
$761$	−13023.6	−0.620376	−0.310188	−	0.950675i	$-0.600392\pi$
$761$	−13023.6	−0.620376	−0.310188	+	0.950675i	$0.600392\pi$
$762$	4363.81	0.207459
$763$	40061.6	1.90082
$764$	−59058.9	−2.79670
$765$	17512.0	0.827646
$766$	−37480.6	−1.76792
$767$	34905.1	1.64322
$768$	6410.36	0.301190
$769$	26526.2	1.24390	0.621949	−	0.783057i	$-0.286341\pi$
$769$	26526.2	1.24390	0.621949	+	0.783057i	$0.286341\pi$
$770$	30702.0	1.43692
$771$	−1330.13	−0.0621316
$772$	−75313.9	−3.51115
$773$	1143.17	0.0531914	0.0265957	−	0.999646i	$-0.491533\pi$
$773$	1143.17	0.0531914	0.0265957	+	0.999646i	$0.491533\pi$
$774$	14439.8	0.670577
$775$	−8042.65	−0.372775
$776$	15551.6	0.719420
$777$	−4790.67	−0.221190
$778$	13816.0	0.636668
$779$	−5283.83	−0.243020
$780$	7143.30	0.327912
$781$	−14984.4	−0.686536
$782$	36554.1	1.67157
$783$	−9735.03	−0.444319
$784$	38341.0	1.74658
$785$	22994.9	1.04551
$786$	398.904	0.0181023
$787$	41354.8	1.87311	0.936556	−	0.350519i	$-0.113995\pi$
$787$	41354.8	1.87311	0.936556	+	0.350519i	$0.113995\pi$
$788$	59598.4	2.69429
$789$	−4176.42	−0.188447
$790$	−44009.3	−1.98200
$791$	−21267.0	−0.955965
$792$	23553.1	1.05672
$793$	−33995.8	−1.52235
$794$	−44139.1	−1.97284
$795$	−3572.83	−0.159390
$796$	−21926.8	−0.976349
$797$	8795.77	0.390918	0.195459	−	0.980712i	$-0.437380\pi$
$797$	8795.77	0.390918	0.195459	+	0.980712i	$0.437380\pi$
$798$	2901.71	0.128721
$799$	−3391.71	−0.150175
$800$	330.692	0.0146146
$801$	−11916.7	−0.525662
$802$	23712.6	1.04404
$803$	−25702.7	−1.12955
$804$	9530.57	0.418056
$805$	−28766.1	−1.25947
$806$	60511.7	2.64446
$807$	3277.77	0.142978
$808$	44265.6	1.92730
$809$	2152.09	0.0935272	0.0467636	−	0.998906i	$-0.485109\pi$
$809$	2152.09	0.0935272	0.0467636	+	0.998906i	$0.485109\pi$
$810$	−30709.3	−1.33212
$811$	24307.1	1.05245	0.526225	−	0.850346i	$-0.323607\pi$
$811$	24307.1	1.05245	0.526225	+	0.850346i	$0.323607\pi$
$812$	114458.	4.94666
$813$	−3710.38	−0.160060
$814$	22363.4	0.962944
$815$	−892.325	−0.0383519
$816$	3732.48	0.160126
$817$	−2791.87	−0.119554
$818$	32166.0	1.37489
$819$	−49647.7	−2.11823
$820$	31222.5	1.32968
$821$	45118.9	1.91798	0.958988	−	0.283446i	$-0.0914775\pi$
$821$	45118.9	1.91798	0.958988	+	0.283446i	$0.0914775\pi$
$822$	−2077.23	−0.0881406
$823$	38608.9	1.63526	0.817631	−	0.575742i	$-0.195287\pi$
$823$	38608.9	1.63526	0.817631	+	0.575742i	$0.195287\pi$
$824$	35111.9	1.48444
$825$	706.171	0.0298009
$826$	−83784.4	−3.52934
$827$	−17908.8	−0.753023	−0.376512	−	0.926412i	$-0.622876\pi$
$827$	−17908.8	−0.753023	−0.376512	+	0.926412i	$0.622876\pi$
$828$	−43854.4	−1.84063
$829$	42616.2	1.78543	0.892716	−	0.450620i	$-0.148797\pi$
$829$	42616.2	1.78543	0.892716	+	0.450620i	$0.148797\pi$
$830$	−31543.1	−1.31913
$831$	−4396.12	−0.183513
$832$	30505.2	1.27112
$833$	−41612.2	−1.73082
$834$	−159.778	−0.00663389
$835$	−16794.4	−0.696043
$836$	−9049.71	−0.374391
$837$	8253.60	0.340844
$838$	−52338.1	−2.15750
$839$	9077.30	0.373520	0.186760	−	0.982406i	$-0.440201\pi$
$839$	9077.30	0.373520	0.186760	+	0.982406i	$0.440201\pi$
$840$	−8628.23	−0.354407
$841$	30545.6	1.25243
$842$	42489.2	1.73905
$843$	−2539.66	−0.103761
$844$	−46268.4	−1.88699
$845$	15171.6	0.617654
$846$	6090.55	0.247515
$847$	25106.6	1.01851
$848$	33217.3	1.34515
$849$	−1570.84	−0.0634997
$850$	−14339.3	−0.578630
$851$	−20953.2	−0.844027
$852$	8368.47	0.336501
$853$	−18315.0	−0.735163	−0.367581	−	0.929991i	$-0.619814\pi$
$853$	−18315.0	−0.735163	−0.367581	+	0.929991i	$0.619814\pi$
$854$	81601.7	3.26973
$855$	6080.09	0.243199
$856$	−33447.7	−1.33554
$857$	−13738.7	−0.547615	−0.273807	−	0.961785i	$-0.588283\pi$
$857$	−13738.7	−0.547615	−0.273807	+	0.961785i	$0.588283\pi$
$858$	−5313.12	−0.211407
$859$	9424.20	0.374330	0.187165	−	0.982328i	$-0.440070\pi$
$859$	9424.20	0.374330	0.187165	+	0.982328i	$0.440070\pi$
$860$	16497.4	0.654134
$861$	4974.82	0.196912
$862$	−24499.5	−0.968048
$863$	−40557.8	−1.59977	−0.799887	−	0.600151i	$-0.795107\pi$
$863$	−40557.8	−1.59977	−0.799887	+	0.600151i	$0.795107\pi$
$864$	−339.365	−0.0133628
$865$	−16950.0	−0.666263
$866$	38890.7	1.52605
$867$	−229.191	−0.00897778
$868$	−97040.4	−3.79466
$869$	21869.2	0.853698
$870$	−8229.48	−0.320696
$871$	47191.1	1.83583
$872$	52556.2	2.04103
$873$	10317.9	0.400010
$874$	12691.4	0.491182
$875$	46135.2	1.78246
$876$	14354.4	0.553642
$877$	−12966.5	−0.499258	−0.249629	−	0.968342i	$-0.580309\pi$
$877$	−12966.5	−0.499258	−0.249629	+	0.968342i	$0.580309\pi$
$878$	−54172.2	−2.08226
$879$	−2669.91	−0.102450
$880$	13711.5	0.525243
$881$	−15640.0	−0.598098	−0.299049	−	0.954238i	$-0.596669\pi$
$881$	−15640.0	−0.598098	−0.299049	+	0.954238i	$0.596669\pi$
$882$	74723.8	2.85270
$883$	−10326.2	−0.393548	−0.196774	−	0.980449i	$-0.563047\pi$
$883$	−10326.2	−0.393548	−0.196774	+	0.980449i	$0.563047\pi$
$884$	72078.9	2.74239
$885$	4024.65	0.152867
$886$	−44974.4	−1.70535
$887$	−2173.72	−0.0822846	−0.0411423	−	0.999153i	$-0.513100\pi$
$887$	−2173.72	−0.0822846	−0.0411423	+	0.999153i	$0.513100\pi$
$888$	−6284.81	−0.237505
$889$	−34651.3	−1.30727
$890$	−20378.4	−0.767513
$891$	15260.2	0.573776
$892$	54045.0	2.02866
$893$	−1177.58	−0.0441280
$894$	3017.57	0.112889
$895$	−3406.46	−0.127224
$896$	−75205.2	−2.80405
$897$	4978.10	0.185300
$898$	49444.3	1.83739
$899$	−46574.9	−1.72787
$900$	17203.1	0.637152
$901$	−36051.4	−1.33301
$902$	−23223.0	−0.857252
$903$	2628.60	0.0968706
$904$	−27899.9	−1.02648
$905$	−7967.24	−0.292641
$906$	−7919.85	−0.290419
$907$	29981.6	1.09760	0.548800	−	0.835954i	$-0.315085\pi$
$907$	29981.6	1.09760	0.548800	+	0.835954i	$0.315085\pi$
$908$	−64268.7	−2.34893
$909$	29368.6	1.07161
$910$	−84901.3	−3.09280
$911$	−725.040	−0.0263684	−0.0131842	−	0.999913i	$-0.504197\pi$
$911$	−725.040	−0.0263684	−0.0131842	+	0.999913i	$0.504197\pi$
$912$	1295.90	0.0470521
$913$	15674.5	0.568183
$914$	14672.3	0.530980
$915$	−3919.80	−0.141623
$916$	−15588.5	−0.562291
$917$	−3167.54	−0.114069
$918$	14715.4	0.529065
$919$	−1750.94	−0.0628488	−0.0314244	−	0.999506i	$-0.510004\pi$
$919$	−1750.94	−0.0628488	−0.0314244	+	0.999506i	$0.510004\pi$
$920$	−37737.8	−1.35237
$921$	4922.39	0.176111
$922$	−4696.63	−0.167761
$923$	41436.9	1.47769
$924$	8520.46	0.303358
$925$	8219.48	0.292168
$926$	21369.9	0.758378
$927$	23295.5	0.825376
$928$	1915.03	0.0677413
$929$	5500.62	0.194262	0.0971311	−	0.995272i	$-0.469033\pi$
$929$	5500.62	0.194262	0.0971311	+	0.995272i	$0.469033\pi$
$930$	6977.16	0.246011
$931$	−14447.5	−0.508592
$932$	−19770.0	−0.694838
$933$	−7093.28	−0.248900
$934$	−39921.0	−1.39856
$935$	−14881.3	−0.520504
$936$	−65132.1	−2.27448
$937$	−55288.1	−1.92762	−0.963811	−	0.266586i	$-0.914104\pi$
$937$	−55288.1	−1.92762	−0.963811	+	0.266586i	$0.914104\pi$
$938$	−113275.	−3.94303
$939$	−3587.37	−0.124675
$940$	6958.42	0.241445
$941$	47729.6	1.65350	0.826748	−	0.562573i	$-0.190188\pi$
$941$	47729.6	1.65350	0.826748	+	0.562573i	$0.190188\pi$
$942$	9551.86	0.330378
$943$	21758.6	0.751388
$944$	−37417.9	−1.29010
$945$	−11580.3	−0.398631
$946$	−12270.6	−0.421724
$947$	−27191.6	−0.933061	−0.466530	−	0.884505i	$-0.654496\pi$
$947$	−27191.6	−0.933061	−0.466530	+	0.884505i	$0.654496\pi$
$948$	−12213.5	−0.418434
$949$	71076.5	2.43124
$950$	−4978.55	−0.170027
$951$	3301.17	0.112563
$952$	−87062.4	−2.96398
$953$	2941.45	0.0999822	0.0499911	−	0.998750i	$-0.484081\pi$
$953$	2941.45	0.0999822	0.0499911	+	0.998750i	$0.484081\pi$
$954$	64738.2	2.19704
$955$	−33718.3	−1.14251
$956$	10427.5	0.352770
$957$	4089.42	0.138132
$958$	63393.8	2.13795
$959$	16494.5	0.555405
$960$	3517.32	0.118251
$961$	9696.35	0.325479
$962$	−61842.2	−2.07263
$963$	−22191.3	−0.742580
$964$	−67222.1	−2.24593
$965$	−42998.7	−1.43438
$966$	−11949.2	−0.397990
$967$	45733.8	1.52089	0.760445	−	0.649402i	$-0.224981\pi$
$967$	45733.8	1.52089	0.760445	+	0.649402i	$0.224981\pi$
$968$	32937.0	1.09363
$969$	−1406.46	−0.0466276
$970$	17644.4	0.584050
$971$	57658.9	1.90562	0.952812	−	0.303561i	$-0.0981757\pi$
$971$	57658.9	1.90562	0.952812	+	0.303561i	$0.0981757\pi$
$972$	−26581.5	−0.877162
$973$	1268.74	0.0418025
$974$	54053.7	1.77823
$975$	−1952.80	−0.0641431
$976$	36443.1	1.19520
$977$	46156.4	1.51144	0.755718	−	0.654897i	$-0.227288\pi$
$977$	46156.4	1.51144	0.755718	+	0.654897i	$0.227288\pi$
$978$	−370.663	−0.0121191
$979$	10126.5	0.330587
$980$	85371.5	2.78275
$981$	34869.1	1.13485
$982$	71859.3	2.33516
$983$	10931.6	0.354694	0.177347	−	0.984148i	$-0.443248\pi$
$983$	10931.6	0.354694	0.177347	+	0.984148i	$0.443248\pi$
$984$	6526.38	0.211437
$985$	34026.3	1.10068
$986$	−83038.9	−2.68205
$987$	1108.71	0.0357556
$988$	25025.4	0.805835
$989$	11496.8	0.369644
$990$	26722.7	0.857881
$991$	−15279.8	−0.489788	−0.244894	−	0.969550i	$-0.578753\pi$
$991$	−15279.8	−0.489788	−0.244894	+	0.969550i	$0.578753\pi$
$992$	−1623.61	−0.0519654
$993$	2958.75	0.0945549
$994$	−99462.9	−3.17382
$995$	−12518.6	−0.398860
$996$	−8753.88	−0.278491
$997$	−7648.81	−0.242969	−0.121485	−	0.992593i	$-0.538766\pi$
$997$	−7648.81	−0.242969	−0.121485	+	0.992593i	$0.538766\pi$
$998$	37065.8	1.17565
$999$	−8435.07	−0.267141

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	47.4.a.a.1.1	✓	3
3.2	odd	2		423.4.a.b.1.3		3
4.3	odd	2		752.4.a.c.1.2		3
5.4	even	2		1175.4.a.a.1.3		3
7.6	odd	2		2303.4.a.a.1.1		3
47.46	odd	2		2209.4.a.a.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
47.4.a.a.1.1	✓	3	1.1	even	1	trivial
423.4.a.b.1.3		3	3.2	odd	2
752.4.a.c.1.2		3	4.3	odd	2
1175.4.a.a.1.3		3	5.4	even	2
2209.4.a.a.1.1		3	47.46	odd	2
2303.4.a.a.1.1		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 47 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	47.a (trivial)

\(f(q)\)	\(=\)	\(q-4.90952 q^{2} -0.777884 q^{3} +16.1033 q^{4} +9.19383 q^{5} +3.81903 q^{6} -30.3255 q^{7} -39.7835 q^{8} -26.3949 q^{9} +O(q^{10})\)	\(q-4.90952 q^{2} -0.777884 q^{3} +16.1033 q^{4} +9.19383 q^{5} +3.81903 q^{6} -30.3255 q^{7} -39.7835 q^{8} -26.3949 q^{9} -45.1373 q^{10} +22.4298 q^{11} -12.5265 q^{12} -62.0257 q^{13} +148.883 q^{14} -7.15173 q^{15} +66.4910 q^{16} -72.1639 q^{17} +129.586 q^{18} -25.0550 q^{19} +148.051 q^{20} +23.5897 q^{21} -110.119 q^{22} +103.176 q^{23} +30.9469 q^{24} -40.4735 q^{25} +304.516 q^{26} +41.5350 q^{27} -488.341 q^{28} -234.381 q^{29} +35.1115 q^{30} +198.714 q^{31} -8.17058 q^{32} -17.4477 q^{33} +354.290 q^{34} -278.807 q^{35} -425.046 q^{36} -203.083 q^{37} +123.008 q^{38} +48.2488 q^{39} -365.763 q^{40} +210.889 q^{41} -115.814 q^{42} +111.430 q^{43} +361.194 q^{44} -242.670 q^{45} -506.542 q^{46} +47.0000 q^{47} -51.7223 q^{48} +576.634 q^{49} +198.705 q^{50} +56.1351 q^{51} -998.822 q^{52} +499.576 q^{53} -203.917 q^{54} +206.215 q^{55} +1206.45 q^{56} +19.4898 q^{57} +1150.70 q^{58} -562.752 q^{59} -115.167 q^{60} +548.091 q^{61} -975.591 q^{62} +800.437 q^{63} -491.814 q^{64} -570.254 q^{65} +85.6600 q^{66} -760.831 q^{67} -1162.08 q^{68} -80.2586 q^{69} +1368.81 q^{70} -668.059 q^{71} +1050.08 q^{72} -1145.92 q^{73} +997.040 q^{74} +31.4836 q^{75} -403.469 q^{76} -680.193 q^{77} -236.878 q^{78} +975.010 q^{79} +611.307 q^{80} +680.353 q^{81} -1035.37 q^{82} +698.827 q^{83} +379.873 q^{84} -663.463 q^{85} -547.067 q^{86} +182.321 q^{87} -892.335 q^{88} +451.477 q^{89} +1191.39 q^{90} +1880.96 q^{91} +1661.47 q^{92} -154.577 q^{93} -230.747 q^{94} -230.351 q^{95} +6.35576 q^{96} -390.906 q^{97} -2830.99 q^{98} -592.031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9}+O(q^{10}) \) 3 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 - 6 * q^5 - 8 * q^6 - 45 * q^7 - 39 * q^8 - 42 * q^9	\( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9} - 40 q^{10} + 2 q^{11} + 12 q^{12} - 80 q^{13} + 162 q^{14} + 14 q^{15} + 89 q^{16} - 39 q^{17} + 181 q^{18} - 24 q^{19} + 232 q^{20} + 24 q^{21} - 14 q^{22} + 120 q^{23} + 192 q^{24} - 171 q^{25} + 316 q^{26} + 64 q^{27} - 408 q^{28} - 184 q^{29} + 116 q^{30} - 4 q^{31} - 7 q^{32} - 208 q^{33} + 218 q^{34} - 156 q^{35} - 343 q^{36} - 589 q^{37} + 42 q^{38} + 60 q^{39} - 432 q^{40} - 92 q^{41} - 54 q^{42} - 250 q^{43} + 466 q^{44} - 78 q^{45} - 816 q^{46} + 141 q^{47} - 120 q^{48} + 30 q^{49} + 137 q^{50} + 317 q^{51} - 900 q^{52} + 459 q^{53} + 106 q^{54} + 448 q^{55} + 1032 q^{56} + 216 q^{57} + 684 q^{58} + 579 q^{59} - 240 q^{60} + 267 q^{61} - 244 q^{62} + 1044 q^{63} - 87 q^{64} - 424 q^{65} + 16 q^{66} - 540 q^{67} - 1334 q^{68} + 642 q^{69} + 1236 q^{70} + 749 q^{71} + 357 q^{72} - 1924 q^{73} + 950 q^{74} + 473 q^{75} - 402 q^{76} - 288 q^{77} - 152 q^{78} + 805 q^{79} + 448 q^{80} + 291 q^{81} - 938 q^{82} + 712 q^{83} + 372 q^{84} - 1038 q^{85} - 1294 q^{86} + 1216 q^{87} - 2190 q^{88} + 835 q^{89} + 764 q^{90} + 2040 q^{91} + 1596 q^{92} - 1500 q^{93} - 235 q^{94} - 312 q^{95} - 1432 q^{96} - 2243 q^{97} - 2989 q^{98} + 554 q^{99}+O(q^{100}) \) 3 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 - 6 * q^5 - 8 * q^6 - 45 * q^7 - 39 * q^8 - 42 * q^9 - 40 * q^10 + 2 * q^11 + 12 * q^12 - 80 * q^13 + 162 * q^14 + 14 * q^15 + 89 * q^16 - 39 * q^17 + 181 * q^18 - 24 * q^19 + 232 * q^20 + 24 * q^21 - 14 * q^22 + 120 * q^23 + 192 * q^24 - 171 * q^25 + 316 * q^26 + 64 * q^27 - 408 * q^28 - 184 * q^29 + 116 * q^30 - 4 * q^31 - 7 * q^32 - 208 * q^33 + 218 * q^34 - 156 * q^35 - 343 * q^36 - 589 * q^37 + 42 * q^38 + 60 * q^39 - 432 * q^40 - 92 * q^41 - 54 * q^42 - 250 * q^43 + 466 * q^44 - 78 * q^45 - 816 * q^46 + 141 * q^47 - 120 * q^48 + 30 * q^49 + 137 * q^50 + 317 * q^51 - 900 * q^52 + 459 * q^53 + 106 * q^54 + 448 * q^55 + 1032 * q^56 + 216 * q^57 + 684 * q^58 + 579 * q^59 - 240 * q^60 + 267 * q^61 - 244 * q^62 + 1044 * q^63 - 87 * q^64 - 424 * q^65 + 16 * q^66 - 540 * q^67 - 1334 * q^68 + 642 * q^69 + 1236 * q^70 + 749 * q^71 + 357 * q^72 - 1924 * q^73 + 950 * q^74 + 473 * q^75 - 402 * q^76 - 288 * q^77 - 152 * q^78 + 805 * q^79 + 448 * q^80 + 291 * q^81 - 938 * q^82 + 712 * q^83 + 372 * q^84 - 1038 * q^85 - 1294 * q^86 + 1216 * q^87 - 2190 * q^88 + 835 * q^89 + 764 * q^90 + 2040 * q^91 + 1596 * q^92 - 1500 * q^93 - 235 * q^94 - 312 * q^95 - 1432 * q^96 - 2243 * q^97 - 2989 * q^98 + 554 * q^99

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