LMFDB - Embedded newform 47.4.a.a.1.3

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			1.3
Root			$1.43163$ 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)

$f(q)$	$=$	$q+1.51882 q^{2} -5.95044 q^{3} -5.69320 q^{4} -6.17438 q^{5} -9.03763 q^{6} -3.35636 q^{7} -20.7975 q^{8} +8.40778 q^{9} +O(q^{10})$	\(q+1.51882 q^{2} -5.95044 q^{3} -5.69320 q^{4} -6.17438 q^{5} -9.03763 q^{6} -3.35636 q^{7} -20.7975 q^{8} +8.40778 q^{9} -9.37775 q^{10} +20.2172 q^{11} +33.8770 q^{12} -5.57597 q^{13} -5.09770 q^{14} +36.7403 q^{15} +13.9580 q^{16} -26.5077 q^{17} +12.7699 q^{18} -25.3539 q^{19} +35.1519 q^{20} +19.9718 q^{21} +30.7062 q^{22} -90.2723 q^{23} +123.754 q^{24} -86.8770 q^{25} -8.46888 q^{26} +110.632 q^{27} +19.1084 q^{28} -123.275 q^{29} +55.8018 q^{30} +129.587 q^{31} +187.579 q^{32} -120.301 q^{33} -40.2604 q^{34} +20.7234 q^{35} -47.8671 q^{36} -213.578 q^{37} -38.5079 q^{38} +33.1795 q^{39} +128.411 q^{40} -124.700 q^{41} +30.3336 q^{42} -424.723 q^{43} -115.100 q^{44} -51.9128 q^{45} -137.107 q^{46} +47.0000 q^{47} -83.0566 q^{48} -331.735 q^{49} -131.950 q^{50} +157.733 q^{51} +31.7451 q^{52} +361.680 q^{53} +168.030 q^{54} -124.828 q^{55} +69.8038 q^{56} +150.867 q^{57} -187.233 q^{58} +836.472 q^{59} -209.170 q^{60} -194.845 q^{61} +196.818 q^{62} -28.2195 q^{63} +173.234 q^{64} +34.4282 q^{65} -182.715 q^{66} +902.163 q^{67} +150.914 q^{68} +537.160 q^{69} +31.4751 q^{70} +690.711 q^{71} -174.860 q^{72} -698.209 q^{73} -324.386 q^{74} +516.957 q^{75} +144.345 q^{76} -67.8561 q^{77} +50.3936 q^{78} -449.718 q^{79} -86.1823 q^{80} -885.319 q^{81} -189.396 q^{82} -543.425 q^{83} -113.704 q^{84} +163.669 q^{85} -645.076 q^{86} +733.543 q^{87} -420.465 q^{88} +725.592 q^{89} -78.8461 q^{90} +18.7150 q^{91} +513.938 q^{92} -771.099 q^{93} +71.3844 q^{94} +156.545 q^{95} -1116.18 q^{96} -214.741 q^{97} -503.844 q^{98} +169.981 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.51882	0.536983	0.268491	−	0.963282i	$-0.413475\pi$
$2$	1.51882	0.536983	0.268491	+	0.963282i	$0.413475\pi$
$3$	−5.95044	−1.14516	−0.572582	−	0.819848i	$-0.694058\pi$
$3$	−5.95044	−1.14516	−0.572582	+	0.819848i	$0.694058\pi$
$4$	−5.69320	−0.711649
$5$	−6.17438	−0.552253	−0.276127	−	0.961121i	$-0.589051\pi$
$5$	−6.17438	−0.552253	−0.276127	+	0.961121i	$0.589051\pi$
$6$	−9.03763	−0.614933
$7$	−3.35636	−0.181226	−0.0906132	−	0.995886i	$-0.528883\pi$
$7$	−3.35636	−0.181226	−0.0906132	+	0.995886i	$0.528883\pi$
$8$	−20.7975	−0.919126
$9$	8.40778	0.311399
$10$	−9.37775	−0.296550
$11$	20.2172	0.554155	0.277077	−	0.960848i	$-0.410634\pi$
$11$	20.2172	0.554155	0.277077	+	0.960848i	$0.410634\pi$
$12$	33.8770	0.814955
$13$	−5.57597	−0.118961	−0.0594807	−	0.998229i	$-0.518944\pi$
$13$	−5.57597	−0.118961	−0.0594807	+	0.998229i	$0.518944\pi$
$14$	−5.09770	−0.0973155
$15$	36.7403	0.632420
$16$	13.9580	0.218094
$17$	−26.5077	−0.378180	−0.189090	−	0.981960i	$-0.560554\pi$
$17$	−26.5077	−0.378180	−0.189090	+	0.981960i	$0.560554\pi$
$18$	12.7699	0.167216
$19$	−25.3539	−0.306136	−0.153068	−	0.988216i	$-0.548915\pi$
$19$	−25.3539	−0.306136	−0.153068	+	0.988216i	$0.548915\pi$
$20$	35.1519	0.393011
$21$	19.9718	0.207534
$22$	30.7062	0.297572
$23$	−90.2723	−0.818395	−0.409197	−	0.912446i	$-0.634191\pi$
$23$	−90.2723	−0.818395	−0.409197	+	0.912446i	$0.634191\pi$
$24$	123.754	1.05255
$25$	−86.8770	−0.695016
$26$	−8.46888	−0.0638802
$27$	110.632	0.788560
$28$	19.1084	0.128970
$29$	−123.275	−0.789368	−0.394684	−	0.918817i	$-0.629146\pi$
$29$	−123.275	−0.789368	−0.394684	+	0.918817i	$0.629146\pi$
$30$	55.8018	0.339599
$31$	129.587	0.750789	0.375395	−	0.926865i	$-0.377507\pi$
$31$	129.587	0.750789	0.375395	+	0.926865i	$0.377507\pi$
$32$	187.579	1.03624
$33$	−120.301	−0.634598
$34$	−40.2604	−0.203076
$35$	20.7234	0.100083
$36$	−47.8671	−0.221607
$37$	−213.578	−0.948972	−0.474486	−	0.880263i	$-0.657366\pi$
$37$	−213.578	−0.948972	−0.474486	+	0.880263i	$0.657366\pi$
$38$	−38.5079	−0.164390
$39$	33.1795	0.136230
$40$	128.411	0.507591
$41$	−124.700	−0.474995	−0.237498	−	0.971388i	$-0.576327\pi$
$41$	−124.700	−0.474995	−0.237498	+	0.971388i	$0.576327\pi$
$42$	30.3336	0.111442
$43$	−424.723	−1.50627	−0.753135	−	0.657866i	$-0.771459\pi$
$43$	−424.723	−1.50627	−0.753135	+	0.657866i	$0.771459\pi$
$44$	−115.100	−0.394364
$45$	−51.9128	−0.171971
$46$	−137.107	−0.439464
$47$	47.0000	0.145865
$47$	47.0000	0.145865
$48$	−83.0566	−0.249754
$49$	−331.735	−0.967157
$50$	−131.950	−0.373212
$51$	157.733	0.433078
$52$	31.7451	0.0846588
$53$	361.680	0.937370	0.468685	−	0.883365i	$-0.344728\pi$
$53$	361.680	0.937370	0.468685	+	0.883365i	$0.344728\pi$
$54$	168.030	0.423443
$55$	−124.828	−0.306034
$56$	69.8038	0.166570
$57$	150.867	0.350576
$58$	−187.233	−0.423877
$59$	836.472	1.84575	0.922876	−	0.385097i	$-0.125832\pi$
$59$	836.472	1.84575	0.922876	+	0.385097i	$0.125832\pi$
$60$	−209.170	−0.450062
$61$	−194.845	−0.408973	−0.204486	−	0.978869i	$-0.565552\pi$
$61$	−194.845	−0.408973	−0.204486	+	0.978869i	$0.565552\pi$
$62$	196.818	0.403161
$63$	−28.2195	−0.0564338
$64$	173.234	0.338348
$65$	34.4282	0.0656968
$66$	−182.715	−0.340768
$67$	902.163	1.64503	0.822513	−	0.568746i	$-0.192571\pi$
$67$	902.163	1.64503	0.822513	+	0.568746i	$0.192571\pi$
$68$	150.914	0.269132
$69$	537.160	0.937196
$70$	31.4751	0.0537428
$71$	690.711	1.15454	0.577270	−	0.816553i	$-0.304118\pi$
$71$	690.711	1.15454	0.577270	+	0.816553i	$0.304118\pi$
$72$	−174.860	−0.286215
$73$	−698.209	−1.11944	−0.559720	−	0.828682i	$-0.689091\pi$
$73$	−698.209	−1.11944	−0.559720	+	0.828682i	$0.689091\pi$
$74$	−324.386	−0.509582
$75$	516.957	0.795907
$76$	144.345	0.217862
$77$	−67.8561	−0.100427
$78$	50.3936	0.0731533
$79$	−449.718	−0.640471	−0.320235	−	0.947338i	$-0.603762\pi$
$79$	−449.718	−0.640471	−0.320235	+	0.947338i	$0.603762\pi$
$80$	−86.1823	−0.120443
$81$	−885.319	−1.21443
$82$	−189.396	−0.255064
$83$	−543.425	−0.718659	−0.359329	−	0.933211i	$-0.616995\pi$
$83$	−543.425	−0.718659	−0.359329	+	0.933211i	$0.616995\pi$
$84$	−113.704	−0.147691
$85$	163.669	0.208851
$86$	−645.076	−0.808841
$87$	733.543	0.903955
$88$	−420.465	−0.509338
$89$	725.592	0.864187	0.432093	−	0.901829i	$-0.357775\pi$
$89$	725.592	0.864187	0.432093	+	0.901829i	$0.357775\pi$
$90$	−78.8461	−0.0923456
$91$	18.7150	0.0215589
$92$	513.938	0.582410
$93$	−771.099	−0.859776
$94$	71.3844	0.0783270
$95$	156.545	0.169065
$96$	−1116.18	−1.18666
$97$	−214.741	−0.224780	−0.112390	−	0.993664i	$-0.535851\pi$
$97$	−214.741	−0.224780	−0.112390	+	0.993664i	$0.535851\pi$
$98$	−503.844	−0.519347
$99$	169.981	0.172563
$100$	494.608	0.494608
$101$	1390.95	1.37034	0.685171	−	0.728382i	$-0.259727\pi$
$101$	1390.95	1.37034	0.685171	+	0.728382i	$0.259727\pi$
$102$	239.567	0.232556
$103$	−1545.53	−1.47850	−0.739249	−	0.673432i	$-0.764819\pi$
$103$	−1545.53	−1.47850	−0.739249	+	0.673432i	$0.764819\pi$
$104$	115.966	0.109340
$105$	−123.314	−0.114611
$106$	549.326	0.503351
$107$	−1327.49	−1.19938	−0.599688	−	0.800234i	$-0.704709\pi$
$107$	−1327.49	−1.19938	−0.599688	+	0.800234i	$0.704709\pi$
$108$	−629.849	−0.561179
$109$	−973.505	−0.855457	−0.427729	−	0.903907i	$-0.640686\pi$
$109$	−973.505	−0.855457	−0.427729	+	0.903907i	$0.640686\pi$
$110$	−189.591	−0.164335
$111$	1270.88	1.08673
$112$	−46.8482	−0.0395245
$113$	−365.992	−0.304687	−0.152343	−	0.988328i	$-0.548682\pi$
$113$	−365.992	−0.304687	−0.152343	+	0.988328i	$0.548682\pi$
$114$	229.139	0.188253
$115$	557.375	0.451961
$116$	701.831	0.561753
$117$	−46.8816	−0.0370445
$118$	1270.45	0.991137
$119$	88.9695	0.0685363
$120$	−764.105	−0.581274
$121$	−922.266	−0.692912
$122$	−295.934	−0.219611
$123$	742.018	0.543947
$124$	−737.763	−0.534299
$125$	1308.21	0.936078
$126$	−42.8603	−0.0303040
$127$	−1034.31	−0.722682	−0.361341	−	0.932434i	$-0.617681\pi$
$127$	−1034.31	−0.722682	−0.361341	+	0.932434i	$0.617681\pi$
$128$	−1237.52	−0.854552
$129$	2527.29	1.72493
$130$	52.2901	0.0352780
$131$	−98.9237	−0.0659771	−0.0329885	−	0.999456i	$-0.510502\pi$
$131$	−98.9237	−0.0659771	−0.0329885	+	0.999456i	$0.510502\pi$
$132$	684.898	0.451611
$133$	85.0968	0.0554799
$134$	1370.22	0.883351
$135$	−683.084	−0.435485
$136$	551.293	0.347596
$137$	−1607.67	−1.00257	−0.501286	−	0.865282i	$-0.667140\pi$
$137$	−1607.67	−1.00257	−0.501286	+	0.865282i	$0.667140\pi$
$138$	815.848	0.503258
$139$	3175.56	1.93775	0.968875	−	0.247549i	$-0.0796251\pi$
$139$	3175.56	1.93775	0.968875	+	0.247549i	$0.0796251\pi$
$140$	−117.983	−0.0712239
$141$	−279.671	−0.167039
$142$	1049.06	0.619968
$143$	−112.730	−0.0659230
$144$	117.356	0.0679145
$145$	761.149	0.435931
$146$	−1060.45	−0.601120
$147$	1973.97	1.10755
$148$	1215.94	0.675336
$149$	−1155.85	−0.635509	−0.317754	−	0.948173i	$-0.602929\pi$
$149$	−1155.85	−0.635509	−0.317754	+	0.948173i	$0.602929\pi$
$150$	785.163	0.427389
$151$	−1941.03	−1.04608	−0.523042	−	0.852307i	$-0.675203\pi$
$151$	−1941.03	−1.04608	−0.523042	+	0.852307i	$0.675203\pi$
$152$	527.297	0.281378
$153$	−222.871	−0.117765
$154$	−103.061	−0.0539278
$155$	−800.118	−0.414626
$156$	−188.898	−0.0969481
$157$	−1023.76	−0.520414	−0.260207	−	0.965553i	$-0.583791\pi$
$157$	−1023.76	−0.520414	−0.260207	+	0.965553i	$0.583791\pi$
$158$	−683.038	−0.343922
$159$	−2152.16	−1.07344
$160$	−1158.19	−0.572267
$161$	302.986	0.148315
$162$	−1344.64	−0.652128
$163$	728.517	0.350073	0.175036	−	0.984562i	$-0.443996\pi$
$163$	728.517	0.350073	0.175036	+	0.984562i	$0.443996\pi$
$164$	709.939	0.338030
$165$	742.784	0.350459
$166$	−825.363	−0.385907
$167$	−1594.21	−0.738706	−0.369353	−	0.929289i	$-0.620421\pi$
$167$	−1594.21	−0.738706	−0.369353	+	0.929289i	$0.620421\pi$
$168$	−415.363	−0.190750
$169$	−2165.91	−0.985848
$170$	248.583	0.112150
$171$	−213.170	−0.0953306
$172$	2418.03	1.07194
$173$	−1111.94	−0.488665	−0.244332	−	0.969692i	$-0.578569\pi$
$173$	−1111.94	−0.488665	−0.244332	+	0.969692i	$0.578569\pi$
$174$	1114.12	0.485408
$175$	291.591	0.125955
$176$	282.192	0.120858
$177$	−4977.38	−2.11369
$178$	1102.04	0.464053
$179$	924.477	0.386026	0.193013	−	0.981196i	$-0.438174\pi$
$179$	924.477	0.386026	0.193013	+	0.981196i	$0.438174\pi$
$180$	295.550	0.122383
$181$	1695.96	0.696462	0.348231	−	0.937409i	$-0.386783\pi$
$181$	1695.96	0.696462	0.348231	+	0.937409i	$0.386783\pi$
$182$	28.4246	0.0115768
$183$	1159.41	0.468340
$184$	1877.43	0.752208
$185$	1318.71	0.524073
$186$	−1171.16	−0.461685
$187$	−535.911	−0.209570
$188$	−267.580	−0.103805
$189$	−371.321	−0.142908
$190$	237.763	0.0907848
$191$	−1985.24	−0.752079	−0.376039	−	0.926604i	$-0.622714\pi$
$191$	−1985.24	−0.752079	−0.376039	+	0.926604i	$0.622714\pi$
$192$	−1030.82	−0.387464
$193$	172.859	0.0644696	0.0322348	−	0.999480i	$-0.489738\pi$
$193$	172.859	0.0644696	0.0322348	+	0.999480i	$0.489738\pi$
$194$	−326.152	−0.120703
$195$	−204.863	−0.0752335
$196$	1888.63	0.688277
$197$	18.9634	0.00685829	0.00342915	−	0.999994i	$-0.498908\pi$
$197$	18.9634	0.00685829	0.00342915	+	0.999994i	$0.498908\pi$
$198$	258.171	0.0926636
$199$	2200.21	0.783762	0.391881	−	0.920016i	$-0.371824\pi$
$199$	2200.21	0.783762	0.391881	+	0.920016i	$0.371824\pi$
$200$	1806.82	0.638808
$201$	−5368.27	−1.88382
$202$	2112.60	0.735850
$203$	413.757	0.143054
$204$	−898.003	−0.308200
$205$	769.943	0.262318
$206$	−2347.37	−0.793928
$207$	−758.990	−0.254848
$208$	−77.8297	−0.0259448
$209$	−512.584	−0.169647
$210$	−187.291	−0.0615443
$211$	236.491	0.0771599	0.0385800	−	0.999256i	$-0.487717\pi$
$211$	236.491	0.0771599	0.0385800	+	0.999256i	$0.487717\pi$
$212$	−2059.12	−0.667079
$213$	−4110.04	−1.32214
$214$	−2016.21	−0.644044
$215$	2622.40	0.831843
$216$	−2300.86	−0.724787
$217$	−434.940	−0.136063
$218$	−1478.58	−0.459366
$219$	4154.65	1.28194
$220$	710.673	0.217789
$221$	147.806	0.0449888
$222$	1930.24	0.583555
$223$	1396.42	0.419334	0.209667	−	0.977773i	$-0.432762\pi$
$223$	1396.42	0.419334	0.209667	+	0.977773i	$0.432762\pi$
$224$	−629.584	−0.187794
$225$	−730.443	−0.216428
$226$	−555.874	−0.163612
$227$	3034.35	0.887211	0.443605	−	0.896222i	$-0.353699\pi$
$227$	3034.35	0.887211	0.443605	+	0.896222i	$0.353699\pi$
$228$	−858.915	−0.249487
$229$	4938.80	1.42517	0.712587	−	0.701584i	$-0.247523\pi$
$229$	4938.80	1.42517	0.712587	+	0.701584i	$0.247523\pi$
$230$	846.551	0.242695
$231$	403.774	0.115006
$232$	2563.81	0.725529
$233$	4759.80	1.33831	0.669153	−	0.743125i	$-0.266657\pi$
$233$	4759.80	1.33831	0.669153	+	0.743125i	$0.266657\pi$
$234$	−71.2045	−0.0198922
$235$	−290.196	−0.0805544
$236$	−4762.20	−1.31353
$237$	2676.02	0.733443
$238$	135.128	0.0368028
$239$	3666.84	0.992419	0.496210	−	0.868203i	$-0.334725\pi$
$239$	3666.84	0.992419	0.496210	+	0.868203i	$0.334725\pi$
$240$	512.823	0.137927
$241$	6119.79	1.63573	0.817864	−	0.575412i	$-0.195158\pi$
$241$	6119.79	1.63573	0.817864	+	0.575412i	$0.195158\pi$
$242$	−1400.75	−0.372082
$243$	2280.98	0.602160
$244$	1109.29	0.291045
$245$	2048.26	0.534116
$246$	1126.99	0.292090
$247$	141.373	0.0364183
$248$	−2695.07	−0.690070
$249$	3233.62	0.822982
$250$	1986.93	0.502658
$251$	−4257.22	−1.07057	−0.535285	−	0.844671i	$-0.679796\pi$
$251$	−4257.22	−1.07057	−0.535285	+	0.844671i	$0.679796\pi$
$252$	160.659	0.0401611
$253$	−1825.05	−0.453517
$254$	−1570.93	−0.388068
$255$	−973.901	−0.239169
$256$	−3265.45	−0.797228
$257$	−6251.15	−1.51726	−0.758630	−	0.651522i	$-0.774131\pi$
$257$	−6251.15	−1.51726	−0.758630	+	0.651522i	$0.774131\pi$
$258$	3838.49	0.926255
$259$	716.844	0.171979
$260$	−196.006	−0.0467531
$261$	−1036.47	−0.245809
$262$	−150.247	−0.0354286
$263$	−5008.06	−1.17418	−0.587091	−	0.809521i	$-0.699727\pi$
$263$	−5008.06	−1.17418	−0.587091	+	0.809521i	$0.699727\pi$
$264$	2501.96	0.583276
$265$	−2233.15	−0.517665
$266$	129.247	0.0297918
$267$	−4317.60	−0.989635
$268$	−5136.19	−1.17068
$269$	4565.07	1.03471	0.517356	−	0.855770i	$-0.326916\pi$
$269$	4565.07	1.03471	0.517356	+	0.855770i	$0.326916\pi$
$270$	−1037.48	−0.233848
$271$	3111.82	0.697526	0.348763	−	0.937211i	$-0.386602\pi$
$271$	3111.82	0.697526	0.348763	+	0.937211i	$0.386602\pi$
$272$	−369.996	−0.0824790
$273$	−111.362	−0.0246885
$274$	−2441.75	−0.538364
$275$	−1756.41	−0.385147
$276$	−3058.16	−0.666955
$277$	−4349.10	−0.943364	−0.471682	−	0.881769i	$-0.656353\pi$
$277$	−4349.10	−0.943364	−0.471682	+	0.881769i	$0.656353\pi$
$278$	4823.09	1.04054
$279$	1089.54	0.233795
$280$	−430.995	−0.0919888
$281$	−1640.50	−0.348271	−0.174135	−	0.984722i	$-0.555713\pi$
$281$	−1640.50	−0.348271	−0.174135	+	0.984722i	$0.555713\pi$
$282$	−424.769	−0.0896972
$283$	−6268.38	−1.31667	−0.658333	−	0.752727i	$-0.728738\pi$
$283$	−6268.38	−1.31667	−0.658333	+	0.752727i	$0.728738\pi$
$284$	−3932.36	−0.821628
$285$	−931.510	−0.193607
$286$	−171.217	−0.0353995
$287$	418.537	0.0860817
$288$	1577.13	0.322684
$289$	−4210.34	−0.856980
$290$	1156.05	0.234087
$291$	1277.80	0.257410
$292$	3975.04	0.796649
$293$	4720.69	0.941248	0.470624	−	0.882334i	$-0.344029\pi$
$293$	4720.69	0.941248	0.470624	+	0.882334i	$0.344029\pi$
$294$	2998.10	0.594737
$295$	−5164.69	−1.01932
$296$	4441.88	0.872226
$297$	2236.66	0.436984
$298$	−1755.52	−0.341257
$299$	503.356	0.0973573
$300$	−2943.14	−0.566407
$301$	1425.52	0.272976
$302$	−2948.07	−0.561729
$303$	−8276.76	−1.56927
$304$	−353.891	−0.0667666
$305$	1203.05	0.225856
$306$	−338.500	−0.0632378
$307$	−9475.50	−1.76155	−0.880774	−	0.473536i	$-0.842977\pi$
$307$	−9475.50	−1.76155	−0.880774	+	0.473536i	$0.842977\pi$
$308$	386.318	0.0714692
$309$	9196.57	1.69312
$310$	−1215.23	−0.222647
$311$	7724.72	1.40845	0.704227	−	0.709975i	$-0.251294\pi$
$311$	7724.72	1.40845	0.704227	+	0.709975i	$0.251294\pi$
$312$	−690.050	−0.125213
$313$	−7719.80	−1.39409	−0.697043	−	0.717029i	$-0.745501\pi$
$313$	−7719.80	−1.39409	−0.697043	+	0.717029i	$0.745501\pi$
$314$	−1554.91	−0.279454
$315$	174.238	0.0311657
$316$	2560.33	0.455791
$317$	10756.5	1.90583	0.952914	−	0.303241i	$-0.0980687\pi$
$317$	10756.5	1.90583	0.952914	+	0.303241i	$0.0980687\pi$
$318$	−3268.73	−0.576420
$319$	−2492.28	−0.437432
$320$	−1069.61	−0.186854
$321$	7899.15	1.37348
$322$	460.181	0.0796425
$323$	672.074	0.115775
$324$	5040.30	0.864248
$325$	484.424	0.0826801
$326$	1106.48	0.187983
$327$	5792.79	0.979638
$328$	2593.43	0.436581
$329$	−157.749	−0.0264346
$330$	1128.15	0.188190
$331$	3699.02	0.614250	0.307125	−	0.951669i	$-0.400633\pi$
$331$	3699.02	0.614250	0.307125	+	0.951669i	$0.400633\pi$
$332$	3093.83	0.511433
$333$	−1795.72	−0.295509
$334$	−2421.32	−0.396672
$335$	−5570.30	−0.908471
$336$	278.768	0.0452620
$337$	9917.45	1.60308	0.801540	−	0.597942i	$-0.204015\pi$
$337$	9917.45	1.60308	0.801540	+	0.597942i	$0.204015\pi$
$338$	−3289.62	−0.529384
$339$	2177.81	0.348916
$340$	−931.798	−0.148629
$341$	2619.88	0.416053
$342$	−323.766	−0.0511909
$343$	2264.65	0.356501
$344$	8833.15	1.38445
$345$	−3316.63	−0.517569
$346$	−1688.83	−0.262405
$347$	−7738.48	−1.19719	−0.598593	−	0.801054i	$-0.704273\pi$
$347$	−7738.48	−1.19719	−0.598593	+	0.801054i	$0.704273\pi$
$348$	−4176.21	−0.643299
$349$	−8939.40	−1.37110	−0.685552	−	0.728024i	$-0.740439\pi$
$349$	−8939.40	−1.37110	−0.685552	+	0.728024i	$0.740439\pi$
$350$	442.873	0.0676358
$351$	−616.881	−0.0938082
$352$	3792.32	0.574237
$353$	7812.51	1.17795	0.588977	−	0.808150i	$-0.299531\pi$
$353$	7812.51	1.17795	0.588977	+	0.808150i	$0.299531\pi$
$354$	−7559.73	−1.13501
$355$	−4264.71	−0.637599
$356$	−4130.94	−0.614998
$357$	−529.408	−0.0784852
$358$	1404.11	0.207289
$359$	4671.36	0.686754	0.343377	−	0.939198i	$-0.388429\pi$
$359$	4671.36	0.686754	0.343377	+	0.939198i	$0.388429\pi$
$360$	1079.65	0.158063
$361$	−6216.18	−0.906281
$362$	2575.85	0.373988
$363$	5487.89	0.793498
$364$	−106.548	−0.0153424
$365$	4311.00	0.618214
$366$	1760.94	0.251491
$367$	−2391.88	−0.340204	−0.170102	−	0.985426i	$-0.554410\pi$
$367$	−2391.88	−0.340204	−0.170102	+	0.985426i	$0.554410\pi$
$368$	−1260.03	−0.178487
$369$	−1048.45	−0.147913
$370$	2002.88	0.281418
$371$	−1213.93	−0.169876
$372$	4390.02	0.611859
$373$	13119.4	1.82117	0.910583	−	0.413326i	$-0.135633\pi$
$373$	13119.4	1.82117	0.910583	+	0.413326i	$0.135633\pi$
$374$	−813.950	−0.112536
$375$	−7784.43	−1.07196
$376$	−977.480	−0.134068
$377$	687.380	0.0939042
$378$	−563.968	−0.0767391
$379$	−8968.11	−1.21546	−0.607732	−	0.794142i	$-0.707920\pi$
$379$	−8968.11	−1.21546	−0.607732	+	0.794142i	$0.707920\pi$
$380$	−891.239	−0.120315
$381$	6154.63	0.827589
$382$	−3015.22	−0.403853
$383$	−10643.6	−1.42001	−0.710005	−	0.704197i	$-0.751307\pi$
$383$	−10643.6	−1.42001	−0.710005	+	0.704197i	$0.751307\pi$
$384$	7363.82	0.978602
$385$	418.969	0.0554614
$386$	262.541	0.0346191
$387$	−3570.98	−0.469051
$388$	1222.56	0.159964
$389$	−4554.56	−0.593639	−0.296819	−	0.954934i	$-0.595926\pi$
$389$	−4554.56	−0.593639	−0.296819	+	0.954934i	$0.595926\pi$
$390$	−311.149	−0.0403991
$391$	2392.91	0.309501
$392$	6899.24	0.888939
$393$	588.640	0.0755546
$394$	28.8019	0.00368278
$395$	2776.73	0.353702
$396$	−967.738	−0.122805
$397$	−2676.50	−0.338362	−0.169181	−	0.985585i	$-0.554112\pi$
$397$	−2676.50	−0.338362	−0.169181	+	0.985585i	$0.554112\pi$
$398$	3341.71	0.420867
$399$	−506.364	−0.0635336
$400$	−1212.63	−0.151579
$401$	−2570.96	−0.320169	−0.160084	−	0.987103i	$-0.551177\pi$
$401$	−2570.96	−0.320169	−0.160084	+	0.987103i	$0.551177\pi$
$402$	−8153.42	−1.01158
$403$	−722.572	−0.0893149
$404$	−7918.94	−0.975203
$405$	5466.30	0.670673
$406$	628.420	0.0768177
$407$	−4317.94	−0.525878
$408$	−3280.44	−0.398054
$409$	−2904.50	−0.351145	−0.175572	−	0.984467i	$-0.556178\pi$
$409$	−2904.50	−0.351145	−0.175572	+	0.984467i	$0.556178\pi$
$410$	1169.40	0.140860
$411$	9566.34	1.14811
$412$	8798.99	1.05217
$413$	−2807.50	−0.334499
$414$	−1152.77	−0.136849
$415$	3355.31	0.396882
$416$	−1045.94	−0.123272
$417$	−18896.0	−2.21904
$418$	−778.521	−0.0910974
$419$	−10586.1	−1.23428	−0.617141	−	0.786852i	$-0.711709\pi$
$419$	−10586.1	−1.23428	−0.617141	+	0.786852i	$0.711709\pi$
$420$	702.049	0.0815630
$421$	8155.31	0.944098	0.472049	−	0.881572i	$-0.343514\pi$
$421$	8155.31	0.944098	0.472049	+	0.881572i	$0.343514\pi$
$422$	359.187	0.0414336
$423$	395.166	0.0454223
$424$	−7522.03	−0.861561
$425$	2302.91	0.262842
$426$	−6242.40	−0.709965
$427$	653.969	0.0741166
$428$	7557.65	0.853535
$429$	670.796	0.0754926
$430$	3982.94	0.446685
$431$	170.047	0.0190043	0.00950217	−	0.999955i	$-0.496975\pi$
$431$	170.047	0.0190043	0.00950217	+	0.999955i	$0.496975\pi$
$432$	1544.21	0.171981
$433$	−8139.98	−0.903424	−0.451712	−	0.892164i	$-0.649186\pi$
$433$	−8139.98	−0.903424	−0.451712	+	0.892164i	$0.649186\pi$
$434$	−660.594	−0.0730634
$435$	−4529.17	−0.499212
$436$	5542.35	0.608786
$437$	2288.76	0.250540
$438$	6310.15	0.688381
$439$	−9489.42	−1.03168	−0.515838	−	0.856686i	$-0.672519\pi$
$439$	−9489.42	−1.03168	−0.515838	+	0.856686i	$0.672519\pi$
$440$	2596.11	0.281284
$441$	−2789.15	−0.301172
$442$	224.491	0.0241582
$443$	−8297.13	−0.889862	−0.444931	−	0.895565i	$-0.646772\pi$
$443$	−8297.13	−0.889862	−0.444931	+	0.895565i	$0.646772\pi$
$444$	−7235.39	−0.773370
$445$	−4480.08	−0.477250
$446$	2120.91	0.225175
$447$	6877.81	0.727761
$448$	−581.437	−0.0613176
$449$	3521.42	0.370125	0.185063	−	0.982727i	$-0.440751\pi$
$449$	3521.42	0.370125	0.185063	+	0.982727i	$0.440751\pi$
$450$	−1109.41	−0.116218
$451$	−2521.07	−0.263221
$452$	2083.66	0.216830
$453$	11550.0	1.19794
$454$	4608.62	0.476417
$455$	−115.553	−0.0119060
$456$	−3137.65	−0.322223
$457$	−2750.17	−0.281504	−0.140752	−	0.990045i	$-0.544952\pi$
$457$	−2750.17	−0.281504	−0.140752	+	0.990045i	$0.544952\pi$
$458$	7501.13	0.765294
$459$	−2932.60	−0.298218
$460$	−3173.25	−0.321638
$461$	−4820.72	−0.487035	−0.243518	−	0.969896i	$-0.578301\pi$
$461$	−4820.72	−0.487035	−0.243518	+	0.969896i	$0.578301\pi$
$462$	613.258	0.0617562
$463$	12993.3	1.30422	0.652108	−	0.758126i	$-0.273885\pi$
$463$	12993.3	1.30422	0.652108	+	0.758126i	$0.273885\pi$
$464$	−1720.68	−0.172157
$465$	4761.06	0.474814
$466$	7229.27	0.718647
$467$	2516.52	0.249359	0.124679	−	0.992197i	$-0.460210\pi$
$467$	2516.52	0.249359	0.124679	+	0.992197i	$0.460210\pi$
$468$	266.906	0.0263627
$469$	−3027.99	−0.298122
$470$	−440.754	−0.0432563
$471$	6091.83	0.595959
$472$	−17396.5	−1.69648
$473$	−8586.69	−0.834707
$474$	4064.38	0.393847
$475$	2202.67	0.212770
$476$	−506.521	−0.0487738
$477$	3040.93	0.291896
$478$	5569.26	0.532912
$479$	9282.30	0.885426	0.442713	−	0.896663i	$-0.354016\pi$
$479$	9282.30	0.885426	0.442713	+	0.896663i	$0.354016\pi$
$480$	6891.72	0.655339
$481$	1190.90	0.112891
$482$	9294.84	0.878357
$483$	−1802.90	−0.169845
$484$	5250.64	0.493111
$485$	1325.89	0.124135
$486$	3464.39	0.323350
$487$	10070.8	0.937063	0.468532	−	0.883447i	$-0.344783\pi$
$487$	10070.8	0.937063	0.468532	+	0.883447i	$0.344783\pi$
$488$	4052.28	0.375897
$489$	−4335.00	−0.400890
$490$	3110.93	0.286811
$491$	−8493.09	−0.780627	−0.390313	−	0.920682i	$-0.627633\pi$
$491$	−8493.09	−0.780627	−0.390313	+	0.920682i	$0.627633\pi$
$492$	−4224.45	−0.387100
$493$	3267.75	0.298523
$494$	214.719	0.0195560
$495$	−1049.53	−0.0952987
$496$	1808.78	0.163743
$497$	−2318.28	−0.209233
$498$	4911.28	0.441927
$499$	−11520.7	−1.03354	−0.516769	−	0.856125i	$-0.672866\pi$
$499$	−11520.7	−1.03354	−0.516769	+	0.856125i	$0.672866\pi$
$500$	−7447.89	−0.666160
$501$	9486.27	0.845939
$502$	−6465.93	−0.574878
$503$	15802.2	1.40077	0.700385	−	0.713766i	$-0.253012\pi$
$503$	15802.2	1.40077	0.700385	+	0.713766i	$0.253012\pi$
$504$	586.895	0.0518698
$505$	−8588.25	−0.756776
$506$	−2771.92	−0.243531
$507$	12888.1	1.12896
$508$	5888.56	0.514296
$509$	−18133.1	−1.57905	−0.789525	−	0.613718i	$-0.789673\pi$
$509$	−18133.1	−1.57905	−0.789525	+	0.613718i	$0.789673\pi$
$510$	−1479.18	−0.128430
$511$	2343.44	0.202872
$512$	4940.58	0.426454
$513$	−2804.95	−0.241407
$514$	−9494.35	−0.814742
$515$	9542.67	0.816506
$516$	−14388.3	−1.22754
$517$	950.206	0.0808318
$518$	1088.75	0.0923497
$519$	6616.52	0.559601
$520$	−716.019	−0.0603836
$521$	1617.14	0.135985	0.0679926	−	0.997686i	$-0.478341\pi$
$521$	1617.14	0.135985	0.0679926	+	0.997686i	$0.478341\pi$
$522$	−1574.21	−0.131995
$523$	7545.61	0.630873	0.315436	−	0.948947i	$-0.397849\pi$
$523$	7545.61	0.630873	0.315436	+	0.948947i	$0.397849\pi$
$524$	563.192	0.0469526
$525$	−1735.09	−0.144239
$526$	−7606.32	−0.630516
$527$	−3435.05	−0.283934
$528$	−1679.17	−0.138402
$529$	−4017.91	−0.330230
$530$	−3391.75	−0.277977
$531$	7032.87	0.574766
$532$	−484.473	−0.0394823
$533$	695.322	0.0565061
$534$	−6557.64	−0.531417
$535$	8196.42	0.662359
$536$	−18762.7	−1.51199
$537$	−5501.05	−0.442063
$538$	6933.51	0.555622
$539$	−6706.74	−0.535955
$540$	3888.93	0.309913
$541$	12142.0	0.964923	0.482461	−	0.875917i	$-0.339743\pi$
$541$	12142.0	0.964923	0.482461	+	0.875917i	$0.339743\pi$
$542$	4726.28	0.374560
$543$	−10091.7	−0.797562
$544$	−4972.30	−0.391885
$545$	6010.79	0.472429
$546$	−169.139	−0.0132573
$547$	17852.1	1.39543	0.697715	−	0.716375i	$-0.254200\pi$
$547$	17852.1	1.39543	0.697715	+	0.716375i	$0.254200\pi$
$548$	9152.77	0.713480
$549$	−1638.21	−0.127354
$550$	−2667.66	−0.206817
$551$	3125.51	0.241654
$552$	−11171.6	−0.861401
$553$	1509.41	0.116070
$554$	−6605.48	−0.506570
$555$	−7846.91	−0.600149
$556$	−18079.1	−1.37900
$557$	10350.7	0.787388	0.393694	−	0.919242i	$-0.371197\pi$
$557$	10350.7	0.787388	0.393694	+	0.919242i	$0.371197\pi$
$558$	1654.81	0.125544
$559$	2368.24	0.179188
$560$	289.259	0.0218275
$561$	3188.91	0.239992
$562$	−2491.62	−0.187015
$563$	−12353.6	−0.924761	−0.462380	−	0.886682i	$-0.653005\pi$
$563$	−12353.6	−0.924761	−0.462380	+	0.886682i	$0.653005\pi$
$564$	1592.22	0.118873
$565$	2259.77	0.168264
$566$	−9520.52	−0.707027
$567$	2971.45	0.220087
$568$	−14365.0	−1.06117
$569$	−24894.7	−1.83416	−0.917081	−	0.398700i	$-0.869461\pi$
$569$	−24894.7	−1.83416	−0.917081	+	0.398700i	$0.869461\pi$
$570$	−1414.79	−0.103963
$571$	10062.0	0.737442	0.368721	−	0.929540i	$-0.379796\pi$
$571$	10062.0	0.737442	0.368721	+	0.929540i	$0.379796\pi$
$572$	641.796	0.0469141
$573$	11813.1	0.861253
$574$	635.681	0.0462244
$575$	7842.59	0.568798
$576$	1456.52	0.105361
$577$	4187.40	0.302121	0.151060	−	0.988525i	$-0.451731\pi$
$577$	4187.40	0.302121	0.151060	+	0.988525i	$0.451731\pi$
$578$	−6394.74	−0.460183
$579$	−1028.59	−0.0738283
$580$	−4333.37	−0.310230
$581$	1823.93	0.130240
$582$	1940.75	0.138224
$583$	7312.14	0.519448
$584$	14521.0	1.02891
$585$	289.465	0.0204579
$586$	7169.86	0.505434
$587$	−4333.47	−0.304704	−0.152352	−	0.988326i	$-0.548685\pi$
$587$	−4333.47	−0.304704	−0.152352	+	0.988326i	$0.548685\pi$
$588$	−11238.2	−0.788189
$589$	−3285.53	−0.229844
$590$	−7844.22	−0.547359
$591$	−112.840	−0.00785386
$592$	−2981.13	−0.206966
$593$	−6152.44	−0.426055	−0.213027	−	0.977046i	$-0.568332\pi$
$593$	−6152.44	−0.426055	−0.213027	+	0.977046i	$0.568332\pi$
$594$	3397.08	0.234653
$595$	−549.331	−0.0378494
$596$	6580.47	0.452259
$597$	−13092.2	−0.897536
$598$	764.506	0.0522792
$599$	−7257.67	−0.495059	−0.247529	−	0.968880i	$-0.579619\pi$
$599$	−7257.67	−0.495059	−0.247529	+	0.968880i	$0.579619\pi$
$600$	−10751.4	−0.731539
$601$	−15872.7	−1.07731	−0.538654	−	0.842527i	$-0.681067\pi$
$601$	−15872.7	−1.07731	−0.538654	+	0.842527i	$0.681067\pi$
$602$	2165.11	0.146583
$603$	7585.19	0.512260
$604$	11050.7	0.744445
$605$	5694.42	0.382663
$606$	−12570.9	−0.842669
$607$	−15068.0	−1.00757	−0.503783	−	0.863830i	$-0.668059\pi$
$607$	−15068.0	−1.00757	−0.503783	+	0.863830i	$0.668059\pi$
$608$	−4755.87	−0.317230
$609$	−2462.04	−0.163821
$610$	1827.21	0.121281
$611$	−262.071	−0.0173523
$612$	1268.85	0.0838075
$613$	12024.1	0.792252	0.396126	−	0.918196i	$-0.370354\pi$
$613$	12024.1	0.792252	0.396126	+	0.918196i	$0.370354\pi$
$614$	−14391.6	−0.945921
$615$	−4581.50	−0.300397
$616$	1411.23	0.0923056
$617$	4814.15	0.314117	0.157059	−	0.987589i	$-0.449799\pi$
$617$	4814.15	0.314117	0.157059	+	0.987589i	$0.449799\pi$
$618$	13967.9	0.909178
$619$	−9575.51	−0.621764	−0.310882	−	0.950448i	$-0.600624\pi$
$619$	−9575.51	−0.621764	−0.310882	+	0.950448i	$0.600624\pi$
$620$	4555.23	0.295068
$621$	−9987.00	−0.645354
$622$	11732.4	0.756315
$623$	−2435.35	−0.156613
$624$	463.121	0.0297110
$625$	2782.25	0.178064
$626$	−11725.0	−0.748600
$627$	3050.10	0.194273
$628$	5828.47	0.370353
$629$	5661.46	0.358883
$630$	264.636	0.0167355
$631$	21583.8	1.36171	0.680855	−	0.732418i	$-0.261608\pi$
$631$	21583.8	1.36171	0.680855	+	0.732418i	$0.261608\pi$
$632$	9352.98	0.588673
$633$	−1407.23	−0.0883607
$634$	16337.2	1.02340
$635$	6386.25	0.399103
$636$	12252.7	0.763914
$637$	1849.75	0.115054
$638$	−3785.31	−0.234893
$639$	5807.35	0.359523
$640$	7640.94	0.471929
$641$	−11164.6	−0.687951	−0.343975	−	0.938979i	$-0.611774\pi$
$641$	−11164.6	−0.687951	−0.343975	+	0.938979i	$0.611774\pi$
$642$	11997.4	0.737536
$643$	8679.03	0.532298	0.266149	−	0.963932i	$-0.414249\pi$
$643$	8679.03	0.532298	0.266149	+	0.963932i	$0.414249\pi$
$644$	−1724.96	−0.105548
$645$	−15604.4	−0.952596
$646$	1020.76	0.0621690
$647$	9386.78	0.570374	0.285187	−	0.958472i	$-0.407944\pi$
$647$	9386.78	0.570374	0.285187	+	0.958472i	$0.407944\pi$
$648$	18412.4	1.11621
$649$	16911.1	1.02283
$650$	735.752	0.0443978
$651$	2588.08	0.155814
$652$	−4147.59	−0.249129
$653$	25776.8	1.54475	0.772376	−	0.635166i	$-0.219068\pi$
$653$	25776.8	1.54475	0.772376	+	0.635166i	$0.219068\pi$
$654$	8798.18	0.526049
$655$	610.792	0.0364361
$656$	−1740.56	−0.103594
$657$	−5870.38	−0.348593
$658$	−239.592	−0.0141949
$659$	−176.671	−0.0104433	−0.00522164	−	0.999986i	$-0.501662\pi$
$659$	−176.671	−0.0104433	−0.00522164	+	0.999986i	$0.501662\pi$
$660$	−4228.82	−0.249404
$661$	443.366	0.0260892	0.0130446	−	0.999915i	$-0.495848\pi$
$661$	443.366	0.0260892	0.0130446	+	0.999915i	$0.495848\pi$
$662$	5618.14	0.329842
$663$	−879.514	−0.0515196
$664$	11301.9	0.660538
$665$	−525.420	−0.0306390
$666$	−2727.36	−0.158683
$667$	11128.4	0.646014
$668$	9076.16	0.525700
$669$	−8309.34	−0.480206
$670$	−8460.26	−0.487833
$671$	−3939.21	−0.226634
$672$	3746.30	0.215055
$673$	−33702.8	−1.93038	−0.965192	−	0.261544i	$-0.915768\pi$
$673$	−33702.8	−1.93038	−0.965192	+	0.261544i	$0.915768\pi$
$674$	15062.8	0.860826
$675$	−9611.38	−0.548062
$676$	12330.9	0.701578
$677$	7298.49	0.414334	0.207167	−	0.978306i	$-0.433576\pi$
$677$	7298.49	0.414334	0.207167	+	0.978306i	$0.433576\pi$
$678$	3307.70	0.187362
$679$	720.748	0.0407360
$680$	−3403.89	−0.191961
$681$	−18055.7	−1.01600
$682$	3979.11	0.223414
$683$	27754.4	1.55490	0.777448	−	0.628948i	$-0.216514\pi$
$683$	27754.4	1.55490	0.777448	+	0.628948i	$0.216514\pi$
$684$	1213.62	0.0678419
$685$	9926.35	0.553674
$686$	3439.59	0.191435
$687$	−29388.0	−1.63206
$688$	−5928.30	−0.328509
$689$	−2016.72	−0.111511
$690$	−5037.36	−0.277926
$691$	19119.2	1.05257	0.526287	−	0.850307i	$-0.323584\pi$
$691$	19119.2	1.05257	0.526287	+	0.850307i	$0.323584\pi$
$692$	6330.48	0.347758
$693$	−570.519	−0.0312731
$694$	−11753.3	−0.642868
$695$	−19607.1	−1.07013
$696$	−15255.8	−0.830849
$697$	3305.50	0.179634
$698$	−13577.3	−0.736259
$699$	−28322.9	−1.53258
$700$	−1660.08	−0.0896360
$701$	−29188.2	−1.57264	−0.786322	−	0.617817i	$-0.788017\pi$
$701$	−29188.2	−1.57264	−0.786322	+	0.617817i	$0.788017\pi$
$702$	−936.929	−0.0503734
$703$	5415.03	0.290515
$704$	3502.30	0.187497
$705$	1726.79	0.0922480
$706$	11865.8	0.632541
$707$	−4668.53	−0.248342
$708$	28337.2	1.50420
$709$	−11377.0	−0.602639	−0.301320	−	0.953523i	$-0.597427\pi$
$709$	−11377.0	−0.602639	−0.301320	+	0.953523i	$0.597427\pi$
$710$	−6477.32	−0.342380
$711$	−3781.13	−0.199442
$712$	−15090.5	−0.794297
$713$	−11698.1	−0.614442
$714$	−804.073	−0.0421452
$715$	696.040	0.0364062
$716$	−5263.23	−0.274715
$717$	−21819.3	−1.13648
$718$	7094.93	0.368775
$719$	−8170.53	−0.423796	−0.211898	−	0.977292i	$-0.567964\pi$
$719$	−8170.53	−0.423796	−0.211898	+	0.977292i	$0.567964\pi$
$720$	−724.602	−0.0375060
$721$	5187.35	0.267943
$722$	−9441.24	−0.486657
$723$	−36415.5	−1.87318
$724$	−9655.42	−0.495637
$725$	10709.8	0.548623
$726$	8335.11	0.426095
$727$	19612.6	1.00054	0.500270	−	0.865870i	$-0.333234\pi$
$727$	19612.6	1.00054	0.500270	+	0.865870i	$0.333234\pi$
$728$	−389.224	−0.0198154
$729$	10330.8	0.524858
$730$	6547.63	0.331971
$731$	11258.4	0.569642
$732$	−6600.77	−0.333294
$733$	−16619.4	−0.837451	−0.418725	−	0.908113i	$-0.637523\pi$
$733$	−16619.4	−0.837451	−0.418725	+	0.908113i	$0.637523\pi$
$734$	−3632.82	−0.182684
$735$	−12188.0	−0.611650
$736$	−16933.2	−0.848053
$737$	18239.2	0.911599
$738$	−1592.40	−0.0794268
$739$	−19520.6	−0.971686	−0.485843	−	0.874046i	$-0.661487\pi$
$739$	−19520.6	−0.971686	−0.485843	+	0.874046i	$0.661487\pi$
$740$	−7507.68	−0.372956
$741$	−841.231	−0.0417050
$742$	−1843.74	−0.0912206
$743$	162.092	0.00800345	0.00400172	−	0.999992i	$-0.498726\pi$
$743$	162.092	0.00800345	0.00400172	+	0.999992i	$0.498726\pi$
$744$	16036.9	0.790243
$745$	7136.65	0.350962
$746$	19925.9	0.977935
$747$	−4569.00	−0.223790
$748$	3051.04	0.149141
$749$	4455.53	0.217359
$750$	−11823.1	−0.575625
$751$	30012.2	1.45827	0.729135	−	0.684370i	$-0.239923\pi$
$751$	30012.2	1.45827	0.729135	+	0.684370i	$0.239923\pi$
$752$	656.028	0.0318123
$753$	25332.3	1.22598
$754$	1044.00	0.0504250
$755$	11984.7	0.577704
$756$	2114.00	0.101700
$757$	−25232.6	−1.21148	−0.605742	−	0.795661i	$-0.707124\pi$
$757$	−25232.6	−1.21148	−0.605742	+	0.795661i	$0.707124\pi$
$758$	−13620.9	−0.652683
$759$	10859.9	0.519351
$760$	−3255.73	−0.155392
$761$	−26393.1	−1.25723	−0.628613	−	0.777719i	$-0.716377\pi$
$761$	−26393.1	−1.25723	−0.628613	+	0.777719i	$0.716377\pi$
$762$	9347.76	0.444401
$763$	3267.43	0.155031
$764$	11302.4	0.535217
$765$	1376.09	0.0650362
$766$	−16165.7	−0.762521
$767$	−4664.15	−0.219573
$768$	19430.9	0.912956
$769$	−12734.0	−0.597140	−0.298570	−	0.954388i	$-0.596510\pi$
$769$	−12734.0	−0.597140	−0.298570	+	0.954388i	$0.596510\pi$
$770$	636.337	0.0297818
$771$	37197.1	1.73751
$772$	−984.118	−0.0458798
$773$	29097.4	1.35389	0.676947	−	0.736032i	$-0.263303\pi$
$773$	29097.4	1.35389	0.676947	+	0.736032i	$0.263303\pi$
$774$	−5423.66	−0.251873
$775$	−11258.1	−0.521811
$776$	4466.06	0.206601
$777$	−4265.54	−0.196944
$778$	−6917.55	−0.318774
$779$	3161.62	0.145413
$780$	1166.33	0.0535399
$781$	13964.2	0.639794
$782$	3634.40	0.166197
$783$	−13638.2	−0.622464
$784$	−4630.37	−0.210932
$785$	6321.09	0.287401
$786$	894.036	0.0405715
$787$	−4595.00	−0.208124	−0.104062	−	0.994571i	$-0.533184\pi$
$787$	−4595.00	−0.208124	−0.104062	+	0.994571i	$0.533184\pi$
$788$	−107.962	−0.00488070
$789$	29800.2	1.34463
$790$	4217.34	0.189932
$791$	1228.40	0.0552173
$792$	−3535.18	−0.158608
$793$	1086.45	0.0486519
$794$	−4065.12	−0.181695
$795$	13288.2	0.592811
$796$	−12526.2	−0.557764
$797$	−29030.2	−1.29021	−0.645107	−	0.764092i	$-0.723187\pi$
$797$	−29030.2	−1.29021	−0.645107	+	0.764092i	$0.723187\pi$
$798$	−769.074	−0.0341164
$799$	−1245.86	−0.0551633
$800$	−16296.3	−0.720203
$801$	6100.62	0.269107
$802$	−3904.82	−0.171925
$803$	−14115.8	−0.620343
$804$	30562.6	1.34062
$805$	−1870.75	−0.0819073
$806$	−1097.46	−0.0479606
$807$	−27164.2	−1.18491
$808$	−28928.2	−1.25952
$809$	−25943.4	−1.12747	−0.563734	−	0.825957i	$-0.690636\pi$
$809$	−25943.4	−1.12747	−0.563734	+	0.825957i	$0.690636\pi$
$810$	8302.30	0.360140
$811$	18828.1	0.815222	0.407611	−	0.913156i	$-0.366362\pi$
$811$	18828.1	0.815222	0.407611	+	0.913156i	$0.366362\pi$
$812$	−2355.60	−0.101805
$813$	−18516.7	−0.798782
$814$	−6558.15	−0.282387
$815$	−4498.14	−0.193329
$816$	2201.64	0.0944520
$817$	10768.4	0.461124
$818$	−4411.40	−0.188559
$819$	157.351	0.00671344
$820$	−4383.43	−0.186678
$821$	23631.4	1.00456	0.502278	−	0.864706i	$-0.332495\pi$
$821$	23631.4	1.00456	0.502278	+	0.864706i	$0.332495\pi$
$822$	14529.5	0.616515
$823$	−37638.6	−1.59417	−0.797084	−	0.603868i	$-0.793625\pi$
$823$	−37638.6	−1.59417	−0.797084	+	0.603868i	$0.793625\pi$
$824$	32143.0	1.35893
$825$	10451.4	0.441056
$826$	−4264.08	−0.179620
$827$	3452.86	0.145185	0.0725924	−	0.997362i	$-0.476873\pi$
$827$	3452.86	0.145185	0.0725924	+	0.997362i	$0.476873\pi$
$828$	4321.08	0.181362
$829$	−2949.59	−0.123575	−0.0617875	−	0.998089i	$-0.519680\pi$
$829$	−2949.59	−0.123575	−0.0617875	+	0.998089i	$0.519680\pi$
$830$	5096.11	0.213119
$831$	25879.1	1.08031
$832$	−965.950	−0.0402503
$833$	8793.53	0.365760
$834$	−28699.5	−1.19159
$835$	9843.27	0.407953
$836$	2918.24	0.120729
$837$	14336.4	0.592043
$838$	−16078.3	−0.662789
$839$	−21099.3	−0.868212	−0.434106	−	0.900862i	$-0.642936\pi$
$839$	−21099.3	−0.868212	−0.434106	+	0.900862i	$0.642936\pi$
$840$	2564.61	0.105342
$841$	−9192.18	−0.376899
$842$	12386.4	0.506965
$843$	9761.70	0.398827
$844$	−1346.39	−0.0549108
$845$	13373.1	0.544438
$846$	600.184	0.0243910
$847$	3095.46	0.125574
$848$	5048.35	0.204435
$849$	37299.7	1.50780
$850$	3497.70	0.141141
$851$	19280.2	0.776634
$852$	23399.3	0.940898
$853$	−5832.07	−0.234099	−0.117049	−	0.993126i	$-0.537344\pi$
$853$	−5832.07	−0.234099	−0.117049	+	0.993126i	$0.537344\pi$
$854$	993.260	0.0397994
$855$	1316.19	0.0526466
$856$	27608.4	1.10238
$857$	−3182.86	−0.126866	−0.0634331	−	0.997986i	$-0.520205\pi$
$857$	−3182.86	−0.126866	−0.0634331	+	0.997986i	$0.520205\pi$
$858$	1018.82	0.0405382
$859$	34773.9	1.38122	0.690610	−	0.723227i	$-0.257342\pi$
$859$	34773.9	1.38122	0.690610	+	0.723227i	$0.257342\pi$
$860$	−14929.8	−0.591980
$861$	−2490.48	−0.0985776
$862$	258.270	0.0102050
$863$	−13655.6	−0.538636	−0.269318	−	0.963051i	$-0.586798\pi$
$863$	−13655.6	−0.538636	−0.269318	+	0.963051i	$0.586798\pi$
$864$	20752.3	0.817137
$865$	6865.52	0.269867
$866$	−12363.1	−0.485123
$867$	25053.4	0.981382
$868$	2476.20	0.0968290
$869$	−9092.01	−0.354920
$870$	−6878.98	−0.268068
$871$	−5030.44	−0.195695
$872$	20246.4	0.786273
$873$	−1805.49	−0.0699962
$874$	3476.20	0.134536
$875$	−4390.82	−0.169642
$876$	−23653.2	−0.912293
$877$	−43458.4	−1.67330	−0.836651	−	0.547736i	$-0.815490\pi$
$877$	−43458.4	−1.67330	−0.836651	+	0.547736i	$0.815490\pi$
$878$	−14412.7	−0.553992
$879$	−28090.2	−1.07788
$880$	−1742.36	−0.0667443
$881$	26874.5	1.02772	0.513862	−	0.857873i	$-0.328214\pi$
$881$	26874.5	1.02772	0.513862	+	0.857873i	$0.328214\pi$
$882$	−4236.21	−0.161724
$883$	−9065.53	−0.345503	−0.172752	−	0.984965i	$-0.555266\pi$
$883$	−9065.53	−0.345503	−0.172752	+	0.984965i	$0.555266\pi$
$884$	−841.491	−0.0320163
$885$	30732.2	1.16729
$886$	−12601.8	−0.477840
$887$	42928.3	1.62502	0.812510	−	0.582947i	$-0.198101\pi$
$887$	42928.3	1.62502	0.812510	+	0.582947i	$0.198101\pi$
$888$	−26431.1	−0.998841
$889$	3471.53	0.130969
$890$	−6804.42	−0.256275
$891$	−17898.6	−0.672982
$892$	−7950.11	−0.298419
$893$	−1191.63	−0.0446545
$894$	10446.1	0.390795
$895$	−5708.07	−0.213184
$896$	4153.58	0.154867
$897$	−2995.19	−0.111490
$898$	5348.39	0.198751
$899$	−15974.9	−0.592649
$900$	4158.56	0.154021
$901$	−9587.32	−0.354495
$902$	−3829.05	−0.141345
$903$	−8482.49	−0.312602
$904$	7611.70	0.280046
$905$	−10471.5	−0.384623
$906$	17542.3	0.643272
$907$	−13748.8	−0.503330	−0.251665	−	0.967814i	$-0.580978\pi$
$907$	−13748.8	−0.503330	−0.251665	+	0.967814i	$0.580978\pi$
$908$	−17275.1	−0.631383
$909$	11694.8	0.426724
$910$	−175.504	−0.00639331
$911$	−15009.6	−0.545874	−0.272937	−	0.962032i	$-0.587995\pi$
$911$	−15009.6	−0.545874	−0.272937	+	0.962032i	$0.587995\pi$
$912$	2105.81	0.0764586
$913$	−10986.5	−0.398248
$914$	−4177.00	−0.151163
$915$	−7158.66	−0.258643
$916$	−28117.5	−1.01422
$917$	332.023	0.0119568
$918$	−4454.08	−0.160138
$919$	40518.6	1.45439	0.727195	−	0.686431i	$-0.240823\pi$
$919$	40518.6	1.45439	0.727195	+	0.686431i	$0.240823\pi$
$920$	−11592.0	−0.415409
$921$	56383.4	2.01726
$922$	−7321.79	−0.261530
$923$	−3851.39	−0.137346
$924$	−2298.76	−0.0818439
$925$	18555.0	0.659551
$926$	19734.5	0.700341
$927$	−12994.5	−0.460403
$928$	−23123.9	−0.817974
$929$	14399.3	0.508531	0.254265	−	0.967135i	$-0.418166\pi$
$929$	14399.3	0.508531	0.254265	+	0.967135i	$0.418166\pi$
$930$	7231.17	0.254967
$931$	8410.77	0.296082
$932$	−27098.5	−0.952404
$933$	−45965.5	−1.61291
$934$	3822.13	0.133901
$935$	3308.92	0.115736
$936$	975.017	0.0340486
$937$	−48380.8	−1.68680	−0.843400	−	0.537286i	$-0.819450\pi$
$937$	−48380.8	−1.68680	−0.843400	+	0.537286i	$0.819450\pi$
$938$	−4598.95	−0.160087
$939$	45936.3	1.59646
$940$	1652.14	0.0573265
$941$	−31933.6	−1.10628	−0.553139	−	0.833089i	$-0.686570\pi$
$941$	−31933.6	−1.10628	−0.553139	+	0.833089i	$0.686570\pi$
$942$	9252.38	0.320020
$943$	11256.9	0.388734
$944$	11675.5	0.402548
$945$	2292.67	0.0789214
$946$	−13041.6	−0.448223
$947$	−50096.9	−1.71904	−0.859519	−	0.511103i	$-0.829237\pi$
$947$	−50096.9	−1.71904	−0.859519	+	0.511103i	$0.829237\pi$
$948$	−15235.1	−0.521955
$949$	3893.19	0.133170
$950$	3345.46	0.114254
$951$	−64006.2	−2.18248
$952$	−1850.34	−0.0629935
$953$	12685.9	0.431205	0.215602	−	0.976481i	$-0.430829\pi$
$953$	12685.9	0.431205	0.215602	+	0.976481i	$0.430829\pi$
$954$	4618.61	0.156743
$955$	12257.6	0.415338
$956$	−20876.0	−0.706255
$957$	14830.2	0.500931
$958$	14098.1	0.475458
$959$	5395.91	0.181693
$960$	6364.68	0.213978
$961$	−12998.3	−0.436316
$962$	1808.77	0.0606205
$963$	−11161.2	−0.373485
$964$	−34841.2	−1.16406
$965$	−1067.30	−0.0356036
$966$	−2738.28	−0.0912036
$967$	35048.9	1.16556	0.582780	−	0.812630i	$-0.301965\pi$
$967$	35048.9	1.16556	0.582780	+	0.812630i	$0.301965\pi$
$968$	19180.8	0.636874
$969$	−3999.14	−0.132581
$970$	2013.79	0.0666585
$971$	29484.5	0.974462	0.487231	−	0.873273i	$-0.338007\pi$
$971$	29484.5	0.974462	0.487231	+	0.873273i	$0.338007\pi$
$972$	−12986.1	−0.428527
$973$	−10658.3	−0.351172
$974$	15295.6	0.503187
$975$	−2882.54	−0.0946822
$976$	−2719.65	−0.0891946
$977$	−25899.3	−0.848100	−0.424050	−	0.905639i	$-0.639392\pi$
$977$	−25899.3	−0.848100	−0.424050	+	0.905639i	$0.639392\pi$
$978$	−6584.07	−0.215271
$979$	14669.4	0.478893
$980$	−11661.1	−0.380103
$981$	−8185.02	−0.266389
$982$	−12899.4	−0.419183
$983$	20316.8	0.659210	0.329605	−	0.944119i	$-0.393084\pi$
$983$	20316.8	0.659210	0.329605	+	0.944119i	$0.393084\pi$
$984$	−15432.1	−0.499956
$985$	−117.087	−0.00378751
$986$	4963.11	0.160302
$987$	938.676	0.0302719
$988$	−804.863	−0.0259171
$989$	38340.7	1.23272
$990$	−1594.04	−0.0511738
$991$	52179.6	1.67259	0.836297	−	0.548276i	$-0.184716\pi$
$991$	52179.6	1.67259	0.836297	+	0.548276i	$0.184716\pi$
$992$	24307.8	0.777997
$993$	−22010.8	−0.703417
$994$	−3521.04	−0.112355
$995$	−13584.9	−0.432835
$996$	−18409.6	−0.585675
$997$	13574.2	0.431193	0.215596	−	0.976483i	$-0.430830\pi$
$997$	13574.2	0.431193	0.215596	+	0.976483i	$0.430830\pi$
$998$	−17497.8	−0.554993
$999$	−23628.5	−0.748322

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.51882 q^{2} -5.95044 q^{3} -5.69320 q^{4} -6.17438 q^{5} -9.03763 q^{6} -3.35636 q^{7} -20.7975 q^{8} +8.40778 q^{9} +O(q^{10})\)	\(q+1.51882 q^{2} -5.95044 q^{3} -5.69320 q^{4} -6.17438 q^{5} -9.03763 q^{6} -3.35636 q^{7} -20.7975 q^{8} +8.40778 q^{9} -9.37775 q^{10} +20.2172 q^{11} +33.8770 q^{12} -5.57597 q^{13} -5.09770 q^{14} +36.7403 q^{15} +13.9580 q^{16} -26.5077 q^{17} +12.7699 q^{18} -25.3539 q^{19} +35.1519 q^{20} +19.9718 q^{21} +30.7062 q^{22} -90.2723 q^{23} +123.754 q^{24} -86.8770 q^{25} -8.46888 q^{26} +110.632 q^{27} +19.1084 q^{28} -123.275 q^{29} +55.8018 q^{30} +129.587 q^{31} +187.579 q^{32} -120.301 q^{33} -40.2604 q^{34} +20.7234 q^{35} -47.8671 q^{36} -213.578 q^{37} -38.5079 q^{38} +33.1795 q^{39} +128.411 q^{40} -124.700 q^{41} +30.3336 q^{42} -424.723 q^{43} -115.100 q^{44} -51.9128 q^{45} -137.107 q^{46} +47.0000 q^{47} -83.0566 q^{48} -331.735 q^{49} -131.950 q^{50} +157.733 q^{51} +31.7451 q^{52} +361.680 q^{53} +168.030 q^{54} -124.828 q^{55} +69.8038 q^{56} +150.867 q^{57} -187.233 q^{58} +836.472 q^{59} -209.170 q^{60} -194.845 q^{61} +196.818 q^{62} -28.2195 q^{63} +173.234 q^{64} +34.4282 q^{65} -182.715 q^{66} +902.163 q^{67} +150.914 q^{68} +537.160 q^{69} +31.4751 q^{70} +690.711 q^{71} -174.860 q^{72} -698.209 q^{73} -324.386 q^{74} +516.957 q^{75} +144.345 q^{76} -67.8561 q^{77} +50.3936 q^{78} -449.718 q^{79} -86.1823 q^{80} -885.319 q^{81} -189.396 q^{82} -543.425 q^{83} -113.704 q^{84} +163.669 q^{85} -645.076 q^{86} +733.543 q^{87} -420.465 q^{88} +725.592 q^{89} -78.8461 q^{90} +18.7150 q^{91} +513.938 q^{92} -771.099 q^{93} +71.3844 q^{94} +156.545 q^{95} -1116.18 q^{96} -214.741 q^{97} -503.844 q^{98} +169.981 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9}+O(q^{10}) \) 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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