LMFDB - Embedded newform 87.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [87,4,Mod(1,87)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(87, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("87.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	87.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:	$5.13316617050$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	87.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.70156 q^{2} -3.00000 q^{3} +5.70156 q^{4} +9.10469 q^{5} +11.1047 q^{6} -5.59688 q^{7} +8.50781 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.70156 q^{2} -3.00000 q^{3} +5.70156 q^{4} +9.10469 q^{5} +11.1047 q^{6} -5.59688 q^{7} +8.50781 q^{8} +9.00000 q^{9} -33.7016 q^{10} +9.29844 q^{11} -17.1047 q^{12} -77.1203 q^{13} +20.7172 q^{14} -27.3141 q^{15} -77.1047 q^{16} -61.8062 q^{17} -33.3141 q^{18} -39.3141 q^{19} +51.9109 q^{20} +16.7906 q^{21} -34.4187 q^{22} -37.4031 q^{23} -25.5234 q^{24} -42.1047 q^{25} +285.466 q^{26} -27.0000 q^{27} -31.9109 q^{28} +29.0000 q^{29} +101.105 q^{30} +26.5969 q^{31} +217.345 q^{32} -27.8953 q^{33} +228.780 q^{34} -50.9578 q^{35} +51.3141 q^{36} -279.105 q^{37} +145.523 q^{38} +231.361 q^{39} +77.4609 q^{40} -194.570 q^{41} -62.1515 q^{42} +309.889 q^{43} +53.0156 q^{44} +81.9422 q^{45} +138.450 q^{46} +109.325 q^{47} +231.314 q^{48} -311.675 q^{49} +155.853 q^{50} +185.419 q^{51} -439.706 q^{52} -560.303 q^{53} +99.9422 q^{54} +84.6594 q^{55} -47.6172 q^{56} +117.942 q^{57} -107.345 q^{58} +147.461 q^{59} -155.733 q^{60} +278.125 q^{61} -98.4500 q^{62} -50.3719 q^{63} -187.680 q^{64} -702.156 q^{65} +103.256 q^{66} -970.795 q^{67} -352.392 q^{68} +112.209 q^{69} +188.623 q^{70} -519.684 q^{71} +76.5703 q^{72} -264.744 q^{73} +1033.12 q^{74} +126.314 q^{75} -224.152 q^{76} -52.0422 q^{77} -856.397 q^{78} +875.066 q^{79} -702.014 q^{80} +81.0000 q^{81} +720.214 q^{82} +141.297 q^{83} +95.7328 q^{84} -562.727 q^{85} -1147.07 q^{86} -87.0000 q^{87} +79.1093 q^{88} +970.342 q^{89} -303.314 q^{90} +431.633 q^{91} -213.256 q^{92} -79.7906 q^{93} -404.673 q^{94} -357.942 q^{95} -652.036 q^{96} -2.93122 q^{97} +1153.68 q^{98} +83.6859 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 6 q^{3} + 5 q^{4} - q^{5} + 3 q^{6} - 24 q^{7} - 15 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - q^2 - 6 * q^3 + 5 * q^4 - q^5 + 3 * q^6 - 24 * q^7 - 15 * q^8 + 18 * q^9	\( 2 q - q^{2} - 6 q^{3} + 5 q^{4} - q^{5} + 3 q^{6} - 24 q^{7} - 15 q^{8} + 18 q^{9} - 61 q^{10} + 25 q^{11} - 15 q^{12} - 71 q^{13} - 29 q^{14} + 3 q^{15} - 135 q^{16} - 98 q^{17} - 9 q^{18} - 21 q^{19} + 59 q^{20} + 72 q^{21} + 8 q^{22} - 62 q^{23} + 45 q^{24} - 65 q^{25} + 302 q^{26} - 54 q^{27} - 19 q^{28} + 58 q^{29} + 183 q^{30} + 66 q^{31} + 249 q^{32} - 75 q^{33} + 131 q^{34} + 135 q^{35} + 45 q^{36} - 539 q^{37} + 195 q^{38} + 213 q^{39} + 315 q^{40} - 101 q^{41} + 87 q^{42} - 155 q^{43} + 42 q^{44} - 9 q^{45} + 72 q^{46} + 526 q^{47} + 405 q^{48} - 316 q^{49} + 94 q^{50} + 294 q^{51} - 444 q^{52} - 698 q^{53} + 27 q^{54} - 74 q^{55} + 385 q^{56} + 63 q^{57} - 29 q^{58} + 455 q^{59} - 177 q^{60} + 44 q^{61} + 8 q^{62} - 216 q^{63} + 361 q^{64} - 764 q^{65} - 24 q^{66} - 1551 q^{67} - 327 q^{68} + 186 q^{69} + 691 q^{70} + 126 q^{71} - 135 q^{72} - 760 q^{73} + 331 q^{74} + 195 q^{75} - 237 q^{76} - 341 q^{77} - 906 q^{78} - 158 q^{79} - 117 q^{80} + 162 q^{81} + 973 q^{82} - 934 q^{83} + 57 q^{84} - 197 q^{85} - 2403 q^{86} - 174 q^{87} - 290 q^{88} - 691 q^{89} - 549 q^{90} + 319 q^{91} - 196 q^{92} - 198 q^{93} + 721 q^{94} - 543 q^{95} - 747 q^{96} + 532 q^{97} + 1142 q^{98} + 225 q^{99}+O(q^{100}) \) 2 * q - q^2 - 6 * q^3 + 5 * q^4 - q^5 + 3 * q^6 - 24 * q^7 - 15 * q^8 + 18 * q^9 - 61 * q^10 + 25 * q^11 - 15 * q^12 - 71 * q^13 - 29 * q^14 + 3 * q^15 - 135 * q^16 - 98 * q^17 - 9 * q^18 - 21 * q^19 + 59 * q^20 + 72 * q^21 + 8 * q^22 - 62 * q^23 + 45 * q^24 - 65 * q^25 + 302 * q^26 - 54 * q^27 - 19 * q^28 + 58 * q^29 + 183 * q^30 + 66 * q^31 + 249 * q^32 - 75 * q^33 + 131 * q^34 + 135 * q^35 + 45 * q^36 - 539 * q^37 + 195 * q^38 + 213 * q^39 + 315 * q^40 - 101 * q^41 + 87 * q^42 - 155 * q^43 + 42 * q^44 - 9 * q^45 + 72 * q^46 + 526 * q^47 + 405 * q^48 - 316 * q^49 + 94 * q^50 + 294 * q^51 - 444 * q^52 - 698 * q^53 + 27 * q^54 - 74 * q^55 + 385 * q^56 + 63 * q^57 - 29 * q^58 + 455 * q^59 - 177 * q^60 + 44 * q^61 + 8 * q^62 - 216 * q^63 + 361 * q^64 - 764 * q^65 - 24 * q^66 - 1551 * q^67 - 327 * q^68 + 186 * q^69 + 691 * q^70 + 126 * q^71 - 135 * q^72 - 760 * q^73 + 331 * q^74 + 195 * q^75 - 237 * q^76 - 341 * q^77 - 906 * q^78 - 158 * q^79 - 117 * q^80 + 162 * q^81 + 973 * q^82 - 934 * q^83 + 57 * q^84 - 197 * q^85 - 2403 * q^86 - 174 * q^87 - 290 * q^88 - 691 * q^89 - 549 * q^90 + 319 * q^91 - 196 * q^92 - 198 * q^93 + 721 * q^94 - 543 * q^95 - 747 * q^96 + 532 * q^97 + 1142 * q^98 + 225 * 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 $

$2$	−3.70156	−1.30870	−0.654350	−	0.756192i	$-0.727058\pi$
$2$	−3.70156	−1.30870	−0.654350	+	0.756192i	$0.727058\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	5.70156	0.712695
$5$	9.10469	0.814348	0.407174	−	0.913351i	$-0.366514\pi$
$5$	9.10469	0.814348	0.407174	+	0.913351i	$0.366514\pi$
$6$	11.1047	0.755578
$7$	−5.59688	−0.302203	−0.151101	−	0.988518i	$-0.548282\pi$
$7$	−5.59688	−0.302203	−0.151101	+	0.988518i	$0.548282\pi$
$8$	8.50781	0.375996
$9$	9.00000	0.333333
$10$	−33.7016	−1.06574
$11$	9.29844	0.254871	0.127436	−	0.991847i	$-0.459325\pi$
$11$	9.29844	0.254871	0.127436	+	0.991847i	$0.459325\pi$
$12$	−17.1047	−0.411475
$13$	−77.1203	−1.64533	−0.822666	−	0.568524i	$-0.807514\pi$
$13$	−77.1203	−1.64533	−0.822666	+	0.568524i	$0.807514\pi$
$14$	20.7172	0.395493
$15$	−27.3141	−0.470164
$16$	−77.1047	−1.20476
$17$	−61.8062	−0.881777	−0.440889	−	0.897562i	$-0.645337\pi$
$17$	−61.8062	−0.881777	−0.440889	+	0.897562i	$0.645337\pi$
$18$	−33.3141	−0.436233
$19$	−39.3141	−0.474698	−0.237349	−	0.971424i	$-0.576279\pi$
$19$	−39.3141	−0.474698	−0.237349	+	0.971424i	$0.576279\pi$
$20$	51.9109	0.580382
$21$	16.7906	0.174477
$22$	−34.4187	−0.333550
$23$	−37.4031	−0.339091	−0.169545	−	0.985522i	$-0.554230\pi$
$23$	−37.4031	−0.339091	−0.169545	+	0.985522i	$0.554230\pi$
$24$	−25.5234	−0.217081
$25$	−42.1047	−0.336837
$26$	285.466	2.15325
$27$	−27.0000	−0.192450
$28$	−31.9109	−0.215379
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	101.105	0.615304
$31$	26.5969	0.154095	0.0770474	−	0.997027i	$-0.475451\pi$
$31$	26.5969	0.154095	0.0770474	+	0.997027i	$0.475451\pi$
$32$	217.345	1.20067
$33$	−27.8953	−0.147150
$34$	228.780	1.15398
$35$	−50.9578	−0.246098
$36$	51.3141	0.237565
$37$	−279.105	−1.24012	−0.620061	−	0.784553i	$-0.712892\pi$
$37$	−279.105	−1.24012	−0.620061	+	0.784553i	$0.712892\pi$
$38$	145.523	0.621237
$39$	231.361	0.949933
$40$	77.4609	0.306191
$41$	−194.570	−0.741141	−0.370570	−	0.928804i	$-0.620838\pi$
$41$	−194.570	−0.741141	−0.370570	+	0.928804i	$0.620838\pi$
$42$	−62.1515	−0.228338
$43$	309.889	1.09901	0.549507	−	0.835489i	$-0.314815\pi$
$43$	309.889	1.09901	0.549507	+	0.835489i	$0.314815\pi$
$44$	53.0156	0.181646
$45$	81.9422	0.271449
$46$	138.450	0.443768
$47$	109.325	0.339291	0.169646	−	0.985505i	$-0.445738\pi$
$47$	109.325	0.339291	0.169646	+	0.985505i	$0.445738\pi$
$48$	231.314	0.695569
$49$	−311.675	−0.908673
$50$	155.853	0.440819
$51$	185.419	0.509094
$52$	−439.706	−1.17262
$53$	−560.303	−1.45214	−0.726071	−	0.687620i	$-0.758656\pi$
$53$	−560.303	−1.45214	−0.726071	+	0.687620i	$0.758656\pi$
$54$	99.9422	0.251859
$55$	84.6594	0.207554
$56$	−47.6172	−0.113627
$57$	117.942	0.274067
$58$	−107.345	−0.243019
$59$	147.461	0.325386	0.162693	−	0.986677i	$-0.447982\pi$
$59$	147.461	0.325386	0.162693	+	0.986677i	$0.447982\pi$
$60$	−155.733	−0.335084
$61$	278.125	0.583775	0.291887	−	0.956453i	$-0.405717\pi$
$61$	278.125	0.583775	0.291887	+	0.956453i	$0.405717\pi$
$62$	−98.4500	−0.201664
$63$	−50.3719	−0.100734
$64$	−187.680	−0.366562
$65$	−702.156	−1.33987
$66$	103.256	0.192575
$67$	−970.795	−1.77017	−0.885086	−	0.465428i	$-0.845901\pi$
$67$	−970.795	−1.77017	−0.885086	+	0.465428i	$0.845901\pi$
$68$	−352.392	−0.628439
$69$	112.209	0.195774
$70$	188.623	0.322069
$71$	−519.684	−0.868665	−0.434332	−	0.900753i	$-0.643016\pi$
$71$	−519.684	−0.868665	−0.434332	+	0.900753i	$0.643016\pi$
$72$	76.5703	0.125332
$73$	−264.744	−0.424465	−0.212232	−	0.977219i	$-0.568073\pi$
$73$	−264.744	−0.424465	−0.212232	+	0.977219i	$0.568073\pi$
$74$	1033.12	1.62295
$75$	126.314	0.194473
$76$	−224.152	−0.338315
$77$	−52.0422	−0.0770228
$78$	−856.397	−1.24318
$79$	875.066	1.24623	0.623117	−	0.782128i	$-0.285866\pi$
$79$	875.066	1.24623	0.623117	+	0.782128i	$0.285866\pi$
$80$	−702.014	−0.981094
$81$	81.0000	0.111111
$82$	720.214	0.969931
$83$	141.297	0.186860	0.0934298	−	0.995626i	$-0.470217\pi$
$83$	141.297	0.186860	0.0934298	+	0.995626i	$0.470217\pi$
$84$	95.7328	0.124349
$85$	−562.727	−0.718074
$86$	−1147.07	−1.43828
$87$	−87.0000	−0.107211
$88$	79.1093	0.0958305
$89$	970.342	1.15569	0.577843	−	0.816148i	$-0.303895\pi$
$89$	970.342	1.15569	0.577843	+	0.816148i	$0.303895\pi$
$90$	−303.314	−0.355246
$91$	431.633	0.497224
$92$	−213.256	−0.241668
$93$	−79.7906	−0.0889667
$94$	−404.673	−0.444031
$95$	−357.942	−0.386569
$96$	−652.036	−0.693210
$97$	−2.93122	−0.00306825	−0.00153412	−	0.999999i	$-0.500488\pi$
$97$	−2.93122	−0.00306825	−0.00153412	+	0.999999i	$0.500488\pi$
$98$	1153.68	1.18918
$99$	83.6859	0.0849571
$100$	−240.062	−0.240062
$101$	1773.40	1.74713	0.873565	−	0.486708i	$-0.161803\pi$
$101$	1773.40	1.74713	0.873565	+	0.486708i	$0.161803\pi$
$102$	−686.339	−0.666252
$103$	1357.23	1.29837	0.649184	−	0.760632i	$-0.275111\pi$
$103$	1357.23	1.29837	0.649184	+	0.760632i	$0.275111\pi$
$104$	−656.125	−0.618638
$105$	152.873	0.142085
$106$	2074.00	1.90042
$107$	118.814	0.107348	0.0536738	−	0.998559i	$-0.482907\pi$
$107$	118.814	0.107348	0.0536738	+	0.998559i	$0.482907\pi$
$108$	−153.942	−0.137158
$109$	1033.18	0.907899	0.453949	−	0.891027i	$-0.350015\pi$
$109$	1033.18	0.907899	0.453949	+	0.891027i	$0.350015\pi$
$110$	−313.372	−0.271626
$111$	837.314	0.715985
$112$	431.545	0.364082
$113$	431.459	0.359188	0.179594	−	0.983741i	$-0.442522\pi$
$113$	431.459	0.359188	0.179594	+	0.983741i	$0.442522\pi$
$114$	−436.570	−0.358672
$115$	−340.544	−0.276138
$116$	165.345	0.132344
$117$	−694.083	−0.548444
$118$	−545.836	−0.425833
$119$	345.922	0.266476
$120$	−232.383	−0.176780
$121$	−1244.54	−0.935041
$122$	−1029.50	−0.763986
$123$	583.711	0.427898
$124$	151.644	0.109823
$125$	−1521.44	−1.08865
$126$	186.455	0.131831
$127$	−1897.09	−1.32551	−0.662755	−	0.748836i	$-0.730613\pi$
$127$	−1897.09	−1.32551	−0.662755	+	0.748836i	$0.730613\pi$
$128$	−1044.05	−0.720955
$129$	−929.667	−0.634516
$130$	2599.07	1.75349
$131$	2027.28	1.35209	0.676046	−	0.736860i	$-0.263692\pi$
$131$	2027.28	1.35209	0.676046	+	0.736860i	$0.263692\pi$
$132$	−159.047	−0.104873
$133$	220.036	0.143455
$134$	3593.46	2.31662
$135$	−245.827	−0.156721
$136$	−525.836	−0.331545
$137$	−1010.04	−0.629882	−0.314941	−	0.949111i	$-0.601985\pi$
$137$	−1010.04	−0.629882	−0.314941	+	0.949111i	$0.601985\pi$
$138$	−415.350	−0.256210
$139$	−2087.35	−1.27371	−0.636857	−	0.770982i	$-0.719766\pi$
$139$	−2087.35	−1.27371	−0.636857	+	0.770982i	$0.719766\pi$
$140$	−290.539	−0.175393
$141$	−327.975	−0.195890
$142$	1923.64	1.13682
$143$	−717.098	−0.419348
$144$	−693.942	−0.401587
$145$	264.036	0.151221
$146$	979.965	0.555497
$147$	935.025	0.524623
$148$	−1591.33	−0.883829
$149$	728.820	0.400720	0.200360	−	0.979722i	$-0.435789\pi$
$149$	728.820	0.400720	0.200360	+	0.979722i	$0.435789\pi$
$150$	−467.559	−0.254507
$151$	1045.01	0.563189	0.281594	−	0.959534i	$-0.409137\pi$
$151$	1045.01	0.563189	0.281594	+	0.959534i	$0.409137\pi$
$152$	−334.477	−0.178484
$153$	−556.256	−0.293926
$154$	192.637	0.100800
$155$	242.156	0.125487
$156$	1319.12	0.677013
$157$	−1838.74	−0.934698	−0.467349	−	0.884073i	$-0.654791\pi$
$157$	−1838.74	−0.934698	−0.467349	+	0.884073i	$0.654791\pi$
$158$	−3239.11	−1.63095
$159$	1680.91	0.838395
$160$	1978.86	0.977767
$161$	209.341	0.102474
$162$	−299.827	−0.145411
$163$	1986.07	0.954364	0.477182	−	0.878805i	$-0.341658\pi$
$163$	1986.07	0.954364	0.477182	+	0.878805i	$0.341658\pi$
$164$	−1109.35	−0.528208
$165$	−253.978	−0.119831
$166$	−523.019	−0.244543
$167$	184.097	0.0853045	0.0426523	−	0.999090i	$-0.486419\pi$
$167$	184.097	0.0853045	0.0426523	+	0.999090i	$0.486419\pi$
$168$	142.851	0.0656026
$169$	3750.54	1.70712
$170$	2082.97	0.939743
$171$	−353.827	−0.158233
$172$	1766.85	0.783263
$173$	−1060.75	−0.466169	−0.233084	−	0.972457i	$-0.574882\pi$
$173$	−1060.75	−0.466169	−0.233084	+	0.972457i	$0.574882\pi$
$174$	322.036	0.140307
$175$	235.655	0.101793
$176$	−716.953	−0.307059
$177$	−442.383	−0.187862
$178$	−3591.78	−1.51245
$179$	3090.07	1.29030	0.645148	−	0.764058i	$-0.276796\pi$
$179$	3090.07	1.29030	0.645148	+	0.764058i	$0.276796\pi$
$180$	467.198	0.193461
$181$	−3324.07	−1.36506	−0.682531	−	0.730857i	$-0.739121\pi$
$181$	−3324.07	−1.36506	−0.682531	+	0.730857i	$0.739121\pi$
$182$	−1597.72	−0.650717
$183$	−834.375	−0.337042
$184$	−318.219	−0.127497
$185$	−2541.16	−1.00989
$186$	295.350	0.116431
$187$	−574.702	−0.224740
$188$	623.323	0.241811
$189$	151.116	0.0581590
$190$	1324.95	0.505903
$191$	−4448.15	−1.68512	−0.842558	−	0.538606i	$-0.818951\pi$
$191$	−4448.15	−1.68512	−0.842558	+	0.538606i	$0.818951\pi$
$192$	563.039	0.211635
$193$	354.653	0.132272	0.0661359	−	0.997811i	$-0.478933\pi$
$193$	354.653	0.132272	0.0661359	+	0.997811i	$0.478933\pi$
$194$	10.8501	0.00401542
$195$	2106.47	0.773576
$196$	−1777.03	−0.647607
$197$	−2386.43	−0.863078	−0.431539	−	0.902094i	$-0.642029\pi$
$197$	−2386.43	−0.863078	−0.431539	+	0.902094i	$0.642029\pi$
$198$	−309.769	−0.111183
$199$	2957.89	1.05366	0.526832	−	0.849970i	$-0.323380\pi$
$199$	2957.89	1.05366	0.526832	+	0.849970i	$0.323380\pi$
$200$	−358.219	−0.126649
$201$	2912.39	1.02201
$202$	−6564.36	−2.28647
$203$	−162.309	−0.0561177
$204$	1057.18	0.362829
$205$	−1771.50	−0.603547
$206$	−5023.87	−1.69917
$207$	−336.628	−0.113030
$208$	5946.34	1.98223
$209$	−365.559	−0.120987
$210$	−565.870	−0.185947
$211$	−5067.01	−1.65321	−0.826605	−	0.562782i	$-0.809731\pi$
$211$	−5067.01	−1.65321	−0.826605	+	0.562782i	$0.809731\pi$
$212$	−3194.60	−1.03494
$213$	1559.05	0.501524
$214$	−439.798	−0.140486
$215$	2821.44	0.894980
$216$	−229.711	−0.0723604
$217$	−148.859	−0.0465679
$218$	−3824.39	−1.18817
$219$	794.231	0.245065
$220$	482.691	0.147923
$221$	4766.52	1.45082
$222$	−3099.37	−0.937009
$223$	−3734.21	−1.12135	−0.560675	−	0.828036i	$-0.689458\pi$
$223$	−3734.21	−1.12135	−0.560675	+	0.828036i	$0.689458\pi$
$224$	−1216.45	−0.362847
$225$	−378.942	−0.112279
$226$	−1597.07	−0.470070
$227$	5393.99	1.57714	0.788571	−	0.614943i	$-0.210821\pi$
$227$	5393.99	1.57714	0.788571	+	0.614943i	$0.210821\pi$
$228$	672.455	0.195326
$229$	−908.554	−0.262179	−0.131089	−	0.991371i	$-0.541848\pi$
$229$	−908.554	−0.262179	−0.131089	+	0.991371i	$0.541848\pi$
$230$	1260.54	0.361382
$231$	156.127	0.0444692
$232$	246.727	0.0698206
$233$	5044.02	1.41822	0.709108	−	0.705100i	$-0.249098\pi$
$233$	5044.02	1.41822	0.709108	+	0.705100i	$0.249098\pi$
$234$	2569.19	0.717749
$235$	995.370	0.276301
$236$	840.758	0.231901
$237$	−2625.20	−0.719514
$238$	−1280.45	−0.348737
$239$	822.622	0.222640	0.111320	−	0.993785i	$-0.464492\pi$
$239$	822.622	0.222640	0.111320	+	0.993785i	$0.464492\pi$
$240$	2106.04	0.566435
$241$	−957.597	−0.255951	−0.127976	−	0.991777i	$-0.540848\pi$
$241$	−957.597	−0.255951	−0.127976	+	0.991777i	$0.540848\pi$
$242$	4606.74	1.22369
$243$	−243.000	−0.0641500
$244$	1585.75	0.416053
$245$	−2837.70	−0.739976
$246$	−2160.64	−0.559990
$247$	3031.91	0.781036
$248$	226.281	0.0579390
$249$	−423.890	−0.107883
$250$	5631.69	1.42472
$251$	3408.72	0.857197	0.428599	−	0.903495i	$-0.359007\pi$
$251$	3408.72	0.857197	0.428599	+	0.903495i	$0.359007\pi$
$252$	−287.198	−0.0717929
$253$	−347.791	−0.0864245
$254$	7022.21	1.73470
$255$	1688.18	0.414580
$256$	5366.07	1.31008
$257$	4903.44	1.19015	0.595075	−	0.803670i	$-0.297122\pi$
$257$	4903.44	1.19015	0.595075	+	0.803670i	$0.297122\pi$
$258$	3441.22	0.830392
$259$	1562.11	0.374769
$260$	−4003.39	−0.954921
$261$	261.000	0.0618984
$262$	−7504.09	−1.76948
$263$	8416.48	1.97332	0.986658	−	0.162804i	$-0.0520539\pi$
$263$	8416.48	1.97332	0.986658	+	0.162804i	$0.0520539\pi$
$264$	−237.328	−0.0553278
$265$	−5101.38	−1.18255
$266$	−814.477	−0.187740
$267$	−2911.03	−0.667236
$268$	−5535.05	−1.26159
$269$	5282.51	1.19732	0.598662	−	0.801002i	$-0.295699\pi$
$269$	5282.51	1.19732	0.598662	+	0.801002i	$0.295699\pi$
$270$	909.942	0.205101
$271$	−7049.65	−1.58020	−0.790102	−	0.612975i	$-0.789973\pi$
$271$	−7049.65	−1.58020	−0.790102	+	0.612975i	$0.789973\pi$
$272$	4765.55	1.06233
$273$	−1294.90	−0.287073
$274$	3738.74	0.824327
$275$	−391.508	−0.0858502
$276$	639.769	0.139527
$277$	507.867	0.110162	0.0550808	−	0.998482i	$-0.482458\pi$
$277$	507.867	0.110162	0.0550808	+	0.998482i	$0.482458\pi$
$278$	7726.44	1.66691
$279$	239.372	0.0513649
$280$	−433.539	−0.0925319
$281$	−8776.99	−1.86331	−0.931657	−	0.363339i	$-0.881637\pi$
$281$	−8776.99	−1.86331	−0.931657	+	0.363339i	$0.881637\pi$
$282$	1214.02	0.256361
$283$	−1061.07	−0.222877	−0.111438	−	0.993771i	$-0.535546\pi$
$283$	−1061.07	−0.222877	−0.111438	+	0.993771i	$0.535546\pi$
$284$	−2963.01	−0.619093
$285$	1073.83	0.223186
$286$	2654.38	0.548801
$287$	1088.99	0.223975
$288$	1956.11	0.400225
$289$	−1092.99	−0.222468
$290$	−977.345	−0.197902
$291$	8.79365	0.00177145
$292$	−1509.45	−0.302514
$293$	−6565.00	−1.30898	−0.654491	−	0.756070i	$-0.727117\pi$
$293$	−6565.00	−1.30898	−0.654491	+	0.756070i	$0.727117\pi$
$294$	−3461.05	−0.686574
$295$	1342.59	0.264977
$296$	−2374.57	−0.466281
$297$	−251.058	−0.0490500
$298$	−2697.77	−0.524422
$299$	2884.54	0.557917
$300$	720.187	0.138600
$301$	−1734.41	−0.332125
$302$	−3868.16	−0.737045
$303$	−5320.20	−1.00871
$304$	3031.30	0.571898
$305$	2532.24	0.475396
$306$	2059.02	0.384661
$307$	6787.60	1.26185	0.630926	−	0.775843i	$-0.282675\pi$
$307$	6787.60	1.26185	0.630926	+	0.775843i	$0.282675\pi$
$308$	−296.722	−0.0548938
$309$	−4071.69	−0.749613
$310$	−896.356	−0.164225
$311$	−10607.3	−1.93404	−0.967019	−	0.254704i	$-0.918022\pi$
$311$	−10607.3	−1.93404	−0.967019	+	0.254704i	$0.918022\pi$
$312$	1968.37	0.357171
$313$	2519.55	0.454994	0.227497	−	0.973779i	$-0.426946\pi$
$313$	2519.55	0.454994	0.227497	+	0.973779i	$0.426946\pi$
$314$	6806.22	1.22324
$315$	−458.620	−0.0820328
$316$	4989.24	0.888186
$317$	−5471.65	−0.969459	−0.484729	−	0.874664i	$-0.661082\pi$
$317$	−5471.65	−0.969459	−0.484729	+	0.874664i	$0.661082\pi$
$318$	−6221.99	−1.09721
$319$	269.655	0.0473284
$320$	−1708.76	−0.298509
$321$	−356.443	−0.0619772
$322$	−774.887	−0.134108
$323$	2429.85	0.418578
$324$	461.827	0.0791884
$325$	3247.13	0.554210
$326$	−7351.57	−1.24898
$327$	−3099.55	−0.524176
$328$	−1655.37	−0.278666
$329$	−611.879	−0.102535
$330$	940.116	0.156823
$331$	1867.25	0.310070	0.155035	−	0.987909i	$-0.450451\pi$
$331$	1867.25	0.310070	0.155035	+	0.987909i	$0.450451\pi$
$332$	805.612	0.133174
$333$	−2511.94	−0.413374
$334$	−681.446	−0.111638
$335$	−8838.79	−1.44154
$336$	−1294.64	−0.210203
$337$	−10112.3	−1.63457	−0.817285	−	0.576233i	$-0.804522\pi$
$337$	−10112.3	−1.63457	−0.817285	+	0.576233i	$0.804522\pi$
$338$	−13882.9	−2.23411
$339$	−1294.38	−0.207377
$340$	−3208.42	−0.511768
$341$	247.309	0.0392744
$342$	1309.71	0.207079
$343$	3664.13	0.576807
$344$	2636.48	0.413225
$345$	1021.63	0.159428
$346$	3926.43	0.610075
$347$	−6032.03	−0.933188	−0.466594	−	0.884472i	$-0.654519\pi$
$347$	−6032.03	−0.933188	−0.466594	+	0.884472i	$0.654519\pi$
$348$	−496.036	−0.0764090
$349$	8001.09	1.22719	0.613594	−	0.789622i	$-0.289723\pi$
$349$	8001.09	1.22719	0.613594	+	0.789622i	$0.289723\pi$
$350$	−872.291	−0.133217
$351$	2082.25	0.316644
$352$	2020.97	0.306017
$353$	−11369.0	−1.71419	−0.857095	−	0.515159i	$-0.827733\pi$
$353$	−11369.0	−1.71419	−0.857095	+	0.515159i	$0.827733\pi$
$354$	1637.51	0.245855
$355$	−4731.56	−0.707395
$356$	5532.47	0.823652
$357$	−1037.77	−0.153850
$358$	−11438.1	−1.68861
$359$	4888.10	0.718618	0.359309	−	0.933219i	$-0.383012\pi$
$359$	4888.10	0.718618	0.359309	+	0.933219i	$0.383012\pi$
$360$	697.149	0.102064
$361$	−5313.40	−0.774662
$362$	12304.3	1.78646
$363$	3733.62	0.539846
$364$	2460.98	0.354369
$365$	−2410.41	−0.345662
$366$	3088.49	0.441087
$367$	1649.48	0.234611	0.117306	−	0.993096i	$-0.462574\pi$
$367$	1649.48	0.234611	0.117306	+	0.993096i	$0.462574\pi$
$368$	2883.96	0.408523
$369$	−1751.13	−0.247047
$370$	9406.26	1.32164
$371$	3135.95	0.438842
$372$	−454.931	−0.0634061
$373$	12769.2	1.77256	0.886280	−	0.463151i	$-0.153281\pi$
$373$	12769.2	1.77256	0.886280	+	0.463151i	$0.153281\pi$
$374$	2127.29	0.294117
$375$	4564.31	0.628533
$376$	930.117	0.127572
$377$	−2236.49	−0.305531
$378$	−559.364	−0.0761126
$379$	−11762.9	−1.59424	−0.797122	−	0.603818i	$-0.793645\pi$
$379$	−11762.9	−1.59424	−0.797122	+	0.603818i	$0.793645\pi$
$380$	−2040.83	−0.275506
$381$	5691.28	0.765284
$382$	16465.1	2.20531
$383$	2282.30	0.304491	0.152245	−	0.988343i	$-0.451350\pi$
$383$	2282.30	0.304491	0.152245	+	0.988343i	$0.451350\pi$
$384$	3132.16	0.416244
$385$	−473.828	−0.0627234
$386$	−1312.77	−0.173104
$387$	2789.00	0.366338
$388$	−16.7125	−0.00218673
$389$	2444.93	0.318670	0.159335	−	0.987225i	$-0.449065\pi$
$389$	2444.93	0.318670	0.159335	+	0.987225i	$0.449065\pi$
$390$	−7797.22	−1.01238
$391$	2311.75	0.299003
$392$	−2651.67	−0.341657
$393$	−6081.83	−0.780630
$394$	8833.53	1.12951
$395$	7967.20	1.01487
$396$	477.141	0.0605485
$397$	8906.54	1.12596	0.562980	−	0.826470i	$-0.309655\pi$
$397$	8906.54	1.12596	0.562980	+	0.826470i	$0.309655\pi$
$398$	−10948.8	−1.37893
$399$	−660.108	−0.0828239
$400$	3246.47	0.405809
$401$	−9730.86	−1.21181	−0.605905	−	0.795537i	$-0.707189\pi$
$401$	−9730.86	−1.21181	−0.605905	+	0.795537i	$0.707189\pi$
$402$	−10780.4	−1.33750
$403$	−2051.16	−0.253537
$404$	10111.2	1.24517
$405$	737.480	0.0904831
$406$	600.798	0.0734412
$407$	−2595.24	−0.316072
$408$	1577.51	0.191417
$409$	−75.4913	−0.00912666	−0.00456333	−	0.999990i	$-0.501453\pi$
$409$	−75.4913	−0.00912666	−0.00456333	+	0.999990i	$0.501453\pi$
$410$	6557.32	0.789861
$411$	3030.13	0.363663
$412$	7738.33	0.925340
$413$	−825.321	−0.0983326
$414$	1246.05	0.147923
$415$	1286.46	0.152169
$416$	−16761.7	−1.97551
$417$	6262.04	0.735379
$418$	1353.14	0.158336
$419$	−5408.69	−0.630625	−0.315313	−	0.948988i	$-0.602109\pi$
$419$	−5408.69	−0.630625	−0.315313	+	0.948988i	$0.602109\pi$
$420$	871.617	0.101263
$421$	14091.3	1.63127	0.815637	−	0.578564i	$-0.196387\pi$
$421$	14091.3	1.63127	0.815637	+	0.578564i	$0.196387\pi$
$422$	18755.9	2.16356
$423$	983.925	0.113097
$424$	−4766.95	−0.545999
$425$	2602.33	0.297016
$426$	−5770.93	−0.656344
$427$	−1556.63	−0.176418
$428$	677.426	0.0765062
$429$	2151.30	0.242111
$430$	−10443.7	−1.17126
$431$	−14739.4	−1.64727	−0.823635	−	0.567121i	$-0.808057\pi$
$431$	−14739.4	−1.64727	−0.823635	+	0.567121i	$0.808057\pi$
$432$	2081.83	0.231856
$433$	−3554.21	−0.394467	−0.197234	−	0.980357i	$-0.563196\pi$
$433$	−3554.21	−0.394467	−0.197234	+	0.980357i	$0.563196\pi$
$434$	551.012	0.0609434
$435$	−792.108	−0.0873073
$436$	5890.76	0.647055
$437$	1470.47	0.160966
$438$	−2939.90	−0.320716
$439$	−13317.5	−1.44786	−0.723929	−	0.689875i	$-0.757666\pi$
$439$	−13317.5	−1.44786	−0.723929	+	0.689875i	$0.757666\pi$
$440$	720.266	0.0780394
$441$	−2805.07	−0.302891
$442$	−17643.6	−1.89868
$443$	−602.583	−0.0646266	−0.0323133	−	0.999478i	$-0.510287\pi$
$443$	−602.583	−0.0646266	−0.0323133	+	0.999478i	$0.510287\pi$
$444$	4774.00	0.510279
$445$	8834.66	0.941130
$446$	13822.4	1.46751
$447$	−2186.46	−0.231356
$448$	1050.42	0.110776
$449$	−16961.3	−1.78275	−0.891376	−	0.453266i	$-0.850259\pi$
$449$	−16961.3	−1.78275	−0.891376	+	0.453266i	$0.850259\pi$
$450$	1402.68	0.146940
$451$	−1809.20	−0.188896
$452$	2459.99	0.255992
$453$	−3135.02	−0.325157
$454$	−19966.2	−2.06401
$455$	3929.88	0.404914
$456$	1003.43	0.103048
$457$	13470.4	1.37882	0.689410	−	0.724371i	$-0.257870\pi$
$457$	13470.4	1.37882	0.689410	+	0.724371i	$0.257870\pi$
$458$	3363.07	0.343113
$459$	1668.77	0.169698
$460$	−1941.63	−0.196802
$461$	5394.71	0.545025	0.272513	−	0.962152i	$-0.412145\pi$
$461$	5394.71	0.545025	0.272513	+	0.962152i	$0.412145\pi$
$462$	−577.912	−0.0581968
$463$	13083.9	1.31331	0.656654	−	0.754192i	$-0.271971\pi$
$463$	13083.9	1.31331	0.656654	+	0.754192i	$0.271971\pi$
$464$	−2236.04	−0.223718
$465$	−726.469	−0.0724498
$466$	−18670.7	−1.85602
$467$	5391.09	0.534197	0.267099	−	0.963669i	$-0.413935\pi$
$467$	5391.09	0.534197	0.267099	+	0.963669i	$0.413935\pi$
$468$	−3957.36	−0.390874
$469$	5433.42	0.534951
$470$	−3684.42	−0.361595
$471$	5516.23	0.539648
$472$	1254.57	0.122344
$473$	2881.48	0.280107
$474$	9717.33	0.941628
$475$	1655.31	0.159896
$476$	1972.30	0.189916
$477$	−5042.73	−0.484047
$478$	−3044.98	−0.291369
$479$	15518.9	1.48033	0.740163	−	0.672428i	$-0.234748\pi$
$479$	15518.9	1.48033	0.740163	+	0.672428i	$0.234748\pi$
$480$	−5936.58	−0.564514
$481$	21524.6	2.04041
$482$	3544.60	0.334963
$483$	−628.022	−0.0591635
$484$	−7095.82	−0.666399
$485$	−26.6878	−0.00249862
$486$	899.480	0.0839531
$487$	−9407.95	−0.875390	−0.437695	−	0.899123i	$-0.644205\pi$
$487$	−9407.95	−0.875390	−0.437695	+	0.899123i	$0.644205\pi$
$488$	2366.23	0.219497
$489$	−5958.22	−0.551002
$490$	10503.9	0.968407
$491$	−705.615	−0.0648553	−0.0324277	−	0.999474i	$-0.510324\pi$
$491$	−705.615	−0.0648553	−0.0324277	+	0.999474i	$0.510324\pi$
$492$	3328.06	0.304961
$493$	−1792.38	−0.163742
$494$	−11222.8	−1.02214
$495$	761.934	0.0691846
$496$	−2050.74	−0.185647
$497$	2908.61	0.262513
$498$	1569.06	0.141187
$499$	9363.34	0.840001	0.420000	−	0.907524i	$-0.362030\pi$
$499$	9363.34	0.840001	0.420000	+	0.907524i	$0.362030\pi$
$500$	−8674.56	−0.775876
$501$	−552.291	−0.0492506
$502$	−12617.6	−1.12181
$503$	2275.13	0.201676	0.100838	−	0.994903i	$-0.467848\pi$
$503$	2275.13	0.201676	0.100838	+	0.994903i	$0.467848\pi$
$504$	−428.554	−0.0378757
$505$	16146.3	1.42277
$506$	1287.37	0.113104
$507$	−11251.6	−0.985606
$508$	−10816.4	−0.944685
$509$	2648.37	0.230623	0.115311	−	0.993329i	$-0.463213\pi$
$509$	2648.37	0.230623	0.115311	+	0.993329i	$0.463213\pi$
$510$	−6248.90	−0.542561
$511$	1481.74	0.128274
$512$	−11510.4	−0.993541
$513$	1061.48	0.0913557
$514$	−18150.4	−1.55755
$515$	12357.2	1.05732
$516$	−5300.55	−0.452217
$517$	1016.55	0.0864756
$518$	−5782.26	−0.490460
$519$	3182.25	0.269143
$520$	−5973.81	−0.503787
$521$	−10097.1	−0.849061	−0.424530	−	0.905414i	$-0.639561\pi$
$521$	−10097.1	−0.849061	−0.424530	+	0.905414i	$0.639561\pi$
$522$	−966.108	−0.0810065
$523$	−8221.06	−0.687346	−0.343673	−	0.939089i	$-0.611671\pi$
$523$	−8221.06	−0.687346	−0.343673	+	0.939089i	$0.611671\pi$
$524$	11558.6	0.963629
$525$	−706.964	−0.0587704
$526$	−31154.1	−2.58248
$527$	−1643.85	−0.135877
$528$	2150.86	0.177281
$529$	−10768.0	−0.885017
$530$	18883.1	1.54760
$531$	1327.15	0.108462
$532$	1254.55	0.102240
$533$	15005.3	1.21942
$534$	10775.3	0.873211
$535$	1081.77	0.0874183
$536$	−8259.34	−0.665577
$537$	−9270.22	−0.744952
$538$	−19553.5	−1.56694
$539$	−2898.09	−0.231595
$540$	−1401.60	−0.111695
$541$	22669.4	1.80154	0.900770	−	0.434297i	$-0.143003\pi$
$541$	22669.4	1.80154	0.900770	+	0.434297i	$0.143003\pi$
$542$	26094.7	2.06801
$543$	9972.21	0.788119
$544$	−13433.3	−1.05873
$545$	9406.81	0.739345
$546$	4793.15	0.375692
$547$	−9705.61	−0.758650	−0.379325	−	0.925263i	$-0.623844\pi$
$547$	−9705.61	−0.758650	−0.379325	+	0.925263i	$0.623844\pi$
$548$	−5758.83	−0.448914
$549$	2503.12	0.194592
$550$	1449.19	0.112352
$551$	−1140.11	−0.0881492
$552$	954.656	0.0736103
$553$	−4897.63	−0.376616
$554$	−1879.90	−0.144168
$555$	7623.48	0.583061
$556$	−11901.1	−0.907770
$557$	4989.90	0.379585	0.189792	−	0.981824i	$-0.439219\pi$
$557$	4989.90	0.379585	0.189792	+	0.981824i	$0.439219\pi$
$558$	−886.050	−0.0672213
$559$	−23898.7	−1.80824
$560$	3929.09	0.296490
$561$	1724.10	0.129754
$562$	32488.6	2.43852
$563$	−2332.02	−0.174570	−0.0872852	−	0.996183i	$-0.527819\pi$
$563$	−2332.02	−0.174570	−0.0872852	+	0.996183i	$0.527819\pi$
$564$	−1869.97	−0.139610
$565$	3928.30	0.292504
$566$	3927.62	0.291679
$567$	−453.347	−0.0335781
$568$	−4421.38	−0.326614
$569$	−23481.5	−1.73004	−0.865022	−	0.501733i	$-0.832696\pi$
$569$	−23481.5	−1.73004	−0.865022	+	0.501733i	$0.832696\pi$
$570$	−3974.84	−0.292083
$571$	7973.60	0.584387	0.292193	−	0.956359i	$-0.405615\pi$
$571$	7973.60	0.584387	0.292193	+	0.956359i	$0.405615\pi$
$572$	−4088.58	−0.298867
$573$	13344.5	0.972902
$574$	−4030.95	−0.293116
$575$	1574.85	0.114219
$576$	−1689.12	−0.122187
$577$	−15243.9	−1.09985	−0.549924	−	0.835215i	$-0.685343\pi$
$577$	−15243.9	−1.09985	−0.549924	+	0.835215i	$0.685343\pi$
$578$	4045.76	0.291144
$579$	−1063.96	−0.0763672
$580$	1505.42	0.107774
$581$	−790.821	−0.0564695
$582$	−32.5503	−0.00231830
$583$	−5209.94	−0.370109
$584$	−2252.39	−0.159597
$585$	−6319.41	−0.446624
$586$	24300.8	1.71306
$587$	5044.41	0.354693	0.177347	−	0.984148i	$-0.443249\pi$
$587$	5044.41	0.354693	0.177347	+	0.984148i	$0.443249\pi$
$588$	5331.10	0.373896
$589$	−1045.63	−0.0731485
$590$	−4969.66	−0.346776
$591$	7159.30	0.498298
$592$	21520.3	1.49405
$593$	−89.3318	−0.00618620	−0.00309310	−	0.999995i	$-0.500985\pi$
$593$	−89.3318	−0.00618620	−0.00309310	+	0.999995i	$0.500985\pi$
$594$	929.306	0.0641917
$595$	3149.51	0.217004
$596$	4155.41	0.285591
$597$	−8873.66	−0.608333
$598$	−10677.3	−0.730146
$599$	7097.18	0.484112	0.242056	−	0.970262i	$-0.422178\pi$
$599$	7097.18	0.484112	0.242056	+	0.970262i	$0.422178\pi$
$600$	1074.66	0.0731211
$601$	6290.97	0.426978	0.213489	−	0.976945i	$-0.431517\pi$
$601$	6290.97	0.426978	0.213489	+	0.976945i	$0.431517\pi$
$602$	6420.03	0.434652
$603$	−8737.16	−0.590057
$604$	5958.18	0.401382
$605$	−11331.1	−0.761448
$606$	19693.1	1.32009
$607$	−18590.2	−1.24309	−0.621544	−	0.783379i	$-0.713494\pi$
$607$	−18590.2	−1.24309	−0.621544	+	0.783379i	$0.713494\pi$
$608$	−8544.73	−0.569958
$609$	486.928	0.0323996
$610$	−9373.25	−0.622150
$611$	−8431.18	−0.558247
$612$	−3171.53	−0.209480
$613$	−17971.9	−1.18414	−0.592069	−	0.805887i	$-0.701689\pi$
$613$	−17971.9	−1.18414	−0.592069	+	0.805887i	$0.701689\pi$
$614$	−25124.7	−1.65139
$615$	5314.50	0.348458
$616$	−442.765	−0.0289603
$617$	−1586.93	−0.103545	−0.0517726	−	0.998659i	$-0.516487\pi$
$617$	−1586.93	−0.103545	−0.0517726	+	0.998659i	$0.516487\pi$
$618$	15071.6	0.981018
$619$	−15260.6	−0.990911	−0.495455	−	0.868633i	$-0.664999\pi$
$619$	−15260.6	−0.990911	−0.495455	+	0.868633i	$0.664999\pi$
$620$	1380.67	0.0894339
$621$	1009.88	0.0652581
$622$	39263.6	2.53108
$623$	−5430.88	−0.349252
$624$	−17839.0	−1.14444
$625$	−8589.11	−0.549703
$626$	−9326.26	−0.595451
$627$	1096.68	0.0698518
$628$	−10483.7	−0.666155
$629$	17250.4	1.09351
$630$	1697.61	0.107356
$631$	−8338.07	−0.526043	−0.263022	−	0.964790i	$-0.584719\pi$
$631$	−8338.07	−0.526043	−0.263022	+	0.964790i	$0.584719\pi$
$632$	7444.89	0.468579
$633$	15201.0	0.954481
$634$	20253.6	1.26873
$635$	−17272.4	−1.07943
$636$	9583.81	0.597520
$637$	24036.5	1.49507
$638$	−998.144	−0.0619387
$639$	−4677.16	−0.289555
$640$	−9505.79	−0.587108
$641$	−1971.38	−0.121474	−0.0607370	−	0.998154i	$-0.519345\pi$
$641$	−1971.38	−0.121474	−0.0607370	+	0.998154i	$0.519345\pi$
$642$	1319.39	0.0811096
$643$	22321.5	1.36901	0.684507	−	0.729007i	$-0.260018\pi$
$643$	22321.5	1.36901	0.684507	+	0.729007i	$0.260018\pi$
$644$	1193.57	0.0730329
$645$	−8464.33	−0.516717
$646$	−8994.26	−0.547793
$647$	−28115.8	−1.70842	−0.854210	−	0.519928i	$-0.825959\pi$
$647$	−28115.8	−1.70842	−0.854210	+	0.519928i	$0.825959\pi$
$648$	689.133	0.0417773
$649$	1371.16	0.0829316
$650$	−12019.4	−0.725294
$651$	446.578	0.0268860
$652$	11323.7	0.680170
$653$	14472.2	0.867289	0.433644	−	0.901084i	$-0.357227\pi$
$653$	14472.2	0.867289	0.433644	+	0.901084i	$0.357227\pi$
$654$	11473.2	0.685989
$655$	18457.7	1.10107
$656$	15002.3	0.892897
$657$	−2382.69	−0.141488
$658$	2264.91	0.134187
$659$	−6799.54	−0.401931	−0.200965	−	0.979598i	$-0.564408\pi$
$659$	−6799.54	−0.401931	−0.200965	+	0.979598i	$0.564408\pi$
$660$	−1448.07	−0.0854032
$661$	5028.70	0.295906	0.147953	−	0.988994i	$-0.452732\pi$
$661$	5028.70	0.295906	0.147953	+	0.988994i	$0.452732\pi$
$662$	−6911.72	−0.405788
$663$	−14299.6	−0.837630
$664$	1202.13	0.0702584
$665$	2003.36	0.116822
$666$	9298.11	0.540983
$667$	−1084.69	−0.0629676
$668$	1049.64	0.0607961
$669$	11202.6	0.647412
$670$	32717.3	1.88654
$671$	2586.13	0.148787
$672$	3649.36	0.209490
$673$	6053.04	0.346698	0.173349	−	0.984860i	$-0.444541\pi$
$673$	6053.04	0.346698	0.173349	+	0.984860i	$0.444541\pi$
$674$	37431.2	2.13916
$675$	1136.83	0.0648244
$676$	21383.9	1.21666
$677$	8724.62	0.495294	0.247647	−	0.968850i	$-0.420343\pi$
$677$	8724.62	0.495294	0.247647	+	0.968850i	$0.420343\pi$
$678$	4791.22	0.271395
$679$	16.4057	0.000927234 0
$680$	−4787.57	−0.269993
$681$	−16182.0	−0.910564
$682$	−915.431	−0.0513983
$683$	−4392.89	−0.246104	−0.123052	−	0.992400i	$-0.539268\pi$
$683$	−4392.89	−0.246104	−0.123052	+	0.992400i	$0.539268\pi$
$684$	−2017.36	−0.112772
$685$	−9196.13	−0.512943
$686$	−13563.0	−0.754867
$687$	2725.66	0.151369
$688$	−23893.9	−1.32405
$689$	43210.7	2.38926
$690$	−3781.63	−0.208644
$691$	−22695.9	−1.24948	−0.624741	−	0.780832i	$-0.714795\pi$
$691$	−22695.9	−1.24948	−0.624741	+	0.780832i	$0.714795\pi$
$692$	−6047.92	−0.332236
$693$	−468.380	−0.0256743
$694$	22327.9	1.22126
$695$	−19004.6	−1.03725
$696$	−740.180	−0.0403110
$697$	12025.7	0.653521
$698$	−29616.5	−1.60602
$699$	−15132.0	−0.818808
$700$	1343.60	0.0725476
$701$	−7055.27	−0.380134	−0.190067	−	0.981771i	$-0.560871\pi$
$701$	−7055.27	−0.380134	−0.190067	+	0.981771i	$0.560871\pi$
$702$	−7707.57	−0.414393
$703$	10972.7	0.588684
$704$	−1745.13	−0.0934261
$705$	−2986.11	−0.159523
$706$	42082.9	2.24336
$707$	−9925.51	−0.527987
$708$	−2522.27	−0.133888
$709$	12058.3	0.638727	0.319363	−	0.947632i	$-0.396531\pi$
$709$	12058.3	0.638727	0.319363	+	0.947632i	$0.396531\pi$
$710$	17514.2	0.925768
$711$	7875.59	0.415412
$712$	8255.49	0.434533
$713$	−994.806	−0.0522522
$714$	3841.35	0.201343
$715$	−6528.96	−0.341495
$716$	17618.2	0.919587
$717$	−2467.86	−0.128541
$718$	−18093.6	−0.940456
$719$	25400.6	1.31750	0.658750	−	0.752362i	$-0.271086\pi$
$719$	25400.6	1.31750	0.658750	+	0.752362i	$0.271086\pi$
$720$	−6318.13	−0.327031
$721$	−7596.25	−0.392370
$722$	19667.9	1.01380
$723$	2872.79	0.147773
$724$	−18952.4	−0.972873
$725$	−1221.04	−0.0625492
$726$	−13820.2	−0.706496
$727$	−797.068	−0.0406625	−0.0203312	−	0.999793i	$-0.506472\pi$
$727$	−797.068	−0.0406625	−0.0203312	+	0.999793i	$0.506472\pi$
$728$	3672.25	0.186954
$729$	729.000	0.0370370
$730$	8922.28	0.452368
$731$	−19153.1	−0.969086
$732$	−4757.24	−0.240209
$733$	7975.10	0.401865	0.200933	−	0.979605i	$-0.435603\pi$
$733$	7975.10	0.401865	0.200933	+	0.979605i	$0.435603\pi$
$734$	−6105.67	−0.307036
$735$	8513.11	0.427226
$736$	−8129.39	−0.407138
$737$	−9026.88	−0.451166
$738$	6481.93	0.323310
$739$	−17739.6	−0.883035	−0.441517	−	0.897253i	$-0.645560\pi$
$739$	−17739.6	−0.883035	−0.441517	+	0.897253i	$0.645560\pi$
$740$	−14488.6	−0.719745
$741$	−9095.74	−0.450932
$742$	−11607.9	−0.574312
$743$	2085.57	0.102977	0.0514887	−	0.998674i	$-0.483603\pi$
$743$	2085.57	0.102977	0.0514887	+	0.998674i	$0.483603\pi$
$744$	−678.844	−0.0334511
$745$	6635.68	0.326325
$746$	−47266.0	−2.31975
$747$	1271.67	0.0622865
$748$	−3276.70	−0.160171
$749$	−664.988	−0.0324408
$750$	−16895.1	−0.822561
$751$	−34644.3	−1.68334	−0.841669	−	0.539994i	$-0.818427\pi$
$751$	−34644.3	−1.68334	−0.841669	+	0.539994i	$0.818427\pi$
$752$	−8429.47	−0.408765
$753$	−10226.2	−0.494903
$754$	8278.50	0.399848
$755$	9514.47	0.458632
$756$	861.595	0.0414496
$757$	−11557.8	−0.554923	−0.277461	−	0.960737i	$-0.589493\pi$
$757$	−11557.8	−0.554923	−0.277461	+	0.960737i	$0.589493\pi$
$758$	43541.0	2.08639
$759$	1043.37	0.0498972
$760$	−3045.30	−0.145348
$761$	29295.6	1.39548	0.697742	−	0.716349i	$-0.254188\pi$
$761$	29295.6	1.39548	0.697742	+	0.716349i	$0.254188\pi$
$762$	−21066.6	−1.00153
$763$	−5782.60	−0.274370
$764$	−25361.4	−1.20097
$765$	−5064.54	−0.239358
$766$	−8448.06	−0.398487
$767$	−11372.2	−0.535368
$768$	−16098.2	−0.756373
$769$	−5777.96	−0.270948	−0.135474	−	0.990781i	$-0.543256\pi$
$769$	−5777.96	−0.270948	−0.135474	+	0.990781i	$0.543256\pi$
$770$	1753.90	0.0820861
$771$	−14710.3	−0.687133
$772$	2022.08	0.0942695
$773$	10733.4	0.499424	0.249712	−	0.968320i	$-0.419664\pi$
$773$	10733.4	0.499424	0.249712	+	0.968320i	$0.419664\pi$
$774$	−10323.7	−0.479427
$775$	−1119.85	−0.0519049
$776$	−24.9382	−0.00115365
$777$	−4686.34	−0.216373
$778$	−9050.05	−0.417044
$779$	7649.35	0.351818
$780$	12010.2	0.551324
$781$	−4832.25	−0.221398
$782$	−8557.07	−0.391305
$783$	−783.000	−0.0357371
$784$	24031.6	1.09473
$785$	−16741.2	−0.761169
$786$	22512.3	1.02161
$787$	−7268.94	−0.329237	−0.164619	−	0.986357i	$-0.552639\pi$
$787$	−7268.94	−0.329237	−0.164619	+	0.986357i	$0.552639\pi$
$788$	−13606.4	−0.615111
$789$	−25249.4	−1.13929
$790$	−29491.1	−1.32816
$791$	−2414.82	−0.108548
$792$	711.984	0.0319435
$793$	−21449.1	−0.960504
$794$	−32968.1	−1.47354
$795$	15304.2	0.682745
$796$	16864.6	0.750941
$797$	7518.00	0.334130	0.167065	−	0.985946i	$-0.446571\pi$
$797$	7518.00	0.334130	0.167065	+	0.985946i	$0.446571\pi$
$798$	2443.43	0.108392
$799$	−6756.97	−0.299179
$800$	−9151.26	−0.404432
$801$	8733.08	0.385229
$802$	36019.4	1.58590
$803$	−2461.70	−0.108184
$804$	16605.1	0.728381
$805$	1905.98	0.0834497
$806$	7592.49	0.331804
$807$	−15847.5	−0.691275
$808$	15087.8	0.656913
$809$	−41676.1	−1.81119	−0.905595	−	0.424143i	$-0.860576\pi$
$809$	−41676.1	−1.81119	−0.905595	+	0.424143i	$0.860576\pi$
$810$	−2729.83	−0.118415
$811$	−15200.3	−0.658144	−0.329072	−	0.944305i	$-0.606736\pi$
$811$	−15200.3	−0.658144	−0.329072	+	0.944305i	$0.606736\pi$
$812$	−925.417	−0.0399948
$813$	21148.9	0.912332
$814$	9606.43	0.413643
$815$	18082.6	0.777184
$816$	−14296.7	−0.613337
$817$	−12183.0	−0.521700
$818$	279.436	0.0119441
$819$	3884.70	0.165741
$820$	−10100.3	−0.430145
$821$	−10265.1	−0.436362	−0.218181	−	0.975908i	$-0.570012\pi$
$821$	−10265.1	−0.436362	−0.218181	+	0.975908i	$0.570012\pi$
$822$	−11216.2	−0.475925
$823$	−6168.86	−0.261280	−0.130640	−	0.991430i	$-0.541703\pi$
$823$	−6168.86	−0.261280	−0.130640	+	0.991430i	$0.541703\pi$
$824$	11547.1	0.488180
$825$	1174.52	0.0495656
$826$	3054.98	0.128688
$827$	9610.39	0.404094	0.202047	−	0.979376i	$-0.435241\pi$
$827$	9610.39	0.404094	0.202047	+	0.979376i	$0.435241\pi$
$828$	−1919.31	−0.0805562
$829$	12791.4	0.535902	0.267951	−	0.963433i	$-0.413653\pi$
$829$	12791.4	0.535902	0.267951	+	0.963433i	$0.413653\pi$
$830$	−4761.92	−0.199143
$831$	−1523.60	−0.0636018
$832$	14473.9	0.603116
$833$	19263.5	0.801248
$834$	−23179.3	−0.962391
$835$	1676.15	0.0694676
$836$	−2084.26	−0.0862268
$837$	−718.116	−0.0296556
$838$	20020.6	0.825299
$839$	37025.1	1.52354	0.761769	−	0.647848i	$-0.224331\pi$
$839$	37025.1	1.52354	0.761769	+	0.647848i	$0.224331\pi$
$840$	1300.62	0.0534233
$841$	841.000	0.0344828
$842$	−52159.7	−2.13485
$843$	26331.0	1.07578
$844$	−28889.9	−1.17824
$845$	34147.5	1.39019
$846$	−3642.06	−0.148010
$847$	6965.53	0.282572
$848$	43202.0	1.74948
$849$	3183.21	0.128678
$850$	−9632.70	−0.388704
$851$	10439.4	0.420514
$852$	8889.04	0.357434
$853$	37893.8	1.52105	0.760527	−	0.649306i	$-0.224941\pi$
$853$	37893.8	1.52105	0.760527	+	0.649306i	$0.224941\pi$
$854$	5761.97	0.230879
$855$	−3221.48	−0.128856
$856$	1010.85	0.0403623
$857$	2766.91	0.110287	0.0551434	−	0.998478i	$-0.482438\pi$
$857$	2766.91	0.110287	0.0551434	+	0.998478i	$0.482438\pi$
$858$	−7963.15	−0.316850
$859$	−34378.3	−1.36551	−0.682754	−	0.730648i	$-0.739218\pi$
$859$	−34378.3	−1.36551	−0.682754	+	0.730648i	$0.739218\pi$
$860$	16086.6	0.637848
$861$	−3266.96	−0.129312
$862$	54558.9	2.15578
$863$	−17326.9	−0.683448	−0.341724	−	0.939800i	$-0.611011\pi$
$863$	−17326.9	−0.683448	−0.341724	+	0.939800i	$0.611011\pi$
$864$	−5868.32	−0.231070
$865$	−9657.78	−0.379624
$866$	13156.1	0.516239
$867$	3278.96	0.128442
$868$	−848.731	−0.0331887
$869$	8136.74	0.317630
$870$	2932.04	0.114259
$871$	74868.0	2.91252
$872$	8790.12	0.341366
$873$	−26.3810	−0.00102275
$874$	−5443.03	−0.210656
$875$	8515.29	0.328993
$876$	4528.36	0.174656
$877$	−14753.1	−0.568046	−0.284023	−	0.958817i	$-0.591669\pi$
$877$	−14753.1	−0.568046	−0.284023	+	0.958817i	$0.591669\pi$
$878$	49295.5	1.89481
$879$	19695.0	0.755741
$880$	−6527.63	−0.250053
$881$	−6902.00	−0.263943	−0.131972	−	0.991253i	$-0.542131\pi$
$881$	−6902.00	−0.263943	−0.131972	+	0.991253i	$0.542131\pi$
$882$	10383.2	0.396394
$883$	−25055.7	−0.954915	−0.477458	−	0.878655i	$-0.658442\pi$
$883$	−25055.7	−0.954915	−0.477458	+	0.878655i	$0.658442\pi$
$884$	27176.6	1.03399
$885$	−4027.76	−0.152985
$886$	2230.50	0.0845768
$887$	1065.02	0.0403157	0.0201579	−	0.999797i	$-0.493583\pi$
$887$	1065.02	0.0403157	0.0201579	+	0.999797i	$0.493583\pi$
$888$	7123.71	0.269207
$889$	10617.8	0.400573
$890$	−32702.0	−1.23166
$891$	753.173	0.0283190
$892$	−21290.8	−0.799181
$893$	−4298.01	−0.161061
$894$	8093.32	0.302775
$895$	28134.1	1.05075
$896$	5843.44	0.217875
$897$	−8653.62	−0.322114
$898$	62783.5	2.33309
$899$	771.309	0.0286147
$900$	−2160.56	−0.0800208
$901$	34630.2	1.28047
$902$	6696.87	0.247208
$903$	5203.23	0.191753
$904$	3670.77	0.135053
$905$	−30264.6	−1.11164
$906$	11604.5	0.425533
$907$	−30648.4	−1.12201	−0.561005	−	0.827812i	$-0.689585\pi$
$907$	−30648.4	−1.12201	−0.561005	+	0.827812i	$0.689585\pi$
$908$	30754.1	1.12402
$909$	15960.6	0.582376
$910$	−14546.7	−0.529910
$911$	−30520.2	−1.10997	−0.554983	−	0.831861i	$-0.687275\pi$
$911$	−30520.2	−1.10997	−0.554983	+	0.831861i	$0.687275\pi$
$912$	−9093.89	−0.330185
$913$	1313.84	0.0476251
$914$	−49861.7	−1.80446
$915$	−7596.72	−0.274470
$916$	−5180.18	−0.186854
$917$	−11346.4	−0.408606
$918$	−6177.05	−0.222084
$919$	−44926.8	−1.61262	−0.806310	−	0.591493i	$-0.798539\pi$
$919$	−44926.8	−1.61262	−0.806310	+	0.591493i	$0.798539\pi$
$920$	−2897.28	−0.103827
$921$	−20362.8	−0.728531
$922$	−19968.9	−0.713275
$923$	40078.2	1.42924
$924$	890.166	0.0316930
$925$	11751.6	0.417720
$926$	−48431.0	−1.71873
$927$	12215.1	0.432789
$928$	6303.01	0.222960
$929$	5122.97	0.180925	0.0904624	−	0.995900i	$-0.471166\pi$
$929$	5122.97	0.180925	0.0904624	+	0.995900i	$0.471166\pi$
$930$	2689.07	0.0948151
$931$	12253.2	0.431346
$932$	28758.8	1.01076
$933$	31822.0	1.11662
$934$	−19955.5	−0.699104
$935$	−5232.48	−0.183016
$936$	−5905.12	−0.206213
$937$	30750.7	1.07212	0.536062	−	0.844178i	$-0.319911\pi$
$937$	30750.7	1.07212	0.536062	+	0.844178i	$0.319911\pi$
$938$	−20112.1	−0.700090
$939$	−7558.64	−0.262691
$940$	5675.16	0.196919
$941$	21790.9	0.754902	0.377451	−	0.926030i	$-0.376801\pi$
$941$	21790.9	0.754902	0.377451	+	0.926030i	$0.376801\pi$
$942$	−20418.7	−0.706237
$943$	7277.54	0.251314
$944$	−11369.9	−0.392012
$945$	1375.86	0.0473616
$946$	−10666.0	−0.366576
$947$	30011.6	1.02983	0.514914	−	0.857242i	$-0.327824\pi$
$947$	30011.6	1.02983	0.514914	+	0.857242i	$0.327824\pi$
$948$	−14967.7	−0.512794
$949$	20417.1	0.698385
$950$	−6127.22	−0.209256
$951$	16414.9	0.559717
$952$	2943.04	0.100194
$953$	−9552.71	−0.324704	−0.162352	−	0.986733i	$-0.551908\pi$
$953$	−9552.71	−0.324704	−0.162352	+	0.986733i	$0.551908\pi$
$954$	18666.0	0.633473
$955$	−40499.1	−1.37227
$956$	4690.23	0.158674
$957$	−808.964	−0.0273251
$958$	−57444.1	−1.93730
$959$	5653.09	0.190352
$960$	5126.29	0.172344
$961$	−29083.6	−0.976255
$962$	−79674.8	−2.67029
$963$	1069.33	0.0357826
$964$	−5459.80	−0.182415
$965$	3229.00	0.107715
$966$	2324.66	0.0774273
$967$	310.364	0.0103212	0.00516062	−	0.999987i	$-0.498357\pi$
$967$	310.364	0.0103212	0.00516062	+	0.999987i	$0.498357\pi$
$968$	−10588.3	−0.351571
$969$	−7289.56	−0.241666
$970$	98.7866	0.00326995
$971$	42733.1	1.41233	0.706164	−	0.708048i	$-0.250424\pi$
$971$	42733.1	1.41233	0.706164	+	0.708048i	$0.250424\pi$
$972$	−1385.48	−0.0457194
$973$	11682.6	0.384920
$974$	34824.1	1.14562
$975$	−9741.38	−0.319973
$976$	−21444.7	−0.703309
$977$	−46297.0	−1.51604	−0.758021	−	0.652230i	$-0.773834\pi$
$977$	−46297.0	−1.51604	−0.758021	+	0.652230i	$0.773834\pi$
$978$	22054.7	0.721096
$979$	9022.67	0.294551
$980$	−16179.3	−0.527378
$981$	9298.65	0.302633
$982$	2611.88	0.0848762
$983$	−50156.9	−1.62742	−0.813711	−	0.581270i	$-0.802556\pi$
$983$	−50156.9	−1.62742	−0.813711	+	0.581270i	$0.802556\pi$
$984$	4966.10	0.160888
$985$	−21727.7	−0.702845
$986$	6634.61	0.214289
$987$	1835.64	0.0591985
$988$	17286.6	0.556641
$989$	−11590.8	−0.372666
$990$	−2820.35	−0.0905419
$991$	48947.5	1.56899	0.784495	−	0.620135i	$-0.212922\pi$
$991$	48947.5	1.56899	0.784495	+	0.620135i	$0.212922\pi$
$992$	5780.71	0.185018
$993$	−5601.74	−0.179019
$994$	−10766.4	−0.343551
$995$	26930.6	0.858048
$996$	−2416.84	−0.0768880
$997$	−27637.1	−0.877911	−0.438956	−	0.898509i	$-0.644651\pi$
$997$	−27637.1	−0.877911	−0.438956	+	0.898509i	$0.644651\pi$
$998$	−34659.0	−1.09931
$999$	7535.83	0.238662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.4.a.b.1.1	✓	2
3.2	odd	2		261.4.a.a.1.2		2
4.3	odd	2		1392.4.a.k.1.2		2
5.4	even	2		2175.4.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.4.a.b.1.1	✓	2	1.1	even	1	trivial
261.4.a.a.1.2		2	3.2	odd	2
1392.4.a.k.1.2		2	4.3	odd	2
2175.4.a.f.1.2		2	5.4	even	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:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	87.a (trivial)

\(f(q)\)	\(=\)	\(q-3.70156 q^{2} -3.00000 q^{3} +5.70156 q^{4} +9.10469 q^{5} +11.1047 q^{6} -5.59688 q^{7} +8.50781 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.70156 q^{2} -3.00000 q^{3} +5.70156 q^{4} +9.10469 q^{5} +11.1047 q^{6} -5.59688 q^{7} +8.50781 q^{8} +9.00000 q^{9} -33.7016 q^{10} +9.29844 q^{11} -17.1047 q^{12} -77.1203 q^{13} +20.7172 q^{14} -27.3141 q^{15} -77.1047 q^{16} -61.8062 q^{17} -33.3141 q^{18} -39.3141 q^{19} +51.9109 q^{20} +16.7906 q^{21} -34.4187 q^{22} -37.4031 q^{23} -25.5234 q^{24} -42.1047 q^{25} +285.466 q^{26} -27.0000 q^{27} -31.9109 q^{28} +29.0000 q^{29} +101.105 q^{30} +26.5969 q^{31} +217.345 q^{32} -27.8953 q^{33} +228.780 q^{34} -50.9578 q^{35} +51.3141 q^{36} -279.105 q^{37} +145.523 q^{38} +231.361 q^{39} +77.4609 q^{40} -194.570 q^{41} -62.1515 q^{42} +309.889 q^{43} +53.0156 q^{44} +81.9422 q^{45} +138.450 q^{46} +109.325 q^{47} +231.314 q^{48} -311.675 q^{49} +155.853 q^{50} +185.419 q^{51} -439.706 q^{52} -560.303 q^{53} +99.9422 q^{54} +84.6594 q^{55} -47.6172 q^{56} +117.942 q^{57} -107.345 q^{58} +147.461 q^{59} -155.733 q^{60} +278.125 q^{61} -98.4500 q^{62} -50.3719 q^{63} -187.680 q^{64} -702.156 q^{65} +103.256 q^{66} -970.795 q^{67} -352.392 q^{68} +112.209 q^{69} +188.623 q^{70} -519.684 q^{71} +76.5703 q^{72} -264.744 q^{73} +1033.12 q^{74} +126.314 q^{75} -224.152 q^{76} -52.0422 q^{77} -856.397 q^{78} +875.066 q^{79} -702.014 q^{80} +81.0000 q^{81} +720.214 q^{82} +141.297 q^{83} +95.7328 q^{84} -562.727 q^{85} -1147.07 q^{86} -87.0000 q^{87} +79.1093 q^{88} +970.342 q^{89} -303.314 q^{90} +431.633 q^{91} -213.256 q^{92} -79.7906 q^{93} -404.673 q^{94} -357.942 q^{95} -652.036 q^{96} -2.93122 q^{97} +1153.68 q^{98} +83.6859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 6 q^{3} + 5 q^{4} - q^{5} + 3 q^{6} - 24 q^{7} - 15 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - q^2 - 6 * q^3 + 5 * q^4 - q^5 + 3 * q^6 - 24 * q^7 - 15 * q^8 + 18 * q^9	\( 2 q - q^{2} - 6 q^{3} + 5 q^{4} - q^{5} + 3 q^{6} - 24 q^{7} - 15 q^{8} + 18 q^{9} - 61 q^{10} + 25 q^{11} - 15 q^{12} - 71 q^{13} - 29 q^{14} + 3 q^{15} - 135 q^{16} - 98 q^{17} - 9 q^{18} - 21 q^{19} + 59 q^{20} + 72 q^{21} + 8 q^{22} - 62 q^{23} + 45 q^{24} - 65 q^{25} + 302 q^{26} - 54 q^{27} - 19 q^{28} + 58 q^{29} + 183 q^{30} + 66 q^{31} + 249 q^{32} - 75 q^{33} + 131 q^{34} + 135 q^{35} + 45 q^{36} - 539 q^{37} + 195 q^{38} + 213 q^{39} + 315 q^{40} - 101 q^{41} + 87 q^{42} - 155 q^{43} + 42 q^{44} - 9 q^{45} + 72 q^{46} + 526 q^{47} + 405 q^{48} - 316 q^{49} + 94 q^{50} + 294 q^{51} - 444 q^{52} - 698 q^{53} + 27 q^{54} - 74 q^{55} + 385 q^{56} + 63 q^{57} - 29 q^{58} + 455 q^{59} - 177 q^{60} + 44 q^{61} + 8 q^{62} - 216 q^{63} + 361 q^{64} - 764 q^{65} - 24 q^{66} - 1551 q^{67} - 327 q^{68} + 186 q^{69} + 691 q^{70} + 126 q^{71} - 135 q^{72} - 760 q^{73} + 331 q^{74} + 195 q^{75} - 237 q^{76} - 341 q^{77} - 906 q^{78} - 158 q^{79} - 117 q^{80} + 162 q^{81} + 973 q^{82} - 934 q^{83} + 57 q^{84} - 197 q^{85} - 2403 q^{86} - 174 q^{87} - 290 q^{88} - 691 q^{89} - 549 q^{90} + 319 q^{91} - 196 q^{92} - 198 q^{93} + 721 q^{94} - 543 q^{95} - 747 q^{96} + 532 q^{97} + 1142 q^{98} + 225 q^{99}+O(q^{100}) \) 2 * q - q^2 - 6 * q^3 + 5 * q^4 - q^5 + 3 * q^6 - 24 * q^7 - 15 * q^8 + 18 * q^9 - 61 * q^10 + 25 * q^11 - 15 * q^12 - 71 * q^13 - 29 * q^14 + 3 * q^15 - 135 * q^16 - 98 * q^17 - 9 * q^18 - 21 * q^19 + 59 * q^20 + 72 * q^21 + 8 * q^22 - 62 * q^23 + 45 * q^24 - 65 * q^25 + 302 * q^26 - 54 * q^27 - 19 * q^28 + 58 * q^29 + 183 * q^30 + 66 * q^31 + 249 * q^32 - 75 * q^33 + 131 * q^34 + 135 * q^35 + 45 * q^36 - 539 * q^37 + 195 * q^38 + 213 * q^39 + 315 * q^40 - 101 * q^41 + 87 * q^42 - 155 * q^43 + 42 * q^44 - 9 * q^45 + 72 * q^46 + 526 * q^47 + 405 * q^48 - 316 * q^49 + 94 * q^50 + 294 * q^51 - 444 * q^52 - 698 * q^53 + 27 * q^54 - 74 * q^55 + 385 * q^56 + 63 * q^57 - 29 * q^58 + 455 * q^59 - 177 * q^60 + 44 * q^61 + 8 * q^62 - 216 * q^63 + 361 * q^64 - 764 * q^65 - 24 * q^66 - 1551 * q^67 - 327 * q^68 + 186 * q^69 + 691 * q^70 + 126 * q^71 - 135 * q^72 - 760 * q^73 + 331 * q^74 + 195 * q^75 - 237 * q^76 - 341 * q^77 - 906 * q^78 - 158 * q^79 - 117 * q^80 + 162 * q^81 + 973 * q^82 - 934 * q^83 + 57 * q^84 - 197 * q^85 - 2403 * q^86 - 174 * q^87 - 290 * q^88 - 691 * q^89 - 549 * q^90 + 319 * q^91 - 196 * q^92 - 198 * q^93 + 721 * q^94 - 543 * q^95 - 747 * q^96 + 532 * q^97 + 1142 * q^98 + 225 * q^99

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