LMFDB - Embedded newform 177.4.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, 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("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.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:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.254436$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.254436 q^{2} -3.00000 q^{3} -7.93526 q^{4} -10.8225 q^{5} -0.763309 q^{6} -23.2950 q^{7} -4.05451 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.254436 q^{2} -3.00000 q^{3} -7.93526 q^{4} -10.8225 q^{5} -0.763309 q^{6} -23.2950 q^{7} -4.05451 q^{8} +9.00000 q^{9} -2.75362 q^{10} +51.0608 q^{11} +23.8058 q^{12} +51.8661 q^{13} -5.92710 q^{14} +32.4674 q^{15} +62.4505 q^{16} -0.0421754 q^{17} +2.28993 q^{18} -85.2082 q^{19} +85.8790 q^{20} +69.8851 q^{21} +12.9917 q^{22} -8.50543 q^{23} +12.1635 q^{24} -7.87442 q^{25} +13.1966 q^{26} -27.0000 q^{27} +184.852 q^{28} -101.142 q^{29} +8.26087 q^{30} +271.338 q^{31} +48.3257 q^{32} -153.182 q^{33} -0.0107310 q^{34} +252.110 q^{35} -71.4174 q^{36} -9.54238 q^{37} -21.6800 q^{38} -155.598 q^{39} +43.8797 q^{40} -185.363 q^{41} +17.7813 q^{42} +277.532 q^{43} -405.181 q^{44} -97.4021 q^{45} -2.16409 q^{46} +309.877 q^{47} -187.351 q^{48} +199.659 q^{49} -2.00354 q^{50} +0.126526 q^{51} -411.571 q^{52} +273.504 q^{53} -6.86978 q^{54} -552.603 q^{55} +94.4499 q^{56} +255.625 q^{57} -25.7341 q^{58} -59.0000 q^{59} -257.637 q^{60} +752.893 q^{61} +69.0383 q^{62} -209.655 q^{63} -487.308 q^{64} -561.319 q^{65} -38.9751 q^{66} +751.435 q^{67} +0.334673 q^{68} +25.5163 q^{69} +64.1458 q^{70} -651.607 q^{71} -36.4906 q^{72} +842.428 q^{73} -2.42793 q^{74} +23.6233 q^{75} +676.149 q^{76} -1189.46 q^{77} -39.5898 q^{78} -368.828 q^{79} -675.868 q^{80} +81.0000 q^{81} -47.1631 q^{82} -545.888 q^{83} -554.557 q^{84} +0.456442 q^{85} +70.6143 q^{86} +303.425 q^{87} -207.026 q^{88} +634.328 q^{89} -24.7826 q^{90} -1208.22 q^{91} +67.4928 q^{92} -814.015 q^{93} +78.8440 q^{94} +922.162 q^{95} -144.977 q^{96} +316.368 q^{97} +50.8004 q^{98} +459.547 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * 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 $

$2$	0.254436	0.0899568	0.0449784	−	0.998988i	$-0.485678\pi$
$2$	0.254436	0.0899568	0.0449784	+	0.998988i	$0.485678\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.93526	−0.991908
$5$	−10.8225	−0.967990	−0.483995	−	0.875071i	$-0.660815\pi$
$5$	−10.8225	−0.967990	−0.483995	+	0.875071i	$0.660815\pi$
$6$	−0.763309	−0.0519366
$7$	−23.2950	−1.25781	−0.628907	−	0.777481i	$-0.716497\pi$
$7$	−23.2950	−1.25781	−0.628907	+	0.777481i	$0.716497\pi$
$8$	−4.05451	−0.179186
$9$	9.00000	0.333333
$10$	−2.75362	−0.0870773
$11$	51.0608	1.39958	0.699791	−	0.714347i	$-0.253276\pi$
$11$	51.0608	1.39958	0.699791	+	0.714347i	$0.253276\pi$
$12$	23.8058	0.572678
$13$	51.8661	1.10654	0.553272	−	0.833001i	$-0.313379\pi$
$13$	51.8661	1.10654	0.553272	+	0.833001i	$0.313379\pi$
$14$	−5.92710	−0.113149
$15$	32.4674	0.558869
$16$	62.4505	0.975789
$17$	−0.0421754	−0.000601709 0	−0.000300854	−	1.00000i	$-0.500096\pi$
$17$	−0.0421754	−0.000601709 0	−0.000300854		1.00000i	$0.500096\pi$
$18$	2.28993	0.0299856
$19$	−85.2082	−1.02885	−0.514424	−	0.857536i	$-0.671994\pi$
$19$	−85.2082	−1.02885	−0.514424	+	0.857536i	$0.671994\pi$
$20$	85.8790	0.960157
$21$	69.8851	0.726199
$22$	12.9917	0.125902
$23$	−8.50543	−0.0771089	−0.0385544	−	0.999257i	$-0.512275\pi$
$23$	−8.50543	−0.0771089	−0.0385544	+	0.999257i	$0.512275\pi$
$24$	12.1635	0.103453
$25$	−7.87442	−0.0629954
$26$	13.1966	0.0995411
$27$	−27.0000	−0.192450
$28$	184.852	1.24764
$29$	−101.142	−0.647639	−0.323819	−	0.946119i	$-0.604967\pi$
$29$	−101.142	−0.647639	−0.323819	+	0.946119i	$0.604967\pi$
$30$	8.26087	0.0502741
$31$	271.338	1.57206	0.786029	−	0.618189i	$-0.212133\pi$
$31$	271.338	1.57206	0.786029	+	0.618189i	$0.212133\pi$
$32$	48.3257	0.266964
$33$	−153.182	−0.808049
$34$	−0.0107310	−5.41278e−5 0
$35$	252.110	1.21755
$36$	−71.4174	−0.330636
$37$	−9.54238	−0.0423988	−0.0211994	−	0.999775i	$-0.506748\pi$
$37$	−9.54238	−0.0423988	−0.0211994	+	0.999775i	$0.506748\pi$
$38$	−21.6800	−0.0925518
$39$	−155.598	−0.638863
$40$	43.8797	0.173450
$41$	−185.363	−0.706071	−0.353035	−	0.935610i	$-0.614850\pi$
$41$	−185.363	−0.706071	−0.353035	+	0.935610i	$0.614850\pi$
$42$	17.7813	0.0653265
$43$	277.532	0.984263	0.492131	−	0.870521i	$-0.336218\pi$
$43$	277.532	0.984263	0.492131	+	0.870521i	$0.336218\pi$
$44$	−405.181	−1.38826
$45$	−97.4021	−0.322663
$46$	−2.16409	−0.00693647
$47$	309.877	0.961708	0.480854	−	0.876801i	$-0.340327\pi$
$47$	309.877	0.961708	0.480854	+	0.876801i	$0.340327\pi$
$48$	−187.351	−0.563372
$49$	199.659	0.582096
$50$	−2.00354	−0.00566686
$51$	0.126526	0.000347397 0
$52$	−411.571	−1.09759
$53$	273.504	0.708842	0.354421	−	0.935086i	$-0.384678\pi$
$53$	273.504	0.708842	0.354421	+	0.935086i	$0.384678\pi$
$54$	−6.86978	−0.0173122
$55$	−552.603	−1.35478
$56$	94.4499	0.225382
$57$	255.625	0.594005
$58$	−25.7341	−0.0582595
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−257.637	−0.554347
$61$	752.893	1.58030	0.790148	−	0.612916i	$-0.210004\pi$
$61$	752.893	1.58030	0.790148	+	0.612916i	$0.210004\pi$
$62$	69.0383	0.141417
$63$	−209.655	−0.419271
$64$	−487.308	−0.951774
$65$	−561.319	−1.07112
$66$	−38.9751	−0.0726895
$67$	751.435	1.37018	0.685092	−	0.728456i	$-0.259762\pi$
$67$	751.435	1.37018	0.685092	+	0.728456i	$0.259762\pi$
$68$	0.334673	0.000596840 0
$69$	25.5163	0.0445188
$70$	64.1458	0.109527
$71$	−651.607	−1.08918	−0.544588	−	0.838704i	$-0.683314\pi$
$71$	−651.607	−1.08918	−0.544588	+	0.838704i	$0.683314\pi$
$72$	−36.4906	−0.0597285
$73$	842.428	1.35067	0.675334	−	0.737512i	$-0.263999\pi$
$73$	842.428	1.35067	0.675334	+	0.737512i	$0.263999\pi$
$74$	−2.42793	−0.00381406
$75$	23.6233	0.0363704
$76$	676.149	1.02052
$77$	−1189.46	−1.76041
$78$	−39.5898	−0.0574701
$79$	−368.828	−0.525271	−0.262635	−	0.964895i	$-0.584592\pi$
$79$	−368.828	−0.525271	−0.262635	+	0.964895i	$0.584592\pi$
$80$	−675.868	−0.944554
$81$	81.0000	0.111111
$82$	−47.1631	−0.0635158
$83$	−545.888	−0.721916	−0.360958	−	0.932582i	$-0.617550\pi$
$83$	−545.888	−0.721916	−0.360958	+	0.932582i	$0.617550\pi$
$84$	−554.557	−0.720323
$85$	0.456442	0.000582448 0
$86$	70.6143	0.0885411
$87$	303.425	0.373914
$88$	−207.026	−0.250785
$89$	634.328	0.755490	0.377745	−	0.925910i	$-0.376700\pi$
$89$	634.328	0.755490	0.377745	+	0.925910i	$0.376700\pi$
$90$	−24.7826	−0.0290258
$91$	−1208.22	−1.39183
$92$	67.4928	0.0764849
$93$	−814.015	−0.907629
$94$	78.8440	0.0865121
$95$	922.162	0.995914
$96$	−144.977	−0.154132
$97$	316.368	0.331158	0.165579	−	0.986197i	$-0.447051\pi$
$97$	316.368	0.331158	0.165579	+	0.986197i	$0.447051\pi$
$98$	50.8004	0.0523635
$99$	459.547	0.466528
$100$	62.4856	0.0624856
$101$	−1615.25	−1.59132	−0.795662	−	0.605740i	$-0.792877\pi$
$101$	−1615.25	−1.59132	−0.795662	+	0.605740i	$0.792877\pi$
$102$	0.0321929	3.12507e−5 0
$103$	1261.07	1.20638	0.603190	−	0.797598i	$-0.293896\pi$
$103$	1261.07	1.20638	0.603190	+	0.797598i	$0.293896\pi$
$104$	−210.291	−0.198277
$105$	−756.329	−0.702954
$106$	69.5892	0.0637651
$107$	733.783	0.662967	0.331483	−	0.943461i	$-0.392451\pi$
$107$	733.783	0.662967	0.331483	+	0.943461i	$0.392451\pi$
$108$	214.252	0.190893
$109$	582.848	0.512171	0.256086	−	0.966654i	$-0.417567\pi$
$109$	582.848	0.512171	0.256086	+	0.966654i	$0.417567\pi$
$110$	−140.602	−0.121872
$111$	28.6271	0.0244790
$112$	−1454.79	−1.22736
$113$	−1661.78	−1.38342	−0.691712	−	0.722174i	$-0.743143\pi$
$113$	−1661.78	−1.38342	−0.691712	+	0.722174i	$0.743143\pi$
$114$	65.0401	0.0534348
$115$	92.0496	0.0746406
$116$	802.585	0.642398
$117$	466.795	0.368848
$118$	−15.0117	−0.0117114
$119$	0.982479	0.000756838 0
$120$	−131.639	−0.100141
$121$	1276.20	0.958831
$122$	191.563	0.142158
$123$	556.090	0.407650
$124$	−2153.14	−1.55934
$125$	1438.03	1.02897
$126$	−53.3439	−0.0377163
$127$	1560.58	1.09039	0.545193	−	0.838310i	$-0.316456\pi$
$127$	1560.58	1.09039	0.545193	+	0.838310i	$0.316456\pi$
$128$	−510.595	−0.352583
$129$	−832.597	−0.568264
$130$	−142.820	−0.0963548
$131$	2608.59	1.73980	0.869898	−	0.493232i	$-0.164185\pi$
$131$	2608.59	1.73980	0.869898	+	0.493232i	$0.164185\pi$
$132$	1215.54	0.801510
$133$	1984.93	1.29410
$134$	191.192	0.123257
$135$	292.206	0.186290
$136$	0.171001	0.000107818 0
$137$	−2284.40	−1.42459	−0.712296	−	0.701879i	$-0.752345\pi$
$137$	−2284.40	−1.42459	−0.712296	+	0.701879i	$0.752345\pi$
$138$	6.49227	0.00400477
$139$	−288.324	−0.175937	−0.0879687	−	0.996123i	$-0.528038\pi$
$139$	−288.324	−0.175937	−0.0879687	+	0.996123i	$0.528038\pi$
$140$	−2000.56	−1.20770
$141$	−929.632	−0.555242
$142$	−165.792	−0.0979787
$143$	2648.32	1.54870
$144$	562.054	0.325263
$145$	1094.60	0.626908
$146$	214.344	0.121502
$147$	−598.977	−0.336073
$148$	75.7212	0.0420557
$149$	591.503	0.325220	0.162610	−	0.986690i	$-0.448009\pi$
$149$	591.503	0.325220	0.162610	+	0.986690i	$0.448009\pi$
$150$	6.01061	0.00327176
$151$	−2398.78	−1.29278	−0.646391	−	0.763007i	$-0.723722\pi$
$151$	−2398.78	−1.29278	−0.646391	+	0.763007i	$0.723722\pi$
$152$	345.477	0.184355
$153$	−0.379579	−0.000200570 0
$154$	−302.642	−0.158361
$155$	−2936.55	−1.52174
$156$	1234.71	0.633693
$157$	135.478	0.0688684	0.0344342	−	0.999407i	$-0.489037\pi$
$157$	135.478	0.0688684	0.0344342	+	0.999407i	$0.489037\pi$
$158$	−93.8432	−0.0472517
$159$	−820.511	−0.409250
$160$	−523.003	−0.258419
$161$	198.134	0.0969886
$162$	20.6093	0.00999520
$163$	1649.77	0.792763	0.396381	−	0.918086i	$-0.370266\pi$
$163$	1649.77	0.792763	0.396381	+	0.918086i	$0.370266\pi$
$164$	1470.91	0.700357
$165$	1657.81	0.782184
$166$	−138.894	−0.0649412
$167$	3500.76	1.62214	0.811070	−	0.584949i	$-0.198886\pi$
$167$	3500.76	1.62214	0.811070	+	0.584949i	$0.198886\pi$
$168$	−283.350	−0.130124
$169$	493.092	0.224439
$170$	0.116135	5.23951e−5 0
$171$	−766.874	−0.342949
$172$	−2202.29	−0.976298
$173$	2524.80	1.10958	0.554790	−	0.831991i	$-0.312799\pi$
$173$	2524.80	1.10958	0.554790	+	0.831991i	$0.312799\pi$
$174$	77.2023	0.0336361
$175$	183.435	0.0792364
$176$	3188.77	1.36570
$177$	177.000	0.0751646
$178$	161.396	0.0679614
$179$	2717.45	1.13470	0.567351	−	0.823476i	$-0.307968\pi$
$179$	2717.45	1.13470	0.567351	+	0.823476i	$0.307968\pi$
$180$	772.911	0.320052
$181$	2116.30	0.869077	0.434539	−	0.900653i	$-0.356911\pi$
$181$	2116.30	0.869077	0.434539	+	0.900653i	$0.356911\pi$
$182$	−307.416	−0.125204
$183$	−2258.68	−0.912384
$184$	34.4853	0.0138168
$185$	103.272	0.0410416
$186$	−207.115	−0.0816473
$187$	−2.15351	−0.000842141 0
$188$	−2458.96	−0.953925
$189$	628.966	0.242066
$190$	234.631	0.0895892
$191$	−162.049	−0.0613899	−0.0306950	−	0.999529i	$-0.509772\pi$
$191$	−162.049	−0.0613899	−0.0306950	+	0.999529i	$0.509772\pi$
$192$	1461.92	0.549507
$193$	−1737.71	−0.648099	−0.324050	−	0.946040i	$-0.605044\pi$
$193$	−1737.71	−0.648099	−0.324050	+	0.946040i	$0.605044\pi$
$194$	80.4956	0.0297899
$195$	1683.96	0.618413
$196$	−1584.35	−0.577385
$197$	−1898.32	−0.686548	−0.343274	−	0.939235i	$-0.611536\pi$
$197$	−1898.32	−0.686548	−0.343274	+	0.939235i	$0.611536\pi$
$198$	116.925	0.0419673
$199$	−925.010	−0.329509	−0.164754	−	0.986335i	$-0.552683\pi$
$199$	−925.010	−0.329509	−0.164754	+	0.986335i	$0.552683\pi$
$200$	31.9269	0.0112879
$201$	−2254.30	−0.791077
$202$	−410.979	−0.143150
$203$	2356.10	0.814609
$204$	−1.00402	−0.000344585 0
$205$	2006.09	0.683469
$206$	320.862	0.108522
$207$	−76.5489	−0.0257030
$208$	3239.06	1.07975
$209$	−4350.80	−1.43996
$210$	−192.437	−0.0632354
$211$	2553.68	0.833186	0.416593	−	0.909093i	$-0.363224\pi$
$211$	2553.68	0.833186	0.416593	+	0.909093i	$0.363224\pi$
$212$	−2170.32	−0.703106
$213$	1954.82	0.628836
$214$	186.701	0.0596383
$215$	−3003.58	−0.952756
$216$	109.472	0.0344843
$217$	−6320.84	−1.97736
$218$	148.298	0.0460733
$219$	−2527.28	−0.779809
$220$	4385.05	1.34382
$221$	−2.18748	−0.000665817 0
$222$	7.28378	0.00220205
$223$	−2867.05	−0.860949	−0.430475	−	0.902603i	$-0.641654\pi$
$223$	−2867.05	−0.860949	−0.430475	+	0.902603i	$0.641654\pi$
$224$	−1125.75	−0.335792
$225$	−70.8698	−0.0209985
$226$	−422.816	−0.124448
$227$	−5544.65	−1.62120	−0.810598	−	0.585603i	$-0.800858\pi$
$227$	−5544.65	−1.62120	−0.810598	+	0.585603i	$0.800858\pi$
$228$	−2028.45	−0.589198
$229$	−3010.94	−0.868857	−0.434429	−	0.900706i	$-0.643050\pi$
$229$	−3010.94	−0.868857	−0.434429	+	0.900706i	$0.643050\pi$
$230$	23.4208	0.00671443
$231$	3568.39	1.01638
$232$	410.079	0.116048
$233$	132.563	0.0372724	0.0186362	−	0.999826i	$-0.494068\pi$
$233$	132.563	0.0372724	0.0186362	+	0.999826i	$0.494068\pi$
$234$	118.770	0.0331804
$235$	−3353.63	−0.930923
$236$	468.180	0.129135
$237$	1106.48	0.303265
$238$	0.249978	6.80827e−5 0
$239$	7104.31	1.92276	0.961380	−	0.275224i	$-0.0887519\pi$
$239$	7104.31	1.92276	0.961380	+	0.275224i	$0.0887519\pi$
$240$	2027.60	0.545338
$241$	124.665	0.0333211	0.0166605	−	0.999861i	$-0.494697\pi$
$241$	124.665	0.0333211	0.0166605	+	0.999861i	$0.494697\pi$
$242$	324.713	0.0862534
$243$	−243.000	−0.0641500
$244$	−5974.40	−1.56751
$245$	−2160.80	−0.563463
$246$	141.489	0.0366709
$247$	−4419.42	−1.13846
$248$	−1100.14	−0.281690
$249$	1637.67	0.416798
$250$	365.886	0.0925627
$251$	−6840.81	−1.72027	−0.860135	−	0.510066i	$-0.829621\pi$
$251$	−6840.81	−1.72027	−0.860135	+	0.510066i	$0.829621\pi$
$252$	1663.67	0.415878
$253$	−434.294	−0.107920
$254$	397.068	0.0980877
$255$	−1.36933	−0.000336277 0
$256$	3768.55	0.920056
$257$	−374.942	−0.0910049	−0.0455024	−	0.998964i	$-0.514489\pi$
$257$	−374.942	−0.0910049	−0.0455024	+	0.998964i	$0.514489\pi$
$258$	−211.843	−0.0511192
$259$	222.290	0.0533298
$260$	4454.21	1.06246
$261$	−910.275	−0.215880
$262$	663.719	0.156506
$263$	7174.93	1.68222	0.841112	−	0.540861i	$-0.181901\pi$
$263$	7174.93	1.68222	0.841112	+	0.540861i	$0.181901\pi$
$264$	621.079	0.144791
$265$	−2959.98	−0.686152
$266$	505.038	0.116413
$267$	−1902.98	−0.436182
$268$	−5962.83	−1.35910
$269$	−2997.57	−0.679425	−0.339713	−	0.940529i	$-0.610330\pi$
$269$	−2997.57	−0.679425	−0.339713	+	0.940529i	$0.610330\pi$
$270$	74.3479	0.0167580
$271$	5993.77	1.34353	0.671763	−	0.740766i	$-0.265537\pi$
$271$	5993.77	1.34353	0.671763	+	0.740766i	$0.265537\pi$
$272$	−2.63388	−0.000587141 0
$273$	3624.67	0.803571
$274$	−581.233	−0.128152
$275$	−402.074	−0.0881672
$276$	−202.478	−0.0441586
$277$	−8759.33	−1.89999	−0.949994	−	0.312267i	$-0.898912\pi$
$277$	−8759.33	−1.89999	−0.949994	+	0.312267i	$0.898912\pi$
$278$	−73.3600	−0.0158268
$279$	2442.05	0.524020
$280$	−1022.18	−0.218168
$281$	−4991.90	−1.05976	−0.529878	−	0.848074i	$-0.677762\pi$
$281$	−4991.90	−1.05976	−0.529878	+	0.848074i	$0.677762\pi$
$282$	−236.532	−0.0499478
$283$	−470.303	−0.0987866	−0.0493933	−	0.998779i	$-0.515729\pi$
$283$	−470.303	−0.0987866	−0.0493933	+	0.998779i	$0.515729\pi$
$284$	5170.67	1.08036
$285$	−2766.49	−0.574991
$286$	673.829	0.139316
$287$	4318.05	0.888105
$288$	434.931	0.0889881
$289$	−4913.00	−1.00000
$290$	278.506	0.0563946
$291$	−949.105	−0.191194
$292$	−6684.89	−1.33974
$293$	1777.56	0.354425	0.177212	−	0.984173i	$-0.443292\pi$
$293$	1777.56	0.354425	0.177212	+	0.984173i	$0.443292\pi$
$294$	−152.401	−0.0302321
$295$	638.525	0.126022
$296$	38.6896	0.00759726
$297$	−1378.64	−0.269350
$298$	150.500	0.0292558
$299$	−441.143	−0.0853243
$300$	−187.457	−0.0360761
$301$	−6465.13	−1.23802
$302$	−610.336	−0.116294
$303$	4845.76	0.918752
$304$	−5321.29	−1.00394
$305$	−8148.15	−1.52971
$306$	−0.0965786	−1.80426e−5 0
$307$	4138.06	0.769289	0.384645	−	0.923065i	$-0.374324\pi$
$307$	4138.06	0.769289	0.384645	+	0.923065i	$0.374324\pi$
$308$	9438.70	1.74617
$309$	−3783.21	−0.696503
$310$	−747.164	−0.136891
$311$	7356.20	1.34126	0.670630	−	0.741792i	$-0.266024\pi$
$311$	7356.20	1.34126	0.670630	+	0.741792i	$0.266024\pi$
$312$	630.874	0.114475
$313$	6696.92	1.20937	0.604685	−	0.796465i	$-0.293299\pi$
$313$	6696.92	1.20937	0.604685	+	0.796465i	$0.293299\pi$
$314$	34.4706	0.00619518
$315$	2268.99	0.405850
$316$	2926.75	0.521020
$317$	7828.34	1.38701	0.693507	−	0.720450i	$-0.256065\pi$
$317$	7828.34	1.38701	0.693507	+	0.720450i	$0.256065\pi$
$318$	−208.768	−0.0368148
$319$	−5164.37	−0.906424
$320$	5273.87	0.921307
$321$	−2201.35	−0.382764
$322$	50.4125	0.00872478
$323$	3.59369	0.000619066 0
$324$	−642.756	−0.110212
$325$	−408.415	−0.0697071
$326$	419.762	0.0713144
$327$	−1748.54	−0.295702
$328$	751.557	0.126518
$329$	−7218.60	−1.20965
$330$	421.807	0.0703627
$331$	−791.925	−0.131505	−0.0657525	−	0.997836i	$-0.520945\pi$
$331$	−791.925	−0.131505	−0.0657525	+	0.997836i	$0.520945\pi$
$332$	4331.77	0.716074
$333$	−85.8814	−0.0141329
$334$	890.721	0.145922
$335$	−8132.37	−1.32633
$336$	4364.36	0.708617
$337$	3609.06	0.583378	0.291689	−	0.956513i	$-0.405783\pi$
$337$	3609.06	0.583378	0.291689	+	0.956513i	$0.405783\pi$
$338$	125.460	0.0201898
$339$	4985.33	0.798720
$340$	−3.62199	−0.000577735 0
$341$	13854.8	2.20023
$342$	−195.120	−0.0308506
$343$	3339.14	0.525646
$344$	−1125.26	−0.176366
$345$	−276.149	−0.0430938
$346$	642.401	0.0998142
$347$	−4501.38	−0.696389	−0.348194	−	0.937422i	$-0.613205\pi$
$347$	−4501.38	−0.696389	−0.348194	+	0.937422i	$0.613205\pi$
$348$	−2407.76	−0.370889
$349$	4505.21	0.690998	0.345499	−	0.938419i	$-0.387710\pi$
$349$	4505.21	0.690998	0.345499	+	0.938419i	$0.387710\pi$
$350$	46.6725	0.00712785
$351$	−1400.38	−0.212954
$352$	2467.55	0.373639
$353$	−6156.62	−0.928282	−0.464141	−	0.885761i	$-0.653637\pi$
$353$	−6156.62	−0.928282	−0.464141	+	0.885761i	$0.653637\pi$
$354$	45.0352	0.00676157
$355$	7051.99	1.05431
$356$	−5033.56	−0.749376
$357$	−2.94744	−0.000436960 0
$358$	691.417	0.102074
$359$	8811.87	1.29547	0.647734	−	0.761867i	$-0.275717\pi$
$359$	8811.87	1.29547	0.647734	+	0.761867i	$0.275717\pi$
$360$	394.918	0.0578166
$361$	401.436	0.0585269
$362$	538.462	0.0781794
$363$	−3828.61	−0.553582
$364$	9587.56	1.38056
$365$	−9117.14	−1.30743
$366$	−574.690	−0.0820752
$367$	13290.6	1.89036	0.945181	−	0.326546i	$-0.105885\pi$
$367$	13290.6	1.89036	0.945181	+	0.326546i	$0.105885\pi$
$368$	−531.168	−0.0752420
$369$	−1668.27	−0.235357
$370$	26.2761	0.00369197
$371$	−6371.28	−0.891591
$372$	6459.43	0.900284
$373$	10006.7	1.38908	0.694541	−	0.719453i	$-0.255608\pi$
$373$	10006.7	1.38908	0.694541	+	0.719453i	$0.255608\pi$
$374$	−0.547931	−7.57563e−5 0
$375$	−4314.08	−0.594075
$376$	−1256.40	−0.172324
$377$	−5245.82	−0.716641
$378$	160.032	0.0217755
$379$	−3895.86	−0.528013	−0.264006	−	0.964521i	$-0.585044\pi$
$379$	−3895.86	−0.528013	−0.264006	+	0.964521i	$0.585044\pi$
$380$	−7317.60	−0.987855
$381$	−4681.74	−0.629535
$382$	−41.2312	−0.00552244
$383$	−3038.64	−0.405398	−0.202699	−	0.979241i	$-0.564971\pi$
$383$	−3038.64	−0.405398	−0.202699	+	0.979241i	$0.564971\pi$
$384$	1531.78	0.203564
$385$	12872.9	1.70406
$386$	−442.136	−0.0583009
$387$	2497.79	0.328088
$388$	−2510.47	−0.328478
$389$	−13537.3	−1.76444	−0.882222	−	0.470834i	$-0.843953\pi$
$389$	−13537.3	−1.76444	−0.882222	+	0.470834i	$0.843953\pi$
$390$	428.459	0.0556305
$391$	0.358720	4.63971e−5 0
$392$	−809.518	−0.104303
$393$	−7825.76	−1.00447
$394$	−483.002	−0.0617597
$395$	3991.63	0.508457
$396$	−3646.63	−0.462752
$397$	−3491.32	−0.441371	−0.220686	−	0.975345i	$-0.570829\pi$
$397$	−3491.32	−0.441371	−0.220686	+	0.975345i	$0.570829\pi$
$398$	−235.356	−0.0296415
$399$	−5954.78	−0.747148
$400$	−491.761	−0.0614702
$401$	12667.7	1.57754	0.788769	−	0.614689i	$-0.210719\pi$
$401$	12667.7	1.57754	0.788769	+	0.614689i	$0.210719\pi$
$402$	−573.577	−0.0711627
$403$	14073.3	1.73955
$404$	12817.5	1.57845
$405$	−876.619	−0.107554
$406$	599.477	0.0732796
$407$	−487.241	−0.0593407
$408$	−0.513002	−6.22485e−5 0
$409$	10431.3	1.26111	0.630557	−	0.776143i	$-0.282826\pi$
$409$	10431.3	1.26111	0.630557	+	0.776143i	$0.282826\pi$
$410$	510.421	0.0614827
$411$	6853.19	0.822488
$412$	−10006.9	−1.19662
$413$	1374.41	0.163753
$414$	−19.4768	−0.00231216
$415$	5907.85	0.698808
$416$	2506.47	0.295408
$417$	864.971	0.101578
$418$	−1107.00	−0.129534
$419$	−6955.06	−0.810923	−0.405462	−	0.914112i	$-0.632889\pi$
$419$	−6955.06	−0.810923	−0.405462	+	0.914112i	$0.632889\pi$
$420$	6001.67	0.697265
$421$	−2977.09	−0.344643	−0.172321	−	0.985041i	$-0.555127\pi$
$421$	−2977.09	−0.344643	−0.172321	+	0.985041i	$0.555127\pi$
$422$	649.748	0.0749507
$423$	2788.90	0.320569
$424$	−1108.92	−0.127014
$425$	0.332107	3.79049e−5 0
$426$	497.377	0.0565681
$427$	−17538.7	−1.98772
$428$	−5822.76	−0.657602
$429$	−7944.97	−0.894142
$430$	−764.220	−0.0857069
$431$	−9721.43	−1.08646	−0.543230	−	0.839584i	$-0.682799\pi$
$431$	−9721.43	−1.08646	−0.543230	+	0.839584i	$0.682799\pi$
$432$	−1686.16	−0.187791
$433$	814.588	0.0904079	0.0452040	−	0.998978i	$-0.485606\pi$
$433$	814.588	0.0904079	0.0452040	+	0.998978i	$0.485606\pi$
$434$	−1608.25	−0.177877
$435$	−3283.80	−0.361945
$436$	−4625.05	−0.508027
$437$	724.732	0.0793333
$438$	−643.033	−0.0701491
$439$	15373.9	1.67143	0.835716	−	0.549162i	$-0.185053\pi$
$439$	15373.9	1.67143	0.835716	+	0.549162i	$0.185053\pi$
$440$	2240.53	0.242757
$441$	1796.93	0.194032
$442$	−0.556573	−5.98947e−5 0
$443$	5462.95	0.585897	0.292949	−	0.956128i	$-0.405364\pi$
$443$	5462.95	0.585897	0.292949	+	0.956128i	$0.405364\pi$
$444$	−227.164	−0.0242809
$445$	−6864.99	−0.731307
$446$	−729.480	−0.0774482
$447$	−1774.51	−0.187766
$448$	11351.9	1.19715
$449$	−9408.54	−0.988901	−0.494451	−	0.869206i	$-0.664631\pi$
$449$	−9408.54	−0.988901	−0.494451	+	0.869206i	$0.664631\pi$
$450$	−18.0318	−0.00188895
$451$	−9464.80	−0.988204
$452$	13186.6	1.37223
$453$	7196.34	0.746388
$454$	−1410.76	−0.145838
$455$	13075.9	1.34727
$456$	−1036.43	−0.106437
$457$	1254.04	0.128362	0.0641809	−	0.997938i	$-0.479557\pi$
$457$	1254.04	0.128362	0.0641809	+	0.997938i	$0.479557\pi$
$458$	−766.091	−0.0781596
$459$	1.13874	0.000115799 0
$460$	−730.438	−0.0740366
$461$	−15647.0	−1.58081	−0.790407	−	0.612582i	$-0.790131\pi$
$461$	−15647.0	−1.58081	−0.790407	+	0.612582i	$0.790131\pi$
$462$	907.927	0.0914299
$463$	18387.3	1.84564	0.922821	−	0.385228i	$-0.125877\pi$
$463$	18387.3	1.84564	0.922821	+	0.385228i	$0.125877\pi$
$464$	−6316.34	−0.631959
$465$	8809.65	0.878575
$466$	33.7287	0.00335290
$467$	13135.8	1.30161	0.650805	−	0.759245i	$-0.274431\pi$
$467$	13135.8	1.30161	0.650805	+	0.759245i	$0.274431\pi$
$468$	−3704.14	−0.365863
$469$	−17504.7	−1.72344
$470$	−853.286	−0.0837429
$471$	−406.435	−0.0397612
$472$	239.216	0.0233280
$473$	14171.0	1.37756
$474$	281.530	0.0272808
$475$	670.965	0.0648126
$476$	−7.79623	−0.000750713 0
$477$	2461.53	0.236281
$478$	1807.59	0.172965
$479$	−4918.54	−0.469173	−0.234587	−	0.972095i	$-0.575374\pi$
$479$	−4918.54	−0.469173	−0.234587	+	0.972095i	$0.575374\pi$
$480$	1569.01	0.149198
$481$	−494.926	−0.0469162
$482$	31.7193	0.00299745
$483$	−594.403	−0.0559964
$484$	−10127.0	−0.951072
$485$	−3423.88	−0.320558
$486$	−61.8280	−0.00577073
$487$	12272.0	1.14189	0.570944	−	0.820989i	$-0.306577\pi$
$487$	12272.0	1.14189	0.570944	+	0.820989i	$0.306577\pi$
$488$	−3052.61	−0.283166
$489$	−4949.32	−0.457702
$490$	−549.786	−0.0506873
$491$	−9165.57	−0.842437	−0.421218	−	0.906959i	$-0.638397\pi$
$491$	−9165.57	−0.842437	−0.421218	+	0.906959i	$0.638397\pi$
$492$	−4412.72	−0.404351
$493$	4.26569	0.000389690 0
$494$	−1124.46	−0.102413
$495$	−4973.43	−0.451594
$496$	16945.2	1.53400
$497$	15179.2	1.36998
$498$	416.681	0.0374938
$499$	−4308.93	−0.386561	−0.193281	−	0.981144i	$-0.561913\pi$
$499$	−4308.93	−0.386561	−0.193281	+	0.981144i	$0.561913\pi$
$500$	−11411.1	−1.02064
$501$	−10502.3	−0.936543
$502$	−1740.55	−0.154750
$503$	−10004.0	−0.886795	−0.443397	−	0.896325i	$-0.646227\pi$
$503$	−10004.0	−0.886795	−0.443397	+	0.896325i	$0.646227\pi$
$504$	850.049	0.0751274
$505$	17481.0	1.54039
$506$	−110.500	−0.00970816
$507$	−1479.28	−0.129580
$508$	−12383.6	−1.08156
$509$	−11329.0	−0.986538	−0.493269	−	0.869877i	$-0.664198\pi$
$509$	−11329.0	−0.986538	−0.493269	+	0.869877i	$0.664198\pi$
$510$	−0.348406	−3.02503e−5 0
$511$	−19624.4	−1.69889
$512$	5043.61	0.435348
$513$	2300.62	0.198002
$514$	−95.3989	−0.00818650
$515$	−13647.9	−1.16776
$516$	6606.88	0.563666
$517$	15822.6	1.34599
$518$	56.5586	0.00479738
$519$	−7574.41	−0.640616
$520$	2275.87	0.191930
$521$	−578.251	−0.0486251	−0.0243125	−	0.999704i	$-0.507740\pi$
$521$	−578.251	−0.0486251	−0.0243125	+	0.999704i	$0.507740\pi$
$522$	−231.607	−0.0194198
$523$	−22559.8	−1.88618	−0.943088	−	0.332543i	$-0.892093\pi$
$523$	−22559.8	−1.88618	−0.943088	+	0.332543i	$0.892093\pi$
$524$	−20699.8	−1.72572
$525$	−550.305	−0.0457472
$526$	1825.56	0.151327
$527$	−11.4438	−0.000945922 0
$528$	−9566.31	−0.788486
$529$	−12094.7	−0.994054
$530$	−753.126	−0.0617240
$531$	−531.000	−0.0433963
$532$	−15750.9	−1.28363
$533$	−9614.07	−0.781298
$534$	−484.188	−0.0392376
$535$	−7941.33	−0.641745
$536$	−3046.70	−0.245517
$537$	−8152.34	−0.655120
$538$	−762.691	−0.0611189
$539$	10194.7	0.814691
$540$	−2318.73	−0.184782
$541$	−8649.61	−0.687386	−0.343693	−	0.939082i	$-0.611678\pi$
$541$	−8649.61	−0.687386	−0.343693	+	0.939082i	$0.611678\pi$
$542$	1525.03	0.120859
$543$	−6348.89	−0.501762
$544$	−2.03816	−0.000160635 0
$545$	−6307.84	−0.495777
$546$	922.247	0.0722867
$547$	−9720.33	−0.759801	−0.379901	−	0.925027i	$-0.624042\pi$
$547$	−9720.33	−0.759801	−0.379901	+	0.925027i	$0.624042\pi$
$548$	18127.3	1.41306
$549$	6776.04	0.526765
$550$	−102.302	−0.00793124
$551$	8618.09	0.666322
$552$	−103.456	−0.00797713
$553$	8591.87	0.660693
$554$	−2228.69	−0.170917
$555$	−309.816	−0.0236954
$556$	2287.92	0.174514
$557$	14288.6	1.08694	0.543472	−	0.839428i	$-0.317110\pi$
$557$	14288.6	1.08694	0.543472	+	0.839428i	$0.317110\pi$
$558$	621.345	0.0471391
$559$	14394.5	1.08913
$560$	15744.4	1.18807
$561$	6.46053	0.000486210 0
$562$	−1270.12	−0.0953323
$563$	−2067.53	−0.154771	−0.0773856	−	0.997001i	$-0.524657\pi$
$563$	−2067.53	−0.154771	−0.0773856	+	0.997001i	$0.524657\pi$
$564$	7376.87	0.550749
$565$	17984.5	1.33914
$566$	−119.662	−0.00888652
$567$	−1886.90	−0.139757
$568$	2641.94	0.195165
$569$	−3196.60	−0.235516	−0.117758	−	0.993042i	$-0.537571\pi$
$569$	−3196.60	−0.235516	−0.117758	+	0.993042i	$0.537571\pi$
$570$	−703.894	−0.0517244
$571$	−15856.8	−1.16215	−0.581073	−	0.813852i	$-0.697367\pi$
$571$	−15856.8	−1.16215	−0.581073	+	0.813852i	$0.697367\pi$
$572$	−21015.1	−1.53617
$573$	486.148	0.0354435
$574$	1098.67	0.0798911
$575$	66.9753	0.00485750
$576$	−4385.77	−0.317258
$577$	18096.9	1.30569	0.652846	−	0.757491i	$-0.273575\pi$
$577$	18096.9	1.30569	0.652846	+	0.757491i	$0.273575\pi$
$578$	−1250.04	−0.0899567
$579$	5213.13	0.374180
$580$	−8685.94	−0.621835
$581$	12716.5	0.908036
$582$	−241.487	−0.0171992
$583$	13965.3	0.992082
$584$	−3415.63	−0.242020
$585$	−5051.87	−0.357041
$586$	452.277	0.0318829
$587$	1903.98	0.133877	0.0669383	−	0.997757i	$-0.478677\pi$
$587$	1903.98	0.133877	0.0669383	+	0.997757i	$0.478677\pi$
$588$	4753.04	0.333354
$589$	−23120.3	−1.61741
$590$	162.464	0.0113365
$591$	5694.97	0.396379
$592$	−595.926	−0.0413723
$593$	16287.4	1.12790	0.563948	−	0.825810i	$-0.309282\pi$
$593$	16287.4	1.12790	0.563948	+	0.825810i	$0.309282\pi$
$594$	−350.776	−0.0242298
$595$	−10.6328	−0.000732611 0
$596$	−4693.73	−0.322588
$597$	2775.03	0.190242
$598$	−112.243	−0.00767550
$599$	−13497.9	−0.920715	−0.460357	−	0.887734i	$-0.652279\pi$
$599$	−13497.9	−0.920715	−0.460357	+	0.887734i	$0.652279\pi$
$600$	−95.7807	−0.00651705
$601$	13860.8	0.940755	0.470378	−	0.882465i	$-0.344118\pi$
$601$	13860.8	0.940755	0.470378	+	0.882465i	$0.344118\pi$
$602$	−1644.96	−0.111368
$603$	6762.91	0.456728
$604$	19034.9	1.28232
$605$	−13811.7	−0.928139
$606$	1232.94	0.0826479
$607$	13140.0	0.878641	0.439321	−	0.898330i	$-0.355219\pi$
$607$	13140.0	0.878641	0.439321	+	0.898330i	$0.355219\pi$
$608$	−4117.75	−0.274666
$609$	−7068.29	−0.470315
$610$	−2073.18	−0.137608
$611$	16072.1	1.06417
$612$	3.01206	0.000198947 0
$613$	−10495.5	−0.691535	−0.345768	−	0.938320i	$-0.612381\pi$
$613$	−10495.5	−0.691535	−0.345768	+	0.938320i	$0.612381\pi$
$614$	1052.87	0.0692028
$615$	−6018.26	−0.394601
$616$	4822.69	0.315441
$617$	9157.76	0.597533	0.298766	−	0.954326i	$-0.403425\pi$
$617$	9157.76	0.597533	0.298766	+	0.954326i	$0.403425\pi$
$618$	−962.586	−0.0626552
$619$	15266.3	0.991286	0.495643	−	0.868526i	$-0.334932\pi$
$619$	15266.3	0.991286	0.495643	+	0.868526i	$0.334932\pi$
$620$	23302.3	1.50942
$621$	229.647	0.0148396
$622$	1871.68	0.120655
$623$	−14776.7	−0.950266
$624$	−9717.19	−0.623396
$625$	−14578.7	−0.933036
$626$	1703.94	0.108791
$627$	13052.4	0.831359
$628$	−1075.06	−0.0683111
$629$	0.402454	2.55117e−5 0
$630$	577.312	0.0365090
$631$	6869.98	0.433422	0.216711	−	0.976236i	$-0.430467\pi$
$631$	6869.98	0.433422	0.216711	+	0.976236i	$0.430467\pi$
$632$	1495.42	0.0941210
$633$	−7661.03	−0.481040
$634$	1991.81	0.124771
$635$	−16889.3	−1.05548
$636$	6510.97	0.405938
$637$	10355.5	0.644114
$638$	−1314.00	−0.0815390
$639$	−5864.46	−0.363059
$640$	5525.89	0.341297
$641$	14205.2	0.875306	0.437653	−	0.899144i	$-0.355810\pi$
$641$	14205.2	0.875306	0.437653	+	0.899144i	$0.355810\pi$
$642$	−560.102	−0.0344322
$643$	32050.2	1.96569	0.982843	−	0.184442i	$-0.0590479\pi$
$643$	32050.2	1.96569	0.982843	+	0.184442i	$0.0590479\pi$
$644$	−1572.25	−0.0962038
$645$	9010.75	0.550074
$646$	0.914366	5.56892e−5 0
$647$	27275.5	1.65736	0.828679	−	0.559725i	$-0.189093\pi$
$647$	27275.5	1.65736	0.828679	+	0.559725i	$0.189093\pi$
$648$	−328.415	−0.0199095
$649$	−3012.59	−0.182210
$650$	−103.916	−0.00627063
$651$	18962.5	1.14163
$652$	−13091.4	−0.786347
$653$	−6694.40	−0.401182	−0.200591	−	0.979675i	$-0.564286\pi$
$653$	−6694.40	−0.401182	−0.200591	+	0.979675i	$0.564286\pi$
$654$	−444.893	−0.0266004
$655$	−28231.3	−1.68411
$656$	−11576.0	−0.688976
$657$	7581.85	0.450223
$658$	−1836.67	−0.108816
$659$	1828.61	0.108092	0.0540460	−	0.998538i	$-0.482788\pi$
$659$	1828.61	0.108092	0.0540460	+	0.998538i	$0.482788\pi$
$660$	−13155.2	−0.775854
$661$	3994.07	0.235025	0.117512	−	0.993071i	$-0.462508\pi$
$661$	3994.07	0.235025	0.117512	+	0.993071i	$0.462508\pi$
$662$	−201.494	−0.0118298
$663$	6.56243	0.000384410 0
$664$	2213.31	0.129357
$665$	−21481.8	−1.25267
$666$	−21.8513	−0.00127135
$667$	860.253	0.0499387
$668$	−27779.5	−1.60901
$669$	8601.14	0.497069
$670$	−2069.17	−0.119312
$671$	38443.3	2.21175
$672$	3377.25	0.193869
$673$	1511.93	0.0865980	0.0432990	−	0.999062i	$-0.486213\pi$
$673$	1511.93	0.0865980	0.0432990	+	0.999062i	$0.486213\pi$
$674$	918.276	0.0524788
$675$	212.609	0.0121235
$676$	−3912.81	−0.222623
$677$	105.944	0.00601442	0.00300721	−	0.999995i	$-0.499043\pi$
$677$	105.944	0.00601442	0.00300721	+	0.999995i	$0.499043\pi$
$678$	1268.45	0.0718503
$679$	−7369.81	−0.416535
$680$	−1.85065	−0.000104366 0
$681$	16634.0	0.935998
$682$	3525.15	0.197925
$683$	−5370.44	−0.300870	−0.150435	−	0.988620i	$-0.548067\pi$
$683$	−5370.44	−0.300870	−0.150435	+	0.988620i	$0.548067\pi$
$684$	6085.34	0.340174
$685$	24722.8	1.37899
$686$	849.597	0.0472854
$687$	9032.81	0.501635
$688$	17332.0	0.960433
$689$	14185.6	0.784364
$690$	−70.2623	−0.00387658
$691$	−11938.4	−0.657246	−0.328623	−	0.944461i	$-0.606584\pi$
$691$	−11938.4	−0.657246	−0.328623	+	0.944461i	$0.606584\pi$
$692$	−20035.0	−1.10060
$693$	−10705.2	−0.586805
$694$	−1145.31	−0.0626449
$695$	3120.37	0.170306
$696$	−1230.24	−0.0670001
$697$	7.81778	0.000424849 0
$698$	1146.29	0.0621599
$699$	−397.688	−0.0215192
$700$	−1455.60	−0.0785952
$701$	20482.4	1.10358	0.551791	−	0.833982i	$-0.313945\pi$
$701$	20482.4	1.10358	0.551791	+	0.833982i	$0.313945\pi$
$702$	−356.309	−0.0191567
$703$	813.089	0.0436219
$704$	−24882.3	−1.33209
$705$	10060.9	0.537469
$706$	−1566.47	−0.0835053
$707$	37627.4	2.00159
$708$	−1404.54	−0.0745564
$709$	−9508.25	−0.503653	−0.251826	−	0.967772i	$-0.581031\pi$
$709$	−9508.25	−0.503653	−0.251826	+	0.967772i	$0.581031\pi$
$710$	1794.28	0.0948424
$711$	−3319.45	−0.175090
$712$	−2571.89	−0.135373
$713$	−2307.85	−0.121220
$714$	−0.749934	−3.93075e−5 0
$715$	−28661.4	−1.49913
$716$	−21563.7	−1.12552
$717$	−21312.9	−1.11011
$718$	2242.06	0.116536
$719$	7683.75	0.398547	0.199274	−	0.979944i	$-0.436142\pi$
$719$	7683.75	0.398547	0.199274	+	0.979944i	$0.436142\pi$
$720$	−6082.81	−0.314851
$721$	−29376.7	−1.51740
$722$	102.140	0.00526489
$723$	−373.995	−0.0192379
$724$	−16793.4	−0.862044
$725$	796.432	0.0407982
$726$	−974.138	−0.0497984
$727$	−28207.9	−1.43903	−0.719515	−	0.694477i	$-0.755636\pi$
$727$	−28207.9	−1.43903	−0.719515	+	0.694477i	$0.755636\pi$
$728$	4898.75	0.249395
$729$	729.000	0.0370370
$730$	−2319.73	−0.117612
$731$	−11.7051	−0.000592239 0
$732$	17923.2	0.905001
$733$	−21546.7	−1.08574	−0.542870	−	0.839817i	$-0.682662\pi$
$733$	−21546.7	−1.08574	−0.542870	+	0.839817i	$0.682662\pi$
$734$	3381.61	0.170051
$735$	6482.40	0.325316
$736$	−411.031	−0.0205853
$737$	38368.9	1.91769
$738$	−424.468	−0.0211719
$739$	13254.6	0.659780	0.329890	−	0.944019i	$-0.392988\pi$
$739$	13254.6	0.659780	0.329890	+	0.944019i	$0.392988\pi$
$740$	−819.490	−0.0407095
$741$	13258.2	0.657293
$742$	−1621.08	−0.0802046
$743$	−13311.7	−0.657280	−0.328640	−	0.944455i	$-0.606590\pi$
$743$	−13311.7	−0.657280	−0.328640	+	0.944455i	$0.606590\pi$
$744$	3300.43	0.162634
$745$	−6401.51	−0.314810
$746$	2546.07	0.124957
$747$	−4913.00	−0.240639
$748$	17.0887	0.000835326 0
$749$	−17093.5	−0.833889
$750$	−1097.66	−0.0534411
$751$	30067.0	1.46093	0.730467	−	0.682948i	$-0.239302\pi$
$751$	30067.0	1.46093	0.730467	+	0.682948i	$0.239302\pi$
$752$	19352.0	0.938423
$753$	20522.4	0.993199
$754$	−1334.73	−0.0644667
$755$	25960.7	1.25140
$756$	−4991.01	−0.240108
$757$	−40542.2	−1.94654	−0.973269	−	0.229666i	$-0.926236\pi$
$757$	−40542.2	−1.94654	−0.973269	+	0.229666i	$0.926236\pi$
$758$	−991.247	−0.0474983
$759$	1302.88	0.0623078
$760$	−3738.91	−0.178453
$761$	−35970.2	−1.71343	−0.856713	−	0.515793i	$-0.827498\pi$
$761$	−35970.2	−1.71343	−0.856713	+	0.515793i	$0.827498\pi$
$762$	−1191.20	−0.0566310
$763$	−13577.5	−0.644216
$764$	1285.90	0.0608931
$765$	4.10798	0.000194149 0
$766$	−773.140	−0.0364683
$767$	−3060.10	−0.144060
$768$	−11305.7	−0.531195
$769$	29540.6	1.38525	0.692627	−	0.721296i	$-0.256453\pi$
$769$	29540.6	1.38525	0.692627	+	0.721296i	$0.256453\pi$
$770$	3275.34	0.153292
$771$	1124.83	0.0525417
$772$	13789.2	0.642854
$773$	31936.1	1.48598	0.742989	−	0.669304i	$-0.233408\pi$
$773$	31936.1	1.48598	0.742989	+	0.669304i	$0.233408\pi$
$774$	635.529	0.0295137
$775$	−2136.63	−0.0990324
$776$	−1282.72	−0.0593388
$777$	−666.870	−0.0307900
$778$	−3444.38	−0.158724
$779$	15794.5	0.726439
$780$	−13362.6	−0.613409
$781$	−33271.6	−1.52439
$782$	0.0912714	4.17373e−6 0
$783$	2730.82	0.124638
$784$	12468.8	0.568003
$785$	−1466.21	−0.0666640
$786$	−1991.16	−0.0903590
$787$	19554.7	0.885705	0.442853	−	0.896594i	$-0.353966\pi$
$787$	19554.7	0.885705	0.442853	+	0.896594i	$0.353966\pi$
$788$	15063.7	0.680993
$789$	−21524.8	−0.971232
$790$	1015.61	0.0457391
$791$	38711.2	1.74009
$792$	−1863.24	−0.0835950
$793$	39049.6	1.74867
$794$	−888.319	−0.0397043
$795$	8879.94	0.396150
$796$	7340.19	0.326842
$797$	−19917.3	−0.885202	−0.442601	−	0.896719i	$-0.645944\pi$
$797$	−19917.3	−0.885202	−0.442601	+	0.896719i	$0.645944\pi$
$798$	−1515.11	−0.0672110
$799$	−13.0692	−0.000578668 0
$800$	−380.537	−0.0168175
$801$	5708.95	0.251830
$802$	3223.11	0.141910
$803$	43015.1	1.89037
$804$	17888.5	0.784675
$805$	−2144.30	−0.0938840
$806$	3580.75	0.156484
$807$	8992.72	0.392266
$808$	6549.06	0.285142
$809$	−23172.3	−1.00704	−0.503520	−	0.863984i	$-0.667962\pi$
$809$	−23172.3	−1.00704	−0.503520	+	0.863984i	$0.667962\pi$
$810$	−223.044	−0.00967525
$811$	−9763.58	−0.422744	−0.211372	−	0.977406i	$-0.567793\pi$
$811$	−9763.58	−0.422744	−0.211372	+	0.977406i	$0.567793\pi$
$812$	−18696.3	−0.808017
$813$	−17981.3	−0.775685
$814$	−123.972	−0.00533809
$815$	−17854.6	−0.767386
$816$	7.90163	0.000338986 0
$817$	−23648.0	−1.01266
$818$	2654.11	0.113446
$819$	−10874.0	−0.463942
$820$	−15918.8	−0.677939
$821$	24006.4	1.02050	0.510249	−	0.860026i	$-0.329553\pi$
$821$	24006.4	1.02050	0.510249	+	0.860026i	$0.329553\pi$
$822$	1743.70	0.0739884
$823$	36909.5	1.56329	0.781643	−	0.623726i	$-0.214382\pi$
$823$	36909.5	1.56329	0.781643	+	0.623726i	$0.214382\pi$
$824$	−5113.02	−0.216166
$825$	1206.22	0.0509034
$826$	349.699	0.0147307
$827$	−17834.8	−0.749912	−0.374956	−	0.927043i	$-0.622342\pi$
$827$	−17834.8	−0.749912	−0.374956	+	0.927043i	$0.622342\pi$
$828$	607.435	0.0254950
$829$	−7164.05	−0.300142	−0.150071	−	0.988675i	$-0.547950\pi$
$829$	−7164.05	−0.300142	−0.150071	+	0.988675i	$0.547950\pi$
$830$	1503.17	0.0628625
$831$	26278.0	1.09696
$832$	−25274.8	−1.05318
$833$	−8.42070	−0.000350252 0
$834$	220.080	0.00913758
$835$	−37886.9	−1.57021
$836$	34524.7	1.42830
$837$	−7326.14	−0.302543
$838$	−1769.62	−0.0729480
$839$	−30792.5	−1.26708	−0.633538	−	0.773712i	$-0.718398\pi$
$839$	−30792.5	−1.26708	−0.633538	+	0.773712i	$0.718398\pi$
$840$	3066.54	0.125959
$841$	−14159.4	−0.580564
$842$	−757.480	−0.0310030
$843$	14975.7	0.611851
$844$	−20264.1	−0.826444
$845$	−5336.47	−0.217254
$846$	709.596	0.0288374
$847$	−29729.2	−1.20603
$848$	17080.4	0.691680
$849$	1410.91	0.0570345
$850$	0.0845001	3.40980e−6 0
$851$	81.1620	0.00326933
$852$	−15512.0	−0.623747
$853$	31914.3	1.28104	0.640519	−	0.767942i	$-0.278719\pi$
$853$	31914.3	1.28104	0.640519	+	0.767942i	$0.278719\pi$
$854$	−4462.47	−0.178809
$855$	8299.46	0.331971
$856$	−2975.13	−0.118794
$857$	−48527.1	−1.93425	−0.967126	−	0.254296i	$-0.918156\pi$
$857$	−48527.1	−1.93425	−0.967126	+	0.254296i	$0.918156\pi$
$858$	−2021.49	−0.0804341
$859$	−5570.19	−0.221249	−0.110624	−	0.993862i	$-0.535285\pi$
$859$	−5570.19	−0.221249	−0.110624	+	0.993862i	$0.535285\pi$
$860$	23834.2	0.945047
$861$	−12954.1	−0.512748
$862$	−2473.48	−0.0977345
$863$	−45442.4	−1.79244	−0.896221	−	0.443607i	$-0.853699\pi$
$863$	−45442.4	−1.79244	−0.896221	+	0.443607i	$0.853699\pi$
$864$	−1304.79	−0.0513773
$865$	−27324.6	−1.07406
$866$	207.261	0.00813280
$867$	14739.0	0.577350
$868$	50157.5	1.96136
$869$	−18832.7	−0.735160
$870$	−835.518	−0.0325594
$871$	38974.0	1.51617
$872$	−2363.16	−0.0917737
$873$	2847.32	0.110386
$874$	184.398	0.00713657
$875$	−33498.9	−1.29425
$876$	20054.7	0.773498
$877$	−23890.4	−0.919865	−0.459932	−	0.887954i	$-0.652126\pi$
$877$	−23890.4	−0.919865	−0.459932	+	0.887954i	$0.652126\pi$
$878$	3911.69	0.150357
$879$	−5332.69	−0.204627
$880$	−34510.3	−1.32198
$881$	−9587.92	−0.366657	−0.183329	−	0.983052i	$-0.558687\pi$
$881$	−9587.92	−0.366657	−0.183329	+	0.983052i	$0.558687\pi$
$882$	457.204	0.0174545
$883$	−50410.3	−1.92122	−0.960612	−	0.277893i	$-0.910364\pi$
$883$	−50410.3	−1.92122	−0.960612	+	0.277893i	$0.910364\pi$
$884$	17.3582	0.000660429 0
$885$	−1915.57	−0.0727586
$886$	1389.97	0.0527054
$887$	7413.01	0.280614	0.140307	−	0.990108i	$-0.455191\pi$
$887$	7413.01	0.280614	0.140307	+	0.990108i	$0.455191\pi$
$888$	−116.069	−0.00438628
$889$	−36353.8	−1.37150
$890$	−1746.70	−0.0657860
$891$	4135.92	0.155509
$892$	22750.8	0.853982
$893$	−26404.1	−0.989450
$894$	−451.499	−0.0168908
$895$	−29409.5	−1.09838
$896$	11894.3	0.443484
$897$	1323.43	0.0492620
$898$	−2393.87	−0.0889584
$899$	−27443.6	−1.01813
$900$	562.370	0.0208285
$901$	−11.5351	−0.000426516 0
$902$	−2408.19	−0.0888957
$903$	19395.4	0.714771
$904$	6737.69	0.247890
$905$	−22903.5	−0.841258
$906$	1831.01	0.0671426
$907$	−13964.2	−0.511216	−0.255608	−	0.966781i	$-0.582276\pi$
$907$	−13964.2	−0.511216	−0.255608	+	0.966781i	$0.582276\pi$
$908$	43998.3	1.60808
$909$	−14537.3	−0.530442
$910$	3326.99	0.121196
$911$	48214.6	1.75348	0.876740	−	0.480965i	$-0.159714\pi$
$911$	48214.6	1.75348	0.876740	+	0.480965i	$0.159714\pi$
$912$	15963.9	0.579624
$913$	−27873.5	−1.01038
$914$	319.072	0.0115470
$915$	24444.5	0.883179
$916$	23892.6	0.861826
$917$	−60767.1	−2.18834
$918$	0.289736	1.04169e−5 0
$919$	−41258.5	−1.48095	−0.740475	−	0.672084i	$-0.765399\pi$
$919$	−41258.5	−1.48095	−0.740475	+	0.672084i	$0.765399\pi$
$920$	−373.216	−0.0133745
$921$	−12414.2	−0.444149
$922$	−3981.17	−0.142205
$923$	−33796.3	−1.20522
$924$	−28316.1	−1.00815
$925$	75.1407	0.00267093
$926$	4678.41	0.166028
$927$	11349.6	0.402126
$928$	−4887.74	−0.172897
$929$	2682.75	0.0947451	0.0473726	−	0.998877i	$-0.484915\pi$
$929$	2682.75	0.0947451	0.0473726	+	0.998877i	$0.484915\pi$
$930$	2241.49	0.0790338
$931$	−17012.6	−0.598888
$932$	−1051.92	−0.0369708
$933$	−22068.6	−0.774376
$934$	3342.22	0.117089
$935$	23.3063	0.000815184 0
$936$	−1892.62	−0.0660922
$937$	12243.8	0.426883	0.213441	−	0.976956i	$-0.431533\pi$
$937$	12243.8	0.426883	0.213441	+	0.976956i	$0.431533\pi$
$938$	−4453.83	−0.155035
$939$	−20090.8	−0.698230
$940$	26612.0	0.923390
$941$	29701.6	1.02895	0.514476	−	0.857505i	$-0.327986\pi$
$941$	29701.6	1.02895	0.514476	+	0.857505i	$0.327986\pi$
$942$	−103.412	−0.00357679
$943$	1576.59	0.0544443
$944$	−3684.58	−0.127037
$945$	−6806.96	−0.234318
$946$	3605.62	0.123921
$947$	11974.6	0.410899	0.205450	−	0.978668i	$-0.434134\pi$
$947$	11974.6	0.410899	0.205450	+	0.978668i	$0.434134\pi$
$948$	−8780.24	−0.300811
$949$	43693.5	1.49457
$950$	170.718	0.00583033
$951$	−23485.0	−0.800793
$952$	−3.98347	−0.000135614 0
$953$	23437.4	0.796655	0.398327	−	0.917243i	$-0.369591\pi$
$953$	23437.4	0.796655	0.398327	+	0.917243i	$0.369591\pi$
$954$	626.303	0.0212550
$955$	1753.77	0.0594248
$956$	−56374.6	−1.90720
$957$	15493.1	0.523324
$958$	−1251.46	−0.0422053
$959$	53215.1	1.79187
$960$	−15821.6	−0.531917
$961$	43833.6	1.47137
$962$	−125.927	−0.00422043
$963$	6604.04	0.220989
$964$	−989.249	−0.0330514
$965$	18806.3	0.627353
$966$	−151.238	−0.00503726
$967$	8006.86	0.266270	0.133135	−	0.991098i	$-0.457496\pi$
$967$	8006.86	0.266270	0.133135	+	0.991098i	$0.457496\pi$
$968$	−5174.38	−0.171809
$969$	−10.7811	−0.000357418 0
$970$	−871.160	−0.0288363
$971$	−13457.8	−0.444780	−0.222390	−	0.974958i	$-0.571386\pi$
$971$	−13457.8	−0.444780	−0.222390	+	0.974958i	$0.571386\pi$
$972$	1928.27	0.0636309
$973$	6716.51	0.221297
$974$	3122.45	0.102721
$975$	1225.25	0.0402454
$976$	47018.5	1.54204
$977$	25366.7	0.830659	0.415329	−	0.909671i	$-0.363666\pi$
$977$	25366.7	0.830659	0.415329	+	0.909671i	$0.363666\pi$
$978$	−1259.29	−0.0411734
$979$	32389.3	1.05737
$980$	17146.5	0.558903
$981$	5245.63	0.170724
$982$	−2332.05	−0.0757829
$983$	5778.72	0.187500	0.0937500	−	0.995596i	$-0.470115\pi$
$983$	5778.72	0.187500	0.0937500	+	0.995596i	$0.470115\pi$
$984$	−2254.67	−0.0730450
$985$	20544.5	0.664572
$986$	1.08535	3.50552e−5 0
$987$	21655.8	0.698391
$988$	35069.2	1.12925
$989$	−2360.53	−0.0758954
$990$	−1265.42	−0.0406239
$991$	24959.9	0.800079	0.400039	−	0.916498i	$-0.368996\pi$
$991$	24959.9	0.800079	0.400039	+	0.916498i	$0.368996\pi$
$992$	13112.6	0.419684
$993$	2375.78	0.0759244
$994$	3862.14	0.123239
$995$	10010.9	0.318961
$996$	−12995.3	−0.413426
$997$	29815.3	0.947101	0.473550	−	0.880767i	$-0.342972\pi$
$997$	29815.3	0.947101	0.473550	+	0.880767i	$0.342972\pi$
$998$	−1096.35	−0.0347738
$999$	257.644	0.00815966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.c.1.4	✓	8
3.2	odd	2		531.4.a.f.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.4	✓	8	1.1	even	1	trivial
531.4.a.f.1.5		8	3.2	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+0.254436 q^{2} -3.00000 q^{3} -7.93526 q^{4} -10.8225 q^{5} -0.763309 q^{6} -23.2950 q^{7} -4.05451 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.254436 q^{2} -3.00000 q^{3} -7.93526 q^{4} -10.8225 q^{5} -0.763309 q^{6} -23.2950 q^{7} -4.05451 q^{8} +9.00000 q^{9} -2.75362 q^{10} +51.0608 q^{11} +23.8058 q^{12} +51.8661 q^{13} -5.92710 q^{14} +32.4674 q^{15} +62.4505 q^{16} -0.0421754 q^{17} +2.28993 q^{18} -85.2082 q^{19} +85.8790 q^{20} +69.8851 q^{21} +12.9917 q^{22} -8.50543 q^{23} +12.1635 q^{24} -7.87442 q^{25} +13.1966 q^{26} -27.0000 q^{27} +184.852 q^{28} -101.142 q^{29} +8.26087 q^{30} +271.338 q^{31} +48.3257 q^{32} -153.182 q^{33} -0.0107310 q^{34} +252.110 q^{35} -71.4174 q^{36} -9.54238 q^{37} -21.6800 q^{38} -155.598 q^{39} +43.8797 q^{40} -185.363 q^{41} +17.7813 q^{42} +277.532 q^{43} -405.181 q^{44} -97.4021 q^{45} -2.16409 q^{46} +309.877 q^{47} -187.351 q^{48} +199.659 q^{49} -2.00354 q^{50} +0.126526 q^{51} -411.571 q^{52} +273.504 q^{53} -6.86978 q^{54} -552.603 q^{55} +94.4499 q^{56} +255.625 q^{57} -25.7341 q^{58} -59.0000 q^{59} -257.637 q^{60} +752.893 q^{61} +69.0383 q^{62} -209.655 q^{63} -487.308 q^{64} -561.319 q^{65} -38.9751 q^{66} +751.435 q^{67} +0.334673 q^{68} +25.5163 q^{69} +64.1458 q^{70} -651.607 q^{71} -36.4906 q^{72} +842.428 q^{73} -2.42793 q^{74} +23.6233 q^{75} +676.149 q^{76} -1189.46 q^{77} -39.5898 q^{78} -368.828 q^{79} -675.868 q^{80} +81.0000 q^{81} -47.1631 q^{82} -545.888 q^{83} -554.557 q^{84} +0.456442 q^{85} +70.6143 q^{86} +303.425 q^{87} -207.026 q^{88} +634.328 q^{89} -24.7826 q^{90} -1208.22 q^{91} +67.4928 q^{92} -814.015 q^{93} +78.8440 q^{94} +922.162 q^{95} -144.977 q^{96} +316.368 q^{97} +50.8004 q^{98} +459.547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * q^99

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