LMFDB - Embedded newform 169.4.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.22799$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.22799 q^{2} -9.74867 q^{3} -3.03607 q^{4} +8.20685 q^{5} +21.7199 q^{6} -8.35495 q^{7} +24.5882 q^{8} +68.0366 q^{9} +O(q^{10})$	\(q-2.22799 q^{2} -9.74867 q^{3} -3.03607 q^{4} +8.20685 q^{5} +21.7199 q^{6} -8.35495 q^{7} +24.5882 q^{8} +68.0366 q^{9} -18.2848 q^{10} +9.69898 q^{11} +29.5976 q^{12} +18.6147 q^{14} -80.0059 q^{15} -30.4937 q^{16} +44.6219 q^{17} -151.585 q^{18} -87.7418 q^{19} -24.9166 q^{20} +81.4497 q^{21} -21.6092 q^{22} +107.053 q^{23} -239.703 q^{24} -57.6475 q^{25} -400.052 q^{27} +25.3662 q^{28} -14.0430 q^{29} +178.252 q^{30} -171.090 q^{31} -128.766 q^{32} -94.5522 q^{33} -99.4170 q^{34} -68.5678 q^{35} -206.564 q^{36} +413.954 q^{37} +195.488 q^{38} +201.792 q^{40} -258.282 q^{41} -181.469 q^{42} +61.0718 q^{43} -29.4468 q^{44} +558.367 q^{45} -238.512 q^{46} +68.7115 q^{47} +297.273 q^{48} -273.195 q^{49} +128.438 q^{50} -435.004 q^{51} +328.701 q^{53} +891.312 q^{54} +79.5981 q^{55} -205.433 q^{56} +855.367 q^{57} +31.2876 q^{58} -147.144 q^{59} +242.904 q^{60} -97.8083 q^{61} +381.186 q^{62} -568.442 q^{63} +530.839 q^{64} +210.661 q^{66} -677.597 q^{67} -135.475 q^{68} -1043.62 q^{69} +152.768 q^{70} -786.767 q^{71} +1672.90 q^{72} -997.675 q^{73} -922.285 q^{74} +561.987 q^{75} +266.390 q^{76} -81.0345 q^{77} +383.897 q^{79} -250.258 q^{80} +2062.99 q^{81} +575.448 q^{82} -519.718 q^{83} -247.287 q^{84} +366.205 q^{85} -136.067 q^{86} +136.901 q^{87} +238.481 q^{88} -683.222 q^{89} -1244.03 q^{90} -325.019 q^{92} +1667.90 q^{93} -153.088 q^{94} -720.085 q^{95} +1255.30 q^{96} +347.294 q^{97} +608.675 q^{98} +659.886 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * 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 $

$2$	−2.22799	−0.787713	−0.393856	−	0.919172i	$-0.628859\pi$
$2$	−2.22799	−0.787713	−0.393856	+	0.919172i	$0.628859\pi$
$3$	−9.74867	−1.87613	−0.938066	−	0.346455i	$-0.887385\pi$
$3$	−9.74867	−1.87613	−0.938066	+	0.346455i	$0.887385\pi$
$4$	−3.03607	−0.379509
$5$	8.20685	0.734043	0.367022	−	0.930212i	$-0.380377\pi$
$5$	8.20685	0.734043	0.367022	+	0.930212i	$0.380377\pi$
$6$	21.7199	1.47785
$7$	−8.35495	−0.451125	−0.225562	−	0.974229i	$-0.572422\pi$
$7$	−8.35495	−0.451125	−0.225562	+	0.974229i	$0.572422\pi$
$8$	24.5882	1.08666
$9$	68.0366	2.51987
$10$	−18.2848	−0.578215
$11$	9.69898	0.265850	0.132925	−	0.991126i	$-0.457563\pi$
$11$	9.69898	0.265850	0.132925	+	0.991126i	$0.457563\pi$
$12$	29.5976	0.712009
$13$	0	0
$13$	0	0
$14$	18.6147	0.355357
$15$	−80.0059	−1.37716
$16$	−30.4937	−0.476465
$17$	44.6219	0.636612	0.318306	−	0.947988i	$-0.396886\pi$
$17$	44.6219	0.636612	0.318306	+	0.947988i	$0.396886\pi$
$18$	−151.585	−1.98494
$19$	−87.7418	−1.05944	−0.529720	−	0.848173i	$-0.677703\pi$
$19$	−87.7418	−1.05944	−0.529720	+	0.848173i	$0.677703\pi$
$20$	−24.9166	−0.278576
$21$	81.4497	0.846370
$22$	−21.6092	−0.209414
$23$	107.053	0.970522	0.485261	−	0.874369i	$-0.338725\pi$
$23$	107.053	0.970522	0.485261	+	0.874369i	$0.338725\pi$
$24$	−239.703	−2.03871
$25$	−57.6475	−0.461180
$26$	0	0
$27$	−400.052	−2.85149
$28$	25.3662	0.171206
$29$	−14.0430	−0.0899214	−0.0449607	−	0.998989i	$-0.514316\pi$
$29$	−14.0430	−0.0899214	−0.0449607	+	0.998989i	$0.514316\pi$
$30$	178.252	1.08481
$31$	−171.090	−0.991247	−0.495624	−	0.868537i	$-0.665060\pi$
$31$	−171.090	−0.991247	−0.495624	+	0.868537i	$0.665060\pi$
$32$	−128.766	−0.711339
$33$	−94.5522	−0.498770
$34$	−99.4170	−0.501467
$35$	−68.5678	−0.331145
$36$	−206.564	−0.956314
$37$	413.954	1.83929	0.919643	−	0.392754i	$-0.128478\pi$
$37$	413.954	1.83929	0.919643	+	0.392754i	$0.128478\pi$
$38$	195.488	0.834534
$39$	0	0
$40$	201.792	0.797653
$41$	−258.282	−0.983825	−0.491912	−	0.870645i	$-0.663702\pi$
$41$	−258.282	−0.983825	−0.491912	+	0.870645i	$0.663702\pi$
$42$	−181.469	−0.666697
$43$	61.0718	0.216590	0.108295	−	0.994119i	$-0.465461\pi$
$43$	61.0718	0.216590	0.108295	+	0.994119i	$0.465461\pi$
$44$	−29.4468	−0.100892
$45$	558.367	1.84970
$46$	−238.512	−0.764492
$47$	68.7115	0.213247	0.106623	−	0.994299i	$-0.465996\pi$
$47$	68.7115	0.213247	0.106623	+	0.994299i	$0.465996\pi$
$48$	297.273	0.893911
$49$	−273.195	−0.796486
$50$	128.438	0.363278
$51$	−435.004	−1.19437
$52$	0	0
$53$	328.701	0.851896	0.425948	−	0.904748i	$-0.359941\pi$
$53$	328.701	0.851896	0.425948	+	0.904748i	$0.359941\pi$
$54$	891.312	2.24615
$55$	79.5981	0.195146
$56$	−205.433	−0.490218
$57$	855.367	1.98765
$58$	31.2876	0.0708322
$59$	−147.144	−0.324687	−0.162344	−	0.986734i	$-0.551905\pi$
$59$	−147.144	−0.324687	−0.162344	+	0.986734i	$0.551905\pi$
$60$	242.904	0.522645
$61$	−97.8083	−0.205296	−0.102648	−	0.994718i	$-0.532732\pi$
$61$	−97.8083	−0.205296	−0.102648	+	0.994718i	$0.532732\pi$
$62$	381.186	0.780818
$63$	−568.442	−1.13678
$64$	530.839	1.03680
$65$	0	0
$66$	210.661	0.392888
$67$	−677.597	−1.23555	−0.617774	−	0.786356i	$-0.711965\pi$
$67$	−677.597	−1.23555	−0.617774	+	0.786356i	$0.711965\pi$
$68$	−135.475	−0.241600
$69$	−1043.62	−1.82083
$70$	152.768	0.260847
$71$	−786.767	−1.31510	−0.657549	−	0.753411i	$-0.728407\pi$
$71$	−786.767	−1.31510	−0.657549	+	0.753411i	$0.728407\pi$
$72$	1672.90	2.73824
$73$	−997.675	−1.59958	−0.799788	−	0.600282i	$-0.795055\pi$
$73$	−997.675	−1.59958	−0.799788	+	0.600282i	$0.795055\pi$
$74$	−922.285	−1.44883
$75$	561.987	0.865236
$76$	266.390	0.402067
$77$	−81.0345	−0.119932
$78$	0	0
$79$	383.897	0.546731	0.273366	−	0.961910i	$-0.411863\pi$
$79$	383.897	0.546731	0.273366	+	0.961910i	$0.411863\pi$
$80$	−250.258	−0.349746
$81$	2062.99	2.82989
$82$	575.448	0.774971
$83$	−519.718	−0.687307	−0.343654	−	0.939096i	$-0.611665\pi$
$83$	−519.718	−0.687307	−0.343654	+	0.939096i	$0.611665\pi$
$84$	−247.287	−0.321205
$85$	366.205	0.467301
$86$	−136.067	−0.170610
$87$	136.901	0.168704
$88$	238.481	0.288888
$89$	−683.222	−0.813723	−0.406862	−	0.913490i	$-0.633377\pi$
$89$	−683.222	−0.813723	−0.406862	+	0.913490i	$0.633377\pi$
$90$	−1244.03	−1.45703
$91$	0	0
$92$	−325.019	−0.368321
$93$	1667.90	1.85971
$94$	−153.088	−0.167977
$95$	−720.085	−0.777675
$96$	1255.30	1.33457
$97$	347.294	0.363530	0.181765	−	0.983342i	$-0.441819\pi$
$97$	347.294	0.363530	0.181765	+	0.983342i	$0.441819\pi$
$98$	608.675	0.627402
$99$	659.886	0.669909
$100$	175.022	0.175022
$101$	554.794	0.546575	0.273288	−	0.961932i	$-0.411889\pi$
$101$	554.794	0.546575	0.273288	+	0.961932i	$0.411889\pi$
$102$	969.184	0.940819
$103$	1137.13	1.08781	0.543905	−	0.839147i	$-0.316945\pi$
$103$	1137.13	1.08781	0.543905	+	0.839147i	$0.316945\pi$
$104$	0	0
$105$	668.445	0.621272
$106$	−732.341	−0.671050
$107$	−1556.14	−1.40596	−0.702980	−	0.711210i	$-0.748147\pi$
$107$	−1556.14	−1.40596	−0.702980	+	0.711210i	$0.748147\pi$
$108$	1214.59	1.08216
$109$	71.6448	0.0629572	0.0314786	−	0.999504i	$-0.489978\pi$
$109$	71.6448	0.0629572	0.0314786	+	0.999504i	$0.489978\pi$
$110$	−177.344	−0.153719
$111$	−4035.50	−3.45075
$112$	254.774	0.214945
$113$	−980.872	−0.816573	−0.408286	−	0.912854i	$-0.633874\pi$
$113$	−980.872	−0.816573	−0.408286	+	0.912854i	$0.633874\pi$
$114$	−1905.75	−1.56570
$115$	878.565	0.712405
$116$	42.6355	0.0341259
$117$	0	0
$118$	327.836	0.255760
$119$	−372.814	−0.287191
$120$	−1967.20	−1.49650
$121$	−1236.93	−0.929324
$122$	217.916	0.161714
$123$	2517.90	1.84579
$124$	519.441	0.376187
$125$	−1498.96	−1.07257
$126$	1266.48	0.895455
$127$	−2177.31	−1.52130	−0.760649	−	0.649164i	$-0.775119\pi$
$127$	−2177.31	−1.52130	−0.760649	+	0.649164i	$0.775119\pi$
$128$	−152.575	−0.105358
$129$	−595.369	−0.406351
$130$	0	0
$131$	−1919.81	−1.28041	−0.640207	−	0.768202i	$-0.721152\pi$
$131$	−1919.81	−1.28041	−0.640207	+	0.768202i	$0.721152\pi$
$132$	287.067	0.189288
$133$	733.079	0.477940
$134$	1509.68	0.973257
$135$	−3283.17	−2.09311
$136$	1097.17	0.691778
$137$	744.013	0.463980	0.231990	−	0.972718i	$-0.425476\pi$
$137$	744.013	0.463980	0.231990	+	0.972718i	$0.425476\pi$
$138$	2325.17	1.43429
$139$	2820.00	1.72078	0.860392	−	0.509633i	$-0.170219\pi$
$139$	2820.00	1.72078	0.860392	+	0.509633i	$0.170219\pi$
$140$	208.177	0.125672
$141$	−669.846	−0.400080
$142$	1752.91	1.03592
$143$	0	0
$144$	−2074.69	−1.20063
$145$	−115.249	−0.0660062
$146$	2222.81	1.26001
$147$	2663.29	1.49431
$148$	−1256.79	−0.698025
$149$	−2894.60	−1.59151	−0.795755	−	0.605619i	$-0.792926\pi$
$149$	−2894.60	−1.59151	−0.795755	+	0.605619i	$0.792926\pi$
$150$	−1252.10	−0.681557
$151$	−494.004	−0.266235	−0.133118	−	0.991100i	$-0.542499\pi$
$151$	−494.004	−0.266235	−0.133118	+	0.991100i	$0.542499\pi$
$152$	−2157.42	−1.15125
$153$	3035.92	1.60418
$154$	180.544	0.0944717
$155$	−1404.11	−0.727618
$156$	0	0
$157$	50.7450	0.0257955	0.0128977	−	0.999917i	$-0.495894\pi$
$157$	50.7450	0.0257955	0.0128977	+	0.999917i	$0.495894\pi$
$158$	−855.318	−0.430667
$159$	−3204.39	−1.59827
$160$	−1056.77	−0.522154
$161$	−894.419	−0.437826
$162$	−4596.32	−2.22914
$163$	751.801	0.361261	0.180631	−	0.983551i	$-0.442186\pi$
$163$	751.801	0.361261	0.180631	+	0.983551i	$0.442186\pi$
$164$	784.161	0.373370
$165$	−775.976	−0.366119
$166$	1157.93	0.541401
$167$	−2974.32	−1.37820	−0.689102	−	0.724664i	$-0.741995\pi$
$167$	−2974.32	−1.37820	−0.689102	+	0.724664i	$0.741995\pi$
$168$	2002.70	0.919714
$169$	0	0
$170$	−815.901	−0.368099
$171$	−5969.66	−2.66966
$172$	−185.418	−0.0821976
$173$	1633.74	0.717982	0.358991	−	0.933341i	$-0.383121\pi$
$173$	1633.74	0.717982	0.358991	+	0.933341i	$0.383121\pi$
$174$	−305.013	−0.132891
$175$	481.642	0.208050
$176$	−295.758	−0.126668
$177$	1434.46	0.609156
$178$	1522.21	0.640980
$179$	3392.65	1.41664	0.708320	−	0.705892i	$-0.249454\pi$
$179$	3392.65	1.41664	0.708320	+	0.705892i	$0.249454\pi$
$180$	−1695.24	−0.701976
$181$	−3801.07	−1.56095	−0.780473	−	0.625190i	$-0.785021\pi$
$181$	−3801.07	−1.56095	−0.780473	+	0.625190i	$0.785021\pi$
$182$	0	0
$183$	953.501	0.385163
$184$	2632.23	1.05462
$185$	3397.26	1.35012
$186$	−3716.06	−1.46492
$187$	432.787	0.169243
$188$	−208.613	−0.0809290
$189$	3342.42	1.28638
$190$	1604.34	0.612584
$191$	1265.51	0.479419	0.239710	−	0.970845i	$-0.422948\pi$
$191$	1265.51	0.479419	0.239710	+	0.970845i	$0.422948\pi$
$192$	−5174.98	−1.94517
$193$	2212.32	0.825109	0.412555	−	0.910933i	$-0.364637\pi$
$193$	2212.32	0.825109	0.412555	+	0.910933i	$0.364637\pi$
$194$	−773.767	−0.286357
$195$	0	0
$196$	829.438	0.302273
$197$	4357.25	1.57585	0.787923	−	0.615774i	$-0.211157\pi$
$197$	4357.25	1.57585	0.787923	+	0.615774i	$0.211157\pi$
$198$	−1470.22	−0.527696
$199$	1090.63	0.388507	0.194254	−	0.980951i	$-0.437772\pi$
$199$	1090.63	0.388507	0.194254	+	0.980951i	$0.437772\pi$
$200$	−1417.45	−0.501145
$201$	6605.67	2.31805
$202$	−1236.08	−0.430544
$203$	117.329	0.0405658
$204$	1320.70	0.453273
$205$	−2119.68	−0.722170
$206$	−2533.50	−0.856881
$207$	7283.49	2.44559
$208$	0	0
$209$	−851.007	−0.281652
$210$	−1489.29	−0.489384
$211$	−727.448	−0.237344	−0.118672	−	0.992934i	$-0.537864\pi$
$211$	−727.448	−0.237344	−0.118672	+	0.992934i	$0.537864\pi$
$212$	−997.958	−0.323302
$213$	7669.93	2.46730
$214$	3467.06	1.10749
$215$	501.207	0.158986
$216$	−9836.58	−3.09859
$217$	1429.45	0.447176
$218$	−159.624	−0.0495922
$219$	9726.01	3.00102
$220$	−241.665	−0.0740595
$221$	0	0
$222$	8991.05	2.71820
$223$	1015.45	0.304931	0.152465	−	0.988309i	$-0.451279\pi$
$223$	1015.45	0.304931	0.152465	+	0.988309i	$0.451279\pi$
$224$	1075.83	0.320903
$225$	−3922.14	−1.16212
$226$	2185.37	0.643225
$227$	−5292.59	−1.54749	−0.773747	−	0.633494i	$-0.781620\pi$
$227$	−5292.59	−1.54749	−0.773747	+	0.633494i	$0.781620\pi$
$228$	−2596.95	−0.754330
$229$	3010.03	0.868597	0.434298	−	0.900769i	$-0.356996\pi$
$229$	3010.03	0.868597	0.434298	+	0.900769i	$0.356996\pi$
$230$	−1957.43	−0.561171
$231$	789.979	0.225008
$232$	−345.292	−0.0977136
$233$	−2373.96	−0.667482	−0.333741	−	0.942665i	$-0.608311\pi$
$233$	−2373.96	−0.667482	−0.333741	+	0.942665i	$0.608311\pi$
$234$	0	0
$235$	563.905	0.156532
$236$	446.740	0.123222
$237$	−3742.49	−1.02574
$238$	830.624	0.226224
$239$	783.439	0.212035	0.106018	−	0.994364i	$-0.466190\pi$
$239$	783.439	0.212035	0.106018	+	0.994364i	$0.466190\pi$
$240$	2439.68	0.656169
$241$	−3517.40	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$241$	−3517.40	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$242$	2755.86	0.732040
$243$	−9310.02	−2.45777
$244$	296.953	0.0779117
$245$	−2242.07	−0.584656
$246$	−5609.86	−1.45395
$247$	0	0
$248$	−4206.80	−1.07715
$249$	5066.56	1.28948
$250$	3339.67	0.844877
$251$	1416.96	0.356326	0.178163	−	0.984001i	$-0.442985\pi$
$251$	1416.96	0.356326	0.178163	+	0.984001i	$0.442985\pi$
$252$	1725.83	0.431417
$253$	1038.30	0.258013
$254$	4851.01	1.19835
$255$	−3570.02	−0.876718
$256$	−3906.78	−0.953804
$257$	−1482.63	−0.359861	−0.179930	−	0.983679i	$-0.557587\pi$
$257$	−1482.63	−0.359861	−0.179930	+	0.983679i	$0.557587\pi$
$258$	1326.47	0.320088
$259$	−3458.56	−0.829748
$260$	0	0
$261$	−955.438	−0.226591
$262$	4277.31	1.00860
$263$	7229.64	1.69505	0.847526	−	0.530753i	$-0.178091\pi$
$263$	7229.64	1.69505	0.847526	+	0.530753i	$0.178091\pi$
$264$	−2324.87	−0.541992
$265$	2697.60	0.625329
$266$	−1633.29	−0.376479
$267$	6660.50	1.52665
$268$	2057.23	0.468901
$269$	−494.904	−0.112174	−0.0560870	−	0.998426i	$-0.517862\pi$
$269$	−494.904	−0.112174	−0.0560870	+	0.998426i	$0.517862\pi$
$270$	7314.87	1.64877
$271$	−4205.58	−0.942696	−0.471348	−	0.881947i	$-0.656232\pi$
$271$	−4205.58	−0.942696	−0.471348	+	0.881947i	$0.656232\pi$
$272$	−1360.69	−0.303323
$273$	0	0
$274$	−1657.65	−0.365483
$275$	−559.123	−0.122605
$276$	3168.50	0.691020
$277$	4208.55	0.912878	0.456439	−	0.889755i	$-0.349125\pi$
$277$	4208.55	0.912878	0.456439	+	0.889755i	$0.349125\pi$
$278$	−6282.92	−1.35548
$279$	−11640.4	−2.49782
$280$	−1685.96	−0.359841
$281$	−4740.83	−1.00646	−0.503228	−	0.864153i	$-0.667855\pi$
$281$	−4740.83	−1.00646	−0.503228	+	0.864153i	$0.667855\pi$
$282$	1492.41	0.315148
$283$	−3742.82	−0.786176	−0.393088	−	0.919501i	$-0.628593\pi$
$283$	−3742.82	−0.786176	−0.393088	+	0.919501i	$0.628593\pi$
$284$	2388.68	0.499091
$285$	7019.87	1.45902
$286$	0	0
$287$	2157.93	0.443828
$288$	−8760.81	−1.79249
$289$	−2921.89	−0.594726
$290$	256.773	0.0519939
$291$	−3385.66	−0.682030
$292$	3029.01	0.607053
$293$	5254.97	1.04778	0.523889	−	0.851787i	$-0.324481\pi$
$293$	5254.97	1.04778	0.523889	+	0.851787i	$0.324481\pi$
$294$	−5933.77	−1.17709
$295$	−1207.59	−0.238334
$296$	10178.4	1.99867
$297$	−3880.10	−0.758069
$298$	6449.14	1.25365
$299$	0	0
$300$	−1706.23	−0.328364
$301$	−510.251	−0.0977090
$302$	1100.64	0.209717
$303$	−5408.51	−1.02545
$304$	2675.58	0.504786
$305$	−802.698	−0.150696
$306$	−6764.00	−1.26363
$307$	−252.464	−0.0469344	−0.0234672	−	0.999725i	$-0.507471\pi$
$307$	−252.464	−0.0469344	−0.0234672	+	0.999725i	$0.507471\pi$
$308$	246.026	0.0455151
$309$	−11085.5	−2.04088
$310$	3128.34	0.573154
$311$	2561.20	0.466986	0.233493	−	0.972359i	$-0.424984\pi$
$311$	2561.20	0.466986	0.233493	+	0.972359i	$0.424984\pi$
$312$	0	0
$313$	−695.893	−0.125668	−0.0628342	−	0.998024i	$-0.520014\pi$
$313$	−695.893	−0.125668	−0.0628342	+	0.998024i	$0.520014\pi$
$314$	−113.059	−0.0203194
$315$	−4665.12	−0.834444
$316$	−1165.54	−0.207489
$317$	−5747.37	−1.01831	−0.509155	−	0.860675i	$-0.670042\pi$
$317$	−5747.37	−1.01831	−0.509155	+	0.860675i	$0.670042\pi$
$318$	7139.35	1.25898
$319$	−136.203	−0.0239056
$320$	4356.52	0.761053
$321$	15170.3	2.63777
$322$	1992.75	0.344881
$323$	−3915.21	−0.674452
$324$	−6263.39	−1.07397
$325$	0	0
$326$	−1675.00	−0.284570
$327$	−698.442	−0.118116
$328$	−6350.69	−1.06908
$329$	−574.081	−0.0962010
$330$	1728.87	0.288397
$331$	4244.76	0.704874	0.352437	−	0.935836i	$-0.385353\pi$
$331$	4244.76	0.704874	0.352437	+	0.935836i	$0.385353\pi$
$332$	1577.90	0.260839
$333$	28164.0	4.63477
$334$	6626.76	1.08563
$335$	−5560.94	−0.906946
$336$	−2483.70	−0.403265
$337$	−7122.49	−1.15130	−0.575648	−	0.817698i	$-0.695250\pi$
$337$	−7122.49	−1.15130	−0.575648	+	0.817698i	$0.695250\pi$
$338$	0	0
$339$	9562.20	1.53200
$340$	−1111.82	−0.177345
$341$	−1659.40	−0.263523
$342$	13300.3	2.10292
$343$	5148.28	0.810440
$344$	1501.65	0.235359
$345$	−8564.84	−1.33657
$346$	−3639.95	−0.565563
$347$	−3367.90	−0.521033	−0.260517	−	0.965469i	$-0.583893\pi$
$347$	−3367.90	−0.521033	−0.260517	+	0.965469i	$0.583893\pi$
$348$	−415.640	−0.0640248
$349$	−6079.71	−0.932491	−0.466246	−	0.884655i	$-0.654394\pi$
$349$	−6079.71	−0.932491	−0.466246	+	0.884655i	$0.654394\pi$
$350$	−1073.09	−0.163884
$351$	0	0
$352$	−1248.90	−0.189110
$353$	−3230.99	−0.487162	−0.243581	−	0.969881i	$-0.578322\pi$
$353$	−3230.99	−0.487162	−0.243581	+	0.969881i	$0.578322\pi$
$354$	−3195.96	−0.479840
$355$	−6456.88	−0.965340
$356$	2074.31	0.308815
$357$	3634.44	0.538809
$358$	−7558.78	−1.11590
$359$	−5345.81	−0.785908	−0.392954	−	0.919558i	$-0.628547\pi$
$359$	−5345.81	−0.785908	−0.392954	+	0.919558i	$0.628547\pi$
$360$	13729.2	2.00999
$361$	839.632	0.122413
$362$	8468.73	1.22958
$363$	12058.4	1.74353
$364$	0	0
$365$	−8187.78	−1.17416
$366$	−2124.39	−0.303398
$367$	−12171.3	−1.73117	−0.865583	−	0.500765i	$-0.833052\pi$
$367$	−12171.3	−1.73117	−0.865583	+	0.500765i	$0.833052\pi$
$368$	−3264.43	−0.462419
$369$	−17572.6	−2.47911
$370$	−7569.05	−1.06350
$371$	−2746.28	−0.384312
$372$	−5063.86	−0.705776
$373$	−4463.95	−0.619664	−0.309832	−	0.950791i	$-0.600273\pi$
$373$	−4463.95	−0.619664	−0.309832	+	0.950791i	$0.600273\pi$
$374$	−964.244	−0.133315
$375$	14612.9	2.01228
$376$	1689.49	0.231726
$377$	0	0
$378$	−7446.87	−1.01330
$379$	−5369.06	−0.727679	−0.363839	−	0.931462i	$-0.618534\pi$
$379$	−5369.06	−0.727679	−0.363839	+	0.931462i	$0.618534\pi$
$380$	2186.23	0.295134
$381$	21225.9	2.85416
$382$	−2819.54	−0.377645
$383$	−195.084	−0.0260269	−0.0130135	−	0.999915i	$-0.504142\pi$
$383$	−195.084	−0.0260269	−0.0130135	+	0.999915i	$0.504142\pi$
$384$	1487.40	0.197665
$385$	−665.038	−0.0880351
$386$	−4929.02	−0.649949
$387$	4155.12	0.545779
$388$	−1054.41	−0.137963
$389$	9120.52	1.18876	0.594381	−	0.804183i	$-0.297397\pi$
$389$	9120.52	1.18876	0.594381	+	0.804183i	$0.297397\pi$
$390$	0	0
$391$	4776.89	0.617845
$392$	−6717.38	−0.865507
$393$	18715.6	2.40223
$394$	−9707.91	−1.24131
$395$	3150.59	0.401325
$396$	−2003.46	−0.254236
$397$	6613.46	0.836071	0.418036	−	0.908431i	$-0.362719\pi$
$397$	6613.46	0.836071	0.418036	+	0.908431i	$0.362719\pi$
$398$	−2429.92	−0.306032
$399$	−7146.54	−0.896678
$400$	1757.89	0.219736
$401$	−8589.26	−1.06964	−0.534822	−	0.844965i	$-0.679621\pi$
$401$	−8589.26	−1.06964	−0.534822	+	0.844965i	$0.679621\pi$
$402$	−14717.4	−1.82596
$403$	0	0
$404$	−1684.39	−0.207430
$405$	16930.7	2.07726
$406$	−261.407	−0.0319542
$407$	4014.93	0.488975
$408$	−10696.0	−1.29787
$409$	−7515.59	−0.908612	−0.454306	−	0.890846i	$-0.650113\pi$
$409$	−7515.59	−0.908612	−0.454306	+	0.890846i	$0.650113\pi$
$410$	4722.62	0.568863
$411$	−7253.14	−0.870489
$412$	−3452.39	−0.412833
$413$	1229.38	0.146474
$414$	−16227.5	−1.92642
$415$	−4265.25	−0.504513
$416$	0	0
$417$	−27491.2	−3.22842
$418$	1896.03	0.221861
$419$	−4557.33	−0.531361	−0.265680	−	0.964061i	$-0.585597\pi$
$419$	−4557.33	−0.531361	−0.265680	+	0.964061i	$0.585597\pi$
$420$	−2029.45	−0.235778
$421$	−2225.19	−0.257599	−0.128800	−	0.991671i	$-0.541112\pi$
$421$	−2225.19	−0.257599	−0.128800	+	0.991671i	$0.541112\pi$
$422$	1620.75	0.186959
$423$	4674.90	0.537355
$424$	8082.17	0.925719
$425$	−2572.34	−0.293593
$426$	−17088.5	−1.94352
$427$	817.183	0.0926142
$428$	4724.54	0.533574
$429$	0	0
$430$	−1116.68	−0.125235
$431$	−396.028	−0.0442599	−0.0221299	−	0.999755i	$-0.507045\pi$
$431$	−396.028	−0.0442599	−0.0221299	+	0.999755i	$0.507045\pi$
$432$	12199.1	1.35863
$433$	5188.06	0.575802	0.287901	−	0.957660i	$-0.407043\pi$
$433$	5188.06	0.575802	0.287901	+	0.957660i	$0.407043\pi$
$434$	−3184.79	−0.352246
$435$	1123.52	0.123836
$436$	−217.519	−0.0238928
$437$	−9392.99	−1.02821
$438$	−21669.4	−2.36394
$439$	11329.9	1.23176	0.615882	−	0.787838i	$-0.288800\pi$
$439$	11329.9	1.23176	0.615882	+	0.787838i	$0.288800\pi$
$440$	1957.18	0.212056
$441$	−18587.2	−2.00705
$442$	0	0
$443$	15625.2	1.67579	0.837897	−	0.545828i	$-0.183785\pi$
$443$	15625.2	1.67579	0.837897	+	0.545828i	$0.183785\pi$
$444$	12252.1	1.30959
$445$	−5607.10	−0.597308
$446$	−2262.41	−0.240198
$447$	28218.5	2.98588
$448$	−4435.14	−0.467724
$449$	−13686.6	−1.43855	−0.719276	−	0.694725i	$-0.755526\pi$
$449$	−13686.6	−1.43855	−0.719276	+	0.694725i	$0.755526\pi$
$450$	8738.49	0.915414
$451$	−2505.07	−0.261550
$452$	2978.00	0.309896
$453$	4815.89	0.499492
$454$	11791.8	1.21898
$455$	0	0
$456$	21031.9	2.15989
$457$	−12673.1	−1.29720	−0.648602	−	0.761127i	$-0.724646\pi$
$457$	−12673.1	−1.29720	−0.648602	+	0.761127i	$0.724646\pi$
$458$	−6706.32	−0.684205
$459$	−17851.1	−1.81529
$460$	−2667.38	−0.270364
$461$	−9154.85	−0.924911	−0.462456	−	0.886642i	$-0.653032\pi$
$461$	−9154.85	−0.924911	−0.462456	+	0.886642i	$0.653032\pi$
$462$	−1760.06	−0.177241
$463$	−6910.59	−0.693655	−0.346827	−	0.937929i	$-0.612741\pi$
$463$	−6910.59	−0.693655	−0.346827	+	0.937929i	$0.612741\pi$
$464$	428.223	0.0428443
$465$	13688.2	1.36511
$466$	5289.16	0.525784
$467$	2920.19	0.289359	0.144679	−	0.989479i	$-0.453785\pi$
$467$	2920.19	0.289359	0.144679	+	0.989479i	$0.453785\pi$
$468$	0	0
$469$	5661.29	0.557386
$470$	−1256.37	−0.123303
$471$	−494.696	−0.0483958
$472$	−3618.02	−0.352823
$473$	592.334	0.0575804
$474$	8338.21	0.807989
$475$	5058.10	0.488593
$476$	1131.89	0.108992
$477$	22363.7	2.14667
$478$	−1745.49	−0.167023
$479$	17088.6	1.63006	0.815030	−	0.579419i	$-0.196720\pi$
$479$	17088.6	1.63006	0.815030	+	0.579419i	$0.196720\pi$
$480$	10302.1	0.979630
$481$	0	0
$482$	7836.73	0.740567
$483$	8719.39	0.821421
$484$	3755.40	0.352686
$485$	2850.19	0.266846
$486$	20742.6	1.93602
$487$	13813.0	1.28527	0.642634	−	0.766174i	$-0.277842\pi$
$487$	13813.0	1.28527	0.642634	+	0.766174i	$0.277842\pi$
$488$	−2404.93	−0.223086
$489$	−7329.06	−0.677774
$490$	4995.31	0.460541
$491$	−11848.4	−1.08903	−0.544513	−	0.838752i	$-0.683286\pi$
$491$	−11848.4	−1.08903	−0.544513	+	0.838752i	$0.683286\pi$
$492$	−7644.53	−0.700492
$493$	−626.625	−0.0572450
$494$	0	0
$495$	5415.59	0.491743
$496$	5217.17	0.472294
$497$	6573.40	0.593274
$498$	−11288.2	−1.01574
$499$	5224.46	0.468696	0.234348	−	0.972153i	$-0.424705\pi$
$499$	5224.46	0.468696	0.234348	+	0.972153i	$0.424705\pi$
$500$	4550.95	0.407049
$501$	28995.7	2.58569
$502$	−3156.97	−0.280682
$503$	7884.39	0.698902	0.349451	−	0.936955i	$-0.386368\pi$
$503$	7884.39	0.698902	0.349451	+	0.936955i	$0.386368\pi$
$504$	−13977.0	−1.23529
$505$	4553.12	0.401210
$506$	−2313.32	−0.203241
$507$	0	0
$508$	6610.45	0.577345
$509$	4026.56	0.350637	0.175318	−	0.984512i	$-0.443905\pi$
$509$	4026.56	0.350637	0.175318	+	0.984512i	$0.443905\pi$
$510$	7953.95	0.690602
$511$	8335.53	0.721609
$512$	9924.86	0.856681
$513$	35101.3	3.02098
$514$	3303.29	0.283467
$515$	9332.23	0.798499
$516$	1807.58	0.154214
$517$	666.432	0.0566918
$518$	7705.64	0.653603
$519$	−15926.8	−1.34703
$520$	0	0
$521$	6196.12	0.521030	0.260515	−	0.965470i	$-0.416108\pi$
$521$	6196.12	0.521030	0.260515	+	0.965470i	$0.416108\pi$
$522$	2128.70	0.178488
$523$	7899.39	0.660452	0.330226	−	0.943902i	$-0.392875\pi$
$523$	7899.39	0.660452	0.330226	+	0.943902i	$0.392875\pi$
$524$	5828.67	0.485928
$525$	−4695.37	−0.390329
$526$	−16107.6	−1.33521
$527$	−7634.36	−0.631040
$528$	2883.25	0.237646
$529$	−706.752	−0.0580876
$530$	−6010.22	−0.492580
$531$	−10011.2	−0.818171
$532$	−2225.68	−0.181382
$533$	0	0
$534$	−14839.5	−1.20256
$535$	−12771.0	−1.03203
$536$	−16660.9	−1.34262
$537$	−33073.8	−2.65780
$538$	1102.64	0.0883609
$539$	−2649.71	−0.211746
$540$	9967.94	0.794355
$541$	6146.22	0.488441	0.244220	−	0.969720i	$-0.421468\pi$
$541$	6146.22	0.488441	0.244220	+	0.969720i	$0.421468\pi$
$542$	9369.98	0.742574
$543$	37055.4	2.92854
$544$	−5745.79	−0.452847
$545$	587.979	0.0462133
$546$	0	0
$547$	5555.49	0.434252	0.217126	−	0.976144i	$-0.430332\pi$
$547$	5555.49	0.434252	0.217126	+	0.976144i	$0.430332\pi$
$548$	−2258.87	−0.176084
$549$	−6654.54	−0.517321
$550$	1245.72	0.0965775
$551$	1232.16	0.0952663
$552$	−25660.8	−1.97861
$553$	−3207.44	−0.246644
$554$	−9376.59	−0.719085
$555$	−33118.8	−2.53300
$556$	−8561.70	−0.653052
$557$	−7580.71	−0.576669	−0.288335	−	0.957530i	$-0.593102\pi$
$557$	−7580.71	−0.576669	−0.288335	+	0.957530i	$0.593102\pi$
$558$	25934.6	1.96756
$559$	0	0
$560$	2090.89	0.157779
$561$	−4219.10	−0.317523
$562$	10562.5	0.792799
$563$	−13594.6	−1.01766	−0.508831	−	0.860866i	$-0.669922\pi$
$563$	−13594.6	−1.01766	−0.508831	+	0.860866i	$0.669922\pi$
$564$	2033.70	0.151834
$565$	−8049.88	−0.599400
$566$	8338.97	0.619281
$567$	−17236.2	−1.27664
$568$	−19345.2	−1.42906
$569$	5650.14	0.416285	0.208143	−	0.978099i	$-0.433258\pi$
$569$	5650.14	0.416285	0.208143	+	0.978099i	$0.433258\pi$
$570$	−15640.2	−1.14929
$571$	6297.53	0.461547	0.230773	−	0.973008i	$-0.425874\pi$
$571$	6297.53	0.461547	0.230773	+	0.973008i	$0.425874\pi$
$572$	0	0
$573$	−12337.0	−0.899454
$574$	−4807.84	−0.349609
$575$	−6171.32	−0.447586
$576$	36116.5	2.61259
$577$	−17838.9	−1.28707	−0.643537	−	0.765415i	$-0.722534\pi$
$577$	−17838.9	−1.28707	−0.643537	+	0.765415i	$0.722534\pi$
$578$	6509.93	0.468473
$579$	−21567.2	−1.54801
$580$	349.903	0.0250499
$581$	4342.22	0.310061
$582$	7543.20	0.537243
$583$	3188.06	0.226477
$584$	−24531.1	−1.73819
$585$	0	0
$586$	−11708.0	−0.825348
$587$	455.091	0.0319993	0.0159997	−	0.999872i	$-0.494907\pi$
$587$	455.091	0.0319993	0.0159997	+	0.999872i	$0.494907\pi$
$588$	−8085.92	−0.567105
$589$	15011.7	1.05017
$590$	2690.50	0.187739
$591$	−42477.4	−2.95649
$592$	−12623.0	−0.876355
$593$	16240.6	1.12466	0.562330	−	0.826913i	$-0.309905\pi$
$593$	16240.6	1.12466	0.562330	+	0.826913i	$0.309905\pi$
$594$	8644.82	0.597140
$595$	−3059.63	−0.210811
$596$	8788.21	0.603992
$597$	−10632.2	−0.728891
$598$	0	0
$599$	−6704.05	−0.457296	−0.228648	−	0.973509i	$-0.573430\pi$
$599$	−6704.05	−0.457296	−0.228648	+	0.973509i	$0.573430\pi$
$600$	13818.3	0.940214
$601$	26413.2	1.79271	0.896354	−	0.443339i	$-0.146206\pi$
$601$	26413.2	1.79271	0.896354	+	0.443339i	$0.146206\pi$
$602$	1136.83	0.0769666
$603$	−46101.4	−3.11343
$604$	1499.83	0.101039
$605$	−10151.3	−0.682164
$606$	12050.1	0.807758
$607$	−3326.37	−0.222427	−0.111214	−	0.993797i	$-0.535474\pi$
$607$	−3326.37	−0.222427	−0.111214	+	0.993797i	$0.535474\pi$
$608$	11298.2	0.753621
$609$	−1143.80	−0.0761068
$610$	1788.40	0.118705
$611$	0	0
$612$	−9217.27	−0.608801
$613$	26258.7	1.73014	0.865072	−	0.501647i	$-0.167272\pi$
$613$	26258.7	1.73014	0.865072	+	0.501647i	$0.167272\pi$
$614$	562.486	0.0369708
$615$	20664.1	1.35489
$616$	−1992.50	−0.130325
$617$	27352.9	1.78474	0.892370	−	0.451304i	$-0.149041\pi$
$617$	27352.9	1.78474	0.892370	+	0.451304i	$0.149041\pi$
$618$	24698.3	1.60762
$619$	−13056.5	−0.847795	−0.423897	−	0.905710i	$-0.639338\pi$
$619$	−13056.5	−0.847795	−0.423897	+	0.905710i	$0.639338\pi$
$620$	4262.98	0.276137
$621$	−42826.6	−2.76743
$622$	−5706.33	−0.367851
$623$	5708.28	0.367091
$624$	0	0
$625$	−5095.82	−0.326132
$626$	1550.44	0.0989905
$627$	8296.19	0.528417
$628$	−154.065	−0.00978961
$629$	18471.4	1.17091
$630$	10393.8	0.657302
$631$	6552.91	0.413419	0.206709	−	0.978402i	$-0.433725\pi$
$631$	6552.91	0.413419	0.206709	+	0.978402i	$0.433725\pi$
$632$	9439.35	0.594109
$633$	7091.66	0.445289
$634$	12805.1	0.802137
$635$	−17868.8	−1.11670
$636$	9728.76	0.606557
$637$	0	0
$638$	303.458	0.0188308
$639$	−53528.9	−3.31388
$640$	−1252.16	−0.0773373
$641$	−5764.77	−0.355218	−0.177609	−	0.984101i	$-0.556836\pi$
$641$	−5764.77	−0.355218	−0.177609	+	0.984101i	$0.556836\pi$
$642$	−33799.2	−2.07780
$643$	12279.5	0.753120	0.376560	−	0.926392i	$-0.377107\pi$
$643$	12279.5	0.753120	0.376560	+	0.926392i	$0.377107\pi$
$644$	2715.52	0.166159
$645$	−4886.10	−0.298279
$646$	8723.04	0.531274
$647$	−29024.3	−1.76362	−0.881810	−	0.471605i	$-0.843675\pi$
$647$	−29024.3	−1.76362	−0.881810	+	0.471605i	$0.843675\pi$
$648$	50725.3	3.07512
$649$	−1427.15	−0.0863182
$650$	0	0
$651$	−13935.2	−0.838962
$652$	−2282.52	−0.137102
$653$	−17167.1	−1.02879	−0.514396	−	0.857552i	$-0.671984\pi$
$653$	−17167.1	−1.02879	−0.514396	+	0.857552i	$0.671984\pi$
$654$	1556.12	0.0930415
$655$	−15755.6	−0.939880
$656$	7875.97	0.468758
$657$	−67878.5	−4.03073
$658$	1279.05	0.0757787
$659$	18313.0	1.08251	0.541254	−	0.840859i	$-0.317950\pi$
$659$	18313.0	1.08251	0.541254	+	0.840859i	$0.317950\pi$
$660$	2355.92	0.138945
$661$	6885.30	0.405155	0.202577	−	0.979266i	$-0.435068\pi$
$661$	6885.30	0.405155	0.202577	+	0.979266i	$0.435068\pi$
$662$	−9457.28	−0.555238
$663$	0	0
$664$	−12779.0	−0.746867
$665$	6016.27	0.350828
$666$	−62749.1	−3.65087
$667$	−1503.34	−0.0872706
$668$	9030.25	0.523040
$669$	−9899.29	−0.572090
$670$	12389.7	0.714413
$671$	−948.641	−0.0545781
$672$	−10488.0	−0.602056
$673$	6238.26	0.357307	0.178653	−	0.983912i	$-0.442826\pi$
$673$	6238.26	0.357307	0.178653	+	0.983912i	$0.442826\pi$
$674$	15868.8	0.906890
$675$	23062.0	1.31505
$676$	0	0
$677$	25482.4	1.44663	0.723316	−	0.690517i	$-0.242617\pi$
$677$	25482.4	1.44663	0.723316	+	0.690517i	$0.242617\pi$
$678$	−21304.5	−1.20678
$679$	−2901.62	−0.163997
$680$	9004.34	0.507795
$681$	51595.7	2.90331
$682$	3697.12	0.207581
$683$	−27426.5	−1.53652	−0.768261	−	0.640136i	$-0.778878\pi$
$683$	−27426.5	−1.53652	−0.768261	+	0.640136i	$0.778878\pi$
$684$	18124.3	1.01316
$685$	6106.00	0.340582
$686$	−11470.3	−0.638394
$687$	−29343.8	−1.62960
$688$	−1862.31	−0.103197
$689$	0	0
$690$	19082.4	1.05283
$691$	13886.3	0.764486	0.382243	−	0.924062i	$-0.375152\pi$
$691$	13886.3	0.764486	0.382243	+	0.924062i	$0.375152\pi$
$692$	−4960.14	−0.272480
$693$	−5513.31	−0.302213
$694$	7503.65	0.410425
$695$	23143.3	1.26313
$696$	3366.14	0.183324
$697$	−11525.0	−0.626314
$698$	13545.5	0.734535
$699$	23143.0	1.25229
$700$	−1462.30	−0.0789567
$701$	15744.4	0.848301	0.424151	−	0.905592i	$-0.360573\pi$
$701$	15744.4	0.848301	0.424151	+	0.905592i	$0.360573\pi$
$702$	0	0
$703$	−36321.1	−1.94861
$704$	5148.60	0.275632
$705$	−5497.33	−0.293676
$706$	7198.61	0.383744
$707$	−4635.28	−0.246574
$708$	−4355.12	−0.231180
$709$	28689.1	1.51967	0.759833	−	0.650118i	$-0.225281\pi$
$709$	28689.1	1.51967	0.759833	+	0.650118i	$0.225281\pi$
$710$	14385.9	0.760410
$711$	26119.0	1.37769
$712$	−16799.2	−0.884237
$713$	−18315.6	−0.962027
$714$	−8097.48	−0.424427
$715$	0	0
$716$	−10300.3	−0.537627
$717$	−7637.49	−0.397807
$718$	11910.4	0.619070
$719$	−12619.1	−0.654538	−0.327269	−	0.944931i	$-0.606128\pi$
$719$	−12619.1	−0.654538	−0.327269	+	0.944931i	$0.606128\pi$
$720$	−17026.7	−0.881315
$721$	−9500.63	−0.490738
$722$	−1870.69	−0.0964265
$723$	34290.0	1.76384
$724$	11540.3	0.592392
$725$	809.544	0.0414700
$726$	−26866.0	−1.37340
$727$	−12644.5	−0.645061	−0.322531	−	0.946559i	$-0.604534\pi$
$727$	−12644.5	−0.645061	−0.322531	+	0.946559i	$0.604534\pi$
$728$	0	0
$729$	35059.5	1.78121
$730$	18242.3	0.924900
$731$	2725.14	0.137884
$732$	−2894.89	−0.146173
$733$	−19109.0	−0.962903	−0.481451	−	0.876473i	$-0.659890\pi$
$733$	−19109.0	−0.962903	−0.481451	+	0.876473i	$0.659890\pi$
$734$	27117.6	1.36366
$735$	21857.2	1.09689
$736$	−13784.7	−0.690370
$737$	−6572.01	−0.328471
$738$	39151.6	1.95283
$739$	14093.6	0.701543	0.350771	−	0.936461i	$-0.385919\pi$
$739$	14093.6	0.701543	0.350771	+	0.936461i	$0.385919\pi$
$740$	−10314.3	−0.512381
$741$	0	0
$742$	6118.67	0.302727
$743$	17587.4	0.868397	0.434198	−	0.900817i	$-0.357032\pi$
$743$	17587.4	0.868397	0.434198	+	0.900817i	$0.357032\pi$
$744$	41010.7	2.02087
$745$	−23755.6	−1.16824
$746$	9945.63	0.488117
$747$	−35359.9	−1.73193
$748$	−1313.97	−0.0642293
$749$	13001.5	0.634263
$750$	−32557.3	−1.58510
$751$	−16587.0	−0.805948	−0.402974	−	0.915211i	$-0.632023\pi$
$751$	−16587.0	−0.805948	−0.402974	+	0.915211i	$0.632023\pi$
$752$	−2095.27	−0.101605
$753$	−13813.5	−0.668514
$754$	0	0
$755$	−4054.22	−0.195428
$756$	−10147.8	−0.488191
$757$	33819.5	1.62376	0.811882	−	0.583822i	$-0.198443\pi$
$757$	33819.5	1.62376	0.811882	+	0.583822i	$0.198443\pi$
$758$	11962.2	0.573202
$759$	−10122.1	−0.484068
$760$	−17705.6	−0.845066
$761$	28131.2	1.34002	0.670011	−	0.742351i	$-0.266290\pi$
$761$	28131.2	1.34002	0.670011	+	0.742351i	$0.266290\pi$
$762$	−47290.9	−2.24825
$763$	−598.589	−0.0284015
$764$	−3842.18	−0.181944
$765$	24915.4	1.17754
$766$	434.645	0.0205018
$767$	0	0
$768$	38085.9	1.78946
$769$	−23735.0	−1.11301	−0.556506	−	0.830844i	$-0.687858\pi$
$769$	−23735.0	−1.11301	−0.556506	+	0.830844i	$0.687858\pi$
$770$	1481.70	0.0693463
$771$	14453.7	0.675146
$772$	−6716.75	−0.313136
$773$	−9536.25	−0.443719	−0.221860	−	0.975079i	$-0.571213\pi$
$773$	−9536.25	−0.443719	−0.221860	+	0.975079i	$0.571213\pi$
$774$	−9257.55	−0.429917
$775$	9862.92	0.457144
$776$	8539.34	0.395032
$777$	33716.4	1.55672
$778$	−20320.4	−0.936403
$779$	22662.1	1.04230
$780$	0	0
$781$	−7630.84	−0.349619
$782$	−10642.8	−0.486685
$783$	5617.94	0.256410
$784$	8330.73	0.379498
$785$	416.457	0.0189350
$786$	−41698.1	−1.89227
$787$	17696.2	0.801527	0.400763	−	0.916182i	$-0.368745\pi$
$787$	17696.2	0.801527	0.400763	+	0.916182i	$0.368745\pi$
$788$	−13228.9	−0.598047
$789$	−70479.4	−3.18014
$790$	−7019.47	−0.316128
$791$	8195.14	0.368376
$792$	16225.4	0.727961
$793$	0	0
$794$	−14734.7	−0.658584
$795$	−26298.0	−1.17320
$796$	−3311.24	−0.147442
$797$	−32566.7	−1.44739	−0.723696	−	0.690118i	$-0.757558\pi$
$797$	−32566.7	−1.44739	−0.723696	+	0.690118i	$0.757558\pi$
$798$	15922.4	0.706325
$799$	3066.04	0.135755
$800$	7423.05	0.328056
$801$	−46484.1	−2.05048
$802$	19136.8	0.842572
$803$	−9676.44	−0.425248
$804$	−20055.3	−0.879721
$805$	−7340.36	−0.321384
$806$	0	0
$807$	4824.66	0.210453
$808$	13641.4	0.593940
$809$	11622.3	0.505089	0.252544	−	0.967585i	$-0.418733\pi$
$809$	11622.3	0.505089	0.252544	+	0.967585i	$0.418733\pi$
$810$	−37721.3	−1.63629
$811$	6494.39	0.281195	0.140597	−	0.990067i	$-0.455098\pi$
$811$	6494.39	0.281195	0.140597	+	0.990067i	$0.455098\pi$
$812$	−356.218	−0.0153951
$813$	40998.8	1.76862
$814$	−8945.22	−0.385172
$815$	6169.92	0.265181
$816$	13264.9	0.569074
$817$	−5358.55	−0.229464
$818$	16744.7	0.715725
$819$	0	0
$820$	6435.49	0.274070
$821$	34897.8	1.48349	0.741743	−	0.670685i	$-0.234000\pi$
$821$	34897.8	1.48349	0.741743	+	0.670685i	$0.234000\pi$
$822$	16159.9	0.685695
$823$	1502.61	0.0636426	0.0318213	−	0.999494i	$-0.489869\pi$
$823$	1502.61	0.0636426	0.0318213	+	0.999494i	$0.489869\pi$
$824$	27959.9	1.18208
$825$	5450.70	0.230023
$826$	−2739.05	−0.115380
$827$	27887.8	1.17262	0.586308	−	0.810088i	$-0.300581\pi$
$827$	27887.8	1.17262	0.586308	+	0.810088i	$0.300581\pi$
$828$	−22113.2	−0.928124
$829$	30843.7	1.29221	0.646107	−	0.763247i	$-0.276396\pi$
$829$	30843.7	1.29221	0.646107	+	0.763247i	$0.276396\pi$
$830$	9502.93	0.397412
$831$	−41027.8	−1.71268
$832$	0	0
$833$	−12190.5	−0.507053
$834$	61250.1	2.54307
$835$	−24409.8	−1.01166
$836$	2583.71	0.106890
$837$	68445.0	2.82653
$838$	10153.7	0.418560
$839$	−36598.8	−1.50599	−0.752997	−	0.658023i	$-0.771393\pi$
$839$	−36598.8	−1.50599	−0.752997	+	0.658023i	$0.771393\pi$
$840$	16435.9	0.675110
$841$	−24191.8	−0.991914
$842$	4957.71	0.202914
$843$	46216.8	1.88825
$844$	2208.58	0.0900741
$845$	0	0
$846$	−10415.6	−0.423282
$847$	10334.5	0.419241
$848$	−10023.3	−0.405898
$849$	36487.6	1.47497
$850$	5731.15	0.231267
$851$	44314.8	1.78507
$852$	−23286.4	−0.936362
$853$	21578.4	0.866155	0.433077	−	0.901357i	$-0.357428\pi$
$853$	21578.4	0.866155	0.433077	+	0.901357i	$0.357428\pi$
$854$	−1820.67	−0.0729534
$855$	−48992.1	−1.95964
$856$	−38262.7	−1.52779
$857$	−31199.6	−1.24359	−0.621795	−	0.783180i	$-0.713596\pi$
$857$	−31199.6	−1.24359	−0.621795	+	0.783180i	$0.713596\pi$
$858$	0	0
$859$	8035.71	0.319179	0.159590	−	0.987183i	$-0.448983\pi$
$859$	8035.71	0.319179	0.159590	+	0.987183i	$0.448983\pi$
$860$	−1521.70	−0.0603366
$861$	−21037.0	−0.832680
$862$	882.346	0.0348641
$863$	−8741.47	−0.344801	−0.172400	−	0.985027i	$-0.555152\pi$
$863$	−8741.47	−0.344801	−0.172400	+	0.985027i	$0.555152\pi$
$864$	51513.2	2.02837
$865$	13407.9	0.527030
$866$	−11558.9	−0.453566
$867$	28484.5	1.11578
$868$	−4339.90	−0.169707
$869$	3723.41	0.145349
$870$	−2503.20	−0.0975475
$871$	0	0
$872$	1761.62	0.0684128
$873$	23628.7	0.916049
$874$	20927.5	0.809934
$875$	12523.7	0.483863
$876$	−29528.8	−1.13891
$877$	−1744.13	−0.0671552	−0.0335776	−	0.999436i	$-0.510690\pi$
$877$	−1744.13	−0.0671552	−0.0335776	+	0.999436i	$0.510690\pi$
$878$	−25242.8	−0.970276
$879$	−51229.0	−1.96577
$880$	−2427.24	−0.0929800
$881$	−5688.81	−0.217549	−0.108775	−	0.994066i	$-0.534693\pi$
$881$	−5688.81	−0.217549	−0.108775	+	0.994066i	$0.534693\pi$
$882$	41412.2	1.58098
$883$	−3940.14	−0.150165	−0.0750827	−	0.997177i	$-0.523922\pi$
$883$	−3940.14	−0.150165	−0.0750827	+	0.997177i	$0.523922\pi$
$884$	0	0
$885$	11772.4	0.447147
$886$	−34812.8	−1.32004
$887$	−36880.4	−1.39608	−0.698039	−	0.716060i	$-0.745944\pi$
$887$	−36880.4	−1.39608	−0.698039	+	0.716060i	$0.745944\pi$
$888$	−99225.8	−3.74978
$889$	18191.3	0.686295
$890$	12492.6	0.470507
$891$	20008.9	0.752328
$892$	−3082.97	−0.115724
$893$	−6028.88	−0.225922
$894$	−62870.5	−2.35202
$895$	27843.0	1.03987
$896$	1274.75	0.0475296
$897$	0	0
$898$	30493.5	1.13317
$899$	2402.62	0.0891343
$900$	11907.9	0.441033
$901$	14667.2	0.542327
$902$	5581.26	0.206026
$903$	4974.27	0.183315
$904$	−24117.9	−0.887334
$905$	−31194.8	−1.14580
$906$	−10729.7	−0.393457
$907$	17912.0	0.655741	0.327871	−	0.944723i	$-0.393669\pi$
$907$	17912.0	0.655741	0.327871	+	0.944723i	$0.393669\pi$
$908$	16068.7	0.587288
$909$	37746.3	1.37730
$910$	0	0
$911$	51246.0	1.86373	0.931864	−	0.362807i	$-0.118181\pi$
$911$	51246.0	1.86373	0.931864	+	0.362807i	$0.118181\pi$
$912$	−26083.3	−0.947045
$913$	−5040.74	−0.182721
$914$	28235.5	1.02182
$915$	7825.24	0.282726
$916$	−9138.67	−0.329640
$917$	16039.9	0.577627
$918$	39772.0	1.42993
$919$	25496.5	0.915182	0.457591	−	0.889163i	$-0.348712\pi$
$919$	25496.5	0.915182	0.457591	+	0.889163i	$0.348712\pi$
$920$	21602.4	0.774140
$921$	2461.19	0.0880552
$922$	20396.9	0.728564
$923$	0	0
$924$	−2398.43	−0.0853924
$925$	−23863.4	−0.848243
$926$	15396.7	0.546401
$927$	77366.2	2.74114
$928$	1808.26	0.0639646
$929$	−45048.6	−1.59095	−0.795477	−	0.605983i	$-0.792780\pi$
$929$	−45048.6	−1.59095	−0.795477	+	0.605983i	$0.792780\pi$
$930$	−30497.2	−1.07531
$931$	23970.6	0.843830
$932$	7207.51	0.253315
$933$	−24968.3	−0.876127
$934$	−6506.16	−0.227931
$935$	3551.82	0.124232
$936$	0	0
$937$	−2280.50	−0.0795099	−0.0397550	−	0.999209i	$-0.512658\pi$
$937$	−2280.50	−0.0795099	−0.0397550	+	0.999209i	$0.512658\pi$
$938$	−12613.3	−0.439060
$939$	6784.03	0.235770
$940$	−1712.06	−0.0594054
$941$	−31174.8	−1.07999	−0.539994	−	0.841669i	$-0.681573\pi$
$941$	−31174.8	−1.07999	−0.539994	+	0.841669i	$0.681573\pi$
$942$	1102.18	0.0381220
$943$	−27649.7	−0.954823
$944$	4486.98	0.154702
$945$	27430.7	0.944256
$946$	−1319.71	−0.0453568
$947$	28394.0	0.974318	0.487159	−	0.873313i	$-0.338033\pi$
$947$	28394.0	0.974318	0.487159	+	0.873313i	$0.338033\pi$
$948$	11362.4	0.389277
$949$	0	0
$950$	−11269.4	−0.384871
$951$	56029.2	1.91049
$952$	−9166.83	−0.312078
$953$	−33916.8	−1.15286	−0.576429	−	0.817147i	$-0.695554\pi$
$953$	−33916.8	−1.15286	−0.576429	+	0.817147i	$0.695554\pi$
$954$	−49826.0	−1.69096
$955$	10385.9	0.351915
$956$	−2378.57	−0.0804693
$957$	1327.80	0.0448501
$958$	−38073.2	−1.28402
$959$	−6216.19	−0.209313
$960$	−42470.3	−1.42784
$961$	−519.228	−0.0174290
$962$	0	0
$963$	−105874.	−3.54284
$964$	10679.1	0.356794
$965$	18156.2	0.605666
$966$	−19426.7	−0.647044
$967$	10792.9	0.358920	0.179460	−	0.983765i	$-0.442565\pi$
$967$	10792.9	0.358920	0.179460	+	0.983765i	$0.442565\pi$
$968$	−30413.9	−1.00986
$969$	38168.1	1.26536
$970$	−6350.19	−0.210198
$971$	7431.13	0.245599	0.122799	−	0.992432i	$-0.460813\pi$
$971$	7431.13	0.245599	0.122799	+	0.992432i	$0.460813\pi$
$972$	28265.8	0.932744
$973$	−23560.9	−0.776288
$974$	−30775.1	−1.01242
$975$	0	0
$976$	2982.54	0.0978164
$977$	−12422.6	−0.406790	−0.203395	−	0.979097i	$-0.565198\pi$
$977$	−12422.6	−0.406790	−0.203395	+	0.979097i	$0.565198\pi$
$978$	16329.1	0.533891
$979$	−6626.56	−0.216329
$980$	6807.08	0.221882
$981$	4874.47	0.158644
$982$	26398.2	0.857840
$983$	38791.6	1.25866	0.629328	−	0.777140i	$-0.283330\pi$
$983$	38791.6	1.25866	0.629328	+	0.777140i	$0.283330\pi$
$984$	61910.8	2.00574
$985$	35759.3	1.15674
$986$	1396.11	0.0450926
$987$	5596.53	0.180486
$988$	0	0
$989$	6537.89	0.210205
$990$	−12065.9	−0.387352
$991$	−1066.51	−0.0341863	−0.0170932	−	0.999854i	$-0.505441\pi$
$991$	−1066.51	−0.0341863	−0.0170932	+	0.999854i	$0.505441\pi$
$992$	22030.6	0.705113
$993$	−41380.8	−1.32244
$994$	−14645.4	−0.467329
$995$	8950.66	0.285181
$996$	−15382.4	−0.489369
$997$	−30651.6	−0.973665	−0.486833	−	0.873495i	$-0.661848\pi$
$997$	−30651.6	−0.973665	−0.486833	+	0.873495i	$0.661848\pi$
$998$	−11640.0	−0.369198
$999$	−165603.	−5.24470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.k.1.4	✓	9
3.2	odd	2		1521.4.a.bh.1.6		9
13.2	odd	12		169.4.e.h.147.6		36
13.3	even	3		169.4.c.l.22.6		18
13.4	even	6		169.4.c.k.146.4		18
13.5	odd	4		169.4.b.g.168.13		18
13.6	odd	12		169.4.e.h.23.13		36
13.7	odd	12		169.4.e.h.23.6		36
13.8	odd	4		169.4.b.g.168.6		18
13.9	even	3		169.4.c.l.146.6		18
13.10	even	6		169.4.c.k.22.4		18
13.11	odd	12		169.4.e.h.147.13		36
13.12	even	2		169.4.a.l.1.6	yes	9
39.38	odd	2		1521.4.a.bg.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.4	✓	9	1.1	even	1	trivial
169.4.a.l.1.6	yes	9	13.12	even	2
169.4.b.g.168.6		18	13.8	odd	4
169.4.b.g.168.13		18	13.5	odd	4
169.4.c.k.22.4		18	13.10	even	6
169.4.c.k.146.4		18	13.4	even	6
169.4.c.l.22.6		18	13.3	even	3
169.4.c.l.146.6		18	13.9	even	3
169.4.e.h.23.6		36	13.7	odd	12
169.4.e.h.23.13		36	13.6	odd	12
169.4.e.h.147.6		36	13.2	odd	12
169.4.e.h.147.13		36	13.11	odd	12
1521.4.a.bg.1.4		9	39.38	odd	2
1521.4.a.bh.1.6		9	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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22799 q^{2} -9.74867 q^{3} -3.03607 q^{4} +8.20685 q^{5} +21.7199 q^{6} -8.35495 q^{7} +24.5882 q^{8} +68.0366 q^{9} +O(q^{10})\)	\(q-2.22799 q^{2} -9.74867 q^{3} -3.03607 q^{4} +8.20685 q^{5} +21.7199 q^{6} -8.35495 q^{7} +24.5882 q^{8} +68.0366 q^{9} -18.2848 q^{10} +9.69898 q^{11} +29.5976 q^{12} +18.6147 q^{14} -80.0059 q^{15} -30.4937 q^{16} +44.6219 q^{17} -151.585 q^{18} -87.7418 q^{19} -24.9166 q^{20} +81.4497 q^{21} -21.6092 q^{22} +107.053 q^{23} -239.703 q^{24} -57.6475 q^{25} -400.052 q^{27} +25.3662 q^{28} -14.0430 q^{29} +178.252 q^{30} -171.090 q^{31} -128.766 q^{32} -94.5522 q^{33} -99.4170 q^{34} -68.5678 q^{35} -206.564 q^{36} +413.954 q^{37} +195.488 q^{38} +201.792 q^{40} -258.282 q^{41} -181.469 q^{42} +61.0718 q^{43} -29.4468 q^{44} +558.367 q^{45} -238.512 q^{46} +68.7115 q^{47} +297.273 q^{48} -273.195 q^{49} +128.438 q^{50} -435.004 q^{51} +328.701 q^{53} +891.312 q^{54} +79.5981 q^{55} -205.433 q^{56} +855.367 q^{57} +31.2876 q^{58} -147.144 q^{59} +242.904 q^{60} -97.8083 q^{61} +381.186 q^{62} -568.442 q^{63} +530.839 q^{64} +210.661 q^{66} -677.597 q^{67} -135.475 q^{68} -1043.62 q^{69} +152.768 q^{70} -786.767 q^{71} +1672.90 q^{72} -997.675 q^{73} -922.285 q^{74} +561.987 q^{75} +266.390 q^{76} -81.0345 q^{77} +383.897 q^{79} -250.258 q^{80} +2062.99 q^{81} +575.448 q^{82} -519.718 q^{83} -247.287 q^{84} +366.205 q^{85} -136.067 q^{86} +136.901 q^{87} +238.481 q^{88} -683.222 q^{89} -1244.03 q^{90} -325.019 q^{92} +1667.90 q^{93} -153.088 q^{94} -720.085 q^{95} +1255.30 q^{96} +347.294 q^{97} +608.675 q^{98} +659.886 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * q^99

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