LMFDB - Embedded newform 6045.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.230224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 $ x^5 - 2x^4 - 4x^3 + 6x^2 + 3x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.424945$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24437 q^{2} +1.00000 q^{3} +3.03718 q^{4} +1.00000 q^{5} -2.24437 q^{6} +3.68511 q^{7} -2.32782 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.24437 q^{2} +1.00000 q^{3} +3.03718 q^{4} +1.00000 q^{5} -2.24437 q^{6} +3.68511 q^{7} -2.32782 q^{8} +1.00000 q^{9} -2.24437 q^{10} -5.85660 q^{11} +3.03718 q^{12} -1.00000 q^{13} -8.27074 q^{14} +1.00000 q^{15} -0.849890 q^{16} +4.18442 q^{17} -2.24437 q^{18} -6.31873 q^{19} +3.03718 q^{20} +3.68511 q^{21} +13.1444 q^{22} -6.24936 q^{23} -2.32782 q^{24} +1.00000 q^{25} +2.24437 q^{26} +1.00000 q^{27} +11.1924 q^{28} -2.86569 q^{29} -2.24437 q^{30} -1.00000 q^{31} +6.56310 q^{32} -5.85660 q^{33} -9.39137 q^{34} +3.68511 q^{35} +3.03718 q^{36} +3.60315 q^{37} +14.1815 q^{38} -1.00000 q^{39} -2.32782 q^{40} -4.16862 q^{41} -8.27074 q^{42} +4.81592 q^{43} -17.7876 q^{44} +1.00000 q^{45} +14.0259 q^{46} +9.19622 q^{47} -0.849890 q^{48} +6.58005 q^{49} -2.24437 q^{50} +4.18442 q^{51} -3.03718 q^{52} -10.8549 q^{53} -2.24437 q^{54} -5.85660 q^{55} -8.57826 q^{56} -6.31873 q^{57} +6.43166 q^{58} -3.44574 q^{59} +3.03718 q^{60} -7.47293 q^{61} +2.24437 q^{62} +3.68511 q^{63} -13.0302 q^{64} -1.00000 q^{65} +13.1444 q^{66} +14.5140 q^{67} +12.7088 q^{68} -6.24936 q^{69} -8.27074 q^{70} +10.6883 q^{71} -2.32782 q^{72} +12.1297 q^{73} -8.08679 q^{74} +1.00000 q^{75} -19.1911 q^{76} -21.5822 q^{77} +2.24437 q^{78} +6.92326 q^{79} -0.849890 q^{80} +1.00000 q^{81} +9.35591 q^{82} -9.03569 q^{83} +11.1924 q^{84} +4.18442 q^{85} -10.8087 q^{86} -2.86569 q^{87} +13.6331 q^{88} -17.3029 q^{89} -2.24437 q^{90} -3.68511 q^{91} -18.9804 q^{92} -1.00000 q^{93} -20.6397 q^{94} -6.31873 q^{95} +6.56310 q^{96} -3.17344 q^{97} -14.7680 q^{98} -5.85660 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 5 * q^9	$ 5 q + 5 q^{3} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 5 q^{9} - 9 q^{11} + 6 q^{12} - 5 q^{13} - 26 q^{14} + 5 q^{15} - 4 q^{16} - 17 q^{17} - 2 q^{19} + 6 q^{20} - q^{21} + 8 q^{22} - 17 q^{23} + 6 q^{24} + 5 q^{25} + 5 q^{27} - 16 q^{28} - 6 q^{29} - 5 q^{31} - 8 q^{32} - 9 q^{33} - 16 q^{34} - q^{35} + 6 q^{36} - 3 q^{37} + 4 q^{38} - 5 q^{39} + 6 q^{40} + 9 q^{41} - 26 q^{42} + 8 q^{43} - 30 q^{44} + 5 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} + 30 q^{49} - 17 q^{51} - 6 q^{52} - 51 q^{53} - 9 q^{55} - 18 q^{56} - 2 q^{57} + 12 q^{58} - 6 q^{59} + 6 q^{60} - 23 q^{61} - q^{63} - 12 q^{64} - 5 q^{65} + 8 q^{66} + 2 q^{68} - 17 q^{69} - 26 q^{70} + 9 q^{71} + 6 q^{72} - 16 q^{73} + 12 q^{74} + 5 q^{75} - 38 q^{76} - 13 q^{77} + 21 q^{79} - 4 q^{80} + 5 q^{81} + 8 q^{82} - 44 q^{83} - 16 q^{84} - 17 q^{85} + 12 q^{86} - 6 q^{87} - 12 q^{88} - 65 q^{89} + q^{91} - 34 q^{92} - 5 q^{93} - 24 q^{94} - 2 q^{95} - 8 q^{96} + 11 q^{97} + 22 q^{98} - 9 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 5 * q^9 - 9 * q^11 + 6 * q^12 - 5 * q^13 - 26 * q^14 + 5 * q^15 - 4 * q^16 - 17 * q^17 - 2 * q^19 + 6 * q^20 - q^21 + 8 * q^22 - 17 * q^23 + 6 * q^24 + 5 * q^25 + 5 * q^27 - 16 * q^28 - 6 * q^29 - 5 * q^31 - 8 * q^32 - 9 * q^33 - 16 * q^34 - q^35 + 6 * q^36 - 3 * q^37 + 4 * q^38 - 5 * q^39 + 6 * q^40 + 9 * q^41 - 26 * q^42 + 8 * q^43 - 30 * q^44 + 5 * q^45 - 4 * q^46 - 2 * q^47 - 4 * q^48 + 30 * q^49 - 17 * q^51 - 6 * q^52 - 51 * q^53 - 9 * q^55 - 18 * q^56 - 2 * q^57 + 12 * q^58 - 6 * q^59 + 6 * q^60 - 23 * q^61 - q^63 - 12 * q^64 - 5 * q^65 + 8 * q^66 + 2 * q^68 - 17 * q^69 - 26 * q^70 + 9 * q^71 + 6 * q^72 - 16 * q^73 + 12 * q^74 + 5 * q^75 - 38 * q^76 - 13 * q^77 + 21 * q^79 - 4 * q^80 + 5 * q^81 + 8 * q^82 - 44 * q^83 - 16 * q^84 - 17 * q^85 + 12 * q^86 - 6 * q^87 - 12 * q^88 - 65 * q^89 + q^91 - 34 * q^92 - 5 * q^93 - 24 * q^94 - 2 * q^95 - 8 * q^96 + 11 * q^97 + 22 * q^98 - 9 * 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 $

$2$	−2.24437	−1.58701	−0.793503	−	0.608566i	$-0.791745\pi$
$2$	−2.24437	−1.58701	−0.793503	+	0.608566i	$0.791745\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.03718	1.51859
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.24437	−0.916259
$7$	3.68511	1.39284	0.696421	−	0.717634i	$-0.254775\pi$
$7$	3.68511	1.39284	0.696421	+	0.717634i	$0.254775\pi$
$8$	−2.32782	−0.823007
$9$	1.00000	0.333333
$10$	−2.24437	−0.709731
$11$	−5.85660	−1.76583	−0.882916	−	0.469530i	$-0.844423\pi$
$11$	−5.85660	−1.76583	−0.882916	+	0.469530i	$0.844423\pi$
$12$	3.03718	0.876759
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−8.27074	−2.21045
$15$	1.00000	0.258199
$16$	−0.849890	−0.212472
$17$	4.18442	1.01487	0.507436	−	0.861690i	$-0.330594\pi$
$17$	4.18442	1.01487	0.507436	+	0.861690i	$0.330594\pi$
$18$	−2.24437	−0.529002
$19$	−6.31873	−1.44962	−0.724808	−	0.688951i	$-0.758072\pi$
$19$	−6.31873	−1.44962	−0.724808	+	0.688951i	$0.758072\pi$
$20$	3.03718	0.679135
$21$	3.68511	0.804157
$22$	13.1444	2.80239
$23$	−6.24936	−1.30308	−0.651541	−	0.758614i	$-0.725877\pi$
$23$	−6.24936	−1.30308	−0.651541	+	0.758614i	$0.725877\pi$
$24$	−2.32782	−0.475164
$25$	1.00000	0.200000
$26$	2.24437	0.440157
$27$	1.00000	0.192450
$28$	11.1924	2.11516
$29$	−2.86569	−0.532145	−0.266073	−	0.963953i	$-0.585726\pi$
$29$	−2.86569	−0.532145	−0.266073	+	0.963953i	$0.585726\pi$
$30$	−2.24437	−0.409763
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	6.56310	1.16020
$33$	−5.85660	−1.01950
$34$	−9.39137	−1.61061
$35$	3.68511	0.622898
$36$	3.03718	0.506197
$37$	3.60315	0.592354	0.296177	−	0.955133i	$-0.404288\pi$
$37$	3.60315	0.592354	0.296177	+	0.955133i	$0.404288\pi$
$38$	14.1815	2.30055
$39$	−1.00000	−0.160128
$40$	−2.32782	−0.368060
$41$	−4.16862	−0.651029	−0.325515	−	0.945537i	$-0.605538\pi$
$41$	−4.16862	−0.651029	−0.325515	+	0.945537i	$0.605538\pi$
$42$	−8.27074	−1.27620
$43$	4.81592	0.734420	0.367210	−	0.930138i	$-0.380313\pi$
$43$	4.81592	0.734420	0.367210	+	0.930138i	$0.380313\pi$
$44$	−17.7876	−2.68158
$45$	1.00000	0.149071
$46$	14.0259	2.06800
$47$	9.19622	1.34141	0.670703	−	0.741726i	$-0.265993\pi$
$47$	9.19622	1.34141	0.670703	+	0.741726i	$0.265993\pi$
$48$	−0.849890	−0.122671
$49$	6.58005	0.940007
$50$	−2.24437	−0.317401
$51$	4.18442	0.585936
$52$	−3.03718	−0.421181
$53$	−10.8549	−1.49103	−0.745516	−	0.666487i	$-0.767797\pi$
$53$	−10.8549	−1.49103	−0.745516	+	0.666487i	$0.767797\pi$
$54$	−2.24437	−0.305420
$55$	−5.85660	−0.789704
$56$	−8.57826	−1.14632
$57$	−6.31873	−0.836936
$58$	6.43166	0.844518
$59$	−3.44574	−0.448597	−0.224299	−	0.974520i	$-0.572009\pi$
$59$	−3.44574	−0.448597	−0.224299	+	0.974520i	$0.572009\pi$
$60$	3.03718	0.392099
$61$	−7.47293	−0.956811	−0.478406	−	0.878139i	$-0.658785\pi$
$61$	−7.47293	−0.956811	−0.478406	+	0.878139i	$0.658785\pi$
$62$	2.24437	0.285035
$63$	3.68511	0.464280
$64$	−13.0302	−1.62878
$65$	−1.00000	−0.124035
$66$	13.1444	1.61796
$67$	14.5140	1.77316	0.886581	−	0.462573i	$-0.153074\pi$
$67$	14.5140	1.77316	0.886581	+	0.462573i	$0.153074\pi$
$68$	12.7088	1.54117
$69$	−6.24936	−0.752335
$70$	−8.27074	−0.988543
$71$	10.6883	1.26847	0.634235	−	0.773140i	$-0.281315\pi$
$71$	10.6883	1.26847	0.634235	+	0.773140i	$0.281315\pi$
$72$	−2.32782	−0.274336
$73$	12.1297	1.41967	0.709837	−	0.704366i	$-0.248769\pi$
$73$	12.1297	1.41967	0.709837	+	0.704366i	$0.248769\pi$
$74$	−8.08679	−0.940071
$75$	1.00000	0.115470
$76$	−19.1911	−2.20137
$77$	−21.5822	−2.45952
$78$	2.24437	0.254124
$79$	6.92326	0.778928	0.389464	−	0.921042i	$-0.372660\pi$
$79$	6.92326	0.778928	0.389464	+	0.921042i	$0.372660\pi$
$80$	−0.849890	−0.0950205
$81$	1.00000	0.111111
$82$	9.35591	1.03319
$83$	−9.03569	−0.991796	−0.495898	−	0.868381i	$-0.665161\pi$
$83$	−9.03569	−0.991796	−0.495898	+	0.868381i	$0.665161\pi$
$84$	11.1924	1.22119
$85$	4.18442	0.453864
$86$	−10.8087	−1.16553
$87$	−2.86569	−0.307234
$88$	13.6331	1.45329
$89$	−17.3029	−1.83411	−0.917053	−	0.398764i	$-0.869439\pi$
$89$	−17.3029	−1.83411	−0.917053	+	0.398764i	$0.869439\pi$
$90$	−2.24437	−0.236577
$91$	−3.68511	−0.386305
$92$	−18.9804	−1.97885
$93$	−1.00000	−0.103695
$94$	−20.6397	−2.12882
$95$	−6.31873	−0.648288
$96$	6.56310	0.669843
$97$	−3.17344	−0.322214	−0.161107	−	0.986937i	$-0.551506\pi$
$97$	−3.17344	−0.322214	−0.161107	+	0.986937i	$0.551506\pi$
$98$	−14.7680	−1.49180
$99$	−5.85660	−0.588611
$100$	3.03718	0.303718
$101$	−3.12063	−0.310514	−0.155257	−	0.987874i	$-0.549621\pi$
$101$	−3.12063	−0.310514	−0.155257	+	0.987874i	$0.549621\pi$
$102$	−9.39137	−0.929885
$103$	−15.7521	−1.55210	−0.776051	−	0.630670i	$-0.782780\pi$
$103$	−15.7521	−1.55210	−0.776051	+	0.630670i	$0.782780\pi$
$104$	2.32782	0.228261
$105$	3.68511	0.359630
$106$	24.3623	2.36628
$107$	−18.0842	−1.74826	−0.874132	−	0.485688i	$-0.838569\pi$
$107$	−18.0842	−1.74826	−0.874132	+	0.485688i	$0.838569\pi$
$108$	3.03718	0.292253
$109$	4.42045	0.423402	0.211701	−	0.977334i	$-0.432100\pi$
$109$	4.42045	0.423402	0.211701	+	0.977334i	$0.432100\pi$
$110$	13.1444	1.25327
$111$	3.60315	0.341996
$112$	−3.13194	−0.295940
$113$	8.79345	0.827218	0.413609	−	0.910455i	$-0.364268\pi$
$113$	8.79345	0.827218	0.413609	+	0.910455i	$0.364268\pi$
$114$	14.1815	1.32822
$115$	−6.24936	−0.582756
$116$	−8.70362	−0.808111
$117$	−1.00000	−0.0924500
$118$	7.73350	0.711927
$119$	15.4201	1.41355
$120$	−2.32782	−0.212500
$121$	23.2998	2.11816
$122$	16.7720	1.51847
$123$	−4.16862	−0.375872
$124$	−3.03718	−0.272747
$125$	1.00000	0.0894427
$126$	−8.27074	−0.736816
$127$	−9.43797	−0.837485	−0.418742	−	0.908105i	$-0.637529\pi$
$127$	−9.43797	−0.837485	−0.418742	+	0.908105i	$0.637529\pi$
$128$	16.1184	1.42468
$129$	4.81592	0.424018
$130$	2.24437	0.196844
$131$	−21.3910	−1.86894	−0.934468	−	0.356047i	$-0.884124\pi$
$131$	−21.3910	−1.86894	−0.934468	+	0.356047i	$0.884124\pi$
$132$	−17.7876	−1.54821
$133$	−23.2852	−2.01909
$134$	−32.5746	−2.81402
$135$	1.00000	0.0860663
$136$	−9.74056	−0.835246
$137$	−13.4142	−1.14605	−0.573027	−	0.819537i	$-0.694231\pi$
$137$	−13.4142	−1.14605	−0.573027	+	0.819537i	$0.694231\pi$
$138$	14.0259	1.19396
$139$	−4.33281	−0.367504	−0.183752	−	0.982973i	$-0.558824\pi$
$139$	−4.33281	−0.367504	−0.183752	+	0.982973i	$0.558824\pi$
$140$	11.1924	0.945927
$141$	9.19622	0.774461
$142$	−23.9885	−2.01307
$143$	5.85660	0.489754
$144$	−0.849890	−0.0708241
$145$	−2.86569	−0.237983
$146$	−27.2235	−2.25303
$147$	6.58005	0.542713
$148$	10.9434	0.899544
$149$	−15.1941	−1.24475	−0.622374	−	0.782720i	$-0.713832\pi$
$149$	−15.1941	−1.24475	−0.622374	+	0.782720i	$0.713832\pi$
$150$	−2.24437	−0.183252
$151$	−1.86397	−0.151688	−0.0758438	−	0.997120i	$-0.524165\pi$
$151$	−1.86397	−0.151688	−0.0758438	+	0.997120i	$0.524165\pi$
$152$	14.7088	1.19305
$153$	4.18442	0.338290
$154$	48.4385	3.90328
$155$	−1.00000	−0.0803219
$156$	−3.03718	−0.243169
$157$	−3.99082	−0.318502	−0.159251	−	0.987238i	$-0.550908\pi$
$157$	−3.99082	−0.318502	−0.159251	+	0.987238i	$0.550908\pi$
$158$	−15.5383	−1.23616
$159$	−10.8549	−0.860848
$160$	6.56310	0.518858
$161$	−23.0296	−1.81499
$162$	−2.24437	−0.176334
$163$	−20.7289	−1.62361	−0.811807	−	0.583926i	$-0.801516\pi$
$163$	−20.7289	−1.62361	−0.811807	+	0.583926i	$0.801516\pi$
$164$	−12.6609	−0.988647
$165$	−5.85660	−0.455936
$166$	20.2794	1.57399
$167$	12.0640	0.933537	0.466769	−	0.884379i	$-0.345418\pi$
$167$	12.0640	0.933537	0.466769	+	0.884379i	$0.345418\pi$
$168$	−8.57826	−0.661827
$169$	1.00000	0.0769231
$170$	−9.39137	−0.720286
$171$	−6.31873	−0.483205
$172$	14.6268	1.11528
$173$	−11.2698	−0.856823	−0.428412	−	0.903584i	$-0.640927\pi$
$173$	−11.2698	−0.856823	−0.428412	+	0.903584i	$0.640927\pi$
$174$	6.43166	0.487583
$175$	3.68511	0.278568
$176$	4.97747	0.375191
$177$	−3.44574	−0.258998
$178$	38.8341	2.91074
$179$	−7.14972	−0.534395	−0.267197	−	0.963642i	$-0.586098\pi$
$179$	−7.14972	−0.534395	−0.267197	+	0.963642i	$0.586098\pi$
$180$	3.03718	0.226378
$181$	−15.0378	−1.11775	−0.558874	−	0.829253i	$-0.688766\pi$
$181$	−15.0378	−1.11775	−0.558874	+	0.829253i	$0.688766\pi$
$182$	8.27074	0.613068
$183$	−7.47293	−0.552415
$184$	14.5474	1.07245
$185$	3.60315	0.264909
$186$	2.24437	0.164565
$187$	−24.5065	−1.79209
$188$	27.9306	2.03705
$189$	3.68511	0.268052
$190$	14.1815	1.02884
$191$	23.5003	1.70043	0.850213	−	0.526440i	$-0.176473\pi$
$191$	23.5003	1.70043	0.850213	+	0.526440i	$0.176473\pi$
$192$	−13.0302	−0.940375
$193$	10.1850	0.733133	0.366566	−	0.930392i	$-0.380533\pi$
$193$	10.1850	0.733133	0.366566	+	0.930392i	$0.380533\pi$
$194$	7.12237	0.511356
$195$	−1.00000	−0.0716115
$196$	19.9848	1.42749
$197$	7.99973	0.569957	0.284979	−	0.958534i	$-0.408013\pi$
$197$	7.99973	0.569957	0.284979	+	0.958534i	$0.408013\pi$
$198$	13.1444	0.934129
$199$	21.3061	1.51035	0.755174	−	0.655524i	$-0.227552\pi$
$199$	21.3061	1.51035	0.755174	+	0.655524i	$0.227552\pi$
$200$	−2.32782	−0.164601
$201$	14.5140	1.02374
$202$	7.00384	0.492789
$203$	−10.5604	−0.741194
$204$	12.7088	0.889797
$205$	−4.16862	−0.291149
$206$	35.3535	2.46320
$207$	−6.24936	−0.434361
$208$	0.849890	0.0589292
$209$	37.0063	2.55978
$210$	−8.27074	−0.570735
$211$	−1.60355	−0.110393	−0.0551966	−	0.998476i	$-0.517579\pi$
$211$	−1.60355	−0.110393	−0.0551966	+	0.998476i	$0.517579\pi$
$212$	−32.9683	−2.26427
$213$	10.6883	0.732352
$214$	40.5876	2.77451
$215$	4.81592	0.328443
$216$	−2.32782	−0.158388
$217$	−3.68511	−0.250162
$218$	−9.92111	−0.671943
$219$	12.1297	0.819649
$220$	−17.7876	−1.19924
$221$	−4.18442	−0.281475
$222$	−8.08679	−0.542750
$223$	−14.8574	−0.994927	−0.497464	−	0.867485i	$-0.665735\pi$
$223$	−14.8574	−0.994927	−0.497464	+	0.867485i	$0.665735\pi$
$224$	24.1857	1.61598
$225$	1.00000	0.0666667
$226$	−19.7357	−1.31280
$227$	10.3037	0.683878	0.341939	−	0.939722i	$-0.388916\pi$
$227$	10.3037	0.683878	0.341939	+	0.939722i	$0.388916\pi$
$228$	−19.1911	−1.27096
$229$	−23.3043	−1.53999	−0.769996	−	0.638049i	$-0.779742\pi$
$229$	−23.3043	−1.53999	−0.769996	+	0.638049i	$0.779742\pi$
$230$	14.0259	0.924838
$231$	−21.5822	−1.42001
$232$	6.67080	0.437960
$233$	12.5679	0.823352	0.411676	−	0.911330i	$-0.364944\pi$
$233$	12.5679	0.823352	0.411676	+	0.911330i	$0.364944\pi$
$234$	2.24437	0.146719
$235$	9.19622	0.599895
$236$	−10.4653	−0.681235
$237$	6.92326	0.449714
$238$	−34.6083	−2.24332
$239$	−15.6445	−1.01196	−0.505979	−	0.862546i	$-0.668869\pi$
$239$	−15.6445	−1.01196	−0.505979	+	0.862546i	$0.668869\pi$
$240$	−0.849890	−0.0548601
$241$	0.629091	0.0405233	0.0202616	−	0.999795i	$-0.493550\pi$
$241$	0.629091	0.0405233	0.0202616	+	0.999795i	$0.493550\pi$
$242$	−52.2933	−3.36154
$243$	1.00000	0.0641500
$244$	−22.6967	−1.45300
$245$	6.58005	0.420384
$246$	9.35591	0.596511
$247$	6.31873	0.402051
$248$	2.32782	0.147817
$249$	−9.03569	−0.572614
$250$	−2.24437	−0.141946
$251$	−25.9444	−1.63759	−0.818797	−	0.574084i	$-0.805358\pi$
$251$	−25.9444	−1.63759	−0.818797	+	0.574084i	$0.805358\pi$
$252$	11.1924	0.705052
$253$	36.6000	2.30102
$254$	21.1823	1.32909
$255$	4.18442	0.262039
$256$	−10.1151	−0.632197
$257$	4.72534	0.294758	0.147379	−	0.989080i	$-0.452916\pi$
$257$	4.72534	0.294758	0.147379	+	0.989080i	$0.452916\pi$
$258$	−10.8087	−0.672919
$259$	13.2780	0.825056
$260$	−3.03718	−0.188358
$261$	−2.86569	−0.177382
$262$	48.0091	2.96601
$263$	−4.84655	−0.298851	−0.149425	−	0.988773i	$-0.547742\pi$
$263$	−4.84655	−0.298851	−0.149425	+	0.988773i	$0.547742\pi$
$264$	13.6331	0.839059
$265$	−10.8549	−0.666810
$266$	52.2606	3.20430
$267$	−17.3029	−1.05892
$268$	44.0815	2.69271
$269$	−16.0462	−0.978354	−0.489177	−	0.872184i	$-0.662703\pi$
$269$	−16.0462	−0.978354	−0.489177	+	0.872184i	$0.662703\pi$
$270$	−2.24437	−0.136588
$271$	−22.4447	−1.36342	−0.681709	−	0.731623i	$-0.738763\pi$
$271$	−22.4447	−1.36342	−0.681709	+	0.731623i	$0.738763\pi$
$272$	−3.55630	−0.215632
$273$	−3.68511	−0.223033
$274$	30.1064	1.81879
$275$	−5.85660	−0.353167
$276$	−18.9804	−1.14249
$277$	12.5995	0.757032	0.378516	−	0.925595i	$-0.376434\pi$
$277$	12.5995	0.757032	0.378516	+	0.925595i	$0.376434\pi$
$278$	9.72442	0.583232
$279$	−1.00000	−0.0598684
$280$	−8.57826	−0.512649
$281$	−2.88952	−0.172374	−0.0861872	−	0.996279i	$-0.527468\pi$
$281$	−2.88952	−0.172374	−0.0861872	+	0.996279i	$0.527468\pi$
$282$	−20.6397	−1.22908
$283$	−13.1482	−0.781581	−0.390791	−	0.920480i	$-0.627798\pi$
$283$	−13.1482	−0.781581	−0.390791	+	0.920480i	$0.627798\pi$
$284$	32.4624	1.92629
$285$	−6.31873	−0.374289
$286$	−13.1444	−0.777243
$287$	−15.3618	−0.906780
$288$	6.56310	0.386734
$289$	0.509375	0.0299633
$290$	6.43166	0.377680
$291$	−3.17344	−0.186031
$292$	36.8401	2.15590
$293$	22.1884	1.29626	0.648130	−	0.761530i	$-0.275551\pi$
$293$	22.1884	1.29626	0.648130	+	0.761530i	$0.275551\pi$
$294$	−14.7680	−0.861290
$295$	−3.44574	−0.200619
$296$	−8.38748	−0.487512
$297$	−5.85660	−0.339835
$298$	34.1011	1.97542
$299$	6.24936	0.361410
$300$	3.03718	0.175352
$301$	17.7472	1.02293
$302$	4.18343	0.240729
$303$	−3.12063	−0.179276
$304$	5.37022	0.308003
$305$	−7.47293	−0.427899
$306$	−9.39137	−0.536869
$307$	−18.0797	−1.03186	−0.515931	−	0.856630i	$-0.672554\pi$
$307$	−18.0797	−1.03186	−0.515931	+	0.856630i	$0.672554\pi$
$308$	−65.5492	−3.73501
$309$	−15.7521	−0.896106
$310$	2.24437	0.127471
$311$	2.79692	0.158599	0.0792993	−	0.996851i	$-0.474732\pi$
$311$	2.79692	0.158599	0.0792993	+	0.996851i	$0.474732\pi$
$312$	2.32782	0.131787
$313$	−5.17590	−0.292559	−0.146280	−	0.989243i	$-0.546730\pi$
$313$	−5.17590	−0.292559	−0.146280	+	0.989243i	$0.546730\pi$
$314$	8.95685	0.505465
$315$	3.68511	0.207633
$316$	21.0272	1.18287
$317$	2.76470	0.155281	0.0776405	−	0.996981i	$-0.475261\pi$
$317$	2.76470	0.155281	0.0776405	+	0.996981i	$0.475261\pi$
$318$	24.3623	1.36617
$319$	16.7832	0.939679
$320$	−13.0302	−0.728411
$321$	−18.0842	−1.00936
$322$	51.6869	2.88040
$323$	−26.4402	−1.47117
$324$	3.03718	0.168732
$325$	−1.00000	−0.0554700
$326$	46.5233	2.57669
$327$	4.42045	0.244451
$328$	9.70378	0.535802
$329$	33.8891	1.86837
$330$	13.1444	0.723574
$331$	−14.0239	−0.770824	−0.385412	−	0.922745i	$-0.625941\pi$
$331$	−14.0239	−0.770824	−0.385412	+	0.922745i	$0.625941\pi$
$332$	−27.4430	−1.50613
$333$	3.60315	0.197451
$334$	−27.0759	−1.48153
$335$	14.5140	0.792982
$336$	−3.13194	−0.170861
$337$	9.45575	0.515088	0.257544	−	0.966267i	$-0.417087\pi$
$337$	9.45575	0.515088	0.257544	+	0.966267i	$0.417087\pi$
$338$	−2.24437	−0.122077
$339$	8.79345	0.477595
$340$	12.7088	0.689234
$341$	5.85660	0.317153
$342$	14.1815	0.766850
$343$	−1.54757	−0.0835607
$344$	−11.2106	−0.604434
$345$	−6.24936	−0.336454
$346$	25.2935	1.35978
$347$	−14.5089	−0.778878	−0.389439	−	0.921052i	$-0.627331\pi$
$347$	−14.5089	−0.778878	−0.389439	+	0.921052i	$0.627331\pi$
$348$	−8.70362	−0.466563
$349$	10.4730	0.560607	0.280303	−	0.959911i	$-0.409565\pi$
$349$	10.4730	0.560607	0.280303	+	0.959911i	$0.409565\pi$
$350$	−8.27074	−0.442090
$351$	−1.00000	−0.0533761
$352$	−38.4375	−2.04872
$353$	−30.1731	−1.60595	−0.802975	−	0.596013i	$-0.796751\pi$
$353$	−30.1731	−1.60595	−0.802975	+	0.596013i	$0.796751\pi$
$354$	7.73350	0.411031
$355$	10.6883	0.567277
$356$	−52.5521	−2.78526
$357$	15.4201	0.816116
$358$	16.0466	0.848088
$359$	31.0913	1.64093	0.820467	−	0.571694i	$-0.193713\pi$
$359$	31.0913	1.64093	0.820467	+	0.571694i	$0.193713\pi$
$360$	−2.32782	−0.122687
$361$	20.9264	1.10139
$362$	33.7502	1.77387
$363$	23.2998	1.22292
$364$	−11.1924	−0.586639
$365$	12.1297	0.634897
$366$	16.7720	0.876687
$367$	−23.3757	−1.22020	−0.610100	−	0.792325i	$-0.708871\pi$
$367$	−23.3757	−1.22020	−0.610100	+	0.792325i	$0.708871\pi$
$368$	5.31127	0.276869
$369$	−4.16862	−0.217010
$370$	−8.08679	−0.420412
$371$	−40.0015	−2.07677
$372$	−3.03718	−0.157471
$373$	5.75597	0.298033	0.149016	−	0.988835i	$-0.452389\pi$
$373$	5.75597	0.298033	0.149016	+	0.988835i	$0.452389\pi$
$374$	55.0016	2.84406
$375$	1.00000	0.0516398
$376$	−21.4071	−1.10399
$377$	2.86569	0.147591
$378$	−8.27074	−0.425401
$379$	26.0327	1.33721	0.668606	−	0.743617i	$-0.266891\pi$
$379$	26.0327	1.33721	0.668606	+	0.743617i	$0.266891\pi$
$380$	−19.1911	−0.984484
$381$	−9.43797	−0.483522
$382$	−52.7434	−2.69859
$383$	−10.5334	−0.538234	−0.269117	−	0.963107i	$-0.586732\pi$
$383$	−10.5334	−0.538234	−0.269117	+	0.963107i	$0.586732\pi$
$384$	16.1184	0.822538
$385$	−21.5822	−1.09993
$386$	−22.8589	−1.16349
$387$	4.81592	0.244807
$388$	−9.63833	−0.489312
$389$	8.70257	0.441238	0.220619	−	0.975360i	$-0.429192\pi$
$389$	8.70257	0.441238	0.220619	+	0.975360i	$0.429192\pi$
$390$	2.24437	0.113648
$391$	−26.1500	−1.32246
$392$	−15.3171	−0.773633
$393$	−21.3910	−1.07903
$394$	−17.9543	−0.904526
$395$	6.92326	0.348347
$396$	−17.7876	−0.893859
$397$	1.34428	0.0674674	0.0337337	−	0.999431i	$-0.489260\pi$
$397$	1.34428	0.0674674	0.0337337	+	0.999431i	$0.489260\pi$
$398$	−47.8187	−2.39693
$399$	−23.2852	−1.16572
$400$	−0.849890	−0.0424945
$401$	25.2611	1.26148	0.630739	−	0.775995i	$-0.282752\pi$
$401$	25.2611	1.26148	0.630739	+	0.775995i	$0.282752\pi$
$402$	−32.5746	−1.62468
$403$	1.00000	0.0498135
$404$	−9.47793	−0.471544
$405$	1.00000	0.0496904
$406$	23.7014	1.17628
$407$	−21.1022	−1.04600
$408$	−9.74056	−0.482230
$409$	26.2201	1.29650	0.648249	−	0.761428i	$-0.275502\pi$
$409$	26.2201	1.29650	0.648249	+	0.761428i	$0.275502\pi$
$410$	9.35591	0.462056
$411$	−13.4142	−0.661674
$412$	−47.8420	−2.35701
$413$	−12.6979	−0.624824
$414$	14.0259	0.689333
$415$	−9.03569	−0.443545
$416$	−6.56310	−0.321782
$417$	−4.33281	−0.212179
$418$	−83.0557	−4.06239
$419$	1.54677	0.0755647	0.0377823	−	0.999286i	$-0.487971\pi$
$419$	1.54677	0.0755647	0.0377823	+	0.999286i	$0.487971\pi$
$420$	11.1924	0.546131
$421$	14.0034	0.682486	0.341243	−	0.939975i	$-0.389152\pi$
$421$	14.0034	0.682486	0.341243	+	0.939975i	$0.389152\pi$
$422$	3.59896	0.175195
$423$	9.19622	0.447135
$424$	25.2682	1.22713
$425$	4.18442	0.202974
$426$	−23.9885	−1.16225
$427$	−27.5386	−1.33269
$428$	−54.9250	−2.65490
$429$	5.85660	0.282759
$430$	−10.8087	−0.521241
$431$	−13.4072	−0.645803	−0.322902	−	0.946433i	$-0.604658\pi$
$431$	−13.4072	−0.645803	−0.322902	+	0.946433i	$0.604658\pi$
$432$	−0.849890	−0.0408903
$433$	18.9525	0.910798	0.455399	−	0.890287i	$-0.349497\pi$
$433$	18.9525	0.910798	0.455399	+	0.890287i	$0.349497\pi$
$434$	8.27074	0.397008
$435$	−2.86569	−0.137399
$436$	13.4257	0.642975
$437$	39.4880	1.88897
$438$	−27.2235	−1.30079
$439$	7.41093	0.353704	0.176852	−	0.984237i	$-0.443409\pi$
$439$	7.41093	0.353704	0.176852	+	0.984237i	$0.443409\pi$
$440$	13.6331	0.649933
$441$	6.58005	0.313336
$442$	9.39137	0.446702
$443$	14.9487	0.710236	0.355118	−	0.934822i	$-0.384441\pi$
$443$	14.9487	0.710236	0.355118	+	0.934822i	$0.384441\pi$
$444$	10.9434	0.519352
$445$	−17.3029	−0.820238
$446$	33.3455	1.57896
$447$	−15.1941	−0.718655
$448$	−48.0178	−2.26863
$449$	6.24347	0.294648	0.147324	−	0.989088i	$-0.452934\pi$
$449$	6.24347	0.294648	0.147324	+	0.989088i	$0.452934\pi$
$450$	−2.24437	−0.105800
$451$	24.4140	1.14961
$452$	26.7073	1.25621
$453$	−1.86397	−0.0875769
$454$	−23.1252	−1.08532
$455$	−3.68511	−0.172761
$456$	14.7088	0.688805
$457$	24.3435	1.13874	0.569369	−	0.822082i	$-0.307187\pi$
$457$	24.3435	1.13874	0.569369	+	0.822082i	$0.307187\pi$
$458$	52.3034	2.44398
$459$	4.18442	0.195312
$460$	−18.9804	−0.884968
$461$	−5.65062	−0.263176	−0.131588	−	0.991305i	$-0.542008\pi$
$461$	−5.65062	−0.263176	−0.131588	+	0.991305i	$0.542008\pi$
$462$	48.4385	2.25356
$463$	−12.5060	−0.581204	−0.290602	−	0.956844i	$-0.593856\pi$
$463$	−12.5060	−0.581204	−0.290602	+	0.956844i	$0.593856\pi$
$464$	2.43552	0.113066
$465$	−1.00000	−0.0463739
$466$	−28.2070	−1.30667
$467$	−36.5120	−1.68957	−0.844787	−	0.535103i	$-0.820273\pi$
$467$	−36.5120	−1.68957	−0.844787	+	0.535103i	$0.820273\pi$
$468$	−3.03718	−0.140394
$469$	53.4856	2.46973
$470$	−20.6397	−0.952038
$471$	−3.99082	−0.183887
$472$	8.02105	0.369199
$473$	−28.2049	−1.29686
$474$	−15.5383	−0.713700
$475$	−6.31873	−0.289923
$476$	46.8335	2.14661
$477$	−10.8549	−0.497011
$478$	35.1120	1.60598
$479$	−6.18562	−0.282628	−0.141314	−	0.989965i	$-0.545133\pi$
$479$	−6.18562	−0.282628	−0.141314	+	0.989965i	$0.545133\pi$
$480$	6.56310	0.299563
$481$	−3.60315	−0.164290
$482$	−1.41191	−0.0643107
$483$	−23.0296	−1.04788
$484$	70.7658	3.21663
$485$	−3.17344	−0.144099
$486$	−2.24437	−0.101807
$487$	9.29435	0.421167	0.210584	−	0.977576i	$-0.432464\pi$
$487$	9.29435	0.421167	0.210584	+	0.977576i	$0.432464\pi$
$488$	17.3956	0.787463
$489$	−20.7289	−0.937394
$490$	−14.7680	−0.667152
$491$	33.6091	1.51676	0.758379	−	0.651814i	$-0.225992\pi$
$491$	33.6091	1.51676	0.758379	+	0.651814i	$0.225992\pi$
$492$	−12.6609	−0.570796
$493$	−11.9913	−0.540059
$494$	−14.1815	−0.638058
$495$	−5.85660	−0.263235
$496$	0.849890	0.0381612
$497$	39.3877	1.76678
$498$	20.2794	0.908742
$499$	29.9395	1.34027	0.670137	−	0.742237i	$-0.266235\pi$
$499$	29.9395	1.34027	0.670137	+	0.742237i	$0.266235\pi$
$500$	3.03718	0.135827
$501$	12.0640	0.538978
$502$	58.2287	2.59887
$503$	−22.2912	−0.993916	−0.496958	−	0.867775i	$-0.665550\pi$
$503$	−22.2912	−0.993916	−0.496958	+	0.867775i	$0.665550\pi$
$504$	−8.57826	−0.382106
$505$	−3.12063	−0.138866
$506$	−82.1439	−3.65174
$507$	1.00000	0.0444116
$508$	−28.6648	−1.27180
$509$	−2.35518	−0.104392	−0.0521958	−	0.998637i	$-0.516622\pi$
$509$	−2.35518	−0.104392	−0.0521958	+	0.998637i	$0.516622\pi$
$510$	−9.39137	−0.415857
$511$	44.6993	1.97738
$512$	−9.53468	−0.421377
$513$	−6.31873	−0.278979
$514$	−10.6054	−0.467784
$515$	−15.7521	−0.694121
$516$	14.6268	0.643910
$517$	−53.8586	−2.36870
$518$	−29.8007	−1.30937
$519$	−11.2698	−0.494687
$520$	2.32782	0.102082
$521$	−33.7090	−1.47682	−0.738410	−	0.674353i	$-0.764423\pi$
$521$	−33.7090	−1.47682	−0.738410	+	0.674353i	$0.764423\pi$
$522$	6.43166	0.281506
$523$	−7.13675	−0.312068	−0.156034	−	0.987752i	$-0.549871\pi$
$523$	−7.13675	−0.312068	−0.156034	+	0.987752i	$0.549871\pi$
$524$	−64.9682	−2.83815
$525$	3.68511	0.160831
$526$	10.8774	0.474278
$527$	−4.18442	−0.182276
$528$	4.97747	0.216616
$529$	16.0545	0.698022
$530$	24.3623	1.05823
$531$	−3.44574	−0.149532
$532$	−70.7215	−3.06616
$533$	4.16862	0.180563
$534$	38.8341	1.68052
$535$	−18.0842	−0.781848
$536$	−33.7858	−1.45933
$537$	−7.14972	−0.308533
$538$	36.0136	1.55266
$539$	−38.5367	−1.65989
$540$	3.03718	0.130700
$541$	−27.1983	−1.16934	−0.584672	−	0.811270i	$-0.698777\pi$
$541$	−27.1983	−1.16934	−0.584672	+	0.811270i	$0.698777\pi$
$542$	50.3741	2.16375
$543$	−15.0378	−0.645332
$544$	27.4628	1.17746
$545$	4.42045	0.189351
$546$	8.27074	0.353955
$547$	−18.0630	−0.772319	−0.386159	−	0.922432i	$-0.626199\pi$
$547$	−18.0630	−0.772319	−0.386159	+	0.922432i	$0.626199\pi$
$548$	−40.7414	−1.74039
$549$	−7.47293	−0.318937
$550$	13.1444	0.560478
$551$	18.1075	0.771406
$552$	14.5474	0.619177
$553$	25.5130	1.08492
$554$	−28.2780	−1.20142
$555$	3.60315	0.152945
$556$	−13.1595	−0.558089
$557$	−22.3940	−0.948863	−0.474432	−	0.880292i	$-0.657346\pi$
$557$	−22.3940	−0.948863	−0.474432	+	0.880292i	$0.657346\pi$
$558$	2.24437	0.0950116
$559$	−4.81592	−0.203692
$560$	−3.13194	−0.132349
$561$	−24.5065	−1.03466
$562$	6.48515	0.273560
$563$	−29.6334	−1.24890	−0.624450	−	0.781065i	$-0.714677\pi$
$563$	−29.6334	−1.24890	−0.624450	+	0.781065i	$0.714677\pi$
$564$	27.9306	1.17609
$565$	8.79345	0.369943
$566$	29.5094	1.24037
$567$	3.68511	0.154760
$568$	−24.8805	−1.04396
$569$	−38.5726	−1.61705	−0.808523	−	0.588464i	$-0.799733\pi$
$569$	−38.5726	−1.61705	−0.808523	+	0.588464i	$0.799733\pi$
$570$	14.1815	0.594000
$571$	−7.30063	−0.305522	−0.152761	−	0.988263i	$-0.548816\pi$
$571$	−7.30063	−0.305522	−0.152761	+	0.988263i	$0.548816\pi$
$572$	17.7876	0.743736
$573$	23.5003	0.981741
$574$	34.4776	1.43907
$575$	−6.24936	−0.260616
$576$	−13.0302	−0.542926
$577$	−6.12569	−0.255016	−0.127508	−	0.991838i	$-0.540698\pi$
$577$	−6.12569	−0.255016	−0.127508	+	0.991838i	$0.540698\pi$
$578$	−1.14323	−0.0475519
$579$	10.1850	0.423274
$580$	−8.70362	−0.361398
$581$	−33.2975	−1.38141
$582$	7.12237	0.295232
$583$	63.5728	2.63291
$584$	−28.2357	−1.16840
$585$	−1.00000	−0.0413449
$586$	−49.7989	−2.05717
$587$	30.3997	1.25473	0.627364	−	0.778726i	$-0.284134\pi$
$587$	30.3997	1.25473	0.627364	+	0.778726i	$0.284134\pi$
$588$	19.9848	0.824160
$589$	6.31873	0.260359
$590$	7.73350	0.318383
$591$	7.99973	0.329065
$592$	−3.06228	−0.125859
$593$	23.3694	0.959666	0.479833	−	0.877360i	$-0.340697\pi$
$593$	23.3694	0.959666	0.479833	+	0.877360i	$0.340697\pi$
$594$	13.1444	0.539320
$595$	15.4201	0.632161
$596$	−46.1472	−1.89026
$597$	21.3061	0.872000
$598$	−14.0259	−0.573560
$599$	−8.77999	−0.358741	−0.179370	−	0.983782i	$-0.557406\pi$
$599$	−8.77999	−0.358741	−0.179370	+	0.983782i	$0.557406\pi$
$600$	−2.32782	−0.0950327
$601$	−36.4279	−1.48592	−0.742962	−	0.669334i	$-0.766579\pi$
$601$	−36.4279	−1.48592	−0.742962	+	0.669334i	$0.766579\pi$
$602$	−39.8312	−1.62340
$603$	14.5140	0.591054
$604$	−5.66121	−0.230352
$605$	23.2998	0.947272
$606$	7.00384	0.284512
$607$	−12.5666	−0.510062	−0.255031	−	0.966933i	$-0.582086\pi$
$607$	−12.5666	−0.510062	−0.255031	+	0.966933i	$0.582086\pi$
$608$	−41.4704	−1.68185
$609$	−10.5604	−0.427929
$610$	16.7720	0.679079
$611$	−9.19622	−0.372039
$612$	12.7088	0.513725
$613$	3.56411	0.143953	0.0719764	−	0.997406i	$-0.477069\pi$
$613$	3.56411	0.143953	0.0719764	+	0.997406i	$0.477069\pi$
$614$	40.5774	1.63757
$615$	−4.16862	−0.168095
$616$	50.2395	2.02421
$617$	42.6971	1.71892	0.859461	−	0.511202i	$-0.170800\pi$
$617$	42.6971	1.71892	0.859461	+	0.511202i	$0.170800\pi$
$618$	35.3535	1.42213
$619$	9.51204	0.382321	0.191161	−	0.981559i	$-0.438775\pi$
$619$	9.51204	0.382321	0.191161	+	0.981559i	$0.438775\pi$
$620$	−3.03718	−0.121976
$621$	−6.24936	−0.250778
$622$	−6.27731	−0.251697
$623$	−63.7632	−2.55462
$624$	0.849890	0.0340228
$625$	1.00000	0.0400000
$626$	11.6166	0.464293
$627$	37.0063	1.47789
$628$	−12.1208	−0.483674
$629$	15.0771	0.601163
$630$	−8.27074	−0.329514
$631$	−24.9354	−0.992664	−0.496332	−	0.868133i	$-0.665320\pi$
$631$	−24.9354	−0.992664	−0.496332	+	0.868133i	$0.665320\pi$
$632$	−16.1161	−0.641064
$633$	−1.60355	−0.0637355
$634$	−6.20500	−0.246432
$635$	−9.43797	−0.374534
$636$	−32.9683	−1.30728
$637$	−6.58005	−0.260711
$638$	−37.6677	−1.49128
$639$	10.6883	0.422823
$640$	16.1184	0.637135
$641$	28.9189	1.14223	0.571114	−	0.820871i	$-0.306511\pi$
$641$	28.9189	1.14223	0.571114	+	0.820871i	$0.306511\pi$
$642$	40.5876	1.60186
$643$	43.2219	1.70451	0.852253	−	0.523130i	$-0.175236\pi$
$643$	43.2219	1.70451	0.852253	+	0.523130i	$0.175236\pi$
$644$	−69.9451	−2.75622
$645$	4.81592	0.189627
$646$	59.3416	2.33476
$647$	−36.7339	−1.44416	−0.722079	−	0.691811i	$-0.756813\pi$
$647$	−36.7339	−1.44416	−0.722079	+	0.691811i	$0.756813\pi$
$648$	−2.32782	−0.0914453
$649$	20.1803	0.792147
$650$	2.24437	0.0880313
$651$	−3.68511	−0.144431
$652$	−62.9575	−2.46560
$653$	47.6251	1.86372	0.931858	−	0.362824i	$-0.118187\pi$
$653$	47.6251	1.86372	0.931858	+	0.362824i	$0.118187\pi$
$654$	−9.92111	−0.387946
$655$	−21.3910	−0.835814
$656$	3.54287	0.138326
$657$	12.1297	0.473225
$658$	−76.0595	−2.96511
$659$	−15.3216	−0.596844	−0.298422	−	0.954434i	$-0.596460\pi$
$659$	−15.3216	−0.596844	−0.298422	+	0.954434i	$0.596460\pi$
$660$	−17.7876	−0.692380
$661$	−31.6405	−1.23067	−0.615337	−	0.788264i	$-0.710980\pi$
$661$	−31.6405	−1.23067	−0.615337	+	0.788264i	$0.710980\pi$
$662$	31.4748	1.22330
$663$	−4.18442	−0.162509
$664$	21.0334	0.816256
$665$	−23.2852	−0.902962
$666$	−8.08679	−0.313357
$667$	17.9087	0.693429
$668$	36.6404	1.41766
$669$	−14.8574	−0.574421
$670$	−32.5746	−1.25847
$671$	43.7660	1.68957
$672$	24.1857	0.932985
$673$	−35.7896	−1.37959	−0.689795	−	0.724005i	$-0.742299\pi$
$673$	−35.7896	−1.37959	−0.689795	+	0.724005i	$0.742299\pi$
$674$	−21.2222	−0.817448
$675$	1.00000	0.0384900
$676$	3.03718	0.116815
$677$	25.7520	0.989730	0.494865	−	0.868970i	$-0.335218\pi$
$677$	25.7520	0.989730	0.494865	+	0.868970i	$0.335218\pi$
$678$	−19.7357	−0.757946
$679$	−11.6945	−0.448793
$680$	−9.74056	−0.373534
$681$	10.3037	0.394837
$682$	−13.1444	−0.503324
$683$	27.0789	1.03615	0.518073	−	0.855336i	$-0.326649\pi$
$683$	27.0789	1.03615	0.518073	+	0.855336i	$0.326649\pi$
$684$	−19.1911	−0.733791
$685$	−13.4142	−0.512531
$686$	3.47331	0.132611
$687$	−23.3043	−0.889115
$688$	−4.09300	−0.156044
$689$	10.8549	0.413538
$690$	14.0259	0.533955
$691$	1.09780	0.0417621	0.0208810	−	0.999782i	$-0.493353\pi$
$691$	1.09780	0.0417621	0.0208810	+	0.999782i	$0.493353\pi$
$692$	−34.2283	−1.30116
$693$	−21.5822	−0.819842
$694$	32.5633	1.23609
$695$	−4.33281	−0.164353
$696$	6.67080	0.252856
$697$	−17.4433	−0.660711
$698$	−23.5053	−0.889687
$699$	12.5679	0.475362
$700$	11.1924	0.423031
$701$	36.2479	1.36907	0.684533	−	0.728982i	$-0.260006\pi$
$701$	36.2479	1.36907	0.684533	+	0.728982i	$0.260006\pi$
$702$	2.24437	0.0847082
$703$	−22.7673	−0.858687
$704$	76.3128	2.87615
$705$	9.19622	0.346350
$706$	67.7194	2.54865
$707$	−11.4999	−0.432497
$708$	−10.4653	−0.393311
$709$	1.32041	0.0495889	0.0247944	−	0.999693i	$-0.492107\pi$
$709$	1.32041	0.0495889	0.0247944	+	0.999693i	$0.492107\pi$
$710$	−23.9885	−0.900273
$711$	6.92326	0.259643
$712$	40.2781	1.50948
$713$	6.24936	0.234040
$714$	−34.6083	−1.29518
$715$	5.85660	0.219025
$716$	−21.7150	−0.811527
$717$	−15.6445	−0.584254
$718$	−69.7802	−2.60417
$719$	1.51410	0.0564664	0.0282332	−	0.999601i	$-0.491012\pi$
$719$	1.51410	0.0564664	0.0282332	+	0.999601i	$0.491012\pi$
$720$	−0.849890	−0.0316735
$721$	−58.0483	−2.16183
$722$	−46.9664	−1.74791
$723$	0.629091	0.0233961
$724$	−45.6724	−1.69740
$725$	−2.86569	−0.106429
$726$	−52.2933	−1.94079
$727$	−18.4895	−0.685737	−0.342869	−	0.939383i	$-0.611399\pi$
$727$	−18.4895	−0.685737	−0.342869	+	0.939383i	$0.611399\pi$
$728$	8.57826	0.317932
$729$	1.00000	0.0370370
$730$	−27.2235	−1.00759
$731$	20.1518	0.745342
$732$	−22.6967	−0.838893
$733$	−2.53520	−0.0936397	−0.0468198	−	0.998903i	$-0.514909\pi$
$733$	−2.53520	−0.0936397	−0.0468198	+	0.998903i	$0.514909\pi$
$734$	52.4636	1.93647
$735$	6.58005	0.242709
$736$	−41.0152	−1.51184
$737$	−85.0025	−3.13111
$738$	9.35591	0.344396
$739$	−22.7066	−0.835277	−0.417638	−	0.908613i	$-0.637142\pi$
$739$	−22.7066	−0.835277	−0.417638	+	0.908613i	$0.637142\pi$
$740$	10.9434	0.402288
$741$	6.31873	0.232124
$742$	89.7779	3.29585
$743$	15.8438	0.581252	0.290626	−	0.956837i	$-0.406136\pi$
$743$	15.8438	0.581252	0.290626	+	0.956837i	$0.406136\pi$
$744$	2.32782	0.0853419
$745$	−15.1941	−0.556668
$746$	−12.9185	−0.472980
$747$	−9.03569	−0.330599
$748$	−74.4307	−2.72146
$749$	−66.6423	−2.43506
$750$	−2.24437	−0.0819527
$751$	−12.8835	−0.470127	−0.235064	−	0.971980i	$-0.575530\pi$
$751$	−12.8835	−0.470127	−0.235064	+	0.971980i	$0.575530\pi$
$752$	−7.81577	−0.285012
$753$	−25.9444	−0.945465
$754$	−6.43166	−0.234227
$755$	−1.86397	−0.0678368
$756$	11.1924	0.407062
$757$	5.53850	0.201300	0.100650	−	0.994922i	$-0.467908\pi$
$757$	5.53850	0.201300	0.100650	+	0.994922i	$0.467908\pi$
$758$	−58.4270	−2.12216
$759$	36.6000	1.32850
$760$	14.7088	0.533546
$761$	33.9230	1.22971	0.614854	−	0.788641i	$-0.289215\pi$
$761$	33.9230	1.22971	0.614854	+	0.788641i	$0.289215\pi$
$762$	21.1823	0.767353
$763$	16.2899	0.589732
$764$	71.3748	2.58225
$765$	4.18442	0.151288
$766$	23.6409	0.854181
$767$	3.44574	0.124418
$768$	−10.1151	−0.364999
$769$	−7.36812	−0.265701	−0.132851	−	0.991136i	$-0.542413\pi$
$769$	−7.36812	−0.265701	−0.132851	+	0.991136i	$0.542413\pi$
$770$	48.4385	1.74560
$771$	4.72534	0.170179
$772$	30.9337	1.11333
$773$	−18.2574	−0.656673	−0.328337	−	0.944561i	$-0.606488\pi$
$773$	−18.2574	−0.656673	−0.328337	+	0.944561i	$0.606488\pi$
$774$	−10.8087	−0.388510
$775$	−1.00000	−0.0359211
$776$	7.38719	0.265185
$777$	13.2780	0.476346
$778$	−19.5318	−0.700248
$779$	26.3404	0.943742
$780$	−3.03718	−0.108749
$781$	−62.5973	−2.23991
$782$	58.6901	2.09875
$783$	−2.86569	−0.102411
$784$	−5.59232	−0.199726
$785$	−3.99082	−0.142438
$786$	48.0091	1.71243
$787$	7.95677	0.283628	0.141814	−	0.989893i	$-0.454706\pi$
$787$	7.95677	0.283628	0.141814	+	0.989893i	$0.454706\pi$
$788$	24.2966	0.865532
$789$	−4.84655	−0.172542
$790$	−15.5383	−0.552829
$791$	32.4048	1.15218
$792$	13.6331	0.484431
$793$	7.47293	0.265372
$794$	−3.01706	−0.107071
$795$	−10.8549	−0.384983
$796$	64.7105	2.29360
$797$	26.1755	0.927185	0.463593	−	0.886049i	$-0.346560\pi$
$797$	26.1755	0.927185	0.463593	+	0.886049i	$0.346560\pi$
$798$	52.2606	1.85000
$799$	38.4808	1.36135
$800$	6.56310	0.232041
$801$	−17.3029	−0.611369
$802$	−56.6951	−2.00197
$803$	−71.0388	−2.50691
$804$	44.0815	1.55464
$805$	−23.0296	−0.811686
$806$	−2.24437	−0.0790544
$807$	−16.0462	−0.564853
$808$	7.26426	0.255556
$809$	3.59793	0.126497	0.0632483	−	0.997998i	$-0.479854\pi$
$809$	3.59793	0.126497	0.0632483	+	0.997998i	$0.479854\pi$
$810$	−2.24437	−0.0788590
$811$	50.8047	1.78399	0.891996	−	0.452042i	$-0.149304\pi$
$811$	50.8047	1.78399	0.891996	+	0.452042i	$0.149304\pi$
$812$	−32.0738	−1.12557
$813$	−22.4447	−0.787170
$814$	47.3611	1.66001
$815$	−20.7289	−0.726102
$816$	−3.55630	−0.124495
$817$	−30.4305	−1.06463
$818$	−58.8474	−2.05755
$819$	−3.68511	−0.128768
$820$	−12.6609	−0.442136
$821$	−9.31785	−0.325195	−0.162598	−	0.986692i	$-0.551987\pi$
$821$	−9.31785	−0.325195	−0.162598	+	0.986692i	$0.551987\pi$
$822$	30.1064	1.05008
$823$	−21.3415	−0.743917	−0.371959	−	0.928249i	$-0.621314\pi$
$823$	−21.3415	−0.743917	−0.371959	+	0.928249i	$0.621314\pi$
$824$	36.6680	1.27739
$825$	−5.85660	−0.203901
$826$	28.4988	0.991601
$827$	−3.84819	−0.133815	−0.0669074	−	0.997759i	$-0.521313\pi$
$827$	−3.84819	−0.133815	−0.0669074	+	0.997759i	$0.521313\pi$
$828$	−18.9804	−0.659616
$829$	−35.2364	−1.22381	−0.611906	−	0.790930i	$-0.709597\pi$
$829$	−35.2364	−1.22381	−0.611906	+	0.790930i	$0.709597\pi$
$830$	20.2794	0.703909
$831$	12.5995	0.437073
$832$	13.0302	0.451742
$833$	27.5337	0.953986
$834$	9.72442	0.336729
$835$	12.0640	0.417491
$836$	112.395	3.88726
$837$	−1.00000	−0.0345651
$838$	−3.47152	−0.119922
$839$	−56.7675	−1.95983	−0.979916	−	0.199409i	$-0.936098\pi$
$839$	−56.7675	−1.95983	−0.979916	+	0.199409i	$0.936098\pi$
$840$	−8.57826	−0.295978
$841$	−20.7878	−0.716821
$842$	−31.4289	−1.08311
$843$	−2.88952	−0.0995205
$844$	−4.87028	−0.167642
$845$	1.00000	0.0344010
$846$	−20.6397	−0.709607
$847$	85.8624	2.95027
$848$	9.22545	0.316803
$849$	−13.1482	−0.451246
$850$	−9.39137	−0.322121
$851$	−22.5174	−0.771886
$852$	32.4624	1.11214
$853$	−4.21265	−0.144238	−0.0721192	−	0.997396i	$-0.522976\pi$
$853$	−4.21265	−0.144238	−0.0721192	+	0.997396i	$0.522976\pi$
$854$	61.8067	2.11498
$855$	−6.31873	−0.216096
$856$	42.0967	1.43883
$857$	−10.4612	−0.357349	−0.178675	−	0.983908i	$-0.557181\pi$
$857$	−10.4612	−0.357349	−0.178675	+	0.983908i	$0.557181\pi$
$858$	−13.1444	−0.448741
$859$	−15.4196	−0.526110	−0.263055	−	0.964781i	$-0.584730\pi$
$859$	−15.4196	−0.526110	−0.263055	+	0.964781i	$0.584730\pi$
$860$	14.6268	0.498770
$861$	−15.3618	−0.523530
$862$	30.0907	1.02489
$863$	55.2741	1.88155	0.940776	−	0.339028i	$-0.110098\pi$
$863$	55.2741	1.88155	0.940776	+	0.339028i	$0.110098\pi$
$864$	6.56310	0.223281
$865$	−11.2698	−0.383183
$866$	−42.5363	−1.44544
$867$	0.509375	0.0172993
$868$	−11.1924	−0.379893
$869$	−40.5468	−1.37546
$870$	6.43166	0.218054
$871$	−14.5140	−0.491787
$872$	−10.2900	−0.348463
$873$	−3.17344	−0.107405
$874$	−88.6256	−2.99781
$875$	3.68511	0.124580
$876$	36.8401	1.24471
$877$	24.0059	0.810621	0.405311	−	0.914179i	$-0.367163\pi$
$877$	24.0059	0.810621	0.405311	+	0.914179i	$0.367163\pi$
$878$	−16.6328	−0.561331
$879$	22.1884	0.748396
$880$	4.97747	0.167790
$881$	−12.0869	−0.407219	−0.203609	−	0.979052i	$-0.565267\pi$
$881$	−12.0869	−0.407219	−0.203609	+	0.979052i	$0.565267\pi$
$882$	−14.7680	−0.497266
$883$	−16.6523	−0.560393	−0.280197	−	0.959943i	$-0.590400\pi$
$883$	−16.6523	−0.560393	−0.280197	+	0.959943i	$0.590400\pi$
$884$	−12.7088	−0.427445
$885$	−3.44574	−0.115827
$886$	−33.5504	−1.12715
$887$	45.7235	1.53524	0.767622	−	0.640903i	$-0.221440\pi$
$887$	45.7235	1.53524	0.767622	+	0.640903i	$0.221440\pi$
$888$	−8.38748	−0.281465
$889$	−34.7800	−1.16648
$890$	38.8341	1.30172
$891$	−5.85660	−0.196204
$892$	−45.1247	−1.51089
$893$	−58.1084	−1.94452
$894$	34.1011	1.14051
$895$	−7.14972	−0.238989
$896$	59.3981	1.98435
$897$	6.24936	0.208660
$898$	−14.0126	−0.467608
$899$	2.86569	0.0955761
$900$	3.03718	0.101239
$901$	−45.4214	−1.51321
$902$	−54.7939	−1.82444
$903$	17.7472	0.590590
$904$	−20.4695	−0.680807
$905$	−15.0378	−0.499872
$906$	4.18343	0.138985
$907$	38.5458	1.27989	0.639947	−	0.768419i	$-0.278956\pi$
$907$	38.5458	1.27989	0.639947	+	0.768419i	$0.278956\pi$
$908$	31.2941	1.03853
$909$	−3.12063	−0.103505
$910$	8.27074	0.274172
$911$	17.6977	0.586352	0.293176	−	0.956058i	$-0.405288\pi$
$911$	17.6977	0.586352	0.293176	+	0.956058i	$0.405288\pi$
$912$	5.37022	0.177826
$913$	52.9185	1.75135
$914$	−54.6356	−1.80719
$915$	−7.47293	−0.247048
$916$	−70.7794	−2.33862
$917$	−78.8281	−2.60313
$918$	−9.39137	−0.309962
$919$	−48.1926	−1.58973	−0.794864	−	0.606788i	$-0.792458\pi$
$919$	−48.1926	−1.58973	−0.794864	+	0.606788i	$0.792458\pi$
$920$	14.5474	0.479612
$921$	−18.0797	−0.595746
$922$	12.6821	0.417662
$923$	−10.6883	−0.351810
$924$	−65.5492	−2.15641
$925$	3.60315	0.118471
$926$	28.0681	0.922374
$927$	−15.7521	−0.517367
$928$	−18.8078	−0.617396
$929$	−9.72470	−0.319057	−0.159529	−	0.987193i	$-0.550997\pi$
$929$	−9.72470	−0.319057	−0.159529	+	0.987193i	$0.550997\pi$
$930$	2.24437	0.0735957
$931$	−41.5776	−1.36265
$932$	38.1711	1.25033
$933$	2.79692	0.0915669
$934$	81.9463	2.68136
$935$	−24.5065	−0.801448
$936$	2.32782	0.0760871
$937$	−13.1771	−0.430477	−0.215239	−	0.976561i	$-0.569053\pi$
$937$	−13.1771	−0.430477	−0.215239	+	0.976561i	$0.569053\pi$
$938$	−120.041	−3.91948
$939$	−5.17590	−0.168909
$940$	27.9306	0.910995
$941$	17.1774	0.559966	0.279983	−	0.960005i	$-0.409671\pi$
$941$	17.1774	0.559966	0.279983	+	0.960005i	$0.409671\pi$
$942$	8.95685	0.291830
$943$	26.0512	0.848344
$944$	2.92850	0.0953145
$945$	3.68511	0.119877
$946$	63.3022	2.05813
$947$	−8.16245	−0.265244	−0.132622	−	0.991167i	$-0.542340\pi$
$947$	−8.16245	−0.265244	−0.132622	+	0.991167i	$0.542340\pi$
$948$	21.0272	0.682932
$949$	−12.1297	−0.393747
$950$	14.1815	0.460110
$951$	2.76470	0.0896516
$952$	−35.8951	−1.16337
$953$	−55.8342	−1.80865	−0.904323	−	0.426848i	$-0.859624\pi$
$953$	−55.8342	−1.80865	−0.904323	+	0.426848i	$0.859624\pi$
$954$	24.3623	0.788760
$955$	23.5003	0.760453
$956$	−47.5152	−1.53675
$957$	16.7832	0.542524
$958$	13.8828	0.448533
$959$	−49.4329	−1.59627
$960$	−13.0302	−0.420548
$961$	1.00000	0.0322581
$962$	8.08679	0.260729
$963$	−18.0842	−0.582755
$964$	1.91066	0.0615383
$965$	10.1850	0.327867
$966$	51.6869	1.66300
$967$	−13.6622	−0.439345	−0.219673	−	0.975574i	$-0.570499\pi$
$967$	−13.6622	−0.439345	−0.219673	+	0.975574i	$0.570499\pi$
$968$	−54.2377	−1.74327
$969$	−26.4402	−0.849382
$970$	7.12237	0.228686
$971$	30.3046	0.972521	0.486260	−	0.873814i	$-0.338361\pi$
$971$	30.3046	0.972521	0.486260	+	0.873814i	$0.338361\pi$
$972$	3.03718	0.0974177
$973$	−15.9669	−0.511875
$974$	−20.8599	−0.668395
$975$	−1.00000	−0.0320256
$976$	6.35117	0.203296
$977$	−2.48529	−0.0795116	−0.0397558	−	0.999209i	$-0.512658\pi$
$977$	−2.48529	−0.0795116	−0.0397558	+	0.999209i	$0.512658\pi$
$978$	46.5233	1.48765
$979$	101.336	3.23873
$980$	19.9848	0.638391
$981$	4.42045	0.141134
$982$	−75.4312	−2.40711
$983$	−6.34926	−0.202510	−0.101255	−	0.994861i	$-0.532286\pi$
$983$	−6.34926	−0.202510	−0.101255	+	0.994861i	$0.532286\pi$
$984$	9.70378	0.309345
$985$	7.99973	0.254893
$986$	26.9128	0.857077
$987$	33.8891	1.07870
$988$	19.1911	0.610551
$989$	−30.0964	−0.957010
$990$	13.1444	0.417755
$991$	39.2060	1.24542	0.622710	−	0.782452i	$-0.286032\pi$
$991$	39.2060	1.24542	0.622710	+	0.782452i	$0.286032\pi$
$992$	−6.56310	−0.208379
$993$	−14.0239	−0.445035
$994$	−88.4004	−2.80389
$995$	21.3061	0.675448
$996$	−27.4430	−0.869566
$997$	25.2551	0.799838	0.399919	−	0.916551i	$-0.369038\pi$
$997$	25.2551	0.799838	0.399919	+	0.916551i	$0.369038\pi$
$998$	−67.1952	−2.12703
$999$	3.60315	0.113999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.s.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.s.1.1	✓	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24437 q^{2} +1.00000 q^{3} +3.03718 q^{4} +1.00000 q^{5} -2.24437 q^{6} +3.68511 q^{7} -2.32782 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.24437 q^{2} +1.00000 q^{3} +3.03718 q^{4} +1.00000 q^{5} -2.24437 q^{6} +3.68511 q^{7} -2.32782 q^{8} +1.00000 q^{9} -2.24437 q^{10} -5.85660 q^{11} +3.03718 q^{12} -1.00000 q^{13} -8.27074 q^{14} +1.00000 q^{15} -0.849890 q^{16} +4.18442 q^{17} -2.24437 q^{18} -6.31873 q^{19} +3.03718 q^{20} +3.68511 q^{21} +13.1444 q^{22} -6.24936 q^{23} -2.32782 q^{24} +1.00000 q^{25} +2.24437 q^{26} +1.00000 q^{27} +11.1924 q^{28} -2.86569 q^{29} -2.24437 q^{30} -1.00000 q^{31} +6.56310 q^{32} -5.85660 q^{33} -9.39137 q^{34} +3.68511 q^{35} +3.03718 q^{36} +3.60315 q^{37} +14.1815 q^{38} -1.00000 q^{39} -2.32782 q^{40} -4.16862 q^{41} -8.27074 q^{42} +4.81592 q^{43} -17.7876 q^{44} +1.00000 q^{45} +14.0259 q^{46} +9.19622 q^{47} -0.849890 q^{48} +6.58005 q^{49} -2.24437 q^{50} +4.18442 q^{51} -3.03718 q^{52} -10.8549 q^{53} -2.24437 q^{54} -5.85660 q^{55} -8.57826 q^{56} -6.31873 q^{57} +6.43166 q^{58} -3.44574 q^{59} +3.03718 q^{60} -7.47293 q^{61} +2.24437 q^{62} +3.68511 q^{63} -13.0302 q^{64} -1.00000 q^{65} +13.1444 q^{66} +14.5140 q^{67} +12.7088 q^{68} -6.24936 q^{69} -8.27074 q^{70} +10.6883 q^{71} -2.32782 q^{72} +12.1297 q^{73} -8.08679 q^{74} +1.00000 q^{75} -19.1911 q^{76} -21.5822 q^{77} +2.24437 q^{78} +6.92326 q^{79} -0.849890 q^{80} +1.00000 q^{81} +9.35591 q^{82} -9.03569 q^{83} +11.1924 q^{84} +4.18442 q^{85} -10.8087 q^{86} -2.86569 q^{87} +13.6331 q^{88} -17.3029 q^{89} -2.24437 q^{90} -3.68511 q^{91} -18.9804 q^{92} -1.00000 q^{93} -20.6397 q^{94} -6.31873 q^{95} +6.56310 q^{96} -3.17344 q^{97} -14.7680 q^{98} -5.85660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 5 * q^9	\( 5 q + 5 q^{3} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 5 q^{9} - 9 q^{11} + 6 q^{12} - 5 q^{13} - 26 q^{14} + 5 q^{15} - 4 q^{16} - 17 q^{17} - 2 q^{19} + 6 q^{20} - q^{21} + 8 q^{22} - 17 q^{23} + 6 q^{24} + 5 q^{25} + 5 q^{27} - 16 q^{28} - 6 q^{29} - 5 q^{31} - 8 q^{32} - 9 q^{33} - 16 q^{34} - q^{35} + 6 q^{36} - 3 q^{37} + 4 q^{38} - 5 q^{39} + 6 q^{40} + 9 q^{41} - 26 q^{42} + 8 q^{43} - 30 q^{44} + 5 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} + 30 q^{49} - 17 q^{51} - 6 q^{52} - 51 q^{53} - 9 q^{55} - 18 q^{56} - 2 q^{57} + 12 q^{58} - 6 q^{59} + 6 q^{60} - 23 q^{61} - q^{63} - 12 q^{64} - 5 q^{65} + 8 q^{66} + 2 q^{68} - 17 q^{69} - 26 q^{70} + 9 q^{71} + 6 q^{72} - 16 q^{73} + 12 q^{74} + 5 q^{75} - 38 q^{76} - 13 q^{77} + 21 q^{79} - 4 q^{80} + 5 q^{81} + 8 q^{82} - 44 q^{83} - 16 q^{84} - 17 q^{85} + 12 q^{86} - 6 q^{87} - 12 q^{88} - 65 q^{89} + q^{91} - 34 q^{92} - 5 q^{93} - 24 q^{94} - 2 q^{95} - 8 q^{96} + 11 q^{97} + 22 q^{98} - 9 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 5 * q^9 - 9 * q^11 + 6 * q^12 - 5 * q^13 - 26 * q^14 + 5 * q^15 - 4 * q^16 - 17 * q^17 - 2 * q^19 + 6 * q^20 - q^21 + 8 * q^22 - 17 * q^23 + 6 * q^24 + 5 * q^25 + 5 * q^27 - 16 * q^28 - 6 * q^29 - 5 * q^31 - 8 * q^32 - 9 * q^33 - 16 * q^34 - q^35 + 6 * q^36 - 3 * q^37 + 4 * q^38 - 5 * q^39 + 6 * q^40 + 9 * q^41 - 26 * q^42 + 8 * q^43 - 30 * q^44 + 5 * q^45 - 4 * q^46 - 2 * q^47 - 4 * q^48 + 30 * q^49 - 17 * q^51 - 6 * q^52 - 51 * q^53 - 9 * q^55 - 18 * q^56 - 2 * q^57 + 12 * q^58 - 6 * q^59 + 6 * q^60 - 23 * q^61 - q^63 - 12 * q^64 - 5 * q^65 + 8 * q^66 + 2 * q^68 - 17 * q^69 - 26 * q^70 + 9 * q^71 + 6 * q^72 - 16 * q^73 + 12 * q^74 + 5 * q^75 - 38 * q^76 - 13 * q^77 + 21 * q^79 - 4 * q^80 + 5 * q^81 + 8 * q^82 - 44 * q^83 - 16 * q^84 - 17 * q^85 + 12 * q^86 - 6 * q^87 - 12 * q^88 - 65 * q^89 + q^91 - 34 * q^92 - 5 * q^93 - 24 * q^94 - 2 * q^95 - 8 * q^96 + 11 * q^97 + 22 * q^98 - 9 * q^99

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